Thermodynamics, Fluid Dynamics, and Heat Transfer

Chapter 2 Thermodynamics, Fluid Dynamics, and Heat Transfer 2. Introduction In this chapter we will review fundamental concepts from Thermodynamics, Fluid Dynamics, and Heat Transfer. Each section first begins with a review of the fundamentals. Subsequently, a review of important equations and solutions to fundamental problems from each of the three fields. This chapter is only intended to provide the necessary reference material for the course. It is not intended as a substitute for the basic texts used in the thermo-fluids courses. During this course extensive reference will be made to the following texts dealing with thermo-fluid fundamentals. These are: ) Fundamentals of Fluid Mechanics, Potter and Wiggert 2) Fundamentals of Engineering Thermodynamics, Moran and Shapiro 3) Fundamentals of Heat and Mass Transfer, Incropera and DeWitt Where possible, the use of robust design models or correlations which span a wide range of flow conditions will be encouraged. These comprehensive models allow for greater flexibility in the design and optimization of thermal systems. Whereas piecewise models, i.e. those which consider each different flow region separately, tend to detract from the integrated design approach developed in the notes. 3

4 Mechanical Equipment and Systems 2.2 Thermodynamics 2.2. First Law of Thermodynamics The First Law of Thermodynamics, better known as the conservation of energy will be utilized for both open and closed systems throughout this course. We shall begin by examining the different ways of stating the First Law for both open and closed systems. The First Law of Thermodynamics for a closed system states E 2 E = Q 2 + W 2 2.) This is understood to imply that the change in energy of a closed system is related to the net heat input and the net work done on the system. In terms of instantaneous transfer rates, the First Law may be written on a per unit time basis de dt = Q + Ẇ 2.2) The First Law of Thermodynamics may also be written for an open system containing a number of inlets and outlets de cv dt = ṁ i u i + p i v i + V ) i 2 2 + gz i ṁ e u e + p e v e + V ) e 2 2 + gz e + Q cv +Ẇcv 2.3) This equation states that the accumulation of energy within the control volume must equal the net inflow of energy into the control volume minus the net outflow of energy from the control volume plus the increase in energy due to work and heat transfers. The sign convention adopted in these notes is that any work done on a system is considered positive, while any work done by the system is considered negative. This convention reflects the notion that work done on a system increases the energy of the system, i.e. the use of a pump or compressor. Contemporary texts in Thermodynamics have preferred the use of the heat engine convention which reflects that useful work done by the system is considered positive. Either convention may be applied so long as consistency is applied throughout the analysis of a problem. 2.2.2 Second Law of Thermodynamics The Second Law of Thermodynamics deals with the irreversibility of thermodynamic processes. A reversible process is one in which there is no production of entropy. For a closed system the second law of thermodynamics states that 2 δq T S 2 S 2.4) We define the entropy production S gen as the difference between the entropy change and the entropy transfer such that

Fundamentals 5 S gen = S 2 S ) 2 δq T 0 2.5) The Second Law of Thermodynamics for an open system containing a number of inlets and outlets becomes ds cv dt = Q j T j + m i s i m e s e + Ṡgen 2.6) This equation states that the rate of entropy accumulation within the control volume is balanced by the net transfer of entropy through heat exchanges with the surroundings plus the net flow of entropy into the control volume and the rate of entropy production within the control volume. It should be noted that for a steady state analysis the entropy production rate is not zero except for reversible processes). Whereas the rate of accumulation of entropy within the control volume is zero for a steady state process. 2.2.3 Exergy The first and second laws of thermodynamics may be combined to develop a new relation which governs a new quantity Exergy. Exergy is a measure of the potential of a thermodynamic system to do work. Unlike energy, exergy can be destroyed. Exergy analysis, sometimes called availability analysis, is used quite frequently in the design and analysis of thermal systems. Exergy is defined as E = E U o ) + p o V V o ) T o S S o ) 2.7) Here E = U + P.E. + K.E.), the energy of the system, U is internal energy, S is entropy, and V is the volume of the system. The reference or dead state as it is referred is denoted by the subscript ) o. We may also define exergy as an intensive property, that is on a per unit mass basis, such that e = [u + V 2 /2 + gz) u o ] + p o v v o ) T o s s o ) 2.8) The change in exergy between any two states is merely E 2 E = E 2 E ) + p o V 2 V ) T o S 2 S ) 2.9) For a closed system the exergy balance yields E 2 E = 2 T ) o δq [W p o V 2 V )] E d 2.0) T b The term E d = T o S gen, is the exergy which is destroyed due to irreversibilities in the system. For an open system with a number of inlets and outlets the exergy balance yields:

