Asymptotes and Asymptotic Analysis for Development of Compact Models for Microelectronics Cooling M. Michael Yovanovich Distiniguished Professor Emeritus Principal ScientificAdvisor Microelectronics Heat Transfer Laboratory University of Waterloo, ON, Canada email: [email protected].
Abstract– In the thermal analysis of microelectronic sys- F (Pr) = Prandtl number function tems various levels of compact models are required to bridge g = gravitational constant, m/s2 thegapbetweentheuseofsimplecorrelationequations G = body gravity function which give quick, partial solutions and the use of costly, time consuming, numerical methods which give complete G = dimensionless reciprocal flux, 1/q∗ solutions with much detail. There are various levels of com- h = heat transfer coefficient, W/m2K pact models: some are based on simple combinations of two 2 HB = Brinell hardness, N/m or more correlation equations; some are based on resistor networks; and some are based on various combinations of k = thermal conductivity, W/mK asymptotic solutions. This paper shows that many different Kn = Knudsen number, Λ/D complex physical problems, when they are nondimensional- L = length of tube, pipe, duct, m ized, exhibit similar trends, i.e., that the complex solution L = optical path length, m varies smoothly between two asymptotic solutions. Rules λ are presented for the method of combining the asymptotes = arbitrary length scale, m and how to calculate the “fitting” parameter. One exam- m,L n = exponents of dimensionless solutions ple shows how to develop a compact model for steady con- M = gas parameter, αβΛ duction across a gas layer for very small and very large Knudsen numbers. A compact model for predicting the Nu = Nusselt number, hL/k complex elastic-plastic contact between a hard sphere and p = fitting parameter a softer substrate is presented. This elastic-plastic contact P = perimeter, m model is required for development of the thermal spreading- constriction model. Another example shows how to model P =gaspressure,Pa the effect of very small and very large Prandtl numbers for Pr = Prandtl number, ν/α forced and natural convection from isothermal plates. Ex- Q = heat transfer rate, W amples are given from forced and natural convection in short QL∗ = dimensionless heat flow, Q /kAθ0 and long ducts and pipes of arbitrary cross section for uni- 2 L form wall temperature and wall heat flux. The simple com- q =heatflux, W/m pact models are found to be accurate when compared with q∗ = dimensionless heat flux numerical results and experimental data, and they are easy r =radius,m to implement. Re = Reynolds number, U /υ Keywords– Asymptotes, asymptotic analysis, compact Ra = Rayleigh number, gβL∆T 3/αν models, steady and transient conduction, radiation, rar- L S∗ = dimensionless shape factor, Q /kAθ0 efaction, contact mechanics, elastic, plastic, elastic-plastic, L L forced and natural convection. t =time,s T =temperature,K Nomenclature U = mean velocity, m/s A = flow area, m2 w = axial velocity, m/s a = contact radius, m a, b = semi major and minor axes of ellipse, m Greek a, b = half dimension of rectangle, m α = thermal diffusivity, m2/s AR = aspect ratio α = accommodation coefficient ci = constants, i =1...5 β = gas parameter cP = specificheat,J/kgK γ = ratio of specificheats,cp/cv D =gapthickness,m γ = symmetry parameters Dh = hydraulic diameter 4A/p, m ∆ = thermal boundary layer thickness, m E1,E2 = Young’s modulus, Pa δ = momentum boundary layer thickness, m f =frictionfactor,2τ/ρv2 ² = aspect ratio, b/a < 1 F =load,N Λ = gas mean free path, m 2 2 F0 = dimensionless time, αt/A µ = dynamic viscosity, Ns/m µ∗ = dimensionless gas parameter, M/D rameters such as Reynolds number or modified Reynolds ν = kinematic viscosity, m2/s for forced convection and Rayleigh and modified Rayleigh ν1, ν2 = Poisson’s ratio number for pipe and channel flows, and the Prandtl num- ρ = fluid density, kg/m3 ber if fluids other than air are used. ρ = radius of curvature, m There is a need for the development simple “compact τ = wall shear stress, Pa models” which incorporate one or more asymptotic solu- θ = temperature excess, T Ti,K tions and which are not limited to particular ranges of the − φ = dimensionless dependent parameter independent parameters. These compact models can there- ξ = dimensionless independent parameter fore be used over broader ranges of the parameters and are Stephan-Boltzman constant, σ = more useful and valuable than the simple correlation equa- 5.67E 8,W/m2K4 tions alone. − This paper will be restricted to several relatively simple, Subscripts but important, compact models which have been applied 0 = reference values to some microelectronics cooling problems. 0 = corresponding to very small value = corresponding to very large value II. Asymptotes and Simple Compact Model 1∞, 2 = surfaces 1 and 2 development √A = based on square root of area Many complex physical systems exhibit a smooth transi- bl = boundary layer tion between two asymptotic solutions [1,2,3]. The smooth c =critical transition means that there are no discontinuities and no Dh = based on hydraulic diameter sudden changes in slope within the transition region. e =entrance These asymptotes appear in steady and transient inter- fd = fully developed nal and external conduction, steady conduction across a = based on arbitrary length gas layer between two large parallel isothermal plates, radi- RL =radiation ation through a highly porous substance between two large t =thermal isothermal walls, contact between a “rigid” sphere and a x = local value softer substrate, natural and forced internal and external w=wall convection. There are many other examples from conduc- λ =optical tion, fluid flow, and mass transfer; however, they will not be considered in this paper. I. Introduction Since the various problems have different dependent and HE present demands on the thermal analysts who independent parameters and different symbols are used, it Tmust daily deal with microelectronics cooling issues is is necessary to present the asymptotes and the method of daunting. They are required to provide, in very short time, combining them in a general form based on dimensionless reliable estimates of temperatures within devices, packages, dependent and independent parameters. boardsandsystemswhichareoftencooledbyairbyeither The dimensionless dependent parameter is called φ and natural or forced flow. Although experiments and numer- the independent parameter is called ξ. The functional de- ical methods can be employed to find the temperatures pendence φ(ξ) can be relatively simple or quite complex and provide much detail about the temperature distribu- depending on the physical problem. tion throughout the system; these methods are both time The dimensionless parameter φ has two asymptotes cor- consuming and costly. Certainly in the early stages of de- responding to very small and very large values of the di- sign, experiments are not feasible and numerical methods mensionless independent parameter ξ [1,2,3]: may require information that is not readily available. Fig- m φ0 = C0 ξ as ξ 0 ure 1 illustrates that the single correlation equations fall in φ → (1) the lower left corner, the numerical results lie in the upper → ( φ = C ξn as ξ or ξ 1 right corner where much detailed information is available ∞ ∞ →∞ → at some high cost, and the compact models form the rel- The asymptotes φ0 and φ are based on analytical solu- ative large “bridge” between the very low cost and high tions and they consist of a constant∞ which is a positive, real cost regions. There are many different types of compact number. The two constants are denoted as C0 as ξ 0 models ranging from relatively simple models to relatively and as C for ξ or ξ 1. The exponents m and→ n complex models. can have∞ the values→∞ such as→ 0, 1, 1/2, 1/4, 1/3 for example It is apparent that alternative simpler methods are re- [1,2,3]. quired in the first stages of device, package and board devel- It is known from analysis, experiments and numerical opment to give quick and acceptable estimates of temper- work that φ frequently transitions in a smooth manner be- aturelevels.Oneverysimplemethodisbasedontheuse tween the two asymptotes. of correlation equations which have been developed from The plots in Figs. 2 through 5 illustrate four basic, but experimental or numerical data. Often the correlations different, trends. Figure 2 shows both asymptotes φ0 and are limited to the particular ranges of the independent pa- φ increasing with increasing values of ξ, and the solution ∞
2 Compact Modeling
Numerical Models •Highly detailed results •Time consuming sign •Steep learning curve final de tion detail
u Complex Models
sol Compact Models gn si Simple Models yde r Correlations ina
• quick, easy to use m - constrained• constrained to limited to limited rangerange of data of data preli
cost
Fig. 1. Overview of Simple, Compact and Detailed Models
Fig. 2. Asymptotes and Compact Model 1 Fig. 3. Asymptotes and Compact Model 2
φ is concave upwards. For example, external forced and natural convection from single isothermal convex bodies downwards. For example, internal forced and natural con- exhibit this trend. vection within ducts and channels exhibit this trend. Figure 3 shows both asymptotes φ0 and φ decreasing with increasing values of ξ, and the solution∞φ is concave Figure 5 shows the asymptote φ0 to be constant indepen- upwards. For example, steady conduction in spherical wall dent of ξ and the asymptote φ decreases with increasing and external transient conduction from isothermal convex values of ξ, and the solution φ∞is concave downwards.An bodies exhibit this trend. example from tribology is the temperature rise of a heated Figure 4 shows both asymptotes φ0 and φ increasing contact area which moves with constant velocity in the sur- with increasing values of ξ, and the solution ∞φ is concave face of a semi-infinite body [4].
