CONCENTRATION CHANGE GEORGE H. HORNE Richfield Oil Corporation, Los Angeles, Calif.
A
VARIETY of problems may be idealized by a system consisting of a thoroughly mixed constant quantity of material into which and from which streams of material are simultaneously added and withdrawn a t equal rates. I n such a system, knowledge of the composition of the effluent stream may be desired after a change is made in the composition of the incoming stream.
tor. How long after a change is made in the composition of the incoming sbream will it be necessary to wait before a dependable sample can be taken? Presume that a satisfactory sample may be obtained after the composition of the effluent stream has approached that of the incoming stream to within 5 per cent of the difference between the effluent and incoming streams after the change was made-i. e., when the fraction of the total concentration change remaining to occur is reduced t o 0.05. The corresponding value of Rt/V = 3.00 may be read from the graph. From the conditions of the problem, R = 100 gallons per minute and V = 1000 gallons. Then for Rt/V = 3.00, t must be 30 minutes.
Let R = rate of introduction and withdrawal streams V = quantity of material subject to mixing t = elapsed time after initial conditions CO = concentration in mixed quantity and in effluent stream at time of change in incoming stream (i. e., at t = 0) C = concentration in mixed quantity and in effluent stream at time t after change in incoming stream CI = concentration in incoming stream after change is made Making a material balance around the system in differential form: ClRdt = quantity of solute entering the system during time interval dt CRdt = quantity of solute leaving the system during time interval dt VdC = change in quantity of solute in the system during time interval dt The difference between the quantity of solute entering and leaving the system must equal the change in the quantity of solute in the system: CiRdt
- CRdt
VdC
(1)
Solving the differential equation: Rt = In -V
c-c (r ,$
(2)
Thus Equation 2 represents in generalized form the variation of C, the concentration in the effluent stream, with t , the elapsed time after a change in the concentration of the incoming stream. The significance of the terms of Equation 2 are as follows: Rt = total quantity of material flowing through the system during time t Rt/V = equivalent number of times the quantity of material subject to mixing is replaced during time t CO- C1 = total concentration change C - C1 = portion of concentration change remaining to occur after time’ c -ci -= fraction of total concentration change remaining to co - c1 occur after time t FIGURE 1
R and V must have the same units of quantity; R and t must have the same units of time. For example, if R is ex-
pressed in gallons per minute, V must be in gallons and t must be in minutes. Thus the ratio Rt/V is dimensionless. For convenience Figure 1 was prepared from Equation 2. The utility of this graph is illustrated by the following example : Given a gasoline liquid-vapor separator normally containing 1000 gallons of liquid. The rate a t which liquid is simultaneously introduced and withdrawn is 100 gallons per minute. The only satisfactory sampling connection is on the discharge of a pump removing liquid from the separa-
The foregoing derivation and discussion are based upon perfect mixing of the quantity of material held in the system; i. e., the concentration in the effluent stream is that of the mixed mean concentration of the material held in the system. If mixing is incomplete, it may be shown for values of Rt/V greater than 1.0 that Equation 2 and Figure 1 are conservative in that they give values of t greater than are actually necessary. 1042
