CONTINUUM THEORY OF TRANSPORT THROUGH CAPILLARY MEMBRANES Dissertation for the Degree of Ph. D. MICHIGAN STATE UNWERSITY PM I LE 1975 This is to certify that the thesis entitled CONTINUUM THEORY OF TRANSPORT THROUGH CAPILLARY MEMBRANES presented by Ping I Lee has been accepted towards fulfillment of the requirements for Ph.D. degreein Chemistry Mflfiég DateEé‘t’é/77f 0-7839 H G SENS' 803' BINDEW 1N8. usuav BINDERS Mgmsrop‘r. menu] AA“ \- We 0 \fl‘ ABSTRACT O=Y9 V — ¢2 = - 2 36 ° Y & (2.38) o=1 v —— where gs = q/T + 2 Ease o=l Yua = Qua - xa . (2.39) Although it would seem that from postulate II we should relate all the conjugate driving forces and fluxes, 20 one does not have to consider all the interactions indi- cated by (2.20). It can be shown on the basis of symmetry (Curie's theorem) that, for isotrOpic systems, coupling can occur only between driving forces and fluxes of the same order or between those which differ in tensorial characters by even integers. This implies that only 23; is related to ja . By postulate II, the linear phenomenological equations are 9 XII 1 GB o=1,...,v , (2.40) I A.) Q II u M C ' I B B where the 9's are the phenomenological, or Onsager co— efficients. These coefficients are not all independent 9&8 = 0 , B=1,...,v . (2.41) II MC o 1 Furhter, due to the requirement of positive entrOpy pro- duction, it has been shown (Bartelt and Horne, 1969) that v E 9&8 = o, o=1,...,v . (2.42) Thus, for the 0-1 independent fluxes jl"'°’jv’ the linear phenomenological equations are n y(u' - HQ) a=1,...,v-l . (2.43) 21 If the fluxes and forces of (2.43) satisfy Onsager's (1931) condition, then the matrix of Onsager coefficients is symmetric; i.e. QOB = 98a for o,8=l,...v-l However, since total experimental verification of the Onsager Reciprocal Relations (ORR) is still an Open ques- tion (Miller, 1960, 1969), we only accept them as postu~ lates. We discuss applicability of the ORR in membrane transport systems in the next chapter. For most purposes it is more convenient to use (2.40) and consider the phenomenological coefficients as conductance coefficients. However, sometimes it is useful to invert (2.43) so that the forces can be eXpressed as linear functions of the fluxes. The result is - 2(ué- u ) = E R jB (2.44) with the resistance coefficient ROB = [GIGS/[0| , a,B=1,...,v-l , (2.45) where [Q] is the determinant of only those phenomenological coefficients which appear in (2.43) and IQIQB is the appro- priate cofactor. If the matrix of 0&8 is symmetric, then the matrix of ROB is also symmetric. The friction coefficient description of Bearman and Kirkwood (1958) and Bearman (1959) is equivalent to this resistance coefficient formulation. 22 The expression for the chemical potential gradient, which appears in (2.40),(2.43) and (2.44) has the form (Horne, 1966) —' g —' = on Magua Qua vayp + RTYlnaa + Mag? + zaFYO . (2.46) The Gibbs-Duhem equation including the external forces can be expressed as v Yp — p3 = z p VLI' . (2.47) In (2.46), v: is the partial molar volume at infinite dilu— tion and is related to the partial molar volume of component a by v"" = zim v , (2.48) C! O. x1+1 this implies the equality of v: and Va at thermodynamic ideality. Due to the arbitrariness of choosing the fluxes and forces in (2.38), one can define the fluxes and forces differently in order to suit particular purposes. Scatter- good and Lightfoot a965, 1968) and Lightfoot (1974) have chosen, instead of all the diffusional fluxes summed to zero, all the forces summed to zero. They end up with a set of Stefan-Maxwell equations and a set of Stefan—Maxwell 23 diffusivities show a smaller composition dependence than the usual phenomenological coefficients and they reduce to the more familiar Fickian diffusion coefficient for ideal binary solutions. For the purpose of this Thesis we use only the conventional linear phenomenological equations (2.43) and (2.44). In fact it can be shown that the Stefan-Maxwell equations are equivalent to (2.44). For more general discussion about the physical implication of various phenomenological coefficients, see Pitts (1962), de Groot and Mazur (1962), Haase (1969), Horne (1966) and Lightfoot (1974). CHAPTER III CONTINUOUS AND DISCONTINUOUS APPROACHES OF MEMBRANE TRANSPORT PROCESS A. Introduction The system used in most passive membrane transport experiments consists essentially of two large reservoirs (regions I 8 II) containing an isotrOpic, 0 component solution, connected by a small capillary, porous wall or another homogeneous phase as a membrane (region III). In general the reservoirs may differ in pressure, solute con— centration and electrical potential (we consider only isothermal systems in this thesis). The membrane may be itself charged or uncharged. However, no chemical reac- tions occur in the three regions. There are two types of treatment of passive mem- brane tranSport processes. If one considers the region III as so small that it can be almost disregarded, then in passing from region I to II the state variables (or the thermodynamic prOperties) suffer discontinuous jumps and the system is usually referred to as a discontinuous system. In this case the membrane is treated as a black box, and no detailed knowledge of the structure or func- tion is required. All flows and driving forces refer to 24 25 regions I and II, while the membrane merely appears as a barrier which sustains finite differences in pressure, concentration and electrical potential. The transport equations are in finite difference form, and the phenome- nological coefficients appearing in them can be used to characterize the membrane. In some cases it is possible to analyze the flow inside the membrane in terms of the differential transport equation obtained in Chapter II where the state variables (or thermodynamic prOperties) are continuous functions of Space coordinates and of time. This is usually referred to as a continuous system. In order to be applicable ex- perimentally, the differential transport equations for the continuous system have to be integrated across the membrane for some model membrane structure. The final working equations are expressed in terms of the differences in state variables (or thermodynamic properties) of regions I and II. These are the same as those for the discontinuous systems. However, by going from a continuous to a discon- tinuous formulation, one can obtain an explicit knowledge of the empirical phenomenological coefficients in terms of more fundamental prOperties of the solution and the mem- brane, such as diffusion coefficients, viscosity coeffi- cient, charge density, pore radius, concentration, dielectric constant, etc. More importantly, one can also gain a better 26 understanding of the mechanism of various membrane processes. This point will be emphasized in the next few chapters. There are generally three classes of membranes. de Groot and Mazur (1962) distinguish only two classes, while Mears, et a1. (1967) classify four types. However, three classes are enough for the present purpose: (a) MacrOporous membranes-~relatively large capillaries or pores compared to the mean free path of the molecules (say greater than 25°A radius). Species are transported through these pores primarily by convective flow. (b) Microporous membranes--the dimension of the pores is smaller than the mean free path of the molecules (very small pores). Species are transported through these pores by convection as well as by diffusion. (c) Homogeneous membranes--a separate homogeneous phase, sometimes con- sidered as a solvent through which permeants are trans- ported by diffusion. In cases (a) and (c), the fluid may be treated as a continuum and the flow in the membrane may be described by local macroscopic transport equations. From these the phenomenological equations describing trans— port between regions I and II as a discontinuous system can be derived by integration with the inclusion of appropriate boundary conditions at the membrane/solution interfaces. Relevant boundary conditions include equilibrium distribu— tion coefficients and pressure and concentration discon- tinuities. These boundary conditions are generally due to 27 membrane structural and chemical factors and cannot be accounted for by an inert continuous membrane model which is Open to both the solute and solvent (Eiiit one dimen- sional models or capillary models). Many authors (Kobatake and Fujita, 1964; Gross and Osterle, 1968; Fair and Osterle, 1971; Chen, 1971) have attempted to derive discontinuous membrane equations from continuous local transport equations. However, they fail to include these structural boundary con- ditions, which give rise to the ordinary osmotic phenomenon. In case (b), the mean free path of the molecules is larger than the pore dimension, and the collisions between fluid molecules and bounding surfaces become more important. There are fewer molecules in the flow cross section, and the continuum description of the transport becomes less precise. It seems reasonable to suppose that, even then, the continuum description should retain some validity in a statistical sense. In fact, Levitt (1973), who has done molecular dynamics calculations on kinetics of diffusions and convection in small pores, has shown that the continuum hydrodynamic theory can be extrapolated, at least qualita- tively, to pores 3.2 A in radius! Therefore, the discon- tinuous membrane equation can still be obtained from local transport equations although they have to be used with caution for membranes of class (b). In the following sections, we illustrate the mechanism of ordinary osmosis and the boundary conditions 28 that describe this. We then discuss transport equations and reference frames for these three different types of membrane. We also derive the discontinuous membrane trans- port equations from the local equations and compare them with the widely used Kedem-Katehalsky formulation. B. Mechanism of Ordinarnysmosis The passage of water through semi-permeable porous membranes is of great interest in many fields. Two mecha- nisms of osmotic flow have been considered. These are (1) diffusion of solvent down a gradient of chemical potential and (2) bulk flow through pores under a hydrostatic pressure gradient. Discussions of semi~permeabi1ity have usually been concerned with non-ionic solutes whose molecules are suf- ficiently large to be excluded from the pores by mechanical sieving. Two kinds of experiments are common: (1) eXperi- ments with an osmotically induced volume flow and (ii) self-diffusion experiments with isotOpically labelled water. Some conceptual difficulties have resulted from the following experimental observations: (a) A hydrostatic pressure difference and an equal osmotic pressure differ- ence produce the same flow through a semi-permeable mem- brane (Mauro, 1957, 1960). (b) The volume flow produced by a difference in total chemical potential may differ 29 from (and can be much greater than) the self-diffusive transfer produced by an equal potential difference arising from an isot0pic concentration difference (Durbin, Frank and Solomon, 1956). Chinard (1952) gave a careful and detailed account of the case in favor of diffusion as the main mechanism of solvent transport. This is consistent with the homogeneous membrane discussed in the last section, which is capable of water transport due to the chemical potential gradient. However, Chinard's viewpoint cannot explain the result (b) above observed for porous membranes. On the other hand, the result (b) is consistent with the vieWpoint (Pappen— heimer, 1953; Koefoed-Johnson and Ussing, 1953) that osmotic transfer takes the form of a pressure induced bulk flow. A thorough, but less fundamental comparison of the two theories as applied to plant membranes has been given by Ray (1960) and later on improved by Dianty (1963). However, they all consider rapid water diffusion at the pore exit. This notion of joint diffusion-viscous flow has been criticized by Philip (1969). It is not obvious how the concentration difference across a semi-permeable porous membrane could induce bulk flow when there is no measurable pressure difference across it. However, there is ample experimental evidence that 30 this is so, and the mechanism of the phenomenon has been proposed by Mauro (1957, 1960), who acknowledged his debt to Onsager and by Longsworth (1960), who acknowledged his debt to Kirkwood. Thermodynamically, the total chemical potential across a semi-permeable membrane separating pure water and a solution at the same hydrostatic pressure also at steady state, and in the absence of external fields must vary continuously across the membrane (Fig. 3.1). At each point in the pore the total chemical potential con- tains contributions due to the water concentration (or activity) and the hydrostatic pressure. If the mechanism of semi-permeability is the total exclusion of solute from the pores of the membrane, the steep change (or discon- tinuity) of the water concentration in the interfacial layer at the pore opening between the pure water phase and the solution phase must be accompanied by a steep change in the hydrostatic pressure. The pressure change in the interfacial layer is sustained by the interface. The pressure change along the pore will cause a flow of water through it; i;g;, the osmotic flow. Hence, an osmotic pressure difference (or concentration difference) across the semi-permeable membrane produces water flow by exactly the same kind of mechanism as a hydrostatic pres« sure difference. 