Nomenclature a, b = stoichiometric coefficients for water vapor, carbon C.4 = concentration of water vapor in nitrogen, mol/cm3 D = diffusion coefficient, cm2/sec dL, = diameter of graphite rod, cm E = activation energy, cal/mol k,, = Arrhenius preexponential factor, cm/hr k , = diffusion reaction rate constant, cm/hr ks = surface controlling reaction rate constant, cm/hr k s = mean reaction rate constant, cm/hr N r 4 = moles of carbon P.\* = equilibrium partial pressure of water vapor in ni-
trogen, cm Hg P = total pressure in saturator, cm Hg R = ideal gas law constant Sex = exterior surface area of graphite rod, cm2 t = time, hr T = reaction temperature, "K Sc = Schmidt number Re = Reynoldsnumber y.\ = mole fraction of water vapor Literature Cited Ergun, S., Menster. M.. "The Chemistry and Physics of Carbon," Vol. I , pp 203-263, P. L. Walker, Jr., Ed., Marcel Dekker, Inc., New York. N.Y., 1965.
Johnstone. H. F., Chen. C. F., Scott. D . , Ind. Eng. Chem., 4 4 (7). 1564 (1952). Mayers, M. A . , J . Amer. Chem. Soc., 56, 1879 (1934). Parker, A . S.. Hottel, H. C., Ind. Eng. Chem., 28, 1334 (1936). Powell, R. W.. Trans. lnst. Chem. Eng., (London), 18, 36, (1940). Riede, 8. E.. M.S. Thesis, Newark College of Engineering, Newark, N.J., 1969. Scott, G . S., Ind. Eng. Chem., 33, 1284 (1941). Shah, M. J., Ind. €ng. Chem., 59 (5), 74 (1967). Tu, C. M., Davis. H . , Hottel. H. C., Ind. Eng. Chem., 26, 749 (1934). Yagi, S..Kunii, D.. "5th Symposium (International) on Combustion,'' p 231, Reinhold. New York, N.Y.. 1955. Yang, K. H.. Hougen, 0. A , , Chem. Eng. Progr., 46, 146 (1950).
Recezuedfor reuieu March 11, 1974 Accepted September 5, 1974
Supplementary Material Available. Additional experimental details will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the ,Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D. C. 20036. Remit check or money order for $3.00 for photocopy or $2.00 for microfiche, referring to code number PROC-75-70.
A Simplified Analytical Design Method for Differential Extractors with Backmixing. I. Linear Equilibrium Relationship
H. R . C . Pratt University of Melbourne, Victoria, Australia
Approximate solutions are presented to Miyauchi's equations (1957) for mass transfer with backmixing in differential extractors. Given values of the Peclet numbers P x and P,, and of H o x (or the mass transfer coefficient), these permit the column length to be calculated directly for the case of a linear equilibrium relationship. Comparisons of exact and approximate solutions indicated that, with a high degree of backmixing in both phases and values of the extraction factor between 0.50 and 2.0, the results are accurate to well within 5% when the length > 4 ft and Nox > 2.
Introduction General. In spite of a voluminous literature on the theory of two-phase mass transfer, the precise design of contacting equipment from first principles is unreliable in the absence of experimental data for the particular system and contactor to be used. This is a result of the complexity of the interacting factors which control performance. Liquid-liquid extraction is no exception in this regard, to the extent that the extensive published data for droplet contactors ( e . g . , packed columns) are virtually useless for design purposes outside the exact range of conditions reported. This is largely due to the effect of backmixing (longitudinal dispersion) of the phases, which reduces concentration gradients and adversely affects performance. In an extensive programme of work some years ago by the author and coworkers on packed columns (Gayler and Pratt. 1951, 1957a,b; Pratt, 1955), the need was recognized to consider separately the questions of interfacial 74
Ind. Eng. C h e m . , Process Des. Develop., Vol. 14, No. 1 , 1 9 7 5
area of contact of the phases, individual phase mass transfer coefficients, and "longitudinal mixing" of the phases, and methods were devised for predicting these in the absence of interfacial (Marangoni) effects. The backmixing correction was, however, based on a relatively crude graphical method and more recently Eguchi and Nagata (1958), Sleicher (1959) and Miyauchi (1957) proposed a more realistic diffusional model. This yields a rather complex analytical solution which can be used to obtain the concentration profile directly, but requires trial and error to calculate the column length required for a specified duty. To overcome this difficulty, a simple approximate solution to the model is presented which gives the length directly. This possesses useful advantages over other proposed methods, summarized later. Theoretical Basis. Mass balances over a differential length of contactor, taking into account axial diffusion, yield the following equations for one-dimensional steadystate countercurrent flow (Miyauchi, 1957; Miyauchi and
Vermeulen, 1963)
1
t
I-X-JUW
-i
where E,, E , are effective turbulent diffusion coefficients. Integration of (1) gives the expression for the number of “true” overall transfer units, No,.
