The calculation of the conductivity and heat capacity as a function of the temperature for numerical schemes.
When a current passes through a conductor with non-zero finite conductivity, energy is dissipated in the conductor. This energy loss can be used to calculate the temperature rise using the heat capacity based on the temperature at the previous time step. We allow our conductivity to change by updating it to the value corresponding to the calculated temperature at the end of our time step. In the range of our interest (copper at room temperature), this is valid because of the very small changes of the conductivity relative to the values of A(r, t). A further improvement would take into consideration that for most of the temperature range, the conductivity of copper changes inversely as the absolute temperature. Thus, one could try extrapolating the temperature rise over the time step, and set the conductivity to the average temperature. A nonlinear behavior would best be accounted for by using predictor corrector schemes.
However, these minor improvements yield small changes in this particular problem because of the crude assumptions earlier, namely that the radius of the conductor is negligible when compared to its length and that the problem is invariant along the z-axis.
This above is given by the following formulas. Given a small volume element in cylindrical coordinates with cross-sectional areaDF and length l with a current IDF (= J * DF), we can calculate the temperature at every time step. Initially, at time t0, the volume element is considered at some temperature T0. The thermal energy stored in this element can be obtained using the reciprical heat capacity (multiplicative inverse of the heat capacity, since these are the actual tabulated values), see Holiday and Resnick [B.1], and is
(B.1) E0 = 1/(T0 * Rcp0).
The electrical conductivity,s, and reciprical heat capacity , that is (heat capacity)-1, Rcp, change as a function of the temperature (see figures B.1 and B.2), and their values as tabulated in many engineering texts, see Touloukian and Buyco [B.2], Kittel [B.3] and CRS [B.4]. The discreet values are stored in an array and intermediate values can be obtained by interpolation.
The dissipated power per unit length at a time tn+1 is given as
(B.2) Pn+1 = Jn+12 DF / sn .
The increment in energy in a time intervalDt is
(B.3)D En+1 = Pn+1 * Dt = Jn+12 DF l Dt / sn.
The total energy, En+1, at time tn+1 can be expressed as
(B.4) En+1 = En + DE n+1
The temperature, Tn+1, at time tn+1 is obtained from
(B.5) Tn+1 = En+1/ Rcp (Tn)
The conductivity at the next time step is obtained by evaluating the conductivity as a function of the temperature from the tables:
(B.6)sn+1 = s(Tn+1).
Likewise, the heat capacity is given as
(B.7) Rcp n+1 = Rcp(Tn+1).
The conductivity is calculated in this manner in the finite difference and finite element schemes presented. The interested reader is refered to the finite difference program for the program coding.
Figure B.1. Plot of the electrical conductivity of copper
as a function of the temperature.
Figure B.2. Plot of the reciprical heat capacity of copper
as a function of the temperature.
[B.1] Holliday, D. and Resnick, R., Fundamentals of Physics, John Wiley and Sons, NY (1974)
[B.2] Touloukian, Y. S. and Buyco, E. H., Specific Heat, Metallic Elements and Allows, Thermophysical Properties of Matter, Vol. 4.IFI/Plenum, NY-Washington (1970)
[B.3] Kittel, C. Introduction to Solid State Physics, 4th edition, John Wiley and Sons, NY (1971)
[B.4] The Chemical Rubber Company Handbook of Physical Material Properties. Chemical Rubber Company, 27th ed.