Figure $2$: Part $($a$)$

A spherical metal particle of radius $a_p$ and thermal conductivity $k_p$ is immersed in an infinite fluid reservoir

of thermal conductivity $k_f.$ Very far away from the particle, the temperature in the fluid reservoir is $T_{}.$

$($a$)$ A small portion of the particle is irradiated by a laser, resulting in a constant heat generation $H_V$ in a

spherical region of radius $(x_0,y_0,z_0)(H_V=0)(x_0,y_0,z_0)=(0,0,0),a_0=a_p{H}_V{T}_p(r)T_f(r)T_p(r)T_f(r)r=a_p{a}_0$ located $at$ a point $(x_0,y_0,z_0)$ relative $to$ the metal particle's center. There

$isno$ heat generation $(H_V=0)in$ the rest $of$ the metal particle that $is$ not irradiated.

Use the integral form $of$ the energy conservation equation $to$ determine the total rate $of$ heat transferred

$(energ\frac{y}{t}$ime$)$ from the particle $to$ the fluid $at$ steady state. $(Note$: The temperature profiles inside and

outside $of$ the particle cannot $be$ solved for analytically and you $do$ not need them $to$ complete this part.$)$

$(b)$ The laser $is$ recalibrated $so$ that $(x_0,y_0,z_0)=(0,0,0),a_0=a_p,$ and heat generation $H_V$ occurs uniformly

throughout the particle. There $is$ a temperature profile $T_p(r)$ inside the particle and a temperature profile

$T_f(r)in$ the surrounding fluid.

Use the differential form $of$ the interfacial enery conservation equation $to$ determine the four boundary

conditions $on{T}_p(r)$ and $T_f(r)at$ the flui$\frac{d}{p}$article interface $r=a_p.$