TA,
s v
V X, a
The magnitude of G ( j u ) , called the amplitude ratio, is
p.4; 7,
= steady component of perturbed flow rate = rate of production of i t h species = Laplace transform variable = fluctuating component of flow rate about = volume of flow reactor = deviation variable of ith species about its equilibrium concentration = defined in Equation 18 = deviation variable for TA, = relaxation time
SUBSCRIPT and 1 G ( j w ) l us. w plot is the frequency response data used in extracting rate constants.
= inlet stream concentration in flow reactor
Literature Cited
Nomenclature
ith chemical species. Also used to designate con(:entration of i t h species in flow reactor A , = steady-state concentration of i t h species = deviation variable (= A , - A , ) a, b,, = defined in Equation 13 C, = concentration of ith species in batch reactor 6, = equilibrium value of C, G (s) = transfer function, defined in Equation 17 k, = rate constant K,, = ratio of rate constants, k,/k, P, = pole of transfer function q = volumetric flow rate A,
0
=
Arbesman, El. W,, 11.9.thesis, Northwestern Univerjity, 1969. Czerlinski, G. H., “Chemical Relaxation,’’ pp. 2-3, Marcel Dekker, New York, 1966. Czerlinski, G. H., J . Theoret. Biol. 7, 435 (1964). Draper, N. R., Smith, H., “Applied Regression Analysis,” pp. 263-84, \Gley, New York, 1966. Eigen, AT., Discussions Faraday Soc. 17, 194 (1954). Eigen, M., DeMaeyer, L., “Technique of Organic Chemistry,” A. Weissberger, ed., Part 11, Vol. 8, pp. 914-19, Wiley, New York. 1963. Eigen, hl., et al., Progr. Reaction Kinetics 2, 287 (1963). Hulburt, H . M., Kim, Y. c f . , I n d . Eng. C h e m 68, No. 9, 20 (1966). RECEIVED for review December 16, 1968 February 27, 1969 ACCEPTED
PARAMETRIC PUMPING Separations via Direct Thermal Mode N O R M A N H . SWEED A N D R I C H A R D H . WILHELM’ Department of Chemical Engineering, Princeton Uni:nizersity, Princeton, h’. J . 08640
A computational investigation of separations by direct thermal mode, liquid phase parametric pumping i s presented. Calculations are performed using the new STOP-GO algorithm, a modification of the method of characteristics. Using the toluene-heptane-silica gel system, we have determined the effect of displacement, cycle time, phase angle, and reservoir volume on separations. Separation factor increases exponentially with number of cycles, and in the case of equilibrium operation, there i s almost no limit to separation capability. W e have simulated experimental separations presented previously by the authors. A graphical calculation procedure also is presented for equilibrium operation.
P A R A M E T R I C pumping is a dynamic separation principle based on periodic, synchronous, coupled transport actions. It is not restricted to any particular type of mixture nor limited to any particular form of energy. To date both gaseous and liquid mixtures have been separated at the expense of mechanical, thermal, or chemical energy. Because so much of current thinking on separations is based on the equilibrium stage or phase exchange concept, it is useful to put parametric pumping into proper perspective. Let us compare and contrast parametric pumping with Deceased.
equilibrium phase exchange operations (staged or continuous) such as distillation or solvent extraction.
1. The conventional operations require the presence of at least t v o distinct phases. So does parametric pumping. 2. These operations all require a n equilibrium function relating the compositions of the two phases. This function is usually dependent on several parameters, the most common of which are temperature, pressure, and the concentrations of the other components present,. Parametric pumping (parapumping) also requires a n equilibrium function. Thus far there have been no distinctions between the conventional operations and those involving parapumping, VOL.
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Table 1.
Requirements for Parametric Pumping
Conventional Phase Exchange Processes At least two
General At least two
This paper Two: liquid, solid
Equilibrium function
Any function
Any function
Toluene-heptane on silica gel
Relative motion between phases
Unidirectional
Alternating
Alternating flow in a packed bed
State variables
All steady
One or more forced periodically. Others periodic in response
Temperature forced to vary periodically. Others periodic in response
No. of phases
Parametric Pumping
because all phase exchange operations may be parapumpedi.e., all may be operated using the parapump principles.
3. I n conventional operations the relative motion between the phases is unidirectional. I n distillation, for example, the vapor always moves upward relative to the liquid; it never reverses it8s direction. I n parametric pumping the direction of relative motion reverses periodically. 4. Convent>ionalcontinuous processes are usually operated near steady state: The state variables-concentration, temperature, pressure-do not change with time. Parapumping is different in that the state variables all vary periodically with time. One or more of the parameters, 8, in the equilibrium function are driven periodically; the others are periodic in response. -4 direct consequence of the periodicity of the variables is a periodic interphase flux.
