Integral form of the cylindrical perfect conductors solution for the dual-probe heat-pulse method

The dual-probe heat-pulse (DPHP) method is useful for measuring soil thermal properties. Measurements are made with a sensor that has two parallel rods: one for introducing a pulse of heat and one for measuring change in temperature. Thermal properties are obtained by fitting a solution to the temperature data. The cylindrical perfect conductors solution of Knight et al. (https://doi.org/10.2136/vzj2011.0112) allows for accurate determination of those properties because it accounts for the finite radius and finite heat capacity of the rods. In this note, we show how their solution can be written in integral form. Specifically, we rearrange their Laplace-domain solution and perform contour integration in the complex plane to express their solution as the integral of a real function over the positive real axis. This form of their solution is easier to evaluate and thus improves its accessibility for routine use in determining thermal properties with the DPHP method.