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.
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Contributed by: Clay Gruesbeck (November 2019)
Open content licensed under CC BY-NC-SA
Details
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:
here:
and
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.
Reference
[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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