It is the synchronous coupling of the alternating velocity with this interphase flus that produces the parametric pumping separation. To understand the parapumping principle bet'ter, consider a specific application. Let us suppose we want t'o separate into its components a liquid solut,ionof toluene and n-heptane. This system consists of only one phase; therefore a second phase must be sought. We could choose the vapor in equilibrium with the iiiixt'ure and thus attempt to parapump dist,illation. However, it is difficult, though not impossible, to alternate experimentally the relative velocity of a vaporliquid system because of the effects of gravity. (To parapump distillat,ion we might invert the column periodically or possibly use individual stages not in colunin arrangeinent. ) I t is far easier, however, to use as t,he second 1)hase a solid adsorbent such as silica gel. With t,he adsorbent fixed in a column, the liquid can then be made to f l o s back arid forth over the bed with no difficulty. Of course, there must be a distributioii function relating the concent1,ations between the phases. The next specification we must provide is the parameter t o be varied. Whatever affects the distribution function is acceptable; but for several reasons temperature is a good choice. Over an easily obtainable temperature range (4' to 70' C.) the amount of toluene adsorbed at a fixed fluid composition varies more than loyo. The greater this variation is, the better will be the separation. Table I summarizes the requirements for parapumping. How the temperature is to vary is yet another area for our specificat,ion. R e could introduce heat at one end of t'he column and have it transported into the bed convectively 222
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FUNDAMENTALS
by the moving fluid. Cooling could be effected by introducing cold fluid a t the other end. Temperature oscillations within the bed would thus depend on the thermal properties of the bed and fluid, the rate and duration of fluid flow, and the rate of interphase heat transfer. This method of causing temperature oscillations, known as the "recuperative mode," is potentially reversible thermodynamically; however, it restricts the temperature oscillations to a particular form once the system has been specified. The freedom to vary the temperatures a t will is sacrificed for the sake of thermal efficiency. Clearly, we could, if we desired, vary the temperature directly by introducing heat through the column walls. I n this "direct mode," the temperature of the fluid in the column jacket is changed as we choose; the temperature oscillations are nearly independent of flow. Although direct heating may be less efficient thermodynamically, it has definite advantages and so we consider it only in further discussions. Background
During the past decade a number of workers have studied parametric pumping in one of its specific forms. Thermal parametric pumping was conceived by Wilhelni about 1962. Separations of a NaCI-H20 liquid system using ion exchange resins have been reported by Rice (1966), \Whelm, Rice, and Bendelius (1966), Rolke (1967), and Kilhelm, Rice, Rolke, and Sweed (1968). The ratio of the KaC1 concentrations in t'he column effluents in these studies was about 1.2, for open, recuperative mode operation. l\lcAndrew (1967) st'udied recuperative thermal parametric pumping as applied to a gaseous methane-nitrogen system. Jeiiczewski and Meyers (1968) presented a brief note on the direct thermal mode for separating gas mixtures. Earlier, Skarstrom (1969) described a system for drying air over a silica gel bed using pressure as the varied parameter. This we consider to be parametric pumping, although it was not so described a t the time. Recently, in a manner similar to the work of Skarstrom, -ilexis (1967) applied the same idea to upgrade hydrogen by removal of hydrocarbons. Both applications are counterparts of the direct thermal mode. Experimental direct thermal mode liquid phase separations have been reported for the toluene-n-heptane-silica gel syst'em by Wilhelm and Sweed (1968), Kilhelm et al. (1968), and h e e d (1968). Separations of a benzene-hexane mixture were presented by Wakao et al. (1968). Here we present a connputatioiial study of direct therinal mode parapurnping, iiicluding a new iiuinerical algorithm for solving the model equations, 8 simple graphical solution of the model, a discussion of calculated t'oluene-heptane system behavior, and a simulation of the experimental separations presented previously by Wilhelm and Sweed (1968) and Wilhelm et al. (1968). Mathematical Model
The model equations for parametric pumping, discussed in detail by Wilhelm et al. (1968), are presented below:
ao"+ at
A(+/*
- $Jf)
=
0
Equation 1 is a fluid phase material balance, Equation 2 is an interphase transfer rate expression, and Equation 3 is an equilibrium function where 0 ( t ) is the parameter (temperature) to be varied periodically. If we choose certain values for 7 and/or X, we can simplify greatly the solution of the parametric pumping equations. For example, if 7 = 00, the separation is greatly reduced. The rate of axial diffusion is so great that any potential concentration gradient in the column is dissipated immediately. The high diffusivity causes the column to behave like a single stirred tank. If we choose X = 0, we again obtain no separation. With the interphase flux eliminated, the solid adsorbent behaves like an inert packing. Fluid moves back and forth over the adsorbent but does not interact with it; thus, because the fluid composition is not affected by this movement (except for axial mixing), no separation in the fluid is possible. We have solved tn-o limiting-though trivial-examples of parametric pumping without any calculations. At the other extremes of these variables (7 = 0, X = 00 ), solutions cannot be obtained so easily. However, a graphical technique is available which takes advantage of these limits (discussed below). First, however, let us consider how to solve the equations without such strong restrictions. Because the authors have encountered experimental separation factors in excess of IO5 (Tl’ilhelm and Sweed, 1968), it became apparent that axial dispersion did not seriously retard separations. R e may therefore eliminate the second-order term in Equation 1. Thus, for 7 = 0, Equations 1 and 2 can be rewritten
Here R represents any interphase transfer rate expression. Equation 4 is hyperbolic and so it may be solved using the method of characteristics (Acrivos, 1956). Method of Characteristics
This technique solves the coupled partial differential equations ( P D E ) (4 and 5 ) by first converting them into two ordinary differential equations (ODE), the latter being easier to solve than the former. We convert each P D E into an ODE by assuming that the left-hand side of each represents a total derivative. Thus from Equation 4
By the chain rule the total derivative may be expanded
