The Parametric Pump as a Chemical Reactor George P. Apostolopoulos Department of Chemical Engineering, University of Rochester, Rochester, New York 74627
A near-equilibrium approach was used to study a reversible catalytic reaction taking place in a parametric pump. The simple “stop-go” method was used to investigate numerically the separation phenomena that resulted from a thermal energy flux coupled with an alternating velocity field. Two kinds of enrichment take place simultaneously, a change in the reactant-product concentration ratio and an increase in the total concentration at one end of the column, as in regular parametric pumping. These two enrichments are interrelated as shown in this work. A near-equilibrium approach may also give useful results in the case of regular parametric pumping.
Introduction The idea of parametric pumping was first introduced by Wilhelm in 1966. A thermal parametric pump operating in the recuperative mode with ion-exchange resin as packing material was used to separate NaCl solutions. Since that time parametric pumping has been studied extensively as a separation process (Butts, e t al., 1972; Sweed, et al., 1969; Wilhelm, et al., 1968). Pigford, e t al. (1969), introduced an equilibrium theory of the parametric pump which assumes perfect local equilibrium although the difference in composition at the two ends of the column is large. The equilibrium theory gives a better insight to the whole phenomenon and clarifies the origin of the separation. Deviations from the predictions of the equilibrium theory have been observed and should be expected (Pigford, et al., 1969; Rhee and Amundson, 1970; Wilhelm, et al., 1968). The effects of parametric pumping on chemical reactions have not been studied in depth. Kim and Hulburt (1971) were the first to study these effects. Their system was continuous and without flow reversal but it can be characterized as parametric pumping. However, the problem of determining whether a given system is going to give improved results is still open and constitutes one of the aims of this work. In principle, chemical parametric pumping is similar to classical parametric pumping but there are some important differences which will be developed by analyzing the problem with a “near-equilibrium’’ approach.
The Kinetic Model A reversible catalytic chemical reaction is considered. Such reactions involve a two-phase equilibrium and a surface reaction that can be affected by parametric pumping. The simplest case to be studied is shown in Figure 1. Such a kinetic model may apply, for example, to isomerization reactions. The equipment can be the same as a “classical” parametric pump operating in the direct mode, see Figure 2, where a step change in temperature is accompanied by flow reversal. Heat resistances are considered negligible. The kinetics are assumed to be first order and the rate constants are dimensionalized so that everything is referred to a unit of free volume. Assuming also that there is no axial dispersion, the design equations can be written as
where u L = 0 if i is a component of the stationary phase. The reaction terms r, are
~4
=
k34~3
-
k43~4
The boundary conditions are similar to those used in the “classical” parametric pump (Wilhelm, et al. 1968). Equations 1 can be integrated easily using simple numerical methods such as the stop-go method (Sweed and Wilhelm, 1969). Some results are discussed later in this paper. Near-Equilibrium Theory The main advanl;age of parametric pumping is that one may obtain concentrations which exceed the ones obtained by operating at steady state under the most favorable temperature condition. As one would expect, not all reaction systems (sets of hi’s) will show improvement due to parametric pumping. There is no simple way that one can tell a priori what is going to happen and good results for a particular system would be fortuitous. One therefore needs a better insight to the whole phenomenon in order to derive criteria for improved conversions. Suppose ’chat the system in question is described by reactions of the form A i
kif
t;;;
Also assume that the reaction rate for a species is given by
The “near equilibrium” approach is based on the fact that one can express the concentration of a species i at a given point and at a given moment as where ci* would be the concentration of i if it were in equilibrium with all species present at the same point at the same moment. The departure from equilibrium is represented by t L and it is assumed to be a small number. Obviously both cL* and t L in general are functions of time and position. As a first approximation in the following development t L is assumed constant during each half-cycle. The simplification is based on similar calculations by Giddings (1965) in his analysis of the dynamics of gas chromatography. At equilibrium Ind. Eng. Chern., Fundam., Vol. 14, No. 1, 1975
11
Further, all terms ZiZ, will be neglected since all ti’s were assumed to be small. These simplifications result in a set of (n - 1) linearly independent equations in the n variable t i . The nth independent equation is obtained from the equation for conservation of mass
Figure 1. The kinetic model considered.
