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BTEX10

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

Model number: 0079

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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

$\large \frac{\partial C_p}{\partial t} = \frac{-F_p \cdot L}{V_p} \cdot \frac{\partial C_p}{\partial x}- \frac{G_p}{V_p} \cdot C_p +D_p \cdot \frac{\partial^2 C_p}{\partial x^2}$

Left Boundary Conditions

$\large -{\frac {{\it F_p}\cdot L \cdot \left( {\it C_p}-{\it C_{in}} \right) }{{\it V_p}}}+{D_{p} \cdot \it {\frac {\partial}{\partial x}}C_p=0$ .

Right Boundary Conditions

$\large {\it {\frac {\partial }{\partial x}}C_p=0$$\large {\it C_{out}={\it C_{p}$ .

Initial Conditions

$\large C_p=C_p0$   or
$\large C_p=C_p0(x)$ .

The equations for this model may also 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.

References


W.C. Sangren and C.W. Sheppard.  A mathematical derivation of the
exchange of a labelled substance between a liquid flowing in a
vessel and an external compartment.  Bull Math BioPhys, 15, 387-394,
1953.

Gonzalez-Fernandez JM. Theory of the measurement of the dispersion of
an indicator in indicator-dilution studies. Circ Res 10: 409-428, 1962.

C.A. Goresky, W.H. Ziegler, and G.G. Bach. Capillary exchange modeling:
Barrier-limited and flow-limited distribution. Circ Res 27: 739-764, 1970.

J.B. Bassingthwaighte. A concurrent flow model for extraction
during transcapillary passage.  Circ Res 35:483-503, 1974.

B. Guller, T. Yipintsoi, A.L. Orvis, and J.B. Bassingthwaighte. Myocardial
sodium extraction at varied coronary flows in the dog: Estimation of
capillary permeability by residue and outflow detection. Circ Res 37: 359-378, 1975.

C.P. Rose, C.A. Goresky, and G.G. Bach.  The capillary and
sarcolemmal barriers in the heart--an exploration of labelled water
permeability.  Circ Res 41: 515, 1977.

J.B. Bassingthwaighte, C.Y. Wang, and I.S. Chan.  Blood-tissue
exchange via transport and transformation by endothelial cells.
Circ. Res. 65:997-1020, 1989.

Poulain CA, Finlayson BA, Bassingthwaighte JB.,Efficient numerical methods
for nonlinear-facilitated transport and exchange in a blood-tissue exchange
unit, Ann Biomed Eng. 1997 May-Jun;25(3):547-64.



Related Models

Blood Tissue Exchange (BTEX) models

Key Terms

plasma, piston flow or plug flow