d4/ -__ a+/ - I a+/ dz dt at dz dt Clearly for Equations 6 and 7 to be identical dz
-= aj(t) dt
Therefore, Equation 4,rewritten
is v d i d along the characteristic lines obtained by solving
Equation 8 z = /cxf(t)dt
Treating Equation 5 similarly,
is valid along the characteristic lines (lob)
z= /dz
Equations 9b and 10b form a network of characteristic lines in the z - t domain along which the ordinary differential Equations Sa and loa, respectively, are solved. These ODE’Sare initial value problems for which computer solutions are well known (Lapidus, 1962). I n effect the characteristic lines describe the trajectories in space and time which their respective phases follow-the fluid displacement follows Equation 9b, the solids remain a t a constant position (Equation l o b ) . If we approximate the continuous z - t domain with N Z position and N T time increments, we must solve 2.NZ.IVT ODE’s, one a t each time and position step. In practice only two equations are solved a t each position and time step to avoid handling large sets of simultaneous equations. The number of equations increases by 2 for each additional component. The method of characteristics is a very good technique for solving the modified parapump equations because it eliminates all axial diffusion except that due to finite differencing. The authors have simplified the method, however, making it easier to use. This new STOP-GO method is described in detail below. STOP-GO Method
The STOP-GO method is superior to the method of characteristics in several respects. First it reduces by a factor of one half the number of ODE’s which must be solved. At the same time it provides a very clear physical picture of what the calculations actually mean. The calculational algorithm is presented in detail after this qualitative description. I n the STOP-GO method we displace the fluid during one time step a distance
j,
t,+At
af(t)
allowing no interphase transfer t o occur. No axial mixing occurs either. When the flow (GO) ceases, transfer between the phases begins (STOP). This transfer is assumed to occur in the same real time step used for flow but subsequent to it for calculational purposes. Therefore, the interphase transfer rate expression is integrated from time t, to t, At. The solution of the parametric pumping equations by this method requires that only N Z . N T ODE’S be solved-that is, only the rate of transfer from the solid phase need be calculated from a differential equation; the fluid phase concentration change may be obtained from an algebraic material balance. After the rate and mass balance equations are solved and the new values of and +8 calculated, another flow without transfer occurs, followed by a transfer without flow.
+
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By displacing less than one entire section each time step we
~
BEFORE GO
A F T E R GO B E F O R E STOP
TIMESTEP j
Figure 1.
A F T E R STOP TIME S T E P j
+I
Schematic for STOP-GO computational algorithm
This method allows use of any complexity of interphase rate expression or equilibrium function because the equations solved are always initial value ODE'S which can be solved readily. Also each additional component requires one O D E and one algebraic mateiial balance equation. I t is often possible with simple rate expresqions to obtain an analytical solution of the rate equation, eliminating recourse to nunierical schemes. For the lumped parameter rate expression used in this work (Equation 2 ) a quasianalytical solution does exist. As developed by Rolke (1967), the solution is based on a locally linearized isotherm and involves exponential decay toward the equilibrium composition. The STOP-GO method reduces the number of required equations by half because it substitute3 for Equation 9a a simple renumbering of the position increments. This method may be implemented in a very straightforward manner now that we understand physically what we are doing computationally. K e have found no instance of numerical instability in our studies using this algorithm. Even with A'Z and LYT both as small as 10, the algorithm correctly computes trends in separation caused by varying the system parameters. I t does not calculate the magnitude of separation accurately with these values, however, because it is impossible t o represent the concentration profile by so few points. But, because of this stability, one can scan system behavior to study trends rapidly in only 4y0of the time that would be required for more precise calculation (NZ = NT = 50). The requirement for N Z to be at least 50 comes not from stability considerations but from the fact that using fewer axial position increments causes a flattening of the concentration gradient in a way similar to diffusion. Further, N T must be large enough not to invalidate the assumption of locally linear equilibrium in the quasianalytical solution of the rate expression. We have used X Z = 50 in all coniputations. Computer studies using this method indicate t h a t the procedure can be unstable only if the solution of the ODE'S in the STOP portion of the calculation is unstable. [Lapidus (1962) discusses instabilities in numerical ODE solutions.] Computational Algorithm for STOP-GO Method
1. Divide the column into iYZ equal position increments. 2. Divide the time domain into ArT increments such that the displacement during each time step equals the length of one position increment-that is,
AZ
af(tj)Atj
or
introduce diffusion to the model. The mixing effect due t o incomplete displacement produces a spreading of concentration waves identical to t h a t of axial dispersion. T o date no quantitative relationship between fraction displaced and 7 has been found. 3. Initialize the fluid and solid compositions in the N Z increments to some physically realizable values. [The model is now set up and we can begin the operational steps of STOP-GO.] 4 . Refer to Figure 1. I n the GO step we displace the fluid relative t o the solid. No calculations are required because the fluid originally opposite solid section i is now opposite section i 1. Each fluid section is displaced exactly one step ahead. Thus the fluid +I,, began the j t h time step opposite &,, and ended the GO portion of the calculation opposite rbs, It is clear that a simple renumbering of the subscripts is all that is required for the GO part. 5. After GO is completed, solve the transfer rate expression for each level of the column These S Z equations are entirely independent of one another and represent only batch, partial equilibration between adjacent phases. Also in this STOP portion, since each section is closed, whatever material the solid phase loses the fluid phase gains. The solids lose solute according to the ODE rate expression; the fluid gains it by a simple algebraic balance. Figure 1 also shows the STOP portion of the calculation. 6. The procedure now continues from step 4 until the half cycle is completed-Le., NT time steps.
