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Flow measurement by the dilution method with incomplete mixing
International Journal of Applied Radiation and Isotopes. Vol. 29. pp. 525 529 Pergamon Press Ltd.. 1978. Printed in Great Britain
Flow Measurement by the Dilution Method with Incomplete Mixing* B. J. B A R R Y Institute of Nuclear Sciences, Department of Scientific and Industrial Research, Private Bag, Lower Hutt, New Zealand
(Received 12 December 1977) If complete mixing cannot be attained in a flow measurement by the dilution method the flow can be determined provided its variation across the steam is known. This enables the flow to be calculated either by determining the flow weighted average concentration or by applying a correction factor if the arithmetic average concentration is used. The errors to be expected from these methods are less than the errors in the measurement of the variation of flow across the stream provided a reasonable degree of mixing is attained. INTRODUCTION THE TRACER dilution method of flow measurement is a well known technique and has been described in detail. ~11 Briefly, a tracer such as a radioactive isotope or a fluorescent dye is injected into the stream to be measured either at a continuous constant flow rate or instantaneously, and at a suitable point downstream its concentration is measured. The extent of dilution of the tracer is proportional to the flow in the stream. The requirements for selecting the measuring point are that, (a) for a continuous injection the concentration at steady state should be constant across the stream, and (b) for an instantaneous injection the time integral of the concentration should be constant across the stream. These conditions are both described as complete mixing. Complete mixing is often difficult, sometimes impossible, to achieve. Thus a method to be used in cases of incomplete mixing would extend the range of the technique. A recent paper ~21 describes a method involving the use of a correction factor to an approximate flow calculated from the arithmetic mean of the cross stream concentrations. A more direct approach, which although it uses the same information has certain advantages, is described here. It will be discussed only in terms of the continuous injection method but all points made are applicable to the instantaneous injection method also as a comparison of the next section and Appendix 1 will show. Similarly the correction factor method will be discussed only in terms of a continuous injection although it was originally propounded for an instantaneous injection.
width at the measuring point be divided into n sections, q, and c, being the flow and concentration respectively in the rth section. Then assuming complete vertical mixing, no sinks for the tracer, Qi '~ Q, and that steady state concentrations have been achieved at the measuring point, we have by mass balance
QjCi = y-q,c,, (where the summation is from r = 1 to n; this will be the case in the remainder of this paper unless otherwise indicated.) 1. The aoerage concentration method We have
Q~Ci = y-q,c, y-q, y,q,
_ Eq,c, Q, Yq, since Q = ]~q'' Thus
Q = Q~C~/(~q~'c']. I\ ~q,l
(1)
Let
Y,q,c, Y-q,
=C.
I
THEORY Consider an injection of tracer of concentration Ci at a flow Q~ into a stream of flow Q. Let the channel * INS contribution No. 876. A.R.Z. 29--9,'10~A
Now, when mixing is complete all the c, "are equal so that equation (1) reduces to the usual form
QICj Q = T
525
(2)
B. J. Barry
526
But if mixing is incomplete this simplification cannot be made. Instead a flow weighted average concentration, which as comparison of equations (1) and (2) shows is the concentration which would have occurred had mixing been complete, must be calculated. Determining flow by this process will be called the average concentration method (ACM) to distinguish it from the normal dilution method and from direct methods such as velocity-cross section area determinations. At first sight the ACM is a pointless exercise as knowledge of the q, implies a prior knowledge of the flow (--- Eq,); this has also been pointed out by RI~MAR.~3~ However two important points should be noted, each arising from the appearance of q, in both the numerator and denominator of the expression for C. Firstly, errors in the numerator and denominator are correlated• Thus, for example, an erroneously high value of one of the q, will cause both numerator and denominator to increase• This can result in a smaller error in C and thus in the Q calculated from it, than in the Q calculated directly from the q,. Secondly, quantities proportional to the q, can be used. Thus, for example, if the q, were found by the velocity-cross-section area method measurement of surface velocities may well be sufficient (assuming the stream depths are known or can be estimated); alternatively the velocity meter used need not be accurately calibrated provided it was consistent within one set of measurements•
2. The correction factor method (2) Setting c, = C(I + ~,), where
On the other hand the correction factor depends on the absolute values of the q, so quantities proportional to the q, should not be used.
ANALYSIS OF ERRORS The effect of random errors in the c, and q, will be considered in this section. For both methods we have
Q = f ( Q i C , cl .... c., ql .... q.). Thus the variance of Q is given by
o'q..
+ E
(4)
For the direct method we have
Thus by calculating the partial differentials in (4) and assigning values to crQ,c,, tr, and aq,, we will be able to compare the errors in the ACM and CFM with that of the direct method• Errors due to the choice of the number and distribution of cross stream sampling points will not be included in the analysis; presumably these would affect the ACM and CFM and the direct method equally so need not be considered further.
1. Average concentration method
Yc, /1
We have
gives Q = QiC, Eq, . T.qrc,
Q,Ci = (~Eq,(1 + ~,), = CQ + CT.q,¢,,
,)
dQ
y.,q,
d(Q~Ci)
Y,q,c, '
dc,
Therefore Q=
QiC~yq,(C,) (~ ~-
(1)
(Eq,c,) ~
and 1 .
(3)
dQ _ QiCl (Eq,c, -- c,F,q,). dq, (Xq,c,) 2 Thus the correction factor method (CFM) depends on the individual q, values and therefore requires a prior knowledge of the flow. However as with the ACM the effects of errors in the q, are partially suppressed, although for a different reason. This is that the correction factor depends on the difference between ~ and C; if the ratio -C/C is close to one the correction factor will necessarily be small so that large fractional errors in it will not result in large fractional errors in Q.
Substituting these values into (4) and dividing by Q2 [as given by equation (1)] gives
-20"2c~ (aQ,c,~ 2 + Z 't,
(aQ~2 \QAcM
=
\Q,cd +
Z(Xq,c, - c,Xq,)e a 2, (Zq'c'Xq') 2
(5)
Flow measurement by the dilution method Later we shall discuss the effects of errors in the c, and q, assuming these to be proportional to the c, and q,. For this purpose we set
527
Substituting these values into (4) and dividing by Q2 [as given by equation (3)] gives ~2
2 2 ¢rQ,c, + ~q,2 cre.+ Z(C -- crl~202 Cr
ae. = ac, and a , = bq,.
\QJcFM
~2
i _ £q, ~ _ _ 1
Then equation (5) becomes As before we will set at. = ac, and aq. = bq,. so that
5:q 2,c 2, ]^cM
\ QiCi/I
(~.q,c,) 2
O'~iC, + a 2 Zq,2 c,2 + b2E(C - c,)2q2,
(~Q~2
+ b2 ~.(Eq,c,. - c, Eq,)2q 2, (~.q,c, gq,)2
(6)
We may then compare the errors of the ACM and the direct method by calculating the ratio
RACM = (a~)A CU/(~q,)a~'"
\ Q kE•
C2
(7'
DATA
We have
QiCi _ Eq,
(c, ) - 1
c
~
(3) "
Then since
Zc, n
- 1
and we can compare the C F M and the direct method by calculating
2. Correction factor method
Q=
-- Eq,
~C
1
gc, = n '
so we get
aQ 1 ~(Q,C,) = -~' dQ
q,
gc,
C
c~Q 63q,
1~ - c, C
and
Because the variation of ~ I Q and R under different conditions is not readily apparent by inspection of the equations examples will be used to illustrate the two methods. Values of a, b, aQ,c, and Q,C, are needed for this, as are cross stream concentration and flow profiles. The value chosen for a is 0.01; this is a realistic value for both radioisotope and chemical tracers. Two values of b will be used; those chosen are 0.1 and 0.2. Such rather imprecise determinations of the q, are taken because it is only under such conditions that the dilution method would be used even if complete mixing were possible. It has been shown ~'~) that the least expected fractional standard deviation associated with the measurement of QiCi using radioactive tracers is 0.002. Since this is a lower limit, a somewhat higher figure of 0.005 will be used here for the ACM. Because of the form of equation (8), testing the C F M requires separate values of aQ,c, and QiCi; the latter will be
TABLE !. Hypothetical results Experiment No. I 2 3 4 5 6 7
q,
I
2
3
4
5
6
7
3
I
I
c,
5 3 2 I I I 1
10 7 4 2 3 2 I
10 9 16 3 I0 15 1
I0 IO 14 4 5 20 I
10 10 12 5 2 I0 10
10 10 10 6 7 5 I
10 10 8 7 9 I I
I0 I0 6 3 3 I I
10 5 4 I 2 I I
5 2 2 I I I 1
TABLE 2. Experimental results Experiment No. 8
q, C,
0.69 8.7
0.37 9.4
0.36 10.2
0.17 10.7
0.22 II.0
0.31 12.0
0.61 13.1
0.45 13.1
0.38 13.1
0.21 13.1
9
q, c,
0.80 5.8
0.59 7.3
0.34 7.5
0.14 8.6
0.17 8.9
0.40 9.8
0.59 10.5
0.45 11.4
0.47 11.9
0.19 12. I
528
B. J. Barry TABLE3. Results
Experiment No.
