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A new method for determining membrane diffusion coefficients from their response to regular forced concentration waves
Journal of Membrane Science, 27 (1986) 105-117 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
105
A NEW METHOD FOR DETERMINING MEMBRANE DIFFUSION COEFFICIENTS FROM THEIR RESPONSE TO REGULAR FORCED CONCENTRATION WAVES
RUSSELL PATERSON and PAUL DORAN Department of Chemistry, (Great Britain)
University
of Glasgow,
Glasgow
G12 899,
Scotland
(Received August 6, 1984; accepted in revised form April 18, 1985)
Summary It is shown that when a membrane separates two permeant solutions, initially at equilibrium, and subsequently one of the solutions is forcibly oscillated, the concentration waves detected on the other side will show reduced amplitude and a change in phase angle. Using mathematical models it is shown that these may be expressed as explicit functions of the permeability and diffusion coefficient of the permeant, respectively. Solutions for cosine and square concentration waves are given. An experimental system was devised to exploit these effects and measure diffusion’coefficients directly from the phase shift of the emergent waves. Input waves were generated using alternating sprays (concentrated and dilute) from two artists’ air brushes. In preliminary experiments, diffusion coefficients for salt through test membranes were easily obtained to a precision of 5%.
Introduction
In network thermodynamics, membranes are described in terms of generalised resistors and capacitors [l] . These reflect the ability of a membrane both to dissipate power and to store chemical energy internally in each local volume during transport processes. For quantitative modelling, the membrane is subdivided conceptually and mathematically into homogeneous “lumps” or slices, each characterized by its own resistance and capacitance. The accuracy of quantitative modelling of diffusion processes depends upon the degree of reticulation of the lumped model; since the model approaches ever more closely to a true continuum of states as the number of lumps (n) is increased. As an example, to model a membrane which was exposed on one side to a source of constant effort, SE, (defined by a constant chemical potential of the permeant) and connected to a closed (homogeneous) volume of solution *Paper presented at the Europe-Japan Stresa, Italy, June 18-22,
0376-7388
/86/$03.50
Congress on Membranes and Membrane Processes,
1984.
0 1986 Elsevier Science Publishers B.V.
106
(or gas) on the other (defined by a chemical capacitor, C,), we may use a bond graph, Fig. la or an equivalent circuit, Fig. lb. These alternative representations show a S-lump model of the membrane. (The bond graph notation, although less familiar, is the more powerful for modelling complex, coupled, transport phenomena [l-3] .)
(a) tb)
SE-l-O+-O-l-0-l-Ct jc,
Fig. 1. (a): A bond graph representation source of effort, SE, (either constant or voir or collecting volume on the other, (b): The equivalent circuit representation
of a 3-lump model of a membrane exposed to L oscillating) on one side and connected to a reserrepresented here by the terminal canacitor. .~_ -~ _~I Ct. of the same membrane system.
For a system which obeys Fick’s laws of diffusion it is convenient to use a pseudo-system in which local efforts (properly defined as chemical potentials) may be replaced by concentrations [4, 51. In these papers it was shown that the chemical resistance, R, for each lump of an n-lump model of a membrane (defined by Fick’s law) is given by eqn. (l), and the corresponding capacitance, C, is defined by the lump volume, R = l/DA a n
(1)
C = &Al/n
(2)
In these equations it was assumed that the membrane was regular, of area A, and thickness I, and had a diffusion coefficient, D, defined by Fick’s first law. (The distribution coefficient, IY, was introduced [5] to define the concentration of the equilibrium solution as a common effort in all phases.) These networks may be used algorithmically [2,4, 51 to compute the system dynamics of a membrane assembly for any experiment or membrane, in which bond graph parameters and initial conditions are defined [ 4-61. For such simple conditions as described here it is only necessary to know the geometry of the membrane/solution system and the diffusion coefficient and distribution coefficient of the membrane involved (unstirred layers of solution may be treated as additional lumps). These methods are of particular interest for simulation of membrane processes and are easily expanded to deal with multicomponent, coupled transport, including energy transduction
[61.
The formal analogy between a network thermodynamic bond graph and an electrical circuit, Fig. 1, has lead us to reexamine the inverse problem, that of membrane characterisation. With the circuit and bond graph repre-
107
sentations clearly in mind, it is obvious that all standard measurements of membrane transport are, in electrical terms, DC methods and typically involve fixed driving forces (the efforts of network theory) and steady state measurements. In particular, the membrane scientist lacks powerful AC methods used routinely to investigate electrical circuitry. Using mathematical models and bond graph simulations, we set out to examine the system dynamics of a simple membrane assembly consisting of a (concentration) source, a membrane, and a collecting vessel, as represented in Fig. 1, when the source of concentration is oscillated with a fixed frequency and amplitude. From the properties of the analogous electrical circuit, Fig. lb, we would expect the membrane to act as an AC filter, reducing the amplitude and causing phase shifts in the emergent waves (detected in the terminal capacitor/collecting volume). In this paper we examine how these effects may be employed for membrane characterisation. A variety of concentration wave forms, frequencies and amplitudes were tested. From the results it became obvious that, with correct selection of experimental conditions, diffusion coefficients might be obtained directly (and repetitively) from the phase shift of the emergent concentration wave. On the basis of these mathematical models, apparatus was designed and new procedures were devised to perform such experiments in the laboratory. To assist with the design of membrane cells and experimental conditions, computer simulations were made, using a multilumped version of the bond graph, Fig. la. -These predicted the evolutionary path of experiments to a steady state of oscillation (a requirement of the mathematical models) and ensured that cell volumes and other geometric factors in the proposed cell designs would not cause unacceptable deviations from the mathematical predictions, which were limited, at least initially, to infinite collecting volumes, VT. In the course of this research we discovered some interesting precedents for the use of forced oscillations, in studies on thermal diffusion, by &rgstrijm [ 71, Thomson (Lord Kelvin) [S] and more recently in gas diffusion
