At high Prandtl numbers Deissler’s Nusselt number is lower than Friend and Metzner’s and to bring them into agreement the eddy diffusivity near the wall must be increased, or the turbulent Prandtl number near the wall must be decreased to a value less than 1. Although the latter change is in the direction of the Jenkins model, the changes are too small to be significant because of the uncertainty in the eddy diffusivity of momentum near the wall. Thus, on the basis of the available experimental information, the most that can be said is that the turbulent Prandtl number near the wall is of order of magnitude 1 and that the mixing model is consistent with this result.
y+
Acknowledgment
-
The authors are grateful for the fellowship assistance of E. I . d u Pont de Nemours 8: Co., Inc., to J. M. Marchello.
literature Cited
Nomenclature
(1) Danckwerts, P. V., A.I.Ch.E. J . 1, 456 (1955). (2) Danckwerts, P. V., Ind. Eng. Chem. 43, 1460 (1951). (3) Deissler, R. G., Natl. Advisory Comm. Aeronaut., T.R.1210
concentration diffusivity flux flux across center line of layer by molecular diffusion flux across center line of layer by mixing length Nusselt number Prandtl number Schmidt number distance coordinate tube radius frequency of mixing or renewal distance from wall
Z (3 eD err ei 0
= y+ coordinate = distance coordinate = defined by Equation 11
= eddy diffusivity of mass
v
= = = =
$
=
eddy diffusivity of heat eddy diffusivity of momentum time kinematic viscosity distribution function
SUBSCRIPTS = value immediately preceding ith mixing i m = space mean value SUPERSCRIPT = time-average value
(1955). ,---, (4) Friend, W. L., Metzner, A. B., A.I.Ch.E. J . 4, 393 (1958). (5) Hanratty, T. J., Ibid., 2, 359 (1956). (6) Harriott, P., Chem. Eng. Sci. 17, 149 (1962). (7) Higbie, R., Trans. Am. Inst. Chem. Engrs. 31, 365 (1955). (8) Jenkins, R., “Heat Transfer and Fluid Mechanics Preprints,” Stanford Univ. Press. Palo Alto. Calif.. 1951. (9) Lubarsky: B., Kaufman, S. J:, Natl: Advisory Comm. Aeronaut., T.N.3336 (1955). (10) Marchello, J. M., Ph.D. thesis. Carnegie Institute of Technology, Pittsburgh, Pa., 1959. (11) Toor, H. L., Marchello, J. M., A.I.Ch.E. J . 4, 97 (1958).
RECEIVED for review March 16, 1962 ACCEPTEDNovember 19, 1962
STABILITY OF NONADIABATIC
PACKED BED REACTORS An Elementavy Tyeatment S H E A N - L I N
L I U , R U T H E R F O R D A R I S , A N D N E A L R. A M U N D S O N
University of Minnesota, Minneapolis 14, Minn.
The problem of stability of a nonadiabatic fixed bed reactor packed with small catalyst particles is considered. The transient equations for the reactor are solved by the method of characteristics. The results show that under certain conditions, there will be nonunique temperature and concentration profiles. For the present case, in the unstable region the particle conditions may jump from one state to another state in a manner different from that which occurred in the adiabatic case developed in a previous publication. For some sets of parameters, the concentration and temperature profiles, and consequently the reactor effluenl, are sensitive not only i o small changes in the inlet conditions but also to a small change in the initial particle temperature. HE
problem of stability of an adiabatic fixed bed reactor was
Tconsidered in a previous paper (3) and it was shown that under certain conditions nonunique steady-state temperature and concentration profiles resulted from different initial states of the reactor. The purpose of this paper is to study the stability of nonadiabatic fixed bed reactors and to exhibit some pathological cases. Prior extensive calculations ( 2 ) on the steadystate equations for the empty tubular reactor showed regions 12
l&EC FUNDAMENTALS
of operation in which the reactor effluent was very sensitive to operating conditions. With the assumption of no resistance to heat or mass transfer between fluid and packing and on the basis of steady-state equations, Barkelew ( 7 ) studied the stability of packed bed reactors and developed some empirical correlations. The same model as in the adiabatic case (3) is considered in this paper, except that in the present case there is transfer of
heat from the gas mixture throught the reactor wall. It is assumed that there is 1 nstantaneous radial transport of heat or mass in the reactor itself and that resistance to heat transfer may be lumped at the reactor wall. I t is shown that as in the adiabatic case because of the existence of multiple stead) states for a single particle there are nonunique concentration and temperature profiles. I n the present case the gas temperature inside the bed may be lower than the inlet gas temperature. and hence in the unstable region the particle conditions may change in a manner somenhat different from the adiabatic case. Summary of Equations
Since the equations for this paper are very similar to those given previously, they are not derived in detail. The transient equations for the interstitial fluid are
\\here p is the partial pressure of A in the gas. t the fluid temperature, C the reactor over-all heat transfer coefficient, and R, the radius of the reactor. The other quantities are listed in the table of nomenclature. For the first-order irreversible chemical reaction, A + B, the mass and heat balances on a particle are
k = ko exp ( - A E / R t p )
From Equations 3’ and 4’, one obtains
As before ( 3 ) , Q I I when plotted as a function of t , gives a sigmoidal shaped curve, while for fixed t and p, Q I is a linear function oft,. I n the adiabatic case for constant parameters it was shown that the straight lines passed through a common point. However, this does not occur here and, furthermore, it is difficult to construct these lines analytically except for the first one corresponding to the gas inlet conditions (cf. line LO in Figure 3). This follows of course. since the relation betneen p and t is more complicated. Therefore. the lines will be constructed from the numerical results of the transient computations. Since the purpose of this paper is to consider the transient behavior and its consequences. the computations are carried out on the transient equations, 1 through 6. Although the numerical values of p and t can be obtained by solving the steady-state equations, 1’ through 4’, there is a question of what particle temperature should be chosen when the particle has multiple steady states. For any set of parameters the straight lines, Q r , are a family and have an envelope. The loci of the envelope can be calculated from the steady-state values ofp, t , and t , as follo\vs: Re\vrite Equation 7 as
The straight lines are (4)
The boundary conditions a t the entrance to the bed are t
=
t,(e)
p
=
p e ( e ) ,x
=
0,
e>o
(5)
In Equation 9 p can be taken as the parameter and t as a function of p , t = t ( p ) . Then, differentiating Equation 9 with respect to p , one obtains
and the initial conditions of the solid and interstitial fluid are (10)
The transient equations, 1 through 6, are solved by the method of characteristics, described in some detail in the previous paper ( 3 ) .
The envelope can be obtained [see ( 4 ) ]by solving Equations 9 and 10 simultaneously for t, and t. Solution of Equation 10 for t, yields
The substitution of this expression into Equation 9 results in
Steady States and EnveJopes
The steady-state equations are
dt d.7
P - =
+
w)
t’ - P P
=
- (1
t, - t =
where
t
+ t p + wt,,
SPP
spp,
(2‘)
The derivative in Equations 11 and 12 can be expressed as a function of p and t , from Equations 1 ’ through 4’. Equation 3 ’ may be written
( 3 ‘1 (4’)
=
P P
1P+*
This may be substituted in Equation 1 ’ to give
@ dl
=
-f(tp)
From Equation 8 t =
tP
-
Pfi/(tP)
Substituting this expression in Equation 2 ’, one obtains Also
dt
-
dE
= w(t,
- tP)
VOL. 2
+ (1 + NO.
1
w)
PMb)
FEBRUARY 1963
(14) 13
dt - = dP
Dividing Equation 14 by Equation 13, there results
dt - -- w ( t m
- tP)
+ (1+
- Pf(tP)
dP
w)
-
P
, a constant
PPf(tP)
P
Therefore the equations of the envelope become
1
. ' 2
= P
adia.
Equations 16 and 17 are used to calculate the envelope. It also can be proved that in the adiabatic case the envelope reduces to a point. For the adiabatic case
1.4-
Therefore from Equations 15, 16, and 17
I
Ua6.25 BTU/hr./ft.
