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Temperature differences between phases in a moving bed reactor
Chef&o/ Engineering Science. Vol. 42, No. Printed in Great Britain.
5, pp. 1175-l
185. 1987. Q
TEMPERATURE
DIFFERENCES MOVING BED
BETWEEN REACTOR
000%2509/87 1987 Pergamon
PHASES
S3.00 + 0.00 Journals Lid.
IN A
J. WEI, R. CWIKLINSKI, J. TOMURO and J. XIAO Department of Chemical Engineering, MIT, Cambridge, MA 02139, U.S.A.
21 January 1986; accepted 30 May 1986)
(Received
Abstract-Pseudo-homogeneous models of packed bed reactors assume equal temperatures and concentrations (or chemical potentials) for the solid and the fluid phases and are simpler than heterogeneous models. An analysis is presented for the degree of temperature departure between these two models under plug flow conditions with no axial dispersion. Fixed bed, cocurrent and countercurrent flow reactors are considered. The analysis yields two important parameters: a, the ratio of solid to gas thermal capacitances, and /3, which is closely related to the number of interphase heat transferance units. In most industrial reactors, where /3 is greater than 50, the average temperature d@erence between phases is small, except for countercurrent reactors where gas and solid heat capacitances are nearly equal. Within this range of rx, temperature differences can persist through the reactor, even with large values of 8. The mnximum temperature difference between phases is attained when the reaction heat effect is released in the worst case of a localized pulse in the reactor stream with the smaller thermal capacitance. These temperature difference measures can be used to estimate the validity of a nseudo-homoeeneous model. This analysis is easily extended to concentration differences between-phases.
1. INTRODLJCI-ION
of fixed bed reactors have been divided into two broad categories by Froment [3]: pseudohomogeneous and heterogeneous models. While a hetModels
erogeneous phase
model
transport,
explicitly a
accounts
for
pseudo-homogeneous
the
inter-
model
as-
sumes negligible resistance to heat and mass transfer so that the fluid and solid temperatures at the same reactor location are equal. The basic model (A. 1 or B. 1) is one-dimensional and assumes plug flow for the phases; and uniform temperature and concentrations in the reactor’s radial direction. Model A.1 uses one temperature and one fluid composition to describe the conditions at any point along the axis of the reactor. Model B.l distinguishes between the temperature and concentrations in the fluid and the solid. The purpose of this study is to identify conditions for which the pseudo-homogeneous simplification is sufficient. Carberry [2] has reviewed the intraparticle and interphase temperature differences for fixed bed reactors. The classical Prater temperature, AT_ = (-AH)D
I
@,--cm) k
I
estimates an internal temperature difference between the surface and center of a reacting particle. Carberry and Luss have separately produced estimates of the external temperature difference between the solid surface and the bulk fluid. In Carberry’s derivation: r(q W 1 +qDa(r-
AT= Ar,
1)’
AT” is the difference between the center temperature of the particle and the bulk fluid temperature, and A TX is the
external
temperature
and
Carberry/Luss
Prater internal
and
external
difference. can
temperature
The
provide
efforts
of
estimates
of
differences
for
the
solid particle in a fixed bed reactor. Presumably, if the two estimated temperature differences are small, a pseudo-homogeneous description would be valid for a fixed bed reactor. However these analyses do not apply to moving bed reactors where the solid phase flows. For this case the Prater analysis can still indicate the solid internal temperature difference because only internal parameters are involved. Carberry and Luss both derived their estimate of the external temperature difference by using a heat balance across a fluid film around the particle: AT, This
is correct
=
4,(--H) I
ha
for fixed bed reactors,
stream flows, this equation additional term, the enthalpy
should
’ but when also
the solid
include
an
conveyed by the solid. Thus their temperature difference estimate is not applicable to moving bed reactors. The analysis here assumes both phases move under plug flow conditions with no axial dispersion. Both cocurrent and countercurrent reactors are considered with the fixed bed reactor as an asymptotic limit. The two temperature difference measures are: (1) an average temperature difference throughout the entire reactor and (2) the maximum temperature difference in a heat input zone. These measures indicate the persistence of a temperature difference between phases in the reactor and the intensity of a locally large temperature difference. They depend on the values of two dimensionless variables: 0, which is related to the number of interphase heat transference units and a, which is the ratio of the heat capacitances of the two phases, as well as the intensity and distribution of heat input. The analysis is kinetics-free, and the solution method uses Green’s functions to treat any arbitrary axial distribution of reaction heat generation (or absorption). The
1175
1176
J. WEI et al.
worst case is encountered when the entire heat output is released at a single point as a delta function, so that the temperature differences between the phases wilI be at a maximum. The temperature difference measures are examined for their sensitivity to the distribution of the reaction heat input as well as the values of the characteristic parameters u and /?. The measures indicate when a pseudo-homogeneous model is valid, and the degree of departure in a given condition. 2. TWO-PHASE
MOVING
BED
REACTORS
AND
GREEN’S
FUNCTION
This section presents a mathematical description of an indealized two-phase moving bed reactor which corresponds to model B.l in the classification of Froment [3]. The two streams are assumed to move under plug flow conditions without axial dispersion. The mass fluxes and heat capacities for each stream are assumed constant throughout the reactor, as are the interphase heat transfer coefficient and surface area. There is no heat exchange with the exterior as the reactor is adiabatic. A general differential enthalpy balance can be written for each stream flowing through the reactor.
dz/(C, + ICZ I), the ratio ofthe totai reaction heat effect to the total flow of thermal capacitance. Thus the dimensionless temperatures are related to the temperatures by: 7i = TJAT,. The length of the reactor (H) was used to make the position variable dimensionless: 5 = z/H. The ratio of stream thermal capacitances, u (= CZ /C, ), is positive for cocurrent, zero for fixed bed, and negative for countercurrent reactors. The parameter /l is a dimensionless ratio of interphase-heat transfer to the total flow of thermal capacitances and equals haH/(C, + IC, I). The reaction heat effect beH comes 2 = qH q dz, a dimensionless axial distri/s 0 I bution of the total heat input. The integral A (0 di s0 equals 1. The parameters (a, /3) characterize the interphase heat transfer behavior in a reactor. Their values are generally not constant within actual reactors, and average values should be used in this analysis. Table 1 lists data available in the general literature for a number of industrially important reactors [4-121, and Fig. 1 gives the (a,& values. Equation (2) can be expressed in vector notation as:
c, ~=ha(T,--7,)+(1-_y(z))q(z)
d
(1) C*Z=
-ha(T,-7-T,)+y(z)q(z).
a$?
= -p(l
The
temperatures
a characteristic
+lal)(72
-
were
7,)+
(I+
made
temperature
(2) lal)r(tJ~(i).
dimensionless
71 I72 I
(3)
=,(l+,.l)I-~_~~/::/t/~:(
or alternatively,
r, and r, are the stream temperatures. C, and C, are the thermal capacitances (defined as UPC,) for each stream. Stream 1 is the fluid phase chosen to flow in the positive z direction, and C, is therefore positive. Stream 2 is the solid phase that can flow in either direction, so that C, can be positive for a cocurrent reactor, zero for a fixed bed reactor, and negative for a countercurrent reactor. The first term on the R.H.S. of eqs (1) and (2) describes the local rate of heat transfer between streams, assumed proportional to the local temperature difference. The parameter (h) is the convective heat transfer coefficient and the parameter (a) is the interfacial surface area per unit reactor volume. The second term on the R.H.S. of the equations provides for heat generation within each stream. The local rate of heat generation per unit reactor volume is q(z), while the local fraction injected into stream 2 is y(z) and the local fraction injected into stream 1 is 1 - y. This analysis can accommodate arbitrary distributions for the heat input. In dimensionless form eqs (3) and (4) become: dr --‘=8(~+l~l)(~~-~~~+~~+l~l~(~-~((r~~~(i~ dS
di
with H increase, AT, = 4 (2) s Cl
*(i)
= MT(<)+@(<)
where a(i)
= (I+
Ial) 1 ;(; (i) Acc-). I-/ L1
(4)
The matrix M is singular, with eigenvalues of zero and - /3(1 + Iu () (1 + a)/~. The description of the system is completed by specifying the boundary conditions. For the countercurrent reactor the boundary conditions are split: E, which equals
rr (0) =
710
and
7‘2
(1) =
721 _
(5)
For cocurrent reactors the boundary conditions TV (0) = rlo and r2 (0) = 720. The solution to eq. (3) is given by
are (6)
i‘= 1 r(i)
=
’ G(5,i’)~(i’)dS’-G(i,i’)r(i’) s0
<‘=O
(7)
where G is a matrix of Green’s functions. G is defined as gu(&<‘) for i,j = i,2 such that --(I-&,+MT)
GT=
IB(<-(-‘)
and eq. (7) satisfies the specified boundary conditions. Note that (-a/a<’ - MT) is the adjoint differentia1 operator of the original differential system, and 6 (c - <‘) is the Dirac delta function. The elements of G represent the temperature profiles produced when the components of the forcing function, 4, are delta function heat inputs. Specifically,
Phases in a moving bed
1177
Table I. Process parameters for industrial reactors Reactor type
h
a
Btu
[ ft” h “F (J/M’sK) Countercurrent TCC (countercurrent section) Blast furnace Oil shale retort Lurgi dry ash coal gasifier Slagging coal gasifier Fixed bed Petroleum reformer Automotive catalytic converter Laboratory scale Cocurrent TCC (cocurrent section) FCC (riser section)
1
[ft-‘1 (m-l)
85 (22.9) 260 (79.2) 10 (3.0)
50 (280)
100 (330)
10 (3.0)
140 (790)
900 (2900)
7 (2.1)
20 (110)
800 (2600)
0.25
(0.076)
0 (0)
0.09 (1800)
60 (340)
900 (2900)
0.25
(0.076)
0 (0)
0.2 (4000)
30 (170)
300 (980)
4600
(1500)
REACTORS
a<
a=0
cocurr.n+ a-0
(1800)
0.01
(200)
0.04 (800) 0.05 (1000) 0.04 (800)
0.04 0.13 0.35
(800) (2600) (7200)
0.12
0.26
(5300)
0.09
(2450)
1.3 (26000)
0 (0)
18 (5.5)
0.09 (1800)
0.006
50 (15.2)
0.37 (7600)
0.16
(120)
(3200)
point < produced by a delta heat input multiplied by CI into stream 2 (the solid) at c’. The G T 1; term of eq. (7) explicitly incorporates the boundary conditions. The construction procedure for the Green’s functions implicitly includes an effect of the boundary conditions
c51.
