ENGINEERING, DESIGN, AND EQUIPMENT 15 and 16 for the given separation gives r p = r B = 3.4. Cross plotting the theoretical stages at the optimum solvent ratio and also the total solvents required a t this point against the reflux ratio indicated that the optimum reflux ratio giving the minimum column volume was in agreement with this calculated value. The curve of total volume was again so flat that very little difference wm observed a t reflux ratios between 3 and 5, so it would appear that the equation can be used as the starting point for a study on a nonideal system. It requires a preliminary set of stagewise calculations to evaluate the relative distribution a t the feed stage unless this value can be estimated satisfactorily by a superficial inspection of the distribution data. The total decrease in the column volume between no reflux and the optimum reflux ratio in this case was only 5%. The sharper minima in the curves of Figure 3 indicate that the soivent ratio is much more critical at t h e higher reflux ratios and that improper control may give a poorer separation than is attainable without reflux. The over-all advantages of reflux in this system may be questionable. On the other hand, in the ideal systems previously studied the column volume was decreased by 12% through the use of reflux. The proposed equation for the calculation of optimum reflux ratio can be applied to effect a saving in column size in ideal systems and also in nonideal systems, if the proper relative distribution is used. Nomenclature
a
= cross-sectional area of extraction column
B
=
total solute product in heavy solvent leaving column
concentration in light phase distribution coefficient, concentration in heavy phase E = extraction factor = L D / H = quantity of component in feed = total quantity of feed h = height of extraction column H = flow rate of heavy solvent L = flow rate of light solvent m = stages below and including feed stage n = stages above and including feed stage P = solute product in light solvent leaving column Q = quantity of solute in feed stage T = reflux ratio R = rejection ratio = ratio of quantity of solute in light solvent product stream to quantity in heavy solvent product stream R‘ = retention ratio = ratio of quantity of solute in heavy solvent product stream to quantity in light solvent product stream = 1 / R p = relative distribution = 0 1 / 0 2 Subscripts 1 and 2 refer to components more and less soluble in the light solvent, respectively. Subscripts P and B refer to the top and bottom of the column, respectively.
D
=
fF
literature cited (1) Asselin, G. F., and Comings, E. W., IND. ENG.CHEM.,42, 1198
(1950).
(2) Bartels, C. R., and Kleiman, G., Chem. Eng. Progr., 45, 689
(1949). (3) Fenske, M. R., IND.ENG.CHEM.,24, 482 (1932). (4) Klinkenberg, A., Ibid.,45, 653 (1953) (5) Scheibel, E, G., Ibid., 46, 16 (1954). RECEIVED for review
January 29, 1966.
A C C E P T ~July D 23, 1965.
Heat Transfer in Packed Beds Analytical Solution of Temperature Profiles in Fixed- and Moving-Bed Reactors and Heat Exchangers ANDREW PUSHENG TING’ Cafalyfic Construction Co., Philadelphiu, Pa.
I
S THE design of nonadiabatic packed reactors, temperature
profile must be known as a function of radial position and height, in order t o account for the effect of temperature upon the reaction rate, and to calculate the heat-transfer surface required and the maximum skin temperature of the reactor tubes. In the case of fixed- or moving-bed heat exchangers, it is also more logical t o size the equipment based upon the bed-edge temperature instead of the average bed temperature. Thus, the knowledge of temperature profile in such heat exchangers is also important if a rational and accurate design is required. Investigations of radial heat-transfer rates in fixed beds have followed two different methods of approach. I n the first method, only the uniform inlet temperature and the bulk mean outlet temperature have been measured. The results have been correiated as heat-transfer coefficients, h, or over-all effective thermal conductivities, kea. Coefficients h are based upon the log mean of the difference between a “constant” wall temperature and the uniform gas temperature at the inlet, and the difference between the “constant” wall temperature and the bulk mean gas temperature a t the exit. The works of Colburn (6),Leva and others ( f I - I 3 ) , Brinn and others ( 2 ) , Hougen and Piret (9),Singer and Wilhelm (15), and Vershoor and Schuit (16) are of this type. The work of Colburn and of Leva is based purely on dimensional 1
Piesent address, Chemical Construction Corp., New York, N.
