Applications of one –diamention heat transfer (Spheres )

Applications of one –diamention heat transfer (Spheres ) Spherical systems may also be treated as one-dimensional when the temperature is a function of radius only. The heat flow is then
(55)
Examples:
A thick-walled tube of stainless steel [18% Cr, 8% Ni, k =19 W/m• ?C] with 2-cm inner diameter
(ID) and 4-cm outer diameter (OD) is covered with a 3-cm layer of asbestos insulation [k =0.2 W/m• ?C]. If the inside wall temperature of the pipe is maintained at 600?C, calculate the heat loss per meter of length. Also calculate the tube–insulation interface temperature.
Solution
Figure blow shows the thermal network for this problem. The heat flow is given by


This heat flow may be used to calculate the interface temperature between the outside tube wall
and the insulation.We have

where Ta is the interface temperature, which may be obtained as Ta =595.8?C
The largest thermal resistance clearly results from the insulation, and thus the major portion of the
temperature drop is through that material.


Convection Boundary Conditions
We have already seen in previous section that convection heat transfer can be calculated from

An electric-resistance analogy can also be drawn for the convection process by rewriting
the equation as
(56)
Where the 1/hA term now becomes the convection resistance

THE OVERALL HEAT-TRANSFER COEFFICIENT
Consider the plane wall shown in Figure 16 exposed to a hot fluid A on one side and a cooler fluid B on the other side. The heat transfer is expressed by

The heat-transfer process may be represented by the resistance network in Figure 2-5, and
the overall heat transfer is calculated as the ratio of the overall temperature difference to
the sum of the thermal resistances:
(57)

Figure 16 Overall heat transfer through a plane wall


Figure 17 Resistance analogy for hollow cylinder with convection boundaries
Observe that the value 1/hA is used to represent the convection resistance. The overall
heat transfer by combined conduction and convection is frequently expressed in terms of
an overall heat-transfer coefficient U, defined by the relation .
(58)
Where A is some suitable area for the heat flow. In accordance with Equation (57), the
Overall heat-transfer coefficient would be

The overall heat-transfer coefficient is also related to the Rvalue of Equation R= ?T/ (q/A) through

For a hollow cylinder exposed to a convection environment on its inner and outer surfaces,
the electric-resistance analogy would appear as in Figure 17 where, again, TA and TB are
the two fluid temperatures. Note that the area for convection is not the same for both fluids
in this case, these areas depending on the inside tube diameter and wall thickness. The
overall heat transfer would be expressed by
(59)
in accordance with the thermal network shown in Figure 17. The terms Ai and Ao represent
the inside and outside surface areas of the inner tube. The overall heat-transfer coefficient
may be based on either the inside or the outside area of the tube. Accordingly,
(60)
(61)
The general notion, for either the plane wall or cylindrical coordinate system, is that