Page 37 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

SOME BASIC DISCRETE SYSTEMS
known). The solution of the remaining equations will give the temperature distribution
for both the fluids, that is, T 2 ,T 3 ,T 4 ,T 5 ,T 6 ,T 7 ,T 8 and T 9 for the incoming hot fluid and
T 11 ,T 12 ,T 13 ,T 14 ,T 15 ,T 16 ,T 17 and T 18 for the coolant. 29
With the calculated exit temperatures T 9 and T 18 , the effectiveness of the heat exchanger
can be calculated.
2.3 Transient Heat Transfer Problem (Propagation
Problem)

In a transient, or propagation, problem, the response of a system changes with time. The
same methodology used in the analysis of a steady state problem is employed here, but
the temperature and element equilibrium relations depend on time. The objective of the
transient analysis is to calculate the temperatures with respect to time.
Figure 2.8 shows an idealized case of a heat treatment chamber. A metallic part is
heated to an initial temperature, T p , and is placed in a heat treatment chamber in which an
inert gas such as nitrogen is present. Heat is transferred from the metallic part to the gas
by convection. The gas in turn loses heat to the enclosure wall by convection. The wall
also receives heat by radiation from the metallic part directly as the gas is assumed to be
transparent. The wall loses heat to the atmosphere by radiation and convection.
The unknown variables in the present analysis are the temperature of the metallic part
T p , the temperature of the gas T g , and the temperature of the enclosure wall T w .
For simplicity, we are using a lumped-parameter approach, that is, the temperature
variation within the metal, gas and wall is ignored.
Let c p , c g and c w be the heat capacities of the metallic part, the gas and the wall
respectively. The heat balance equations with respect to time can be derived as follows:
For the metallic part,
dT p 4 4
c p =− hA p (T p − T g ) + p σA p (T − T ) (2.39)
p
w
dt
For the gaseous part,
dT g
c g = h p A p (T p − T g ) − h g A g (T g − T w ) (2.40)
dt

Gas
Convection + Convection +
Wall, T w
radiation radiation
Gas, T g
Metallic part Metallic Radiation Convection +
T p part Wall radiation Atmosphere

Figure 2.8 Heat treatment chamber and associated heat transfer processes