Patent Grant
7578615
1. Field of the Invention
The invention relates generally to methods and systems for determining wind chill temperature. More specifically, the invention includes methods and systems for determining the wind chill temperature and the time to freeze for the facial temperature.
2. Description of Related Art
The sensation upon exposed skin that the ambient temperature in the presence of wind feels colder than the actual measured temperature is often referred to as the wind chill temperature. The concept of “wind chill” was first proposed by Siple, P. A., and C. F. Passel, “Measurements of dry atmospheric cooling in subfreezing temperatures”, Proc. Amer. Phil. Soc., Vol. 89, No. 1, pp. 177-199, 1945. The Siple and Passel experiments and model have been criticized by investigators as being primitive, flawed and lacking a theoretical basis. However, the Siple and Passel experiments yielded results that have proven useful for six decades. One criticism of the Siple and Passel model is that it incorrectly assumes a constant skin temperature of 33° C. (91.4° F.) during the entire time of skin exposure. This assumption was known to result in predicted values of the wind chill temperatures that would be colder than the actual values.
Despite this criticism, Osczevski, R. J., “The basis of wind chill”, Arctic., Vol. 48, No. 4, pp. 372-382, 1995, gave credence to these predictions when he stated that the test cylinder used by Siple and Passel was nearly the perfect size to represent the human head. This may explain, at least partially, why these predictions have served so well over the intervening years.
Wind chill temperature is an actual temperature that it is not restricted to the winter season. The wind chill temperature sensed by an individual on a windy, cold winter day is, conceptually, no different than that which he senses in front of an electric fan on a hot summer day. From a health or safety standpoint, the wind chill temperature sensed in the winter is the one which causes concern. The reason for this is the subjective nature of this temperature, i.e., the temperature sensed by one person may be quite different from that sensed by another. This is believed to be especially true at very low temperatures where a noticeable difference between the actual and perceived temperatures may be quite difficult.
Individual differences in sensing these low temperatures have led to proposed solutions on how the subjective nature of this temperature could be minimized. Suggestions have been made that the wind chill temperature should be replaced with categories such as “cold”, “very cold” and “extremely cold” or combining it with the heat index to come up with comfort index categories ranging from “minus to plus 10”. However, these approaches are also very subjective. Fortunately, neither of these proposed solutions has prevailed.
The subjective nature of the wind chill temperature has led to suggestions that it should be combined with clothing distributions to define a comfort level to which an individual can more easily relate. Because the comfort level is determined by the warmer temperatures sensed by the clothed surface of the body, this approach could mask a potentially dangerous situation. This could occur if the individual feels so comfortable, as a result of being adequately dressed, that he is unaware that his face may be subjected to the most imminent hazard of extended exposure, i.e., frostbite.
A more recent development is the wind chill model disclosed in Bluestein, M. and Zecher, J., “A new approach to an accurate wind chill factor”, Bull. Amer. Meteor. Soc., Vol. 80, No. 9, pp. 1893-1899, 1999. The National Oceanic & Atmospheric Administration's National Weather Service adopted the Bluestein and Zecher wind chill model on Nov. 1, 2001. Developed as an analytical counterpart to the Siple and Passel experiment, the Bluestein and Zecher model corrected for the constant skin temperature assumption by allowing it to vary, i.e., decrease with increasing exposure time. This was expected to give correct wind chill temperatures that were warmer than the Siple and Passel values. In addition, Bluestein and Zecher used a wind speed reduction at head level based upon the assumption that the free-stream velocity is always 50% greater. With this assumption and the skin temperature correction, the Bluestein and Zecher model does indeed predict wind chill temperatures that are as much as 15° F. (8.33° C.) warmer than the corresponding Siple and Passel values. However, a close examination of the Bluestein and Zecher results shows that essentially all of this warming is due to the wind reduction at head level with, at most, 2° F. (1.1° C.) being due to the varying skin temperature. At very low temperatures and high velocities the Bluestein and Zecher results show no moderation whatsoever. Instead of a moderation, their wind chill temperatures are approximately 1° F. (0.56° C.) colder than the Siple and Passel values. This result calls into question the accuracy of the Bluestein and Zecher model. Unfortunately, the Bluestein and Zecher model assumes that the free-stream velocity is always 50% greater than that at head level. Boundary layer calculations for all individuals in real life situations show that this 33% reduction in the velocity at head level is a unique condition which will almost never exist; in fact analyses will show that in all instances the reduction will be at or near zero. Without this incorrect velocity reduction, Bluestein and Zecher's results are actually no different than those of Siple and Passel.
Various devices have been proposed for determining wind chill temperature using the conventional methods disclosed by Siple and Passel, Bluestein and Zecher and others. Such conventional wind chill temperature devices are disclosed in U.S. Pat. No. 3,753,371 to Anderson, U.S. Pat. No. 3,954,007 to Harrigan, U.S. Pat. No. 4,047,431 to Mulvaney et al., U.S. Pat. No. 4,106,339 to Baer, U.S. Pat. No. 4,261,201 to Howard and PCT Patent Application No. WO 81/00462 to Howard. Generally, these devices are based on measurements of air temperature and wind speed only. Moreover, none of these conventional devices appears to correct for the above-noted errors in the prior art methods. Furthermore, the inventors are not aware of any methods or systems that account for other important factors such as altitude, insolation, and metabolic heat generation.
For the above reasons, it would be highly advantageous to provide a more accurate and complete wind chill model. It would also be advantageous to provide a system and method for wind chill determination based on a more accurate and complete wind chill model.
The present invention includes methods for more accurately determining wind chill temperature, Twc, equivalent temperature, Teq, time to freeze, tf, facial temperature, Tfm+Δt, as a function of time and the altitude correction factor, Δtf/1000. Methods for determining a wind reduction factor, WRF, are also disclosed.
Embodiments of computer-readable media storing computer-executable instructions for performing the various methods of the present invention are also disclosed.
A system for implementing one or more of the methods of determining wind chill temperature, Twc, equivalent temperature, Teq, time to freeze, tf, facial temperature, Tfm+Δt, as a function of time and the altitude correction factor, Δtf/1000, according to the present invention is also disclosed.
Additional features and advantages of the invention will be set forth in the description which follows, and in part will be apparent from the description, or may be learned by the practice of the present invention.
The following drawings illustrate exemplary embodiments for carrying out the invention. Like reference numerals refer to like parts in different views or embodiments of the present invention in the drawings.
The wind chill model disclosed herein is confined to the facial surface since the face of even a properly dressed individual will most likely be uncovered. Furthermore, the individual senses the wind chill temperature by way of the sensory nerves located within the intermediate skin layer (dermis) of the exposed facial surface and not from the surface of the clothed body. The wind chill model disclosed herein consists of an equation for the wind chill temperature and a second equation called the time to freeze. The latter defines the length of the exposure time at the wind chill temperature when facial freezing occurs. The time to freeze feature eliminates much of the controversy associated with wind chill by reducing the subjective nature of the wind chill temperature and at the same time provide a warning to the individual of a safe exposure time.
The wind chill model disclosed herein may be viewed as a complete model in that it is not restricted to the two major heat losses (forced convection, radiation) from the facial surface but is also capable of accounting for the two major heat gains (metabolic, solar) at the surface due to the individual's physical activity and the presence of sunshine. A lesser heat loss (evaporation) resulting from the physical activity is also included. The model incorporates two changes in the forced convection heat loss expression. One aspect of wind chill that has escaped attention in the prior art is the dependency of this heat loss on the ambient pressure and consequently altitude. The reason for this may have been due to both oversight and the belief that the effect of pressure was negligible. However, such is not the case and the present model shows that this effect can be quite significant. Since altitude, rather than ambient pressure, will generally be the known quantity in wind chill calculations, the forced convection term was expressed in terms of altitude. The second change involves defining an accurate value for the wind velocity at head level. Bluestein and Zecher correctly noted that this velocity might be less than that at the NWS 10 m level, but incorrectly used a value that is referred to in this model as the wind reduction factor (WRF). Their value of WRF was 0.33 based upon the velocity profile acquired in a test by Steadman (1971). This WRF reflects their assumption that the NWS 10 m velocity would always be 50% greater than the velocity at head level. Bluestein and Zecher made the error of assuming the boundary layer profile at the Steadman test site as being the same for all individuals at all times irrespective of their location. Clearly, this is not so since the wind-generated boundary layer adjacent to an individual is dependent upon individual's distance from the nearest wind obstruction. The wind chill model of the present invention, which provides a procedure for determining the WRF for each specific case, uses the WRF as a multiple of the NWS 10 m velocity value in the forced convection heat loss term.
This model, which has been validated experimentally, is believed to provide the most accurate prediction of the wind chill temperature and the only means of predicting the corresponding time to freeze. It is applicable to all individuals engaged in a wide range of physical activity in either the presence or absence of sunshine. Finally, the model is adaptable over a wide range of ambient temperatures, wind velocities, altitudes, and geographical locations worldwide.
Development of the Model Equations: The model consists of two analytically derived equations, one for the wind chill temperature, and one for the time to freeze. Each of these equations was derived from a previous equation. Because wind chill is not restricted to humans but applies to inanimate objects as well, the wind chill temperature equation was obtained from its basic counterpart, the equivalent temperature for objects. The time to freeze equation is a curve fit of extensive results from a time dependant facial temperature equation in which facial freezing times were determined over a wide range of ambient conditions. The equations are described below in the order in which they were developed.
Equivalent Temperature Equation: To really understand the meaning and significance of the equivalent temperature and appreciate the scope of its application, its expression was derived analytically. This was done through the use of the two-dimensional surface segment as shown in
The forced convection heat loss, qfc, may be expressed as:
qfc=hfc(Ts−Ta) (1)
Where hfc is forced convection heat transfer coefficient and the radiation heat loss, qr, is:
qr=εσ(Ts4−Ta4) (2)
where ε is the emissivity of the surface and σ is the Stefan-Boltzmann constant. The evaporation heat loss, qe, for a wetted surface is:
qe={dot over (w)}le (3)
where {dot over (w)} is the water evaporation flux rate and le is the latent heat of evaporation of the wetting liquid. The heat addition due to insolation, qi, is:
qi=αG (4)
where α is the thermal absorptivity of the surface and G is the local insolation value. The heat addition due to conduction into the segment, qb, is:
where k and s are the thermal conductivity and total thickness, respectively, of the segment. The natural convection heat loss, qnc, in the absence of velocity is equal to the sum of all the heat losses and gains as shown in
qnc=qfc+qr+qe−qi−qb (6)
which can now be expressed as follows in terms of the previously defined equivalent temperature (Teq):
qnc=hnc(Ts−Teq) (7)
The natural convection heat transfer coefficient (hnc) disclosed in Jakob, M. and Hawkins, G. A., Elements of Heat Transfer and Insulation, John Wily & Sons, p. 107, 1954, can be expressed as:
where the exponent φ=0.25 for a heated vertical plane or cylindrical surface and where the sole source of heat is within the surface itself. In instances where additional heat is being conducted through the surface, from inside to outside, φ is expected to take on other values. The coefficient C1 is a function of air density, thermal conductivity, specific heat, dynamic viscosity and the coefficient of thermal expansion and where L is the length of the segment. The air density (ρ), dynamic viscosity (μ), specific heat (Cp) and thermal conductivity (k) are calculated at the film temperature (Tfilm) which is defined as the average of the surface and air temperatures, that is Tfilm=(Ts+Ta)/2. The expression for C1 as disclosed in Chapman, A. J. Heat Transfer, 3rd ed., Macmillan, p. 384, 1974, is:
where g is the gravitational constant and the coefficient of thermal expansion (β) of the air is β=(Tfilm)−1. Values of C1 were calculated over a wide range of ambient temperatures (−291.4° F.≦Ta≦108.6° F. (−179.67° C.≦Ta≦42.56° C.)) plotted as a function of the film temperature expressed in terms of the surface and ambient temperature (Ts, Ta) and then curve fitted using TableCurve 2D™ to obtain the following expression:
C1=0.3268−9.5500×10−5(Ts+Ta) (8c)
and where the correlation coefficient r2=0.9953. TableCurve 2D™ is available from Systat Software, Inc., 501 Canal Blvd, Suite E, Point Richmond, Calif.
Substituting Eqs. (1), (2), (3), (4), (5), (7), and (8a) into Eq. (6), and solving for the equivalent temperature, Teq, gives:
where C1 is calculated from Eq. (8c). Equation (9) is the most basic equation for the equivalent temperature of a planar surface. The conductive, insolative and radiative heat transfer quantities (k, s, α and ε) in the equation depend upon the material properties of the surface while the forced convective heat transfer coefficient (hfc) does not. Therefore, by substituting an expression for hfc into this equation, it can be made applicable to all two-dimensional surfaces. This was done by selecting convective heat transfer coefficients for a flat surface segment disclosed in Harms et al., Aerodynamic Heating of High-Speed Aircraft, Bell Aerosystems Company, Report No. 7006-3352-001, Vol. 1, pp. 14-16, 1959. Harms et al. expresses the incompressible form of the forced convection coefficient (hfc) in terms of the Nusselt (Nu), Reynolds (Re), and molecular Prandtl (Pr) numbers. The forms are:
for turbulent flow, where the characteristic length (x) in the Re number is the length L of the segment. Because the surface temperature of the segment may differ from the free-stream temperature and because of the absence of a pressure gradient within the surface boundary layer, the density varies across the layer and the flow is considered compressible. A solution for the compressible laminar boundary layer is disclosed in Crocco, L., The laminar boundary layer in gases, Translation, North American Aviation, Aerophysics Laboratory, AL-684, 1948, while van Driest, E. R., “Turbulent boundary layer in compressible fluids,” J. of Aero. Sciences, Vol. 18, pp. 145-160, 1951, discloses a similar solution for a turbulent boundary layer. Harms et al. describes and uses a successful correlation of the Crocco and van Driest results by evaluating all of the transport and fluid properties in Eqs. (10a) and (10b) in terms of the following reference temperature, Tl, as suggested in Eckert, E. R. G. Survey on heat transmission at high speeds, USAF Wright Patterson, WADC TR 54-70, 1954:
In Eq (10c), the first term on the right side represents a static temperature component and the second term is a recovery temperature or a dynamic temperature component expressed as a function of the Mach number (M). By using Tl as the reference temperature, the resulting forced convection heat transfer coefficients for laminar flow (hfc,l) and turbulent flow (hfc,t) are applicable to all velocities (V) and all surface temperatures (Ts). For wind chill calculations, where generally M<0.1, the dynamic term in Tl can be neglected. Thus, it is possible to reduce Eqs. (10a) and (10b) to the following forms of the forced convection coefficient:
for the laminar flow along the segment, and
for turbulent flow. In these equations P is the ambient pressure, and the ambient (Ta) and surface temperatures (Ts) are both expressed in ° F. absolute (or ° R). In this form, values of hfc can be readily expressed in terms of the variables P, V, Ta, and Ts, which are primary variables in this study.
Flow along a portion or possibly the entire length of the segment will be laminar. Substitution of the laminar forced convection coefficient in Eq. (11a) into Eq. (9) gives
which is a detailed expression for the equivalent temperature, Teq. If the surface flow had been turbulent, substitution of Eq. (11b) into Eq. (9) would have resulted in a similar equation for Teq. In either case, it is obvious that an inanimate object cannot “sense” the chilling effect represented by this equivalent temperature. This temperature is not a fictitious temperature but an actual temperature encountered by the object in a cooling environment. Consequently, Eq. (12) becomes useful in determining not only the magnitude of this cooling but also providing guidelines as to what human actions might be taken in certain situations to minimize the cooling. Such is the situation as described in the following paragraphs along with two additional and potentially useful applications of Eq. (12), one of which may already exist.
Consider a motor vehicle that must be parked outdoors overnight in a cold winter environment. The owner is fully aware of the added difficulty of starting a very cold engine the following morning, and therefore is likely to take some action to minimize the overnight cooling of the engine. In doing so, he is attempting to “maximize the engine's Teq”. His first action is likely to be a covering of the engine, whereby any wind-generated forced convection heat loss in Eq. (12) is eliminated and the radiation heat loss is reduced if not completely eliminated. An alternative to covering the engine, but not as effective, would be the parking of the vehicle on the leeward side of a wind obstruction. In this instance, the convective heat loss would be reduced if not completely eliminated and the radiation heat loss would remain unchanged. As far as the other terms in Eq. (12) are concerned, the evaporation ({dot over (w)}le) and solar radiation (αG) terms will play no part, but the conductive heat term (k/s)(Tb−Ts) will. It can be viewed as residual engine heat remaining from the previous day's operation, which will partly counteract the heat losses due to forced convection and radiation, and thus aid in the subsequent starting of the engine. In this case where a conduction of heat takes place from the engine's interior to its surface, the actual value of φ is neither known nor easily determined. However, an estimate of the Teq could be made by assuming φ=0.25 and solving for Teq from Eq. (12). Individuals experienced in winter living in the northern US latitudes can testify to the validity of these actions as it pertains to vehicle operation.
