• Home

• About

• Models

• JSim

• Links

• Search

BTEX10

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

Model number: 0079

Further reading:     Distributed Blood Tissue Exchange Models Explained Download PDF file.

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

 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.