Ind. Eng. Chem. Fundam. 1980, 19, 396-404
396
0 = variable used in algorithm A (see Figure 5)
Kardos, P. W.; Stevens, W. F. AIChE J . 1971, 77(5).1090. Liu, S.L.; Amundson, N. R. Ind. Eng. Chem. Fundam. 1962, 1 200. Luyben, W. L. "Process Modeling, Simulation and Control for Chemical Enoineers". McGraw-Hill: New York. 1973. ChaDter 11. 377. Miche'son. M. L.: Vakil. H. B.:. Foss.. A. S. Ind. €no'. Chem.'Fundam. 1973. 12, 323. Pavlica, R. T., Ph.D. Thesis, University of Delaware, 1970. Sorenson, J. P. Chem. Eng. Sci. 1977, 32, 763. Strangeland. B. E.; Foss, A. S. Ind. Eng. Chem. Fundam. 1970, 9 , 30. Vakil, B.; Michelson, M. L.; Foss, A. S. Ind. Eng. Chem. Fundam. 1973, .-H.--IZ,BZU.
= characteristic time defined by t - $;ldz/u TD -= time required for the temperature wave to traverse
7
distance of
0, . ._
I
L i t e r a t u r e Cited Amundson, N. R. Ind. Eng. Chem. 1956, 48, 26. Anzelius, A. Z . Angew. Math. Mech. 1926, 6 , 291. Bilous, 0.; Amundson, N. R. AIChE J . 1956, 2 , 117. Boreskov, G. K.; Slinko, M. G. Pure ADP/. Chem. 1965, 10, 611. Brinkley. S.R., Jr. J. Appl. Phys. 1947, 18, 582. Crider, J. E.; Foss, A. S. AIChE J. 1966, 72, 514. Crider, J. E.; Foss, A. S. AIChE J. 1968, 74, 77. Georgakis, C.;Aris, R.; Amundson, N. R. Chem. Eng. Sci. 1977a, 32, 1359. Georgokis, C.;Aris, R.; Arnundson, N. R. Chem. Eng. Sci. 1977b, 32, 1371. Georaakis. C.: Aris. R.: Arnundson. N. R. Chem. Ena. Sci. 1977c. 32. 1381. Hoibgrg, J. A., Ph.D. Thesis, University of California, Berkeley, 1969.
Received for review July 12, 1979 Accepted August 12, 1980
The major part of this work was supported by a grant from the Mobil Oil Co.
Novel Integral Heat Balance Approach for the Prediction of Temperature Peaks in Adiabatic Reactors Gerhard K. Glger,' Rajakkannu Mutharasan, and Donald R. Coughanowr +
Department of Chemical Engineering, Drexel University, Philadelphia, Pennsylvania 19 104
When a temperature decrease in the feed gas to an adiabatic packed bed reactor occurs, a temperature wave is initiated which travels through the reactor with increasing temperature peaks. A novel analytical approach is presented, which is based on an integral heat balance, to predict the maximum deviation in temperature from the steady-state value. Such an approach would be useful in estimating temperature excursions in practical industrial reactors. Furthermore, it would provide a rational basis for the design of a control system to prevent excessive temperature excursions. Examples of a hydrodealkylation reactor and a carbon monoxide oxidation reactor are presented in which actual temperature peaks obtained through dynamic simulations are compared with the predicted values, using the proposed analytical procedure.
Introduction
When an adiabatic catalytic fixed-bed reactor is subjected to a step decrease in inlet gas temperature, a temperature wave is initiated which traverses the reactor a t a very low velocity (far slower than the material velocity). During the transient state of the reactor, temperature peaks occur which are in magnitude several times the inlet temperature decrease. Such excursions may cause deactivation of the catalyst, change in selectivity of the catalyst, and even structural damage to the reactor. In general, to obtain a numerical value for the temperature excursion, one would have to solve the partial differential equations which describe the dynamics of the reactor (Giger et al., 1980). A procedure is developed in this paper to predict the magnitude of the temperature excursions without having to solve the dynamic equations. In our earlier paper (Giger et al., 1980) a control strategy was described which reduces the high temperature peaks in a hydrodealkylation reactor. One of the design characteristics of the control algorithm which was presented is the rate of change of the setpoint temperature (at a location within the reactor), which must be lower than some predetermined rate obtained through the response of the reactor subjected to ramp change in inlet temperature. In order to determine the maximum allowable rate of change of setpoint, a number of simulations for different rates of change of feed temperature are needed. Such a
procedure is tedious. It is therefore desirable to have a simplified procedure for the estimation of the resulting maximum temperature for a given ramp change in feed temperature. The procedure for predicting the magnitude of temperature excursions due to step change in feed temperature is extended to the case of a ramp change in temperature. Reactor Model
Consider a two-phase plug-flow tubular reactor with no intraphase temperature gradients. All reactions are assumed to take place at the surface of the catalyst particles. The dynamics can be described by the following partial differential equations (Giger e t al., 1980).
