Question

Obtain a relation for the time required for a lumped system to reach the average temperature $\frac{1}{2}\left(T_i+T_{\infty}\right)$, where $T_i$ is the initial temperature and $T_{\infty}$ is the temperature of the environment.

   Obtain a relation for the time required for a lumped system to reach the average temperature $\frac{1}{2}\left(T_i+T_{\infty}\right)$, where $T_i$ is the initial temperature and $T_{\infty}$ is the temperature of the environment.
 
Introduction To Thermodynamics and Heat Transfer
Introduction To Thermodynamics and Heat Transfer
Yunus A. Cengel 1st Edition
Chapter 11, Problem 13 ↓

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The temperature of a lumped system varies with time according to: $$\frac{T(t) - T_{\infty}}{T_i - T_{\infty}} = e^{-bt}$$ where $b = \frac{hA}{\rho V c_p}$ is the time constant, $h$ is the heat transfer coefficient, $A$ is the surface area, $\rho$ is the density,  Show more…

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Obtain a relation for the time required for a lumped system to reach the average temperature $\frac{1}{2}\left(T_i+T_{\infty}\right)$, where $T_i$ is the initial temperature and $T_{\infty}$ is the temperature of the environment.
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