Mathematical models The membrane is considered to be exposed on one side to forced oscillations in the concentration of the contact solution and to be connected on the other to a closed reservoir or collecting volume, VT, whose initial concentration, cO, is the mean concentration of the ingoing wave. It is assumed also that the membrane is in equilibrium with its contacting solutions at all times, and that (if the membrane is in the form of a clamped sheet) “edge effects” can be neglected [lo, 111. Fick’s laws are assumed with constant diffusion coefficient, D. A further simplification is the assumption that the collecting volume, VT, is effectively infinite. In practice this is not a severe limitation, as shown below. Three wave forms were considered, cosine, square, and triangular. Only
108
the first two will be discussed here because no further information was obtained by the use of triangular waves. For our systems the phase shift altered by only 4” on changing from square to triangular waves, but for triangular waves there was. knificant losr of amnlitude gn the em_e~gep_k o_‘___--____-LL -.-, in ___addition_ _.__.______, a_L_ r-----concentration wave. Cosine waves The fundamental responses to a cosine wave were considered initially. The wave form and boundary conditions for a solution of Fick’s laws were eqns. (3) and (4). In eqn. (3) the source of effort, SE, at any time, t, is defined as the concentration of permeant at the surface of the membrane (x = 0), defined by the cosine function, which oscillates about a mean concentration, cO, with a frequency of w rad-set-‘. Interfacial equilibrium is assumed. With the additional assumption of an infinite homogeneous (stirred) collecting volume, VT , its concentration, cT , is constant and set equal to the mean, cO, eqn. (4). This is obtained at the opposite membrane face (x = 1) SE = c = co [ 1 + cos(o t)]
atx=O
(3)
atx=l
(4)
and cr = c‘,
In this model the quantity of permeant, qT (mol), transferred into the (infinite) collecting volume, is obtained as a function of time. Once initial transients have decayed, this is given by eqn. (5). This was obtained as a solution of Fick’s second law of diffusion, by the standard technique of separation of variables [ 111, under the boundary conditions specified by eqns. (3) and (4) qr =B(w)
cos[wt + ‘$(a) - 7T/2] + 40
(5)
where B(w) is the amplitude of the emerging wave, defined by eqn. (6) B(w)
=
23’2A K (Y co D {B[cosh(2KI)w
cos(XE)]
7%
(6)
and [4(w) - ‘IT /2] is the phase change between input and output waves, eqn. (7), (see also Fig. 2) G(w) = tan-’
(7)
In eqns. (6) and (7), K = [w /(2D)] I/n From eqn. (7), it is clear that the phase change of the emergent wave is defined by only three parameters, the frequency of the input wave, the membrane thickness and the diffusion coefficient, eqn. (7). The corresponding amplitude, eqn. (6), is a more complicated function and depends additionally upon the distribution coefficient of the diffusant, OL,and the membrane area, A. These analyses were made under the simplifying assumption that the
109
collecting volume was infinite. Solutions were also obtained for finite systems [12], but are omitted here since it was shown that negligible error was incurred in these studies where the terminal, collecting volume was = 1 ml. (Limiting criteria are given below in the section on membrane responses to square waves.) Under conditions where the collecting volume, VT, is finite and eqns. (5), (6) and (7) remain valid, it is convenient to define the displacements and amplitude of the corresponding concentration waves by dividing eqn. (5) by VT. It is clear from eqn. (5) that the quantity of permeant or, alternatively its concentration, cT = qT/VT , will oscillate with the same frequency as the forcing wave, eqn. (3), but phase shifted by an amount which depends upon the diffusion coefficient and might be used to determine this in test membranes under laboratory conditions. Bond graph simulations [2, 31 were made for a range of common membranes exposed to cosine concentration waves in cells with various collecting * IT volumes, VT, and exposed membrane areas. That given in Fig. 2 illustrates the calculated response of a Visking dialysis membrane in the cell which was ultimately constructed. The permeant was potassium chloride and the wave was defined by eqn. (3) in which co was 0.065 M and w = 2n/lOO (period, 100 set). The output amplitude was small, approximately 0.2% of the source. Such output amplitudes are typical and so largely preclude their use for the determination of permeability. With a proper choice of frequency, however, phase shifts can be large. This is clearly illustrated in Fig. 2 and shows that if emergent waves could be recorded such shift might be used to determine diffusion coefficients accurately. (In Fig. 2 arbitrary scales of amplitude have been used for illustrative purposes.)
TIME
IS1
Fig. 2. Representation of the phase shift with a source, SE = c, [ 1 + cos(wt)] , and a large collecting volume. Membrane parameters: D = 2.75 x 10-6cm2-sec-‘. 1 = 0.205 mm, A = 0.771 cm*, (Y = 0.702, and c0 = 0.065 M, taken from independent measurements [13]. Arbitrary scales have been used and the input and output amplitudes have been made equal for ease of comparison.
The choice of amplitude and frequency for the input concentration wave determines both the phase shift and the amplitude of the emergent wave. For accurate determinations we require that the phase angle G(W), eqn. (7), be large, favoring the application of high frequencies. The amplitude of the
110
emergent wave, B(w)/V, (expressed as a concentration), is itself frequency dependent, but in the opposite sense. It tends to zero rapidly as frequency is increased. The practical limit to experimental frequency is therefore the sensitivity of the method used for detection of the wave. Since membrane parameters are strongly concentration dependent it is not practical to use large concentration oscillations; once more the most useful application of these methods appeared to be direct and repetitive measurements of diffusion coefficients from the phase shift. For typical membrane conditions large phase shifts were predicted for oscillations with periods in the range lo1000 set which were sufficiently long to encourage us to consider generating concentration waves using microcomputer controlled equipment. Experience has shown that practical frequencies are of the order of D/E’, which is, in turn, equal to the reciprocal of the product RC for a l-lump bond graph model, eqn. (1) [2]. In free response the relaxation time of the l-lump membrane is 7 = RC/4 = 12/4D [ 1, 5 1, while breakthrough times for diffusion equal 12/6D. It is therefore clear that there is an intrinsic time base for a membrane (=RC) and that our choice of frequency should reflect this. Square waves Although performing sinusoidal oscillations in concentration by ingenious use of controlled syringes is feasible, by far the most convenient wave form is the square wave, which may be obtained using sprays, as described below. A solution for source oscillations in the form of a square wave was similar to those given above. The boundary conditions were as before except that the source, SE, was now a square wave represented by a Fourier series, eqn. (8). The corresponding steady state oscillations (also as a summation) are given by eqn. (9) SE=
(-l)(=
co ) . ..
qT
=
4i =1,3,5 c
B(nw) cos[not
;
n
, ...
