Page 159 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

TRANSIENT HEAT CONDUCTION ANALYSIS
t < 0
T = T o 151

Hot metal body Liquid,
T(t) T < T o
a

Figure 6.1 Lumped heat capacity system. A hot metal body is immersed in a liquid
maintained at a constant temperature

in temperature in such systems varies only with respect to time. It is therefore obvious
that the lumped heat capacity analysis is limited to small-sized bodies and/or high thermal
conductivity materials.
Consider a body at an initial temperature T o , immersed in a liquid maintained at a
constant temperature T a , as shown in Figure 6.1. At any instant in time, the convection
heat loss from the surface of the body is at the expense of the internal energy of the body.
Therefore, the internal energy of the body at any time will be equal to the heat convected
to the surrounding medium, that is,
dT
−ρc p V = hA(T (t) − T a ) (6.1)
dt
where ρ is the density, c p is the specific heat and V is the volume of the hot metal body; A is
the surface area of the body; h is the heat transfer coefficient between the body surface and
the surrounding medium; t is the time and T(t) is the instantaneous temperature of the body.
Equation 6.1 is a first-order differential equation in time, which requires an initial
condition to obtain a solution. As mentioned previously, the initial temperature of the body
at time t = 0, is T o . Applying the variable separation concept to Equation 6.1, we get
dT hA
=− dt (6.2)
T(t) − T a ρc p V
Integrating between temperatures T o and T(t), we obtain

dT hA
T(t) t
=− dt (6.3)
T(t) − T a 0 ρc p V
T o