# Validation of a heat conduction model for finite domain, non-uniformly heated, laminate bodies

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s00231-015-1647-7

- Cite this article as:
- Desgrosseilliers, L., Kabbara, M., Groulx, D. et al. Heat Mass Transfer (2016) 52: 1283. doi:10.1007/s00231-015-1647-7

## Abstract

Infrared thermographic validation is shown for a closed-form analytical heat conduction model for non-uniformly heated, laminate bodies with an insulated domain boundary. Experiments were conducted by applying power to rectangular electric heaters and cooled by natural convection in air, but also apply to constant-temperature heat sources and forced convection. The model accurately represents two-dimensional laminate heat conduction behaviour giving rise to heat spreading using one-dimensional equations for the temperature distributions and heat transfer rates under steady-state and pseudo-steady-state conditions. Validation of the model with an insulated boundary (complementing previous studies with an infinite boundary) provides useful predictions of heat spreading performance and simplified temperature uniformity calculations (useful in log-mean temperature difference style heat exchanger calculations) for real laminate systems such as found in electronics heat sinks, multi-ply stovetop cookware and interface materials for supercooled salt hydrates. Computational determinations of implicit insulated boundary condition locations in measured data, required to assess model equation validation, were also demonstrated. Excellent goodness of fit was observed (both root-mean-square error and *R*^{2} values), in all cases except when the uncertainty of low temperatures measured via infrared thermography hindered the statistical significance of the model fit. The experimental validation in all other cases supports use of the model equations in design calculations and heat exchange simulations.

### List of symbols

### Dimensional variables

*d*Fin region insulated boundary condition location (m)

*d**Estimated fin region insulated boundary condition location (m)

*h*Convection heat transfer coefficient (W m

^{−2}K^{−1})*h*_{1}Left-profile heated region convection heat transfer coefficient (W m

^{−2}K^{−1})*h*_{2}Left-profile fin region convection heat transfer coefficient (W m

^{−2}K^{−1})- \(h_{1}^{'}\)
Right-profile heated region convection heat transfer coefficient (W m

^{−2}K^{−1})- \(h_{2}^{'}\)
Right-profile fin region convection heat transfer coefficient (W m

^{−2}K^{−1})*k*Thermal conductivity (W m

^{−1}K^{−1})*L*Heated region length (m)

*L*_{1}Left-profile heated region length (m)

*L*_{2}Right-profile heated region length (m)

*LMTD*Log-mean temperature difference (K)

- \(Q_{fin}^{'}\)
Linear heat flux entering the fin region (W m

^{−1})- \(q_{0}^{{\prime \prime }}\)
Applied finite heat flux (W m

^{−2})*R*Thermal resistance above the highly conductive metal core (m

^{2}K W^{−1})*RMS*Root-mean-square error

*R*^{2}Quality of fit regression parameter

*t*Layer thickness (m)

*T*Temperature of the highly conductive metal core (K)

*T*_{i}Applied temperature heat source (K)

*T*_{inf}Free stream temperature (K)

*T*_{o}Boundary temperature at

*x*=*L*(K)*x*Axial position (m)

*z*Vertical axis position (m)

### Greek symbols

*α*Heated region constant (m

^{−1})*β*Heated region particular solution (K)

*∆T*Temperature uncertainty 95 % confidence limit (K)

*γ*Fin region constant (m

^{−1})*σ*Standard deviation

### Subscripts

*1*High conductivity metal core

*2*Top thermally resistive layer

*3*Bottom thermally resistive layer

*f*Fin region only

*h*Heated region only

*meas*Based on measurement data

*model*Based on two-region fin model results