Question

A 4 cm diameter sphere of copper is initially at a temperature of 95 °C. It is placed in a very large water bath at time t- 0. The water bath is initially at a uniform temperature of 25 °C. The heat transfer coefficient is estimated to be 400 watts / (m2 °C). The specific heat of copper is 0.385 Joule (gram °C) and its density is approximately 9 gram per cc. Estimate the time at which the average temperature of the sphere will be 35 °C.

Answer 1

Answer:112.376 s

Explanation:

Given

$T_i=95^{\circ}C$

$T_f=35^{\circ}C$

$T_\infty \left ( ambient\right )=25^{\circ}C$

$h=400 watts/\left ( m^{2}^{\circ}C\right )$

$c=0.385 J/\left ( m^2^{\circ}C\right )$

$\rho =9 gm/cm^{3}$

Using Newton's law of cooling

$\frac{T_i-T_{\infty}}{T-T_{\infty}}$=$e^{\frac{ht}{\rho L_{c}c}}$

$\frac{95-25}{35-25}$=$e^{\frac{400\times 3\times 10^{-4}\times t}{9\times 2\times 0.385}}$

7=$e^{1.7316\times 10^{-2}\times t}$

Taking log both side

t=112.376sec