Uncoupled, independent fluxes of water and of 2 solutes, across a membrane separating 2 stirred tanks equipped with columns above each to provide observable column heights as measures of their pressures.

Description

   This model is the same as Osm.Uncoupled1.proj except for using column heights
 instead of elasticity. It also allows temperature changes.
   
    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, Pf cm/s, 
      Pf = Lp*RT/Vw
 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, and there is no solute advection..

   The system is composed of two volumes of pressure-dependent size, yet
 stirred instantaneously continually. The slope of the pressure/volume
 relationship is linear and defined by the height of a column of fluid above
 the rigid chambers  The narrow columns of fluid have heights h1 and h2. 

   The pressure is rho*grav*h1 in chamber 1, where rho is fluid density,
grav is acceleration due to gravity;
 
  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 causes a 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 rho* grav*(h2-h1) cm H2O, where rho is the fluid 
 density. g/ml, in the narrow colums..
 
   The model OsmUncoupledA.proj uses an analogous linear chamber elastance, 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 default par. . PermA = 0, PermB > 0.1. See Notes.
        Situation 2 = Model par2           PermB > 0. 

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. Y is volume 1 calculated from the membrane permeability while V1 is calculated from Hydraulic conductivity. Bottom figure shows hydrostatic pressure in V1 and volume 2 (V2) as a function of time. Permeability of A into V2 is one tenth the permeability of B 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.