Temperature Excursions in Adiabatic Packed Bed Reactors Shoichi Kimura and Octave Levenspiel’ Department of Chemical Engineering, Oregon Sfate University, Corvallis, Oregon 9 733 1
This paper distinguishes between the two types of temperature excursions in adiabatic packed bed reactors, the hot spot and the unstable temperature runaway. For the latter a simple graphical procedure using the temperature vs. conversion diagram is developed to tell whether a proposed operating point is stable or not, and to locate the envelope of conditions which must be avoided in design. The method is extended to reactions in the strong pore diffusion regime.
The temperature vs. conversion diagram does for adiabatic packed bed reactor design what the McCabe-Thiele does for distillation: it provides a simple tool for determining performance as well as a direct visualization of the operations. Figure 1 sketches the optimal design for staged operations with exothermic reactions. Points A’, B’, C’, D’, , . . represent reactor conditions when there are no temperature limitations; points A”, B’, C‘, D”, . . . represent the best we can do when there is an upper limit to the allowable temperature, T,,,. Figure 2 sketches the corresponding optimum operations when infinite recycle of fluid is used a t each stage. Methods for finding the best reactor temperatures and distribution of catalyst, in effect the optimal location of points A, B, C, D, . . . and P, Q, R, S, . . . are known (Konoki, 1956, 1960,1961; Levenspiel, 19?2),and this design procedure has also been extended to moderate recycle, and to the injection Of cold feed or cold inert between stages. However, one point has not been considered: whether the chosen design points are stable or not, whether they are liable to a temperature runaway or hot spot behavior. We will treat this problem of temperature excursion here, and present the results primarily in terms of the conversion vs. temperature diagram. First of all, there are two distinct types of temperature excursion in adiabatic packed beds: (1) the hot spot, characteristic of the packed bed with no recycle of gas, thus plug flow of gas through the reactor; (2) temperature runaway, characteristic of the packed bed with large recycle of gas, thus mixed flow of gas. Since we use staged packed beds without recycle when the adiabatic line is steep (moderate adiabatic temperature rise), it is this system which incurs hot spots. On the other hand, staged reactors with large recycle of fluid are used when the adiabatic line is shallow (extreme adiabatic temperature rise) and it is here that the temperature runaway may pose a danger. We treat these two types of temperature excursions separately.
Temperature Excursion in Staged Plug Flow Reactors-The Hot Spot Here a hot spot usually is not mobile and is caused by deviation of flow from the ideal of plug flow, some fluid going faster and some going slower. To illustrate this phenomenon, imagine a packed reactor operating in two ways, plug flow in one case, and two parallel flow velocities in the other. Figure 3 illustrates what happens on the conversion vs. temperature diagram in a bad situation. Point 1on the adiabatic line is the expected operating point. For faster fluid conversion drops (point 2), while for sluggish fluid conversion rises (point 3). The actual hot spot temperature reached (point 3) depends on the size of the stagnancy and the rate of heat transfer to the surrounding faster-moving fluid. Point 4 is the very highest expected temperature for completely stagnant fluid. For steep
adiabatics this maximum temperature is not very much higher than the design point. In a recent paper Jaffe (1976) suggests how to simulate and estimate the hot spot temperature for commercial hydrogenation processes. This type of hot spot is likely to occur at some obstruction in the packed bed: a baffle, an inserted tube, and so on. A temperature probe is likely to create its own hot spot. Luckily such hot spots are localized and not malignant. Also, since the packing density of catalyst is lower near the walls of a reactor, flow is faster there, conversion is lower, so that the fluid is cooler at the walls of a reactor.
Temperature Excursion in Beds with Large RecycleTemperature Runaway The material balance curve, the locus of constant values of weight-time 7, is S-shaped, as shown in Figure 4. For a given feed temperature the energy balance line and S-shaped material balance line for a particular operating 7 intersect a t either one or three points, which give the conditions within the reactor. These are shown in Figure 5 . van Heerden (1953) discusses and gives examples of this type of reacting system. In the case of three solutions, points 1and 3 are stable while point 2 is unstable. At the unstable point we always find that slope of material balance curve
i
slope of energy
) > ( balance line
or
slope of the (constant 7 line)
>- C P A -fir
where C p is~ the mean specific heat of the fluid per mole of entering reactant A, AHr is the heat of reaction per mole of entering reactant A, and the ratio of these two gives the slope of the energy balance line (see Levenspiel, 1972).We use this property to distinguish between stable and unstable operating points. Construction of the Constant T Lines for Mixed Flow on the Conversion vs. Temperature Diagram. The conversion vs. temperature diagram for a given reaction normally has curves of constant rates drawn in, as shown by the dashed lines in Figure 4. To draw in the constant T lines on this chart we note that for mixed flow
So at convenient points on the diagram find X A and -rA, and then the corresponding T from eq 2. From these many 7 values draw the locus of constant T’S. The results will be a set of curves as shown in Figure 4. To Find Whether a Given Operating Point on the Conversion vs. Temperature Diagram Is Stable. The Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 1, 1977
145
A the mixed flau reactor
Cool
Cool
Maximum ollouable temperature
Lacus o i maximum
/ reoctian rate
A
u -T
XA
Reversibfe in this T range
frreversibfe in this T range
Figure 4. Conversion in a mixed flow reactor as a function of T and Tmox
T
T.
Figure 1. Optimal design for staged operations of exothermic reac-
tions.
3 solutions
i Cool
Cool
Cool
lntersection only
at lau Xa
Different energy balance lines
Maximum allauable temperoture
1 acus a i maximum reaction rate
T
TI
Figure 5. Three types of solutions to the energy and material balances
for exothermic reversible reactions.
/
temperature
I
limilatian
I
Siape of material balonce line
temperoture limilatian
I
0
xA2
b
T
Tmox
XAI
Figure 2. Optimal design for staged operations of exothermic reactions when m recycle of fluid is used at each stage.
......, , . ...... 1 . . . . , .. . . . . . . . . . .. I
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Adiabatic line I
I/ -'A2 b
T
Figure 6. Comparison of slopes of the material balance line and the adiabatic line. Operating point M is unstable.
A
adiabatic line
far slugglsh fluid X, ond
fq
XA
for fasfer
XA
L ,temperature iar
equilibrium expece td\
P
I
operating paint
1 Tf
point M. b
T
Figure 3. The shift in plug flow operating point because of stagnancy. Points 3 t o 4 represent hot spot conditions.
procedure is shown in Figure 6 and is as follows: (i) evaluate X Aand ~ r A 1 for the point M in question; (ii) go to the nearest constant r A line, and on it locate point N such that
146
T
Figure 7. The stable points P and Q corresponding to an unstable
and T drop
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Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
(iii) join M and N; this is the slope of the material balance (or constant T ) line; (iv) compare this slope with the slope of the adiabatic, see eq 1. If it is greater, then point M is unstable and one cannot operate a reactor at that condition. If the slope of MN is smaller than the adiabatic then point M is stable. To Locate the Stable Points Corresponding to an Unstable Point M. Simply continue the material balance line from M until it again intersects the adiabatic at P and Q.As shown in Figure 7 the conversion for the stable points is either
diffusion strong port free 'r'diffusion
Fixed concentrotion
t
I
diffusion free strong pore diffusion
XA
m~oiffere T
Figure 9. Shift in the constant T line in the strong pore diffusion regime.
concmtraticins of inert
ddiobotic
This is the general expression for the material balance line, to be compared with the adiabatic line according to eq 1to find whether a proposed operating point is stable or not. For the first-order reversible reaction A e R with rate
XA
-rA Feed
Figure 8. Construction of the region of instability on the X A VS. T diagram.
close to zero or close to equilibrium. Consequently, although you may expect your reactor with large recycle to operate a t point M, it will be either at point P or point Q depending on startup conditions. To Map the Region of Instability. To find the condition in which a reactor cannot operate, where a temperature runaway is expected, draw a number of constant T lines on the conversion vs. temperature diagram. Then from the concept of instability given in eq 1 and Figure 5 we see that the boundary between stable region and unstable region is where the slope of the constant r line equals the slope of the adiabatic line. Thus, drawing tangents and joining the touching points gives the boundary of the unstable region, as shown in Figure 8. The shaded area of Figure 8 represents unattainable operations. For different concentration of inerts the slope of the adiabatic changes and the region of instability likewise changes. For a lot of inerts the unstable region disappears. In general, the region of instability is far from the equilibrium. It is always to the left of the maximum reaction rate line. Therefore in operations where the conditions are close to optimal, as shown by points P', Q', R', . . . in Figure 2, there is no need to worry about temperature runaways. But with a temperature limitation, as shown by points P", Q', R", . . . in Figure 2, this problem will have to be considered. Temperature Runaway Given a Rate Equation for the Reaction. So far we have shown how to locate unstable operations when only a graphical representation of the rate data is available. However, if a rate equation is available then a simple analytic solution to this problem is possible. Rearranging eq 2 we find
= k l C A - k2CR;
k , = k;Oe-Ei/RT
(6)
insertion of eq 6 in eq 5 and combining with eq 1gives for the region of instability
h i T [ E i + ( E l - Edk2rI CpA (7) RT2[1 ( k l + k z ) ~ ] ' -AHr For first-order irreversible kinetics k2 = 0; hence the condition for instability is
+
with these expressions it is easy to tell whether a proposed operating point is liable to temperature runaway. Temperature Runaway When the Catalyst Is in the Strong Pore Diffusion Regime. When operating with catalyst in the strong pore diffusion regime, a first-order reaction A R has an observed rate given
-
For this condition the constant r line on the conversion vs. temperature diagram is shallower than for operations in the diffusion-free regime, with the ratio of slopes given by eq 10 at given temperature
1-1
where * represents operations in the strong pore diffusion regime. To find where the transition to strong pore diffusion occurs, we use the condition of Weisz and Prater (1954)
(4) Differentiating with respect to T a t constant T gives
Inserting eq 11 into the performance expression for mixed flow, eq 2, shows that the transition to strong pore diffusion regime occurs where 1
(12)
Rearrangement then gives the slope of the constant T line
(k)
a(-rA)
(
F
I
X
,
(5)
On combining eq 12 with eq 4 we can show that this transition occurs a t a fixed temperature, independent of T or XA.Figure 9 displays the consequences of the intrusion of strong pore Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 1, 1977
147
diffusion: its onset at fixed temperature, the drop in the slope of the constant T line, and ratio of slopes. For nth-order reaction a treatment similar to that for first-order reactions but using the nth-order rate and the generalized Thiele modulus (see Bischoff, 1967) gives for the onset of strong pore diffusion (13)
We can also show that the dividing line between regimes has a positive slope for n > 1,and negative slope for n < 1.Note that Figure 9 gives a vertical line for this transition for R = 1. In operating under strong pore diffusion the shallower T curve will give a much smaller region of instability where a temperature runaway is possible. Its size and location are found by the procedure described earlier and illustrated in Figure 9.
