Howland, J. L., Vaillencourt, R., J . SOC.Znd. Appl. Math., 9 ( 2 ) , 165-8 (1961). Levenspiel, O., “Chemical Reaction Engineering,” Wiley, New York, N . Y., pp 244-8. Levenspiel, O., Bischoff, K. B., Aduan. Chem. Eng., 4, 95-198, (1963). Levenspiel, O., Smith, W. K., Chem. Eng. Sci., 6, 227 (1957). Moser, J. H., Cupit, C. R., Chem. Eng. Progr., 62 (6), 60-5 (1966).
Place, G., Ridgway, K., Danckwerts, P. V., Trans. Inst. Chem. Eng., 37, 268-76 (1959). Van Deemter, J. J., Chem. Eng. Sci., 13 (3), 143-54 (1961). Van der Laan, E. Th., ibid., 7, 187 (1958). Wen, C. Y., Chung, S. F., Can. J . Chem. Eng., 43 (3) 101 (1966). RECEIVED for review August 7, 1969 ACCEPTED December 7, 1970
Analysis of Nonisothermal Moving Bed for Noncatalytic Solid-Gas Reactions Masaru lshida and C. Y. Wen’ Department o f Chemical Engineering, West Virginia Uniuersity, Morgantown, W . Va. 26506 The design and performance of a moving bed for solid-gas reactions under nonisothermal conditions are discussed based on unreacted-core shrinking model. Numerical solutions based on unsteady-state analysis are compared with graphical solutions based on pseudosteady-state analysis. The pseudosteady-state analysis i s satisfactory if no abrupt temperature change occurs in the bed. For an exothermic reaction, thermal instability may occur. In such cases, the initial temperature of solid will considerably affect the reactor performance. Moreover, the transition of rate controlling steps occurs more readily in a moving bed reactor than in a single particle.
M o v i n g beds, rotary kilns, and fluidized beds have been widely used in industries for solid-gas reactions. In spite of this, the design and performance of these reactors are not thoroughly based on fundamental principles of solidgas reactions. A nonisothermal analysis of both irreversible and reversible reactions between gas and a single particle was presented both from theoretical and experimental points of view in previous papers (Ishida and Shirai, 1969, 1970; Ishida and Wen, 1968; Wen, 1968; Wen and Wang, 1970). The important role played by heat of reaction in solidgas systems, was discussed. If the basic information obtained from a single particle can be applied directly to the design of the multiparticle systems, the optimal conditions of an industrial-size reactor can be selected. Such attempts have been made by several investigators. Meissner and Schora (1960) analyzed a multistage reactor for reduction of iron ore by comparing it with a distillation column. Yagi et al. (1968) analyzed a moving bed reactor in which isothermal solid-gas reaction took place. Horio et al. (1969) extended this study to nonisothermal cases assuming that the temperature of the solid is uniform and the reaction proceeds under the chemical reaction controlling regime. Jackman and Aris (1968) discussed the optimal control of pyrolysis reactors in which a coke layer builds u p decreasing the heat transfer coefficient a t the wall. Since these analyses are made either under
’ To whom correspondence should be addressed. 164
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
isothermal condition or in the absence of diffusional effect within the solids, they can only be applied to restricted cases. This paper will present nonisothermal steady-state analysis of a moving bed utilizing the knowledge gained in the single particle studies. I n these studies, the heat of reaction and simultaneous mass and heat transfer within the particle are taken into consideration. Thermal and transitional instabilities of exothermic reactions are also discussed. The particle follows the unreacted-core shrinking model, while solids move in the bed according to plug flow. Equation Derivation
Consider the following reversible solid-gas reaction
S,
+ aA
p1
Z SI,
+ bB,
(1)
In an unreacted-core shrinking particle, the reactibn rate per unit reaction surface area, m 4 , is assumed to be uniquely determined by a set of intensive properties a t the reaction surface as follows: mA
= mA(TC, C A c , C B ~C, I ~ )
(2)
Furthermore, the particle size is assumed to be constant during the reaction. Figure 1 shows a schematic diagram of a moving bed. The gas at temperature ( T o ) ois fed from z = 0 a t a molar flow rate, G, and leaves the reactor a t t = 2 a t a temperature, ( T o ) l .I n cocurrent flow, the solids, S,
a t temperature ( T J oare introduced a t z = 0 at a molar flow rate, F ( > O ) . They react with the gas component, A , and are discharged at z = Z. I n countercurrent flow, the solids a t temperature ( T J 1are introduced at z = Z a t a flow rate, F ( < O ) , and are discharged a t z = 0. Transport Phenomena within Solid Particles. The basic equations are as follows: Mass Transfer
