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Non-isothermal reaction in catalyst particles
Chemical Engineering Science, 1966, Vol. 21, pp. 383-390. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Non-isothermal reaction in catalyst particles D. J. GUNN Department
of Chemical Engineering,
University College of Swansea
(Received 25 November 1965; in revised form 13 December 1965) Abstract-The interactions between heat transmission, diffusion and reaction inside a porous catalyst particle in the case of a first order irreversible reaction have been studied. When the reaction velocity constant is expressed in the form of a truncated Taylor series, analytical solutions may be obtained, and the problem is thereby reduced to the solution of an ordinary transcendental equation. The analytical expression for temperature and concentration profiles is given in terms of Airy functions when the reaction velocity constant is linear in variation. A comparison of the effectiveness factors calculated in this way with other work shows that for endothermic reactions, and for reactions of low to moderate exothermicity, the linear approximation is satisfactory. For reactions of stronger exothermicity some disagreement is noted and discussed.
THE RATE of chemical
reaction in porous catalyst particles is affected by the resistance to transport of matter and heat in the particle. Even for simple reactions the differential equations that describe the physical situation inside the particle are nonlinear if the restriction of isothermality is avoided; consequently solutions are difficult to find unless some simplifying assumptions are made. Because of the mathematical difficulties, numerical methods have been widely used to obtain solutions for simple first order and second order reactions. An example of this approach may be found in the paper by WEISZ and HICKS [l]. There are some examples of the use of simplifying assumptions to produce equations that may be solved by analytical methods. In one or two instances it is a little difficult to provide mathematical and physical justification for the assumptions made. The asymptotic solutions of PETERSEN [2] however clearly indicate that at high values of the Thiele modulus chemical reaction in porous particles is restricted to the surface layer and the geometry of the particle is not important. This work, and others, are described in two recent books mainly devoted to reaction in catalyst particles 13941. The calculation of the effectiveness been the principal concern in work to passing attention to the calculation of tion and temperature distributions. In
factor has date, with concentrathis paper 383
we attempt to redress the imbalance; analytical expressions for concentration and temperature distributions inside the catalyst particle are obtained, and the effectiveness factor may be calculated from these. The starting point in the derivation is the expression of the velocity constant in terms of a Taylor series about the point R = R,in the catalyst particle. To degrees of succeeding refinement the concentration distribution may be expressed in terms of exponential functions for the temperature invariant velocity constant, in terms of Airy functions for the reaction velocity constant of linear variation, in terms of Weber functions for the velocity constant of quadratic variation, and in terms of as yet untabulated transcendental functions for variations of higher order. A FIRST ORDER IRREVERSIBLEREACTION IN SPHERICAL AND PLATE-LIKE CATALYST PARTICLES
The reaction is, A -+ B. A material balance for the reacting component over an element of catalyst volume leads to a second For the spherical order differential equation. catalyst particle when steady state conditions have been reached, the equation is
-$-& R29E =kC
(
where
R
is the radial
)
co-ordinate,
C is the concen-
D. J. GUNN
tration of reacting component, k is the reaction velocity constant and 9 is the effective diffusivity for the component in the catalyst particle. Similarly a heat balance gives the equation, = AHkC
‘“(R299 ~2 dR
-
-++&(g),R-K))C=O
(2)
The following substitutions are now made: RC -= R&s
where T is temperature, p is the thermal conductivity of the catalyst particle, and AH is theenthalpy change on reaction. Equations (1) and (2) are valid for spherical symmetry, a condition attained when the concentration of reactant at the surface of the particle is uniform. On rearranging and integrating (1) and (2) an equation first derived by Damkiihler is obtained, T-T,=
y
(6)
f
R-R s=r RS
k& -= 9
(C - C,)
4
Equation (6) becomes, where the subscript s indicates conditions at the surface of the particle. To some extent p, 9 and AH are temperature sensitive; the variations are usually minor and are ignored in this analysis for that reason. The reaction velocity constant is often strongly sensitive to temperature; it is assumed that the dependence of k upon temperature follows the Arrhenius equation,
- (1 + ar)f =0
b$
and with the further substitution, -
0_$‘I3
(1+ ar) = Z,
(7) becomes
(8)
(4) where the frequency factor A and the activation energy for the reaction E are assumed independent of temperature. G is the gas constant. The temperature function (4) is analytic for all possible ranges of temperature, and so k may be expressed as a Taylor series about the point R = R,,
The boundary conditions to Eq. (l), C = C, at R = R, and C is finite at R = 0, become
f = 1 at Z = -
J$ 1’3, 0
and
f = 0 at Z = -
The solution to Eq. (8) is f+
Cl s
“+
(%
If the linear terms of the Taylor series alone are considered (1) becomes,
U,(Z)
+
C2U2@>
(9)
s
where U,(Z) and U,(Z) are Airy functions of the first and second kind and C, and C, are constants determined from the boundary conditions. If Z, = -(d/~r~)‘/~ and Z,, = -(4/a2)li3(1 - ~1)the constants may be expressed, 384
Non-isothermal
Cl = UWl(Zs)- ww-J2(wu2(m) C2= MU2W - ~2Gwlm/ulG))
reaction in catalyst particles
The effectiveness factor of Thiele is of interest.
