c
.
Simplified Calculation of Diffusional Processes Number of Transfer Units or Theoretical Plates1 ALLAN P. COLBURN University of Delaware, Newark, Del.
Relatively simple approximate equations are derived and represented graphically by Figures I and 2 for calculating the number of transfer units or theoretical plates for operations such as absorption, distillation, and countercurrent liquidliquid extraction. These equations and graphs are extended to apply where there is curvature in the operating or equilibrium line, or both. In four examples with marked curvature (Figures 3 to 6) the approximate equations give answers with deviations from correct solutions of 2.5 to 6 per cent for transfer units and 2 to 9 per cent for theoretical plates.
Thus, based on the gas stream, the height, 2,is the product of the number of transfer units, N O G , by the height of the transfer unit (H. T. U.)OQ,
Z
(H. T. U.)oa
(1)
or the height is related in terms of the rate of transfer of solute, Ne, the mass transfer coefficient, Koa, the crosssectional area of the tower, 8, and the mean driving force, ('y
-
Y*)m:
Z = No/(Koa) (8)(Y - ~ " 1 % (2) The value of (H. T. U.)oQor KQais usually based on experimental data, and these terms are related as follows: (3)
Expressions 1 to 3 are in terms of over-all resistances based on the gas stream, but analogous equations are available based on fiLm resistances and the liquid stream (4). Both methods consider resistances to transfer in both fluid streams; the application of the method utilizing transfer units was recently discussed (4). At that time a simple equation and convenient chart were presented for calculating the number of transfer units under certain special conditionsthat is, when there is no solute in the inlet scrubbing fluid, and when the operating and equilibrium lines are straight (true only when the solute is rather dilute). A treatment of the general case is greatly to be desired, however; otherwise one is always in doubt as to the effect of some deviation from the conditions stated. The fundamental expression for the number of transfer units for problems involving diffusion in one direction (such as in absorption and extraction) is (3):
ESPITE the widespread use of the processes of absorp tion, distillation, and extraction, and notwithstanding the continued development of the theory underlying these processes, there is great need for convenient and sound methods of calculating their best designs. Present design methods are either limited in application or very laborious, and insufficient attention has been paid to the optimum values of the various factors. The purpose of this paper is to present a simple method of calculating the number of transfer units or theoretical plates for these operations, which applies over a wide range of conditions. A subsequent paper will deal with convenient relations between the optimum operating and cost factors, such as the value of lost solute, the ratio of the rates of the two fluid streams, and the cost of the column. While the height of a column used in a diffusional process such as absorption distillation, or extraction is dependent upon many factors, the calculation of the height for any given conditions can be relatively straightforward. Types of columns now in general use are chiefly of two classes-those in which the changes in composition in the fluid streams occur continuously throughout (as in packed columns) and those in which changes in composition occur more or less in steps (as in plate columns). These two types will be treated separately, and the specific columns mentioned! packed and plate, will be considered as representative of their classes.
D
(4)
I n this form, where concentrations in the gas are employed, An andogous expression in terms of liquid concentrations gives the number of liquid transfer units. In case of diffusion in both directions a t an equal rate (such as in distillation), the equation is simpler by the omission of the ratio (1 y),/(l y), because of the simpler diffusion relation (3). Wiegand (92) showed that if, as an approximation, the arithmetic mean of (1 y) and (1 y*) is employed instead of the logarithmic mean for the term (1 v ) ~Equation , 4 becomes: No0 is the number of over-all gas transfer units.
-
Packed Columns
-
The height of a packed column is usually calculated by either transfer coefficients or transfer units. These quantities are related and should give the same results; the choice of either method is a matter of convenience. 1
= (Noo)
T h e first paper in this series appeared in 1939 (4).
459
-
-
-
460
I
INDUSTRIAL AND ENGINEERING CHEMISTRY
10
100 M
10
FIGURE 1. NUMBEROF TRANSFER UNITS (TABLE I) 2.3 log [(l - P ) M PI NT = (1 - P )
M
Equation 8 can be rearranged as follows:
Y
- Y* = (?/- mzxz) (1 - mz/Rz) + (YZ- mz~z)mz,/Ri
?(
- y2
=
m2x
= (5
- 52)RL
(6)
(9)
The integration of dy/(y - y*) between the limits of y:!and ya (over which region Equations 6 and 7 are assumed to apply) gives the following solution: d,,
dU
Jyz-
y*
IO00
100
FIGURE 2. NUXBEROF THEORETICAL PLATES(TABLE I’i
+
The second term is often negligible. This separation of terms is desirable since a n integration of the first will hold for equal-molar counterdiffusion. I n any case, we wish to obtain one solution of the first term in Equation 5 to be useful in both types of problems. As previously shown (4, the integration of this term is straightforward if the term (y - y*) is linear in y. In the general case, however, the relation is of a higher power, and direct integration to a rational solution rva does not follow. However, if the transfer duty is divided into two parts, it is possible t o integrate each with certain approximations and to add the solutions. This procedure works particularly well when the transfer operation is carried out to very low concentrations of solute. Now, where y1 is the mole fraction of solute in the entering stream and yz is that in the exit stream, let yo be an intermediate concentration such as the geometric mean of yl and I/Z. Where yz is very small, as is usually the case for an optimum degree of transfer, va mill be small enough, regardless of the magnitude of yl, so that the operating and equilibrium lines between y,, and yz can be assumed to be straight. I n this region, then,
Vol. 33, No. 4
(dilute region) Over the range of y. to yl, both the operating and equilibrium lines may be curved. Either or both of these effects can be approximately represented by adding a term in the square of y in the representation of (y - y*). The (yz mars) term now becomes negligible compared to y and can be dropped: dY
(y
- mzxz) (1 - mz/Rz) + a(y - mzzJ2
-
(7)
where m2 is Henry’s law constant for the dilute region and Rz is the ratio of the rate of flow of the scrubbing stream to that of the stream being scrubbed at the dilute end of the column. The value of (y - y*) in the dilute region then becomes: y - y* = ?I - r / t i L = y - ( g - yl)ml/Rz - ~ ~ 2 x 2 (8)
Owing to the small absolute value of the term a(ya - V W P ) compared with (1 - mz/R2),this term can be dropped. The term [I - m*/Rz a(yl - rn2rz)l is equal t o (y1 - Y*)/ (yl - mzzz)or, ignoring the value of rnzzEcompared to V I , to (1 - y:/yl); Equation 11 then becomes:
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
April, 1941
The desired integral is now the sum of Equations 10 and 12. If the second term in the logarithm of Equation 10 is dropped during the addition, which is usually justifiable owing to its unimportance compared to the first term, and then returned later so that the final form will reduce to the equation for the dilute case, a simple solution is obtained:
=4 + + (1
y1
(1
Y1fdY Y ) (Y - Y * )
Following the procedure utilized by Wiegand (22) of employing the arithmetic mean in place of the logarithmic mean of (1 Y),,Equation 14 becomes:
+
NoG=sy,-
dY
y1
(Y
Equation 13 is readily represented graphically, as shown by Figure 1. I n case of dilute conditions where Equations 6 and 7 apply, the ratio of (1 - mz/R2)/(1 yT/yJ becomes unity2, and if there is no solute in the entering scrubbing fluid, ma22 = 0; then Equation 13 reduces to Equation 13 of the previous paper (4). The successful application of Equation 13 results from the fact that in many practical transfer problems most of the transfer units are located in the dilute region and the additional factors make only a small correction, which, however, brings the results closely into line with the exact solution for cases where there is curvature of the operating or equilibrium lines, or both. Inasmuch as the derivation of Equation 13 involved a number of approximations, and it is difficult to estimate how much error these may have introduced, the degree of reliability of this equation is best checked by a number of examples. Before doing so, however, it is advisable to study the possibility of utilizing units other than mole fraction in this equation. The DEFINITION OF UNITS IN TRANSFER CALCULATIONS. original definition (3) of number of transfer units by Equation 4 was based on mole fraction units, inasmuch as these are the fundamentally correct units appearing in the diffusion equations for gases. For liquids, on the other hand, the case is not 80 clear, but for sake of uniformity, mole fraction units are used analogously to define liquid transfer units. I n case of distillation these are the units almost invariably utilized, so that the definition is fortunate for that operation. While there is some reason to suggest that mole fraction units be utilized in other operations for the sake of uniformity and since the theory of phase equilibrium generally utilizes those units, nevertheless the actual available equilibrium data may be expressed in other units, such as weight per cent in extraction. Therefore the problem is suggested of substituting other units in Equation 4 and deriving the correct expression in terms of Equation 4, so that one will be able to obtain the answer, regardless of what units are used. Units for which this is particularly desirable are mole ratio, weight fraction, and weight ratio. Mole Ratio. Defining Y as the ratio of moles of solute to moles of solute-free stream, Y = y/(l y); then y = Y/(l Y ) , (1 y) = 1/(1 Y ) ,and dy = dY/(l Y)2. Equation 4 can be transformed :
-
+
-
+
-
+
461
+ Y * ) - lJ5 log 1 Yz
+
(15)
As illustrated later, Equation 15 gives the same answer as Equation 5. Weight Units. By a similar procedure the following equations are obtained in weight units, which are identical with Equation 5 in mole fraction units:
-
=
1.15 log ? i!-@! 1 rHz
+
(16)
where r = ratio of molecular weight - of solute-free stream to that of solute H = weight ratio of solute to solute-free stream (for example, humidity units such as Ib. water vapor/ Ib. drv air) w = weight f"raction of solute in stream under eonsideration (particularly useful in extraction work where reference is being made t o a triangular diagram in weight-fraction units) Note that H = w/(l - w). While Equations 5, 15, 16, and 17 all give the same answer, this is true only if the integrals are correctly evaluated. It is of interest that Equation 13 is a close approximation in any of these units.