El
@]
q
kzl
IOO-
Cl
50-
Q 2or
(J/M’sK)
70 (230) 20 (70) 100 (330)
Co”nt%rcurre”+
200-D
0
(J/M*sK)
30 (170) 30 (170) 70 (400)
Fired Bed
500-
f
[&I
18 (5.5)
I
1000
[&I
300 (980)
PHASE
0
Cl (gas)
40 (230)
200 (1100)
TWO
(m)
C2 (solid)
c
10
LI -THERMAL
CAPAClTANCE
RATIO
Fig. 1. Values of E and p for industrial reactors: 1, Lurgi dry ash coal gasifier [14]; 2, Lurgi slagging coal gasifier [14]; 3, Thermofor Catalytic Cracker (TCC~ountercurrent section of regenerator kiln [ 111; 4, TCC-ocurrent section of regenerator kiln [lo]: 4, blast furnace [lo]; 6, oil shale retortAirect heated [S]; 7, fluidized catalytic cracker-riser cracking [4,6]; 8, automobile catalytic converter [13]; 9, petroleum reformer [4,9]. gr t (c, [‘) and qrI (<, c’) are the temperature profiles of gas and solid streams at point < resulting from a delta function heat input into stream 1 (the gas) at c’, and g2* and grz are the solid and gas temperature profiles at
Equation (7) divides the temperature profiles into two components: (1) a contribution from the heat input 4, and (2) the contribution from the boundary conditions (G r 1:). When I is a distributed function, the first contribution is a linear combination of the Green’s functions. Tables 2 and 3 present the Green’s functions for the cocurrent and countercurrent reactor cases along with the corresponding boundary condition terms, G r 1:. The Green’s functions change at <‘, the location of the delta function heat input. The components of G for 0 < < -C <’ have a superscript “L” to indicate “left” or gas stream before heat source. For <’ < c -C 1, the components of G have the superscript “R” to indicate right or gas stream after heat source. The cocurrent Green’s functions have the property gf; = 0 so that the temperatures of both streams before the heat sources are unaffected by the heat input. At < = c’, g:r and g:2 jump to the value of one due to the delta function heat inputs. Enthalpy balance requires that gT1 (l,<‘)+a&, s:,(Li’)+~gr&(l,i’)
(kc’)
= 1 = a-
1178
.f.
WE1
Table 2. Green’s functions for the cocurrent reactor
1 _eE(i-_(‘) sf1 (Li’)
= 0
SF, (i, i’) =
l+a a+eE(c--i’)
&i(i,i’)
= 0
$75 (L-7i’) =
l+a
boundary conditions term:
The cocurrent
Green’s
function
have the property
s41(0, i’) = s4, (035’) = 0 s:, (I, i’) = s;, (1, i’) = 0 so that the inlet temperatures are not affected internal heat input. At c = [‘, 9:1 (iPi)
by the
et al.
The decay parameter E governs the behavior of the Green’ function. It is useful to define here the characteristic decay length cd = 11/E 1. In dimensional variables, E = - huH(l/C1 + l/C,). For both cocurrent and countercurrent reactions, E is directly proportional to the extent of interphase heat transfer. For a cocurrent reactor, E is negative. For a countercurrent reactor, C, is negative, so that the sign of E depends on whether Ci is greater than IC2 I. When Cr = [C,l, or when the two streams have the same heat capacitance, E vanishes and temperature differences do not decay. See Fig. 2. Both/l and E are similar but not identical to the notion of number-of-heat-transfer-units (NTU) from heat exchanger analysis [6]. The NTU of a reactor is haH&,+,,, where C,, is the smaller of C, or C2_ The NTU of a reactor approaches 1E 1when Ia 1is far from 1. A large magnitude for fE I indicates quick damping of temperature differences between phases that originate from heat inputs or boundary conditions, but does not ensure against large local temperature differences. The Green’s functions are not only functions of /I, the dimensionless heat transfer coeficient, but also of a, the thermal capacitance ratio. The mathematical limit as a approaches zero (either from + co or - co) corresponds to a fixed bed reactor. In the fixed bed limit, C,+Oanda-+OsothatE-+ -co.Thesecondhalfof eq. (2) reduces to 0 = -ha(T, -Tl)+yq, or T2 = Tl + y q/ha. Substitution yields
= 841 (L i) + 1
C+yq+(l-_y)q=q
s:z (i, 5) = s:z (I, i) + 1. The jumps are due to delta Enthalpy balance requires:
function
heat inputs.
s:i (1, i’) + la/ s:i (0, c’) = 1 9~2(l,i’)+Iar94,(0,i’)
Tl (z) =
= (q/C, ) dz’ + T,, s0
T,(z) =
= (q/C, ) dz’ + Tlo + ?q(z)lha. s0
= lat.
Table 3. Green’s functions for the countercurrent reactor
d-1(i, i’) =
a(1
gt2Gc’) = -
-eEl)
e-EC’_e-E
SPl(i,i’) =
1 +aeeE
1+Or
eeEc+
a(1 -eEC)
aeeE
gP2K,iT
1 + aeeE
l+a
I=
a+eEC
sh(i;i’) = -- l+a
e-Ei’+ae-E
g~22(i,i,)
l+a l+aeE((-1)
l+a
1-eEE(T-l)
R
g21 (L i’) =
=
1
I
-e-El’)
1 +aewE
l+ae-Ec’
1 +aevE
l-;~(~-l)
a(l-epEi’) 1 +aeeE
boundary conditions term: O=
1 +aepE a(1
l+a
l+aepE
GT
l+aemEi’
l+aeEK-1)
Tzle- E l+aeeE
a(eEc-(a+eEC)
1) I
rlOeFE l+ae-E
/eE+aeEcl eE-eEc
1179
Phases in a moving bed COUNTERCURRENT
COCURRENT
IOPC,
)
.Ol
/=-
I/ . .
J_
n ---,
?_--_
I
IO
0. I
100 Q
E
4s
Fig. 2. Values of E and id, the temperature difference decay parameter and decay length, with a for countercurrent and cocurrent reactors. Gas temperature at a point z depends only on the integral of total heat input from gas entrance to z and on the inlet temperature; and solid temperature at z is r,(z) plus the effect of local heat input to the solid, yq(z)/ha. In dimensionless form: i c i dz’
‘51(i) = and 52
=
=1
l+YUP
which agrees with the criterion of Carberry
3.
TWO
MEASUXES
OF THE
TEMPERATURE
G+tW(&-F). Solid temperature would be driven to be higher than gas temperature when the fraction of heat go into solid divided by solid heat capacitance (y/C,) is greater than the fraction of heat going into gas divided by gas heat capacitance ((1 - y)/C,). In a countercurrent reactor, the temperature difference is given by
and Luss.
INTERPHASE
DIFFERENCE
The temperature difference between phases at any position can be obtained from eq. (7) directly: 72-71
For a cocurrent
=
(-
1,1).7(i).
reactor, this expression
5 72-71
=
i
-(l-~(c’))e~(~-~‘)
+
Y (c;‘) -
~K’)(l+lall 0
reduces to:
(
>
a
d<‘+(t,,--r,,)eEc.
(9)
The second term on the R.H.S. contains boundary condition information. The exponential decay term is governed by E, the negative eigenvalue of the matrix M. In regions of zero heat input, any temperature difference decays exponentially with a characteristic decay length 5, = 11/E 1.Values of (a, p) that yield large magnitudes of E and short decay length would favor the pseudo-homogeneous assumption in a detailed reactor model. The first term on the R.H.S. ofeq. (9) is a convolution integral of a function of the heat distributions A([‘) with an exponential decay term governed by E. The factor (1 + 1a-1) [y (<')/a - (1 - y (<'))I describes how the heat input affects the temperature difference between streams. In dimensional terms, this factor becomes:
e-E
l+aemE
(l+a)(z,,--rlo)eEc.
(10)
The first term refers to heat input before the measurement point, the second term refers to heat input after the measurement point, and the third term contains the boundary condition. Two measures of the temperature difference between streams will be examined. The first is the average absolute temperature difference (ATD) over the entire length of the reactor. ATD
=
i
O’lfz--Tk
t di.
(11)
Note that this measure is dimensionless. To obtain the actual average temperature, multiply the ATD with the AT,, the characteristic temperature increase specific to the actual reaction system. Large values of this measure reflect low magnitudes of E (or correspondingly large cr values) which allow temperature differences between the phases to persist within the reactor.