With backmiximg absent, E , = 0 and eq 3 reduces to the normal expression for the number of piston-flow transfer units. N o X p . When backmixing occurs the operating line is no longer straight, as shown in Figure 1, and the driving force is reduced. Consequently the “true” value of No, differs from Noirp in that it includes the effect of changes in both driving force ( e - cy*), and in transport represented by the term E,/Ur(dc,/dz) in eq 3. A third definition termed by Miyauchi and Vermeulen (1963) the “measured” or “interior apparent” value, No,M, which uses the correct driving force but omits the additional transport term. has been proposed by Geankoplis, et al (1950, 1951), and Gier and Hougen (1953). There appears to be no justification for the use of this definition, however. Assuming a linear equilibrium line represented by e,* =
ii?CY
+
q
Figure 1. Y-X diagram for extraction column. The driving force ( X - X * ) is typically represented by AC for piston flow and by AB with backmixing. T h e “true” operating line corresponds t o P,B = P,B = 4.0, “Vox = 4.00, F = 0.50; for piston flow, .Yoxp = 1.924.
(4)
eq 1 and 2 can be expressed in dimensionless form as F R A C T I O N A L LENGTH.2
dY
P,B-dZ
I3 = 0 1’) = 0
(5) (6)
-
Figure 2. Concentration profiles in extraction column, for conditions of Figure 1: ABCDE. HJKLM, X , Y profiles, respectively, with backmixing; AFGE, HPQM, X , Y profiles. respectively, for piston flow.
where the dimensionless concentrations, which have values of Xo = 1.0 a t Z = 0, the X-phase inlet, and F = 0 at Z = 1.0, the Y phase inlet, are defined by
(16a)
ai = = cy
- c,1_ I -
C Y ” * - C,’
li/(CY -
c,fl
cy’)
- (il?CY*
+
CY-
dZ’
diX
dZ
-
7
I. = A ,
FP,,)
+
-\-,,,P,P,B?(l
-
+
z=o:
PyP,B2
A2a2eh2Z+ A,a,ex?
+ Alex:z +
h?
-
ax? - 13x -
)
= 0
2-0
z=1:
Ala4ex:z
= 0: P = Yo
= 0:x’ =
(14)
(15)
x,
(18)
(19)
2.1
-
where the A, are the roots of the characteristic equation
and the aiare given by
(16b)
(S)
(E) z = 1 : (E)
F)
A, + A ? P ’ ? ~+ A,e*iz
+
X i ’P,B)
--
The solutions to eq 5 , 6, and 9 are as follows. X =
-
-c h,’P,B)
/l--d ? X - dX dZ’ id2 = 0
where 13 = L\‘,J,B(P,
(1
Equation 16b is obtained by elimination of NoX between (16a) and (15). The coefficients A , are obtained from the following boundary conditions (Miyauchi, 1957)
q)
Elimination of Y between eq 5 and 6 gives
_d’x __
F(l
= P,BI’,
(20)
Z=i
These show that concentration “jumps” due to backmixing occur a t the phase inlets, i.e., from XO = 1.0 to Xo at 2 = 0, and from F = 0 to Yl a t Z = 1.0, but that no jump occurs a t the phase outlets (see Figures 1 and 2 ) . Substitution of eq 13 and 14 in eq 17 to 20 leads to four Ind. Eng. Chern., Process Des. Develop., Vol. 14, No. 1, 1975
75
simultaneous equations in the A t , the solutions to which
are as follows (Miyauchi, 1957)
The present method follows Mecklenburgh and Hartlands’ method initially, but arrives finally a t simple analytical formulas covering the whole column.