31 pore (I through ve' £1 membrane Z/,//"J[/:’mmembrane __l£i___ 1(7 VWP l ‘ RT naw RTlna ' ”_’____________ _____2LEg ”",,.1. = e e “w “w +pr+RT£naw m ‘- .——-’ a, ’ ..." I —--- 5; E3 membrane I K\~ interfacial layer solution side 1 water side Fig. 3.1--Pressure and concentration profiles in a semi— permeable membrane at steady state pore osmotic through pressure E3 membrane #Z/////ffi—- membrane e~ Mir—g"? Q ' e 1:. _| W solution side water side t\“ interfacial layer Fig. 3.2—-Pressure and concentration profiles in a semi— permeable membrane at osmotic equilibrium 32 Molecularly, the total reflection of the solute molecules and the partial transmission of the solvent molecules at the membrane/solution interface causes an asymmetry in the momentum transfer by the thermal motions of the molecules. Since the average momentum transfer is prescribed by the hydrostatic pressure of the phase, the asymmetry (or deficiency) in momentum due to the finite size of the pore gives rise to a sharp change in the pres- sure sustained by the interface in the interfacial layer and a pressure gradient within the pore. Since the pore is filled with water this pressure gradient inside it causes a flow in exactly the same way as a directly ap- plied hydrostatic pressure gradient. In the self-diffusion of water through porous membranes no asymmetry in momentum transfer accompanies the gradient of labelled water, therefore no bulk flow occurs. This is the explanation of the result (b) men- tioned in this section. The pressure and concentration profiles in a semi- permeable membrane at osmotic equilibrium are shown in Fig. 3.2. In comparison with the profiles for steady- state osmotic flow in Fig. 3.1. We see that the sharp changes in pressure and concentration in the interfacial layer between the pure water phase and the solution phase are always sustained by the interface, and it is only the 33 gradient of pressure or concentration inside the pore that gets equilibrated. It must be emphasized that the inter- facial layer, across which the sharp changes of pressure and concentration occur, is not a diffusion layer which could be removed by effective stirring; its thickness is determined by the membrane pore and surface structure and by the dimensions and mean free path of the molecules. It is now widely accepted that osmotic flow of water through a membrane which contains pores is a pres- sure induced bulk flow. The mechanism given here can be generalized to membranes which separate solutions of dif- ferent concentrations and to cases of incomplete solute exclusion. Renkin (1954) has considered the case of in- complete solute exclusion and has calculated the perme- ability of pores to molecules of various sizes. By using this treatment and assuming that flow in the pores could be described by Poiseville's law, Durbin, Frank and Solomon (1956) have shown that the study of the permeation of non-ionic solutes of graded sizes in molecular diameters permits an estimation of effective total area and radii of the pores in a membrane. For membranes with adsorption of solute and solvent in the pores, the adsorption force field affects the steady state pressure distribution during osmosis because the adsorption force also contri- butes to the total chemical potential (Banin and Low, 34 1971). In the case of charged porous membranes, ordinary osmosis is still effective, but additional phenomena due to the charge occur simultaneously. This is discussed in detail in the next chapter. For more detailed discussion of the mechanism of ordinary osmosis see Mauro (1957, 1960), Longsworth (1960), Mears (1966) and Philip (1969). C. Mechanical Restraints and Reference Frames In the nonequilibrium thermodynamic study of transport processes in free solution, the local center of mass is the usual reference frame for diffusional flows (de Groot and Mazur, 1962). Other available reference frames are the local center of volume or any of the in- dividual components of the system, particularly the sol- vent. In membrane transport, the most convenient refer- ence frame, both eXperimentally and theoretically is the one fixed on the membrane itself because the membrane does not move. Therefore it is advantageous to transform the reference frame from the local center of mass, upon which almost all the transport equations are based, to the membrane framework. In doing so, it is necessary to ascertain whether or not such changes of reference frame preserve Onsager Reciprocal Relations in the local phe- nomenological equations. Coleman and Truesdell (1960) have shown that the transformations of fluxes and forces 35 have to follow certain transformational prOperties in order to preserve the reciprocal relations. Kirkwood, et a1. (1960) gave a detailed discussion on the importance of reference frames in testing the Onsager Reciprocal Re- lations for isothermal diffusion in liquids. We demon- strate in the next section the effect of changing refer- ence frames on the Onsager Reciprocal Relations in a mem— brane transport system. For the time being we consider the effect of mechanical restraints and reference frames on the total entrOpy production and the phenomenological equations. It is a common practice, without justification, to consider the membrane system to be in a state of mechanical equilibrium (Katchalsky and Curran, 1965; Hanley, 1967, 1969) such that, according to Prigogine's theorem (Prigo- gine, 1955), in the entropy production ¢2 of Eq. (2.38) the barycentric velocity u occurring in the definition of the diffusion flux ja can be replaced arbitrarily by another velocity. In this case the membrane velocity is a natural choice because it is essentially zero. Accord- ing to de Groot and Mazur (1962), the mechanical equilib— rium state is the state in which both the acceleration du/dt and the velocity gradient Vu vanish and therefore also the stress tensor may be neglected. Bartelt and 36 Horne (1970) derive necessary and sufficient conditions for mechanical equilibrium. At mechanical equilibrium, the Navier-Stokes equation (2.16) has the form 93 - Yp = 0 . (3.1) The Gibbs-Duhem equation (2.47), for the mechanical equilibrium state, becomes = 0 . (3.2) Based on (3.2), Prigogine's theorem follows immediately from (2.38): V ¢2 = - X j ° 95' (2.38) (3.3) 37 where ua is an arbitrary reference velocity. When the membrane is taken as the reference frame, 92 = u = 0 with 9m the membrane component velocity. Eq. (3.3) re- duces to , (3.4) where g (3.5) u "D as Eq. (3.4) and its integrated form are used in a large number of membrane transport literatures (see, for example Katchalsky and Curran, 1965) without questioning the validity of the mechanical equilibrium assumption. Generally, in macroporous membranes (class a) and some- times in microporous membranes (class b) where viscous flow dominates, mechanical equilibrium does not hold. In order to demonstrate this, we distinguish between the cases when the membrane can be taken as a component and when it cannot be. Mikulecky and Caplan (1966) and Mikulecky (1969) have considered the membrane as a component for macro- porous membranes, but the entrOpy production they ob- tained for the membrane system is the same as the one which excludes the membrane as a component. This is be- cause they make the trivial assumption that the partial mass density of the membrane, pm , is zero. 38 Besides, although they intended to derive the entrOpy pro— duction for stationary situations in which mechanical equilibrium does not necessarily hold, they implicitly adopt the requirement of mechanical equilibrium (see their equations (5) and (6)). Therefore the validity of their final results is questionable. Hanley (1967, 1969) later discussed the cases in which the membrane may or may not be taken as a component. He also reconciled the continuous and discontinuous approach for the case that the membrane is treated as a component. However, he and most other authors have failed to recognize that in either case the membrane component is fixed in space by an external con- straint which is generally not accounted for in the trans- port equations. More than a decade ago, in dealing with diffusion in porous media, Vink (1961) and Evans, Watson and Mason (1962) simultaneously, but independently, introduced the idea of an external constraint on the lattice component of the porous media. The external constraint acts only on the lattice and arises simply from whatever clamping system the experimenter uses to keep his porous diaphragm from being moved along just like any other diffusing Species. This is described mathematically as if a sepa— rate body force acted on each constituent of the lattice to keep it stationary. Aranow (1963), Scattergood and 39 Lightfoot (1968) and Lightfoot (1974) later applied this to membrane transport systems. Based on this we shall derive, from the equation of motion for each component, criteria for the applica— bility of mechanical equilibrium. (1) Membrane as A Separate Phase Membrane and solution are considered as separate phases. This is suitable for macroporous membranes (class a) and with some reservation for microporous membranes (class b). The membrane merely behaves as a stationary boundary and the entropy production occurs only in the fluid phase. This system allows viscous bulk flow. For stationary incompressible fluid and slow flow, the component equation of motion (in this case the Stokes equation) can be obtained from (2.18). For simplicity we assume the membrane is uncharged and the gravitational force can be neglected. Of course it is still isothermal. Membraneyphase: The equation of motion is an um cmyum + m~m + m~m 0 , (3 6) where the subscript m stands for the membrane. The body force exerted by the clamping support on the membrane is transmitted to the membrane matrix 40 and can be considered as uniformly distributed. We may then write this body force locally (Lightfoot, 1974) 25m = cm Yp - (3.7) 0 Eu = 2p . (3.8) Since this phase only has one component, the total fric- tional and thermal force F5 = O . Therefore (3.6) reduces to n V g = 0 . (3.9) By the requirement of no acceleration across the membrane and the physical boundary condition of no movement, the solution of (3.9) is simply u = 0 . (3.10) For this membrane phase alone, (3.7) fulfills the requirement of mechanical equilibrium (3.1). Solutiongphase: The equation of motion for each species is 2 * _ _ naV g cayqx+ Gaga 4- caX — 0 , o—l,...,v . (3.11) 41 Summing over all components and using the previously ob- tained relations V Z c Vu = Yp r (2.19) V nvzg - 3p + Z c x = o . (3.12) Therefore, unless the external forces exactly balance the pressure gradient, mechanical equilibrium generally does not exist due to the presence of appreciable viscous flow and velocity gradients. In the case of no external forces, mechanical equilibrium is impossible in the solution phase with the presence of pressure gradient. This has a very important bearing on the entropy production. For the system as a whole, the entropy production occurs only in the solution phase Due to the general non- existence of mechanical equilibrium in the solution phase, Prigogine's theorem cannot be applied. The entrOpy pro- duction for this membrane system is given by (2.38) 42 (2.38) where the diffusion flux ja is still referred to the barycentric velocity. Since at steady state only ga = p g (10. is constant, the integration of (2.38) over the membrane volume is not so easy as the integration of (3.4). This is discussed in more detail in the reconcilation of the continuous and discontinuous approaches in the next section. (ii) Membrane as A Component Membrane and solution are considered as a homo- geneous phase. This is suitable for homogeneous membranes (class c) and for some very fine micrOporous membranes (class b). The membrane component is interspersed among the components of the permeating fluid in the molecular level. The system approximates a thermodynamic mixture or solution, and the entropy production occurs in this single phase. This is essentially a diffusion system and contains no mechanism for viscous flow other than a simple diffusion mechanism. The equation of motion for the membrane component is the same as (3.6) 2 _ * _ an u cmyum + cmEm + mem - 0 . (3.6) 43 Again, we assume an uncharged membrane and neglect the gravitational force for simplicity. Since the membrane component partakes in the tranSport processes by fric- tional and other interactions, the body force on the mem- brane component is still gm: (1/cm)yp . (3.7) The equation of motion for the solution is the same as (3.11) naV u - c Vu + c F* + c X = 0 , d=1,...,v . (3.11) where Xa , in this case, contains only the electrical force. Summing over all components including the membrane component and using (2.19), we obtain nV u = 0 , (3.14) v where we use the relation 2 Gaza = O for electroneutrality o=l of the whole system. By the requirement of no acceleration across the membrane, the solution of (3.14) is g = constant , (3.15) in agreement with the outcome of the diffusion mechanism. Eq. (3.14) implies the validity of (3.1) for this 44 homogeneous membrane-solution phase, hence the mechanical equilibrium requirement is fulfilled. The entrOpy production for this system reduces to one similar to (3.4) due to the mechanical equilibrium condition , (3.16) where the membrane component is also included. However, the membrane component is fixed in space by the external mechanical restraint, and um = 0. This implies = p u = 0 , and eq. (3.16) reduces