Plate Columns The derivation of the number of theoretical plates for a transfer process in the range of dilute solute, so that the operating and equilibrium lines can be considered straight, has been carried out by Sore1 (16, 19) and by Thormann ( I ? ) for distillation; by Kremser (9), by Souders and Brown ( I @ ) , and by Silver (18) for absorption; and by Hunter and Nash (8) and by Underwood (18) for extraction. Thormann employed an equation of the type y* = mx
+c
for the equilibrium line and arrived a t an equation not Iirnited to the dilute region, but applicable over any region where the equilibrium curve can be approximated by a straight line. Silver derived also an equation utilizing the fractional plate efficiency, E, which, on rearrangement, is simiIar to Equation 18 except for the replacement of the term (log R/m)by the term log E
R/m
+ (1 - E ) R / m
The equations of all of these investigators were expressed in various forms which can be rearranged to give the following (based on change in gas strengths) : 2 This is true where the value of ys is negligible: otherwise it"wou1dbe true only if the term yi were replaoed by YI us.
-
(dilute region-i.
e., constant m/R)
462
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
-4comparison of this equation with Equation 10 for the number of transfer units in the dilute region and a comparison of analogous equations for liquid strengths gives the following relations, respectively:
Vol. 33, No. 4
EXAMPLE B INVOLVISG A MARKEDCURVATURE OF OPLINE BUT A STRAIGHT EQUILIBRIUM LINE, This case is represented by a very slightly soluble gas present in high strength in the inlet gas. Suppose the inlet gas contains 0.50 mole fraction solute and the remainder air, and is 1 - m/R Np I - R/m Np scrubbed with water containing no solute initially. The exit (19) KG- 2 . 3 log R/rri ' Or Il'ot = 2 . 3 log m/R gas is to contain only 0.001 mole fraction solute. The value of m is constant a t 1000, and a t the dilute end of the column For cases where a large number of the plates are concenthe value of mz/Rz is chosen to be 0.4. This means that R, trated in the dilute region but where the operating or equilibis 2500 and that the exit liquid contains 0.0004 mole fraction rium lines, or both, are slightly curved in the region covered solute. A table is constructed of liquid and gas concentraby the problem, Equation 18 can obviously be modified tions, and the values are plotted on Figure 4. A graphical analogously to Equation 10: integration of dy/(y - y*) results in a value of 10.76. Incidentally, a graphical integration of d Y / ( Y - Y*)results in a value of 11.46. By Equation 5 the number of tranefer units is 10.76 0.35 = 11.11. By Equation 15 for mole A plot of this equation is given by Figure 2. Equation 20 ratio units, the number of transfer units is calculated as is not recommended to be generally used in concentrated 11.46 - 0.35 = 11.11, which is a check on the graphical regions where the custoniary graphical procedure is entirely procedure as well as the correctness of Equations 5 and 15. satisfactory, or where, in distillation of ideal mixtures, the To use Equation 13, we find that y:/yl = 0.4/0.5 = 0.8. exact but cumbersome equations of Smoker (14) can be used. Equation 13 then gives 11.3, to which must be added 0.35 to obtain 11.65 transfer units. This answer is about 5 per Application to Absorption and Desorption cent high, but is on the safe side and close enough to justify the simple and rapid use of Equation 13 rather than the labor The number of transfer units for absorption of one of the of a graphical integration. If Equation 13 had been used with components of a gas mixture is, in mole fraction units, mole ratio units, the answer would have been 11.62 - 0.35 or 11.3 2.3 [(?/I - m 2 r d ( l - m 2 / R z ) 2 + r,lz/Rz]+ 1.15 log _-_yz (21) transfer units, again very close to 1 - yl = 1 - ms/Rz log (Yz - mzzz) (1 - Y,*/Yl) the results using mole fractions. By a comb&ation of stepwise or in mole ratio units, graphical procedure and Equation 2.3 [(Yi - mzxz) (1 - mz/Rz)' + m z , ~ z ]- 1.16 log MY! (22) 18, the exact number of theoretical 1 y2 = 1 - rnz/RZ log ( Y z - m a , ) (1 - Y,*/YJ plates required is 8.2, compared to 7.4 by the convenient approximate equation (Equation 20) which is 9 per cent low. The first term of either of these expressions is readily evaluated by Figure 1. Analogous equations are used for desorption-i. e., reApplication to Distillation moval of a gas from a liquid; these are given in Table I. For mixtures which follow Raoult's law, special analytical EXAMPLE A INVOLVING CURVEDEQGILIBRIUM CURVEOwexpressions are available; they are reviewed below. I n adING TO H E - ~ OF T SOLUTION OF SOLVENT.A gas mixture comdition, it is desirable to have convenient analytical solutions posed of 7 mole per cent acetone and 93 mole per cent nearly useful for any mixtures in the regions of the dilute bottoms saturated air a t 25" C. and 1 atmosphere absolute pressure is and of the concentrated tops, where graphical methods are scrubbed by water a t 25" C. and initially free of acetone. less accurate. The exit gas is to contain only 0.01 mole per cent acetone, and BIKARYMIXTURESWHICH FOLLOW RAOULT'SLAW. In sufficient water is to be applied to the column so that a t the case of total reflux, the number of transfer units is readily dilute end m2/Rz = 0.5. At 25' C. and 1 atmosphere prescalculated by the equation derived by Chilton and Colburn sure the value of m for acetone in water is 1.75 ( 1 ) . Owing (8): to the condensation and dilution of the acetone in water, the waterwillrise 11.7"C. and leave thetower therefore a t 36.7'C., NOG= 2 .3 log yY I ( 1 - Y2)M + 2.3 log & LE I and will contain 2.1 mole per cent acetone. At interme(23) 01 - 1 (1 - Y2) diate values of liquid concentration the temperatures are (distillation, constant 01, total reflux) calculated and then the values of equilibrium concentration of acetone in air, plotted on Figure 3. An exact graphical where y2 and y1 refer to vapor compositions in the column. solution of the value of dy/(y - y*) results in a value of 13.77, Liquid compositions can be used if they represent values in whence, by Equation 5, the number of transfer units is 13.77 the column; i. e., the composition of liquid leaving the column 0.04 = 13.81. To evaluate the ansv,-er by Equation 13, the is used for xl,and this is calculated as one theoretical step value of yT/yl = 5.95/7 = 0.85. Equation 13 then gives above the still composition. 14.12 so that the number of transfer units becomes 14.16 by The analogous equation for number of theoretical plates this method. This answer is about 2.5 per cent higher than a t total reflux is that derived by Fenske (7) and Underwood the correct value of 13.81, which is sufficiently close. It is of (19): interest that if mole ratio units had been used in Equation 13 (i. e., Y1 = 0.0753, YZ= 0.0001, Y f / Y 1 = 0.84), the result would have been 14.12, or by Equation 15 the number of log a. transfer units would be 14.08. This is practically the same (distillation, constant 01, total reflux) answer as was obtained with mole fraction units. By the combination of stepwise graphical procedure and where 2, and z1refer to liquid compositions in the column. Equation 18, the exact number of theoretical plates is 11.2, If 22 and 21 refer to liquid from the condenser and liquid in t h e while the convenient approximate equation, Equation 20, still, N P should be replaced by ( N p 1). gives 10.2 plates or 8 per cent low. ERATING
+
+
+
+
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
April, 1941
For mixtures which follow Raoult’s law, it is possible t o obtain solutions for finite reflux, but these are so complicated that usually it is more desirable to use graphical solution in the body of the distillation diagram and analytical solution a t the ends. as described below. However, for some problems involving very low relative volatilities and large nimbers of plates, general analytical equations are desirable. By utilizing the relation
51
1
a = - (l
(1
e
+ ( a - 1)s
lxl =Lx1
a
[ X - G F i p -
&x
(a
- l)R
= - - . e1 - - = -
Noo=
b 2
R (1
-a - a)R
R 2(1
+
(1
- R)ZD R
+ a - (1 - R ) X D - a)R 2R -
+
ll/lu*u=l 1
=
For example, given a feed a t the boiling point containing 0.801 mole fraction low boiler to be enriched to 0.999 mole fraction low boiler; thus, based on the high boiler, z1 = 0.199 and xD = 0.001; a = 0.333; R = 0.5. I n the above, a = -0.0015, b = -0.499, q = 0.505, e = - 1.5, (e b/2)/q = - 2.48. Then by general Equation 27, NOG = n = 1.71 10.99 = 12.7 transfer units. A number of integrations were made to obtain number of theoretical plates, which were based on daerential action in t h e column. These would (25) therefore lead to transfer units (R 1 ) ~ l)xw] instead of plates. Thus, Lewis (a 1)R (10) i n 1922 e x D r e s s e d t h e number of plates 6y the following equation which was integrated graphically for any (26) problem:
-
-
(R
Enriching column (based on high boiler) : XI
- R)xD. b - a)R ’
1-a’
Stripping column (based on low boiler) :
= XD
R = L / vin enriching
and the equations of the operating lines, the number of transfer units can be directly integrated. The substitutions result in the following equations for the stripping and enriching columns, in which the former is based on the liquor film. The latter is based on the gas film, although eventually with liquid concentrations for convenience, and is based on the high boiler.
