Page 171 - Mechanical Engineers' Handbook (Volume 4)

160 Heat-Transfer Fundamentals

of actual the heat-transfer rate from a fin to the heat-transfer rate that would occur if the
entire fin surface could be maintained at a uniform temperature equal to the temperature of
the base of the fin. For this case, Newton’s law of cooling can be written as
q hA (T T )
b
ƒ

ƒ
where A is the total surface area of the fin and T is the temperature of the fin at the base.
b
ƒ
The application of fins for heat removal can be applied to either forced or natural convection
of gases, and while some advantages can be gained in terms of increasing the liquid–solid
or solid–vapor surface area, fins as such are not normally utilized for situations involving
phase change heat transfer, such as boiling or condensation.
1.5 Transient Heat Conduction
Given a solid body, at a uniform temperature, T , immersed in a fluid of different temperature
i
T , the surface of the solid body will be subject to heat losses (or gains) through convection

from the surface to the fluid. In this situation, the heat lost (or gained) at the surface results
from the conduction of heat from inside the body. To determine the significance of these
two heat-transfer modes, a dimensionless parameter referred to as the Biot number is used.
This dimensionless number is defined as Bi hL/k, where L V/A or the ratio of the
volume of the solid to the surface area of the solid, and really represents a comparative
relationship of the importance of convections from the outer surface to the conduction oc-
curring inside. When this value is less than 0.1, the temperature of the solid may be assumed
uniform and dependent on time alone. When this value is greater than 0.1, there is some
spatial temperature variation that will affect the solution procedure.
For the first case, Bi 0.1, an approximation referred to as the lumped heat-capacity
method may be used. In this method, the temperature of the solid is given by
T T exp exp( BiFo)
t
T T t
i
where is the time constant and is equal to C V/hA. Increasing the value of the time
t
p
constant, , will result in a decrease in the thermal response of the solid to the environment
t
and hence, will increase the time required for it to reach thermal equilibrium (i.e., T T ).

In this expression, Fo represents the dimensionless time and is called the Fourier number,
2
2
the value of which is equal to tA /V . The Fourier number, along with the Biot number,
can be used to characterize transient heat conduction problems. The total heat flow through
the surface of the solid over the time interval from t 0 to time t can be expressed as
Q VC (T T )[1 exp( t/ )]

i
t
p
Transient Heat Transfer for Infinite Plate, Infinite Cylinder, and Sphere Subjected to
Surface Convection
Generalized analytical solutions to transient heat-transfer problems involving infinite plates,
cylinders, and finite diameter spheres subjected to surface convection have been developed.
These solutions can be presented in graphical form through the use of the Heisler charts, 3
illustrated in Figs. 3–11 for plane walls, cylinders, and spheres. In this procedure, the solid
is assumed to be at a uniform temperature, T , at time t 0 and then is suddenly subjected
i
to or immersed in a fluid at a uniform temperature T . The convection heat-transfer coeffi-

cient, h, is assumed to be constant, as is the temperature of the fluid. Combining Figs. 3 and
4 for plane walls, Figs. 6 and 7 for cylinders, and Figs. 9 and 10 for spheres allows the
resulting time-dependent temperature of any point within the solid to be found. The total