Laminar tube flow through an abrupt contraction

Abstract

The general equations of motion were solved numerically for the laminar isothermal flow of Newtonian fluids from a large tube of circular cross section through an abrupt contraction into a coaxial tube of smaller diameter and through the flow‐development region of the smaller tube. The ratio of the diameter of the large tube to that of the smaller tube was varied from one to eight (the latter in one case). Solutions were obtained for the case where the larger tube is real, with no slip at the wall, and for the case where it is a frictionless “stream” tube. The results are presented as charts giving excess pressure losses attributable to contracted and developing flow in terms of equivalent smaller‐tube diameters as functions of the tube‐contraction ratio and the Reynolds number, which was varied from 0.01 to as high as 500 in one case. Both radial‐ and axial‐velocity profiles are presented. The computed results are shown to be in satisfactory agreement with some experimental data. The results are presented in a manner convenient for use in the design of equipment in which contracted Newtonian flow occurs, such as fiber spinnerettes and heat exchangers, and in the analysis of experimental data for contracted flow.

Nonisothermal laminar contracted flow

“…Concavities, such as those noted in these three figures, have been observed experimentally (Burke and Berman, 1969) and in previous isothermal-flow computations (Christiansen et al, 1972; Vrentas et al, 1966; Wang and Longwell, 1964) and are discussed by Christiansen et al rounding the concavity moves closer to the wall, while, with cooling, the opposite occurs. and, in some cooling cases, is negative in consequence of our use of fully developed flow at the tube-wall temperature as the reference.…”

“…However, most pertinent are the numerical solutions of the general equations of motion-including axial-diff usion, radial-convection, and inertial terms-by Wang and Longwell (1964) for laminar Newtonian flow between parallel plates, preceded by plates of the same spacing but with frictionless surfaces; by Vrentas, Duda, and Bargeron (1966) for Newtonian flow from a stream tube through the entrance region of a real tube of the same diameter; and by Christiansen and Carter (1969) and by Christian- We present here results from an extension of our investigations (Christiansen and Carter, 1969;Christiansen et al, 1972) to nonisothermal temperature-dependent ST-RT and RT-RT contracted Newtonian flow. In these studies, there is heat exchange with the surface of the smaller tube, which is at a higher or lower constant temperature than the initial temperature of the fluid.…”

“…As shown in Figure 11 where L,, is plotted versus z / D with N R , as a parameter, Leq increases greatly with N R e at higher NR,. However, based on isothermal-flow computations (Christiansen et al, 1972), data for N R e = 0.1 are expected to be essentially the same as those for N R e = 0.01; and those for N R e = 1.0 would be only slightly higher. Since most nonisothermal-flow computations were made for ST-RT contracted flow at B = 2, the effect of f3 and the RT-RT boundary conditions on the results is not certain.…”

“…Many solutions of boundary-layer equations, including some for non-Newtonian fluids (Chen et al, 1970;Collins and Schowalter, 1963;Michiyoshi et al, 1966), and some numerical solutions of simplified equations of motion (Christiansen and Lemmon, 1965;Hornbeck, 1965;Lemmon, 1963) have been reported for laminar, isothermal flow through a real-tube entrance region for which a flat entrance velocity and constant fluid properties were assumed (see Christiansen et al, 1972, andLemmon, 1965, for comparisons of Newtonian flow results). More recently, numerical solutions of somewhat restricted equations of motion and energy (radial flow, inertial terms, axial diffusion, and viscous dissipation were neglected) for temperature-dependent Newtonian and non-Newtonian flow accompanied by heat transfer have been reported (Newtonian and power-law flow: Christiansen and Craig, 1962; Christiansen and Jensen, 1969;Christiansen et al, 1966;Jensen, 1963;Bingham flow: Jensen, 1963; Powell-Eying flow: Christiansen and Jensen, 1962;Jensen, 1963) for the case where the fluid in fully developed flow in a real tube enters a region in which the tube wall is abruptly at a higher or lower constant temperature than that of the fluid.…”

“…The computations were restricted largely to ST-RT contracted flow-an approximation of flow into the tubes in the tube sheet of shell-and-tube heat exchangers-and to / 3 = 2, which approximates the contraction ratio for flow in the tube sheets of many heat exchangers. However, computations can be made at higher values of 8; for example, in isothermal Newtonian flow, values of /3 up to 8 were employed (Christiansen et al, 1972;Kelsey, 1971). The N R e and Npe ranges were restricted since the computations became unstable at values greater than about 100, depending on the case.…”