Conduction Heat Transfer refrigeration systems, and domestic heaters, to name only some applications. The particular design of the ns may be very different (plates, pins, tubes, etc.), depending on the particular technical application, mounting conditions, weight restrictions, fabrication technology, and cost. The radiators may be utilized either to extend the surfaces of the solid bodies through which the heat transfer takes place, or as intermediate heat transfer elements between different working uids (heat exchangers). They may be made of ns with variable cross sections but, in any situation, they fulll the same function: they convey the largest part of the heat that is transferred from the nned body to its surrounding uid environment.

2.6.1 THE GENERAL EQUATION OF HEAT CONDUCTION IN FINS The heat conduction equation is obtained by writing the heat transfer rate balance for a control volume. For simplicity, we shall consider that the 1D n with variable cross section shown in Fig. 2.11 is made of a linear, isotropic, and homogeneous substance and that there is no internal heat generation. The heat balance for the dx slice is then qx (cid:136) qx (cid:135)dx (cid:135) dqconv : Figure 2.11 The heat transfer rate balance for a 1-D control volume within a n of variable cross section. Fouriers law (2.6) may be used to compute the longitudinal heat ux that enters the control volume: qx (cid:136) (cid:255)kAc(cid:133)x (cid:134) dT dx : [Ac(cid:133)x (cid:134) is the n cross-sectional area.] Taylors linearization scheme gives a simpler expression for the heat ux that leaves the control volume, namely, qx (cid:135)dx (cid:136) qx (cid:135) dqx dx

12 Heat transfer from ns with constant crosssection. Principles of Heat Transfer which combined with (53) yields qx (cid:135)dx (cid:136) (cid:255)kAc(cid:133)x (cid:134) dT dx (cid:255) k d dx (cid:20) Ac(cid:133)x (cid:134) dT dx (cid:21) dx : (cid:133)2:27(cid:134) Finally, if we substitute the lateral convection heat transferred from the side wall of the control volume, dqconv (cid:136) hdAS (cid:133)T (cid:255) T1(cid:134); (cid:133)2:28(cid:134) and (2.25) and (2.27) in the balance equation (2.24), we obtain d 2T dx 2 (cid:135) (cid:18) 1 Ac dAc dx (cid:19) dT dx (cid:255) (cid:18) 1 Ac h k

6.2 FINS WITH CONSTANT CROSS-SECTIONAL AREA For these ns (Fig. 2.12), Ac(cid:133)x (cid:134) (cid:136) Ac (cid:136) constant; the outer surface area is AS (cid:133)x (cid:134) (cid:136) Px (cid:136) const, where P is the wet perimeter of the n cross-section; and (2.29) reduces then to d 2T dx 2 (cid:255) hP kAc (cid:133)T (cid:255) T1(cid:134) (cid:136) 0; (cid:133)2:30(cid:134) or, in nondimensional form, d 2y dx 2 (cid:255) m2y (cid:136) 0; m2 (cid:136)