November 1955
Y.
analysis. With modifications Hougen, Pigford, Schuit, and Wilhelm use the analytical solution of Graetz (8),which is a study of heat transfer with liquid flows in empty tubes. Wilhelm gives solutions of point temperature as well as bulk temperature. The Graetz equations require the use of a uniform jacket temperature and assumptions of an infinite thermal conductivity for the tube wall and infinite heat transfer coefficients for film resistances on both sides of the tube wall. The second method is based upon the measurement of temperature profiles within the beds. Coberly and Marshall (a), Felix and Neill (7), Bunnell, Irvin, Olson, and Smith (9), Irvin, Olson, and Smith (IO), Schuler, Stallings, and Smith ( l a ) , and Argo and Smith (I) made studies in this way. Results were reported in terms of effective thermal conductivities, k,, of the gas-solids bed. The knowledge of temperature profiles permitted evaluation of k , as a function of radial position. It was found that the resistance to heat transfer increased greatly near the wall. Coberly and Marshall, and Felix and Neill accounted for this b y postulating that an additional resistance to heat transfer existed a t the wall over that in the bed proper. These investigators could reproduce their measured temperature profiles by utilizing a wall heat-transfer coefficient and a constant effective thermal conductivity throughout the bed. Smith handled the problem by allowing the effective thermal conductivity to vary in a con-
INDUSTRIAL AND ENGINEERING CHEMISTRY
2293
ENGINEERING, DESIGN, AND EQUIPMENT ItLP
the bed and along the tube skin need be calculated directly. Otherwise, numerical or graphical methods have t o be used and hundreds of points cmay have to be calculated from the uuitor, before the bottom of the heat exchanger or the catalyst tube can be -wk.t reached. Because of the exponential expression in the Laplace transformation, the equations presented possess extremely fast convergence. Usually only one term-at most two-in a series is necessary for engineering accuracy (see examples given below). 1 Using the equations given in this tFFLYtUT bFFLVINT paper, the total time required for solvFigure 1. Simple flow diagrams for three major cases ing an actual problem is usually about 2 to 4 hours. For a series of calCase 1. Can- or countercurrent shell tluid culations with only one independent Case 2. Shell fluid a t constant temperature Case 3. Constant heot flux along tube height variable, such as space velocity, it takes only 15 minutes to show the trend of the change in design - for each successive value of the variable. tinuous manner as the wall was approached rather than by postulating an additional resistance next to the wall. This paper presents a design method of nonadiabatic movingDifferential equation and bed reactors, wherein fixed beds are considered as moving beds auxiliary conditions are defined with negligible solids rates, heat exchangers as reactors without If the conduction in the direction of the height is neglected, heats of reaction, and solids beds in shells as solids beds in tubes a heat balance around a ring-shaped differential bed of height by principles of equivalent diameter and conformal mapping dz, thickness dr, and radius r in a cylindrical nonadiabatic reactor of temperature profile. All three mechanisms of heat transfer gives (conduction, convection, and radiation) are included in a single term, the effective thermal conductivity. The author made a mathematical study of temperature profiles in a gas-solids bed in exchanger or reactor tubes being heated or cooled externally by a shell fluid or other kind of heat source (For definitions of terms see nomenclature.) Over limited or sink, such as a furnace at a constant heat flux along the tube temperature ranges, the heat of reaction may be considered as a height (Figure 1). The shell fluid may be a gas or liquid which linear function of temperature, flows in a con- or counter-current direction outside the tube, q = po at (2) with respect to the reacting gas stream in the tube, or an evaporating liquid or condensing vapor a t a constant temperature. The Laplace