HEAT EXCHANGER APPARATUS FOR THE REMOVAL OF HEAT FROM ELECTRONIC COMPONENTS

TECHNICAL FIELD

Embodiments are related to heat transfer devices, systems and methods including heat exchangers. Embodiments further relate to devices, systems and methods for the removal of heat from electronic components. Embodiments also relate to double pipe heat exchanger (DPHE) devices.

BACKGROUND

A wide variety of engineering applications such as electronics, solar collectors and internal combustion engines produce heat. This heat can be a positive or negative aspect for a particular application. There are also many thermal engineering devices used in both cooling and heating processes such as heat exchangers. The efficiency of these devices may be limited due to many inherent factors such as the nature of heat dissipation, operation mode, etc. In order to remove heat and keep the device working under a proper operating temperature and/or increase the efficiency, the need for new ideas, techniques or even designs has become crucially important in recent decades due to the worldwide rapid growth in energy consumption.

BRIEF SUMMARY

The following summary is provided to facilitate an understanding of some of the innovative features unique to the disclosed embodiments and is not intended to be a full description. A full appreciation of the various aspects of the embodiments disclosed herein can be gained by taking the entire specification, claims, drawings, and abstract as a whole.

It is, therefore, one aspect of the disclosed embodiments to provide for improved heat transfer devices, systems and methods.

It is another aspect of the disclosed embodiments to provide for an improved heat exchanger.

It is a further aspect of the disclosed embodiments to provide for devices, systems and methods for the removal of heat from electronic components.

It is also an aspect of the disclosed embodiments to provide for a double pipe heat exchanger (DPHE) apparatus.

The aforementioned aspects and other objectives and advantages can now be achieved as described herein. In an embodiment, a heat exchanger apparatus can include a convergent interface separating two counterflows, and at least two concentric conduits including an inner conduit and an outer conduit. The outer conduit can comprise an outer conduit radius that is maintained as invariant, and the inner conduit can comprise an inner conduit radius that is adjustable to form the convergent interface separating the two counterflows, wherein heat transfer with respect to the two counterflows can occur at the convergent interface.

In an embodiment of the heat exchanger apparatus, the heat exchanger apparatus can further comprise a double conduit heat exchanger comprising the at least two concentric conduits including the inner conduit and the outer conduit.

In an embodiment of the heat exchanger apparatus, the overall heat transfer coefficient and the pressure drop can increase as the contraction ratio, the Reynolds number, and the Prandtl number increase in the double conduit heat exchanger.

In an embodiment of the heat exchanger apparatus, the convergent interface can be controllable independent of a length of the at least two concentric conduits.

In an embodiment of the heat exchanger apparatus, each of the at least two concentric conduits can comprise a pipe.

An embodiment of the heat exchanger apparatus can further include a microchannel comprising the at least two concentric conduits.

An embodiment of the heat exchanger apparatus can further include a double pipe heat exchanger that includes the convergent interface and the at least two concentric conduits.

An embodiment of the heat exchanger apparatus can further include a compact heat exchanger comprising the at least two concentric conduits.

An embodiment of the heat exchanger apparatus can further include a radiator comprising the convergent interface and the at least two concentric conduits.

An embodiment of the heat exchanger apparatus can further include an electronic cooling device that includes the convergent interface and the at least two concentric conduits.

An embodiment of the heat exchanger apparatus can further include a solar collector that includes the convergent interface and the at least two concentric conduits.

In an embodiment of the heat exchanger apparatus, the convergent interface can comprise: a straight wall profile, a concave wall profile, or a convex wall profile of the at least two concentric conduits.

In an embodiment, the heat exchanger apparatus can comprise at least one of: a fluid or a nanofluid, wherein the fluid or the nanofluid flow through the at least two counterflow concentric conduits.

In an embodiment of the heat exchanger apparatus, the heat exchanger apparatus can be operable at Reynolds numbers including low Reynolds numbers as well as high Reynolds numbers and operable at Prandtl numbers including low Prandtl numbers and high Prandtl numbers.

In an embodiment of the heat exchanger apparatus, the Reynolds numbers can include a Reynolds number range from 200 to 2100 and the Prandtl numbers can include a Prandtl number range from 0.7 to 7.

In an embodiment of the heat exchanger apparatus, the heat exchanger apparatus can be operable with a heat transfer performance of up to 32% and a thermal-hydraulic performance of up to 20% compared to a plain heat exchanger.

In an embodiment, a heat exchanger apparatus can include a convergent interface separating two counterflows; and a double conduit heat exchanger comprising at least two concentric conduits including an inner conduit and an outer conduit, wherein the outer conduit comprises an outer conduit radius that is maintained as invariant, and the inner conduit comprises an inner conduit radius that is adjustable to form the convergent interface separating the two counterflows, wherein heat transfer with respect to the two counterflows occurs at the convergent interface, and wherein the convergent interface is controllable independent of a length of the at least two concentric conduits.