P
C/C
(ao/Q),,,., ~
I 2 3 4 5 6 7 8 9
0,976 0.950 0.883 0.855 0.805 0.654 0.473 0.941 0.907
0.93 0.83 0.82 0.72 0.77 0.85 0.81 1.0l 1.04
0007 0,008 0.013 0,016 0.021 0.035 0.053 0.008 0.01 I
b=0.1 R~c., ~ (tto/Q)c, ,, 0.18 0.22 0.36 0.43 0.56 0.95 1.43 0.25 ' 0.32
0.006 0.013 0.018 0,030 0.031 0.041 0.067 0.007 0.010
h=0.2 Re, ,~
(tre/Q)~c, ,
R ,,.,,
(~o/Q)c, ,~
R o ,~
0.16 0.34 0,49 0.81 0.82 I.I 1 1.80 0,20 0.28
0.008 0.012 0.024 0.030 0.040 0,070 0.106 0.013 0.020
0. I I 0.16 0.33 0.40 0.53 0,93 1.42 0.19 0.28
0.010 0,024 0.035 0.059 0.061 0.082 0.133 0.012 0.019
0.13 0,32 0.47 0.80 0.8 I 1.10 1.79 0.18 0.27
Plots of P vs R for each method are given in Figs. 1 and 2.
taken as 5~q,c, and ~Q,c, will be set at 0.005 of this value. To test the two methods thoroughly the effects of different mixing conditions should be assessed. A range of such conditions is shown in Table 1 which gives the results of hypothetical experiments in which injections of tracer were made at seven different points to give the cross stream concentration profiles shown at a common measuring point (whose cross stream flow profile is shown at the top of the table). In addition the results of two experiments carried out in the Wainuiomata River (Wainuiomata, New Zealand) using la~I as the tracer and a Gudey meter and measuring rod for velocity-cross-section area measurements are shown in Table 2.
1.8
4
1.6
b:0.1 o b=02
1"4
1"2
,,= n,-
1-0
6 G
0-8
0"6
#9#8
0"4
0"2
RESULTS
0 0'/*
For each of experiments 1-9 Table 3 gives the degree of mixing P (see Appendix 2), C./C, (oQ/Q)AcM,RAc M, (oQ/Q)crM, and RCFM as calculated from the appropriate equations.
o'.s o'-6 017 o'8
o'9
1:o
Degree of N i x i n g FIG. 2. Ratio of the standard deviations of the CFM and the direct method as a function of the degree of mixing.
1-6
1"~.
'I~
& b=0"l o b=0"2
1"2
1"0 's0.8 n," 0-6
0'4
0.2
0 0"/.
05
0-6
0-7
0"8
09
10
D e g r e e of Mixing FIG. 1. Ratio of the standard deviations of the ACM and the direct method as a function of the degree of mixing.
DISCUSSION The results given in Table 3 and Figs. I and 2 show that both the methods can give more accurate values of Q than a direct measurement based on the same qr values. More specifically, the following points can be made: 1. A degree of mixing in excess of 0.65 for the ACM and 0.70 for the CFM is necessary for these methods to be more accurate than the direct method. As inspection of the concentration profiles in Table 1 reveals, these are not difficult degrees of mixing to obtain. 2. As expected the measurement of Q becomes more accurate as the q, measurements become more accurate. However this may be only marginal for the higher degrees of mixing as the relative advantage of the ACM and CFM over the direct method increases with increasing P.
