with measurement of In (Vg/cm3). In this work SO was less than 0.01. The 95% confidence limit for any estimate of us is therefore
Table V. Hydrogen-Bond Energy AH'b of n-Propyl Alcohol in Polar Solvents
estimate f t X SE(afi,) 1-Octadecene 1-Octadecyl bromide 1-Octadecyl cyanide
7.11 6.95 12.84
(A31
where t is taken from the Student t table and depends upon the number of degrees of freedom associated with SO. Assuming the residual error SOis independent of a particular retention determination, the associated degrees of freedom are very considerable and it may be safely taken as 2.0 in all cases.
7.99 6.95 13.35
Appendix. Errors in Determination of AHB In this work values for a f i , were based on k replicate determinations of V, at each of two temperatures (TI and T2).I t can be shown that the variance of a f i , is then given by var.
(a,) = 2u02R2/k[(Tl- T2)/T1T2)I2
(AI)
where uo2 is the residual error which may be associated with measurement of In ( V,/cm3) if it is assumed that temperature is known without appreciable error. Taking a coefficient of variation for V, as 1%and using temperatures of 310 and 320 K by way of illustration, then for four replicates at each temperature, the standard error of hH, [var. ( afi,)]1/2is obtained as 0.59 k J mol-I. In this work up to six replicates were made a t each temperature and in general the coefficient of variation for V , was 0.01 so that the standard error (SE) for any value of msmay be given as
(a) (TiT2IAT)So
SE(ARJ = 8.31
where So is an estimate of the residual error
(A21
(00) associated
L i t e r a t u r e Cited Anderson, A., Shymanska, M., Izv. Akad. Nauk Latv. USSR, 5, 537 (1969). Bazant. V.. Chvalovsky, V., Rathousky. J., "Organosilicon Compounds", Czechoslovak Academy of Science, Prague, 1965. Brown, I., Chapman, I. L., Nicholson, G. J., Aust. J. Chem.. 21, 1125 (1968). Green, J. T., Ph D. Thesis, University of Manchester Institute of Science and Technology, 1968. Hammers, W. E., de Ligny, C. L., Recueil, 88, 961 (1969). Hayes, J. R., Ph.D. Thesis, University of Manchester Institute of Science and Technology, 1969. logansen, A. V., Kurkchi, G. A., Levina, 0. V., "Gas Chromatography", p 35, A. B. Littlewood, Ed., Institute of Petroleum, London, 1967. Kovats, E., Weisz, P. E.,Ber. Bunsenges. Phys. Chem., 69, 812 (1965). Langer, S.H., Purnell, J. H., J. Phys. Chem., 70, 804 (1966). Littlewood, A. E.,Anal. Chem., 36, 1441 (1964). McClellan, A. t., "Tables of Experimental Dipole Moments", W. H. Freeman and Co., London, 1963. Meyer, E. F., J. Chem. Educ., 50, 191 (1973). National Bureau of Standards Circular 537, 1953. Smith, J. W., "Electrical Dipole Moments", Butterworths, London, 1955.
Received for review J u n e 7,1974 Accepted F e b r u a r y 13,1975 T h e a u t h o r s t h a n k Professor R. N. Haszeldine for m a k i n g available facilities a t U.M.I.S.T., a n d also Mr. J. M. B a t h e r u n d e r whose auspices t h e w o r k was performed.
Transport Resistances Influencing the Estimation of Optimum Frequencies in Parametric Pumps Richard G . Rice Department of Chemical Engineering. Univers/ty of Oueensland, St. b o a , Brlsbane 4067, Oueensland, Austraha
The selection of frequency and amplitude in the closed direct-thermal-mode parametric pump is studied with the view of maximizing ultimate separation. Models of the ultimate separation which include the dissipative effects owing to axial dispersion, interparticle thermal and mass diffusion, film mass transfer resistance, and solid pore diffusion are presented. Analytical solutions to the periodic convective-diffusion equations with linearized adsorption isotherms indicate there exists an optimum frequency which depends on the type of mass or heat transfer constraint. The comparison with some literature experimental results was favorable, and shows that pore diffusion appears to be a most significant dissipative effect in parametric pumping.