transformation will be taken with respect to the Assumptions involved in the derivation of equations in this height variable, x , paper are rodlike gas flow, uniform gas velocity across the radius, constant effective thermal conductivity, existence of a gas film between the bed and the tube wall, and negligible thermal L [t(r,x ) ] = h(r, p ) = e-%(r, z)dz (3) conduction in the direction of flow of the solids or of the gas. T h e equations thus derived can, however, also be applied to On applying the transformation t o Equation 1, one obtains cases of nonuniform gas fronts by using average effective thermal conductivities. d2h 1 dh Both the constant effective thermal conductivities for cases (4) F~ f ; + ‘(h) = of uniform gas velocities and the average effective thermal conductivities for cases of nonuniform gas velocities include the Let effects due t o conduction, convection, and radiation. The B2 = a A p part of effective conductivity due t o convection and conduction he can be calculated from Argo and Smith’s or Coberly and Marshall’s equation and that due to radiation from Damkoehler’s x = rB equation (6). The radiation effect usually accounts for no more and than a few per cent of the total effective thermal conductivity. This “anomaly” is due to the very short radiation paths and very (7) close temperature levels between the particles. However, its effect may account for a major part of heat transfer across the film between the tube wall and solids bed because of longer Then b y substituting in Equation 4 radiation paths and much higher temperature gradients next to the wall. The use of constant or average effective thermal conductivities enables the differential equations for heat transfer t o be solved The general solution can be written directly, as this is a Bessel’s analytically. This, in turn, gives the great advantage of time equation of zero order. Thus, saving in process design of heat exchangers and reactors. From h’ = AJo(2) A’Yo(2) analytical equations only a few representative key points within (9) N
P
FILV
-
I hY
I
+
Lrn
&
+
+
2294
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 11
ENGINEERING. DESIGN. AND EQUIPMENT Because t, and therefore h', are finite a t the center of the reactor, A' must be equal to zero, and the general solution of the Laplace transform for all cases is
Then
Equation 10 is used to obtain the Laplace transform of the cross-sectional average bed temperature, tav.
& J"
hr dr =
2A
-
RB
J 1 ( R B ) - -% Ic,pB2
(12)
Figure 2.
Graphical solutions of
Case 1. Con- and Countercurrent Shell Fluid. The rate of heat transfer is
when z
> 0 and
r =
R.
A heat balance around the upper portion of the reactor gives
B for con- and countercurrent jacket fluid
the inverse transform of Equation 20 must be found. This is done by determining the poles of h ( r , p ) in Equation 20 and evaluating the residues of h(r,p)ePza t these poles by Cauchy's residue theorem. The sum of these residues is the inverse transform. The character of the poles, their position and multiplicity, are determined by the parameters in the function. CASE l A . Heat of reaction is a linear function of the bed temperature. qo # 0, a # 0. Expand by Taylor's series,
CfWJ
Combining Equations 14, 15, and 16, k , at
=
E
-t
+
where p k ' s are solutions of the left side of Equation 21. tuting Equations 5 and 21 in Equations 20,
Substi-
when z > 0 and r = R. The transformation of Equation 17 is
when r = R and z > 0. Substituting Equations 10, 11, and 12 in Equation 18, a
If pk # 0 and pk # cgJthe first term has simple poles a t a 0, cG - and pk's, and the second term has simple poles a t 0 and The sum of all the residues and hence the inverse transform is given by
5'
Substituting Equations 5 and 19 in Equation 10,
t(r, z) =
Equation 20 is the solution of the general problem in the form of the Laplace transform. In order to find the function itself,
November 1955
.
. -
INDUSTRIAL AND ENGINEERING CHEMISTRY
(Continued)
2295
ENGINEERING. DESIGN. AND EQUIPMENT 1.”
and
=
-..E.+ 1 - b
EVALUATION O F p k ’ S FOR CASE 1. The pk’S are solutiolis Equation 21, which can be expressed in the form of
2
Ra
(2)
J2(RBk)
2R2bRBJI(RBI,)
+ , .]