J. WEI et al.
I180
Another dimensionless measure temperature difference (MTD). MTD
is the maximum
= max (tZ - tl ]
(12)
Its evaluation requires locating the maximum temperature difference. The MTD measure indicates the existence of large localized temperature differences, which may quickly decay within the reactor and therefore has little affect on the ATD. The maximum temperature difference may occur at a reactor boundary because of specified stream input conditions. Restricting the MTD search to regions of the reactor where L(c) is nonzero usually separates this boundary condition effect from the MTD measure. The maximum temperature difference can be estimated for an actual reactor system by multiplying the MTD with AT,, the appropriate characteristic temperature increase of the reaction system. 4. COCURRENT
REACTOR
We give first the computation results of a particular reactor where the heat input function 2 ([‘) is a square function, with the value of 50 between the limits 0.49 and 0.51, and the value of zero elsewhere. The inlet conditions are 71,, = 2.222 and TV,, = 1.333, and both streams flow to the right. To exaggerate the temperature differences, let fi = 1. Let y = 1, so that a11the heat is injected into the second stream. Three temperatures profiles are given in Fig. 3 for the o! values of 10, 1 and 0.1. The upper profile corresponds to OL= 10; so that the thermal capacitance of stream 2 is ten times that of stream 1. Right of the inlet, the stream temperatures converge because of the interphase heat transfer. Stream 1 has a smaller thermal capacitance which causes it to undergo the larger temperature change. Since all heat is injected in stream 2, the value of rZ rises a
=I0
E
=-IF.!.1
I
ri
64-
E=-4
0=1
2 =-‘I___ 0 ,r2,
1
,
I
I
I
I
J
r2 _______--1
,
=I
]
,
,
J
sharply at the heating zone (0.49 -C [ -C 0.51). This same pattern for the converging temperatures is repeated in the profile right of the heat input region. The middle profile corresponds to 0: = 1; the streams having equal thermal capacitances. For this value of a, the temperature difference decay coefficient, E, has its smallest magnitude. The rate at which the temperatures of the two streams converge is therefore smallest when a equals 1. The lower profile of Fig. 3 corresponds to an 0:value of 0.1, where the thermal capacitance of stream 2 is one tenth that ofstream 1. In the heat-transfer-only regions of the reactor (where i = 0), the temperatures of the streams converge and stream 2 undergoes the larger temperature change because of its smaller thermal capacitance. The entire reaction heat effect is inserted into stream 2 for all three profiles. The functional form of the heat input is not a delta function but is quite intense, delivered within the central two percent of the reactor’s length. The low value of 8, equal to 1, hardly moderates the temperature rise of stream 2 in the heat input zone. The change in the temperature difference between streams therefore reflects the influence of the point distribution factor (1 + 01)[v/cl - (1 - y)]. The reaction heat effect contributes most to the temperature difference between streams when it is injected into the stream of smaller thermal capacitance. Hence the largest temperature difference between the streams is observed in the a = 0.1 profile where stream 2, receiving the entire heat input, has the smaller thermal capacitance. Analytical expressions can be given for the temperature difference from eq. (9). We concentrate here on three special cases: (a) Pulse heat release at a point [ *, so that ,I (<‘) = 6 (<’ - <*) a delta function. This would be the worst case where the temperature difference would be at the maximum. (b) Square heat release so A([‘) = &for
c* - A < <’ < <* + A and = Oelsewhere.
(c) Uniform everywhere.
heat
IO E
(a) a
= .I
E =-12.1
so
that
n(c’)
and
1 +Or =-----e CL
7*-~~
E(C-PI
for
= 0
n
(b)
~~-7~
1+ol
=pp
u
2
c*
= 1.333.
l+cr 1 a 2AE
(13)
i -=Zr*
1 2AE
[
Ce
E(i-P+A)_eEti-i*-A)
for c*+A<<
Fig. 3. Dimensionless temperature profiles for a cocurrent reactor; a values of 10,1, and 0.1; conditions of b = 1, y = 1, 2 = 50 for A = 0.01, and <’ = 0.5. Inlet 710 = 2.222, 520
= I
When A -+ 0, case (b) becomes case (a); when A + 0.5, case (b) becomes case (c). When y = 1, the heat release terms are:
14 12
release
1 (14)
[eE(C--C*+A)_ 11 for <*
-A
(15)
Phases in a moving bed
(c)
72-7,
l+a
=
yE(eEi-‘).
1
1181
1.6 1.4
The boundary term (720 - slo)e EC should be added to the eqs (13k( 15) to obtain the entire r2 -t1. The average difference, temperature ATD
= s
0’ IT,-T,ldL
second term from (~20 here:
contains boundary
two
conditions
terms.
The
1 is E (eE-
1.0 c3
.6
1)
The first term from heat release E is given
71o).
1.2
2
.4 .8 . .2
(a) ATD
= q
;
(eE(l-i*)
_ 1)
(16)
0
0. I a
(b) ATD
= !$?
&
(eE(l--i*+A)
_e.%--*--)_2~~) (c) ATD=+$
(17)
(eE-1-E).
0.2 THERMAL
0.5
I 2 CAPACITANCE
5 RATIO
IO
Fig. 4. Variation of the average absolute temperature difference (ATD) with changes in TVfor a cocurrent reactor. Parametric values of B. Conditions of Fig. 3. Crosses (+) indicate the values for the fixed bed asymptotic limits (a = 0), l/8.
(IV
maximum temperature difference, MTD = 172 occurs either at the entrance [ = 0, exit c = 1, or at the point of heat release. Computations shows that from the heat release term, we have:
the characteristic temperature rise. ATD is not sensitive to A, the half width of the heating zone, or <*, the center position of the heating zone. The value of E has a minimum value of 4/3at a = 1, and rises to 12.18 at a = 0.1 or 10. When jl= 50, the value of IE I is 200 or greater, so that ATD for all cases reduce to [ (1 + a)/a]
(a) MTD
= e
(-- l/E).
(b) MTD
= e
&
(c) MTD
= F
i
The -
71 Lax
at <=<*
(19
(e2AE - 1) (eE - 1)
at < = <* + A (20)
at < = 1.
(21)
The boundary term (boo - 710) eET has a maximum of at c = 0. For a fixed bed, a is zero and E goes 720 -*IO to negative infinity. The values of z2 - 71, ATD, and MTD for cases (a) (b) and (c) are given in Table 4. The ATD is sensitive to the values of a and /3,which governs the value of E, see Fig. 4. The fixed bed limit for ATD is l/p, so that for most industrial reactors with b greater than 50, we have ATD less than 0.02 or 2 y0 of
The value of MTD for A = 0.01 is given in Fig. 5. The fixed bed limit for MTD = I/2A/?, which is very sensitive to the value of A but not to the value of <*. In the limit of no interphase heat transfer, j? = 0, all MTD becomes (1 + a)/a. Even when fl= 50, MTD can be exceedingly large. 5. COUNTERCURRENT REAaORS We give first the results of a countercurrent reactor with an intense exothermic reaction where the heat input is a rectangular distribution, where A is constant
Table 4. Temperature differences for a fixed bed for different input heat distributions ATD
52-T1 (a) pulse (b)
1
=ooat[=[* = 0 elsewhere 1 = __ b-2A
MTD
s
*
at 1
1
B
2AB
1
1
1
B
B
B
square c*-A
a -
(c) uniform
TtlERMAL
CAPACITANCE
RATIO
Fig. 5. Variation of the maximum temperature difference (MTD) with changes in IXfor a cocurrent reactor. Parametric values of /3_ Conditions of Fig. 3. Crosses (+) indicate the values for the fixed bed asymptotic limits (c( = 0) + 50/b.
1182
J. WEI et al.
in the interval 0.49 < 5 < 0.51 and zero elsewhere. 2 = 1 is assumed, indicating that stream 2 receives the entire reaction heat effect. Outside this interval onIy heat transferoccurs between the streams. The inlet conditions are: stream 1 enters from the left with 7 10 = 2.222 and stream 2 enters from the right with 7 21 = 1.333. A very low value of fi = 1 was selected to the temperature difference between exaggerate streams. Fig. 6 presents the temperature profiles for three values of u. The upper profile of Fig. 6 corresponds to a = - 10, or the thermal capacitance of solid stream 2 is ten times that of gaseous stream 1. The disparity in size of the thermal capacitances causes two effects. The larger stream (stream 2) leaves the reactor with most of the heat of reaction. The second effect is that the temperature difference decays rapidly along the reactor length in those regions where 2 is 0. Note also there that stream 1, the stream of smaller thermal capacitance, experiences the larger temperature change. The middle temperature profile is for a = - 1, the case of countercurrent streams of equal thermal capacitance. In the regions of the reactor where 2 equals 0, the temperature profiles are linear and the temperature difference between streams is constant. Recall that for o! = - 1, both eigenvalues of the system matrix of eq. (3) are zero and the associated Green’s functions have not an exponential but a linear dependence on 5. When two streams have equal thermal capacitances, any heat transferred between them produces the same magnitude of temperature change in each stream. In a countercurrent system this implies a constant temperature difference wherever il = 0. The heat transfer rate between streams is proportional to this constant temperature difference. Consequently the temperature profiles are linear with the same slope. For countercurrent reactors with a near - 1, any temperature dif4 Ti
a =-IO
r2
2 0
E=
-9.9 7-2
=I
I
217
I
a =-I
I
I
IT21
t
I
1
I
1-21
I
I
I
E=O
ference produced by a reaction heat effect will persist the reactor. Employing the pseudothrough homogeneous approach in a detailed model would be highly inaccurate in these cases. The lower temperature profile of Fig. 6 corresponds to an u of - 0.1, the thermal capacitance of solid stream 2 is one tenth that of as gaseous stream 1. As before, most of the reaction heat leaves with the stream of larger thermal capacitance, stream 1 in this case. In the regions of the reactor where i = 0, the temperature difference decays along the reactor length while stream 2, the stream of smaller thermal capacitance, undergoes the larger temperature change. The most prominent feature of the u = - 0.1 temperature profile is the large temperature increase for stream 2 in the heat input zone (0.49 < 5 ,( 0.51). The reaction heat input is not a delta function but is concentrated within the central two percent of the reactor length. The value of fl is too small to effectively moderate the temperature difference between streams within the reaction zone. The change in temperature difference is therefore close to the value of the point distribution temperature difference factor. The physical significance of the a = -0.1 profile is that when any reactor heat input is injected into the stream of smaller thermal capacitance, it contributes most to the temperature differences between streams. Here the value of y is 1. Stream 2 receives the entire heat input and because of its small thermal capacitance, it undergoes large temperature increase in the heat input region. Finally, note the effect of a on the location of the maximum temperature difference. At a = - 10, the MTD occurs at the upper end of the heat input zone, < = 0.51. However, at a = - 1, the temperature difference at c = 0.49 has grown and equals that at [ = 0.51. At a = - 0.1, clearly the MTD is at c = 0.49, the lower edge of the heat input zone. The location of the MTD depends on the specific functional forms of the heat distribution functions. However, near a = - 1, there is a characteristic shift in MTD location from one extreme end of the heat input zone to the other. For the case of y = 1, the temperature difference can be derived from eq. (10) as:
T2
4 TI -_;,
2
01
I
I
I
I
t2-7,
=
I
5 (&i-i’)
_
eE’)
dc’
0 L
-
AK’)
(I+
Iat)
1 $
s 5
X
1
_eE(I+c--i’)+eEC a
e -e + 0
.I
.2
.3
5 DIMENSIONLESS
.4
.5
.6
.7
REACTOR
.0
.9
1.0
POSITION
Fig. 6. Dimensionless temperature profiles for a countercurrent reactor; c( values of - 10, - 1 and - 0.1: conditions of fi = 1, y = 1, 1 = 50 for A = 0.02 and c* = 0.5. Inlet rl,, = 2.222, 721 = 1.333.
d<’ >
1 +orewE
(1 +a)
(TZ1 -71,)e”i.
(22)
The first term gives the contributions of heat sources to the left of the measure point, the second term gives the contribution of heat sources to the right of the measure point, and the third term gives the boundary condition. For a pulse heat input at i*,
Phases in a moving bed
T*-71
or
=
eCE
(l+lal)
- (1 + fal) I :fE for
The value of ATD ATD
= (1+
(@(I-S*)
1 + ccc-s
(
_ &)
~eE(l+i-r*)+eE~
1183
(23)
>
[ < c*.
is
lal) 1 :ie_E
k (l+
l/a) (eE(‘-i’)-
eE) (24)
-10
and MTD=
-(l+(crl)
e--E l+olepE
(AeE+eEi’) c2
(25)
-2 -5 a -THERMAL
-I
-0.5
-0.2
CAPACITANCE
-0.1
RATIO
Fig. 8. Variation of the average absolute temperature difference (ATD) with changes in a for a countercurrent reactor. Five parametric values of A, the heat input interval. Conditions of Fig. 6 except B = 50.
01
Figure 7 shows the value of ATD for the square function heat input as functions of a and /3.When #?is greater than 50, ATD is generally below l/p except in the neighborhood of CL= - 1. The value of E vanishes in this neighborhood, so that temperature differences persist throughout the reactor without decay. In this unusual neighborhood, ATD is also sensitive to the values of A and [*, see Figs 8 and 9. The value of MTD is particularly sensitive to the values of fl for small values of Icrl, see Fig. 10. When heat is injected into the stream with the smaller heat capacity, and when interphase heat transfer is slow, we should anticipate local “hot spots”. The dry ash Lurgi coal gasifier has a = - 0.1 and p = 50, so that the value of E = - 495. The combustion heat effect would give rise to a characteristic temperature increase of 650K. The decay length cd = 0.002, which amounts to 0.002 ft or 0.24 in. in a 10 ft reactor.
1.6
z*=
1.0
.5
.9
.8 .7 .6 .5 .4 .3
I
t
-10
‘! -5 -2 a -THERMAL
-I
-0.5
-0.2
CAPACITANCE
-
0.1
RATIO
Fig. 9. Variation of the ATD with a for a countercurrent reactor. Five parametric values of [*. Conditions of Fig. 6 except p = 50.
+
/ 8=0
1.4
/
+
.2 0 -10
%iERMAl_ a
-2
c--
-
&AC
- 0.5
ITANCE
y-“” I -a2
-0.1
RATIO
Fig. 7. Variation of the average absolute temperature difference (ATD) with changes in OL for a countercurrent reactor. Parametric values of /3.Conditions of Fig. 6. Crosses (+) indicate the values for the fixed bed asymptotic limits (a = O),
-10
-5
a-THERMAL
-2
-I - 0.5 CAPACITANCE
-0.2
-0.1 RATIO
Fig. 10. Variation of the maximum temperature difference (MTD) with changes in a for a countercurrent reactor. Parametric values of fi. Conditions of Fig. 6. Crosses (+) indicate the values for the fixed bed asymptotic limits (a = O), 50/b.
J. WEL et al.
1184
Therefore, except for the immediate neighborhood of the reactor zone, interphase temperature differences are negligible. The value of ATD is 13 K. The values of MTD are: A MTD
0.02
0.05
0.10
650 K
260 K
130K
Even for a combustion zone as narrow as A = 0.02 or 2.4in., the temperature difference will decline from 650 K to 10 K in a distance of 1 in. The effect of axial dispersion would significantly decrease the value of MTD. 6.
CONCLUSIONS
A kinetics-free analysis has been developed for the interphase heat transfer in two-phase moving bed reactors. The temperature difference between the streams in a reactor can be easily estimated from a few easily calculated parameters and any arbitrary heat generation function, including the worst case where all the heat is injected at a single point in the reactor as a delta function and into the stream with the smaller heat capacity. A heterogeneous model is necessary when temperature difference produced by a reaction heat effect persists throughout the reactor, and when the reaction heat input is so intense that a locally large temperature difference beteen streams is produced, commonly referred to as a “hot spot.” Two dimensionless temperature difference measures have been developed: LZ,the ratio of solid to gas thermal capacitance, and /?l,a dimensionless interphase heat transfer coefficient. The average temperature difference (ATD) is an integral of the absolute temperature difference between the streams over the entire reactor. Its magnitude indicates the tendency of a temperature difference to persist within the reactor. Temperature differences between streams are produced by the reaction heat effect and by the inlet stream temperatures. The maximum temperature difference (MTD) indicates the magnitude of the maximum difference between the stream temperatures within the reactor. The temperature differences are governed by the exponential decay coefficient E, and the reciprocal characteristic decay length, cd. A temperature difference between streams, produced by either the reaction heat effect or dictated by boundary conditions, decays very slowly along the reactor length when the magnitude of E is small. For countercurrent reactors, E has a very small magnitude (and c., is large) when both streams have nearly the same thermal capacitance. The worst possible situation occurs when a has the value of - 1, counterflow of streams of equal thermal capacitances. E is zero and a temperature difference between streams does not decay. For cocurrent reactors, E has a minimum magnitude when the value of CLis one. For most industrial reactors, /I is greater than 50. This ensures large magnitudes of E in most counterand all cocurrent reactors. The corresponding ATD is
small except for countercurrent reactors with -0.5 < CL-Z - 2. The maximum temperature difference can indicate the existence of large local temperature differences between the streams. It has a larger magnitude when the reaction heat effect is released in the stream of the smaller thermal capacitance. The homogeneous model for temperature of moving bed reactors is adequate for most industrial reactors where /?is greater than 50, except for countercurrent reactors where the heat capacitances of the two streams are nearly equal. If the heat release is very localized, the heterogeneous model for temperature should be used in the reaction zone. Axial dispersion of heat would decrease the value of MTD. The fixed bed is prone to large “hot spots” when heat is released into the stationary solid phase, and can be relieved by moving the solid phase. The analysis of this paper is easily extended to concentration differences between phases in a moving bed when we define u’ as the ratio of mass flow ratio of ( VP)ZI( VP)l, and p’ as the dimensionless mass transfer ratio kaH/( ( VP), + ( VP),). NOTATION
solid surface area per unit reactor volume, l/m thermal capacities of the gas and solid streams-product of mass flux and heat capacity, J/m2 K s molecular concentrations, at surface and particle center effective intraparticle diffusivity, m2/s Damkohler number, reaction rate/external mass transport rate coefficient for the exponential decay of the temperature difference between solid and gas -B(l+ lal) (l+ l/cc) matrix of Green’s function solid-gas heat transfer coefficient, J/m2 K s reactor length, m identity matrix effective particle thermal conductivity, J/MKs characteristic length of particle, m model matrix in + = M T + a(<) overall reaction rate, defined by Carberry
VI
gas and solid temperatures, K heat generation due to reaction per unit reactor volume, J/m3 s ratio of mass to thermal Biot numbers position variable along reactor length, m dimensionless half-length of reaction zone enthalpy change of reaction, J/mol the characteristic temperature increase, K
s H
q(z) dz/W, + IC, I)
0
particle’s internal temperature difference between surface and center, K
Phases in a moving bed overall temperature difference (AT + AT,), K external temperature difference, between particle’s surface and bulk gas, K boundary condition temperature difference; zIo -tzl for countercurrent, TIo - rzo for coourrent forcing function vector for t = Mz -t@(T) thermal capacitance ratio, C,/C, dimensionless heat transfer coefficient, haHl(C, + IC, I) fraction of heat effect directed to solid Dirac delta function dimensionless reactor position, z/H characteristic decay constant, 11/E 1 overall effectiveness factor, global reaction rate/kinetics controlled rate distribution heat reaction
is H
=
a(z)H
d.4 dz
0
dimensionless temperatures; made dimensionless by a characteristic temperaH ture increase: z1 = Ti qt.4 dzl(C, /s 0 + 1C, 1); rl-gas, T,-solid REFERENCES
[l]
American Petroleum Institute, 1966, Technical Data Book; Petroleum Refining, N.Y.
1185
c23 Carberry, J. J.. 1975, On the relative importance of external-internal temperature gradients in hetero-
geneous catalysis. Ind. Engng Chem. Fundam. 14, 12%131. c31 Froment, G. F., 1972, Analysis and design of fixed bed _.-. catalvtic reactors. Advanvces in Chemistry Seric__,Nn~ 109, A.C.S., Washington, D.C., pp. l-34.. c41 Gates. B. C., Katzer, J. R. and Schuit, G. C., 1979. Che&ry of.Catalytic Processes. McGraw-Hill, N.Y.’ IIs1 Greenberg, M. D., 1971. Applications of Green’s Functions in Science and Engineering. Prentice-Hall.
N.J.
I31 Kays, W. and London, A. L., 1955, Compact Heat
Exchangers, p. 15. McGraw-Hill, N.Y. 1969, Ind. Engng Chem. Fundam. 8, 596. ES1 Lewis, A. E. and Braun, R. L., 1981, Retorting and combustion processes in surface oil shale retorts. J. Energy 5, 355-361. Nelson, W. L., 1958, Perroleum Refinery Engineering. McGraw-Hill, N.Y. Perry, R. H. and Chilton, C. H., 1950, Chemical Enoineer’s Handbook. 3rd edn. McGraw-Hill, N.Y. &&er, C. D., Wei. J., Weekman, V. W. Jr. and Gross, B., 1983, A reaction engineering case history: coke burning in thermofor catalytic cracking reactors. Adu. Chem. Engng, Vol. 12. A&demic Press, N.Y. Cl21 Satterfield, C. N., 1980, Heterogeneous Carolysis in Practice. McGraw-Hill, N.Y. Cl31 Wei, J., 1975, Catalysis for motor vehicle emissions. Advances in Catalysis, Vol. 24, pp. 57-125. Academic Press, N.Y. Cl41 Yoon, H.. Wei, J. and Denn. M. M., 1977, Modeling and Analysis of Moving Bed Coal Gasjfiers. Electric Power Research Institute, Palo Alto, California, A5-590. c71 Lee, J. C. M. and Luss, D.,
5, pp. 1175-l
185. 1987. Q
TEMPERATURE
DIFFERENCES MOVING BED
BETWEEN REACTOR
000%2509/87 1987 Pergamon
PHASES
S3.00 + 0.00 Journals Lid.
