CHAPTER 6 THE DESIGN OF FIXED BED CATALYTIC REACTORS-I
6.8 Effectiveness factors for non-isothermal catalyst pellets [Smith, 1981]
For the simple chemical reaction AoB, neglecting the temperature dependence of De, the material balance equation for a spherical shell of (spherical) catalyst pellet can be written as:
2 A 2 A 2
e e
r r r
dC dC
4 r D 4 r D 4 r r k C
dr dr '
S S S '
ư ẵ ư ẵ
đ ắ đ ắ
¯ ¿ ¯ ¿ UP 1 A (Eq. 6.97)
Dividing by 4 rS 2'r and taking the limit as'ro0,
2
1 p A
A A
2
e
k C
d C 2 dC
dr r dr D 0
U (Eq. 6.98)
with boundary conditions dC / drA 0 at r=0 and CA=CAs at r=Rs. The energy balance over the same spherical shell volume element can be written as:
2 2 2
e e P 1
r
dT dT
4 r 4 r 4 r r k C H
dr dr
S O S O S ' U '
ư ẵ ư ẵ
đ ắ đ ắ
¯ ¿ ¯ ¿ A r (Eq. 6.99)
In the last three equations,UPis the density of the pellet itself. Neglecting the temperature dependence of Oe, the effective thermal conductivity, dividing by 4 rS 2'r and taking the limit as'ro0, we get:
2
1 P A r
2 e
d T 2 dT k ȡ
C ǻ H r dr Ȝ
dr
0. (Eq. 6.100) with at r=0 and T = Ts at r = Rs. Note that the exponential dependence of the reaction rate on the temperature makes the pair of equations non-linear.
dT/dr 0
^ `
1 1,0 a
k k exp E / RT (Eq. 6.101)
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6.8.1 Calculating the maximum temperature rise
To find an analytical relationship between the reactant concentration and the temperature at any point in the pellet, we eliminate k1UpCA from the two equations.
2 2
A A e
e 2
r
Ȝ
d C 2 dC d T 2 dT
D dr r dr ǻH dr2 r dr
ư ẵ ư ẵ
° ° ° °
đ ắ đ ắ
° ° °
¯ ¿ ¯ °¿. (Eq. 6.102) This equation can be re-written as
2 A e
2 e 2
r
Ȝ
1 d dC 1 d dT
D r r
dr dr ǻH dr dr
r r
ư ẵ ư
đ ắ đ
¯ ¿ ¯
2 ẵ
ắ¿. (Eq. 6.103) We then integrate both sides, twice. For the first integration, we use the symmetry boundary condition:
and at r=0. That makes the lower bound of the integration equal to zero, for both sides of the equation. We integrate again using the surface boundary conditions.
dC / drA 0 dT/dr 0
r e
s p A
e
ǻ H D
(T T ) (C C )
Ȝ A,s p (Eq. 6.104) The maximum temperature rise would occur when all reactant has been consumed.
r e
s max A,s
e
ǻ H D
(T T ) C
Ȝ
(Eq. 6.105) This result is valid for any particle geometry at steady state and is not limited to first order kinetics.
6.8.2 Effectiveness factors for non-isothermal catalyst pellets
The two differential equations (Eqs. 6.100 and 6.102) must be solved numerically. We will illustrate the nature of the problem for a spherical geometry. The problem is usually presented by defining three new parameters.
1. Thiele – type modulus, evaluated at surface conditions:
s s s 1 s p e
3( ) ) r (k ) ȡ / D (Eq. 6.106) 2. The Arrhenius number
Ȗ E/R Tg s, and, (Eq. 6.107) 3. The heat of reaction parameter
R e A,s e s
( ǻ H ) D C ȕ ȜT
(Eq. 6.108)
The solution of the system of equations shows contours [cf. Smith, 1981, p. 502, Fig. 11-13] for the effectiveness factor for a non-isothermal pellet plotted against the modified Thiele modulus, for a range of values of E and J = 20. Clearly, E = 0 traces the line for an isothermal pellet; values of E < 0 trace contours for endothermic reactions.
The contours indicate that the value of K no longer ranges simply between zero and unity. During exothermic reactions, heat generated inside the pellet may give rise temperature increases inside the pellet; thermal conduction rates are never instantaneous, so temperatures may rise compared to the surface. Values of K >1 represent higher rates of reaction inside the pellet compared to the surface. Since reactant concentrations are higher at the surface of the pellet, K >1 arises due to higher temperatures boosting the rate inside the pellet.
Large increases in internal temperatures due to highly exothermic reactions may cause deactivation by sintering; they may also give rise to undesired side reactions which would reduce selectivity.
The contours also show that for certain values of the parameters, three possible values of K can be calculated;
in other words, multiple solutions are possible due to the non-linear heat generation curve. Only the highest
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and lowest states, corresponding to large and small temperature gradients respectively, represent stable states, whilst the middle value corresponds to a meta-stable state.
In Eq. 6.108, we defined E as ȕ ( ǻ H ) D Cr e A,s/ȜeTs. Combining with'Tmax ǻ H D Cr e A,s/Ȝe in Eq. 6.105, we getȕ ǻ Tmax/Ts. For many industrial applications E is a small number; typically E 0.1. Since the thermal conductivities of solids, including porous solid matrices are high, for standard catalysts at steady state conditions, 'Tmax in catalyst pellets is usually not very large. Usually external heat and mass transfer resistances have a more significant effect on reaction rates. The multiple steady-state behaviour deduced from the calculations is therefore not likely to be observed in common catalytic reactions.
For highly exothermic reactions such as hydrogenation, however, 'Tmax may be quite significant. In one example, the hydrogenation of benzene, which has a heat of reaction of about 210 kJ mol-1 gives rise to a maximum temperature rise of about 35 °C from the centre to the surface of the pellet. This is a gradient that would affect the reaction rate quite significantly, without giving rise to pellet instability.
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CHAPTER 7