The above paragraph clearly shows an application of the equivalent temperature in Eq. (12) to a stationary vehicle. But Eq. (12) would also apply if in its development the segment in
Eq. (12) offers the real possibility of being used to predict the onset of icing on aircraft surfaces, see e.g., Wilson, J. R., “Icing research heats up”, Aerospace America, pp. 38-43, May Issue, 2006. This can be viewed as the counterpart to the onset of facial freezing as discussed in detail herein. Therefore, the analytical development that was necessary to arrive at a “time to freeze” as disclosed herein, could be repeated to determine a corresponding onset time for ice formation. Such a development is possible because Eq. (12) realistically treats all modes of heat transfer that contribute to the equivalent temperature of the wing and fuselage surfaces prior to takeoff (V=0) and in flight (V>0). The laminar forced convection term in Eq. (12) could be replaced by the turbulent forced convection term in the event high angle of attack flight causes localized turbulence. In either convection term, the ambient pressure (P) would be expressed as a function of altitude (H) making Eq. (12) readily adaptable to all flight altitudes. The radiation heat loss term of Eq. (12) would remain unchanged and would be the only means of aircraft surface cooling prior to takeoff. The evaporative heat loss term, qe, (see Eq. 3) would not exist while the solar insolation heat gain term (αG) would correct for the absence or presence of sunshine. A key term is the heat conduction term, k/s (Tb−Ts), which represents heat flow from inside the aircraft structure to the outer surfaces. This heat flow would delay ice formation; after ice formation it would be the necessary heat flow required to loosen the ice such that it could be swept from the surfaces by windshear. In its final form Eq. (12) would calculate the equivalent surface temperature (Teq) as a function of flight conditions (H, V, Ta) and the initial surface temperature (Ts). This surface temperature will decrease due to the surface cooling during flight. Since it is desired to know the time when the value of Ts decreases to the point that the water component in the air freezes on the surface, presumably at 32° F., an expression for Ts as a function of time (t) would have to be determined. Temperature distributions along surfaces in laminar or turbulent flows are available in forms of analytical solutions with respect to wing de-icing. More accurate solutions may be obtained using computational fluid dynamics (CFD). This development would parallel the development of a similar expression for the facial temperature (Tf) as a function of time as developed in this application. With the final expression Ts vs. t, the predicted onset of ice formation would take place at the time (t) when Ts=32° F.
Wind Chill Temperature Equation: Eq. (12) is the basic equation for the equivalent temperature of a two-dimensional surface. The wind chill model developed herein essentially modifies this equation for the equivalent temperature (Teq) to derive a corresponding equation for the wind chill temperature (Twc) as it applies to humans. It is natural to assume that the human head can best be simulated by a cylindrical surface. Osczevski correctly recognized that the real sensation of wind chill relates to cooling of the exposed face, see Osczevski, R. J., “The basis of wind chill”, Arctic., Vol. 48, No. 4, pp. 372-382, 1995. Osczevski used a cylindrical face simulator in a wind tunnel to obtain forced convection heat transfer coefficients. The Siple and Passel experiment also used a cylinder. A comparison of Siple and Passel's results with those of Osczevski showed an agreement that prompted Osczevski to observe that “[a]lthough the cylinder used by Siple and Passel was too small to represent a human body or even a head, it was nearly the perfect size to represent a face in the wind.” The agreement was partly due to Siple and Passel's fortuitous choice of cylinder size. Buettner had also shown some agreement between measured and computed facial temperatures due to forced convection cooling of a cylinder representing the human head, see Buettner, K., “Effects of extreme heat and cold on human skin, I., Analysis of temperature changes caused by different kinds of heat application, II., Surface temperature, pain and heat conductivity in experiments with radiant heat”, J. Appl. Physiology, Vol. 3, No. 12, pp. 691-713, 1951. The Bluestein and Zecher wind chill model also assumed the face to be the surface of a cylinder.
The assumption that the human head can be viewed as a cylinder was also made by the inventors in the wind chill model of the present invention, and this assumption resulted in changes in the heat loss/gain terms of Eq. (12). There was a significant change in the forced convection term (qfc) where the characteristic dimension (L) of the two-dimensional surface now becomes the vertical length or height of the cylinder. The coefficients in the radiation (qr), evaporation (qe), insolation (qi) and conduction (qb) terms take on special meanings, not because of the surface change, but because they now apply to humans. The following sections discuss each of these changes after defining the cylindrical equivalent of a human head.
The cylinder is viewed as being vertical with its longitudinal axis normal to the wind. Its length (L) and diameter (D) must be specified. Because it can be demonstrated that an adult human head can be closely approximated by a 7 in. (17.78 cm) diameter cylinder that is 8.5 in. (21.59 cm) in length, the selected dimensions for a cylindrical model of a human head were D=7 in. and L=8.5 in. These are essentially the same dimensions as used by Bluestein and Zecher in the development of their wind chill model. The flat surface segment shown in
A cross-sectional sketch of the skin drawn to scale, may be found in World Book Encyclopedia, Vol. 17, 404d, p. 405, 1978. The World Book Encyclopedia states that the thickness of the dermis varies from 3 mm (0.0098 ft) on an individual's back to 1.6 mm (0.0052 ft) on the eyelid. From this it was assumed that the thickness of the dermis on the face was sd=2.5 mm. Based upon the sketch, the corresponding thickness of the epidermis was se=0.87 mm and that of the subcutaneous fat was ssf=1.96 mm. From these values, the cylindrical wall thickness corresponding to total skin thickness in
The outer part of the epidermis consists of layers of lifeless fat cells that provide the body with a protective covering and a barrier to prevent loss of water through the skin. The inner part, at the epidermis-dermis interface, consists of live cells including nerve cells through which the effects of the wind chill are sensed. The dermis/subcutaneous fat interface is the location of the glands that produce the sweat in the evaporative heat loss.
It is important to note that the ratio of the facial skin height (L) of 7 in. to the total skin thickness (s) of 5.33 mm (0.2098 in.), i.e., L/s, is approximately 33. Since this skin thickness (s) is small relative to the facial skin height, it is reasonable to assume that conduction through the skin occurs exclusively in one direction. The reasonableness of this assumption and the accuracy of this one-dimensional treatment will be verified below when comparing the facial freezing time as calculated here with an infinite series solution of the one-dimensional transient heat conduction in a plane wall.
Forced Convection Heat Loss (qfc): The laminar forced convection coefficient (hfc, l, s) for the two-dimensional segment shown in Eq. (11a) and used in Eq. (12) for the equivalent temperature, Teq, must now be replaced with the corresponding equation for a cylinder. At this point it should be noted that for a cylinder there is no need for an equivalent expression of the turbulent forced convection coefficient (hfc, t, s) in Eq. (11b) since laminar flow will extend circumferentially outward to about 80 degrees on either side of the wind stagnation point. This laminar region is essentially the entire facial portion of the head that is directly exposed to the wind.
From Harms et al., the incompressible form of the laminar forced convection coefficient (hfc) for this laminar stagnation region on a cylinder may be expressed as:
where the characteristic length (x) in the Re number is the cylinder diameter (D). Equation (13a) is a special case of the general equation for forced convection over a cylinder, NuD=h D/k=C (ReD)m(Pr)n. Selecting the exponent m to be 0.5 reduces the general relation to laminar flow. Furthermore, selecting the coefficient C to be 1.14 reduces the general relation to the laminar flow to the stagnation point. Thus, this special case applies to laminar stagnation heat transfer for all velocities. Neglecting the compressibility effects and the usage of the reference temperatures (Tl) of Eq. (10c), as done before, Eq. (13a) reduces to the following form for the laminar forced convection coefficient for a cylinder as used in this study:
where again the temperatures (Ts and Ta) are expressed in ° F. absolute (or ° R). Note the similarity of Eq. (13b) to that for the two-dimensional segment as shown in Eq. (11a).
Suppose the diameter (D) of a cylinder is equal to the length (L) of a segment. Then the ratio of the coefficients (hfc,l,c/hfc,l,s), is 3.36, which shows that the forced convection cooling of the cylinder with its longitudinal axis normal to the wind direction is 3.36 times greater than that for a two-dimensional surface aligned so as to be parallel to the wind. Perhaps this explains why a person facing into the wind on a cold winter day may instinctively turn his head to the side to lessen the cold sensation.
Eq. (13b), like Eq. (11a), clearly shows the dependency of the forced coefficient on the wind velocity (V) and the ambient pressure (P). This velocity, V, must be that at head level, and must also be expressed in ft/sec. However, in dealing with wind chill, it is more convenient to express velocity, V, in mph. Since the velocity at head level may under certain situations be less than that at the National Weather Service (NWS) standard 10 m level, the concept of a wind reduction factor (WRF), as discussed more fully below, was introduced.
Making these changes, the velocity in ft/sec at head level becomes (1.467) (1−WRF) V where V is the velocity in mph at the NWS 10 m (32.81 ft) level. The ambient pressure, P, is a function of altitude. Expressed in lbf/ft2 in terms of its sea level value (2116.8 lbf/ft2 (1 atm)) and altitude (H) in ft using the correlation disclosed in John, J. E. A. and Haberman, W. L., Introduction to Fluid Mechanics, 2nd ed., Prentice Hall, pp. 24-26, 1980, the pressure is 2116.8 [1−(6.92×10−6)H]5.21. Putting these expressions for V and P into Eq. (13b), the final form of the forced convection coefficient for the cylindrical surface becomes:
It can be shown that the absolute humidity is a function of ambient pressure, ambient temperature, and the relative humidity. The above expression for the convective heat transfer coefficient is a function of the ambient pressure and ambient temperature, and therefore accounts for the effects of humidity.
Radiation Heat Loss (qr): The radiation heat loss (qr) as shown in Eq. (2) and used in Eq. (12) remains the same but is expressed in terms of a radiation coefficient (hr) which applies to humans. Upon expansion, Eq. (2) becomes:
qr=εσ(Ts+Ta)(Ts2+Ta2)(Ts−Ta)=hr(Ts−Ta) (14a)
where the radiation heat loss coefficient is:
hr=εσ(Ts+Ta)(Ts2+Ta2) (14b)
The surface emissivity of the human head was determined to be ε=0.8 based upon a dynamic model disclosed in Fiala, D., Lomas, K. J. and Stohrer, M., “A computer model of human thermoregulation for a wide range of environmental conditions: the passive system”, J. Appl. Phys., Vol. 98, pp. 1957-1972, 1999. The Fiala et al. model was developed for evaluating the human response to a cold, cool, neutral, warm, or hot environment. With this value of ε and with the Stefan-Boltzmann constant α=1.714×10−9 Btu/hr-ft2−° R4 (5.670×10−−8 W/m2-° K4), the radiation heat loss coefficient becomes:
hr=(1.3712×10−9)(Ts+Ta)(Ts2+Ta2) (14c)
Evaporation Heat Loss (qe): The evaporation heat loss (qe) from the skin segment as shown in Eq. (3) is the total heat loss resulting from the evaporation of sweat from the skin surface and from a natural diffusion of water through the skin layer. In this expression, {dot over (w)} is the total water evaporation flux rate and le is the latent heat of evaporation for water. It is known that le=539.49 cal/gm (971.08 Btu/lbm) so that qe=971 {dot over (w)}, see, e.g., Eshbach, O. W., Handbook of Engineering Fundamentals, John Wiley & Sons, Inc., pp. 1-142 and 7-119, 1952.
The problem here is the determination of this total water evaporation flux rate which is based upon the sweat rate and the diffusion rate. From the 1993 ASHRAE (American Society of Heating, Refrigerating and Air Conditioning Engineers) Handbook, Fundamentals, sweating is shown to be a thermoregulatory process in which a risen body core temperature (TCR) may be lowered to its equilibrium or neutral value (98.2° F. or 36.78° C.) based upon warm signals from the core and skin. Since information was lacking in the calculation of both the sweat rate and the diffusion rate, these values were obtained from Vogel, H. C. A., The Nature Doctor, Instant Improvement, Inc., p. 316, 1991. Vogel discloses that the diffusion rate varies from 1.7 to 2.6 pints/day (0.85 to 1.3 quarts/day). This represents a heat loss from the skin that persists at all times in the absence of perspiration. Using an average value of 2.2 pints/day (1.1 quarts/day), this value of the diffusion rate must be considered at all times in the calculation of wind chill.
Another source of water loss may be from sweat. Clayman, C. B., The American Medical Association, Home Medical Encyclopedia, Vol. 2, Random House, p. 1070, 1989, discloses that the sweat rate is 1.1 pints/day (0.55 quarts/day) in a cold climate and 5.0 pints/day (2.5 quarts/day) in a warm climate. Clayman notes that these are average values and as such represent cases where individuals are performing a minimum of physical activity. Based on these values, the total water loss rate, diffusion plus sweat, would be 3.3 pints/day (1.65 quarts/day) in a cold climate during minimum physical activity. But the sweat rate depends upon the severity of the physical activity. Vogel has stated that an individual perspiring under extreme conditions in a steam or sauna bath or living in the tropics can experience a total (diffusion and sweating) water loss rate that is 10 times greater than the average diffusion rate of 2.2 pints/day (1.1 quarts/day). Because even the most extreme wintertime activities are expected to result in a total water loss rate much less than this, a rather arbitrary assumption was made that this total water loss rate for maximum physical activity would be 5 times, not 10 times, the average diffusion rate. Thus, the total water loss rate from the skin surface for maximum physical activity would be 5×2.2=11.0 pints/day (5.5 quarts/day).
Swerdlow, J. K., “Unmasking Skin”, National Geographic, pp. 36-63, November, 2002, discloses that the total skin area of an average person is 21 ft2 (1.95 m2), which is close to values quoted by other sources. It was assumed that diffusion and sweating occur over the entire 21 ft2 (1.95 m2). If it does not, then the resultant water evaporation flux rate would be conservative. For minimum physical activity in a cold climate where the total water loss rate is 3.3 pints/day (1.65 quarts/day), the minimum evaporation flux rate is {dot over (w)}min=0.00655 lbm/hr-ft2 (0.032 kg/hr-ft2) and for maximum physical activity in a cold climate where the total water loss rate is 11.0 pints/day (5.5 quarts/day), the maximum evaporation flux rate is {dot over (w)}max=0.0283 lbm/hr-ft2 (0.1066 kg/hr-ft2). The choice as to which evaporation flux rate to use was determined by the degree of strenuousness of the individual's activity.
Table 1 below, shows values of the metabolic heat rate (Mact) required for various activities and which will be used later to determine the metabolic heat gain. The following assumptions were made in order to specify the evaporation flux rate for each different activity: (1) when Mact≦40 Btu/hr2-ft2 (126.18 W/m2), the physical activity is at a minimum and {dot over (w)}=0.00655 lbm/hr-ft2 (0.032 kg/hr-ft2), and (2) when Mact>40 Btu/hr2-ft2 (126.18 W/m2), the physical activity is at a maximum and {dot over (w)}=0.02183 lbm/hr-ft2 (0.1066 kg/hr-ft2). With the evaporation flux rate ({dot over (w)}) known, along with the known value of the latent heat of evaporation (le) for water, the evaporation heat loss, qe=(971){dot over (w)} can be calculated. This value of qb provides an estimate of the evaporation heat loss.
Solar Heat Gain (qi): The solar heat (qi) received by the skin segment as shown in Eq. (4) depends on the insolation value (G). If sunshine is present, G>0, in darkness G=0. In the present case, where the major concern is an excessive wind chill leading to possible facial freezing, insolation or solar radiation heating of the face has a moderating effect. Because the heating of the face is maximized on a clear day when the sun's rays are perpendicular to the facial surface, this maximum value depends on the latitude of the individual's location. The value also depends on the season of the year. Consider, for example, the case of an individual at a northern latitude angle (LAT). In the summer, when the sun is high in the sky the angle of incidence of the sun's rays upon an upright individual's face may be small, resulting in relatively little solar heating of the face. Contrast this with the winter season when the sun is low in the sky and the sun's rays may be nearly perpendicular to the face, thus maximizing the facial heating. It is fortunate that this radiation heating is at its maximum in the winter season when its benefit can best be utilized. Finally, the insolation value at a given location is dependent upon the daily number of hours of sunshine at the location.
New England Solar Electric Inc., The Solar Electric Independent Home Book, New England Solar Electric Inc., Chap. 8, pp. A18-A39, 1998, discloses insolation data by the month for 221 cities across the 50 states as compiled by the U.S. Department of Energy. The data in this reference was used to devise a method for selecting the correct value of insolation to be used in the calculation of wind chill. This was done by dividing the portion of the lower 48 states that would most likely be subjected to wind chill into three latitude (LAT) regions of 50 each. These regions are 35°<LAT<40°, 40°<LAT<45° and 45°<LAT<50°, where LAT is the latitude angle of a particular location within a region. The combined span of these three regions may be visualized by noting that the 35° latitude line follows the southern boundary of Tennessee while the 50° latitude line lies close to the southern boundary of the Canadian provinces. Within each region, the average monthly insolation (Gav˜KW-hr/m2) was determined for four or more cities over a four month period from November through February. During this four month period, the nearly normal angle of incidence of the sun's rays on a collecting surface, in this case, an individual's head, was LAT+15°.
