Thermal Conductivity of Solids
CE 328
Spring 2004
Objective:
The goal of this experiment is to measure the thermal conductivity of several solid materials.
System:
Solids of known and unknown thermal conductivity are available in the shapes of cylinders and spheres. The objects are first cooled to ice temperature and then immersed in a circulating water bath whose temperature is maintained at a uniform (but higher) temperature.
Measurements:
A thermocouple located at the center of each spherical or cylindrical body measures the temperature at that point. This temperature is recorded as a function of time. Bring a floppy disk for this lab.
Theory:
The thermal conductivity of the material can be determined by comparing the measured temperature to that predicted from a mathematical analysis of heat conduction in the solid body. The thermal conductivity is a parameter in the theory that can be varied to match the experimental temperature profile.
Fourier’s Law of Heat Conduction states that the heat flux by conduction is proportional to the temperature gradient.
(1)
As an example, consider the problem of heat conduction in a cylinder of infinite length. Application of Fourier's Law of Heat Conduction to the unsteady flow of heat from the object to the water produces the energy balance
for t ³ 0 and 0 ≤ r ≤ R (2)
where
T = temperature of the solid at radius r and time t
t = time
r = radial distance from the center
a = k/(rCp) = thermal diffusivity of the solid
k = thermal conductivity of the solid
r = density of the solid
Cp = heat capacity of the solid
R = radius of the cylinder
For a cylinder initially at a uniform temperature, To, suddenly immersed into a bath of constant,
uniform temperature, T¥, the initial and boundary conditions for this partial differential equation are:
T = To for t = 0 and 0 ≤ r ≤ R (3)
at r = 0 for all t > 0 (4)
at r = R for all t > 0 (5)
where h is the heat transfer coefficient for the transfer of heat from the solid to the fluid. This coefficient must be determined experimentally.
The solution to the above partial differential equation with the initial and boundary conditions given can be written as
(6)
where
Jk(x) = kth order of Bessel function of x
bn = nth root of the equation
(7)
The solution can be written in terms of dimensionless variables as:
(8)
where
t == dimensionless time
h = r/R = dimensionless radial position
Bi = = Biot number (dimensionless)
q = = unaccomplished temperature change (dimensionless)
Expression (7) has been evaluated and plotted by several authors. Similar solutions for other shapes, e.g., spheres and flat plates, as well as the solution for a cylinder of finite length can also be found in the literature. Appendix 2 gives the expression to be used for spheres. The references given below provide a start for locating such solutions and graphical presentations of them.
Experimental work:
Students will record the thermal history of several samples (initially in thermal equilibrium with ice), immersed in a circulatory water bath that is held at constant temperature. They will monitor the temperature using a thermocouple embedded in the center of the sample. This thermocouple, as well as thermocouples located in the ice bath, and in the circulatory water bath are connected to a data acquisition board controlled using LabView software. A basic LabView program for recording the temperature measurements from these thermocouples is provided. It is described in more detail below. Bring a floppy disk for this lab.
The unknown heat transfer coefficient, h, must first be determined by recording the thermal history of a sample of known shape, thermal conductivity, density and heat capacity. We postulate that this coefficient is independent of the material; it depends only on the shape of the sample and the conditions prevailing in the water bath. Thus, once h is known for the system, the thermal conductivity of any solid can be determined.
Calculations
1. The aluminum alloy (Type 6262-T9) sphere and cylinder are to be used to determine the heat transfer coefficient, h. The remaining unknown materials are believed to be stainless steel (Type 304), "free cutting" brass, and poly-methyl-methacrylate (Lucite® or Plexiglas®). This can be done using a graphical presentation of the solution of the appropriate heat transfer problem or by evaluating the solution numerically. Use the full temperature vs. time curve to fit the best possible value of the Biot modulus to match all of the data. The heat transfer coefficient should be calculated. The error in q should be calculated by propogation of errors from your experimental data, and compared (after making the appropriate corrections) to the sum of squares error resulting from your curve fit.
2. Because neither the Biot modulus nor the dimensionless time can be calculated without knowing the thermal conductivity, a trial and error procedure is necessary to determine the unknown thermal conductivities. For any particular value of the unaccomplished temperature change (q), there correspond an infinite number of (Bi,t) pairs that would lead to that value for q. The thermal conductivity can be calculated from both the Biot number and from the dimensionless time. The resulting two values of k will be equal for the actual Biot number and dimensionless time. That is the unique (Bi,t) pair that gives the measured unaccomplished temperature change using a single value for the thermal conductivity.
3. An Excel file will be given to you that you can use to solve for the heat transfer coefficient of Al, and for the thermal conductivity of the other materials. For a description of the calculation method, see Appendix 4.
4. Copy a chart for center temperature profiles of spheres and cylinders from your heat transfer text. Draw your results on it. Discuss how well your results fit this theory.
Pre-lab Assignment:
Write a detailed operating procedure. Determine whether you will use Maple, Excel, or charts to determine values of h and k. Be prepared to discuss the calculations/programming needed. Also, for error determination in theta, be prepared to discuss how you will estimate the error in Tcenter.
Some Ideas to Consider in the Reports
1. How do the measured values for k compare with those given in the literature?
2. Do h and k appear to change with unaccomplished temperature?
3. How did you determine the "best" value for h and k from your work?
4. What are some of the sources of error (both systematic and random) in your work? How can you quantify these sources, and what is their effect on your final results?
5. What is the error that you determined for q using propogation of errors? Did it change for the different materials and/or shapes? How does it compare to the sum of square errors, (SSQ) that you determined for your best fits? (You need to make some calculations to the SSQ errors or to theta to make a comparison.)