3 The parameter p is a fitting parameter whose value can be found in a simple manner. To find a value of p we can select an intermediate value of ξ = ξi for which φ(ξi) is known through analytical, exper- imental or numerical methods. For the intermediate value of ξ we can write
m p n p 1/p φ(ξi)=[(C0ξi ) +(C ξi ) ] (4) ∞ or 1 1 p 1 p 1/p = m + n (5) φ(ξi) C0ξi C ξi ·µ ¶ µ ∞ ¶ ¸ In both relationships the “fitting” parameter p is unknown. It can, however, be calculated by numerical methods or by means of computer algebra systems such as Maple, Math- ematica, and Mathcad.
B. Alternative Forms of Simple Compact Models
Fig. 4. Asymptotes and Compact Model 3 Often the approximate solution is presented in a form which is based on one of the two asymptotes. If this is
done, then the two models can expressed as [1,2,3]:
p 1/p φ0 [1 + (φ /φ0) ] model 1 φ = ∞ (6) p 1/p φ [1 + (φ0/φ ) ] ∞ ∞ p 1/p φ0/ [1 + (φ0/φ ) ] model 2 φ = ∞ (7) p 1/p φ / [1 + (φ /φ0) ] ∞ ∞ Several examples from steady and transient conduction, contact mechanics, forced and natural convection heat transfer will be presented in the following sections to illus- trate several features of the asymptotes, asymptotic anal- ysis and the development of simple compact models. III. Steady Conduction Across Stationary Gas Layer The first example illustrates the importance of asymp- totes for steady conduction across a stationary gas layer Fig. 5. Asymptotes and Compact Model 4 placed between two infinitely large parallel isothermal plates. The absolute temperatures of the bounding sur- A. Combined or Simple Compact Models faces are T1 and T2 where T2
2γ N2 (Teagan and Springer, 1967) β = (10) * (γ +1)Pr 101 interpolated model, G = 1+ M where γ = cp/cv,andPr = ν/α is the Prandtl number of the gas. The third gas parameter is called the molecular G mean free path and its defined as [5,6] 100 T P Λ = Λ 0 (11) 0 T P µ 0 ¶µ ¶ The molecular mean free path Λ0 is the value at the ref- T P 10-1 erence temperature 0 and reference gas pressure 0.Its -3 -2 -1 0 1 2 seen from the relationship that increasing the gas tempera- 10 10 10 M* 10 10 10 ture or decreasing the gas pressure increases the molecular mean free path. Some typical values of gas properties are listed in Table 1. Fig. 6. Comparison of Data and Compact Model for Conduction Across Gas Layer Gas parameter values are for T0 =288KandP0 =760torr
TABLE I ? ? We now set φ = q , ξ = M , φ0 =1andφ =1/ξ.Plots Typical Values of Gas Parameters ∞ of the two asymptotes show that φ0 > φ as ξ 0, and the solution is cave downwards. This trend∞ corresponds→ to the α γ Pr k Λ g 0 example shown in Fig. 5 and, therefore, the simple compact Gas [—] [—] [—] W/(m K) nm model is given by Eq. (3). The dimensionless parameters, Argon 0.90 1.67 0.67 0.0177· 66.6 the constants and the exponents for the asymptotes are Nitrogen 0.78 1.41 0.69 0.0259 62.8 listed in Table 2. Helium 0.55 1.67 0.67 0.150 186 TABLE II Parameters for Conduction Across Stationary Gas Layer (1 atm).
The heat flux is given by the following relationship φξC0 mC np [5,6,9,10] qD M ∞ T1 T2 10 11 1 q = k − (12) kg (T1 T2) D − g D + M − This relationship has two asymptotes depending on the rel- ative magnitude of M/D or M/D. The asymptotes are The alternative relationship for the heat flux is T1 T2 M 1 D + M D M q0 = kg − as 0continuum D D = = + (17) → q kg (T1 T2) kg (T1 T2) kg (T1 T2) q = − − − T1 T2 M q = kg − as free molecules This relationship clearly shows how the two asymptotes ∞ M D →∞ (13) for the heat flux combine to give the compact model for conduction across the gas layer. The non-dimensional form The two heat flux asymptotes q0 and q depend on the ∞ is dimensionless parameter M/D whichcanbeexpressedas 1 1 1 M 1 [5,6] ? = ? + ? =1+ =1+ ? (18) M Λ q q0 q D M M? = = αβ = αβKn (14) ∞ D D This form is consistent with that given by the general model where Kn = Λ/D is called the Knudsen number for the gas for combining asymptotes. This relationship shows that the layer. The Knudsen number depends on the temperature “fitting” parameter must be p = 1 in Eq. (3). and the pressure of the confined gas. The simple compact model was compared against data We can nondimensionalized the heat flux such that obtained for Argon, Nitrogen and Helium or a wide range of qD the parameter M/D. Song [5] and Song et al. [6] defined q? = (15) G =1/q? which was called the dimensionless gap resis- kg (T1 T2) − tance and M? = M/D which was called the gas rarefac- The dimensionless heat flux has the following two dimen- tion parameter, and compared the simple gas-layer model sionless asymptotes: expressed as ? q? =1 as Kn 0 G =1+M (19) 0 → q? = (16) against the experimental data obtained for Argon, Nitrogen ? D 1 q = = ? as Kn and Helium as shown in Fig. 6 for a wide range of the ∞ M M →∞ 5 dimensionless gas-layer parameter M?. The Argon and Comparison of this problem with the general model shows ? ? ? Nitrogen data [9] and the Argon and Helium data [10] are that φ = q , φ0 = q0, φ = q ,andξ = `/L.These seen to lie in the continuum regime for M ? < 0.1, the free parameters and the corresponding∞ ∞ constants and the expo- molecule regime for M? > 10 and in the transition. The nents are listed in Table 3. Plots of the asymptotes with agreement between the simple gas layer conduction model and all data is excellent. TABLE I II Parameters for Radiation Through Porous Layer of IV. Simple Model for Radiation Through Porous Material Substance φξC mC np In this example we examine steady heat transfer by radi- 0 q ` ∞ ation through a porous layer of thickness L. The problem 4/31 1 01 σ (T 4 T 4) L and its solution is given in Gebhart [11]. 1 − 2 The optical path length Lλ may be very small, i.e., Lλ/L << 1, or very large, i.e., Lλ/L >> 1relativeto respect to ξ show that φ < φ as ξ 0, and the solu- the layer thickness. If Lλ/L << 1, the material is said to 0 tion is concave downwards. The∞ approximate→ solution is be opaque, and if Lλ/L >> 1, the material is said to be therefore the second model, Eq. (3). nonopaque. If the material is assumed to be gray, Lλ is not Theapproximatemodelforq? is, therefore, a function of λ and we set Lλ = `. There are two models corresponding to the magnitude of the parameter `/L.The p p 1/p layer is assumed to be in contact with two isothermal black 1 1 1 ? = ? + ? (26) “infinitely” large surfaces whose absolute temperatures are q q0 q ·µ ¶ µ ∞ ¶ ¸ T and T with T >T . 1 2 1 2 A number of analyses have indicated the behavior of q? A. Thin Layer Asymptote: Nonopaque Material in the intermediate range which lies between the asymp- totes. An approximate result by Probstein [11] which is in For very thin layers the net radiation transport across agreement with the exact analysis is the layers is given by [11]
