The Lumped Capacitance Method

• Non-dimensionalization of Heat Equation and Initial/Boundary Conditions:
θ T − T
Dimensionless temperature difference:
*
∞
θ ≡ =
θ T − T
Dimensionless coordinate:
Dimensionless time: *
Fo → the Fourier Number
The Biot Number:
( )
θ
*
= f x *
, Fo,
Bi
∂ θ
∂
2 * *
=
∂θ
x*2
∂F o
Bi
x
x * ≡
L
αt
t ≡
L
2
≡
k
hL
solid
≡
Fo
i
i
∞
Initial Condition:
*( *,0) 1
∂θ
*
Boundary Conditions: =0
∂x
θ
x
∂θ
*
∂x
=
*
x * = 0
*
x * = 1
− Bi θ
=
*(1, t
*)