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C HAPTER 1 Introduction 1.1 CONVECTIVE HEAT TRANSFER Heat convection is the term applied to t he process involved when energy is transferred from a surface to a fluid flowing over it as a result o f a difference between the temperatures o f the surface and the fluid. I n c onvection therefore as indicated in Fig. 1.1, there is always a surface, a fluid flowing,relative to this surface, and a temperature difference between the surface and the fluid and the concern is with the rate o f h eat transfer between the surface a nd t he fluid. Convective heat transfer occurs extensively i n practice. T he cooling o f the cutting tool during a machining operation, the cooling o f t he electronic components i n a computer, the generation and condensation o f s team i n ~,;!perfu.alpower plant, t he heating and cooling o f buildings, cooking, and the t hermafcontrol o'f a re-entering spacecraft all, for example, involve convective h eat transfer. S ome examples o f situations in which convective h eat t ransfer is important are shown i n Fig. 1.2. Convection i s one o f t he three so-called modes o f h eat transfer, the other two are conduction a nd radiation [1],[2],[3],[4]. I n m ost real situations, the overall heat transfer is accomplished' b y a combination o f a t l east two o f t hese modes o f h eat transfer. However, i t i s possible, i n m any s uch cases, to consider the modes separately and then combine the solutions -for each o f t he modes i n o rder to obtain the overall heat transfer rate. F or example, heat transfer from one fluid to another fluid through the walls o f a p ipe occurs i n m any p ractical devices. In, this, case, heat i s transferred b y convection f ronithe h otter fluid to t he o ne surface o f t he pipe. H eat is then 'transferred'by conduction t hrough,the walls o f t he pipe. Finally, heat is,transferred by convection from the other surf~ce to t he c older fluid. T hese h eat tJ:ansfer processes are shown i n F ig. 1.3., T he over~ h eat transfer rate can becalcul~ted by considering the three processes separately \and then combining the results. -I / 2 Introduction to Convective Heat Transfer Analysis Convective Heat Transfer F rom Surface t o Fluid o r F rom Fluid to Surface F luidFlowFluid at ---~ Temperature T f Temperature Tw F IGURE 1.1 Convective heat transfer. i Electronic Component C oolant---.... Substrate Rolling o f M etal Sheet Cooling o f an Electronic Device Water Flow From Engine Heat Transfer in an Automobile Radiator -iFIGURE 1.2 Some. situations that involve convective heat transfer. This book is concerned with a description of some methods o f determining convective heat transfer rates i n various flow situations, realizing that in many cases the~e methods will need to b e c ombined with calculations for the other modes o f hea~ transfer in order to p redict the overall heat transfer rate. In some cases i t is not possible to consider the modes separately. For example, i f a gas, such as water vapor o r c arbon dioxide, which absorbs and generates thermal . radiation, flows over a s urface at a higher temperature, heat is transferred from the surface to the gas b y b oth convection and radiation. I n this case, the radiant heat exchange influences the temperature distribution in the fluid. Therefore, because the convective heat transfer r ate depends on this temperature distribution in the fluid, the radiant and convective modes interact with each other and cannot be considered separately. However, even i n c ases such as this, the calculation procedures developed' for convection by itself form the basis o f the calculation o f the convective part o f the overall h eat transfer rate. '. . CHAPTER 1: Introduction 3 F IGURE 1.3 Combined conduction and convection. Convective heat transfer rates depend on the details o f the flow field about the surface involved as well as on the properties o f t he fluid. The determination o f convective heat transfer rates is therefore, i n general, a n extremely difficult task since it involves the determination o f both the velocity and temperature fields. I t is only in comparatively recent times that any widespread success has been achievedjn the development o f methods o f calculating convective heat transfer rates. The transfer o f h eat b y convection involves the transfer o f energy from the surface to the fluid on a molecular scale and then the diffusion o f this heat through the fluid by bulk mixing due to the fluid motion. T he basic heat transport mechanism i n convection i s still conduction which is, o f course, governed by Fourier's law. This law states that the heat transfer rate, q, i n a ny direction, n, p er unitarea.:measured normal to n i s given by: q= - k - . aT an (1.1) k being the coefficient o f conductivity which is a property o f t he fluid involved. Conduction o f h eat is always important close to the surface over which the fluid is flowing. However, when the flow is turbulent,the rate at which heat is "convected" by the turbulent eddies i s usually much greater than the rate o f h eat conduction, and, therefore, i n such flows, conduction can often b e neglected except in a region lying close to the surface. At a solid surface, where the velocity is effectively zero, Fourier's law always, o f course, applies as indicated in Fig. 1.4. N onnal to Surface, n Velocity Profile qw- -dn n =O - dTI - F IGURE 1.4 Heat transfer at wall. 4 Introduction to Convective Heat Transfer Analysis / 1.2 . FORCED, F REE, AND COMBINED CONVECTION \ Since convective heat transfer rates depend on the details o f the flow field they will b e strongly dependent on how the flow is generated. I t is important, therefore, to distinguish between forced and free (or natural) convection [5],[6],[7],[8],[9],[10]. In the case o f forced convection, the fluid motion is caused by some external means such as a fan o r pump. In the case o f free convection, the flow is generated by the body forces that occur as a result o f the density changes arising from the tempera. ture changes in the flow field. These body forces are actually generated by pressure. gradients imposed on the whole fluid. The most common source o f this imppsed pressure field is gravity, the pressure gradient then being the normal hydrostatic pressure gradient existing in any fluid bulk. The bodyJorces in this case are usually termed buoyancy forces. Another source o f imposed pressure gradients which can cause free-convective.fl.2.w~we the centrifugal forces w hich arise when there is an o~erall rotary motion such .as-exists i n a rotating machine. I n all flows involving heat transfer and, therefore, temperature changes, the buoyancy forces arising from the gravitational field will, o f course, exist. The term forced convection is only applied to flows in which the effects o f these buoyancy forces are negligible. In some flows in