(10) For the spherical particle, 3a
The Airy functions are available in the form of tables [5]. Equations (9) and (10) contain the unknown temperature gradient. Some further arrangement gives, AH%,. P
E - - a/(Cl(Ul(Zs) GT,Zt-
where
y is the effectiveness
Equation (11) is a transcendental equation which when solved, gives the temperature gradient (dT/dr),. The primed quantities represent derivatives with respect to Z. A solution may also be obtained in terms of Bessel functions of order one third. A similar procedure may be followed for the flat pellet in which end effects may be neglected. The concentration distribution is, (12)
where U, and U, are Airy functions of the first and second kind as before, and Z is defined as before. X, takes the place of R, in the expressions for 4, Z and cc. The boundary conditions to the differential equation,
=
kC
and
Y=;
+ ZSUXZS)) +
+ &U,(Z)
factor
6 =
(AHzPC,/p) . (E/G?“:), and for the flat pellet,
C2W2(Z*) + w; (Z&) (11)
C - = &U,(Z) C,
(15)
y=-&$
(16)
THE REACTION VELOCITYCONSTANTOF QUADRATIC
VARIATION
A more accurate solution is obtained if terms of higher order than the first are considered in the Taylor series. The expansion up to and including quadratic terms is k = kX1 + ar + /It-‘)
(17)
where
fi=&((g):(*-;)+(g),)
(18)
From Eq. (2), (19)
so that
(13)
AHC,k,R,Z -IP
C = C, at x = x,, and dC/dx = 0 at x = 0,
(20)
On substituting f for CR/C,R, Eq. (1) becomes,
become, C
($1I’;1-
-=latZ=-
CS
f
a)
$ - (1 + ar + fir’)f
= 0
(21)
The further substitutions,
and
d(C/Csl -=()atZ= dZ
Z-a -
I=tp’
Equation (12) contains the unknown temperature gradient. The equation analogous to (11) is, AH%,. P
E 7 = - WJJXZS) GT,
+ C,U;(Zs))Z,
and
w2 = 2J(&W2,
transform Eq. (21) into,
$+ ip+;-;w2 1 f=O
(14) 385
(22)
D. J. GUNN
where
P = 3(a2J(@)
The solution
- &/a>
(23)
- I)
to (22) is
f=S!$ S S
CP,(W)
A singularity in distributions at R finite amount of A first step in the
the temperature or concentration = 0 will always be found when a heat or matter is injected there. integration of Eqs. (1) and (2) is,
(24)
+ C&(-W)
R2!$R2~!&C5
where D, is the Weber function of order p. The boundary conditions to Eq. (1) become,
(27)
so that f = 1 at W = a&2&@)), and
f = 0 at W = 2J(2J(@))(~
(T-
+ D).
+ CzDp(-
W,) +
+ 2. BJPJ(4B))(cl~;(K)
- Gqi
- WI) (25)
Equation (25) is a transcendental equation which, when solved, gives the temperature gradient. Values of the Weber functions are available in the form of tables [6]. We have not yet considered restrictions on the solutions that may arise from the use of a truncated Taylor series as an approximation to the reaction velocity constant. Before we do so, however, it is necessary to look more closely at the derivation of Eq. (3). When 9 and p are constant with temperature, both Eqs. (1) and (2) on expansion give an equation of the form of (26) d2C 2 dC kC ----_=O dR2+RdR SB
‘5
(c - C,) +
2-2
(28)
s
The terms ccand /3 both contain the unknown temperature gradient. An expression for the temperature gradient may be obtained as before to give, 6 = - a/(clL$(wS)
TJ=
(26)
It may be proved [7] that the only possible singular points of the integrals of a linear differential equation in which the coefficient of the highest derivative equals one, and the other coefficients are analytic functions of the independent variable, are the singular points of the coefficients themselves. This theorem is sufficient to ensure that provided a series of the type (5) is chosen to represent k, the only possible singularities of the solutions to (26) will occur at R = 0. The existence of singularities in the coefficients does not necessarily mean that the integrals will always possess singularities, and the question may be decided by the boundary or initial conditions. 386
The last term on the right hand side of (28) is a source term corresponding to an injection of a finite amount of heat or matter at R = 0. If C, is set equal to zero, Eq. (3) is obtained, but this equation can still be satisfied when a finite amount of heat and an associated amount of reactant are injected at R = 0. Thus the Damkohler equation, although necessary, is not a sufficient condition for the absence of singularities in both the temperature and concentration distributions. To remove this possibility it is sufficient to stipulate that either T or C be bounded in the range 0 < R < R,. This clause is clearly derived from the physical nature of chemical reaction in the particle and is therefore expressed in the boundary conditions. A consideration of the conditions of diffusion, heat transmission and reaction in the limit as R + 0 gives stronger results. Consider a region of the catalyst particle centred on R = 0, of small radius compared to the radius of the particle R,. As the radius is small the reaction velocity constant k will nowhere be very different from the value at the centre of the particle k,, the temperature T will nowhere be very different from T,,, and the concentration will not differ much from C,. Material and thermal balances for this region are,
so that
4aR2
4 w 3 nR3koC,
(29)
4nR2
w 4 xR3AHk,Co
(30)
dC -M-dR
RkC0 o 39
(31)
Non-isothermal
dT dR=
- $AHk&
reaction in catalyst particles
(32)
The conditions in the centre of the particle may be investigated by taking the limit as R -+ 0. If this is done in Eqs. (31) and (32), the right-hand sides vanish because k and C are both bounded. Hence,
LtE=LtdT-=
,t-+,,dR
0)
R+OdR
THE CALCULATION OF THE TEMPERATURE GRADIENT
and
The calculation of the dimensionless temperature gradient CIin terms of 4 and 6 from Eqs. (1 l), (14) and (25) is clearly difficult. However the inverse problem of calculating 6 from values of o! and Cpis straightforward, and this has been done for a range of values in terms of Airy functions for both spherical and plate-like particles. The range of 4 considered has been limited by the range of the tabulated values of Airy functions. The results are shown in Figs. 1 and 2. Figure 1 shows the dependence of 6 upon $ with the dimensionless temperature gradient a as a parameter for the spherical particle, and Fig. 2 shows the relationships for the flat pellet. A better comparison
and It !dT= R+~ R dR
- AHk,C, 3P
.