IN A
J. WEI, R. CWIKLINSKI, J. TOMURO and J. XIAO Department of Chemical Engineering, MIT, Cambridge, MA 02139, U.S.A.
21 January 1986; accepted 30 May 1986)
(Received
Abstract-Pseudo-homogeneous models of packed bed reactors assume equal temperatures and concentrations (or chemical potentials) for the solid and the fluid phases and are simpler than heterogeneous models. An analysis is presented for the degree of temperature departure between these two models under plug flow conditions with no axial dispersion. Fixed bed, cocurrent and countercurrent flow reactors are considered. The analysis yields two important parameters: a, the ratio of solid to gas thermal capacitances, and /3, which is closely related to the number of interphase heat transferance units. In most industrial reactors, where /3 is greater than 50, the average temperature d@erence between phases is small, except for countercurrent reactors where gas and solid heat capacitances are nearly equal. Within this range of rx, temperature differences can persist through the reactor, even with large values of 8. The mnximum temperature difference between phases is attained when the reaction heat effect is released in the worst case of a localized pulse in the reactor stream with the smaller thermal capacitance. These temperature difference measures can be used to estimate the validity of a nseudo-homoeeneous model. This analysis is easily extended to concentration differences between-phases.
1. INTRODLJCI-ION
of fixed bed reactors have been divided into two broad categories by Froment [3]: pseudohomogeneous and heterogeneous models. While a hetModels
erogeneous phase
model
transport,
explicitly a
accounts
for
pseudo-homogeneous
the
inter-
model
as-
sumes negligible resistance to heat and mass transfer so that the fluid and solid temperatures at the same reactor location are equal. The basic model (A. 1 or B. 1) is one-dimensional and assumes plug flow for the phases; and uniform temperature and concentrations in the reactor’s radial direction. Model A.1 uses one temperature and one fluid composition to describe the conditions at any point along the axis of the reactor. Model B.l distinguishes between the temperature and concentrations in the fluid and the solid. The purpose of this study is to identify conditions for which the pseudo-homogeneous simplification is sufficient. Carberry [2] has reviewed the intraparticle and interphase temperature differences for fixed bed reactors. The classical Prater temperature, AT_ = (-AH)D
I
@,--cm) k
I
estimates an internal temperature difference between the surface and center of a reacting particle. Carberry and Luss have separately produced estimates of the external temperature difference between the solid surface and the bulk fluid. In Carberry’s derivation: r(q W 1 +qDa(r-
AT= Ar,
1)’
AT” is the difference between the center temperature of the particle and the bulk fluid temperature, and A TX is the
external
temperature
and
Carberry/Luss
Prater internal
and
external
difference. can
temperature
The
provide
efforts
of
estimates
of
differences
for
the
solid particle in a fixed bed reactor. Presumably, if the two estimated temperature differences are small, a pseudo-homogeneous description would be valid for a fixed bed reactor. However these analyses do not apply to moving bed reactors where the solid phase flows. For this case the Prater analysis can still indicate the solid internal temperature difference because only internal parameters are involved. Carberry and Luss both derived their estimate of the external temperature difference by using a heat balance across a fluid film around the particle: AT, This
is correct
=
4,(--H) I
ha
for fixed bed reactors,
stream flows, this equation additional term, the enthalpy
should
’ but when also
the solid
include
an
conveyed by the solid. Thus their temperature difference estimate is not applicable to moving bed reactors. The analysis here assumes both phases move under plug flow conditions with no axial dispersion. Both cocurrent and countercurrent reactors are considered with the fixed bed reactor as an asymptotic limit. The two temperature difference measures are: (1) an average temperature difference throughout the entire reactor and (2) the maximum temperature difference in a heat input zone. These measures indicate the persistence of a temperature difference between phases in the reactor and the intensity of a locally large temperature difference. They depend on the values of two dimensionless variables: 0, which is related to the number of interphase heat transference units and a, which is the ratio of the heat capacitances of the two phases, as well as the intensity and distribution of heat input. The analysis is kinetics-free, and the solution method uses Green’s functions to treat any arbitrary axial distribution of reaction heat generation (or absorption). The
1175
1176
J. WEI et al.
worst case is encountered when the entire heat output is released at a single point as a delta function, so that the temperature differences between the phases wilI be at a maximum. The temperature difference measures are examined for their sensitivity to the distribution of the reaction heat input as well as the values of the characteristic parameters u and /?. The measures indicate when a pseudo-homogeneous model is valid, and the degree of departure in a given condition. 2. TWO-PHASE
MOVING
BED
REACTORS
AND
GREEN’S
FUNCTION
This section presents a mathematical description of an indealized two-phase moving bed reactor which corresponds to model B.l in the classification of Froment [3]. The two streams are assumed to move under plug flow conditions without axial dispersion. The mass fluxes and heat capacities for each stream are assumed constant throughout the reactor, as are the interphase heat transfer coefficient and surface area. There is no heat exchange with the exterior as the reactor is adiabatic. A general differential enthalpy balance can be written for each stream flowing through the reactor.
dz/(C, + ICZ I), the ratio ofthe totai reaction heat effect to the total flow of thermal capacitance. Thus the dimensionless temperatures are related to the temperatures by: 7i = TJAT,. The length of the reactor (H) was used to make the position variable dimensionless: 5 = z/H. The ratio of stream thermal capacitances, u (= CZ /C, ), is positive for cocurrent, zero for fixed bed, and negative for countercurrent reactors. The parameter /l is a dimensionless ratio of interphase-heat transfer to the total flow of thermal capacitances and equals haH/(C, + IC, I). The reaction heat effect beH comes 2 = qH q dz, a dimensionless axial distri/s 0 I bution of the total heat input. The integral A (0 di s0 equals 1. The parameters (a, /3) characterize the interphase heat transfer behavior in a reactor. Their values are generally not constant within actual reactors, and average values should be used in this analysis. Table 1 lists data available in the general literature for a number of industrially important reactors [4-121, and Fig. 1 gives the (a,& values. Equation (2) can be expressed in vector notation as:
c, ~=ha(T,--7,)+(1-_y(z))q(z)
d
(1) C*Z=
-ha(T,-7-T,)+y(z)q(z).
a$?
= -p(l
The
temperatures
a characteristic
+lal)(72
-
were
7,)+
(I+
made
temperature
(2) lal)r(tJ~(i).
dimensionless
71 I72 I
(3)
=,(l+,.l)I-~_~~/::/t/~:(
or alternatively,
r, and r, are the stream temperatures. C, and C, are the thermal capacitances (defined as UPC,) for each stream. Stream 1 is the fluid phase chosen to flow in the positive z direction, and C, is therefore positive. Stream 2 is the solid phase that can flow in either direction, so that C, can be positive for a cocurrent reactor, zero for a fixed bed reactor, and negative for a countercurrent reactor. The first term on the R.H.S. of eqs (1) and (2) describes the local rate of heat transfer between streams, assumed proportional to the local temperature difference. The parameter (h) is the convective heat transfer coefficient and the parameter (a) is the interfacial surface area per unit reactor volume. The second term on the R.H.S. of the equations provides for heat generation within each stream. The local rate of heat generation per unit reactor volume is q(z), while the local fraction injected into stream 2 is y(z) and the local fraction injected into stream 1 is 1 - y. This analysis can accommodate arbitrary distributions for the heat input. In dimensionless form eqs (3) and (4) become: dr --‘=8(~+l~l)(~~-~~~+~~+l~l~(~-~((r~~~(i~ dS
di
with H increase, AT, = 4 (2) s Cl
*(i)
= MT(<)+@(<)
where a(i)
= (I+
Ial) 1 ;(; (i) Acc-). I-/ L1
(4)
The matrix M is singular, with eigenvalues of zero and - /3(1 + Iu () (1 + a)/~. The description of the system is completed by specifying the boundary conditions. For the countercurrent reactor the boundary conditions are split: E, which equals
rr (0) =
710
and
7‘2
(1) =
721 _
(5)
For cocurrent reactors the boundary conditions TV (0) = rlo and r2 (0) = 720. The solution to eq. (3) is given by
are (6)
i‘= 1 r(i)
=
’ G(5,i’)~(i’)dS’-G(i,i’)r(i’) s0
<‘=O
(7)
where G is a matrix of Green’s functions. G is defined as gu(&<‘) for i,j = i,2 such that --(I-&,+MT)
GT=
IB(<-(-‘)
and eq. (7) satisfies the specified boundary conditions. Note that (-a/a<’ - MT) is the adjoint differentia1 operator of the original differential system, and 6 (c - <‘) is the Dirac delta function. The elements of G represent the temperature profiles produced when the components of the forcing function, 4, are delta function heat inputs. Specifically,
Phases in a moving bed
1177
Table I. Process parameters for industrial reactors Reactor type
h
a
Btu
[ ft” h “F (J/M’sK) Countercurrent TCC (countercurrent section) Blast furnace Oil shale retort Lurgi dry ash coal gasifier Slagging coal gasifier Fixed bed Petroleum reformer Automotive catalytic converter Laboratory scale Cocurrent TCC (cocurrent section) FCC (riser section)
1
[ft-‘1 (m-l)
85 (22.9) 260 (79.2) 10 (3.0)
50 (280)
100 (330)
10 (3.0)
140 (790)
900 (2900)
7 (2.1)
20 (110)
800 (2600)
0.25
(0.076)
0 (0)
0.09 (1800)
60 (340)
900 (2900)
0.25
(0.076)
0 (0)
0.2 (4000)
30 (170)
300 (980)
4600
(1500)
REACTORS
a<
a=0
cocurr.n+ a-0
(1800)
0.01
(200)
0.04 (800) 0.05 (1000) 0.04 (800)
0.04 0.13 0.35
(800) (2600) (7200)
0.12
0.26
(5300)
0.09
(2450)
1.3 (26000)
0 (0)
18 (5.5)
0.09 (1800)
0.006
50 (15.2)
0.37 (7600)
0.16
(120)
(3200)
point < produced by a delta heat input multiplied by CI into stream 2 (the solid) at c’. The G T 1; term of eq. (7) explicitly incorporates the boundary conditions. The construction procedure for the Green’s functions implicitly includes an effect of the boundary conditions
c51.
