Temperature Measurements and Gamma Distribution a Perfect Wedding

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Temperature measurements and gamma

distribution : A perfect wedding

Flavio Floriani - October 02, 2014

TEMPERATURE MEASUREMENT AND GAMMA DISTRIBUTION

Let’s introduce a method to estimate the errors created by placing the thermocouples in

temperature measurement. It will be shown how the errors depend on temperature gradient. We will

associate a probability density function (PDF) to the error calculated from experimental data and we

will give a statistical instrument to evaluate the results of temperature test on electrical products.

An overview

Since the 1800s heat propagation has been studied by many physicists; its behavior can be

described by some partial differential equations (PDE) :

The Laplace’s equation if we want to study heat propagation when the heat source is outside the

object and when is not dependent by time (time invariant)

The Poisson equation if we want to study heat propagation when the heat source is also contained in

the object and when it is not dependent by time (time invariant)

The heat equation when we need to study time-dependent phenomena .

We’ll start with a quick introduction to the time-independent phenomena by studying the Laplace

and Poisson equations and then elaborate to understand the technique used to measure the

temperature on a solid object (thermocouple method)

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We will focus on the measurement errors and we investigate them.

Therefore

T is the temperature (scalar field)

f is a function which satisfy the equation [2]

a is the thermal diffusivity

t is the time

Now build a domain where we will solve the Laplace equation on a two dimensional plane:

Simply construct a rectangular shape with a spherical surface on the left end side (imagine that it

could be an object in contact with a heat source on the spherical side), and impose these boundary

conditions:

At each boundary δ we impose Dirichlet conditions; the temperature values have a decreasing linear

trend along the x axis: at the spherical side the temperature has the highest value whereas the

lowest value is at the right side (don’t pay too much attention to the numerical values, just focus on

the qualitative representation given in the picture below in particular to the length of the arrows

which represent the temperature gradient)



Suppose now we want to solve the Poisson equation by also changing the boundary condition; let’s

take the same object, impose the Neumann conditions at the spherical side and also parameterize a

heat source; again don’t focus on the numerical values, they can be arranged for each particular

physics object and situation

Let’s think about the temperature measurement: we want to measure the highest temperature of the

object by placing a thermocouple; the hot junction will be placed on the object and it will be fixed by





And obviously its cumulative density function (CDF) is:

Choosing the parameters k and θ it’s possible to obtain functions like these:

Figure 1 – Low gradient zone





So now suppose that the maximum ΔT measured is the closest value to the ‘true’ temperature. We

can also evaluate the average value of ΔT as usual:

And then take the difference between the maximum ΔT recorded and the average ΔT:

And also calculate the variance of ΔT values:

Now we finally get the values of µ (average error) and s2 which are required to get the values of k

and θ in formulas [5] and [6]. So we can parameterize the gamma function and with a calculus

program like ‘Matlab’ or ‘ Excel’ we can evaluate the probability density function as indicated in

figure 1 or figure 2 (note that k shall be chosen as the nearest integer to the calculated value).

With this data you can check the results of a temperature measurement, compare the recorded value

with the limit and decide if more investigation is needed. An example:

Suppose you measure 75 °C and the given limit for the conformity is 77 °C. If you estimated a

probability density function for that fixing method you can know what the confidence degree of your

reading is and decide if you need further investigation or not.

Over time you can supervise the calculated probability density function by fitted data of new

repeated measurements:



Figure 3 – Fitted data (green cross) of measurements repeated after one year compared

with probability density function (blue line) calculated on first data set

Finally we notice a relationship between the temperature gradient and the temperature

measurement errors:

Further proof of this relationship is given by the same evaluation performed on the ambient

temperature: if you do these measurements on ambient temperature (or on any homogenous

temperature object) you will get both parameters going to zero… and try to take the limits for µ and

s2 go to zero (sure, they really can never be zero due to instrumental errors) and see what happens

to the probability density function…

Now you can stop here or if you want to be more precise (need more math) you should calculate the

convolution between the estimated probability density function and the Gaussian distribution whichdescribes the uncertainty of your measurement system at each measurement point:



Figure 4 – Convolution of two PDFs

As shown in the picture above, the red curve looks like a Gaussian shape and it makes sense due to

central limit theorem; furthermore now you obtain a non-zero probability to do a positive error (to

measure a temperature higher than the ‘true’ temperature), and this also makes sense because the

symmetry of Gaussian function. Notice that the Gaussian curve shall to be centered on the measured

value (because we assume that the measured value is the most probable of a Gaussian curve). You

should make this procedure any time you take a measurement and it appears too complicated to

manage or does not make any sense for the purpose of this article.

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Temperature Measurements and Gamma Distribution a Perfect Wedding