# Analytical solution to the unsteady one-dimensional conduction problem with two time-varying boundary conditions: Duhamel’s theorem and separation of variables

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## Abstract

An analytical resolution of the time-dependent one-dimensional heat conduction problem with time-dependent boundary conditions using the method of separation of variables and Duhamel’s theorem is presented. The two boundary conditions used are a time-dependent heat flux at one end and a varying temperature at the other end of the one-dimensional domain. It is put forth because the author found that the prescribed resolution method using separation of variables and Duhamel’s theorem presented in heat conduction textbooks is not directly applicable to problems with more than one time-dependent boundary condition. The analytical method presented in this paper makes use of one of the property of the heat conduction equation: the apparent linearity of the solutions. For that reason, in order to solve a problem with two time-dependent boundary conditions, the author first separates the initial problem into two independent but complementary problems, each with only one time-dependent boundary condition. Doing that, both simpler problems can be solved independently using a prescribed method that is known to work and the final solution can be obtained by joining the two independent solutions from the simpler separated problems. Every step of the resolution method is presented in this paper, along with a numerical validation of the final solution of three test case problems.

## Keywords

Heat Conduction Equation Auxiliary Problem Heat Conduction Problem Resolution Method Transient Solution## List of symbols

## Dimensional variables

*C*_{p}Heat capacity (J kg

^{−1}K^{−1})*k*Thermal conductivity (W m

^{−1}K^{−1})*L*Length of the solid rod (m)

*m*Summation index

*n*Summation index

*p*Summation index

*q*″Heat flux (W m

^{−2})*t*Time (s)

*T*Temperature (K)

*x*Coordinate (m)

## Greek symbols

*α*Thermal diffusivity of the solid (m

^{2}s^{−1})*β*Eigenvalues from the method of separation of variables

- Δ
*t* Numerical time step (s)

- Δ
*x* Distance between nodes (m)

*θ*Modified temperature variable (K)

*ρ*Density (kg m

^{−3})*τ*Extra time variable in Duhamel’s theorem (s)

- T
Period (s)

*ϕ*Temperature variable in auxiliary problem (K)

## Subscripts

- 1
Complementary problem 1

- 2
Complementary problem 2

*c*Heat flux coefficient in the cosine summation

*e*Eastern boundary of the control volume

*E*Eastern node

*i*Initial

*L*Position at

*x*=*L**m*Summation index

*n*Summation index

*N*Node N

*p*Summation index

*P*Present node

*s*Surface

*s*Stationary solution

*s*Heat flux coefficient in the sine summation

*t*Transient solution

*w*Western boundary of the control volume

*W*Western node

## Superscript

- 0
Previous time step

## Notes

### Acknowledgments

The author is grateful to the Natural Science and Engineering Research Council of Canada and to Dalhousie University for their financial support.

## References

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