Device for Determining and/or Monitoring the Volume Flow Rate and/or Mass Flow Rate of a Medium to be Measured

Abstract

A ultrasonic flow measuring device, which is distinguished by a low energy consumption. A control/evaluation unit ascertains a plurality of sampled values α, with i=1, 2, 3, . . . ) of a received measuring signal at defined points in time (t) of a predetermined time range and interpolates the sampled values by a continuous function (f(t)), wherein the continuous function (f(t)) is formed by a sum of a predetermined number (n ε N) of wavelets (W) and wherein each wavelet (W) corresponds to the product of a sampled value with a sinc function

Description

The invention will now be explained in greater detail on the basis of the appended drawings, the figures of which show as follows:

FIG. 1 a schematic presentation of a diagram, in which the scanned amplitude values of a measuring signal are plotted against time (→ state of the art);

FIG. 2 a schematic presentation of a diagram, wherein the way in which the device of the invention works is visualized and wherein the scanned amplitude values of the measuring signal are plotted against time; and

FIG. 3 a schematic presentation of a preferred embodiment of the device of the invention.

In FIG. 1, the amplitude values α_n of six sampled values, which were sampled at time intervals T, are plotted against time. The linear connections between neighboring measurement points are shown by dashed lines. The solid curve f(t) characterizes a curve, which was found using one of the known reconstruction algorithms, e.g. according to Lagrange, or according to Levenberg-Markart. Disadvantage of the known methods is the high calculational effort and the high energy consumption of the control/evaluation unit, respectively the microprocessor or DSP, involved therewith.

According to the invention, the control/evaluation unit 11 ascertains a plurality of sampled values (α, with i=1, 2, 3, . . . ) of a received measuring signal at defined points in time (t) within a predetermined time range. Then, the sampled values in the predetermined time range are interpolated by a continuous function (f(t)), with the continuous function (f(t)) being formed by a sum of a predetermined number (n ε N) of wavelets (W) and with each wavelet (W) corresponding to the product of a sampled value with a sinc function

$\left(\frac{\sin \left(x\right)}{x}\right)$

and with a Gaussian bell curve (e^−αχ2, α ε R).

In the time domain, the formula has the following form:

$f\left(t\right)=\sum _{n=-\infty }^{+\infty }a_n·g\left[\pi \left(\frac{t}{T}-n\right)\right]=\sum _{n=-\infty }^{+\infty }a_n·\frac{\sin \left[\pi \left(\frac{t}{T}-n\right)\right]}{\pi \left(\frac{t}{T}-n\right)}{}^{-{\alpha \left[\pi \left(\frac{t}{T}-n\right)\right]}^2}$

By means of the above calculation, an efficient and rapid interpolation can be achieved at relatively small calculational capacity.

Let us consider the case of oversampling and assume that the oversampling rate is equal to σ, where σ is a whole number and greater than or equal to 2. As a result, each sampling range of duration T is divided into σ sampling range portions of duration T/σ.

In the following, let us concentrate on the time range −T/2<t≦+T/2. Of course, also other values of t can be calculated, analogously, by shifting the sampling range correspondingly. The formula for the function f(iT/σ) is, for i in the range −(σ/2+1)≦i≦+(σ/2), wherein i ≠0:

$f\left(\frac{\mathrm{iT}}{\sigma }\right)=\sum _{n=-\infty }^{+\infty }a_n·\frac{\sin \left[\pi \left(\frac{i}{\sigma }-n\right)\right]}{\pi \left(\frac{i}{\sigma }-n\right)}{}^{-{\alpha \left[\pi \left(\frac{i}{\sigma }-n\right)\right]}^2}=\sum _{n=-\infty }^{+\infty }a_nc\left(\frac{i}{n},n\right)$

c(i/σ,n) converges with increasing n very rapidly to zero, so that, in practice, the approximation can be made, that c(i/σ,n) is equal to zero, as soon as the absolute value of n reaches a predetermined limit value. This limit value depends, in the final analysis, on the required, respectively necessary, accuracy of measurement and is referenced in the following as MaxSamples. It has been found, that a desired accuracy of measurement can be achieved in practice, when the value of MaxSamples lies in the range of about 3 to 10.

In this way, the following equation results:

$f\left(\frac{\mathrm{iT}}{\sigma }\right)=\sum _{n=-\mathrm{Max}\mathrm{Samples}}^{+\mathrm{Max}\mathrm{Samples}}a_n·c\left(\frac{i}{\sigma },n\right)$

The coefficients c(i/σ,n) are calculated once and then stored in a table of size (2×MaxSamples+1)×(σ−1). In this way, essentially a processing of the measured values in real time can be achieved.

	n	i/σ = −0.25	i/σ = 0.25	i/σ = 0.5
	−5	X1	X2	X3
	.	.	.	.
	.	.	.	.
	.	.	.	.
	0	Y1	Y2	Y3
	.	.	.	.
	.	.	.	.
	.	.	.	.
	5	Z1	Z2	Z3

It is essential, on the basis of the amplitude values sampled in a time range T, to ascertain that point in time t_max, at which a relative maximum amplitude value (or also a relative minimum amplitude value) of the function f(t) occurs. For example, this value can lie between two sequentially following, sampled values. If t_max is known, then it is also possible to determine the actual maximum f(t_max) (or minimum) of the function f(t).

The manner of proceeding is as follows (see also FIG. 2): First a maximum (or minimum) is determined in the time range by comparing the sampled values with one another. Let us assume that a₀ corresponds to the maximum value (or the minimum value). Additionally, it is assumed, that at least MaxSamples are present before and behind a₀. For practical applications, the parameter MaxSamples lies in the range of 3-10.

In the case of the abscissa value t_max, the first derivative f′(t) of the function f(t) is equal to zero. The formula is, consequently: f′(t_max)=0. Therefore, a₀ is the maximum (or the minimum) of the time range, respectively time interval [a_i], with the following holding: −T<t_max<T.

The value t_max can be found by a linear interpolation of the first derivative according to the following formula:

$t_{\max }=-\frac{f^{\prime }\left(0\right)}{f^{″}\left(0\right)}$

As already stated, the following interpolation wavelet is used in the invention:

$g\left(x\right)=\frac{\sin \left(x\right)}{x}{}^{-\alpha x^2}$

The first derivative g′(x) and the second derivative g″(x) of the Function g(x) are:

$g^{\prime }\left(x\right)=\left[-\frac{\sin \left(x\right)}{x^2}+\frac{\cos \left(x\right)}{x}-2\alpha \sin \left(x\right)\right]{}^{-\alpha x^2}$ $g^{″}(x)=[2\frac{\sin \left(x\right)}{x^3}-2\frac{\cos }{x^2}+\left(2\alpha -1\right)\frac{\sin \left(x\right)}{x}4\alpha ·\cos \left(x\right)+4\alpha x·\sin \left(x\right)]{}^{-\alpha x^2}$

A limit value calculation with x=0 leads to g(0)=1, g′(0)=0, g″(0)=(−2α−1)/3.

Therewith, there results for f(t), f′(t), f″(t) the following mathematical equations:

$f\left(t\right)=\sum _{-\infty }^{+\infty }a_n·g\left[\pi \left(\frac{t}{T}-n\right)\right]$ $f^{\prime }\left(t\right)=\frac{\pi }{T}\sum _{-\infty }^{+\infty }a_n·g^{\prime }\left[\pi \left(\frac{t}{T}-n\right)\right]$ $f^{″}\left(t\right)=\frac{{\pi }^2}{T^2}\sum _{-\infty }^{+\infty }a_n·g^{″}\left[\pi \left(\frac{t}{T}-n\right)\right]$

As a result, the following hold for t=0:

$f^{\prime }\left(0\right)=-\frac{1}{T}\sum _{n\ne 0}{\left(-1\right)}^n\frac{a_n}{n}{}^{-{\mathrm{\alpha \pi }}^2n^2}$ $f^{″}\left(0\right)=-\frac{1}{T^2}\left[a_0\left(2\alpha +\frac{1}{3}\right){\pi }^2+2\sum _{n\ne 0}{\left(-1\right)}^na_n·\left(2{\mathrm{\alpha \pi }}^2+\frac{1}{n^2}\right){}^{-{\mathrm{\alpha \pi }}^2n^2}\right]$ $t_{\max }=-\frac{f^{\prime }\left(0\right)}{f^{″}\left(0\right)}=T\frac{\sum _{n=1}^{\infty }{\left(-1\right)}^n\frac{a_{-n}-a_n}{n}{}^{-{\mathrm{\alpha \pi }}^2n^2}}{a_0\left(2\alpha +\frac{1}{3}\right){\pi }^2+2\sum _{n=1}^{\infty }{\left(-1\right)}^n\left(a_{-n}+a_n\right)·\left(2{\mathrm{\alpha \pi }}^2+\frac{1}{n^2}\right){}^{-{\mathrm{\alpha \pi }}^2n^2}}$

If it is additionally assumed, that only the sampled values within the sub time range of a_{−Maxsamples} to a_Maxsamples are relevant, then a_n=0, when |n|>MaxSamples. The last equation is then:

$t_{\max 1}=T\frac{\sum _{n=1}^{\mathrm{Max}\mathrm{Samples}}{\left(-1\right)}^n\frac{a_{-n}-a_n}{n}{}^{-{\mathrm{\alpha \pi }}^2n^2}}{a_0\left(2\alpha +\frac{1}{3}\right){\pi }^2+2\sum _{n=1}^{\mathrm{Max}\mathrm{Samples}}{\left(-1\right)}^n\left(a_{-n}+a_n\right)·\left(2{\mathrm{\alpha \pi }}^2+\frac{1}{n^2}\right){}^{-{\mathrm{\alpha \pi }}^2n^2}}$

When the range [a_i] is vertically shifted into the range [a_i−a₀], then the last formula can be simplified and one obtains:

$t_{\max 2}=T\frac{\sum _{n=1}^{\mathrm{Max}\mathrm{Samples}}{\left(-1\right)}^n\frac{a_{-n}-a_n}{n}{}^{-{\mathrm{\alpha \pi }}^2n^2}}{2\sum _{n=1}^{\mathrm{Max}\mathrm{Samples}}{\left(-1\right)}^n\left(a_{-n}+a_n-2a_0\right)·\left(2{\mathrm{\alpha \pi }}^2+\frac{1}{n^2}\right){}^{-{\mathrm{\alpha \pi }}^2n^2}}$

In order to find the optimum value for the coefficient α, the equality t_max1=t_max2 should hold, since the abscissa value of the maximum should not change in the case of a translational shifting, since it must be invariant relative to a translation. However, in the strict theory, t_max1≠t_max2 (even when MaxSamples=∞). The reason is, that the function which interpolates the shifted region [a_i−a₀] is not a simple translation of the function which interpolates the region [a_i]. Therefore, the assumption, that t_max1≈t_max2, is only valid for the residuum formulated as follows:

$\mathrm{residue}\left(\alpha ,\mathrm{Max}\mathrm{Samples}\right)=2\alpha +\frac{1}{3}+4\sum _{n=1}^{\mathrm{Max}\mathrm{Samples}}{\left(-1\right)}^n\left(2\alpha +\frac{1}{n^2{\pi }^2}\right){}^{-{\mathrm{\alpha \pi }}^2n^2}$

The absolute value of this residuum is an indicator of how good the selected wavelet is for practical use. The analysis of the residuum by means of a mathematical simulation program (e.g. Mathcad) enables the finding of an optimum value for α for a predetermined number of (2×MaxSamples) around a₀. Optimum values for α lie in the order of magnitude of 0.01 to ca.0.04. If a value for α is inserted, which lies away from the optimum value, then the interpolation function between two sampled values is not ‘smooth’, respectively it has significant harmonics beyond the Nyquist limit, but small deviations from the optimum value (e.g. 5%) have little influence on the result. This last shows how good the proposed device is for performing practical measurements. Especially, rounding errors affect the measurement results minimally.

In practice, two coefficient tables [c_i] and [d_i] (0<i≦MaxSamples), are constructed. Then, the following relationship is obtained:

$t_{\max }=T\frac{\sum _{n=1}^{\mathrm{Max}\mathrm{Samples}}\left(a_{-n}-a_n\right)·c_n}{\sum _{n=1}^{\mathrm{Max}\mathrm{Samples}}\left(a_{-n}+a_n-2a_0\right)·d_n}\mathrm{with}$ $c_n=-{\left(-1\right)}^n\frac{{}^{-{\mathrm{\alpha \pi }}^2\left(n^2-1\right)}}{n}\mathrm{and}$ $d_n=-2·{\left(-1\right)}^n·\left(2{\mathrm{\alpha \pi }}^2+\frac{1}{n^2}\right){}^{-{\mathrm{\alpha \pi }}^2\left(n^2-1\right)}$

The coefficient domains [c_i] and [d_i] are normalized, so that the following holds: c₁=1. The optimum value for α, which depends on the number of sampled values MaxSamples, is read into a table. The coefficients are calculated and stored in a table. During operation, this means for the microprocessor, respectively the control/evaluation unit, of the ultrasonic flow measuring device, that only simple calculative operations, like additions, 2×MaxSamples multiplications and a division need to be performed.

As already mentioned above a number of times, a travel-time difference method is used for measuring flow by means of an ultrasonic flow measuring device. An ultrasonic pulse is radiated into the pipeline, respectively into the measuring tube, in the stream direction (Up) of the medium being measured, received by an ultrasonic transducer, and, thereafter, a plurality of sampled values [up_i] are collected by a high-speed A/D converter within a predetermined time range. The same signal is then sent into the pipeline, respectively into the measuring tube, counter to the stream direction (Down), likewise received by an ultrasonic transducer, and sampled by the A/D converter. Within a predetermined time range, likewise a plurality of sampled values [dn_i] are collected. The time difference between the two measurement signals is proportional to the flow velocity of the medium in the pipeline. The two sampled values are correlated with one another according to the formula:

${\mathrm{corr}}_i=\sum _{n=-\infty }^{+\infty }{\mathrm{up}}_n{\mathrm{dn}}_{n+i}$

The maximum value of this range corresponds to the time difference of the two aforementioned ultrasonic measuring signals. For a highly accurate calculation of the volume flow, this correlation method is, however, much too inaccurate. Consequently, the above method of the invention is applied to the correlation function.