Flow measurement by the dilution method 3. T h e accuracy of the A C M a n d the C F M a n d their relative a d v a n t a g e over the direct m e t h o d b o t h increase as the degree of mixing increases. 4. The C F M is marginally more accurate t h a n the A C M in experiments 1 (b = 0.1), 8 a n d 9. These are the cases when -C/C is closest to one so the correction factor is small. Otherwise the A C M is more accurate. 5. T h e A C M does not require absolute values of the q, whereas the C F M does. T h u s a systematic error would be introduced in the C F M if, say, only surface velocities could be m e a s u r e d across the stream. 6. The results of experiments 8 a n d 9 for the A C M fit well o n the P vs R curve for the hypothetical results (see Fig. 1). This indicates t h a t the relative a d v a n t a g e of the A C M over the direct m e t h o d is largely independent of the cross stream flow profile except insofar as the flow profile affects the value of P. The situation is not so clearcut for the C F M . Here the P vs R curve (see Fig. 2) is not s m o o t h because the value of-C/C is a m a j o r influence o n R; bearing this in mind the results for experiments 8 a n d 9 are compatible with the hypothetical results.
529 APPENDIX
I
Consider an instantaneous injection of an amount A of tracer. Then the amount of tracer, A,, which passes through section r at the measuring point is A, =
q, f c, d t ,
where the integration is over the time of passage of the tracer. Then, A = ZA,,
= zq, fc,
dt,
_ Y~q,Sc,dt Q. Zq, "'" Q =
A/(Eq'~c'dt~. /\ y.q, ]
If complete mixing is obtained (.fc, dt = const.) the usual equation results A
SUMMARY If it is not feasible to o b t a i n complete mixing in a flow determination by the dilution m e t h o d then the flow can be calculated by the average concentration m e t h o d or the correction factor m e t h o d provided there is some, not necessarily very accurate, way of determining the variation of flow across the stream. In the case of the average c o n c e n t r a t i o n method, only the relative variation need be known. The accuracies of the techniques are determined by the particular conditions of the experiment a n d may be calculated from the equations developed here.
Q = j"c dt" Otherwise the flow weighted average of Sc,dt must be found just as the flow weighted average of the steady state concentrations had to be found for the continuous injection case. A similar error analysis to the one applied to that case can thus be made for the instantaneous injection. APPENDIX 2 Various definitions of the degree of mixing P appear in the literature, some examples being given in the references J 5-7) The definition used here is P = 1- ~
1
[Z(c, - C)2q2,]I/2.
Acknowledoement--I wish to thank Dr K. R. LASSEY of the Institute of Nuclear Sciences for his assistance in the error analysis.
REFERENCES 1. DINCER T. Isotopes in Hydrology, p. 93. IAEA, Vienna (1967). 2. GRIESSEIER H., MAKOWSKi J. and MOSEy E. Archwm. Hydrotech. 23, 79 (1976). 3. RlMMAR G. M. Trudy G. G. 1. 36(90), 18-48, NEL Translation No. 749, National Engineering Laboratory, Glasgow (1960). 4. CLAYTON C. G., SP^CKMAN R. and BALL A. M. Radioisotope Tracers in Industry and Geophysics, p. 563. IAEA, Vienna (1967). 5. COBBE. D. and BAILEYJ, F. Surface Water Techniques, Chapter 14. U.S.G.S. Washington (1965). 6. YOTSUKURAN. and COBB E. D. Geological Survey Professional Paper 582-C, Washington (1972). 7. WARD P. R. B. J. Hydraul. Div. Am. Soc. cir. Enars 99, HY7, 1069 (1973).
This gives P -- 1 for complete mixing (c, = C for all r) and tends to zero as mixing becomes more incomplete. For little mixing it gives numbers which intuitively may be felt to be too high (see, for example, experiments 6 and 7 in Table 1). The definition is similar to that of YOTSUKURA and COB# 6~ except that it uses the r.m.s, rather than the sum of the absolute deviations of the c, from C. This was done because the r.m.s, is mathematically more manageable and the P so defined can be incorporated in the equations developed to express the errors of the ACM. Thus by substituting C = Eq, c,/T,q, and Q = Eq, in the expression for P and re-arranging we find (1 -- e)2 = Eq2(c,~'q, -- Y:qr cr)2
(Zq, c,Y~q,)2 Then from equation (6) we get
(~Q~
o~,c,
~q~,c~
Q ]ACM (Q~C,)2 + a2 ~
+ b2(l -- p)2.
Flow Measurement by the Dilution Method with Incomplete Mixing* B. J. B A R R Y Institute of Nuclear Sciences, Department of Scientific and Industrial Research, Private Bag, Lower Hutt, New Zealand
(Received 12 December 1977) If complete mixing cannot be attained in a flow measurement by the dilution method the flow can be determined provided its variation across the steam is known. This enables the flow to be calculated either by determining the flow weighted average concentration or by applying a correction factor if the arithmetic average concentration is used. The errors to be expected from these methods are less than the errors in the measurement of the variation of flow across the stream provided a reasonable degree of mixing is attained. INTRODUCTION THE TRACER dilution method of flow measurement is a well known technique and has been described in detail. ~11 Briefly, a tracer such as a radioactive isotope or a fluorescent dye is injected into the stream to be measured either at a continuous constant flow rate or instantaneously, and at a suitable point downstream its concentration is measured. The extent of dilution of the tracer is proportional to the flow in the stream. The requirements for selecting the measuring point are that, (a) for a continuous injection the concentration at steady state should be constant across the stream, and (b) for an instantaneous injection the time integral of the concentration should be constant across the stream. These conditions are both described as complete mixing. Complete mixing is often difficult, sometimes impossible, to achieve. Thus a method to be used in cases of incomplete mixing would extend the range of the technique. A recent paper ~21 describes a method involving the use of a correction factor to an approximate flow calculated from the arithmetic mean of the cross stream concentrations. A more direct approach, which although it uses the same information has certain advantages, is described here. It will be discussed only in terms of the continuous injection method but all points made are applicable to the instantaneous injection method also as a comparison of the next section and Appendix 1 will show. Similarly the correction factor method will be discussed only in terms of a continuous injection although it was originally propounded for an instantaneous injection.
width at the measuring point be divided into n sections, q, and c, being the flow and concentration respectively in the rth section. Then assuming complete vertical mixing, no sinks for the tracer, Qi '~ Q, and that steady state concentrations have been achieved at the measuring point, we have by mass balance
QjCi = y-q,c,, (where the summation is from r = 1 to n; this will be the case in the remainder of this paper unless otherwise indicated.) 1. The aoerage concentration method We have
Q~Ci = y-q,c, y-q, y,q,
_ Eq,c, Q, Yq, since Q = ]~q'' Thus
Q = Q~C~/(~q~'c']. I\ ~q,l
(1)
Let
Y,q,c, Y-q,
=C.
I
THEORY Consider an injection of tracer of concentration Ci at a flow Q~ into a stream of flow Q. Let the channel * INS contribution No. 876. A.R.Z. 29--9,'10~A
Now, when mixing is complete all the c, "are equal so that equation (1) reduces to the usual form
QICj Q = T
525
(2)
B. J. Barry
526
But if mixing is incomplete this simplification cannot be made. Instead a flow weighted average concentration, which as comparison of equations (1) and (2) shows is the concentration which would have occurred had mixing been complete, must be calculated. Determining flow by this process will be called the average concentration method (ACM) to distinguish it from the normal dilution method and from direct methods such as velocity-cross section area determinations. At first sight the ACM is a pointless exercise as knowledge of the q, implies a prior knowledge of the flow (--- Eq,); this has also been pointed out by RI~MAR.~3~ However two important points should be noted, each arising from the appearance of q, in both the numerator and denominator of the expression for C. Firstly, errors in the numerator and denominator are correlated• Thus, for example, an erroneously high value of one of the q, will cause both numerator and denominator to increase• This can result in a smaller error in C and thus in the Q calculated from it, than in the Q calculated directly from the q,. Secondly, quantities proportional to the q, can be used. Thus, for example, if the q, were found by the velocity-cross-section area method measurement of surface velocities may well be sufficient (assuming the stream depths are known or can be estimated); alternatively the velocity meter used need not be accurately calibrated provided it was consistent within one set of measurements•
2. The correction factor method (2) Setting c, = C(I + ~,), where
On the other hand the correction factor depends on the absolute values of the q, so quantities proportional to the q, should not be used.
