Module 5: Learning objectives - E-Courses

Heat and Mass Transfer Prof. Pradip Dutta
Indian Institute of Science Bangalore
Note: Once spatial variability of temperature is included, there is existence of
seven different independent variables.
T = T ( x,
t,
Ti
, T∞
, h,
k,
α)
How may the functional dependence be simplified?
• The answer is Non-dimensionalisation. We first need to understand
the physics behind the phenomenon, identify parameters governing the
process, and group them into meaningful non-dimensional numbers.
Non-dimensionalisation of Heat Equation and Initial/Boundary Conditions:
The following dimensionless quantities are defined.
θ T − T∞
Dimensionless temperature difference: θ = =
θi
Ti
− T∞
*
* x
Dimensionless coordinate: x =
L
* αt
Dimensionless time: t = = Fo 2
L
hL
The Biot Number: Bi =
ksolid
The solution for temperature will now be a function of the other non-dimensional quantities
( , , )
*
*
θ = f x Fo Bi
Exact Solution:
2
*
( ζ Fo)
cos(
ζ )
θ ∑ exp −
x
*
= Cn n
n
n 1
∞
=
4sinζ
= ζ tanζ
= Bi
n
Cn n n
2ζ
n + sin(
2ζ
n )
The roots (eigenvalues) of the equation can be obtained from tables given in standard
textbooks.
The One-Term Approximation Fo > 0.
2
Variation of mid-plane ( 0)
temperature with time ( )
* x =
Fo
* T − T∞
2
θ0 = ≈ C1
exp( − ζ 1 Fo)
Ti
− T∞
From tables given in standard textbooks, one can obtain C1 and ζ 1 as a function of Bi.
Variation of temperature with location ( ) and time ( ):
*
x Fo
* *
*
θ = θ 0 = cos( ζ 1x
)
Change in thermal energy storage with time: