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Laplace heat transfer equations for thermocouple

See the following attachment and answer questions related to Laplace equation for temperature transfer, transient response, and graph plotting. All questions outline in attachments for better view of figure and equations. Text is mentioned below.

Questions 4

A thermocouple is a device commonly used to measure the temperature in, for example, boilers and engines. It has the advantage of directly giving out a voltage that is approximately proportional to the measured temperature. The thermo-voltage is, however, only a few millivolts.

In the block diagram of FIGURE 4, the output of the thermocouple, vT(t), is fed into an amplifier to give a sufficiently large signal to drive a signal processing stage. The signal processing stage includes an analogue-to-digital converter (ADC) to give an output to drive a digital display. A low-pass filter (LPF) is placed between the amplifier and the signal processing stage to remove noise that may be picked up from adjacent 'noisy' equipment and from 'mains' interference.

[FIG.4] (See Attachment! img001.jpg)

The thermocouple should ideally have a step change output to a step change in temperature. In practice this is not the case because the mass of the device introduces a thermal delay as it warms up or cools down to a new temperature. It can be shown that the temperature of the thermocouple is related to a step change in temperature by the equation:

[Eq.1] (See Attachment! img001.jpg)

The time constant depends upon the mass (in) and surface area (A) of the thermocouple, the thermal specific heat (c) of the material it is made from and also the heat transfer coefficient (h) between the thermocouple and its surroundings. Thus:

[Eq.] (See Attachment! img002.jpg)

4. (a) In FIGURE 5(a) the thermocouple is shown with an initial temperature the same as that for the ambient air, TA. It is then immersed into a tub of hot water at a constant temperature of Tw. Thus the applied step change in temperature is Del-T = Tw — TA. (See Attachment! img002.jpg)

(i) Write down the Laplace equation for equation (1).
(ii) Hence obtain the equation of the transient response of the thermocouple.
(iii) Plot the transient response using the data in TABLE B over a time period of about 4t.

[TABLE B] (See Attachment! img002.jpg)


Solution Preview

This problem is about Laplace transformation of the ordinary differential equation (ODE). So we will need to first transform given ODE into Laplace format and then take inverse Laplace to get the solution. The solution is provided in the attached docx file named "Solution_Laplace heat transfer_thermocouple_552311.docx". Please find the attachment for better view of equations and graph.

Ordinary Differential Equation (ODE) for change of temp in thermocouple with respect to time is given as,

τ∙d/dt ∆T(t)+ ∆T(t)= ∆T_step------------------------(1)
Taking Laplace transform of (1), we get,

L[τ∙d/dt ∆T(t) ]+ ...

Solution Summary

Solution for this problem can be found by Laplace transformation of the ordinary differential equation (ODE) which explined here step wise including how transform given ODE into Laplace format and then take inverse Laplace to obtained the required form.