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)=T1−TsT1−T0(1)(1)Θs(ξ,τ)=T1−TsT1−T0
Dimensionless fluid temperature:
Θs(ξ,τ)=T1−TsT1−T0(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
∂Θs∂τ=1ξ2∂∂ξ(ξ2∂Θs∂ξ)(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=−3B∂Θs∂ξ∣∣∣ξ=1(9)(9)dΘfdτ=−3B∂Θs∂ξ|ξ=1
in which
B=ρfC^pfVfρsC^ps