1)‘%os(n6) t)
(8)
n + @(nw) - n/2] (n
l)(n -I)‘2 +qo
(9)
The symbols have the same significance as before but now $(nw) is the value of o(w) for the nth contributory wave of eqn. (B), and similarly for B(nw). The results are shown in Fig. 3. Once more an infinite collecting volume, VT, was assumed. Parallel derivations for limited collecting volumes were made [12] but are not presented here, since it was shown that the two solutions effectively coincide when &?DAd/w VT@l. For the test system this function was 0.002 and the change in #, even for a collecting volume of 2 ml, as used in this work, was only 0.09”~. These predictions were verified by bond graph simulations. The calculations were performed using the methods described earlier for the SE-C
111
a-~I 0
50
100
150
50
100
150
200
250
300
250
300
7E-03
-7E-03
0
200
TIME
350
IS1
Fig. 3. Concentration waves in the collecting volume, generated by a square wave source. X, experimental; -, from the mathematical model, eqns. (8) and (9); and A, using a 20lump bond graph, as in Fig. la. Membrane parameters as in Fig. 2. Amplitudes are expressed as experimental conductances (mmho).
model [3, 41, but now using the cosine, square, or other wave expressions for SE, as above. Lumped models were constructed according to the membrane and cell parameters of the model to be predicted and resistances and capacitances defined by eqns. (1) and (2). The state space equations were integrated numerically to simulate the experiment and provide predictions of permeant concentration as a function of time in the collecting volume. It is worth noting that the whole experiment was modelled including the initial period in which transients exist. Additional information as to the outcome of experiments with other wave forms or the time dependent profiles of permeant concentration across the membrane are obtained easily from the same bond graph, as a matter of course. Experimental The diffusion cell was made from a rectangular block of perspex of dimensions 3.3 X 4 X 8 cm3. This was drilled horizontally and vertically to provide two intersecting cylinders, Fig. 4. The vertical one was plugged at the bottom to hold a “Spin fin” stirrer (Be1 Art) which was driven magnetically from outside the cell. The upper portion of the vertical tube was threaded and sealed with a teflon screw top which held a pair of platinised platinum wires which served as conductance electrodes. The bottom of this screw top was a hemispherical hollow to improve mixing in the cell. It also contained a capillary to aid the removal of entrapped bubbles. The horizontal hole passed through the block. On one side, Fig. 4B, it defined the exposed membrane area, on the other it too was threaded and sealed with a teflon screw, which held a thermistor temperature probe.
Sb
4
5cm
’
Th
Fig. 4. A diagrammatic representation of the experimental cell. (A), (B) and (C) represent sections of the cell viewed from the top, side and front. Symbols: E, platinum (conductivity) electrodes; M, membrane; Mg, motor driven magnet; Sb, perspex spray box to which the cell is clamped when in use; St, teflon coated magnetic stirrrer; Te, flanged template defining membrane area exposed to spray input; and Th, thermistor. The sprays themselves are not shown. They are placed some 18 cm from the exposed membrane surface and aimed at its center.
The cell, which held a solution volume of 2 ml, was bolted to a large perspex (spray) box. The membrane aperture was defined by the coincident holes in the cell and a flanged template, Fig. 4. Three sprays (Badger Air Brush Coy), attached to thermostatted reservoirs of the solutions to be sprayed, were directed into this box, each aimed at the exposed membrane area. In these demonstration experiments dilute potassium chloride with concentrations c,, and 2c, and water were used. The cell was filled initially with salt at the mean concentration c *. (Water might also be used, in which case the steady state of oscillation was reached after a longer period. This was partly compensated by reducing the number of sprays to two only, water and potassium chloride at 2c,.) The square wave was obtained by turning the air brushes on and off alternately at regular intervals, using solenoids attached to the nitrogen gas lines (20 psi) which operated them. The switch-
113
ing was performed by a small computer and the concentration waves in the cell (collecting) volume were detected by a Wayne Kerr conductance bridge (precision 0.01%) and recorded. (Initially a precision chart recorder was used, subsequently the data has been collected digitally.) Tests showed that concentration changes at the surface of the membrane were complete within 0.1 set on switching. Estimates were obtained by replacing the test membrane with a perspex sheet which held two platinised platinum wire electrodes, pressed level with its surface on the exposed membrane area. Adiabatic cooling required that the solutions to be sprayed should be thermostatted at a much higher temperature than required in the diffusion cell itself. In this work these solutions were maintained at 40” C. To minimise temperature variations which occurred on switching from one spray to another, the nitrogen line and solution tubes leading to the air-brush sprays were thermostatted for the distance of =l m from thermostat to spray. Three sprays were required. In the preliminary state the membrane was sprayed with the Y^^_ _-._-_-L.._L:_.- co, eq_ad to &&, in *he fGled ce;_Gnce this sy;ys*em w88 rIlea COIlCeIlbrablOIl, in thermal (and chemical) equilibrium as indicated by steady temperature and conductivity readings, the experiment was begun, by switching between the lower and upper concentrations at regular intervals. The three spray guns were directed at the centre of the exposed membrane area, Fig. 4, at a distance of al8 cm. A thermistor, Fig. 4, was used to measure solution temperature within the cell. This was affected strongly by temperature variations in the sprayed solutions, and great care was needed to ensure that the impinging sprays were at 25 + O.l”C. This was achieved by thermostatting the gas and solution lines at 40°C right upto the air brush jets; this also prevented significant temperature fluctuations when switching sprays. As an additional precaution, the cell block was thermostatted independently by circulating water at 25 + 0.05”C. Membranes were prepared from Visking dialysis tubing and preequilibrated with water and solution before use. Results and discussion The results of experiments using Visking dialysis tubing are summarised in Table 1. Square waves were produced with amplitudes of around 0.05 M by spraying alternately with water and potassium chloride solutions of 0.1 M. The concentration of salt in the collecting volume was obtained from the conductance of the solution, using standard calibration procedures. For the very dilute salt solutions detected in this work the conductance and concentrations were virtually proportional (after correction for the conductivity of the original water). Conductivities were recorded continuously during the experiment. Suitable periods of oscillation for Visking membranes were calculated to be close to 200 sec. The actual choice of frequency is a compromise. Since accurate determinations depend upon large phase angles a high frequency is favored. Ampli-
11.4 TABLE
1
Diffusion coefficients for potassium chloride in Visking dialysis membranes: A comparison of experimental data obtained using regular, square concentration waves and those predicted from independent measurements using eqn. (9) Disc Thickness
Period T (set)
imm)
Phase angle, @
Diffusion coefficient,
obs. (degrees)
obs. b (cmz-se@
CdC.a
0.213 0.210 0.204 0.196 0.205 0.205
60 200 200 200 200 400
125.1 38.7 34.6 52.7 36.2 17.9
113 37.4 36.7 49.9 38.2 20.3
0.205
400
23.8
19.0
D
calca 106)
%
2.84 3.00 1.81 2.90
2.94 2.83 1.91 2.75
+3.5 -5.8 +5.4 -5.3
-
-
-
X
Square waves (0.065 M -f 0.065 M KCl) at 25” C, with small variations from one experiment to another, were used in al1 cases. Membrane area: 0.771 cm3; voIume of collecting vessel, VT :2.06 cm3. sFrom eqn. (9). bBy independent measurements
[ 131.
