Semianalytical Solution for Dual‐Probe Heat‐Pulse Applications that Accounts for Probe Radius and Heat Capacity

Abstract

The dual‐probe heat‐pulse (DPHP) method is useful for measuring soil thermal properties. Measurements are made with a sensor that has two parallel cylindrical probes: one for introducing a pulse of heat into the soil (heater probe) and one for measuring change in temperature (temperature probe). We present a semianalytical solution that accounts for the finite radius and finite heat capacity of the heater and temperature probes. A closed‐form expression for the Laplace transform of the solution is obtained by considering the probes to be cylindrical perfect conductors. The Laplace‐domain solution is inverted numerically. For the case where both probes have the same radius and heat capacity, we show that their finite properties have equal influence on the heat‐pulse signal received by the temperature probe. The finite radius of the probes causes the heat‐pulse signal to arrive earlier in time. This time shift increases in magnitude as the probe radius increases. The effect of the finite heat capacity of the probes depends on the ratio of the heat capacity of the probes (C0) and the heat capacity of the soil (C). Compared with the case where C0/C = 1, the magnitude of the heat‐pulse signal decreases (i.e., smaller change in temperature) and the maximum temperature rise occurs later when C0/C > 1. When C0/C < 1, the magnitude of the signal increases and the maximum temperature rise occurs earlier. The semianalytical solution is appropriate for use in DPHP applications where the ratio of probe radius (a0) and probe spacing (L) satisfies the condition that a0/L ≤ 0.11.

A Dual‐Probe Heat‐Pulse Sensor with Rigid Probes for Improved Soil Water Content Measurement

“…Copyrighted by SSSA sumes that heat is conducted from an infinitely long line heat source into a homogeneous, isotropic medium of infinite extent. In practice, however, the sensor's probes are finite in diameter, finite in length, and have thermal properties that differ from those of the soil (Knight et al, 2012). The probes are typically constructed from stainless steel tubing that houses the active electronic components of the probes: a resistant heater wire for the heater probe and a thermistor or thermocouple for the temperature probe.…”

“…Furthermore, before the work of Knight et al (2012), the influence of the dimensions and thermal properties of the probes of the C-DPHP sensor had not been thoroughly analyzed. Data obtained with C-DPHP sensors show consistent disparities between measured and actual thermal properties.…”

“…However, we attribute the observed bias and offset in volumetric heat capacity and water content mainly to the presence of the probes. Considering that the ILS model does not account for the probes and the sensor body, it cannot fully represent the DPHP sensor because it neglects the contribution of the sensor components to heat transfer in the sensor-soil system (Knight et al, 2012). Knight et al (2012) hypothesized that the major contributing properties for these disparities are the finite radius and finite heat capacity of the probes, such that the probes can be simulated as infinitely long rods with infinite thermal conductivity and finite heat capacity.…”

“…Considering that the ILS model does not account for the probes and the sensor body, it cannot fully represent the DPHP sensor because it neglects the contribution of the sensor components to heat transfer in the sensor-soil system (Knight et al, 2012). Knight et al (2012) hypothesized that the major contributing properties for these disparities are the finite radius and finite heat capacity of the probes, such that the probes can be simulated as infinitely long rods with infinite thermal conductivity and finite heat capacity. They introduced the semi-analytical identical cylindrical perfect conductors (ICPC) solution, proposing it instead of the ILS model for more adequate representation of the DPHP sensor and the sensor-soil system.…”

“…In the first, the optimal probe length was determined by evaluating R-DPHP sensors with different probe lengths. Measurements were also made with C-DPHP sensors, and the results were evaluated using both the ILS and ICPC models to confirm the theoretical findings of Knight et al (2012). The second experiment focused exclusively on evaluation of the ICPC model in conjunction with the R-DPHP sensor that was determined to have probes of optimal length.…”