Thermal Energy Storage Systems and Applications. Ibrahim Dincer

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СКАЧАТЬ rel="nofollow" href="#ulink_4f6af398-73e1-5f84-8731-01a4d969e8ca">Figure 1.16 A wall subject to convection heat transfer on both sides.

      (b) Newton's Law of Cooling

      Newton's law of cooling states that the heat transfer from a solid surface to a fluid is proportional to the difference between the surface and fluid temperatures, and the surface area. This is a particular type of convection heat transfer, and is expressed as

(1.85)

where h is referred to as the convection heat transfer coefficient (the heat transfer coefficient, the film coefficient, or the film conductance). It encompasses all effects that influence the convection mode and depends on conditions in the boundary layer, which is affected by factors such as surface geometry, the nature of the fluid motion, and thermal and physical properties (Figure 1.16).

In Eq. (1.85), a radiation term is not included. The calculation of radiation heat transfer is discussed later. In many heat transfer problems, the radiation effect on the total heat transfer is negligible compared with the heat transferred by conduction and convection from a surface to a fluid. When the surface temperature is high, or when the surface loses little heat by natural convection, then the heat transfer due to radiation is often of a similar magnitude to that lost by convection.

      To better understand Newton's law of cooling, consider the heat transfer from a high‐temperature fluid A to a low‐temperature fluid B through a wall of thickness x (Figure 1.16). In fluid A, the temperature decreases rapidly from T_A to T_s1 in the region of the wall, and similarly in fluid B from T_s2 to T_B. In most cases, the fluid temperature is approximately constant throughout its bulk, apart from a thin film (Δ_A or Δ_B) of fluid near each solid surface. The heat transfer per unit surface area from fluid A to the wall and that from the wall to fluid B can be expressed as

(1.86)

(1.87)

      Also, the heat transfer in thin films is by conduction only, as given below:

      (1.88)

(1.89)

Equating Eqs. (1.86)–(1.89), the convection heat transfer coefficients can be found to be h_A = k_A/Δ_A, and h_B = k_B/Δ_B. Thus, the heat transfer in the wall per unit surface area becomes

(1.90)

For the case of steady‐state heat transfer, 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)СКАЧАТЬ