Heat Transfer between Flowing Liquids in Cylindrical Tubes

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This Demonstration shows the velocity and temperature profiles of two liquids at different initial temperatures flowing concurrently in laminar flow in a cylindrical tube and the surrounding cylindrical annulus.


The properties of the two liquids are assumed to be constant, and their velocity profiles are assumed to be independent of axial location. In the inlet region , the liquids in the annular and the tube regions have different uniform temperatures and , respectively; in the region , the external annulus wall is thermally insulated and the wall of the circular tube is assumed to be thermally thin.

Similar arrangements occur in various industrial applications, including heat exchangers and tubular reactors. You can vary the ratio of the properties of the fluid in the cylinder to the fluid in the annulus, the relative dimension of the tubular to axial radii and the pressure gradient in the two flow regions to see the temperature and velocities in the two flow regions.


Contributed by: Clay Gruesbeck (November 2019)
Open content licensed under CC BY-NC-SA


The dimensionless horizontal distance is

, where is the length of the tube. The dimensionless radial distance is


where is the external annular radius, and the dimensionless temperature is


The temperature profile of the two liquids is obtained by solving a partial differential equation with two different sets of parameters and velocity profiles, one for each flow region:



where is the ratio of the radii of the tube to the annulus, is the dimensionless temperature, is the velocity, is the Péclet number , is the thermal diffusivity , and , and are the fluid thermal conductivity, density and heat capacity, respectively. The initial conditions are:



with boundary conditions


Here and are the radii of the tube and the annulus respectively.

Analytic solutions for fully developed laminar flow in the tube and the cylinder are shown in [1]:


and the maximum velocity is

Here is the horizontal pressure and


The average (cup) temperatures of the two fluids are:


These equations are solved with the built-in Mathematica function NDSolve. Plots of the temperature and velocity contours, as well as the average temperatures, are shown.


[1] R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, 2nd ed., New York: John Wiley and Sons, 2002.


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