Introduction Parametric pumping is the name given to a dynamic separation principle which couples periodic adsorptiondesorption mass transfer with periodic fluid motion so as to effect a separation of fluid components. The usual design is comprised of a packed adsorbent bed surrounded by a jacket which is alternately heated or cooled (directthermal-mode, see Sweed and Wilhelm, 1969). The fluid to be separated is pulsed back and forth through the packing in phase with periodic heating or cooling. An up202
Ind. Eng. Chern., Fundam., Vol. 14, No. 3, 1975
to-date review of research in the area has recently appeared (Sweed, 1971). To date, research has been mainly concerned with modelling the transient response of the parapump, with various degrees of parameter estimation to obtain a good fit of theory and experiment. No doubt such work was necessary to understand specific systems, but would be difficult to apply to the design of entirely different systems. A very readable account of the formulation and solution of transient nondispersive models for parametric pumping has appeared in a recent book by Aris and Amundson (1973).
On considering the construction of a direct-thermalmode parametric pump, one naturally inquires: what pulsation frequency should be used? There are countless ways of pulsing the fluid, including: infusion pumps, motor driven lever arm-cam arrangements, and recently (Jones, 1974) a leak-proof and rather novel technique was demonstrated whereby a mercury-filled reservoir was alternatively raised and lowered. The purchase of a motor gear reducer is usually inevitable if one is to obtain cycle times reported in the literature; commonly for liquid separations this would be in the range of 8 to 140 min (Sweed and Wilhelm, 1969), and for gaseous separations of the order of 2 min (Jenczewski and Myers, 1970). It is very likely that many rather unsuccessful parapumps have not been reported in the literature (there are exceptions, for example, Rice and Mackenzie, 1973) and this is unfortunate since much could be learned from the poor performers as regards proper frequency selection. We set out here to develop design relations based on first principles which will allow an a priori estimation of frequency and amplitude to produce the best separation possible. The approach we take here is to develop steady-state models of the separation, that is, when the time-average solute concentration in the parapump reservoirs no longer change. This limit occurs owing to certain destructive effects inherent in the process, for example: axial dispersion, fluid-packing mass and heat transfer resistance, inter- and intraparticle heat and mass diffusion, as well as effects arising from the type of adsorption isotherm obtaining. Recently, the ultimate steady-state for single solute separation has been modelled with the view of elucidating the effect of axial dispersion (Rice, 1973). The combined effects of axial dispersion and interparticle heat and mass diffusion (Rice and Foo, 1974) has also been studied. Both of these works invoked the concept of equilibrium between bulk fluid and solid phases, and both demonstrated there should exist an optimum frequency for maximum separation. The current work considers the consequence of a state of nonequilibrium between moving fluid and solid, as well as limitations imposed by diffusion within a porous solid adsorbent. Our first nonequilibrium model will thus consider the combined effects of axial dispersion, interparticle heat and mass diffusion and finite mass transfer resistance outside the particles. Our second nonequilibrium model will consider the combined effects of axial dispersion, finite mass transfer resistance and intraparticle mass diffusion (pore diffusion).
sumed equal, the combined energy balance for both phases is taken as
It is this radial dependence of temperature which induces a concentration dependence on radial position. The reservoirs are sufficiently small so that terms arising from axial temperature gradients can be neglected. The solute balance on the solid phase assumes that unconvected solute within the solid pore structure is in equilibrium with the solid phase, hence
k,a(C - C * ) (3) A linearized version of the equilibrium isotherm such that concentration and temperature do not appear as products can be written as a deviation from the steady-state profile
where the partial derivatives are evaluated at the (arbitrary) steady temperature and initial or average column composition. The equilibrium isotherm thus takes the simplified form (deviation from steady-state)
.
