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Then, the heat transferred from an object's surface to its surroundings per unit area is
(1.96)
Note that if the emissivity of the object at Ts is much different from the emissivity of the object at Ta, then this gray object approximation may not be sufficiently accurate. In this case, it is a good approximation to take the absorptivity of object 1 when receiving radiation from a source at Ta as being equal to the emissivity of object 1 when emitting radiation at Ta. This results in
There are numerous applications for which it is convenient to express the net radiation heat transfer (radiation heat exchange) in the following form:
After combining Eqs. (1.97) and (1.98), the radiation heat transfer coefficient can be found as follows:
(1.99)
Here, the radiation heat transfer coefficient is seen to strongly depend on temperature, whereas the temperature dependence of the convection heat transfer coefficient is generally weak.
The surface within the surroundings may also simultaneously transfer heat by convection to the surroundings. The total rate of heat transfer from the surface is the sum of the convection and radiation modes:
(1.100)
1.6.4 Thermal Resistance
There is a similarity between heat flow and electricity flow. While electrical resistance is associated with the conduction of electricity, thermal resistance is associated with the conduction of heat. The temperature difference providing heat conduction plays a role analogous to that of the potential difference or voltage in the conduction of electricity. Below we give the thermal resistance for heat conduction, based on Eq. (1.84), and similarly the electrical resistance for electrical conduction according to Ohm's law:
(1.101)
(1.102)
It is also possible to write the thermal resistance for convection, based on Eq. (1.85), as follows:
(1.103)
In a series of connected objects through which heat is transferred, the total thermal resistance can be written in terms of the overall heat transfer coefficient. The heat transfer expression for a composite wall is discussed next.
1.6.5 The Composite Wall
In practice, there are many cases in the form of a composite wall, for example, the wall of a cold storeroom. Consider that we have a general form of the composite wall as shown in Figure 1.17. Such a system includes any number of series and parallel thermal resistances because of the existence of layers of different materials. The heat transfer rate is related to the temperature difference and resistance associated with each element as follows:
(1.104)
Therefore, the one‐dimensional heat transfer rate for this system can be written as
(1.105)
Figure 1.17 A composite wall with many layers in series.
where, ∑Rt = Rt,t = 1/HA. Therefore, the overall heat transfer coefficient becomes
(1.106)
1.6.6 The Cylinder
A practical common object is a hollow cylinder, and a commonly encountered problem is the case of heat transfer through a pipe or cylinder. Consider that we have a cylinder of internal radius r1 and external radius r2, whose inner and outer surfaces are in contact with fluids at different temperatures (Figure 1.18). In a steady‐state form with no heat generation, the governing heat conduction equation is written as
Figure 1.18 A hollow cylinder.
Based on Fourier's law, the rate at which heat is transferred by conduction across the cylindrical surface in the solid is expressed as
(1.108)
where A = 2𝜋rL is the area normal to the direction of heat transfer.
To determine the temperature distribution in the cylinder, it is necessary to solve Eq. (1.107) under appropriate boundary conditions, by assuming that k is constant. By integrating Eq. (1.107) twice, the following heat transfer equation is obtained:
(1.109)
If we now consider a composite hollow cylinder,
