August 1950
INDUSTRIAL AND ENGINEERING CHEMISTRY
It caused a marked improvement in the action of the anionic product, but had no apparent effect on the nonionic product. This observation is consistent with the anticipated specificity of detergent action which a t present is based on the thought that the reaction of the detergent with the surface material contributes far more to its cleaning efficiency than does a reaction between the reagent and the radioelement. Additional evidence pointing in the same direction is the fact that Sequestrene A.A. which is a chelating reagent was much better for removing P3*, with which there should be no chemical combination, than it was for removing Ba140,with which it forms a complex ion, thus displacing the barium ion equilibrium toward the solution. I n the second experiment air-dried P32,BaI40, and 1131 were removed from a representative group of materials by Mulsor 224 which had proved t o be a n average detergent in the previous experiment. The results, presented in Table IV, show that Mulsor can compete with the standard reagent in only a few instances, most of these being in the removal of I13l. It should also be noted that on a particular material, one or another of the elements may be removed more efficiently than the others. There is no consistent relation between the indexes for the different elements when one uses the same detergent for removing contamination from different surfaces. The same observation was made with respect t o the removal of P32,BaI4o,and from glass, stainless steel, and lead, again pointing up the fact that reagent decontamination involves the interaction of a t least three major variables-the surface, the radioelement, and the decontamination reagent. The development of detergents which have outstanding cleaning properties promises to be a matter of fitting the reagent t o a specific job, a t least until the mechanism of the action is better understood. Despite the unfavorable comparison of many detergents with the standard reagent, the first experiment demonstrated that a detergent may be found that is just as good as the standard reagent for removing a particular element from a particular surface material. Also, a large number of cleaning problem exist for which a decontamination index of 3 or 4 is quite adequate, and
1481
Mulsor, a n average detergent when used with lucite, gave an index of this magnitude on several of the materials. Mulsor on asphalt tile and on Koroseal tile was virtually useless. If these materials are selected for use in a radiochemical laboratory t o take advantage of their proved wearing qualities, one should be prepared to adopt a maintenance policy of replacing the contaminated areas. SUMAIA RY
The corrosion resistance and decontamination properties of several available paints, plastics, and resins have been studied under standardized conditions. It is concluded that some of these may be used to advantage in place of glass, stainless steel, or lead for many common functions, and that they may often be decontaminated by mild reagents, such as detergents. The combination of the contaminating conditions, the surface material, and the decontamination reagent are interdependent variables which leads to a high degree of specificity in cleaning efficiency. ACKNOWLEDGMENT
The authors gratefully acknowledge the assistance of Maryann Huddleston and George A. West for aid in performing these tests. They also wish to acknowledge Arthur D. Little, Inc., for making available to them some information obtained during their extensive studies of materials to be used in the construction of some of the new Atomic Energy Commission facilities. LITERATURE CITED
(1) Tompkins, P. C., a n d Bizsell, 0. M., IND.ENG.CHEM., 42, 1469 (1950). (2) U. S. Atomic E n e r g y Commission, Isotopes Division, “Radioisotopes Catalog,” September 1947. RECEIVED August 26, 1949. Presented before the Division of Paint, VarCnExInish, and Plastics Chemistry a t the 117th Meeting of the AMBRICAN CAL SOCIETY, Detroit, Mich. This paper is based on work done in the Biology Division, Oak Ridge National Labmatory under Contract No. W7405-Eng-26 for the Atomic Energy Commission.
Solid-Fluid Heat Exchange in Moving Beds WILLIAM D. MUNRO AND NEAL R. AMUNDSON University of M i n n e s o t a , Minneapolis 14, M i n n .
A solution of the problem of heat exchange between solid and fluid is presented for parallel or counterflow exchangers. It is assumed that the solid consists of uniform spheres. The resistance to heat transfer by conduction in the spheres is taken into consideration. Heat generation within the spheres may occur either at a constant rate or as a linear function of the temperature. An exact solu-
B
ECAUSE of its great industrial application, the general problem of heat exchange between a fluid and a solid has received widespread treatment in the literature. The fluid usually moves through the exchanger in a single direction. The solid, on the other hand, may be stationary, moving in a direction parallel, either concurrent or countercurrent, t o the fluid, or moving in almost random motion as in a fluidized bed. The fixed-bed operation has received the most attention because of the use of checker-brick regenerators and because all of the older catalytic processes were of this type. A rather complete bibliography of the theoretical work in this field is given by
tion of the theoretical equations is obtained by means of the Laplace transformation. Numerous special cases are obtained, among which is the use of the equations in catalytic reactor design for a simple reaction. Numerical examples are included for comparison with approximate treatments. The examples show that approximate treatments predict exchangers whose transfer area is too small.