In an embodiment, a method of operating a heat exchanger apparatus, can involve adjusting an inner conduit radius to form a convergent interface separating two counterflows, wherein at least two concentric conduits include the inner conduit and an outer conduit, wherein the outer conduit comprises an outer conduit radius that is maintained as invariant, and the inner conduit comprises the inner conduit radius that is adjustable to form the convergent interface separating the two counterflows, wherein heat transfer with respect to the two counterflows occurs at the convergent interface.

In an embodiment, a heat exchanger apparatus can comprise a double conduit heat exchanger comprising at least two concentric conduits including an inner conduit and an outer conduit with a porous layer on a convergent interface between the inner conduit and the outer conduit. The porous layer may be a porous medium such as, for example, a metal foam.

In an embodiment, the porous layer can be located at the convergent interface in the outer conduit, or the porous medium may be located at the convergent interface in the inner conduit.

In an embodiment, the porous medicum may be located at the convergent interface in the inner conduit and the outer conduit.

In an embodiment, the fluid can flow through the porous layer and the conduit space simultaneously in the at least two concentric conduits including the inner conduit and outer conduit.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying figures, in which like reference numerals refer to identical or functionally-similar elements throughout the separate views and which are incorporated in and form a part of the specification, further illustrate the present invention and, together with the detailed description of the invention, serve to explain the principles of the present invention.

FIG. 1A illustrates a schematic diagram of a double-pipe heat exchanger, in accordance with an embodiment;

FIG. 1B illustrates a schematic diagram of the double-pipe heat exchanger shown in FIG. 1A with geometrical considerations, in accordance with an embodiment;

FIG. 1C illustrates a 2-D axisymmetric view of the double-pipe heat exchanger shown in FIG. 1A and FIG. 1B with boundary conditions depicted, in accordance with an embodiment;

FIG. 2A illustrates a graph indicative of grid independence study results at Cr=0 and Re=800 for an air case, in accordance with an embodiment;

FIG. 2B illustrates a graph indicative of grid independence study results at Cr=0 and Re=800 for a water case, in accordance with an embodiment;

FIG. 3A illustrates a graph depicting a comparison of local Nusselt number results along the pipe section at Cr=0 for the air case, in accordance with an embodiment;

FIG. 3B illustrates a graph depicting a comparison of local Nusselt number results along the pipe section at Cr=0 for the water case, in accordance with an embodiment;

FIG. 4 illustrates a graph depicting data indicative of a comparison of the pressure drop results along each section of the convergent double pipe heat exchanger (C-DPHE) with the analytical solution for both air and water cases at Cr=0, in accordance with an embodiment;

FIG. 5 illustrates a graph depicting data indicative of the overall heat transfer coefficient versus the dimensionless length of heat exchanger for the air flow case at Re=1000, Cr=0.05, and 0.35, in accordance with an embodiment;

FIG. 6A illustrates a graph depicting data indicative of the overall heat transfer coefficient at different inlet Reynolds numbers for various contraction ratios for the air case, in accordance with an embodiment;

FIG. 6B illustrates a graph depicting data indicative of the overall heat transfer coefficient at different inlet Reynolds numbers for various contraction ratios for the water case, in accordance with an embodiment;

FIG. 7A Illustrates a graph depicting data indicative of pressure drop variation versus the inlet Reynolds number for contraction ratios in the air case, in accordance with an embodiment;

FIG. 7B illustrates a graph depicting data indicative of pressure drop variation versus the inlet Reynolds number for contraction ratios in the water case, in accordance with an embodiment;

FIG. 8A illustrates a graph depicting data indicative of the rate of heat exchanged along the C-DPHE at different contraction ratios for different inlet Reynolds numbers in the air case, in accordance with an embodiment;

FIG. 8B illustrates a graph depicting data indicative of the rate of heat exchanged along the C-DPHE at different contraction ratios for different inlet Reynolds numbers in the water case, in accordance with an embodiment;

FIG. 9A illustrates a graph depicting data indicative of the total pumping power versus contraction ratio for different inlet Reynolds numbers in the air case, in accordance with an embodiment;

FIG. 9B illustrates a graph depicting data indicative of the total pumping power versus contraction ratio for different inlet Reynolds numbers in the water case, in accordance with an embodiment;

FIG. 10 illustrates graphs, which depict data indicative of heat transfer enhancement and total pumping power increase over regular DPHE versus contraction ratio at Re=200, and 1000 for the air case and the water case, in accordance with an embodiment;