From Eq. (26) it then follows that Lt d2T AHk& R-+OdRf-p
: 2AHk&
3P
5 AHk& =:P’
and Lt d2C
-=R-.OdR2
k&, 2k,C,, f-=36 9
If Eqs. (1) and (2) are differentiated with respect to R, it can be seen that third and higher derivatives of C and Tare not defined at R = 0 when the linear approximation to k is used. These restrictions are not severe, and the analysis is sufficiently encouraging to compare effectiveness factors expressed in terms of Airy functions with results obtained in other ways.
-lOO(
5 k,Co
This analysis shows that provided C, T and the approximation to k are bounded throughout the range 0 < R f R,, then the solutions to Eqs. (1) and (2) are analytic and regular throughout the range 0 < R < R,, and the functions T and C together with their first and second derivatives are finite and continuous at R = 0. If the limiting values of k, C and T at R = 0 are ko, Co and T,, the limiting values of (dT/dR) and (dC/dR) are both zero, and the limiting values of (d2C/dRz) and (d2T/dR2) are ~k,C,J9) and $(AHk,C,/p). The general nature of the conclusion is easily seen to be valid for all reactions no matter how complex, and for all regular shapes of particle. The values of C and T are always bounded, the first spatial derivatives at the origin of symmetry are zero and the values of the second derivatives are defined and continuous throughout the particle. The numerical constants in the limiting value of the second derivatives will depend upon the shape of the particle. 387
-IO<
8-+
I
01
I I
4 FIG. 1. Dependence of the dimensionless temperature gradient
TVupon + and 6 for _the spherical
particle.
D. J.
8
FIG 2. Dependence of the dimensionlesstemperature gradient CLupon C$and 6 for the flat pellet.
FIG. 3.
GUNN
with previous work can be made if the effectiveness factors for both geometries are calculated from Eqs. (15) and (16). The results of the calculations are shown as Figs. 3 and 4. TINKLER and METZNER [S] solved the same problem on an analogue computer and their results have been presented in a form suitable for comparison with this work. The comparison is made in Figs. 3 and 4. For endothermic reactions the agreement is very good. No difference between the results for the irreversible endothermic reaction of TINKLER and METZNER, and the effectiveness factors calculated for Airy functions could be found that was significant compared to the error of transcription. For exothermic reactions agreement was good up to 6 = 2. At values of 6 = 4 and 5, however, there is disagreement. From Fig. 3 it can be seen that effectiveness factors calculated from Airy functions are low. It is probable that this is due to temperature gradients inside the particle in excess of that found at the surface. This is possible in a spherical particle because the absolute value of
Comparisonof effectivenessfactors for the sphericalparticle,........... dataof TINKLERandMETZNER,388
presentwork.
Non-isothermal
reaction in catalyst particles
multiple steady states can also be found for flat pellets and the nature of the disagreement between the results of TINKLER and METZNER and the present work may be due to this cause. The tabulated values of Airy functions were not sufficiently extensive to support a search for multiple solutions, and for this reason no conclusions can be drawn concerning upper and lower steady states. The tabulated values of Weber functions have not been carried into ranges of the variables applicable to this work. The values may be extended by means of the recurrence relations, D,+ I(Y) + @,(Y) + PD,- I(Y) = 0 d D,(Y) + ~YD,(Y) dy
;
- PD,- I(Y) = 0
(33)
DJY) - +YD,(Y) + D,+ i(y) = 0
FIG. 4. Comparison flat pellet,
of effectiveness factors for the ---..........- data of TINKLER and METZNER, present work.