The values of Gav are those corresponding to this angle of incidence. The values of Gav were then used to obtain the average values (GLAT) over each latitude region. These values are shown in Table 2, below, where GLAT has also been expressed as G in English units (Btu/hr-ft2) for use in this model. It should be noted that the values for G (37.78 and 42.66 Btu/hr-ft2 (119.17 and 134.58 W/m2)) in the two northern regions, where severe wind chill and facial freezing is a possibility, are only 12 to 13% of the value of 240 cal/s-m2 (318.53 Btu/hr-ft2) used by Steadman (1971) for an upright human after he corrected his direct insolation value for a low angle solar altitude.
The thermal absorptivity (α) of the human skin was determined from results of a study disclosed in Buettner, K., The effects of natural sunlight on human skin, Proceedings of the first International Conference, Sponsored jointly by the Skin and Cancer Hospital, Temple University Health Sciences Center and the International Society of Biometeorology, Pergamon Press, 237-249, 1969. In this study, the reflectivity of sunshine from white skin was found to be 35%. Assuming no transmissivity, the absorptivity was 65% and therefore the thermal absorption coefficient (α) was 0.65. For dark skin, the absorptivity would be expected to be greater than 0.65; however, since this value was not known an α=0.65 was used in this model. With G determined, the solar heat gain is qe=(0.65) G.
Metabolic Heat Gain (qb): What was previously described as heat conduction (qb) into the segment in Eq. (5) now becomes the metabolic heat flow from deep inside the body, called the core, into the multi-layered skin segment. This is more clearly understood in terms of the two-component model disclosed in Gagge, A. P., Stolwijk, J. A. J, and Nishi, Y., “An effective temperature scale based on a simple model of human physiological regulatory response”, ASHRAE Transactions, Vol. 77, No. 1, p. 247, 1971, in which the human body is represented as two concentric cylinders. Described in the 1993 ASHRAE Handbook, the inner cylinder represents the body core (skeleton, muscle, internal organs) and the outer annular cylinder, the skin layer. The metabolic heat (M) produced within the core of the body is the heat required for an individual's activity (Mact) plus that required for shivering (Mshv), should that occur. Typical values of Mact for various activities were obtained from the 1993 ASHRAE Handbook and OSHA Technical manual, Occupational Safety and Health Administration, Heat stress, Section III, U.S. Department of Labor, Chap. 4, 2003, and are shown in Table 1, above. Shivering is a thermoregulatory process through which the core temperature (TCR) may be restored to its equilibrium or neutral value of 98.2° F. after a lowering. Although (Mshv) can be theoretically determined based on cold signals from the core and the skin, it was not considered here because of the uncertainty of its occurrence and its anticipated minimal effect on the final wind chill temperature.
If a portion of the heat produced within the core is expended as work (W) performed by the muscles, then the net heat production (Mact−W) is dissipated to the environment through the skin layer or is stored in the core causing the core temperature (TCR) to rise. The possible rise in core temperature was not considered because most wintertime activities are not so extreme that the metabolic heat cannot be completely dissipated through the skin. Therefore, the core temperature was assumed to remain constant. Calculations using large values of Mact verify this. In addition, work performed by the muscles cannot be specifically defined for the general populace when predicting wind chill temperatures; consequently, it too was neglected (W=0). Based on the two-cylinder concept of the human body, the metabolic heat flow (qb) becomes a conductive heat transfer of heat from the core cylinder at a constant temperature (TCR) to the skin cylinder whose outer surface is at the facial temperature (Tf). In
qb=K(TCR−Tf) (15)
This expression is similar to Eq. (5) except that the k/s term is replaced by a more complex conductance term (K). The 1993 ASHRAE Handbook states that this heat is transferred from the core to the skin by conductance (κ) at the interface between them and by convection due to the skin blood flux rate ({dot over (m)}b,sk) within the skin segment. Therefore the complex conductance term becomes:
K=(κ+Cp,b{dot over (m)}b,sk)
with κ specified as 0.93 Btu/hr-ft2-° F. (5.28 W/m2-° C.) and with the blood specific heat Cp,b=1.0 Btu/lbm-° F. (4.186 KJ/kg-° C.). The conductance term becomes:
and so it is directly proportional to the skin blood flux rate which is limited to the range, 0.1 lbm/hr-ft2<{dot over (m)}b,sk<18 lbm/hr-ft2 (0.49 kg/hr-m2<{dot over (m)}b,sk<87.88 kg/hr-m2). Increasing activity (Mact), increases the skin blood flux rate and thus increases the metabolic heat flow from the core to the skin segment.
The above conductance term (K) is defined differently in a thermoregulatory model developed by Havenith in which he determined the human response to heat and cold exposure, see Havenith, G., “Individualized model of human thermoregulation for the simulation of heat stress response”, J. Appl. Physiology, Vol. 90, pp. 1943-1954, 2001. Havenith also used the two cylinder model developed by Gagge et al., but rather than defining the interface as in
Havenith determined the thermal resistance (R) of the shell, that is the inverse of the thermal conductance (1/K), by adding in series the resistance due to the skin blood flux rate (Rb), the resistance due to the muscle insulation and the muscle blood flux rate (Rm) and the resistance of the fat layer and two skin layers (Rsk) to get the following equation:
K=R−1=(Rb+Rm+Rsk)−1 (17a)
where each of the three resistances is expressed in SI units (m2-° C./W). The resistance due to blood flux rate is:
where Havenith refers to η as a countercurrent heat exchange efficiency. This efficiency (η) is a measure of the arterial blood's ability to carry heat to the skin segment. The greater the value of η, the smaller the values of Rb and R, the greater the conductance (K) and the greater the heat flow to the skin. The blood vessels in the skin are both arterial and venous in nature. The arteries carry heated blood from the core to the inner part of the epidermis but not into its outer part, which consists of multiple layers of dead cells. The veins carry the cooled returning blood to the core. The flow in the veins is counter to that in the arteries, and as a result, some of the heat in the incoming arterial flow directed toward the epidermis is absorbed by the outgoing venous flow leaving the epidermis. This heat exchange mechanism is the same principle employed in counterflow heat exchanges where their efficiency, ηcf, is a measure of the ability of the colder flow to remove heat from the warmer. Consequently, the greater the heat removal efficiency (ηcf) of the veins, the lesser the heat transport efficiency (η) of the arteries; therefore η can be expressed as η=1−ηcf. Havenith assumed η=0.5 and then calculated the specific heat of the blood (Cp, b) on the basis of what is called a “standard man” whose fat content is 15% of his total mass. Knowing the specific heat for fat as 2.51 J/gm-° C. (0.6 Btu/lbm-° F.) and that for other body tissue (skin, skeleton, muscles) as 3.65 J/gm-° C. (0.87 Btu/lbm-° F.), Cp,b=(0.15) (2.51)+(1−0.15)(3.65)=3.48 J/gm-° C. or 3480 J/kg-° C. (0.83 Btu/lbm-° F.). With these values, Eq. (17b) becomes:
where the blood flux rate ({dot over (m)}b,sk) is expressed in kg/sec-m2.
The resistance due to the muscle may be expressed as:
where 0.05 is called the maximal muscle insulation and the denominator relates muscle blood flux rate to energy consumption through the metabolic heat rate (Mact) expressed in W/m2 for a given activity.
The resistance due to the subcutaneous fat layer and the two other layers (dermis, epidermis) of the skin is:
Rsk=0.0048(s−2)+0.0044 (17e)
where s is the total thickness of all three layers expressed in mm. With s=5.33 mm (0.0175 ft) from
Rsk=0.02038 (17f)
Substituting Eqs. (17c), (17d) and (17f) into Eq. (17a) the thermal resistance (R) of the core is:
Expressing the blood flux rate ({dot over (m)}b,sk) in kg/hr-m2 rather than kg/sec-m2 and then expressing the conductance (K) as the inverse of Eq. (17g), K becomes:
Aside from the difference in units, this conductance term by Havenith is quite different from the conductance term of Eq. (16b) as obtained from 1993 ASHRAE Handbook, Fundamentals. Each is a function of the skin blood flux rate, which in turn is dependent upon the metabolic heat (Mact) required by the activity. The following paragraphs describe the procedure used to determine which of these expressions for K would be used in the development of this model.
It has already been stated that the core temperature (TCR) would not rise but remain constant, since all of the metabolic heat (qb) flowing to the skin could be completely dissipated through the skin such that qb=Mact. Then from Eq. (15),
qb=K(TCR−Tf)=KΔT=Mact (17i)
With this equation, it was possible to determine the conductance (K) term and hence the blood flux rate ({dot over (m)}b,sk) that satisfies each value of ΔT and Mact. This required establishing a range of values for ΔT and Mact. With TCR=98.2° F. (36.78° C.) and with an initial facial temperature Tf=91.4° F. (33° C.), the initial and minimum temperature differential is ΔT=98.2° F.−91.4° F.=6.8° F. (3.78° C.) at the moment of exposure to the wind. Assume, during continued exposure, the face was allowed to cool down to 40° F. (4.44° C.). The final and maximum temperature differential would be ΔT=98.2° F.−40° F.=58.2° F. (32.33° C.). Thus, the temperature differential range is 6.8° F.<ΔT<58.2° F. (3.78° C.<ΔT<32.33° C.).
Over this range of ΔT, individuals may be engaged in various physical activities requiring differing amounts of metabolic heat (Mact). At this point, the range of Mact was assumed to be that which existed during the Adamenko and Khairullin (1972) experiment, see Adamenko, V. N. and Khairullin, K. Sh., “Evaluation of conditions under which unprotected parts of the human body may freeze in urban air during winter”, Boundary-Layer Meteorology, Vol. 2, pp. 510-518, 1972. In this experiment, the facial components (cheeks, nose, ears) of 40 people were instrumented to record the component temperature while the people were engaged in different levels of activity while exposed to ambient temperatures ranging from 10° C. to −40° C. (50° F. to −40° F.) and winds up to 15 m/s (33.55 mph). Their estimate of heat production by the human body, that is Mact, for all 40 people involved in the experiment ranged from 0.08 cal-cm−2-min−1 (17.7 Btu/hr-ft2) to 0.60 cal-cm−2-min−1 (132.7 Btu/hr-ft2). Therefore, the range of Mact considered here was 17.7 Btu/hr-ft2<Mact<132.7 Btu/hr-ft2 (55.84 W/m2<Mact<418.62 W/m2).
Determining which expression for K to use in this model went beyond determining K itself for a given ΔT and Mact from Eq. (17i); rather it depended upon the magnitude of the blood flux rate {dot over (m)}b,sk, which is characteristic of this equation as well as Eq. (16b). Consider first the ASHRAE expression for K in Eq. (16b). Substituting it into Eq. (17i) and solving for {dot over (m)}b,sk in units of lbm/hr-ft2 gives:
At the moment of exposure when ΔT=6.8° F. (3.78° C.), the activity level would likely be at the lowest level, that is Mact=17.7 Btu/hr-ft2 (55.84 W/m2). With these values, the blood flux rate from Eq. (17j) is {dot over (m)}b,sk=1.67 lbm/hr-ft2 (8.15 kg/hr-m2). Now after extended exposure, when ΔT=58.2° F. (32.33° C.), the activity level is likely to be at its maximum value of Mact=132.7 Btu/hr-ft2 (418.62 W/m2) and the corresponding blood flux rate is {dot over (m)}b,sk=1.35 lbm/hr-ft2 (6.59 kg/hr-m2). Thus the blood flux rate from the ASHRAE expression for K decreases as the activity level increases.
Determining the corresponding blood flux rates for the Havenith expression for K in Eq. (17h) required an iterative procedure, since this K is a function of both {dot over (m)}b,sk and Mact. For the minimum values of ΔT and Mact, the blood flux rate was found to be {dot over (m)}b,sk=41 kg/hr-m2 (8.4 lbm/hr-ft2). For the maximum value of ΔT and Mact, the blood flux rate was {dot over (m)}b,sk=43 kg/hr-m2 (8.81 lbm/hr-ft2). The Havenith values differ from the ASHRAE values in two respects. First, the larger blood flux rate for the Havenith K occurs at the higher activity level, whereas for the ASHRAE K it occurs at the lower activity level. Secondly, the Havenith blood flux rates are much larger than the ASHRAE values. These much larger values of blood flux rates may be the result of the shell concept employed by Havenith. The shell consists of muscle in addition to the three skin layers. Equation (17c) relates primarily to the resistance of the skin blood flux rate. Equation (17h) relates to skin blood flux rate plus muscle resistance and muscle blood flux rate. From this it is clear that the Havenith values include both the skin and muscle blood flux rates. This may explain why the Havenith values of {dot over (m)}b,sk are so much larger than the ASHRAE values. Furthermore, Havenith has stated that the skin blood flux rate decreases with increasing Mact, as already demonstrated by the ASHRAE result, while the muscle blood flux rate increases with increasing Mact. Intuitively, it might be reasoned that the muscle blood flux rate could be much greater than that in the skin. If that is so, then as Mact increases, the decrease in the skin blood flux rate is more than offset by the increase in the muscle blood flux rate. The net effect is that the larger total blood flux rate (8.81 lbm/hr-ft2 (43 kg/hr-m2)) would indeed occur at the higher values of Mact as demonstrated.
The above discussion suggests that either expression for K could be used to calculate the metabolic heat flow. However, if the above assumption that the total blood flux rate in the Havenith approach is that of both the skin and the muscle, then the use of Havenith's expression might be more appropriate. The reason for this is that the metabolic heat rate (Mact) now determines the blood flux rate in the muscle as per Eq. (17d), and therefore its effect on muscle resistance (Rm) and consequently its contribution to the value of K. This aspect of core to skin heat transfer is not present in the ASHRAE conductance term. Nevertheless, the average value of the skin blood flux rate ({dot over (m)}b,sk) of 1.5 lbm/hr-ft2 (7.32 kg/hr-m2) as determined in the ASHRAE calculations was used in the evaluation of the Havenith expression for K from Eq. (17i).
The selection of the more appropriate expression for K was determined in another way, although in an unusual manner. Consider the two average blood flux rates of 1.5 lbm/hr-ft2 (7.32 kg/hr-m2) for the skin and 8.61 lbm/hr-ft2 (42.04 kg/hr-m2) for the muscle and skin. From the difference of these two values, the muscle blood flux rate alone becomes 7.1 lbm/hr-ft2 (34.67 kg/hr-m2). With a blood density of 62.4 lbm/ft3 (999.52 kg/m3), these two blood flux rates (1.5, 7.1) correspond to blood flux rate velocities of 0.29 and 1.37 in./hr (0.74 to 3.48 cm/hr). Based on one of the inventors' personal experience with influenza inoculation, pain accompanied with shivering was felt in the center of the underarm 5 to 6 hours after inoculation. This is defined as the start of the body reaction time to the influenza strain injected into the arm. The distance between the syringe's insertion point in the arm and the center of the underarm was approximately 6.5 in. (16.51 cm). With this approximate distance, the skin blood velocity (0.29 in./hr (0.74 cm/hr)) and the muscle blood velocity (1.37 in./hr (3.48 cm/hr)) correspond to approximate body reaction times of 22.4 and 4.7 hours, respectively. Assuming no other effects that might dramatically alter this body reaction time, the time of 4.7 hours appears realistic since it is very close to the 5 to 6 hour reaction time experienced. Consequently, it lends credence to the validity of Havenith's value of K and for that reason this K was selected in the calculation of qb.
Equation (17h) shows K to be dependent upon {dot over (m)}b,sk and Mact, both expressed in SI units. As already stated, the average {dot over (m)}b,sk of 1.5 lbm/hr-ft2 (7.32 kg/hr-m2) obtained from the ASHRAE calculations was used in this expression for K after converting it to units of 7.32 kg/hr-m2. Although {dot over (m)}b,sk takes on this constant value, Mact depends on the individual's activity and must be specified from Table 1, above, where Mact is listed for various wintertime activities. Listed in units of Btu/hr-ft2, it is convenient to maintain Mact in these units when substituting them into Eq. (17h). To do so, the values of Mact in Table 1 must be multiplied by 3.1546 to convert them to W/m2. Making these substitutions in Eq. (17h) and then converting the entire equation to English units, the equation for K becomes:
This expression for K is now used in Eq. (15) to calculate the metabolic heat transfer (qb).