6 Mechanical Equipment and Systems de cv dt = T ) o T j Q j ) dv cv Ẇ cv p o + ṁ i e i ṁ e e e E dt d 2.) e = h h o ) T o s s o ) + V 2 /2 + gz 2.2) is the flow exergy. Table Dimensionless Groups Group Biot Number Reynolds Number Prandtl Number Peclet Number Grashof Number Rayleigh Number Ra Nusselt Number Definition Bi hl k s Re ρv L µ Pr ν α Pe V L α RePr gβ T L3 Gr ν 2 gβ T L3 αν GrPr Nu q/a)l k f T hl k f Stanton Number St Nu RePr Colburn Factor j Nu RePr /3 Friction Coefficient C f τ 2 ρv 2 Fanning Friction Factor f p/l)a/p) 2 ρv 2

Fundamentals 7 2.3 Dimensionless Groups Before proceeding to the review of fluid dynamics and heat transfer models, a brief discussion on the use of dimensionless quantities is required. A number of important dimensionless quantities appear throughout the text. The student should familiarize himself or herself with these parameters and their use. Table summarizes the most important groups that will be encountered during this course. 2.4 Fluid Dynamics 2.4. Conservation Equations Conservation of mass and momentum for a control volume will be applied throughout the course. Here we will merely state the general form as previously discussed in fluid mechanics courses. Conservation of Mass dm CV dt = ṁ i ṁ e 2.3) Conservation of Momentum F = Ve ρ V e A e ) Vi ρ V i A i ) 2.4) In addition, we will also apply Bernoulli s equation for a number of incompressible flows. Bernoulli s Equation P γ + V 2 2g + z = P 2 γ + V 2 2 2g + z 2 + h L 2.5) 2.4.2 Internal Flows When analyzing flow in ducting or piping systems as well as flow through mechanical equipment, a number of design models and correlations are required for relating the mass flow rate to the pressure drop of the working fluid. The most common method is through the definition of the friction factor. The Fanning friction factor will be adopted for this course. It is defined as follows: A p f = τ = P L 2 ρu2 = 2 ρu2 D h 4 p L 2.6) 2 ρu2

8 Mechanical Equipment and Systems D h = 4A P 2.7) A is the cross-sectional area and P is the perimeter of the duct. In fully developed laminar flows the friction factor takes the following form: f = C Re Dh 2.8) C is a constant which is a function of the shape and aspect ratio of the duct. Table 2 summarizes a number of values for common duct shapes. Apparent friction factors for developing flows may be computed from the following formula f app Re Dh = [ 3.44 L ) 2 + fre Dh )2 ] /2 2.9) L = L D h Re Dh 2.20) In circular tubes the flow is developing in a region L < 0.058. The entrance length for flow development is L e = 0.058DRe D 2.2) Table 2 Typical values of fre Dh = C for Non-Circular Ducts Shape fre Dh = C Equilateral Triangle 3.33 Square 4.23 Pentagon 4.74 Hexagon 5.05 Octagon 5.4 Circle 6 Elliptic 2: 6.82 Elliptic 4: 8.24 Elliptic 8: 9.5 Rectangular 2: 5.55 Rectangular 4: 8.23 Rectangular 8: 20.58 Parallel Plates 24

Fundamentals 9 For turbulent flows the friction factor is predicted using the Colebrook relation. This correlation is the basis for the Moody diagram ǫ/d = 2 log fd 3.7 + 2.5 ) 2.22) Re D fd the subscript d denotes the Darcy friction factor defined as: D p f d = L 2.23) 2 ρu2 The entrance length for turbulent flow in a tube is L e = 4.4DRe D ) /6 2.24) In non-circular ducts we use the concept of the hydraulic diameter D = D h = 4A/P to compute an equivalent duct diameter. 2.4.3 External Flows A number of important design equations for external fluid flows are required to relate the free stream velocity to the overall drag force. The three most common geometries are the flat plate, the cylinder, and the sphere. Flat Plate For laminar boundary layer flows, 000 < Re L < 500, 000, the important parameters are the boundary layer thickness and the friction coefficient: δx) = 5x Re /2 x 2.25) C f,x = 0.664 2.26) Re /2 x C f =.328 2.27) Re /2 L For turbulent boundary layer flows, 500, 000 < Re L < 0 7, the boundary layer thickness and friction coefficient are: δx) = 0.38x Re /5 x C f,x = 0.059 Re /5 x C f = 0.074 Re /5 L 2.28) 2.29) 2.30)

20 Mechanical Equipment and Systems If the boundary layer is composed of a combined laminar-turbulent flow, Re L > 500, 000, the friction coefficient is computed from the integrated value: C f = 0.074 742 2.3) Re /5 Re L L Finally, a number of useful models for predicting drag on flat plates, cylinders, and spheres in low Reynolds number flows are also provided. These models will provide the building blocks for analysing a fluid component or system. Flat Plate 0.0 < Re L < 500, 000 C f = 2.66 Re 7/8 L +.328 Re /2 L 2.32) Cylinder 0. < Re D < 250, 000 C D = 0 +.0 2.33) Re 2/3 D Sphere 0.0 < Re D < 250, 000 C D = 24 Re D + 6 + Re /2 D + 0.4 2.34) C D, C f = F/A 2 ρu2 2.35) Note care must be taken to ensure the correct characteristic area A is chosen based upon the geometry. 2.5 Heat Transfer 2.5. Conduction -Dimensional Steady Conduction Steady one-dimensional conduction in plane walls, cylinders, and spheres is easily analyzed using the resistance concept. The thermal resistance is defined such that T = QR t 2.36) For a multi-component system containing j layers, the following thermal resistance results are useful.

Fundamentals 2 Plane Wall R t = h i A + t j k j A + h o A 2.37) Cylinder R t = + lnr oj /r ij ) + 2.38) 2πr i L)h i 2πk j L) 2πr o L)h o Sphere R t = 4πr 2 i )h i + ) + 2.39) 4πk j r ij r oj 4πro)h 2 o Multi-Dimensional Steady Conduction In two or three dimensions, heat transfer by means of conduction is best analyzed using shape factors. Many multi-dimensional solutions of practical interest have been obtained and are outlined below. The conduction shape factor S, is defined such that: R = Sk 2.40) R is the thermal resistance and k is the thermal conductivity of the medium. The shape factor S, is only a function of the geometry of the system. The overall heat transfer rate is then related to an appropriate temperature difference: Q = Sk T 2.4) T is the temperature difference between two isothermal surfaces. A number of useful shape factors are tabulated in the handout. Transient Conduction Transient conduction in finite and semi-infinite regions are also of interest. The following solutions are useful for modelling a number of thermal systems. Semi-Infinite Regions Isothermal Wall Tx,t) T s T i T s ) x = erf 2 αt 2.42)

22 Mechanical Equipment and Systems q s t) = kt s T i ) παt 2.43) Isoflux Wall Tx,t) T i = 2q s αt/π exp k ) x 2 4αt q ) sx x k erfc 2 αt 2.44) Surface Convection Tx,t) T i T T i T s t) T i = 2q s k ) /2 αt 2.45) π ) [ )][ x hx = erfc 2 exp αt k + h2 αt x erfc k 2 2 αt + h )] αt k 2.46) T s t) T i T T i ) h 2 αt h αt = exp erfc k 2 k q s t) ) ) αt h 2 kt Ti) = exp αt h αt erfc k 2 k ) 2.47) 2.48) Finite Regions Transient conduction from finite one dimensional and multi-dimensional regions may be analyzed using the following solutions. In the solutions below = T T f, i = T i T f, and Q i = ρc p V T i T f ). The notation adopted in this section follows that of Yovanovich 999). Plane Wall i = A n exp δnfo) 2 cosδ n X) 2.49) n= A n = The eigenvalues δ n are determined from 4 sinδ n ) 2δ n + sin2δ n ) 2.50) δ n sinδ n ) = Bi cosδ n ) 2.5) In the expressions above, Fo = αt/l 2, X = x/l, and Bi = hl/k. The heat flow at the surface of the wall is determined from