aca -- -1- a e at
( C D d -)axa
rL ar
(31
ar
If the heat capacity of the solid is much smaller than the heat of reaction and the left-hand side of Equation 4 is also negligibly small (pseudosteady-state approximation for heat transfer), Equation 4 can be integrated as
where 6, I = R/[(h,&h,) - 111 is the corresponding equivalent thickness of gas film for heat transfer. Here, h, represents an overall heat transfer coefficient including both the convective and the radiative modes of heat transfer. From Equations 5 and 6, the reaction rate, m A , T , , can be eliminated as
atr=R Heat Transfer
(4)
aT
at r = r,
he- = m.4AH.4 ar k, aT = h,(T, - T ) dr
atr=R
Equation 3 is based on equimolal counterdiffusion The case with unidirectional diffusion where b = 0 was discussed elsewhere (Ishida, 1969). Since for solid-gas reactions, the left-hand side of Equation 3 is negligibly small (pseudosteady-state approximation for mass transfer), i t can be integrated as
where 6,, { = R / [ ( h , , R / D e A )- 11) is the thickness of solid product layer whose diffusion resistance is equivalent to that of gas film around the external surface of the solid particle (Ishida and Shirai, 1969).
Hence, Equations 2, 5 , and 7 are the basic transport equations for a solid particle with negligible heat capacity reacting with a gaseous reactant. However, the pseudosteady-state approximation does not always hold for heat transfer (Wen, 1968; Ishida and Shirai, 1970),particularly in a moving bed where the surrounding gas temperature, T o , changes with the solid conversion. Since it is difficult to solve the partial differential equation, Equation 4, a more simplified heat balance equation with the following two assumptions is used (Beverage and Goldie, 1968; Wen and Wang, 1970). The temperature within the unreacted-core is uniform a t the reaction temperature, T,. The temperature profile within product layer can be approximated by the following pseudosteady-state profile.
Then, the accumulation rate of heat per unit particle, d H / d t , is
-F
where
A = [ 3 ( R+ 6h)(R'- rj) - 2(Rd- d ) ] / ( R+ 6 h - r0 A' = ( R - rc)' (2R + r J ( R + 6h)/(R+ 6 h Instead of Equation 4, the heat balance equation for a solid particle in the bed becomes
F
G
Figure 1. Schematic diagram o f a moving bed for solidgas reactions indicating both cocurrent and countercurrent flows
4a R'h,(T, - To) (10) Finally, from Equations 8, 9, and 10, we get Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
165
T, + %n-CScA rcA dT, + A' dT o+ Mn-rlCaoc.5ddt [ dt dt ( R + dh)A dr, dr, ( T , - T o )- + 4 n r ; a C ~ ( - A H ~) + R + f i h - rc dt dt redh 4xRh, - ( T , - T o )= 0 ( 1 1 ) R + 6 h - rc Hence, Equations 2, 5 , and 11 are the basic equations for solid-gas reaction in a particle in which the heat capacity of solid is also taken into account. Axial Transport Phenomena in Moving Bed. I t is assumed that there is no mixing of solid particle and that the radial profiles for temperature and concentration of the reactant gas are flat. Thus, Mass transfer
CE,
d'xa,, dGa dFs -_ _ _ +adz' dz dz
=o
(12)
where X is the conversion of solid reactant S and F / 1 t r ) C , is the velocity of solid particles flowing in the reactor. Analysis of Isothermal Reactions
For simplicity, the following relations are assumed. a = b,
E, = 0, m A =
(17)
The last equation shows that the reaction is irreversible and its rate is proportional to the initial solid concentration, CS(>,and the reactant gas concentration, CAc,(Ishida and Wen, 1968). From Equations 12, 14, and 15,
xA0/ (xAo)o= B + [ U P /(Ga)o](r,/R)'
(18)
where
B = 1 - [ U F / ( G ~for ) ~ cocurrent ] flow B = 1 for countercurrent flow From Equation 5 , we have
dr, -
dt
-k,C4, 1 + 4 q ( r c / R )[l - rc/ ( R + S m ) ]
(19)
where & = Rak,Cs,/ De* is the ratio of the chemical reaction rate to the diffusion rate. Hence, the effectiveness factor, q\, defined as the ratio of actual reaction rate represented by Equation 19 to the rate without diffusion effect, can be expressed as
vs = 1 / 1 1 + 4 d r J R ) [1 - r , / ( R + 6 d ] l
The following relations hold for moving bed:
(20)
Since Equation 20 is precisely the same equation derived for an isothermal single particle (Ishida and Wen, 1968; Ishida, 1969), v 7 for a first-order reaction is not affected by the change of the concentration of A in the reactor. The time required for complete conversion can be obtained as
{M (3)' ' (
(-
1
4s
1 -
4s
-
k ) In
kd ( k + 1)' k'R In -~ +k'+1 R+6, kJ+l
--
+ k ) [ tan-'
2-k (3)' ' k + tan-'
~-
+
1 p]) (a')
where
k = [B(Ga)o/aF]'' When a F / (GJo is very small and ( x ~ =) (xa) ~ ,, Equation 21 can be reduced to the following equation for a single particle.
Figure 2. Relation between overall gas conversion and reaction time (isothermal case) 166
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
Figure 2 shows the ratio of Equation 2 1 to Equation 22 as a function of the conversion of the reactant gas, 1 - [ (xA)l/(x.4)a],for various rate controlling steps. When gas film diffusion is the rate controlling step, no difference is seen between countercurrent and cocurrent operation. For other cases, the countercurrent reactor is always more effective than the cocurrent reactor.
Analysis of Nonisothermal Reactions
I
0
z I2 (Cad0 QA)Q aCso R2
0 .36
1 .o
0.3
0 0.07
0.1
0.2
w'
0.8
0.06
0
cr:
h
3
u \
1
2 0.6
0.05
$
,o 0
d2 a4
0.04
X
Since equations for nonisothermal reactions cannot be integrated analytically, two approaches are used here. The first is a numerical method based on unsteady-state equation given in Equation 11. The second is a graphical method based on the pseudosteady-state equation given in Equation 7 . Conditions given b y Equation 1 7 as well as U = E h = 0 are assumed in the following discussion. Numerical Solution. Two examples in Figures 3a and b are calculated results of an endothermic reaction for cocurrent flow and countercurrent flow. respectively, both under the same operating conditions. Since from Equation 5 , the reaction rate of solid, m,, is proportional to ( x s - x , , ) , the rate in cocurrent flow is highest just after the initial period (Figure 3a). AS the reaction proceeds along the reactor length 2 , x a and T decrease rapidly owing to the endothermicity of the reaction, giving rise to the decreasing reaction rate. On the other hand, in countercurrent flow (Figure 3b), the solids introduced from the top ( z = 2) of the reactor immediately come in contact with the low temperature and low concentration gas, resulting in a low initial rate. However, as the solids move downward, they encounter the gas a t a higher temperature and a higher concentration resulting in a much shorter reaction time than in a cocurrent flow. Figure 4 shows the calculated result of an exothermic reaction taking place in a cocurrent flow reactor indicating that the concentration of the reactant gas decreases as the reaction proceeds along the axial distance, z. However, the gas-phase temperature increases with an increase in the reaction rate, in contrast t o the case of endothermic reaction (Figure 3a). Reaction Diagram. From the foregoing discussion, the problem of a moving bed without solids mixing can be replaced by that of a single particle if the temperature and the concentration of reactant gas are allowed to change
0.2
n "
0
0.2
0.8
0.6
0.4 2
1.0 1.o
I2
-
0.5
0
0.2
0
Lo--
1.5
2.0
2.42
0.6
0.8
1.0
Figure 3. Examples of concentration and temperature profiles in a moving bed reactor for an endothermic reaction a. Cocurrent flow
0.8
b. Countercurrent flow 0
h
Since CD, is assumed to be independent of temperature (Ishida and Shirai, 1969), Equation 2 1 holds not only for the isothermal case but also for the nonisothermal case when the chemical reaction rate is much faster than the diffusion rate-i.e., $& + a . This case, therefore. gives the minimum reaction time. Hence, from Equation 16, the minimum length of reactor for complete conversion is obtained as
. 2 0
XQ
v
0.6
0
h
40.4 XQ X
0.2
'/L In
kJ+ 1
___
(k+l)j
Rk + ___
R+s,
In
k' kJ+l
~-
n "
+
2-k (3)' '[tan-' (3)"h + tan-'
0.4 z I2
--i(3)' ]1 - 1
Figure 4. Concentration and temperature profiles in the reactor for an exothermic reaction (cocurrent flow) Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
167
Figure 5. Reaction diagram for an endothermic reaction (cocurrent flow) Unsteady-state path for gas ( Cand ) solid (+) Pseudosteady-statepath for gas (-+.)