R2 dT/dR is a maximum at the surface, but the temperature gradient is not necessarily a maximum there. The same pattern of agreement and disagreement is found in the results for the flat pellet. The results for endothermic reactions agree, while agreement for exothermic reactions is found only uptos = 2. The absolute value of the temperature gradient in the flat pellet is a maximum at the surface; one would therefore anticipate that the use of this maximum gradient would lead to an overestimate of the rate of reaction in the particle, but Fig. 4 shows that the results of TINKLER and METZNERare greater for large values of 6 ! A comparison of the results for the spherical particle with those of WEISZ and HICKS [l] gives the same pattern of agreement. WEISZ and HICKS found from their results that multiple solution for the steady state effectiveness factors could be found for a range of small values of 4. Some further evidence in support has been recently advanced by MCGUIRE and LAPIDUS [9] who found two steady state temperature distributions in porous catalyst particles for this reaction system, and by WEI [lo] in a study of the stability of temperature profiles in catalyst particles. It is possible that
The comparisons with other work show that for endothermic reactions and for reactions of low to moderate exothermicity, the expression for concentration and temperature distributions are probably accurate. For reactions of stronger exothermicity, the rates of reaction predicted are too small. This is due in part, at least for spherical particles, to the increase of temperature gradient inside the particle from the surface condition. But part of the difference is also possibly due to the existence of multiple steady states at low values of the Thiele modulus. Some consideration of the use of Weber functions to describe the temperature and concentration profiles may throw further light on the question.
389
NOTATION A C 9
DP E
A: k P
Arrhenius frequency factor Concentration moles/unit volume Effective diffusivity Weber function of order p Arrhenius activation energy Gas constant Enthalpy change on reaction Reaction velocity constant Weber function tiarameter Dimensionless radial co-ordinate = (R - Rs)/Rs Radial co-ordinate Temperature Airy function
R’ T
u
D. 1. GLJNN p Thermal conductivity I# Thiele modulus R112k,/g
Greek letters a /i 8 7
Dimensionless temperature gradient E/(GT#s) @T/d& Coefficient in second order term of Taylor series defined by (20) AHC&@/pE/(GTP) Effectiveness factor
Subscripts 0 s
Conditions at the centre of the particle Surface conditions.
REFERENCES WEISZ P. B. and HICKS J. S., Chem. Engng Sci. 1962 17265. PETERSEN E. E., Chem. Engng Sci. 1962 17 987. SATIYERFIELD C. N. and SHERWOODT. K., The Role of Diffusion in Catalysis, Addison Wesley 1963. PETEMENE. E. Chemical Reaction Analysis, Prentice-Hall 1965. SMIRNOVA. D. Tables of Airy Functions, Pergamon Press 1960. KIREYEVAI. Y. and KARPOV K. A. Tables of Weber Functions, Pergamon Press 1961.
T~ICOMIF. G. Di’rential
Equations, Blackie 1960.
TINKLERJ. D. and METZNERA. B., Znd. Engng Chem., 1961 53 663.
MCGUIRE M. L. and LAP~DUSL., A.Z.Ch.E. JL, 1965 11 85.
WEI J., Chem. Engng Sci. 1965 20 729.
R&unB-Dans le cas dune reaction irreversible du premier ordre l’auteur a Ctudie les interactions entre la transmission de chaleur, la diffusion et la reaction a l’interieur d’une particule poreuse de catalyseur. On peut obtenir les solutions analytiques quand la constante de vitesse de la reaction est represent&e par une serie de Taylor tronqu&e, et, de cette facon, le probleme se reduit a la solution dune equation transcendante ordinaire. Quand la constante de vitesse de la reaction varie lineairement, l’expression analytique de la temperature et les profils de concentration sont don&s a l’aide des fonctions d’hiry. Une comparaison des facteurs d’efficacite calcules par cette methode avec d’autres travaux montre que pour les reactions endothermiques et pour les reactions peu ou moyennement exothermiques, l’approximation lineaire est satisfaisante. Pour les reactions fortement exothermiques on peut noter et discuter quelques differences. Zusammenfassung-Die Wechselwirkung zwischen Warmeleitung, Stofftransport und chemischer Reaktion innerhalb eines poriisen Katalysators wurden fiir den Fall einer irreversiblen Reaktion 1. Ordnung untersucht. Wenn die Reaktionsgeschwindigkeitskonstante durch die ersten Glieder einer Taylor-Reihe ausgedriickt wird, erhllt man eine analytische Losung, wobei das Problem auf die Losung einer gewtihnlichen transzendenten Gleichung reduziert wird. Der analytische Ausdruck zur Beschreibung von Temperatur- und Konzentrationsprofilen wird als Airy-Funktion angegeben, wenn die Reaktionsgeschwindigkeitskonstante von den Variablen linear abhangig ist. Vergleicht man den so bestimmten Porenausnutzungsgrad mit Literaturangaben, so sieht man, da5 diese linearisierende Nlherungslosung fur endotherme oder schwach exotherme Reaktionen ausreicht; bei starkerer Exothermitlt treten Abweichungen auf, die ebenfalls diskutiert werden.