(21)
Derivation of Equations Characteristic Equation. Equation 15 is divided through by L3 and the substitutions B = L / d p and NO, = L/HQX made in eq 10 to 12. A modified characteristic equation is thus obtained
where
AI?
p‘x’ - y’ = 0
cylh’?-
-
(28)
where the roots A,’ = A,/L, and the coefficients are
= ( P , - P,) l d ,
cy‘
(29)
(31) The characteristic equation has three real roots, Xz’, and X4’, which are given as
A3’,
DA3 = A,X,(n2c’J
D,,
-
= -X,X,(a?ex3
(25)
a,&)
-
n,e’?l
(26)
where k = 0, 120. and 240” f o r X?’. A j ’ .
The above equations are modified if F = 1.0, and if P, or
and X,’
P,= m (Miyauchi and Vermeulen, 1963). The essence of the design problem is that the desired contactor length, L, appears in both the exponential terms and in the A t in eq 13 and 14 via the roots A,, necessitating a tedious iterative solution. Miyauchi and Vermeulen (1963) attempted to overcome this by defining a parameter ?Jo.,D,the “number of overall dispersion units,” relating the easily calculable N o X p to NO,as follows. Ay”Xp-l - ll;lx-l=
(27)
yn x D - 1
They then proposed a method of correlating Noxn in terms of correction factors relating this to P,B and P,B uia the relation for NO, (that is, kOxa a, not L a ; see Pratt, 1971). However, as shown in the Appendix. the relationship between N0,p and NO, is more complex than that indicated by eq 27, since N O ~isD a function of both NO,Pand NO,. Stemerding and Zuidenveg (1963) proposed an alternative. empirical correlation for N O , D of acceptable accuracy when F does not differ too greatly from unity, and Sleicher (19593 also gave an empirical formula for the “column efficiency” factor E, ( = No,p/No,). These methods require trial and error solution since they all use Peclet numbers based on column length, but Watson and Cochran (1971) proposed a method resembling that of Stemerding and Zuiderweg (1963) which overcomes this difficulty. An alternative approach was suggested by Rod (19643, who described a stepwise graphical method of calculating the true operating line. This gives the extractor length directly when backmixing is absent in one phase. but requires trial and error solution for one of the inlet “jump” concentrations when it occurs in both. Mecklenburgh and Hartland (1967a) showed that, under some conditions, the order of accuracy of guessing this concentration when solving eq 1 and 2 by the equivalent numerical method could tax a 39 bit word computer to the limit. They further proposed an approximate design method in which the column was divided into a central and two end regions, for each of which a different simplified solution could be used.
-
76
-
-
Ind. Eng. Chem., Process Des. Develop., Vol. 1 4 , No. 1, 1975
p’ q’ =
= (cy’ ’3)?
(CY’ ’3)3
+
,j’
’3
;’
CY',^' ’6
= cos-1
21
+
respectively.
’2
($)
taking the angle between 0 and x . This solution applies only if qf3
-
c
pl?
(33)
0
which is normally the case in the present application. The roots A,(= A,’L) have the following properties: (i) A2 is large and positive, increasing with P,; and X3 is large and negative, increasing numerically with increase in P , ; (ii) X 4 = 0 when F = 1.0, is small and negative when F < 1.0, and small and positive when F > 1.0; (iii) Xz and lh31 both increase slowly as F increases. Anomalous Roots When P, or P, m . Although P , increases as the backmixing in the Y phase diminishes, the correct roots for zero backmixing are not approached if P, is increased to very large values in eq 30 since I X 3 / becomes very large. It is therefore necessary to divide eq 28 by P, and to let P, m , when A3 vanishes and it reduces to a quadratic; thus
-
-
X’!
where
-
,3,1A’
-
x
‘iI
= o
by‘ = ( F ‘H,,, + P , ’dF) )Xt
= PJI
-
F)/H~,~,
(34) (351 (36)
The roots hz’ and Xq‘ are therefore given by (37)
Similarly, when P, =
03,
hz vanishes and eq 28 becomes
A‘?