The above defined expressions for hfc and hr from Eqs. (13c) and (14c), the expressions for quantities qe and qi, along with qb from Eq. (15) and its conductance term from Eq. (17l) are the replacements for their counterparts in Eq. (12) when the latter is applied to humans. The surface temperature (Ts) in the forced convection, radiation, and metabolic heat terms is replaced with the facial temperature (Tf) and the equivalent temperature (Teq) now becomes the wind chill temperature (Twc). The above quantities and Eq. (8a) relating to the natural convection coefficient were substituted into Eq. (6) and the following equation for the wind chill temperature (Twc) was obtained in the same manner as Eq. (9) for the equivalent temperature (Teq),
where in this study, the dimensions D=7 in. (0.5833 ft, 17.78 cm) and L=8.5 in. (0.7083 ft, 0.2159 m) must be expressed in ft, and where all temperatures (Twc, Tf, Ta, TCR) are expressed in ° F. absolute (or ° R) and where C1 is determined from Eq. (8c).
Equation (18a) is the final form of the wind chill temperature. At this point, exponent φ is unknown. As already stated, for natural convection from a heated vertical surface, plane or cylinder, φ=0.25 if the sole source of the heat is within the surface itself. This is not the case for a human where the facial surface heat is being supplemented by the metabolic heat from the body core. In this case, φ will take on a value different from 0.25. Natural heat convection (qnc) will occur in a no wind (V=0 mph (0 Km/hr)) environment when the facial surface heats the adjacent air layer, causing upward convection. In the absence of forced convection (qfc) heating, the air layer will be heated by radiation (qr) from the facial surface, which is not only being heated by the metabolic heat (qb) but also by solar radiation (qi). The evaporative heat loss (qe) will have a cooling effect on the adjacent air. Therefore qnc=qr+qb+qi−qe. With the components of qnc as shown in Eq. (18a) and with Eq. (8a) substituted in Eq. (7), where Ts and Teq have been replaced by Tf and Twc, Eq. (7) becomes,
(C1)L−φ(Tf−Twc)(1+φ)=(1.3712×10−9)(Tf+Ta)(Tf2+Ta2)(Tf−Ta)+{1.721+[1.7611+0.0854(Mact)]−1}−1(TCR−Tf)+(0.65)G−(971){dot over (w)} (18b)
which upon solving for Twc becomes,
For specified values of Ta, Mact and G, Eq. (18a) with V=0 and Eq. (18c) are two equations in two unknowns (Twc, φ). To simplify the solution of these quantities, sunshine was assumed to be absent (G=0). This absence means that the values of φ so determined will reflect the lowest values of Twc whenever Eq. (18a) is applied even if the effect of sunshine is subsequently included. In addition, by assuming the individual's usual physical activity as “walking about”, then from Table 2 Mact=31 Btu/hr-ft2 (97.79 W/m2) and {dot over (w)}=0.00655 lbm/hr-ft2 (0.032 kg/hr-m2). By making these assumptions and noting that Tf=91.4° F. (551.09° R) and TCR=98.2° F. (557.89° R), values of Twc and φ from Eqs. (18a) and (18c) were determined as functions of Ta, the only remaining variable. A plot of φ as a function of Ta was curve fitted using TableCurve 2D to give the following expression for φ:
φ=a+bexp[−(Ta/C)] (18d)
where a=0.46259934, b=0.077254543 and c=−59.573525, and where the correlation coefficient is r2=0.998884458. The wind chill temperature (Twc) at a given ambient temperature (Ta) can now be determined from Eq. (18a) after determining φ from Eq. (18d).
Equation (18a) can be viewed as a complete expression for the wind chill temperature, since it considers all heat losses and gains that contribute to it. It is applicable to any individual engaged in one of a variety of physical activities from merely standing to being engaged in a strenuous activity. Knowing the specific location of the individual, the solar radiation (G) is known as well as the altitude (H). Thus, the wind reduction factor (WRF) can be determined as described below.
Equation (18a) is a detailed expression for the wind chill temperatures (Twc) on the facial surface that reveals certain features of this temperature that might not be immediately apparent. These features become evident when one considers the particular situation where there is no sunshine (G=0), no wind reduction (WRF=0), and where there is negligible heat conduction (K=0) and evaporation (w=0). These features are as follows:
Facial Temperature Equation: The wind chill model disclosed herein predicts the wind chill temperature immediately upon exposure and the time to freeze if exposure continues. The determination of this time to freeze required the development of a time dependent facial temperature (Tf) equation for the outer surface of the facial skin segment as shown in
(1) Lumped Capacitance Approach: In the lumped capacitance approach, all heat losses (qfc, qr, qe) from the segment and the heat gains (qb, qi) to the segment are lumped together in a determination of the facial temperature. This temperature variation was determined from the time variation of the average skin segment temperature (TSG) defined as TSG=(TCR+Tf)/2 in
ĖSG=ĖIN−ĖOUT (19)
With A as the surface area of the skin segment, using the heat loss and heat gain terms from Eq. (18a) in Eq. (19) gives,
where ρ is the skin density, Cp is the skin specific heat and the skin segment thickness s=0.0175 ft (5.33 mm), as previously determined. Rationalizing Eq. (20) and noting that TCR remains constant as the segment cools, then,
Rewriting Eq. (21b) with time as the dependant variable and integrating gives,
Eq. (21c) could be used to get the facial temperature (Tf) at any time (t) after exposure. Unfortunately, the integral on the right side is not known to exist, at least at the time of this writing. Thus, the facial temperature was determined through a numerical integration of Eq. (21b).
In Eq. (21b), the skin density (ρ) and the specific heat at constant pressure (Cp) were taken to be the same as those for water at a facial temperature of 91.4° F. (33° C.). Choosing the values to be those for water was based on a finding by Meyer (1971) that the product kρCp for the skin should lie within the range of 15-60 Btu2/hr-ft4-° F.2 (1.741×106-6.964×106 kg2/s5-° C.2), see Meyer, G. E., W., Analytical Methods in Conduction Heat Transfer, McGraw-Hill Book Company, pp. 202 and 491, 1971. For water ρ=62.42 lbm/ft3 (999.84 kg/m3), Cp=1 Btu/lbm-° F. (4.1868 KJ/kg-° C.) and the thermal conductivity k=0.36 Btu/lbm-° F. (0.62 W/m-° C.). Consequently, the product kρCp=22.47 Btu2/hr-ft4-° F.2 (2.608×106 kg2/s5-° C.2) for water lies within the above range. This value of the product, kρCp, is substantiated by similar values determined by Buettner, Yuan et al. and Valvano et al., where their values were 14.98, 17.39 and 17.37 Btu2/hr-ft4-° F.2 (1.739×106, 2.019×106 and 2.016×106 kg2/s5-° C.2, respectively). See, Yuan, D. Y., Valvano, J. W., Rudie, E. N. and Xu, L. X., “2-D finite difference modeling of microwave heating in the prostate”, http://www/ece/utexas.edu/˜valvano/research/ASME95.pdf, 1995, and Valvano, J. W., Nho, S. and Anderson, G. T., “Analysis of the Weinbaum-Jiji model of blood flow in the canine kidney cortex for self-heated thermistors”, http://www/ece/utexas.edu/˜valvano/research/ASME94.pdf, 1999. With the value of ρ=62.42 lbm/ft3 (999.84 kg/m3) and Cp=1 Btu/lbm-° F. (4.1868 KJ/kg-° C.) substituted into Eq. (21b), this expression for the temperature gradient when written in an incremental form becomes,
With this equation, a step-by-step calculation of Tf can be made for any combination of Ta and V starting at time zero when m=0 and where Tjm=91.4° F. (33° C.). Calculations with D=7 in. (0.5833 ft, 0.1778 m) have shown that the optimum time increment Δt=1 sec.
(2) Infinite Series Approach: Eq. (22) represents what is referred to as a lumped capacitance approach to heat transfer. It is a preferred method of treating transient conduction and indeed the only simple means when multiple modes of heat transfer exist. However, there is an error associated with a lumping of these multiple modes. It is small if the Biot number (Bi) here is defined as hfc(s/2)/k≦0.1. This ratio implies that if only an external forced convection heat loss was present, and none of the other heat transfer modes, it would initiate a one-dimensional conduction of heat through the skin thickness (s) driven by the temperature difference (TCR−Ta). With the known values for the skin thickness (s) as 5.33 mm (0.0175 ft) and thermal conductivity (k) as 0.36 Btu/hr-ft-° F. (0.62 W/m-° C.) and with the forced convection heat transfer coefficient (hfc) calculated from Eq. (13c), the Biot number can be calculated. With the Biot number, the magnitude of the error associated with the lumped capacitance approach can be determined.
Since the hfc is velocity (V) dependent, a calculation of Bi as a function of velocity is shown in
The one-dimensional facial temperature expression uses the following existing exact infinite series solution disclosed in Incropera, F. P. and DeWitt, D. P., Introduction to Heat Transfer, John Wiley & Sons, Chap. 5, 1985,
With T as the skin temperature at any axial location (x) in the direction of heat flow, θ is the temperature difference (T−Ta) with Ta being the ambient temperature. θ* is the temperature difference θ normalized by the maximum temperature difference (TCR−Ta) where TCR is the core temperature (98.2° F. (36.78° C.)). Thus,
In Eq. (23a), the Fourier number, Fo=4αt/s2 where α is the skin's thermal diffusivity (k/ρCp) and is shown plotted in
where the discrete values (eigenvalues) of ζn are positive roots of the transcendental equation,
ζn tan ζn=Bi (23d)
This infinite series solution can be approximated by the first term of the series solution when Fo≧0.2.
The normalized mid-segment temperature is,
which yields the mid-segment temperature as,
Tx*=o=To=θo*=(TCR−Ta)+Ta (23f)
The normalized surface temperature is,
which yields the surface or facial temperature as,
Tx*=1=Tf=θ*(TCR−Ta)+Ta (23h)
Equation (23h) is the facial temperature equation based on an infinite series heat conduction solution. Facial temperatures using this equation were compared with those of Eq. (22) to determine if a significant error is encountered as a result of using the lumped capacitance approach.
Time to Freeze Equation: In addition to providing a more precise prediction of the wind chill temperature (Twc), a key feature of the wind chill model of the present invention is its capability of determining the exposure time when facial freezing will occur. The wind chill temperature, although a true sensed temperature, becomes subjective in nature, since individuals may differ greatly in their perception of its actual magnitude. When the time to freeze (tf) is specified, along with the wind chill temperature, it frees the individual from becoming overly concerned with the actual value of the temperature. Instead it becomes a warning to the individual of a limiting time after exposure when facial freezing will occur. It has the added benefit of allowing preplanning of outdoor activities, so as not to exceed this time limit.
Ideally the dependence of facial temperature (Tf) on time (t) would be obtained through integration of Eq. (21c) to get a closed form solution for the time to freeze (tf). Unfortunately, the integral shown in Eq. (21c) does not exist. Instead, the time to freeze (tf) was determined from Eq. (22) through step-by-step calculations of the time for the facial temperature to decrease from the initial value of 91.4° F. to 32° F. (33° C. to 0° C.). In this manner, considerable data was generated on the time to freeze. This data was then curve fitted to obtain an explicit equation for the time to freeze (tf). In the generation of this data, the evaporative heat loss (qe) and the metabolic heat gain (qb) were neglected in Eq. (22) since indications were that their effects would be offsetting. Only the primary heat losses (qfc, qr) were considered along with the solar heat gain (qi), where the latter is based on values of G≧0 depending upon the presence or absence of sunshine. As a further simplification, sea level altitude (H=0 ft (0 m)) was assumed with the intent of developing a subsequent correction to tf for any other altitude.
Four equations for tf were derived, one when sunshine is absent (G=0) and three others when sunshine is present (G>0), the latter three values of G being those for the latitude regions of Table 2, above. Calculations were made over the widest range of ambient temperature (−140° F.≦Ta≦25° F. (−95.56° C.≦Ta≦−3.89° C.)) and wind velocities (0 mph≦V≦160 mph (0 Km/hr≦V≦257.49 Km/hr)) anticipated worldwide. The following procedure was used:
The eleven dashed curves of
where a=12.213472, b=−4.7287903, c=0.71377035, d=−0.040374904, e=0.0020284147, f=−0.042770886, g=0.0048025828, h=−0.016212428, i=−2.3703809×10−5 and where the correlation coefficient is r2=0.99984977. The curve fitted values of tf vs. log V using Eq. (24) are shown as solid lines in
As already noted, it was essential to use Eq. (22) in the determination of Eq. (24) because the integral in Eq. (21c) was nonexistent. If the integral had consisted of only one rather than five heat loss/gain terms, a closed form solution of Eq. (21c) would have been possible. This means that if either of the primary heat losses (radiation, forced convection) is present, a closed-form solution for the time to freeze (tf) is possible.
Closed Form Solution of tf-Radiation Only: When an individual is exposed to a cold environment, a radiation heat loss will always exist when Tf>Ta even in the absence of a forced convection heat loss (V=0 mph (0 Km/hr)). Considering only the radiation heat loss as defined in Eq. (2), Eq. (21c) becomes,
Integrating from the initial exposure time, ti=0, when the initial facial temperature is Tfi=91.4° F. (551.09° R) to the time to freeze, tfr, when the final facial temperature Tff=32° F. (491.69° R) gives,
Inserting the numerical values for the facial temperature limits into Eq. (25b) gives,
With the values of ρ, Cp, s, ε, and σ already known, Eq. (25c) provides an exact value for the times to freeze, tf, at any ambient temperature (Ta) when only the radiation heat loss is present.
Closed Form Solution of tf-Forced Convection Only: A forced convection heat loss is not likely to exist in the absence of a radiation heat loss. However, if it does, and with the convection heat loss as defined by Eq. (1), Eq. (21c) becomes,
Integrating as before gives,
Inserting the numerical values for the facial temperature limits into Eq. (26b) along with the expression for hfc from Eq. (13c) gives,
With values of ρ, Cp, s and D known and ambient conditions (Ta, V) specified along with altitude (H) and wind reduction factor (WRF), Eq. (26c) provides an exact value for the time to freeze when only a convection heat loss is present.
Wind Chill Temperature: Comparisons were made between the predictions of the wind chill temperatures using Eq. (18a) and the Siple and Passel results. The Siple and Passel experiment was conducted at or near sea level (H=0) conditions at Little America, Antarctica. Because their experiment was conducted out in the open, well above ground level, there was no reduction in the wind speed (WRF=0). Neither the solar heat gain (G=0) nor the evaporative heat loss ({dot over (w)}=0) were of concern, since Siple and Passel, when referring to insolation and evaporation, stated that these factors “were almost missing in the Antarctic winter when the experiments were carried on”. Finally, the metabolic heat gain was not present (qb=0) because it did not exist in their experiment. Since the radiation heat loss term in Eq. (18a) and the experiment are identical, except for an emissivity factor, any major difference in the wind chill temperatures as predicted by Eq. (18a) and the Siple and Passel results would be due to a difference in the forced convection coefficients. Determining this difference meant extracting the Siple and Passel forced convection coefficient from their test results and comparing it with the forced convection coefficient of Eq. (13c) as used in this model. This is described in the following paragraphs.
Determination of the inherent forced convection coefficient from the Siple and Passel results began with an equation that expressed their results as wind chill temperatures. This equation,
Twc=91.4−(0.04544)[10.45−0.447V+6.6858V0.5](91.4−Ta) (27a)
was obtained from Marks' Standard Handbook for Mechanical Engineers, 9th ed., McGraw-Hill, Chap. 12, p. 113, 1989, where it is shown as Eq. (12.4.47). In this equation, V is in mph and Twc and Ta are in ° F. This equation can be written in terms of what has been called the wind chill index (WCI), which represents the heat loss rate ({dot over (q)}) from their test container when the container surface temperature was assumed to be at a constant 91.4° F. (33° C.). Although Siple and Passel erred in their assumption that this temperature of 91.4° F. remained constant, a simple test using a hand held thermometer in contact with the skin will verify that this was a good choice for the initial skin temperature of a human suddenly exposed to a cold environment. Their so-called wind chill index (WCI) from Eq. (27a) is,
WCIE=[10.45−0.447V+6.6858V0.5](91.4−Ta)={dot over (q)}E (27b)
and has English units of Btu/hr-ft2. It is shown as Eq. (12.4.45) in Mark's. Siple and Passel, who initiated the term WCI, expressed it as,
WCISI=[10.45−V+10V0.5](33−Ta)={dot over (m)}SI (27c)
where it has SI units in Kcal/hr-m2 and is shown as Eq. (12.4.44) in Mark's. Equation (27a), as shown in Eq. (12.4.47), can be expressed in terms of WCIE as,
Twc=91.4−0.04544(WCIE) (27d)
and its counterpart in SI units from Eq. (12.4.44) in Mark's can be written as,
Twc=33−0.04544(WCISI) (27e)
Since there is an equivalency in Twc irrespective of the units in which it is expressed, then from Eqs. (27d) and (27e), WCIE=WCISI. This means that {dot over (q)}E={dot over (q)}SI where from Eqs. (27b) and (27c),
{dot over (q)}E=[10.45−0.447V+6.6858V0.5](91.4−Ta)=hE(91.4−Ta) (27f)
and,
{dot over (q)}SI=[10.45−V+10V0.5](33−Ta)=hSI(33−Ta) (27g)
These two equations demonstrated that over the same temperature range, hE must be made equivalent to hSI through a unit conversion factor. To convert from SI units in which the Siple and Passel results are expressed in English units, hE must be multiplied by 1/4.88 or 0.2049. Thus, the actual Siple and Passel WCI or heat flow rate is not the expression in Eq. (27b) but rather,
WCI=(0.2049)[10.45−0.447V+6.6858V0.5](91.4−Ta) (27h)
The presence of this coefficient (0.2049) indicates that Eq. (12.4.45) in Mark's is in error. It also changes the existing coefficient (0.5556) in the WCI expression in Eq. (69) of 1993 ASHRAE Handbook, Fundamentals to 0.1138. Consider the case where V=4 mph (1.79 m/s) and Ta=0° F. (−17.78° C.). The error in Mark's will be demonstrated based on the heat flux rate followed by the convective heat transfer coefficient equivalencies. Substituting the values in English units in Eq. (27b) yields {dot over (q)}E=2013.87 Btu/hr-ft2 (6352.95 W/m2). Similarly, substituting the values in SI units in Eq. (27c) yields {dot over (q)}SI=1119.15 Kcal/hr-m2 or 1301.32 W/m2 (412.64 Btu/hr-ft2). They are equivalent when Eq. (27b) is multiplied by a factor of 0.2049. Similarly, substituting the values in English units in Eq. (27f) yields hE=22.03 Btu/hr-ft2-° F. (125.10 W/m2-° C.). Substituting the values in SI units in Eq. (27g) yields hSI=25.63 W/m2-° C. (4.51 Btu/hr-ft2-° F.). They are equivalent when Eq. (27f) is multiplied by a factor of 0.2049.
