Non-Steady-State Probe Science

Hukseflux is specialised in design and manufacture of non-steady-state (hot wire, needle) probes for thermal conductivity measurement. Main applications are in soil thermal resistivbity measurments, analysis of foodstuff, powders, slurries etc.

All non-steady-state probes are based on the same phenomenon: that one can determine the thermal conductivity of a medium from the temperature response to heating. After an initial transition period, the temperature rise close to the heater depends only on the thermal conductivity of the surrounding medium, and no longer on heat capacity. Generally, this method avoids the necessity of reaching a real thermal equilibrium with constant temperatures. Non-steady-state techniques are fast and also there is no need for careful sample preparation. Sensors based on this principle are therefore suitable for quick experiments and also for field use.

Some conventional sensor designs have a temperature measurement at a large distance from the heater (typically some centimeters away, sometimes using a physically separated heater and probe). Other designs measure the temperature rise of the heater itself.

The central equation governing these probes is determined by the temperature field around a heating wire that is switched on at t = 0 remaining constant from that moment on:

Below we treat the working principle of TP02 (also that of STP01) and of TP01.

TP01 and STP01 are generally used in long term monitoring as part of meteorological field observations.

TP02 is typically used as a sensor for analysis of soil samples and foodstuff in the laboratory. If necessary, TP02 can also be used in field experiments.

TP02 and STP01 both incorporate a heating wire and a differential temperature sensor. STP in fact has 4 such combinations and can measure at 4 depths. They both can be used for thermal conductivity measurements of the surrounding material.

The main principle is that after an initial transient period, the temperature rise only depends on the heating power, Q,  and the thermal conductivity , l.

U - U₀ = A . ln t  + B

equation 1.1

A = Q / 4. pi . l

equation 1.2 (with pi denoting the constant pi)

And B is a constant depending on the sensor size, the properties of the surrounding material. From these equations l can easily be deducted also without knowing B.

The difference between TP02 and STP01 is that with STP01 the distance heating wire to sensor is larger than with TP02. This means that the transient period will be larger. For more details about this technique please consult the TP02 manual.

TP01 Thermal Properties sensor

As indicated in the introduction, TP01 design is a modification of the well known non-steady state probe.

TP01 uses a new technique which depends heavily on a very sensitive temperature gradient sensor. A differential temperature sensor (2 thermopiles) measures the radial differential temperature around the central heating wire with record breaking sensitivity. This technique is easier to employ than conventional techniques because the interpretation of the signals is very easy.

A thermopile essentially is a number of thermocouples in series. A thermocouple delivers an output signal that is proportional to the differential temperature between the hot joints and the cold joints. Multiple thermocouples in series, a thermopile, will produce a proportionally larger signal. In case of TP01 the hot joints are located near the heating wire (at 1 mm distance, r_h ) and the cold joints are located far away from the heater (at about 5mm, r_c). There are two rows of each 20 thermocouples (copper – constantan) , which results in a sensitivity of about 1.5 mV when the medium at 1 mm from the heater differs 1 degree Celsius from the medium at 5 mm from the heater.

This sensitivity is not equaled by any other sensor that is known to us. It opens the possibility to reduce the sensor dimensions considerably and to use low heater power, which is essential for accurate measurements in humid materials. In humid material it is recommended that the heater power remains low to avoid local transport of moisture by evaporation.

Table 1.1 comparison of TP01 to conventional techniques, showing why TP02 is very suitable for field experiments where the surrounding is often thermally unstable.

With TP01 one measures not the absolute temperature, but the differential temperature at two different radii r_h and r_c.

D T ~ Q / l

equation 1.3

For TP01, the thermopile output U varies linearly with D T,

U =  E _T . D T + U₀

equation 1.4

With U₀ the sensor output at t = 0 and E_T the thermopile sensitivity for thermal gradients. This step requires the assumption that the thermal mass and the conductivity of the sensor are quite low. In this situation only the parameters of the medium play a role. The validity of this assumption is treated in the appendices. It can be used in the thermal conductivity range from 0.3 to 4 W/m.K.

U₀ is caused by a variety of factors: temperature gradients in the medium and offset of the electronics are the most common ones. The assumption is that these offsets do not vary during the experiment.

Also it is assumed that the medium properties do not change. This is the reason why the heater power must be low. In case of high power, especially in moist media, local moisture transport might take place. With the TP01 typical heater power of 0.3 W/m, the temperature rise will not be higher than 1 degree during a typical measurement of 2 minutes. This results in negligible moisture transport.

Finally, there is the implicit assumption that the sensor does not move during the measurement and that the sensor dimensions are stable.

For large t, the integral, hereafter referred to as the function F (a.t) approaches a constant value, so that

D T = (Q / 4 . pi . l ) ln (r_c²/ r_h²) for (4 .a . t) / r_c² >> 1

equation 1.5

and

U - U₀ = E _T . (Q / 4 . p i. l ) ln (r_c²/ r_h²) 

equation 1.6

It should be noted that the while the differential temperature has stabilized, the absolute temperature is still rising with ln(4 . a . t / r²).

We can define a new constant, only depending on sensor geometry and thermopile sensitivity:

E _l = E _T . (1 / 4 . pi ) ln (r_c²/ r_h²)

equation 1.7

The sensitivity and geometry are not exactly equal from one sensor to the other. One can expect E_T to be an individual sensor property.

The pulse response of TP01 scales with Q / l (the power divided by the thermal conductivity) for the amplitude, and with a (the thermal diffusivity) for the time response.

Volumetric heat capacity is determined by:

C_v= l / a 

equation 1.8

The approach for using TP01 as a sensor for thermal diffusivity and volumetric heat capacity has been to measure the signal amplitude U₀ - U₁₈₀ and to establish how much time it takes after switching the heater off, to return to U₀+ 0.37 (U₀ - U₁₈₀). This is equivalent to determining the 63% or 1/e response time t _63%.

Based on the measurement results, we now state an estimated accuracy for the thermal diffusivity measurement with TP01 of +/- 20% and specify C_v as equal to l / a. Combined with the earlier established accuracy of +/- 5% for l, we cover all the observations. In addition we observed that the capability to detect changes in a meaningful way is much better than the accuracy. We specify a resolution of C_v of 10%.

For more details: please consult the TP01 manual.

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