+
Graphical Solution. If we make certain assumptions about the nature of the physical parapumping problem (local solid-fluid equilibrium), we can obtain a simple graphical solution of the model equations. This is desirable because it allows parapump separations to be calculated using a ruler and pencil and paper instead of a large digital computer. Moreover, the graphical method is valid for all isotherm shapes, even nonlinear ones. If we let the rate of interphase transfer be so great that adjacent fluid and solid are always in equilibrium-Le., X = m-and we remove all axial diffusion-i.e., 7 = 0-the parapump equations are 841 a+f af(t)-+-+-=
l&EC
FUNDAMENTALS
0
0)
(12)
Equation 12 is the equilibrium function relating the composition of the fluid phase, &*, to that of the solid, +s. If rewritten so that the solid composition is a function of the fluid (solving 12 for & ) we get
4s
= F(+j*,
6)
(13)
We have assumed that equilibrium will prevail between adjacent phases; therefore if we know one composition, say +*, we also know the other, + j . I n Equation 11 we therefore whichever have only one dependent variable-either +s or 41, we choose. Eliminating with the following substitution +8
(14)
and inserting into Equation 11 we obtain a+f
224
at
+f* = +f* (+s,
af(t)-+ az For nonconstant f ( t ) the time steps will not all be equal.
a+*
at
a2
(
1+-
aF
a+)
a+j a~ae -+--=O at ae at
(15)
If we now restrict 0 to square wave variation, then ae/at is
zero at all but t r o instants in the cycle. Thus for almost all time the parametric pumping equation is simply
each temperature studied.
The dimensionless isotherm
+,*[ 3 is related to the dimensional one, Cj*, by the equation +f* =
As in the method of characteristics, we consider the left side of Equation 16 to represent a total derivative
where Co is some characteristic fluid concentration such as the initial one. The solid phase concentration is made dimensionless,
+*
(17) Thus n.e have the ODE
which is valid along characteristic lines described by dz - -dt 1
fff( t )
+ (aF/aw
The solution of Equation 18a indicates that 4, is constant along the characteristic line. And because +I does not vary along these lines, the denominator of 18b is also constant along the line. From initial conditions, therefore, the solution of 18b is
Equation 19 describes the trajectory in the z - t domain of constant concentrations. For f ( t ) a square wave with magnitude V ,
This indicates that the constant composition lines are straight and that their slope depends on the velocity and on the initial composition of the bed. Equation 20 describes the movement of concentration waves through a column with constant temperature. I t does not account for Ivhat occurs n hen the temperature switcher from cold to hot or hot to cold every half cycle. At these times solute is redistributed betveen the phases. This new distribution can be obi ained graphically rising a material balance line and a plot of the adsorption isotherms a t the tKo operating teniperatures. Or the new distribution can be calculated from the equation9 for the isotherm. An illustration of the procedure for solving the parapunip equation. graphically is presented below for one cycle using the toluene-heptane-silica gel equilibria. Computer Exploration and Simulation. 'l'o investigate the behavior of a paramt.tric pumping system, we formulated a mathematical model arid developed a specific computational algoritIiIn-STOP-GOto solve it. We present no^ computer explorations of the direct thermal mode of parametric puniping for the toluene-heptane-silica gel system Considered previously. First we study a wide range of possible operating conditions and qysteni parameters to get a feeling for which of these are most influential; second, we simulate the specific experinients described by Kilhelm et a / . (1968). Also, we compare the similaritieq and differences between predicted and experimental findings. l?QUILIBRIC\I DATA. T o implement the computational algorithm for a particular system, certain experimental data must be obtained. I n particular, we must have an algebraic expression for +,*[ 1, the toluene adsorption isotherm, a t
Cf*/CO
C*P&
= co
where C, is the concentration of toluene in the particle measured in milliliters per gram of dry silica gel. Experimental equilibrium data of Robins (1967) were correlated to fit an isotherm in which the Langmuir adsorption on the solid surfaces is added to the solute that is present in the pores of the adsorbent. Thus
+ ml. toluene in pores c, = ml. toluene adsorbed grams dry silica gel
It has been assumed that a t equilibrium the pore fluid is identical to the bulk fluid. The modified Langmuir isotherm is
C8 =
A -C 1 B*C
+
___
+ D-C
where A , B , and D are constants at any given temperature. For the temperatures shown the Robins data give: 0"
65" C.
D
B 70.7 22.2
A 9.20 2.01
c.
