I
STANFORD W. BRIGGS
School of Chemical Engineering, Purdue University, Lafayette, Ind.
Rigorous Application of Absorption Transfer Unit This concept can be used without altering the basic definition. In some Cases, alterations give rise t o significant errors
T m
TRANSFER-UNiT CONCEPT for masstransfer calculations was introduced by Chilton and Colburn (7, 2). When the transfer is unicomponent, the number of transfer units based on the gas film is defined
Information needed to perform the indicated integration can be obtained if the equilibrium relation and relative phase resistances are known. The differential form of the definition of the height of a transfer unit may be written dZ
HI,, =
d ln(1
ln(1 The equation applies in most absorption operations and in some other operations in which substantially only solutetransfer takes place between phases and in which an inert gas is present. The corresponding equation based on the liquid film generally applies under similar circumstances.
+ Y)
(5)
+ Y i ) - ln(1 + Y)
where 2 = distance in column from point of gas entrance. The group dY/(1 Y)is substituted for d ln(1 Y),and the equation is rearranged to
+
+
V’dY = V’(1 Y)[ln(l
+
+ YO - 1 4 1 + Y)l
dZ
Ht,,
(6) These equations are seldom used without basic change. A rigorous calculation method for the case of double-phase resistance has been developed but is not widely known ( 4 ) . The method is presented in a similar manner in this article. A number of calculations are presented to show the applicability and limitations of certain “substitute” formulas for both double-phase and single-phase resistance situations.
By similar reasoning the corresponding equation for the liquid phase is L’dX = L’(1
+ X) Dn(1 f X) - ln(1 +Xi)]dz HtZ1
+
+ xi) Y,)- ln(1 + Y)+ - In(1 + X) -
The equation also might be written ln(1 In(1
(3)
I n terms of mole ratios the equation is
V’Ntyl where k,la = SZ
x, = 0
Hi,i/”t,i
INDUSTRIAL AND ENGINEERING CHEMISTRY
S = cross-section of column
= 5.00 Yi = 1.20 x,
Calculate the number of transfer units based on the gas film. Solution. From a material balance the following relation between X and Y is derived : X = O.SOO(Y
- 0.050)
For each of several values of Y,values of (1 Y),and ln(1 X) are calculated. The modified operating line, ln(1 Y ) us. ln(1 A’), is plotted in Figure 1. For each of several values of Yi,values of Xi are determined from the equilibrium relation, and values of ln(1 Yi) and ln(1 X i ) are calculated. The modified equilibrium line,
+
+
+ +
(4)
+
L‘/Vf = 2.00
+ Yi) - ln(1 + Y)--
+ Xi) - ln(1 + X)
+ + +
Yb 1.000 Y , = 0.050
X,In
98.8
+
~
Since both V‘dY and L‘dX are equal to dn, the two equations may be equated. The following equation is thus obtained :
Rigorous Calculation Method, Double-Phase Resistance
Equation 1 has been converted by Drew (3) to the following form:
+
(7)
ln(1 in(1
+ +
A modified operating line, In(1 Y) X),and a modified equiYi)us. ln(1 Xi), librium line, ln(1 are plotted on a common graph. For each of several pairs of values of X and Y satisfying the material balance, a point is found on the modified operating line, and through this point a “connecting line” is drawn with a slope equal to the right hand side of Equation 8 (or 9). The intersections of this connecting line with the modified operating line and modified equilibrium line determine a pair of values of ln(1 Y) and In(1 Ys). 1 The quantity ln(1 Y , - ln(1 Y: evaluated and plotted us. ln(1 Y). The area under the curve between he appropriate limits is the number of transfer units. The evaluation of the number of transfer units is illustrated in the following example : A solute is absorbed from a gas in a packed column. The following information is given : us. ln(1
+
+
+
+
In(1 Yi)us. ln(1 XJ, is also plotted in Figure 1. For each of several points on the modified operating line, the connecting line is drawn with a slope of - L'Htu1 (1 X ) V'Htzl(1 Y)' At Y = 0.500. for examale. X = 0.225.
+ +
Equation 21 involves an approximation but the error is slight. When the equilibrium line is substantially a straight line with a slope of m on X,Y coordinates, the following equations apply :
connecting line is, therefore, drawn through the point 0.405, 0.203, which is on the modified operating line, with a slope of -8.17. The intersection of the connecting line with the modified equilibrium line determines the value of Y J = 0.260. To avoid the ln(1 confusion which might arise from plotting
+
negative numbers,
1
+ Y ) -ln(l + Yi) 1 + Y,) - ln(1 + Y ) is ln(1
instead of ln(1 Y ) in F i g u e 2. For plotted us. ln(1 Y ) = 0.405, the point at which ln(1
+
it is found that
+
+
1 Y ) - ln(1
+
ln(1 Yi) 6.90. The number of transfer units based on the gas film is equal to the area under the curve between the appropriate limits in Figure 2. This value is 5.75. =
Equation 16 is an adaption of one proposed by Wiegand ( 5 ) . It is developed by replacing (1 .- y),lm in Equation 1 by 1
-
3 - $,
Calculations for a number of cases in which two-phase resistance occurs have been made by the rigorous method and by the substitute formula methods. The results are summarized in Table I.