Eq. (27h) represents the actual heat flow from the Siple and Passel container. Eq. (27h) may be expressed as,
WCI=hfc,r(91.4−Ta) (27i)
where hfc,r is a combined forced convection and radiation heat transfer coefficient measured during their tests and where from Eq. (27h),
hfc,r=(0.2049)[10.45−0.447V+6.6858V0.5] (27j)
Since the radiation coefficient in the Siple and Passel tests was expected to take on the same form as that in Eq. (14c) for the human head, their radiation coefficient, assuming their emissivity of ε=1 rather than ε=0.8, was,
hr=(1.714×10−9)(91.4+Ta)└(91.4)2+Ta2┘ (27k)
Consequently, the Siple and Passel forced convection coefficient (hfc) was determined from,
hfc=hfc,r−hr (271)
where upon using Eqs. (27j) and (27k), it became,
hfc=(0.2049)└10.45−0.447V+6.6858V0.5┘−(1.714×10−9)(91.4+Ta)[(91.4)2+Ta2] (27m)
A comparison was made of the forced convection coefficient values using this deduced expression of hfc from the Siple and Passel experiment with the corresponding values from the analytical expression (hfc) shown in Eq. (13c). The comparison was made for an ambient temperature (Ta) of 0° F. (459.69° R) over the velocity range 0<V<45 mph (0<V<72.42 Km/hr) that existed during the Siple and Passel experiment. In addition to the previously mentioned conditions (WRF=H=0) in Antarctica, the surface temperature (Ts) in Eq. (13c) was set at 91.4° F. (33° C.) and the diameter D at 2.26 in (0.1883 ft, 0.0574 m) so as to match the container diameter used in the Siple and Passel experiment. Calculated values of the two hfc coefficients are plotted in
From
The wind chill temperature was then calculated for the head (D=7 in.) using the hfc (curve 2) from
Facial Temperature: No direct comparison of facial temperature as predicted by Eq. (22) was possible since no other model predictions of these temperatures were available. However, the predicted values showed good agreement with several sets of test data. The initial interest in the facial temperature stemmed from Siple and Passel's assumption of a constant body surface temperature which limited the determination of wind chill to individuals momentarily exposed to the elements. It does not apply when individuals are subjected to an extended exposure when the surface temperature must necessarily decrease. Bluestein and Zecher in their calculations of a more accurate wind chill considered the time dependency of this temperature. Aside from their correction of this constant surface temperature, Bluestein and Zecher presumably made other corrections to what they and numerous investigators have referred to as the flawed Siple and Passel experiment. The inference was that these flaws introduced error or somehow invalidated the experiment's results. Looking at each of these flaws, the simple human error of misidentifying the heat transfer coefficient as a cooling rate does nothing to invalidate the results. Neither does a particular curve fit of the data invalidate the data upon which it was based. Granted the data itself had inaccuracies, which seem to have been due to conditions beyond the control of the experimenters. Furthermore, it was stated that Siple and Passel ignored thermal gradients through the water and the container and did not consider the difference between radiation and convection heat loss. It should be obvious that in their experiment these gradients naturally developed, and that the radiative and convective heat transfer actually took place as part of the whole thermal process of container cool-down. The fact that these gradients and heat losses were an inherent part of the experiment makes the claim that they were not considered totally irrelevant. One concludes from all this that the only valid flaw was Siple and Passel's assumption of a constant body temperature during the entire exposure time.
Bluestein and Zecher's development of a more accurate wind chill model was intended to correct the flaws in the Siple and Passel experiment. They recognized the fact that Siple and Passel's assumption of a constant 91.4° F. (33° C.) facial temperature would lead to a lower (colder) wind chill temperature than that which would actually exist. Their iterative heat transfer analysis of the surface of a cylinder simulating the human head provided a means of computing this temperature as it decreased with exposure time. Consequently, their results would have been expected to yield considerably higher (warmer) wind chill temperatures than those of Siple and Passel. But this was not the case. This can be verified by a comparison of the tabulated results in Bluestein and Zecher's paper. Table 1 of their paper shows the Siple and Passel wind chill temperature from Eq. (27a) based on their assumption of a constant 91.4° F. facial temperature. Table 2 of their paper shows the final Bluestein and Zecher results based on a varying facial temperature. These Table 2 results also include a wind reduction factor (WRF) of 0.33 to account for the fact that the NWS 10 m velocity was assumed to be 50% greater than that at head level. The portion of these Table 2 results that can be attributed to the varying facial temperature can be determined by removing the wind reduction effects from these results. This can be done for a given temperature (Ta) and velocity (V) by using the WRF of 0.33 to reduce the velocity in Eq. (27a) and then comparing the computed wind chill temperature with the corresponding value at the same Ta and V in Table 2. The difference in these values reflects what Bluestein and Zecher found as an increase or moderation in the wind chill temperature as a result of the facial temperature variation. For example, consider the case where Ta=−40° F. and V=40 mph. Using a WRF=0.33 in Eq. (27a) yields a Siple and Passel value of Twc=−105.97° F. The corresponding Bluestein and Zecher value in their Table 2 is Twc=−106.7° F. This slight difference of −0.73° F. is not even a moderation but rather a decrease in the wind chill temperature. For all other combinations of Ta and V, the Bluestein and Zecher results show, at most, a 2° F. moderation in the wind chill temperature. Consequently, without any wind reduction, the Siple and Passel and Bluestein and Zecher results are nearly identical, as illustrated in
This means that the Bluestein and Zecher results reflect a negligible moderation in the wind chill temperature due to a decreasing facial temperature. Assuming Bluestein and Zecher's calculations of the decreasing facial temperature are correct, then their choice of the Churchill-Bernstein forced convection equation may have been the reason that their results show this negligible moderation. Bluestein (1998) chose the Churchill-Bernstein equation because it was appropriate for the Reynolds and Prandtl numbers that existed during the Siple and Passel experiment, see Bluestein, M., “An evaluation of the wind chill factor: its development and applicability”, J. of Biomech. Eng., Vol. 120, pp. 255-258, 1998. Perhaps the choice of the Churchill-Bernstein equation was influenced by the fact that this equation yields values of the forced convection coefficient that are nearly identical to those of Eq. (6) in Oscevski's paper. Oscevski had already shown that in applying Eq. (6) to the Siple and Passel cylinder he achieved good agreement with his own test data as shown by curves 3 and 4 in
Predicted facial temperatures using Eq. (22) were verified using the computed and measured facial temperatures found by Adamenko and Khairullin, by Buettner (1951), by LeBlanc, J., Blais, B., Barabe, B. and Cote, J., “Effects of temperature and wind on facial temperature, heart rate, and sensation”, J. Appl. Physiology, Vol. 40, No. 2, pp. 127-131, 1976, (hereinafter, LeBlanc et al. (1976)), by the inventors through home freezer experiments and those found during a Discovery Channel™ experiment. The previously discussed experiment by Adamenko and Khairullin, which was conducted in northern Russia, was extensive in that it consisted of instrumenting three facial components (cheeks, nose, ears) on 40 people during their exposure to temperatures (Ta) ranging from 10° C. to −40° C. (50° F. to −40° F.) and wind velocities (V) up to 15 m/s (33.55 mph). The temperature of each component after a half to one hour was expressed in terms of the ambient temperature and wind velocity. In each case, the component temperature was 33° C. (91.4° F.) prior to exposure. These test data provided information on facial temperature variation and the ambient conditions when facial freezing would occur. This information was expected to be useful in determining the exposure time when facial freezing occurs, since this facial freezing time is a key element in our model that makes the concept of wind chill temperature less subjective.
The results of the Adamenko and Khairullin experiment were described by Schaefer, J. T., “The effect of wind and temperature on humans”, National Weather Service, Central Region Technical Attachment 88-05, pp. 1-2, 1988, as further described in Schwerdt, R. W., “Letters to the editor”, Bull. Amer. Meteor. Soc., Vol. 76, No. 9, pp. 1631-1636, 1995. Schwerdt made a convincing argument for informing the listening public of the relationship between the wind chill temperature and facial freezing. The Adamenko and Khairullin results were presented as a least square equation of all measured temperatures for each of the three facial components in terms of the ambient conditions. An average of these equations provided an equation for the approximate facial temperature (Tf) after a half to a one hour exposure to the ambient conditions. This equation is,
Tf=0.4Ta−3.3V0.5+16 (28)
where the temperatures (Tf, Ta) are in ° C. and the wind velocity (V) is in m/s. As an example, this equation shows that if the face is to cool down from 33° C. to a condition of facial frostbite (Tf=0° C.) within a one half to one hour exposure in calm conditions (V=0 m/s), then the ambient temperature must be Ta=−39.9° C. It should be noted that Eq. (28) for the face temperature is similar to each of the three component equations in that they all differ only in the value of the constant. These values are 19, 17, and 12 for the cheek, nose, and ear, respectively; the lower the value, the more readily the component freezes. Adamenko and Khairullin concentrated their efforts on determining the nose temperature, because the nose is more difficult to protect than the cheeks and the ears, and consequently is more vulnerable to freezing. The use of Eq. (28) for the face temperature, where the constant (16) is slightly less than that for the nose (17), provides a degree of conservatism in that it forewarns of the possibility of freezing before it is likely to occur.
Equation (28) is expressed as follows in English units so as to put it in a more practical form for facial temperature calculations,
Tf=0.4(Ta−32)−3.97V0.5+60.8 (29)
where in this equation, Tf and Ta are in ° F. and V is in mph. The initial facial temperature upon exposure to the wind is now 91.4° F. (33° C.). Equation (29) represents the face temperature after a half to one hour cool down from 91.4° F. at ambient conditions (Ta, V). Schwerdt notes that “The higher the wind speed, the faster (closer to a half hour than an hour) the skin will freeze, assuming all conditions except wind speed are equal”. Consequently, for the higher wind speeds of interest in wind chill predictions, the facial temperature Tf in Eq. (29) was assumed to be reached after a half hour of exposure. It should also be emphasized that although Eq. (29) permits the calculation of the face temperature after 30 min., it cannot be used to determine the temperature at some other time within this 30 minute interval. This is because the actual variation of the facial temperature with time is not linear as one might infer in an application of Eq. (29). Rather it is nonlinear and because of this nonlinearity, Eq. (22) for the surface temperature was developed using the conservation of energy principle. Before comparing the computed values of the facial surface temperature (Tf) from Eq. (22) with the Adamenko and Khairullin test results, a comparison was made with four other studies and experiments.
Buettner Study: A study by Buettner (1951) determined the temperature decay of warm bare skin when impacted by a cold stream of air. His expression for the decrease in temperature, here expressed as a facial temperature, is the classic solution for surface convection over a semi-infinite solid, that is,
where the thermal diffusivity (α) is k/ρCp, t is the time of exposure to the air stream of velocity V and h is the heat transfer coefficient. A time dependent calculation of this temperature was made for the case where the initial face temperate (Tf) was 86° F. (30° C.), where in the calculation of h, Ta=−40° F. (−40° C.), V=13.41 mph (21.58 Km/hr), P=2116.8 lbf/ft2 or 1 atm (sea level conditions) and where the values of ρ and Cp were the same as those used in the derivation of Eq. (22). With the forced convection coefficient, h=7.16 Btu/hr-ft2-° F. (40.66 W/m2-° C.), for a cylinder in crossflow, the decay in facial temperature as shown in
Leblanc et al. Experiment: The second comparison was made with the results of a study conducted by LeBlanc et al. In this study, each of 25 subjects was placed at the end of a wind tunnel blowing air directly into their faces at a temperature of −20° C. to 24° C. (−4° F. to 75.2° F.) and at speeds up to 30 mph (48.28 Km/hr). In each case the initial face temperature (Tfi) was 30° C. (86° F.). After 10 min. of exposure to these conditions, temperatures were measured on the nose, forehead, and cheek.
Home Freezer Experiments: The third comparison was made with the results of three home freezer experiments conducted by the inventors. In all of these experiments, different objects were used to simulate the human head. Each object was heated to an approximate temperature of the 91° F. to 92° F. (32.78° C. to 33.33° C.), this being the accepted temperature of the human face when suddenly exposed to a cold environment. The heated object was then placed in a freezer where the simulated ambient temperature (Ta) varied from −18° F. to 8° F. (−27.78° C. to −13.33° C.). Immediately upon placement in the freezer, the surface temperature (Ts) of the object and the freezer temperature were recorded at 5 minute intervals. Values of T, were plotted as a function of time (t), curve fitted and then compared with a surface temperature prediction using Eq. (22). In the absence of any air movement (V=0 mph (0 Km/hr)) within the freezer, Eq. (22) predicts the surface temperature only as a result of a radiation heat loss.
In the first experiment, the head was simulated using a 10 in. (25.40 cm) diameter, 2.5 in. (6.35 cm) deep wooden bowl filled with 2.25 lbs (1.02 kg) of tap water heated to a temperature greater than 92° F. (33.33° C.). A thermometer was used to measure the water temperature 5 mm (0.0164 ft) below the water surface at the bowl center. When the water had cooled down to the desired 92° F. to 91° F. (33.33° C. to 32.78° C.) temperature range, the bowl was placed in the freezer and the water temperature (Ts) and the freezer temperature (Ta) were recorded in 5 minute intervals over a one hour time period. During this one hour period, the average freezer temperature was 9° F. (−12.78° C.), while the water surface temperature exhibited the decay shown in
The water in the bowl froze at 60 min. after placement in the freezer. In these experiments where only radiation heat losses exist, the closed form solution for the time to freeze (tfr), as shown in Eq. (25c), was used to determine the exposure time at the intersection of the Eq. (22) temperature decay curve with the surface freeze line (Ts=32° F. (0° C.)). In this case, tfr=64 min. These results compare favorably with the actual 60 min. and support the capability of predicting the time to freeze using the closed form solution of Eq. (25c).
Two other similar experiments were conducted by simulating the head using food items (chicken, pot roast) that might be found in a home freezer.
In each of these cases, the agreement with Eq. (22) is not as good as in the first experiment. The more rapid decrease in surface temperature is probably due to heat conduction through the outside surface of these meat products, whereas in the first experiment this was prevented by the insulative effect of the wood container. Even though neither of these cases seem to represent only a radiative heat loss,
Discovery Channel™ Experiment: The fourth comparison was made with the results of an experiment conducted for the Discovery Channel™ (2006) and described in an April 2006 show entitled “I Shouldn't Be Alive: Lost in the Snow.” A subsequent program entitled “Science of Survival” described the experiences of a family who ventured into the wilderness in mid-winter became lost, stranded, and narrowly escaped with their lives in the bitterly cold environment. Because of the family's inexperience with survival techniques, the program illustrated the many survival measures that could have been taken to insure their survival.
The “Lost in the Snow” episode emphasized the importance of preventing a drastic drop in the body core temperature (TCR) since a 13° F. drop can prove fatal. A dexterity test was conducted in a controlled, refrigerated chamber where the subject, Les Stroud a wilderness expert, sat clad only in shorts. The subject ingested a pill with a temperature sensor to transmit the body core temperature. During the test, the subject's skin and finger temperatures were monitored, the latter being important at the time when visually the subject might first suffer loss of dexterity.
The ambient temperature (Ta) in the chamber was 23° F. (−5° C.). For the first 30 min., the experiment was conducted in still air (V=0 mph). At 30 min., an air fan instantaneously increased the air speed to 5 mph (8.05 km/hr) for a period of 6 min. At that point, still air conditions were resumed. The test was continued at these conditions until the subject became extremely cold and uncomfortable.