Fundamentals 23 Q Q i = n= 2Bi 2 δ 2 nbi 2 + Bi + δ 2 n) ) exp δ 2 nfo) 2.52) Next if Fo > 0.24, the series solutions for temperature and heat flow reduce to single term approximations = A exp δ Fo) 2 cosδ X) i 2.53) ) Q 2Bi 2 = exp δ Q i δbi 2 2 + Bi + δ Fo) 2 ) 2 2.54) δ =.5708 [ +.5708/ Bi) 2.39 ] 0.4675 2.55) Finally, if the Biot number is small Bi < 0.2), spatial effects are no longer significant and the lumped capacitance model applies. For a plane wall this results in i = exp BiFo) 2.56) Infinite Cylinder Q Q i = exp BiFo) 2.57) i = A n exp δnfo)j 2 0 δ n R) 2.58) n= A n = The eigenvalues δ n are determined from 2J δ n ) δ n J 2 0δ n ) + J 2 δ n )) 2.59) δ n J δ n ) = J 0 δ n )Bi 2.60) In the expressions above, Fo = αt/a 2, R = r/a, and Bi = ha/k. The heat flow at the surface of the cylinder is determined from Q Q i = n= 4Bi 2 δ 2 nbi 2 + δ 2 n) ) exp δ 2 nfo) 2.6) Next if Fo > 0.2, the series solutions for temperature and heat flow reduce to single term approximations

24 Mechanical Equipment and Systems = A exp δ Fo)J 2 0 δ R) i 2.62) ) Q 4Bi 2 = exp δ Q i δbi 2 2 + δ Fo) 2 ) 2 2.63) δ = 2.4048 [ + 2.4048/ 2Bi) 2.238 ] 0.4468 2.64) Finally, if the Biot number is small Bi < 0.2), spatial effects are no longer significant and the lumped capacitance model applies. For an infinite cylinder this results in i = exp 2BiFo) 2.65) Sphere Q Q i = exp 2BiFo) 2.66) i = n= The eigenvalues δ n are determined from A n exp δ 2 nfo) sinδ nr) δ n R A n = 4[sinδ n) δ n cosδ n )] 2δ n sin2δ n ) 2.67) 2.68) δ n cosδ n ) = Bi) sinδ n ) 2.69) In the expressions above, Fo = αt/a 2, R = a/l, and Bi = ha/k. The heat flow at the surface of the sphere is determined from Q Q i = n= 6Bi 2 δ 2 nbi 2 Bi + δ 2 n) ) exp δ 2 nfo) 2.70) Next if Fo > 0.8, the series solutions for temperature and heat flow reduce to single term approximations Q Q i = = A exp δ Fo) 2 sinδ R) i δ R 6Bi 2 δbi 2 2 Bi + δ) 2 2.7) ) exp δ 2 Fo) 2.72)

Fundamentals 25 δ = 3.459 [ + 3.459/ 3Bi) 2.34 ] 0.4322 2.73) Finally, if the Biot number is small Bi < 0.2), spatial effects are no longer significant and the lumped capacitance model applies. For a sphere this results in 2.5.2 Convection i = exp 3BiFo) 2.74) Q Q i = exp 3BiFo) 2.75) Convective heat transfer models for internal and external flows are required for modelling heat exchangers, heat sinks, electronic enclosures, etc. A number of useful design models and correlations are now presented for internal and external flows. Internal Forced Convection Circular and Non-Circular Ducts In laminar flow, Muzychka and Yovanovich 200) proposed the following model for developing laminar flows: { Nu A z ) = { )} ) fre 5 m/5 C A 3 8 + πǫ γ C C 2 fre A z { C4 fpr) z } ) 5 3 + } m /m 2.76) and m = 2.27 +.65Pr /3 2.77) z = z ARe A Pr 2.78) and fre A = 2 [ ǫ /2 + ǫ) 92ǫ π ) tanh ] 2.79) π 5 2ǫ

26 Mechanical Equipment and Systems In the above model, the characteristic length scale is the square root of the crosssectional duct area. The parameter γ is chosen based upon the duct geometry. The lower bound value is for ducts that have re-entrant corners, i.e. angles less than 90 degrees. The upper bound is for ducts with rounded corners, rectangular or elliptical shapes. The coefficients are tabulated in Table 3 for various conditions. For turbulent flows the most popular expression is the correlation developed by Gneilinski 976). Nu Dh = f/8)re Dh Pr.07 + 2.7f/8) /2 Pr 2/3 ) 2.80) f = 0.79 ln Re dh.64) 2 2.8) Table 3 Coefficients for General Model Boundary Condition Isothermal C 2 = 0.409, C 3 = 3.24 fpr) = Isoflux C 2 = 0.50, C 3 = 3.86 fpr) = 0.564 [ +.664Pr /6 ) 9/2] 2/9 0.886 [ +.909Pr /6 ) 9/2] 2/9 Nusselt Type Local C = C 4 = Average C = 3/2 C 4 = 2 Shape Parameter Upper Bound γ = /0 Lower Bound γ = 3/0 External Forced Convection Flate Plate For a flat plate in laminar boundary layer flow, 000 < Re L < 500, 000, the Nusselt number is obtained from the following expressions: Nu x = Re x Pr) /2 fpr) 2.82)

