Transient Conduction in a Sphere with Convective Boundary Conditions

 Why do you need a thermometer to cook a turkey?
Common sense tells us that when you put a cold turkey into a hot
oven it takes time for the center of the turkey to heat up sufficiently
to cook the meat.
 Why do you burn your mouth when you eat pizza?
Although the crust has cooled enough for you to pick it up, the
sauce and cheese below the surface of the pizza are still very hot.
 The general problem:
The analysis described on this page provides a simple mathematical model
for these everyday problems. The details of these two situations are
rather complicated: the ``material'' has very nonuniform thermal
properties, both heat and
moisture are being exchanged with the surroundings, and the initial
temperature is not uniform. To a first approximation, however,
we can think of the turkey as a sphere of uniform material, and
the pizza a infinite slab of material. These assumptions reduce
the problem to that of analyzing onedimensional transient heat
conduction with convective boundary conditions.
The rate at which heat is transfered
to or from the object is also influenced by the convective
boundary condition, i.e. the resistance to heat flow at the surface
of the object.
The most generic example of this situation is the heating or cooling
of a sphere of uniform material. We consider the case where the sphere
is initially at a uniform temperature, and at an instant it is
immersed in a stream of flowing fluid at a different temperature.


 Overview
 Describes the physical background and some of the nomenclature.
 Lumped Analysis
 Transient solution for a highly conductive or small radius sphere.
In this case (Bi < 0.1)
there is a neglible temperature gradient inside the sphere and
the temperature is governed by a simple exponential decay.
 Transient with Radial Temperature Variation
 Transient solution when the radial temperature gradient is important.

The remainder of this page deals with the physical experiment depicted
in the sketch below. A sphere of uniform material is intially at some
temperature T_{h}. (Suppose, for example, that it has been stored
inside a warm oven for a sufficient period of time.) The sphere is then
moved to a different environment in which a flow of fluid (say air or water)
at another temperature T_{c} < T_{h} washes over the
sphere.
The moving fluid provides convective cooling of the sphere. The thermophysical
properties of the sphere and the effectiveness of the convective cooling
determine the internal temperature distribution of the sphere as it cools.
If the material has a high thermal conductivity (i.e. if it is a ``good''
conductor of heat) and if the convective cooling is relatively weak
(e.g. if the fluid velocity is low) then the internal temperature of
the sphere will be relatively uniform. In other words there will be
little difference in temperature between the surface and the center
of the sphere.
On the other hand if the material has a low conductivity and if the
convective cooling is strong then the internal temperature of the sphere
will not be uniform. This is the case qualitatively represented by the
color distribution in the sphere on the right hand side of the preceding
sketch. A nonuniform temperature distribution means that there is
a significant difference in temperature between the center (red <> warm)
and the surface (blue/green <> cool) of the sphere.
The following section introduces some terms that enable a quantitative
analysis of the cooling sphere.
Terminology
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When the sphere has a small Bi number, specifically when Bi < 0.1, the
radial temperature variation is negligible compared to the temperature
difference between the surrounding fluid and the surface of the sphere.
In this case the sphere responds as a lumped mass.
The transient temperature response is
theta = exp(  t/tau )
where tau = rho*c*V/(h*A) is the time constant for the sphere.
Substituting the definitions of Bi and Fo into the temperature
response formula gives
theta = exp(  3*Bi*Fo )
The following figure shows the dimensions temperature response.
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For large Bi numbers the thermal resistance of the surface is small
compared to the internal thermal resistance of the sphere. The result is a
small temperature drop at the surface (Tfluid  T(R,t)), and a
relatively large temperature variation between the surface and the
center of the sphere.
The following plot shows the radial temperature variation at
three different times (three different Fo values) for a sphere
with Bi = 10. The intial temperature variation (Fo = 0) is
uniform  this is the initial condition.
The next plot displays the transient cooling as measured
by a two thermocouples, one at the center and the other on the surface
of the sphere.
Now imagine that you could install lots and lots of thermocouples
throughout the interior of the sphere. Ignoring the practical difficulties
of this arrangement, such an experiment would create a much more complete
picture of the temperature response.
The following animation simulates the temperature response of all
points in the sphere. The temperature is indicated by the color:
dark red is hot (theta = 1) and dark blue is cold (theta = 0).
The color bar on the right gives the mapping between temperature
and color.
N.B. The background is green because, well, because I haven't
figured out how to tweak the background to be a neutral color like
gray. It's not hard, I know, but .... Until that is fixed, be aware
that the surrounding fluid is at theta = 0. (I could set the background
color to be dark blue, but then the sphere would disappear as t
approached infinity).
To start the animation press the small right triangle in the lower
left corner of the image. The buttons in the lower right corner of
the image can be used to advance or rewind one frame at at time.
The slider bar on the bottom of the image can also be used to move
forward and backward through the animation.
Send questions and comments via email to gerry
A description of the MATLAB code used to generate these results is available
on this page.
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