tude of the emergent wave becomes much reduced as frequency is increased and so sensitivity of detection sets the upper limit. Experience showed that frequencies of the order of II/l2 were most useful. (As noted above,. this equals the product of membrane resistance R and capacitance C for the membrane, treated as a Fickian system by bond graph methods [3,4], and close to the reciprocal of the relaxation time, 7, of the membrane [l, 51 for which r = RC/4.) Most experiments were made with a period of 200 set, but to test the frequency response, the range of oscillations was varied between 60 and 400 sec. Because experimental conditions were chosen so that the infinite volume solution held to a good approximation, the phase of the concentration wave is found to depend only very weakly upon the distribution coefficient, (Y. (Amplitude depends upon it linearly.) This separation allows us to determine diffusion coefficients independently of distribution coefficients or the related permeability, P = Dol. Since eqn. (9) cannot be transformed to an explicit expression for D, the diffusion coefficient was obtained by a method of successive approximations. An approximate value of the diffusion coefficient, D*, was obtained from $I* taken from the square wave experiment as if it were a cosine wave, as Fig. 2. A typical square wave experiment is shown in Fig. 3. #J* was obtained by measuring the time-lag, tL , from the mid-point of a square wave to the next peak of the output wave such that
115 fp"- s/2 = -2llt=/7
Since we require only an approximate starting value for D* we may neglect the tanh(K1) term, eqn, (7), so that @* = --Kl + n/4 or, rearranging, D” = w12/2(2ntL/r
- n/4)*
With these approximations, D* is always larger than D, the true diffusion coefficient. Choosing a time tmax, when the output wave is at a maximum, the corresponding concentration, cT = qT/VT, was calculated using eqn. (9). Since D* is greater than D, the guessed value of D* was stepped down progressively to produce a series of cr. The experimental diffusion coefficient was taken as the value of D* which reproduced the maximum in cr. Values of (11may be set to unity if unknown in the calculation procedures just described. The only effect is to produce calculated values proportional to the actual. Errors introduced by the simplifying assumptions of no effect of a! on phase and the infinite volume assumption amount to =0.06” in rpand cause a change in estimated D of 0.2%, which is not a significant error, relative to the estimated intrinsic uncertainties of 0.4%. From independent time-lag determinations of diffusion coefficients [ 131, observed phase shifts could be compared with those calculated using eqn. (9). The agreement, Table 1, was usually within one or two degrees, and, on conversion of $* to diffusion coefficient, the forced square wave and time-lag methods agree to within 5-6%. This may be regarded as substantive proof of the validity of the method, since the agreement was within the experimental error of each method and could well be reduced with further refinement. The observed output wave for a Visking membrane is shown in Fig. 3 and compared with that calculated from independently obtained diffusion data, using both the present analysis and an equivalent bond graph simulation, the latter by techniques described previously [3, 41. The basic system of oscillating waves has therefore been shown to provide an extremely rapid and repetitive method for determining diffusion coefficients which may be easily modified to follow the course of diffusion of almost any diffusant. The accuracy of this demonstration is good (<5%) and could be improved further with refinement. Its sole limitation is the requirement that the detection method must be very sensitive indeed due to the low ratio of the amplitude of output to source (=1/500 here). It is also to be noted that the output is further attenuated the thicker the membrane, eqn. (9). Although the amplitude-to-permeability relationship was not .pursued, here, it is to be noted that the simulation, Fig. 3, reproduced both the observed output amplitude and phase shift from independently measured diffusion coefficients and distribution coefficients. As a final word on the wave form, it is to be noted that, should the distribution coefficient be a function of concentration (for example, as for charged membranes obeying Donnan conditions), all wave forms, except the
116
square wave, will be distorted on transposition from the solution into the membrane phase. Although we have shown that phase shift (but, importantly, not amplitude) is virtually unaffected by the wave form, nevertheless, this must be an additional factor in favor of the square wave and our present experimental system. Acknowledgement We are greatful to the University of Glasgow for a post-graduate research scholarship awarded to Paul Doran. List of symbols distribution coefficient of permeant, defined as cmemhr&cmlution phase shift angle, defined in eqn. (5) and Fig. 2. 4 T period of wave, set-’ radial velocity, rad-set-’ membrane area (exposed), cm2 c capacitance of chemical capacitor, eqn. (2) initial or mean concentration in the collecting volume, (terminal capaciCO tor), equals qo/vr, mol-cm-3 concentration in collecting volume at any time, equals qT /VT, mokm-3 CT D diffusion coefficient, cm2-set-l e network effort (equals concentration here) 1 membrane thickness, cm K [w /W)l Ya n number of lumps in bond graph simulation model charge on the chemical capacitor, mol i chemical resistance, eqn. (1) VT volume of collecting vessel, cm’ a
A”
References 1 G. Oster, A.S. Perelson and A. Katchalsky,
Network thermodynamics: Dynamic modelling of biophysical systems, Q. Rev. Biophys., 1 (1973) 6. 2 D. Karnopp and R. Rosenberg, System Dynamics: A Unified Approach, WileyInterscience, New York, NY, 1975. in: E.E. Bittar (Ed.), Membrane Struc3 R. Paterson, Network thermodynamics, ture and Function, John Wiley, New York, NY, Vol. 1, 1980, p. 2. 4 R. Paterson and Lutfullah, Simulation of membrane processes using network thermodynamics, in: D. Naden and M. Streat (Eds.), Ion Exchange Technology, Ellis Horwood, Chichester, 1984, p. 242 (Proceedings of Sot. Chem. Ind. Conference, “IEX ‘84”, Cambridge, England). 5 R. Paterson and Lutfullah, Simulation of transport processes using bond graph methods, Part I, Gas diffusion through planar membranes and systems obeying Fick’s laws, J. Membrane Sci., 23 (1985) 59. 6 R. Paterson, Lutfullah and P. Doran, in preparation.
117 7 8
9
10 11 12 13
A.J. Angstrom, Neue Methode das .Warmeleitungsvermogen der Kiirper zu bestimmen, Ann. Phys. Leipzig, 513 (1861) 114. W.W. Thomson (Lord Kelvin), On the reduction of observations of underground temperature; with application to Professor Forbes’ Edinburgh Observations and the Calton Hill Series, Trans. R. Sot. Edinburgh, 405 (1861) 22. K. Evnochides and E.J. Henley, Simultaneous measurement of.vapour, diffusion and solubility coefficients in polymers by frequency response techniques, J. Polym. Sci., A-2, 1987 (1970) 8. R.M. Barrer, J.A. Barrie and M.G. Rogers, Permeation through a membrane with mixed boundary conditions, Trans. Faraday Sot., 58 (1962) 2473. H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd edn., Clarendon Press, Oxford, 1984, p. 68. P. Doran, Ph.D. Thesis, Glasgow University, 1985. P. Doran and R. Paterson, in preparation.