Thiele ( 1 4 ) in a review article. The case of heat release in a fixed-bed catalytic converter was considered by Brinkley (a), Wilhelm, Johnson, and Acton ( I T ) , and Wilhelm and Singer (18). Other treatments of heat transfer in packed beds are those of Arthur and Linnett ( I ) , Damkohler ( 4 , 6), and Grossman (10).
The problem of a solid moving countercurrent to a fluid has received some attention because of its importance in blast furnace operations, kiln-type heaters, and the newly developed catalytic processes for cracking petroleum. Furnas (7-9) discussed this problem in an approximate manner. Love11 and Karnofsky
1482
INDUSTRIAL AND ENGINEERING CHEMISTRY
( 1 1 ) considered the exact problem, but obtained an approximate solution to the equations describing the process. Problems of heat exchange in fluid-solid systems can be considered from two points of viea-. The older and approximate theories assumed a negligible resistance to heat transfer within the particles. Saunders and Ford ( 1 3 ) were able to show that this assumption is valid provided the particles T5-hich make up the bed are small enough, and it should also be true if the thermal conduct,ivity of the solid is sufficient.ly high. Lovell and Karnofsky ( 1 1 ) examined the problem assuming that the particles are uniform spheres in which the conduction through the particle is a fact,or, the physical properties being independent of the temperature. They solved the problem by a modification of the Schmidt graphical method. It is the purpose of this present paper t o generalize the problem somewhat and obtain an exact’ solution of the equations involved. From this generalized problem it will be shown how some other problems of interest to chemical engineers can be solved. The authors presume that the solid moves as a whole in a downward direction while the fluid moves countercurrent to it,. The problem of concurrent flow offers no additional difficult?; as will be shown later. The solid particles consist of uniform spheres which enter the top of the exchanger in a steady stream and a t a constant temperature. In their flew down the exchanger the particles meet’the fluid which has entered the bottom a t a constant temperature. The assumption is made that the reactor is adiabatic, the only heat transferred being that between the solid and fluid. This is an essential simplification since it eliminates the radius of the exchanger as a variable in the system. It is assumed further that heat conduction, in the direction of the flow of fluid, in the fluid is negligible. It is necessary in t,his analysis that the fluid velocity profile be uniform. Heat may be generated in the spheres as occasioned by chemical reaction, and there may be a surface resistance to heat tzansfer at the soliclfluid interface. DIFFERENTIAL EQUATION AND 4UXILIARY CONDITIONS
If one considers a single sphere where T denotes the radius variable and where q ia the amount of heat generated in the sphere per unit time per unit volume, the general equation for conduction of heat is
Vol. 42, No. 8
when T = R, where T is the temperatuie of the turbulent fluid in the neighborhood of the sphere. For no surface film resistance the surface of the sphere is a t the same temperature as the fluid and hence
t = T (5’) when r = R. This condition is a special case of Equation 5 and can be obtained by letting hr become infinite. One must now find a relation between the fluid temperature and the sphere temperature. The rate a t n.hich heat leaves a single sphere is
and hence the rate a t which heat leaves a pound of spheres is
If one considers an element of exchanger of thickness A x , each pound of spheres is in the section for Az/V hours. G, pounds per hour pass through the section and hence the rate of heat release from the spheres as they pass through the section is 3G,k, at --
(z,), ”
Rp s l y
=R
The rate a t which heat is picked up by the fluid is
-GCp
dt
-
dz
IZ
so that
- d_T Equation 6 neglects heat conduction in the fluid in the direction of flow. Relations similar t o this one have been obtained by Ebel (6) and Lovell and Karnofsky (11). The latter authors use an integral relation which can be shown to be equivalent to this one but not quite so easy to use in obtaining the solution. Equation 6 gives the relation between solid and fluid temperatures in the form of an ordinary differential equation. Consequently the fluid temperature must be specified a t some point of the exchanger. The natural temperature to use for this purpose would be that of the fluid entering a t the bottom of the exchanger. However, for this analysis it is convenient t o use the outlet fluid temperature a t the top of the exchanger. Hence
(7)
2’ = To From the assumption that the spheres move with a uniform velocity, V , VO = 2, n-here 2 is the variable distance from the top of the reactor. Hence
I n general, 9 arises from a catalytic ieaction and, therefore, is a rather complicated function of the temperature as well as the catalyst activity, activation energy, heat of reaction, concentration of reactants, and time of contact. Certainly over limited temperature ranges the function p (-anbe approximated by a linear function as y=n+bt
when z = 0. This, as will be seen later, causes no difficulty SOLUTION OF THE GENER4L PRORLEWI AND SPECIAL CASES