El
@]
q
kzl
IOO-
Cl
50-
Q 2or
(J/M’sK)
70 (230) 20 (70) 100 (330)
Co”nt%rcurre”+
200-D
0
(J/M*sK)
30 (170) 30 (170) 70 (400)
Fired Bed
500-
f
[&I
18 (5.5)
I
1000
[&I
300 (980)
PHASE
0
Cl (gas)
40 (230)
200 (1100)
TWO
(m)
C2 (solid)
c
10
LI -THERMAL
CAPAClTANCE
RATIO
Fig. 1. Values of E and p for industrial reactors: 1, Lurgi dry ash coal gasifier [14]; 2, Lurgi slagging coal gasifier [14]; 3, Thermofor Catalytic Cracker (TCC~ountercurrent section of regenerator kiln [ 111; 4, TCC-ocurrent section of regenerator kiln [lo]: 4, blast furnace [lo]; 6, oil shale retortAirect heated [S]; 7, fluidized catalytic cracker-riser cracking [4,6]; 8, automobile catalytic converter [13]; 9, petroleum reformer [4,9]. gr t (c, [‘) and qrI (<, c’) are the temperature profiles of gas and solid streams at point < resulting from a delta function heat input into stream 1 (the gas) at c’, and g2* and grz are the solid and gas temperature profiles at
Equation (7) divides the temperature profiles into two components: (1) a contribution from the heat input 4, and (2) the contribution from the boundary conditions (G r 1:). When I is a distributed function, the first contribution is a linear combination of the Green’s functions. Tables 2 and 3 present the Green’s functions for the cocurrent and countercurrent reactor cases along with the corresponding boundary condition terms, G r 1:. The Green’s functions change at <‘, the location of the delta function heat input. The components of G for 0 < < -C <’ have a superscript “L” to indicate “left” or gas stream before heat source. For <’ < c -C 1, the components of G have the superscript “R” to indicate right or gas stream after heat source. The cocurrent Green’s functions have the property gf; = 0 so that the temperatures of both streams before the heat sources are unaffected by the heat input. At < = c’, g:r and g:2 jump to the value of one due to the delta function heat inputs. Enthalpy balance requires that gT1 (l,<‘)+a&, s:,(Li’)+~gr&(l,i’)
(kc’)
= 1 = a-
1178
.f.
WE1
Table 2. Green’s functions for the cocurrent reactor
1 _eE(i-_(‘) sf1 (Li’)
= 0
SF, (i, i’) =
l+a a+eE(c--i’)
&i(i,i’)
= 0
$75 (L-7i’) =
l+a
boundary conditions term:
The cocurrent
Green’s
function
have the property
s41(0, i’) = s4, (035’) = 0 s:, (I, i’) = s;, (1, i’) = 0 so that the inlet temperatures are not affected internal heat input. At c = [‘, 9:1 (iPi)
by the
et al.
The decay parameter E governs the behavior of the Green’ function. It is useful to define here the characteristic decay length cd = 11/E 1. In dimensional variables, E = - huH(l/C1 + l/C,). For both cocurrent and countercurrent reactions, E is directly proportional to the extent of interphase heat transfer. For a cocurrent reactor, E is negative. For a countercurrent reactor, C, is negative, so that the sign of E depends on whether Ci is greater than IC2 I. When Cr = [C,l, or when the two streams have the same heat capacitance, E vanishes and temperature differences do not decay. See Fig. 2. Both/l and E are similar but not identical to the notion of number-of-heat-transfer-units (NTU) from heat exchanger analysis [6]. The NTU of a reactor is haH&,+,,, where C,, is the smaller of C, or C2_ The NTU of a reactor approaches 1E 1when Ia 1is far from 1. A large magnitude for fE I indicates quick damping of temperature differences between phases that originate from heat inputs or boundary conditions, but does not ensure against large local temperature differences. The Green’s functions are not only functions of /I, the dimensionless heat transfer coeficient, but also of a, the thermal capacitance ratio. The mathematical limit as a approaches zero (either from + co or - co) corresponds to a fixed bed reactor. In the fixed bed limit, C,+Oanda-+OsothatE-+ -co.Thesecondhalfof eq. (2) reduces to 0 = -ha(T, -Tl)+yq, or T2 = Tl + y q/ha. Substitution yields
= 841 (L i) + 1
C+yq+(l-_y)q=q
s:z (i, 5) = s:z (I, i) + 1. The jumps are due to delta Enthalpy balance requires:
function
heat inputs.
s:i (1, i’) + la/ s:i (0, c’) = 1 9~2(l,i’)+Iar94,(0,i’)
Tl (z) =
= (q/C, ) dz’ + T,, s0
T,(z) =
= (q/C, ) dz’ + Tlo + ?q(z)lha. s0
= lat.
Table 3. Green’s functions for the countercurrent reactor
d-1(i, i’) =
a(1
gt2Gc’) = -
-eEl)
e-EC’_e-E
SPl(i,i’) =
1 +aeeE
1+Or
eeEc+
a(1 -eEC)
aeeE
gP2K,iT
1 + aeeE
l+a
I=
a+eEC
sh(i;i’) = -- l+a
e-Ei’+ae-E
g~22(i,i,)
l+a l+aeE((-1)
l+a
1-eEE(T-l)
R
g21 (L i’) =
=
1
I
-e-El’)
1 +aewE
l+ae-Ec’
1 +aevE
l-;~(~-l)
a(l-epEi’) 1 +aeeE
boundary conditions term: O=
1 +aepE a(1
l+a
l+aepE
GT
l+aemEi’
l+aeEK-1)
Tzle- E l+aeeE
a(eEc-(a+eEC)
1) I
rlOeFE l+ae-E
/eE+aeEcl eE-eEc
1179
Phases in a moving bed COUNTERCURRENT
COCURRENT
IOPC,
)
.Ol
/=-
I/ . .
J_
n ---,
?_--_
I
IO
0. I
100 Q
E
4s
Fig. 2. Values of E and id, the temperature difference decay parameter and decay length, with a for countercurrent and cocurrent reactors. Gas temperature at a point z depends only on the integral of total heat input from gas entrance to z and on the inlet temperature; and solid temperature at z is r,(z) plus the effect of local heat input to the solid, yq(z)/ha. In dimensionless form: i c i dz’
‘51(i) = and 52
=
=1
l+YUP
which agrees with the criterion of Carberry
3.
TWO
MEASUXES
OF THE
TEMPERATURE
G+tW(&-F). Solid temperature would be driven to be higher than gas temperature when the fraction of heat go into solid divided by solid heat capacitance (y/C,) is greater than the fraction of heat going into gas divided by gas heat capacitance ((1 - y)/C,). In a countercurrent reactor, the temperature difference is given by
and Luss.
INTERPHASE
DIFFERENCE
The temperature difference between phases at any position can be obtained from eq. (7) directly: 72-71
For a cocurrent
=
(-
1,1).7(i).
reactor, this expression
5 72-71
=
i
-(l-~(c’))e~(~-~‘)
+
Y (c;‘) -
~K’)(l+lall 0
reduces to:
(
>
a
d<‘+(t,,--r,,)eEc.
(9)
The second term on the R.H.S. contains boundary condition information. The exponential decay term is governed by E, the negative eigenvalue of the matrix M. In regions of zero heat input, any temperature difference decays exponentially with a characteristic decay length 5, = 11/E 1.Values of (a, p) that yield large magnitudes of E and short decay length would favor the pseudo-homogeneous assumption in a detailed reactor model. The first term on the R.H.S. ofeq. (9) is a convolution integral of a function of the heat distributions A([‘) with an exponential decay term governed by E. The factor (1 + 1a-1) [y (<')/a - (1 - y (<'))I describes how the heat input affects the temperature difference between streams. In dimensional terms, this factor becomes:
e-E
l+aemE
(l+a)(z,,--rlo)eEc.
(10)
The first term refers to heat input before the measurement point, the second term refers to heat input after the measurement point, and the third term contains the boundary condition. Two measures of the temperature difference between streams will be examined. The first is the average absolute temperature difference (ATD) over the entire length of the reactor. ATD
=
i
O’lfz--Tk
t di.
(11)
Note that this measure is dimensionless. To obtain the actual average temperature, multiply the ATD with the AT,, the characteristic temperature increase specific to the actual reaction system. Large values of this measure reflect low magnitudes of E (or correspondingly large cr values) which allow temperature differences between the phases to persist within the reactor.