/ O F
pe '0.07 atrn. te = 1400 O R
w = o
A
t,=560°R
U.6.25 BTIUhr./sa. ft./OF. pe = .07atrn. te ~ 1 4 0 0 O R tai=l20OoR
----2.0
PARTICLE TEMPERATURE IN
OR
x 10-3
Figure 3. Heat generation and heat rejection curves for a single particle with envelope Stable case
x , REACTOR LENGTH, FEET
Figure 1.
Transient temperatures vs. reactor length
U 6.25 BIU/hr./sq. ft./OF: pe = .07atrn. t, = 1400 O R tpi=12000R ,t 560 OR
t ' -0
X,
Figure 2. 14
REACTOR LENGTH, FEET
Transient partial pressures vs. reactor length
I&EC F U N D A M E N T A L S
----- tp .25
,50 .75 1.0 x, REACTOR LENGTH, FEET
1.25
1.5
Figure 4. Effect of influent temperature on steady-state temperature profiles Stable case
Hence the envelopes degenerate into a point when w = 0. From Equation 9 it i:, obvious that for any fixed t the slope of the straight line QI is ( / @ - I and for first-order irreversible chemical reaction this is always greater than or equal to (@pp,) Therefore. as in the adiabatic case. if
-'.
inflection point
each particle in the bed has a unique steady state. The local stability of the Farticle behavior was discussed in detail in the previous paper 13) and the conditions f x stability can be applied to the present case because the transient equations for the particle are identical in the two cases.
U.7.5 BTJJJhr./.sq. ft./OF:
Example
To examine the transient behavior of the bed and the effect of parameters on the temperature and concentration profiles. the transient equations, 1 through 6, are solved by the method of characteristics. This numerical technique was described in the previous paper ( 3 ) , and is not repeated here. Let us consider an example using the same parameters as in the adiabatic case and in addition specify the radius of the reactor
R* = 0 . 2 foot [In the previous paper ( 3 ) ,the quantity Q, the particle radius, should be 0.1875 inch and u should be 408 feet per minute.] I n this example the reactor wall over-all heat transfer coefficient, ambient temperature, and the initial and influent conditions are not specified. As before, zl-l is assumed to be so small as to be neglected and the partial pressure capacity term is neglected. Then the transformed transient equations become
250
(2)
=
(' - ") + ('
p exp (12.98 - 22,000/tP) lo3' 1 f exp (12.98
1
- 22,000/tD)
P
pp
=
1 f exp (12.98 - 22,000jtP)
where I and I1 refer to the characteristics 0 = constant and x = constant. respectively. I t is assumed that 0
.25
.50
.75
1.0
0 = 0,x
1.25
1.5
x, IREACTOR LENGTH, FEET
Figure 5. Effect of influent temperature on steady-state partial pressure profiles Stable case
t,=
0
.25
'
1
1400 O R
I I .75 1.0 x , REACTOR LENGTH, FEET
.50
I
1
1.25
1.5
Figure 6. Effect of influent partial pressure on steadystate temperature profiles Stable case
The values used for t p z ,te, ,be, U , and tu are given on the graphs. The increments used are as follows:
?re
U =7.5 BTU/hr./sq.ft./OF:
10 .-
> 0,pt = O,p,, = 0,ti,, = t = t , (constant) e > 0; p = j e ,t = t , (constant)
x = 0.