to (3.4) It has to be emphasized here that although the membrane component also contributes to the entropy production through frictional and other interactions with the solu— tion components, it does not appear in the final entrOpy 45 production equation due to the mechanical equilibrium condition. At steady state, ga == papa is constant by the continuity equation. The integration of (3.4) across the membrane gives the form of the entropy production which is widely used in discontinuous membrane transport theory. However, the previous analysis indicates that it is strictly applicable in homogeneous membranes. For very fine microporous membranes, it can only be used as a good approximation. D. Reconciliation of Continuous and Discontinuous Treatments-—Comments on Kedem-Katchalsky Theory More than a decade ago Kedem and Katchalsky (1958, 1961) derived the "practical" integrated flow equations for describing solute and water transport across uncharged membranes from nonequilibrium thermodynamic considerations. These equations have since become very popular alongside 46 the Nernst-Planck equation and the Goldman equation as standard working models for physiologists and biOphysicists. Nevertheless, there remain several aSpects of the Kedem— Katchalsky equations which are a source of confusion and should be clarified. In particular, it is necessary to point out: (a) These equations are one dimensional and are strictly applicable only to homogeneous membranes with thermodynamically ideal binary solutions; (b) For porous membranes, the Kedem-Katchalsky equations can be used only when the barycentric velocity is linearly re- lated to the external forces; and (c) The reciprocal re- lation in Kedem-Katchalsky's theory is strictly valid only when the system is thermodynamically ideal and the partial molar volumes of solute and solvent are equal. For a porous membrane in a binary solution, the reciprocity of the local coefficients is the natural outcome of the de- pendence of fluxes and cannot be tested by independent experiments. First, we outline Kedem—Katchalsky theory briefly. They started from the entropy production (3.17) 47 which is essentially the same as (3.4), since = 0 ' o 2. MM. We At steady state, (2.4) gives or, for the one dimensional case considered here, Na = constant (3.18) at any point in the system. Integrating (3.17) across the membrane from surface A to surface B and evaluating the entrOpy production per unit area of the membrane as a whole, we obtain B .... I V B 4)2 = A ¢2dx = - uglNa(ua _ u ,A a ) V or o2 = ail NaAué , (3.19) where the x component of Na is denoted by Na and x is the ~ direction of flow across the membrane. This rearranges, for a binary nonelectrolyte solution, to 2 _ = ' = o2 £1 NaAua JVAp + JDAn (3.20) 48 where 1 stands for the solvent and 2 for the solute, and where the total volume flow Jv and the exchange flow Jd are defined by (.1 II V 1 1 2 2 The quantities v1 and v2 are partial molar volumes in the external phase and (c: - c:)/£n (cg/c QCD co In (3.20), Ap is the change in pressure and An is the change in osmotic pressure across the membrane, where _ A _ B _ An - RT (c2 c2) - RTAc2 . The phenomenological equations are = A JV- LpAp + LpD n = AW JD LDpAp + LD (3.25) with LpD = LDp , v N + v N (3.21) J = (NZ/6'2) - (Nl/El) . (3.22) ) , a=0,1 . (3.23) (3.24) where the L's are phenomenological coefficients. Two other transport coefficients defined from these four phenome- nological coefficients have appears in membrane transport 49 literature frequently. These are the reflection coeffi- cient 3 introduced by Staverman (1952) and defined by 6'=-L L 3.27 pD/p ( ) and the solute permeability coefficient w defined by _ _ -2 - w - (LD 0 Lp) c2 . (3.28) The set of coefficients Lp, 3 and w is more con- venient for description of membrane systems than the set L , L and L because the former set of coefficients can p pD D' more easily be related to the transport characteristics of greatest interest: (1) Lp measures the mechanical filtration capacity or the hydraulic permeability of the membrane. (ii) The reflection coefficient 3 can be considered as a measure of the membrane permselectivity. When 3&1 all the solute is "reflected" from the membrane; this is a semipermeable membrane. 3(x) + w(x,r) , (4.6) where P(x) is the externally applied pressure and (dd/dx) is the externally measurable electric potential gradient between the ends of the capillary. This gradient includes both any applied gradient and any gradient due to diffu- sion. P'(x,r) and w(x,r) are, respectively, the additional pressure and electrical potential due to the fixed wall charge and the concentration polarization of the electrical double layer in the capillary. As the concentration varia- tion vanishes, 4(x) is only the externally applied poten- tial and w(x,r) and P'(x,r) reduces to 0(r) and P'(r); which include only the potential and pressure distribution due to the fixed wall charges. The radial component of the Navier-Stokes equation, (4.3), then becomes v o = (3P'/3r) + F( 2 caza)(3w/3r) , (4.7) d=l Using (4.5) and (4.6) and integrating (4.4) twice, we ob- tain u = u + u + u , x p eo co (4n)'1(r2-a2)(dP/dx) :1 II 84 Ir ‘1 Ir V ue0 = (F/n)(d¢/dx) a r dr 0 (8:1 caza)rdr r r uco = n'1 ; r-ldr 5 [(3P'/3x) \J + F( 2 c z )(Bw/ax)]rdr , (4.8) (1:1 0. G. where the boundary conditions are such that uX vanishes at the wall (r = a) and is finite at the center (r = 0). The double integrals can be evaluated only after w(x,r) and ca(x,r) are obtained. The up term is the axial velocity due to an externally applied pressure gradient, the u term is the electroosmotic flow velocity, and the 80 uco term is the capillary osmotic flow veloctiy. We show in Section D that uco is proportional to the axial con- centration gradient. C. Boltzmann Equation We now assume that there is no radial flow of any component within the capillary. This is equivalent to assuming that, for each value of x, the system is in equilibrium in the radial direction. Mechanical equilib- rium in the radial direction is represented by (4.3) or (4.7). For equilibrium with respect to movement of com- ponent a, it is necessary and sufficient that its chemical potential u& be constant. Thus, 85 (Bud/8r) = 0 , o=1,...,v . (4.9) In order to exploit this assumption, we use the explicit formula for the chemical potential by combining (2.32), (2.33), (2.34) and (2.39) “a = ua(T,p) + RT 2n xa fa+ zaF¢ (4.10) where the pure solvent standard state is used. We have neglected any polarization effects in the eXpression for u& because such effects are very small for the application we envisage. Elaboration of this point may be found in Sanfield (1968) and Horne and Chen (1973). If we now further assume that the activity coef- ficients are constants in the radial direction, then (4.9) and (4.10) yield, after some algebra (Horne and Chen, 1973), _ 0 0 gal x - xa(x1/xa) a [eXp(- zaW)], a=2,...,v (4.11) where W = Fw/RT , (4.12) and the ratio gal is defined by 961 = VZ/v: , a=2,...,v , (4.13) with v: the limiting partial molar volume of a at infinite dilution defined by (2.48), and where x3 is the mole frac- tion of a when w is zero. We shall show in the next sec- tion that this corresponds to zero surface charge density. 86 For the application at hand, therefore, x2 is the mole frac- tion of a in the compartments on either side of the capillary. When there is a composition difference between the compart- ments, then x3 is a function of the axial variable x. Horne and Chen (1973) have eliminated x from (4.11) l by defining a correction parameter E by x = x0 (1—8) (4 14) l ’ 1 ° 0 and using the fact that Z xa = l. The found, essentially, a=l M II II MC xg[exp(-zaw) - 1] + o (12) , (4.15) a 2 where I is the ionic strength, 0 I = (1/2) 2 C32: (4.16) a=2 with cg the molar concentration of a when 9 is zero. For the case that za T < l , g = -V:142 , (4.17) For sufficiently dilute solutions, 6 << 1 , and we have the usual Boltzmann equation, xa = x2 [exp(- saw)] , d=l,...,v , (4.18) or, for constant molar volume, _ 0 _ _ ca - ca [exp( zaW)2 , d—l,..., . (4.19) 87 The net change per unit volume at any point is then cgzateXp(- zaw)) , (4.20) "MC 0 N 9 ll IIMC 2 a a 2 \J E caza = -218 . (4.21) For a symmetric binary electrolyte with z = |z_I = z , and c2 = C? = c , (4.20) becomes 0 Z Caz = - 2cz[sinh (29)] , (4.22) X c z = - 2czZW (4.23) for |zw| < 0.245. D. Electrical Potential Distribution: Poisson-Boltzmann Equation The classical treatment of the diffuse double layer relies on the Poisson-Boltzmann equation, which in turn gives rise to the Gouy-Chapman type double layer. 88 Alhtough this is a relatively simple model, most rigorous quantitative theories are based on it. There are several simplifying physical assumptions involved: (1) the di- electric constant is independent of position; (2) the ions are point charges that interact coulombically with the charged wall; (3) the charges on the capillary wall are uniformly distributed on its surface; and (4) the eXpo— nential term in the Boltzmann distribution contains the average potential w(x,r) instead of the potential of the mean force. A number of corrections to these simplifications in the Poisson-Boltzmann equation have been prOposed, in- cluding corrections for ionic volume, dielectric satura— tion, ion polarization, self—atmosphere effect of the counterion and the discreteness of surface charge. Haydon- (1964) and Overbeek and Wiersema (1967), however, have sug- gested that these corrections at least partially compensate each other and that it is therefore not advisable to con- sider one or two corrections and to leave out others. They suggest use of the Poisson-Boltzmann equation inas- much as refinements in double layer theory are still under development. We follow this suggestion here. Consequently, any experimental test of our final equation is to some ex- tent a test of the Poisson-Boltzmann equation, along with our other assumptions. 89 For a circular capillary with fixed wall charge density and fixed end concentrations in an electrolyte solution, the potential distribution inside the capillary is governed by Poisson's equation, 2 2 -1 v (a ¢/ax ) + r (a/ar)r(a¢/ar) = - (F/e) 2 caza , (4.24) d=2 where 6 is the dielectric permitivity of the medium. By (4.6) and (4.20), the Poisson—Boltzmann equation is v (829/3x2)+-r l(B/Br)r(3‘P/8r) = -(F2/€RT) X chGIexp(—zaW)] , o=2 (4.25) where we have required 0 of (4.6) to be linear in x , thus, (d0/dx) = constant . (4.26) The Poisson-Boltzmann equation in the form of (4.25) is quite insoluble. In order to render it tractable, we make two further simplifications. First, we neglect (82W/8x2)--the effect of this can be accessed after an explicit formula for W(x,r) is obtained since the Ed are functions of x. Second, we linearize according to (4.21). Then we find the linearized Poisson-Boltzmann equation, r‘1(a/ar)r(aw/ar) = K24 (4.27) where the parameter K , defined by 90 K=IF(2I/€RT)l/2 , (4.28) is the reciprocal of the Debye length. The ionic strength I is defined by (4.16). The boundary conditions for (4.27) are (aw/ar)r=o I) O s (aw/8r)r=a (oF/ERT) , (4.29) where 0 is the surface charge density on the wall. In order to simplify the analysis of flow in charged capil- laries, consideration of surface phenomena is minimized by assuming that the surface charge density 0 is the charge density of the fluid at some distance from the wall. At this distance from the wall, which is of the order of molecular dimensions, the fluid is assumed to be stationary. The effective capillary radius, a, is then measured from the center up to this stationary layer. The solution of (4.27) is, with (4.29), W = (OF/KCRT)[10(Kr)]/[Il(Ka)] (4.30) where I0 and Il are modified Bessel functions of the first kind of, respectively, order zero and order one. Much has been written on the validity of (4.27) as the correct form of the Poisson-Boltzmann equation rather than (4.25). We present here a brief analysis of 91 the physical conditions in which (4.25) reduces to (4.27) for a symmetric, binary electrolyte. As our starting point, we note from (4.23) that sinh z? = 2? to better than 1% accuracy as long as IzWI 5 0.245. This condition is met for all values of Ka such that [Io(xa)]/[I1(Ka)]:50.245 Z(2€RT)l/2(Cl/2/0) , where we have used (4.30) and (4.28). For T = 298°K, R = 3,314J molle-1 , z = l, and s = 7 X 10-10 Clem-1 I the condition becomes [I0(Ka)]/[Il(Ka)] 5 0.456 x 10‘3(c1/2/o) , with both c and o in SI units. For 0 = 10-4'Cm-2 , the condition is met for all concentrations c greater than 5 x 10.3 molar = 5 x 10.2 mol m-3 . For 0 = 10"3 C m_2 , the condition is met for all concnetrations c greater than 5 x 10"3 molar.. For 0 = 10-2 C m-2 , the condition is met only for concen- trations c greater than 0.5 molar. For a wide range of surface charge densities and concentrations, then, (4.27) is valid regardless of theoretical doubts concerning (4.25). Moreover, for values of Ka large enough for (4.21) to hold, the x-derivative term of (4.25) is approximately, (BZT/BXZ) = — saw (d2nI/dx)2 . (4.31) 92 Even for large gradients of the ionic strength I, the x- derivative term is very small compared to KZW because K is very large, 107 to 109 m-1 . E. Pressure, Electroosmotic and Capillarnysmotic Flows In this section we combine the results obtained from previous sections to formulate an analytical expression for the capillary osmosic flow in a charged circular capil- lary. Since the presence of the tangential concentration gradient together with the ion