a(fuCi) + a(fU I= R i ( C , P ) i = 1, 2 , 3, ..., n, (3) a% at The boundary conditions are given by z = 0 , t 1 0: 2
= 0 , t L 0: 2
Ciba-Geigy, Basel, Switzerland. 0196-4313/80/1019-0396$01,00/0
0 1980 American
= ToG
ci = CiO (i = 1, 2, 3, ..., n,)
L 0 , t = 0:
Chemical Society
Ta = TBB"%)
Ind. Eng. Chern. Fundam., Vol. 19, No. 4, 1980
397
I
O
2.t: CISTANCE ALONG T-IE REACTOR, Z
Figure 2. Simplified temperature profile when heat transfer resistance is neglected.
1 . 1 01 02 03 04 05 06 07 CB 09 10
00
3IMENSIOVLESS DISTANCE, p
velocity a t which the location z * ( t ) moves through the reactor is given by
Figure 1. Transient temperature profiles in the CO oxidation reactor for a step decrease of 25 K in inlet gas temperature.
(4) Table I.
CO Oxidation Reactor Parameters
L = 0.762 m v = 0.4 m/s f = 0.4 U,8= 1.091 X l o 5 W/(m3 K ) C p G = 2.872 X l o 4 J/(kmol K ) C = 2.628 x lo-* kmol/m3 $sCps= 1.544 X l o 6 J / ( K m 3 ) ToG= 473.2 K k = 2.896 x l o 3 s-l E = 3.887 x l o 7 J/kmol A H r = - 2.829 X l o 8 J / k m o l ?= k exp(-E/RTS)C kmol/(s m3) C, = 7.88 x kmol/m3
A typical response to a step decrease in inlet temperature for a CO oxidation reactor is shown in Figure 1. The transients were calculated using the procedure outlined by Giger e t al. (1980). The reactor parameters are given in Table I. Integral Heat Balance for Step Changes in Inlet Feed Gas Temperature A typical temperature profile during the transient of the reactor after a step decrease in inlet gas temperature has taken place (see, for example, in Figure 1 the curve a t t = 120 min) reaches the initial steady state (ISS) asymptotically for large reactor distance. This behavior can be explained by the fact that the thermal capacity of the solid phase is much greater than that of the reactant gases. Furthermore, the concentration profiles show similar characteristics except for the overshoot. Hence, considering the reactor as an overall system, the only change that would be observed during the transient state is the decrease in the feed temperature. Since the outlet gas temperature (and concentration) remain constant at the initial steady state value for a significant time period following the depression in feed gas temperature, the catalyst bed during this period of time continuously loses heat to the flowing gas medium. Consider the reactor a t some time t after a depression in inlet gas temperature has taken place. The section of the reactor up to certain distance from the inlet, z * ( t ) ,is cooled down to the final steady state (FSS) temperature. It is obvious that a larger and larger part of the reactor will cool down to the final steady state value as time progresses. When heat transfer resistance is neglected, the
where
UOBL u,= CC,Gfv
A similar expression was reported by Crider and Foss (1966) for a liquid-phase packed-bed reactor where the thermal capacity of the flowing medium is comparable to that of the packing. The expression for wave velocity given by Crider and Foss reduces to eq 4 for a gas-phase reactor. In the presence of heat transfer resistance, eq 4 is a good approximation for the wave velocity (see Appendix B). In order to develop an expression for the prediction of the wave peak, consider initially the simplified temperature profile shown in Figure 2 in which the vertical wave front has reached a location z = z*(t) a t time t. The profile in Figure 2 is based on the assumption that the resistance to heat transfer is negligible. In a later stage of development, this assumption will be replaced by a more realistic one. The location of the wave front, z *(t),in Figure 2 is given by
z * ( t ) = v,t
(5)
From Figure 1, it is seen that after a decrease in feed gas temperature, the outlet temperature remains a t the initial state until the thermal wave breaks through some time later. Based on this observation, the decrease in energy input to the reactor which results from a decrease in feed gas temperature is given by
E,(t) = AT&cfvCCPGt
(6)
where AT, is the magnitude of the temperature drop in feed gas a t time zero. Since the temperature of the gas leaving the reactor a t time t has not changed, the decrease in energy to the reactor, E,(t), must be accounted for by a loss of heat from the contents of the reactor to give a temperature profile such as one of those shown in Figure 1 at a specific value of t. Replacing t in eq 6 by z * / V , from eq 5 and using eq 4 to eliminate V , gives an alternative expression for E l in terms of z*
398
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
