ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT a liquid product predominantly in the gasoline boiling range. Injection of alkali shifted the make-up of products toward the heavy ends, and thus is a t,ool for controlling product distribution. Because massive iron catalysts can be operated over a wide range of conditions and are resistant to disintegration and oxidation, use of these materials in the oil circulation process appears promising. Other forms of massive iron such as steel wool are being investigated. Literature Cited
(1) Anderson, R. B., Seligman, B., and associates, IND.ENG.C H E M . , 44,391-401 (1952). (2) Crowell, J. H., Benson, H. E,, and associates, Ibid., 42, 237G-84 (1950). (3) Duftschmid, F., and associates, U. S.Patent 2,159,077 (1Iay 23,
1939); 2,207,581 (July 9, 1940); 2,287,092(June 23, 1942); and 2,318,602 (May 11, 1943). (4) Hall, C. C., Gall, D., and Smith, S. L., presented before the Institute of Petroleum, London, March 26, 1952. (5) Kastens, VI.L., and associates, IND. ENG.C H E m , 44,450 (1952). (6) Pichler, If., U. S. Bur. Ilines, Special Report, reproduced in T.O.M. Reel 259, Frames 467-664, 1947. (7) Pichler, H., and Merkel, I%., U. S. Bur. Mines, Tech. Paper 718. 1949. (8) Scheuermann, A., in report to Zorn, IT., U. S. Dept. of Commerce, Washington 25, D. C., OTS, PB-97,368 (FIAT Final Rept. 1267),1949. (9) Storch, €1. H., and associates, "Fischer-Tropsch and Related Syntheses," John Wiley & Sons, New York, 1951. RECEIVEDfor review April 20, 1954. ACCEPTEDJuly 6, 1054. Presented as part of the Symposium on Synthetic Liquid Fuels and Chemicals before'the Division of Gas and Fuel Chemistry a t the 126th hIeetiny of t h e QXERICAK CHEMICAL. SOCIETY, Kansas City, Mo.
Heat Transfer in Cross-Flow Heat Exchangers and Packed Evaluation of Equations for Penetration of Heat or Solutes A, KLINKENBERG N . V . De Bataafsche Petroleum Moafschappij (Royal Dufch/Shell Group), The Hague, The Netherlands
M
,4NY investigators ( 1 , 2, 5-12, 14-26) have used the set of equations
with boundary conditions
Y
=
0 ; TI = To
z=o;
T2=0
Pa) (2b)
These equations describe the following important processes in chemical engineering :
1. Heat transfer in a double cross-flow exchanger-Le., one with both fluids unmixed laterally: Temperatures T I and 7'2 of fluids 1 and 2 depend on the dimensionless groups Y and Z , which are proportional t o the coordinates in the directions of the two flows. These groups represent numbers of transfer units. The fluids are entering a t temperatures TOand 0, respectively. This process was examined by Nusselt ( 1 8 )in 1911. a packed bed, initially at zero 2. Heating-or cooling-of temperature, by passage of a fluid entering a t 2'0: I n this case T I and 2'2 are the ternperaturcs of the fluid and the solid in a plane situated Y transfer units from the inlet plane, a t a time given by group 2. The unit of 2 is the mean holding time per transfer unit. At the bed entrance ( Y = 0), 2 is simply proportional to the time elapsed since the moment of introducing the hot fluid. I n general, however, 2 is proportional t o the time elapsed since the fluid, introduced a t Y = 0, Z = 0, reached the point under consideration. If the heat content of the fluid in the bed is negligible with regard to that of the solid-Le., if the fluid is a gas-the time of passage of the fluid through the bed can be neglected with regard t o the time during which fluid is introduced in the bed. Since the simplifying assumptions include absence of axial heat conduction, for Y # 0, 2 < 0 both temperatures are identically equal to zero. The fact that the two physical processes described above obey the same mathematics was not noticed when the heating or cooling of a bed was first examined-Anzelius ( I ) , Schumann (2O),
November 1954
and Furnas ( 6 , 6)-but (10).
it has been stated very clearly by Hausen
3. Mass transfer in a packed bed, initially free of solute, when a solute concentration TOis entering from zero time onward: The transfer process may be adsorption (in chromatography), extraction (in partition chromatography), absorption (in gas chromatography), or ion exchange. For the applicability of the above equations, the concentrations in stagnant and moving phase when in equilibrium must be proportional. With these mass transfer processes this limitation is very real, especially for chromatography and ion exchange. The significance of the variables Y and 2 in these cases has been examined by Klinkenberg
(14).