distribution (non- electroneutrality) result in the axial polarization of w(x,r) and P'(x,r), the following physical conditions should be satisfied: whenever o = 0 or I = constant, (8P'(x,r)/3x) = 0 and (30(x,r)/8x) = 0 . (4.32) This implies that whenever the wall charge density is zero or the ionic strength is the same in the compartments on either side of the capillary, the axial polarization ef- fect vanishes. Combining equations (4.7), (4.19) and (4.20), we find \) -RT(3 Z ca/ar) + (8P'/8r) = 0 , (4.33) o=2 93 this implies v P'(x,r) - RT 2 ca(x,r) = f(x) , (4.34) a=2 where f(x) is an unknown function of axial coordinates. Introduction of (4.19) into (4.34) and eXpansion of the exponential yields 0 P'(x,r) - RT 2 cg(x) - RTIW a=2 2 + 0(43) = f(x) . (4.35) In (4.5) and (4.6) we have tacitly assumed that the polarization terms P'(x,r) and ¢(x,r) are complicated functions of x and r . They are not further separable in the form of (4.5) and (4.6). This is clearly the case in (4.30). Hence f(x) can be identified from (4.35) as \) f(x)=-RT Z co(x) (4.36) d=2 a and (4.34) becomes 0 v 0 P'(x,r)-RT X c (x,r)-tRT Z c (x) = 0 . (4.37) a=2 a a=2 a Differentiating (4.37) with respect to x and utilizing (4.12) and (4.20), we obtain (8P'/3x) = (szz/RT)(dI/dx) + (ZIsz/RT)(Bw/3x) , (4.38) 94 where on the right hand side we retain only up to the square term in 0. (4.38) satisfies the physical restriction in (4.32), which further confirms the choice of f(x) in (4.36). Gross and Osterle (1968) and Fair and Osterle (1971) set f(x) = 0 arbitrarily. Therefore their formulation does not satisfy the physical restrictions in (4.32). Instead, - 0 they have an extra term containing 2RTc, (or RT 2 c2) d=2 which they denote as n, the solute partial pressure given by the Van't Hoff equation for equilibrium osmotic pressure. In fact their results are erroneous since whenever there exists only a concentration variation across an uncharged capillary, their equation predicts a center of mass move- ment caused by the solute partial pressure gradient (or rather ordinary osmotic pressure gradient from the Van't Hoff equation). This seemingly correct prediction is in fact wrong. For a circular capillary Open to both solute and solvent, there can be no ordinary osmotic flow. The only mechanism that can give rise to an osmotic pressure is a momentum deficiency due to a sharp change of solute concentrations at the capillary openings (Mauro, 1957; Longsworth, 1960; Meares, 1966; Philip, 1969). This can- not be taken into account structurally in a,continuous theory like this, but it can be taken care of mathematically by a boundary condition as discussed in Chapter III. 95 For the discussions in this chapter, we stick to our original assumption in Section B that the capillary is open to both solute and solvent so that no ordinary os- motic effect will occur. (4.38) and (4.21), together with the last of (4.8), gives r r (F2/2nRT)(dI/dx) ; (dr)r' 1 I 0 (drmp2 . (4.39) c: II CO Substituting (4.21) and (4.30) into (4.8) and (4.39), and performing the integration we obtain x p eo co up = (4n)-l(r2-a2)(dP/dX) ueo = [O/fiKIl(Ka)][IO(Ka) - 10(Kr)](d¢/dx) uco = [02/4nK281i(Ka)]{(Kr)2[Ig(Kr)-Ii(Kr)]-(Kr)Io(Kr)Il(Kr) - (Ka)2[Ig(Ka)-Ii(Ka)] + (Ka)IO(Ka)Il(Ka))(dQnI/dx) . (4.40) For the first time the general analytical expression of barycentric flow in a charged circular capillary is written down including capillary osmosis. The first term, up, represents the well known Poiseville flow due to ex- ternal pressure gradient. The second term, u , repre- eo sents the capillary osmotic flow caused by the double layer 96 polarization due to an external concentration gradient. It has been shown by Derjaguin, et a1. (1969), that in general the rate of capillary osmosis and of electro- osmosis due to diffusion potential are of the same order of magnitude. We observe that whenever there is no concentration variation, (4.40) reduces to the ordinary electrokinetic flow equation for circular capillaries (Newman, 1973; Sorensen and Koefoed, 1974). Furthermore, if 0 = 0 (zero wall charge) we get the usual pressure flow equation. The velocity profiles in the capillary for the above mentioned three different cases are shown in Fig. 4.1 to Fig. 4.3. Fig. 4.1 shows the familiar parabolic velocity profile in the capillary due to external pressure gradient. Fig. 4.2 shows the electroosmotic velocity profile due to the external electric field as a function of Ka, the ratio of capillary radius to Debye length. Fig. 4.3 shows the capillary osmotic velocity profile due to the electrolyte concentration gradient across the capillary as a function of Ka. In the presence of an electrolyte concentration gradient, a diffusion potential will occur, so the capil- lary osmosis must be accompanied by electroosmosis. Therefore the capillary osmotic velocity can only be measured by short circuiting two reversible electrodes placed at both ends of the capillary. In the most general 97 Fig. 4.l—-Poiseuille Flow Y1 = - 4n up(r2 - a2)-l(dP/dx)- 1 98 r/a Fig. 99 4.2--Electroosmotic Flow _ -1 - Y2 — nKueo o (d¢/dx) 1 100 Ka = 10 1.0 K8 = 5 K8 = 2 Y2 0.5 0 l l l l 0 0.2 0.4 0.6 0.8 1.0 r/a g) 101 Fig. 4.3--Capillary Osmotic Flow Y = 4m<2 2 - -l 3 so uco (dinI/dx) 102 Ka=2 Ka=5 Ka=10 l J 02 0.4 r/ a 103 case the barycentric velocity is a mixture of these three aforementioned flows. For large values of Ka, the diffuse double layer is relatively thin, and the velocity variation occurs near the wall where the cylindrical geometry is not important. In this case there tends to be a velocity discontinuity at the wall (see Fig. 4.2 and Fig. 4.3), these are the so called electroosmotic slip and capillary osmotic slip respectively. Asymptotic expansions for Ka >> 1 show that 10(Kr)/I0(Ka) , IO(Kr)/Il(Ka) and I1(Kr)/Il(Ka) are negligible, except in the double layer region very close to the wall. Asymptotic expansion also yields [10(Ka)/I1(Ka)] = 1 + (1/2Ka) + (3/8K2a2) + ... , (4.41) Hence, in the limit of large Ka, the electroosmotic velocity in (4.40) reduces to ueo = (o/nK)(d¢/dx) , (4.42) which is the same as the classical electroosmotic slip velocity given by Helmholtz (1879) and Smoluchowski (1914). The capillary osmotic velocity in (4.40) reduces to, in the limit of large Ka, uco = - (02/8nK2€)(d2nI/dx) , (4.43) 104 which is consistent with the capillary osmotic slip ve- locity for flat double layers given by Derjaguin, et a1. (1947, 1961, 1965, 1969, 1971, 1972, 1974). This last statement will be justified in the next section. One thing we like to stress here is that the capillary osmotic velocity contains charge density square term [see (4.40) and (4.43)], which implies that as long as the axial concentration gradient is fixed, the capillary osmotic flow will be in one direction only no matter what the sign of the fixed charges on the capillary wall is. This is also consistent with the results given by Der- jaguin, et.al. (1969). F. Comparison with Derjaguin's Formulation Derjaguin, et a1. (1961, 1969) and Dukhin and Derjaguin (1964) derived, by the method of discontinuous nonequilibrium thermodynamics, the formula for capillary osmosis of dilute electrolyte solutions along a flat sur- face. Their capillary osmotic velocity is expressed as, for a binary electrolyte, uco = - (v+€++v-E_)(%)RT(d£nc/dx) (4.44) + - 0 0 where v and v are the number of cations and anions per molecule with c = (c2/v+) = (cg/v-), where c2 and c9 are the ion concnetrations in the bulk. Also 105 a: = (cg(x))’1 6m[ci(h,x) - Cg(x)]hdh , (4.45) where ci(h,x) are the ion concentrations at a distance h from the slip plane and cg(x)§i are the moments of ad- sorption of ions relative to the slip plane with x the coordinate along the flat surface. Combining (4.44) and (4.45), their capillary osmotic velocity on a flat sur- face becomes _ _ -1‘ I"0 - — ° uco — n RT(d£nc/dx) o [ E ca(h,x) Z ca(x)]hdh . (4.46) a—+ a—+ In fact, in our previous derivation, if we use (4.37) and the Boltzmann distribution from (4.19) without ex- panding the exponentials, we obtain, instead of (4.38), \) \) (3P'(x,r)/3x) = [ 2 c (x,r) - 2 c0(x)]RT(d£nI/dx) a=2 a d=2 a V - F 2 caza(aw(x,r)/3x) . (4.47) o=2 Substituting (4.47) into (4.8), the general form of the capillary osmotic velocity in a circular capillary is obtained -1 r _1 r \) \) uCO==n RT(dinI/dx) ; (dr)r 5 (dr)r[a£2 cacao-a:2 ca(x)] . (4.48) 106 This is similar to Derjaguin's formula for flat surface, (4.45). But (4.48) results from the Boltzmann distribu- tion and a hychodynamic approach. The point we wish to demonstrate here is that when a Boltzmann distribution is assumed and linearization is applied as we did in previous sections, (4.46) due to Derjaguin, et a1. (1961, 1969) becomes identical with our limiting equation (4.43). 4 Similar to (4.19), the Boltzmann distribution for a flat surface is c (h x) = c0 ex {- z W} (4 49) a ' a ' P d ' with h the distance from the shear plane. According to Overbeek (1952), when linearization is applicable, the potential distribution near a flat surface obtained from solving Poisson-Boltzmann equation with boundary conditions similar to (4.29) reads W(h,x) = (OF/KERT) eXp{- Kh} , (4.50) where K is still defined by (4.28). Substitution of (4.49) and (4.50) into (4.46) yields the capillary osmotic slip velocity uco = - (oz/BnK26)(d2nI/dx) , (4.51) where we have used, for this binary electrolyte case 107 2+v_zE]/dx)/2 = (dine/dx). (4.52) (dinI/dx) = (d£n[c(v+z+ (4.51) is exactly the same as our limiting equation (4.43). This implies that our equation is consistent with that given by Derjaguin, et a1. (1961, 1969) and it serves as a check of the validity of our more general expressions (4.40) and (4.48) for the multicomponent electrolytes capillary osmosis in circular charged capillaries. G. Conclusion A general analytical expression for the capillary osmotic velocity of multicomponent electrolyte solution in a charged circular capillary has been derived from hydro- dynamic consideration by taking into account the concentra- tion polarization of the electrical double layer near the capillary wall. In the limit of very large radius to Debye length ratio our equation reduces to one which is consistent with that obtained by Derjaguin, et a1. (1961, 1969) for flat surfaces from thermodynamic considerations. It is possible to obtain a similar expression for nonelectrolyte systems by taking into account the concen- tration polarization of the mobile adsorption layer near the wall, as long as the distribution of surface molecular force field can be formulated. Some experimental evidence of capillary osmosis for nonelectrolytes has been observed by Cleland (1966). 108 Capillary Osmosis is a general phenomenon whenever there are a mobile ionic or molecular adsorption layer and a tangential concentration gradient present. Study of this process can be valuable for analysis of the structure of ionic double layers and of adsorbed molecular layers at- solid and solution interface. For diffusion through porous media, it is necessary to take into account this capillary osmotic process. It can be very important in cases of transport of electrolyte solutions through porour charged membranes generated only by concentration gradients. A detailed account of its relation with anomalous osmosis in porous charged membrane will be presented in the next chapter. CHAPTER V ANOMALOUS OSMOSIS THROUGH CHARGED POROUS MEMBRANES A. Introduction Osmotic transport of a nonelectrolyte solution through a membrane or of an electrolyte solution through an uncharged membrane occurs if the membrane acts to some extent as a barrier to the solute (see Chapter III) and if there is a difference of concentration across the mem- brane. The rate of transport is, in those cases, prOpor- tional to the difference in the chemical potential of the solvent; 112;! the rate is roughly proportional to the concentration difference of solute on the two sides of the membrane. Moreover, the direction of transport is toward the more concentrated solution. However, for a charged porOus membrane which allows convective transfer of the solution and which separates electrolyte solutions of different concentrations, the rate of osmotic transport appears to be greater and exhibits anomalous behavior. When the concentration ratio of the two solutions (both maintained at atmospheric pressure) is fixed and flow rate is measured for different mean concentrations, 109 110 the plot of flow rate against the logarithm of concentra- tion is an N-shaped curve (see Figs. 5.1 and 5.2). The flow rate for medium concentrations is higher than for more concentrated solutions (anomalous positive osmosis). In some cases, the flow occurs toward the less concentrated solution (anomalous negative osmosis). These phenomena, now known collectively as "anomalous osmosis" do not occur in strictly semipermeable membranes. Anomalous osmosis was first described by Dutrochet (1835) and later by Graham (1854). Since then various transport theories have been develOped (sf. Loeb, 1922; Sollner, 1930; Grim and Sollner, 1957; Scthgl, 1955; Kobatake and Fujita, 1964; Toyoshima, Kobatake and Fujita, 1967; Fujita and Kobatake, 1968; Tusaka, et al., 1969; Kedem and Katchalsky, 1961; Dorst, et al., 1964) to de- scribe the mechanism of anomalous osmosis. They are generally of three types: (a) Loeb (1922), Sollner (1930), Grim and Sollner (1957), and Kobatake and Fujita (1964) recognized the electrochemical nature of this phenomenon. Their theories are based on the idea that electro- osmosis, caused by the diffusion potential across the membrane, is superposed on ordinary osmotic flow which is due to the difference in solute concentration. Grim and Sollner (1957) carried 111 out careful and exact measurements of anomalous osmosis of various electrolyte solutions across oxyhemoglobin-coated collodion membranes, which have clearly defined isoelectric points. By ad- justing the PH of the electrolyte solution, the membrane can be negatively or positively charged. The total osmotic flow is composed of a normal component and an abnormal component. The normal flow due to ordinary osmosis was estimated by using the electrolyte as its own reference under condi- tions of zero net charge on the membrane. Kobatake and Fujita formulated a more quantitative theory by considering a charged capillary model for the porous membrane. They obtained an eXplicit con- centration dependence of the elctroosmotic coef- ficient. Their theory is successful in predicting the shape of the curve for the anomalous osmosis data obtained by Grim and Sollner (1957). However, their theory cannot OOpe with the experimental ob- servation that, for KCl solutions, the osmotic flow is in only one direction (toward the more con- centrated solution) for both positively and nega- tively charged membranes. That is they predict (incorrectly) that the direction of flow is de- termined by the sign of the charge. 