c = C(s) + C ( r ,f ) c* = E*(.\-) + c * ( I , , 1 ) T = T + T(Y,/)
(1 -
> 1, both Models I and 11 collapse to a single design curve independent of AIR when eq 24 and 46 are presented as a function of PeMAIR. We now corroborate the theoretical predictions with some experimental results. Comparison with Literature Experiments Few reported parametric pump studies provide sufficient data to compare with the current models, since for Model I one needs: AIR, PeM, Pr, Sc, and RID, as well as the adsorbent parameters t , k, and at. Model I1 requires values of A fR, PeM, Sc, Sp/S,and RID,, in addition to c, k , and at. The most difficult system parameter to estiis possibly the most important. Fortunately mate, Sp/a>, one particular study of the separation of aqueous NaCl using ion retardation resin (Sweed and Gregory, 1971) presents a value of Spalong with the necessary adsorption equilibria. Table I lists the physical properties given by Sweed and Gregory, along with estimates of parameters not explicitly presented. Figures 2 and 3 compare the experimental separations with predictions based on Model I and Model II. In Figure 2, constant amplitude, variable frequency results are compared and Figure 3 combines the variable amplitude, variable frequency experimental results in a single plot since published earlier (i.e., Model I with k , a).The paramfor AIR >> 1; the two Models can be shown to depend eters used in this figure correspond to the aqueous oxalic only on PeMAIR, when all other parameters are fixed. acid system on activated carbon and would be typical of The comparison is gratifying, considering the complexiliquid separations. For a specified amplitude ratio, the ty of the models. When the amplitude-frequency product mass resistance-thermal diffusion Model (Model I), and is small one expects that pore diffusion should be controlthe pore diffusion model asymptotically approach the ling, as indeed the experimental data seem to show. As simple equilibrium model a t low frequency (low Peclet amplitude-frequency product increases, solute penetration number). This is certainly expected and lends creditabilinto the porous adsorbent would be slight, with adsorpity to the rather complicated analysis required in develtion/desorption taking place mainly on the outside suroping Models I and 11. Furthermore, the curves show that face. One would expect then that thermal diffusion and the peak values of separation diminish as the magnitude of pore diffusion increases, as expected. When Sp = 9, mass transfer resistance would be controlling and the experimental data in Figure 3 seem to show this trend. Real the pore diffusion model predicts lower separations than experimental systems are thus expected to fall somewhere Model I, even though Model I included thermal diffusion in between the two models and hence the area separating limitations not included in Model 11. Pore diffusion is thus the two curves can be classified as a region of probable seen to be the most likely rate-limiting step unless Peclet separation for design estimation. number is large. At large Peclet number, thermal peneThe relative magnitudes of the destructive terms in the tration is slight, hence Model I (which includes thermal
-
206
Ind. Eng. Chem., Fundam., Vol. 74, No. 3, 1975
Table 11. Relative Magnitudes of Destructive Termsa in Models I and I1
0.1
1.0009 1.092 10.18 918.56
0.00206 1.686 107.16 4,975.0 Parameters as in Table I, A I R = 87.3.
1 .o 10.0 100 .o
0.00646 5.101 416.8 32,276.6
3.13 3.02 3 .89 6.69
denominator of eq 24 and 46 are compared in Table I1 for the Sweed-Gregory parameters. Because of the similarities of the model structures and the fact that both Models have the same Taylor dispersion term (RD)in the denominator, it is thus possible to generalize as to which destructive effect produces the largest constraint. For the Sweed-Gregory experiments, it seems obvious from Table I1 that pore diffusion controls. Furthermore, it would be expected that order 25-fold increase in separation would have occurred had the amplitude-frequency product been selected in the vicinity of optimum. Transient models which neglect this dominant resistance cannot be expected to give good predictions using transport coefficients (such as eq 29) from the literature and must therefore depend on parameter estimation techniques. Clearly such expedience is quite acceptable after a successful experiment has been performed. However, one must be very cautious in extrapolating such results to new and unknown systems. It is suggested the current models may be useful in defining an acceptable operating region. Comments and Conclusions The dissipative influences which control the ultimate separation in parametric pumps are treated in the form of two distinct models. If the adsorbent is very porous it would seem Model I1 (film transfer and pore diffusion controlling) works best in estimating design values of displacement amplitude and frequency. However, when adsorbent particles are relatively small and nonporous, it would appear thermal diffusion becomes increasingly important, especially a t higher frequency. Under these conditions, Model I (interparticle thermal-mass diffusion and film transfer controlling) is recommended. Both models use purely sinusoidal potentials to describe the coupling of velocity and temperature waves. While the first harmonics only approximate the commonly used square wave potential, there is some justification for using the sinusoidal approximation if one of the two potentials is nearly sinusoidal. It has been shown under such conditions (Rice, 1973) that only products of first harmonics contribute to the time-average flux. The linearized isotherms used in this work decouple temperature and concentration. This effectively means that the isotherms are parallel, hence an average mass transfer driving force is assumed to exist. This naturally precludes the occurrence of a pinch-type limiting effect on separation. This important thermodynamic limitation will be treated in a later communication. Appendix Functions used to compute ultimate separations are presented below.