? 1 4 4 ` q = (27) q = σ T1 T2 for >> 1(20) 3 ` ∞ − L 1+ 4 L where σ is the Stefan-Boltzmann¡ ¢ constant. This indicates that the “fitting” parameter is p =1. B. Thick Layer Asymptote: Opaque Material V. Steady Conduction in a Spherical Enclosure For very thick layers, the radiation transport in the gray material is by local radiation and absorption. The layer is Steady conduction occurs in a spherical wall having radii locally nearly in radiant equilibrium. The transport pro- a, b with a 4 ` 4 4 r = a, T = T1 and r = b, T = T2 (29) q0 = σ T T (23) 3 L 1 − 2 The solution, in a convenient nondimensional form, is showing the `/L dependence.¡ To illustrate¢ the transport rates, we introduce the dimensionless radiation flux T T2 1/r 1/b − = − a r b (30) q T1 T2 1/a 1/b ≤ ≤ q? = (24) − − σ (T 4 T 4) 1 − 2 Alternative forms of the solution are given in texts on con- The dimensionless radiation flux asymptotes are duction [12] and heat transfer [13,14]. The heat transfer rate into the spherical wall at the inner 4 ` ` q? = as 0 boundary is found from 0 3 L L q? = → (25) 2 dT ? ` Q =4πa k (31) q =1 as − dr ∞ L →∞ · ¸r=a 6 ? The conduction [12] and heat transfer texts [13,14] present AplotofQ versus L/√Ai shows that this relationship the result in the following forms: has two asymptotes corresponding to the very thin and the verythickwalls;theyare 4πk (T1 T2) ab 4πk (T1 T2) Q = − or Q = − (32) b a 1/a 1/b √A L − − Q? = i as 0 0 L √A → whichdonotshowtheverythinandverythickwallasymp- Q? = i (39) totes explicitly. L Q? =2√π as If the spherical wall is infinitely thick, i.e., b/a , ∞ √A →∞ →∞ i then the relationship for the spherical wall goes to the re- lationship for the isolated sphere: For this example the two asymptotes are linearly combined to give the complete model for the spherical wall. Q =4πak (T1 T2)(33)Comparison with the general model shows that φ = − ? ? ? Q , φ0 = Q0, φ = Q and ξ = √Ai/L. Plots of the two The traditional relationships for Q do not reveal other ∞ ∞ asymptotes with respect to ξ show that φo > φ as ξ 0. very important features of the solution that can be used Since the solution is concave upwards, the first∞ model,→ Eq. to model other systems such as steady conduction in an (2), must be use with p = 1, which is in agreement with enclosure formed by two concentric isothermal cubes, for the analytical relationship. example. The dimensionless parameters and the corresponding If we introduce the wall thickness L = b a into the constants and exponents are listed in Table 4. relationship for Q, and after some algebraic manipulations− we obtain the new, more interesting, relationship: TABLE IV ParametersforSteadyConductioninSphericalWall T1 T2 T1 T2 Q =4πa2k − +4πa2k − (34) L a µ ¶ µ ¶ φξC0 mC np Q? √A /L 112∞π 01 The new relationship shows that the heat transfer rate into i √ the spherical wall consists of two terms which become very important depending on the relative magnitude of the wall thickness L to the inner radius a. The new relationship clearly shows that the heat transfer rate into the spherical VI. Model for Enclosures Formed by Two wall has two asymptotes which are: Isothermal Concentric Boundaries The analytical result found for the enclosure formed 2 T1 T2 L Q0 =4πa k − as 0 by two concentric isothermal spheres can be used to de- L a → Q = µ ¶ velop approximate models for other enclosures such as one 2 T1 T2 L formed by concentric cubes for which there is no analyt- Q =4πa k − as ∞ a a →∞ ical solution. The three-dimensional temperature field in µ ¶ the enclosure formed by two concentric cubes is complex, (35) The relationship for Q can be made nondimensional with and the heat flux distribution over all surfaces of the inner respect to some system length scale denoted as to give: isothermal cube is also highly nonuniform. The maximum L heat flux values occur along the edges and the corners of Q Q? = L (36) the cube. Aik (T1 T2) Thesimplecompactmodelforthecube-in-cubeenclo- − 2 sure can be expressed as where Ai =4πa is the surface area of the inner sphere. The nondimensional form of the relationship is, therefore, p 1/p √A Q? = i +(Q? )p (40) Q? = L + L (37) " L ∞ # L a µ ¶ There are three system length scales; the two radii and the The nondimensional heat transfer rate from an isolated ? wall thickness. We cannot select b or L because they both cube was found by numerical methods to be Q =3.391 ∞ become infinitely large in one limit. Therefore, we must which is close to the analytical value for the isothermal ? select a length scale associated with the inner sphere. We sphere which is Q =2√π =3.545 [15,16] which is ap- ∞ can select its radius or the square root of the inner surface proximately 4.5% larger. If the inner cube has side dimen- area which is more convenient for subsequent modeling. If sions Li and the outer cube has side dimensions Lo,then the geometric parameters for the approximate model are we select √Ai =2√πa set to [15,16], then we get the analytical relationship: L Lo Li √Ai 2√6 Ai = √6Li L = − = ? √Ai L 2 L Lo/Li 1 Q = +2√π for 0 < < (38) − L √Ai ∞ p (41) 7 The simple compact model for the cube-in-cube enclosure equation which, in spherical coordinates, is [12] can be expressed in terms of the side dimension ratio as 1 ∂θ 2 ∂θ 1 ∂θ p 1/p 2 r = ,r>a,t>0(43) 2√6 r ∂r ∂r α ∂t Q? = +(3.391)p (42) µ ¶ Lo/Li 1 "à − ! # The initial condition is By trial and error the fitting parameter value was found t =0,ra, θ(r, 0) = 0 (44) to be p =1.070. The comparisons between the numerical ≥ values and those obtained from the approximate models The boundary conditions are with p =1andp =1.070 are listed in Table 5. for t>0, θ(r, t)=θ0 at r = a (45) TABLE V Comparisons of Numerical Values and Values from Two for t>0, θ(r, t) 0asr (46) Approximate Models → →∞ The solution is given in [12]: Lo/Li numerical model 1 model 2 θ0a r a 1.2 27.52 27.89 27.24 θ = erfc − t>0,r a (47) r 2√αt ≥ 1.5 12.77 13.19 12.71 µ ¶ 2.0 7.87 8.29 7.93 where erfc( ) is the complementary error function. The 5.0 4.45 4.62 4.45 · 10 3.89 3.94 3.84 temperature gradient on the surface of the sphere is [12] 50 3.52 3.49 3.46 ∂θ θ θ = 0 + 0 ,t>0(48) −∂r a √π√αt · ¸r=a All values from the first compact model, with the ex- ception of the last point, lie above the numerical values. The temperature gradient consists of two terms: a term The maximum and average differences between the numer- whichisconstantwithrespecttotimeanditsrelatedto ical values and the first approximate model with p = 1 are the sphere radius a, and a second term which is related to 5.4% and 2.3%, respectively. The maximum and average the thermal penetration thickness √αt.Thetemperature differences between the numerical values and the second gradient, therefore, consists of two asymptotes: the small compact model with p =1.070arereducedtoabout1.7% penetration depth asymptote where √αt/a << 1andthe and 0.03%, respectively. thick penetration depth asymptote where √αt/a >> 1. The simple compact model developed for the cube-in- The instantaneous heat transfer rate from the sphere sur- cube enclosure can be used for any enclosure in which face into the surroundings is obtained from Fourier’s Law the gap thickness is uniform. For example, it has been of Conduction: used for enclosures formed by two concentric finite circular 2 ∂θ 2 θ0 2 θ0 cylinders, and other geometries. The asymptote for very Q = k4πa =4πa k +4πa k ,t>0 −∂r r=a a √π√αt thin gaps is unchanged; however, the value for the very µ ¶ (49) thick asymptote must be modified slightly to account for The above expression can be written in terms of two asymp- the shape and aspect ratio of the inner body. The agree- totic heat transfer rates: ment between the numerical values and the simple compact model values where found to be close. Q = Q + Q0 (50) If the gap thickness varies with position, a more complex ∞ model is required to account for the regions of the enclosure 2 θ0 √αt Q0 =4πa k as 0 where the gap thickness is very small. √π√αt a → Q = µ ¶ (51) VII. Transient Conduction External to θ0 √αt Q =4πa2k as Isothermal Spheres ∞ a a →∞ µ ¶ A sphere of radius a is placed in a large stationary The solution for the external transient conduction from medium whose thermophysical properties, k,thethermal the surface of an isothermal sphere into a medium of large conductivity, and α,thethermaldiffusivity are constant. extent shows a striking resemblance to steady conduction Initially, the temperature of the sphere and the surround- through a spherical wall of thickness L on a sphere of radius ings is Ti everywhere. Suddenly the surface temperature of a.Theinfinitely thick layer asymptote where L/a the sphere is raised to uniform and constant temperature is identical to the steady-state solution asymptote where→∞ T0 such that T0 >Ti for all t>0. √αt/a . The temperature field external to the sphere r>ais The→∞ layer thickness in the steady conduction problem transient and radial, T (r, t). The temperature excess θ = and the thermal penetration depth in the transient con- T (r, t) Ti is the solution of the one-dimensional diffusion duction problem play similar roles. − 