112 (b) Scthgl (1955) and Toyoshima, Kobatake and Fujita (1967) used a one-dimensional treatment and ig- nored the interactions between ions and solvent. They also made arbitrary assumptions on activity and mobility of ions in the membrane. They assumed that the pressure gradient set up inside the mem- brane together with the electrostatic potential gradient combine to produce observed flow. Toyoshima, Kobatake and Fujita (1967) succeeded in predicting some experimental observations, but they did not resolve the discrepancy mentioned in (a). (b) Kedem and Katchalsky (1961), Dorst, et a1. (1964) and Tasaka, et a1. (1969) used the discontinuous non- equilibrium thermodynamic approach. Anomalous osmosis was attributed to cross terms in the one dimensional phenomenological equations. The ob- served flow behavior was attributed to frictional interaction between solvent and ions. This "black box" type of theory suffers the same difficulties mentioned in Chapter III. Although these three types of theories are satis- factory in some respects, they are all inadequate in one way or another. Moreover, the conditions of the numerous eXperiments so far reported are usually not well defined. 113 All previous authors have omitted the capillary osmotic contribution described in the last Chapter. The concentra- tion gradient imposed across the membrane in general causes concentration polarization of the electrical double layer along the pore wall and sets up an additional center of mass movement by capillary osmosis. In the absence of externally applied gradients of pressure and electric po- tential (in fact this is the usual experimental conditions for studying anomalous osmosis), the rate of capillary osmosis is of the same order of magnitude as the rate of electroosmosis due to the diffusion potential. The possible contribution of electroosmosis to the mechanism of anomalous osmosis was realized long ago (Loeb, 1922; Sollner, 1930; Grim and Sollner, 1957; Kobatake and Fujita, 1964), but this is the first time that capillary osmotic contribution has been considered. In this chapter, we use the continuous approach and the capillary osmosis results of Chapter IV and derive, without recourse to most of the restrictive simplifications required by previous workers, a phenomenological theory for the capillary membrane model which correctly predicts the direction and magnitude of flow for KCl solutions through positively and negatively charged porous membranes. The system consists of a moderately charged mem- brane which separates two aqueous uni-univalent electrolyte 114 solutions of different concentrations (with CB > CA) at the same temperature and in the absence of an external hydrostatic pressure difference. The membrane is assumed to contain a bundle of charged capillaries of equal radius a which is large enough to permit a diffuse double layer on the capillary walls but small compared to the thickness 2 of the membrane. We assume the unstirred layer thick- ness on the two sides of the membrane has been minimized by effective stirring and can be ignored here. B. Phenomenological Equations of Anomalous Osmosis We confine our discussion to the system which separates two aqueous solutions containing the solvent molecules and the same single uni-univalent electrolyte. Positive ions and negative ions are denoted by + and -, respectively. The solvent (water) is denoted by w. The one dimensional modified Nernst—Planck equation is, from Appendix A, No = - zawacaF(8¢/8x) - Da(8ca/8x) — Ba(3p/8x) + cauX , d=+,- . (5.1) As shown in the appendix, this equation is valid only for dilute solutions. The absolute mobility ma is related to the ionic conductance Ad by ma = (Ad/IzaIFZ). By (A.19), 115 Da the diffusion coefficient of a is related to the binary Fickian diffusion coefficient D and to the mobilities by -l D = D+zawaRT(w+-mj(z+w -z_w_) , d=+,- . (5.2) O. + The pressure term coefficnet is defined by (A.23) as —v_w_)(z w --z_00_)"l B = Vd(c+v + + . a +c_v_)(D/vRT) + c z wa(v+w + a a + (5.3) The electric current and the solute flux of the electrolyte component both relative to the capillary wall are defined by i=FZzN, (5.4) N=)N. (5.5) The volume flow rate of the liquid which permeates through unit area of the membrane is defined by J = N+v V + N_v_ + N v (5.6) + w w where v+, v_ and vw are the partial molar volumes. With the help of (2.7) and (2.9), (5.6) can be rearranged to JV = (v+-M+vw/Mw)(N+-c+ux)+-(v_-M_vw/Mw)(N_-c_ux) + 11x (5.7) 116 By (5.4) and (5.1), for the uni-univalent case, i = - (w+c++w_c_)F2(3¢/8x) - FD+(3c+/8x)+FD_(8c_/3x) - F(B+-B_)(8p/8x) + F(c+-c_)uX , (5.8) with B+ - B_ = (c+w+-c_w_)(v+w+-v_w_)(w++w_)‘l . (5.9) Likewise, Ns = - (c+w+-c_w_)F(3¢/3x) - D+(3C+/3X) - D_(3c_/3x) e (B++B_)(3p/8x) + (c++c_)ux , (5.10) with B++B_ = (c+v++c_v_)(D/RT) + (c+(1)++c__(1)_)(v+0)+-v__00_)(00++(1)_)-l . (5.11) Substitution of the Boltzmann equation (4.19) for the ion concentration with the exponential expanded up to the linear term, separation of the electric potential and pressure according to (4.5) and (4.6) and use of (4.30) and (4.38) for the electrical potential and pressure distribution in the charged capillary transforms (5.8) and (5.10) into with + 117 c F2{(w++w_)-(w+-w_)W}(d¢/dx) {-F2[(w++w_)-(D++D_)/RT][rIl(Kr)/Il(Ka) an(Kr)IO(Ka)/Ii(Ka)](O/2€)-F(D+-D_) F(D++b_)w —FRTC [(w+-s_)(w++w_)‘l-9](v+w -v_w_)92 + (F2c o/s)[(w+-w_)(w++w_)-l-W](v+w -v_w_)W[rIl(Kr)/I1(Ka) + an(Kr)Io(Ka)/Ii(Ka)]}(dc /dx) Fc [(w+-w_)(w++w_)_l-W](v w -v_w_)(dP/dx)—2c F‘i’uX , + + (5.12) - c F{(w+-w_y_u%fw_)W}(d¢/dx) + {[(w++w_)-(D++D_)/RT](OW/26)[rIl(Kr)/I1(Ka) - an(Kr)Io(Ka)/Ii(Ka)]-(D++D_)+(D+-D_)W - [Dc (v++v_)-Dc (v+-v_)W+RTc[(w++w_)-(w+-w_)W] X (v+w+-v_w_)(w++w_)-11\P2 - (Fc o/s)[((v++v_)-(v+—v_)W)D/RT+(w++w_)-(w+- w_)9](v+w+-v_w_)(w++w_)’1w x [rIl(Kr)/Il(Ka)-an(Kr)IO(Ka)/Ii(Ka)]}(dc /dx) - {Dc (v++v_)-Dc (v+-v_)W+RTc [(w++w_) - (w+-w_)w](v+w+-v_w_)(w++w_)‘l}(dP/dx) + 2c ux (5.13) K = (2F2c /eRT)l/2 (5.14) 118 and T = Fw/RT = 0I0(Kr)/KsIl(Ka) , (5.15) where we have used (aw/ax) = (o/2€)[rIl(Kr)/Il(Ka) - aIO(Kr)IO(Ka)/Ii(Ka)](dinc /dx) (5.16) For this uni-univalent electrolyte, the center of mass velocity ux is given by (4.40) _ -1 2 2 ux - (4n) (r -a )(dP/dx)+[o/nKIl(Ka)][10(Ka)-I0(Kr)](d0/dx) + [02/4nK28Ii(Ka)]{(Kr)2[I3(Kr)-Ii(Kr)] - (Kr)Io(Kr)Il(I o (5.43) where K = /; K‘l , B = cA/cB b = (D++D_)g—(2nsK)-1(Ka)2 . d = (D.._-D._)Fa+(oR'I'.-12/2K26)(v+u)+-v__uo_)(0)+--u)_)(<1)+,+m_)-l ) 2 f = (w++w_)F a , g = (w+-w_)oF-02(nK)’l . A = -(3K/2agfl(bf/g)-d] , B = -(3K2/8azg)[(bf/g)-d]-b/f . 5 = 3Kb/2af , E = - 3K2b/8a2g , f = - 3Kb/2ag[(bf/g)-d) . G = - d+(3x?b/8azg)[(bf/g)-d]+bf/g . (5.44) For the case of KCl , D+-D_ = RT(w+-w_) z 0, and the above procedure yields 129 1 2 z = - (8n>' a An+<2oKRT/n){b(3f>‘l(cBy)'3/2(e‘3/2-1) GI X 2 - (3Kb/8af)(cBy)' (8‘2-1)+<3K2b/40af>(cBY>‘5/2 0 , (6.8) BCIV -Da(3ca/3r) = Pm(ca—CO) at r = 0 for x > 0, (6.9) where cb is the inlet concentration, c0 is the dialysate concentration, P is the permeability coefficient of the m tubular membrane, and where the derivative is zero at the center of the tube by symmetry. BCII indicates that for downstream the concentration of the solution approaches the dialysate concentration and BCIV serves as a defini- tion of the permeability coefficient. The dimensionless variables c = (ca'°0)/(Cb—CO) , (6.10) c = r/a . (6.11) z = x/aPe where Pe = 2uoa/Da , (6.12) Nsh = p a/D , (6.13) W m on transform the convective diffusion equation (6.5) and the boundary conditions (6.6)-(6.8) to (l-C2)(BC/32) = c’IIa/acIcIac/ac) + PéZIaZc/azz> , (6.14) 153 BCI c = l at z = 0 for 0 i c i 1 , (6.15) BCII c = 0 when 2 + w for 0 3'; i l , (6.16) BCIII (ac/3;) = 0 at g = 0 for z > 0 , (6.17) BCIV -(3c/3§) = Nshwc at C = 1 for z > 0 , (6.18) where the Peclet number is a measure of convective to dif- fusive effects and the wall Sherwood number Nsh describes w the transport conductance of the membrane. C. The Graetz Problem and Its Extension On the assumption that the Peclet number is large, which implies that axial convection dominates over axial diffusion since the Peclet number is the ratio of convective to diffusive effects, the second derivative with respect to z in (6.14) is usually neglected. This neglect is equivalent to the neglect of the contribution of axial diffusion. (6.14) then reduces to (l-C2)(BC/32) c‘lIa/acI (ac/ac) . (6.19) Neglect of axial diffusion changes the problem considerably. Firstly, there are no dimensionless parameters in the prob- lem. Secondly, the original partial differential equation is elliptic, while without axial diffusion the equation becomes parabolic. In the elliptic problem, all boundary conditions including the one at x + w have to be Specified. 154 In the parabolic problem, there is no upstream propagation of effects, and the boundary condition (6.16) is not neces- sary. Furthermore, if constant wall concentration obtains, then the Sherwood number N8 + w , and the boundary con- h ditions reduce to w BCI c = l at z = 0 for 0 i c i l (6.20) BCII (ac/3;) = 0 at c = 0 for z > 0 (6.21) BCIII c = 0 at C = l for z > 0 . (6.22) (6.19)-(6.22) constitute the classical Graetz problem. Graetz (1883, 1885) treated this problem by the method of separation of variables: C(C.2) = R(C)Z(2) . (6.23) Substitution of (6.23) into (6.19) gives <1-c2IRId2/dzI = IZ/c>c(dR/ch or z'l(d2/dz> = [(1-62)CR1-1(d/dC)C(dR/dc) = - 12 (5.24) where A are the eigenvalues. The function Z(z) can be determined from (dZ/dz) = - 122 (6.25) with the solution (apart from a multiplicative constant) 155 z = exp(-lzz) (6.26) The function R(;) can be determined from c‘l(d/dc)c(dR/d6) + 12(1-62)R = o (6.27) and the boundary conditions (dR/dC) = 0 at C = 0 , R = 0 at C = l . (6.28) (6.27) and (6.28) constitute a Sturm-Liouville problem be- cause the linear, second-order, ordinary differential equation (6.27) is self-adjoint with homogeneous boundary conditions (6.28). The eigenfunctions of the Sturm- Liouville problem are orthogonal, I1 2 { 0 n + m C(1-I; )R (2;)R (Dd; = (6.29) 0 n m l n = m ' where R.n and Rm are eigenfunctions corresponding to eigen- values An and Am. The general solution is then _ ” 2 c — Z AnIexp(-lnz)]Rn(c) . (6.30) n=1 This is known as the Graetz series, with An the linear combination coefficient corresponding to the nth eigen- value and eigenfunction. 1' d. a. I l“ ‘ 34 - ,v..-. . —:,.‘ 156 Only two boundary conditions have been used. The remaining one, (6.20), together with (6.30), gives 1 = nil Aan(§) . (6.31) With the use of the orthogonality property (6.29), the coefficients can be obtained by 1 1 ...I _2 I -22 An_- o C(1 6 )Rn(;)dC/ 0 C(l ; )Rn(C)dC . (6.32) This completes the solution of the classical Graetz problem. From this solution one can calculate other related quantities for the problems of interest. Inspection of the solution (6.30) shows that fewer terms are needed for large 2 and more terms for small 2 in order to obtain proper convergence. LéveQue (1928) used a boundary layer treatment (which is equivalent to singular perturbation) to obtain a simple equation valid for small 2. For more detailed discussion of the Lévedue solution and its extension see Newman (1973). Other extensions of the Graetz problem involve constant mass flux or finite wall permeability boundary conditions, which require extra mathematical manipulations. However, the forms of the solution are still the same as (6.30). The references for these extensions have been given in Section A. The most 157 important extension of the Graetz problem, however, is the inclusion of axial diffusion. This is the topic of the next section. D. Exact Solution with Axial Diffusion As pointed out in Section A, it is not always justifiable to neglect the axial diffusion. Schneider (1957) analyzed the effect of axial conduction on entrance- heat transfer and concluded that it is appreciable if the Peclet number Pe < 100. This was confirmed later by Singh (1958) and Hsu (1967). The axial diffusion effect can be very important in gases, for which the diffusion coeffi- 1 cmz/sec and the cients are usually of the order of 10- Peclet number may well be much smaller than 100. This can make the last term on the right hand side of (6.14) comparable to or greater than the other terms. Conse- quently, the axial diffusion contribution cannot be neglected. Tan and Hsu (1970) recognized the necessity of including axial diffusion for gas flow problems, but they solved the differential equation by a Runge-Kutta numerical scheme for a constant wall concentration bound- aryecondition. The starting equations are (6.14)—(6.18). With the inclusion of the axial diffusion term, the method of separation of variables no longer works. 