which a forced velocity exists, the effects o f these buoyancy (orces will, however, not b e negligible and such flows are termed combined- or mixed free and forced convective flows. The various types o f convective heat transfer are illustrated in Fig. 1.5. ~ --------....... ~ Forced Flow Forced Convection (Buoyancy Forces Negligible) Natural- or Free Convection (Negligible or Zero Forced Flow) Buoyancy Forces -...... -----... ~ ~ t Combined- or Mixed Convection (Buoyancy Forces and Forced Flow Important) Forced Flow F IGURE 1.5 Forced, free, and combined convection. CHAPTER 1: Introduction 5 F IGURE 1.6 External Flow Internal Flow External and internal flows. F low-- F IGURE 1.7 A flow having some characteristics o f both an external and an internal flow. H eat transfer involving a change o f p hase is classified as convective heat transfer even though when the solid phase is involved, the overall process involves combined and interrelated convection and conduction. H eat t ransfer during boiling, condensation, and solidification (freezing) all, thus, involve convective heat transfer. 1.3 EXTERNAL AND INTERNAL F LOWS Convective flows are conventionally broken down into those that involve external flow and those that involve internal flow. These two types o f flow are illustrated i n Fig. 1.6. . E xternal flows involve a flow, t hat is essentially infinite in extent, over the outer surface o f a body. Internal flows involve t he flow through a duct or channel. I t i s not always possible to clearly place a convective heat transfer problem into one o f these categories since some problems have several o f t he characteristics o f both an internal and an external flow. An example o f s uch a case i s shown in Fig. 1.7. I n predicting heat transfer rates one o f t he first steps i n creating a model o f the flow is to decide whether it involves internal o r external flow because there are differeI\t assumptions that c an b e conveniently adopted i n t he two types o f flow. 1.4 T HE CONVECTIVE HEAT TRANSFER C OEFFICIENT Consider a fluid flowing over a surface w hich is maintained a t a temperature that is different from that o f t he b ulk o f the fluiq as shown i n Fig. 1.8. I f t he temperature a t s ome point o n t h¢ s urface i s Tw a nd i f t he rate a t w hich h eat is being transferred locally a t this point f rom t he surface to the fluid per unit surface 6 Introduction to Convective Heat Transfer Analysis Fluid at Temperature Tj ~ ::,....f"'" I II Temperature Tw F IGUREl.S Convective heat transfer from a surface. area is q, then i t is usual to define a quantity, h, such that: q = h(Tw - Tj) (1.2) In this equation T j is some convenient fluid temperature which will be more precisely defined at.,a.later stage [11],[12],[13],[14],[15]. T he heat transfer rate, q, is taken as positive i n the direction wall-to-fluid so that it will have the same sign as (Tw - T j ) and h will always, therefore, be positive. A number o f names have been applied to h including "convective h eat transfer coefficient", " heat transfer coefficient", "film coefficient", " film conductance", and " unit thermal convective conductance". The heat transfer coefficient, h, has the units W /m2 -K or, since its definition only involves temperature differences, W/m2°C, i n the SI system o f units. I n the imperial system o f units, h h as the units Btulft2-hr-°F. It m ust be clearly understood that h is not only dependent on the fluid involved in the process but is also strongly dependent on, among other things, the shape o f the surface over which the fluid is flowing and on the velocity o f the fluid. A large part o f the present book will b e concerned with a discussion o f some o f the analytical and numerical methods that can b e u sed to try to predict h for various flow sItUations. Basically, these methods involve the simultaneous application o f the principles governing viscous fluid flow, i.e., the principles o f conservation o f mass and momentum and the principle o f conservation o f energy. Although considerable success has been achieved with these methods, there still remain many cases in which experimelltal results have to b e u sed to arrive at working relations for the prediction o f h. , Consideration will now b e given to how the fluid temperature, T j, used in Eq. (1.2), is defined, the definition used depending o f course, on the flow situation. Consider first, incompressible flow over the outside o f a body, as shown in Fig. 1.9. In this case, T j is most conveniently taken as the temperature, T 1 , o f the fluid in the freestream ahead o f the body. This is, o f course, a very obvious choice for I FIGURE 1.9 Fluid temperature in external flows. C HAPTER 1: Introduction 7 Outer Inviscid Region. Temperature Effectively Constant and Equal to T 1 Boundary Layer F IGURE 1.10 Boundary layer on a surface. F IGURE 1.11 Fluid temperature in internal flows. Tf when, as i s the case of the majority of problems of engineering importance, a boundary layer-type flow exists, i.e., there exists adjacent to the surface o f the body a comparatively thin region in which very rapid changes in velocity and temperature occur in the direction normal to the surface and where outside this region the fluid temperature i s effectively equal to that existing in the freestream, as shown i n Fig. 1.10. In t he case o f internal flows, such as flow through a pipe, the fluid temperature will, in general, vary \continuously across the whole flow area, as indicated in Fig. 1.11. In such cases, T f i s taken as some mean temperature.,]'he-most commonly used mean temperature is the so-called bulk temperature, Tb:~a:sically, this is defined . such that: . _ Rate at which enthalpy crosses Mass flow rate X speclfic enthalpy - the sel t d secti'on 0f d uct ec e i.e., (1.3) where Cp is the specific heat a t t he constant pressure o f the fluid, the specific enthalpy having thus been s et equal to cpT. The specific heat is, in general, temperat1ire dependent. I t is assumed to be constant here having been evaluated a t some suitable . .. mean temperature. Now, i f a n elemental portion o f t he c~oss-sectional area, dA, o f the duct as shown . in Fig. 1.12 is considered and i f the vel09ity and temperature at the place where d4 ./ 8 Introduction to Convective Heat Transfer Analysis ~ F low Cross-section o f D uct- Area =A F IGURE 1.12 E lement o f d uct cross-section. is taken are u and T respectively then the rate at which enthalpy passes thrbugh d A will be given by: \ d H = p udAcpT (1.4) Therefore, integrating this equation to get the total rate at which enthalpy passes through the cross-section o f the duct gives: H= L p ucpT d A (1.5) Substituting this into Eq. (1.3) then gives the following, because, as discussed. above, C p is being treated as a constant: Tb = L p uT dAlin (1.6) B y the same reasoning as that used to deduce Eq. (1.5), it can be shown that: ./ m= L