NOL =
x on feed plate, x2
Note t h a t values of x and of a refer to the high boiler
CYX
1/* =
=
463
-
(1 (1
- R)XD - a)R
n
These differential equations are of the same type; a general solution of both of them follows:
=LA %+I
- Xn
(28)
Unfortunately, by the method of substitution used, the de(2x1 b 9) (2x2 b - n) n nominator turned out t o be x,* + 1 - 2% + I instead of z,* - x,, and as a result the driving force considered a t any where q = 1/62 - 4a, and the other factors are defined below. plate was greater (in the stripping column, for example) For stripping columns (mixtures of constant a): than was required by the ratio of (z: + 1 - x,,+ 1) t o (z,* 2,) which can be shown, in the case of straight equilibrium n = NOL(for open steam) lines, to be equal to m / R . Thus Lewis’ answer can be multi= NOL 2: (for closed steam) n plied by m / R t o give transfer units if this ratio is constant. To obtain the number of plates, Equation 19 can be used, 2 1 = x on feed plate, xz = xw or N , = n ( m / R ) (.lR / m ) / ( 2 . 3 log m / R ) . For example, a R = L/V in stripping section value of 10 by Lewis for a case where m / R = 1.25 would mean NOL = 12.5 and N P = 11.2. It is noted that Noa = 12.5/ 1.25 = 10, or Lewis’ method using liquid strengths apparently solves for gas-film transfer units. Peters ( l a ) and Dodge and Huffman (6) obtained analytical solutions for equations of the type used by Lewis, and their b 01 t R - (R 1 ) ~ w solutions therefore also utilize a displaced driving force. It e - 2 = - 2(a - l ) R 2R should be stated that as m / R approaches unity, the error is For example, given a feed a t the boiling point containing negligible, and also the numbers of plates and transfer units 0.1 mole fraction low boiler which is t o be stripped to 0.001 become equal. mole fraction with closed steam, with a = 3 and R = 2. For plate columns an analytical solution was made by Then a = -0.00025, b = -0.2505, q = 0.2522, e - b/2 = Smoker (14) for binary mixtures of constant a. It is usually -0.625 : easier, however, to use a stepwise graphical solution in the main portion of the diagram 2.3 log (-0.0153) (2.3) (0.625) (0.2017) (-0.5007) and the equations given below n == 2.055 + 10.88 = 13.94 2 (-0.0002495) + 0.2522 log (-0.3027) (0.0037) at the ends. EQUATIONS FOR ANY MIX0 405 TURH. Special equations for the dilute ends of the diagram NOL = n = 12.94 = 11.72 transfer units 0.333 are given below. They apply for mixtures which do not follow Raoult’s law as well as for those which do. For an enriching column (mixtures of constant a): Stripping Columns. I n stripping columns the stream being scrubbed is the liquor stream, so the basis chosen is t h a t of n = NOQ the liquor. There are two chief types of operation, t h a t 72
2.3 2
= -log
a a
+ bxt + x12 + bxl + $2
+
2.3 ( e
- b / 2 ) log (2x1 + b - n) (2x2 + b + n)
+ +
+
(27)
-
+
3Gf
-
-
’;;“““R“/a“
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
464
where open steam is used, and that utilizing closed steam. The former is exactly analogous to the process of desorption where the scrubbing liquor contains no initial solvent. The number of transfer units in case of open steam is therefore:
(distillation, stripping, open steam)
Vol. 33, No. 4
S o t e that NOG= KoL(R/tn)
(30)
Figure 1 can be conveniently utilized to solve Equation 29. The number of theoretical plates in a stripping column with open steam is analogously:
(distillation, stripping, open steam)
b'IGURE 3. EXAMPLE ABSORPTIOS O F A C E TONE, W I T H A C U R V E DE Q U I LIBRIUM LINE OWING TO HEAT O F ABSOHPTION Dashed line is a tangent to t h e equilibrium cur,ve a t t h e origin.
Note that Figure 2 can be utilized to solve this equation. Where closed steam is employed, the vapors entering the column are in equilibrium with the liquid leaving the still, and the liquid leaving the column is somewhat richer in solute than that leaving the still. We are, however, now interested in the final composition of the liquid, and therefore define x2 as mole fraction of solute in the liquid leaving the still. -4calculation of the number of transfer units between z2 and z1 now includes the action of the still, which is that of one theoretical plate. As Equation 19 shows, the number of transfer units represented by one theoretical plate is (2.3 log mz/Rz)/(l- Rz/mz). To obtain the required number of transfer units in the column, this quantity must be subtracted:
- 2.3 log mdRa 1 - Rdmz
(32)
(distillation, stripping, closed steam)
Analogously the number of theoretical plates (not counting the still) is: FIQURE 4. ExA M P L E OF ABSORPTION O F A
SLIGHTLY SOLUBLE GAS O p e r a t i n g l i n e is curved since a large p r o p o r t i o n pf t h e t o t a l gas i s a b sorbed.
1
=
cgqz
+
( x i - zzjmz) (1 - R2/m,)2
-
-
W m z ] (33) (Lz zz/mz)(l Z , " / L d (distillation, stripping, closed steam)
Enriching Columns. These are analogous to absorption columns if the less volatile constituent is treated as the solute. Then:
(distillation, enriching; compositions and the value of m refer to the less volatile constituent)
In the usual case of total condensers, x2 = y 2 = mole fraction of less volatile constituent in the product. Analogously for plate columns: (Yl - mzx2z) (l - m z / R % ) 2 (yz - m2xz) (1 - yT/y1)
+
7nz/Rz] ('5)
(distillation, enriching; compositions and the value of m refer to the less volatile constituent) Multicomponent Mixtures. Where a relatively sharp separation is being made between the key components, Brown (8) showed that the compositions of the liquid and vapor leaving the feed plate can be closely estimated, and that the temperature change from plate to plate is approximately linear except for a few plates at each end of the column. WITH A CURVED E Q U I L I B R I U M Under these conditions the analytical solution for number LINE of plates is used by Brown, employing a value of equilibrium ratio, K , which represents an average of the value at the feed plate and the value at the plate a t the end of the section being calculated. These equations in revised form are given in Table I. Example C, Stripping Column with Curved Equilibrium Line. An aqueous solution containing 5 weight per cent methanol is fed a t the boiling point to the top of a stripping
FIGURE5 .