Conclusion In adiabatic packed bed reactors we encounter two types of temperature excursions: the hot spot and the temperature runaway. The hot spot is characteristic of reactors having no recycle of fluid. It is caused by fluid stagnancy and is localized, usually near flow obstructions. Since the adiabatic line on the conversion vs. temperature diagram is steep for operations with no fluid recycle, the temperature rise for hot spots may not be too serious, and it can be estimated directly from the conversion vs. temperature diagram. The temperature runaway results from trying to operate a reactor with large recycle at an unstable point. We show that the conversion vs. temperature chart is a convenient tool for identifying these unstable points, for finding the corre-
sponding stable points, and for finding the envelope of operating conditions which must be avoided in design. Extension to strong pore diffusion kinetics is direct. Finally, simple analytical expressions are developed for identifying unstable operating points when a rate equation for the reaction is available. Note that the treatment in this paper is limited to steadystate operations.
Nomenclature CAO= concentration of reactant in feed, mol/m3 C P =~ specific heat of the fluid per mole of entering reactant A, cal/mol K ZI = effective diffusion coefficient in a porous structure, m2/s E = activation energy, cal/mol Eobs = observed activation energy, cal/mol -AHr = heat of reaction per mole of reactant A, cal/mol k l , k z = first-order reaction rate constant, s-1 k o b s = observed reaction rate constant R = ideal gas law constant, cal/mol K - r A = rate of reaction based on unit mass of catalyst, molkg of catalyst s T = temperature, K X A = fraction of reactant A converted into product T = weight time, kg of cat s/m3 feed Literature Cited Bischoff, K. B., Chem. Eng. Sci., 22, 525 (1967). Jaffe, S.B., lnd. Eng. Chem., Process. Des. Dev., 15, 410 (1976). Konoki, K., Chem. Eng. (Jpn.),21, 408. 780 (1956). Konoki, K., Chem. Eng. (Jpn.), 24, 569 (1960). Konoki, K., Chem. Eng. (Jpn.), 25, 31 (1961). Levenspiel, 0.. "Chemical Reaction Engineering," 2nd ed, Chapters 8 and 14, Wiley. New York, N.Y., 1972. van Heerden, C., lnd. Eng. Chem., 45, 1242 (1953). Weisz, P. B., Prater, C. D., Adv. Catal., 6 , 143 (1954).
Received for reuieu: J u n e 1, 1976 Accepted September 15,1976
Thermodynamics of the Reaction between Sulfur Dioxide and Methane John J. Helstrom and Glenn A. Atwood' The Department of Chemical Engineering, The University of Akron, Arkon, Ohio 44325
The equilibrium composition of the reaction products for the reaction of methane and sulfur dioxide in an inert diluent are presented in this work. The compositions were calculated using data available in the literature for a total system pressure of 1 atm, reaction temperatures between 400 and 1000 K, methane to sulfur dioxide ratios from 0.1 to 2.0, sulfur dioxide concentrations of 0.003 and 0.1 atm, and various initial concentrations of H 2 0and COP.The calculations show that the equilibrium yield of elemental sulfur is maximized at methane to sulfur dioxide ratio of 2. Below 800 K, lowering the temperature increases the yield of elemental sulfur and above 800 K increasing the temperature increases the sulfur. This work is unique in that all of the likely reaction products were included as well as the seven sulfur species which are known to exist in the vapor state.
Since the 1930's there has been an interest in the reactions of methane with sulfur or sulfur dioxide to produce hydrogen sulfide, carbon disulfide, or sulfur. Work on the last reaction has increased greatly during the past decade because of its possible utility in controlling air pollution caused by the sulfur dioxide emitted from fossil fuel power plants and ore smelting operations. However, the equilibrium compositions 148
Ind. Eng. Chem., ProcessDes. Dev., Vol. 16, No. 1, 1977
for the reduction of sulfur dioxide with methane have been calculated for only a limited number of reactant compositions and temperatures. Lepsoe (1937) presented equilibrium compositions for the reaction of hydrogen with sulfur dioxide and calculated the equilibrium constant for the reaction CH4 2,902 COS Sp HzO. Thacker and Miller (1944) provided equilibria for methane, sulfur, and CS2 but they neither
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