4. More complicated processes: The treatment of mass trapsion fer processes in the presence of chemical reactions-e.g,, exchange processes-gives rise to more complicated mathematics. The present problem is a special case thereof, which authors in this field-Hiester and Vermeulen ( I d ) , Gilliland and Baddour (Y), and Goldstein (8, 9)-have been forced t o study first. Understanding the problem is also basic for the study of heat regenerators-e.g., Hansen (IO).
The usual assumptions in all these treatments include constancy of density and viscosity of a fluid during the transfer process and uniformity of velocity over the cross section of the bed or duct ( 1 4 ) . It is also necessary to neglect diffusion or thermal conduction in the direction of flow. For these reasons extreme accuracy in the solutions is never required. A quick method o f T Tz finding -'. and - accurate t o within 0.001 should be sufficient To To for all chemical engineering purposes. Duplications and Impracticality of Solutions Are Justification for Renewed Qiseussion
It appears that investigators, especially when operating in different fields, have often worked along independent lines so t h a t the literature on the subject reveals a great many alternative treatments and duplications. T h e exact solution t o Equations 1 and 2 is often in a form ill suited for numerical evaluation, Ap-
INDUSTRIAL AND ENGINEERING CHEMISTRY
228.5
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT proximations have therefore becn developed to cover certain ranges of variables. The many approximations are not equally good. Authors may fall short in mentioning the accuracy of their solution, so that i t may not) he so good as an older one or it may be far better than they seem t'o t,hink. I n recent gears the author has had opportunity to review t,wo new treatments--one incorrect and the other impractical. Some recent literature also requires comment,. It is therefore felt, t,hat a comprehensive summary of the various lines of t'hought is w 4 l worth presenting and may save others a considerable anlourit, of work, This paper presents a revie17 of the following:
Nusselt (18, Tables 3 and 4)covered the range Y = 0 - (0.2) 2; 2 = 0 - (0.1) - 1or, alternately, the range Y = 0 - (0.1) 2. His nomenclature is converted into 1; 2 = 0 - (0.2) the nomenclature of this paper by the following substitutions:
-
ltrplace:
by either:
or :
Y
z
I'
-~
UO !
General solution in terms of Bessel functions Nethods of solution via the corresponding equations in finite differences Expansions in series Asymptotic approximations related t o error functions Since many different sets of symbols are in use, a comparison of symbols is given in Table I.
%) - ;)
7' l o
-;
100 (1 -
100
100 ( I
100 4
I'
7 0
The Susselt values appear to be accurate to within a f c ~ vuiiita of the fourth decimal placcl. Methods of Solution via Corresponding Equations in Finite Differences
Table I. Tliis Papes
Furnas (6, 6 )
Comparison of Symbols
Hiester and Scliumann Vermeulon (1 2) ($0)
Goldstein (8)
Brinkley (3)
T h e diffeientisl equations may be ieplaced by equations in finite differences and the latter solved hp a step by step pioc.cidiirc. 1i:xanipleP are the following: 1. Graphical ticatinent by Hausen (10, 11) and Ledou.: (16') 2. Solution with an electrical analog computor by Tipler (8$) 3. Numerical solution with a digital computor by Saundcix and Smoleniec (19)
Z
Z
Y
a
2
General Solution in Terms of Integrals of Bessel Functions
The general solution has been derived either by classical methods (1, 6, 6 , 10, 11, 17, 1 8 ) or by the method of Laplace transformation ( 1 5 ) . It can be nritten
Methods 1 and 2 are not very accurate. 'They arc useful in the treatment of more complex problems w c h 5s the heat rcgenerat,or. Method 3 is thc method of choice for accurate computation of the solutions covering the entire field from I' = 0, Z = 0 to high values of these variables. Such solutions can be ohtained at high speed in a computor but, as far as thr author is a,\varr, h a w not been published. Expansions in Series r 7
- --
212
- e - F z I o (2
To
dF2)
1he series expansions are generally valid, but, they bccome impractical a t high valucs of Y and Z. (There is no need t o comhiIle a low Y Tvith a high 2 when T , / T o and TliTo amrevery near t o unity, or a high Y with a low 2 when 7'I/To and T 2 / Y barc very close t o zero.) The following forms are given. Double Power Series. According to Riiinie and Poole (9)and Smith ( d l ), developing the exponentials :mci the Ressel functions in 1i:quations 3 and 4 leads t80