t as a result of a decrease in feed temperature is represented by area abe'ha in Figure 3, which is the same as area A in Figure 2. As an approximation, the difference between the decrease in energy to the reactor as a result of a decrease in feed gas temperature and the loss of energy of the catalyst in cooling is proportional to the area he'fgh (labeled area B ) in Figure 3. This difference in energy is transported by the gas stream to produce the overshoot which is centered near location z*. As shown in Figure 3, the overshoot is approximated by a triangular shape having a height which is labeled T,,, centered a t z* and having a width of d(z*). Based on this geometry for the overshoot, area B should be equal to area C which is enclosed by triangle gcdeg in Figure 3. Area C can be expressed by area C(z*) = area cegc DISTANCE ALONG THE REACTOR, Z Figure 3. Transient temperature profile with heat transfer resistance included.
The heat lost by the bed up to location z*(t) for the simplified profile in Figure 2 can be expressed as
Referring to Figure 2, the area A lying between the initial steady state (ISS) and the dotted line (which is parallel to ISS and offset below by ATo)represents El. Due to the temperature dependence of the reaction rate, less heat is produced a t lower temperature and the final steady state (FSS) is below the dotted line as shown in Figure 2. The entire area between ISS and FSS is proportional to the heat given up by the cooling of the catalyst between 0 and z*. The difference between E,(t) and E,(t), which is represented by area B in Figure 2, is given by Ez(z*(t))- El(Z*(t))=
+ area cedc
(loa)
Since the triangles cegc and cedc have the same base, ce, and the same height of d/2, we may simplify the above to area C(z*) = 2larea of cedc)
(lob)
If we make the assumption that the ISS and FSS profiles are parallel in the vicinity of z * , one can show that the point e is the midpoint of the line bf. Hence the area C can be written as the sum of area cdb and area bde, which have the same height of d / z ; thus area Ck*) =
The above equation can be rewritten as
The area B(z*) is given by
A c ~ s C C , " ~ z * ( t ) ( T--I sTFSs(z*) s ( ~ * ) - AT,) dz (9) The difference in heat expressed by eq 9 is transported by the gas stream to produce a peak which is centered around the location z*. The peak is not shown in Figure 2 since a refinement in temperature profile will not be considered. T o refine the initial assumption that the temperature profile is a flat wave front at location z* as shown in Figure 2, we now take the more realistic case for which the heat transfer resistance is finite. This means that the transition from the FSS to the ISS profile takes place over a finite distance. This transition distance is a strong function of the heat transfer coefficient. For the present analysis, it is reasonable to assume that the transition from FSS to ISS (labeled as gd in Figure 3 ) takes place linearly over a distance, d(z*), given by
The above relationship is derived from a transient analysis of the reactor neglecting the heat effects of the reaction. Details of the derivation are given in Appendix A. Referring to Figure 3, the energy corresponding to the area adegha is the heat lost by the bed and can be shown from geometrical considerations to be equal to area abefgha when the ISS and FSS profiles are parallel in the vicinity of z*. As was the case for the simple profile of Figure 2, the decrease in energy to the reactor up to time
Again, when d(z*) is small, the above integral can be written as
From an earlier discussion, areas B(z*)and C(z*) must be equal. Hence we can solve for the maximum temperature. Thus
Tmax(z*)= TIS&*)
AT0
-
-+ 2
For small changes in feed gas temperature, the ratio of overshoot in temperature to change in inlet temperature can be expressed as
Taking the limit as AT0 further simplified to
-
0, the above expression can be
-~-------
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
(14)
i
I5C
1~
399
a LIUMERICAL SiMUL A T I O h C A L C U L A T k D W T H THE PROPOSED PROCEOURE
~
where @(z*) is the ratio of overshoot to the inlet temperature change and the term dTss(z)/dTo represents the changes in steady-state temperature a t location z for changes in inlet temperature. The physical significance of such a term is the sensitivity of the reactor to feed temperature changes. In terms of dimensionless reactor length, the above equation simplifies to
50I
I
*/2
I
~
380 i --
L--_-L-.-t
04
5IVEW0NLLESS DISTANCE,
where
08
06
0
7
Figure 4. Calculated and predicted relative temperature overshoot for the CO oxidation reactor for a step decrease of 25 K.