The primary purpose of this paper is to obtain an exact analytical solution of Equations 3, 4,5 , 6, and 7. The method of solution will be that of the Laplace transformation, details of which can be found in the books of Marshall and Pigford ( 1 2 ) or Churchill ( 3 ) . The transform will be taken with respect to the distance variable, 2. There are two dependent variables in the system, t(r,s)and T ( z ) ,whose transforms will be denoted by e - ~ z t ( T , x ) d z= h
L [ l ( r , z ) ]= h ( r , p ) =
The differential Equation 2 t h n i ieduces to
L [ l ’ ( z ) ]= H ( p ) Since the spheres enter the exchanger a t a constant temperature, t = la
=
X”
e-p”T(x)dx = H
On applying the transform to Equations 3 and 4,one obtains
(4)
when z = 0. If one assumes there is a surface film resistance to heat transfer between the spheres and the fluid, this condition is described by
The general solution of this equation can be written directly, since the left-hand side is equivalent to a form of Bessel’s equation of order one half, and the right-hand side is a constant with respect t o r. Thus,
August 1950
INDUSTRIAL AND ENGINEERING CHEMISTRY
It is clear that B must be zero since a t the center of a sphere the temperature must remain finite. If one makes use of the relation (181,
case E, no heat generation, A = 0, n = 0, p # 1; caseF, A = 0, a = 0, p = 1-it is apparent thM singularities will occur only a t the zeros of the denominators. Case A. Heat generation a linear function of the temperature, A # 0, a # 0. It can be shown that the poles are all simple poles, and the possibility of complex poles is not excluded. Examination of 11 reveals that the first term has poles a t p = b/C,p,V and the zeros of D. The second term has poles a t p = 0 and p = b/C,p,V. The residue at the latter pole is a
then Equation 9 reduces to
where A is a new constant whose value must be determined from the auxiliary conditions. The transforms of Equations 5 and 6 are ks($)
=
hf(H - h )
1483
+b 610 exp (*--) C,P,V
while the residue of the first term at the same pole is the negative of this. The residue of the second term a t p = 0 is -a/b. The sum of the residues a t the poles of the first term at the zeros of D is given by Churchill ( 3 , page 170, Equation 8). The sum of all the residues and hence the inverse transforni is given by Equation 13 (below), where dn = 2(1 -
e)
- 3p + € ( A - at)
+
when r = R, and
3pr dh R (6)
'
pH
1To
when r = R, respectively. These two equations may be solved simultaneously, resulting in
when r = R. When Equation 10 is substituted in this relation, an equation in A and known quantities is obtained. After solving for A and substituting in Equation 10there results
1 CapeVp(to - T O )+ a + bTo) sin h = r (____(C,p,Vp - b ) D
and wn are the roots of D = 0. Equation 13 gives the temperature in a sphere as a function 01 the radius and the position in the exchanger. In itself this formula is not of great interest, although it. solves the problem originally posed in the paper. A much more interesting formula can be obtained by developing the relation between the rate of heat release from the sphere surface and the inlet and outlet fluid temperatures. This can be done by integrating Equation 6 as shown in Equation 14. From Equation 13 one can ralculate the temperature gradient a t the sphere surface
+
+
CapaVtop a p ( C 6 ~ , V p- b ) (11)
where
D
=
Y
( [ - 38
+
(e
- 1 ) ( A - a*)]sin [3P -
E
(A
w
+
- w 2 ) ] w COS a )
where (12)
1
z=-
w:
[3@ - € ( A
- @:)I2
+
Equation 11 is the solution of the general problem in the form of the Laplace transform. In order to find the function itself, Substitution in Equation 14 gives Equation 15 which is the inverse transform of Equation 11 must be found. This is a general formula relating the temperatures of the fluid done by the well known method discussed fully in Churchill and solid streams and the physical constants and control vari(8,page 168). In brief, one must determine the polcs of h(r,p) ables in the system. The form is somewhat complicated but the in Equation 11 and evaluate the residues of h(r,p)ePz a t these convergence is rapid since each term in the series behaves approximately as w ~ - ~ .Because of this rapid convergence, sufficient poles. The aum of the residues is the inverse transform. The character of the poles, their position and multiplicity, are deteraccuracy for engineering purposes should result from the use of mined by the parameters in the function. only a few terms of the series. The function h(r.m) . ,. of Eauation 11. for which the inverse transform is to ( A - @:)(to - TO)^, f R2(a TO) sin 3 , $ ( A - 0')2 t(r,z) = - (L + (13) be determined, is analytic except for b k.r n = 1 d,w: sin w,, R poles, all of which lie in a left-half plane. The nature of these poles will depend on the values of the parameters. From the cases discussed below-case A, heat ( g ) r = R d x = f g d z = L r d T = T - To (14) generation a linear function of the temperature, A # 0, a # 0; easa B, heat generation proportional t o the temperature, A # 0, a # 0; case c, $ ( A - w:)x R2 constant heat generation, A = 0, a f 0, T - To = 6p [ ( A - w:)(to - TO) - ( a bTo) fn e k , n=l 6 = 1; caSeD, A = 0, a # 0, i3 = 1;
2
+
Tf
2
+
+
1 [I
INDUSTRIAL AND ENGINEERING CHEMISTRY
1484
If the average temperature of the particles leaving the bottom of the exchanger is desired it can be obtained from the formula
Vol. 42, No. 8
B. Heat generation proportional to the temperature,
Case
a = 0. This case is not essentially different from case A and the result can be obtained by setting a = 0 in Equations A # 0,
13 and 15.
By integrating by parts, after substituting Equation 13 in this expression, one obtains
CHARACTER OF POLES FOR CASEB. The only variation here will be removal of the pole a t p = 0 in the second term. The rest of the analysis for case A is still valid. Case C. Constant heat generation, A = 0, a # 0, p # 1. It is apparent that application of Equation 13 in this case is not possible. Equation 11 is still applicable and reduces to
where CHARACTER OF POLES FOR CASE 8. The second term of b = 0 and p = C,P,V' Only the first term requires further discussion. If p = ~-' ( z A ) R2 this term becomes h(r,p) obviously has simple poles a t p
,-
~
+
It is clear that there are poles in both terms a t p = 0 and poles of the first term at the zeros of E. I t is evident on closer examination that the poles a t p = 0 for both terms are of the second order. The residue of the first term a t p = 0 is t o - Tc
p which is analytic except for a simple pole a t z of m
zn
g(z) =
[ - 8en3
=
0 and the zeros
+ 4 ( ~- l)nz + ( 6 p + 4 e
-
+
U1.2
6 k e ( p - 1) +
1
aR2(2 106 - 36) 30 k,(6 -
+
a yx __k,(P - 1)
The residue of the last term a t p = 0 is
-263. - 2)n - A ]
ic
+ a yz
n=O
The analysis for these double poles follows Churchill (S, page 171;. The residues corresponding to the nonzero roots of E can be evaluated in the usual manner to give the final inverse transform as
where
+
In case the function +(a)has only real negative zeros, the same will be true of f(z) by a t,heorem of Laguerre (16). The zeros of $(a)will all be real and negative with the possible except'ion of those of the factor of the third degree polynomial. These, for appropriate values of the parameters, may include two positive or two complex values. In either of these cases the theorem of Laguerre does not apply. It can be shown by an extension of the theorem, however, that the folloning condit,ions hold. If all of the roots of an associated cubic equation 8ez3
+
+ ( 2 0 ~+ 4)2'
aR2(2 10s - 38) 30 ks(p - 1)'
where gn
=
and
wn
2(l -
- 3p - ea:
e)
- (6P - 2eA - 6 ) ~ A = 0
are real, then g ( z ) = 0 will have as many real positive roots as t'he cubic, and all the rest will be real and negative. Investigat'ion of the algebraic signs shows t'here will be a t most two positive roots in case A > 0 and 60 - 2eA - 6 > 0. In all cases there d l be one or none. In case the cubic has two complex roots, y(z) ~ i l have l a t most two complex zeros, the rest' being real and negative. Therefore in all cases y(z) will have a t most two positive or two complex zeros, the rest being real and negative. Since p is relat,ed to z by a shift to the right or left, the nature of the poles in terms of p will depend on the value of 4. The pole a t z = 0 occurs for p = If there are two complex poles C,P,V' in terms of z , t.he same will be true of p . On the other hand, the existence of positive real poles in terms of z does not necessarily imply any positive poles in terms of p since A may be negative. In any case, however, there viill be a t most two poles to the right ofp=and the expansion in Equation 13 is valid. For CSPSVJ some values of the parameters this may involve positive exponents on e in a finite number of terms.
+
+ (E - 1 1 4 ( I + 2e) + 3a 3P + €4
(17)
+ (38 + e w ; ) ~cos~ ~ an = 0
(18)
are the roots of
[ - 3p
+ (1 -
E ) w , ~sin ] wit
As shown belon-, these roots nil1 be all real and negative in case P < 1. If p > 1 there will be one positive real root, and the rest will be negative. To find the relation between inlet and outlet temperatures, again a substitution is made in Equation 14 giving
~
6p
2
_ -, d X [(To - t o ) 4
n= 1
+ F] ha
R 2 - 1)
(19)
+ 2e)w*
(20)
where
Fn-l = - (3p
+ ea:)? + 3(3P +
- (1
August 1950
INDUSTRIAL AND ENGINEERING CHEMISTRY
The average temperature of the spheres can be calculated as in the previous case.
~ ( =~to 1
CHARACTER OF POLES FOR CASEC. The second term of h ( r , p ) has a pole of order two at p = 0. In the first term, let p = 2, R2 and it becomes
to
1485
- To +
P - 1
- $&
m
c
6(to -
n=l
(3p
+ sw;t)2 - 3(& +
sw:)2
+
w,2
(1
+ 2s)
where
(n
+ 1)[4en2+ (6e + 2)n - 3(P -
n=O
Therefore, there will be a pole of order two a t p = 0 in the first term and simple poles at the zeros of
where
If +(n)has only real negative zeros, the Laguerre theorem cited will apply, and the same will be true of g(z). All zeros of +(n) will be real and negative except possibly the roots of the quadratic equation
+ (66 + 2)2 - 3(p - 1) = 0 whose discriminant is 4[(3e + 12epI. The discriminant is always positive and hence the roots alwa s real. If p < 1 both 4e22
CHARACTER OF POLES FOR CASEE. I n both the first term and the second term of h(r,p), cancellation of the factor p reduces the order of the pole t o one a t p = 0. Otherwise the analysis for case C is valid, and the poles will be all negative for p < 1, one positive the rest negative for p > 1. Case F. A = 0, a = 0, p = 1. This is a rather special case but will be included for later reference. The preceding formulas cannot be used for p = 1, and it can be shown for this case that there is a pole of second order in the first term a t p = 0. There are poles of the first order a t the zeros of Equation 18 with 6 = 1 and a simple pole of the second term at p = 0. The whole inverse transform can be written
t(r,x) = 15 x - Rz(1 5 s )
[
-
+
5r2 2R2( 1 56)
+
1- +
15 (1
14
76) (fa
-
T o )+
1)2
roots are negative and all the zeros of g(tYwil1 be real and negative. I n case p > 1, the quadratic has one positive root. By an extension of the theorem of Laguerre i t can be shown that g ( z ) will have exactly one positive zero; the rest are real and negative. The first term of h(r,p) then will have a pole of order t k o at p = 0, its remaining poles being simple and all negative if p < 1, one positive, the rest negative if p > 1.
Case D. A = 0, a # 0, p = 1. It is apparent from Equation 19 that this formula is not valid for ,5 = 1. It can be shown quite easily that there is a pole of the third order for p = 1 . The standard method can be used for the evaluation of the residue but it will not be done here.
and wn are the roots of Equation 18 with p = 1. Hence
T - to To - tc =
+
15yx Rz(1 5s)
Case E. KO heat generation, A = 0, a = 0, /I# 1. It can be shown that Equation 9 for this case has a pole of first order
a t p = 0 whose residue is to T o The remaining poles are a t p- 1' the zeros of Equation 18 in the first term while the second term has a pole at p = 0 whose residue is to. Hence the complete * inverse transform can be written
+
- $*":z
m
c
n=l
CHARACTER OF POLE FOR CASED. Analysis similar t o that of case C shows that the pole a t zero is of order three. A11 remaining poles are real and negative.
+
(3
+
4
)
2
1-e - 3(3
+
Em:)
+ 4 ( 1 + 28)
(22)
and the average particle temperature may be evaluated as above. This concludes the special cases which will be treated here, although there are many more which could be handled in t h e same manner but these would require more than a coincidental choice of parameter values. In so far as the authors are aware these problems are new, not only in their physical aspect, but also there is no evidence of such problems in the operational mathematics literature, although their solution is straightforward and offers no difficulty. The problem of concurrent flow of solid and fluid can be handled in the same manner as countercurrent flow. The only change required in the fundamental equations is in Equation 6 which must be replaced by
where gn is defined in Equation 17 and un are the roots of Equation 18. The relation between inlet and outlet temperatures is m
T -
to To - tc =
+
6p
z:
n= 1
(3p
+
E d ) 2
-
1-e 3(3p
- -wRrx Rg
+
EWE)
+ w: ( 1 +
2E) (21)
The average temperature of the leaving_.particles can be com. puted as before to give
The analysis from this point proceeds as before and will not be pursued further.