8 The dimensionless heat transfer rate at the sphere sur- 3.19 S? 4.20. The “fitting” parameter depends on ≤ √A ≤ face is defined as the aspect ratio, and its values are in the narrow range[17]: Q Q? = L (52) 0.87 p 1.10. The smaller values of p correspond to L Akθ0 bodies≤ with≤ high aspect ratios and the larger values of p where represents some arbitrary length scale which is correspond to bodies with small aspect ratios. relatedL to the dimensions of the sphere gives VIII. Elastic-Plastic Model for Contact Between Hemisphere and Flat Q? = L + L (53) L a √π√αt The mechanical contact between a rigid hemisphere hav- ing radius of curvature ρ and a flat substrate with radius of If we choose the recommended length scale [12], i.e. = curvature ρ is a circular area whose radius is a.The √ L A, we obtain the relationship: elastic properties→∞ of the hemisphere and the substrate are 1 Young’s modulus: E1 and E2, and Poisson’s ratio: ν1 and Q? =2√π + (54) ν . The subscripts 1 and 2 denote the hemisphere and sub- √A π αt/A 2 √ strate respectively. The Brinell hardness of the substrate is The dimensionless time αt/A is oftenp denoted as Fo.Com- much smaller than the Brinell hardness of the hemisphere. parison with the general model leads to the following re- If the contact strain defined as ²c = a/ρ << 1, the defor- ? √ mation of the hemisphere and the substrate is elastic, and lationships: φ = Q√ , φ0 =1/√π Fo, φ =2√π and A ∞ the classical elasticity theory of Hertz can be used to calcu- ξ = Fo. The dimensionless parameters and the correspond- late the magnitude of the contact radius. The Hertz elastic ing constants and exponents are listed in Table 6. Plots of contact model in [18,19] shows that the contact radius is TABLE VI given by the relationship: Parameters for Transient Conduction External Isothermal 3 F ρ 1/3 Sphere a = (56) e 4 E µ 0 ¶ φξC0 mC np ? ∞ where F is the steady axial mechanical load supported by Q√A Fo 1/√π 1/22√π 01 − the contact area, and E0 is the effective modulus of the contact [18,19]: the asymptotes show that φ0 > φ as ξ 0andtheso- 1 1 ν2 1 ν2 ∞ → = − 1 + − 2 (57) lution is concave upwards. The model must be as shown E E E in Fig. 2 and Eq. (2) must be used. This is in agreement 0 1 2 with the analytical solution. For a particular contact, the geometry and elastic prop- The first term on the right hand side of the relation- erties are fixed and the mechanical load is the variable. ship is the dimensionless shape factor S? valid for steady Therefore, the elastic contact radius relationship can be √A conduction from an isolated isothermal convex body in a expressed as medium of large extent. The following general simple com- 1/3 pact model with “fitting” parameter p canbeusedforex- 1/3 3 ρ ae = ceF where ce = (58) ternal transient conduction from arbitrary isothermal con- 4 E µ 0 ¶ vex bodies into a medium of large extent: This relationship shows how the contact radius increases p 1/p with the load. p 1 Q? S? As the load increases, the contact strain increases, and √A = √A + (55) " Ã√π αt/A! # the maximum shear zone which lies on the axis of the con- ³ ´ tact at points some distance below the contact area, grows p The “fitting” parameter will depend on the body shape and in some complex manner [19,20,21]. For very large contact its aspect ratio. This more general simple compact model strains, ²c < 1, the deformation of the substrate becomes was used to model transient conduction from the isother- fully plastic. mal convex bodies shown in Fig. 7. The figure shows oblate The contact radius under these conditions no longer de- and prolate spheroids, cuboids, the circular and square pends on the elastic properties, but does depend on the disks, the rectangular strip, the sphere and the cube. The plastic property of the substrate. The plastic yield is re- aspect ratio of the bodies lies in the range: 0 AR 10. lated to the Brinell hardness HB of the substrate, and its The numerical values[17] of Q? for the various≤ bodies≤ are √A assumed to be a fully work-hardened solid. 6 3 shown in Fig. 8 for the wide range: 10− αt/A < 10 . From a force balance applied to the contact area we find All numerical values approach and lie on≤ the asymptote the relationship for the plastic contact radius [19,21]: corresponding to very short times and follow the trend of the sphere for Fo > 10 4. The values of the dimen- F 1/2 − a = (59) sionless shape factor lie in the relatively narrow range[17]: p πH µ B ¶ 9 OBLATE SPHEROID SPHERE PROLATE SPHEROID (AR=0.5) (AR=1) (AR=1.93) PROLATE SPHEROID (AR=10) CIRCULAR DISK RECTANGULAR STRIP (AR=0) (AR=0) SQUARE DISK CUBE TALL CUBOID TALL CUBOID (AR=0.1) (AR=1) (AR=2) (AR=10) Fig. 7. Ellipsoids and Cuboids for External Transient Conduction circular disk (AR=0) square disk (AR=0.1) rectangular strip (AR=0) cube (AR=1) 2 10 tall cuboid (AR=2) tall cuboid (AR=10) oblate (AR=0.5) √ sphere (AR=1) Q* ≡Q A √A θ prolate (AR=1.93) kA 0 1 1 prolate (AR=10) 10 √π √ Fo√A half-space sphere Q* = 2√π + 1 √A √π √ Fo√A 100 10***-6 10 ***-5 10-4 10-3 10-2 10-1 100 101 102 103 ≡ αt Fo√A (√A)2 Fig. 8. Comparison of Numerical Values and Compact Model for Transient Conduction from Ellipsoids and Cuboids 10 This relation can be written as 102 mild-steel Tabor (1951) 1 1/2 a = c F 1/2 where c = (60) elastic-plastic model, p = 5 p p p πH µ B ¶ which clearly shows how the plastic contact radius grows c with the mechanical load. From pure elastic deformation a 1 to pure plastic deformation, there is a smooth transition / 10 from the elastic contact asymptote to the plastic contact a asymptote which are defined as [20]: ae for ²<<²c a = (61) ( ap for ²>>²c 100 10-2 10-1 100 101 102 ξ A model for the elastic-plastic contact can be developed in =F/Fc the following manner. The elastic and plastic contact radii are equal when the mechanical load is Fc [20] Fig. 9. Comparison of Elastic-Plastic Compact Model and Data for Contact Between Hard Sphere and Mild Steel Substrate 1/3 1/2 ae = ap gives ceFc = cpFc (62) Solving for Fc gives the relationship: The elastic-plastic transition can be modeled accurately by means of the following simple compact model based on 6 ce the contact radius [20]: Fc = (63) cp µ ¶ p p 1/p a = ae + ap with p =5 (68) which can be expressed as [20] with the “fitting”¡ parameter¢ value p =5obtainedby 9π3 H 2 comparison of the simple compact model predictions and F = ρ2H B (64) c 16 B E some accurate experimental data [21] for a particular µ ¶ µ 0 ¶ hemisphere-substrate contact. This equation shows the important relationship between The above relationship for contact radius leads to the the critical mechanical load Fc and the radius of curvature following relationship in terms of the applied load and the ρ, the elastic properties E0 and the Brinell hardness HB. elastic and plastic coefficients ce and cp: The critical contact radius a can be found from substitu- c 1/5 tion of F = F in the relationship for a or the relationship 5 5/3 5 5/2 c e a = ceF + cpF (69) for a . In both cases, the critical contact radius is [20] p ³ ´ The plot in Fig. 9 shows the very good agreement between 3π ρHB ac = (65) the simple compact model and the experimental data of 4 E0 [21] for a mild steel substrate. The experimental points The two asymptotes for the contact radius can now be es- are seen to fall in the elastic and plastic regions, as well as tablished by defining the relative load on the contact area: in the complex elastic-plastic region for which there is no analytical solution. The model and experimental data are F plotted as dimensionless contact radius versus the dimen- ξ = (66) Fc sionless contact load. The asymptotes for the contact radius are [20] IX. Prandtl Number Functions: Forced and Natural Convection from Isothermal Plates ae as ξ 0 We next consider the local Prandtl number functions for a = → (67) both laminar boundary layer forced and natural convec- ( ap as ξ →∞ tion from isothermal