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Should T 1 T 2 , then (by symmetry) it results that the vertical midplane is an adiabatic surface, that is, dT adxj x0 0. Hence the problem may be reduced (Figs. 2.9a, 2.9b), and the solution is T x qL 2 a2k T S . 2.6 Heat Transfer from Extended Surfaces (Fins) By heat transfer from extended surfaces is usually understood the global heat transfer process Ð by conduction, inside a solid, ®nned body (inclusively the ®ns), and by convection andaor radiation from this one to its ambient (e.g., air). The most common applications are those where such extended surfaces are used to enhance the heat transfer rate from a solid body to its surround- ing environment. In this context, such an extended, ®nned surface is called radiator. For the particular cases in Fig. 2.10a there are two options for enhancing the heat transferred from the solid body to its surroundings: either by improving (increasing) the heat transfer coef®cient h,orbyincreasing the total heat transfer area, that is, by extending it through ®nning. In many situations the ®rst option is not affordable since it would imply a (larger) pump or even another, thermally more effective ¯uid (e.g., water, or dielectric ¯uids instead of air). Figure 2.10b shows a possible solution to the second, usually preferred option. A key feature that a ®nned structure must possess is a higher thermal conductivity than that of the substrate, or it may diminish the heat transferred rather than increasing it. Ideally, the ®nned structure should be made of such a material as to allow for (almost) isothermal operation, thus maximizing the heat transfer rate. Finned surfaces are extensively used in electrical machine design, electronic and electric devices and circuits, internal combustion engines, Figure 2.10 Heat transfer from extended surfaces. 468 Principles of Heat Transfer 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 ful®ll 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 q x q xdx dq conv X 2X24 Fourier's law (2.6) may be used to compute the longitudinal heat ¯ux that enters the control volume: q x ÀkA c x dT dx X 2X25 [A c x is the ®n cross-sectional area.] Taylor's linearization scheme gives a simpler expression for the heat ¯ux that leaves the control volume, namely, q xdx q x dq x dx dxY 2X26 Figure 2.11 The heat transfer rate balance for a 1-D control volume within a ®n of variable cross section. 2. Conduction Heat Transfer 469 Heat Transfer which combined with (53) yields q xdx ÀkA c x dT dx À k d dx A c x dT dx ! dxX 2X27 Finally, if we substitute the lateral convection heat transferred from the side wall of the control volume, dq conv hdA S T ÀT I Y 2X28 and (2.25) and (2.27) in the balance equation (2.24), we obtain d 2 T dx 2 1 A c dA c dx dT dx À 1 A c h k dA S dx T À T I 0X 2X29 2.6.2 FINS WITH CONSTANT CROSS-SECTIONAL AREA For these ®ns (Fig. 2.12), A c xA c constant; the outer surface area is A S xPx const, where P is the wet perimeter of the ®n cross-section; and (2.29) reduces then to d 2 T dx 2 À hP kA c T À T I 0Y 2X30 or, in nondimensional form, d 2 y dx 2 À m 2 y 0Y m 2 hP kA c Y yxT xÀT I X 2X31 The solution to this standard, second-order Euler ordinary differential equation is of the form yxC 1 e mx C 2 e Àmx Y 2X32 where the integration constants C 1 and C 2 may be determined by imposing the boundary conditions prescribed for x 0 and x L. Table 2.2 Figure 2.12 Heat transfer from ®ns with constant cross- section. 470 Principles of Heat Transfer Heat Transfer summarizes some frequently encountered types of ®ns with isothermal bases Ð that is, T 0T b ,ory0y b T 0 À T I . The ®ns' performance in enhancing the heat transferred from the ®nned body to its surroundings is evaluated against several quality indicators: ef®cacy e f , thermal resistance, R thYf ; ef®ciency Z f , and overall super®cial ef®ciency Z ov . Table 2.3 summarizes the de®nitions of these quantities and their actual forms for the ®ns listed in Table 2.2. Two common types of radiators are shown in Fig. 2.13. Table 2.3 Effectivity E f def q f hA cYb y b R thYb R thYf E f kP hA c 1a2 (in®nite ®n) Thermal resistance R thYf def y b q f R thYb def y b q b Ef®ciency Z f def q f q max q f hA f y b Z f tanhmL mL (insulated tip) Z f tanhmL c mL c (active tip) Overall ef®ciency Z ov def q t q max q t hA t y b Z ov 1 À A f A t 1 À Z f A cYb A c 0, ®n basis area; A f , ®n lateral area; A t ÀA f A b , radiator total area (®nned and un®nned surface). L c L ta2, corrected length for the active-tip ®n, acceptable for ht ak ` 0X0625. q f , heat ¯ux rate transmitted by the ®n; q t hA b y b hA f Z f y b , total heat ¯ux rate transmitted by the ®n; q b , heat ¯ux rate transmitted to the ®n (through the area covered by its base). R thYb , convection thermal resistance (what would be without the ®n). Table 2.2 a Tip condition Temperature, yay b Heat transfer rate, q f Convection hyLÀk dy dx xL coshmL À x h mk sinhmL À x coshmL h mk sinhmL M sinhmL h mk coshmL coshmL h mk sinhmL Adiabatic dy dx xL 0 coshmL À x coshmL M tanhmL Temperature yLy L y L y b sinhmxsinhmL À x sinhmL M coshmLÀ y L y b sinhmL Asymptotic yL 3 L3I 0 e Àmx M a y T xÀT I ; y b y0T 0ÀT I ; m 2 hP akAc; M hPkA c p . 2. Conduction Heat Transfer 471 Heat Transfer 2.7 Unsteady Conduction Heat Transfer In many applications heat transfer is a dynamic, time-dependent process. For instance, the onset of an electric current or the onset of a time-dependent magnetic ®eld in an electroconductive body, or a change in the external thermal conditions of the body, are examples where the thermal steady state (if any) is reached asymptotically, through a transient regime. In these circumstances, the temperature ®eld inside the body is obtained by solving the time-dependent energy balance equation. 2.7.1 LUMPED CAPACITANCE MODELS When the thermal properties of the body under investigation and the thermal conditions of its surface are such that the temperature inside the body varies uniformly in time, and the body is Ð at any moment Ð almost isothermal, then the lumped capacitance method is a very convenient, simpler, yet satisfactory accurate tool of thermal analysis. Let us assume that a uniformly heated, isothermal (T i ) iron chunk is immersed at t 0 in a cooling ¯uid with T I ` T i (Fig. 2.14). The temperature inside the body decreases smoothly, monotonously, to eventually reaching the equilibrium value, T I . Heat is transferred inside the body by conduction, and by convection from the body to the surrounding ¯uid reservoir. If the thermal resistance of the body is small as compared to the thermal resistance of the ¯uid, then the heat transfer process is such that the instantaneous temperature ®eld inside the body is uniform, which implies that the internal temperature gradients are negligibly small. The energy balance equation then takes the particular form À E out E st Y 2X33 Figure 2.13 Radiators. 472 Principles of Heat Transfer Heat Transfer which means ÀhA S T ÀT I rVc dT dt Y 2X34 or, put in nondimensional form, rVc hA S dy dt ÀyY y T ÀT I X 2X35 Time integration from the initial state t 0, y0y i to the current state t Y yt, that is, rVcahA S y y i dyay À t t i dt, where y i T i À T I , yields rVc hA S ln y i y tY or y i y T ÀT I T i À T I e ÀhA S arVct X 2X36 The group rVcahA S s is the thermal time constant (seconds), and it may be re-written as t t 1 hA S rVcR t C t X 2X37 From a practical point of view, it is particularly useful to outline the analogy that exists between the heat ¯ux problem described by (2.36) and that of the electrical current in the R C circuit shown in Fig. 2.15. The heat transferred to the ¯uid in the time span 0Y t, Q def t 0 qdt hA S t 0 ydtY 2X38 is a measure of the change in the internal energy undergone by the system (body) from the initial (at t 0) to the current state (t t): ÀQ DE st X 2X39 Figure 2.14 The lumped capacitance model for the cooling of a uniformly heated, isothermal iron chunk. 