158 Nevertheless, we can seek a solution of the same form as in the case of no axial diffusion, i.e. c(6,2) = 21 BnIexp(-6§z)1 Yn(c) , (6.33) n:- where 8n and Yn(;) are the eigenvalues and eigenfunctions of: 6‘1(d/d6)6(dYn/dc) + sin-c2 + (Bn/Pe)2] Yn = o (6.34) with boundary conditions (BYn/Bc) = 0 at C = 0 -(aY/a;) = Nsthn at c = 1 . (6.35) By the transformation a = Bncz and wn(a) = Yn(c) exp (Bncz/Z) , (6.36) we obtain the CHF equation (or Kummer's equation) 2 2 1 2 _ £(d wh/da ) + (l-€)(de/d€)-{§~(Bn/4)[1+(Bn/Pe) I}wn — O o (6.37) This has the solution (apart from a multiplcation constant), under the boundary condition (6.17), _ 1_ 2 2 Wn - M(§ (Sn/4)[1+(Bn/Pe) J . l . 6n; ) , (6.38) 159 where M is Kummer's function, defined by (see, e.g., Abramowitz and Stegun, 1964) (a)2y2 (anyn = 3! M(a,b,y) 1+ b+W+ ...‘i'W‘l'... (6.39) with (a)n (b) II H a(a+l)(a+2)...(a+n-1) , (a)O I H (6.40) n b(b+l)(b+2)...(b+n—l) , (b)0 - Instead of M(a,b,y), the notation 1Fl (a,b,y) is also widely used. The advantages of expressing the solution to this problem in terms of Kummer function are (1) (2) (3) (4) The properties of the functions (e.g., derivatives, recurrence relations, convergence properties) are well known, and the numerical values of the function are tabulated. The asymptotic forms of the function are available. Direct power series solutions obtained by the Fro- benius method suffer from rapid error build-up due to recurrence relations which evaluate coefficients from previous coefficients. Moreover, fast con- vergence is not guaranteed. (Both these remarks apply only in the case that the general term cannot be found). The Kummer function can be evaluated quickly, and its convergence properties are known. 160 When the eigenvalue 8n is sufficiently large, the following asymptotic form can be used instead of the series from M(a,b,y) = P(b)sin(afl)eXp[(b—2a)(%sinh20- cosh20)] x [(b-2a)cosh0]l-b[fl(%b-a)sinh201-1/2[1+O(|%b-2|_l)] (6.41) where coshze = y/(2b-4a) . Combination of (6.36) and (6.38) gives 2 l 2 2 Yn = exP(-Bn§ /2) M(§--Bn/4)[l+(8n/Pe) 1, 1181,11; ) . (6.42) Applying the boundary conditions (6.35) to (6.42), we ob- tain the transcendental equation {1+%enI1-(sn/Pe)21-Nsh }M(%-(Bn/4)I1+(en/Pe)21, 1.8n) W = {l-§enI1+(en/Pe)21}M(%~(en/4)I1+(sn/Pe)21, 1,8n) . (6.43) The eigenvalues are those values of 8n which satisfy this equation. We have solved this equation by a half-interval method (Carnahan, et al., 1969) for various values of Pe and Nshw' Eigenvalues up to the 10th have been calculated to an accuracy of at least 9 significant figures on a CDC 6500 computer. In the range of parameters considered here, computer calculations show that function converges 161 to the 9th decimal place in less than 50 terms, and for some parameters convergence is achieved in less than ten terms. In order to assure convergence, we used 100 terms for every Kummer function calculated. Sideman, Luss and Peck (1965), who used a Frobenius method for the no axial diffusion case, had to calculate 300 terms in order to assure convergence. It has not been necessary in our work to use the asymptotic formula (6.41) because the computation time needed for evaluating the more general expression is reasonably short. Eigenvalues calculated for different values of Pe and Nshw are tabulated in Table 6.1. For the case of Pe = w (no axial diffusion) and Nshw = 0 (constant wall concentration), our eigen- values are exactly the same as the most accurate ones reported by Brown (1960). This serves as a check on the accuracy of our calculations and the solution of the transcendental equation (6.43). The solution to this problem is, then 0 II °° 2 2 1 2 2 nngnexm-anmxphsn; /2)M(-2--(Bn/4)[l+(Bn/Pe) 1,1,8nr, ) . (6.43) Application of boundary condition (6.15) leads to the re- quirement that 162 EUES AND LINEAR COMBINATION COEFFICIENTS HHERE P IS THE PECLET NOov NSHH IS THE HAIL SHERWOOD NO. TABLE 6.l-cEIGENVA (NO AXIAL DIFFUSION). NSHH 00 PE L.C.COEFFICIENT EIGENVALUE I. IQ 33222211 0000000000 112222222 000000000 1334567800 1 NSHH (NO AXIAL DIFFUSION), = no PE LoC.COEFFICIENT EIGENVALUE 1001111112 127Av6 0.... .. -.a .4 -.2 .4 -.3 1112222222 0000000000 0 o o o o o a o 9 o —.= =-...:.=.:.=.= ...— M9M?HMW&#$ 9834924296 AGMQSWGMANM l‘ ‘ ‘ ‘ IESLVEENNN 1234567890 I. (NO AXIAL DIFFUSION), NSHH 00 PE L.c.coerr1c15~t .13464218 EIGENVALUE 100000010]. ll ‘1‘! 1112222222 0000000000 12345611890 1.. 163 (CONTINUED) TABLE 6.1 S L.C.COEEFICIENT (NO AXIAL DIFFUSION), NSHH = w PE EIGENVALUE 1000000001 1122222222 0000000000 12314567890 1 20 NSHH (NO AXIAL DIFFUSION). z 00 PE L.C.COEFFICIENT EIGENVALUE 1000011111 1112222222 12175567800 1 (NO AXIAL DIFFUSION), NSHH PE L.C.COEFFICIENT EIGENVALUE 1000000000 0000000000 l l ( 1122222222 I,” IV, (- (III 123456789. 1 164 (CONTINUED) TABLE 6.1 NSHU = 109 PE LoC.COEEFICIENT EIGENVALUE 1433332222 9 20"“ .1979“ I .3 111222222 1234561189 0 1 NSHU 109 PE L.C.COEFFICIENT EIGENVALUE .IOOOIIIIIII I]€l19§£2)§£2 0000000000 6687012456 1468111111 0 0 I O O O I O O 0 1236567800 1 NSHH = 109 PE LoC.COEFFICIENT .140789SOSE001 -.667468723E*0 EIGENVALUE 000000111 .430915217E0 5 9 h 0 5 1 1111222222 1236567890 1 OOONO‘WI‘UNI‘ 2 fl OOQQOM¢UN~ 2 H O OONO‘U‘J‘UNH 2 H TABLE 6.] (CONTINUED) PE 3 109 NSHH = S EIGENVALUE LoCoCOEFFICIENT .22":I\ ‘[ -' :‘ol Olt’ci‘bsa 5’7E‘01 .5107073fl95‘01 -.79h0]4 37E‘00 .7gih .9;45001 .Sunovyno7g.oo .av‘qr 56:32:00] -.§U"‘Hl VI 'VQEOOO .10“. .Ile=+oz .c, p2t+00 .117'9“0155602 -.Zg!h "I! ‘2E*00 .1299!2248;002 .131J49Ioaeooo .141353452;.02 -.1JyE6V593E000 0151,93335E902 01975744'8E900 ol61h306725‘02 ~o7u 38376E'01 PE = 109 NSHU = 20 EIGENVALUE LoCoCOEFFICIENT .194469705E‘01 0134716874F’01 .489 097605‘01 -.548532ITPE‘00 .69 74 1356.01 .32 716618E‘00 876854786E‘01 -.206942774E‘00 .103164 BZE‘OZ .539238737E‘00 0116899 GZE’OZ -. 73835464E‘01 0129329049E‘02 .7506170025'01 0140743695E‘02 ~0539480010E'01 0151346842E°02 .48008l532E'0* 0161284238E002 v.26§72 llE-o PE = 10! NSHH = m EIGENVALUE LoCoCOEFFICIENT 0259003069:.°1 0154482887E901 .SSQ. 8364:*01 -o99?091824E900 . 7lnpqu1ugool .834827499Eooo .945”? Zflg’OI -o743207154E900 o109§.’010;‘02 .664788758E000 .12211!043;‘02 -.610516342E‘00 .13ar 44=ooz .550461882E000 .14509317lg‘02 -o519711 63E900 .1561'!fl43;90? .423382882E000 .165ll.’ I 3'. —.02 -0434981718E.oo 166 (CONTINUED) TABLE 6.1 59 NSHH PE L.C.COEFFICIENT EIGENVALUE 1544433333 0000000000 9 o u .. ._ . q . a . EEEEEEEEEE 0448210278 0687 543.563 0545091178 0353074638 0470429449 0K¥DOATAV6§§J 0190202491 0155408821 1&{126112446 111111222 000000000 0 O 6 O O O O O O EEEEEEEEE 026386482 668042565 733510716 813669285 834634722 843425821 988739207 525664011 0356789111 0 o o o o o o o 0 0 1234567890 1 59 NSHH PE LoC.COEFFICIENT EIGENVALUE 1001111112 0000000000 0 o o v v . . c o . FEEEEEEEEE 7137330851 1961518 494 6639888610 6559190197 1111111222 0000000000 9 O O G O 6 O O 9 O EEEEEEEEE 4329667734 5613122987 0541032801 3358967613 4876759820 6715119932 2862050207 5736765011 1356789111 0 O O C O O O O O 0 1236567890 1 NSHH 3 EIGENVALUE .196825988 50 PE LoC.COEFFIC1ENT 1 000001111 0 000000000 8 6 5 0 3 3 l. 7 79398199 49255988 86966146 54583105 90583307 47765011 6789111 0 O O O O O 0 .40262713 .5 1 3 3 8 8 2 S 3 1234567890 1 166 (CONTINUED) TABLE 6.1 59 NSHH PE L.C.COEFF1CIENT EIGENVALUE N 1544433333 0000000000 0 v p 9 . . 9 . 9 . EEEEEEEEEE 0448210278 0687 543543 0545091178 03530. 74638 0470429449 0050619653 0190202491 0155408821 1<{146Iik446 111111222 000000000 6 O O O O O O O O EEEEEEEEE 026386482 668042565 733510716 813669285 834634722 843425821 98.8739207 525664011 0356789111 0 o o o o o o o b 0 1234567890 1 59 NSHH PE L.C.COEFFICIENT EIGENVALUE 1001111112 0000000000 00 o c v . . o v . EEEEEEEEEE 7137330851 1961518 494 6639888610 6559190197 1888790184 9958095620 4138638147 2677334345 11919A233223, O O O O O O O 0 1111111222 0000000000 9 O O Q 6 O O O O 4 EEEEEEEEE 4329667734 5613122987 0541032801 3358967613 4876759820 6715119932 2862050207 5736765011 1356789111 0 O O O O O O O O 0 1234567890 1 3 EIGENVALUE .19682598 6271 .402 NSHH .5 59 PE L.C.COEFF1C1ENT .100X2001i111 0 000000000 .Lvooo....v. 1 111111222 3 5847 4699 S923 8659 3658 1191 8489 689 13388253 90583307 47765011 6789111 O O O O O O I 1234567890 1 167 (CONTINUED) TABLE 6.1 NSHH 59 PE L.C.COEFFICIENT EIGENVALUE 1000000001 0000000000 0 O O O O O O O 9 . EEEEEEEEEE 312235:3510?. 3911624751 1618701622 3478920084 1958846447 .30671184{}b8 0889885115 5357695200 1853211117 0 o o o o o o o o 0 1111111222 0000000000 0 O O 0 O O O O O O EEEEEEEEEE 0752774382 4774615430 9116446616 6IZZISRVAV47. 3624585280 1679035465 1586026307 1:557R3L50131 2456789111 o o o o o o o o o 0 1234567890 1 20 8 EIGENVALUE 59 NSHH PE L.C.COEFFICIENT 1000011111 0000000000 3157621 1111111222 0000000000 0 O O O O 9 O O O O EEEEEEEEEE 1215859541 6993434022 6735660143 6670949030 .I9IAYI472918 .J§?£58nl?14?. 2236272307 8946765011 1356789111 c o o o o o o o o 0 1234567890 1 59 NSHH PE L.C.COEFFICIENT .157996866 -.104878236 EIGENVALUE 1 100000000 0000000000 .86389647 9 8 6 8 9 S -.7 6 6 S S a 3 1111111222 0000000000 0 O o 6 o O O o o O EEEEEEEEEE 8211786824 8191405962 3053407407 0421949336 3959389565 5067493674 8175770630 1259i:!09n¥£c 2457899111 6 o o o o . o o o . 1234567890 1 167 (CONTINUED) TABLE 6.1 59 NSHU PE L.C.COEEFICIENT EIGENVALUE 1000000001 0000000000 9991:9999... EEEEEEEEEE 3IZZJSRXEIO?. .3911162h5LPI .161851916?5c 3478920084 1958846447 «JOAYLIBAJEOB 085K288fi7115 5357695200 1853211117 0 O O O O O O O O 0 1111111222 0000000000 .99991199999 .tEFEEtEFEkEt 07§5§Ll43855 475I461éfQJ0 9115644£3916 6|23l5=¥£947. .5677358figfi90 1679035465 1§5$QOPZXQO7 1|&#£87§X¥!1 .(4fiZEI86‘511 O I O O O O O O O O .1?594555189nv 1 NSHH 20 EIGENVALUE 59 PE LoC.COEFFICIENT 1000011111 .UOnXYOOEXZUO 1111111222 00:200062900 .99991299999 FEEtFEEtFEEEt .1?7158:X254I. .69053254n526 6752565XYI41. .66732940K250 .191331492918 3:11580Z?!4?. 925567545J07 8013676E1911 11E397801151 O O O O O O O O O O I§£J4§zggg90 l 59 NSHH PE L.c.coerr1crenf 0157994866E. -.106878236 EIGENVALUE .2385303 F .EFFhELEFEu 75 828 520 176 928 795 345 0096£81 9863096477 0453 -.750 .66 60 54 51 43 39 1111111222 00:200652900 911999129999 EFEhEtEFEkEt 89%llié9689fi4 813?1A£259AXC “25346¥!907 ISEJ4éZEf8QKV 1 168 TABLE 6.1 (CONTINUED) 29 NSHH PE L.c.cosrr1c16~t EIGENVALUE 15 44 4 33333 0000000000 00.0.00... 111111111 000000000 1234567890 1 29 NSHH PE LoC9C0EFFICIENT EIGENVALUE 1001111112 245569112& 1318533229 .......... 1111111111 0000000000 9 6 O O 6 O O O 0 6 1234567890 1 29 NSHH PE LoCoCOEFFICIENT EIGENVALUE 1 000001111 c . 5297320007 [amnwlfillwmw ..JML mwrflmmm. I WW. 6mm9m0filmm fWWHMSMtTM 4873510809 130397519A¥D3 . . . . O O O . . . - - c . u 1 111111111 V1111 ) 0 718115): .7549" 1234567890 1 ~ fl u- OOO‘IO‘U'R‘UNF' Z OOOQG‘M’UN— Z OOONOU‘J‘UNN 2 PE PE PE 29 29 29 TABLE 6.1 NSHH = 5 EIGENVALUE .167604193E .ZBBQQOIHQE 37)]! .QQVJ" .5191 6' L m 0 Evgu 5H m n1 m 0600000000 I-lHHI‘l-lD-H-Md-‘I—l m m m oooooooo+§ ...7' .7ssufln 435 NSHH = 20 EIGENVALUE .14?§08§g8§4 NSHH = m EIGENVALUE .186754496E+01 .312312007E001 .4400195773E001 o472062018E‘01 .536393020E001 .590201686E001 .641185041E‘01 .688411546E*01 .792685580E‘01 .7 42 SOSOEOOI 169 (CONTINUED) L.C.COEFFICIENT .152230693E‘0 -o841060575E*0 .524050013E00 -.348?20229E9 .2469786 E0 -.IBQRQhQAQE 0 01“ 7.! "t‘ 0 - -.11248h330E .97573h380E -.7006]3747E- WHOOOOOOOD‘ L.c.coarr1cxe~f .137772b48E601 -.554581736E 00 .275920727E t-n—r-n-u-ooco 170 nil Bnexp(-BnC2/2)M(%-(Bn/4)[l+(Bn/Pe)2],1,BnZ;2) = 1 . (6.44) do not constitute a Sturm-Liouville system, and the eigen— functions (6.42) are therefore not orthogonal. The usual way of finding the linear combination coefficients Bn fails. We demonstrate in the next section that by employing an "Overdetermined Collocation" method the values of Bn can easily be calculated. The advantages of this "Overdeter- mined Collocation" have been described in Section A. E. Overdetermined Collocation Method-- Least Square Scheme In this section we utilize an approximate but direct method to evaluate the linear combination coeffi- cients in (6.44) where the eigenfunctions are not ortho- gonal. Of several different methods for obtaining approxi- mate solutions, the "Overdetermined Collocation" (Lee, 1966) is applied here, not only because of its simplicity but also because it is formulated in such a way that the boundary conditions pertinent to the problem are satisfied particularly well on the boundaries. This is similar to the least-squares method often used in solving integral equations (Hildebrand, 1965). The usual method of collocation consists in using a truncated series solution of the differential equation 171 to satisfy the boundary conditions at a selected finite set of boundary points. It is hOped that the solution thus found will also meet the boundary conditions at boundary points between those of the selected set. The accuracy of the solution found in this way depends on how well the boundary conditions are met at the inter- mediate boundary points. Usually, the solution satisfies only the selected collocation