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Flat Solid Wall If we consider a one-dimensional heat ow along the x direction in the plane wall shown in Fig 91a, direct application of Eq (91) can be made, and then integration yields q kA (T x 2 T1) (93)
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where the thermal conductivity is considered constant, x is the wall thickness, and T1 and T2 are the wall-face temperatures Note that q / A q , where q is the heat transfer rate through an area A Equation (93) can be written in the form q T2 T1 x / kA T2 Rth T1 thermal potential difference thermal resistance (94)
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where x / kA assumes the role of a thermal resistance Rth The relation of Eq (94)
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FIGURE 91 One-dimensional heat conduction through a plane wall (a) and electrical analogue (b) (From Rohsenow, Hartnett, and Ganic,5 p 1-3)
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HEAT AND MASS TRANSFER
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is quite like Ohm s law in electric circuit theory The equivalent electric circuit for this case is shown in Fig 91b The electrical analogy may be used to solve more complex problems involving both series and parallel resistances Typical problems and their analogous electric circuits are given in many heat transfer textbooks1 4 In treating conduction problems it is often convenient to introduce another property which is related to the thermal conductivity, namely, the thermal diffusivity , k c where is the density and c is the speci c heat (95)
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Cylindrical Solid Wall The rate of heat transfer through a cylindrical solid wall of length L is calculated from the equation q 2 Lk T1 T2 ln (r2 / r1) (96)
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where r2 and r1 are outside and inside radii, and ln is logarithm with base e Similarly for the spherical solid wall q 4 kr1r2(T1 T2) r2 r1 (97)
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An important case of heat transfer is that from a hot uid on one side of a solid wall to a cooler uid on the other side The wall may be a cylindrical or a at wall of a single material or composite of different materials The rate of heat transfer is calculated from q UA(Th Tc) (98)
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where Th and Tc are the temperatures of the hot and cold uids, respectively Flat Composite Wall Equation (98) is used to calculate the rate of heat transfer from a hot uid successively through the hot-side lm, layers of solid material of the wall, and cold-side lm to the cold uid (Fig 92) For this case, U of Eq (98) is
FIGURE 92 Composite wall
CHAPTER NINE
(1 /hh)
(xA / kA)
1 (xB / kB)
(1 /hc)
(99)
where hh and hc are the hot- and cold-side lm coef cients of heat transfer (see the section Convection for a de nition of h) Cylindrical Composite Wall Equation (98) is also used to calculate the rate of heat transfer from a hot to a cold uid through a composite cylindrical pipe wall (Fig 93) For this case, the product of UA of Eq (98) is UA 1 2r1Lhh ln (r2 / r1) 2L kA 1 ln (r3 / r2) 2L kB 1 2rn LhC (910)
Spherical Composite Wall UA 1 4r2hh 1 1 r2 r1 4kr1r2 1 4r2hc n (911)
General Conduction Equation The differential equation for temperature distributions in solids is given by Eq (1023)
2 Mass Transfer by Diffusion
As mentioned above, heat transfer will occur whenever there exists a temperature difference in a medium Similarly, whenever there exists a difference in the concentration or density of some chemical species in a mixture, mass transfer must occur Hence, just as a temperature gradient constitutes the driving potential for heat transfer, the existence of a concentration gradient for some species in a mixture provides the driving potential for transport of that species Therefore, the term mass transfer describes the relative motion of species in a mixture due to the presence of concentration gradients Since the same physical mechanism is associated with heat transfer by conduction (ie, heat diffusion) and mass transfer by diffusion, the corresponding rate
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