CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



599 



If indicator is introduced into the inflow to the 

 primary system at constant rate, /, concentration of 

 indicator at the outflow from the primary system, 

 from equation i8, is Co{t) = I H(t)/F, providing 

 there is no recirculation. As indicator recirculates, 

 the output from the primary system, F Co{i), is 

 distributed through the recirculating system by the 

 function /(<), so that the first recirculating indicator 

 appears at the input in concentration 



C,{t) 



F Jo F Jf, 



g{s) ds 



For repeated distribution of C,(/) through the primary 

 and recirculating system, the general equation for 

 concentration at output is 



= i 



C,(t) = / Ci{t - s)-h{s)ds 



(29) 



and the concentration at input is, as for sudden- 

 injection. 



•'0 



CM = / c, u- s) g(s) dt 

 Jo 



(27) 



where C,(o) = I/F, and C, = o for / < o 

 Combining equations 27 and 29, 



CM = h(s) C, (t - s - r) g(r) dr ds (30) 



Jo Jo 



To return to the argument developed for equation 

 28 and in figure 9, the average concentration of 

 indicator throughout the entire system at any time 

 is the mass of indicator in the system. It, divided by 

 the total volume, I'r . Since, as was argued for equa- 

 tion 28, the concentration function in the system 

 behaves as though it were a damped oscillation and, 

 at any point in the system, approaches /-//Fj- , the 

 asymptotic behavior of the concentration at outflow 

 is 



lim CM = - + 



(-»» r Jo 



r i-(t-s 



Jo f'r 



his) ds 



- + — / hO) ds-~ 



F y T Jo f" 7- Jo 



= I/F + C,(t) - II/Vt 



sh(s) ds 



(3O 



where is the mean transit time through the primary 



system and F is the flow through the primary system. 



Stephenson (30) has pointed out that advantage 



can be taken of the asymptotic behavior of the con- 



centration function. If, in addition to constant-in- 

 jection of one indicator directly into the inflow of 

 the primary system, there is also constant-injection of 

 a second indicator into some point of the recirculating 

 system, then, from equation 31, the concentration of 

 the second indicator at the primary outflow will 

 have as its limit for large I 



lim A'„(/) 



K,(t) - li/Vr 



(32) 



where K„ and A', are concentrations of the second 

 indicator. 



Equations 31 and 32 determine F and i, and 

 therefore V, the volume of the primary system, despite 

 recirculation, where it is possible to inject indicators 

 at constant rate into the primary and into the re- 

 circulating system, and to measure the concentra- 

 tion of both indicators at the inflow and at the out- 

 flow of the primary system. In this case, recirculation 

 is an essential part of the scheme and is not corrected. 

 Vt , the total volume, which appears in equations 31 

 and 32, is estimated independently, or simply from 

 the limit equation 28. 



Solution of equations 31 and 32 yields 



and 



F = I/[Co(t) - C\{l) + K,(t) - KM)] (33) 



i = Vt [A'.(0 - KM]/I (34) 



where C, and A', are limits of the respective concen- 

 trations for large t. 



Another way of treating the problem of recircula- 

 tion is to solve the observed concentration-time curves 

 for the frequency function of transit times, h{t), 

 through the primary system. This solution again 

 requires that there be two separate injections. Let 

 these injections be a) at the inflow to the primary 

 system and b) at the outflow from the primary 

 system. Injection may be sudden or constant. Con- 

 sider the case of sudden-injection. 



A quantity of indicator, q,, , is introduced into the 

 inflow. During the first circulation the concentration 

 at outflow is Cait) = q F/h{t). Indicator, leaving the 

 primary system at rate F Ca(t), is distributed through 

 the recirculating system in a manner described by 

 the function /(/) and re-enters the primary system 

 where it is distributed again in the manner described 

 by h{t). The frequency function, g(t), describes the 

 over-all distribution of transit times through both 

 the primary and recirculating systems. Therefore, 

 indicator reappears at the outflow site at concentra- 

 tion Ca(t) = Jo Ca (t — r) g{r) dr. The complete 