and solid (--&
Figure 6. Reaction diagram for an endothermic reaction (countercurrent flow) Unsteady-state path for gas (+)
and solid
Pseudosteady-statepath for gas (-+-)
(e)
and solid (-*-)
with the conversion of solid. Hence, the reaction diagram proposed in the analysis of nonisothermal dehydration of gypsum (Ishida and Shirai, 1969, 1970) can also be applied to this problem. The reaction diagrams corresponding to Figure 3a and b are shown in Figure 5 and Figure 6, respectively. They have the coordinates of dimensionless temperature and mole fraction. The ordinate also corresponds to the conversion of reactant gas A . In each diagram, two bold solid curves appear: One is the operating curve connecting closed circles; the other is the curve connecting open circles. The closed circle on the abscissa representing the starting point of the operating curve corresponds to the gas inlet a t z = 0. This point corresponds to solid conversion of zero ( r c / R= 1.0) for cocurrent flow and 1.0 ( r ( / R = 0 ) for countercurrent flow. The upper closed circle corresponds t o the outlet for gas a t z = 2 , a t which the solid conversion is 1.0 ( r , / R = 0 ) for cocurrent flow and 0 ( r < / R= 1.0) for countercurrent flow. Similarly, any point on the operating curve which may be called an operating point represents a certain gas conversion corresponding to the values shown on the ordinate and a certain solid reactant conversion shown by parameter ( I ; / R 1. 168
The curve connecting open circles shows the relation between T , and xAC a t the reaction surface of the solid. Each point on this curve also corresponds to a certain r , / R and is called a reaction point. Those for r,fR = 1.0, 0.96, 0.9, 0.8, 0.5, and 0.2 are indicated by large circles, and those for r , / R = 0.7, 0.6, 0.4, 0.3, and 0.1 by small circles. On these figures, the difference between the reaction point and the operating point a t a given r,/R projected on the abscissa represents the difference between the reaction temperature and the surrounding gas temperature, while the difference projected on the ordinate shows the difference in mole fraction of gas reactant A which is proportional to the reaction rate, mA. Moreover, the operating point and the reaction point a t each r,/R are related by Equation 5 . Equation 5 is shown as a sigmoid curve for each r,/R on the reaction diagram. In Figures 5 and 6, these sigmoid curves are drawn by thin, solid lines for r,/R = 1.0, 0.96, 0.9, 0.8, 0.5, and 0.2. They show the characteristics of the reaction surface and are called the reaction curve. Hence, a reaction point exists on each of the corresponding reaction curves. If the reaction point is located at the upper part of the reaction curve--i.e., x A C = 0, the rate controlling step is diffusion. On the other hand, if it is located a t the lower part of the reaction curve--i.e., x A C = x A o , the rate controlling step is the chemical reaction. Accordingly, it can be concluded that the reaction in Figure 5 proceeds in the intermediate regime between chemical reaction controlling and diffusion controlling, since each reaction point exists near the middle of each sigmoid curve. On the other hand, in Figure 6 the rate controlling step is chemical reaction at the start of reaction-Le., r , / R = 1.0 which subsequently shifts to diffusion a t about r , / R = 0.5. Graphical Solution. By using the following approximations, the problem of the moving bed can also be solved graphically. Since in Equation 11 the terms containing heat capacity make it difficult to solve, they are approximated by one term as
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
4.rrRhP where
rcdh
R
+
-
cs = (cs
dh
- re
(Tc- T o )= 0 (23)
+ ch)/2
is the average heat capacity of solid. By substituting Equation 23 into Equation 13, the relation between incremental conversion, AX,and incremental gas temperature, A T o , is obtained as follows.