390
Non-isothermal reaction in catalyst particles D. J. GUNN Department
of Chemical Engineering,
University College of Swansea
(Received 25 November 1965; in revised form 13 December 1965) Abstract-The interactions between heat transmission, diffusion and reaction inside a porous catalyst particle in the case of a first order irreversible reaction have been studied. When the reaction velocity constant is expressed in the form of a truncated Taylor series, analytical solutions may be obtained, and the problem is thereby reduced to the solution of an ordinary transcendental equation. The analytical expression for temperature and concentration profiles is given in terms of Airy functions when the reaction velocity constant is linear in variation. A comparison of the effectiveness factors calculated in this way with other work shows that for endothermic reactions, and for reactions of low to moderate exothermicity, the linear approximation is satisfactory. For reactions of stronger exothermicity some disagreement is noted and discussed.
THE RATE of chemical
reaction in porous catalyst particles is affected by the resistance to transport of matter and heat in the particle. Even for simple reactions the differential equations that describe the physical situation inside the particle are nonlinear if the restriction of isothermality is avoided; consequently solutions are difficult to find unless some simplifying assumptions are made. Because of the mathematical difficulties, numerical methods have been widely used to obtain solutions for simple first order and second order reactions. An example of this approach may be found in the paper by WEISZ and HICKS [l]. There are some examples of the use of simplifying assumptions to produce equations that may be solved by analytical methods. In one or two instances it is a little difficult to provide mathematical and physical justification for the assumptions made. The asymptotic solutions of PETERSEN [2] however clearly indicate that at high values of the Thiele modulus chemical reaction in porous particles is restricted to the surface layer and the geometry of the particle is not important. This work, and others, are described in two recent books mainly devoted to reaction in catalyst particles 13941. The calculation of the effectiveness been the principal concern in work to passing attention to the calculation of tion and temperature distributions. In
factor has date, with concentrathis paper 383
we attempt to redress the imbalance; analytical expressions for concentration and temperature distributions inside the catalyst particle are obtained, and the effectiveness factor may be calculated from these. The starting point in the derivation is the expression of the velocity constant in terms of a Taylor series about the point R = R,in the catalyst particle. To degrees of succeeding refinement the concentration distribution may be expressed in terms of exponential functions for the temperature invariant velocity constant, in terms of Airy functions for the reaction velocity constant of linear variation, in terms of Weber functions for the velocity constant of quadratic variation, and in terms of as yet untabulated transcendental functions for variations of higher order. A FIRST ORDER IRREVERSIBLEREACTION IN SPHERICAL AND PLATE-LIKE CATALYST PARTICLES
The reaction is, A -+ B. A material balance for the reacting component over an element of catalyst volume leads to a second For the spherical order differential equation. catalyst particle when steady state conditions have been reached, the equation is
-$-& R29E =kC
(
where
R
is the radial
)
co-ordinate,
C is the concen-
D. J. GUNN
tration of reacting component, k is the reaction velocity constant and 9 is the effective diffusivity for the component in the catalyst particle. Similarly a heat balance gives the equation, = AHkC
‘“(R299 ~2 dR
-
-++&(g),R-K))C=O
(2)
The following substitutions are now made: RC -= R&s
where T is temperature, p is the thermal conductivity of the catalyst particle, and AH is theenthalpy change on reaction. Equations (1) and (2) are valid for spherical symmetry, a condition attained when the concentration of reactant at the surface of the particle is uniform. On rearranging and integrating (1) and (2) an equation first derived by Damkiihler is obtained, T-T,=
y
(6)
f
R-R s=r RS
k& -= 9
(C - C,)
4
Equation (6) becomes, where the subscript s indicates conditions at the surface of the particle. To some extent p, 9 and AH are temperature sensitive; the variations are usually minor and are ignored in this analysis for that reason. The reaction velocity constant is often strongly sensitive to temperature; it is assumed that the dependence of k upon temperature follows the Arrhenius equation,
- (1 + ar)f =0
b$
and with the further substitution, -
0_$‘I3
(1+ ar) = Z,
(7) becomes
(8)
(4) where the frequency factor A and the activation energy for the reaction E are assumed independent of temperature. G is the gas constant. The temperature function (4) is analytic for all possible ranges of temperature, and so k may be expressed as a Taylor series about the point R = R,,
The boundary conditions to Eq. (l), C = C, at R = R, and C is finite at R = 0, become
f = 1 at Z = -
J$ 1’3, 0
and
f = 0 at Z = -
The solution to Eq. (8) is f+
Cl s
“+
(%
If the linear terms of the Taylor series alone are considered (1) becomes,
U,(Z)
+
C2U2@>
(9)
s
where U,(Z) and U,(Z) are Airy functions of the first and second kind and C, and C, are constants determined from the boundary conditions. If Z, = -(d/~r~)‘/~ and Z,, = -(4/a2)li3(1 - ~1)the constants may be expressed, 384
Non-isothermal
Cl = UWl(Zs)- ww-J2(wu2(m) C2= MU2W - ~2Gwlm/ulG))
reaction in catalyst particles
The effectiveness factor of Thiele is of interest.
