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 one-dimensional 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.



Contents:

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.




Overview

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 Th. (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 Tc < Th washes over the sphere.

Schematic of sphere cooling experiment

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 non-uniform 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

  • Biot number, dimensionless surface resistance 

    	                 h R
	           Bi = -----
	                  k
    where h is the heat transfer coefficient, R is the radius of the sphere, and k is the thermal conductivity of the sphere.

  • Fourier number, dimensionless time 

    	                 a t
	           Fo = -----
	                  2
	                 R
    where a (alpha) is the thermal diffusivity of the sphere, t is the time, and R is the radius of the sphere.

  • theta, dimensionless temperature 

    	                    T(r,t) - Tfluid
	           theta = -----------------
	                     Ti - Tfluid
    where T(r,t) is the temperature in the sphere, Tfluid is the temperature of the surrounding fluid, and Ti is the uniform, initial temperature of the sphere.


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    Lumped Analysis

    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.
    Transient response of a lumped mass


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    Transient with Radial Temperature Variation

    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.

    Radial temperature distribution at different times for Bi = 10

    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.

    Temperature response at center and surface for Bi = 10

    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.

    colorbar for 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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