ANALYSIS OF ERRORS The effect of random errors in the c, and q, will be considered in this section. For both methods we have
Q = f ( Q i C , cl .... c., ql .... q.). Thus the variance of Q is given by
o'q..
+ E
(4)
For the direct method we have
Thus by calculating the partial differentials in (4) and assigning values to crQ,c,, tr, and aq,, we will be able to compare the errors in the ACM and CFM with that of the direct method• Errors due to the choice of the number and distribution of cross stream sampling points will not be included in the analysis; presumably these would affect the ACM and CFM and the direct method equally so need not be considered further.
1. Average concentration method
Yc, /1
We have
gives Q = QiC, Eq, . T.qrc,
Q,Ci = (~Eq,(1 + ~,), = CQ + CT.q,¢,,
,)
dQ
y.,q,
d(Q~Ci)
Y,q,c, '
dc,
Therefore Q=
QiC~yq,(C,) (~ ~-
(1)
(Eq,c,) ~
and 1 .
(3)
dQ _ QiCl (Eq,c, -- c,F,q,). dq, (Xq,c,) 2 Thus the correction factor method (CFM) depends on the individual q, values and therefore requires a prior knowledge of the flow. However as with the ACM the effects of errors in the q, are partially suppressed, although for a different reason. This is that the correction factor depends on the difference between ~ and C; if the ratio -C/C is close to one the correction factor will necessarily be small so that large fractional errors in it will not result in large fractional errors in Q.
Substituting these values into (4) and dividing by Q2 [as given by equation (1)] gives
-20"2c~ (aQ,c,~ 2 + Z 't,
(aQ~2 \QAcM
=
\Q,cd +
Z(Xq,c, - c,Xq,)e a 2, (Zq'c'Xq') 2
(5)
Flow measurement by the dilution method Later we shall discuss the effects of errors in the c, and q, assuming these to be proportional to the c, and q,. For this purpose we set
527
Substituting these values into (4) and dividing by Q2 [as given by equation (3)] gives ~2
2 2 ¢rQ,c, + ~q,2 cre.+ Z(C -- crl~202 Cr
ae. = ac, and a , = bq,.
\QJcFM
~2
i _ £q, ~ _ _ 1
Then equation (5) becomes As before we will set at. = ac, and aq. = bq,. so that
5:q 2,c 2, ]^cM
\ QiCi/I
(~.q,c,) 2
O'~iC, + a 2 Zq,2 c,2 + b2E(C - c,)2q2,
(~Q~2
+ b2 ~.(Eq,c,. - c, Eq,)2q 2, (~.q,c, gq,)2
(6)
We may then compare the errors of the ACM and the direct method by calculating the ratio
RACM = (a~)A CU/(~q,)a~'"
\ Q kE•
C2
(7'
DATA
We have
QiCi _ Eq,
(c, ) - 1
c
~
(3) "
Then since
Zc, n
- 1
and we can compare the C F M and the direct method by calculating
2. Correction factor method
Q=
-- Eq,
~C
1
gc, = n '
so we get
aQ 1 ~(Q,C,) = -~' dQ
q,
gc,
C
c~Q 63q,
1~ - c, C
and
Because the variation of ~ I Q and R under different conditions is not readily apparent by inspection of the equations examples will be used to illustrate the two methods. Values of a, b, aQ,c, and Q,C, are needed for this, as are cross stream concentration and flow profiles. The value chosen for a is 0.01; this is a realistic value for both radioisotope and chemical tracers. Two values of b will be used; those chosen are 0.1 and 0.2. Such rather imprecise determinations of the q, are taken because it is only under such conditions that the dilution method would be used even if complete mixing were possible. It has been shown ~'~) that the least expected fractional standard deviation associated with the measurement of QiCi using radioactive tracers is 0.002. Since this is a lower limit, a somewhat higher figure of 0.005 will be used here for the ACM. Because of the form of equation (8), testing the C F M requires separate values of aQ,c, and QiCi; the latter will be
TABLE !. Hypothetical results Experiment No. I 2 3 4 5 6 7
q,
I
2
3
4
5
6
7
3
I
I
c,
5 3 2 I I I 1
10 7 4 2 3 2 I
10 9 16 3 I0 15 1
I0 IO 14 4 5 20 I
10 10 12 5 2 I0 10
10 10 10 6 7 5 I
10 10 8 7 9 I I
I0 I0 6 3 3 I I
10 5 4 I 2 I I
5 2 2 I I I 1
TABLE 2. Experimental results Experiment No. 8
q, C,
0.69 8.7
0.37 9.4
0.36 10.2
0.17 10.7
0.22 II.0
0.31 12.0
0.61 13.1
0.45 13.1
0.38 13.1
0.21 13.1
9
q, c,
0.80 5.8
0.59 7.3
0.34 7.5
0.14 8.6
0.17 8.9
0.40 9.8
0.59 10.5
0.45 11.4
0.47 11.9
0.19 12. I
528
B. J. Barry TABLE3. Results
Experiment No.