CHARACTER OF POLEFOR CASEF. There is a simple pole a t p = 0 in the second term. The first term becomes, with the substitution of case C,
INDUSTRIAL AND ENGINEERING CHEMISTRY
1486
Vol. 42, No. 8
nofsky ( 1 1 ) also considered the case in which conduction iri the solid is neglected and the problem is considered as an ordiriary countercurrent heat exchanger in which the area for heat exchange is the total external area of thP particles in the bed at one time. I n this method one can write
C,G,(t - t o ) = CpG(T - To) = h / A ( T - t ) - (To - hi log T - - t e To - to where A , the area factor, can be expressed as
3G,O
A =-
&Pa
-
These equations can be combined to give
and
Equations 21’ and 22’ indicate that the behavior for values of j3 # 1 and p = 1 is different and this was true also in the more general case as shown by Equations 21 and 22. There are now three methods of solving the same problem and it should be of mme interest to compare the results numerically. Let us consider the following three cases: P
Case
T - to or0 yx Figure 1. Plot of -US. - or - - for Cases I, 11, Tu - t o R 2 R2 and 111 11,111,1111, = Furnas (7-9) approximate solution 12, 111, 1 1 1 9 = Lovell and Karnofsky (11) graphical method 1 8 , 111, 1118, = Authors’ exact solution
where g ( z ) reduces in this case t o m
+ 5)! (n + l ) ( n + 2)(2en + 5 , + 1)
Zn
g ( z ) = -4z2 n=O
(2n
e
These cases were considered in the aforementionel papers. One must first compute the roots of Equation 18. This is done most easily by Sewton’s method. Cases I and I11 are straightforward since all the roots are negative-ie., p , < 0. Thus, from the relation p = - w 2 ( r / R 2it) is seen that the positive roots of Equation 18 can be calculated. For Case 11, all of the roots in p are negative save one and these negative roots can be calculated as in the other two cases. If there is to be a positive root in p there must be an imaginary root in w. Hence in Equation 18 if we let w = iw‘ an analogous equation will be obtained containing hyperbolic functions and some sign changes. This equation can then be used in Newton’s method to find the root in w’. The complete procedure outlined here has been carried out for the three cases and the roots obtained are shown in Table I. It should be noted from Table I that the roots are approaching odd multiples of 7r/2. This is true in general for problems of this type. This fact can usually be utilized to compute the roots as a first approximation.
TABLE I. ROOTSOF EQUATION 18 with
Case WI
w2
ws lo4 WS
Hence g ( z ) will have a zero of order two a t z = 0. The remaining zeros are real and negative, since +(n)has only negative zeros. Therefore, the first term of h ( r , p ) has a pole of order two a t p = 0, wit,h its remaining poles all simple, real, and negative. HEAT TRANSFER IN A PEBBLE-TYPE HEATER
The problem of straight heat exchange between a fluid and a solid without heat release has been considered widely as stated previously. This situation is that of case E or case F with the use of Equations 21 and 22. Corresponding to these formulas, Love11 and Karnofsky ( 1 1 ) used a modified Schmidt,method to solve the same problem. Furnas (7-9) and Lovell and Kar-
YB
w7 W8
I
5.115
1;:Ei;
14.386 17.493 20.601 23.724
....
Case I1
2.59ldrl 4,755 7.932
11.066
14.197 17.331 20.466 23.602
Case 111
2 124 5.514 8.561 11.599 14,655
17,728 20 815 23,913
Equations 21 and 22 are then used to calculate the temperatures or the exchanger length. This can be done in a straightforward manner. The convergence of the series is rapid enough so that for engineering purposes only three or four terms are sufficient to ensure accuracy of a few per cent. In the calculations all the roots shown in Table I have been used and the remainder of the series has been approximated by an integral approximation. Figure 1 shows a comparison of the three methods and it i R seen
INDUSTRIAL AND ENGINEERING CHEMISTRY
August 1950
that in general the stepwise method of Lovell and Karnofsky is not sufficiently accurate to justify its use over that presented here. The amount of time required to make calculations using Equations 21 and 22 is not excessive, for a competent computer can calculate five roots of Equation 18 in about 2 hours and can perform the remaining calculations in another 2 hours. Let us re-do the problem of Lovell and Karnofsky on the burning of lime. It is a rather curious fact in the light of t h e special situation which results when p = 1 that this is precisely the problem chosen by Lovell and Karnofsky to illustrate their graphical method. However, the graphical method does not distinguish among the parametric values so that no difficulty arises there. Limestone is calcined in a continuous countercurrent vertical lime kiln. The calcium carbonate is fed in at the top in 2-inch pieces a t a superficial mass velocity of 2230 pounds per square foot per hour. Flue gas and carbon dioxide from the calcination rise through the bed a t a superficial mass velocity of 2500 pounds per square foot per hour. If the limestone enters a t 100" F. and the gas leaves a t 200' F., a t what level in the kiln is the gas temperature 400" F.? Average specific heat of gas = C, = 0.25 B.t.u./lb./" F. Average specific heat of CaC03 = C, = 0.28 B.t.u./lb./" F. Density of CeCOI = pa = 162 lb./cu. ft. Heat transfer coefficient = h/ = 78 B.t.u./hr./sq. ft./' F. Thermal conductivity of CaC03 = k , = 1.3 B.t.u./hr./sq. ft./ ( O
F./ft.)