ec, =
c
jsi
or
or n
n
EXj* + C X j * F j = j;
1
1
1.1
or
Criteria for Improved Conversion Assume that the above reaction scheme is taking place in a packed bed. In an infinitesimal section of the bed the total number of moles is conserved. This means dci E= 0 dt
Figure 2. Parametric pump operation in the direct mode.
(14)
which is equivalent to eq 8. For a particular species i one can write CkfiCj* - Ci*Ckij = 0
(5)
(15)
(6 )
Let us now make the following definitions of distribution functions
J
j
By substituting eq 4 into eq 3 and using eq 5 one gets
-
-
Yi = C i * C k i j ( E j - E i ) f
where the overbar indicates a constant value. In general, for each component i one can write y.
*
= -al +c .
at
u.-a c i
(16)
ax
where u L= 0 if i is a component of the stationary phase. For an infinitesimal section of the bed it is obvious that the total concentration c only changes due to inflow or outflow of material; thus
Equations 16 represent three equations in four unknowns, c1, c2, c3, c4, which can be solved in terms of one unknown. If c1 is chosen as the unknown, one obtains c2 = C d f l
Also it is convenient to define the equilibrium fractions as Ci* xi* = (9) C
From the definition it can be seen that x l * is a known constant since it is a function of the Kl,’s of the system. Substitution of eq 4 and 9 into eq 8 yields
- acj* xi* a t
(10)
These concentrations may now be substituted into eq 14 and after differentiation the result is
(17a) Substituting from eq 4 , 5 , and 9 into eq 7 yields Similarly if yields Equations 6 and 11 are two expressions for the reaction rate which may be equated and rearranged to yield 12
Ind. Eng. Chem., Fundam., Vol. 14, No. 1, 1975
c4
is chosen as unknown, the manipulation
at the top it must be true that in Figure 3
(OB)- (DC)> (EF) - (GH) which means the displacement of moles to the top is greater than to the bottom end. The above equation can be rewritten in terms of characteristics to give
if the equality prevails then no effect is observed (Zr’s are zeros). Inequality 23 affects the value of the term a In c/ax in eq 12 and therefore the various A’s through the F’s. This means that the criteria are not completelytindependent of each other and it may be difficult to satisfy all of them. Numerical Examples
L
X
Figure 3. Typical characteristics in the solution of eq 17a and 17b
where
x, =
I/
1 +1
+ f 3
+.I
fifi
x,
(18a)
fl
V
=
1 +-I
l + f fi’f3
+J-2L
If one now considers the characteristics of the hyperbolic system eq 17a and 17b it is easy to see in Figure 3 that along the characteristics of A1 one obtains clXl = constant
(19)
c4A4 = constant
(19a)
and for A4
In order to have separation of A1 and A4 the concentration waves must propagate with different speeds during one or both of the half-cycles. This requirement implies that the slopes of the characteristics must be different. I
i
(20) To have good separation, the characteristics of A1 and A4 should move gradually to opposite ends of the column. This means, in reference to Figure 3, that in order to concentrate A4 a t the top end it must be true that d4
AdC
(21)
where the superscripts h and c indicate the hot and cold half-cycle, respectively. During the hot half-cycle the upward displacement occurs. Similarly, in order to have A1 moving to the bottom it is necessary that Aih
k 2 1 for all cases where data have been reported.) During the cold half-cycle the opposite changes predominate. Inequality 30 shows that as the separation proceeds the inequality is less and less satisfied and therefore deviations from the equilibrium theory must be expected. This effect is distinct from axial dispersion or nonlinear kinetics which may also cause some deviation from predictions of the equilibrium theory. €1
5
3
9
17 21 NO O F C Y C L E S
25
Figure 9. Enrichment of the product during parametric pumping operation.