J. WEI et al.
I180
Another dimensionless measure temperature difference (MTD). MTD
is the maximum
= max (tZ - tl ]
(12)
Its evaluation requires locating the maximum temperature difference. The MTD measure indicates the existence of large localized temperature differences, which may quickly decay within the reactor and therefore has little affect on the ATD. The maximum temperature difference may occur at a reactor boundary because of specified stream input conditions. Restricting the MTD search to regions of the reactor where L(c) is nonzero usually separates this boundary condition effect from the MTD measure. The maximum temperature difference can be estimated for an actual reactor system by multiplying the MTD with AT,, the appropriate characteristic temperature increase of the reaction system. 4. COCURRENT
REACTOR
We give first the computation results of a particular reactor where the heat input function 2 ([‘) is a square function, with the value of 50 between the limits 0.49 and 0.51, and the value of zero elsewhere. The inlet conditions are 71,, = 2.222 and TV,, = 1.333, and both streams flow to the right. To exaggerate the temperature differences, let fi = 1. Let y = 1, so that a11the heat is injected into the second stream. Three temperatures profiles are given in Fig. 3 for the o! values of 10, 1 and 0.1. The upper profile corresponds to OL= 10; so that the thermal capacitance of stream 2 is ten times that of stream 1. Right of the inlet, the stream temperatures converge because of the interphase heat transfer. Stream 1 has a smaller thermal capacitance which causes it to undergo the larger temperature change. Since all heat is injected in stream 2, the value of rZ rises a
=I0
E
=-IF.!.1
I
ri
64-
E=-4
0=1
2 =-‘I___ 0 ,r2,
1
,
I
I
I
I
J
r2 _______--1
,
=I
]
,
,
J
sharply at the heating zone (0.49 -C [ -C 0.51). This same pattern for the converging temperatures is repeated in the profile right of the heat input region. The middle profile corresponds to 0: = 1; the streams having equal thermal capacitances. For this value of a, the temperature difference decay coefficient, E, has its smallest magnitude. The rate at which the temperatures of the two streams converge is therefore smallest when a equals 1. The lower profile of Fig. 3 corresponds to an 0:value of 0.1, where the thermal capacitance of stream 2 is one tenth that ofstream 1. In the heat-transfer-only regions of the reactor (where i = 0), the temperatures of the streams converge and stream 2 undergoes the larger temperature change because of its smaller thermal capacitance. The entire reaction heat effect is inserted into stream 2 for all three profiles. The functional form of the heat input is not a delta function but is quite intense, delivered within the central two percent of the reactor’s length. The low value of 8, equal to 1, hardly moderates the temperature rise of stream 2 in the heat input zone. The change in the temperature difference between streams therefore reflects the influence of the point distribution factor (1 + 01)[v/cl - (1 - y)]. The reaction heat effect contributes most to the temperature difference between streams when it is injected into the stream of smaller thermal capacitance. Hence the largest temperature difference between the streams is observed in the a = 0.1 profile where stream 2, receiving the entire heat input, has the smaller thermal capacitance. Analytical expressions can be given for the temperature difference from eq. (9). We concentrate here on three special cases: (a) Pulse heat release at a point [ *, so that ,I (<‘) = 6 (<’ - <*) a delta function. This would be the worst case where the temperature difference would be at the maximum. (b) Square heat release so A([‘) = &for
c* - A < <’ < <* + A and = Oelsewhere.
(c) Uniform everywhere.
heat
IO E
(a) a
= .I
E =-12.1
so
that
n(c’)
and
1 +Or =-----e CL
7*-~~
E(C-PI
for
= 0
n
(b)
~~-7~
1+ol
=pp
u
2
c*
= 1.333.
l+cr 1 a 2AE
(13)
i -=Zr*
1 2AE
[
Ce
E(i-P+A)_eEti-i*-A)
for c*+A<<
Fig. 3. Dimensionless temperature profiles for a cocurrent reactor; a values of 10,1, and 0.1; conditions of b = 1, y = 1, 2 = 50 for A = 0.01, and <’ = 0.5. Inlet 710 = 2.222, 520
= I
When A -+ 0, case (b) becomes case (a); when A + 0.5, case (b) becomes case (c). When y = 1, the heat release terms are:
14 12
release
1 (14)
[eE(C--C*+A)_ 11 for <*
-A
(15)
Phases in a moving bed
(c)
72-7,
l+a
=
yE(eEi-‘).
1
1181
1.6 1.4
The boundary term (720 - slo)e EC should be added to the eqs (13k( 15) to obtain the entire r2 -t1. The average difference, temperature ATD
= s
0’ IT,-T,ldL
second term from (~20 here:
contains boundary
two
conditions
terms.
The
1 is E (eE-
1.0 c3
.6
1)
The first term from heat release E is given
71o).
1.2
2
.4 .8 . .2
(a) ATD
= q
;
(eE(l-i*)
_ 1)
(16)
0
0. I a
(b) ATD
= !$?
&
(eE(l--i*+A)
_e.%--*--)_2~~) (c) ATD=+$
(17)
(eE-1-E).
0.2 THERMAL
0.5
I 2 CAPACITANCE
5 RATIO
IO
Fig. 4. Variation of the average absolute temperature difference (ATD) with changes in TVfor a cocurrent reactor. Parametric values of B. Conditions of Fig. 3. Crosses (+) indicate the values for the fixed bed asymptotic limits (a = 0), l/8.
(IV
maximum temperature difference, MTD = 172 occurs either at the entrance [ = 0, exit c = 1, or at the point of heat release. Computations shows that from the heat release term, we have:
the characteristic temperature rise. ATD is not sensitive to A, the half width of the heating zone, or <*, the center position of the heating zone. The value of E has a minimum value of 4/3at a = 1, and rises to 12.18 at a = 0.1 or 10. When jl= 50, the value of IE I is 200 or greater, so that ATD for all cases reduce to [ (1 + a)/a]
(a) MTD
= e
(-- l/E).
(b) MTD
= e
&
(c) MTD
= F
i
The -
71 Lax
at <=<*
(19
(e2AE - 1) (eE - 1)
at < = <* + A (20)
at < = 1.
(21)
The boundary term (boo - 710) eET has a maximum of at c = 0. For a fixed bed, a is zero and E goes 720 -*IO to negative infinity. The values of z2 - 71, ATD, and MTD for cases (a) (b) and (c) are given in Table 4. The ATD is sensitive to the values of a and /3,which governs the value of E, see Fig. 4. The fixed bed limit for ATD is l/p, so that for most industrial reactors with b greater than 50, we have ATD less than 0.02 or 2 y0 of
The value of MTD for A = 0.01 is given in Fig. 5. The fixed bed limit for MTD = I/2A/?, which is very sensitive to the value of A but not to the value of <*. In the limit of no interphase heat transfer, j? = 0, all MTD becomes (1 + a)/a. Even when fl= 50, MTD can be exceedingly large. 5. COUNTERCURRENT REAaORS We give first the results of a countercurrent reactor with an intense exothermic reaction where the heat input is a rectangular distribution, where A is constant
Table 4. Temperature differences for a fixed bed for different input heat distributions ATD
52-T1 (a) pulse (b)
1
=ooat[=[* = 0 elsewhere 1 = __ b-2A
MTD
s
*
at 1
1
B
2AB
1
1
1
B
B
B
square c*-A
a -
(c) uniform
TtlERMAL
CAPACITANCE
RATIO
Fig. 5. Variation of the maximum temperature difference (MTD) with changes in IXfor a cocurrent reactor. Parametric values of /3_ Conditions of Fig. 3. Crosses (+) indicate the values for the fixed bed asymptotic limits (c( = 0) + 50/b.
1182
J. WEI et al.
in the interval 0.49 < 5 < 0.51 and zero elsewhere. 2 = 1 is assumed, indicating that stream 2 receives the entire reaction heat effect. Outside this interval onIy heat transferoccurs between the streams. The inlet conditions are: stream 1 enters from the left with 7 10 = 2.222 and stream 2 enters from the right with 7 21 = 1.333. A very low value of fi = 1 was selected to the temperature difference between exaggerate streams. Fig. 6 presents the temperature profiles for three values of u. The upper profile of Fig. 6 corresponds to a = - 10, or the thermal capacitance of solid stream 2 is ten times that of gaseous stream 1. The disparity in size of the thermal capacitances causes two effects. The larger stream (stream 2) leaves the reactor with most of the heat of reaction. The second effect is that the temperature difference decays rapidly along the reactor length in those regions where 2 is 0. Note also there that stream 1, the stream of smaller thermal capacitance, experiences the larger temperature change. The middle temperature profile is for a = - 1, the case of countercurrent streams of equal thermal capacitance. In the regions of the reactor where 2 equals 0, the temperature profiles are linear and the temperature difference between streams is constant. Recall that for o! = - 1, both eigenvalues of the system matrix of eq. (3) are zero and the associated Green’s functions have not an exponential but a linear dependence on 5. When two streams have equal thermal capacitances, any heat transferred between them produces the same magnitude of temperature change in each stream. In a countercurrent system this implies a constant temperature difference wherever il = 0. The heat transfer rate between streams is proportional to this constant temperature difference. Consequently the temperature profiles are linear with the same slope. For countercurrent reactors with a near - 1, any temperature dif4 Ti
a =-IO
r2
2 0
E=
-9.9 7-2
=I
I
217
I
a =-I
I
I
IT21
t
I
1
I
1-21
I
I
I
E=O
ference produced by a reaction heat effect will persist the reactor. Employing the pseudothrough homogeneous approach in a detailed model would be highly inaccurate in these cases. The lower temperature profile of Fig. 6 corresponds to an u of - 0.1, the thermal capacitance of solid stream 2 is one tenth that of as gaseous stream 1. As before, most of the reaction heat leaves with the stream of larger thermal capacitance, stream 1 in this case. In the regions of the reactor where i = 0, the temperature difference decays along the reactor length while stream 2, the stream of smaller thermal capacitance, undergoes the larger temperature change. The most prominent feature of the u = - 0.1 temperature profile is the large temperature increase for stream 2 in the heat input zone (0.49 < 5 ,( 0.51). The reaction heat input is not a delta function but is concentrated within the central two percent of the reactor length. The value of fl is too small to effectively moderate the temperature difference between streams within the reaction zone. The change in temperature difference is therefore close to the value of the point distribution temperature difference factor. The physical significance of the a = -0.1 profile is that when any reactor heat input is injected into the stream of smaller thermal capacitance, it contributes most to the temperature differences between streams. Here the value of y is 1. Stream 2 receives the entire heat input and because of its small thermal capacitance, it undergoes large temperature increase in the heat input region. Finally, note the effect of a on the location of the maximum temperature difference. At a = - 10, the MTD occurs at the upper end of the heat input zone, < = 0.51. However, at a = - 1, the temperature difference at c = 0.49 has grown and equals that at [ = 0.51. At a = - 0.1, clearly the MTD is at c = 0.49, the lower edge of the heat input zone. The location of the MTD depends on the specific functional forms of the heat distribution functions. However, near a = - 1, there is a characteristic shift in MTD location from one extreme end of the heat input zone to the other. For the case of y = 1, the temperature difference can be derived from eq. (10) as:
T2
4 TI -_;,
2
01
I
I
I
I
t2-7,
=
I
5 (&i-i’)
_
eE’)
dc’
0 L
-
AK’)
(I+
Iat)
1 $
s 5
X
1
_eE(I+c--i’)+eEC a
e -e + 0
.I
.2
.3
5 DIMENSIONLESS
.4
.5
.6
.7
REACTOR
.0
.9
1.0
POSITION
Fig. 6. Dimensionless temperature profiles for a countercurrent reactor; c( values of - 10, - 1 and - 0.1: conditions of fi = 1, y = 1, 1 = 50 for A = 0.02 and c* = 0.5. Inlet rl,, = 2.222, 721 = 1.333.
d<’ >
1 +orewE
(1 +a)
(TZ1 -71,)e”i.