2. Conduction Heat Transfer 473 Heat Transfer Although this result is reported here for a cooling process, such as the metallurgical process of annealing, where the internal temperature decreases (that is, Q b 0, the relation (2.39) is also true for heating processes, where Q ` 0Y that is, the internal energy of the body increases. The Limits of Applicability for the Lumped Capacitance Model It is important to recognize that, although very convenient, the lumped capacitance models have a limited validity and, subsequently, applicability criteria for them are needed. The plate of ®nite thickness, L, in Fig. 2.16 is assumed to be initially isothermal, T i . The face at x L is in contact with a ¯uid reservoir at T I T i b T I , while the face at x 0 is maintained at T i . The heat ¯ux balance for the control surface at x L is then kA L T 1 À T 2 hAT 2 À T I Y or T 1 À T 2 T 2 À T I LakA 1ahA R cond R conv BiX 2X40 The nondimensional quantity Bi hLak is called the Biot number.This group plays an important role in the evaluation of the internal conduction heat transfer processes with surface convection conditions, and it may be Figure 2.15 The analogy between the heat ¯ux problem of the lumped capacitance model and the electrical current in an R C circuit. Figure 2.16 The Biot criterion used to assess the validity of the lumped capacitance model. 474 Principles of Heat Transfer Heat Transfer used to assess the validity of the lumped capacitance method for a particular case. The concept of characteristic length, L c , and the Bi-criterion may be used to decide whether this assumption is valid or not. Essentially, Bi ( 1 means that the (internal) conduction thermal resistance of the body is much smaller the convection thermal resistance from this one to the ¯uid; hence, the lumped capacitance model may be safely used. In contrast when Bi ) 1 the (internal) conduction thermal resistance of the body is larger than the convection thermal resistance from the body to the ¯uid, and therefore lumped capacitance models must be used with caution. Consequently, if Bi hL c ak ` 0X1, then the lumped capacitance model is consistent. This interpretation is correct, of course, in linear, isotropic, and homogeneous substances. Figure 2.17 gives a qualitative image of the temperature ®eld inside a plate of ®nite thickness for different ranges of the Bi number. As apparent, the proper evaluation of L c is crucial to the success of the lumped capacitance method, and for simple problems it is not too dif®cult to ®nd it. For instance, in the previous problem (Fig. 2.16) L c L. For bodies of more complex geometry L c may be taken as the size of the body in the direction of the temperature gradient (heat ¯ux ¯ow). Sometimes L c is conveniently approximated by L c V aA S Y where V is the volume of the body and A s its external surface area. This simple de®nition yields hA S rVc t h rcL c t hL c k k rc S 1 L 2 c t hL c k a L 2 c t Bi FoY 2X41 where Fo aaL 2 c t is the Fourier number Ð a nondimensional time. It should be noticed that, unlike the Bi number, Fo is not a constant, but rather a dynamic quantity. If we use this notation, (2.36) becomes y y i T ÀT I T i À T I e ÀBi Fo X 2X42 Figure 2.17 The temperature ®eld inside a plate of ®nite thickness for different ranges of Biot number. 