Fundamentals 27 Nu L = 2Re L Pr) /2 fpr) 2.83) for the constant surface temperature, T s, boundary condition fpr) = 0.564 [ +.664Pr /6 ) 9/2] 2/9 2.84) and for the constant heat flux, q s, boundary condition fpr) = 0.886 [ +.909Pr /6 ) 9/2] 2/9 2.85) In turbulent boundary layer flow, 500, 000 < Re L < 0 7, the following equations are often used: Nu x = 0.0296Re 4/5 x Pr /3 2.86) Nu L = 0.037Re 4/5 L Pr/3 2.87) For a combined laminar/turbulent boundary layer, Re L > 500, 000, the following integrated expression is useful: Cylinder Pe D > 0.2 Nu L = 0.037Re 4/5 L 87)Pr/3 2.88) For a cylinder in crossflow Churchill and Bernstein 977) proposed the following correlation of experimental data: Nu D = 0.3 + [ ) ] 5/8 4/5 0.62Re/2 D Pr/3 ReD + 2.89) [ + 0.4/Pr) 2/3 ] /4 282, 000 Spheroids 0 < Re A < 2 05 and Pr > 0.7 For a sphere or spheroidal shaped body Yovanovich 988) recommends the following model Nu A = 2 π + [ ) ] /2 P 0.5 Re /2 A A + 0.35Re 0.566 A Pr /3 2.90) A is the surface area and P is the maximum equitorial perimeter.

28 Mechanical Equipment and Systems Internal Natural Convection Parallel Plates The Nusselt number for laminar natural convection flow between parallel isothermal plates is obtained from the following correlation developed by Bar-Cohen and Rohsenow 984) [ ] /2 576 Nu b = [Ra b b/l)] + 2.87 2.9) 2 [Ra b b/l)] /2 The Nusselt number for laminar natural convection flow between parallel isoflux plates is obtained from the follow correlation developed by Bar-Cohen and Rohsenow 984) [ ] /2 48 Nu b = [Ra + 2.5 2.92) b b/l)]2 [Ra b b/l)]2/5 Ra b = gβ Tb 3 /αν) and Ra b = gβq b 4 /kαν), and b is the plate spacing. Circular and Non-Circular Ducts For laminar natural convection in vertical isothermal ducts, Yovanovich et al.200) recommend: Nu A = 2 Ra A A/L ) ) 2 A fre A P n + 0.6 Ra A ) /4 n A L /n and 2.93) n =.2 ǫ /9 2.94) fre A = 2 [ ǫ /2 + ǫ) 92ǫ π ) tanh ] 2.95) π 5 2ǫ In the above model, the characteristic length scale is the square root of the crosssectional duct area. External Natural Convection Flate Plate For a vertical isothermal wall the following correlation is recommended for laminar

Fundamentals 29 flow Gr L < 0 9 : Nu x = 0.503Ra /4 x fpr) 2.96) Nu L = 4 3 Ra/4 L fpr) 2.97) ) /4 Pr fpr) = 2.98) Pr + 0.986Pr /2 + 0.492) A correlation which is valid for both the laminar and turbulent regions 0 < Ra L < 0 2 was proposed by Churchill and Chu 975). Their correlation takes the following form: Horizontal Cylinder Nu L = 0.825 + ) 0.387Ra /6 2 L 2.99) [ + 0.492/Pr) 9/6 ] 8/27 A correlation which is valid for both the laminar and turbulent regions 0 5 < Ra L < 0 2 was proposed by Churchill and Chu 975). Their correlation takes the following form: Sphere Nu D = 0.60 + ) 0.387Ra /6 2 L 2.00) [ + 0.559/Pr) 9/6 ] 8/27 For a sphere with Ra < 0, the following correlation is recommended: Nu D = 2 + Other Three Dimensional Bodies 0.589Ra /4 D [ + 0.469/Pr) 9/6 ] 4/9 2.0) For three dimensional bodies in any orientation, Yovanovich 987) recommends the following correlation for 0 < Ra A < 08 : Nu A = 2 π + Ra /4 A fpr) 2.02) fpr) = and A is the surface area of the body. 0.67 [ + 0.492/Pr) 9/6 ] 4/9 2.03)