(24)
CASE1B. Constant heat of reaction. qa # 0, a = 0.
Of
T o facilitate the calculation of pk’s, a graphical method is used. Value of y is plotted against (EB)2in Figure 2. The left side of Equation 30, a function of (RB)z, is plotted as a set of curves shown. The right side of Equation 30 is a linear function of ( R B ) aand, therefore, should be a straight line for any particular case. After this straight line is drawn on Figure 2, the abscissa of its intersections with the curves gives the values of (RBk)2, from which pk’s can be calculated. Case 2. Constant Jacket Fluid Temperature. From Equation 13
when r = R. Its Laplace transform ii
when r = R.
If p k # 0, the first term has poles a t 0 of order 2 and a t p k ’ s of order 1, and the second term has a pole a t 0 of order 2. The inverse of transform is
Substituting Equations 10 and 11in Equation 32,
Jo(RB)
-
$X
133)
BJl(RB)
Then
(34) Expand by Taylor’s series,
Jo(RB) -
k6 p
X BJi ( R B ) =
Equation 35 can be solved graphically. RBJ1(RB) J o ( R B ) is plotted against (RBI2 in Figure 3.
CASE1c. No heat of reaction. po = 0, a = 0. ‘(Reactors” of this class are actually heat exchangers. If p k # 0,
The abscissas of its intersections with the curves are (RBb)z, from which Pk’s can be calculated for Case 2. CASE 2.4. Heat of reaction is a linear function of the bed temperature.
t(r, 2 ) = ___ R + 1 - h
2296
Draw a horizontal line for
qo # 0, a # 0.
If a
pk
# 0 and
term of h has simple poles a t 0, 2 and
INDUSTRIAL AND ENGINEERING CHEMISTRY
pk’s,
pk
#
a
-&, the first
and the second tern1
Vol. 47, No. 11
ENGINEERING, DESIGN, AND EQUIPMENT t(r, z )
=:
+A g
TO
-
J O(rBk)exP
m
k=l
P k
pkz
__
RB~_)R L-J o (c/ + k, x (41)
t,>(Z)
=
exp
m
4T
-
pkZ
k=l
(42) Case 3. Constant Heat Flux along Tube Wall. For a constant heat of reaction, qO 0, a = 0. From Equation 10,
+
Figure 3:
Graphical solutions of
+~ G P
h = AJo ( T B )
B for constant-temperature jacket fluid
(43)
-@
When r = R,
I t s Laplace transformation is given by
when r
=
R.
Since from Equation 43,
and when r = R,
The first term has poles a t p = 0 of order 2 and simple poles a t roots P k ’ s of J1(RB) = 0-Le., a t RBk = 3.83, 7.02, 10.17,The second term has a pole a t p = 0 of order 2. The sum of all the residues and hence the inverse of the transform is
CASE 2B. Constant heat of reaction. qo # 0, a # 0, the first term of h has poles at 0 of order 2 and order 1, and the second term has a pole of 0 of order 2. of all the residues and hence the inverse transform is pk
= 0. If a t pk’s of The sum given by
and
t,,(z)
=
(y
(49)
Use of equations and charts i s illustrated
(39) and
The following examples illustrate the use of the equations and charts that have been presented. Example for Case 2C. A vertical tube-in-shell catalyst cooler is to be designed under the following conditions. Shell diameter Pipes
CASE 2C. No heat of reaction. The “reactor” in this case reduces to a heat exchanger. qO = 0, a = 0 . If p k f 0, November 1955
12 feet 6 inches
Diameter Schedule Total No.
Catalyst settling in tubes. 200,000 lb./hr. Density 50 lb./ cu. foot k 0.16 B.t.u./hr., ft., O F . c. 0.29 B.t.u./lb., O F.