points and oscillates be- tween them. It is therefore desirable to have a solution which minimizes the difference between the real and the mathced boundary values. The method of Overdetermined Collocation is an extension of the usual method of collo- cation. It involves writing more boundary equations than there are unknown coefficients and solving the overdeter- mined system of equations by a least-squares scheme. We illustrate this method by solving (6.44) for Bn' After truncation of the infinite series at the Nth term and division of the dimensionless radial co- ordinate into m-l divisions such that 0 i :1 < :2 < ... < c 3 1 with m > N , (6.45) m (6.44) reduces to N 2 1 2 2 nil BnexP('BnCi/2)M(-§-(Bn/4)[1+(Bn/Pe) 1, Lanai) = 1 , i=l,2,...,m (6.46) 172 or y. B = 1 , (6.47) where _ _ 2 l_ 2 2 Yin - exP( Suzi/2)M(§ (Sn/4)Il+(Bn/Pe) 1, 1. anci). (6.48) Define the residue as N s. = 2 Y. B -1 , (6.49) and also define the residue squares by m g(Bl,B2,...,bN) = Z 5151 . (6.50) i=1 Minimizing g by (Bg/BBk) = o , (6.51) we obtain Y . , k=l,2,...,N . (6.52) This is similar to the Galerkin method used in elasticity problems (Sokolinkoff, 1956). The difference is that the weighting functions in (6.52) are the Yin themselves. (6.52) is the required system of N equations for determining the N 173 unknowns Bn‘ The final result is in a form which is very convenient for computer calculations. The solutions Bn obtained minimize the residues in the least-square sense. (6.52) was solved on a CDC 6500 computer for various values of Fe and Nsh with a Gauss-Jordan reduc- tion algorithm to invert thewmatrix. We have used N = 10 and m = 8. All the Bn are calculated to nine significant figures. They appear in Table 6.1. F. Physical Analysis From the results of Sections D and E, particularly (6.43), the local radial concentration distributions at certain fixed axial coordinates are calculable. These are shown in Fig. 6.2 to Fig. 6.3. The local bulk concentration is defined as 1 1 E(c.z) = 5 uxc(c.z)cdc/ g uxcdc . (6.53) Substitution of (6.3) into (6.53) yields 1 1 E(;.z) = 5 (1—c2)c(6,z)cdc/ 5 (l-C2)Cdc , (6.54) which can be further simplified by the use of (6.43) with truncation — N 2 1 2 2 C(C,2) = Z Bnexp(-an) 6 C(l-C )exP(-Bnc /2) n=1 x M(§-(sn/4)(1+(6n/Pe)21, 1, BnC2)d . (6.55) 174 Fig. 6.2--Radial concentration distribution for NSh 175 ———Pe:10 3 2:0.2 4 2:0.5 2 —v—Pe l 0.5 0.75 1.0 l 0.25 176 Fig. 6.3-—Radial concentration distribution for Nsh w 177 0.75 1.0 0.5 0.25 178 The local bulk concentrations have also been calculated. The integrals in (6.55) were computed by the use of a lS-point Gauss-Legendre quadrature formula (Carnahan, 1969). The results are shown in Fig. 6.4 to Fig. 6.6. As expected the local bulk concentrations decrease with axial distance and with increasing Peclect number. We observe that if axial diffusion is neglected for small Peclect number, the local bulk concentration may have up to 400% error near the netrance region (iLELJ small 2). In general neglect of axial diffusion usually lends to underestimation of the local bulk concentration. One can also define an overall Sherwood number by use of a total mass transport coefficient: Nsh = (hDa/D) = - (ac/3;)C=1/E (6.56) where the total mass transfer coefficient is defined as hD = NCW/(E40) = - 9(ac/ag)c=1/aE (6.57) with N2;W the radial diffusional flux at the wall. Again, the overall Sherwood number is a dimensionless mass trans- fer coefficient which characterizes the rate of mass trans- port for the whole system. Substitution of (6.43) into (6.56) gives 179 Fig. 6.4--Local bulk concentration as a function of re- duced axial distance from the entrance for N = 1, Pe = 2, 5, and w. Shw 180 NA 0._ 0.0 0.0 «.0 «.0 N Hmn—m n ”ma—N 8 ”01—. —H;Imz N0 V0 0.0 0.0 0; IO 181 Fig. 6.5-~Local bulk concentration as a function of re- duced axial distance from the entrance for Nsh = 5, Pa = 2, 5, and w. w 182 NA 0.— md 00 v.0 Nd O T . _ O N N.O #0 0.0 N ”mam mHmn—N nonmn: . m HBIwz m 0 0; l0 183 Fig. 6.6--Local bulk concentration as a function of re- duced axial distance from the entrance for) Nsh = w, Pa = 2, 5, and m. w 184 N.— N ”mam m 8 0; l0 185 N _ _ 2 _. - Nsh - nil Bnexp( BnZ)exp( Bn/Z) [2kM(k+l.l.Bn) (2k+8n)M(k.l,Bn)]/ N 2 ,1 2 2 2 2 “£1 Bnexp(-enz) 0 C(l-C )exp(—Bnc /2)M(k,1,8nc )dc (6.58) with k = -1- - (a /4) mm /Pe)2] (6 59) 2 n n ' ' where we have truncated the infinite series to N terms for the numerical calculation. The overall Sherwood numbers are calculated for various Fe and Nsh values and are shown in Fig. 6.7 to Fig. 6.9. It is seen that the Sherwood num- ber increases with increasing Peclect number for a fixed wall Sherwood number and, not surprisingly, increases with increasing wall Sherwood number for fixed Peclet number. The figures also indicate that the mass transfer rate is highest near the entrance region and generally decreases to a constant value. The overall picture can be summarized qualitatively: (1) Whenever the membrane permeability increases (in— creasing Nshw) the total mass tansfer rate through the membrane also increases (increasing overall Sherwood number Nah)“ (2) The local bulk concentration decreases with axial coordinates due to the fact that solutes are Fig. 186 6.7--Overa11 Sherwood number as a function of re- duced axial distance from the entrance for P8 = 2' Nshw = 1' 5' CD. 187 To _ :.-.,v Fig. 188 6.8--Overall Sherwood number as a function of re- duced axial distance from the entrance for Fe = 10, Nshw = l, 5, w. 189 Imz 188 Fig. 6.8—-Overall Sherwood number as a function of re- duced axial distance from the entrance for Fe = 10, N8 = l, 5, m. hW 189 Imz 190 Fig. 6.9--Overall Sherwood number as a function of re- duced axial distance from the entrance for Pa = m, N = l, 5, w. shw 191 mlo 9 TE 192 diffusing through the membrane into the dialysate along the membrane wall. (3) The presence of axial diffusion (small Pe) tends to decrease the overall mass transfer rate (de- creasing Nsh) along the tabular membrane and also tends to reduce the size of local concentration gradients. G. Discussion In this chapter we have treated tubular membrane transport with axial diffusion and with a boundary con- dition of finite wall permeabilities. The solutions are expressed terms of-Kummer functions, and numerical values are obtained by the use of an "Overdetermined Collocation" method. In addition we have used a boundary condition of finite wall permeability which is more general than the constant wall concentration and the constant wall mass flux conditions. In fact these are the limiting cases of our boundary condition. We have also used the boundary condition that the entrance concentration is uniform over the cross section of the tubular membrane. Rigorously, the axial diffusion effect, which tends to propagate up- stream, will change the entrance concentration profile. Nevertheless under some experimental conditions (e.g., 193 in hollow-fiber artificial kidney), the uniform entrance concentration condition can be considered as a very good approximation. Although only selected results for only a few values of Pe and Nshw are presented here, they suffice to demonstrate the general trends. For other values of Pe and N , one can use the method developed here systemat— shw ically. From the results in previous sections it is clear that axial diffusion is important for small Peclet numbers. This effect is significant for the prediction of performance in artificial kidney systems Operated at low blood flow rate or in gas separation through tubular glass membranes. Further extensions of this approach can be made by taking into consideration of chemical reactions at the mem- brane surface. This should be a good model for the hollow- fiber membrane/enzyme reactor Operated at low flow rate such that Pe < 100 (Waterland, et al., 1974; Lewis and Middleman, 1974). Further improvement in the results can be attained by considering non—uniform entrance concentra— tions and extending the problem to an infinite domain in- stead of the semi-infinite one considered here. One can also extend the approach to non-Newtonian flows (which do not have parabolic velocity profile) such as polymer solutions. Moreover, one could take into account osmotic 194 pressure and convection across the membrane in the radial direction. In either case the neglect of axial diffusion can only be justified when the Peclet number is very large. BIBLIOGRAPHY BIBLIOGRAPHY Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, NBS Applied Mathematics Series 55 (U.S. Government Printing Office, Washington, D.C., 1964). Aranow, R. H., Proc. Nat. Acad. Sci. 50, 1066 (1963). Banin, A., and P. R. Low, Isr. J. Chem. 2, 645 (1971). Bartelt, J. L., Ph.D. Thesis, Michigan State University, 1968. Bartelt, J. L., and F. H. Horne, J. Chem. Phys. 51, 210 (1969). Bartelt, J. L., and F. H. Horne, Pure and Applied Chemistry 22, 349 (1970). Bearman, R. J., and J. G. Kirkwood, J. Chem. Phys. _8, 136 (1958). Bearman, R. J., and J. Chem. Phys. 31, 751 (1959). Beck, R. E., and J. S. Schultz, Biochem. BiOphys. Acta 2 5, 273 (1972). Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport Phenomena (John Wiley & Sons, Inc., New York, N.Y., 1960). Bodnrescu, M. V., VDI-Forschungsheft 450, 19 (1955). Bowman, F., Introduction to Bessel Functions (Dover, New York, N.Y., 1958). Bresler, G. H., and R. P. Wendt, J. Phys. Chem. 13, 264 (1969). Bresler, G. H., and R. P. Wendt, Science 163, 944 (1969). Brown, G. M., AIChE J. g, 179 (1960). Burgreen, D., and F. Nakache, J. Phys. Chem. 69, 4017 (1965). 195 196 Carman, P., Disc. Farady Soc. 3, 72 (1948). Carnahan, B., H. A. Luther, and J. O. Wilkes, Applied Numeri- cal Methods (John Wiley & Sons, Inc., New York, N.YT, 1969). Chen, J., Ph.D. Thesis, Michigan State University, 1971. Chinard, F. P., Am. J. Physiol. 313, 578 (1952). Cleland, R. L., Trans. Farady Soc. 33, 336 (1966). Coleman, B. D., and Truesdell, J. Chem. Phys. 33, 28 (1960). Cook, G. M. W., D. H. Heard, and G. V. F. Seaman, Nature 191, 44 (1961). Davis, H. R., and G. V. Parkinson, Appl. Sci. Res. 33, 20 (1970). Davis, E. J., Can. J. Chem. Eng. 33, 562 (1973). de Groot, S. R., Thermodynamics of Irreversible Processes (North-Holland, Amsterdam, 1951). 7 de Groot, S. R., and P. Mazur, Nonequilibrium Thermodynamics (North-Holland, Amsterdam, 1962). Derjaguin, B. V., G. G. Sidorenkov, E. A. Zubashchenko, and G. V. Kieseleva, Koll. Zhurn. 3, 335 (1947). Derjaguin, B. V., S. S. Dukhin and A. A. Korotkova, Koll. Zhurn. 33, 53 (1961). Derjaguin, B. V., J. C. Dzyaloschinsky, M. M. Koptelova, and L. P. Pitayevsky, Disc. Farady Soc. 53, 246 (1965). Derjaguin, B. V., S. S. Dukhin and M. M. KOptelova, K011. Zhurn. 33, 359 (1959). Derjaguin, B. V., and M. M. Koptelova, Koll. Zhurn. 33, 692 (1969). Derjaguin, B. V. in Research in Surface Forces, Vol. III, B. V. Derjaguin Ed. (Plenum Press, New York, N.Y., 1971). Derjaguin, B. V., S. S. Dukhin, and M. M. Koptelova, J. Colloid Interface Sci. 33, 984 (1972). 197 Derjaguin, B. V., and S. S. Dukhin in Surface and Colloid Science, Vol. 7, E. Matijevic Ed.T(John Wiley & Sons, Inc., New York, N.Y., 1974). Dianty,CL, Adv. Bot. Res. 1, 279 (1963). Dorst, W., A. J. Staverman, and R. Caramazza, Rec. Trav. Chim. 33, 1329 (1964). Dresner, L., J. Phys. Chem. 31, 1635 (1963). Dukhin, S. S., and B. V. Derjaguin, Dokl. Akad. nauk SSSR 159, 401, 636 (1964). Durbin, R. P., H. Frank, and A. K. Solomon, J. Gen. Physiol. 33, 535 (1956). Dutrochet, R. J. H., Ann. Chim. Phys. 33, 337 (1835). Eschrich, W., R. F. Evert, and J. H. Young, Planta 107, 279 (1972). Evans, R. E., III, G. M. Watson, and E. A. Mason, J. Chem. Phys. 33, 1894 (1962). Fair, J. C., and J. F. Osterle, J. Chem. Phys. 33, 3307 (1971). Fitts, D. D., Nonequilibrium Thermodynamics (McGraw-Hill, ' New York, N.Y., 1962). Flanagan, J. B., and L. Marcoux, J. Phy. Chem. 13, 718 (1974). Fujita, H. and Y. Kobatake, J. of Colloid Interface Sci. 31, 609 (1968). Graetz, L. B. and Ann. Physik. Chemie 13, 79 (1883); 33, 337 (1885). Graham, T., Phil. Trans. Roy Soc. London 144, 177 (1854). Gyarmati, I., Nonequilibrium Thermodynamics (Springer- Verlag, New York, N.Y., 1970). Grim, E., K. Sollner, J. Gen. Physiol. 33, 887 (1957). Gross, R. J., and J. F. Osterle, J. Chem. Phys. 33, 228 (1968). 198 Haase, R., Thermodynamics of Irreversible Processes (Addison-Wesley Publishing Co., Reading, Massa- chusetts, 1969). Hanley, H. J. M., J. Chem. Gd. 33, 717 (1969). Hanley, H. J. M. ianransport Phenomena in Fluids, H. J. M. Hanley Ed. (Marcel Deffer, New York, N.Y., 1969). Happel, J., and H. Brenner, Low Reynolds number Hydrody- namics (Prentice-Hall, Englewood C1iffsj‘N.J., 1965). Haydon, D. A. in Recent Progress in Surface Science, J. F. Danielle, K. G. A. Pankhurst, and A. G. Riddiford Ed. (Academic Press, New York, N.Y., 1964). Helfferich, F., Ion Exchange (McGraw-Hill, New York, N.Y., 1962). Helmholtz, H., Ann. Phys. 1, 338 (1879). Hildebrand, F. B., Advanced Calculus for Applications (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1962). Hildebrand, F. 3., Methods of Applied Mathematig§'(Prentice— Hall, Inc., Englewood Cliffs, N.J., 1965). Hildreth, D., J. Phys. Chem. 13, 2006 (1970). Horne, F. H., J. Chem. Phys. 33, 3069 (1966). Horne, F. H., and J. S. Chen, Proc. IIIrd Int. Conf. on Chemical Thermodynamics, Vol. IV, p. 6 (G. Gistel, Vienna, 1973). Hsu, C., Appl. Sci. Res. 11, 359 (1967). Hsu, C., AIChE J. 17, 133 (1971). Ingle, S. E., Ph.D. Thesis, Michigan State University, 1971. Jones, A. 8., Int. J. Heat Mass Transfer 13, 619 (1971). Katchalsky, A., and P. F. Curran, Nonequilibrium Thermo- dynamics in Biophysics (Harvard University Press, Cambridge, Mass., 1965). Kaufman, T. G., and E. F. Leonard, A.I.Ch.E. J. 13, 110 (1968). 