pUdA (1.7) Therefore, i f the density'P is also assumed to be constant, Eq. (1.6) can be written as: Tb = ~A:.:-- _ _ J uTdA L (1.8) u dA The bulk temperature is, o f course, equal to the temperature that would be attained i f the fluid at the particular section of the dUct being considered was discharged into a container and, without any heat transfer occurring, was mixed until a uniform temperature was obtained. For this n~ason i t is sometimes referred to as the "mixing cup temperature". E XAMPLE 1 .1. A ir flows through a pipe with a diameter D. T he velocity distribution in . the pipe is approximately given b y u = 20[(R - r)/Rpn mis, r b eing the radial distance to the point at which u is the velocity and R is the radius o f t he pipe. The temperature distribution in the flow is approximately given by 7 0-40[(R- r)/R]lnoC. Find the mean temperature in the flow. / CHAPTER I: Introduction 9 T ru .1 _ __ _______ _ FIGUREE1.1 Solution. Using: Tb = :::.!A~ _ _ f u TdA u dA L and noting as indicated in Fig. E1.1 that: dA = 2'1Trdr it follows that: i.e., Tb = i 1 u T(rlR) d (rl R) ::...;O::.....,-~_ _ __ Ll u (rIR)d(rIR) q sing the given expressions for the velocity and temperature distributions then gives: Tb = 70-20X40-=O~1-------20 r [1 - (rIR)f'fl(rIR)d(rIR) [1 - (rIR)]ln(rIR)d(rIR) L i.e., writing: X =\1 - (rIR) ;' / 10 Introduction to Convective Heat Transfer Analysis T .t\BLE 1 .1 i 'JYpical values f or t he m ean h.eat t ransfer coefficient M ean h eat t ransfer coefficient Flow situation a nd fluid Forced convection in a ir Forced convection in water Forced convection in liquid metals Free convection in air Free convection in water Boiling in water Condensing in s team ( W/m2-K) 1 0-200 40-10,000 10,000-100,000 3:..20 2 0-200 2,000-100,000 5,000-50,000 the above equation becomes: fO X2I7(1 - X)dX Tb = 70 = 70 _ 20 X 40~1-,:-~.- - - - 20 flO x ln (1 - X) dX T b 2 0 x 40[7/9 - 7116] 20[7/8 - 7/15] i.e., Tb = 36.7°C Eq. (1.2) defined the local heat transfer coefficient, h, i n terms o f the local rate o f h eat transfer rate per unit area, the local wall temperature and, in those cases where it is changing, the local fluid temperature. In general, all o f these quantities, i.e., h, q, Tw, a nd Tf ' vary with position on the surface. For the majority o f applications it is convenient to define, therefore, a mean or average heat transfer coefficient, It, such that i f Q is the total heat transfer rate from the entire surface o f area A then: (1.9) where T w is the mean surface temperature and T f i s the mean fluid temperature, the latter being defined in some suitable manner. I f there is no possibility o f any ambiguity, the bars on It, T w , and T f to indicate mean quantities are usually omitted. Typical values o f the mean convective heat transfer coefficient for various flow situations are listed in Table 1.1. A pipe with a diameter o f 2 c m is kept at a surface temperature o f 40°C. Find the heat transfer rate per m length o f this pipe i f i t is (a) placed in an air flow in which the temperature is 50°C and ( b) placed in a tank o f water kept at a temperature o f 30°C. T he heat transfer coefficients i n these two situations, which involve (a) forced convection in air and (b) free convection in water, are estimated to be 20 W/m2K and 70 W/m2K, respectively. E XAMPLE 1 .2. CHAPTER 1: Introduction 11 Solution. The definition o f the mean heat transfer coefficient gives: In the situation being considered, Tw = 40°C and, since a I -m length o f the pipe is being considered, A = 7TDL = 1T X 0.02. Hence: Q= h 1T X 0.02 X (40 - Tf ) For case (a), this gives: Q = 20 X 1T X 0.02 X (40 -: 50) = - 12.57 W The negative sign indicates that the heat transfer is from the air to the cylinder. For case (b), this gives: Q = 70 X 1T X 0.02 X (40 - 30) = 43.98 W This result is positive because in this case, the heat transfer is from the cylinder to the water. 1.5 APPLICATION OF D IMENSIONAL ANALYSIS TO CONVECTION A s previously mentioned, the major a im o f t he p resent book is to describe some typical analytical and numerical methods for determining h. However, there exist m any c ases o f g reat practical importance for w hich i t i s not possible to o btain even approximate theoretical solutions. I n s uch c ases, the prediction o f h eat transfer rates has to b e based on the results o f p revious experimental s tudies: I n o rder that these experimental results b e o f t he m ost u sefulness they must b e e xpressed in terms o f g eneral variables that then allow t hem to b e a pplied to a m uch w ider range o f conditions than those under which the actual measurements were obtained. For example, i f m easurements o f t he forced convective h eat t ransfer rates from cylinders to air a nd w ater a re made, the results m ust b e e xpressed i f p ossible)n t erms o f v ariables that will allow them to be u sed to predict t he h eat t ransfer rate"fiom c ylinder to another fluid, e.g., fuel oil. Therefore, i n o rder to b e a ble to make the most u se o f e xperimental results, they must b e e xpressed i n t erms o f t he " general" v ariables governing the problem. O ne w ay o f d etermining t hese " general" variables is b y t he application o f d imensional analysis [16],[17],[18],[19]. T he h eat transfer coefficient, h, i s t he variable whose value is being sought. COll-sider a series o f b odies o f t he s ame g eometrical shape, e.g., a series o f e lliptical cylinders (see Fig. 1.13), b ut o f d ifferent size p laced i n various fluids. I t c an b e deduced, either b y p hysical argument o r b y c onsidering available experimental results, that if, i n t he case o f g as flows, t he v elocity is low enough for compressibility effects to b e ignored, h will d epend on: a • T he conductivity, k, o f t he fluid w ith w hich t he body is exchanging heat. • T he viscosity, 1J." o f t he fluid w ith wh1ch t he b ody is exchanging heat. · 12 I ntroduction ,to Convective H eat Transfer Analysis F IGURE 1.13 Series o f bodies with same geometrical shape. • The specific heat, cP ' o f the fluid with which the body is exchanging heat. The re~on for ~sing th~ ~l?ecific ~eat a t const~t pressure may b~ ~ueried~ For.liquids, . which m ay b e assnnR;d to be mcompresslble under the conditions here bemg considered, all specific heats are the same. For gases, Cp is used basically pecause i t is the changes o f enthalpy o f the fluid that are o f importance and the enthalpy o f the fluid per unit mass at any point is C pT. • The density, p, o f the fluid with which the body is exchanging heat. • The size o f t he body as specified b y some characteristic dimension, e. Since a series of bodies havmg- the same geometrical shape is being considered, any convenient dimension can b e used for e. • The magnitude o f the forced fluid velocity, U, relative to the body. The manner in which this is defined will depend on the nature o f t he problem