where the Bessel function is givcn by
The numerical evaluation of Equation 3 oi 4 ieyuire.: tedious graphical integrations ( 5 , 6, 18) Equation 5 is easy to use. For low values of the argument u the function Io ( u ) has been tabulated by Jahnke and Emde ( I S ) ; for high values an asymptotic approximation is adequate (Equation 17). Furnas (5, 6) plotted
2' 6 and against To
2 foi constmt Y and
obtained S-shaped curvcs showing the break through a t the end of a bed with dimensionless length Y , as a function ol time 2. T h e parameter, Y, had the following values: 9-10-(2)-22-25-(5)50-( 10)-100-(50)-500 (the figures in parenthese- indicate the size of the steps). The range of 2 was such that Y1/To and T 2 / T o covered the range from about 0.02 to about 0.98. The values of T1/To and T2/To as read from the graphs of Furnas niay be in error by about 0.02. 2286
It is more practical to usc 01115' one of thew equations i n conjunction with Equation 5 . Equations 7 and 8 may be derived in a straightforward manner from Equations 1 and 2 by inserting in 1:quation 1 double poxer series v i t h unknown coefficients and making these fit, the boundary conditions of Equation 2. (In the equations as given in t,hc cited literature there are some misprints.) Exponentials and Double Power Series. I n accordance mit>h t,he x o r k of Yusselt ( I S ) , developing in E q u a h n s 3 and 4 thc .Hessel functions only lends to
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 11
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Error Function with Corrected Argument. Klinkenberg ( 1 4 ) modified the limit of integration of the error function by using
Equations 9 and 10 are preferable to Equations 7 and 8 because of faster convergence. Exponentials and Series in Powers and Derivatives of Bessel Functions. Schumann (20)derived the following:
The solutions have been stated to h r accurate t o within 0.006 for Y = 2, 0.002 for Y = 4, and 0.001 for Y = 8. The equations should not be used for Y < 2 and for Z < 1. 1 . . . . . is the expansion in series of Since 4; & -_ 8
Jq,
du +
one may also try
71=
m
znLw,,( Y Z )
-
(12) Expanding in series one finds for 2 = Y for both cases
n=l
where iMn( Y Z ) is the nth derivative of Z 1 ( 2 d / Y z )with respect to Y Z . He gave numerical solutions in the range Y = 0-10, Z = 0-91/2 in the form of graphs. Attention is drawn to an actually
TI To
nonexisting inflexion point in the - curves for Y = 1 and Y
Ti To
=
1 2
1 + 4----= (1 d7rY
so that the difference a t Z = Y between either approximation and thr exact solution (Equations 18 and 15) is given by 1 / 4 8 Y d a
*.. . . . .
The following was found: Deviation between Equation 19 or 20 and Exact Solution a t Y = Z Exact First term of expansion O.OlG0 0.0118 0,0042 0.0047 0.0014 0.0016
Y = Z 1 2 4
Asymptotic Approximations Related to Error Functions
Probability Integral or Error Function with Argument ( 4According to Walter (65) and also Klinkenberg ( 1 4 ) ,
T versus Z and _" versus Z, both a t constant Y , upon To To increasing Y approach from either side to
T
1 2 = 5 [ I + erf ( 4- ~ dY11 TO
(14)
This approach is very slow-viz., in first approximation in proportion t o Y-112. This is easily demonstrated a t the point8 Y = 2. At such points, in view of the symmetry of Equation 1 , Bee also Equations 9 and 11,
More complete data for Y = 2 are given in Table 11. I n the case of Equations 19 and 20 the error a t various values of Z # Y was found to surpass the error a t Z = Y only insignificantly-unless Y or 2 is very small-whereas, in the case of Equations 21 and 22, the error for Z # Y might be higher. Consequently, the field of application of Equations 21 and 22 begins a t somewhat higher Y and Z values. Error Function with Added Terms. Onsager, as cited by Thomas ( l d , ZS), ralculated the exact value of
Equation 5 and subdivided this difference in the ratio of Y1" t o order to obtain two parts, which he added to and subtracted from the value given by Equation 14:
3 (Walter To
Also
4- ~ ~ Y ) +I yl14
yll4
Tz
+ Z1/4 e -Y-zZo( 2 &%)
1
To= 5 [I + erf (dZ-
1 approximation) = 2
9 75 +- + . . .) ( 1 + 8u-1 + 128u2 1024u3 Ti 1 1 1 9 (exact) = -. + -= ( 1 + + 512Y2 + . .) To 2 4dTY 16Y
G
--
Equation 14, therefore, should not be used.
,
(17) (18)
(25
21'4
yl,c+
eu Zo(u) = _ _
November 1954
+ erf (
(16)
-
In view of the expansion (IS,page 100)
-
1 =
-___.
+ 21 e-ZYIo(2Y)
-TP 7 from 0
Ti
21j4in
gf
so that
T 1 2 (exact) = To 2
.) (24)
Value of
the curves
, ,
=
2, where the curves are considerably in error [compare Hausen (10 , f I ) and Saunders and Smoleniec (IQ)]. Exponentials and Series in Powers and Higher Order Bessel Functions. Goldstein (8) receiitly developed some solutions to Equations 1 and 2, including the folloffing:
dy),
1 -+ 48Y
zlf,e - u - ~ ~ 0 ( 2 d E ) ( 2 6 )
The difference between the exact solution and the right-hand member of Equation 25 or 26 has been expressed by Onsager in the form of a series involving, in the terms given, the use of six other Bessel functions. According to Thomas (WS), Equations 25 and 26 are valid for "very high" values of Y and Z, whereas the complete equations are valid for "high" values of Y and 2. There is no indication
INDUSTRIAL AND ENGINEERING CHEMISTRY
2287
---
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT how very high Y and 2 should be for Equation? 25 and 26 to become applicable. T h e complete equations are difficult to evaluate as such, but on expanding the various Bessel functions in the added series with the aid of expansions of the type of Equation 17 it is found that this series becomes a double poner series in powers of
with the first term
This first term diminishes very rapidly for increasing I' and Z. A check of the simplified Equation 26 was accordingly carried out for low values of Y and 2-viz., Y = 2, Z = 0--4 (Table 11)-and the equation was found--rrith t,he exception of I' = 2, 2 = 0-to be very accurate. I n viem- of Equation 5, Equations 25 and 26
It is concluded that Gilliland and Baddour have vcry much reduced the accuracy of the Onsager equation by ucglccting higher terms in the expansion of 1,(2 This iiitroduces appreciable errors in a region of ion- Y and 2 !\-here Onsager's Equation 26 is still very accurate. There v a s no need to do this, is a tabulated function (15,page 130). since 11(2 D e v e l o p e n t of Onsager Equation by Expansion of both Error Function and Bessel Function. Thomas ($2, 23) took Equation 26 as a starting point and developed the error function using a serniconvergent series ( 13,page 3 I )
d-).
dm)
and also the 10(2\/Y7) using Equation 17. Both expansions n'ere broken off after the first term The reiult is given as
are equally accurate. Error Function with Added Term. Hiester and T-ernleulen ( 1 2 ) and Gilliland and Baddour ( 7 ) modified Equat,ions 25 and 26 by expanding the Bessel function by means of Equation 17, hut b y breaking off after t8hefirst term. This gives The equations viere supplementcld b l the equation correbponding to Equation 18
Hiester and T'ermeulen used t,hese equations for 1' > 10, Z > 10 and say that the result is accurate to m-ithin 1% for z/FZ 2 8. Gilliland and Baddour also used them for very lo;^ values of I' and Z. They derived correction factors by comparing n-ith the exact values calculated by Brinkley and Bririlrley (4). The correction factor t o be applied to then plotted against
Tz
--
To
from Equat,ion 28 ~ r a s
d? for various round
(\/Z-
values oi
471, EXAIIPLE Y
= 1, Z = 1;
3 (according to Equation 28) TO
=
0.5 - >~= = 4dx Correction factor from graph Corrected value
0.3590 0.962 0.346
Onsager's Equation 26, however, gives 0.3457, which, pince Y Z, is also the exact value. ;Y =
=
0.25, Z = 1; Gilliland and Baddour, Equation 28 0.6351 Correction factor 0.820 0.530 Corrected value Onsager, Equation 26 0.5485 Exact, Equation 9 and 5 0,5458
Table II. Sum of Equa- Equs- Col. 2 tion 10, tion 5, and 3 , TI T I - 1 2 3 To TO Ta
Walter
3 0.1353 0.1871 0.2118 0.2183 0 2070 0.1896 0.1677 0.1477 0.1221
0.0228 0.1587 0.2790 0.3944 0.6000 0.5933 0 G736 0 7408 0.7063
1
0 0.5 1.0 1.5 2.0 2 5 3.0 3.5 4.0
2288
2
n 6,0819 0.1826 0.2902 0,3065 0,4964 0.5853 0,6634 0.7300
4
0.1363 0.2890 0.3944 0.5066 0 ,6035 0 , 6858 0.7530 0.8081 0.8521
I
Goldstein 11-alter solution 0,078650 Fraction of e-131,3, Equation 12 0.021415 Onsager, Equation 25 Series expansion 0.002175 Exact solution 0,102240
~
Onsagcr O.Oi8650
0.02357O 0.102229 0.0000 11 0.102240
Numerical Evaluation for Y = 2
Esacr,
z
The range of validity of Equations 30 and 31 is incorrectly stated, hccause Equation 29 can only be used for 4 3 - 1 / y >> 1. This means, however, that both temperature ratios will have to be extremelj- close to either 0 or 1. In the intermediate yange, Equations 30 and 31 ivere indeed found to be highly incorrect, as shovin in Table 111. Also a t Y = 10 errors were found t o he large. Goldstein Series. Goldstein ( 8 ) derived t,v,-o expansions (his Equat,ions 86 and 99) to be used for high values of Y andlor Z. Equation 86 ie related to Onsager's equation, the difference being t,hat Goldst,ein divided e - y - z I,, ( 2 d Y Z ) int,o equal parts, which he added t>oand subt>ra,ctedfrom Walter's solution, JThereiLs Onsager divided in the ratio of Yir4 t o 21!4.The remaining ~onithe exact solution was developed in poxrers of 2)-z and (21'2- IrL"2)2, Tj-ith this procedure the additional series is much more important than in thc Onsager case. Thus, for 1; = 9, Z = 5 t,he following was found:
5
L)
Onsager Difference c0i.G and 2,
To a Column No. 6 7 0 0228 +0.0228 0 0812 -0,0007 0.1823 - 0.0003 0 2901 - 0,0001 0.0000 0 3965 0.49s9 0.0000 0 5854 + 0 . 0001 0.0000 0 6634 0.0000 0,7300
Iiiinkenberg E(lua- Difference Equation 20, coi. 8 tio$:0, TZ and 2, To To 8
9
10
t0.0027 t0.0048 +0.0054 + O . 0048
0 .'2$60 0.1874 0.3882 0.2956 0.5006 0.4013 0.5988 0.4992 + O . 0033 0,6810 0.5880 4-0.0027 0.7507 0.8086 0.6651 -0.0017 0 , 7 3 0 8 ~ 0 , 0 0 0 8 0.8513 0,0846
INDUSTRIAL AND ENGINEERING CHEMISTRY
Difference c o i . 10 and 4, 11 -0 0030 -0 0062 - 0,0059 -0 0047 -0.0036
-0 0023 -0.0016 -0.0008
Vol. 46, No. 11
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Table 111.
Computations for Y = 2
0.50
1.00 Equation 3 1
Equation 32
3 (calcd.) TO
0.342
0.682
1.543
0.600
E* (exact) TO
0.269
0.394
0.507
0.603
It is concluded that the Onsager equation is preferable because it is faster. Equation 99 by Goldstein does not simplify matters either. Conclusions
h critical survey has been given of various methods for evaluating the solutions of a set of equations frequently met in heat and mass transfer problems-viz., in double cross-flow heat exchangers and in percolators (heating of packed beds, chromatography, etc.). The solutions give twa temperatures or concentrations expressed in two dimensionless parameters, Y and 2, representing two length coordinates or a length and a time coordinate for the cross-flo~heat exchangers and the percolators, respectively. T h e following solutions are preferred: For Y and 2 values above a certain limit which is somewhere in the region 2 t o 4 and depends on the accuracy desired, a solution presented by Klinkenberg (14)aiid given by Equations 19 and 20. The author has presented a nomograph therefore. At somewhat higher I’ and Z values a simpler variant of this solution (Equations 21 and 22) is adequate. For Y and 2 values below this limit, but excluding very low values (depending again on the accuracy desired) a solution by Onsager (Equations 25 and 26)) which requires a little more work. When a t least one parameter is very small a series expansion which then converges w r y fast. Depending on which parameter is smallest, Equation 9 or 10 is used in conjunction with Equation 5. Several of the graphs given in the literature are not very axcurate. Computed solutions in table form are not yet available, except for very low Y and 2 values (f7). Brinkley and Brinkley (3) mention having computed Equation 4 for 1/Y = 0 - (0.1) 5 and 4 2 = 0 - (0.1) - 5 in five significant figures (results unpublished). Nomenclature
TI Ts 2’@
I’
Cross-Flow Heat Exchanger Percolator = local outlet temperature, = temperature or concentration in fluid fluid 1 = local outlet temperature, = temperature or concentration in solid fluid 2 = inlet temperature, fluid 1 = inlet temperature or con(inlet temperature fluid centration of fluid 2 = zero) (original temperature or concentration in solid = zero) = dimensionless length eo- = dimensionless length coordinate in flow direcordinate tion of fluid 1
November 1954
2 Values 2.50
2.00
1.50
~
-0.654
0.686
Z
3.00 3.50 Equation 30 +0.275 0.753
4.00
0.564
0.713
0.808
0.852
= dimensionless length eo- = dimensionless time coor-
ordinate in flow direcdinate, if necessary cortion of fluid 2 rected for place (f4) erf (u)= error function I,(u) = modified Bessel function of 1st kind, nth order k = uositive whole number
n T U
= positive whole number = limit of both T I and T P ,Equation 14 = any variable: variable of integration
A
=
I uI
>>
computed value minus exact value of sign
= value of u regardless = large with respect t o
literature Cited
(1) Anselius, A , , 2. angew. Math. u.Mech., 6,291-4 (1926). (2) Rinnie, A. M., and Poole, E. G . C., Proc. Cambridye Phil. SOC.,
33,403 (1937). (3) Brinkley, S. R., and Brinkley, R. F., Math. Tables and Other Aids t o Computation, 2, 221 (1947). (4) Brinkley, 9. R., and Brinkley, R. F., unpublished table, announced in (3). (5) Furnas, C. C., Trans. Am. Inst. Chem. Enars., 24, 142.46 (1930). (6) Furnas, C. C., U. S. Bur. Mines, Bull. 361, 1932. (7) Gilliland, E. R., and Baddour, R. F., IND.ENG.CHEM.,45, 330-7 (1953). (8) Goldstein, S., Proc. Roy. Soc. (London),A219, 151-71 (1953). (9) Ibid., pp. 171-85. (10) Hausen, H., “Warmeubertragung im Gegenstrom, Gleichstrom und Kreusstrom,” Springer-Verlag, Berlin, 1950 (Vol. 8 of “Technische Physik in Einzeldarstellungen,” W. Meissner, ed.). (11) Hausen, H., 2. angew. Math. 21. Mech., 11, 105-14 (1931). (12) Hiester, N. K., and Vermeulen, T., Chem. Eng. Proor.. 48, 50516 (1952). (13) Jahnke, E., and Emde, F., “Funktionentafeln,” 1st ed., 1909. (14) Klinkenberg, A , , IND.ENG.CHEM.,40, 1902-4 (1948). (15) Kronig, R., and Van Gijn, G., Physica, 12, 118-28 (1946). (16) Ledoux, E., IXD. ENG.CHEM., 40, 1970-7 (1948). (17) Kusselt, W., Tech. Mech. u. Thermodynam., 1, 417-22 (1930). (18) Nusselt, W., 2. Ver. deut. Ing., 55,2021-4 (1911). (19) Saunders, 0. A , , and Smoleniec, S., Proc. 7th Intern. Congr. Appl. Mech. (London), 1948, Sect. 111, 91-105. (20) Schumann, T. E. W., J . Franklin Inst., 208,405-16 (1929). (21) Smith, D. M., Engineerino, 138, 479-82, 6 0 6 7 (1934). (22) Thomas, H. C., in “Ion Exchange,” F. C. Nachod, ed., Academic Press, New York, 1949. (23) Thomas, H. C., J . Am. Chem. Soc., 66, 1664-6 (1944). (24) Tipler, W., Proc. 7th Intern. Congr. Appl. Mech. (London), 1948, Sect. 111, 196-210. (25) Walter, J. E., J . Chem. Phus., 13, 332-6 (1945). RECEIVEDfor review August 11, 1953.
ACCEPTED July 2, 1954:
INDUSTRIAL AND ENGINEERING CHEMISTRY
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