2. Conduction Heat Transfer 475 Heat Transfer 2.7.2 GENERAL CAPACITIVE THERMAL ANALYSIS Although the Bi-criterion may be useful in deciding whether the lumped capacitance model is satisfactorily accurate, there are many situations when its validity is questionable Ð for instance, the presence of internal heat sources, (nonlinear) radiative heat transfer, etc. Figure 2.18 shows a schematic of a plate whose initial temperature T i (at t 0) is such that T i T T I and T i T T surf . The imposed heat ¯ux, q HH S , and the convection, q HH conv , and radiation, q HH rad , heat ¯uxes related to the body surface, A Sh and A S convYrad , respectively, are assumed to be such that, globally, the total combined conduction±radiation heat ¯ows from the body to the enclosure walls. The heat ¯ux balance for the body (the control volume here) may be written as q HH S A Sh E g Àq HH conv q HH rad A S convYrad rVc dT dt Y 2X43 or, by using the heat ¯ux de®nitions (1.17, 1.19, 1.25), as q HH S A Sh E g ÀhT ÀT I seT 4 À T 4 surf A SconvYrad rVc dT dt X 2X44 Although usually this nonlinear ordinary differential equation has no exact solution, in certain speci®c cases it may be analytically integrable. Two such circumstances are listed next. (a) In the absence of internal heat sources E g and imposed heat ¯ux q HH S , if the convection heat ¯ux is negligibly small with respect to the radiative heat ¯ux, q HH conv ( q HH rad , then (2.44) takes the simpler form rVc dT dt esA S rad T 4 À T 4 surf 0Y 2X45 Figure 2.18 The heat ¯ux balance for the general capacitive thermal analysis. 476 Principles of Heat Transfer Heat Transfer which is solved exactly by T T i dT T 4 À T 4 surf rVc esA S rad t 0 dtY 2X46 yielding an explicit de®nition for the time rather than for the temperature, that is, t rVc 4eA S rad  ln T surf T T surf À t À ln T surf T i T surf À T i 2 tan À1 T T surf 23 À tan À1 T i T surf 2345@A X 2X47 (b) If the radiation heat transfer component is negligibly small, q HH rad (q HH conv Y q HH S , and the conduction heat transfer coef®cient h is constant, then (2.44) becomes dy H dt Ay H 0Y y H y À B A Y A hA S conv rVc Y B q HH S A Sh E g rVc Y 2X48 admitting the analytic solution y H y H i e ÀAt Y or T ÀT I T i À T I e ÀAt BaA T i À T I 1 Àe ÀAt Âà X 2X49 2.7.3 UNSTEADY HEAT CONDUCTION DRIVEN BY TEMPERATURE GRADIENTS Outside the limits of validity for the lumped capacitance approach, the thermal problem may be solved by integrating (2.44), which may imply the solution to the domain effects due to temperature gradients. A simple, introductory model is the 1D heat transfer conduction problem of a plate of ®nite thickness L, made of a linear, isotropic, and homogeneous thermo- conductive substance k, and without internal heat sources. At t 0 the face x L, assumed to be initially isothermal, T i , is exposed to a ¯uid reservoir of temperature T I , while the face x 0 is thermally insulated, that is, no heat ¯ux is crossing it. The heat transferred from the plate to the ¯uid is conveyed by conduction, inside the body, and by convection, within the ¯uid. The latter process is characterized by the constant convection heat transfer 2. Conduction Heat Transfer 477 Heat Transfer [...]... 0.1412 0. 199 5 0.24 39 0.2814 0.3142 0.4417 0 .94 08 1.2558 1 .98 98 2.1 795 2.3572 2.38 09 1.0025 1.0050 1.0075 1.0 099 1.0124 1.0246 1.1143 1.2071 1.50 29 1.5677 1.6002 1.6015 0.1730 0.2445 0. 298 9 0.3450 0.3852 0.5423 1.1656 1.5708 2.5704 2.8363 3.0788 3.1102 1.0030 1.0060 1.0 090 1.0120 1.01 49 1.0 298 1.1441 1.2732 1.7870 1 .92 49 1 .99 62 1 .99 90 In many practical situations it is important to know the total heat... 4 79 2 Conduction Heat Transfer Table 2.4 Plane wall In®nite cylinder Sphere Bi z1 (rad) C1 z1 (rad) C1 z1 (rad) C1 0.01 0.02 0.03 0.04 0.05 0.1 0.5 1.0 5.0 10.0 50.0 100.0 0. 099 8 0.1410 0.1732 0. 198 7 0.2217 0.3111 0.6533 0.8603 1.3138 1.42 89 1.5400 1.5552 1.0017 1.0033 1.00 49 1.0066 1.0082 1.0160 1.0701 1.1 191 1.2402 1.2620 1.2727 1.2731 0.1412 0. 