+
jjy‘xI
$y’
=
(1 ’H?., + P,/d,)
+
>y‘
= 0
(381
where (39)
Y,‘
= P y ( l - F)/H,,d,
and X3’, X4’ are given by
1’ = B , (4 1)
In these expressions X 4 ‘ is retained as the root which vanishes when F = 1.0. Of the remaining roots, Xz’ or X3’ become zero respectively when P,or P, = m . General Case (Backmixing in Both Phases). (a) F # 1.0. Following Mecklenburgh and Hartland (196713) eq 13 and 14 are expanded, writing out the A , from eq 21 to 26 in full. On dividing. numerator and denominator by exp(Xz), all terms containing exp(-Xz) and exp(h3) can be disregarded due to property (i) of the roots. This gives
xC,&
Y -
+
X = B,
(40)
X,X,a3e’~e-’2(’-z) + X ~ ( X ~ U , ~ ’ ~ ’ X,a3e’iZ) C,e% + c, (42)
+
+
B2eX?’
+
B,a,exzz
+
B,a3ex3’
Bge’3‘
+
+
B , (2 +
B,Z
&)
(50) (51)
where the B , are given in determinantal form by Miyauchi and Vermeulen (1963). (To retain the present system of numbering of the X, and B,, Miyauchi and Vermeulen’s coefficients Bz, B3, and B4 have been changed to B4, B z , and B3, respectively.) On expanding and simplifying as before, eq 50 and 51 become
x = x,x,~, + X,X,a,
+
C,
C,
+
+
X3a3e-x2(1-z)+ ~,e’3’ X,X,a, + c3 + c,
-
X,X~,Z
-
(52)
~~a~a,e-’2(~-’) +
where
-
Since Xz is large and positive, the term containing exp[X z ( 1 - Z)] can be neglected except when 2 1.0. Hence eq 53 becomes, on replacing the X I by X,’L and putting B = L/dF,and L/No, = Ho,
Y = x 2 ‘ x 3 ‘ a 3 ~+ c,‘ + x2‘a3 [eAi’Lz X,’X,’a,L
Considering first eq 43, since Xz is large and positive the term containing exp[-Xp(l - Z)] can be neglected except near the X-phase outlel, where 2 1.0. Substituting X,’L = A, and B = L / d , , therefore gives, noting that the terms (1 f A,/P,B) become (1 f X,’d,,/P,)and are consequently unchanged in value, as are the a,
-
(46) where the C h ’ are obtained by replacing the X I by AL’, and are numerically equal to Ch/L2. When 2 = 0, Y = Yo and eq 46 becomes
Solving for L and inserting the values of C1. and Cz’ gives
Hence the length is obtained directly as a function of the required solvent phase outlet composition YO An exactly similar treatment can be applied to eq 42, disregarding the term in exp(A3Z) which is negligible except when 2 0, and putting X = X1 when Z = 1. The resulting expression for L reduces to eq 48 on substitution of X1 in terms of YO using the overall material balance
+
C,’
+
- X,’(LZ C,‘
+
H ~ ~ ) ]
(56) where the C k ’ are obtained by replacing the A, by A,’ in eq 55 and are numerically equal to Ck/L. When 2 = 0, Y = Yo and eq 56 becomes, on solving for L and rearranging
An identical expression is obtained starting from eq 52. Backmixing in One P h a s e Only. When backmixing is absent in one of the phases, either Xz or X3 vanishes and the expressions for the concentration profile take somewhat simpler forms (Miyauchi and Vermeulen, 1963). However, useful simplification can still be effected by the present method, and the results are summarized below. (a) P x Finite, P, = a; F # 1.0. The values of the roots Xz‘ and X4‘ are given by eq 37 and the length of column by