ExAMPLE OR STRIPPINQ A DILUTE METHANOLWATERSOLUTION
5
x
I100
April, 1941
INDUSTRIAL AND ENGINEERING CHEMISTRY
465
TABLEI. SUMMARY OF EQUATIONS TRANSFER UNITS
TKEORETICAL PLATES log [(I P)M PI NP = log W P )
+
-
NT
=
2.3 log [(l - P ) M
(1
- P)
+ PI
.GI
NT Absorption Case I, constant m / R
NOG
Case 11, varying m / R
NOG
-
NOG
+ + 1.16 log 11 + Yz
1.15 log
1--Y
- -Y1
p L
1
Y1
Desorption Case I, constant R/wk
NOL
CMe 11, varying R / W L
NOL
xi
-
VZ/?IL
y1
-
mx2
1 - 1.15 iog 1 - x1 51
1 + XI + 1.15 log 1 + xz
XOL
Distillation, Enrichinga Case I, constant m / R
NOQ
Case 11, varying m / R
NOG
Multicomponent mixt.
NOG
yz
- mxa
( m - mzxa) (1 - m / R z ) -YZ - mzxz 1 - Y I /-Y1
- mm
UI
v z - mzxz
Distillation, Stripping, Closed Steamb Case I, constant R / m Case 11. varying R / W L
+
Multicomponent mixt.
2.3 log m / R 1 R/m 2.3 log mdRz 1 Rz/mz 2.3 log mdR2 1 Rz/mz
x,
-
22
xi xa
- XZ/WL
- xz/wl.
- xdma
- xa/wzz
Distillation, Stripping, Open Steamb Case I, constant R/7n
NOL
Case 11, varying R/wi
NOL
Extraction, Stripping
- vdm wa - oB/m vz/rn:) (1 - Rdmz w/ma I - W?/WI WI
Case I, constant R / m
NOL
Case 11, varying R / m
NOL
Case I, constant m / R
NOQ
Case 11, varying m / R
NOG
( -
-e
WI
wa
-
)
Extraction, Enriching
a
2:
va
- mw2 - mwg
- e‘
Concentrations and m are based on high boiler or “heavy key”. Concentrations and m are based on low boiler or “light key”.
Note: Equations for varying m / R or R / m are approximate and hold best for large values of M (say 120 or larger).
column supplied a t the bottom with open steam. The concentration of the bottoms is to be 0.001 of the feed, and the reflux ratio, L / V , is to be 6. To utilize the convenience of a straight operating line, a fictitious molecular weight of methanol of 42 is used (with this value the molar latent heats of methanol and water are equal), and the feed strength becomes 2.2 fictitious mole per cent. A plot of the equilibrium line, from equilibrium data of Cornel1 and Montonna (6), and of the operating line is shown by Figure 5. Over this range the equilibrium line is not greatly curved, though a comparison with the dashed line which is drawn tangent to the equilibrium line a t the origin shows it to be definitely curved. An exact solution results in 28.5 transfer units. The approximate equation gives 30.2, which is 6 per cent high and sufficiently good agreement. By combination of the graphi-
cal stepwise procedure and Equation 18, the exact number of theoretical plates required is 26.5, whereas the approximate Equation 29 gives 27.2, which is only 3 per cent high and an excellent check.
Application to Extraction The case of countercurrent extraction without reflux is analogous to that of desorption or to the stripping section of a distillation column, but the extraction equilibrium data are usually given in units of weight fraction or concentration per unit volume. The former units are preferable for general use with this process inasmuch as they are readily represented on a triangular diagram. I n these units the number of transfer units, based on the stream of solution being extracted (considered as the “liquid”) is:
INDUSTRIAL AND ENGINEERING CHEMISTRY
466
=
where
2.3 (WI - v2/m2) (1 - Rz/md2 + R a / m 2 ]+ - R2/mzlog [ (WZ- vz/m2) (1 - w:/wJ - (1 - ?-)we 1-w 0 = 1.15 log ' - 1.15 (1 - T ) 1% 1 (1 - T)wl) l - w1
0.07 = 20.72. The use of Equation 36 with weight fraction units gives 2.15, or a number of transfer units of 21.5 0.14 - 0.07 = 21.57. This answer is 4 per cent high, which is sufficiently close for most purposes. By combination of stepwise graphical procedure and Equation 18, the exact number of theoretical plates required by this problem is 17.8 compared to 18.1 by the convenient proximate Equation 37, which is 2 per cent high and an excellent check.
