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2.6 Heat Transfer from Extended Surfaces Fins By heat transfer from extended surfaces is usually understood the global heattransfer process Ð by conduction, inside a solid, ®nned body in

Should T1  T2, then (by symmetry) it results that the vertical midplane is

an adiabatic surface, that is, dT =dxjxˆ0ˆ 0 Hence the problem may bereduced (Figs 2.9a, 2.9b), and the solution is T …x† ˆ _qL2=2k ‡ TS

2.6 Heat Transfer from Extended Surfaces (Fins)

By heat transfer from extended surfaces is usually understood the global heattransfer process Ð by conduction, inside a solid, ®nned body (inclusively the

®ns), and by convection and=or radiation from this one to its ambient (e.g.,air) The most common applications are those where such extended surfacesare 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 calledradiator

For the particular cases in Fig 2.10a there are two options for enhancingthe heat transferred from the solid body to its surroundings: either byimproving (increasing) the heat transfer coef®cient h, or by increasing thetotal heat transfer area, that is, by extending it through ®nning In manysituations the ®rst option is not affordable since it would imply a (larger)pump or even another, thermally more effective ¯uid (e.g., water, ordielectric ¯uids instead of air) Figure 2.10b shows a possible solution tothe second, usually preferred option

A key feature that a ®nned structure must possess is a higher thermalconductivity than that of the substrate, or it may diminish the heat transferredrather than increasing it Ideally, the ®nned structure should be made of such

a material as to allow for (almost) isothermal operation, thus maximizing theheat transfer rate

Finned surfaces are extensively used in electrical machine design,electronic and electric devices and circuits, internal combustion engines,

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 beutilized either to extend the surfaces of the solid bodies through which theheat transfer takes place, or as intermediate heat transfer elements betweendifferent working ¯uids (heat exchangers) They may be made of ®ns withvariable 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 ®nnedbody 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 ratebalance for a control volume For simplicity, we shall consider that the 1D ®nwith 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

The heat transfer

rate balance for

which combined with (53) yields

d2T

dx2‡ 1

Ac

dAcdx

dASdx

…T ÿ T1† ˆ 0: …2:29†

2.6.2 FINS WITH CONSTANT CROSS-SECTIONAL AREA

For these ®ns (Fig 2.12), Ac…x† ˆ Ac ˆ constant; the outer surface area is

AS…x† ˆ Px ˆ const, where P is the wet perimeter of the ®n cross-section;and (2.29) reduces then to

y…x† ˆ C1emx‡ C2eÿmx; …2:32†where the integration constants C1 and C2 may be determined by imposingthe boundary conditions prescribed for x ˆ 0 and x ˆ L Table 2.2

summarizes some frequently encountered types of ®ns with isothermalbases Ð that is, T …0† ˆ Tb, or y…0† ˆ yb ˆ T0ÿ T1.

The ®ns' performance in enhancing the heat transferred from the ®nnedbody to its surroundings is evaluated against several quality indicators:

ef®cacy ef, thermal resistance, Rth;f; ef®ciency Zf, and overall super®cialef®ciency Zov Table 2.3 summarizes the de®nitions of these quantities andtheir actual forms for the ®ns listed in Table 2.2 Two common types ofradiators are shown in Fig 2.13

qf Rth;bdefˆyb

qbEf®ciency Zfdefˆqqf

fyb Zf ˆtanh…mL†mL (insulated tip)

Zf ˆtanh…mLc†

mLc (active tip)Overall ef®ciency Zovdefˆ qt

L c ˆ L ‡ t=2, corrected length for the active-tip ®n, acceptable for ht=k < 0:0625.

q f , heat ¯ux rate transmitted by the ®n; q t ˆ 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 th;b , convection thermal resistance (what would be without the ®n).

2.7 Unsteady Conduction Heat Transfer

In many applications heat transfer is a dynamic, time-dependent process Forinstance, the onset of an electric current or the onset of a time-dependentmagnetic ®eld in an electroconductive body, or a change in the externalthermal conditions of the body, are examples where the thermal steady state(if any) is reached asymptotically, through a transient regime In thesecircumstances, the temperature ®eld inside the body is obtained by solvingthe time-dependent energy balance equation

2.7.1 LUMPED CAPACITANCE MODELS

When the thermal properties of the body under investigation and the thermalconditions of its surface are such that the temperature inside the body variesuniformly in time, and the body is Ð at any moment Ð almost isothermal,then the lumped capacitance method is a very convenient, simpler, yetsatisfactory accurate tool of thermal analysis

Let us assume that a uniformly heated, isothermal (Ti) iron chunk isimmersed at t ˆ 0 in a cooling ¯uid with T1< Ti(Fig 2.14) The temperatureinside the body decreases smoothly, monotonously, to eventually reachingthe equilibrium value, T1 Heat is transferred inside the body by conduction,and by convection from the body to the surrounding ¯uid reservoir If thethermal resistance of the body is small as compared to the thermal resistance

of the ¯uid, then the heat transfer process is such that the instantaneoustemperature ®eld inside the body is uniform, which implies that the internaltemperature gradients are negligibly small The energy balance equation thentakes the particular form

