Model: SIMPIPE, flow through a pipe | Interagency Modeling and Analysis Group

1. Simple fluid displacement without dispersion

Consider the lowly pipe. Unless it is clogged or leaks, what comes in goes out. This is mass balance--nothing lost, nothing gained. Fig. 1 shows an idealized pipe, one that is frictionless so that the fluid does not stick to the walls. Sticking only to itself (internal viscosity) but not to walls allows a flat velocity profile. (A velocity profile is defined as the map of velocities across a cross section of the pipe.)

web fig1 — Figure 1. Fluid displacement without dispersion, a uniform velocity profile, often called piston flow.

web fig2 — Figure 2. Brief pulse input of 10 mmole in 1.0 second into a pipe. C in is the concentration at the inflow, a pulse beginning at t = 2 s and ending at t = 3 s, and C *out* is that at the outflow end of the pipe, a pulse beginning at t = 17 s and ending at 18 s. The delay, t0 , is the transit time. The upper panel records the amount of material in the pipe, as if by external detection. F = 0.0167 ml/g/s = 1 ml/g/min , V = 0.25 ml/g, and t = *V/F* = 15 s.

xsim fig3 — Figure 3. Input and output concentration-time curves for a dispersionless pipe with finite pulse input. Same F and V as in Figure 2. The total injected, q 0 = 10 mmole is also the same, but the input pulse is 4.0 seconds, four times as long as that in Figure2.

xsim fig4 — Figure 4. Responses of the same system as in Fig. 2 to a dispersed input. (Input is a lagged normal density curve [Bassingthwaighte, Ackerman, and Wood, 1966] with s = 0.96, t = 1.15 seconds, and tc = 4.85 seconds.)

web fig5 — Figure 5. Effect of diminishing volume on the system responses to a dispersed input (lower panel). The system content fails to achieve a plateau when V is small (upper panel).

2. Fluid displacement with dispersion

web-11 — Figure 6. Responses to a pulse input in a system with dispersion. When the volume is sufficiently large relative to the volume of fluid containing the input, a plateau is reached even when there is internal dispersion. However, internal dispersion abbreviates the plateau, and may even reduce the maximum tracer content when the volume is smaller. F = 1 ml/g/ min.

web-12 — Figure 7. Tracer content of a dispersive system at various volumes when the input is a pulse of 1.0 second duration (upper panel), or a dispersed input form (lower panel). With a small system volume, the peak, q ( t ), reaches the plateau representing the total dose only when the input concentration-time curve is shorter than the shortest transit time.

The notation defined originally by Zierler (Meier and Zierler, 1954; Zierler, 1962 and 1965) and formalized by an international agreement (Bassingthwaighte et al., 1986), summarizes what has been represented for transport through an idealized pipe. When the input is the Dirac function, d ( t ), then the system responses are given formal names; these names are completely general. When any other input occurs, the system responses can be defined from the formal descriptors by minor arithmetic manipulations, usually a convolution.

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References

Bassingthwaighte, J. B., F. H. Ackerman, and E. H. Wood. Applications of the lagged normal density curve as a model for arterial dilution curves. Circ. Res. 18:398-415, 1966.

Bassingthwaighte, J. B. Physiology and theory of tracer washout techniques for the estimation of myocardial blood flow: Flow estimation from tracer washout. Prog. Cardiovasc. Dis. 20:165-189, 1977.

Bassingthwaighte, J. B., and C. A. Goresky. Modeling in the analysis of solute and water exchange in the microvasculature. In: Handbook of Physiology. Sect. 2, The Cardiovascular System. Vol IV, The Microcirculation , edited by E. M. Renkin and C. C. Michel. Bethesda, MD: Am. Physiol. Soc., 1984, pp. 549-626.

Bassingthwaighte, J. B., F. P. Chinard, C. Crone, C. A. Goresky, N. A. Lassen, R. S. Reneman, and K. L. Zierler. Terminology for mass transport and exchange. Am. J. Physiol. 250 ( Heart Circ. Physiol. 19):H539-H545, 1986.

Bassingthwaighte, J. B., G. R. Raymond, and J. I. S. Chan. Principles of tracer kinetics. In: Nuclear Cardiology: State of the Art and Future Directions , edited by B. L. Zaret and G. A. Beller. St. Louis: Mosby-Year Book, 1993, pp. 3-23.

King, R. B., A. Deussen, G. R. Raymond, and J. B. Bassingthwaighte. A vascular transport operator. Am. J. Physiol. 265 ( Heart Circ. Physiol. 34):H2196-H2208, 1993.

Meier, P., and K. L. Zierler. On the theory of the indicator-dilution method for measurement of blood flow and volume. J. Appl. Physiol. 6:731-744, 1954.

Stephenson, J. L. Theory of the measurement of blood flow by the dilution of an indicator. Bull. Math. Biophys. 10:117-121, 1948.

Zierler, K. L. Circulation times and the theory of indicator-dilution methods for determining blood flow and volume. In: Handbook of Physiology, Sec. 2: Circulation . Washington, D.C.: American Physiological Society, 1962, pp. 585-615.

Zierler, K. L. Theoretical basis of indicator-dilution methods for measuring flow and volume. Circ. Res. 10:393-407, 1962.

Zierler, K. L. Equations for measuring blood flow by external monitoring of radioisotopes. Circ. Res. 16:309-321, 1965.