flat plates. In both examples the lo- For practical applications we can say that the contact is cal Prandtl number function is developed from analytical elastic provided ξ < 0.05, and the contact is plastic pro- asymptotes obtained for very small and very large values vided ξ > 20. The elastic-plastic transition lies in the prac- of the Prandtl number. tical interval: 0.05 ξ 20. The plots of the elastic and ≤ ≤ A. Prandtl Number Function for Forced Convection plastic asymptotes show that ae >ap as ξ 0andthe solution is concave upwards in the transition→ from elastic The boundary layer continuity, momentum and energy to plastic deformation. The first model, Eq. (2), must be equations for the forced laminar flow past an isothermal used for this problem. flat plate have been solved for very small and very large 11 values of the Prandtl number, i.e., Pr 0andPr . texts [13,14]. The average value of the Nusselt number is The analytical solutions have the following→ asymptotes→∞ and given by the relationship: the value corresponding to Pr = 1 [22,23]: Nu 0.6674 Pr1/3 L =2F (Pr)= (77) 1 1/4 Pr1/2 as Pr 0 √ReL 1+(0.0468/Pr)2/3 √π → Nux = 0.3321 at Pr =1 (70) where £ ¤ √Rex hL UL NuL = and ReL = (78) 0.3387 Pr1/3 as Pr k ν →∞ The summary of the dimensionless parameters, the con- From these asymptotes we have the following relationships stants and the exponents for the local Prandtl number func- for the dependent and independent dimensionless parame- tion for forced convection from an isothermal flat plate are ters: listed in Table 6. Nux φ = and ξ = Pr (71) TABLE VI I √Rex Parameters for Forced Convection Past Isothermal Flat where the local Nusselt and Reynolds numbers are defined Plate as hx Ux Nux = and Rex = (72) φξC0 mC npp0 k ν ∞ Nu The constants and the corresponding exponents for forced x Pr 0.564 0.50.338 1/34.613 4.5 convection are [22,23] √Rex 1 C =0.5642 and m = 0 2 (73) B. Prandtl Number Function for Natural Convection 1 C =0.3387 and n = The laminar boundary layer continuity, momentum and ∞ 3 energy equations for the buoyancy-induced laminar flow Plots of the two asymptotes with respect to the Prandtl past a vertical isothermal flat plate have been solved for very small and very large values of the Prandtl number, number show that φ0 < φ as ξ 0, and the solution is concave downwards. Therefore,∞ the→ second model, Eq. (3), i.e., Pr 0andPr . The analytical solutions have → →∞ must be used. The equation for the “fitting” parameter is the following asymptotes and the value corresponding to Pr =1[22,23]: 1 1 p 1 p 1/p = + (74) 0.6004 Pr1/4 for Pr 0 0.3321 0.3387 0.5642 → ·µ ¶ µ ¶ ¸ Nux 1/4 = 0.401 at Pr =1 (79) By means of computer algebra systems we obtain the value Rax 0.5027 for Pr p =4.612607 rounded to 6 decimals. If one uses the ap- →∞ proximate value p0 =9/2=4.5, the model for the local where Nux is the local Nusselt number and Rax is the local Prandtl number function can be expressed as: Rayleigh number which are defined as 1/3 Nux 0.3387 Pr h(x)x gβθ x3 = F (Pr)= (75) w 2/9 Nux = and Rax = (80) √Rex 1+(0.0468/P r)3/4 k αν where θw = Tw T is the constant temperature difference which is applicable in the£ range: 0 12 given by Eq. (3) must be used. Using the value φ =0.401 the compact model for the Prandtl number function de- at ξi = 1, gives the following relationship for the “fitting” veloped above. We seek a general relationship between the parameter p: dimensionless heat transfer rate Q? andtheRayleighnum- ber Ra : L 1 1 p 1 p 1/p L = + (83) ? Q gβ (Tw T ) 3 0.401 0.6004 0.5027 Q = L Ra = − ∞ (87) ·µ ¶ µ ¶ ¸ L Ak (Tw T ) L αν L − ∞ Solving for p we obtain the value p =2.265478 rounded to 6 decimals. Churchill and Chu [25] chose the approximate where denotes the arbitrary length scale of the convex body. L value p0 =9/4=2.50 and recommended the following relationship for the local Prandtl number function: The dimensionless heat transfer rate from the surface of the convex body into its surroundings has the following Nu 0.503 asymptotes [15,16,27]: x = F (Pr)= (84) Ra1/4 9/16 4/9 x 1+(0.492/P r) S? as Ra 0 Q? = L L → (88) £ ¤ which is valid for 0 13 body-gravity function based on = √A where the total wettedcuboidsurfaceareais L A =2(HL + HW + LW)(91) is [27] 3/4 0.625 L4/3W + H (L + W)4/3 G =21/8 (92) √A 7/6 " (HW + HL + LW ) # The numerator consists of two terms. The first term is the contribution of the top and bottom horizontal surfaces, and the second term corresponds to the contribution of the vertical side and end surfaces. The denominator is associated with the total surface area of the cuboid. Fig. 11. Compact Model and Air Data for Natural Convection from The calculated values for several cuboids show that they Cube are in the range [27]: 0.776 G 1.525 (93) ≤ √A ≤ The smallest value corresponds to a horizontal very thin square disk with relative dimensions of H =0.01,W = 1,L= 1. The largest value corresponds to a horizontal very long square bar relative dimensions of H =1,W =1,L = 100. The value for a horizontal cube is Gcube =0.985. C. Dimensionless Shape Factor There is no analytical solution for the dimensionless shape factor for an isothermal cuboid. Some numerical val- ues are available that show that S? changes slowly with √A its shape and the aspect ratios of the sides. For the cube, a numerical value is reported to be S√A =3.373 which com- pares with the analytical value for the isothermal sphere having the same surface area which is S? =2√π =3.545. Fig. 12. Compact Model and Air Data for Natural Convection from √A Horizontal Long Square Cuboid The difference is approximately 5%. The numerical values for square cuboids with relative dimensions: H =1,W = 1,L =2, 3, 4, 5areS? =3.406, 3.465, 3.532, 3.598, √A The model has been compared with air data with many which show the slow change with the aspect ratio. The di- different body shapes over broad ranges of the Rayleigh mensionless shape factors for isothermal ellipsoids, prolate number and the agreement is very good to excellent. Fig- and oblate spheroids, right circular cylinders and rectan- ures 11 through 13 show comparisons of the simple com- gular plates are given in [15]. pact model for air data for the cube, long square cuboid, For this problem we note the dimensionless dependent and the vertical thin cuboids. Figure 14 shows the predic- and independent parameters are φ = Q? and ξ = Ra , √A √A tions of the compact model for cuboids and the compact ? and the asymptotes are φ0 = S√ and φ = Nu√A.Since model for natural convection in vertical channels which is A ∞ φ0 > φ as ξ 0, the model is concave upwards and not presented in this paper. the firstmodelgivenbyEq.(2)mustbeused.The“∞ → fit- ting” parameter can be found by experimental or numerical XI. Natural Convection in Vertical Isothermal Ducts and Channels methods for some intermediate values of ξ = Ra√A.From numerous experiments in air, Pr =0.71, for many different Convective heat transfer in vertical ducts of arbitrary body shapes, its found that p = 1 gives acceptable accuracy shape and channels is of great interest to thermal analysts 10 for the entire range: 0 14 Fig. 10. Schematics of Various Cuboids nal ducts, the elliptical duct and the rectangular duct are showninFig.15. The fluid enters the lower end at some temperature T0 and moves up the duct whose temperature is Tw >T0 due to buoyancy-induced forces. The fluid velocity at the inlet is assumed to be zero. Its well known from experimental and numerical work that heat transfer from the isothermal wall to the fluid is different when the duct is either very short or when its very long. One criterion for characterizing short and long ducts is to consider the dimensionless geometric parameter: L L? = (95) L where is some characteristic length associated with the duct crossL section dimensions. In the past, the hydraulic Fig. 13. Compact Model and Air Data for Natural Convection from radius and more frequently the hydraulic diameter D = Vertical Thin Cuboids h 4A/P have been used to model the convective heat trans- fer. It can be shown by scaling analysis that there are two asymptotic solutions for this problem which are [29]: ? Nubl as L 0 Nu = → (96) ? L ( Nufd as L →∞ where Nubl is called the Nusselt number for boundary layer flow and Nufd is defined as the Nusselt number for fully- developed flow. The very short duct asymptote is given by following relationship [29]: 1/4 Nubl =0.60 RaL (97) if the duct length L is introduced into the Nusselt and Rayleigh numbers. The relationship becomes [29]: 1/4 √A Fig. 14. Compact Models for Cuboids, Parallel Plates, and Heat Nubl =0.6 Ra√A (98) Sinks à L ! 