The intent here was to compare the recorded time varying skin and finger temperatures with the predicted temperature using Eq. (22). There was an audible difficulty in gathering information off the video recording of the program plus the fact that essential information like the initial skin and finger temperatures was not provided. This resulted in an incomplete comparison of the measured and predicted skin temperature. For example, the 10° F. drop in finger temperature over a 15 minute period during the “still air” portion of the test and the measured 43° F. temperature at the end of the 40 minute test period quite clearly indicated that the initial finger temperature was close to the usual 91.4° F. This permitted a comparison of the measured and predicted finger temperature drop over the entire 40 minute test period. Such was not the case for the skin temperature. The first indication of skin temperature is the 81° F. at the end of the 30 minute “still air” portion of the test and its drop to 55° F. at the end of the 6 minute period when the velocity was 5 mph. No information was given on the initial skin temperature at the start of the “still air” portion. As a result, the only measured/predicted temperature drop comparison possible for the skin was during the 6 minute period.
Before the above comparison was made, it should be noted that one visual observation was supported by recorded data. The subject is shown to be shivering badly during the test. As discussed in the section METABOLIC HEAT GAIN (qb) “Shivering is a thermoregulatory process through which the core temperature (TCR) may be restored to its equilibrium or neutral value of 98.2° F. after a lowering.” Since the core temperature was shown to remain constant during the test, a lowering of the core temperature must have taken place before the onset of shivering which restored it to its initial value.
The recorded time varying finger temperatures were compared with the temperature decay curve using Eq. (22) with Ta=23° F., V=0 mph, Mact=31 Btu/hr-ft2 since the subject was involved in minor physical activity with his hands and assuming the following: no evaporative cooling ({dot over (w)}=0), no solar heat flux (G=0), sea level conditions (H=0 ft) and an initial finger temperature (Tf) of 91.4° F. This temperature decay curve is shown as the dashed curve of
The third curve of
Adamenko and Khairullin Experiment: Having established the validity of Eq. (22), it was then used to calculate values of Tf after 30 min. for a comparison with the Adamenko and Khairullin values of Tf from Eq. (29). In doing so, the values of H, Mact, {dot over (w)}, and G used in Eq. (22) had to be specified. Schwerdt notes that the Adamenko and Khairullin experiment was conducted in four Russian cities each at or near sea level. Thus, H was set equal to zero. The forty people instrumented were instructed to walk slowly during the test when the temperature on each of the facial components was recorded. Walking slowly suggests that the participants in the test were engaged in minimum physical activity. Therefore, the associated metabolic heat generation may be that for “walking about” as shown in Table 1, above, in which case Mact=31 Btu/hr-ft2 (97.79 W/m2) was used in the calculation of the metabolic heat flow. For minimum physical activity, {dot over (w)}=0.00655 lbm/hr-ft2 (0.032 kg/hr-m2) was used. The value of the solar radiation (G) during the test is unknown, although Adamenko and Khairullin stated that “[s]ufficiently intense solar short-wave radiation was also taken into account.”
Since the value of G during their experiment is unknown, the approach taken here was to determine if an assumed value of G in Eq. (22) would, at a given ambient temperature (Ta) and a velocity (V), yield values of Tf at 30 min. that are in reasonably good agreement with Adamenko and Khairullin values from Eq. (29). This approach was restricted by the fact that the selected value of G could not exceed the maximum value expected for the four Russian cities where the tests were conducted. Because the latitude at the Russian test sites was higher than that of the northernmost region (45°-50°) in the US, where G=37.78 Btu/hr-ft2 (119.17 W/m2) in Table 2, the maximum value of G was estimated to be 32.0 Btu/hr-ft2 (100.95 W/m2). Results of a calculation with an assumed value of G=5 Btu/hr-ft2 (15.77 W/m2) are shown in
With Eq. (22) validated, it can be combined with Eq. (18a) to determine the time variation in the wind chill temperature experienced by an individual during an extended exposure. Consider an individual that is exposed to ambient conditions of Ta=0° F. (−17.78° C.) and V=20 mph (32.19 Km/hr). Assume the absence of sunshine (qi=0). Realistically, the evaporative heat loss (qe) and the metabolic heat gain (qb) are always present. In view of making a cursory comparison with the Bluestein and Zecher results, it was also assumed that qe=qb=0.
Time to Freeze: In addition to providing a more precise prediction of the wind chill temperature (Twc), a key feature of the wind chill model of the present invention is its capability of determining the exposure time when facial freezing will occur. The wind chill temperature, although a true sensed temperature, becomes subjective in nature, since individuals may differ greatly in their discernment of its actual magnitude. For example, an individual is probably more likely to detect the difference between two warmer wind chill temperatures like 30° F. (−1.11° C.) and 15° F. (−9.44° C.) than two colder and potentially more dangerous ones like −20° F. (−28.89° C.) and −40° F. (−40° C.). When the time to freeze (tf) is specified along with the wind chill temperature, it frees the individual from having to be concerned with the actual value of the temperature. Instead it becomes a warning to the individual of a limiting time after exposure when facial freezing will occur. It has the added benefit of allowing the individual to preplan outdoor activities so as not to exceed this time limit. This was the basis for developing the time to freeze expression of Eq. (24).
There is no known experimental data or any existing equation for facial freezing time against which the predictions of Eq. (24) can be compared. However, actions taken by some contestants in the World Downhill Ski Championships during the 2003/2004 racing season confirmed the validity of Eq. (24) through the use of Eq. (22), from which it was derived. These races were conducted in the European Alps at what was considered a colder than normal temperature (Ta), believed to have been less than 10° F. (−12.22° C.). On this course, the contestants reached a velocity (V) of 80 mph (128.74 Km/hr) in about 6 seconds after leaving the starting gate. The total time to run the course was expected to be just under two min. The local racers, who were familiar with the course, were apparently aware from past experiences of the possibility of facial freezing under these conditions (Ta, V). As a result, these racers took precautions to prevent or minimize freezing by applying tape strips to their faces. At the finish line, none of the racers appeared to have experienced frostbite. The warming effect of sunshine could not have been a factor in the absence of frostbite, since the race course was in the shadow cast by the mountain. Altitude, which will be shown later to have the effect of delaying facial freezing, may have had some effect. The average altitude was estimated to be 8,000 ft (2,438.43 m) based on an estimated altitude of 10,000 ft (3,048.04 m) at the starting gate and a 4,000 ft (1,219.21 m) elevation drop to the finish line. The very strenuous activity of racing downhill at a speed of 80 mph would generate a higher than normal metabolic heat flow rate to the face, and that might have had a slight effect on forestalling frostbite.
Table 1, above, shows the normal heat generation (Mact) for downhill skiing as 96 Btu/hr-ft2 (302.84 W/m2) and that for cross-country skiing as 83 Btu/hr-ft2 (261.83 W/m2). The average velocity of a non-racing downhill skier might be 25 mph (40.23 Km/hr), whereas that for a cross-country skier might be 3 mph (4.83 Km/hr). It is postulated that the metabolic heat generation varies as the square of the velocity, so that 2 VdV/dMact=[2(14)(25−3]/(96−83)=47.38 (mph)2 (hr-ft2/Btu)=3.669×105 ft4/Btu-s (3 m4/J-s). With this gradient, the metabolic heat generation for the downhill racer would be Mact=217.89 Btu/hr-ft2 (687.35 W/m2). This value is much larger than that (96 Btu/hr-ft2 (302.84 W/m2)) for the more casual, non-racing downhill skier because of the much greater muscle exertion required to execute direction changing at the higher speed. Finally using the guidelines previously set forth, the racers' evaporation flux rate would be {dot over (w)}=0.02183 lbm/hr-ft2 (0.1066 kg/hr-m2), which, although a maximum, most certainly did little to enhance the possibility of frostbite.
None of the racers experienced facial freezing upon completion of the race at about 1 minute 50 seconds (1.83 min.), including the locals who anticipated this possibility.
Having verified Eq. (24), a prediction of the time to freeze using this equation was compared against two existing approximations. A case in point was the pre-dawn weather forecast for the morning of Jan. 5, 2004 in Minneapolis, Minn. The weather conditions were stated as Ta=−4° F. (−20° C.), V=13 mph (20.92 Km/hr) with a Twc=−23° F. (−30.56° C.), the latter correctly being the value of wind chill temperature from the current wind chill model. Intended to warn people walking to and waiting at bus stops during the dark morning hours of the potential severity of this Twc, the comment was made that under these conditions “freezing would occur in 30 min.”. This comment suggests the meteorologist's familiarity with an equation like Eq. (29) which, as previously discussed, expresses facial temperature (Tf) after 30 min. of exposure to a given set of ambient conditions (Ta, V). Because the above values of Ta=−4° F. and V=13 mph in Eq. (29) result in a facial temperature of exactly 32° F. (0° C.) after 30 min., the comment was likely made to warn the listeners of the possibility of facial freezing, should their exposure time equal or exceed 30 min. Unfortunately, this was misleading advice for two reasons. First, Eq. (29) was derived from the Adamenko and Khairullin test results where their experiment was conducted under partially sunny conditions. Consequently, Eq. (29) would reflect an increase in the facial freezing time (tf) as a result of solar radiation. Since sunshine was absent during the dark morning hours, a freezing time of 30 min. would be expected to be an over-prediction. Secondly, as the results of
Aside from using Eq. (29) to determine the freezing time, the Minneapolis meteorologist could have used an equation developed by Environment Canada (2002) to approximate the “min. to frostbite” for 5% of the population that is most susceptible to frostbite. The equation is,
tf={[−24.5(0.667V+4.8)]+2111}(−4.8−Ta)−1.668 (31a)
where Ta is in ° C. and V is in km/hr. The equation is limited to −50° C.≦Ta≦15° C. (−58° F.≦Ta≦59° F.) and 20 km/hr≦V≦80 km/hr (12.43 mph≦V≦49.71 mph). Expressed in English units, the equation is,
tf={[−24.5(1.067V+4.8)]+2111}(12.978−0.556Ta)−1.668 (31b)
where Ta is in ° F. and V is in mph. With the weather forecast conditions of −4° F. and V=13 mph, Eq. (31b) predicts the time to freeze as tf=17.70 min. Again an approximate value greatly over predicts the time to freeze. This coupled with the fact that it does not seem to apply to the remaining 95% of the populace makes any application of Eq. (31b) pointless. The above applications clearly show that neither of the currently available approximations, Eq. (29) and Eq. (31b), can be used to predict the onset of facial freezing.
Since neither of the two above expressions can correctly predict the freezing time even when applied within their specific ranges (Ta, V) of applicability, they certainly cannot be expected to predict the time to freeze when either the ambient temperature or wind velocity or both lie outside of these ranges. An extreme example of this is the earth's coldest recorded temperature of Ta=−128° F. (−88.89° C.) in Antarctica. Only Eq. (24) permits a calculation of the exact time to freeze (tf) for personnel located there who may have been exposed to this temperature. It is not known with certainty whether windy or calm conditions prevailed. The assumption was made that there was no wind reduction (WRF=0) and that this temperature occurred during hours of darkness (G=0 Btu/hr-ft2 (W/m2)). Also the altitude would have been near sea level (H=0 ft (0 m)). Calculation of the time to freeze (tf) for individuals who may have been exposed to this severe temperature was made for two velocities V=1 mph (1.61 Km/hr) and V=4 mph (6.44 Km/hr). From Eq. (24) the time to freeze when Ta=−128° F. (−88.89° C.) was tf=4.45 min. when V=1 mph and tf=2.53 min. when V=4 mph. These are very low values of tf which could not have been accurately determined by either of the above approximate expressions.
To better visualize the effect of velocity, the values of tf were also calculated using Eq. (22) to determine these values from the intercept point of the facial temperature decay curves with the freeze line (Tf=32° F. 0° C.)). These results are shown in
One of the objectives of the model developed here was to define the facial freezing time corresponding to a given wind chill temperature. This time should assist a user to “quantify” the value of the wind chill temperature, particularly at its extreme lower values where sensing it becomes very subjective. However, as already demonstrated, unless the announced freezing time is accurate, the listening public may experience more harm than good as a result of it. It is the inventors' belief that only Eq. (24) provides this accuracy, since it was based on a theoretical development of the facial temperature, which, in turn, has been substantiated by experimental data. It should be noted that Eq. (24) along with its constants in Table 4 were derived for sea level conditions (H=0 ft (0 m)). However, the time to freeze is altitude dependent. This will be discussed below.
Effects of Heat Losses/Heat Gains on the Wind Chill Temperature and Time to Freeze: Each of the three heat losses (qfc, qr, qe) and each of the two heat gains (qi, qb) were individually examined to determine their contributions to the wind chill temperature and the time to freeze.
Wind Chill Temperature: Eq. (18a) was used to calculate the effect of each heat loss and heat gain on the wind chill temperature (Twc). Calculations were made for an assumed Boston, Mass. resident casually walking on a sunny, cold winter day when the ambient temperature (Ta) is 0° F. (−17.78° C.). Boston is approximately at sea level and is in the mid latitude region, H=0 ft (0 m) and G=42.66 Btu/hr-ft2 (134.58 W/m2) from Table 2. From Table 1 the metabolic heat generation for “walking about” is Mact=31 Btu/hr-ft2 (97.79 W/m2) and therefore the water evaporation flux rate becomes {dot over (w)}=0.00655 lbm/hr-ft2 (0.032 kg/hr-m2). It was assumed that the initial facial temperature (Tf) was 91.4° F. (33° C.) and that there was no wind speed reduction at head level, thus WRF=0.
Several comments and conclusions can be drawn from
Second, the radiation heat loss (qr) is at a maximum at the moment of exposure when the facial temperature Tf is 91.4° F. (33° C.). Only when Tf decreases in time and approaches the ambient temperature (Ta) will the radiation heat loss decrease to the point that the forced convection heat loss (qfc) may dominate.
Third, the effect of the evaporation heat loss (qe) on Twc is nearly offset by the metabolic heat (qb) gain. This might have been expected since both depend upon the level (Mact) of physical activity. Although this was shown to be true for the low level activity of “walking about” considered here, it is likely to be true for all activity levels. From this it can be concluded that (qe) and (qb) can probably be safely neglected if only an approximation of Twc is desired.
Fourth, the effect of the solar heat gains (qi) on Twc is much greater than that of (qe) and (qb). Since qi is constant for any geographical location but qr decreases as Tf decreases with increasing exposure time, it is possible that after an extended period of time the solar heat gain may completely offset the radiation heat loss.
Fifth, the negligible effect of qe and qb on Twc during ambient conditions, when wind chill is of concern, would seem to contradict the statement made by some investigators that physical activity will have a moderating or warming effect on the wind chill sensed by an individual. Actually, there may not be a contradiction, since two different phenomena are being considered here: one, the heat loss from a fully clothed body, the other, the heat loss from the exposed face. For the fully clothed individual participating in a physical activity, it might be reasoned that the potential heat loss (qe) due to perspiration over the clothed portion of the body may not exist if the sweat is primarily being absorbed and retained by the clothing. In addition, the metabolic heat (qb) flowing towards the skin surface is probably accumulating in the sense that it is largely prevented from leaving the skin surface due to the insulating effect of the clothing. This additive effect of qe and qb would result in an overall warming effect which would raise the individual's comfort level as claimed. What must be considered here is that this warming effect in no way represents a moderation of the wind chill temperature (Twc) sensed by the individual, since this temperature is sensed within the skin's dermis layer of the exposed face and not within the clothed body. Thus, physical activity can help an individual feel warmer in a wind chilling environment without really affecting the wind chill temperature.
Time to Freeze: Eq. (22), rather than Eq. (24), was used to calculate the effect of each heat loss and gain on the time to freeze (tf). Calculations were made for the same set of conditions used to obtain the effects of heat losses and gains on the wind chill temperature (Twc) as shown in
Finally,
Several comments and conclusions can be made concerning the effect of heat losses and gains on the time to freeze (tf). First, the effect of the heat loss due to evaporation (qe) is nearly offset by the heat gain due to body metabolism (qb) at the lower ambient temperatures and higher velocities when the danger of facial freezing is the greatest. This is true for a low level of physical activity and may be true for higher levels as well. At low levels of physical activity, like walking, an individual will experience a decrease rather than an increase in the time to freeze, so that he may want to avoid such activity. At higher, more energetic levels of activity, the individual may benefit as a result of an increase in tf. These effects on an individual would take place over the entire range of ambient temperature and wind velocity. As the ambient temperature increases and the velocity decreases, the warming effect of qb becomes much greater than the cooling effect of qe. This difference reaches its maximum under calm conditions (V=0 mph (0 Km/hr)).