(b) P, Finite, P, = m ; F # 1.0. The values of X3’ and for this case are obtained from eq 41 and the length is given by
X4‘
+
Id = F ( 1 - S’) (4 9) (b) F = 1 0. When F = 1.0 the term y becomes zero. so that A 4 = 0 and eq 15 reduces to a quadratic which can be solved for Xz and AS. The solutions to eq 5, 6, and 9 then become
(c) Either P, or P, = m : F = 1.0. In both these cases vanishes and the residual root becomes Az’ or X3’ = 3=(1/Hox + P,/d,)),the negative sign relating to X3’. Substituting these values in the simplified equations for the column length gives, for both cases Xq‘
Ind. Eng. Chem., Process Des. Develop., Vol. 14, No. 1 , 1975
77
Y“2 L -
+
2 x
i
-2- + HQx
Table I. Comparison of Results by Two Methodsa
n
Calculated length, ft
q 1 - YO)
M and H methodb
d~
F
~~~
Accuracy of Solutions The above equations were tested for accuracy by calculating a considerable number of examples, giving particular attention to the general case of backmixing in both phases, and to the use of relatively short columns in order to find the limits of their applicability. These were mostly based upon the tabulation of computed solutions of eq 13, 14, 50, and 51 as given by McMullen, et al. (1958), although exact solutions were calculated from Miyauchi’s equations (1957) for cases where P, or P, = m . Only the extraction factor range F = 0.50-2.00 was considered, since values outside this are seldom used in practice. The procedure used was to specify values of P,, P\, Ho,,do, and F, and to obtain X 1 from McMullen’s data for a specific value of NO,. The value of L was then calculated from eq 48, 57, 58, 59, or 60, and compared with the exact value given by the product of NO,and Ho,. With values of P, and P, from 0.10 down to 0.025, which is as low as is likely to be encountered in practice, the error in the calculated length generally started to become appreciable for lengths of 4.0 f t and below when Ho, > 2.0 ft. Both length and NO, appeared to be relevant variables, and it was noted that the calculated length tended to be slightly low for F = 1.0 and high for F = 1.0. As Ho, diminishes, X2‘ and 1x3’1 increase, and the accuracy improves, even down to lengths of 2.0 ft with Ho, = 0.5 ft, a value which has been observed in practice. An empirical plot of percentage error against Lcalcd(X2’ - X3’) suggested that the error would be well within 5% if this parameter exceeded 7 to 8 with NO, > 2. The accuracy was generally slightly improved with backmixing absent in one of the phases.
b
78
Ind. Eng. Chem., Process Des. Develop., Vol. 14, No. 1, 1975
Exact value ~~
~
16.00 16.00 16.00 8.00 4.00
15.99 16.25 15.99 7.99 3.89
a Parameter values: P x / d D= 0.25; Pb/dp = 1.00; Ho,= 2.00 ft. Mecklenburgh and Hartland (1967).