P
C/C
(ao/Q),,,., ~
I 2 3 4 5 6 7 8 9
0,976 0.950 0.883 0.855 0.805 0.654 0.473 0.941 0.907
0.93 0.83 0.82 0.72 0.77 0.85 0.81 1.0l 1.04
0007 0,008 0.013 0,016 0.021 0.035 0.053 0.008 0.01 I
b=0.1 R~c., ~ (tto/Q)c, ,, 0.18 0.22 0.36 0.43 0.56 0.95 1.43 0.25 ' 0.32
0.006 0.013 0.018 0,030 0.031 0.041 0.067 0.007 0.010
h=0.2 Re, ,~
(tre/Q)~c, ,
R ,,.,,
(~o/Q)c, ,~
R o ,~
0.16 0.34 0,49 0.81 0.82 I.I 1 1.80 0,20 0.28
0.008 0.012 0.024 0.030 0.040 0,070 0.106 0.013 0.020
0. I I 0.16 0.33 0.40 0.53 0,93 1.42 0.19 0.28
0.010 0,024 0.035 0.059 0.061 0.082 0.133 0.012 0.019
0.13 0,32 0.47 0.80 0.8 I 1.10 1.79 0.18 0.27
Plots of P vs R for each method are given in Figs. 1 and 2.
taken as 5~q,c, and ~Q,c, will be set at 0.005 of this value. To test the two methods thoroughly the effects of different mixing conditions should be assessed. A range of such conditions is shown in Table 1 which gives the results of hypothetical experiments in which injections of tracer were made at seven different points to give the cross stream concentration profiles shown at a common measuring point (whose cross stream flow profile is shown at the top of the table). In addition the results of two experiments carried out in the Wainuiomata River (Wainuiomata, New Zealand) using la~I as the tracer and a Gudey meter and measuring rod for velocity-cross-section area measurements are shown in Table 2.
1.8
4
1.6
b:0.1 o b=02
1"4
1"2
,,= n,-
1-0
6 G
0-8
0"6
#9#8
0"4
0"2
RESULTS
0 0'/*
For each of experiments 1-9 Table 3 gives the degree of mixing P (see Appendix 2), C./C, (oQ/Q)AcM,RAc M, (oQ/Q)crM, and RCFM as calculated from the appropriate equations.
o'.s o'-6 017 o'8
o'9
1:o
Degree of N i x i n g FIG. 2. Ratio of the standard deviations of the CFM and the direct method as a function of the degree of mixing.
1-6
1"~.
'I~
& b=0"l o b=0"2
1"2
1"0 's0.8 n," 0-6
0'4
0.2
0 0"/.
05
0-6
0-7
0"8
09
10
D e g r e e of Mixing FIG. 1. Ratio of the standard deviations of the ACM and the direct method as a function of the degree of mixing.
DISCUSSION The results given in Table 3 and Figs. I and 2 show that both the methods can give more accurate values of Q than a direct measurement based on the same qr values. More specifically, the following points can be made: 1. A degree of mixing in excess of 0.65 for the ACM and 0.70 for the CFM is necessary for these methods to be more accurate than the direct method. As inspection of the concentration profiles in Table 1 reveals, these are not difficult degrees of mixing to obtain. 2. As expected the measurement of Q becomes more accurate as the q, measurements become more accurate. However this may be only marginal for the higher degrees of mixing as the relative advantage of the ACM and CFM over the direct method increases with increasing P.