105
A NEW METHOD FOR DETERMINING MEMBRANE DIFFUSION COEFFICIENTS FROM THEIR RESPONSE TO REGULAR FORCED CONCENTRATION WAVES
RUSSELL PATERSON and PAUL DORAN Department of Chemistry, (Great Britain)
University
of Glasgow,
Glasgow
G12 899,
Scotland
(Received August 6, 1984; accepted in revised form April 18, 1985)
Summary It is shown that when a membrane separates two permeant solutions, initially at equilibrium, and subsequently one of the solutions is forcibly oscillated, the concentration waves detected on the other side will show reduced amplitude and a change in phase angle. Using mathematical models it is shown that these may be expressed as explicit functions of the permeability and diffusion coefficient of the permeant, respectively. Solutions for cosine and square concentration waves are given. An experimental system was devised to exploit these effects and measure diffusion’coefficients directly from the phase shift of the emergent waves. Input waves were generated using alternating sprays (concentrated and dilute) from two artists’ air brushes. In preliminary experiments, diffusion coefficients for salt through test membranes were easily obtained to a precision of 5%.
Introduction
In network thermodynamics, membranes are described in terms of generalised resistors and capacitors [l] . These reflect the ability of a membrane both to dissipate power and to store chemical energy internally in each local volume during transport processes. For quantitative modelling, the membrane is subdivided conceptually and mathematically into homogeneous “lumps” or slices, each characterized by its own resistance and capacitance. The accuracy of quantitative modelling of diffusion processes depends upon the degree of reticulation of the lumped model; since the model approaches ever more closely to a true continuum of states as the number of lumps (n) is increased. As an example, to model a membrane which was exposed on one side to a source of constant effort, SE, (defined by a constant chemical potential of the permeant) and connected to a closed (homogeneous) volume of solution *Paper presented at the Europe-Japan Stresa, Italy, June 18-22,
0376-7388
/86/$03.50
Congress on Membranes and Membrane Processes,
1984.
0 1986 Elsevier Science Publishers B.V.
106
(or gas) on the other (defined by a chemical capacitor, C,), we may use a bond graph, Fig. la or an equivalent circuit, Fig. lb. These alternative representations show a S-lump model of the membrane. (The bond graph notation, although less familiar, is the more powerful for modelling complex, coupled, transport phenomena [l-3] .)
(a) tb)
SE-l-O+-O-l-0-l-Ct jc,
Fig. 1. (a): A bond graph representation source of effort, SE, (either constant or voir or collecting volume on the other, (b): The equivalent circuit representation
of a 3-lump model of a membrane exposed to L oscillating) on one side and connected to a reserrepresented here by the terminal canacitor. .~_ -~ _~I Ct. of the same membrane system.
For a system which obeys Fick’s laws of diffusion it is convenient to use a pseudo-system in which local efforts (properly defined as chemical potentials) may be replaced by concentrations [4, 51. In these papers it was shown that the chemical resistance, R, for each lump of an n-lump model of a membrane (defined by Fick’s law) is given by eqn. (l), and the corresponding capacitance, C, is defined by the lump volume, R = l/DA a n
(1)
C = &Al/n
(2)
In these equations it was assumed that the membrane was regular, of area A, and thickness I, and had a diffusion coefficient, D, defined by Fick’s first law. (The distribution coefficient, IY, was introduced [5] to define the concentration of the equilibrium solution as a common effort in all phases.) These networks may be used algorithmically [2,4, 51 to compute the system dynamics of a membrane assembly for any experiment or membrane, in which bond graph parameters and initial conditions are defined [ 4-61. For such simple conditions as described here it is only necessary to know the geometry of the membrane/solution system and the diffusion coefficient and distribution coefficient of the membrane involved (unstirred layers of solution may be treated as additional lumps). These methods are of particular interest for simulation of membrane processes and are easily expanded to deal with multicomponent, coupled transport, including energy transduction
[61.
The formal analogy between a network thermodynamic bond graph and an electrical circuit, Fig. 1, has lead us to reexamine the inverse problem, that of membrane characterisation. With the circuit and bond graph repre-
107
sentations clearly in mind, it is obvious that all standard measurements of membrane transport are, in electrical terms, DC methods and typically involve fixed driving forces (the efforts of network theory) and steady state measurements. In particular, the membrane scientist lacks powerful AC methods used routinely to investigate electrical circuitry. Using mathematical models and bond graph simulations, we set out to examine the system dynamics of a simple membrane assembly consisting of a (concentration) source, a membrane, and a collecting vessel, as represented in Fig. 1, when the source of concentration is oscillated with a fixed frequency and amplitude. From the properties of the analogous electrical circuit, Fig. lb, we would expect the membrane to act as an AC filter, reducing the amplitude and causing phase shifts in the emergent waves (detected in the terminal capacitor/collecting volume). In this paper we examine how these effects may be employed for membrane characterisation. A variety of concentration wave forms, frequencies and amplitudes were tested. From the results it became obvious that, with correct selection of experimental conditions, diffusion coefficients might be obtained directly (and repetitively) from the phase shift of the emergent concentration wave. On the basis of these mathematical models, apparatus was designed and new procedures were devised to perform such experiments in the laboratory. To assist with the design of membrane cells and experimental conditions, computer simulations were made, using a multilumped version of the bond graph, Fig. la. -These predicted the evolutionary path of experiments to a steady state of oscillation (a requirement of the mathematical models) and ensured that cell volumes and other geometric factors in the proposed cell designs would not cause unacceptable deviations from the mathematical predictions, which were limited, at least initially, to infinite collecting volumes, VT. In the course of this research we discovered some interesting precedents for the use of forced oscillations, in studies on thermal diffusion, by &rgstrijm [ 71, Thomson (Lord Kelvin) [S] and more recently in gas diffusion