advantages over purely empirical methods such as that of Stemerding and Zuiderweg (1963). Apart from design, the present equations are also suitable for the interpretation of overall column data ( i e . , without measured profiles), provided values of P x and P, are available from other sources, e.g., tracer experiments. Thus, by iteration on the approximate solution it has been found possible to obtain values of Ho,, and hence ko,, data for which are particularly scarce. The method as described is limited to the case where the equilibrium line is effectively straight over the full concentration range. A modified method for use when the equilibrium relationship is nonlinear will form the subject of a further communication. Worked Examples Example No. 1. Calculate the length of contactor required to meet the following specification: crO = 120.0 g/L; cxl = 34.43 g/l.; c \ l = 8.00 g/l.; LTr/U, = 0.800; P, = 0.025; P, = 0.100; Ho,= 2.00 ft; d,, = 0.10 ft. The equilibrium data are closely represented by
c,* = 0 . 6 2 5 ~+~ 0.500 (i) Material Balance. From the equilibrium relation
Discussion The foregoing closed solutions to the backmixing equation are subject to the same limitation on accuracy as those of Mecklenburgh and Hartland in regard to the omission of terms containing exponentials of - X z and X3. However, apart from this they are exact, unlike those of the above workers, who divided the column into three sections in each of which the profiles were approximated by straight lines. For the limiting case of F = 1.0 the characteristic equation reduces to a quadratic and the profiles are linear, so that both methods become identical. For all other cases, however, the profiles are nonlinear and, further, an approximation was used by Mecklenburgh and Hartland to obtain X4; this, in turn, led to an approximate quadratic equation for the remaining two roots. The effect of these approximations is illustrated by the following table, in which typical results by the two methods are compared with the exact values. As expected the results for F = 1.0 are the same, but for other values Mecklenburgh and Hartland’s method gives appreciable errors which, unlike the present method, do not effectively vanish as the length increases; further, this error is negative for F < 1.0 and positive for F > 1.0. (See Table I.) The present method is particularly suitable for use with small electronic calculators, e . g . , the Compucorp 324G, which can also be programmed to solve the cubic characteristic equation. It is undoubtedly preferable to the graphical method of Rod (1964), especially when backmixing occurs in both phases since the latter method then requires trial and error solution. I t also possesses obvious
~
14.76 16.25 17.19 7.45 3.71
0.50 1.00 2.00 0.50 0.50
Present method
Cx1*
= ~uc,,’+
= 0.625 X 8.00
+
0.500 = 5.50
From eq 7
Since F = 0.625 ance gives
P
X
0.800 = 0.500, the overall material bal-
= F(l - SI)= 0.500 (1 - 0.2527) = 0.3736
(ii) Characteristic Equation. The coefficients in eq 28 are given by eq 29 to 31 as CY’ =
(0.025 - O.lOO)/O. 100 = -0.750
p’ = 0.025 + 0,500
X
2.00 x 0 . 1 0
0.10 +
0.025 x 0 . 1 0 =
o. 625
0.102 y9 =
0.025
X
O.lOO(1.0 - 0.500) = 2 . 0 x 0.102
Substitution in eq 32 gives the roots Xi‘ as X,’
= +O. 5614525
A,’
= -1.2202247
X,’ = -0.0912278 Calculating a4 from eq 16b
o. 0625
-
(1
-
Sp ) = 1.3649111 Pr
-$) = 0.9087722
* ai = 0.750964 Hence substitution in eq 48 gives
I -=
I
d(dX;/dZ) [X(1 - F ) + FX’]
x
-0.0912278 0.56145 x 0.75096 (1.220225 0.091228)(0.500 - 0.37361 0.50’ x 1.22022 (0.091228 + 0.561453)(1 - 0.3736) = 7.99 ft
Since L(A2’ - As’) = 14.25 and No, = 7.9912.00 = 4.0, this result should be of adequate accuracy. and in fact it agrees almost exactly with the correct value of 8.00 ft. Example No. 2. Calculate the column length required for the following duty. where the concentrations are expressed in X-Y units: XO = 1.000; X1= 0.360; F = 0.0; F = 1.00; PI = 0.040; P , = 0.040; Ho, = 2.00 ft; d , , = 0.10 ft. From the overall material balance I“’ = l . O O ( 1
--
= (0.040
/j! -
j ’
0.40
---
[ X ( 1 - F) +
0.040) ‘0.10 = 0 . 0
+ 1.00 x 0.40 2 . 0 0 x 0.10
----_I___._
+
(S)? = 20.00
0.0
Equation 28 therefore reduces to a quadratic, the roots of which are 72’ =- 0.748331: > s ’
= --0.748331
From eq 58 the length is therefore
-
dZ (A3)
(K p,R) +
The first integral on the left can be identified as W,,P, the number of piston flow transfer units between the inlet “jump” concentration X O , and the outlet X I ; it is related as follows to the total piston flow transfer units = LY’(,rp + AYo,)xp
(A4)
N00,p refers to the number between X = XO and X = X O .Designating the second and third integrals is eq A3
where
by I,’ and I,, respectively, this can be rearranged to
Hence the number of “dispersion transfer units” represented by the righthand side of eq A5 is a complex function involving both NO, and N1orp as well as the two Peclet numbers and the extraction factor. The applicability of eq A5 can easily be verified using examples from the calculated profiles of McMullen. et al. (1958).Thus for the following case
F = 0.50
P,B = 2 . 0
S,, = 4.0
PyB = 8 . 0
the integrals are found to be A’owp = 1.81
(O.%o
1
+ -___ 1 FX‘]
0.360) = 0 . 6 4 0
The coefficients in eq 28 are N’
dY
(-No, +) ’P,B
x_ 0.748331 -2_ _ ___
0.748331‘
)
= 10.12 ft
This result should be sufficiently accurate since L(h2’ As’) = 15.1 and NO, = 5.0. Mecklenburgh and Hartland ( 1 9 6 7 ~ )who . obtained an identical result by their own approximate method, in fact gave the correct value as 10.00 ft.
