1. Technique for Heat Flux Determination
John Heim
Vanderbilt University

2. Motivation Accurate characterization of heating loads is crucial.
Heat flux can provide meaningful information concerning the environment that a full-scale structure will experience.
Temperature does not typically scale well.
In controlled tunnel test predictions of a heat flux on a test article can provide meaningful information concerning the environment that a full-scale structure will experience, whereas temperature measurements typically do not scale well from tunnel to flight conditions.
Direct measurements of heat flux in not trivial because it is difficult to measure.

3. Standard Techniques Calorimeter Heat Flux Gauge
Uses thermopiles to measure temperature differences which can be converted to heat fluxes.
Lagging response times, flow disturbance, and Calibration
Calorimeter
Lagging response times, flow disturbance.
Complicated because heat load is not uniform across a surface due to lateral heat conduction.

4. Standard Techniques Thermochromatic Liquid Crystals Inverse Method
Layers of crystals in thin films aligning based on temperature gradients.
Calibration, difficult to apply, and difficult to measure accurately.
Inverse Method
Appropriate data reduction is a balancing act between introducing smoothing bias and amplification of noise
Solutions may contain unacceptable errors.
Say Generally about the inverse method

5. New Heat Flux Determination Approach
Measure a different quantity
HEATING RATE- the time rate of change of temperature for a given location and time.
Data reduction is inherently more stable.
Inverse method techniques
We propose that by measuring the heating rate it will be possible to calculate the Heat Flux more accurately because the data will be more stable.
Thermographic Phosphors will be used to determine the heating rate and inverse techniques will be used to calculate the heat flux.

6. New Problem No method exists to measure heating rate directly.
Describe a technique to measure heating rate.
Demonstrate a stable method for estimating heat flux from a measured heating rate.
Two things I will be talking about from here on out, THE Method we are proposing for measuring a heating rate and Data reduction method to convert that data into a heat flux.

7. Thermographic Phosphors
Rare-earth-doped ceramics that fluoresce when exposed to light.
Intensity, frequency line shift, and decay rate are all temperature dependent.
The strong dependence of the decay rate on temperature will be leveraged to acquire a heating rate.
MEANS that when the phosphors are exposed to light they luminescence which is caused by the absorption of radiation at a particular wavelength. That is followed by nearly immediate reradiation usually at a different wavelength which can be measured.
Because Intensity, frequency line shift, and decay rate are all temperate dependent thermographic phosphors is oftened used for remote temperature sensing.
We will use the fact that many phosphors have a strong dependency of decay rate on temperature to acquire our heating rate.

8. Heating Rate Measure the intensity of phosphor emission.
Temperature is not constant; the normalized intensity is:
The heating rate can now be found:
Measuring the intensity of the phosphor is similar to measuring two temperatures in time and differentiating to obtain a hating rate, but the change in intensity is an exponential function it is not differentiated therefore the measurements noise is not amplified significantly during the conversion to a heating rate.
Because the temperature is not constant we use a first-order Taylor series expansion of the decay rate to introduce the derivative of “”TOW””
dq/ddt is a temperature dependent material property dt/dT is what we are measuring. Any noise found in dt/dT is not amplified it is just multiplied by a factor.
This approach requires three intensity measurements compared to two temperature measurements required for a finite-difference heating rate collection. However, the integration process is more stable then a differencing of noisy data.

9. Phosphor Activity
Excitation
Intensity
Phosphors Decay
Time

10. Theory The infinite series solution for the normalized temperature is:
The solution for the heating rate :
Assume one-dimensional conduction in a slab of length L.
The temperature has been normalized to the initial temperature T0 such that = T –T0
The infinite series solution is found by using a integral transform technique
The spatial and temporal coordinate have been no-dimensionalized ___
The heat flux, Q, at ay-thu = 0 is a continuous function of KS EYE and is presumed known.
The solution for the heating rate is found analytically be differentiation the solution for the normalized temperature.
In the first equation we are at the surface and sie is the time coordinate.
If the heat flux Q( ) is known then equation (1) then the equation is a Volterra equation of the first kind for a known temperature The solution is unstable for discrete temperature measurements with noise.
Equation (2) suggests that the nature of the the solution for heat flux is not as ill-conditioned because it is a Volterra equation of the second kind.

11. Data Reduction ProceduresQ q Yn
QYn QY Hd
QHd H Hn
QHn

12. Triangular Heat Flux From Surface Temperature
To compare the different methods for predicting heat flux a known analytic function was chosen as the exact heat flux Q.
Solutions to both the temperature and the heating rate were calculated to provide the measurement data from which the estimates will be derived.
Normally distributed noise was added to the measurements to evaluate how the estimators behave when measurement error exists in both he temperature and heating rate.
Initially the estimates were obtained from the discrete measurements by assuming a piecewise constant heat flux over each time step. This reduces the basis to be unity over the time step and zero everywhere else.
The first test case is the triangular that flux. This is shown in the first figure. BOTH
Without additional noise as seen by the solid line. The noise is added and is show as the line with the dots.
The temperature histories are used to predict heat fluxes by using the equation (1) shown earlier.
This is inherently an unstable process and the estimate from noisy data Qyn in the second figure shows that small errors become amplified in the solution.
The second graph shows the heat flux history estimates derived from the temperature in the first figure.
The errors seen in the solution from exact data arise from the piecewise-constant approximation. Qy

13. Triangular Heat Flux From
Heating Rate
The new approach requires a measured heating rate which in this case is similar to the measured temperature and is shown in the first graph.
The exact solution to the heating rate equation is accompanied by Hn which contains a normally distributed random noise with a standard deviation of .01. As well as Hd is a central finite difference of the noisy temperature measurement Yn(from the pervious graph).
This approach produces a very noisy signal.
The heat flux estimates are then calculated directly by inverting The second equation we was earlier.
The results are show in the second graph. They show that the Volterra equations of the second kind is is not nearly as sensitive to noise as the Volterra equation of the first kind.
In fact the error in the noisy estimate, Qhn appears to be damped and the solution contains no bias and almost no noise.
Qhd which was produced from a signal whose noise is nearly comparable to the signal, reproduces the original heat flux with reduced errors.

14. Square Heat Flux
The show the same proces used on a square flux. The exact heat flux in not shoun because it is virtually identical to Qhn which is the the estimate from noisy data. Again there are no bias and the measurements errors are damped.

15. Exact Heat Flux vs. Estimated Heat Flux

16. Conclusion
Improved heat flux estimates by reducing instabilities inherent in temperature to heat flux data reduction methods.
By measuring heat rate the integral equation for heat flux becomes a Volterra equation of the second kind.
More stable solutions which can accommodate more noise then a temperature approach.