Mathematical models The membrane is considered to be exposed on one side to forced oscillations in the concentration of the contact solution and to be connected on the other to a closed reservoir or collecting volume, VT, whose initial concentration, cO, is the mean concentration of the ingoing wave. It is assumed also that the membrane is in equilibrium with its contacting solutions at all times, and that (if the membrane is in the form of a clamped sheet) “edge effects” can be neglected [lo, 111. Fick’s laws are assumed with constant diffusion coefficient, D. A further simplification is the assumption that the collecting volume, VT, is effectively infinite. In practice this is not a severe limitation, as shown below. Three wave forms were considered, cosine, square, and triangular. Only
108
the first two will be discussed here because no further information was obtained by the use of triangular waves. For our systems the phase shift altered by only 4” on changing from square to triangular waves, but for triangular waves there was. knificant losr of amnlitude gn the em_e~gep_k o_‘___--____-LL -.-, in ___addition_ _.__.______, a_L_ r-----concentration wave. Cosine waves The fundamental responses to a cosine wave were considered initially. The wave form and boundary conditions for a solution of Fick’s laws were eqns. (3) and (4). In eqn. (3) the source of effort, SE, at any time, t, is defined as the concentration of permeant at the surface of the membrane (x = 0), defined by the cosine function, which oscillates about a mean concentration, cO, with a frequency of w rad-set-‘. Interfacial equilibrium is assumed. With the additional assumption of an infinite homogeneous (stirred) collecting volume, VT , its concentration, cT , is constant and set equal to the mean, cO, eqn. (4). This is obtained at the opposite membrane face (x = 1) SE = c = co [ 1 + cos(o t)]
atx=O
(3)
atx=l
(4)
and cr = c‘,
In this model the quantity of permeant, qT (mol), transferred into the (infinite) collecting volume, is obtained as a function of time. Once initial transients have decayed, this is given by eqn. (5). This was obtained as a solution of Fick’s second law of diffusion, by the standard technique of separation of variables [ 111, under the boundary conditions specified by eqns. (3) and (4) qr =B(w)
cos[wt + ‘$(a) - 7T/2] + 40
(5)
where B(w) is the amplitude of the emerging wave, defined by eqn. (6) B(w)
=
23’2A K (Y co D {B[cosh(2KI)w
cos(XE)]
7%
(6)
and [4(w) - ‘IT /2] is the phase change between input and output waves, eqn. (7), (see also Fig. 2) G(w) = tan-’
(7)
In eqns. (6) and (7), K = [w /(2D)] I/n From eqn. (7), it is clear that the phase change of the emergent wave is defined by only three parameters, the frequency of the input wave, the membrane thickness and the diffusion coefficient, eqn. (7). The corresponding amplitude, eqn. (6), is a more complicated function and depends additionally upon the distribution coefficient of the diffusant, OL,and the membrane area, A. These analyses were made under the simplifying assumption that the
109
collecting volume was infinite. Solutions were also obtained for finite systems [12], but are omitted here since it was shown that negligible error was incurred in these studies where the terminal, collecting volume was = 1 ml. (Limiting criteria are given below in the section on membrane responses to square waves.) Under conditions where the collecting volume, VT, is finite and eqns. (5), (6) and (7) remain valid, it is convenient to define the displacements and amplitude of the corresponding concentration waves by dividing eqn. (5) by VT. It is clear from eqn. (5) that the quantity of permeant or, alternatively its concentration, cT = qT/VT , will oscillate with the same frequency as the forcing wave, eqn. (3), but phase shifted by an amount which depends upon the diffusion coefficient and might be used to determine this in test membranes under laboratory conditions. Bond graph simulations [2, 31 were made for a range of common membranes exposed to cosine concentration waves in cells with various collecting * IT volumes, VT, and exposed membrane areas. That given in Fig. 2 illustrates the calculated response of a Visking dialysis membrane in the cell which was ultimately constructed. The permeant was potassium chloride and the wave was defined by eqn. (3) in which co was 0.065 M and w = 2n/lOO (period, 100 set). The output amplitude was small, approximately 0.2% of the source. Such output amplitudes are typical and so largely preclude their use for the determination of permeability. With a proper choice of frequency, however, phase shifts can be large. This is clearly illustrated in Fig. 2 and shows that if emergent waves could be recorded such shift might be used to determine diffusion coefficients accurately. (In Fig. 2 arbitrary scales of amplitude have been used for illustrative purposes.)
TIME
IS1
Fig. 2. Representation of the phase shift with a source, SE = c, [ 1 + cos(wt)] , and a large collecting volume. Membrane parameters: D = 2.75 x 10-6cm2-sec-‘. 1 = 0.205 mm, A = 0.771 cm*, (Y = 0.702, and c0 = 0.065 M, taken from independent measurements [13]. Arbitrary scales have been used and the input and output amplitudes have been made equal for ease of comparison.
The choice of amplitude and frequency for the input concentration wave determines both the phase shift and the amplitude of the emergent wave. For accurate determinations we require that the phase angle G(W), eqn. (7), be large, favoring the application of high frequencies. The amplitude of the
110
emergent wave, B(w)/V, (expressed as a concentration), is itself frequency dependent, but in the opposite sense. It tends to zero rapidly as frequency is increased. The practical limit to experimental frequency is therefore the sensitivity of the method used for detection of the wave. Since membrane parameters are strongly concentration dependent it is not practical to use large concentration oscillations; once more the most useful application of these methods appeared to be direct and repetitive measurements of diffusion coefficients from the phase shift. For typical membrane conditions large phase shifts were predicted for oscillations with periods in the range lo1000 set which were sufficiently long to encourage us to consider generating concentration waves using microcomputer controlled equipment. Experience has shown that practical frequencies are of the order of D/E’, which is, in turn, equal to the reciprocal of the product RC for a l-lump bond graph model, eqn. (1) [2]. In free response the relaxation time of the l-lump membrane is 7 = RC/4 = 12/4D [ 1, 5 1, while breakthrough times for diffusion equal 12/6D. It is therefore clear that there is an intrinsic time base for a membrane (=RC) and that our choice of frequency should reflect this. Square waves Although performing sinusoidal oscillations in concentration by ingenious use of controlled syringes is feasible, by far the most convenient wave form is the square wave, which may be obtained using sprays, as described below. A solution for source oscillations in the form of a square wave was similar to those given above. The boundary conditions were as before except that the source, SE, was now a square wave represented by a Fourier series, eqn. (8). The corresponding steady state oscillations (also as a summation) are given by eqn. (9) SE=
(-l)(=
co ) . ..
qT
=
4i =1,3,5 c
B(nw) cos[not
;
n
, ...