On the other hand, from Equation 12
Ax,, = - ( a F / G ) A X (25) By eliminating AX from Equations 24 and 25, we get - A X A ~=
[ ( e p+ T s F / G ) / ( - A H a ) ] A T ,
(26)
Thus, the operating curve may be approximated by a straight line as given in Equation 26. The results of Equation 26 are shown by dotted lines in Figures 5 and 6.
1.0
0.8
20.6 0
3
i 5 0.4
3 X
0.2
1
0
I
2
I
1
I
0.6
0.4
0.2
0.8
1.0
IZ
Figure 7. Effect of heat loss through the reactor wall on concentration and temperature profiles (cocurrent flow)
The reaction temperature, T,, and the mole fraction of reactant gas, x A c a, t the reaction surface can be obtained graphically by Equations 5 and 7 for each operating point in Figures 5 and 6. Equation 5 is shown as a sigmoid curve (reaction curves) in Figures 5 and 6, while the tie lines given in Equation 7 based on pseudosteady-state approximation are shown by straight lines passing through each operating point. Both the reaction curves and the tie lines are thin, solid lines in Figures 5 and 6 for r,/ R = 1.0, 0.96, 0.9, 0.8, 0.5, and 0.2. The reaction points which are the intersection of reaction curve and tie line are indicated by squares. Finally, from Equations 5 and 16, the length of the reactor is calculated by the following equation.
z = (1 -F~
C
L'
"dt
JRS dr,~dr, = ('C/')['
aFRL
CD,.4(1 - 67-1
XAo
- XAc
'6,)l d
X
(2)
heat for the reaction from the gas phase becomes insufficient although that of gas reactant may be still quite adequate. Thus, a portion of the heat reaction must be provided by the heat content of the solid causing the solid particle temperature to decrease. Such a temperature change cannot be shown by the pseudosteadystate analysis, but has been observed experimentally in the burning of carbon-cement sphere. The reaction temperature increased rapidly a t the final stage (Ishida and Shirai, 1969). The numerical solution and the graphical solution are compared also in Figures 3a and b in terms of the reaction time and the solid conversion. In Figure 3a, the dimensionless time, (CA,D,4t/aC5,R'),to complete the reaction is 0.88 by the numerical solution, whereas it is 0.86 by the graphical solution indicating a good agreement. In Figure 3b, the reaction time is 0.36 by the numerical solution and 0.31 by the graphical solution. Since the reaction time for 99.2% conversion of solid reactanti.e., r , / R = 0.2-is 0.30 by the numerical solution and 0.29 by the graphical solution, the difference in the time required for completion of the reaction is primarily owing to the final stage where the graphical solution based on the pseudosteady-state approximation cannot properly represent the phenomena. Effects of Heat Loss and Axial Dispersions. So far, an adiabatic case without gas mixing was discussed. I n this section, the effects of heat loss through the reactor wall as well as the axial dispersions are considered briefly. Figure 7 shows the effect of heat loss through the reactor wall. Case i, an adiabatic reactor, is the same example used in Figure 3a. If the effect of heat loss becomes noticeable, as shown in Case ii, the temperature in the gas phase decreases more rapidly, reaching a lower level than in Case i. As a result, the reaction rate is decreased and a longer reactor is required to complete the reaction. Figure 8 shows the effects of the axial dispersion and the axial thermal conduction on concentration and temperature profiles in the reactor. Case i, which neglects these effects, is the same example shown in Figure 3b. If the effect of gas dispersion becomes noticeable, as in
(27)
where aFR'iCD,, (1 - e T ) corresponds to height of transfer unit (H.T.U.) and the integral corresponds to number of transfer unit (N.T.U.). The latter can be obtained graphically by reading the difference (x?, - ~ 4 0 on , the reaction diagram. Comparison of Numerical Solution and Graphical Solution. From Figures 5 and 6, the graphical solution based on pseudosteady-state equation exhibits tendencies similar to the numerical solution based on unsteady-state equation. But there are a few points where the difference between the two solutions becomes noticeable. For example, a t r , / R = 1.0 in Figure 5, the initial solid temperature is 0.045, whereas that for graphical solution is 0.0506. This deviation is owing to the pseudosteady-state approximation which, by neglecting the heat capacity of solid, cannot properly take into account the initial solid temperature. Another deviation is recognized at r, / R = 0.2 in Figure 6. The reaction temperature for pseudosteady-state solution continues to increase, while that of the unsteadystate solution decreases after reaching a maximum a t r , / R = 0.3. This is because as r, decreases, the supply of
1.0
0.07
0.8
0.06
w'
0
P!