FIG. 11A illustrates a graph depicting data indicative of heat transfer enhancement along with the resulting total pumping power increase for various contraction ratios for the air case, in accordance with an embodiment;

FIG. 11B illustrates a graph depicting data indicative of heat transfer enhancement along with the resulting total pumping power increase for various contraction ratios for the water case, in accordance with an embodiment;

FIG. 12A illustrates a graph depicting data indicative of PF (Performance Factor) versus the contraction ratio at different inlet Reynolds numbers for the air case, in accordance with an embodiment;

FIG. 12B illustrates a graph depicting data indicative of PF (Performance Factor) versus the contraction ratio at different inlet Reynolds numbers for the water case, in accordance with an embodiment;

FIG. 13A illustrates a schematic diagram of a heat exchanger apparatus comprising a metal foam layer at the convergent interface in an outer conduit, in accordance with an embodiment;

FIG. 13B illustrates a schematic diagram of a heat exchanger apparatus comprising a metal foam layer at the convergent interface in an inner conduit, in accordance with an embodiment; and

FIG. 13C illustrates a schematic diagram of a heat exchanger apparatus comprising a metal foam layer at the convergent interface in both the inner conduit and the outer conduit, in accordance with an embodiment.

DETAILED DESCRIPTION

The particular values and configurations discussed in these non-limiting examples can be varied and are cited merely to illustrate one or more embodiments and are not intended to limit the scope thereof.

Subject matter will now be described more fully hereinafter with reference to the accompanying drawings, which form a part hereof, and which show, by way of illustration, specific example embodiments. Subject matter may, however, be embodied in a variety of different forms and, therefore, covered or claimed subject matter is intended to be construed as not being limited to any example embodiments set forth herein; example embodiments are provided merely to be illustrative. Likewise, a reasonably broad scope for claimed or covered subject matter is intended. Among other things, for example, subject matter may be embodied as methods, devices, components, or systems. Accordingly, embodiments may, for example, take the form of hardware, software, firmware, or any combination thereof (other than software per se). The following detailed description is, therefore, not intended to be interpreted in a limiting sense.

Throughout the specification and claims, terms may have nuanced meanings suggested or implied in context beyond an explicitly stated meaning. Likewise, phrases such as “in one embodiment” or “in an example embodiment” and variations thereof as utilized herein do not necessarily refer to the same embodiment and the phrase “in another embodiment” or “in another example embodiment” and variations thereof as utilized herein may or may not necessarily refer to a different embodiment. It is intended, for example, that claimed subject matter include combinations of example embodiments in whole or in part. In addition, identical reference numerals utilized herein with respect to the drawings can refer to identical or similar parts or components.

In general, terminology may be understood, at least in part, from usage in context. For example, terms such as “and,” “or,” or “and/or” as used herein may include a variety of meanings that may depend, at least in part, upon the context in which such terms are used. Typically, “or” if used to associate a list, such as A, B, or C, is intended to mean A, B, and C, here used in the inclusive sense, as well as A, B, or C, here used in the exclusive sense. In addition, the term “one or more” as used herein, depending at least in part upon context, may be used to describe any feature, structure, or characteristic in a singular sense or may be used to describe combinations of features, structures, or characteristics in a plural sense. Similarly, terms such as “a,” “an,” or “the”, again, may be understood to convey a singular usage or to convey a plural usage, depending at least in part upon context. In addition, the term “based on” may be understood as not necessarily intended to convey an exclusive set of factors and may, instead, allow for existence of additional factors not necessarily expressly described, again, depending at least in part on context.

FIG. 1A illustrates a schematic diagram of a heat exchanger apparatus 100, in accordance with an embodiment. The heat exchanger apparatus 100 shown in FIG. 1A can be configured as a double-pipe heat exchanger, and can include a convergent interface separating two counterflows, and at least two concentric conduits including an inner conduit and an outer conduit as discussed in greater detail herein. The outer conduit comprises an outer conduit radius that is maintained as invariant, and the inner conduit comprises an inner conduit radius that is adjustable to form the convergent interface separating the two counterflows, wherein heat transfer with respect to the two counterflows occurs at the convergent interface.

The heat exchanger apparatus 100 (i.e., a double-pipe heat exchanger) is also shown in FIG. 1B and FIG. 1C. FIG. 1B illustrates a schematic diagram of the heat exchanger apparatus 100 shown in FIG. 1A with geometrical considerations, in accordance with an embodiment. FIG. 1C illustrates a 2-D axisymmetric view of the heat exchanger apparatus 100 shown in FIG. 1A and FIG. 1B with boundary conditions depicted, in accordance with an embodiment. Note that in the figures depicted and described herein, identical or similar parts or elements are indicated by identical reference numerals. In addition, it can be appreciated that the various arrows shown in FIG. 1A, FIG. 1B, and FIG. 1C with respect to the heat exchanger apparatus 100 represent the flow of fluid (e.g., air, liquid, water, etc) and the direction of the flow of fluid through the heat exchanger apparatus 100.