conditions, T’ (g = 0) = 0 and C’ (g = 0) = 0.5, one can obtain the steady-state temperature profile as T o illustrate the procedure discussed here, two examples of a carbon monoxide oxidation reactor and a hydrodealkylation reactor are treated. Example 1. CO-Oxidation Reactor. Assuming only one reaction is taking place (CO oxidation with air over a cupric oxide catalyst) and that the gas flow velocity is constant over the entire reactor length, the original model eq 1,2, and 3 reduce to the following under steady-state conditions
Tsi(g) = where q = 0.5ay
+ 8.
T,(g) =
ff -(egq
4
- 1)
(25)
In terms of the original variables
To + To%(eqq- 1)
(26)
The sensitivity of the temperature profile to changes in inlet temperature is obtained as
Substitution of the above in eq 15 gives the relative temperature overshoot function, $(?*I Note that a t steady state T” N TG = T. At the dilute CO concentrations considered here, complete conversion of carbon monoxide in the reactor takes place. Let us define the following dimensionless deviation variables
Linearization of the steady-state equations given by eq 16 and 17 about T’ = 0, C’ = 0 gives
dC’--_ P + P C ’ + - Pr T’ _ dg
2
2
(21)
where L(l a= /j=
-- f ) k ( - - A H ) C o exp(-E/RT,,)
CCPCufTo
(22)
L ( l - f ) k exp(-E/RTo) _.___
L’f
y = E/RTo
(23) (24)
Application of’ the Laplace transform with the boundary
(1
-
t)
-
(&)”’:( $)I 1-
(28)
The above expression is plotted as a function of dimensionless distance, g*, in Figure 4 for the parameters given in Table I. Since the approximation for the imaginary Bessel function used in the Appendix to calculate d(g*) is only valid for large distances, g*, eq 28 gives unrealistic values for the temperature overshoot near the inlet of the reactor. Therefore these results are neglected and the function +(g*) is shown only for large values of the variable g*. In any case, the maximum overshoot occurs in the exit section of the reactor and eq 28 is useful in evaluating the maximum overshoot. Example 2. The second example is a hydrodealkylation reactor and parameters for this reactor, which are taken from Pavlica (1970), are summarized in Table 11. The first reaction is a typical hydrocracking reaction which is a very fast reaction which essentially goes to completion in the entrance section of the reactor. The second reaction is the
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
400
I 1
where
A
/I
A NUMERICAL SIMdLATION
1
CALCULATED WITH THE PROPOSED PROCEDURE
IOPI
= -L(1
P2 = -LO DIMENSIONLESS DISTANCE, 7
Figure 5. Calculated and predicted relative temperature overshoot for the hydrodealkylation reactor for a step decrease of 27.8 K.
exp(-El/RTo)/(fu)
(38)
- f)kzCH, exp(-E2/RTo)/(fu)
(39)
- f)klC10°.5
Y1 = E,/RTo
(40)
Yz = W R T O
(41)
The relative overshoot function is then found to be
dd = -0
I
*
I
t?
l
l
l
!
l
2 tl 4 4
ri
j
, ri/ 4
/
0
Q, Q, at, Z p I
1
I t;a
*
I-
TIME
& DISTbhCE ALONG THE REACT0R.Z
Figure 6. Transient temperature profiles for a ramp change in inlet gas temperature.
hydrodealkylation reaction of toluene with only about 30 70 conversion. The steady-state equations for the reactor are dT_ dv
where (43)
P1
cz = P2 + ff2Yz
s 3=
where T" TG = T. Use of the following deviation dimensionless variables
=
c1
ffl(1- P2/Pl)PlYl PI
s 4
- PZ
-
(44)
(47)
ff2Y2
=
((01- CulYZ)/(P2 + ffzYz) ff2 P1
- NYz)(P2
+ a272 - 2%)
- Pz - ff2Yz
(48) (33)
c2/ = cz - czo czo ~
(34)
and the subsequent linearization and solution yields the following steady-state temperature profile
The overshoot predicted by the above equation is compared with the numerical solution in Figure 5. The predicted relative temperature excursion agrees over the entire reactor length with the theoretical result within about 20% accuracy. It should be pointed out that the magnitude of the temperature overshoot in the present reactor is much smaller than in the CO reactor, which follows from the fact that steady-state temperature profile in the CO reactor is much more temperature dependent (see Figure 1). On the other hand, the steady-state temperature profiles in the hydrodealkylation reactor are almost parallel to each other (see Giger et al., 1980),indicating less severe temperature dependence.
