The temperature of a gas stream is to be measured by a thermocouple whose junction can be approximated as a 1.2-mm-diameter sphere. the properties of the junction are k = 35 w/m⋅k, rho = 8500 kg/m3, and cp = 320 j/kg⋅k, and the heat transfer coefficient between the junction and the gas is h = 110 w/m2⋅k. determine how long it will take for the thermocouple to read 98.5 percent of the initial temperature difference

First compute the characteristic length and the Biot number to see if the lumped analysis is applicable

Lc = V/A = (pie*D3/6) / (pie * D2)= 1.2/6 = 0.0012/6= 0.0002m

Bi = hLc/K = (110W/m2.oC)(0.0002)(moC/35W)= 110*0.0002/35 = 0.0006 less than 0.1

Since the Biot number is less than 0.1, we can use the lumped parameter analysis.

In such an analysis, the time to reach a certain temperature is given by

t = -1/bIn(T-Tinfinite/T - Tinfinite)

From the data in the problem we can compute the parameter, b, and then compute the time for the ratio (T – Tinfinite/(Ti – Tinfinite) to reach the desired value.

b = hA/pCpV = h/pCpLc = 110/8500*0.0002 *320*s

b = 110/544s = 0.2022/s

The problem statement is interpreted to read that the measured temperature difference T – Tinfinite has eliminated 98.5% of the transient error in the initial temperature reading Ti – Tinfinite so the value of value of (T – Tinfinite)/(Ti – Tinfinite) to be used in this equation is 0.015

t = -1/bIn(T-Tinfinite/T - Tinfinite)

t = -s/0.1654 (In0.015)

t = (-s*-4.1997)/0.2022

t = 20.77s

It will take the thermocouple 20.77s to reach 98.5% of the initial temperature

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