Fraction void space = f = 0.50
1487
sphere. The condition which holds a t any point in the sphere is given by
where it has been assumed that the reaction $akes place inside the sphere in the pore volume. The condition a t the surface of the sphere analogous to Equation 5 is
where T = R. This equation states that there is a resistance t o mass transfer a t the sphere surface. Spheres which enter a t the top may have a uniform distribution of A in the void volume of the sphere so that c = CO, when 2 = 0. The outlet concentration in the fluid stream is given by C = CO,when z = 0. The relation between the concentration change in the fluid and the rate of transfer from the spheres is given by the equation
This equation replaces Equation 6. It is clear that this system is covered by case B with the proper redefinition of the constants. This can be carried out without difficulty to give
n= 1
where CY
=
k, Cap,
1.3 _ _ (0.28)(162) = 0'0287
= ~
Mn-' and
I' - t o
To
- to
- 400
- 100 = 200 - 100
From Figure 1 onc can read off the corresponding values of aO/R2 or (rz)l(Ra) as Curve 11 [Furnas (7-9)approximate equation] = 0.135 Curve IZ[Lovell and Karnofsky (11) graphical method] = 0.20 Curve I s (this paper) = 0.245 (extrapolated) From these values the lengths of exchanger required are obtained as
z1 = 0.90 feet z2 = 1.33 feet
z3 = 1.63 feet I t is clear from these values that the two approximate values are in error by an appreciable amount and that this error is in a direction which would lead to underdesigning-Le., exchangers would be too small. It is hoped that in the future charts may be prepared to give a wider range of operating variables and physical constants. CHEMICAL REACTION IN MOVING BED REACTOR
The theory presented in this paper can also be applied to a catalytic reactor system in which the porous catalyst in the bed is moving in continuous manner as in the new processes for catalytic cracking of petroleum, provided the assumed reaction mechanism is of simple type. As a first case let the velocity of a chemical reaction A -P B be governed by the equation
where C is the concentration of reactant A and where this equation is written for a unit volume of catalyst as it occurs in a
- ~ : [ 3 4 &+ S ( Y + W , ~ ) ] ' [34s2 S(Y ~,2)1[3d+VI
+
wn
+
- (y
+
w?)[w:(~
+ 29) + y ]
is a root of
[ - 34*2 - (s
- 1)(y
+ d)]sin + w
[3+,2
+ s(y +
w2)1w
corn = 0
A numerical example for this problem will not be carried out. As a second case for chemical reaction one might assume that a chemical reaction of zero order takes place in the sphere void volume. A little examination will show that this would reduce to a problem equivalent to case C. Redefinition of the constants will give the answer directly. SUMMARY
The problem of heat transfer and heat release in a fluidmoving solid exchanger has been considered for the case in which the solid is in the form of uniform spheres. It has been assumed that conduction in the fluid in the direction of flow is negligible and that the exchanger itself is adiabatic, so that no temperature gradient in a radical direction in the fluid exists. The former assumption is probably not objectionable but the latter will seriously limit the applicability of the formulas. If no heat is generated in the solid the results of this paper can be compared with the approximate treatments of Furnas (7-9) and Lovell and Karnofsky (11). Three cases considered here show that the approximate methods can be in error by a sufficient amount t o invalidate their use for design purposes. Generally, approximate methods are used when the amount of work involved is considerably less than a more rigorous method. For the problems considered here it is clear that a t least in some cases the approach of negligible resistance to heat transfer by conduction can lead to serious error. The graphical method has no advantage over the precise solution since it not only gives results which are in error in the wrong direction but is probably more time consuming than the use of Equations 21 and 22. It is undoubtedly true that each of the methods has its own particular domain of application depending upon the parameters of the system, but this can only be substantiated by further data from the field.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1488
NOMENCLATURE
a = rate of heat release, B.t.u./hr./cu. ft. b = rate of heat release, B.t.u./hr./cu. f t . / ” F. c = concentration in fluid in spheres, moles/cu. f t . co = initial concentration in fluid in spheres, moles/cu. ft. c = concentration in fluid stream in reactor, moles/cu. ft. co = outlet concentration in fluid stream from reactor, moles/ cu. ft. c, = specific heat of solid, B.t.u./lb./” F. c, = specific heat of fluid, B.t.u./lb./” F. D = diffusivity of reactant in the fluid inside the solid, sq. ft./hr. void volume in bed fG, = = fractional mass rate of flow of solids, Ib./hr./sq. ft. G = mass rate of flow of fluid, lb./hr./sq. ft. hi = heat transfer coefficient at solid-fluid interface, B.t.u./hr./ sq. ft./“ F. h = Laolace transform of t H = Laplace transform of T i
