Laplace transform解传热偏微分方程

Cooling of a Sphere in Contact with a Well-Stirred Fluid

A homogeneous solid sphere of radius R, initially a t a uniform temperature $T_1$, is suddenly immersed at time $t=0$ in a volume $V_f$ of well-stirred fluid of temperature $T_0$ in an insulated tank. It is desired to find the thermal diffusivity $\alpha_x=\dfrac{k_s}{\rho_s \hat C_{ps}}$ of the solid by observing the change of the fluid temperature $T_f$ with time. We use the following dimensionless variables
Dimensionless solid temperature:

$$\Theta_s(\xi,\tau)=\dfrac{T_1-T_s}{T_1-T_0} \tag{1}$$
Dimensionless fluid temperature:
$$\Theta_s(\xi,\tau)=\dfrac{T_1-T_s}{T_1-T_0} \tag{2}$$
Dimensionless radial coordinate:
$$\xi=\dfrac{r}{R} \tag{3}$$
Dimensionless time:
$$\tau=\dfrac{\alpha_st}{R^2} \tag{4}$$

Solution:

For the solid sphere,
Heat transfer equation

$$\dfrac{\partial \Theta_s}{\partial \tau}=\dfrac{1}{\xi^2} \dfrac{\partial}{\partial \xi} \left(\xi^2 \dfrac{\partial \Theta_s}{\partial \xi}\right) \tag{5}$$
Initial condition
At $\tau=0$, $\Theta_s=0$ $\tag{6}$
Boundary condition
At $\xi=1$, $\Theta_s=\Theta_f$ $\tag{7}$
At $\xi=0$, $\Theta_s=Finite$ $\tag{8}$

For the fluid,

$$\dfrac{d \Theta_f}{d \tau}=-\dfrac{3}{B}\dfrac{\partial \Theta_s}{\partial \xi} \bigg |_{\xi=1} \tag{9}$$
in which

B=ρfC^pfVfρsC^psVs B = ρ f C ^ p f V f ρ s C ^ p s V s