In a particular case of lumped analysis, the temperature distribu
In a particular case of lumped analysis, the temperature distribution was found to be T = α + βe-γτ, where τ = time (in seconds) and α, β and γ are constants. What is the time constant corresponding to the above relation?
A. γ
B. γ<sup>2</sup>
C. γ<sup>-1</sup>
D. γ<sup>-2</sup>
Please scroll down to see the correct answer and solution guide.
Right Answer is: C
SOLUTION
Concept:
In lumped system analysis the temperature distribution is given by,
\(\frac{{T - {T_\infty }}}{{{T_i} - \;{T_\infty }}} = {e^{ - \frac{{hAt}}{{\rho Vc}}}}\;\) ---(1)
Where, \(T =\) temperature at time‘t’
\(\frac{{hA}}{{\rho Vc}} = \frac{1}{{Time\;constant}}\)
\({T_i},\;{T_\infty }\) = initial temperature and surrounding temperature respectively.
Calculation:
Comparing eq.1 with given relation we get,
\(\frac{{T - \;\alpha }}{\beta } = {e^{ - \gamma \tau }}\)
\(\gamma = \frac{{hA}}{{\rho Vc}}\)
\({\rm{Time\;constant}} = \frac{{\rho Vc}}{{hA}}\)
\(\therefore {\rm{Time\;constant}} = \frac{1}{\gamma }\)