Page 160 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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Note that the temperature changes from T o to T(t) as the time changes from 0 to t.
Integration of the above equation results in a transient temperature distribution as follows:
TRANSIENT HEAT CONDUCTION ANALYSIS

T − T a hAt
ln =− (6.4)
T o − T a ρc p V
or
" hA #
T − T a − ρc p V t
= e (6.5)
T o − T a
The quantity ρC p V/hA is referred to as the time constant of the system because it
has the dimensions of time. When t = ρC p V/hA, it can be observed that the temperature
difference (T (t) − T a ) has a value of 36.78% of the initial temperature difference (T o − T a ).
The lumped heat capacity analysis gives results within an accuracy of 5% when
h(V/A)
< 0.1 (6.6)
k s
where k s is the thermal conductivity of the solid. It should be observed that (V/A) represents
a characteristic dimension of the body. The above non-dimensional parameter can thus be
rewritten as hL/k s , which is known as the Biot number. The Biot number represents a ratio
between conduction resistance within the body to convection resistance at the surface of the
hot body (Readers should consult Chapter 1 for the meaning of conduction and convection
resistance).
Owing to the variability of the convection heat transfer coefficient, which can often vary
as much as 25% in many heat transfer problems, a lumped system analysis is often consid-
ered as a realistic approximation even if the Biot number is slightly above 0.1. However, for
higher Biot numbers, this method is certainly not valid. In such situations, numerical meth-
ods such as the finite element method are ideal in obtaining solutions with better accuracy.

6.3 Numerical Solution

Heat conduction solutions for many geometric shapes of practical interest cannot be found
using the charts available for regular geometries (Holman 1989). Because of the time-
dependent boundary, or interface conditions, prevalent in many transient heat conduction
problems, analytical or lumped solutions are also difficult to obtain. In such complex
situations, it is essential to develop approximate time-stepping procedures to determine the
transient temperature distribution.

6.3.1 Transient governing equations and boundary and initial
conditions

The transient heat conduction equation for a stationary medium is given by (Chapter 1)
∂ ∂T ∂ ∂T ∂ ∂T ∂T
k x (T ) + k y (T ) + k z (T ) + G = ρc p (6.7)
∂x ∂x ∂y ∂y ∂z ∂z ∂t