=
2-3 minute and A x
=
2-8 foot
The steady state was obtained when e = 15 minutes and the computing time was of the order of one hour (Univac Scientific Model 1103). Results and Discussions
The transient behavior of gas temperature and particle temperature is shown in Figure 1 and the corresponding partial pressure profiles are shown in Figure 2. From the steady-state values of p , t , and t,, the straight lines, Lg, L1, . . .. Lg in Figure 3, are drawn. The envelope in Figure 3 is obtained from Equations 16 and 17 and it shows clearly that the straight lines are tangential to the envelope. The gas mixture is heated along Lo, L1, . . . until Lj, where it obtains a maximum temperature, 1826" R., and then the gas temperature decreases along L E ,Li, . . .Lg. The particle temperature increases along the steady states ao, a,, . . ., and ar, M here it obtains a maximum value, 1868" R., and then decreases along a3, Q, . . .ag. The maximum gas temperature and the maximum particle temperature are not a t the same point of the bed. This is shown by points t,,, and tPmaxin Figure 1, Figure 4 shows the effect of changes in the inlet gas temperature upon the steady-state gas and particle temperature profiles. for the case in which each particle has a single state as in Figure 3. The corresponding partial pressure profiles are plotted in Figure 5. These profiles were obtained by integrating the transient equations. \.\'hen the inlet gas temperature is relatively low-for example, t, = 1300' R. in VOL. 2
NO. 1
FEBRUARY 1963
15
Figure 4-the gas temperature decreases along the bed, and the chemical conversion is negligibly small. There is a considerable change in the profiles as t, changes from 1350' to 1400' R. Figure 6 shows the effect of changes in the inlet partial pressure of component A upon the temperature profiles for the case in which each particle has a single steady state. The corresponding partial pressure profiles are shown in Figure 7. The higher the inlet partial pressure, the higher is the maximum gas temperature and the chemical reaction is completed in a shorter length of bed, as expected. The effect of changes in the ambient temperature on the temperature and partial pressure profiles is shown in Figures
U =7.5 BTU/hr./sq.ft./
-----
8 and 9, respectively, for the case in which each particle has a single steady state. For the range of ambient temperature, 480' 750'R. changes in ambient temperature do not have a great effect on the profiles. I n Figures 10 and 11 are shown the effects of changes in the over-all wall heat transfer coefficient upon the temperature and partial pressure profiles, respectively, for the case in which each particle has a single steady state. There is a considerable change in the profiles as C decreases from 10 to 8.75 B.t.u.,hr./ sq. ft.1' F. -4s in the adiabatic case, the possibility of nonunique temperature and concentration profiles depends upon the existence of multiple steady states for single particles. For the case shown in Figure 3, all the straight lines intersect the s-curve
-
I
OF:
U.7.5 BTU/hr./sq.frJ0E
to 1400 O R tpi=1500 OR t, = 5 6 0 OR
E
e
z W
.06
a = I v) v)
W
E
.04
i
x, REACTOR LENGTH, FEET X,
REACTOR LENGTH, FEET
Figure 7. Effect of influent partial pressures state partial pressure profiles
on steady-
Figure 9. Effect of wall temperature on steady-state partial pressure profiles Stable case
Stable case
2.0-
U.7.5 BTU/hr./sq,ft./OF:
-t ----- tP
p, =.07 atrn.
2.0 -
t, = 1400 OR
ps= .07 atm. te =I400O R
1.2- t
t I .o
'
0
1.0-
.25
.50
.75
1.0
1.25
I
x , REACTOR LENGTH, FEET
0
I -25
1
I
50
I I .75 1.0 x, REACTOR LENGTH, FEET
L25
1.5
Figure 8. Effecf of wall temperature on steady-sfate temperature profiles
Figure 10. Effect of heat transfer coefficient on steadyslate temperature profiles
Stable case
Stable case
16
l&EC FUNDAMENTALS
at just one point and the sy-stemis absolutely stable. Although it is not shown graphically? unique profiles are obtained from all initial particle states. However, if any particle in the bed has multiple steady states, the system will be unstable for a certain range of initial conditions and nonunique profiles will be obtained. As shown in Figure 12, the first particle a t the bed entrance may have three steady states-ao, bo, and cc-corresponding to an inlet gas temperature of 1300' R., and an inlet partial pressure of 0.13 atm. If the initial particle temperature is lower than 1350' R.: the final steady-state profiles are as shown by curves A and A' in Figures 13 and 14. Figure 12 shows that the straight lines move toward the left in the order Lo, L1, . . .L4, and the particle temperature profile follows the low steady states, an, a , , . . .a+ Curve EA'V. 1 in Figure 12 is the envelope for this case. Since the gas and particle temperatures decrease along the bed, the chemical conversion is extremely small. If the initial particle temperature is higher than 1720' R., the final profiles will be curves F and F' in Figures 13 and 14. As shown in Figure 12, the first particle a t bed entrance is at the high steady state, bo: and the straight lines move toward the right in the order L'5, L'6, L',. Curve E.VV. 2 in Figure 12 is the envelope for these straight lines. T h e particle temperature profile follows the high steady states? bo, hi, b?, and 6 3 , where it attains a maximum value and then decreases following h4,b:. . . .b;.
pe =.07 atm.
c
.""t
t,
= 1400 OR
tpi = 1500 OR
PP
-----
ty
= 560 OR
5
x, REACTOR LENGTH, FEET
Figure 1 1 . Effect of heat transfer coefficient on steadystate partial pressure profiles Stable case
I."
1.6-
1.4-
1.2-
1.0.8-
.6-
I4
0
8
0"
.4-
.2-
-
PARTICLE TEMPERATURE IN
OR
x
10-3
Figure 12. Heat generation and heat rejection curves for unstable reactor showing envelope to heat rejection curve VOL. 2
NO, 1
FEBRUARY 1 9 6 3
17
Nonunique profiles are obtained for the initial particle temperature in the range 1350" R.< t P i < 1720' R. For example, if t P i = 13?0°R., the final profiles will be curves C and C' in Figures 13 and 14. The movement of straight lines is shown in Figure 12 bylines Lo!LI"? . . . Ls". The particle temperature decreases a little folloiving the lower steady states, a0 and a t " , and then jumps to the higher steady states. M:hen such a jump occurs, a new envelope, E.\'V. 3 in Figure 12, is obtained. The curves in Figures 13 and 14 show that the higher the initial
-
U= 6.25 B.TU/ hr./sq. ft./
t
OF,
particle temperature, the earlier the particle jumps from the lower to the higher steady state. It is seen from Figures 13 and 14 that there is a critical change in the profiles as the initial particle temperature changes from 1350" to 1360" R. Figure 15 shows the sensitivity of the profiles in the multiplestate case to the inlet partial pressure. There are large differences in concentration and temperature between curves A and B. The sensitivity of the profiles in the multiple state case in the inlet gas temperature is shown in Figure 16. As shown by curves A and B in Figure 16, a change of 10" in the inlet gas temperature causes a big change in the final profiles. By comparing Figure 15 with Figure 6 and Figure 16 with Figure 4, it is clear that the profiles in the multiple state case are more sensitive to inlet conditions than in the stable case. Some Comments on More Complicated Cases
LVith more complex reactions than the first-order irreversible, similar results will obtain but they cannot be reached so easily. T o see this consider the single reaction Za,A, = 0 among the species A , . A * , . . . A , with the usual convention that products of the reaction have positive stoichiometric coefficients. The rate of the reaction is r I moles of A , produced per unit catalytic surface area per unit time and an intrinsic rate of reaction. r , may be defined by
n
2! x
E
z
ri = a i r
The equations which describe the system are then given by
I .25
I,o 0
I I
50
I -75
I
I 1,25
1.0
1.5
x, REACTOR LENGTH, FEET
Figure 13. Steady-state temperature profiles for unstable case for various initial particle temperatures All other parameters held constant
.20
-
U.6.25 -P
BJU/hr./sq. f t . / O F .
ps=.13 atm.