I,‘ = -2.34
?ioxpl= 1 , 16
I,, = 1.033
The values of NoXp and P o x p were obtained by means of the usual Colburn (1939) relation for a straight equilibrium line, and I,’ and I , were determined by graphical integration after plotting the concentration profiles. Hence from eq A5 1
1
Left hand side = __ 1.16
Appendix Relation between Nor and N o X p . A material balance taken through any cross section of the column and around the Y-phase inlet ( i e . , a t 2 = 1.0)gives, in dimensionless variables
0.50 Right hand side = 2.00
+
-
~
4.0
= 0.612
2.34 -c 4 x 1.16 x 2
The agreement is almost exact in spite of possible errors arising from the need to interpolate the tabulated profiles. In the absence of backmixing the last two terms on the right vanish and eq A1 reduces to the normal piston flow operating line relation. Elimination of Y between eq 5 and A1 gives, on rearrangement
.YnYPYB[ X ( l
-
F)
+ F X ’ ] (A2)
On dividing through by the coefficient of d X / d Z and integrating this becomes
Nomenclature
A ( = coefficient in eq 13 and 14, dimensionless a = interfacial area per unit volume Lz/L3 = L - 1 ai = defined by eq 16a and 16b, dimensionless B = L/d,, dimensionless B , = coefficient in eq 50 and 51, dimensionless c , = concentration of solute in j phase. ML-3 or (mol) L-3 c k = defined by eq 44,45,54, and 55 c k ’ = defined by eq 44, 45, 54, and 55 with the A, replaced by A,’ and putting B = Lid,,, NO,= LjHo, Ind. Eng. Chem., Process Des. Develop., Vol. 14, No. 1 , 1975
79
dp = characteristic length, e.g., size of packing element,
1 = extractant inlet end, outside column (2 = 1.0)
L
* = value at equilibrium with other phase
phase, LZT-1
Subscripts i = number of root of characteristic equation, and of corresponding coefficient in solutions for X and Y, Le. eq 13 and 14 j = X or Y phase x = X phase (feed) y = Yphase (extractant)
E, = effective longitudinal diffusion coefficient in the j t h F = extraction (stripping) factor, mU,/U,, dimensionless Ho, = height of an overall (“true”) transfer unit based on X phase, L kox = overall mass transfer coefficient based on X phase, LT-1 L = total length of contractor, L m = slope of equilibrium line, dcx/dcv,dimensionless No, = number of “true” overall transfer units based on X phase, dimensionless N o x p = number of overall “piston flow” transfer units based on X phase (with superscripts 0 and 1, as defined in Appendix), dimensionless P , = Peclet number for the j t h phase, U,dp/E,, d’imensionless q = intercept of straight equilibrium line on cx axis, ML-3 or (m0l)L-3 U, = superficial velocity of j t h phase, LT-1 X = generalized solute concentration in X (feed) phase, dimensionless Y = generalized solute concentration in Y (extractant) phase, dimensionless Z = z / L , fractional length within column, dimensionless z = length within column measured from X phase inlet, L
Greek Letters a , a’ = defined by eq 10 and 29, respectively p, p’, px‘, p,’ = defined by eq 11, 30, 35, and 39, respectively 7 , yx‘, yv‘, = defined by eq 12, 31, 36, and 40, respectively A,, A t ’ = roots of characteristic eq 15 and 28, respectively.
