Flow with axial dispersion through a one-region pipe of uniform cross-section.

Further reading: Distributed Blood Tissue Exchange Models Explained

Description

   The partial differential equation models flow into, through and out of 
 a pipe with plug flow and axial dispersion (diffusion) along the x-axis
 and instantaneous radial dispersion so that concentration is uniform across
 the cross-section at each x-position. Consumption,Gp, equivalent to loss by 
 a first order reaction or loss by permeation is a uniform fraction per
 unit time along the pipe. (This can be modified by making G a function of
 concentration, Gp(Cp) or of position, Gp(x).) Flow is constant, as are all
 the other parameters.The boundary conditions are 
   (1) At the inflow, the diffusion coefficient, Dp, cm^2/s, times the
 spatial gradient in concentration, dC/dx, balances the difference between
 the inflow concentration and the concentration Cp just inside; 
   (2) At the outflow, the gradient dC/dx is set to zero, as if reflecting
 from an impermeable surface, so that mass is lost into the outflow only
 by flow, Cout = Cp(x=L,t).

 LIMITATIONS: This model cannot approximate Newtonian parabolic flow, where
 the response to a flow-proportiaonal cross-sectional pulse labeling at the
 inflow would give a sharp upstroke and peak at 1/2 the mean transit time
 and then, in the absence of axial dispersion, diminish in proportion to
 1/t^2. See Gonzalez-Fernandez (1962) on this point. 

Equations

Differential Equations

Left Boundary Conditions

Right Boundary Conditions

,

Initial Conditions

 or

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.