(10) For the spherical particle, 3a
The Airy functions are available in the form of tables [5]. Equations (9) and (10) contain the unknown temperature gradient. Some further arrangement gives, AH%,. P
E - - a/(Cl(Ul(Zs) GT,Zt-
where
y is the effectiveness
Equation (11) is a transcendental equation which when solved, gives the temperature gradient (dT/dr),. The primed quantities represent derivatives with respect to Z. A solution may also be obtained in terms of Bessel functions of order one third. A similar procedure may be followed for the flat pellet in which end effects may be neglected. The concentration distribution is, (12)
where U, and U, are Airy functions of the first and second kind as before, and Z is defined as before. X, takes the place of R, in the expressions for 4, Z and cc. The boundary conditions to the differential equation,
=
kC
and
Y=;
+ ZSUXZS)) +
+ &U,(Z)
factor
6 =
(AHzPC,/p) . (E/G?“:), and for the flat pellet,
C2W2(Z*) + w; (Z&) (11)
C - = &U,(Z) C,
(15)
y=-&$
(16)
THE REACTION VELOCITYCONSTANTOF QUADRATIC
VARIATION
A more accurate solution is obtained if terms of higher order than the first are considered in the Taylor series. The expansion up to and including quadratic terms is k = kX1 + ar + /It-‘)
(17)
where
fi=&((g):(*-;)+(g),)
(18)
From Eq. (2), (19)
so that
(13)
AHC,k,R,Z -IP
C = C, at x = x,, and dC/dx = 0 at x = 0,
(20)
On substituting f for CR/C,R, Eq. (1) becomes,
become, C
($1I’;1-
-=latZ=-
CS
f
a)
$ - (1 + ar + fir’)f
= 0
(21)
The further substitutions,
and
d(C/Csl -=()atZ= dZ
Z-a -
I=tp’
Equation (12) contains the unknown temperature gradient. The equation analogous to (11) is, AH%,. P
E 7 = - WJJXZS) GT,
+ C,U;(Zs))Z,
and
w2 = 2J(&W2,
transform Eq. (21) into,
$+ ip+;-;w2 1 f=O
(14) 385
(22)
D. J. GUNN
where
P = 3(a2J(@)
The solution
- &/a>
(23)
- I)
to (22) is
f=S!$ S S
CP,(W)
A singularity in distributions at R finite amount of A first step in the
the temperature or concentration = 0 will always be found when a heat or matter is injected there. integration of Eqs. (1) and (2) is,
(24)
+ C&(-W)
R2!$R2~!&C5
where D, is the Weber function of order p. The boundary conditions to Eq. (1) become,
(27)
so that f = 1 at W = a&2&@)), and
f = 0 at W = 2J(2J(@))(~
(T-
+ D).
+ CzDp(-
W,) +
+ 2. BJPJ(4B))(cl~;(K)
- Gqi
- WI) (25)
Equation (25) is a transcendental equation which, when solved, gives the temperature gradient. Values of the Weber functions are available in the form of tables [6]. We have not yet considered restrictions on the solutions that may arise from the use of a truncated Taylor series as an approximation to the reaction velocity constant. Before we do so, however, it is necessary to look more closely at the derivation of Eq. (3). When 9 and p are constant with temperature, both Eqs. (1) and (2) on expansion give an equation of the form of (26) d2C 2 dC kC ----_=O dR2+RdR SB
‘5
(c - C,) +
2-2
(28)
s
The terms ccand /3 both contain the unknown temperature gradient. An expression for the temperature gradient may be obtained as before to give, 6 = - a/(clL$(wS)
TJ=
(26)
It may be proved [7] that the only possible singular points of the integrals of a linear differential equation in which the coefficient of the highest derivative equals one, and the other coefficients are analytic functions of the independent variable, are the singular points of the coefficients themselves. This theorem is sufficient to ensure that provided a series of the type (5) is chosen to represent k, the only possible singularities of the solutions to (26) will occur at R = 0. The existence of singularities in the coefficients does not necessarily mean that the integrals will always possess singularities, and the question may be decided by the boundary or initial conditions. 386
The last term on the right hand side of (28) is a source term corresponding to an injection of a finite amount of heat or matter at R = 0. If C, is set equal to zero, Eq. (3) is obtained, but this equation can still be satisfied when a finite amount of heat and an associated amount of reactant are injected at R = 0. Thus the Damkohler equation, although necessary, is not a sufficient condition for the absence of singularities in both the temperature and concentration distributions. To remove this possibility it is sufficient to stipulate that either T or C be bounded in the range 0 < R < R,. This clause is clearly derived from the physical nature of chemical reaction in the particle and is therefore expressed in the boundary conditions. A consideration of the conditions of diffusion, heat transmission and reaction in the limit as R + 0 gives stronger results. Consider a region of the catalyst particle centred on R = 0, of small radius compared to the radius of the particle R,. As the radius is small the reaction velocity constant k will nowhere be very different from the value at the centre of the particle k,, the temperature T will nowhere be very different from T,,, and the concentration will not differ much from C,. Material and thermal balances for this region are,
so that
4aR2
4 w 3 nR3koC,
(29)
4nR2
w 4 xR3AHk,Co
(30)
dC -M-dR
RkC0 o 39
(31)
Non-isothermal
dT dR=
- $AHk&
reaction in catalyst particles
(32)
The conditions in the centre of the particle may be investigated by taking the limit as R -+ 0. If this is done in Eqs. (31) and (32), the right-hand sides vanish because k and C are both bounded. Hence,
LtE=LtdT-=
,t-+,,dR
0)
R+OdR
THE CALCULATION OF THE TEMPERATURE GRADIENT
and
The calculation of the dimensionless temperature gradient CIin terms of 4 and 6 from Eqs. (1 l), (14) and (25) is clearly difficult. However the inverse problem of calculating 6 from values of o! and Cpis straightforward, and this has been done for a range of values in terms of Airy functions for both spherical and plate-like particles. The range of 4 considered has been limited by the range of the tabulated values of Airy functions. The results are shown in Figs. 1 and 2. Figure 1 shows the dependence of 6 upon $ with the dimensionless temperature gradient a as a parameter for the spherical particle, and Fig. 2 shows the relationships for the flat pellet. A better comparison
and It !dT= R+~ R dR
- AHk,C, 3P
.
