CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



603 



functions. There are three distribution functions: /) 

 hi{t) is the frequency distribution of injectate, that is, 

 of the total amount of injectate, q, a fraction q /(,■(/) dt 

 is injected between time t and time / + dl; 2) h{t) 

 is the distribution of transit times through the system, 

 defined as we have used it previously throughout 

 this chapter; 5) h„{t) is the frequency distribution 

 at outflow, that is, of the total quantity of injectate, 

 q, which is eliminated eventually from the system, 

 a fraction q ho{t) dt is eliminated between time t 

 and time t + dt. lio{t) is the integral of the convolution 

 of the injectate distribution, h,{t), upon the distribu- 

 tion of transit times of fluid particles, /;(/). That is, 

 hoU) = /o '''•■ (t — -f) /i(s) dt. Remember that the 

 area under a frequency distribution curve is unity; 

 Jo" h,it) dt = j^ hit) dt = Jo" h,{t) dt = Jo"JJ lu it - 's) 

 h{s) ds dt = I . 



In figure 10^, injection is truly instantaneous. 

 The entire injectate is introduced during the zero-th 

 time interval, or hi(t) = i for / = o and /(,(/) = o 

 for t ^ o. The area of the column representing 

 injectate is unity. In our example, /;(/) is such that 

 for t < 5, h(t) = o, during the fifth time interval 

 20 per cent of the quantity introduced at zero time is 

 eliminated from the system, an additional 40 per 

 cent during the sixth time interval, an additional 30 

 per cent during the seventh, and the final i o per cent 

 during the eighth. The convolution of /;,(/) on h{t) 

 in this case yields a distribution, lh,(t), identical with 

 /;(/), that is, the distribution at outflow represents the 

 true distribution of transit times through the system. 

 This occurs because all products of /(, (/ — s) and 

 h(s), which must appear in the convolution integral, 

 are zero for / < 5 (because h{t) is zero for / < 5) 

 and for s < t (because only /;,(o) = h, (t — t) is not 

 equal to zero). 



Because li^,{t) and h{t) are identical in the case 

 under consideration, the mean transit time through 

 the system t is identical with the mean time, 4 , of 

 the distribution /;„(<) • In the example, /„ = i = 

 J^ / hit) dt = Y.t kit) = (5 X 0.2) + (6X 0.4) + 

 (7 X 0.3) + (8 X o.i) = 6.3. The mean time of the 

 injectate is, of course, zero, in this case. 



In figure loB, injection is delivered as a square 

 wave during the zero and first time intervals. That 

 is, /i,(0 = 0.5 during the zero-th time interval and 

 = 0.5 during the first time interval and /;,(/) = o 

 for t "> I . The mean time of injectate, h , is 

 J^t /(,(/) = (o X 0.5) + (i X 0.5) = 0.5. That 

 fraction, 50 per cent, of the injectate introduced 

 during the zero-th time interval is distributed through 

 the system, as described by h(t), so that 20 per cent 



04 



TIME 



FIG. I o. Histograms of distribution of input and output of in- 

 dicator versus time. Fundamental distribution of transit times 

 through system shown in A. In B and C the input at each time 

 interval (indicated successively by white column, coarsely- 

 hatched column, and finely-hatched column) is distributed 

 thiough the fundamental distribution of transit times (or input 

 is convoluted upon the frequency function of through-put) and 

 the output at any time is the sum of the contributions from each 

 input column. 



of it (or 0.5 X o.Q = o.i of the total injectate) is 

 eliminated during the fifth time interval, 40 per cent 

 of it (or 0.5 X 0.4 = 0.2 of the total injectate) is 

 eliminated during the sixth time interval. The re- 

 maining 50 per cent of injectate, injected during the 

 first time interval, is distributed so that 20 per cent of 

 it is eliminated five time units later, that is, during 

 the sixth time interval, 40 per cent of it is eliminated 

 during the seventh time interval, and so on. The total 

 fraction of injectate eliminated during the sixth 

 time interval is 0.2 (from the first 50% of injectate) 

 -f O.I (from the second 50% of injectate) = 0.3, 

 and so on, summing for each time interval. The mean 

 time of the output distribution, t„ = (5 X o.i) -\- 

 (6 X 0.3) + (7 X 0.35) + (8X 0.2) H- (9 X 0.05) = 

 6.8. The true mean transit time of fluid particles 

 through the system is / = ?„ — ?, = 6.8 — 0.5 = 6.3, 

 which is the result obtained from the data of figure 

 10.4. 



Figure loC emphasizes that the form of the in- 

 jectate distribution and the form of the outflow dis- 

 tribution are irrelevant to the problem of determining 

 flow and volume. In figure loC the injectate distribu- 

 tion has been selected so that it is not the square 

 wave of figure loB, but so that it yields the same 

 value for mean time of injection U- ti = (o X 0.625) "I" 

 (i X 0.25) -f- (2 X 0.125) ~ 05. The distribution 

 function at outflow in figure loC has quite a different 

 form from that shown in figure \oB, because it is 

 the integral of the convolution of a different input 

 on the same distribution of transit times. Never- 