0.29 0.29
Lacking data a t other temperatures, a linear dependence of A and B on temperature was assumed for extrapolation and interpolation. Thus
A
=
B
=
- T X 7.19/65 70.7 - T X 48.5/65 9.20
Kith T in degrees C. Other systeni constants supplied by t,he parapumper include cycle time, displacenient amplitude, dead volume in the reservoirs, and phase angle between the heating and flow waves. Below we describe the effect of each of these on parametric pumping. COMPUTEREXPLORATIOSS. We use separation factor (toluene concentration ratio calculated top reservoir to bot.tom) as the characteristic measure of the extent of separation. We could have chosen individual reservoir concentrations, the difference between them, or some other quantity; but, separation factor has the interest'ing and important) propert,y of being approxiniately exponent.ia1 with increasing numbers of cycles. The data which follow were computed using the STOP-GO algorithm, the toluene-heptane isotherm mentioned above, and the parameter values of Table I1 except where explicitly shown t,o be otherwise. All coniputations were performed on a n IBM 360/65 digital computer, using double precision word lengths for reduced roundoff error. EFFECT OF X. The experimental runs presented by Kilhelm and h e e d (1968) differed from one another in that each had a different cycle time, 7. Cycle time, however, does not appear in the model equations; rather it acts on the system because of the may the equations were made dimensionless. Thus T is hidden in the dimensionless parameters, particularly the transfer coefficient,, A. VOL.
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Table II. Numerical Values Used in Computer Explorations 4-70 Temperature range, "C. Density ratio, p s . ~ ,grams dry 0 . 7 3 X 2.0 = 1.46 silica gel/ml. interstitial fluid 0.5 01 0.0 e 10.0 x 0.3 Dead volume Initial concentration 1.o 41 1.24 48 50 No. of position increments, N Z
If film diffusion cont'rols, X = hm7
where h, is a mass transfer coefficient. [For internal diffusion cont,rol, the lumped parameter expression (Equation 2 ) is only an approximat,ion and no simple relation between X and T exists. However, X does increase with increasing 7.1 It might appear that' X and T are linearly related for film control, but this is not correct. For film control h, is not constant but varies with velocity. And since in our experiment,s displacement' is constant, the velocity varies inversely with 7. Further, h, can be determined from "j-factor" correlations-see, for example, Bird et a?. (1960)-where h , is proport,ional t,o j.~, X velocity. Thus, XW
hmT
displacing all the interstitial liquid in the bed requires a = 0.50. The most striking feature of Figure 3 is that separation factor decreases for increasing a. This is expected, because if we flow very far on a half cycle (e > 0.5), we tend to transfer the contents of one reservoir through the column and into the other reservoir. This mixing reduces separation. Using the same reasoning, small a allows little mixing between the reservoirs. It might appear a t first that we should choose the smallest possible a for our operations. However, the volume of product in a full reservoir is directly related to a and so small e indicates small column capacity. An economic judgment is required to decide which a is best in any particular situation. EFFECT OF RESERVOIR DEADVOLUME. The effect of the dead volume in the reservoirs (measured as fraction of the volume displaced) is shown in Figure 4. Increasing this volume slows the separation rate but does not affect the ultimate separation. This behavior is expected because the volume of the reservoir never appears in the model equations or in the procedure making them dimensionless, but only in calculations of new boundary conditions for the system. For any given system, there is only one set of boundary conditions (concentrations in the reservoirs) which will produce the limiting condition repetitive behavior. I t is the reservoir concentrations that determine the separation limit, not their volumes. Therefore, if the column in the previously mentioned experiments was equipped with large reservoirs, the ultimate separations would be identical to those obtained, if
( j X~ve1ocity)T
(j.w
x
j)
IO*
r
I
I
I
A .20
X=JV
B u t j,%,is proportional to (Reynolds n~mber)-O.~l for Reynolds numbers less than 50. Since the Reynolds number varies directly with velocity and so inversely with 7, we conclude that for film control
4;
Kow that we have determined how T affects X, let us consider how X affects separation. The calculated separation factor us. number of cycles is shown in Figure 2 for several values of X. As expected, this parameter significantly affects the process operation. For small X (short cycles), little transfer occurs between the phases and so the adsorbent has little effect on the fluid concentrations-hence little separation. The limit X = 0 applies to a bed with inert, nonadsorbing packing. At the other extreme, large X implies equilibrium between the phases. As X increases, the amount of interphase solute transfer increases toward a limit imposed by equilibrium. The effects of small and large X are apparent in Figure 2. The sensitivity of the separation factor to X is evident in the middle range. One interpretation of the X dependence is that a finite rate of interphase transfer is equivalent to an axial dispersion effect which of course reduces separation. It is this effective axial dispersion which produces the upper limit to the separation. K e may conclude therefore that if there are no dispersive effects, parametric pumping separations increase without limit. This conclusion is supported by the graphical calculation. EFFECTOF a. Figure 3 shows the effect of the velocity coefficient, a , on separation. a is equal to half of the fraction of bed interstitial volume displaced per half cycle. Thus, 226
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FUNDAMENTALS
60
NUMBER OF CYCLES Figure 2. Calculated effect of X, dimensionless mass transfer coefficient, on toluene separation factor vs. number of cycles System parameters given in Table II
2b
4b
io NUMBER OF CYCLES I
I
I
I
1
IO
20
30
40
50
NUMBER
OF CYCLES
Figure 3. Calculated effect of cz, velocity coefficient, on toluene separation factor vs. number of cycles
e - t
5'0
Figure 5. Calculated effect of E, phase angle between the velocity and temperature waves, on toluene separation factors vs. number of cycles System parameters given in Table I1
System parameters given in Table II
1
0
IO
I
20
I
I
1
30
40
50
NUMBER OF CYCLES
Figure 4. Calculated effect of dead volume in each reservoir on toluene separation factor System parameters given in Table II
the displacement volume, 30 ml., was unchanged. Thesle separations would require many inore cycles to complete, of course, but the output of the system would be a greater volume of product. EFFECT OF PHASE AKGLE,E . The last major variable Lve consider is the effect of the phase angle, e, between the velocity and temperature waves. A 180' phase angle appears identical to a 0' one if one views the column upside down. Hence, if at 0' the flow downward is accompanied by low temperatures, then a t 180' the flow downward will occur with the column warni. We should therefore expect from purely symmetry considerations that the separation magnitude is unaffected by a 180' shift; the direction of separation is reversed. There is a slight discrepancy to this because we have arbitrarily designated that a cycle begins with upflow and so a 180' phase angle differs from the true inverse of 0" by a half cycle. Figure 5 shows the computer separations for several phase angles from 0 to 2n radians. That the direction of separation inverts is obvious. Also, a t n/2 and 3 ~ / 2there is no separation. Figure 6 shows the cosine-like effect of phase angle on the ultimate separation. This is very much like the result obtained by Wilhelm et al. (1968) for the Tinkertoy model of parapumping. Even small deviations from 0 or n cause a rapid decrease in the separation factor. Simulation of Experimental Separations. Using the STOP-GO computer model with the modified Langinuir isotherm for the toluene-heptane-silica gel system, values of p and K determined experimentally, and actual operating conditions-phase angles, initial conditions, temperature VOL.
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PHASE ANGLE Figure 6. Calculated toluene separation factor at limiting conditions as a function of phase angle, E
NUMBER OF CYCLES Figure 7. Experimental and calculated toluene separation factors vs. number of cycles for three cycle times
System parameters given in Table II
System parameters used in simulation found in Table 111
Table 111.
Numerical Values for System Parameters Used in Simulation Studies Unless otherwise stated in Figures 7, 8, or 9, the following values were used:
Temperature range, "C.
15-70 0.73
PS
2.0
h
0.38 0.0
01
e
Initial concentration 1.00
+I
1.24
6s C, ml. toluene/ml. solution No. of position increments, N Z
0.20
50
raiige3, and fluid velocitieb (Table III)-we have successfully simulated parametric pumping separation in the direct thermal mode (Figures 7, 8, and 9). In the experiments of Wilhelni and Sweed (1968) the only parameter other than temperature varied was cycle time. It has been shown that this parameter affects only A, and so we expect that if the model is complete and accurate, simulation of all three runs is possible by varying only this single parameter. Figure 7 shows that this is not sufficient. To obtain a close fit to experiment the computer model has to include another variable parameter, the dead volume in the reservoirs. As mentioned previously, this volume is a capacity term which has the effect of slowing the rate of separation but not affecting its ultimate value. It is not unreasonable, however, to assume a greater capacitance than the actual dead volume of 0.3 because the 228
l&EC
FUNDAMENTALS
mathematical model is incomplete. First, we have assumed that the reservoirs are thoroughly mixed, yet experimentally vie know that they are not. The fluid in the small tubes connecting the syringes to the column did not mix with the fluid in the syringes. Second, the model neglects the flow caused by density changes on heating and cooling. It is not clear just how this affects the separation but probably it reduces it. Third, and possibly most important, the model assumes that interphase transfer resistance can be lumped a t the particle surface and that the particles themselves are compositionally homogeneous. This is probably not a very good assumption especially at short cycle times during which the solute does not have time to diffuse evenly throughout each adsorbent particle. This problem can be overcome by changing the form of the interphase transfer rate expression t o include intraparticle diffusion (Rolke, 1967). Combining these three effects into a single dead volume parameter has allowed us to simulate the process successfully. The values of X shown in Figure 7 were obtained by matching plots of separation factor us. number of cycles with experiment. A more detailed test of the simulation is whether the model can predict accurately the individual reservoir concentrations, not just their ratio. Figure 8 presents such data for the experimental run with cycle time of 8.5 minutes. The agreement is good. Most of the increase in separation factor occurs because of the change in the bottom concentration. If the initial concentration had been lower than 20%, the top reservoir concentration would have had a more important effect on separation.
The effect of phase angle on the separation factor after 23 cycles is presented in Figure 9. Again the computed values agree well with experiment. The values of X and dead volume are those used to simulate the e = 0 run for T = 8.5 minutes of Figure 7. The simulation of both the reservoir concentrations and the phase angle effect required no special or extra parameters. Illustration of Graphical Method. Let us now calculate the separation obtained via parapumping for a single cycle using the graphical method. Initially let the fluid phase concentration in the column and bottom reservoir be 20% toluene with the solid phase in equilibrium a t ambient temperature (25" C. ). Further assume operation between 70" and 4" C., CY = 0.38-i.e., 76% of the column fluid is displaced each half cycle-and dead volume = 0.0. Initially, therefore, using 20% to make concentration = 1.24. dimensionless, 40 = 1.0 and T o begin calculations, it is useful to prepare a table of the following information at each temperature being studied:
Figure 8. Experimental and calculated toluene concentrations in reservoirs vs. number of cycles Experimental data for cycle time of 8.5 minutes.
