Question

In an experiment, the temperature of a hot gas stream is to be measured by a thermocouple with a spherical junction. Due to the nature of this experiment, the response time of the thermocouple to register 42 percent of the initial temperature difference must be within 5 s. The properties of the thermocouple junction are k = 35 W/m•K, rho = 8500 kg/m3, and cp = 320 J/kg•K. If the heat transfer coefficient between the thermocouple junction and the gas is 250 W/m2•K, determine the diameter of the junction.

Answer 1

The diameter of the junction is 5.06 mm

Step-by-step explanation:

Given that:

ρ = 8500 kg/m³

k = 35 W/m.k

`c_p=320J/kg.K`

the heat transfer coefficient between the thermocouple junction and the gas =
`hA_s=250W/m^2.k`

The initial temperature difference (t₀) = 42% = 0.42

t = 5 s

`(T(t)-T_\alpha )/(T_i-T_\alpha ) =1-t_0 = 1-0.42=0.58`

`t=-(\rho c_pL_c)/(hA_s)ln(T(t)-T_\alpha)/(T_i-T_\alpha)`

Substituting values:

`5=-(8500* 320*L_c)/(250)ln(0.58)\\L_c=8.436*10^{-4`

`L_c=(V)/(A)=((4\pi r_o^3)/(3) )/(4\pi r_o^2) = (r_o)/(3)=(D)/(6)\\ Therefore, D=6L_c\\D=6*8.436*10^(-4)=5.06mm`

The diameter of the junction is 5.06 mm