Uncoupled, independent fluxes of water and of 2 solutes, across a membrane separating 2 stirred tanks of equal elasticity.

Description

 Uncoupled, independent fluxes of water and of 2 solutes, A and B, 
 across a membrane separating 2 stirred tanks. Solute activities 
 are assumed unity so concentrations = thermodynamic activity. The model describes 
 a situation similar to that for the simplest expressions of Kedem and Katchalsky
 (1958) but omits all interactions between solutes and between water and 
 any solute. One can think of the solutes passing though the membrane by
 passive permeation with permeability coefficients PermA and Perm B, and
 the water passing through aqueous pores with filtration coefficient or 
 hydraulic conductivity, Lp. The aqueous pores do not permit solute passage. 
     
     Lp is the same as the traditional filtration coefficient Kf. Lp translates 
 to a conventional permeability for water filtration coefficienr, Pf cm/s, 
      Pf = Lp* Vw / RT
 where RT = 19.347*10^6 mmHg*cm^3*mol^(-1) at 37C, Vw is the partial molar volume
 of water, 18 ml/mol or the concentration of water in water is 55.55 M

   The driving forces are the pressure difference for water flux and the
 concentration for the solute fluxes. The pressure difference across the
 membrane is the hydrostatic pressure difference minus the osmotic pressure
 difference. The osmotic pressure is given by Van't Hoff's Eq: 
        p_osm = a.C.RT,    where p_osm is the osmotic pressure, mmHg,
 "a" is the activity coefficient, assumed in this model to equal unity,
  C is concentration, M, and RT is the Gas Constant times Temperature Kelvin.

    In this model the solute doesn't permeate the aqueous pore so there is 
 no consideration of a reflection coefficient, or rather it is assumed to be unity.
 Thus solute concentration in the pore water is zero, andthere is no solute advection..

   The system is composed of two volumes of pressure-dependent size, yet
 stirred instantaneously continually. The pressure/volume relationship is
 expressed via the elasticity of the chambers, Elast, the slope of the
 pressure/volume relationship. An equivalent structure is to use rigid chambers
 from each of which there rises narrow columns of fluid to heights h1 and h2.
 The fluid in the columns is considered to be instantaneously mixed with that
 in the chamber from which it rises. Fluid or volume flux, Jv, from side 1 to 
 side 2 raises difference in the column heighta between the two sides by
 Base*(h2-h1) = Jv, where Base = area of the base of the column, and the
 pressure difference rises to (h2-h1)*rho cm H2O, where rho is the fluid 
 density. g/ml. The linear chamber elastance used in this model, Elast mmHg/ml, 
 gives an equivalent measure for flexible chambers, assuming a linear relationship between  
 the pressure change and the volume change. (1 mmHg = 13.59 cm H2O.)

 Notes: Situation 1:= Model parameter set: par1    PermA = 0, PermB > 0.1. See Notes.
        Situation 2 = Model parameter set: par2     PermB > 0.
 See Notes tab for more discussion.

Figure: Top shows concentration of solute A and solute B as a function of time (A1, B1: conc in volume 1, A2, B2 conc in volume 2). Note change in volume 1 (V1) from initial volume of 1 ml. Bottom figure shows hydrostatic pressure in V1 and volume 2 (V2) as a function of time. Permeability of A into V2 is zero and A2_init and B1_init are 0 mM for both figures.

Equations

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.