The heat exchanger apparatus 100 shown in FIG. 1A, FIG. 1B, and FIG. 1C is an example of a heat exchanger that can include two concentric pipes as an emulation of a convergent double pipe heat exchanger (C-DPHE). In FIG. 1A, a left side view 102 and a right side view 104 of the double-pipe heat exchanger 100 are shown.

Section 1 refers to the inner pipe (pipe section) 108, while section 2 refers to the outer pipe (annular section) 106. The inner and outer pipes radii are r₁and r₂, respectively. The outer pipe radius, r₂, has been kept invariant; however, the inner radius is adjustable to form a convergent interface separating the two counter flows in this DPHE. Taking into account the optimal distribution of imperfection, where imperfection is the resistance to flow in our case, the rate of contraction must be the same in both sections. Here, we introduce the parameter δ, which can represent the increment and decrement in radius at the inlet and outlet of the inner pipe, respectively; thus, the inlet and outlet radii of the inner pipe are:

r
_1i
=r
₁
+δ,r
_1o
=r
₁−δ (1)

where, δ<r₁, and δ<(r₂−r₁)

Moreover, another term can be introduced which is the contraction ratio (Cr):

$\begin{matrix} Cr = \frac{δ}{r_{1}} & (2) \end{matrix}$

where, 0≤Cr<1

The heat transfer can occur at the interface between the pipe and annular sections. The main constraint in our design is to keep the convergent interface surface area (A_c) constant and equal to that of the plain case (A_p), i.e.

A
_p
=A
_c (3)

where, A_p=2πr₁L_pis the surface area of a plain pipe, and L_pis the length of the plain pipe, whereas A_c=π(r_1i+r_1o)√{square root over ((r_1i−r_1o)²+L_c²)} is the surface area of a convergent pipe (frustum), while L_cis the length of the convergent pipe. Consequently, the length of the convergent pipe, L_c, is:

L
_c=√{square root over (L_p²−4δ²)} (4)

where, L_c≤L_pfor 0≤Cr<1

The configuration is symmetric around the z-axis, as such a 2-D axisymmetric model can be adopted in accordance with an embodiment, as seen in FIG. 1C. Continuity, Momentum, and Energy equations have been employed to simulate flow and heat transfer throughout the domain as follows:

∇· custom-character =0 (5)

ρ( custom-character ·∇)=−∇P+μ∇² (6)

custom-character ·∇T=α∇²T (7)

It should be noted that the major assumptions are: (a) flow is laminar, single phase, incompressible, and steady; (b) the fluids thermophysical properties are constant; (c) the gravitational force is neglected; (d) the thermal resistance of the inner pipe wall is disregarded; and (e) heat loss from the C-DPHE to the ambient is ignored.

The boundary conditions for the present configuration can be specified as:

i. The Pipe Section:

$at the inlet : 0 \leq r < r_{1 i}, z = 0 : u = V_{f 1 i}, v = 0, T = T_{f 1 i} at the exit : 0 \leq r < r_{1 o}, z = L : P = P_{1 o}, \frac{\partial T}{\partial z} = 0 at the symmetry axis : r = 0, 0 < z < L : \frac{\partial u}{\partial r} = 0, v = 0, \frac{\partial T}{\partial r} = 0 at the interface : r = r_{1 z}, 0 < z < L : u = 0, v = 0, k_{f 1} \frac{\partial T}{\partial r} |_{1} = k_{f 2} \frac{\partial T}{\partial r} |_{2}$

$\begin{matrix} The annular section : & ii . \end{matrix}$

$at the inlet : r_{1 o} < r < r_{2}, z = L : u = V_{f 2 i}, v = 0, T = T_{f 2 i} at the exit : r_{1 i} < r < r_{2}, z = 0 : P = P_{2 o}, \frac{\partial T}{\partial z} = 0 at the outer wall : r = r_{2}, 0 < z < L : u = 0, v = 0, \frac{\partial T}{\partial r} = 0$

A model that can be used with the embodiments can take into account the variation in contraction ratio (Cr), inlet Reynolds number (Re), and Prandtl number (Pr) where two working fluids have been utilized, Air and Water, while the inlet fluid temperatures can be set to fixed-specific values, T_f1iand T_f2i, respectively. In addition, inlet mean velocity at the entrance of each section is calculated as follows:

$\begin{matrix} V_{f 1 i} = \frac{Re \cdot μ_{f 1 i}}{ρ_{f 1 i} \cdot D_{1 i}}, V_{f 2 i} = \frac{Re \cdot μ_{f 2 i}}{ρ_{f 2 i} \cdot D_{2 i}} & (8) \end{matrix}$

where, D_1i=2r_1i, and D_2i=2(r₂−r_1o)=2r_2i

The major outcome of this model can include the outlet fluid temperatures (T_f1o, T_f2o), the rate of heat exchanged through the interface separating the two counter flows (Q), and the pressure drop across each section (ΔP₁, ΔP₂), respectively. Further calculations have been tacitly executed, such as the logarithmic mean temperature difference (ΔT_lm), the overall heat transfer coefficient (U), and the total pumping power (W_P) as follows:

$\begin{matrix} Δ T_{lm} = \frac{Δ T_{2} - Δ T_{1}}{\ln (Δ T_{2} / Δ T_{1})} & (9) \end{matrix}$

where, for the counterflow arrangement:

ΔT₁=T_f1i−T_f2o (10a)

ΔT₂=T_f1o−T_f2i (10b)

and, the overall heat transfer coefficient (U) is:

$\begin{matrix} U = \frac{Q}{A_{c} Δ T_{lm}} & (11) \end{matrix}$

and, the total pumping power (W_P) is:

W
_P=(ΔP₁{dot over (V)}₁)+(ΔP₂{dot over (V)}₂) (12)

where, {dot over (V)}₁and {dot over (V)}₂are the volumetric flow rates in sections 1 and 2, respectively.

In some embodiments, a model can be simulated numerically employing the finite element analysis (FEA) approach. The aforementioned governing partial differential equations (PDEs) are discretized, fully coupled, and are also described by the boundary conditions across the computational domain boundaries. The convergence in solution takes place when the accuracy of the velocity-pressure and temperature coupling equations reaches 10⁻⁴and 10⁻⁵, respectively.

FIG. 2A illustrates a graph 202 indicative of grid independence study results at Cr=0 and Re=800 for an air case, in accordance with an embodiment. FIG. 2B illustrates a graph 204 indicative of grid independence study results at Cr=0 and Re=800 for a water case, in accordance with an embodiment. A grid independence survey has been performed by means of increasing the grid density until the variance in local heat transfer coefficient was being less than 0.5%. Table 1 included herein presents five grid distribution types utilized in this work, while FIG. 2A and FIG. 2B show typical results for the local heat transfer coefficient (h) along each section of the model at Cr=0 and Re=800 for the air and water cases (note that Table 1, Table 2, and Table 3 are included at the end of the detailed description section herein). As seen in graph 202 of FIG. 2A and graph 204 in FIG. 2B, respective grid distributions 3 and 4 can provide the required accuracy for the cases of air and water, respectively. Based on this information, these two grid distributions were adopted for the remainder of the investigation.

Furthermore, an extensive validation has been carried out to evaluate the accuracy of our results over the range of Reynolds and Prandtl numbers studied in the current work. FIG. 3A illustrates a graph 206 depicting a comparison of local Nusselt number results along the pipe section at Cr=0 for the air case. FIG. 3B illustrates a graph 208 depicting a comparison of local Nusselt number results along the pipe section at Cr=0 for the water case.

Graph 206 of FIG. 3A and graph 208 of FIG. 3B exhibit the local Nusselt number

$(Nu = \frac{h D_{h}}{k_{f}})$

results alone the thermally developing flow in the pipe section compared to the correlation introduced by Shah and Bhatti at Cr=0 for both cases of air and water. Note that the ‘Shah and Bhatti’ refers to the following reference: Shah, R. K., and Bhatti, M. S., 1987, “Laminar Convective Heat Transfer in Ducts,” Handbook of Single-phase Convective Heat Transfer, S. Kakac, R. K. Shah, and W. Aung, eds., Wiley, New York, Chap. 3, which is incorporated herein by reference in its entirety.

On the other hand, heat transfer results for the thermal entrance region along the annular section, at a radius ratio of R=r₁/r₂=0.5, have been tabulated in Tables 2 and 3 for both air and water cases, respectively, and compared against the results of Lundberg et al, which refers to the following reference: Lundberg, R. E., McCuen, P. A., and Reynolds, W. C., 1963, “Heat Transfer in Annular Passages. Hydrodynamically Developed Laminar Flow With Arbitrarily Prescribed Wall Temperatures or Heat Fluxes,” Int. J. Heat Mass Transfer, 6(6), pp. 495-529, which is incorporated herein by reference in its entirety.

FIG. 4 thus illustrates a graph 210 depicting data indicative of a comparison of the pressure drop results along each section of the C-DPHE with the analytical solution for both air and water cases at Cr=0, in accordance with an embodiment.

Moreover, the pressure drop results along each section of the C-DPHE for both air and water cases at Cr=0 have been displayed in FIG. 4 with the analytical solution for the pressure drop for a developed flow:

$\begin{matrix} Δ P = 2 f \frac{ρ V_{f i}^{2}}{D_{h}} L & (13) \end{matrix}$

where, f is the Fanning friction factor, which is:

$\begin{matrix} f = \frac{16}{Re} (for the pipe section) & (14) \end{matrix}$

$\begin{matrix} f = \frac{2 4}{Re} (for the annular section) & (15) \end{matrix}$

As seen in the foregoing figures and tables, an excellent agreement is observed between our results and those existing in the literature.

A comprehensive investigation of the flow and heat transfer within a convergent double pipe heat exchanger (C-DPHE) has been conducted, over a wide range of contraction ratio, Cr (0-0.35), to explore its hydraulic and thermal performance in comparison with a conventional one. To examine the effect of the Prandtl number, two traditional fluids have been considered, air and water. In addition, the present study covers a wide range of Reynolds numbers.

FIG. 5 illustrates a graph 212 depicting data indicative of the overall heat transfer coefficient versus the dimensionless length of heat exchanger for the air flow case at Re=1000, Cr=0.05, and 0.35, in accordance with an embodiment. FIG. 6A illustrates a graph 214 depicting data indicative of the overall heat transfer coefficient at different inlet Reynolds numbers for various contraction ratios for the air case, in accordance with an embodiment. FIG. 6B illustrates a graph 216 depicting data indicative of the overall heat transfer coefficient at different inlet Reynolds numbers for various contraction ratios for the water case, in accordance with an embodiment.

FIG. 5, FIG. 6A, and FIG. 6B illustrate the overall heat transfer coefficient in the C-DPHE, respectively, versus the inlet Reynolds number for various contraction ratios for both working fluids. As expected, the rise in heat transfer can occur when the Reynolds number increases. Also, as expected, the water case has substantially higher values for heat transfer over the air case.

Furthermore, it can be noticed from FIG. 5, FIG. 6A, and FIG. 6B that increasing the contraction ratio at any given Reynolds number leads to an increase in the heat transferred along the inner pipe wall. This stems from accelerating the flow on both sections of the heat exchanger. Indeed, this acceleration develops a higher pressure drop, as illustrated in FIG. 7A and FIG. 7B. These characteristics have been confirmed as shown in the examples depicted in FIG. 8A, FIG. 8B, FIG. 9A and FIG. 9B.

FIG. 7A Illustrates a graph 218 depicting data indicative of pressure drop variation versus the inlet Reynolds number for contraction ratios in the air case, in accordance with an embodiment. FIG. 7B illustrates a graph 220 depicting data indicative of pressure drop variation versus the inlet Reynolds number for contraction ratios in the water case, in accordance with an embodiment.

FIG. 8A illustrates a graph 222 depicting data indicative of the rate of heat exchanged along the C-DPHE at different contraction ratios for different inlet Reynolds numbers in the air case, in accordance with an embodiment. FIG. 8B illustrates a graph 224 depicting data indicative of the rate of heat exchanged along the C-DPHE at different contraction ratios for different inlet Reynolds numbers in the water case, in accordance with an embodiment.

Compared to the regular DPHE, an improvement in heat transfer up to 32% and 23% in both cases for air and water, respectively, has been achieved by adopting the concept of convergence in our current configuration, as shown, for example, in FIG. 10.

Note that FIG. 9A illustrates a graph 226 depicting data indicative of the total pumping power versus contraction ration for different inlet Reynolds numbers in the air case, in accordance with an embodiment. FIG. 9B illustrates a graph 228 depicting data indicative of the total pumping power versus contraction ration for different inlet Reynolds numbers in the water case, in accordance with an embodiment.

A concern in designing any heat exchanger is how much heat can be transferred and further, the associated costs. Therefore, FIG. 8 demonstrates the variation in the total rate of heat exchanged in our C-DPHE, whereas FIG. 9A and FIG. 9B demonstrate the associated total pumping power versus the contraction ratio for different inlet Reynolds numbers for both air and water cases. As expected, the total rate of heat transfer values for water case are substantially higher than that for the air case; however, the required pumping power values for water case are less than those of the air case. Despite the higher pressure drop values for water, the foregoing characteristic is mainly due to the fact that water is denser than air, and its flow velocities are much lower than that for the air at the same Reynolds numbers. Further, it can be concluded that increasing the contraction ratio, particularly at high Reynolds numbers, results in a considerable increase in heat rate and pumping power. However, the spent pumping power is much less than the gained heat rate for both cases, as displayed in FIG. 8A, FIG. 8B, FIG. 9A and FIG. 9B.