15 Fig. 15. Cross Sections of Singly Connected Tubes, Pipes and Ducts when = √A is introduced into both Nusselt and Rayleigh If is √A in the Nusselt and Rayleigh numbers we ob- numbers.L The geometric parameter L/√A appears which tainL the relationship [29]: can be used as the criterion for very short and very long ducts. The independent flow parameter [29]: √A 2 Ra√A √A Nu =2 L (103) √A fd fRe P Ra (99) √A à ! √A L The dimensionless relative duct length parameter appears is often called the modified Rayleigh number or the channel with the Rayleigh number and by itself. Its found to be a Rayleigh number. very important geometric parameter. The fully-developed flow Nusselt number is given by the The fluid flow parameter for full-developed flow is fRe . following general relationship [29]: L This parameter is a constant for a particular duct cross section, and it depends only on the duct shape and its h Ra L A 2 aspect ratio. This important parameter is reviewed and Nu = L =2 L L (100) fd k fRe P discussed in the next section. L µ L¶ We note that φ = Nu√A and ξ = Ra√A√A/L.The when both the Nusselt and Rayleigh numbers are based on asymptotes for very short and very long ducts are the arbitrary length . If the hydraulic diameter is used, i.e., is D ,thenwehaveL 2 h 1/4 2 √A L φ0 =0.6 ξ and φ = ξ (104) D ∞ fRe√A à L ! Ra h Dh L Nufd = (101) and the corresponding constants are, therefore, 8 fReDh 2 The area average heat transfer coefficient in the Nusselt 2 √A number is defined as C0 =0.6andC = (105) ∞ fRe√A à L ! Q h = (102) For convective heat transfer in a particular duct cross sec- PL(Tw T0) − tion and fixed length C = const because the geometric ∞ where Q is the total heat transfer rate to the fluid. parameter L/√A and fRe√A are constants. Plots of the 16 102 101 100 A √ A √ 101 Nu fRe -1 Ellipse 10 Rectangle Rectangle with Circular Segment Ends Elenbaas Rectangle with Semi-Circular Ends 10-2 Model Model 100 101 102 103 104 105 106 107 100 Ra ⋅√A/L 0 0.25 0.5ε 0.75 1 √A Fig. 17. Compact Model Validation for Triangular Duct Fig. 16. fRe for Various Duct Cross Sections two asymptotes show that φ < φ as ξ 0, therefore the 1 0 10 second model given by Eq. (3) must∞ be→ used. The general compact model for this system is [29] 100 A p 1/p √ 1/4 − u √A − 0.6 Ra√A L N Nu√A = · ¸ p -1 ³ 2´ − 10 √ √ Elenbaas + 2 A/P Ra√A A/L /fRe√A Davis and Perona · ³ ´ ³ ´ ¸(106) Dyer -2 Model The compact model was compared against experimental 10 data obtained for air. The polygonal duct shapes such as 100 101 102 103 104 105 106 107 ⋅√ the equilateral triangle, the square, and the circular duct Ra√A A/L require the “fitting” parameter to have the value p =1.25 to give small RMS differences. Fig. 18. Compact Model Validation for Circular Duct Comparisons of the compact model predictions against air data for two rectangular ducts having side dimension The agreement between the data and the model predictions ratios of 2 and 5 showed that the “fitting” parameter de- is seen to be very good for all duct cross sections. pended on the duct aspect, ² =smallerside/larger side The recommended compact model can be used to predict 1. The differences between the data and the predicted val-≤ natural convection in arbitrary constant cross section ducts ues are very small when provided the cross section does not have sharp corners with 1.2 small subtended angles. p = 1/9 (107) ² XII. Models for Friction Factor Reynolds It was further demonstrated in [29] that the compact model Number Product can be used to predict Nusselt number for the parallel plate In this example we consider the exact asymptotic channel when the aspect ratio in the model developed for model and the approximate models for the friction fac- the rectangular duct is set to ² =0.01. The “fitting” pa- tor Reynolds product for steady developing and fully- rameter for parallel plates channels is p =2. developed laminar flow through constant cross-section The fully-developed flow parameter fRe√A developed for pipes of length L, cross-sectional area A and perimeter P . the rectangular duct [32,33,35] is shown plotted with re- The fluid enters the pipe with uniform velocity U.Inthe spect to ² in Fig. 16. The analytical values for the ellipse, entrance length 0 z Le there are two regions, the and the numerical values for the rectangular ducts with core region where the≤ velocity≤ is essentially uniform and segment ends and semi-circular ends are seen to be in very increasing with distance from the entrance z =0,andthe close agreement with the curve for the rectangle over the boundary layer region which lies between the wall and the entire range of ². core region. This very important fluid flow problem is de- The compact model and the air data for the equilateral scribed in [30] and many analytical and numerical results triangle, circle, the rectangle duct with ² =0.5andthe are tabulated in [31] for many different cross sections. New parallel plates channel are shown in Figs. 17 through 20. approaches and methods giving new models for singly and 17 1.000 1.000 0.800 0.800 0.600 0.600 0.400 0.400 1 0.200 0.200 10 0.000 0.000 0.000 0.200 0.400 0.600 0.800 0.000 0.200 0.400 0.600 0.800 N = 3 N = 4 N = 5 N = 6 N →∞ TriangleSquare Pentagon Hexagon Circle 0 y y y A 10 √ b b b a 0 a a Nu 0 x x 0 x 10-1 Semi-Circle Rectangle Ellipse Hyper-Ellipse a ≥ b a ≥ b a ≥ b Elenbaas 2 2 n n x y x y -2 Model + = 1 + = 1 10 a b a b 0 < ∞ 100 101 102 103 104 105 106 107 n < ⋅√ Ra√A A/L Fig. 21. Common Singly Connected Duct Geometries Fig. 19. Compact Model Validation for 2:1 Rectangular Duct 2 10 N= 354 6 →∞ Circular Core 101 A √ Nu N= 354 6 →∞ 100 Model Polygonal Core Elenbaas Miyatake & Fujii 10-1 Ofi & Hetherington 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 N = 3 4 5 6 → ∞ Ra ⋅√A/L √A Polygon-Polygon Fig. 20. Compact Model Validation for Parallel Plates Duct Fig. 22. Doubly Connected Cross Sections doubly connected cross section tubes, pipes and ducts are nel flow. In the two special cases, the local wall shear is presented in [32,33,35]. The new relations to be presented constant with respect to the pipe and channel perimeter, next are capable of predicting pressure drop for fluid flow however, it varies with distance from the entrance. in the singly connected cross sections shown in Fig. 21, and Since, in general τ (x, y, z),theaveragevalueoverthe the doubly connected cross sections shown in Fig. 22. w perimeter is defined as In the general case the axial fluid velocity depends on ax- ial position z and points in the cross-section, i.e., w(x, y, z). 