Inverse heat conduction methods can be used to determine heat flux and temperatures on an inaccessible surface of a wall by measuring the temperature on an accessible boundary (TS method, Figure 1). The noise present in any measure of temperature, however, can cause instabilities in the predicted heat fluxes. It has been shown that the prediction can be greatly improved by measuring temperature at two locations. Altering the wall to include an interior thermocouple cannot be performed in many applications, and installation of an interior thermocouple can result in material inhomogeneities that change the heat flow through the wall. By numerical experiments and a sensitivity analysis it can be shown that incorporating a measurement of the heat flux at the accessible boundary (TS/HFS method, Figure 2) can be used to improve the calculation.


The objective of this work was to develop a method by which stable predictions of the heat transfer on an inaccessible boundary could be obtained without altering the thermal boundary condition that would have existed were a sensor not present. A sensor that measures both heat transfer and temperature on an accessible boundary with minimal impact on the boundary condition is described and results of experiments with this sensor are presented.

Figure 1: Single sensor method

 

Figure 2: Heat flux and temperature method.

A schematic diagram of the sensor is shown on Figure 3. A small resistance heater (“Active heater” in Figure 3) is attached to the accessible boundary of a wall, and its temperature is controlled by an electronic feedback loop to track the temperature of a passive temperature sensor (“Passive sensor” in Figure 3) mounted on the same boundary a short distance away. The passive temperature sensor is very thin so the wall boundary condition is only minimally altered. The active heater is cooled by an efficient and substantial cooling mechanism from behind (for example, circulating chilled water or an impinging air or water jet). By measuring the heat added to the active heater ( in Figure 3), we can determine the heat flux through the wall as shown by the simple analysis of the heat flux sensor performance given below. The passive temperature sensor is cooled through convection by a heat transfer coefficient hs. It can easily be shown that the response of the sensor is given by


Examination of this equation indicates the following properties of the heat flux sensor:
1). If the heat transfer coefficients (hs and hh) and the environment and coolant temperatures (T∞,s and T∞,h) are constant and hh>hs, then the heat supplied to the heater is linearly proportional to the heat transfer through the substrate. By measuring , the heat supplied to the heater (qh''), the heat transfer through the wall (qx'') can be determined.
2). If hh is larger than hs, the heat flux sensor acts to amplify the heat transfer through the wall by an amount without disturbing the wall temperature.
3 ). If hs and hh are constant, then drifts in T∞,s and T∞,h simply result in an offset to . The heat flux sensor can be operated with different T∞,s and T∞,h by using the above equation to correct the output.

A schematic of the circuit used to keep the heater the same temperature as the passive sensor is shown in Figure 4.

Figure 3: Sensor schematic.

 

Figure 4: Circuit schematic.

The concept has been tested using the setup shown in Figure 5. A 5.2 cm x 7.6 cm copper plate 10 mm thick was used for the wall, and heated using a Minco foil heater connected to a variable voltage source. A pair of RTDs for the active and passive sensor were specially made for this application. The RTDs consist of a 2.54 micron thick etched platinum foil sandwiched between two 0.0254 mm thick Kapton films. The passive and active sensor resistances were 988.9 ohms and 98.6 ohms at 20 °C. The RTDs had a nominal TCR of 0.0035/°C with dimension 10 mm x 5 mm. The active sensor was cooled from the back using an impinging jet of water at a constant flow rate (160 ml/min) and temperature (30 °C) to provide a high hh, and a fan was used to provide a constant hs to cool the plate. The air and water inlet temperatures (T∞,=25 °C and T∞,h=30 °C) were measured along with the voltage across the heater (Vout) as was increased to verify that they remained constant. The response is shown on Figure 6, and indicates a linear variation as expected. The sensitivity is seen to be quite high.

Figure 5: Schematic of test rig.

 

Figure 6: Sensor response.

This sensor was then used to estimate the transient heat flux into the inaccessible side of the 10 mm thick copper slab for the case where the input heat flux was suddenly decreased from 6500 W/m2 to 0 W/m2. The heat flux sensor provided heat flux and temperature data on the accessible side of the copper slab, and the inverse heat conduction model and software were used to estimate the heat flux variation with time on the inaccessible side. These results are presented in Figure 7. The estimated heat flux agrees very well with the actual heat flux variation.

Figure 7: Estimated vs. actual heat flux.

Papers describing these results in more detail are given below:

Saidi, A. and Kim, J. (2003), “Heat flux sensor with minimal impact on boundary conditions”, Procedings of the 2003 ASME National Heat Transfer Conference, Las Vegas, NM.

Saidi, A. and Kim, J., “Heat flux sensor with minimal impact on boundary conditions”, Experimental Thermal and Fluid Science, Vol. 28/8 pp 903-908, 2004.

This work was performed in conjunction with Arash Saidi when he was at ATEC, Inc., and was funded by Kirtland Air Force Base.