FIG. 10 illustrates a graph 232 and a graph 234, which depict data indicative of heat transfer enhancement and total pumping power increase over regular DPHE versus contraction ratio at Re=200, and 1000 for the air case, in accordance with an embodiment. FIG. 10 further depicts a graph 236 and a graph 238, which depict data indicative of heat transfer enhancement and total pumping power increase over regular DPHE versus contraction ratio at Re=200, and 1000 for the water case, in accordance with an embodiment.

This enhancement can be accompanied by an increase in required pumping power percentage; however, a relatively low to moderate increase in this percentage has been observed (no more than; 50% in air case, and 20% in water case). Furthermore, it is evident that increasing the percentage of pumping power, at any contraction ratio, decreases the desired enhancement in heat transfer over the case without contraction, as seen in FIG. 11.

FIG. 11A illustrates a graph 240 depicting data indicative of heat transfer enhancement along with the resulting total pumping power increase for various contraction ratios for the air case, in accordance with an embodiment. FIG. 11B illustrates a graph 242 depicting data indicative of heat transfer enhancement along with the resulting total pumping power increase for various contraction ratios for the water case, in accordance with an embodiment.

In other words, spending more power on operating this modified heat exchanger is not necessarily needed while more heat can be exchanged or discharged at lower flow velocities for the air case, as shown in graph 232 and graph 234 of FIG. 10 and graph 240 of FIG. 11A. This tendency intensifies in case of water, as illustrated in graph 236 and graph 238 of FIG. 10 and graph 242 of FIG. 11B. This adds another attractive merit to the C-DPHE configuration.

The effectiveness and/or sustainability of this configuration is captured by means of employing the performance factor (PF). It is defined as the ratio of the total rate of heat exchanged over the total pumping power required for C-DPHE to that of the plain one (Cr=0), which can be presented as follows:

$\begin{matrix} P F = \frac{Q / W_{P}}{Q_{*} / W_{P *}} & (16) \end{matrix}$

The heat exchanger performance is better when this factor is higher than one. FIG. 12A and FIG. 12B highlight the performance factor versus the contraction ratio for different inlet Reynolds numbers.

It is apparent that the C-DPHE has an outstanding thermal hydraulic performance compared to the traditional one, especially at low Reynolds numbers, which is consistent with the notion addressed in FIG. 11A and FIG. 11B.

Moreover, an optimal value for the performance factor has been observed, as represented in graph 244 of FIG. 12A and graph 246 of FIG. 12B, which depends on a certain contraction ratio and the examined Reynolds and Prandtl numbers. As such, the best contraction ratio that provides the optimum performance for the C-DPHE can be specified for each flow velocity. Namely, for the inlet Reynolds numbers of (200, 400, 600, 800, and 1000), the best contraction ratios are (0.235, 0.19, 0.15, 0.12, and 0.09) and (0.28, 0.225, 0.2, 0.18, and 0.155) for the air and water cases, respectively. Consequently, selecting the right contraction ratio plays a crucial role in achieving the optimal performance for the C-DPHE.

The foregoing discussion demonstrates the performance of the disclosed novel convergent double pipe heat exchanger (C-DPHE). The effects of contraction ratio, Reynolds and Prandtl numbers on the flow field and heat transfer throughout this configuration have been discussed. The following conclusion can be drawing from the foregoing.

An excellent agreement has been exhibited between the validation results of the disclosed model and the available data from the literature. The overall heat transfer coefficient and pressure drop can increase as the contraction ratio, Reynolds and Prandtl numbers increase. The C-DPHE can provide a substantial enhancement in heat exchange up to 32% and 23% for air and water cases, respectively. The disclosed heat exchanger does not require a high operating pumping power since it is efficient at low Reynolds and high Prandtl numbers.

Furthermore, the performance factor (PF) results affirm the superiority of the C-DPHE over the regular double pipe heat exchanger configuration. In addition, the optimum performance factor has been established in terms of the specification of the optimal contraction ratio, which is a significant aspect in designing the heat exchanger apparatus disclosed herein.