(Shp
DI =
+
Pep)sinh ( p ) cosh ( p ) (Sh, - Pep)c o s ( p ) s i n (p) sinhz ( p ) + sin2 ( p ) I . _
1.
p - Pe,(l
- Nu,)
Addenda The left-hand side of eq 24 and 46 can be expressed in terms of the separation factor a t infinite time, N., = CBOT/CTOPin the following way. If the isotherm is linear, that is = f(C*, T) = k(T)C* and the rate constant can be expressed as a linear function of temperature so that k ( T ) = iio - k,T then eq 4 shows that
Moreover, if we assume the average column composition can be described by a long-mean average then
Hence, the following equivalence can be inserted into t 2 24 and 46 and Figures 1 , 2 , and 3. AC ATa,p,L/R
In (a,) = ATk,p,L/R
Nomenclature
A = displacement amplitude of velocity a = packing factor, 6(1 - t b ) / D s surface area/volume of column at = thermal adsorption coefficient, defined by eq 5 ber({), bei({) = real and imaginary parts, respectively. of
JocTt/=T) C = bulk flowing fluid concentration, M
A C = concentration, difference between reservoirs, CH,, I
-
CTOP
C* = fluid concentration in equilibrium with solids position q DS = diameter of adsorbent particle 3) = molecular diffusion coefficient = pore diffusion coefficient a* = axial (Taylor) dispersion coefficient hs = solids hold-up, (1 - € b ) ( l - f p ) / f b h p = pore hold-up, fp(f - c b ) / t b Hp = solids capacity, p8k( 1 - t p ) / c p Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1875
COIYI-
207
i = 6 1 Jn(x) = Bessel function, first kind, order n k = slope of equilibrium isotherm, defined by eq 5 k , = mass transfer coefficient (calculated from eq 29) L = length of packed bed Le = Lewis number, cue/% = Sc/Pr Nu, = particle Nusselt number, Rskc/Dp P a = PeM(kp,h, + hp)/Sh PeH = Peclet number for heat, R2w/cue PeM = Peclet number for mass, R2w/D Pep = particle Peclet number for mass, RS2w/Dp P r = Prandtl number, v / a e q = solids composition, mol/gr of compressed solid r = cylindrical radial coordinate for packed bed r, = spherical radial coordinate inside solid particle R = columnradius R, = radius of adsorbent particles Ro = dispersion dissipative term defined by eq 25 RM = film mass transfer dissipative term, defined by eq 26 Rp = pore mass transfer dissipative term defined by eq 47 Sc = Schmidt number, v / Sh = Sherwood number, kcaR2/tbD Sh, = particle Sherwood number, kcaRs2/cbDp t = time T = local bed temperature AT = amplitude of applied temperature u = interstitial fluid velocity WL1 = parameter defined by eq 22b x = axial coordinate of bed zL1 = defined by eq 22a Z R = Real(Z+1) ZI = Imag ( Z + I )
Greek Symbols = effective thermal diffusivity of bed and fluid @ = (1+ HD)/2 t b = bulk voidageof bed t p = pore voidage of solid particles ole
T=vTG
d=zz
A+, = h = p,h,ATa;' v = kinematic viscosity [ = dimensionless cyclindrical coordinate, r / R . ^
Fs = dimensionless spherical coordinate, re/Rs
mP)
= com ressed solid density, g/cm3 = w = frequency of oscillation ps
T+1
