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# BTEX10stat

Flow with axial dispersion through a one-region pipe of uniform cross-section. Statistics on inflow and outflow concentration curves */

Model number: 0331

 Run JSim model Java Applet: JSim Tutorial

## 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

BTEX10,PDE,convection,diffusion,permeation,reaction,distributed,capillary, plasma, piston flow or plug flow, statistics, mean transit time, RD, relative dispersion, skewness, kurtosis