Thermocouples are often used to measure the temperature of a gas traveling through a pipe. Although an estimate of the gas temperature can be gained, the temperature measured is usually off by some magnitude of error. The following example involving a thermocouple demonstrates one kind of error that may be made in thermometry. A simple analysis similar to the one shown below can be used to estimate the magnitude of the errors involved.
The thermocouple well wall of thickness B is in contact with the gas stream on one side only, and the tube thickness is small compared with the diameter. Hence the temperature distribution along this wall will be about the same as that along a bar of thickness of 2B, in contact with the gas stream on both sides. This leads to the use of equation 9.7-13. This equation was derived for heat conduction in a cooling fin (http://www.owlnet.rice.edu/~ceng402/bonus98/tray.html)
> restart;
> T1:=500*F; Tw:=350*F; h:=120*Btu/hr/ft^2/F; k:=60*Btu/hr/ft/F; B:=.08*inches; L:=.2*ft;
temperature indicated by thermocouple =
wall temperature =
heat transfer coefficient =
thermal conductivity =
thickness of well wall =
length of well =
Conversions
> inches:=ft/12;
Equation 9.7-13
> eq1:=Theta=cosh(N*(1-zeta))/cosh(N);
Expressions for the dimensionless parameters
> Theta:=(T1-Ta)/(Tw-Ta);zeta:=z/L;N:=sqrt(h*L^2/(k*B));
dimensionless distance =
dimensionless heat transfer coefficient =
> z:=L; Since we are looking at the temperature at the end of the well
> solve(eq1,Ta); Solving for the temperature of the gas stream traveling through pipe
>
From this result, we can see that the thermocouple reading is 10oF too low. The error would have been slightly greater if the temperature variation in the x-direction had been taken into account.