as indicated in Fig. 1.14. In the case o f external flows, the undisturbed freestream velocity is usually the most convenient to use for U, whereas in the case o f internal flows it is u suallymore convenient to take U as the mean or mass average velocity i n the , duct. • Some quantity which !will act as a measure of the magnitude o f the body force acting on the flow. Consideration here will only be given to buoyancy forces, i.e., to forces arising d ueto gravity. However, the method used here to derive a measure of this force can/easily b e applied when other force fields exist. / Um =Mean Velocity -U --.. Freestream l Velocity Velocity Profile FIGURE 1.14 Definition of forced velocity. CHAPTER 1: Introduction 13 FIGURE 1.15 Control volume considered. To derive an expression for a measure o f the magnitude o f the buoyancy force, consider an elemental volume o f the fluid as shown in Fig. 1.15. First, consider the forces acting on this control volume when the fluid is unheated and at rest. Since the fluid is at rest, the hydrostatic pressure forces must just balance the weight o f the fluid. Hence, i f d A is the horizontal cross-sectional area of the control volume and i f Po is the density o f the fluid in this unheated state, then the force balance requires that: [Po - (p + d p)o]dA = P odAdxg (1.10) Now consider the same control volume when the surface is at a different temperature from the fluid far away from it and the fluid is in motion. The fluid near the surface will also b e at a different temperature from the fluid far away from the surface and its density will as a result also b e different. Let the density o f the fluid in the control volume in this case be p. The situation is shown in Fig. 1.16. Applying the conservation o f momentum principle for the x -direction to the con' trol volume gives: [ ' _( + d ) '] d A + N et shearing fo~ce on.surf~ce o f p p p control volume I II x-dIrectIOn (1.11) = Rate o f momentum change in x-direction through control volume , + p dA d xg Now it is convenient to measure the pressure relativetj} the hydr.o static pressure that would exist i f there was no heating and no fluid motion, i.e., to define: p = p i - Po (1.12) Vertical I Fluid i n M otion Weight = pdAdxg F IGURE 1.16 Generation o f buoyancy forces. 1 4 Introduction t o Convective Heat Transfer Analysis from which it follows that: (p + d p) = (p + d p)' - (p + d p)o + d p)]dA (1.13) Using these two equations and Eq. (1.10) it then follows that: [ p' - (p + d p)]dA = = [po - (p + d p)o]dA + [ p - (p (1.14) P odA d xg + [p - (p + d p)]dA Substituting this result into Eq. (1.11) then gives: Net shearing force on [p - (p+ d p)]dA + surface o f control volume in x-direction Rate of mpmentum change in x-direction through control volume (1.15) + (Po - p)gdAdx = The third term on the left-hand side of this equation is called the buoyancy force. I t arises, of course, as a result of the density change with temperature and is equal to the difference between the weight of the fluid in the control volume with no heating and the weight o f the fluid in the control volume when heated. Its value could have been more simply derived using a less rigorous approach. However, difficulty can sometimes be encountered when such approaches are attempted for body forces. arising due to effects other than gravity. The above discussion shows, therefore, that the buoyancy force per unit volume acting on the fluid in the vertically upward direction is given by: (Po - p)g (1.16) Now attention in the present section is being restricted to incompressible flows in which only density changes with temperature are significant, i.e., the density changes produced by the pressure changes in the flow are assumed to b e negligible. It is convenient, therefore, to express this buoyancy force in terms o f the temperature difference by introducing the coefficient of bulk expansion (sometimes called the coefficient of cubical expansion), f3, o f the fluid. This is defined such that i f a mass M of the fluid occupies a volume Vi at a temperature Ti then, i f its temperature is changed to T, its volume V at this temperature is given by: (1.17) I f the densities o f the fluid at temperatures Ti and T are Pi and p respectively then, SInce: M Pi = V i' P = V M (1.18) CHAPTER 1: Introduction 15 Eq. (1.17) can be written as: Le., (1.19) Applying this result between the temperature o f the unheated fluid T f and T then shows using Eq. (1.16) that the buoyancy force per volume is given by: f 3gp(T - Tf) (1.20) Since the temperature difference T - Tf at any point in the flow will depend on the overall temperature difference ( Tw - T f ) i t is seen that a convenient measure o f the buoyancy force"acting on the fluid is f3gp(Tw - Tf). However, p has already been independently selected as a variable on which h depends so the variable that will be used as a measure o f the buoyancy force will be taken as: (1.21) I t is assumed, therefore, that: h = function [k, IL, Cp , p, e, U, f3 geTw - T f )] (1.22) which can be written as: f [h, k, IL, cp , p, e, V, f 3g(Tw - Tf)] = 0 (1.23) where f is some function. There are thus, eight dimensional variables involved in the convection problem. The theory o f dimensional analysis shows, howevet;_ that fewer generalized dimensionless variables are in fact required to describe ~ problem, the number required being equal to the number o f dimensionless variables, i.e., eight i n the present case, less the number o f basic dimensions required to describe the problem. I n the present situation, four basic dimensions, i.e., mass; M , length, L, time, f , and tetriperature, T are required and, therefore, four (i.e., 8 - 4) dimensionless variables are sufficient to describe the problem. I f i t had been assumed that the changes in the kinetic energy o f the flowing fluid are associated with significant enthalpy changes, then temperature could not have been taken as an independent dimension and only three dimensions would have been required to define the problem. I n this case, an extra dimensionless number will arise, this being the Mach number which is equal to the ratio o f the characteristic velocity to the speed o f sound. The fact that incompressible flow is being assumed indicates that the Mach number is low and this effect therefore negligible. Four basic dimensions will therefore b e assumed i n the present work. ' ./ 16 Introduction to Convective Heat Transfer Analysis T he theory o f dimensional analySis indicates that the dimensionless variables are formed by the products o f powers o f c~rtain o f the original dimensional variables. For this reason, the generalized dimensionless variables are often termed dimensionless products and are denoted by the symbol 'TT. T he four dimensionless variables required in the present problem will, therefore, b e denoted by 'TTl, 'TT2, 'TT3, and 'TT4. Eq. (1.23) can, therefore, be reduced to