n
. $
v
X
$06
0.05
c-" p:
0
n
$0.4
0.04
P
x
0.2
0
0
0.2
0.4
0.6
0.8
1.0
z I2 Figure 8. Effect of axial dispersion and axial thermal conduction on concentration and temperature profiles (countercurrent flow) Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
169
Case ii, there is a discontinuity of mole fraction a t the inlet point (z = 0), followed by a substantial decrease in the mole fraction, x ? ~ . On the other hand, if the effect of axial thermal conduction becomes noticeable, a discontinuity of temperature appears a t the inlet followed by a sharp decrease in gas temperature, T , , as shown in Case iii. These cases can also be treated as a problem of a single particle in which the temperature and the concentration of gas are allowed to change. Instabilities in Exothermic Reactions
Transistional Instabilities. Figure 9 shows the reaction path of the exothermic reaction corresponding to Figure 4. Since the rate constant, h,, the effective diffusivity, D,*, and the thickness of gas film, a, are identical to those of the endothermic reaction in Figure 5 , the reaction curves in Figures 5 and 9 are identical. The slope of the tie lines, however, becomes positive in Figure 9 because of the exothermic heat of reaction. As the result of this positive slope, the reaction curve and the tie line may intersect a t three points including the metastable point similar to that discussed in a single particle (Ishida and Wen, 1968). Such three-intersection points are shown in Figure 9 for r fR = 1.0, 0.96, and 0.9. And in this case, the reaction path is located at the lower point in chemical reaction controlling regime. However, after r,fR becomes less than 0.8, only one point of intersect exists in the diffusion controlling regime. Hence, the rate controlling step shifted from chemical reaction to diffusion a t r , / R between 0.9 and 0.8, resulting in ignition of solid reactant. Although the change of the rate controlling step also occurs in endothermic reactions-eg, in Figure 3b and Figure 6, this generally proceeds gradually. Contrary to endothermic reactions, the shift of rate controlling regime for an exothermic reaction may occur abruptly, as evidenced in the changes of x4< and T, in Figure 4. Particularly, if the heat capacity of solid is very small, the shift in rate controlling regime for an exothermic reaction takes place suddenly exhibiting the so-called transitional instability. The shift for endothermic reactions as shown in Figure 3b and Figure 6 occurs gradually even if the heat capacity of solid is zero. 11.0 -
/ B
Parameter: rclR
0.2
=fWfi, /
0.96
0
6,tR =0.0101 6aF/(G& h t R =0.;1=+0.7 1 -
0.04
cs,F/cpG z.0.l
0.05
0.06
RTIEk Figure 9. Reaction diagram for an exothermic reaction showing ignition during the reaction (cocurrent flow) Unsteady-state path for gas ( f )
0 , A (stable),
0 (metastable):
and solid
(e)
Intersection point of sigmoid reaction curve
and tie line
170
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
Since solids come into contact with the gas of the progressively higher temperature for an exothermic reaction in cocurrent flow, the ignition occurs more easily in a moving bed than in a single particle kept in a gas stream at a constant temperature. Extinction, however, is also possible in a cocurrent flow (Figure 10). For r , / R = 1.0,0.96, and 0.9, there are three intersections and the reaction proceeds through the diffusion controlling regime. But, as r , / R becomes less than 0.8, there is only one intersection appearing in the chemical reaction controlling regime. Thus, the rate controlling step shifts from diffusion to chemical reaction and the extinction occurs. Although the progressively increasing temperature of the gas phase is favorable to ignition. the progressively decreasing concentration of reactant gas depresses the reaction rate and causes the extinction. Therefore, in an exothermic reaction, a competing effect exists between the temperature and the concentration on the rate process. This situation also holds for countercurrent flow in which the extinction occurs because of the progressively decreasing temperature as r, / R decreases. The ignition can occur in countercurrent flow system owing to the increasing concentration of reactant gas. Generally speaking, however. the effect of temperature is greater than that of concentration of reactant gas. By combining these competing effects of temperature and concentration on the rate controlling step, the transitions may occur twice during the course of the reaction. For example, as shown in Figure 11 for cocurrent flow the reaction starts under the chemical reaction controlling regime, shifts a t first to diffusion controlling regime a t r / R = 0.9, and subsequently shifts again to chemical reaction controlling regime a t r / R = 0.6, exhibiting both ignition and extinction in the reaction. In