where Ci* would be the concentration of species if perfect local equilibrium were attained. t l is the equilibrium departure, which is assumed to be small and constant during each half-cycle. Further, assume that the kinetics are linear and that there is no axial dispersion in the bed. Then the system can be described by eq 25 and 26
a Ci + at
u ac. . 2 = yi ax
(25)
and Yi=
ci*C k i , ( E J -
Ei)
(26)
J
where u LB = for i = 1 and u, = 0 for i = 2. Algebraic manipulations similar to those used in the derivation of eq 12 yield
and €2
= -EikiJk2i
where C is the total concentration of species. Now consider the parametric pump operating in the direct mode and assume upward displacement during the hot half-cycle. From eq 14 and 15 one gets
or (28)
where (29)
Equation 28 is similar to the parametric pumping equilibrium equation (Butts, et al., 1972; Pigford, et al., 1969). If one performs the usual analysis by the method of characteristics it is easy to show that in order to have accumulation of A1 at that top end the following condition must be satisfied
fh > fC
(30)
and the superscripts h and c indicate the hot and cold half-cycles, respectively. Note that in the equilibrium theory t l = € 2 = 0.
Conclusions
A method of operating a reactor which can increase the conversion significantly above the most favorable steadystate operating condition and, at the same time, yield a more concentrated product stream should be considered for possible industrial applications. In the simple example presented in this paper operating a catalytic reactor as a parametric pump increased the conversion approximately 15% and the product stream was 2.5 times more concentrated. A “near-equilibrium” approach to the analysis of parametric pumping has been developed which shows that significant deviations from equilibrium can exist and are, in fact, essential for improved conversion and concentration. In addition the “near-equilibrium” approach provides a relatively simple way of evaluating the inequalities for rather general kinetics to determine whether or not improved conversions can be obtained in a particular case. It should be emphasized that, although the particular case studied was very simple, as the complexity of the kinetics increases the potential for greater improvement increases. The present study was restricted to systems with no net change in the number of moles. For reactions where there is such a change pressure parametric pumping may provide similar improved conversions. Nomenclature A, = component of the reaction, Figure 1 c t = concentration of the ith component, M cl* = equilibrium concentration defined by eq 4 C = total concentration c = superscript denoting the colder half-cycle d , = angles in Figure 3 el = angles in Figure 3 fi = distribution coefficient defined by eq 16 h = superscript denoting the hotter half-cycle k , , = reaction constant going from i to j , sec-l L = length of the column ( = 50 cm in the examples) rl = rate of production of species i, mol 1.-l sec-1 t = the time variable, sec T = periodof a cycle ut = velocity of component.i, cm/sec V = plug flow velocity, cm/sec x = distance variable, cm Ind. Eng. Chern., Fundam., Vol. 14, No. 1, 1975
15
xi* =
equilibrium fraction of component i defined by eq
9
Greek Letters = equilibrium departure terms of i defined by eq 4 X i = defined by eq 18 and 18a
?j
Literature Cited Butts, T. J., Gupta, R., Sweed, N. H., Chem. Eng. Sci., 27,855 (1972). Giddings, J. C , "Dynamics of Chromatography," Vol. I , Marchel Decker. New York, N. Y., 1965. Kim, D. K., Hulburt, H. M., Northwestern University (Paper presented at the AlChE meeting in San Francisco, Calif., (Dec 1971). Pigford. R. L., Baker, B., I l l , Blum, D. E,, lnd. Eng. Chem., Fundam., 8, 145 (1969).
