Evaluation of Thermal Contact Resistance Between Two Solid Surfaces Using Photoacoustic Technique

Abstract

Thermal contact resistance is one of the factors that complicates heat transfer between the interfaces of different components. Because heat transfer is influenced by several factors, it is difficult to estimate all the influencing factors for practical applications. In this study, the thermal contact resistance between two solid surfaces was evaluated using the photoacoustic (PA) technique. The samples consisted of a metal foil and metal cylinders, which had various surface roughnesses at the contact faces measured at the same contact pressure. A one-dimensional analytical solution of the phase lag of the PA signal showed a similar trend with the experimental results. However, there were some quantitative differences between the two groups. By comparing the numerical calculation results for the developed heat conduction model of a cylindrical coordinate system with the experimental results, we found that the in-plane distribution of thermal contact resistance at the contact face contributed to the deviation between the experimental results and the numerical results. The thermal contact resistance was evaluated by considering the in-plane distribution of the thermal contact resistance, and reasonable values of thermal contact resistance were obtained.

Appendix A

A cylindrical coordinate system, as shown in Fig. 3, is introduced, and a numerical calculation is performed using the axisymmetric thermal conduction equation:

$$\rho C_{p} \frac{\partial T}{\partial t} = \frac{\partial }{\partial x}\left( {k\frac{\partial T}{\partial x}} \right) + \frac{1}{r}\frac{\partial }{\partial r}\left( {rk\frac{\partial T}{\partial r}} \right) + S_{C} .$$

(A1)

The equation for the thermal flux that passes through the contact surface is expressed as follows:

$$q = \frac{{2\left( {\frac{{k_{1} }}{{\Delta x_{1} }}} \right)\left( {\frac{{k_{2} }}{{\Delta x_{2} }}} \right)\left( {T_{1} - T_{2} } \right)}}{{2R\left( {\frac{{k_{1} }}{{\Delta x_{1} }}} \right)\left( {\frac{{k_{2} }}{{\Delta x_{2} }}} \right) + \left( {\frac{{k_{1} }}{{\Delta x_{1} }}} \right) + \left( {\frac{{k_{2} }}{{\Delta x_{2} }}} \right)}}.$$

(A2)

When Eq. A2 is substituted into the numerical calculation equation, Eq. A3 is obtained, differentiated, and transformed. Figure 14 shows the arrangement of the grids containing the boundary surface.

$$A_{S,ij} T_{ij - 1}^{n + 1} + A_{W,ij} T_{i - 1j}^{n + 1} + A_{p,ij} T_{ij}^{n + 1} + A_{E,ij} T_{i + 1j}^{n + 1} + A_{N,ij} T_{ij + 1}^{n + 1} = B_{ij} .$$

(A3)

Fig. 14

Grid arrangement and thermal flux between grids close to the boundary surface

The terms in Eq. A3 are defined as follows:

$$A_{w,ij} = q_{i - 1,i} \Delta r_{j} r_{j} = - \frac{{2\left( {\frac{{k_{i - 1j} }}{{\Delta x_{i - 1} }}} \right)\left( {\frac{{k_{ij} }}{{\Delta x_{i} }}} \right)}}{{\left( {\frac{{k_{i - 1j} }}{{\Delta x_{i - 1} }}} \right) + \left( {\frac{{k_{ij} }}{{\Delta x_{i} }}} \right)}}\Delta r_{j} r_{j} ,$$

(A4)

$$A_{E,ij} = q_{i,i + 1} \Delta r_{j} r_{j} = - \frac{{2\left( {\frac{{k_{ij} }}{{\Delta x_{i} }}} \right)\left( {\frac{{k_{i + 1j} }}{{\Delta x_{i + 1} }}} \right)}}{{2R\left( {\frac{{k_{i + 1j} k_{ij} }}{{\Delta x_{i + 1} \Delta x_{i} }}} \right) + \left( {\frac{{k_{ij} }}{{\Delta x_{i} }}} \right) + \left( {\frac{{k_{i + 1j} }}{{\Delta x_{i + 1} }}} \right)}}\Delta r_{j} r_{j} ,$$

(A5)

$$A_{S,ij} = \left( {r_{j} - \frac{1}{2}\Delta r_{j} } \right)q_{j - 1,j} \Delta x = \left( {r_{j} - \frac{1}{2}\Delta r_{j} } \right)\frac{{2\left( {\frac{{k_{ij - 1} }}{{\Delta r_{j - 1} }}} \right)\left( {\frac{{k_{ij} }}{{\Delta r_{j} }}} \right)}}{{\left( {\frac{{k_{ij - 1} }}{{\Delta r_{j - 1} }}} \right) + \left( {\frac{{k_{ij} }}{{\Delta r_{j} }}} \right)}}\Delta x,$$

(A6)

$$A_{N,ij} = - \left( {r_{j} + \frac{1}{2}\Delta r_{j} } \right)q_{j,j + 1} \Delta x = - \left( {r_{j} + \frac{1}{2}\Delta r_{j} } \right)\frac{{2\left( {\frac{{k_{ij} }}{{\Delta r_{j} }}} \right)\left( {\frac{{k_{ij + 1} }}{{\Delta r_{j + 1} }}} \right)}}{{\left( {\frac{{k_{ij} }}{{\Delta r_{j} }}} \right) + \left( {\frac{{k_{ij + 1} }}{{\Delta r_{j + 1} }}} \right)}}\Delta x,$$

(A7)

$$A_{p,ij} = - \left( {A_{E,ij} + A_{W,ij} + A_{S,ij} + A_{N,ij} } \right) + \frac{{\rho C_{p} r_{j} \Delta r_{j} \Delta x_{i} }}{\Delta t},$$

(A8)

$$B_{ij} = r_{j} \Delta r_{j} \Delta x_{i} S_{c,ij} + \frac{{\rho C_{p} r_{j} \Delta r_{j} \Delta x_{i}^{{}} }}{\Delta t}T_{ij}^{n} .$$

(A9)

It is possible to perform the calculation considering the contact thermal resistance by solving the equations above.