INDUSTRIAL AND ENGINEERING CHEMISTRY
3 inches
40 918
Diameter 0.10 inch llOOo F. in at t o p 900° F, out at bottom
2297
ENGINEERING. DESIGN. AND EQUIPMENT FEQD
At the intersections of this horizontal line with the curves,
I
2 F 1.
(RBb)2= 4.6, 24.6, 62.3, . . . . . RBk = 2.14, 4.96, 7.9,
... ..
0
From Equation 5,
. .. . .
pk = -0.0383, -0.204, -0.518, 4.85
RBk J a ( R B k )= 0.547, -1.392, 7.01, Ji (EBB)
...
,
-
9.10
Substituting in Equation 42, t,,(z)
-
7550
3000
24.a~
Figure 4.
Temperature profile in methane reformer
Cooling air flowing in tubes 3000 lb./hr. cg 0.28 B.t.u./lb., O F.
200° F. in at top 900° F. out at bottom
Steam generated in water jacket (shell side) 250 lb./sq. inch gage 406' F. What is the required height of these tubes? 3000 - 918 X 0.0513 sq.
G o
fc
=
-684
e - 0.03882
+ 17.1 (0.0383
19.e
4-
=
63.8 lb./sq. foot/hr.
(g+
+ 0.753)
' '
')
Since L ( 5 . 7 0 feet) = -190 F.," or 1090" - 190' = 900' F., the calculated height of the tubes is 5.70 feet. In actual design, a height of 8.0 feet is used to allow for some safety factor. If the method proposed by Brinn and others ( 2 ) is used, the tube length calculated for the same conditions, but without gas flow, would be only 2.2 feet. In the calculation method of Brinn and others, the bed-edge temperature is assumed t o be equal t o the jacket or shell temperature all along the tube. Actually, because of the high film resistance between the tube wall and the bed edge, the bed-edge temperature at the tube bottom is lower, and that a t the top is much lower, than the shell temperature. I n the method proposed here, the film resistance is fully taken care of. Underdesigns can thus be avoided. Example for Case 3. A vertical-tube methane reformer is fired with five horizontal gas burners equally spaced along t h e height. Each burner supplies a constant, b u t different, heat flux to the tube wall. Find ( I ) the cross-sectional average bed temperatures, (2) the bed-edge temperatures, and (3) the skin temperatures at the boundaries of these five zones. Data given: Temperature, feed (top of tube), 700' F.; effluent 1400' F. qo Assumed Q'3 Assumed Heat Flux, B.t.u./Hr., B.t.u./Hr., Reaction, Zone No. % Cu. Ft. % Sq. Ft. Inside ?
Bed temperature at top: 200,000 X 0.29 (1100 - t ) = 3000 X 0.28 ( t - 200) t = 1090OF. From Coberly and Marshall k , = 0.16
1
8 X 0 01477 + 0.00098 X 630.0357 = 0.17 B.t.u./hr. ft., ' F. X 2.42
0.216
1 = 80% + -1
U'
=
(4)equation,
1 = 2.95 X 3.94 0.001 0.001
+---
+
11.4 B.t.u./hr., sq. foot,
a
+ 0.086 = 0.088
F
Tube diameter Inside 4 . 5 inches Outside 5 . 5 inches 24 feet 3 inches Exposed tube height 4 S5 feet Height of each zone R = 0.187 feet kc = he, k,, = 4.36 0.36 = 4.72 B.t.u./hr., ft., steam methane reformer) c = 0.608 B.t.u./lb., ' F. B = 1991 lb./hr. sq. foot
+
+
O
P. (for entire
a = O
T
=
F." (see nomenclature)
406 - 1090
=
4T -
4( -684) X 218 0.28 x 3.27 0.0513
cGR
- 0.29
-684
''O
+
k.,
RU'
=
0'17
0.128 X 11.4
Draw a horizontal line on Figure 3, y = 0.117
2298
=
-17.1
Because of the ragid convergence of the series in this example, one p term is sufficient for design accuracy. Since RBI = 3.83, P I = - keB: -
cG
0.128
0.117
- -
4.72(3.83)2 = -1.639 0.608 X 1991(0.1S7)2
CROSS-SECTIONAL AVERAGERED TEMPERATURES AT ZONE BOUNDARIES.Substituting all data in Equation 49, tx,(4.85 feet)
=
(2
:.186720
4.85
- 44250)0.608
INDUSTRIAL AND ENGINEERING CHEMISTRY
x
1991
Vol. 47, No. 11
ENGINEERING, DESIGN, AND EQUIPMENT T h e actual temperature a t the bottom of the first zone is 520' 700" = 1220° F. Likewise, the temperatures a t the bottoms of other zones are calculated as 1361', 1361", 1389', and 1400' F., respectively. BED-EDGETEMPERATURES AT ZONE BOUNDARIES
+
t(R, 0 foot) = 700" F.