From Eq. (26) it then follows that Lt d2T AHk& R-+OdRf-p
: 2AHk&
3P
5 AHk& =:P’
and Lt d2C
-=R-.OdR2
k&, 2k,C,, f-=36 9
If Eqs. (1) and (2) are differentiated with respect to R, it can be seen that third and higher derivatives of C and Tare not defined at R = 0 when the linear approximation to k is used. These restrictions are not severe, and the analysis is sufficiently encouraging to compare effectiveness factors expressed in terms of Airy functions with results obtained in other ways.
-lOO(
5 k,Co
This analysis shows that provided C, T and the approximation to k are bounded throughout the range 0 < R f R,, then the solutions to Eqs. (1) and (2) are analytic and regular throughout the range 0 < R < R,, and the functions T and C together with their first and second derivatives are finite and continuous at R = 0. If the limiting values of k, C and T at R = 0 are ko, Co and T,, the limiting values of (dT/dR) and (dC/dR) are both zero, and the limiting values of (d2C/dRz) and (d2T/dR2) are ~k,C,J9) and $(AHk,C,/p). The general nature of the conclusion is easily seen to be valid for all reactions no matter how complex, and for all regular shapes of particle. The values of C and T are always bounded, the first spatial derivatives at the origin of symmetry are zero and the values of the second derivatives are defined and continuous throughout the particle. The numerical constants in the limiting value of the second derivatives will depend upon the shape of the particle. 387
-IO<
8-+
I
01
I I
4 FIG. 1. Dependence of the dimensionless temperature gradient
TVupon + and 6 for _the spherical
particle.
D. J.
8
FIG 2. Dependence of the dimensionlesstemperature gradient CLupon C$and 6 for the flat pellet.
FIG. 3.
GUNN
with previous work can be made if the effectiveness factors for both geometries are calculated from Eqs. (15) and (16). The results of the calculations are shown as Figs. 3 and 4. TINKLER and METZNER [S] solved the same problem on an analogue computer and their results have been presented in a form suitable for comparison with this work. The comparison is made in Figs. 3 and 4. For endothermic reactions the agreement is very good. No difference between the results for the irreversible endothermic reaction of TINKLER and METZNER, and the effectiveness factors calculated for Airy functions could be found that was significant compared to the error of transcription. For exothermic reactions agreement was good up to 6 = 2. At values of 6 = 4 and 5, however, there is disagreement. From Fig. 3 it can be seen that effectiveness factors calculated from Airy functions are low. It is probable that this is due to temperature gradients inside the particle in excess of that found at the surface. This is possible in a spherical particle because the absolute value of
Comparisonof effectivenessfactors for the sphericalparticle,........... dataof TINKLERandMETZNER,388
presentwork.
Non-isothermal
reaction in catalyst particles
multiple steady states can also be found for flat pellets and the nature of the disagreement between the results of TINKLER and METZNER and the present work may be due to this cause. The tabulated values of Airy functions were not sufficiently extensive to support a search for multiple solutions, and for this reason no conclusions can be drawn concerning upper and lower steady states. The tabulated values of Weber functions have not been carried into ranges of the variables applicable to this work. The values may be extended by means of the recurrence relations, D,+ I(Y) + @,(Y) + PD,- I(Y) = 0 d D,(Y) + ~YD,(Y) dy
;
- PD,- I(Y) = 0
(33)
DJY) - +YD,(Y) + D,+ i(y) = 0
FIG. 4. Comparison flat pellet,
of effectiveness factors for the ---..........- data of TINKLER and METZNER, present work.