Figure 12 for countercurrent flow, the reaction starts under the diffusion controlling regime, shifts to the chemical reaction controlling regime, and then shifts again to the diffusion controlling regime. Thermal Instability. Thermal instability is also observed in the moving bed. Figure 13 shows an example in countercurrent flow operation. Four reaction paths are drawn for various initial temperatures of solids, while the initial temperature, (RT,/Ek), of the gas is kept constant a t 0.039. I n Case i, the reaction temperature a t first increases a little while in Case ii, it decreases. However, after r , f R becomes less than 0.9, these two reaction paths coincide with each other and follow closely along the chemical reaction controlling regime. For Case iii and Case iv, however, the reaction temperature increases further and the reaction paths follow closely along the diffusion controlling regime. If the tie line is drawn for r , / R = 1.0, there are three intersection points and the metastable point exists between RT, / E , = 0.054 and 0.056, as shown in Figure 13. Thus, the small change of tinitial solid temperature near the metastable temperature makes the reactor performance entirely different indicating that the initial temperature of solids is an important factor in the design of a moving bed reactor. In this paper a graphical method for solid-gas reaction taking place in a moving bed is compared with a numerical method. The graphical method using the reaction diagram is a convenient technique t o determine the length of the reactor and to predict the instabilities. However, it should be used as a first approximation since the heat capacity of solids has been neglected.
1.0
I
I
Parameter: rc/R
I
-/=?--
I
1
I
i
i
0.06
0.05 RTl&
Figure 10. Extinction during the reaction (cocurrent flow) and solid (Q)
Figure 12. Extinction and ignition during the reaction (countercurrent flow)
0 , A(stable), O(metastab1e): Intersection point of sigmoid reaction curve
Unsteady-state path for gas (+) and solid (+) 0 , A (stable), O(metastab1e): Intersection point of sigmoid reaction curve
Unsteady-state path for gos (+) and tie line
and tie line
1.0
I
I
I
Parameter: rc/R
0.8-
a -3 0.6-
.
and iv (T,),
0 3
0.04
aF/(Gn)o =-0.7 ceF/cp G -0,041 i: 0.05 ii:O.054 iii: 0.056 iv:O,O6
0.05
UUI
Figure 1 1 . Ignition and extinction during the reaction (cocurrent flow)
Figure 13. Effect of initial solid temperature on the reaction path (countercurrent flow)
Unsteady-state path for gas (+)
Unsteady-state path for gas ( f )
0 ,A (stable), 0(metastable): Intersection point of sigmoid reaction curve
and solid (+) 0 , A (stable), O(metastab1e): Intersection point of sigmoid reaction curve
and tie line
and tie line
and solid (+)
E,
Nomenclature
a, b = stoichiometric
coefficient for gas components A and B , respectively C = total molar concentration of gases, mole/ T d
L CA, CB,
CI = molar concentrations of reactant gas A , product gas B , and inert gas I , respectively, mole/L’ = initial concentration of solid reactant, mole/L3 concentration of solid product, moleiL” molar heat capacity of gas, Himo1e.T molar heat capacity of reactant solid, S, and product solid S’, respectively, H /mole .T average molar heat capacity of solid, Himole .T effective diffusivity of gas component A in ash layer, L’je diameter of moving bed reactor, L axial thermal conductivity of moving bed, H/LeT activation energy of reaction rate constant, k , , Himole
F G H -AHA
e.%
mA
R R
axial dispersion coefficient of moving bed, L2/e total molar flow rate of solids, Fs of solid reactant S, mole/L% total molar flow rate of gases, GA of reactant A, mole/L’e enthalpy of solids per a single particle, H heat of reaction per mole of reactant gas A , H/mole gas film heat transfer coefficient at external surface of the particle, H / L % T effective thermal conductivity of ash layer, H / LeT gas film mass transfer coefficient for gas component A at external surface of the particle, L / o surface reaction rate constant = exp (-Eh/RT,), L4/mole.0 frequency factor for rate constant, L4/mole. reaction rate of reactant A per unit reacting core surface, mole/L% radius of spherical particle, L gas constant, H/mole.T
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r, r, = radius, radius at the reaction surface, respectively, L T , T,, To = temperature, temperature a t outer surface of solid, and ambient temperature, T t, t* = time, and that required for complete conversion, o U = overall heat transfer coefficient across reactor wall, H / L % T X = fractional conversion of solid reactant S x 4 = mole fraction of gas reactant A 2 , 2 = height, and height of reactor required for complete conversion, L GREEKLETTERS = [(CA,),(-~Ha)(Dr~)o/hc](R/Eii) &, = equivalent thickness for gas film in heat
(B)o
transfer, L 6, = equivalent thickness for gas film in mass t
CT
9.
transfer, L = void fraction of ash layer = void fraction of moving bed = effectiveness factor for surface reaction
0 = refers to the inlet for gas, z = 0 2 = refers to the outlet for gas, t = 2 Literature Cited
Beverage, G. S. G., Goldie, P. J., Chern. Eng. Sci., 23, 913 (1968). Horio, M., Mori, S., Muchi, I., Preprint for 34th Annual Meeting of the Society of Chemical Engineers, Japan, 1969. Ishida, M., Ph.D. Dissertation, Tokyo Institute of Technology, 1969. Ishida, M., Shirai, T., J . Chern. Eng. Japan, 2, 175, 185 (1969). Ishida, M., Shirai, T., ibid., 3, 196, 201 (1970). Ishida, M., Yoshino, K., Shirai, T., ibid., 3, 49 (1970). Ishida, M., Wen, C. Y . , Chem. Eng. Sci., 23, 125 (1968). Jackman, A. P., Aris, R., Proceedings of the 4th European Symposium on Chemical Reaction Engineering, Brussels, 1968. Meissner, H. P., Schora, F. C., Trans. Met. SOC.A I M E , 218, 12 (1960). Wen, C. Y., Ind. Eng. Chem., 60 (9), 34 (1968). Wen, C. Y., Wang, S. C., ibid., 62 ( 8 ) , 30 (1970). Yagi, J., Moriyama, A., Muchi, I., Japan Inst. Met. J . , 32, 209 (1968).
SUBSCRIPTS c = refers to reacting-core surface
o = refers to bulk gas phase
RECEIVED for review April 27, 1970 ACCEPTED October 16, 1970
Transport of Solids by a Viscous Flow in a Horizontal Tube at Constant Pressure Drop Donald H. Davidson' Shell Development Co., Houston, Tex. 77001 This paper discusses particle transport by a viscous flow in a constant pressure drop process. Our experiments demonstrate that it is possible to form a nonretaining bed when a fraction of the tube cross section contains a deposit. Although the average fluid velocity and shear stress decrease as the deposit thickens, calculations for flow in a segmented duct show that the local fluid velocity and velocity gradient near the surface of the bed at the center of the tube increase until 40% of the cross section contains a deposit. The flow rate at which a bed becomes nonretaining i s approximated by equating the lift and net buoyant forces acting on a particle-oneparticle radius removed from the bed surface.
S u r f a c e and body forces which act on discrete particles in a flow system may cause solid material to be removed from the main flow. Over a period of time an accumulation of particles at a solid-fluid interface will produce a flow restriction. The large random fluid motions which exist in a turbulent flow oppose impairment. Although a great deal of theoretical and experimental studies exist on particle motion in a viscous flow, the origin and description of the force system acting on a particle a t a finite Reynolds number are still not well defined. Previous research does provide sufficient evidence that in the vicinity of a solid I Present address, Kennecott Copper, Ledgemont Laboratory, 128 Spring St., Lexington, Mass. 02173.
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interface it is possible to obtain a force which opposes particle motion toward the interface. As this force is expected to be a function of both the local fluid velocity and velocity gradient the rate of impairment will be a strong function of how the flow field changes with particle deposition. This paper discusses solids transport in a horizontal tube by a viscous flow at constant pressure drop. The constant pressure drop process is approached in physiological flows and in those industrial operations which have pressure limitations. Although the average fluid velocity and shear stress above the deposit decrease with increasing bed thickness, our experiments demonstrate that it is possible to form a nonretaining bed in a constant pressure