199 5 0.24 39 0.2814 0.3142 0.4417 0 .94 08 1.2558 1 .98 98... of the local downstream coordinate x , that is, d d x X Traditionally, the velocity ®eld may be used to de®ne its size in the y-direction ujud 99 0X99UI , and the boundary layer thickness is then called the velocity boundary layer thickness d d 99 [12] The wall viscous friction is evaluated by the nondimensional wall friction coef®cient, Cf , through the wall shear stress, def Cf tw X 2 rUI... Transfer, McGraw-Hill, New York, 196 1), used with permission from A Bejan, Heat Transfer, John Wiley, 199 3, Fig 4.12, p 166 Principles of Heat Transfer Figure 2.25 Temperature history in the center of a sphere immersed suddenly in a ¯uid of a different temperature (r0 sphere radius) Ð after Heisler, used with permission from A Bejan, Heat Transfer, John Wiley, 199 3, Fig 4.13, p 167 and p 168 Heat... Heat Transfer, John Wiley, 199 3, Fig 4.10, pp 164, 165 Heat Transfer 484 Figure 2.23 Relationship between the temperature at any radius r and the temperature at the centerline (r 0, Figure 26) of a long cylinder immersed suddenly in a ¯uid of a different temperature (L plate halfthickness) Ð after Heisler, used with permission from A Bejan, Heat Transfer, John Wiley, 199 3, Fig 4.11, p 166 Figure... prescribed temperature Heat Transfer Q sin z1 ~ ~ yY 1À Q Qref z1 0 Heat Transfer Figure 2. 19 Temperature history in the midplane of a plate immersed suddenly in a ¯uid of a different temperature (L plate half-thickness) Ð after Heisler, used with permission from A Bejan, Heat Transfer, John Wiley, 199 3, Fig 4.7, pp 1 59 and 160 481 2 Conduction Heat Transfer Figure 2.20 The absence of Fo from (2.57) indicates... Relationship between the temperature at any radius r and the temperature in the center (r 0, Figure 29) of a sphere immersed suddenly in a ¯uid of a different temperature (L plate half-thickness) Ð after Heisler, used with permission from A Bejan, Heat Transfer, John Wiley, 199 3, Fig 4.14, p 1 69 Heat Transfer Figure 2.27 Total heat transfer between a sphere and the surrounding ¯uid, as a function... surrounding ¯uid, as a function of the total time of exposure, t (After H Grober, S Erk and U Grigull, Fundamentals of Heat Transfer, McGraw-Hill, New York, È 196 1), used with permission from A Bejan, Heat Transfer, John Wiley, 199 3, Fig 4.15, p 1 69 487 2 Conduction Heat Transfer an introductory analysis may be found in ref 2 The central idea of the method is the superposition principle that is applicable... surrounding ¯uid, as a func-tion of the total time of exposure, t (After H Grober, È S Erk, and U Grigull, Fundamentals of Heat Transfer, McGraw-Hill, New York, 196 1), used with permission from A Bejan, Heat Transfer, John Wiley, 199 3, Fig 4 .9, p 162 ( J0 and J1 are Bessel functions of ®rst type and order 0 and 1 ) Sphere: I 2 ~ sin zn r 4sin zn À zn cos zn ~ Y 1 À zn cot zn BiX Y Cn ... Transfer Relationship between the temperature in any plane x and the temperature in the midplane (x 0, Fig 2. 19) of a plate immersed suddenly in a ¯uid of a different temperature (L plate half-thickness) Ð after Heisler, used with permission from A Bejan, Heat Transfer, John Wiley, 199 3, Fig 4.8, p 160 482 Principles of Heat Transfer Figure 2.21 Total heat transfer between a plate and the surrounding . C 1 z 1 (rad) C 1 0.01 0. 099 8 1.0017 0.1412 1.0025 0.1730 1.0030 0.02 0.1410 1.0033 0. 199 5 1.0050 0.2445 1.0060 0.03 0.1732 1.00 49 0.24 39 1.0075 0. 298 9 1.0 090 0.04 0. 198 7 1.0066 0.2814 1.0 099 0.3450 1.0120 0.05. 1.42 89 1.2620 2.1 795 1.5677 2.8363 1 .92 49 50.0 1.5400 1.2727 2.3572 1.6002 3.0788 1 .99 62 100.0 1.5552 1.2731 2.38 09 1.6015 3.1102 1 .99 90 2. Conduction Heat Transfer 4 79 Heat Transfer Figure 2. 19. 0.3852 1.01 49 0.1 0.3111 1.0160 0.4417 1.0246 0.5423 1.0 298 0.5 0.6533 1.0701 0 .94 08 1.1143 1.1656 1.1441 1.0 0.8603 1.1 191 1.2558 1.2071 1.5708 1.2732 5.0 1.3138 1.2402 1 .98 98 1.50 29 2.5704 1.7870 10.0