1)‘%os(n6) t)
(8)
n + @(nw) - n/2] (n
l)(n -I)‘2 +qo
(9)
The symbols have the same significance as before but now $(nw) is the value of o(w) for the nth contributory wave of eqn. (B), and similarly for B(nw). The results are shown in Fig. 3. Once more an infinite collecting volume, VT, was assumed. Parallel derivations for limited collecting volumes were made [12] but are not presented here, since it was shown that the two solutions effectively coincide when &?DAd/w VT@l. For the test system this function was 0.002 and the change in #, even for a collecting volume of 2 ml, as used in this work, was only 0.09”~. These predictions were verified by bond graph simulations. The calculations were performed using the methods described earlier for the SE-C
111
a-~I 0
50
100
150
50
100
150
200
250
300
250
300
7E-03
-7E-03
0
200
TIME
350
IS1
Fig. 3. Concentration waves in the collecting volume, generated by a square wave source. X, experimental; -, from the mathematical model, eqns. (8) and (9); and A, using a 20lump bond graph, as in Fig. la. Membrane parameters as in Fig. 2. Amplitudes are expressed as experimental conductances (mmho).
model [3, 41, but now using the cosine, square, or other wave expressions for SE, as above. Lumped models were constructed according to the membrane and cell parameters of the model to be predicted and resistances and capacitances defined by eqns. (1) and (2). The state space equations were integrated numerically to simulate the experiment and provide predictions of permeant concentration as a function of time in the collecting volume. It is worth noting that the whole experiment was modelled including the initial period in which transients exist. Additional information as to the outcome of experiments with other wave forms or the time dependent profiles of permeant concentration across the membrane are obtained easily from the same bond graph, as a matter of course. Experimental The diffusion cell was made from a rectangular block of perspex of dimensions 3.3 X 4 X 8 cm3. This was drilled horizontally and vertically to provide two intersecting cylinders, Fig. 4. The vertical one was plugged at the bottom to hold a “Spin fin” stirrer (Be1 Art) which was driven magnetically from outside the cell. The upper portion of the vertical tube was threaded and sealed with a teflon screw top which held a pair of platinised platinum wires which served as conductance electrodes. The bottom of this screw top was a hemispherical hollow to improve mixing in the cell. It also contained a capillary to aid the removal of entrapped bubbles. The horizontal hole passed through the block. On one side, Fig. 4B, it defined the exposed membrane area, on the other it too was threaded and sealed with a teflon screw, which held a thermistor temperature probe.
Sb
4
5cm
’
Th
Fig. 4. A diagrammatic representation of the experimental cell. (A), (B) and (C) represent sections of the cell viewed from the top, side and front. Symbols: E, platinum (conductivity) electrodes; M, membrane; Mg, motor driven magnet; Sb, perspex spray box to which the cell is clamped when in use; St, teflon coated magnetic stirrrer; Te, flanged template defining membrane area exposed to spray input; and Th, thermistor. The sprays themselves are not shown. They are placed some 18 cm from the exposed membrane surface and aimed at its center.
The cell, which held a solution volume of 2 ml, was bolted to a large perspex (spray) box. The membrane aperture was defined by the coincident holes in the cell and a flanged template, Fig. 4. Three sprays (Badger Air Brush Coy), attached to thermostatted reservoirs of the solutions to be sprayed, were directed into this box, each aimed at the exposed membrane area. In these demonstration experiments dilute potassium chloride with concentrations c,, and 2c, and water were used. The cell was filled initially with salt at the mean concentration c *. (Water might also be used, in which case the steady state of oscillation was reached after a longer period. This was partly compensated by reducing the number of sprays to two only, water and potassium chloride at 2c,.) The square wave was obtained by turning the air brushes on and off alternately at regular intervals, using solenoids attached to the nitrogen gas lines (20 psi) which operated them. The switch-
113
ing was performed by a small computer and the concentration waves in the cell (collecting) volume were detected by a Wayne Kerr conductance bridge (precision 0.01%) and recorded. (Initially a precision chart recorder was used, subsequently the data has been collected digitally.) Tests showed that concentration changes at the surface of the membrane were complete within 0.1 set on switching. Estimates were obtained by replacing the test membrane with a perspex sheet which held two platinised platinum wire electrodes, pressed level with its surface on the exposed membrane area. Adiabatic cooling required that the solutions to be sprayed should be thermostatted at a much higher temperature than required in the diffusion cell itself. In this work these solutions were maintained at 40” C. To minimise temperature variations which occurred on switching from one spray to another, the nitrogen line and solution tubes leading to the air-brush sprays were thermostatted for the distance of =l m from thermostat to spray. Three sprays were required. In the preliminary state the membrane was sprayed with the Y^^_ _-._-_-L.._L:_.- co, eq_ad to &&, in *he fGled ce;_Gnce this sy;ys*em w88 rIlea COIlCeIlbrablOIl, in thermal (and chemical) equilibrium as indicated by steady temperature and conductivity readings, the experiment was begun, by switching between the lower and upper concentrations at regular intervals. The three spray guns were directed at the centre of the exposed membrane area, Fig. 4, at a distance of al8 cm. A thermistor, Fig. 4, was used to measure solution temperature within the cell. This was affected strongly by temperature variations in the sprayed solutions, and great care was needed to ensure that the impinging sprays were at 25 + O.l”C. This was achieved by thermostatting the gas and solution lines at 40°C right upto the air brush jets; this also prevented significant temperature fluctuations when switching sprays. As an additional precaution, the cell block was thermostatted independently by circulating water at 25 + 0.05”C. Membranes were prepared from Visking dialysis tubing and preequilibrated with water and solution before use. Results and discussion The results of experiments using Visking dialysis tubing are summarised in Table 1. Square waves were produced with amplitudes of around 0.05 M by spraying alternately with water and potassium chloride solutions of 0.1 M. The concentration of salt in the collecting volume was obtained from the conductance of the solution, using standard calibration procedures. For the very dilute salt solutions detected in this work the conductance and concentrations were virtually proportional (after correction for the conductivity of the original water). Conductivities were recorded continuously during the experiment. Suitable periods of oscillation for Visking membranes were calculated to be close to 200 sec. The actual choice of frequency is a compromise. Since accurate determinations depend upon large phase angles a high frequency is favored. Ampli-
11.4 TABLE
1
Diffusion coefficients for potassium chloride in Visking dialysis membranes: A comparison of experimental data obtained using regular, square concentration waves and those predicted from independent measurements using eqn. (9) Disc Thickness
Period T (set)
imm)
Phase angle, @
Diffusion coefficient,
obs. (degrees)
obs. b (cmz-se@
CdC.a
0.213 0.210 0.204 0.196 0.205 0.205
60 200 200 200 200 400
125.1 38.7 34.6 52.7 36.2 17.9
113 37.4 36.7 49.9 38.2 20.3
0.205
400
23.8
19.0
D
calca 106)
%
2.84 3.00 1.81 2.90
2.94 2.83 1.91 2.75
+3.5 -5.8 +5.4 -5.3
-
-
-
X
Square waves (0.065 M -f 0.065 M KCl) at 25” C, with small variations from one experiment to another, were used in al1 cases. Membrane area: 0.771 cm3; voIume of collecting vessel, VT :2.06 cm3. sFrom eqn. (9). bBy independent measurements
[ 131.
