I
EDGAR B. GUTOFF Brown
Co., Berlin, N. H.
Efficiencies of Mixing Tanks in Smoothing Concentration Fluctuations
I N MANY chemical processes, automatic controllers are used to maintain desired concentrations. These controllers may introduce small cyclic fluctuations in concentration which can be undekrable. I n such cases, a mixing tank should be considered to smooth out these fluctuations. For example, a consistency regulator holds the concentration of paper pulp slurry to a pulp dryer to within about f2Yo of the desired value. These small variations, if not smoothed out, will show u p as undesirable moisture variations in the dried pulp. The efficiency of such mixing tanks may be defined as the relative reduction in the magnitude of the concentration fluctuations when the material passes through the tank. As poor mixing will lower the effectiveness of the tank, the efficiency of agitation may be defined as the ratio of efficiency to the theoretical tank efficiency with perfect mixing. Thus, Et
EaEp
(11
sinusoidal function. This approximation should be fairly good when automatic controllers are used to maintain the desired concentration. Either mass or volume units may be used, but mass units will be employed here. Let constant input and outflow rates, lb./min. W = constant holdup in the mixing tank, lb. H = W / F = holdup time, min. ci = instantaneous input concentrations, lb./lb. of fluid stream co = instantaneous outflow concentration. As the tank is well mixed, this is also the tank concentration ; = time average concentration, in and out b = maximum fractional deviation of y from C, or [(ci - ?)/2]msx. A = cycle time of variations, min. t = time, min. Equating input less output to accumulation yields ciF dt
- ~)m,,./~~l
E;, the efficiency of the mixing tank with perfect mixing, may be easily found from a simple differential material balance, if the concentration fluctuations are approximated by a
- c,F dt = d(c,W)
= Wdc,
(2)
Assume a sinusoidal variation of c,
E: = efficiency of mixing tank with perfect mixing Ea = efficiency of agitation
Goet/X
= iet”
cd =
C(l
+ 6 sin 2?rt/X)
(3)
Substituting H, the holdup time, for W / F and rearranging gives
et”,
Multiplying by the integrating factor, converts the left side to a perfect
+ Eb -
(A
I?=
Et = efficiency of mixing tank, equal to 1 - [Go
differential. The right side can be integrated by parts. Integrating yields
sin
CtJH
2Tt
- 2 r cos 2;t)
(5)
with the constant of integration being zero a t steady state. Rearranging to show the variation in the output concentration divided by the maximum input concentration variation gives the desired equation,
Equation 6 is plotted for several values the ratio of the holdup time in of (H/X), the tank to the cycle time of fluctuations, in Figure 1. The maximum value of the function
(Gc) is the ratio
of the
maximum outflow concentration variation to the maximum input variation, and is equal to one minus the efficiency (1 - 4 ) . I t is seen that, when the holdup time in the tank is equal to or greater than the cycle time of the fluctuations ([H/X] gl), then the outflow variations are reduced to under one fifth of the incoming ones, and the efficiency of the tank, 4,is over 80%. VOL. 48, NO. 10
OCTOBER 1956
1817
The maximum outflow variation occurs when the outflow concentration equals the input concentration. As seen from Figure 1 when the ratio of the holdup time to the cycle time is greater than about one half ([HIX] 20.5), this maximum outflow variation occurs when the cycle is almost half over. By equating f / X in Equation 5 to 0.5. this maximum variation, and thus one minus the efficiency, can be approximated by ~
.-m> S
U
c
1-E;=
for ( H I X ) 2
l/2
(7)
As an example of the use of Equation
7 and Figure 1, suppose that an automatic controller holds the concentration of a stream to 14% ( b = 0.04), and that the variation of the concentration with time approximates a sine wave with a cycle time of 10 minutes ( A = 10). Will a mixing tank with excellent agitation and having a holdup time of 5 minute; ( H = 5) reduce these concentration fluctuations to &0.'?7,? S o w the ratio of the holdup time to the cycle time is one half (HIX= 0.5),and, from Figure 1 or Equation 7> it is seen that this system could reduce the concenrration fluctuations to only about 30% of the original value (efficiency about ?O'%) or to 2 ~ 1 . 2 % . I t is desired to reduce the fluctuations to (0.7,.'4.0) or 17.5% of the original value, and so the tank is too small. A mixing tank with a holdup time of 9 minutes (H/X = 0.9) would be needed. Efficiency of Agitation
II
With small mixing tanks. it is a fairly simple matter to obtain excellent agitation. In larger tanks it is more difficult. If the mixing is not perfect and the mixing time is finite. then the output concentration will not be equal to the average tank Concentration. and the tank volume is not fully utilized. The efficiency of agitation is defined as the ratio of the efficiency of the actual mixing tank to the efficiency that the tank would have with perfect mixing. This efficiency of agitation may be determined by measuring the actual tank efficiency and obtaining the efficiency with perfect mixing from Figure 1 or Equation 7. If it is desired to improve the efficiency of a given mixing tank, and thus further reduce the concentration fluctuations, the efficiency of agitation should first be determined as above. If it is lo^', then the tank efficiency can be increased by improving the mixing; otherwise, a larger mixing tank is needed. RECEIVED for review March 3, 1956 ACCEPTED May 12, 1956
18 1 8
INDUSTRIAL AND ENGINEERING CHEMISTRY