(36)
1
+
(extraction without reflux) ma = R~ =
ag
v * / w in dilute region ratio of weight rate of solution to weight rate of
solvent solution in dilute region
For plate columns, ATp
log 1 log R z / m ~
[ -- vdmz) (1 - Rz/mdz + R v2/mz) (1 (W1
(WZ
Summary of Examples ~ / (37) ~ ~ ]
W:/WI)
(extraction without reflux)
If there is no solute in the entering solvent, equations are somewhat simplified.
t ~ 2=
Vol. 33, No. 4
A summary of the exact and approximate solutions of the four typical problems described above is given in the following table :
0 and the
Transfer U n i t s DeviaApprox. t i o n , %
Example
Exact
A. Absorption Absorption Distillation D . Extraction
B. C.
13.8 11.1 28.5 20.7
14.2 11.6 30.2 21.6
+2.5 f4 +6 +4
Theoretical P l a t e s DsviaApprox. tion, % '
Exact
11.1
10.2
8.2 26.5 17.8
27.2
7.4 18.1
-8 -9 +3 +2
The agreement of the results by the convenient approximate equations with the exact answers is well within a 10 per cent deviation and in most cases within one transfer unit or one plate. For problems of these types, therefore, where the exit concentration of the stream being scrubbed is quite low FIGURE 6 , EXAMPLE op (around the optimum exit concentration), the method of soluE~~~~~~~~~ OF A~~~~~ tion may be used with confidence. ACID FROM A WATER SOLUTION BY ISOPROPYL ETHER, WITH CURVED OPERATING AND EQUINomenclature LIBRIUM LINES G = gas, vapor, or solvent molar velocity, lb. moles/(hr.) (sq. ft.), or weight velocity, lb./(hr.) (sq. ft.) = weight ratio unit, such as lb. water/lb. dry air H 13.T. U. = height of a transfer unit, ft. K = y*/x, equilibrium ratio Koa = mass transfer coefficient, lb. moles/(hr.) (cu. ft.) (unit 20
W T . ~KID
IN wmwn ETHER LAYER
Under certain special conditions the composition of the solute compared to the quantity of original carrier associated with it can be greatly increased by use of reflux, as illustrated by Varteressian and Fenske (21). The extraction column is then composed of a stripping and an enriching section similar to distillation. The stripping section is calculated as above. The enriching section is treated analogously to distillation, by basing the number of transfer units or plates on the solvent (or "gas") stream; equations for this case are included in Table I. EXAMPLE D. IWOLVING CURVEDOPERATINGAND EQUILIBRIUM LINES. Consider the extraction by isopropyl ether of acetic acid from an aqueous solution containing 25 weight per cent acid. The exit water is to contain only 0.01 weight per cent acid, and sufficient ether is to be used so that the value of R2/m2in the dilute end is 0.7. To obtain the points for the operating line, a triangular diagram must be used, and the data on this system were recently made available (11). By the method described by Varteressian and Fenske (ZO),points on a rectilinear diagram are obtained from the triangular diagram, except that the chosen value of 0.7 for R 2 / m was used to lay off the operating point on the ternary diagram. The rectilinear diagram is shown by Figure 6. A final graphical solution of the rectilinear diagram based on weight fraction gave a value of the integral in Equation 17 of 20.65, and of the number of transfer units of 20.65 0.14 -
+
L
=
M
m AT,
= = =
N&
=
1'
=
S 21
= =
w
=
x
Y
= = = = = = =
yl*
- y),
NOL = Np = NT = n = = P =
X x* ZD
zw
y
2
=
=
a
=
e
=
e' a,
=
AY)
liquor or solution molar velocity, lb. moles/(hr.) (sq. ft.), or weight velocity, lb./(hr.) (sq. ft.) abscissa of Figures 1 and 2, defined in Table I (dy*/dx) or ( d v * / d w ) = slope of equilibrium line rate of transfer of solute. Ib. moles/hr. over-all number of transfer units based on ga.; over-all number of transfer units based on liquid number of theoretical plates ordinate of Figure 1 number of units parameter of Figures 1 and 2, defined in Table I L / V = reflux ratio. liauor-gas ratio, or solvent-solu' tion ratio, in same un-its a s m ratio of molecular weight of solute-free stream to that of solute cross-sectional area, sq. ft. weight fraction of solute in solvent (corresponds to y in distillation) weight fraction of solute in solution (corresponds to x in distillation) mole fraction of solute in liquor x/(1 - x) = mole ratio of solute in liquor equilibrium value of z corresponding to y value of x in distillate value of x in bottoms mole fraction of solute in gas y/(l - y) = mole ratio of solute in gas equilibrium value of y, corresponding to x = logarithmic mean of (1 - 7 ~ )and (1 - g * ) height of column, ft. g*(l - x)/(l - y " ) ~ ,relative volatility 1 - (1 - T ) w2 1.15 log 2-1 - w - 1.15 (I - r ) log ____1 - ( I - 1') w1 1 - Wl 1.15 log 1-A v 1 - v1
b, c, e, q
=
-
1,:s (I -
T)
constants in equations
log-1 - (1 - T ) v2 1 - (1 - T ) tJ1
April, 1941
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Literature Cited (1) Beare, W. G., McVicar, G. A., and Ferguson, J. B., J . Phus. Chem., 34, 1310-18 (1930). (2) Brown, G . G., Trans. Am. Inst. Chem. Engrs., 32, 321-63 (1936). (3) Chilton, T. H., and Colburn, A. P., IND.ENG.CHEM., 27,255-60, 904 (1935). (4) Colburn, A. P., Trans. Am. Inst. Chem. Engrs., 35, 211-36, 587-91 (1939). (5) Cornell, L. W., and Montonna, R. E., IND. ENG. CHEM.,25, 1331-5 (1933). ( 6 ) Dodge, B. F., and Huffman,J. R., Ibid., 29, 1434-6 (1937). (7) Fenske, M. R., Ibid., 24, 482-5 (1932). (8) Hunter, T. G . , and Nash, A. W., I d . Chemist, 9, 245-8 (1933). (9) Kremser, A., Natl. Petroleum News, 22, No. 21, 42 (1930). (10) Lewis, W. K., IND.ENG.CHEM., 14, 492-6 (1922). (11) Nichols, W. T. et aE., Trans. Am. Inst. Chem. Engrs., 36, 601, GO9 (1940).