+
(16200
+ 13600)0.187
=
P +
4.,2 (16200 13600)e-l.639 X 4.88 = 1368' F. 0.608 X 1991 X 0.187( -1.639)
+
The reactor can be considered as five "reactors" in series, each receiving a constant but different heat flux. Thus, the bed-edge temperatures at other levels can be calculated in the same manner and are shown in Figure 4. TXJBESKIN TEMPERATURES.The film coefficient of heat transfer can be calculated with Felix and Neill's equation or Coberly and Marshall's equation. The heat transfer coefficient due to radiation can be estimated by means of Damkoehler's equation. The skin temperatures thus calculated from bededge temperature and temperature difference across the film and the tube wall, as shown in Figure 4, are average values along t h e horizontal circumferences. If the reformer furnace is SO designed that the maximum heat flux is 10% higher than the average flux, the maximum skin temperature of this reformer tub is
4-0.10 (1i51 - 1361) = 1790°F.
This temperature is too high even for stainless steel 310 or Inconel. Therefore, tubes of a smaller diameter should be used. I n calculations of this type, the heats of reaction in different zones have t o be assumed a t first. They usually have to be adjusted after the average bed temperatures have been calculated, and the calculation has to be repeated a few times to obtain heats of reactions consistent with the temperature profile.
Nomenclature
constantp f a - ~GP\'/z k" solutions of equation J o ( R B ) =
CfWf
- cw
specific heats of shell fluid, reactor fluid, and solids, remectivelv. " , B.t.u./lb./' . . F. csGs $ c,G, differential of z differential of r 7r R2goz TO __CfW/ mass velocities of solids and fluid, respectively, in direction of z axis, lb./sq. ft./hr. Use negative numbers if flows in opposite direction. Laplace transform o f t Bessel functions of the first kind of zero, first, and second order, respectively, dimensionless thermal conductivity, B.t.u./hr. ft:, ' F. average effective thermal conductivity due to con-
CG
=
PO'
=
r
=
R T
= =
heat of reaction generated per unit volume of reactor bed per unit time, B.t.u./cu. ft./hr. Negative if heat is absorbed. a constant heat of reaction generated, B.t.u./cu. f t . / hr., a t datum temperature. Negative if heat is absorbed. constant heat flux supplied by a n external source to bed in tube, B.t.u./hr. sq. f t . inside wall surface. Negative if heat is removed from bed. radius variable, feet inside radius of a reactor tube feet temperature of shell fluid a t ievel 2, using average temperature of bed top as datum of temperature scale, F." temperature of shell fluid at reactor top, 'I' F." average point temperature of bed (average of point temperatures of fluid and of solids). Average temperature of bed top is used as datum of temperature scale throughout paper, ( t = 0 a t z = 0 ) ''O
To
t
= =
'(0
ta4z1
=
U'
=
-p.
