transfer equation is found to be as follows, where interfacial contact resistances are neglected:
(1.110)
where Rt,t = (1/2πr1Lh1) + (ln(r2/r1)/2πk1L) + (ln(r3/r2)/2πk2L) + ⋯(1/2πrnLhn)
1.6.7 The Sphere
The case of heat transfer through a sphere is not as common as the cylinder problem. Consider a hollow sphere of internal radius r1 and external radius r2 (Figure 1.19). Also, consider the inside and outside temperatures to be T1 and T2, respectively, and constant thermal conductivity with no heat generation. We can express the heat conduction across the sphere wall in the form of Fourier's law:
Figure 1.19 Heat conduction in a hollow sphere.
where, A = 4πr2 is the area normal to the direction of heat transfer.
After integrating Eq. (1.111), we obtain the following expression:
(1.112)
If we now consider a composite hollow sphere, the heat transfer equation is determined to be as follows, neglecting interfacial contact resistances:
(1.113)
where,
1.6.8 Conduction with Heat Generation
(a) The Plane Wall
Consider a plane wall, as shown in Figure 1.20a, in which there is uniform heat generation per unit volume. The heat conduction equation becomes
By integrating Eq. (1.114) with the prescribed boundary conditions, T(−L) = T1 and T(L) = T2. The temperature distribution can be obtained as
The heat flux at any point in the wall can be found, depending on x, by using Eq. (1.115) with Fourier's law.
If T1 = T2 ≡ Ts, the temperature distribution is symmetrical about the midplane (Figure 1.20b). Then,
Figure 1.20 Heat conduction in a slab with uniform heat generation: (a) asymmetrical boundary conditions, (b) symmetrical boundary conditions.
At the plane of symmetry dT/dx = 0, and the maximum temperature at the midplane is
After combining Eqs. (1.116) and (1.117), we find the dimensionless temperature as follows:
(1.118)
(b) The Cylinder
Consider a long cylinder (Figure 1.18) with uniform heat generation. The heat conduction equation can be rewritten as
By integrating Eq. (1.119), with the boundary conditions, dT/dr = 0, for the centerline (r = 0) and T(r1) = Ts, the temperature distribution can be obtained as
(1.120)
After combining terms, the dimensionless temperature equation results:
(1.121)
The approach mentioned previously can also be used for obtaining the temperature distributions in solid spheres and spherical shells for a wide range of boundary conditions.
1.6.9 Natural Convection
Heat transfer by natural (or free) convection involving motion in a fluid is due to differences in density and the action of gravity, which causes a natural circulation flow and leads to heat transfer. For many problems involving fluid flow across a surface, the superimposed effect of natural convection is negligibly small. The heat transfer coefficients for natural convection are generally much lower than that for forced convection. When there is no forced velocity of the fluid, heat is transferred entirely by natural convection (when there is negligible radiation). For some practical cases, it is necessary to consider the radiative effect on the total heat loss or gain. Radiation heat transfer may be of the same order of magnitude as natural convection in some circumstances even at room temperatures. Hence, wall temperatures in a room can affect the comfort of occupants.
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