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
401
Table 11. Parameters of Hydrodealkylation Reaction
L = 7.01 m
-
C = 0.794 kmol/m3 CH? = 0.564 kinol/m3 ZsC - 4.02 x l o 6 J / ( K m 3 ) To2=s67 K
v = 0.585 m / s f = 0.4 U,a= 1.005 x l o 6 W / ( m 3 K) COG= 6.237 x l o 4 J / ( k mol K ) reaction 1
ki Ei AHri
Ti C io
3.629 X l o 2 * 3.604 X l o 8 -9.202 X l o 7 k , exp(-E,/RTS)C,'.s 9.337 x 10-3
reaction 2 1.112 x 103 2.523 X l o 8 -1.394 X l o 8 k , exp(-E2/RTS)CH20.s C, 4.004 X 10.'
Integral Heat Balance Analysis for a Ramp Change When the feed gas temperature does not change in a step manner but decreases continuously from one inlet temperature to another, an integral heat balance analysis similar to the earlier development can be made to obtain estimates of temperature peaks. Consider the reactor system at some time, t , after the ramp change in the inlet temperature has begun. It is assumed that the continuous change in inlet temperature may be considered to consist of an infinite number of step changes, as shown in Figure 6. The resulting transient response consists of an infinite number of step responses. Consider the total ramp change, AToto be equivalent to n step changes, each of magnitude ATo/n and occurring a t the time, ti. The total change in the inlet temperature takes place over t , time units. The location of the transition part of the profile zi*(t), corresponding to each step change, travels along the reactor with the velocity given by eq 4. The subscript i is used t o indicate the transient due to the ith step change. Corresponding to the step change a t ti,the location of the transition distance is given by
Zi*(t) = V,(t
-
ti) =
( );
v,
t
--
(49)
T h e above equation holds true for large values of n. For the purpose of clear presentation, n is taken to be 4 in Figure 6; however, the analysis presented here is based upon a very large value for n. The midpoint of the transition profile from initial steady state to final steady state is given by
The decrease in energy to the reactor from the time 0 to t is
for n
-
OD,
the above can be written as
Combining eq 4 and 50 with eq 52 gives another expression for El in terms of z*.
E 2 ( t )=
5 ~ z ' * ( t ) [ T I ~ sTF~&)]AgsCpS i(z) dz
i=l
0
-
(53)
(54)
which simplifies to
E2(t) = & z * ( f ) { T ~ s -s (T~F) ~ ~ ( z ) ] A J 'dz C ~ S(55)
y3)
1 r--
A MJMERICAL SIMULATION
W I T A T H E PROPOSED PROCEDURE
rlL.03
:k
,STEP CHAhGE)
100-
02 04 06 DIMENSIONLESS DISTANCE, 7
08
Figure 7. Calculated and predicted relative temperature overshoot for the CO oxidation reactor for a change of 25 K in inlet gas temperature.
As discussed previously, the difference between El and E2 is accounted for by a temperature peak centered around z* and spread over a distance d'(z*). The distance d'(z*) is larger than the corresponding distance for the case of step change. For Figure 6, one gets Li
Li
where r represents the length of the reactor over which the transition from one steady state to the other takes place as the result of the ramp change. That is r = zl*(t) - z,*(t) = V,t, (57) Also as an approximation, eq 56 can be written as d ( z * ) = d(z*) r (58) where d(z*) is the transition distance at the location z * ( t ) for a step change in feed temperature given by eq 10. Applying an area analysis similar to the previous procedure gives 2L7*
+
4J(7*) =
E l ( z * ( t ) )= AT&iPCpSz*(t) The heat lost by the catalyst bed is given by
units ml.'/(s kmol)O.s J/kmol J/kmol kmol/(s kmol/m
-
d(.rl*) + r
+
The function, 4J(s*), is plotted for different values of r / L in Figures 7 and 8 for the CO reactor and hydrodealkylation reactor, respectively. In the same graphs, the results from numerical procedure are also plotted. It is apparent that the proposed analytical procedure predicts the temperature peaks with reasonable accuracy. It is important to point out that the magnitude of the temperature overshoot is maximum a t the end of the reactor and the procedure discussed in this paper is able to predict the maximum deviation with acceptable accuracy.