The calculation meth-
ods are much the !same as those described in the previous section. The equilibrium and operating lines are plotted using appropriate coordinates. The slopes of the connecting line are given by the following equations, respectively:
Comparison with Substitute Formulas, Interface Concentration Negligible
When the interface concentration is negligible, resistance to mass-transfer is reduced to a single phase. The rigorous calculation method and the substitute formula calculation methods are developed by performing the indicated integration of Equations 4, 12, 13, 14, 15, and 16, respectively, between ap-
Comparison with Substitute Formulas, Double-Phase Resistance
The following substitute formulas are considered for comparison purposes:
xi
-x
LN(I+Y)=0.405
,LN'ltY)=
0
0.2
Ob
0.4
0.4
6.6
LN(I+Y)
LN'(I*X) OR LNWXi)
Figure 1. Point on modified equilibrium line i s found corresponding to point on modified operating line
L N ( I + VI. 0.693
0.049
Figure 2.
Area under curve i s the number of transfer units
VOL. 53, NO. 12
DECEMBER 1961
989
Table I.
Two-Phase Resistance, L’/V’ yb
Ya
X a
xb
0.100 0.500 1.000 3.000 1.000 0.100 0.500 1.000 1.000 0.100 0.500 1.000 1.000
0.050 0.050 0.050 0.050 0.500 0.050 0.050 0.050 0.500 0.050 0.050 0.050 0.500
0 0 0 0 0.225 0 0 0 0.225 0 0 0 0.225
0.025 0.225 0.475 1.475 0.475 0.025 0.225 0.475 0.475 0.025 0.225 0.475 0.475
L’Hty -
V’Ht,
10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 1.00 1.00 1.00 1.00
Nomenclature
Xi
Ntyl
1.20 1.20 1.20 1.20 1.20 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
0.92 4.13 5.75 8.37 1.62 1.18 10.6 22.5 11.9 3.0 27 57 30
Comparison with Substitute Formulas, Interface Concentration Constant
When the interface concentration is constant, resistance to mass-transfer is reduced, or effectively reduced, to a single phase. The rigorous calculation method and the substitute formula calculation methods are developed by performing the indicated integration of Equations 4, 12, 13, 14, 15, and 16, respectively, between appropriate limits. Calculations for a number of cases are summarized in Table 111. ~
~
Table II.
“g13
Nly14
Nly15
Nty16
0.96 4.78 7.25 13.5 2.51 1.24 12.5 32.7 20.2 3.1 31 76 45
1.03 5.16 9.41 26.9 4.25 1.33 16.4 50.5 34.1 3.2 37 105 68
0.94 4.27 6.01 8.96 1.64 1.20 10.8 22.8 12.0 3.00 27.0 57.0 30.0
0.92 4.09 5.69 8.29 1.60 1.18 10.6 22.5 11.9 2.98 26.8 56.7 29.9
iCrty12
0.90 3.99 5.51 7.87 1.52 1.16 10.4 22.3 11.9 3.0 27 57 30
Discussion
Since the rigorous calculation of the number of transfer units for unicomponent transfer is not difficult, it probably should be done in most cases when solute concentrations are high and accurate results are desired. Equation 16 is the preferred one of the substitute formulas when solute concentrations are high. Results obtained with the various formulas become substantially the same when solute concentrations are low. Equation 15 then usually is the preferred formula because both the formula itself and the material balance relationship are simple. Equation 12 may be preferred when solute concentrations are low and equilibrium information is given in
~~
~
~~
~~
Number of Transfer Units by Various Formulas, Y, = 0
yb
Ya
Ntvl
Ntylz
Ntyls
Ntyl4
Ntyls
Ntyla
0.05 0.10 0.10 0.25 0.25 0.25 1.00 1.00 1.00 1.00 3.00 3.00 3.00 3.00 3.00
0.01 0.01 0.05 0.01 0.05 0.10 0.01 0.05 0.10 0.25
1.59 2.26 0.67 3.11 1.52 0.85 4.24 2.65 1.98 1.13 4.94 3.35 2.68 1.85 0.69
1.57 2.22 0.65 3.01 1.44 0.79 3.92 2.35 1.70 0.92 4.32 2.76 2.11 1.32 0.41
1.61 2.30 0.69 3.22 1.61 0.92 4.60 3.00 2.30 1.35 5.71 4.09 3.40 2.48 1.10
1.65 2.39 0.74 3.46 1.81 1.07 5.59 3.95 3.20 2.14 8.70 7.04 6.30 5.23 3.10
1.61 2.30 0.69 3.22 1.61 0.92 4.60 3.00 2.30 1.39 5.71 4.09 3.40 2.48 1.10
1.59 2.26 0.67 3.11 1.52 0.85 4.26 2.67 2.00 1.15 5.02 3.42 2.75 1.90 0.75
0.01
0.05 0.10 0.25 1.00
Table 111. y b
0 0 0.02 0 0 0.05 0 0 0.10 0 0 0.50 0 0 1.0
990
= 2.00
yi
propriate limits. Calculations for a number of cases are summarized in Table 11.
