Page 169 - Mechanical Engineers' Handbook (Volume 4)

158 Heat-Transfer Fundamentals

Table 10 (Continued)
Configuration Finite-Difference Equation for x y
Case 5. Node near a curved surface 2 2
T m,n 1
maintained at a nonuniform temperature a 1 b 1
T m 1,n
2 2
T T 2
1
a(a 1) b(b 1)
2 2

T
m,n 0
a b

where h is the convection heat-transfer coefficient (Section 2), T is the temperature differ-
ence between the solid surface and the fluid, and A is the surface area in contact with the
fluid. The resistance occurring at the surface abounding the solid and fluid is referred to as
the thermal resistance and is given by 1/hA, i.e., the convection resistance. Combining this
resistance term with the appropriate conduction resistance yields an overall heat-transfer
coefficient U. Usage of this term allows the overall heat transfer to be defined as q UA
T.
Table 8 shows the overall heat-transfer coefficients for some simple geometries. Note
that U may be based either on the inner surface (U ) or on the outer surface (U ) for the
1 2
cylinders and spheres.
Critical Radius of Insulation for Cylinders
A large number of practical applications involve the use of insulation materials to reduce
the transfer of heat into or out of cylindrical surfaces. This is particularly true of steam or
hot water pipes where concentric cylinders of insulation are typically added to the outside
of the pipes to reduce the heat loss. Beyond a certain thickness, however, the continued
addition of insulation may not result in continued reductions in the heat loss. To optimize
the thickness of insulation required for these types of applications, a value typically referred
to as the critical radius, defined as r k/h, is used. If the outer radius of the object to be
cr
insulated is less than r then the addition of insulation will increase the heat loss, while for
cr
cases where the outer radii is greater than r any additional increases in insulation thickness
cr
will result in a decrease in heat loss.
Extended Surfaces
In examining Newton’s law of cooling, it is clear that the rate of heat transfer between a
solid and the surrounding ambient fluid may be increased by increasing the surface area of
the solid that is exposed to the fluid. This is typically done through the addition of extended
surfaces or fins to the primary surface. Numerous examples often exist, including the cooling
fins on air-cooled engines, i.e., motorcycles or lawn mowers or the fins attached to auto-
mobile radiators.
Figure 2 illustrates a common uniform cross-section extended surface, fin, with a con-
stant base temperature, T , a constant cross-sectional area, A, a circumference of C 2W
b
2t, and a length, L, which is much larger than the thickness, t. For these conditions, the
temperature distribution in the fin must satisfy the following expression: