CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



389 



in exactly the same manner as particles entering 

 P at any other time. This property is called "station- 

 arity" of flow. /;) Flow is constant, c) Volume is con- 

 stant, d) There are no stagnant pools. Fluid anywhere 

 in the system eventually moves out of the system. 

 (This assumption is needed only for measurement of 

 volume; flow can be measured even if there are stag- 

 nant pools. ) (') Flow of indicator particles is repre- 

 sentative of the flow of total fluid, that is, the dis- 

 tribution of traversal times for indicator particles is 

 the same as for fluid particles. /) Indicator does not 

 recirculate. Each indicator particle, as it leaves at 

 Q^, is counted once and only once. 



Indicator is injected at P. Assumption e demands 

 that indicator and fluid mix completely at P. This 

 means, for example, that for the case of constant- 

 injection at rate /, the concentration of indicator at 

 P is I/F. The concentration of indicator is measured 

 at Q. ss a function of time. 



Consider first the case of sudden-injection. Let q 

 units of indicator be injected at P at time zero and 

 let its concentration at () be c{t). The amount of 

 indicator leaving the system during a small time 

 interval, / to / -|- dt, is the concentration of indicator 

 at 0,1 c{t), multiplied by the volume of fluid leaving 

 the system during this time interval, F dt. We have 

 already seen that since all the indicator must leave 

 the system, q equals the sum of the amounts leaving 

 the system during all such time intervals, or 



= f c(t) (F dt) = f/" 

 Jo Jo 



c{t) dt 



(3) 



I particle, 7 sec. The frequency with which one finds 

 particles of the kind that require 3 sec to traverse the 

 system is thus 2/12. The frequency with which one 

 finds particles of the kind that require 4 sec to trav- 

 erse the system is 4/12, and so on. A plot of tlic 

 frequency (with which each traversal time is found) 

 against traversal time describes a frequency function, 

 shown in figure 4. 



When the concentration of indicator at Q^ is plotted 

 against time, since from equation 6 c{t) = q h{t)/F, it 

 can be considered that one is in fact plotting the fre- 

 quency function of traversal times, /((/), multiplied 

 by the constant q/F. Assumption e states that the 

 distribution of traversal times for indicator is the 

 same as the distribution of traversal times for fluid 

 particles, that is, h(t) for indicator particles is the 

 same h{t) for fluid particles. Thus, from the observed 

 plot of (•(/) of indicator particles versus time, one has 

 really determined /;(/) for fluid particles. This fact 

 is illustrated in figure 5. 



Returning to figure 4, the sum of all frequencies, 

 2/12 -|- 4/12 + 3/12 4- 2/12 -|- 1/12, is I. In terms 

 of hit), since all fluid entering at time zero must 

 eventually leave the system. 





h{t) dt = I 



(7) 



or the area under the curve, /;(/) vs. t, is i. 

 Combining equations 3 and 6, 



hit) = 



c{t) 



whence 



/ 



Jo 



c{t) dt 



(8) 



F = -^ 



/ c{t) dt 

 Jo 



We now introduce the function 



(5) 



hit) = 



F dt) 



(6) 



Since F c(t) is the rate at which indicator leaves the 

 system at time /, /;(/) is the fraction of injected indi- 

 cator leaving the system per unit time at time /; 

 h(l) is the function which describes the distribution of 

 traversal times. It is therefore a frequency function 

 which may be illustrated as follows: 



Suppose that 12 indicator particles are introduced 

 into the system at P at time zero. Let 2 of the 12 

 particles require 3 sec to travel from P to Q; 4 parti- 

 cles, 4 sec; 3 particles, 5 sec; 2 particles, 6 sec; and 



Thus, to determine /;(/) for fluid particles, each 

 experimentally observed c(t) is divided by the area 

 under the curve c{t) versus /. 



To find the volume of fluid present in the svstem 

 at time zero, consider that the particles of fluid which 

 compose the volume can be distinguished by their 

 traversal times. An element of volume, dV, is there- 

 fore made up of all those particles which, initially 

 present at time zero, have traversal times between / 

 and / + dt. The fraction of particles which require 

 times between t and t -{- dt Xo leave is /;(/) dt. Some of 

 these particles have just entered the system at time 

 zero. Others entered the system t units before time 

 zero and are therefore just ready to leave the system 

 at time zero. The rest of the particles making up dV 

 entered the system during all times between zero and 

 / units before zero. 



The rate at which all fluid particles enter the system 