Second, it appears, that because of the offsetting effect of qe and qb at the lower ambient temperatures and the higher velocities where the danger of freezing is the greatest, they can be neglected in the determination of the time to freeze (tf). This might have been expected in light of their similar offsetting effects on wind chill temperature (Twc) as shown in
Third, for the low level of activity considered here, the effect of the solar heat gain (qi) is about 15 times greater than the effect of the metabolic heat gain (qb) at the higher velocity (V=40 mph) and over the entire temperature range (−40° F.≦Ta≦0° F. (−40° C.≦Ta≦−17.78° C.)). Thus, sunshine can greatly increase the individual's exposure time before facial freezing occurs, but the actual value of tf might be quite small. At the higher ambient temperature (0° F.) and calm conditions (V=0 mph), the beneficial effect of sunshine is so great that facial freezing may not occur.
The effects of altitude on the wind chill temperature and time to freeze will now be discussed. Wind chill is primarily the result of a wind driven process that is ambient pressure (P) dependent. One aspect of wind chill that has escaped attention is its dependency on the ambient pressure and consequently altitude (H). The reason for this may have been due to both oversight and the belief that the effect of this pressure was negligible. However, such is not the case. The present model shows that for the same ambient temperature (Ta) and wind velocity (V) at all altitudes, increasing altitude will have a significant effect on the wind chill temperature (Twc) but almost a negligible effect on the time to freeze (tf).
Wind Chill Temperature: The ambient pressure decreases with increasing altitude and Eq. (13c) shows a corresponding reduction of the forced convection coefficient (hfc). Therefore, increasing altitude will result in a moderation of the wind chill temperature Twc. To demonstrate this effect, calculations of wind chill temperature as a function of the altitude were made for the basic wind chill components (qfc, qr). The evaporative (qe) and metabolic (qb) components were not considered because of their negligible and counteracting effects. Also the solar radiation (qi) was not considered, since its absence leads to the more severe wind chill temperatures which are of greater concern. Calculations of the wind chill temperature (Twc) were made using Eq. (18a) with Ta=0° F. (−17.78° C.) and −40° F. (−40° C.) and V ranging from 10 to 40 mph (16.09 to 64.37 Km/hr). The range of altitudes (H) considered was from sea level to the height of Mt. Everest (29,082 ft (8,864.3 m)). The results of these calculations, which are plotted in
Time to Freeze: The results in
These temperature decay curves are shown in
where a=0.17046053, b=−0.037699401, c=0.0011765359, d=0.0022511491, e=6.1594515×10−6, f=−9.6552578×10−5, g=3.3382238×10−6, h=1.7864164×10−8, i=1.2133567×10−9, j=−1.4109763×10−7 and where the correlation coefficient is r2=0.99566554. Each of the nine curves in
The following is a description and demonstration of a method for calculating the wind chill temperature (Twc), the time varying facial temperature (Tf) and the facial time to freeze (tf) for an individual exposed to wintertime ambient conditions (Ta, V) according to the methods of the present invention. The following information must be known or specified before proceeding with the calculation of those three quantities: (1) Ambient temperature (Ta, ° F.), (2) Wind velocity (V, mph) at the NWS 10 m level, (3) Latitude (LAT) at individual's location, (4) Location altitude (H, ft), (5) Presence (G>0) or absence (G=0) of sunshine, (with sunshine, G is determined from Table 2 for the known latitude region (LAT)), (6) Metabolic heat generation (Mact) determined from Table 1 for an individual's known physical activity, with Mact, the evaporation flux rate ({dot over (w)}) can be determined, (7) Distance (D) of an individual from the nearest wind obstruction; with D, the wind reduction factor (WRF) can be determined (see below). The first five of these items are automatically known once the weather conditions and location are specified. The remaining two relate entirely to the individual being examined.
In demonstrating the method used to calculate Twc, Tf and tf, consider the extreme weather conditions that existed on top of Mt. Washington, N.H. on Jan. 22, 2003, when the ambient temperature dipped to −34° F. (−36.67° C.) and the wind speed reached 142 mph (228.52 Km/hr). Mt. Washington has an altitude of 6,200 ft (1,889.78 m) and lies at 440.14 north latitude. At this excessive wind speed, stormy conditions almost certainly prevailed. If these stormy conditions occurred during daytime hours, then sunshine may have been at a maximum (G>0) or a minimum (G≅0). With this description, the following information is known: Ta=−34° F.; H=6,200 ft; V=142 mph; G≅0 or 42.66 Btu/hr-ft2 (0 or 134.58 W/m2) from Table 2 using LAT; LAT=44°.14′.
The calculations were made for an individual presumed to be a member of the weather station located on top of the mountain. The individual was assumed to be fully clothed, standing outside, fully exposed to the wind, yet removed from any turbulence effect produced by the station itself. Under more normal conditions, the individual's activity might be described as “standing relaxed”, using the terminology of Table 1. However, an individual standing in a 142 mph wind is not likely to be relaxed. On the contrary, he will be expending a great deal of energy just to remain upright. This suggests that the metabolic heat generation may be more comparable to that of a downhill skier. Thus from Table 1, Mact=96 Btu/hr-ft2 (302.84 W/m2) and since Mact≧40 Btu/hr-ft2 (126.18 W/m2), {dot over (w)}=0.02183 lbm/hr-ft2 (0.1066 kg/hr-m2). Finally, the leeward distance (D) does not exist, since the individual is not downstream of the station. Rather D must be replaced by the wind/surface contact distance (l) which, near the mountain top, would be expected to be very small. Suppose l=100 ft (30.48 m), but because the surface is not likely to be smooth, l was assumed to be 200 ft (60.96 m) to account for surface roughness and boundary layer buildup. Therefore, D=l=200 ft. With l=200 ft and V=142 mph, the boundary layer edge (δ) from Eq. (33), below, is δ=(200)(0.0589)[(142)(200)]−0.2=1.52 ft (0.46 m). Since δ<y (head height) and where y=5 ft (1.52 m), then from Eq. (35b), below, WRF=0. With the above values of Ta, V, H, G, Mact, {dot over (w)} and WRF known, the values of Twc, Tf vs. time and tf can be determined.
Calculation of the Wind Chill Temperature (Twc): Calculation of Twc from Eq. (18a) must be preceded by a calculation of the exponent φ from Eq. (18d). This may be accomplished for the individual in minimum sunshine (G≅0) and in maximum sunshine (G=42.66 Btu/hr-ft2).
These very cold temperatures are considered to be the correct values of Twc on top of Mt. Washington. They are values that lie well beyond the range of applicability of the currently used wind chill model which, among other things, cannot account for the effects of altitude or sunshine. This can be clearly demonstrated in the following manner.
Calculation of the Time to Freeze: Eq. (24) is the expedient way of calculating the time to freeze (tf). Results from using Eq. (22) for the facial temperature (Tf) were the basis for developing this expression for the time to freeze. Although a lengthier approach, Eq. (22) will be used first to calculate the time dependent decrease in facial temperature (Tf) from its initial value of 91.4° F. (33° C.) to the time of facial freezing (32° F. (0° C.)). Calculation of tf from Eq. (22) starts with the initial facial temperature Tfm=91.4° F. at time zero (m=0) and then calculates the facial temperature, Tfm+Δt, at a later time increment (Δt). Using what has been found to be an optimum time increment Δt=1 second, Tf is calculated after one second. Repeated calculations of Tf, where the value at the end of the time increment becomes the starting value on the next, continues until Tf reaches 32° F. Results of these calculations are shown in
Calculations of tf for Mt. Washington using Eq. (24) give tf=0.897 min. with no sunshine (G=0) and tf=0.9155 min. with sunshine (G>0). Since Eq. (24) is restricted to sea level conditions, these sea level based values of tf must be corrected for the Mt. Washington altitude. From Eq. (32) or
The following observations are made regarding the wind chill model disclosed herein. First, the Siple and Passel experiment was not greatly flawed as has been thought by several investigators. A closer examination of this experiment reveals that the only error of any consequence was their assumption of a constant skin temperature of 33° C. (91.4° F.) during the entire exposure time. In reality, as the exposure time increases, the facial skin temperature will decrease and the wind chill temperature sensed by the individual will increase or become warmer. This was correctly recognized by the critics. However, their argument against the assumption of a constant 33° C. (91.4° C.) temperature seems to be that, over an extended exposure time, the individual was somehow being deprived of feeling warmer. Actually, this argument is irrelevant, because the individual is much more likely to be interested in the wind chill temperature at the initial moment of exposure, when the face temperature is realistically near 91.4° F. (33° C.), than at times later, when his facial temperature may be plummeting and he becomes preoccupied with thoughts on how to avoid facial freezing.
Second, Siple and Passel's assumption of a constant skin temperature was a valid criticism. Bluestein and Zecher's development of their prior art wind chill model was an attempt to show a moderation (warming) of the Siple and Passel wind chill temperatures by allowing the skin temperature to vary rather than to remain constant. But, the Bluestein and Zecher model shows at most a 2° F. (1.11° C.) moderation as a result of the skin temperature variation. At very low temperatures and high velocities, the Bluestein and Zecher values show no moderation. Rather they show a −1° F. cooling relative to the Siple and Passel values. The balance of their total 15° F. (8.33° C.) increase in the wind chill temperature was the result of the head level wind reduction based on their erroneous assumption that the NWS 10 m velocity value is always 50% greater than that at head level. The conclusion reached herein is that without their incorrect wind reduction and with a seemingly incorrect skin temperature correction, the Bluestein and Zecher values of the wind chill temperature are really no different from the Siple and Passel values they were intended to correct.
Another observation is that the wind chill model developed here is complete, accurate and more adaptable than the prior art Bluestein and Zecher model in use as of this writing. Like all previous attempts at wind chill prediction, it is based upon the two basic skin heat loss processes, namely radiation (qr), which is always present, and forced convection (qfc), which is present only under windy conditions. These two heat losses are the only ones that have been considered in previous wind chill analyses. What makes the wind chill model of the present invention complete is that it also incorporates both the beneficial (warming) effect of an individual's physical activity and that of sunshine when it is present, both of which are considered as heat gains at the facial surface. Physical activity produces a metabolic heat flow (qb) from the body core and a solar radiation heat flow (qi) from sunshine. The physical activity produces a third facial surface heat loss due to sweat evaporation (qe). The model is unique in that it includes (a) the very important effect of altitude (H), a variable which until now has not been considered, and (b) the capability to determine in each case a correct wind reduction factor (WRF), rather than universally applying an incorrect one. The greater accuracy of this model stems not only from the addition of the above terms (qb, qi), but also from the means by which they were determined. The metabolic heat (qb) from the body core to the facial surface was based upon an accurately derived thermal conductance term from a human thermoregulation model by Havenith. The solar radiation heat (qi) was based upon monthly insolation data for 221 cities across the 50 states as compiled by the U.S. Department of Energy. The WRF can be analytically determined for each case knowing the individual's location relative to an upwind obstruction, see discussion below. Finally, the model is adaptable over the widest range of ambient temperatures (−140° F.≦Ta≦25° F. (−95.56° C.≦Ta≦−3.89° C.)), wind velocities (0 mph≦V≦160 mph (0 Km/hr≦V≦257.49 Km/hr)) and altitudes (0 ft<H<29,082 ft (0 m<H<8,864.3 m)) worldwide.
Another observation is that verification of this model consisted of comparing this model's predicted values of wind chill temperature (Twc), facial temperature (Tf) and time to freeze (tf) with existing experimental data. First, the predicted values of the wind chill temperature from Eq. (18a) were compared with those of Siple and Passel. This also becomes a comparison with the Bluestein and Zecher values when their incorrect wind reduction is removed. The comparison was made for the following conditions that existed at their location in Antarctica at the time of their experiment: sea level (H=0 ft (0 m)) altitude, no wind reduction (WRF=0) at the height of their container above ground level, no evaporative heat loss (qe=0) and no metabolic heat gain (qb=0), since neither of these applied, and no solar radiation (qi=0) since the experiment was conducted in darkness. Comparison of this model's predicted values of Twc with those for Siple and Passel values in
Still another observation is that the wind reduction at head level depends upon the extent to which an individual's head is immersed in either a wind-generated boundary layer along the surface, upon which the individual is positioned, or within a wind-generated turbulence region on the leeward side of an obstruction upwind of the individual. In either case, the distance (D) between the individual and the obstruction must be known. It is this distance and the known wind velocity (V) that ultimately determines the value of the wind reduction factor (WRF). Considering the entire populace in all weather conditions, there are an infinite number of D and V combinations. Consequently, an infinite number of WRFs are possible. It is for this reason that the across-the-board application of a WRF=0.33 in the prior art wind chill model is in error. This detailed description further describes a procedure, below, for determining a more exact WRF in terms of D and V. Applying this procedure, average values of WRF were determined for both an urban and a rural area over the 20 mph≦V≦70 mph (32.19 Km/hr≦V≦112.65 Km/hr) range. For an urban area, WRF=0 for all velocities. For the rural areas the average values of WRF are 0.055 for V=20 mph and 0.072 for V=70 mph. The fact that these two areas may encompass more than 95% of the populace, and since these values of WRF are small and nearly non-existent, emphasizes the error introduced in the current model when a WRF=0.33 is being used.
Yet another observation is that physical activity produces a metabolic heat flow (qb) to the skin surface and evaporative heat loss (qe) from the surface. Their combined effect on the wind chill temperature (Twc) and the time to freeze (tf) is small and nearly offsetting at low temperatures (Ta) and high wind velocities (V), when facial freezing is of the greatest concern. When freezing is of lesser concern at high temperatures and low velocities, qb does provide a beneficial increase in Twc and tf above the opposing effect of qe. Physical activity will have a completely different effect on the clothed portion of the body. The clothing will act as a barrier to prevent or slow the metabolic heat loss from the body. It will also act as an absorbent of the perspiration preventing its evaporation and its additional heat flow from the body. The additive effect of these two heat losses may explain why physical activity is known to have a warming effect on the clothed body. This warming effect plays no part in the wind chill temperature as sensed by the facial surface, nor in the possibility of facial freezing.
Another observation is that sunshine has a much greater moderating effect on wind chill temperature (Twc) and time to freeze (tf) than physical activity. This greater moderation by sunshine extends over a relatively wide range of ambient conditions. Unfortunately, the benefit of this moderation is the least at low ambient temperatures and high velocities, where facial freezing is a real possibility. However, under calm (V=0 mph) conditions and relatively high temperatures such as Ta=0° F., the effect of sunshine is so great that facial freezing may never occur. The conclusion is that the presence of sunshine results in its greatest beneficial effects at the lower velocities and the higher temperatures.
Furthermore, increasing altitude has the effect of increasing the wind chill temperature (Twc) and the time to freeze (tf). Calculations using this model gave rise to an incremental increase in the Twc per 1000 ft increase in altitude. This gradient increases rapidly with decreasing ambient temperature (Ta) and increasing velocity (V). What this means is that the moderating or beneficial effect of altitude on Twc is the greatest at the lower temperatures and higher velocities where it is needed the most. Even though altitude has this great effect on Twc, its effect on the time to freeze (tf) is just the opposite and quite small. Calculations of a corresponding time to freeze gradient, Δtf per 1000 ft, show that it decreases as the velocity (V) increases and as the ambient temperature (Ta) decreases. Unfortunately, the benefit of this altitude increase is least at velocities and temperatures where it is most needed. Furthermore, the increase in tf due to altitude is very small. At even lower temperatures (Ta) and higher velocities (V), this incremental increase in tf with altitude would be even smaller, and as such can possibly be neglected.
Procedure for Determining Wind Chill Factor: In wind chill calculations, the ambient temperature (Ta) is presumed to be known, but determining the actual wind velocity at head level may be difficult if not impossible except in special situations. This is because the actual wind velocity depends upon whether or not the individual's head is immersed in a wind-generated turbulence. If it is not, the head is exposed to the free-stream velocity (V), which is assumed to be that at the NWS 10 m level. The wind reduction factor (WRF) is defined as the difference between the free-stream velocity and the head level velocity divided by the free-stream velocity. Thus, there would be no wind reduction when the free-stream velocity equals the head level velocity, i.e., WRF=0.
The problem here is that the 10 m level, except in specific instances, does not represent the correct height upon which to determine the WRF. Therefore the current practice of assuming the free-stream velocity as being 50% greater than that at head level in all calculations of the wind chill temperature is incorrect. The following discussion will show why the WRF must be determined from the depth of the velocity boundary layer and not from the height of the NWS 10 m velocity sensor, unless the latter happens to be exactly at the boundary layer edge. Wind reduction becomes necessary when the head is immersed in either: (a) a turbulent region on the leeward side of a wind obstruction, or in (b) a turbulent boundary layer generated by the wind. The latter, which is possibly the more likely situation to occur, fortunately is the one that lends itself more easily to analysis, providing the following information is known: (1) the location of the boundary layer edge relative to the individual's head, and (2) the velocity profile within the boundary layer.
Unfortunately, this information is so dependant upon an individual's surroundings that an evaluation of the WRF may not always be possible. But this is no reason for universally applying an incorrect value of WRF to all individuals in the listening area. Determination of this boundary layer is explained in the following paragraphs.