1 P τ (z)= τ (x, y, z)dP (108) In the core region there is a balance between the pressure w P w and inertia forces, and in the boundary layer region there I0 is a balance between the friction and inertia forces. where the integration is with respect to points in the In the fully-developed region where Le z L,thefluid perimeter of the pipe. velocity no longer depends on distance from≤ ≤ the entrance The overall average value of the wall shear from the pipe and the fluid velocity depends on points in the cross-section entrance to the pipe exit is defined as only, i.e., w(x, y), and there is a balance between the pres- 1 L sure and friction forces only. τ = τ (z)dz (109) w L w The local wall shear τ w(x, y, z) varies with distance from Z0 the pipe entrance and it varies over the perimeter, except The relationship between the overall pressure drop ∆p and for circular pipes and parallel plate channels. There is no the average wall shear can be obtained from the force bal- analytical solution for the local wall shear for the entire ance applied to the entire pipe length, therefore, region 0 z L, even for the relatively simple fluid flows ≤ ≤ such as flow in a circular pipe or flow between two infinitely PL ∆pA= τ PL or ∆p = τ (110) large parallel plates, the so-called two-dimensional chan- w w A 18 The definition of the friction factor-Reynolds number prod- thepipelengthscale . These characteristics are clearly uct gives: seen in the friction factorL and Reynolds number product for the rectangular duct whose side dimensions are 2a 2b × 2τ w ρU 2τ w ? with a b. fRe = L = L =2τ w (111) L ρU 2 µ µU The≥ dimensionless average wall shear was presented in µ ¶µ ¶ [32,33,35]; and taking the first term of the series solution where is some length scale associated with the pipe cross- gives: section.L The friction factor and Reynolds number product ? is related to the dimensionless average wall shear τ w with the factor 2. 24 fReD = (116) The relations above can be used to find relations between h 192² π (1 + ²2) 1 tanh the pressure drop ∆p and the mean fluid velocity U or − π5 2² the mass flow rate m˙ , the geometry of the pipe, the fluid · ³ ´¸ properties, and the friction factor and Reynolds number where the duct aspect ratio is defined as ² = b/a 1. The ≤ product. The relations, based on the arbitrary length scale duct length scale is based on the hydraulic diameter Dh ,are where L 4A 4ab 4b² LP fRe Dh = = = (117) ∆p = µU L P a + b 1+² A 2 µ L¶µ ¶ (112) The relationship based on the hydraulic diameter has two LP fRe limits which are constants: ∆p = ν m˙ L A2 2 µ L¶µ ¶ 14.13 as ² 1 → The friction factor-Reynolds number product can be ob- fReDh = (118) ( 24 as ² 0 tained by means of scale analyses, asymptotic analyses, and → the method of [1] for combining asymptotes. The first limit corresponds to a square duct and the second The scale analyses will be done for very long pipes, i.e., one corresponds to the parallel plates channel. L/Le >> 1wherethefluid flow is fully-developed, and When the duct scale length is chosen as √A,therela- for very short pipes, i.e., L/Le << 1, where the fluid flow tionship becomes is developing in the core and the boundary layer regions. From the scale analyses of [35] the asymptotic relationships 12 fRe = (119) are: √A 192² π ² (1 + ²) 1 tanh C as L/Le √ 5 ∞ →∞ − π 2² · ³ ´¸ fRe C0 (113) L → as L/Le 0 The limits for this relationship are [35]: √ξ → where the dimensionless duct length which is based on the 14.13 as ² 1 → arbitrary length scale is defined as fRe = 12 (120) L √A as ² 0 L √² → ξ = (114) Re LL To demonstrate the superiority of set to √A in place of L where ξ depends on the chosen length scale . set to Dh in fRe , the analytical values for the equilat- L L L If we let φ = fRe , φ0 = C0,andφ = C ,weseethat eral triangle, the square and the circle, and the numerical L ∞ ∞ φ0 > φ as ξ 0. Since the trend of φ with respect to results for the other regular polygonal geometries are listed ξ is concave∞ upwards,→ the asymptotes can be combined in in Table 9. The aspect ratio for the regular polygonal ge- the following manner to give the approximate relationship ometries is ² 1. The conversion is applicable for all pipe lengths: ≈ P fRe√A = fReDh (121) C p 1/p 4√A fRe = 0 +(C )p (115) µ ¶ L √ξ ∞ ·µ ¶ ¸ The values for fReDh fall in the range 13.33 fReDh The “fitting” parameter values have been determined for 16 for 3 N . The relative difference between≤ the tri-≤ many different pipe cross-sections [32,33,35]. angular≤ and the≤∞ circular ducts is approximately 16.7 per- The constant for very short pipes can be found from cent. When the characteristic length scale is changed to asymptotic analysis to be C0 =1.72 for the local value and √A, the relative difference is reduced to 7.1 percent for C0 =3.44 for the average value of fRe .Thesetwovalues the equilateral triangle, and less than 0.1 percent for the are independent of the shape of the pipeL cross-section and remaining polygons N 4. Therefore, the value for the its aspect ratio. circular duct can be used≥ for all regular polygonal ducts, The constant C that appears in the very long pipe except the equilateral triangular duct, with negligible er- asymptote depends∞ on the pipe shape, its aspect ratio and rors. 19 TABLE IX fRe Results for Regular Polygonal Geometries fReP fReP N fRe fRe Dh fReC √A fReC 3 13.33 0.833 15.19 1.071 4 14.23 0.889 14.23 1.004 5 14.73 0.921 14.04 0.990 6 15.05 0.941 14.01 0.988 7 15.31 0.957 14.05 0.991 8 15.41 0.963 14.03 0.989 9 15.52 0.970 14.04 0.990 10 15.60 0.975 14.06 0.992 20 15.88 0.993 14.13 0.996 16 1.000 14.18 1.000 ∞ Fig. 23. fRe for Fully Developed Flow in Rectangular and Elliptical Ducts Steady laminar fully developed flow of a Newtonian fluid occurs in long elliptical ducts whose semi-axes are a, b with b a.Theflow is due to a constant pressure gradient ∆p/L≤ = dp/dz. The analytical closed-form solution of the governing− equation has been obtained using elliptic cylinder coordinates. It is x2 y2 w(x, y)=C 1 (122) − a2 − b2 · ¸ where 1 dp a2b2 C = (123) −2µ dz a2 + b2 The maximum velocity which occurs on the axis is wmax = w(0, 0) = C (124) The mean velocity is Fig. 24. fRe for Fully Developed Flow in Doubly Connected Ducts wmax C wm = = (125) 2 2 If we choose the square root of the cross sectional area, The velocity distribution can be expressed as then 2π3/2 (1 + ²2) fRe = (130) w(x, y) x2 y2 √A √²E(m) =2 1 (126) w − a2 − b2 m · ¸ The numerical values calculated by means of the two alter- The friction factor can be obtained from the following gen- native relationships are tabulated in Table 10. eral relationship: Numerical values for the elliptic and rectangular geome- tries are presented in Table 10 for both definitions of the 2 A characteristic length set to Dh and √A.Alsopre- fRe = ? 2L (127) L L wm b P sentedinTable10aretheratiosofthefRe results for µ ¶ the rectangular duct and the fRe results for the elliptic where A is the cross section area and P is the perimeter. duct at corresponding aspect ratios. It may be seen from For the ellipse these geometric parameters are the last column of Table 10, that √A is more appropri- 4A πb ate than Dh overtheentirerangeof² = b/a. It is seen A = πab P =4aE(m) D = = (128) h P E(m) that the numerical values for the rectangular and elliptic ducts differ by less than 7% when the characteristic length where E(m) is the complete elliptic integral of the second is √A, whereas, if the characteristic length is the hydraulic kindofmodulusm = √1 ²2,and² = b/a 1. If we diameter, the results differ by as much as 31%. choose the hydraulic diameter− as the characteristic≤ length, The simple relationship for the rectangular duct with the then “fitting” parameter p =2canbeusedtoobtainapprox- π 2 imate numerical values for rectangular ducts with semi- fRe =2(1+²2) (129) Dh E(m) circular ends, elliptical ducts ² 1, and the regular polyg- · ¸ ≤ 20 TABLE X Numerical Values of fRe for Elliptical and Rectangular Ducts fReDh fRe√A fReR fReR ² = b/a Rect. Ellip. Rect. Ellip. fReE fReE µ ¶Dh µ ¶√A 0.01 23.67 19.73 1.200 119.56 111.35 1.074 0.05 22.48 19.60 1.147 52.77 49.69 1.062 0.10 21.17 19.31 1.096 36.82 35.01 1.052 0.20 19.07 18.60 1.025 25.59 24.65 1.038 0.30 17.51 17.90 0.978 20.78 20.21 1.028 0.40 16.37 17.29 0.947 18.12 17.75 1.021 0.50 15.55 16.82 0.924 16.49 16.26 1.014 0.60 14.98 16.48 0.909 15.47 15.32 1.010 0.70 14.61 16.24 0.900 14.84 14.74 1.007 0.80 14.38 16.10 0.893 14.47 14.40 1.005 0.90 14.26 16.02 0.890 14.28 14.23 1.004 1.00 14.23 16.00 0.889 14.23 14.18 1.004 Fig. 25. fRe for Developing Flow in Rectangular and Elliptical