Rhee, H., Amundson. N., lnd. Eng. Chem., Fundam., 9, 303 (1970). Sweed, N. H . , Wilhelm, R. H., lnd. Eng. Chem., Fundam., 8, 221 (1969). Wilhelm, R. H., Ind. Eng. Chem., Fundam., 5, 141 (1966). Wilhelm. R. H., Rice, A. W., Rolke, R. W., Sweed, N. H., lnd. Eng. Chem., Fundam., 7,337 (1968).
Receiuedfor reuieu June 21,1973 Accepted August 12,1974
This paper has been presented at the AIChE National meeting at New Orleans, La, (March 1973), The support of the Chemical En-
gineering Department of the University of Rochester is acknowledged gratefully.
A General Method for Predicting Pressure Loss in Venturi Scrubbers Kenneth G. T. Hollands* and Kailash C. Goel Thermal Engineering Group, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada
The differential equations developed by Boll have been demonstrated to predict pressure losses in venturi scrubbers accurately. Boll's equations are nondimensionalized and general solutions are obtained for both the liquid velocity and the pressure distribution in conical and rectangular duct geometries. The solutions are presented in the form of design charts and equations by use of which the pressure loss across a venturi scrubber can be readily predicted. These charts are of general validity in that they are independent of the method used to predict the droplet size produced in the scrubber. Recovery in the diffuser section of the kinetic energy delivered to the liquid in the throat is discussed.
Introduction The venturi scrubber is widely used for the collection of particulates from industrial exhausts. It combines the features of high efficiency of particulate removal, easy maintenance, and low initial cost. The high pressure drop through the device, which results in a high running cost, is one of its few disadvantages. The basic physical processes at work in the venturi scrubber have been thoroughly reviewed by Boll (1973). The dominant mechanism of particulate collection appears to be inertial impaction of the particles on the droplets. This inertial impaction is afforded by the high relative velocity between the gas stream and the droplets, particularly near the point of injection of the liquid. The droplet acceleration and irreversible drag-force work which results from this high relative velocity are responsible for the device's high pressure drop. Recently there have been a series of studies whose purpose it is to predict the performance of venturi scrubbers from first principles and to thereby develop a rationalized design method. Of these studies, three, namely those by Gieseke (1963), Calvert (1970), and Boll (1973) have concerned themselves with the prediction of the pressure drop. Gieseke and Calvert treat the venturi simply as a duct of uniform cross section, and this fact has caused some lack of agreement with experiments on practical venturis. Boll, on the other hand, develops a general method which is capable of handling ducts of variable cross section. Agreement between his predicted pressure drops and experimental values is excellent. Despite this success, the applicability of Boll's method 16
Ind.
Eng. Chem., Fundam., Vol. 1 4 , No. 1, 1975
to the practical design of venturi scrubbers is restricted by the fact that no general solution to his basic equations was given. Rather, Boll solved his equations numerically for particular sets of conditions, namely those conditions for which experimental data were available for comparison. Hence, in order to apply his method, the designer must numerically solve a set of simultaneous differential equations for each design considered. The present investigation presents generalized solutions for the pressure drop, based upon Boll's equations. The solutions apply to any venturi which is made up of a combination of convergent, divergent, and straight ducts provided the ducts are either circular or rectangular in cross section. (In the rectangular duct, only the duct width is assumed to diverge or converge, the breadth of the duct being assumed invariant with axial distance.) First the equations are nondimensionalized and the important set of dimensionless groups applying to the flow in a duct of fixed convergence or divergence is determined. The general solutions for both the dimensionless liquid velocities and pressure drop components applying to such a duct are then given, either in the form of closed-form equations, or charts. By simple addition of their corresponding pressure drops (as determined from these charts and equations), the pressure drop across the assembly of convergent, straight, and divergent nozzles which constitute a venturi can be determined. Thus the present paper presents a set of design charts and equations by use of which a designer may readily arrive at an accurate estimate of the pressure drop in a venturi scrubber. In order to reduce the dimensionless groups to a workable number, it has been found necessary to use a drag