Each point represents
a fresh start from equilibrium a t ambient temperature with Auid phase composition a t 20% (volume) toluene. Calculated data for X = 9.0 and dead volume = 1.6, with other system parameters given in Table 111
I._ 0-4
0
This is easily accomplished using a computer but may be done with a desk calculator. Intervals of 0.01 for 4.f are adequate for moderate separation factors. We next construct on rectilinear graph paper a rectangle representing the z - t domain for one half cycle (Figure 10). Extend the time axis to the right so that a second rectangle identical to the first may be drawn. Set the scale of the z - t plane so that the z-axis covers the range 0.0 to 1.0 and the t-axis goes from 0.0 to 3 for the first rectangle and 3 to 1.0 for the second (Figure 10). Next draw the dashed line corresponding t o the velocityLe., dz/dt = CYVwith origin a t (0,O). This line indicates that the fluid (solvent) initially at the bottom of the column will reach z = aV/2 at the end of the half cycle. Next indicate the initial and boundary conditions (4, = 1.0, temp. = 25" C.) along appropriate axes. At t = 0 the temperature in the bed changes to 70" C., and the solute redistributes itself between the phases according t o the new isotherm. So, using the prepared table we find at
CALCULATED
n/Z
7
,PHASE ANGLE,
3n12
Zr
RADIANS
Figure 9. Experimental and calculated toluene separation factors vs. phase angle, E System parameters used in simulation found in Table 111. All data represent separations 23 cycles after start from equilibrium a t ambient temperature with fluid phase composition a t 20% (volume) toluene
I I
E N 0 CYCLE I
I
4.C
hop = 1.24
= 0.72
SEPN.= 1.72
Figure 10. Graphical solution of parametric pumping based on toluene-heptane-silica gel equilibrium Concentrations shown are dimensionless with $J = 1.0 corresponding to 20% (volume) toluene
VOL.
0
NO.
2
MAY
1 9 6 9
229
+
25" C. that $1 i#* = 2.24. Because at 70" C. there is no change in this sum, we search the table a t this temperature for This occurs where +f = 1.24. The reservoir +f +8 = 2.24. is unaffected by temperature and so remains a t = 1.0. This information is all written along the axes of Figure 10. From Equation 15 we know that concentration waves move with linear velocity
+
+/
_ -dz
CYV
dt
1 i- (d+,/d+j*)
All that remains to be done is to plot lines emanating from the initial and boundary conditions (t = 0, z = 0) with slope dependent on the concentration of +I that exists there. Thus, from z = 0 (+f = 1.0) the slope (velocity) at 70" C. (from the table) is 1.000 and from t = 0 (+/ = 1.24) the slope is 1.02. These lines terminate a t either z = 1.0 or t = 3. I n the former case they make up the reservoir concentration (+/ = 1.24). I n the latter instance they represent the concentration profile after a half cycle. The cycle is completed with the redistribution of solute between fluid and solid at the new temperature, 4" C., followed by calculation for the second half cycle. These calculations are identical to those of the first half, except that now the flow is down the column and the inlet reservoir (boundary condition) is a t the top. With nonlinear equilibria the graphical solution predicts dispersion during the desorption flow and concentration sharpening during the adsorption flow. The separation factor is calculated in the usual manner, dividing the top reservoir concentration by the bottom one. Thus Separation factor =
pumping we stage in time not in position. I n practice this means that any desired separation factor can be obtained with small equipment, the major cost being related to time of operation. Other separation processes such as distillation require equipment to increase in size (number of stages) in order to increase separation. We are concerned too about the speed a t which separations can be effected. From Figure 2 it appears that we should choose the highest possible X to reach our separation in the fewest number of cycles. This is not the shortest time, however, because X is a function of cycle time. If we replot Figure 7 to show separation factor us. time (Figure l l ) , we find that the shortest cycle time is actually preferable. Longer cycle times should be used only when shorter ones do not permit reaching the desired separation. (Thus, the 8.5minute time is best for separation factor lo3, but is unacceptable for 105.) Equipment costs and energy production costs may be reduced using parametric pumping. Parametric pumping does not require a phase change (freezing or boiling) and so does not require heat sources or sinks a t these extreme teinperatures, nor does it require that we add and remove the energy associated with these changes. Also, because parametric pumping operates with any finite temperature difference, we may avoid some of the cost in energy conversion equipment by choosing easily attained temperatures. Further, it is possible via parapumping to separate azeotropes and heat-sensitive compounds for which distillation may be unusable.
1.24
-= 1.72 0.72
Using the STOP-GO model to simulate this single cycle, we find Separation factor =
1.238
-- 1.692 0.7319
which agrees well in both concentrations and separation factor. The graphical method has a number of advantages over any numerical technique. Firsb, and most obvious, it is fast, inexpensive, and easy to use. Second, by eliminating all of the dissipative effects in the system (axial dispersion and xioninstaiitaiieous interphase transfer) , we obtain graphically the maximum possible separation for the given parameters. Thus a means is available for rapidly determining the ultimate separation capability of any adsorbent system. Third, the graphical method allows us to visualize what causes the separation. Last, but most important, the graphical procedure shows that if no dispersive forces exist in the system there is no upper limit to the separation.
CYCLE TIME IIN.