(22)
The first term gives the contributions of heat sources to the left of the measure point, the second term gives the contribution of heat sources to the right of the measure point, and the third term gives the boundary condition. For a pulse heat input at i*,
Phases in a moving bed
T*-71
or
=
eCE
(l+lal)
- (1 + fal) I :fE for
The value of ATD ATD
= (1+
(@(I-S*)
1 + ccc-s
(
_ &)
~eE(l+i-r*)+eE~
1183
(23)
>
[ < c*.
is
lal) 1 :ie_E
k (l+
l/a) (eE(‘-i’)-
eE) (24)
-10
and MTD=
-(l+(crl)
e--E l+olepE
(AeE+eEi’) c2
(25)
-2 -5 a -THERMAL
-I
-0.5
-0.2
CAPACITANCE
-0.1
RATIO
Fig. 8. Variation of the average absolute temperature difference (ATD) with changes in a for a countercurrent reactor. Five parametric values of A, the heat input interval. Conditions of Fig. 6 except B = 50.
01
Figure 7 shows the value of ATD for the square function heat input as functions of a and /3.When #?is greater than 50, ATD is generally below l/p except in the neighborhood of CL= - 1. The value of E vanishes in this neighborhood, so that temperature differences persist throughout the reactor without decay. In this unusual neighborhood, ATD is also sensitive to the values of A and [*, see Figs 8 and 9. The value of MTD is particularly sensitive to the values of fl for small values of Icrl, see Fig. 10. When heat is injected into the stream with the smaller heat capacity, and when interphase heat transfer is slow, we should anticipate local “hot spots”. The dry ash Lurgi coal gasifier has a = - 0.1 and p = 50, so that the value of E = - 495. The combustion heat effect would give rise to a characteristic temperature increase of 650K. The decay length cd = 0.002, which amounts to 0.002 ft or 0.24 in. in a 10 ft reactor.
1.6
z*=
1.0
.5
.9
.8 .7 .6 .5 .4 .3
I
t
-10
‘! -5 -2 a -THERMAL
-I
-0.5
-0.2
CAPACITANCE
-
0.1
RATIO
Fig. 9. Variation of the ATD with a for a countercurrent reactor. Five parametric values of [*. Conditions of Fig. 6 except p = 50.
+
/ 8=0
1.4
/
+
.2 0 -10
%iERMAl_ a
-2
c--
-
&AC
- 0.5
ITANCE
y-“” I -a2
-0.1
RATIO
Fig. 7. Variation of the average absolute temperature difference (ATD) with changes in OL for a countercurrent reactor. Parametric values of /3.Conditions of Fig. 6. Crosses (+) indicate the values for the fixed bed asymptotic limits (a = O),
-10
-5
a-THERMAL
-2
-I - 0.5 CAPACITANCE
-0.2
-0.1 RATIO
Fig. 10. Variation of the maximum temperature difference (MTD) with changes in a for a countercurrent reactor. Parametric values of fi. Conditions of Fig. 6. Crosses (+) indicate the values for the fixed bed asymptotic limits (a = O), 50/b.
J. WEL et al.
1184
Therefore, except for the immediate neighborhood of the reactor zone, interphase temperature differences are negligible. The value of ATD is 13 K. The values of MTD are: A MTD
0.02
0.05
0.10
650 K
260 K
130K
Even for a combustion zone as narrow as A = 0.02 or 2.4in., the temperature difference will decline from 650 K to 10 K in a distance of 1 in. The effect of axial dispersion would significantly decrease the value of MTD. 6.
CONCLUSIONS
A kinetics-free analysis has been developed for the interphase heat transfer in two-phase moving bed reactors. The temperature difference between the streams in a reactor can be easily estimated from a few easily calculated parameters and any arbitrary heat generation function, including the worst case where all the heat is injected at a single point in the reactor as a delta function and into the stream with the smaller heat capacity. A heterogeneous model is necessary when temperature difference produced by a reaction heat effect persists throughout the reactor, and when the reaction heat input is so intense that a locally large temperature difference beteen streams is produced, commonly referred to as a “hot spot.” Two dimensionless temperature difference measures have been developed: LZ,the ratio of solid to gas thermal capacitance, and /?l,a dimensionless interphase heat transfer coefficient. The average temperature difference (ATD) is an integral of the absolute temperature difference between the streams over the entire reactor. Its magnitude indicates the tendency of a temperature difference to persist within the reactor. Temperature differences between streams are produced by the reaction heat effect and by the inlet stream temperatures. The maximum temperature difference (MTD) indicates the magnitude of the maximum difference between the stream temperatures within the reactor. The temperature differences are governed by the exponential decay coefficient E, and the reciprocal characteristic decay length, cd. A temperature difference between streams, produced by either the reaction heat effect or dictated by boundary conditions, decays very slowly along the reactor length when the magnitude of E is small. For countercurrent reactors, E has a very small magnitude (and c., is large) when both streams have nearly the same thermal capacitance. The worst possible situation occurs when a has the value of - 1, counterflow of streams of equal thermal capacitances. E is zero and a temperature difference between streams does not decay. For cocurrent reactors, E has a minimum magnitude when the value of CLis one. For most industrial reactors, /I is greater than 50. This ensures large magnitudes of E in most counterand all cocurrent reactors. The corresponding ATD is
small except for countercurrent reactors with -0.5 < CL-Z - 2. The maximum temperature difference can indicate the existence of large local temperature differences between the streams. It has a larger magnitude when the reaction heat effect is released in the stream of the smaller thermal capacitance. The homogeneous model for temperature of moving bed reactors is adequate for most industrial reactors where /?is greater than 50, except for countercurrent reactors where the heat capacitances of the two streams are nearly equal. If the heat release is very localized, the heterogeneous model for temperature should be used in the reaction zone. Axial dispersion of heat would decrease the value of MTD. The fixed bed is prone to large “hot spots” when heat is released into the stationary solid phase, and can be relieved by moving the solid phase. The analysis of this paper is easily extended to concentration differences between phases in a moving bed when we define u’ as the ratio of mass flow ratio of ( VP)ZI( VP)l, and p’ as the dimensionless mass transfer ratio kaH/( ( VP), + ( VP),). NOTATION
solid surface area per unit reactor volume, l/m thermal capacities of the gas and solid streams-product of mass flux and heat capacity, J/m2 K s molecular concentrations, at surface and particle center effective intraparticle diffusivity, m2/s Damkohler number, reaction rate/external mass transport rate coefficient for the exponential decay of the temperature difference between solid and gas -B(l+ lal) (l+ l/cc) matrix of Green’s function solid-gas heat transfer coefficient, J/m2 K s reactor length, m identity matrix effective particle thermal conductivity, J/MKs characteristic length of particle, m model matrix in + = M T + a(<) overall reaction rate, defined by Carberry
VI
gas and solid temperatures, K heat generation due to reaction per unit reactor volume, J/m3 s ratio of mass to thermal Biot numbers position variable along reactor length, m dimensionless half-length of reaction zone enthalpy change of reaction, J/mol the characteristic temperature increase, K
s H
q(z) dz/W, + IC, I)
0
particle’s internal temperature difference between surface and center, K
Phases in a moving bed overall temperature difference (AT + AT,), K external temperature difference, between particle’s surface and bulk gas, K boundary condition temperature difference; zIo -tzl for countercurrent, TIo - rzo for coourrent forcing function vector for t = Mz -t@(T) thermal capacitance ratio, C,/C, dimensionless heat transfer coefficient, haHl(C, + IC, I) fraction of heat effect directed to solid Dirac delta function dimensionless reactor position, z/H characteristic decay constant, 11/E 1 overall effectiveness factor, global reaction rate/kinetics controlled rate distribution heat reaction
is H
=
a(z)H
d.4 dz
0
dimensionless temperatures; made dimensionless by a characteristic temperaH ture increase: z1 = Ti qt.4 dzl(C, /s 0 + 1C, 1); rl-gas, T,-solid REFERENCES
[l]
American Petroleum Institute, 1966, Technical Data Book; Petroleum Refining, N.Y.
1185
c23 Carberry, J. J.. 1975, On the relative importance of external-internal temperature gradients in hetero-
geneous catalysis. Ind. Engng Chem. Fundam. 14, 12%131. c31 Froment, G. F., 1972, Analysis and design of fixed bed _.-. catalvtic reactors. Advanvces in Chemistry Seric__,Nn~ 109, A.C.S., Washington, D.C., pp. l-34.. c41 Gates. B. C., Katzer, J. R. and Schuit, G. C., 1979. Che&ry of.Catalytic Processes. McGraw-Hill, N.Y.’ IIs1 Greenberg, M. D., 1971. Applications of Green’s Functions in Science and Engineering. Prentice-Hall.
N.J.
I31 Kays, W. and London, A. L., 1955, Compact Heat
Exchangers, p. 15. McGraw-Hill, N.Y. 1969, Ind. Engng Chem. Fundam. 8, 596. ES1 Lewis, A. E. and Braun, R. L., 1981, Retorting and combustion processes in surface oil shale retorts. J. Energy 5, 355-361. Nelson, W. L., 1958, Perroleum Refinery Engineering. McGraw-Hill, N.Y. Perry, R. H. and Chilton, C. H., 1950, Chemical Enoineer’s Handbook. 3rd edn. McGraw-Hill, N.Y. &&er, C. D., Wei. J., Weekman, V. W. Jr. and Gross, B., 1983, A reaction engineering case history: coke burning in thermofor catalytic cracking reactors. Adu. Chem. Engng, Vol. 12. A&demic Press, N.Y. Cl21 Satterfield, C. N., 1980, Heterogeneous Carolysis in Practice. McGraw-Hill, N.Y. Cl31 Wei, J., 1975, Catalysis for motor vehicle emissions. Advances in Catalysis, Vol. 24, pp. 57-125. Academic Press, N.Y. Cl41 Yoon, H.. Wei, J. and Denn. M. M., 1977, Modeling and Analysis of Moving Bed Coal Gasjfiers. Electric Power Research Institute, Palo Alto, California, A5-590. c71 Lee, J. C. M. and Luss, D.,