tude of the emergent wave becomes much reduced as frequency is increased and so sensitivity of detection sets the upper limit. Experience showed that frequencies of the order of II/l2 were most useful. (As noted above,. this equals the product of membrane resistance R and capacitance C for the membrane, treated as a Fickian system by bond graph methods [3,4], and close to the reciprocal of the relaxation time, 7, of the membrane [l, 51 for which r = RC/4.) Most experiments were made with a period of 200 set, but to test the frequency response, the range of oscillations was varied between 60 and 400 sec. Because experimental conditions were chosen so that the infinite volume solution held to a good approximation, the phase of the concentration wave is found to depend only very weakly upon the distribution coefficient, (Y. (Amplitude depends upon it linearly.) This separation allows us to determine diffusion coefficients independently of distribution coefficients or the related permeability, P = Dol. Since eqn. (9) cannot be transformed to an explicit expression for D, the diffusion coefficient was obtained by a method of successive approximations. An approximate value of the diffusion coefficient, D*, was obtained from $I* taken from the square wave experiment as if it were a cosine wave, as Fig. 2. A typical square wave experiment is shown in Fig. 3. #J* was obtained by measuring the time-lag, tL , from the mid-point of a square wave to the next peak of the output wave such that
115 fp"- s/2 = -2llt=/7
Since we require only an approximate starting value for D* we may neglect the tanh(K1) term, eqn, (7), so that @* = --Kl + n/4 or, rearranging, D” = w12/2(2ntL/r
- n/4)*
With these approximations, D* is always larger than D, the true diffusion coefficient. Choosing a time tmax, when the output wave is at a maximum, the corresponding concentration, cT = qT/VT, was calculated using eqn. (9). Since D* is greater than D, the guessed value of D* was stepped down progressively to produce a series of cr. The experimental diffusion coefficient was taken as the value of D* which reproduced the maximum in cr. Values of (11may be set to unity if unknown in the calculation procedures just described. The only effect is to produce calculated values proportional to the actual. Errors introduced by the simplifying assumptions of no effect of a! on phase and the infinite volume assumption amount to =0.06” in rpand cause a change in estimated D of 0.2%, which is not a significant error, relative to the estimated intrinsic uncertainties of 0.4%. From independent time-lag determinations of diffusion coefficients [ 131, observed phase shifts could be compared with those calculated using eqn. (9). The agreement, Table 1, was usually within one or two degrees, and, on conversion of $* to diffusion coefficient, the forced square wave and time-lag methods agree to within 5-6%. This may be regarded as substantive proof of the validity of the method, since the agreement was within the experimental error of each method and could well be reduced with further refinement. The observed output wave for a Visking membrane is shown in Fig. 3 and compared with that calculated from independently obtained diffusion data, using both the present analysis and an equivalent bond graph simulation, the latter by techniques described previously [3, 41. The basic system of oscillating waves has therefore been shown to provide an extremely rapid and repetitive method for determining diffusion coefficients which may be easily modified to follow the course of diffusion of almost any diffusant. The accuracy of this demonstration is good (<5%) and could be improved further with refinement. Its sole limitation is the requirement that the detection method must be very sensitive indeed due to the low ratio of the amplitude of output to source (=1/500 here). It is also to be noted that the output is further attenuated the thicker the membrane, eqn. (9). Although the amplitude-to-permeability relationship was not .pursued, here, it is to be noted that the simulation, Fig. 3, reproduced both the observed output amplitude and phase shift from independently measured diffusion coefficients and distribution coefficients. As a final word on the wave form, it is to be noted that, should the distribution coefficient be a function of concentration (for example, as for charged membranes obeying Donnan conditions), all wave forms, except the
116
square wave, will be distorted on transposition from the solution into the membrane phase. Although we have shown that phase shift (but, importantly, not amplitude) is virtually unaffected by the wave form, nevertheless, this must be an additional factor in favor of the square wave and our present experimental system. Acknowledgement We are greatful to the University of Glasgow for a post-graduate research scholarship awarded to Paul Doran. List of symbols distribution coefficient of permeant, defined as cmemhr&cmlution phase shift angle, defined in eqn. (5) and Fig. 2. 4 T period of wave, set-’ radial velocity, rad-set-’ membrane area (exposed), cm2 c capacitance of chemical capacitor, eqn. (2) initial or mean concentration in the collecting volume, (terminal capaciCO tor), equals qo/vr, mol-cm-3 concentration in collecting volume at any time, equals qT /VT, mokm-3 CT D diffusion coefficient, cm2-set-l e network effort (equals concentration here) 1 membrane thickness, cm K [w /W)l Ya n number of lumps in bond graph simulation model charge on the chemical capacitor, mol i chemical resistance, eqn. (1) VT volume of collecting vessel, cm’ a
A”
References 1 G. Oster, A.S. Perelson and A. Katchalsky,
Network thermodynamics: Dynamic modelling of biophysical systems, Q. Rev. Biophys., 1 (1973) 6. 2 D. Karnopp and R. Rosenberg, System Dynamics: A Unified Approach, WileyInterscience, New York, NY, 1975. in: E.E. Bittar (Ed.), Membrane Struc3 R. Paterson, Network thermodynamics, ture and Function, John Wiley, New York, NY, Vol. 1, 1980, p. 2. 4 R. Paterson and Lutfullah, Simulation of membrane processes using network thermodynamics, in: D. Naden and M. Streat (Eds.), Ion Exchange Technology, Ellis Horwood, Chichester, 1984, p. 242 (Proceedings of Sot. Chem. Ind. Conference, “IEX ‘84”, Cambridge, England). 5 R. Paterson and Lutfullah, Simulation of transport processes using bond graph methods, Part I, Gas diffusion through planar membranes and systems obeying Fick’s laws, J. Membrane Sci., 23 (1985) 59. 6 R. Paterson, Lutfullah and P. Doran, in preparation.
117 7 8
9
10 11 12 13
A.J. Angstrom, Neue Methode das .Warmeleitungsvermogen der Kiirper zu bestimmen, Ann. Phys. Leipzig, 513 (1861) 114. W.W. Thomson (Lord Kelvin), On the reduction of observations of underground temperature; with application to Professor Forbes’ Edinburgh Observations and the Calton Hill Series, Trans. R. Sot. Edinburgh, 405 (1861) 22. K. Evnochides and E.J. Henley, Simultaneous measurement of.vapour, diffusion and solubility coefficients in polymers by frequency response techniques, J. Polym. Sci., A-2, 1987 (1970) 8. R.M. Barrer, J.A. Barrie and M.G. Rogers, Permeation through a membrane with mixed boundary conditions, Trans. Faraday Sot., 58 (1962) 2473. H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd edn., Clarendon Press, Oxford, 1984, p. 68. P. Doran, Ph.D. Thesis, Glasgow University, 1985. P. Doran and R. Paterson, in preparation.