and of course initial and influent compositions for each chemical species as well as the temperature must be specified. The approximations to this system, used on the simpler problem, should still be valid. The essential point to be made here. however, is that the full set of equations needs solving and there is little in the way of simplification which may be done save in special cases. The method of solution of the transient equa. tions used in the simpler case applies in the more complicated case. Similar statements do not obtain for the steady-state analysis. Let the steady state equations be
le=1300°R
z
tw= 5 6 0 O R
t
pi
- j& + &m,r t - t,
=
0, i = I , 2, . . ., s
+ Xr
=
0
(19')
(20')
tpi, O R
A 1350,1340 EL down 8
From Equation 19' it follows that
1360
C 1370
D
1400
F 1720,1800
a
and from 18 ' up
I 1.25
1.5
x, REACTOR LENGTH, FEET
Figure 14. Steady-state partial pressure profiles for unstable case for various initial particle temperatures 18
I&EC FUNDAMENTALS
and therefore the 2s set of Equations 18 ' and 19 ' reduces to two equations, one for a single chemical species in the intraparticle gas and one for the interstitial gas. Thus the reaction rate
expression I at each point in the bed is a function of t,, p, and ~ W C I variables stand now for a particular species. Also involved iin r are all of the influent partial pressures. Equations 19’ and 20‘ may be written
-GAS _----PARTICLt:
p,, where the latter
where R also depends upon the influent conditions. 22 may be solved to give (as in the simple case) PP =
U=6.25 BTU/hc/sq.ftJ°F te = 1 3 0 0 ° R
2.4 I
r--
-.
tpi i1340 “R
Equation
Sit,,$)
so that Equation 23 becomes I
tp
-
=
16
(23’)
X’R[S(t,,P), t p l
Equation 23‘ is the analog of Equation 7 and it is easily seen wherein the difficulty lies; the right-hand side is not a function of t , alone but involves the interstitial partial pressure. One might ask: Under what circumstances will the right-hand side of Equation 23’ be separable into a product of a function of t , and a function ofp in the form
E
.I2
c 0
W [L
.os
3
w
W a
a i
.04
5
8 A simple analysis sholvs that it will occur only if the reaction is first-order and irreversible-i.e., the simple case already analyzed. Thus while the simplicity of our analysis is losr on the more complicated problem, the results arc just more difficult to obtain and it is not to be expected that the pathology of the situation is significantly different. All, however, is not :lost; if a fixed curve and a family of lines are not obtainable we can have a fixed line and family of curves. TVe bvrite Equation 23 in the form QI‘
=
t,
=
t
+ X’R[S(tp.$),
tp] =
QII’
,
.”
0
.75
.50
1.0
1.25
0 1.5
(24)
and the right-hand side is now a family of curves. For any one of these curves it is useful to examine the points a t which the slope is unity. These satisfy Ri
.25
+ X’R? = 1 2.4
where This equation must be solved for t, and substituted in the righthand side of Equation 24 to give one or more points in the plane for each x . The :locus of such points will uniquely characterize a useful property of the family of curves, and since the family depends on the inlet conditions: a parametric study of these does not fill the space with a family of families but merely \vith a family of characterizing loci. The usefulness of such a locus is the fact that if it does not intersect line QI’, a unique profile is possible, whereas if it does there will be some points at which the ultimate profile will depend on the initial state. Since this indeterminateness is undesirable, the characteristic loci will show Inow it can be avoided. ,Moreover, the transient analysis is no inore difficult than with the s i q l e s t of reactions? so that a complete study of this kind may again be made. This topic properly belongs to the extensions of this theory which are currently being explored, but requires this much discussion here to justify the assertion that the elementary treatment we have given does indeed present the characteristic features of the general case. I t becomes evident also that in determining the character of the steady states an analJisis of the transient state may be less tedious and even more straightforward on a computer.
I
U=6.25 BTU/hr./sq. f t . / O F .
pe
-_-_-PARTICLE
=0.13 atm. tpi=134O0R ,1 = 5 6 0 O R
-GAS
X, REACTOR LENGTH,FEET
Figure 16. Steady-state temperature and partial pressure profiles for various influent temperatures Unstable case VOL.