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
402
t L."I
I
I A
NUMERICAL SIMULATION
a:
CALCULATED WITH THE PROPOSED PROCEDURE
a W
~
I:
15t
STEADY STATE
L
t-
Z
0 z w
,,
r
INITIAL STEADY STATE
7'
TIME,
Figure 9. Transient temperature profile in a heat regenerator a t the location q* = z*/L.
DIMENSIONLESS DISTANCE,
and introducing a characteristic time T and a dimensionless distance q , eq A-1 and A-2 may be simplified to
4
Figure 8. Calculated and predicted relative temperature overshoot for the hydrodealkylation reactor for a ramp change of 27.8 K in inlet gas temperature.
As seen in Figures 7 and 8, the relative temperature overshoot can be reduced drastically even for small r / L values. That is, the reactor response is sensitive to step changes in feed gas temperature, while for slow changes in inlet temperature, the peak is decreased to a large extent. For example, in the case for the hydrodealkylation reactor when r / L = 0.2 (which is the case where the ramp change is over a time period equal to the time taken by the temperature wave to traverse 20% of the reactor length), the height of the temperature peak is half of the peak corresponding to the step change. Conclusion In this paper an approximate procedure has been developed to predict temperature peaks which occur in adiabatic packed-bed reactors subjected to both step and ramp changes in temperature of the reactant gases. The procedure predicts within about 20% error the magnitude of the temperature overshoot for the two reactors considered. The results clearly show that the temperature overshoot is greatly reduced when the inlet gas temperature changes as a ramp function, which leads to the inference that inclusion of inert packing ahead of the reactor to introduce thermal inertia should be very effective in reducing temperature overshoots. In a subsequent paper, details of such a design will be presented. Appendix A In order to obtain the expression for d(z*) in eq 10, consider the dynamics of a packed bed heat regenerator, for which the following differential equations describe the process
1
fCCpG U- a:
j :a
+-
= U&(P - F )
(A-2)
The associated boundary conditions are given by F = Tocfor z = 0, t I 0 (A-3) ?B = Tosfor z I 0, t = 0
(A-7)
where
UohL u, = CCpGuf
7
r = t - -
z u
v = zz Use of the Laplace transform to solve eq A-7 and A-8 gives ea(q,7)
= Use-u~7~7e-u8xIo(2( UgUsqx)1/2) dx (A-9)
where Io is the modified Bessel function of first kind of zero order (Mickley et al., 1957, p 173). In Figure 9 is shown P ( q , T ) as a function of T for a given distance v* where q* = z*
/L
As shown previously, t* = z*/V, with V , = LUs/U,. In terms of the new variables, r and q, r* = t* - z * / u or r* = Lq* J V, - Lq*/u. In Figure 9 the value of r* occurs at the point of inflection where the rate of change of temperature is maximum. A reasonable approximation for the time required for the transient to pass the location q* is given by Td in Figure 9 and is determined by
(A-4)
By defining dimensionless temperatures Os = (P- T,")/ ( ToG- Tos)
Taking the derivative of the right-hand side of eq A-9 gives (-4-5)
0' = (F - T,")/(ToG- T,")
(A4
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
403
sumption that the heat transfer resistance between solid and gas is zero. For a reactor having a cross-sectional area of unity, eq B-1 may be written
uCpGCfATot = C,”ps(l
-
~)ATOZ*+ CpGCfAToZ*
(B-2)
Solving this equation for z*/t, which is the wave velocity V,, gives
where
Es = p s ( l - f).
Equation B-3 may be written
(B-4)
REACTOR L ENLT-, Z
Figure 10, Temperature profile in a heat regenerator caused by a step change in inlet gas temperature.
By use of the following approximation, which holds true for large values of the argument (Mickley et al., 1957)
I&) =
ex
An expression similar to this, which is called /3, is given by Crider and Foss (1966). Since the ratio of the thermal capacity of the solid to that of the gas is much greater than 1, eq B-4 simplifies to
(A-12)
____ (2RX)’/2
and use of the fact that V ,