Ducts Fig. 27. fRe for Developing Flow in Circular Annular Ducts onal ducts where ² 1andN 4. ≈ ≥ A. General Compact Model for Developing Flow the general compact model for developing flow is given by 2 1/2 12 192² π fRe = √A √² (1 + ²) 1 5 tanh − π 2² 2 · ¸ 3.44 ³ ´ + √ξ µ ¶ Fig. 26. fRe for Developing Flow in Polygonal Ducts (131) 21 TABLE XI XIII. Forced Convection Heat Transfer in Definitions of Aspect Ratio Tubes, Pipes, Ducts and Channels Geometry Aspect Ratio Convective heat transfer in short and long tubes, pipes, Regular Polygons ² =1 ducts and parallel plates channels is of great importance to thermal analysts. These systems can be characterized b Singly-Connected ² = by the cross sectional area A, perimeter P , and the length a L. The cross sections can be singly or doubly connected 2b Trapezoid ² = as shown in Figs. 21 and 22. If the systems are doubly a + c connected, then there are two perimeters: Pi,Po,theinner 1 r∗ and outer perimeters, and two projected areas: A ,A ,the Annular Sector ² = − i o (1 + r∗)Φ inner and outer projected areas. The flow area is A = (1 r∗) Ao Ai. Circular Annulus ² = − − π(1 + r∗) The walls can be isothermal or have a uniform heat flux imposed. The fluid is a single phase substance which may (1 + e∗)(1 r∗) Eccentric Annulus ² = − be a gas, generally air, or some fluid such as water, oil, π(1 + r ) ∗ or a water-glycol mixture, for example. The local or area- average Nusselt is required for heat transfer calculations. The fluid is assumed to enter one end of the tube with a with p =2. The model predictions are compared against uniform temperature T0 and a uniform velocity U.Asthe numerical values in Figs. 25, 26, and 27. The agreement fluid is transported along the tube, a hydrodynamic and a is seen to be very good for many singly connected ducts thermal boundary layer develops and they both grow until shown in Figs. 25 and 26. The very good agreement for they occupy the entire cross section. The entrance length circular annular ducts is shown in Fig. 27. for the hydrodynamic boundary layer is denoted as Lh and The final results to be considered are those of the circular the entry length for the thermal boundary is denoted as annulus and other annular ducts which are bounded exter- Lt. nally by a polygon or internally by a polygon. It is clear The asymptotic solutions for thermally fully developed from Fig. 27 that excellent agreement is obtained when the flow, L>>Lh,L >> Lt, thermally developing flow, results are re-scaled according to √A and a new aspect ra- L>>Lh,L << Lt, and the combined entry problem, tio defined as r = A /A .Thisdefinition was chosen ∗ i o L< 22 A. Long Tube Asymptote Nu =2Nu (139) By means of asymptotic analysis [36], the very long tube L L ? L>>Lh,L>>Lt asymptote can be expressed as After introducing L , the solution for each wall condition can be compactly written as: fRe Nu = C √A (134) √A 1 8√π²γ f(Pr) µ ¶ Nu = C4 (140) L √L? where C1 is equal to 3.01 for the UWT boundary condition and 3.66 for the UWF boundary condition. These results where the value of C4 = 1 for local conditions and C4 =2 are the average value for fully developed flow in a polygonal for average conditions, and f(Pr)isdefined as: tube when the characteristic length scale is the square root 0.564 of cross-sectional area [32,34,36]. The cross section aspect f(Pr)= (141) 9/2 2/9 ratio is defined as ² = short side/longer side 1. 1+ 1.664Pr1/6 The parameter γ ischosenbaseduponthegeometry.≤ h i Values for γ which define the upper and lower bounds are for the UWT condition, and¡ ¢ fixed at γ =1/10 and γ = 3/10, respectively. Almost all 0.886 of the available data are predicted− within 10 percent by f(Pr)= (142) ± 9/2 2/9 Eq. (134), with few exceptions. 1+ 1.909Pr1/6 By means of another asymptotic analysis [36] for short h i tubes where L< 1/3 C. First Compact Model for Nusselt Number fRe L Nu = C2C3 ? (135) L L The first compact model can be developed for the entire µ ¶ range of dimensionless tube lengths and for Pr .Its →∞ where C2 is 1 for local conditions and 3/2 for average con- general form is [36]: ditions, and C3 takes a value of 0.427 for UWT and 0.517 1 n 1/n for UWF conditions respectively. The friction factor and fRe 3 Nu(z?)= C C +(Nu )n (143) Reynolds number product was discussed in an earlier sec- 2 3 z? fd tion. The results from that section can be used here. Ã( µ ¶ ) ! B. Short Tube Asymptote Now using the result for the fully developed friction fac- tor and the result for the fully developed flow Nusselt num- By means of asymptotic analyses [36] for very short tubes ber with n 5, a new model [32,34,36] was proposed hav- ≈ where L< m =2.27 + 1.65Pr1/3 (146) The above model is valid for 0.1 24 TABLE XI I Coefficients for General Model Boundary Condition 0.564 UWT (T) C1 =3.01,C3 =0.409 f(Pr)= 9/2 2/9 1+ 1.664Pr1/6 h ¡ ¢ i 0.886 UWF (H) C1 =3.66,C3 =0.501 f(Pr)= 9/2 2/9 1+ 1.909Pr1/6 h ¡ ¢ i Nusselt Number Type Local C2 =1 C4 =1 Average C2 =3/2 C4 =2 Shape Parameter Upper Bound γ =1/10 Lower Bound γ = 3/10 − Fig. 31. Nusselt Number for Thermally Developing Flow in Various Fig. 32. Nusselt Number for Thermally Developing Flow in Various UWT Ducts UWF Ducts two basic ways of combining the asymptotes depending on where the approximate solution is concave upwards or con- and a softer substrate, pressure drop for fluid flow through cave downwards. A procedure of calculating the “fitting” short and long tubes of arbitrary cross sectional area, natu- parameter was presented. The approximate solutions for ral convection in vertical isothermal ducts of arbitrary cross many different types of problems were shown to be very sectional area, and finally forced convection heat trans- accurate over broad ranges of the independent parameters. fer in short and long tubes and pipes with isothermal and The examples presented include conduction through a isoflux wall conditions. The last example showed how sev- stationary gas for all values of the Knudsen number, radia- eral asymptotes can be combined to develop a compact tion through a porous layer of material, steady conduction model. through a spherical wall and enclosures with constant gap thickness, transient conduction external to a sphere and Many of the examples have previously been applied to other convex bodies, elastic-plastic contact of a rigid sphere some interesting microelectronics cooling problems. 25 ids, Vol. 11, No. 3, pp. 497-506. 10. Braun, D., and Frohn, A. (1977). “Heat Transfer in Bi- nary Mixtures of Monatomic Gases for High-Temperature Differences and for a Large Knudsen Number Range,” Rar- efied Gas Dynamics, J.L. Potter, ed., pp. 149-159. 11. Gebhart, B., (1971). Heat Transfer, Second Edition, McGraw-Hill, New York, NY, pp. 165-167. 12. Carslaw, H.S., and Jaeger, J.C. (1959). Conduction of Heat in Solids, 2nd ed., Oxford Press, London. 13. Incropera, F.P. and DeWitt, D.P., (1990). Fundamen- tals of Heat and Mass Transfer, Wiley, New York, NY. 14. Bejan, A., (1993). Heat Transfer, Wiley, New York, NY. 15. Yovanovich, M.M. (1995). “Dimensionless Shape Fac- tors and Diffusion Lengths of Three Dimensional Bodies,” 4th ASME/JSME Thermal Engineering Joint Conference, Fig. 33. Nusselt Number for Thermally Developing Flow in Parallel Plate Channels Lahaina, Maui, HI, March 19-24. 16. Yovanovich, M. M., (1987). “New Nusselt and Sher- wood Numbers for Arbitrary Isopotential Bodies at Near XV. Acknowledgments Zero Peclet and Rayleigh Numbers,” AIAA 87-1643, AIAA The continued support of the Natural Sciences and Engi- 22nd Thermophysics Conference, Honolulu, Hawaii, June neering Research Council of Canada, NSERC, and funding 8-10. from Manufacturing and Materials of Ontario, MMO, are 17. Yovanovich, M.M., Teertstra, P.M., and Culham, J.R., greatly appreciated. The assistance of M. Bahrami, W. (1995). “Modeling Transient Conduction from Isothermal Khan, P. Teerstra, Y. 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