=
3, j(z)
J
Bessel function of order one half Bessel function of order minus one half reaction velocity constant, reciprocal hours thermal conductivity of solid, B.t.u./hr./sq. ft./( F./ft.) mass transfer coefficient at solid-fluid interface, moles/hr./ sq. ft./unit concentration difference D/V transform variable corresponding t o z rate of heat release in solid in B.t.u./hr./cu. ft. radius variable, ft. radius of a sphere, ft. =
--o s(z) =
k = k, = Kj =
m = p = q
=
r
= = = = =
R s
4-1
D/KfR
temperature in the solid, F. initial temperature of osolid, O F. T = temperature of fluid, F. 7 ’ 0 = outlet temperature of fluid, F. U = volume rate of flow of fluid through the bed, cu. ft./hr./ sq. ft. B = velocity of solid, ft./hr. x = distance along exchanger, ft. y = kR2/D 1
tc
O
2 CY
Vol. 42, No. 8
= G,/p,U
= thermal diffusivity = k,/C,p, P = GsC,/GCp Y = h-,/C,P,V A = bR2/k, B = k,/hjR @a = void fraction in solid ps = density of solid, Ib./cu. ft. 0 = time, hr. un = roots of transcendental equations LITERATURE CITED
(1) Arthur, 3. R., and Linnett, J. W., J . Chem. Soc., 1947, 416. ( 2 ) Brinkley, 8.R., Jr., J . Applied Phys., 18, No. 6 , 582 (1947). ( 3 ) Churchill, R. V.,“Modern Operational Mathematics in Engi-
neering,” New York, McGraw-Hill Book Co., Inc., 1944. (4) Damkohler, G., in “Der Chemie-Ingenieur,” Eucken, A , Ed., Vol. 3, Part 1, page 441, Leipaig, Akad. Verlagsges., 1937. (5) Damkohler, G., 2. Elektrochem., 42, 846 (1936), (6) Ebel, R. A., Ph.D. thesis, Vniversity of Minnesota, 1949. ( 7 ) Furnas, C. C., IND. EIFG.CHEM.,22,26 (1930). (8) Furnas, C. C., Trans. Am. Inst. Chem. Engrs., 24, 1942 (1930). (9) Furnas, C. C., U . S . Bur. Mines, Bull. 361 (1932). (IO) Grossman, L. M., Trans. Am. Inst. Chem. Engrs., 42, 535 (1946). (11) Lovell, C. L., and Karnofsky, G., IND.ENG.CHEM.,35, 391 (1943). (12) Marshall, W. R., Jr., and Pigford, R. L., “Application of Differential Equations to Chemical Engineering Problems.” Newark, Univ. of Delaware, 1947. (13) Saunders, 0 . A,, and Ford, H., J . rron SteelInst., 141, 291 (1940). (14) Thiele, E. W., ISD. ENG.CHEM., 38, 563 (1946). (15) Titchmarsh, E. C., ”Theory of Functions,” London, Oxford University Press, 1939. (16) Ton Karman, T., and Biot, M., “l\lathematical Methods in Engineering,” New York, McGraw-Hill Book Co., Inc., 1940. (17) Wilhelm, R. H., Johnson, W.C., and Acton, F. S.,ISD. ENG. CHEM.,35, 562 (1943). (18) Wilhelm, R. H., and Singer, E., preprint for Am. Inst. Chem. Engrs. Regional Meeting, Tulsa, Okla., May 1949. RECEIVED February 20, 1950.
Alfin Catalvsts and t Polymerization of J -
d
AVERY A. 3IORTOiY Mussuchusetts I n s t i t u t e of Technology, Cambridge, Muss. Alfin catalysts are special combinations of sodium salts which cause the rapid catalytic polymerization of butadiene to polymers of unusually high molecular weight. These polymerizations show characteristics which are common to all reactions of organosodium compoundsnamely, the tendency to undergo multiple reactions. The reagents are also insoluble aggregates of ions whose behavior is affected by the ions in the aggregate. The history
of the discover). is review-ed. The catalytic polymerieation show-s no property in common with the conventional sodium process for polymerizing butadiene. The present problems center in the elimination of secondary reactions that are known to occur. The theory by which all the reactions occur, including polymerizations induced by organosodium compounds, is discussed. Progress,toward the practical use of the Alfin polymers is being made.
T
from the fact that the reagents are insoluble aggregates of ions, somen-hat as in the crystal of sodium chloride, and are not fully dispersed in a solvent, so that the ions act separately. An understanding of these points is essential for anyone who works with these eytremely reactive reagents which operate only in hetei ogeneoua processes.
HE Alfin catalysts are special combinations of sodium
salts, one from an alcohol and the other from an olefin, which have uniqiie properties as polymerizing agents for butadiene and similar compounds. In order t o appreciate clearly this account of the discovery, special properties, problems, and possible mechanisms, it is helpful to keep in mind two important principles which apply to all reactions and reagents in the field of organosodium compounds, whether for pol~merizationor for some simpler process. One is the tendency for organosodium reactions to consist of multiple steps which tumble over each other as if in cascades until the end is reached. The other is the control which inorganic components of the reagent and other inorganic salts exert on the process, w circumstance that arises
DISCOVERY OF THE CATALYST
The importance of the two ideas mentioned above is better understood 11-ith a background of the chemistry of the organosodium compounds. For example, in the TYurtz reaction, long held as the most typical activity of sodium in organic chemistry,