2
NO. 1
FEBRUARY 1963
19
Conclusions
The problem of stability of a nonadiabatic fixed bed reactor has been considered. It is difficult to analyze this problem graphically by considering the steady-state equations only, as was done in the adiabatic case, because one cannot obtain the relation between t and p analytically or even very easily numerically. HoLvever, the envelope of the straight lines defining the steady states can be calculated from steady-state equations. Since the transient equations for the particle are the same as in the adiabatic case, the conditions for the local stability which appeared in the previous paper still can be applied to the nonadiabatic case. The transient equations \yere solved numerically by the method of characteristics and the effects of certain parameters upon the steady-state concentration and temperature profiles were examined. The results of calculations showed that for some sets of parameters the profiles are sensitive not only to a small change in inlet conditions but also to a small change in the initial particle temperature. The model used in this paper is an elementary one and undoubtedly incorrect in detail; however, its gross features must be correct and more sophisticated models incorporating diffusion, radiation, etc.: will probably not change the structure appreciably, although this is being investigated. The main result of this paper is to make explicit the possibility of the erratic behavior of fixed bed reactors in which the model assumed would apply. Since conclusions of a definitive nature are difficult to obtain in nonlinear problems, no advice or counsel can be presented here. Extensive numerical calculations might lead to empirical generalizations, but at the moment it is felt that each case must be investigated on its own. Acknowledgment
The authors are indebted to the Computing Center of the University of Minnesota and to the National Science Foundation for support. Nomenclature a
A, B a1
= radius of catalyst particle, inches = chemical components = total surface area of particles per unit volume of
h/
= = = = =
H,
=
c/ 6,
d
G
20
bed, sq. ft.;'cu. ft. specific heat of gas mixture, B.t.u./lb.jo F. specific heat of particle, B.t.u./lb./" F. average length of pores, A. fluid mass flow rate? Ib./hr./'sq. ft. film heat transfer coefficient, B.t.u./hr./sq. ft.1' F.
G HTU for mass transfer, ft. = MPa __ k,
l&EC FUNDAMENTALS
Hh
GCf = H T U for heat transfer, ft. = -
AH
= heat of reaction, B.t.u./lb. mole = mass transfer coefficient, mole/hr./atm./sq. ft. = chemical reaction rate coefficient, mole,/hr./atm.
4
k
M
P P
P, P e
P'
pDi
QI, Q I r
r Ti
Rh
s, t t,
4 t, tpi
tile tz U
L-T X
2,
ff
B Y
ah1
sq. ft. molecular weight of gas mixture, lb. lib. mole total pressure, atm. partial pressure of component A in gas phase, atm. partial pressure ofcomponent A inside particle, atm. influent partial pressure of component A in gas phase: atm. = partial pressure of ith species in interstitial gas = partial pressure of ith species in intraparticle gas = abbreviations for Equation 7 = average radius of pores, A = reaction rate based on particular species = radius of reactor. = surface area per unit mass of particle, sq. ft.0lb.; p,S, = 2 a:r = gas temperature, R. = particle temperature, O R. = influent gas temperature, O R. = initial interstitial gas temperature. O R. = initial particle temperature, O R. = t, coordinate of envelope, O R. = ambient temperature, ' R. = average interstitial velocity. ft./ min. = reactor-wall over-all heat transfer coefficient, B.t.u.,'hr.,'sq.ft. o F. = axial variable = = = = =
= x/H, = Q I coordinate of envelope = void fraction of particle = (-AH)=(ko,h/) = fractional void volume of bed
e
=
Pi
= density of gas mixture, lb.;'cu. ft.
time variable
x
= z p J ~ ~ ( - W3hf ,
2
= ordinate in Q - t, diagram = density of intraparticle gas, lb.;cu. ft.
pfU
density of particle, lb. )cu. ft. k, ~ p 8 S o3 ,k,,
P B
=
6
= ap,S,k.'3
6i
=
I*
=
0
=
Hh/H, 2 L7;'Rta,hj
literature Cited
(1) Barkelew, C., Chem. Eng. Progr. SymF. Ser. No. 25. 5 5 , 37 (1959). (2) Bilous, O., Amundson, N. R., A.I.Ch.E. J . 2, 117 (1956) 17) Liu. S.-1.. Amundson. N. R.. IND.ENG.CHEM.FUNDAMENTALS 2, 200 (1962). (4) Taylor, .A. E., "Advanced Calculus," p. 393. Ginn and Co., Piew York, 1955. \-,
RECEIVED for review December 3, 1962 .ACCEPTED December 13, 1962
