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turbulent. Compared to flow along a flat plate, the major difference in pipe flow is that there is a limit to the growth of the boundary‐layer thickness because of the pipe radius.

      Many empirical pipe flow equations have been developed, particularly for water. The velocity V and volumetric flow rate equations of Hazen–Williams are the most widely used, and are as follows:

      (1.81)

      (1.82)

      where R_h is the hydraulic radius of the pipe, P is wetted perimeter (A/P, for example, R_h = D/4 for a round pipe), S is the slope of the total head line, h_f/L, A is the pipe cross‐sectional area, and C is the roughness coefficient. The coefficient C takes different values for the pipes, for example, C = 140 for very badly corroded iron or steel pipes.

1.6 General Aspects of Heat Transfer

Thermal processes involving the transfer of heat from one point to another are often encountered in industries. The heating and cooling of gases, liquids, and solids, the evaporation of water, and the removal of heat liberated by chemical reaction are common examples of processes that involve heat transfer. Engineers, scientists, technologists, researchers, and others need to understand the physical phenomena and practical aspects of heat transfer, and have a good knowledge of the basic laws, governing equations, and related boundary conditions.

      In order to transfer heat, there must be a driving force, which is the temperature difference between the locations where heat is taken and where the heat originates. For example, consider that a long slab of food product is subjected to heating on the left side; the heat flows from the left side to the right side, which is colder. Heat tends to flow from a point of high temperature to a point of low temperature, owing to the temperature difference driving force.

Many of the generalized relationships used in heat transfer calculations have been determined by means of dimensional analysis and empirical considerations. It has been found that certain standard dimensionless groups repeatedly appear in the final equations. It is necessary for people working in heat transfer to recognize the importance of these groups. Some of the most commonly used dimensionless groups that appear frequently in the heat transfer literature are given in Table 1.9.

      In the utilization of these groups, care must be taken to use equivalent units so that all the dimensions cancel out. Any system of units may be used in a dimensionless group as long as all units cancel in the final result.

      Basically, heat is transferred in three ways: conduction, convection, and radiation (the so‐called modes of heat transfer). In many cases, heat transfer takes place by all three of these methods simultaneously.

Figure 1.14 shows the different types of heat transfer processes as modes. When a temperature gradient exists in a stationary medium, which may be a solid or a fluid, the heat transfer occurring across the medium is by conduction, the heat transfer occurring between a surface and a moving fluid at different temperatures is by convection, and the heat transfer occurring between two surfaces at different temperatures, in the absence of an intervening medium (or presence of a nonobscuring medium), is by radiation, where all surfaces of finite temperature emit energy in the form of electromagnetic waves.

Table 1.9 Some of the most important heat transfer dimensionless parameters.

      Source: Olson and Wright [8].

Name	Symbol	Definition	Application
Biot number	Bi	hY/k	Steady‐ and unsteady‐state conduction
Fourier number	Fo	at/Y2	Unsteady‐state conduction
Graetz number	Gz	GY 2 c p/k	Laminar convection
Grashof number	Gr	GβΔTY3/v2	Natural convection
Rayleigh number	Ra	Gr × Pr	Natural convection
Nusselt number	Nu	hY/kf	Natural or forced convection, boiling, or condensation
Pec let number	Pe	UY/a = Re × Pr	Forced convection (for small Pr)
Prandtl number	Pr	c p μ/k = v/a	Natural or forced convection, boiling, or condensation
Reynolds number	Re	UY/v	Forced convection
Stanton number	St	h/ρUcp = Nu/ Re Pr	Forced convection

Figure 1.14 Representations of heat transfer modes: (a) conduction through a solid, (b) convection from a surface to a moving fluid, and (c) radiation between two surfaces.

      1.6.1 Conduction Heat Transfer

      Conduction is a mode of transfer of heat from one part of a material to another part of the same material, or from one material to another in physical contact with it, without appreciable displacement of the molecules forming the substance. For example, the heat transfer in a solid object, subject to cooling in a medium, is by conduction. In solid objects, the conduction of heat is partly due to the impact of adjacent molecules vibrating about their mean positions and partly due to internal radiation. When the solid object is a metal, there are also large numbers of mobile electrons that can easily move through the matter, passing from one atom to another, and they contribute to the redistribution of energy in the metal object. The contribution of the mobile electrons predominates in metals, which explains the relation that is observed between the thermal and electrical conductivities of such materials.

      (a) Fourier's Law of Heat Conduction

      Fourier's law states that the instantaneous rate of heat flow through a homogeneous solid object is directly proportional to the cross‐sectional area A

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