A thermocouple junction, which may be approximated as a sphere, is to be used for temperature measurement in a gas stream. The convection coefficient between the junction surface and the gas is known to be h = 400 W/m2 ?K, and the junction thermophysical properties are k = 20 W/m?K, Cp = 400 J/kg?K, and ? = 8500 kg/m3 . Determine the junction diameter needed for the thermocouple to have a time constant of 1 s. If the junction is at 25 oC and is placed in a gas stream that is at 200 oC, how long will it take for the junction to reach 199 oC?

First, we state our assumptions:[tex]1. Temperature \ of \ the \ junction \ is \ uniform \ at \ any \ instant\\2. Radiation \ is \ negligible\\3. Constant \ properties[/tex]

Given [tex]T_i=25\textdegree C, k=20W/m.K\ , C_p=400J/kg.K \ \rho=8500kg/m^3[/tex]

and gas stream[tex]T_-=25\textdegree C , h=400W/m.K[/tex]

We use the time constant to find the diameter of the junction:

[tex]\tau _t=\frac{1}{hA}\rhoVC_p=\frac{1}{h\pi D^2}\times \frac{\rho \pi \D^3}{6}C_p[/tex][tex]=>D=0.706mm[/tex]

Now, check the validity of the lumped system analysis. With [tex]L_c=r_o/3[/tex]

[tex]Bi=\frac{hL_c}{k}=2.35\times 10^-^4\leq 0.1[/tex] #Lumped Analysis is OK

Bi<<0.1, therefore the lumped approximation is excellent. The time required for the junction to reach [tex]T=199\textdegree C[/tex]:

[tex]\frac{T(t)-T_\infty}{T_i-T_\infty}=e^-^b^t\\b=\frac{hA}{\rho VC}\\\\t=\frac{1}{b}In \frac{T_i-T_\infty}{T(t)-T_\infty}\\t=5.2s[/tex]

It takes 5.2seconds for the junction to reach 199 oC