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 T1T1, is suddenly immersed at time t=0t=0 in a volume VfVf of well-stirred fluid of temperature T0T0 in an insulated tank. It is desired to find the thermal diffusivity αx=ksρsC^psαx=ksρsC^ps of the solid by observing the change of the fluid temperature TfTf with time. We use the following dimensionless variables
Dimensionless solid temperature:

Θs(ξ,τ)=T1TsT1T0(1)(1)Θs(ξ,τ)=T1−TsT1−T0

Dimensionless fluid temperature:
Θs(ξ,τ)=T1TsT1T0(2)(2)Θs(ξ,τ)=T1−TsT1−T0

Dimensionless radial coordinate:
ξ=rR(3)(3)ξ=rR

Dimensionless time:
τ=αstR2(4)(4)τ=αstR2
Solution:

For the solid sphere,
Heat transfer equation

Θsτ=1ξ2ξ(ξ2Θsξ)(5)(5)∂Θs∂τ=1ξ2∂∂ξ(ξ2∂Θs∂ξ)

Initial condition
At τ=0τ=0 , Θs=0Θs=0 (6)(6)
Boundary condition
At ξ=1ξ=1 , Θs=ΘfΘs=Θf (7)(7)
At ξ=0ξ=0 , Θs=FiniteΘs=Finite (8)(8)

For the fluid,

dΘfdτ=3BΘsξξ=1(9)(9)dΘfdτ=−3B∂Θs∂ξ|ξ=1

in which
B=ρfC^pfVfρsC^ps
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