r‘,
Literature Cited Colburn, A. P., Trans. Amer. Inst. Chem. Eng., 35, 211 (1939). Eguchi, W., Nagata, S . . Chem. Eng. Jap.. 22, 218 (1958). Gayler, R . . Pratt, H. R. C., Trans. inst. Chem. Eng., 29, 110 (1951). Gayler. R.. Pratt. H. R. C., Trans. lnsf. Chem. Eng.,35, 267 (1957a). Gayler, R., Pratt, H. R. C.. Trans. Inst. Chem. Eng.. 35, 273 (1957b). Geankoplis. C. J., Hixson. A. N., Ind. Eng. Chem., 42, 1141 (1950). Geankoplis, C. J., Wells, P. J.. Hawk, E. L., Ind. Eng. Chem.. 43, 1848 (1951). Gier, T. E., Hougen, J. O., Ind. Eng. Chem.. 45, 1362 (1953) McMullen. A. K.. Miyauchi, T . . Vermeulen. T . , U.S. Atomic Energy Commission Rept., U.C.R.L. 3911, Suppl. (1958). Mecklenburgh. J. C., Hartland. S., I . Chem. E. Symp. Ser., No. 26, 115 (1967a). Mecklenburgh. J. C., Hartland. S., I . Chem. E. Symp. Ser.. No. 26, 121 (1967b). Mecklenburgh. J. C.. Hartland. S . , I . Chem. E. Symp. Ser.. No. 26, 130 (1967~). Miyauchi. T., U.S. Atomic Energy Commission Rept. U.C.R.L. 3911 (1957). Miyauchi, T., Vermeulen. T.. Ind. Eng. Chem.. Fundam.. 2, 113 (1963). Pratt. H. R. C., Ind. Chem.. 509 (1955). Pratt, H. R. C.. Ind. €ng. Chem., Fundam.. 10. 170 (1971). Rod, V . . Brit. Chem. Eng..9, 300 (1964). Sleicher, E. A..A.I.Ch.E. J.. 5 , 145 (1959). Stemerding, S., Zuiderweg. F. J., Chem. Eng. (London), CE156 (May 1963). Watson, J. S.. Cochran. H. D.. Ind. Eng. Chem., Process Des. Develop.. 10, 83 (1971).
Superscripts 0 = feed inlet end, outside column (2 = 0)
Received for review April 18, 1974 Accepted August 5.1974
Production of Crystalline Pyrophosphoric Acid and Its Salts Chung Y. Shen Detergent and Fine Chemicals Division. Monsanfo lndustrial Chemicak Company Monsanto Company, St. Louis, Missouri 63166
A process for producing high-purity, crystalline, free-flowing pyrophosphoric acid from a liquid condensed phosphoric acid was developed. Kinetic studies show that diffusion is the rate-controlling step. Variables such as temperature, acid concentration, and sojourn time were defined for optimum continuous crystallization of pyrophosphoric acid. The easily handleable acid was shown to be a versatile raw material for preparing potassium, calcium, and other pyrophsophates.
In our earlier publications, we reported that polyphosphates could be produced by selective extraction of a condensed phosphoric acid (Shen, 1967) or by direct reaction of a condensed phosphoric acid with sodium carbonate to form a fast dissolving, effervescent phosphate (Shen, 1968) which saves the energy otherwise required in producing polyphosphates by the conventional calcination of orthophosphates and simplifies the process. The condensed phosphoric acids have a Pz05 concentration above 80
Ind. Eng. Chem., Process Des. Develop., Vol. 14, No. 1, 1975
100% HsP04. Several condensed phosphoric acids have been commercially available for some time (Durgin, et al., 1937). The condensed phosphoric acids contain a mixture of phosphates of various chain lengths. The distribution of these phosphate species depends on the P205/HzO mole ratio (Parks and Van Wazer, 1957). The compositions of the commercially available condensed phosphoric acid and an acid with a P205 content equal to pyrophosphoric acid are shown in Table I.