Wind blowing along a surface experiences a retarding action by friction over a layer called the velocity boundary layer. Within this layer, the velocity increases from zero at the surface to the free-stream value (V) at the boundary layer edge (δ). If an individual's head is within the boundary layer, it will experience a velocity (v) less than the free-stream value (V) and consequently a WRF>0. The thickness (δ) of this boundary layer is a function of the free-stream velocity, the air's kinematic viscosity (ν) and most importantly the length (l) that the wind is in contact with the surface. Another variable affecting the thickness is the surface roughness, but this is not easily determined. What is known, is that this roughness guarantees that the flow in the layer will be turbulent and that an increase in roughness will increase δ. Based on all this, it can be stated that an individual exposed to a free-stream velocity (V) will encounter a turbulent boundary layer thickness (δ) that is dependant upon the individual's surroundings, such as the surface roughness on his windward side and upon the wind/surface contact length (l).
Schlichting, H., “Boundary Layer Theory”, McGraw-Hill, p. 42, 1979, discloses that the variation in the boundary layer thickness (δ) for the turbulent flow is:
where the kinematic viscosity (ν) of air is 150×10−6 ft2/s (13.94×10−6 m2/s). With the velocity in the Reynolds number (Rel) expressed in mph, Eq. (33) becomes:
With reference to
where, in this case, v is the velocity at head level, V is the free-stream velocity at the boundary layer edge (δ), y is the head height above ground level, and where the exponent 1/n depends upon the surface roughness and the free-stream velocity. From its definition and using Eq. (35), the WRF can be determined in the following manner:
Calculating an accurate value of this WRF is complicated by the exponent's dependency upon the surface condition and the magnitude of the wind speed. Consider first the surface condition. Schlichting and also, Eshbach, O. W., Handbook of engineering fundamentals, John Wiley & Sons, Inc., p. 1-142 and 7-119, 1952, state that for smooth surfaces of the type encountered in wind chill calculations such as a sidewalk, street, airport runway, or even a frozen lake, the exponent would be 1/7, borrowing from the “ 1/7 power law” for pipe flow. This value is smaller than the corresponding value (1/4.76) used by Steadman based on measurements in the Saskatoon area. Based on Steadman's values, the WRF at head height (y=5 ft (1.52 m)) is
This WRF of 0.33 corresponds to the universal 50% reduction in the NWS 10 m wind speed used in the prior art Bluestein and Zecher wind chill model.
If the 33 ft value for δ as used in Eq. (37) was actually at the edge of the boundary layer in the flow field during the Steadman measurements, then his larger exponent (1/4.76) could have been expected if the measurements were made over a prairie surface exhibiting a certain amount of roughness. If that was the case, then Eq. (37) does provide a correct WRF for an individual on a similar surface, but only if the individual is at a sufficient distance D from an obstruction such that the boundary layer thickness (δ) at the individual's location is exactly 33 ft (10.1 m). This is the only instance in which the current WRF of 0.33 can be correctly applied. In all other cases, this correction is too severe. This can be demonstrated as follows: For velocities (V) spanning the wind chill range, select values of l and calculate (from Eq. (34). Then calculate WRF from Eq. (36a) using y=5 ft (1.52 m) and Steadman's exponent. The results are plotted in
The above discussion and the results shown in
One way of determining whether or not the 33 ft height represents the boundary layer edge (δ) would be the measurement of the height above ground level where the velocity ceases to increase with height; this height would then be the edge. This approach may have been taken during the Saskatoon measurements to insure that the 33 ft height in Eq. (37) represents the true boundary layer edge. If it does not, then Steadman's velocity profile expression is incorrect and its usage would result in an incorrect value for the WRF. This may no longer be of much importance if the use of a WRF is discontinued, as is suggested in the present model.
There may be certain situations where there is a need to determine a WRF. If Steadman's exponent was incorrectly determined, an alternative would be to use the 1/7 exponent. Although this exponent supposedly applies to smooth surfaces such as an airport runway, a blacktop road, a hard packed sand beach or a snow-free frozen lake surface, it should provide a good approximation of the WRF even for surfaces with minor roughness. However, even the 1/7 exponent has its limitation. Eshbach notes that it is applicable only up to a Reynolds number (Rel) of 3×107, above which the exponent will progressively decrease to ⅛, 1/9 etc., as the Reynolds number increases. Based on the definition of the Reynolds number in Eq. (33), this means that the 1/7 exponent is valid up to a wind/surface contact length (l) of 153 ft (46.63 m) at a velocity (V) of 20 mph and only 44 ft (13.41 m) at a velocity of 70 mph. These are very short distances and one would expect numerous cases where the wind/surface contact length far exceeds these values. If so, the Reynolds number would be much greater than 3×107 and the exponent less than 1/7. The exact value of the exponent is unknown. What is known is that the WRF decreases as the exponent decreases. Suppose the boundary layer edge (δ) is 5 ft above an individual's head at y=5 ft. Then (y/δ)=0.5. With this value in Eq. (36a), the WRF is calculated and shown in
Another aspect pertaining to the computation of a WRF is the location of the NWS 10 m sensor. It is presumed to measure the true free-stream velocity. To do so, it must be located at or above the boundary layer edge that exists at its airport location for all wind conditions. Sims, C., Personal communication, National Weather Service, Duluth, Minn., 2001, provided the following specific information on the NWS 10 m sensor which is part of the ASOS/AWOS (Automated Surface Observing System/Automated Weather Observing System) site: “It will be mounted 30 to 33 feet (9.14 to 10.1 m) above the average ground height within a radius of 500 ft (152.4 m)”. In addition, Sims states: “The sensor height shall not exceed 33 ft except as necessary to: (a) be at least 15 ft (4.57 m) above the height of any obstruction (e.g., vegetation, buildings, etc.) within a 500 ft (152.4 m) radius, and (b) if practical be at least 10 ft (3.1 m) higher than the height of any obstruction outside the 500 ft radius, but within a 1000 ft (304.8 m) radius of the wind sensor. An object is considered to be an obstruction if the included lateral angle from the sensor to the ends of the object is 10 degrees or more.”
First, consider the case where there is no obstruction within 500 ft of the sensor. To determine if the sensor is located above the boundary layer edge at the sensor location, the layer's thickness (δ) was calculated from Eq. (34) using a wind/surface contact length (l) equal to the 500 ft radius. With a wind velocity V=20 mph and 70 mph directed towards the sensor, the corresponding boundary layer thickness would be δ=4.7 ft (1.43 m) and 3.6 ft (1.1 m), respectively. With the sensor at 30 to 33 ft above the ground height, it would be well within the free-stream region. Second, in the situation where there is an obstruction within the 500 ft radius, the concern no longer centers on the wind/surface boundary layer thickness, but rather on the size of the turbulent flow region downstream of the obstruction. The NWS 10 m sensor must be above this region to experience free-stream conditions. Because the vertical dimension of the turbulent region diminishes as the distance downstream of the obstruction increases, the most critical situation is when the obstruction is relatively close to the NWS 10 m sensor. Furthermore, if the wind approaching the obstruction is at some positive angle relative to the ground surface, the height of the turbulent region may be significantly greater than the obstruction itself. Suppose the height of an obstruction close to the NWS 10 m sensor is 25 ft (7.62 m). Conceivably, the height of the turbulent region might be 10 ft (3.1 m) greater at 35 ft (10.67 m), such that a sensor at the 33 ft level would be immersed in the turbulence. The specification that the sensor be “at least 15 ft (4.57 m) above the height of any obstruction” would, in this case, avoid that possibility. Similar reasoning could be made for the case of an obstruction outside the 500 ft radius but inside the 1000 ft radius. It appears that the above guidelines were developed so as to insure that the NWS 10 m sensor would be exposed to free-stream conditions at all times even for relatively larger obstructions.
The ASOS/AWOS specifications on the NWS 10 m sensor guarantee that the free-stream velocity will be measured at the sensor location for all velocities providing the wind/surface contact distance (l) is 1000 ft or less. But this wind/surface contact distance may be much greater than the 1000 ft in cases where the sensor is located alongside an airport's runway. An example is the 6,500 ft (1,981.22 m) runway at the Itasca County Airport in Grand Rapids, Minn. where the sensor is located at a distance of 35 ft (10.67 m) from the side of the runway and about 5,000 ft (1,524 m) from one end of the runway. The runway would constitute a smooth surface such that Eq. (34) would correctly apply. If a 20 mph wind in the direction of the runway approaches the sensor from the 5,000 ft end of the runway, the boundary layer thickness at the sensor location is δ=29.5 ft (8.99 m). When V=70 mph, δ=22.9 ft (6.98 m). This means that over an expected 20 mph to 70 mph velocity range, the sensor would always be exposed to free-stream conditions. However, this would no longer be true if V≦11.3 mph (18.19 Km/hr) because when V=11.3 mph, δ is exactly equal to 33 ft. For example, when V=8 mph (12.87 Km/hr), δ=35.4 ft (10.79 m) and the sensor would lie 2.4 ft (0.73 m) below the boundary layer edge. From Eq. (35), the velocity (v) being recorded by this submerged sensor when V=8 mph, y=33 ft, δ=35.4 ft, and assuming n=9, would be 7.94 mph (12.78 Km/hr). This is within 1% of the actual free-stream value and would not result in any error of consequence in wind chill predictions. Because this particular sensor is probably representative of all sensors nationwide, the conclusion is made that all sensors, at all times, record a velocity that is very close to or exactly the free-stream velocity.
The discussion so far has dealt with wind reduction in instances where the head is immersed in a wind-generated turbulent boundary layer. These cases represent situations that are relatively simple to analyze. In the other cases, where the individual is within the turbulent region downstream of an obstruction, the WRF is more difficult to determine. Only when the individual is very close to the obstruction and totally within the flow separation region (x) as shown in
Individuals in an urban area may be subjected to a combined effect of a boundary layer and a separated region. An individual located at some distance downstream from an obstruction on a clear street with buildings on either side may experience a boundary layer. If the wind/surface distance (l) along the street is 1,500 ft (457.21 m) or less, which is comparable to about two standard city blocks, then from
There are two situations where a wind reduction at head level is the result of a modification of the NWS 10 m velocity value and not the result of the individual being exposed to a wind-generated boundary layer. The first is an increase in the NWS 10 m free-stream value that could occur in a large urban area as a result of what Schwerdt referred to as “air funneling around tall buildings.” This increase could be computed knowing the size, number, and layout of the buildings. The second refers to a case where the NWS 10 m value of the velocity would be decreased. Picture the previous illustration of the wind blowing down the 1,500 ft length of street and then being deflected 900 around a building and continuing to flow down a cross street. The deflected wind, including its boundary layer, would generate a turbulent region on the cross street side of the building, with a subsequent reattachment to the surface at a distance (x) downstream of the turning point. Energy losses incurred by the flow as a result of this turning would be reflected as a reduced value in the free-stream velocity after the turn, as compared with its NWS 10 m value. As before, at distances of l<lmax downstream of the reattachment point (x), WRF=0 from
Finally, there are cases where an individual is exposed to an equivalent wind speed due to his motion even when calm wind (V=0 mph) conditions prevail. These cases include motorcycle riders, snowmobile riders, and downhill skiers. Facial freezing could occur at low ambient temperatures and high speeds of motion in the absence of facial protection such as a helmet. In each case, there is no boundary layer and consequently no wind reduction, therefore, WRF=0 at all times.
Based on the above discussions, the following statements and comments can be made concerning the concept of wind reduction: The WRF=0.33 corresponding to the currently assumed 50% reduction in the NWS 10 m wind speed value is correct only if (a) the wind-generated boundary layer thickness during the Saskatoon velocity profile measurements was δ=33 ft (10.1 m) and if (b) the same boundary layer thickness exists at the location of the individual being considered. This is a singular and highly unlikely situation. It emphasizes the fact that the usage of WRF=0.33 for all calculations of the wind chill temperature, without regard for the individual's position relative to the wind, is inaccurate. A WRF=0.33 will correctly and uniquely apply to an individual at some point within the separation region on the leeward side of an obstruction. This is the only instance, in realistic situations, in which the current wind chill model would come close to predicting the correct wind chill temperature (Twc). A WRF=1.0 represents the special case where the individual in the above separation region is adjacent to the obstruction such that the latter completely blocks him from the oncoming flow.
Calculations of boundary layer values of the WRF as a function of the velocity (V) and an individual's distance (D) from an obstruction assuming Steadman's value of the velocity profile exponent (1/4.76), shows that for: (a) V=20 mph: (1) WRF=0, for D<600 ft and (2) WRF−0.055 (average) for 600 ft<D<1,500 ft. For (b) V=70 mph: (1) WRF=0, for D<850 ft and (2) WRF=0.072 (average) for 850 ft<D<1,500 ft.
The distance range, 600 ft<D<850 ft, may reflect situations in an urban area, while the range 600 ft<D<1,500 ft might be considered as those in a more rural area. Since it is anticipated that these two areas encompass more than 95% of the populace, and since the WRF values are very small or non-existent, it emphasizes the importance and need to remove the equivalent correction (WRF=0.33) used in the current wind chill model.
The above calculations of WRF were based on Steadman's velocity profile exponent (1/4.76). If the usage of this exponent is found to be incorrect, then a more appropriate one of lower value (⅛, 1/9, 1/10, etc.), which is consistent with higher velocities and Reynolds numbers, should be used. Decreasing the value of this exponent decreases the values of WRF even further; this emphasizes the advisability of assuming WRF=0 when catering to the public at large.
It is suggested that the actual NWS 10 m velocity value be used at all times, although the local value of the free-stream velocity in an urban environment may be somewhat larger or smaller. And finally, the WRF=0 for all motion-driven sport activists such as cyclists and skiers under calm conditions (V=0 mph), because of the absence of a wind-generated boundary layer.
The above results show that, except for a few special cases, the WRF=0 or is only slightly greater than zero for almost everybody in the listening area. Where WRF takes on small positive values of D between 600 ft and 1,500 ft from an obstruction, the recommendation is made that WRF=0 in this region also. Doing so could, in cases where facial freezing is a possibility, provide the listener with a degree of safety by informing him of a time to freeze before it actually occurs. For values of D greater than 1,500 ft, WRF may become quite large as shown in
The inventors suggest the following recommendations be considered for implementation of the wind chill model of the present invention to provide the listening public with a more accurate wind chill temperature (Twc) at the moment of exposure, facial temperature (Tf) at any time during the exposure and the time to freeze (tf): (1) Determine the latitude (LAT) for the entire listening area from Table 2; (2) Determine the average altitude (H) of the listening area; (3) Determine from Table 1 the most likely physical activity (Mact) of the inhabitants within the listening area. For large urban or rural areas, the most likely activity would probably be “walking about” so that Mact=31 Btu/hr-ft2 (97.79 W/m2); (4) With the likelihood that 95% of the people in the listening area will experience a wind reduction factor (WRF) equal to or less than 0.072, assume WRF=0. This is not only convenient, but also beneficial to the listeners, since it affords them added safety by informing them of a time to freeze (tf) before it actually occurs. For other situations where individuals are at a known distance from an obstruction, the WRF could be determined and used in the calculations; (5) Unless sunshine is a certainty over the entire listening area and for the entire prediction period, assume G=0 Btu/hr-ft2 (0 W/m2). This too provides the listener with a degree of safety by again informing him of a time to freeze (tf) before it actually occurs. If sunshine is expected to be present over the entire prediction period, G is determined from Table 2; and (6) The meteorologist needs to clarify to the public the real meaning of wind chill. Years of mystifying statements on the subject, such as the erroneous statement that the reason we experience wind chill is due solely to evaporation from the skin surface, has left the public confused. This confusion still reigns as evidenced by a recent (2005) statement by a well-known network TV news commentator that wind chill is a “phony number”, Andy Rooney of CBS 60 min., January 2005. The public needs to be informed that wind chill is not just seasonal, but rather a year-round phenomenon.
The methods and procedures for calculating wind chill temperature, Twc, equivalent temperature, Teq, time to freeze, tf, facial temperature, Tfm+Δt, as a function of time, altitude correction factor, Δtf/1000, and any other related calculations as disclosed herein, may be practiced as a computer program, i.e., computer-executable instructions, suitable for processing by a processor based on selected input variables. Such computer-executable instructions may, of course be stored on any suitable computer-readable medium.
While the foregoing advantages of the present invention are manifested in the detailed description and illustrated embodiments of the invention, a variety of changes can be made to the configuration, design, and construction of the invention to achieve those advantages. Hence, reference herein to specific details of the structure and function of the present invention is by way of example only and not by way of limitation.
This is a divisional patent application of U.S. nonprovisional patent application Ser. No. 11/481,684, filed Jul. 6, 2006, now U.S. Pat. No. 7,481,576 titled: “METHOD AND SYSTEM FOR DETERMINING WIND CHILL TEMPERATURE”, the contents of which are herein incorporated by reference for all purposes.
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| Number | Date | Country | |
|---|---|---|---|
| 20090076764 A1 | Mar 2009 | US |
| Number | Date | Country | |
|---|---|---|---|
| Parent | 11481684 | Jul 2006 | US |
| Child | 12291787 | US |