-~
mole fractions. There is no reason to use the other two formulas at either low or high solute concentrations.
Number of Transfer Units by Various Formulas
Number of Transfer Units by Various Formulas, Yi = Constant Nlvl5 Ntyis Yo Yi Ntul Ntylz N t y ~ Nty14
0.02 0.04 0.04 0.05 0.09 0.09 0.10 0.24 0.24 0.50 0.95 0.95 1.00 2.90 2.90
0.05 0.05 0.05 0.10 0.10
0.10 0.25 0.25 0.25 1.00 1.00 1.00 3.00 3.00 3.00
0.52 1.63 1.11 0.71 2.34 1.63 0.56 3.32 2.76 0.88 3.31 2.43 0.69 4.00 3.31
0.53 1.65 1.12 0.74 2.40 1.66 0.61 3.42 2.81 1.10 3.65 2.55 1.10 4.77 3.67
INDUSTRIAL AND ENGINEERING CHEMISTRY
0.54 1.70 1.16 0.76 2.53 1.77 0.64 4.02 3.38 1.39 6.00 4.41 1.63 13.6 12.0
0.54 1.76 1.22 0.78 2.69 1.91 0.68 4.73 4.05 1.78 10.2 8.4 1.88 32.2 30.3
0.51 1.61 1.10 0.69 2.30 1.61 0.51 3.22 2.71 0.69 3.00 2.31 0.41 3.40 2.99
.52 1.63 1.11 0.71 2.34 1.63 0.56 3.33 2.77 0.89 3.33 2.44 0.75 4.08 3.37
Dimensional Units L = length M = moles e = time
Symbols Htz = height of a transfer unit based on liquid film, L Ht, = height of a transfer unit based on gas film, L kzla = modified volumetric liquid film coefficient, equal to L’NtZl/SZ (Me-lL -3) k,Ia = modified volumetric gas film coefficient, equal to V’Nt,,/SZ (Me-lL -9 L’ = rate of liquid diluent flow (Me-’) m = slope of the equilibrium line n = rate of mass transfer (Me-1) Ntz = number of transfer units based on liquid film N t y = number of transfer units based on gas film s = cross-sectional area of contacting equipment, perpendicular to flow (L2) V‘ = rate of gas diluent flow (Me-’) x = mole fraction o f solute in main body of liquid x, = mole fraction of solute in liquid a t interface mole ratio of solute to liquid diluent in main body of liquid = mole ratio of solute to liquid diluent at interface Y = mole fraction of solute in main body of gas Yi = mole fraction of solute in gas at interface Y = mole ratio of solute to gas diluent in main body of gas Y , = mole ratio of solute to gas diluent at interface y* = mole ratio of solute to gas diluent in a gas phase mixture if it were in equilibrium with the main body of liquid 2 = distance in directibn of gas flow (L) Subscripts a = gas outlet and liquid inlet end of column 6 = gas inlet and liquid outlet end of column f l m = log mean across film lm = log mean 1, 12, 13, 14, 15, 16 = definition of number of transfer units according to Equations 1, 12, 13, 14, 15, and 16, respectively, or to parallel formulas for liquid film
x = x,
literature Cited (1) Chilton, T. H., Colburn, A. P., IND. ENC. CHEM.27, 255 (1935). (2) Colburn, A. P., Trans. Am. h t . Chem.
+.211 11939).
Enprs. 35.
(3) Grew, B., ]bid., 56, 681 (1940). (4)Drew, T. B., “Notes on Diffusion and
Mass Transfer,” Columbia University
Book Store, New York, 1950. (5) Wiegand, J. H., Ibid.,36, 679 (1940). .
I
R&EIVEDfor-review March’lO, 1961 ACCEPTED July 20, 1961