Nomenclature

A surface area, m²

Cr contraction ratio

D diameter, m

f Fanning friction factor

h local convective heat transfer coefficient, W/m²K

k thermal conductivity, W/m K

L length, m

Nu Nusselt number

P pressure, Pa

PF performance factor

Pr Prandtl number

Q heat-transfer rate, W

r radius or radial coordinate, m

R radius ratio

Re Reynolds number

T temperature, K

u velocity component in the axial direction, m/s

U overall heat transfer coefficient, W/m²K

v velocity component in the radial direction, m/s

V velocity, m/s

{dot over (V)} volumetric flow rate, m³/s

W_Ptotal pumping power, W

z axial coordinate, m

Z dimensionless axial coordinate; (z/L)

Z* dimensionless axial station; ((4z/D_h)/Re Pr)

ΔT_lmlogarithmic mean temperature difference, K

Greek Symbols

α thermal diffusivity, m²/s

δ increment or decrement in radius, m

μviscosity, Ns/m²

ρ density, kg/m³

Subscripts

c convergent

f fluid

h hydraulic

i inlet

o outlet

p plain

1 Sec. 1 (pipe section)

2 Sec. 2 (annular section)

* at Cr=0

Table 1, Table 2, and Table 3 are shown below:

TABLE 1

Grid distribution configurations examined in the present work

Grid distribution
Min. element
Max. element
Max. element
Curvature

configuration
size, mm
size, mm
growth rate
factor

1
0.0239 (0.00478)
0.538 (0.335)
1.15 (1.1)
0.3 (0.25)

2
0.012 (0.00179)
0.418 (0.155)
1.13 (1.08)
0.3 (0.25)

3
0.00478 (0.000239)
0.335 (0.0801)
1.1 (1.05)
0.25 (0.2)

4
0.00179 (0.000239)
0.155 (0.0801)
1.08 (1.05)
0.25 (0.2)

5
0.000239 (0.000239)
0.0801 (0.0801)
1.05 (1.05)
0.2 (0.2)

Numbers, in the parentheses, indicate the grid characteristics near the boundaries.

TABLE 2

Comparison of the local Nusselt number results along the annular

section compared with the results of Lundberg et al. for the air

case at R = 0.5, and Cr = 0

Re

Z^{⋆} (= \frac{4 z / D_{h}}{RePr})

Nu Present Work
Nu Lundberg et al.
Deviation (%)

200
0.004
16.7097
16.4
1.888

0.02
10.2588
10.1
1.573

0.04
8.4899
8.43
0.710

0.2
6.3608
6.35
0.170

0.4
6.1938
6.19
0.062

400
0.004
16.6979
16.4
1.817

0.02
10.1754
10.1
0.747

0.04
8.4549
8.43
0.295

0.2
6.3549
6.35
0.077

0.4
6.1901
6.19
0.002

600
0.004
16.5731
16.4
1.056

0.02
10.1559
10.1
0.554

0.04
8.4487
8.43
0.222

0.2
6.3551
6.35
0.081

0.4
6.1904
6.19
0.007

800
0.004
16.5241
16.4
0.757

0.02
10.1503
10.1
0.498

0.04
8.4485
8.43
0.219

0.2
6.3556
6.35
0.088

0.4
6.1907
6.19
0.011

1000
0.004
16.4999
16.4
0.609

0.02
10.1484
10.1
0.479

0.04
8.4515
8.43
0.256

0.2
6.3578
6.35
0.122

0.4
6.1918
6.19
0.030

TABLE 3

Comparison of the local Nusselt number results along the annular section

compared with the results of Lundberg et al. for the water case at

R = 0.5, and Cr = 0

Re

Z^{⋆} (= \frac{4 z / D_{h}}{Re \Pr})

Nu Present Work
Nu Lundberg et al.
Deviation (%)

200
0.004
16.4516
16.4
0.314

0.02
10.1353
10.1
0.349

0.04
8.4357
8.43
0.068

0.2
6.3505
6.35
0.007

400
0.004
16.4348
16.4
0.212

0.02
10.1390
10.1
0.386

0.04
8.4382
8.43
0.097

600
0.004
16.4498
16.4
0.304

0.02
10.1468
10.1
0.464

0.04
8.4438
8.43
0.164

800
0.004
16.4639
16.4
0.390

0.02
10.1587
10.1
0.582

0.04
8.4522
8.43
0.263

1000
0.004
16.4919
16.4
0.561

0.02
10.1725
10.1
0.718

0.04
8.4622
8.43
0.382

Based on the foregoing, it can be appreciated that a number of embodiments, including preferred and alternative embodiments, are disclosed herein. For example, in a preferred embodiment, a heat exchanger apparatus can be configured, which can include a convergent interface separating two counterflows, and at least two concentric conduits including an inner conduit and an outer conduit. The outer conduit can comprise an outer conduit radius that is maintained as invariant, and the inner conduit can comprise an inner conduit radius that is adjustable to form the convergent interface separating the two counterflows, wherein heat transfer with respect to the two counterflows can occur at the convergent interface.