199 Kays, W. M., Convective Heat and Mass Transfer (McGraw- Hill, New York, N.Y., 1966). Kedem, O., and A. Katchalsky, Biochem. Biophys. Acta 31, 229 (1958). Kedem, O., and A. Katchalsky, J. Gen. Physiol. 33, 143 (1961). Kirkwood, J., and B. Crawford, J. Phys. Chem. 33, 1048 (1952). Kirkwood, J. G. in Ion Transport Across Membranes, H. T. Clark Ed. (Academic Press, New York, 1954). Kirkwood, J. G., R. L. Baldwin, P. J. Dunlop, L. J. Gosting and G. Kegeles, J. Chem. Phys. 33, 1505 (1960). Klausner, Y., and S. R. Kraft, Isr. J. Tech. 3, 152 (1965). Klausner, Y., and S. R. Kraft, Trans. Soc. Rheol. 13, 603 (1966). Klotz, I. M., Chemical Thermodynamics (W. A. Benjamin, Inc., New York, 1964); Kobatake, Y., and H. Fujita, Kolloid Z. 196, 58 (1964). Kobatake, Y., and H. Fujita, J. Chem. Phys. 33, 2212, 2219 (1964). Kobatake, Y., N. Takeguchi, Y. Toyoshima and H. Fujita, J. Phys. Chem. 33, 3981 (1965). Kobatake, Y., Y. Toyoshima and N. Takeguchi, J. Phys. Chem. 13, 1187 (1966). Koefoed-Johnsen, V., and H. H. Ussing, Acta Physiol. Scand. 33, 60 (1953). Krylov, V. S., and V. G. Levich,Russ. J. Phys. Chem. 31, 50 (1963). Lakshminaraianaiah, N. Transport Phenomena in Membranes (Academic Press, New York, N.Y., 1969). Lee, T. C., A Direct Method of Analyzing Axi-Symmetric Elasticity Problems, Ph.D. Thesis, University of Iowa, 1966. III | Ill}|' '- 200 Lévedue, M. A., Ann. des Mines, Memoirés 13 (Ser. 12), 201, 305, 381 (1928). Levich, V. G., Physicochemical Hydrodynamics (Prentice- Hall, Inc., Englewood Cliffs, N.J., 1962). Levitt, D. G., Biop. J. 13, 186 (1973). Lewis, W., and S. Middleman, AIChE J. 33, 1012 (1964). Lightfoot, E. N., Transport Phenomena and Living Systems (John Wiley & Sons, Inc., New York, N.Y., 1974)? Loeb, J., J. Gen. Physiol. 3, 89 (1922). Longsworth, L. G., J. Phys. Chem. 33, 1914 (1960). Mason, E. A., P. Wendt, and E. H. Bresler, J. Chem. Soc. Farady. Trans., II 33, 1938 (1972). Mauro, A., Science 126, 252 (1957). Mauro, A., Circulation 31, 845 (1960). Mears, P., Eur. Polym. J. 3, 241 (1966). Mears, P., D. G. Dawson, A. H. Sutton, and J. F. Thain, Ber. Bunsenges. Physik. Chem. 11, 765 (1967). Mikulecky, D. C., and S. R. Caplan, J. Phys. Chem. 13, 3049 (1966). Mikulecky, D. C. in Transport Phenomena in Fluids, H. J. M. Hanley Ed. (Marcel Dekker, New York, 1969). Milekhina, M. M., Koll Zhurn. 33, 173 (1961). Miller, D. G., Chem. Rev. 33, 15 (1960). Miller, D. G. in Transport Phenomena in Fluids, H. J. M. Hanley Ed. (Marcel Dekker, New York, 1969). 19 Morrison, F. A. Jr., and J. F. Osterle, J. Chem. Phys. 33 2111 (1965). Mfiller, I., Arch. Rational Mech. Anal. 33, 1 (1968). Newman, J. in Electroana1ytica1 Chemistry, Vol. 6, A. Bard Ed. (Marcel Dekker, Inc., New York, N.Y., 1973). 201 Newman, J. S., Electrochemical SysEems (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973). Olmsted, J. B., and G. G. Borisy, Ann. Rev. Biochem., 33, 507 (1973). Onsager, L., Phys. Rev. 31, 405 (1931); 33, 2665 (1931). Overbeek, J. Th. G. in Colloid Science, Vol. I, H. R. Kruyt Ed. (Elsevier, Amsterdam, 1952). Overbeek, J. Th. G., and P. H. Wiersema in Electrophoresis, Vol. II, M. Bier Ed. (Academic Press, New York, N.Y. 1967). Pappenheimer, J. R., Physiol. Rev. 33, 387 (1953). Philip, J. R., CIBA Foundation Symposium on Circulatory and Respiratory Mass Transport, G. E. W. Wolstenholme and J. Knigh Ed. (Little, Brown and Co., Boston, 1969), 25. Pickard, w. F., Kolloid-Z. z. Polym. 179, 117 (1961). Prigogine, I., Introduction to Thermodynamics of Irre- versible Processes (Thomas Press, Springfield, Illinois, 1955). Ray, P. M., Pl. Physiol. 33, 783 (1960). Rankin, E. M., J. Gen. Physiol. 33, 225 (1954). Rice, C. L., and R. Whitehead, J. Phys. Chem. 33, 4017 Sanfield, A., Thermodynamics of Charged and Polarized La are (John Wiley & Sons, Inc., New York, N.Y., 1968 . Scattergood, E. M., and E. N. Lightfoot, AIChE J., 110, 175 (1965). Scattergood, E. M., and E. N. Lightfoot, Trans. Farady Soc., 33, 1135 (1968). Sch16g1, R., Z. Phys. Chem. 3, 73 (1955). Schlégl, R., Discussions Farady Soc., 31, 46 (1956). Schneider, P. J., Trans. ASME 13, 766 (1957). 202 Sellars, J. R., M. Tribus, and J. S. Klein, Trans. ASME 13, 441 (1956). Sideman, S., D. Luss, and R. E. Peck, Appl. Sci. Res. A14, 157 (1965). Siegel, R., E. M. Sparrow, and T. M. Hallmant, Appl. Sci. Res. 31, 386 (1958). Singh, S. N; Appl. Sci. Res. 31, 328 (1958). Sioda, R. E., Electrochem. Acta 13,,375, 1559 (1968). Smit, J. A. M., and A. J. Staverman, J. Phys. Chem. 13, 966 (1970). Smoluchowski, M. in Handbuch der Zlektrizitat und des Magnetiomus, W. Graetz Ed. (Barth, Leipzig, 1914). Sokolnikoff, I. 8., Mathematical Theory of glasticity (John Wiley & Sons, New York, N.Y.: 1956). Sollner, K., Z. Klektrochem. 33, 36, 234 (1930). Sparrow, E. M., and R. Siegel, Int. J. Heat Mass Transfer 1, 161 (1960). Spiegler, K. 8., Trans. Farady Soc. 33, 1409 (1958). Staverman, A. J., Rec. Trav. Chim. 13, 344 (1951). Stewart, R. D., E. D. Beretta, J. C. Cerny, and H. I. Mahon, Investigative Urology 3, 614 (1966). Taitel, Y., M. Bentwich, and A. Tamir, Int. J. Heat Mass Transfer 13, 359 (1973). Tamir, A., and Y. Taitel, Chem. Eng. Sci. 28, 1921 (1973). Tan, C. W., Int. J. Heat Mass Transfer 13, 471 (1969). Tan, C. W., and C. Hsu, Int. J. Heat Mass Transfer 13, 1887 (1970). Teore11,EL” Ber. Bunsenges, Physik. Chem. 11, 814 (1967). Timmermans, J., The Physico-Chemical Constants of Binapy Systems in Concentrated Solutions (interscience Pubiishers, Inc., New York, N.Y., 1960). 203 Toyoshima, Y., Y. Kobatake, and H. Fujita, Trans. Farady Soc. 33, 2828 (1967). Truesdell, C., Rational Thermodynamics (McGraw-Hill, New York, N.Y., 1969). Vink, H., Ark. Kemi. 11, 311 (1961). Vink, H., Z. Phys. Chem. Frankfurt 11, 51 (1970). Waterland, L. R., A. S. Michaels, and C. R. Robertson, AIChE J. 33, 50 (1974). Worsse-Schmidt, P. M., Int. J. Heat Mass Transfer 13, 541 (1967). Wroblowa, H. S., and G. Razunney, J. Electroanal. Chem. 33, 329 (1974). APPENDICES APPENDIX A THE MODIFIED NERNST-PLANCK EQUATION Equation (5.1) repeated here, N = - z w c FV¢ - DGYC a a a Q ~ - BaYp + c u (A.1) O. (1" is useful, but it is correct only for extreme dilution. Our use of it in Chapter V is justifiable because our chief purpose there was to obtain the form (particularly the sign) of the concentration gradient contribution to anomalous osmosis. Better numerical estimates of the contribution can be obtained by starting with the more exact equation of this Appendix. Most common transport prOperties for electrolyte solutions (mobility, transference number, conductance) are defined and measured in the Hittorf frame of refer- ence, where the diffusion fluxes are defined by jg E ca(9a - 9w) ' (A.2) where uw is the velocity of the solvent, water. The absolute molar fluxes, E c u , (A.3) 204 205 are related to the Hittorf fluxes by .H _ W‘l .H ova = cal—3 + 2a - (X /M) E M828 1 a=l,...,W-l 8—1 ~ w-l H ~w = cwg - (xw/M) X M838 , (A.4) where M is the mean molecular weight, fi = "54$ )4 M , I (A.5) 1 B B and the velocity 9 of the center of mass is given by _ W s = (V/M) X M N B=l 8” B I (A06) where v is the molar volume of the solution. For an isothermal single strong electrolyte, the linear flux equations in the Hittorf frame are (Haase, 1969; Katchalsky and Curran, 1965) .H 2+ = ' a++Yuj - a+-y“' .H _ , , 2- — - a_+Yu+ - a__yu_ (A.7) where the ads are Onsager coefficients and the u& include external potentials. For an ideal solution (or for a sufficiently dilute solution), 206 ' = yua vayp + RTYana + zaF§¢ . (A.8) Moreover, Vinx = Vinx ~ {1+vclv -(z v -z v )(z -z )-l]}-1V£nc ~ + ~ — w - + + - — + ~ + ' (A.9) where, for an electrolyte of molarity c which contains 0+ moles of cation and v_ moles of anion per mole of electro- 1yte, c+ = v+c , c_ = v_c , v = 0+ + v_ , z+v+ + z_v_ = 0 . (A.10) In terms of more accessible experimental quantities, the Onsager coefficients are, when a+_ = a_+ , a = (c lz/z F2A) + (v c D/vRT) ++ + + + + + _ _ 2 a+_ - a__+ - (c+A+A_/z_F A) + (v_c+D/vRT) _ _ 2 — (c_A+A_/z+F A) + (v+c_D/vRT) 2 2 a = - (c_A_/z_F A) + (v_c_D/vRT) . (A.11) The Ad in these formulas are the single ionic conductances, which are related to the Hittorf transference number ta by o l...|.|.I|I.Inl!|n rflmflfllllllJ 207 >2 ll At , a=+,- (A.12) where A is the equivalent conductance, A = A + A (A.13) and the transference numbers sum to one, t+ + t_ = l . . (A.14) The diffusion coefficient D in (A.11) is the Fickian mutual diffusion coefficient for the binary system. Instead of conductances, earlier workers used mobilities ”a defined by 2 w+ = A+/2+F2 w_ = - A_/z_F w_ = c(v_l_ + v+l+)w+ . (A.15) Thus, z w — z w = A/F2 (A.16) + + - - and w - w = ((1 /z ) + (1 /z )/F2 + - + + - - [(t+/z+) + (t_/z_)](z+w+ - z_w_) . (A.17) 208 Combining (A.7) through (A.17), we find jg = - {1+\)c[vw-(z_v+-z+v_)(z_-z+)-l]}”l X [D +z RTw (w -w )(z w -z w )_1]Vc + + + + - + + — - ~ + - c+z+w+Fy¢ —[czm(vw-vm)(z<»-zu))"l + + + + + - - + + - — + v+(c+v++c_v_)(D/vRT)]Yp g? = - {1+vc[vw-(z_v+—z+v_)(z_-z+)-l]}—l[D -1 + z_RTw_(w+-w_)(z+w+- z_w_) JYC - c_z_w_Fy¢ - [c z w (vcn-firu))(z w —z w )-1 - - - + + - - + + - - + v_(c+v++c3v_)(D/vRT)]2p . (A.18) Now define Da by . -1 _ _ Da = D + zawaRT(w+-w_)(z+w+-z_w_) , a—+, . (A.19) In Order to make contact with the Nernst—Planck equation rearrange the middle two parts of (A.11) D 2 (vRTa+_/v+c_) + (vRTl+1_/v+z+F A) 2 (vRTa+_/v_c+) - (vRTA+l_/v_z_F A) (vRT/v+c_)a+_ + (vRTt_/v+)u)+ (vRT/v_c+)a+_ + (vRTt+/v_)w_ . (A.20) 209 (A.20) and (A.19) yield Da = RTwa + (vRT/v+v_c)a+_ , a=+,- . (A.21) Thus, the Einstein Relation RTwa = Du is valid only when a =0 . Katchalsky and Curran (1965) have calculated a++ , a__ and a+_ for Nacl Solutions. They find that for 0.01 M solutions, a+_ is about 4% of a++ and about 3% of a__ ; for 0.1 M solutions, a+_ is about 11% of a++ and about 7% of a__ ; for 1.0 M solution a+_ is about 21% of a++ and about 14% of a__ . For greatest accuracy, the a+_ term should be retained in (A.21). However, for estimating effects, it is satisfactory to use Einstein Relation. With (A.19) and the further definitions RTwa = D& , a=+,- (A.22) and B = c z w (v w —v w )(z w -z w )—l d a a a + + - - + + - - + Va(c+v++c_v_)(D/0RT) , (A.23) and Dc = {1+vc[v -(z v -z v )(z -z )—1}-1D (A 24) a w — + + - - + a ’ ° 210 The Hittorf diffusion fluxes become +m c - D+YC+ - C+z+w+FY¢ - B+Yp (U. I :1: — nyc_ - c_z_w_Fy¢ - B_yp . (A.25) (U. Note that D: = Du for dilute solutions. Substitution of these into (A.4) yields _ -_ c_ — c c N+ — c+u {9+ (x+/M)[M+D++(v_/v+)M_D_]}yc+ - [c+z+w+-(x+/M)(M+c+z+w++M_c_z_w_)]F1¢ - [B+-(x+/fi)(M+B++M_B_)]yp Z (I c_g-{Df- (x_/M) [M+D: (v+/v_) +M_D_‘f] )yc_ - [c_z_w_-(x_/M)(M+c+z+w++M_c_z_w_)]Fy¢ - [B_-(x_/M)(M+B+-M_B_)]yp . (A.26) For high dilution x << 1 and x_ << 1 and we have + Na = cag - DaYca - cazawaFy¢ - BaYp , (A.27) the modified Nernst-Planck Equation. APPENDIX B DERIVATION OF SOME INTEGRALS INVOLVING MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND In the following we drive equations (5.25), (5.26), (5.29), (5.30), (5.31) and (5.32) which are not available in published tables. We utilize the known relations X 6 on(x)dx = xIl(x) , x 2 2 2 2 6 xIO(x)dx = x [10(x)—Il(x)]/2 , (5.24) x 2 2 2 6 x Io(x)I1(x)dx = x Il(x)/2 , and (dIo(x)/dx) = 11(x) , (dIl(x)/dx) = I0(x) — (Il(x)/2) . (B.1) We also employ integration by parts, b b b éudv = uVIa - éVdu . (B.2) 211 212 (1) Let u = XZIIg(X) — Ii(x)]/2 and dV = 2xdx . Integration by parts yields x x x 6 x3I§(x)dx - 6 x3Ii(x)dx = 3 2x{x2[Ig(x)-Ii(X)J/2}dx x = x4[Ig(x)-Ii(x)]/2 — 6 x3I§(x)dx or x 3 2 X 3 2 4 2 2 2 6 x 10(x)dx - 6 x Il(x)dx = x [10(x)-Il(x)]/2 . (B.3) Similarly, letting u = x3Il(x) and dV = Il(x)dx and inte- grating, by parts, we obtain X X X I 3 2 = 3 _ I 2 _ I 3 2 0 x I1(x)dx x Il(x)IO(x) 2 0 x 10(x)Il(x)dx 0 x Io(x)dx or Ixx3I2(x)dx + Ixx3I2(x)dx = x31 (x)I (x)-x212(x) (B 4) 0 0 0 1 l 0 1 ' where we have used (5.24). Solving (B.3) and (B.4) to- gether, we find I x313(x)dx = (1/3){x4[Ig(x)-Ii(x)]/2 + x3Il(x)IO(x)-x21i(x)} (5.25) and Ixx312(x)dx = (1/3){2x31 (x)I (x)—2x212(x)-x4[I2(x) o 1 1 o 1 o - Ii(x)]/2} . (5.26) 213 (2) Integration by parts with u = x2 and dV = Il(x)dx yields Ix 2 2 Ix 0 x I1(x)dx = x 10(x) — 2 0 xIO(x)dx _ 2 _ - x 10(x) 2xIl(x) , (5.29) where we have used (5.24)- (3) Letting u = x21i(x) and dV = xIO(x)dx and integrating by parts we obtain X X g x31i(x)lo(x)dx = x3xi(x) - 2 5 x3Ii(x)IO(x)dx or 1x32 33 0 x Il(x)Io(x)dx = x Il(x)/3 . (5.30) (4) Integration by parts with u = x2 and dV = xIO(x)dx yields x x I 3 = 3 _ I 2 0 x Io(x)dx x 11(x) 2 0 x Il(x)dx 3 _ _ 2 .x Il(x) — 4xIl(x) 2x I0(x) , (5.31) where (5.29) has been used. (5) Let u = x4Io(x) and dV = Il(x)dx. Integration by parts yields 214 Ixx4I (x)I ( )dx-x412(x)— 4 IXx312( )dx- Ixx4I (x)I (x)dx o o 1 x " o o o x o 1 0 or X 5 x4IO(x)Il(x)dx==(x415(x)/6)+-(x4Ii(x)/3)-(2x3Il(x)Io(x)/3) + (2x21i(x)/3) , (5.32) where (5.25) has been used. APPENDIX C COMPUTER PROGRAMS The three primary computer programs used in Chapter VI are listed in this appendix. Program ROOTS uses the half-interval method to calculate eigenvalues from (6.43). Program OCM utilizes the "Overdetermined Collocation" method to evaluate linear combination co- efficients for non-orthogonal functions. Program BULCON ' calculates local axial bulk concentrations and overall Sherwood numbers according to (6.55) and (6.58). 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