30 Mechanical Equipment and Systems 2.5.3 Radiation Radiative heat transfer transfer is determined using the Stefan-Boltzmann law: q 2 = ǫf 2 σt 4 T 4 2 ) 2.04) ǫ is the surface emissivity, F 2 is the view factor, and σ = 5.670e 8 W/m 2 K 4 ), the Stefan-Boltzmann constant. A number of common two surface enclosure problems are: Parallel Plates q 2 = σt 4 T 4 2 ) ǫ + ǫ 2 2.05) Concentric Cylinders σt 4 T2 4 ) q 2 = + ǫ 2 ǫ ǫ 2 r r 2 ) 2.06) Concentric Spheres q 2 = σt 4 T 4 2 ) ǫ + ǫ 2 ǫ 2 r r 2 ) 2 2.07) Additional enclosure problems are discussed in all basic heat transfer texts. For more information on radiative exchange and radiative properties, the student should refer to the course text on heat transfer.

Fundamentals 3 2.6 References Bar-Cohen, A. and Rohsenow, W.M., Thermally Optimum Spacing of Vertical Natural Convection Cooled Parallel Plates, Journal of Heat Transfer, Vol. 06, 984. Bejan, A. Heat Transfer, 993, Wiley, New York. Bejan, A., Advanced Engineering Thermodynamics, 997, Wiley, New York, NY. Bejan, A., G. Tsatsaronis, and Moran, M., Thermal Design and Optimization, 996, Wiley, New York, NY. Churchill, S.W. and Chu, H.H.S., Correlating Equations for Laminar and Turbulent Free Convection from a Horizontal Cylinder, International Journal of Heat and Mass Transfer, Vol. 8, 975, pp. 323-329. Churchill, S.W. and Bernstein, M., A Correlating Equation for Forced Convection from gases and Liquids to a Circular Cylinder in Cross Flow, Journal of Heat Transfer, Vol. 99, 977, pp. 300-306. Churchill, S.W., A Comprehensive Correlating Equation for Forced Convection from Flat Plates, American Institute of Chemical Engineers, Vol. 22, 976, pp. 264-268. Gnielinski, V., New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow, International Chemical Engineering, Vol. 6, 976, pp. 359-368. Incropera, F.P. and DeWitt, D.P., Fundamentals of Heat and Mass Transfer, 996, Wiley, New York, NY. Moran, M.J. and Shapiro, H.N., Fundamentals of Engineering Thermodynamics, 2000, Wiley, New York, NY. Munson, B.R., Young, D.F., Okiishi, T.H., Fundamentals of Fluid Mechanics, 998, Wiley, New York, NY. Muzychka, Y.S. and Yovanovich, M.M., Forced Convection Heat Transfer in the Combined Entry Region of Non-Circular Ducts, Submitted to the 200 International Mechanical Engineering Congress and Exposition, New York, NY, November, 200. Rohsenow, W.M., Hartnett, J.P., and Cho, Y.I., Handbook of Heat Transfer, 999, McGraw-Hill, New York. Yovanovich, M.M., General Expression for Forced Convection Heat and Mass Transfer from Isopotential Spheroids, AIAA Paper 88-0743, AIAA 26th Aerospace Sciences Meeting and Exhibit, Reno, NV, January -4, 988.

32 Mechanical Equipment and Systems Yovanovich, M.M., On the Effect of Shape, Aspect Ratio, and Orientation Upon natural Convection from Isothermal Bodies of Complex Shape, ASME HTD Vol. 82, 987, pp. 2-29. Yovanovich, M.M., Teertstra, P.M., and Muzychka, Y.S. Natural Convection Inside Vertical Isothermal Ducts of Constant Arbitrary Cross-Section, AIAA Paper 0-0368, AIAA 39th Aerospace Sciences Meeting and Exhibit, Reno, NV, January 8-, 200.