R2 dT/dR is a maximum at the surface, but the temperature gradient is not necessarily a maximum there. The same pattern of agreement and disagreement is found in the results for the flat pellet. The results for endothermic reactions agree, while agreement for exothermic reactions is found only uptos = 2. The absolute value of the temperature gradient in the flat pellet is a maximum at the surface; one would therefore anticipate that the use of this maximum gradient would lead to an overestimate of the rate of reaction in the particle, but Fig. 4 shows that the results of TINKLER and METZNERare greater for large values of 6 ! A comparison of the results for the spherical particle with those of WEISZ and HICKS [l] gives the same pattern of agreement. WEISZ and HICKS found from their results that multiple solution for the steady state effectiveness factors could be found for a range of small values of 4. Some further evidence in support has been recently advanced by MCGUIRE and LAPIDUS [9] who found two steady state temperature distributions in porous catalyst particles for this reaction system, and by WEI [lo] in a study of the stability of temperature profiles in catalyst particles. It is possible that
The comparisons with other work show that for endothermic reactions and for reactions of low to moderate exothermicity, the expression for concentration and temperature distributions are probably accurate. For reactions of stronger exothermicity, the rates of reaction predicted are too small. This is due in part, at least for spherical particles, to the increase of temperature gradient inside the particle from the surface condition. But part of the difference is also possibly due to the existence of multiple steady states at low values of the Thiele modulus. Some consideration of the use of Weber functions to describe the temperature and concentration profiles may throw further light on the question.
389
NOTATION A C 9
DP E
A: k P
Arrhenius frequency factor Concentration moles/unit volume Effective diffusivity Weber function of order p Arrhenius activation energy Gas constant Enthalpy change on reaction Reaction velocity constant Weber function tiarameter Dimensionless radial co-ordinate = (R - Rs)/Rs Radial co-ordinate Temperature Airy function
R’ T
u
D. 1. GLJNN p Thermal conductivity I# Thiele modulus R112k,/g
Greek letters a /i 8 7
Dimensionless temperature gradient E/(GT#s) @T/d& Coefficient in second order term of Taylor series defined by (20) AHC&@/pE/(GTP) Effectiveness factor
Subscripts 0 s
Conditions at the centre of the particle Surface conditions.
REFERENCES WEISZ P. B. and HICKS J. S., Chem. Engng Sci. 1962 17265. PETERSEN E. E., Chem. Engng Sci. 1962 17 987. SATIYERFIELD C. N. and SHERWOODT. K., The Role of Diffusion in Catalysis, Addison Wesley 1963. PETEMENE. E. Chemical Reaction Analysis, Prentice-Hall 1965. SMIRNOVA. D. Tables of Airy Functions, Pergamon Press 1960. KIREYEVAI. Y. and KARPOV K. A. Tables of Weber Functions, Pergamon Press 1961.
T~ICOMIF. G. Di’rential
Equations, Blackie 1960.
TINKLERJ. D. and METZNERA. B., Znd. Engng Chem., 1961 53 663.
MCGUIRE M. L. and LAP~DUSL., A.Z.Ch.E. JL, 1965 11 85.
WEI J., Chem. Engng Sci. 1965 20 729.
R&unB-Dans le cas dune reaction irreversible du premier ordre l’auteur a Ctudie les interactions entre la transmission de chaleur, la diffusion et la reaction a l’interieur d’une particule poreuse de catalyseur. On peut obtenir les solutions analytiques quand la constante de vitesse de la reaction est represent&e par une serie de Taylor tronqu&e, et, de cette facon, le probleme se reduit a la solution dune equation transcendante ordinaire. Quand la constante de vitesse de la reaction varie lineairement, l’expression analytique de la temperature et les profils de concentration sont don&s a l’aide des fonctions d’hiry. Une comparaison des facteurs d’efficacite calcules par cette methode avec d’autres travaux montre que pour les reactions endothermiques et pour les reactions peu ou moyennement exothermiques, l’approximation lineaire est satisfaisante. Pour les reactions fortement exothermiques on peut noter et discuter quelques differences. Zusammenfassung-Die Wechselwirkung zwischen Warmeleitung, Stofftransport und chemischer Reaktion innerhalb eines poriisen Katalysators wurden fiir den Fall einer irreversiblen Reaktion 1. Ordnung untersucht. Wenn die Reaktionsgeschwindigkeitskonstante durch die ersten Glieder einer Taylor-Reihe ausgedriickt wird, erhllt man eine analytische Losung, wobei das Problem auf die Losung einer gewtihnlichen transzendenten Gleichung reduziert wird. Der analytische Ausdruck zur Beschreibung von Temperatur- und Konzentrationsprofilen wird als Airy-Funktion angegeben, wenn die Reaktionsgeschwindigkeitskonstante von den Variablen linear abhangig ist. Vergleicht man den so bestimmten Porenausnutzungsgrad mit Literaturangaben, so sieht man, da5 diese linearisierende Nlherungslosung fur endotherme oder schwach exotherme Reaktionen ausreicht; bei starkerer Exothermitlt treten Abweichungen auf, die ebenfalls diskutiert werden.
390