Flow measurement by the dilution method 3. T h e accuracy of the A C M a n d the C F M a n d their relative a d v a n t a g e over the direct m e t h o d b o t h increase as the degree of mixing increases. 4. The C F M is marginally more accurate t h a n the A C M in experiments 1 (b = 0.1), 8 a n d 9. These are the cases when -C/C is closest to one so the correction factor is small. Otherwise the A C M is more accurate. 5. T h e A C M does not require absolute values of the q, whereas the C F M does. T h u s a systematic error would be introduced in the C F M if, say, only surface velocities could be m e a s u r e d across the stream. 6. The results of experiments 8 a n d 9 for the A C M fit well o n the P vs R curve for the hypothetical results (see Fig. 1). This indicates t h a t the relative a d v a n t a g e of the A C M over the direct m e t h o d is largely independent of the cross stream flow profile except insofar as the flow profile affects the value of P. The situation is not so clearcut for the C F M . Here the P vs R curve (see Fig. 2) is not s m o o t h because the value of-C/C is a m a j o r influence o n R; bearing this in mind the results for experiments 8 a n d 9 are compatible with the hypothetical results.
529 APPENDIX
I
Consider an instantaneous injection of an amount A of tracer. Then the amount of tracer, A,, which passes through section r at the measuring point is A, =
q, f c, d t ,
where the integration is over the time of passage of the tracer. Then, A = ZA,,
= zq, fc,
dt,
_ Y~q,Sc,dt Q. Zq, "'" Q =
A/(Eq'~c'dt~. /\ y.q, ]
If complete mixing is obtained (.fc, dt = const.) the usual equation results A
SUMMARY If it is not feasible to o b t a i n complete mixing in a flow determination by the dilution m e t h o d then the flow can be calculated by the average concentration m e t h o d or the correction factor m e t h o d provided there is some, not necessarily very accurate, way of determining the variation of flow across the stream. In the case of the average c o n c e n t r a t i o n method, only the relative variation need be known. The accuracies of the techniques are determined by the particular conditions of the experiment a n d may be calculated from the equations developed here.
Q = j"c dt" Otherwise the flow weighted average of Sc,dt must be found just as the flow weighted average of the steady state concentrations had to be found for the continuous injection case. A similar error analysis to the one applied to that case can thus be made for the instantaneous injection. APPENDIX 2 Various definitions of the degree of mixing P appear in the literature, some examples being given in the references J 5-7) The definition used here is P = 1- ~
1
[Z(c, - C)2q2,]I/2.
Acknowledoement--I wish to thank Dr K. R. LASSEY of the Institute of Nuclear Sciences for his assistance in the error analysis.
REFERENCES 1. DINCER T. Isotopes in Hydrology, p. 93. IAEA, Vienna (1967). 2. GRIESSEIER H., MAKOWSKi J. and MOSEy E. Archwm. Hydrotech. 23, 79 (1976). 3. RlMMAR G. M. Trudy G. G. 1. 36(90), 18-48, NEL Translation No. 749, National Engineering Laboratory, Glasgow (1960). 4. CLAYTON C. G., SP^CKMAN R. and BALL A. M. Radioisotope Tracers in Industry and Geophysics, p. 563. IAEA, Vienna (1967). 5. COBBE. D. and BAILEYJ, F. Surface Water Techniques, Chapter 14. U.S.G.S. Washington (1965). 6. YOTSUKURAN. and COBB E. D. Geological Survey Professional Paper 582-C, Washington (1972). 7. WARD P. R. B. J. Hydraul. Div. Am. Soc. cir. Enars 99, HY7, 1069 (1973).
This gives P -- 1 for complete mixing (c, = C for all r) and tends to zero as mixing becomes more incomplete. For little mixing it gives numbers which intuitively may be felt to be too high (see, for example, experiments 6 and 7 in Table 1). The definition is similar to that of YOTSUKURA and COB# 6~ except that it uses the r.m.s, rather than the sum of the absolute deviations of the c, from C. This was done because the r.m.s, is mathematically more manageable and the P so defined can be incorporated in the equations developed to express the errors of the ACM. Thus by substituting C = Eq, c,/T,q, and Q = Eq, in the expression for P and re-arranging we find (1 -- e)2 = Eq2(c,~'q, -- Y:qr cr)2
(Zq, c,Y~q,)2 Then from equation (6) we get
(~Q~
o~,c,
~q~,c~
Q ]ACM (Q~C,)2 + a2 ~
+ b2(l -- p)2.