- 14C
--
Conclusions
We have explored parametric pumping system behavior both numerically and graphically, and have simulated experimental separations. Let us now consider the significance of our findings. Plots of separation factor us. number of cycles show that separation factor increases approximately exponentially until the limiting separation is approached. This indicates that separations build on one another from cycle to cycle. The exponent,ial rise is analogous to distillation where separation increases exponentially with number of stages; thus in para230
I & E C
FUNDAMENTALS
600
1000
1500
2000
2500
3000
3500
TIME,( MINUTES 1 Figure 1 1. time
Experimental toluene separation factor vs. Experimental points taken from Figure 7
Acknowledgment
= density, gram dry solid/cc. wet bed = cycle time, sec. = dimensionless fluid phase concentration, C/C, +f = dimensionless fluid phase concentration in equilibrium with the solid, C,*/C, d8 = dimensionless solid phase concentration C,P,K/C, w = frequency, sec.-l
pe T
The authors thank R. W. Rolke, R. A. Gregory, T. J. Butts, and D. L. Guttormson for many valuable discussions, and J. E. Sabadell for assistance with experimental work. Nomenclature
A
constant in equilibrium function constant in equilibrium function B fluid concentration, ml. toluene/ml. fluid C Cf* fluid concentration in equilibrium with adsorbent, ml. toluene/ml. fluid C, = arbitrary fluid concentration, ml. toluene/ml. fluid C, = solid concentration, ml. toluene/gram dry solid D = constant in equilibrium function f ( t ) = dimensionless fluid velocity F = equilibrium function (Equation 13) h, = mass transfer coefficient i = position increment i j = time increment j j~ = “j-factor” N T = number of time increments N Z = number of position increments R = interphase mass transfer rate expression t = time, dimensionless T = temperature, O C . V = magnitude of f ( t ) for square wave velocitp z = distance, dimensionless = = = =
GREEK LETTERS a e 77
0 K
X
dimensionless velocity coefficient phase angle, radians dimensionless axial diffusivity a parametei in the distribution function-e.g., perature = reciprocal of column fluid fraction = dimensionless mass transfer coefficient
= = = =
tem-
+*
literature Cited
Acrivos, A,, Ind. Eng. Chem. 48, 703 (1956). Alexis, R. W,,Chenz. Eng. Progr. Symp. Ser. 63, N o . 74, 51 (1 OK?\ \ * Y ” ’ , .
Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” Riley, Sew York, 1960. Jenczewski, T. J., Meyers, -4. L., A.I.Ch.E. J . 14, 509 (1968). Lapidus, L., “Digital Computation for Chemical Engineers,]’ McGraw-Hill, New York, 1962. hlchndrew, 11. A., Ph.D. dissertation, Princeton University, (1967). Rice, A. R., Ph.D. dissertation, Princeton University, 1966. Robins, G. S., B.S.E. thesis, Princeton University, 1967. Rolke, R. IT., Ph.D. dissertation, Princeton University, 1967. Ann. lV.Y. Acad. Sci. 72, 751 (1959). Skarstrom, C. W., Sweed, N. H., Ph.D. dissertation, Princeton University, 1968. Wakao, S . , Matsumoto, H., Suzuki, K., Kawahara, A,, Kagaku Koguku 32, 169 (1968). Wilhelm, It. H., Rice, A. IT.,Bendelius, A. R., ISD. ESG.CHEX. FUNDAMESTAL~ 6, 141 (1966). Kilhelm. R. H.. Rice. A. IT.. Rolke. R. IT., Sweed. S . H.. IND. EKG.%HEM. ’FL7SDAMEXT.& 7, 387 (1968). Wilhelm, R. H., Sweed, N.H., Science 169, 522 (1968). RECEIVED for review December 27, 1968 ACCEPTED February 17, 1969 Studies supported by NSF Grant GK-1427X. Additional support to one of the authors by XSF graduate fellowships is acknowledged. Computations performed in the Princeton University Computer Center, aided by NSF Grant GP-579.
Editor’s Note: The research reported in this paper was carried out jointly by R. H. Wilhelm and S . H. Sweed and is reported more completely in Professor h e e d ’ s Ph.D. thesis writ,teii a t Princeton in 1968.
PARAMETRIC PUMPING AND THE LIVING CELL HElN L. BOOIJ Laboratory of Medical (‘hemistry, Ilniz’ersity of Leidm, Leden, The iVetherlnnds Wilhelm suggested that parametric pumping might serve as a candidate model for active biological transport. In scaling the model to cell dimensions one encounters several difficulties. It i s not easy to fit a process of this kind into the biological membrane, as in doing so the number of adsorbent sites must be drastically reduced. Perhaps this decrease of effective sites will make the process inefficient on the biological level. The next difficulty has to do with the coupling of the oscillatory fields. While oscillatory electrical fields of high frequency at the cell surface are easily conceivable (and perhaps measurable), oscillatory mass transfer fields of high frequency are less likely in the very few (if any) pores of the biological membrane. Beside the experimental approach indicated by Wilhelm, a computational approach, aimed at scaling the mathematical model to a low number of adsorbent sites, might serve as a basis for verification or rejection of the model.
13 1966 a n unusual group of scientists-biochemists, cellular biologists, electron microscopists, geneticists, engineers, mathematicians, physical chemists, and physiologists-met in Frascati, Italy, for a Symposium on Intracellular Transport, organized by the International Society for Cell Biology
(Warren, 1966). The study of energy transductions in the cell, the relationships among structures and functions, movements within cells, and the transport from cell to cell or from medium to cell requires a n increasingly sophisticated integration of the natural, physical, and mathematical sciences. A t VOL.
8
NO. 2 M A Y
1969
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