467
(12) (13) (14)
Peters, W. A., IND.ENQ.CEEM.,15, 402-3 (1923). Silver, L., Trans. Inst. Chem. Engrs. (London), 12, 64-85 Smoker, E . H., Trans. Am. Innst. Chem. Engrs., 34,
(15) (16)
Sorel, E., “Distillation et rectification industrielles”, 1899, Souders, M., and Brown, G. G., IND. ENG. CHEM.,24,
(1934). 165-72
(1938). 519
(1932).
Thormann, K., “Destillieren und Rektifizieren”, Leipng, Otto SpBmer, 1928. (18) Underwood, A. J. Ti., Ind. Chemist, 10, 128-30 (1934). (19) Underwood, A. J. V., Trans. Inst. Chem. Engrs. (London), 10, (17)
112-58 (1932).
Varteressian, K. A.. and Fenske, M. R.,
IND.ENQ.CHFJM.. 28. 1353-60 (1936). (21) Ibid., 29, 270-7 (1937). (22) Wiegand, J. H., Trans. Am. Inst. Chem. Engrs., 36, 679-81 (1940). (20)
Determination of Unit Conductances for Heat and M a s s Transfer by the Transient Method A. L. LONDON, H. B. NOTTAGEi, AND L. M. K. BOELTER2 Stanford University, Calif.
A method is described which utilizes the transient temperature behavior of a thermal capacitor discharging into a fluid stream, for the determination of the unit conductances for mass and heat transfer. This method yields magnitudes which are in substantial agreement with those obtained b y means of the orthodox steady-state procedure. A rate equation for energy transfer, combining the mechanisms of heat and mass transfer, is employed in the analysis. The relations between the unit conductances for mass transfer, heat transfer, and energy transfer b y the combined mechanisms are considered. The accuracy of the transient method and the magnitude of the errors are discussed. The transient method appears to be sufficiently convenient and accurate so that a future program contemplates an extension of its use to a number of heat and mass transfer problems. HE usual method utilized for the experimental deterT mination of film resistances to heat transfer and mass transfer is the steady-state determination of transfer rates and potentials. The object of this paper is to present a method employing the experimentally determined transient behavior of the temperature of a thermal capacitor discharging through the film resistance, as a means of determining the 1 Present address, Illinois Institute of Teohnology, Chicago, 111. 2
Present address, University of California, Berkeley. Calif.
magnitude of the resistance. For the thermal convection runs, the capacitor was operated dry, while mass transfer conductances were obtained b y discharging the capacitor through a water-wet wick on its surface.
Description of Apparatus The experimental arrangement consisted of the following essentials: provision for a controUed flow of air; a thermal capacitor with a transfer area of the desired shape for which the unit conductances were to be determined, and a relatively high internal conductivity; a means for heating the capacitor t o a temperature different from that of the air stream; and a means for determining the temperature-time history of the capacitor on cooling in the air stream. Figure 1 shows the small wind tunnel used in these tests and the arrangement of the a paratus. Flow control was provided by various thin-plate oritces flanged to the fan suction. The air velocity at nozzle discharge was determined by an impact tube and inclined Ellison draft gage. As it was desired to establish the worth of the transient method of obtaining the unit conductances by comparison with the results of steady-state determinations, the capacitors chosen for test were single right-circular cylinders made of solid bars of hard-drawn copper, placed normal to the air flow at the nozzle discharge, as indicated in Figure 1. Figure 2 shows the construction details, and Table I the constants of the capacitors used. The temperature of the capacitor while cooling in the air stream was determined by a type K-2 Leeds & Northrup otentiometer and a No. 28 gage copper-constantan soft-solderef thermocouple, one junction of which was in the interior of the capacitor (FiFure 2 ) . The reference junction was placed in the air path upstream from the nozzle. Thermocouple calibration was necessary for the wet-wick runs; thus calibration was accomplished in place bv immersion of the capacitor in a water bath. For the thermd convection runs, the time determination was made by observing a stop watch when the potentiometer circuit balanced at predetermined microvoltage settings. As the cool-
