Analytic conduction solutions

By Greg Walker1, James Vere Beck2

1. Vanderbilt University 2. Michigan State University

High-precision analytic conduction in parallelepipeds using Green's functions

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Version 1.1.3w - published on 16 Mar 2015

doi:10.4231/D3FT8DK7F cite this

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Prepared for
Sandia National Laboratories
Albuquerque, NM
Purchase Order No. 30629

James V. Beck
Beck Engineering Consultants Company
1935 Danbury W.
Okemos, MI 48864-1873

Tel. Number: 517-349-6688

June 15, 2002


Several engineers and mathematicians participated in the development of V2000A and subroutine COND3D by Beck Engineering Consultants Co. Most of the coding for this program was done under the 1999-2000 contract from Sandia National Laboratories that was administered by Dr. Kevin Dowding of Sandia. The personnel are listed in alphabetical order with a partial description of the individual contributions.

Donald Amos, (Retired applied mathematician, Sandia Nat. Labs.)
Numerical and analytical integration. Theoretical support.

James V. Beck, (Prof. Emeritus Mech. Eng., MSU; President, Beck Engineering
Consultants Co.) Overall direction and coordination of the project.

Ali Haji-Sheikh, (Prof. Mech. Eng. ,University of Texas Arlington) Develop methods of calculating eigenvalues and writing the associated code. Developing Green=s functions for two-layer composite body.

Robert McMasters IV, (Adjunct Prof. Mech. Eng., MSU) wrote the codes.

David Yen, (Prof. Emeritus Mathematics, MSU) Derivation of many small time transient Green=s functions, derivation of steady state multidimensional Green=s functions.

The program solves one-, two- and three-dimensional transient heat conduction problems in the Cartesian coordinate system. The solutions are generally accurate to about 8 or more significant figures. More precisely the errors in a given quantity are usually less than 1 part in 1010 of the maximum value of that quantity in the body at that time for a single nonhomogeneous term. This is illustrated in more detail later. It suffices at this point to say that the solutions can be given as the summation of up to eight components, one for each of the nonhomogeneous boundary conditions, one for the initial temperature and one for the volume energy generation term. In certain abnormal cases it is possible to have positive and negative values of these components resulting in small differences of large numbers. The final temperatures and heat fluxes coming from the superposition of the components may not have the accuracy indicated above but the components before the summation (compared to the maximum value over the spatial domain) should.
The units are any consistent set, such as the SI units of Kelvin (but many time Celsius is used instead), meters, seconds, watts, joules and grams.
There are two ways to use the program. One is to use V2000A itself and the other is to use the basic part of it, COND3D, as a subroutine. V2000A uses an input file for the data and with COND3D, the data is entered through the user’s program. In any case, the input must be introduced. In most of this report it is assumed that the program is accessed using V2000A and the form of the data input for it is described below. The input statement for COND3D is now briefly described.


The call statement to call the subroutine is


There are a number of input quantities to describe the geometry and boundary conditions (BDY and XL); location (X) and time (TIME) for the calculated temperature; thermal properties (COND and C); initial temperature (TZERO); and volumetric energy generation (VGEN). The remaining symbols, DESCRIP to Q, relate to the output given by the subroutine. The inputs are now described in greater detail.


!BDY(IP,IB) ‑ IP = 1, 2, . . ., 6; IB = 1, 2, 3 (6 rows and 3 columns)

First column: indices indicating the boundary condition of the 1st, 2nd or 3rd kinds:
1 for temperature condition,
2 for heat flux condition,
3 for convection boundary condition
Second column: value of boundary condition driving term:
If 1 in 1stcolumn, given T at the boundary,
If 2 in 1st column, given q at the boundary,
If 3 in 1st column, ambient temperature T given
Third column: heat transfer coefficients if boundary condition of the 3rd kind (3 in 1st column)
If boundary condition of the 1st or 2nd kinds, enter 0.0 but not used.

Rows of BDY(IP,IB) are for:

Row 1: x = 0 surface
Row 2: x = L surface
Row 3: y = 0 surface
Row 4: y = W surface
Row 5: z = 0 surface
Row 6: z = H surface

!TIME ‑ time for calculating t and q=s, (sec or any set of consistent units)

!X(IX), IX = 1, 2, 3 ‑ location x, y and z for T and q=s, (m or any set of consistent units)

!COND ‑ thermal conductivity of body (W/mC or any set of consistent units)

!C - volumetric heat capacity (J/m3C)

!XL(IX), IX = 1, 2, 3 - length dimensions: L, W, and H (m)

!TZERO ‑ initial temperature of body (C, or any set of consistent units)

! VGEN ‑ volumetric energy generation rate (W/m3)

! ACC - integer for number of significant figures, used to truncate series.
It must not be outside range of 4 to 10. The recommended value is 10 and need not be changed from case to case.

!Q(10) – dimensionless partition time. The recommended value is 0.05.

T ‑ Temperature at time and position X(IX) (C).

Q(I) ‑ heat fluxes in x, y and z directions for I = 1, 2, 3 respectively

DESCRP() ‑ string array with verbose description of problem solved

Error ‑ integer code to denote type of error which occurred in subroutine

Error codes are:

1) Point requested outside the body

2) Zero heat transfer coefficient not allowed. Use boundary condition of second kind with q =0.

3) ACC outside of range of 4 to 10 not allowed.

Cite this work

Researchers should cite this work as follows:

  • Greg Walker, James Vere Beck (2015), "Analytic conduction solutions," (DOI: 10.4231/D3FT8DK7F).

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