the following form: (1.24) The theory o f dimensional analysis shows that each o f these 'TT'S m ust contain one prime dimensional variable which does not occur in any o f the other 11"'S, together with other dimensional variables which occur i n all the 'TT's. I n the pres\ent problem there are, o f course, eight dimensional variables and four 'TT'S so four prime variables must be selected and there will also be four other dimensional variables which occur in allthe'TT's. The primeyafiftbles should b e selected so that each dimensionless product characterizes some distinct feature o f the flow. For example, the forced velocity, U, should be used as one prime variable since it determines whether or not the problem involves forced convection. Similarly, the buoyancy variable (3 geTw - T f ) will determine the importance o f free convective effects and should also be used as a prime variable. Also, since h is the variable whose value is required, it should be used as a prime variable. The fourth prime variable will b e taken as cpo This choice is not as obvious as the others but stems from the fact that c p will determine the thermal capacity o f the fluid and will, therefore, influence the relation between the velocity and temperature fields. With these four prime variables, the 'TT'S will have the form: 'TTl 'TT2 'TT3 W4 = [ hk a) /-t b) pC) ed)] (1.25) (1.26) = [ U k a2 /-t b2 pC2ed2 ] = [{3g(Tw - Tf)~3/-tb3pC3~3] (1.27) (1.28) = [c p ka4 /-t b4p C4 ed4 ] T he dimensional variables k, /-t, p, a nd e which are not prime variables occur, o f course, in all four w's. The possibility that some o f the indices of these variables a b hI. C I, . .. , C4, d 4 are zero is not excluded. I t should b e noted that no additional assumption is inherent in setting the index o f the prime variables equal to unity since in place o f W I, for example, the following could have been used: wi = [he k f /-tg phe j ] = [he kea /-t eb pec eed] - we I SO t hat i f W I is dimensionless then so is w i. CHAPTER 1: Introduction 17 The values o f the indices a i, b i , C }, • •• , C4, d 4 are determined by using the fact that the 1T'S are dimensionless. In order to utilize this fact, the dimensions o f all the dimensional variables must b e known in terms o f the basic dimensions. These are ill& follows, the way in which they are found from the definition o f the variable being h,dicated i n some cases. For example, since: (1.29) It follows that: Dimensions o f h = Dimensions o f [Ar = ea X T Energ ; Ime X emperature 1 (~~2}L2tT = M t- 3 T- i Similarly, i t follows that since: · . . DlmenSlOns 0 f k = D·ffienSlOns 0 f [ l Area X Energy T· (~ IL t h)" l me X .temperature eng 1 The following c an b e derived using the same approach: Dimensions o f JL = M L - 1 t -1 Dimensions o f p = M L - 3 Dimensions o f ( = L Dimensions o f U = L t- 1 further, since f3 = V - Vi V i(T - Ti) it follows that f3 h as dimensions l iT so: DimensioDSof pg(T w - '{oj = (~)(~)T = L t- 2 , .1, / / 18 Introduction to Convective Heat Transfer Analysis I Lastly, since i t follows that: Dimensions of cp = Dimensions o f [ M E ;ergy ass x emperature = 1 (~;2}(MT) = L 2r2T - 1 Substituting the relevant of the above results into Eq. (1.25) t hen shows that Dimensi6ns of"'*rf.=: ( Mr3r-I)(MLr3T-I)al(M L - l r l )b1(ML-3)C1(L)d1 .. ..I ,_ (1.30) B ut the 1 T'S are dimensionless so i t is known that: Dimensions o f 1TI = MO LOtOTO (1.31) , Equating the indices o f the various dimensions on the right-hand sides o f Eqs. (1.30) . a nd (1.31) then gives: F or M : F or L : a l F or t: F or T: 1 + al - + h I + CI + dl =0 =0 hI - 3CI (1.32) - 3 - 3al - hI = al = 0 - 1- 0 Solving between these equations then gives: al = - 1, hI = 0, CI = 1TI, 0, d l = 1 (1.33) Therefore, the first dimensionless product, 1 Tl m ust be given by: = (hfJk) = Nu, the Nusselt number the Reynolds number (1.34) I f a similar procedure is applied to the other three 1 ("' s the following are obtained: 1T2 = ( u -e pI JL) = ( U fJ v) = Re, f3g(Tw - (1.35) 1T3 = To)p 2 3 /JL 2 = {3g(Tw - Tj)-e3/v2 e = Gr, the Grashof number 1T4 = (1.36) (1.37) , / ( cpp,lk) = P r, t he Prandtl number where i n these equations v = p.J p is the kinematic viscosity. CHAPTER 1: Introduction 19 Therefore, in the convection problem, i n general: function (Nu, Re, Gr, Pr) = 0 which can b e rewritten as: Nu = function (Re, Gr, Pr) (1.38) (1.39) Thus, i n order to correlate experimental heat transfer results for a particular geometrical situation, only the four dimensionless variables Nu, Re, Gr, and P r n eed to be used. This conclusion can b e r eached i n other ways, it being demonstrated i n the next chapter by using the governing differential equations and considering the conditions under which dynamic similarity will exist. I f the buoyancy forces have a negligible effect on the flow, i.e., i f the flow is purely forced convective, then Eq. (1.39) reduces to: Nu = fuIiction (Re, Pr) (1.40) Similarly, i f there is no forced velocity or i f i t is negligibly small, the flow is purely free convective and: Nu = function (Gr, Pr) (1.41) Available experimental data for various flow situations have been correlated i n terms o f the above dimensionless variables a nd the results fitted b y empirical equations o r simply presented i n graphical form. Despite the reduction i n t he number o f .ignificant variables achieved by the introduction o f the dimensiorilessproducts, a considerable amount o f experimental work has to b e carried out i n order to arrive at a useable correlation for most geometries. F or this reason it is u·sually worthwhile to attempt to carry out some form o f analytical or numerical solution o f the problem, e ven i f the solution i s a very simplified one, because this solution may indicate the general form o f the correlation equation or, at least, indicate where the major emphasis should be placed i n the experimental program. In the above analysis, the dependence o f h o n the fluid properties j L, p, k, c p ' a nd ~ was considered. Now, i n general, these properties are all temperature dependent. (They are also, i n general, pressure dependent, b ut this is not for the moment being . considered, attention being restricted to incompressible flow.) Therefore, since i n any problem involving heat transfer, the fluid temperature will vary through the flow field, t \e question arises as to how this variation c an b e accounted for. I t h as been found both experimentally and numerically that, provided the overall temperature change is not very great, this can b e done b y e valuating these fluid properties a t some .uitable mean temperature. I n the case o f e xternal flows, this mean temperature is usually t aken as the average o f the surface and fluid temperatures, i.e., as ( Tw + r f )/2, while in the case o f internal flows it i s usually adequate to evaluate the fluid properties a t the bulk temperature. T he above analysis was based o n t he assumption that orily one length scale, t , was needed to define the problem. This is, however, not always true. Consider for example, heat transfer from the walls o f s hort pipes, as shown i n Fig. 1.17, to a fluid ftowing through them. 2 0 Introduction to Convective Heat Transfer Analysis - ------------------'Short' pipes 1> F IGURE 1.17 Series o f short pipes. I n this cas~, th.e ~,,-e~~ge h eat t ransfer coefficient for the pipe will depend o n b oth f and D. I f dimensionless- analysis i s applied to this problem, assuming for simplicity that the buoyancy force effects are negligible, then one possible result would be: N uv = function [Rev, Pr, ( fID)] (1.42) where N uv a nd Rev are the Nusselt and Reynolds numbers respectively based on the diameter D . T hus, i n c ases s uch as this, another dimensionless product o f t he form ( fl D ) i s required. O ther factors beside those considered above can influence the value o f h. F or example, as already mentioned, i n h igh speed gas flows, the compressibilty o f t he gas, i.e., the changes i n d ensity caused by the changes in the pressure in the flow, can become important. In this case, h will also be dependent on the speed o f sound i n the gas, a. T his will m ean t hat a n e xtra dimensionless variable will be involved, this being the M ach number, M , defined by: UI a = M, the Mach number (1.43) Buoyancy force effects are seldom important when compressibilty effects are important so in high speed gas flows: N u = function [Re, Pr, M ] (1.44) In free surface flows such as that shown i n Fig. 1.18 the gravitational acceleration g has an effect on h a nd this will again mean that an extra dimensionless variable will be involved, this being the Froude number, Fr, defined by: g flU 2 = Fr, the Froude number In this case, i f b uoyancy forces are negligible, N u = function [Re, Pr, F r] (1.45) (1.46) The surface tension can also sometimes influence the heat transfer rate and in this case another new dimensionless number will be involved in defining the Nusselt number. CHAP1ER 1: Introduction 21 FIGURE 1.18 Gravitational Force Flow with a free surface. I n the j et cooling o f a surface, a cooling fluid is discharged at a mean velocity o f U from a nozzle with a diameter, D, onto the surface that is to be cooled. I f the distance o f the nozzle from the surface is H , and i f the characteristic size o f the surface is e, a nd i f the j et is discharged at a n angle, cp, to the surface, write down the dimensionless variables t hat will be· involved i n describing the heat transfer rate-in this situation. Buoyancy effects (i.e., the effects o f the density changes due to the temperature changes) may b e ignored b ut the overall effect o f the gravity on the flow should b e included. E XAMPLE 1 .3. FIGUREE1.3 Solution. I n this case the mean heat transfer coefficient for the surface o n w hich the j et is impinging, h, will depend on: • • • • • • • • • • T he conductivity, k, o f the fluid T he viscosity, /-t. o f t he fluid T he specific heat, cp , o f the fluid T he density, p, o f the fluid T pe size o f t he surface, T he magnitude o f t he initial j et velocity, U T he nozzle diameter, D T he nozzle to surface distance, H T he angle a t w hich the j et i s discharged, cp T he gravitational acceleration, g e Hence: h = function [k, IL, cp , p, e, u, D, H, cp, g] w hich can b e written as: f [h, k, IL, cp , p, e, 4 D, H, cp, g] = 0 2 2 Introduction to Convective Heat Transfer Analysis There are, thus, 11 dimensional v aqables involved in the problem. The number o f dimensionless variables required is then~fore (11 - the number o f basic dimensions), \ ' i.e., 11 - 4 = 7. Each o f these dimensionless numbers inust contain one prime dimensional variable which does not occur i n any o f the other dimensionless variables together with other dimensional variables which o~cur i n alL o f the dimensionless variables. In the present problem then, seven variables m ust b e selected as prime variables. The following will be used here as prime variables: h , U, cpo t , H, t/>, and g. The dimensionless parameters will then have the form: 1 Tl = [hJClI J-t bl pCI [yll ] 1T2 = [U Jtl2 J-t b2 p C [yl2] 2 1T3 = [CpJtl3J-tb3pC3[yl3] 1T4 = [ t Jtl4 J-t b4pC [yl4 ] 4 1T5 = [HJtlsJ-tbspcs[ylS] 1T6 = [t/>Jtl6 J-t b6pC [yl6] 6 1T7 = [g Jtl7 J-t Ir, pC [yl7] 7 I t will be noted that the nozzle diameter has been used as the characteristic length scale that occurs in all the 1T'S. F rom the discussion given above it will be obvious that: , 1 Tl 1T2 = (hDI k) = Nu, t he Nusselt number = (UDplJ-t) = Re, the Reynolds number 1T3 = (cpJ-tlk) = Pr, the Prandtl number Because both 1T4 and 1T5 h ave a length as the prime variable, it should b e clear that: 1T4 = ( tID) 1T5 = ( HID) N ext consider 1T6. Because t/> is an angle i t is dimensionless. Hence: 1T6 = t/> Lastly consider 1T7. Because the dimensions o f g are U t2, i t will be clear that a7 = 0 because no other term will involve energy. Substituting for the dimensions o f the other terms, as discussed above then gives: b7 = - 2, Hence: C7 = 2, d7 = 3 1T7 = (gp2D3/J-t 2) = ( gDIU 2)(pUDIJ-t)2 = ( gDIU 2)Re2 Because the Reynolds number Re has already been taken as a governing dimensionless number, 1T7 will be taken as U 21gD i.e., as the Froude number, Fr. I n some ways the use o f U in this dimensionless number contradicts the use o f U as a prime variable. However, this is not really a problem because the gravitational acceleration can only affect the heat transfer i f there is a flow, i.e., i f U is nonzero. Hence: function [Nu, Re, Pr, ( tID), (HID), Fr] = 0 which can be rewritten as (1.38) Nu = function [Re, Pr, ( tID), (HID), Fr] (1.39) C HAPTER 1: Introduction 23 T AllLE 1 .2 Dbnensionless n umbers ~i11i'rffl.1 t:Ii ~;;;;;w;;; Name Brlnkman n umber Hckcrt number Froude number Orashof number Mach number Nusselt number Peelet number Prandtl number Rayleigh number Reynolds n umber Stanton number Weber number S ymbol Br Ec Fr Gr D efinition V 2 JL/k(Tw - T f ) V2/c p (Tw - T t ) V 2/gL f3g(Tw - Tf)L3/ V 2 V ia h Uk V Ua v ia f3g(T w V Uv h /pc p V p V2Uu - I nterpretation Dissipationlheat transfer rate Kinetic energy/enthalpy change Inertial force/gravitational force Buoyancy force/viscous force Velocity/velocity o f propagation o f weak disturbances Convective heat transfer ratelconduction heat transfer rate M Nu Pe Pr Ra Re R ePr R ate o f diffusion o f viscous effects/rate o f diffusion o f h eat T f )L3/va G rPr I nertia force/viscous force NulRe Pr Inertia force/surface t ension force St We A list of dimensionless numbers that can arise in the analysis o f convective heat transfer is given i n Table 1.2. 1.6 PHYSICAL INTERPRETATION O F T HE D IMENSIONLESS NUMBERS The dimensionless parameters, such as the Nusselt and Reynolds numbers, can be thought o f as m easures o f the relative importance o f o f certain aspects o f t he flow. For example, i f t he flow through an areadA i s considered, as shown in Fig. 1.19, the rate momentum passes through this area is equal to the mass flow rate times the velocity, i.e., equal to rizV d A, i.e., equal to p VV d A, i.e., equal to p V2 d A. If, therefore, U is a " measure" o f t he velocity, the quantity p U2 is a " measure" o f t he magnitude o f t he momentum flux in the flow. This quantity is often termed the " inertia force". Further, since the Newtonian viscosity law indicates that the viscous shear stresses Because dA is small, the velocity is effectively constant over dA FIGURE 1.19 Flow area considered. 24 Introduction to Convective H eat Transfer Analysis i n the flow are given by an equation o f the form T = IL(8UI8n), n being any chosen direction, i t follows that the quantity IL f /l eis a measure o f the magnitude o f the . viscous stresses in the flow. Now, the Reynolds number c an b e written as: Re = pue = pu2 e IL I LUle i.e., the Reynolds number is a measure o f the ratio o f the magnitude o f t he inertia forces in the flow to the magnitude o f the viscous forces in the flow. This means, therefore, that i f ~e ~eynolds n umber is. relatively low, the. vis.cous forces ~e high compared to the mertIa forces a nd a ny dIsturbances that arIse m the flow tend to b e damped out by the action o f viscosity and laminar flow will tend to exist. I f the Reynolds number is relatively high, however, the viscous forces are low compared to the inertia forces and any disturbances that arise in the flow will tend to grow, i.e., turbulent flow'will ten~-o develop. Next, consider the Nusselt number. The convective heat transfer from a surface will depend on the magnitude o f h (Tw - T f ). Also, i f there was no flow, i.e., i f the heat transfer was purely by conduction, Fourier's law indicates that the quantity k (Tw - Tf )1e would be a measure o f t he heat transfer rate. Now, the Nusselt number can be written as: will _ he _ Nu - k- h (Tw - T f ) k(Tw - Tf)lf i.e., the Nusselt number is a measure o f the ratio o f the magnitude o f t he convective heat transfer rate to the magnitude o f the heat rate that would exist in the same situation with pure conduction. To obtain a similar interpretation o f the Grashof number, i t is necessary to obtain a measure o f the magnitude o f t he flow velocity induced by the buoyancy forces. Now, as discussed above, a measure o f the magnitude o f the buoyancy forces in the flow per unit volume is the quantity f3 g p (Tw - Tf ). Therefore a measure o f the work done by the buoyancy forces on the fluid as it flows over the body, i.e., a measure o f the product o f the buoyancy forces and the distance over which they act, will b e f3gp(Tw - Tf)e. B ut the work done by the buoyancy forces results in the fluid gaining kinetic energy. I f Ub is a measure o f velocity induced by the buoyancy forces then a measure o f the kinetic energy per unit volume will b e pUb 2/2. Therefore, a measure o f the magnitude o f Ub is J f3 g (Tw - T f )e. For the reasons discussed when considering the Reynolds number, a measure o f the ratio o f the inertia forces to the viscous forces in natural convection is the quantity: pUb2 = pUb f IL Ub le IL i.e., using the expression for Ub d erived above, a measure o f o f t he ratio o f t he inertia forces to the viscous forces is the quantity: C HAPTER 1: Introduction 25 Hence, the Grashof number is a measure o f the magnitude o f the ratio of the inertia forces induced in the flow by the buoyancy forces to the viscous stresses, i.e., it is effectively a measure o f the magnitude o f the ratio o f the buoyancy forces to the viscous forces in the flow. Attention will, lastly, be given to the Prandtl number. Consider a steady flow over a surface which is at a different temperature from the fluid flowing over the surface, the situation considered being shown in Fig. 1.20. The viscous stress acting on the surface causes a reduction in the momentum of the fluid flow near the surface. I f, as shown in Fig. 1.20, d u is a measure o f the thickness o f the layer in the flow over which the effects o f this momentum decrease occur, then a measure o f the momentum decrease will be the product o f the mass flux through this layer and the initial fluid velocity, U, i.e., will be p U d u W X U, W being a measure o f the width of the body. This momentum decrease is caused b y the viscous force acting on the surface. A measure o f this force will be the product of a measure o f the shear stress acting on the surface and a measure o f the area o f the surface. A measure o f the shear stress is JL U / d u and a measure o f the surface !,trea is .eW. Hence, equating this measure o f the shear force to the measure o f the momentum decrease gives: JL d u.eW U = p U2d U W This can be rearranged to give: t1%=Jd;f= J& " ,j'., .-. (1.47) Next, consider the enthaply changes in the flow that result from the heat transfer from the surface. I f, as shown in Fig. 1.18, d T is a measure o f the thickness o f the layer in the flow over which the effects o f this enthalpy change occur then a measure o f the enthalpy change will be the product o f t he mass flux through this layer and a measure o f the enthalpy change per unit mass. The fluid has a temperature Tf ahead of the body and a measure o f the surface temperature isrTw ' Therefore, a measure of the enthalpy change p er unit mass will b e CpeTw - Tf J. Hence, measure o f t he enthalpy change will be p UdTW X c p(Tw - T f ). This enthalpy change is caused by the heat transfer at the surface. A measure o f the heat transfer rate at the surface a Temperature Distribution FIGURE 1.20 M omentum a nd temperature changes induced by presence o f surface. 2 6 Introduction to Convective Heat Transfer Analysis wi11 b e the product o f a m easure o f the heat transfer rate per unit area at the surface and a measure of the area o f t he surface. A measure o f the heat transfer rate per unit area from the surface is, b y virtue o f Fourier's law, k (Tw - T j )/Dq and a measure o f the surface area is ew. Hence, equating this measure o f the surface heat transfer rate to the measure o f the enthalpy change gives: k ( Tw - Tj) aT ew = puaTw x c p(Tw - Tj) T his can be rearranged to give: e = Vc;upe = v;C;; iRe Dividing Eq. (1.47) by this equation gives: aT ~ It 1 1 au aT = jp,. Hence, the Prandtl number is a measure o f the ratio o f the rate o f spread o f the effects o f momentum changes in the flow to the rate of spread of the effects of temperature differences, i.e., of the effects o f h eat transfer in the flow. Similar interpretations o f the other commonly used dimensionless numbers can b e derived. Some o fthese interpretations are given in Table 1.2. 