The heat transferred to the ¯uid in the time span ‰0; tŠ,

Although this result is reported here for a cooling process, such as themetallurgical process of annealing, where the internal temperature decreases(that is, Q > 0†, the relation (2.39) is also true for heating processes, where

Q < 0; 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 lumpedcapacitance models have a limited validity and, subsequently, applicabilitycriteria for them are needed

The plate of ®nite thickness, L, in Fig 2.16 is assumed to be initiallyisothermal, Ti The face at x ˆ L is in contact with a ¯uid reservoir at T1

…Ti> T1†, while the face at x ˆ 0 is maintained at Ti The heat ¯ux balancefor the control surface at x ˆ L is then

Rcond

Rconv ˆ Bi: …2:40†The nondimensional quantity Bi ˆ hL=k is called the Biot number Thisgroup plays an important role in the evaluation of the internal conductionheat transfer processes with surface convection conditions, and it may be

model and the

model

used to assess the validity of the lumped capacitance method for a particularcase The concept of characteristic length, Lc, and the Bi-criterion may beused to decide whether this assumption is valid or not Essentially, Bi  1means that the (internal) conduction thermal resistance of the body is muchsmaller the convection thermal resistance from this one to the ¯uid; hence,the lumped capacitance model may be safely used In contrast when Bi  1the (internal) conduction thermal resistance of the body is larger than theconvection thermal resistance from the body to the ¯uid, and thereforelumped capacitance models must be used with caution.

Consequently, if Bi ˆ hLc=k < 0:1, then the lumped capacitance model isconsistent This interpretation is correct, of course, in linear, isotropic, andhomogeneous substances Figure 2.17 gives a qualitative image of thetemperature ®eld inside a plate of ®nite thickness for different ranges ofthe Bi number

As apparent, the proper evaluation of Lc is crucial to the success of thelumped capacitance method, and for simple problems it is not too dif®cult to

®nd it For instance, in the previous problem (Fig 2.16) Lc ˆ L For bodies ofmore complex geometry Lc may be taken as the size of the body in thedirection of the temperature gradient (heat ¯ux ¯ow) Sometimes Lc isconveniently approximated by Lc ˆ V =AS; where V is the volume of thebody and As its external surface area This simple de®nition yields

hASrVct ˆ

hrcLct ˆ

hLck

2.7.2 GENERAL CAPACITIVE THERMAL ANALYSIS

Although the Bi-criterion may be useful in deciding whether the lumpedcapacitance model is satisfactorily accurate, there are many situations whenits validity is questionable Ð for instance, the presence of internal heatsources, (nonlinear) radiative heat transfer, etc

Figure 2.18 shows a schematic of a plate whose initial temperature Ti(at

t ˆ 0) is such that Ti 6ˆ T1and Ti 6ˆ Tsurf The imposed heat ¯ux, q00, and the

convection, q00

conv, and radiation, q00

rad, heat ¯uxes related to the body surface,

AS…h†and AS…conv;rad†, respectively, are assumed to be such that, globally, thetotal combined conduction±radiation heat ¯ows from the body to theenclosure walls The heat ¯ux balance for the body (the control volumehere) may be written as

q00AS…h†‡ _Egÿ …q00

conv‡ q00 rad†AS…conv;rad†ˆ rVcdTdt ; …2:43†

or, by using the heat ¯ux de®nitions (1.17, 1.19, 1.25), as

q00AS…h†‡ _Egÿ ‰h…T ÿ T1† ‡ se…T4ÿ T4

surf†ŠAS…conv;rad†ˆ rVcdTdt : …2:44†Although usually this nonlinear ordinary differential equation has noexact solution, in certain speci®c cases it may be analytically integrable Twosuch circumstances are listed next

(a) In the absence of internal heat sources _Eg and imposed heat ¯ux q00,

if the convection heat ¯ux is negligibly small with respect to theradiative heat ¯ux, q00

The heat ¯ux

balance for the

generalcapacitive

thermalanalysis

which is solved exactly by

…T

Ti

dT

T4ÿ T4 surf ˆesArVc

ÿ ln

Tsurf ‡ Ti

Tsurf ÿ Ti

... y-direction …ujuˆd99 ˆ 0 :99 U1†, and the boundary layerthickness is then called the velocity boundary layer thickness d ˆ d99 [12].The wall viscous friction... Fundamentals of Heat Transfer, McGraw-Hill, New York, 196 1), used with permission from A Bejan, Heat Transfer, John Wiley, 199 3, Fig 4.15, p 1 69

Figure 2.26