Semi-infinite surface temperature methods

The principle of this method is to measure surface temperature on a test object, which can be considered to be a semi-infinite solid. It means that for rather thick material and short enough times we can assume that the transfer of heat is one-dimensional and the conductive heating does not reach the back surface of the material. It can be assumed that the surface temperature doesn’t change within the period of time.

Figure 5. Boundary temperatures of the semi-infinite models.

First of all, consider the equation for one-dimensional heat flux:

= (2.15)

Write down the boundary conditions for this model:

(0, ) = ( ) = ( );

̇ ( ) = ( ^{( )}) = ( ^{( )}) ; 9 , ) = (∞, ) = .

Now, we can get the solution of the equation for one-dimensional heat flux based on the boundary conditions for the heat flux rate to the semi-infinite probe:

̇ ( ) = ^{( )}

√ + ∫ ^{( ) ( )}

( )

(2.16)

Rewrite this equation for the constant heat flux rate:

̇ = ^√ [ (0, ) − ] (2.17)

(0, ) is surface temperature as a function of time, is an initial temperature.

To recreate a heat flux signals we can use several methods, but the simplest one is to use the analytical solution with each sampled data point (Diller T. E. 1999). Cook and Felderman (Cook W. J. et al. 1966) presented an equation which let us understand this conversion:

( ) =

√ ∆ ∑ (2.18)

Modifications are also available to provide more solution stability (Diller T. E. et al. 1997). More complex techniques include the use of parameter estimation techniques (Walker D. G. et al.

1995) and numerical solutions to account for changes in property values with the changing temperature (George W. K. et al. 1991). Because of the noise amplification inherent in the conversion from temperature to heat flux, analog methods have been developed to convert the temperature signal electronically before digitizing the signal (Schultz D. L. et al. 1973).

There are two big categories for the measurement of the surface temperature and following determination of the heat flux. The first category is a point measurement where thermocouples are used, and the second is an optical methods. Of course, to use any of these methods we need to start test procedure and reduce data as much as possible. Let’s consider all categories in detail.

The thin-film gauge is one way to determine heat flux and relates to the first category – point temperature measurements. Aluminum film is usually used in this technique. There are different ways to attach the film to the substrate, for example, painting and vacuum deposition. The main manufactures of substrate are Pyrex, fused Quarts or MACOR.

Figure 6. Thin-film gauge of RWTH Aachen (Simeonides G. et al. 1993)

This device developed in the RWTH Aachen has an almost ideal output signal. The manufacture reached such data because of high temperature-resistance coefficient. There are some properties of this device:

1) Thickness of the sensor is several nanometers.

2) The response time is very short (no more than 1µs).

3) Sensor is very sensitive even to small particles.

4) Sensor is used in the aerothermodynamic applications, internal combustion engines, gas-turbine engines and etc.

Point temperature measurements are often made with coaxial surface thermocouple. The main principle of the application is similar with the previous device – to determine the surface temperature of a body as we assumed as semi-infinite surface. The design of the device is not so complicated. It has thermocouple material, inside which there are thermocouple wire and insulating layer between them.

Figure 7. Sketch of the coaxial surface thermocouple.

The coaxial thermocouple assembly is completed by attaching thermocouple lead wires to the coaxial thermoelements (Simeonides G. et al. 1993). We need to measure the surface temperature to determine the heat flux by the equation 2.8. It can be used short and long wind tunnels. Medtherm Corp. is one of the main manufacture of the coaxial surface thermocouple.

There are some advantages of this method:

1) Fast response time (approximately 50 µs) 2) Good durability.

3) No calibration is required because of self-generating.

And disadvantages:

1) Weak output signal.

2) Complex data reduction.

One more method of point temperature measurements is null-point calorimeter. It was developed by AEDC. Now, the main manufacture is also Medtherm Corp. And it is used for quite big heat flux (over 1000 kW m–2).

Figure 8. Concept of the null-point calorimeter.

If we look on the centerline of the cylinder cavity, we can see the point (0,b). It is a null point.

So, we assumed that measured temperature development in this point is identical to the surface temperature history on the outside surface of the same thermal mass without cavity (Giilhan A.

2007). Detailed thermal analysis was conducted to be sure that this assumption is right. So, according to the analysis, if the ratio of the hole radius to the axial distance is approximately 1,4, then we can assume that temperature in the null-point is almost similar with surface temperature history. As a result of this assumption, we can determine heat flux ratio by inserting measured temperature in the equation 2.8. Need to say that we use Chromel-Alumel thermocouple to measure the temperature in the null-point. Thermocouples and wires mounted in a cavity behind the cylinder. It is necessary in order to protect them.

Optical methods allow us to obtain visual representation of the distribution of the temperature over the surface to the greater extent than quantitative heat flux value. Due to the calculations of the heat flux is too complicated because of much data can be collected, we can’t determine heat flux with pinpoint accuracy. In spite of this fact, these methods are very popular. Let’s consider some optical methods to have a full picture about semi-infinite surface temperature methods.

One of the optical methods is based on using liquid crystals. The main idea of this method is to record color change of the specially prepared molecules, which change their color depending on the temperature. Usually the range of temperature is not so big and the figures are approximately between 25 ℃ and 45℃. But one of the manufactures of this technology, Hallcrest, achieved expansion of the range. Their devices can work with temperature range from 5 ℃ and 150℃.

They can easily be spray-painted onto a blackened surface for testing. Setting the lighting for reproducible color, temperature calibration, image acquisition, and accurately establishing the starting temperature are crucial steps (Diller T. E. 1999). There is a one problem. In spite of rather low cost of the basic material, equipment for temperature measurement is quite expensive.

There are a lot of companies on the market which produce different equipment such as high-quality video camera, calibration system, software for image processing and other equipment.

The main manufacture in this field is Image Therm Engineering.

As we have already discussed for radiation heat transfer, surface temperature can be closely connected with radiation which is emitted by all surfaces. The advent of high-speed infrared scanning radiometers has made it feasible to record the transient temperature field for determination of the heat flux distribution (Simeonides G. et al. 1993). We also need to find the radiation field in order to have dependence between surface temperature and radiation emission.

The main problem is a cost of equipment. To have camera and other necessary equipment, company or institute should have decent monetary funds.

Thermographic phosphors emit radiation in the visible spectrum when illuminated with ultraviolet light (Diller T. E. 1999). There is the dependence between surface temperature and intensity of emissions. But most of technologies in this sphere are under development now. So, there a lot of expensive equipment needed to record the transient optical images. Also, manufacturing companies have some problems with calibration of the devices.