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Eq. (1.86) is equal to Eq. (1.87), and hence to Eq. (1.90):

      (1.91)

      which yields

(1.92)

An analogy can be made with Eq. (1.85), allowing Eq. (1.92) to become

      (1.93)

      where, 1/H = (1/h_A + L/k + 1/h_B). H is the overall heat transfer coefficient and includes various heat transfer coefficients.

      1.6.3 Radiation Heat Transfer

      An object emits radiant energy in all directions unless its temperature is absolute zero. If this energy strikes a receiver, part of it may be absorbed, part may be transmitted, and part may be reflected. Heat transfer from a hot to a cold object in this manner is known as radiation heat transfer. The higher the temperature, the greater is the amount of energy radiated. If, therefore, two objects at different temperatures are placed so that the radiation from each object is intercepted by the other, then the body at the lower temperature will receive more energy than it radiates, and thereby its internal energy will increase; in conjunction with this, the internal energy of the object at the higher temperature will decrease. Radiation heat transfer frequently occurs between solid surfaces, although radiation from gases also takes place. Certain gases emit and absorb radiation at certain wavelengths only, whereas most solids radiate over a wide range of wavelengths. The radiative properties of many gases and solids may be found in heat transfer books.

      Radiation striking an object can be absorbed by the object, reflected from the object, or transmitted through the object. The fractions of the radiation absorbed, reflected, and transmitted are called the absorptivity a, the reflectivity r, and the transmissivity t, respectively. By definition, a + r + t = 1. For many solids and liquids in practical applications, the transmitted radiation is negligible, and hence a + r = 1. A body that absorbs all radiation striking it is called a blackbody. For a blackbody, a = 1 and r = 0.

      (c) The Stefan–Boltzmann Law

This law was found experimentally by Stefan, and proved theoretically by Boltzmann. It states that the emissive power of a blackbody is directly proportional to the fourth power of its absolute temperature. The Stefan–Boltzmann law enables calculation of the amount of radiation emitted in all directions and over all wavelengths simply from the knowledge of the temperature of the blackbody. This law is expressible as follows:

      (1.94)

      where, σ denotes the Stefan–Boltzmann constant, which has a value of 5.669 × 10⁻⁸ W/m² K⁴, and T_s denotes the absolute temperature of the surface.

      The energy emitted by a non‐blackbody becomes

      (1.95)

      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 T_s is much different from the emissivity of the object at T_a, 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 T_a as being equal to the emissivity of object 1 when emitting radiation at T_a. This results in

(1.97)

      There are numerous applications for which it is convenient to express the net radiation heat transfer (radiation heat exchange) in the following form:

(1.98)

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

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