71
average temperature a t level z
= "
W
w/ X
Yo 2
The author wishes to thank the management of the Catalytic Construction Co. for permission to publish this paper. Appreciation is also expressed for the valuable suggestions and advice from J. W. Delaplaine and E. H. Lebeis of the Catalytic Construction Co., R. H. Wilhelrn and Leon Lapidus of Princeton University, and Bernard Epstein of University of Pennsylvania.
R2az/2
QQ
Y
Ac knowledgmenl
+
P
L
&(R,4.85 feet) = 1220
r
Pk
k.,
kw
From Equation 48,
1751
+
duction, convection, and radiation = k,, k,,, B.t.u./hr., ft., a F. = average effective thermal conductivity due to conduction and convection only. For uniform gas velocity front, use Argo and Smith's equation. For other cases, use Coberly and Marshall's equation. B.t.u./hr.,ft., O F . = effective thermal conductivity due t o radiation o$y. Use Damkoehler's equation. B.t.u./hr., ft., F. = Laplace transform of a function = transform variable corresponding to z - a - k,BZ
V
heat transfer coefficient across shell fluid film, tube wall, and reactor fluid film at wall. Reactor fluid film coefficient can be obtained from Felix and Neill's or Cobet;ly and Marshall's equation, B.t.u./sq.ft., hr., F. = downward flow rate of solids and reactor fluid per tube, lb./hr., tube. = downward flow rate of shell fluid per tube, lb./hr., -tube. Negative if upward flow. = 7.B
= Bessel function of second kind of zero order = =
ordinates in Figures 2 and 3 height variable, measured from bed top, feet
literature cited
(1) Argo, W. B., and Smith, J. AI., Chem. Eng. Prog.3 49, 443-51 (1953). (2) Brinn, M. S., Friedman, S. J., Gluckert, F. A., and Pigford, R. L., IND. ENG.CHEDI., 40, 1050-61 (1948). (3) Bunnell, D. G., Irvin, H. B., Olson, R. W., and Smith, J. M., Ibid., 41, 1977-81 (1949). (4) Coberly, C. A . , and Marshall, W. R., Chem. Eng. Prog., 47, 141-50 (1951). ENG.CHEM.,23, 910-3 (1931). (5) Colburn, A. P., IND. (6) Damkoehler, G., in "Der Chemie-Ingenieur," by EukenJakob, vol. 3, part 1, p. 445, Akademische Verlagsgesellschaft M . B. IT., Leipzig, 1937. (7) Felix, J. R., and Neill, W. K., Heat Transfer Symposium, pp. 123-70, Annual Meeting, A.I.Ch.E., Atlantic City, December 1951. (8) Graetz, L., Ann. Physik, 18, 79 (1883); 25, 337 (1885). (9) Hougen, J. O., and Piret, E. L., Chem. Eng. Progr., 47, 295-303 (1951). (10) Irvin, H. B., Olson, R. W., and Smith, J. M., Ibid., 47, 287-94 (1951). (11) Leva, M., IND.ENG.CHEM.,39, 857-62 (1947). (12) Leva, M., Grummer, M., and Clark, E. L., Ibid., 40, 415 (19481. (13) Le& k., Weintraub, AI., Grummer, M., and Clark, E. L., Ibid., 40,747-52 (1948). (14) Schuler, R. W., Stallings, V. P., and Smith, J. M., Chem. Eng. Progr., Sump. Series, N o . 4, 19-30 (1952). (15) Singer, E., and Wilhelm, R. H.. Clzem. Eng. Progr., 46, 343-57 (1950). (16) Vershoor, H., and Schuit, G. C. A., A p p l . Sci. Research, 42, A2, No. 2, 97 (1950). RECBXVED f o r review December 20, 1954.
ACCEPTEDAugust 1 , 1955.
E N D OF E N G I N E E R I N G , DESIGN, AND EQUIPMENT SECTION November 1955
INDUSTRIAL AND ENGINEERING CHEMISTRY
2299
