- #1

gfd43tg

Gold Member

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## Homework Statement

A hot solid sphere of initial radius ##a## with a uniform initial temperature ##T_{0}## is allowed to

cool under stagnant air at ambient temperature, ##T_{\infty}## . Assume the temperature within

the sphere is uniform throughout the cooling process. Show that under pseudo-steady

state conditions, the temperature of the solid sphere decreases with time according to

[tex]T - T_{\infty} = (T_{0} - T_{\infty}) \hspace{0.05 in} exp \Big( - \frac {3kt}{\rho C_{p} a^2} \Big)[/tex]

where ##k## is the thermal conductivity of the surrounding air and ##\rho## and ##C_{p}## are the

density and specific heat of the solid sphere, respectively.

## Homework Equations

## The Attempt at a Solution

I do a general energy balance on the sphere,

[tex] \frac {dE}{dt} = \dot Q_{in} - \dot Q_{out} + \dot Q_{gen} + \dot W_{s} [/tex]

I assume that there is no heat generation, no shaft work, and that no heat enters the sphere

[tex] \rho V C_{p} \frac {dT}{dt} = - \dot Q_{out} [/tex]

From Newton's Law of cooling,

[tex] \dot Q_{out} = h_{\infty}A (T - T_{\infty}) [/tex]

[tex] \rho V C_{p} \frac {dT}{dt} = -h_{\infty}A (T - T_{\infty}) [/tex]

[tex] \frac {dT}{dt} + \frac {h_{\infty}A}{\rho V C_{p}} (T - T_{\infty}) = 0 [/tex]

Now that the assumption is pseudo steady state, I say ##\frac {dT}{dt} = 0##, so I end up with

[tex] \frac {h_{\infty}A}{\rho V C_{p}} (T - T_{\infty}) = 0 [/tex]

And from here, I have no idea how I will be able to get the expression that I am supposed to derive now that I have no derivative to integrate and use boundary conditions to derive the expression.