FIG. 3 is a schematic presentation of the device of the invention embodied as an inline ultrasonic flow measuring device 1. The ultrasonic flow measuring device 1 determines, using the known travel-time difference method, the volume flow, or mass flow, of the medium 4, which is flowing in the stream direction (S, respectively Up) in the pipeline 2.

Essential components of the inline ultrasonic flow measuring device 1 are the two ultrasonic transducers 5, 6 and the control/evaluation unit 11. The two ultrasonic sensors 5, 6 are applied to the pipeline 2 at a distance L from one another by means of a securement apparatus (not shown). Appropriate securement apparatuses are sufficiently know from the state of the art and are also sold by the assignee. The pipeline 2 has a predetermined inner diameter di.

An ultrasonic transducer 5; 6 includes as an essential component at least one piezoelectric element 9; 10, which produces and/or receives the ultrasonic measuring signals. The ultrasonic measuring signals are, in each case, via the coupling elements 7, 8, coupled into, respectively coupled out of, the pipeline 2 through which the medium is flowing. The coupling elements 7, 8 care for a best possible impedance matching of the ultrasonic measuring signals as they move from one medium into the other. SP indicates the sound path, on which the ultrasonic measuring signals propagate in the pipeline 2, respectively medium 4. The illustrated case is that of a so-called two-traverse arrangement, in which the ultrasonic transducers 5, 6 are arranged. A “traverse” refers, in such case, to that portion of the sound path SP, on which an ultrasonic measuring signal crosses the pipeline 2 one time. The traverses can, depending on the arrangement of the ultrasonic transducers 5, 6 and, as required, with the insertion of a reflector element into the sound path SP, run diametrally or chordally in the pipeline, respectively in the measuring tube, 2.

Claims

1-9. (canceled)

10. A device for determining and/or monitoring volume flow and/or mass flow of a medium flowing through a pipeline in a stream direction comprising: at least two ultrasonic transducers, which emit ultrasonic measuring signals into the pipeline and receive ultrasonic measuring signals from the pipeline; anda control/evaluation unit, which ascertains the volume- and/or mass-flow of the medium in the pipeline on the basis of the travel-time difference of the ultrasonic measuring signals in the stream direction (S; Up) and counter to the stream direction, wherein:said control/evaluation unit ascertains a plurality of sampled values (α, with i=1, 2, 3, . . . ) of a received measuring signal at defined points in time (t) of a predetermined time range, interpolates the predetermined time range of the measuring signal by a continuous function (f(t)), the continuous function (f(t)) being formed by a sum of a predetermined number (n ε N) of wavelets (W), wherein each wavelet (W) corresponds to the product of a sampled value with a sine function

11. The device as claimed in claim 10, wherein: said control/evaluation unit determines between the sampled values at least one additional sampled value and approximates this additional sampled value, respectively these additional sampled values, by the continuous function, the continuous function being formed by a sum of a predetermined number (n ε N) of wavelets (W), each wavelet (W) corresponding to the product of a sampled value with a sine function

12. The device as claimed in claim 10, wherein: said control/evaluation unit determines an abscissa value (t), at which an ordinate value of the continuous function (f(t)) reaches a predetermined limit value.

13. The device as claimed in claim 12, wherein: the predetermined limit value of the continuous function (f(t)) is a zero point, a maximum, a minimum or an inflection point.

14. The device as claimed in claim 10, wherein: said control/evaluation unit determines an abscissa value (tmax, tmin) for a maximum and/or minimum on the basis of the first derivative f′(t) of the continuous function f(t).

15. The device as claimed in claim 10, wherein: said control/evaluation unit obtains an abscissa value (tmax), at which the continuous function reaches a maximum, by a linear interpolation of the first derivative of the continuous function (f(t)) according to the following formula:

16. The device as claimed in claim 10, wherein: said control/evaluation unit correlates, with one another, two ultrasonic measuring signals in two time ranges, interpolates the corresponding, discrete collection of correlation points by a continuous function (f(t)), and determines the abscissa value of the continuous function (f(t)), at which the ordinate value reaches a maximum value, the abscissa value being a measure for a time shift between ultrasonic measuring signals sent and received in the stream direction (S, Up) and counter to the stream direction (Down).

17. The device as claimed in claim 10, wherein: said calculating/control unit determines by means of a mathematical simulation program, in each case, an optimum value for the coefficient (α) as a function of the number of measurement points (MaxSamplei).

18. The device as claimed in claim 17, further comprising: a memory unit, in which, in each case, the optimum value for the coefficient (α) is stored as a function of the number of measurement points (MaxSample).