1.7 FLUID PROPERTIES T he calculation of heat transfer rates will almost invariably require the determination of some or all of the fluid properties p, J.L (or v), k, f3, Cp (or Pr), v = J.LI P being the kinematic viscosity. These properties are all, in general, temperature and pressure dependent although the effect of pressure changes on J.L, k, f3, and Cp is often negligible. These properties are usually listed for various fluids in tabular form [1],[2],[20],[21], tables of air and water properties being given in the Appendix. The variation of some of the properties o f air and water with temperature are shown in Figs. 1.21 and 1.22. In many circumstances, it is more convenient to have equations describing the variations of these properties such equations being available for many common fluids [21]. 1.8 CONCLUDING REMARKS This chapter has been concerned with the meaning of "convective heat transfer" and the "heat transfer coefficient". T he dimensionless variables on which convective heat transfer rates depend have also been introduced using dimensional analysis. C HAPTER 1: Introduction 27 2.0 . ----.--....----.-......--r----r-..., Air ~ 150 . ---..,.-....,--.----.--....----.-..., "'S A ir 1.5 , l ~ , .q 1.0 ...... x o ~ 100 i:';> ~ .g 0.5 ~ :> 1;l 50 I O~~-~-L-~-~~-~ 0.0 ~--L_-'-----lL-..--'-_..L......---L_...J - 100 0 100 200 300 400 500 600 T emperature - °C ~ - 100 0 1.0 100 200 300 400 500 600 T emperature - °C 80 Air r-----.--"r---r--~_..,.,-....,____, ~ , '.0 ... E 60 A ir x : [ 40 8 i 1 p .. L ..----L_-'-----lL-..--'-_..L......---L_...J i -S ~ 0.8 r- 0.6 20 J o 0.4 ~---L_~_L-_-,---I---L_-'--~ - 100 0 100 200 300 400 500 600 T emperature - °C - 100 0 1 00' 200 300 400 500 600 T emperature - °C F IGURE 1.21 Variation of some properties of air at standard pressure with temperature. PROBLEMS 1.1. Discuss briefly SIX practical situations i n which convective heat transfer occurs. 1.2. A ir flows through a plane duct, i.e., effectively between two large plates, with a width lof 2w. The flow can be assumed to b e the same a t all ~ections across the duct. The velocity distribution i n t he duct is approximately given by u = 12[(w - Y)/W]l/5 mis, y . being the'l(J.istance from the center-plane to the point a t which u is the velocity and the temperature distribution i n the flow is approximately given b y 6 0 - 40[(w - y)/w] lI5°C. F ind the mean temperature in the flow. 1.3. I n an electrical heater, energy is g~nerated i n a 1.5-mm diameter wire at a rate o f 1200 W. A ir at a temperature of 30°C i s blown over the wire at a velocity that gives a mean heat transfer coefficiept of 50 WIm2°C. I f the surface temperature o f the wire is not to e xceed 200°C, find the length ?f the wire. 1.4. Typical values o f t he mean heat tr~sfer coefficient for a variety o f situations were \ / listed i n Table 1.1. Discuss some o f \the physical reasons why these values vary so greatly from one situation to another. / 28 Introduction to Convective H eat T ransfer Analysis 1100 Water N 2 ~ I 1000 M e Water ~ 900 I e '.& .... x .IJ ~ C ~~ CIJ 14 5 800 700 600 ':r;:: 1=1 > 0 .... 0 CIJ 0 1 S v 0 0 15 ~ 0 100 200 300 100 200 300 Temperature - °C Temperature - °C 0.8 U .§ ~ I ° I Water Water ';> 'J:! 0 0 .g 0.6 u § '8 10 i ~ -a ~ 5 0.4 0 100 200 300 Temperature - °C J 5 0 0 100 200 300 Temperature - °C F IGURE 1.22 Variation o f s ome properties o f w ater w ith temperature. 1.5. I n s ome forced convective flows i t h as been found that the Nusselt number is approximately proportional to the square root o f t he Reynolds number. If, in such a flow, it is found that h h as a v alue o f 15 W /m 2K w hen the forced velocity has a magnitude o f 5 mis, find the heat transfer coefficient i f the forced velocity is increased to 40 mls. 1.6. T he s urface tension a has t he d imensions force p er u nit length. Consider a convective - ---heattr-ansfer-situati6nm-whieh-surface tension-is lmportant.--8how thatthe--additional quantity p U 2 .e/a, t ermed the Weber number, We, is required to determine the Nusselt number. 1.7. If, in the case o f gas flows, the heat transfer coefficient is assumed to depend o n t he speed o f sound, a, i n the gas in addition to the variables considered i n this chapter, find the additional dimensionless n umber on which the Nusselt number will depend. u ------ 1.8. D iscuss the physical interpretation o f (i) the Weber n umber a nd (ii) the Stanton number. 1.9. Discuss why dimensionless variables are used i n p resenting convective heat t ransfer results. CHAPTER 1: Introduction 29 1.10. Using the results given in the Appendix, draw a graph showing how the density at ambient pressure, p, the coefficient o f viscosity, FL, the thermal conductivity, k, and the Prandtl number, Pr, o f air vary with temperature for temperatures between 0 and 500°C. 1.11. Using the definition of f3 in conjunction with the perfect gas law, show that for a perfect gas, f3 = l iT. 1.12. It is often assumed that for gases: FLref where T is the absolute temperature and / Lref is the viscosity o f the gas at some chosen temperature T ref. Using the properties o f air given in the Appendix and taking T ref as 273 K, plot the variation o f the index, n, for air with temperature for temperatures between - 50 and + 200°C. 1.13. Consider heat transfer from a circular cylinder whose axis is normal to a forced flow and which is rotating at an angular velocity, w. I f the surface o f the cylinder is maintained at a uniform temperature, find the dimensionless parameters on which the Nusselt number depends. 1.14. One way o f varying the the Grashof number in experimental studies involving air is to keep the model size fixed and to vary the air pressure. Consider a situation in which the model height is 5 cm, the model surface is at a temperature o f 60°C, and the air at a temperature o f 20°C. Plot the variation o f Grashof number with air pressure for air pressures between 0.1 and 10 times standard atmospheric pressure. 1.15. Discuss the physical meaning o f the Eckert number. REFERENCES 1. Incropera, EP. and DeWitt, D.P., Fundamentals o f H eat a nd Mass Transfer, 3 rd ed., Wiley, New York, 1990. 2.' Mills, A F., Heat Transfer, Irwin, Homewood, IL, 1992. 3. 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McAdams, W.H., Heat Transmission, 3 rd ed\, McGraw-Hill, New York, 1954. 14. Jakob, M., Heat Transfer, Vols. 1 and 2, Wiley, New York, 1949, 1957. 15. Jakob, M . a nd Hawkins, G.A., Elements o f Heat Transfer, 3 rd ed., Wiley, New York, 1957. 16. Bridgeman, P. W., Dimensional Analysis, Yale University Press, New Haven, CT, 1931. 17. Kreith, F. a nd Black, W.Z.M, Basic Heat Transfer, H arper & Row, New York, 1980. 18. Langhaar, H.L., Dimensional Analysis and Theory o f Models, Wiley, New York, 1951. 19. Van Driest, E.R., " On Dimensional Analysis and the Presentation o f D ata in Fluid Flow Problems", J. Appl. Mech., 13, A-34, 1940. 20. C hapman, A.J., Heat Transfer, 3 rd ed., Macmillan, New York, 1974. 21. Ried, R.C., Prausnitz, I.M., a nd Poling, B. E., The Properties o f Gases and Liquids, M cGraw-Hill, New York, 1987.