Literature Cited Aris. R.. Amundson, N. R.. "Mathematical Methods in Chemical Engineering", Vol. 2, Prentice-Hall Inc.. 1973. Baker, B., Pigford. R . L.. lnd. Eng. Chem., Fundam.. 10, 283 (1971). Bird, R. B., Stewart, W. E.. Lightfoot, E. N., "Transport Phenomena", p 647, Wiley, New York, N.Y.. 1960. Jenczewski, T. J., Myers, A . L., lnd. Eng. Chem.. Fundarn.. 9 , 216 (1970). Jones, P. J., B. E. Thesis, University of Queensland, 1974. Perry, R. H., Chilton, C. H., Kirkpatrick. S. D.. Ed., "Chemical Engineers' Handbook," 4th ed, p 5-51, McGraw-Hill, New York, N.Y.. 1963. Rice, R. G., Ind. Eng. Chem., fundam.. 12, 406 (1973). Rice, R. G., Foo, S. C., Ind. Eng. Chem.. Fundarn., 13, 396 (1974). Rice, R. G.. Mackenzie. M.. Ind. Eng. Chem., Fundam., 12, 486 (1973). Sweed, N. W. "Progress in Separation and Purification", Vol. 4, Wiley. New York, N.Y.. 1971. Sweed, N. W., Gregory, R. A.,A.I.Ch.E.J., 17, 171 (1971). Sweed, N. W., Wilhelm. R. H.. Ind. Eng. Chern.. fundarn.. 8, 221 (1969).
Received for review June 19, 1974 Accepted March 5, 1975
A New Probe and Circuit for Measuring Electrolyte Conductivity Soon Jai Khang and Thomas J. Fitzgerald* Department of Chemical Engineering, Oregon State University, Corvallis. Oregon 9733 1
A new probe design used in an operational amplifier circuit gives linear response to changes in electrolyte concentration and is not subject to errors from stray current effects, thus allowing multiple simultaneous conductivity measurements in the same vessel, a s well as the presence of other metal surfaces in contact with the electrolyte solution.
The response to a tracer impulse injection can give useful information about a system. For example, in a flow system the tracer impulse response is the residence time distribution of molecules in the exit stream and is useful in formulating a dynamic model of the system. A good tracer must not interfere with the behavior of the system, should be easily measured a t any point in the system or in the output stream, and should be easy and safe to handle. Electrolyte solutions fulfill all the above requirements in many liquid flow systems. Methods of measuring concentrations of electrolyte solutions by means of conductivity cells and alternating current were developed in late 19th century. Parker (1923) first noted that the cell constant of a conductivity cell changes during measurement. Later, Jones et al. (1928) carefully studied this effect and refined the techniques of measurement of the conductivity of uniform solutions. 208
Ind. Eng. Chem., Fundam.. Voi. 14, No. 3, 1975
Their major contribution was the platinization technique to reduce polarization on the surface of electrodes. More recently, methods for measuring time-varying local concentrations have interested investigators in various fields. Kramers et al. (1953) reported the first probes, shown in Figure l a , which were employed to detect local concentration variations in a mixing tank. Prausnitz and Wilhelm (1956) and Clements and Schnelle (1963) made probes with two exposed platinum wires (Figure l b ) . The probe configurations were further changed to measure concentrations in a much smaller volume element. Lamb et al. (1960) designed a probe which consisted of a point electrode and a wire ring electrode (Figure IC). Since the surface area of a point electrode is much smaller than that of a wire ring electrode, the current flow, and thus the conductance through the probe, is mainly proportional to the concentration near the point electrode. Reith (1965)