References
1 Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, New York: John Wiley and Sons, Inc., 1966.
2. Perry's Chemical Engineers’ Handbook, John H. Perry, Editor, NY.
3. Foust, A.S., Clump, C.W., Maus, L., and Wenzel, L.A., Principles of Unit Operations, New York: John Wiley and Sons, Inc., 1967.
4. Hull, C. and West, C.J., International Critical Tables of Numerical Data, Physics, Chemistry and Technology. New York: McGraw-Hill Book Co., Inc., 1933.
5. McAdams, W.H., Heat Transmission. New York: McGraw-Hill Book Co., Inc., 1948.
6. Ingersol, et al., Heat Conduction. New York: McGraw-Hill Book Co., Inc., 1948.
7. Gurney and Lurie, Ind. Eng. Chem. 15, 1170 (1923).
8. Parker, J.D., Boggs, J.H. and Blick, E.F., Introduction to Fluid Mechanics and Heat Transfer, Addison-Wesley Publishing Co., 1969.
9. Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, 2nd edition, Oxford Science Publications, 1959.
Appendix 1: Analysis of Transient Conduction in an Infinite Cylinder:
for t ³ 0, and 0 £ r £ R
with initial and boundary conditions
T = To for t = 0 and 0 ≤ r ≤ R
at r = 0 for all t > 0
at r = R for all t > 0
This can be made dimensionless using
Substituting these into the equations and boundary conditions gives
with initial and boundary conditions
The solution to this problem is on p. 329 of Carlslaw and Jaeger, and is given by
where
Bessel functions are power series solutions to differential equations. The Bessel function Jn is given by
A sketch of Bessel functions are given below. Note that J0 and J1 have roots within each multiple of p.
Appendix 2: Analysis of transient conduction in a sphere:
for t ³ 0, and 0 £ r £ R
with initial and boundary conditions
T = To for t = 0 and 0 ≤ r ≤ R
at r = 0 for all t > 0
at r = R for all t > 0
This can be made dimensionless using
Substituting these into the equations and boundary conditions gives
with initial and boundary conditions
Making the substitution v = hq, so that
, , ,
, , and
transforms the equation and initial and boundary conditions to:
v=h at t for all h
for all t
at h=1 for all t.
A general solution that includes this case is given on pages 126-127 of Carslaw and Jaeger. The solution given there is
where
and the bn are the roots of
(A)
using this substitution and the trig identity for csc2(bn) this can be rearranged to give
Plugging this into the expression for v gives
Evaluating the integral gives
The unaccomplished temperature change is then .
We are interested in the temperature at the center of the sphere, h = 0. So, we take the appropriate limit of the above expression as h goes to zero to get
(B)
Appendix 3: Evaluation of Infinite Series Solutions in Maple®
Rather than using graphs presented in the literature to evaluate equation (7) for the unaccomplished temperature change, q, one can evaluate the expression directly using Maple® (or Mathematica® or other computer packages). At the axis of the cylinder (h = 0) where the temperature is measured, equation (7) reduces to
(8)
since J0(0) = 1. This sum is somewhat difficult to evaluate, because in each term we must solve equation (6), which was
(9)
However, in Maple®, we can define bn as the nth root of equation (9). By plotting equation (9), or by performing some mathematical manipulations of it, we could convince ourselves that it has one solution (root) in each interval (np to (n+1)p). A plot of bJ1(b)-Bi Jo(b) is shown below.
In the interval shown, 0 to 20p, there are twenty zeroes that appear to be about equally spaced. This is far from a proof, but it at least suggests that there is one zero in each interval (np to (n+1)p). Knowing that, we can define the nth root and then evaluate each term in the sum in Maple® as shown below.
Having done this, we can specify particular values for the dimensionless time and the Biot modulus, and then evaluate the sum. Of course we cannot really sum an infinite number of terms, but it turns out that this series converges nicely using only a few terms. The exact number of terms that are needed will depend on the values of t and Bi that we use, but 10 terms was enough for the values that I looked at. In Maple® this is done as follows.
Now, one can rapidly try different values of t and Bi to find a pair that matches a particular experimental value for q.
One can similarly evaluate the expressions for conduction in a sphere that were given in Appendix 2.
Appendix 4: Evaluation of Infinite Series Solution in Excel®
An Excel Program will be available on the website to evaluate the infinite series solution. Use the correct worksheet for the geometry of the object you are solving for.
Put in your time and temperature data in the first two columns. Tau and theta will automatically be calculated.
For the Al cylinder:
1. Put in k for Al.
2. Guess a value for h.
3. Run the macro to zero the root of the Bessel function (Ctrl+Shift+r)
4. Check the solution to make sure “zero” is near zero, and the roots fall between 0 and pi, pi and 2pi, 2pi and 3 pi, etc.
5. Look at the sum of squares error. (bottom of error^2 column)
6. Repeat steps 1-3 until you have minimized sum of squares error.
7. Now, use this h for other cylinders to determine k.
For other cylinders:
1. Put in h you determined for Al.
2. Guess a value for k.
3. Run the macro to zero the root of the Bessel function (Ctrl+Shift+r)
4. Check the solution to make sure “zero” is near zero, and the roots fall between 0 and pi, pi and 2pi, 2pi and 3 pi, etc.
5. Look at the sum of squares error. (bottom of error^2 column)
6. Repeat steps 1-3 until you have minimized sum of squares error.
7. Now, you have k for your “unknown” cylinder.
For spheres:
Follow the same procedure as for a cylinder, but use the worksheet with the tab labeled sphere.
Appendix 5: LabViewä program for recording temperatures
The front panel of the LabView virtual instrument (data acquisition program) for this experiment is shown below: