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Internal Flows (Forced Convection in Pipes)

UPDATED: 07/19/2022

Internal Forced Flow (Pipe Flow)

Internal (pipe) flows involving heating or cooling are usually treated using convection correlations. For laminar flows the correlations are based on analysis; for transition and turbulent flows they are generally based on experiment. Another “on-screen” laboratory, this internal flow module solves the thermal entry length problem (velocity profile already fully developed when a change in the wall thermal boundary condition is introduced) for laminar, transition and turbulent flows. The analytical solution for laminar flow was given by Leo Graetz.  The thermally (and hydrodynamically) fully developed condition is, of course, the asymptotic limit of this case. (Solution of the third internal flow scenario, the combined-entry length problem, while certainly feasible, requires the solution of the axial momentum and continuity equations, in addition to the energy balance equation solved here.) Either a fixed wall temperature or prescribed wall heat flux may be specified. A single pass, space-wise marching technique, which is implicit in the radial direction and uses backward differencing in the axial direction, is used to solve the discretized form of the governing advection-diffusion equation.

Module Description

Inputs to the model are the Prandtl number of the fluid, the Reynolds number based on diameter, the Length/Diameter ratio for the pipe and whether a constant wall temperature or constant heat flux is applied to the surface. In addition, the user can specify a radial magnification factor so that gradients can be observed more easily in the plot. The temperature distribution as a function of axial location and radius is displayed in the form of a color contour plot and the velocity profile (which is unchanging, of course, since we have assumed in this calculation that the flow is hydrodynamically fully developed) is depicted at the inlet. “Heatlines,” the convection heat transfer analog of streamlines, may be superimposed on the isotherms. Using a scrollbar (just to the bottom left of the contour plot in the screenshot below) the user can take data from the screen for the wall temperature, mixed mean temperature and the surface temperature gradient as a function of axial position along the pipe. Using the former and an integral heat balance on a short length of the pipe or the latter directly, the user can develop a local or overall correlation for Nusselt number. This module allows the user to use the mouse as a probe and “measure” local values of velocity and temperature. For values of Reynolds number greater than critical, it gives the user the illusion of turbulence (the “fuzz” seen below).

Interface for HTT_pipe virtual laboratory.  This VB executable assumes that the velocity profile is fully developed at the beginning of the heated section. Either a step change in wall temperature or surface heat flux may be selected. This animated GIF file shows computed results for a range of Reynolds numbers from laminar to highly turbulent.  A smooth pipe surface is assumed.
The HTTpipe module allows the user to take measurements of the local velocity and temperature anywhere in the flow field using the mouse as a probe.   In addition, a scrollbar on the main form can be used to measure the temperature gradient right at the wall, the wall temperature (which will vary when a prescribed heat flux is applied) and the mixed mean temperature as a function of location along the pipe.  These two quantities are, of course, all that is needed to find the local Nusselt number.
Main Interface of HTTpipe with popup form showing the profiles of fluid mixed mean temperature and wall temperature as a function of position along the length of the pipe. The case considered here uses a uniform heat flux as the thermal boundary condition. The fluid mean temperature (plotted in blue in the pop-up form) rises linearly along the pipe, and the wall temperature (in red) rise as expected. The thermal entry length, which here corresponds to about 20% of the pipe length, is evident.

Like the external flow module, this simulation allows an infinite number of flow and fluid input parameters — with virtually negligible setup time and no hazardous materials! Using either of these forced flow modules the user becomes immediately aware of the tremendous difference in the heat transfer characteristics of ordinary (Pr = ~ 1.0) and extreme Prandtl number fluids (liquid metals Pr << 1 and oils (Pr >>1) and understands physically the reason why. Conditions under which the use of heat transfer enhancement techniques might be justified also become readily evident. An Excel spreadsheet that was developed to aid in verification of this internal flow module and evaluates a broad range of the conventional internal flow correlations may be downloaded here.

To highlight the versatility of HTTpipe and encourage the testing of a wide range of parameters, this secondary form is set up to show the specific conditions to which the model is applicable:

Secondary form showing applicability of the HTTpipe module. Ten cases covering a wide range of parameters can be run at the click of a button.

Note that this module only applies to smooth pipes.  An Excel workbook implementing the conventional procedures to pipes whose inside surface is rough and which may be considered both hydrodynamically and thermally fully developed can be found here.

This module was created based on a lab experiment the author, then an undergraduate engineering student, did a few years back:

Actual laboratory setup that the HTT_pipe virtual laboratory is designed to mimic.

The unheated starting section (where the velocity profile develops) is to the left.   The heating begins roughly at the point along the pipe where the student sits.   From that point on the pipe is wrapped with insulation.

Link to YouTube Introduction to HTT_pipe Video (3:40)


The algorithm and laminar flow results (corresponding to the well-known Graetz problem and not including the mixing length model used in the transition and turbulent regime) are described in: Ribando, R.J., and O’Leary, G.W., “Numerical Methods in Engineering Education: An Example Student Project in Convection Heat Transfer,” Computer Applications in Engineering Education, Vol. 2, No.3, 1994, pp. 165-174.

Notice to International Users (in those countries where decimal points (periods) are used instead of commas to break up long numbers): If, after you have installed this module, it does not work properly, then in the International Setting of the Windows Control Panel, please change the language to English (US).

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