BOOKCOMP, Inc. — John Wiley & Sons / Page 1147 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL NATURAL CONVECTION 1147 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1147], (17) Lines: 662 to 696 ——— 11.54515pt PgVar ——— Normal Page * PgEnds: Eject [1147], (17) asymptotes, an optimal spacing (pore size) where the hot-spot temperature is minimal when the heat generation rate and volume are fixed is found. The same spacing represents the design with maximal heat generation rate and fixed hot-spot temperature and volume. Analytical and numerical results have been developed for optimal spacings in applications with solid components shaped as par- allel plates (Bejan, 1995). Optimal spacings for cylinders in crossflow were deter- mined analytically and experimentally in Stanescu et al. (1996). The spacings of heat sinks with square pin fins and impinging flow were optimized numerically and ex- perimentally in Ledezma et al. (1996). All the dimensionless results developed for optimal spacings (S opt ) have the form S opt L ∼ Be −n L (15.58) where L is the dimension of the given volume in the flow direction, and Be L isanew dimensionless group (the Bejan number) that serves as the forced-convection analog of the Rayleigh number of natural convection (Bhattacharjee and Grosshandler, 1988; Petrescu, 1994), Be L = ∆P · L 2 µ f α f (15.59) In this definition ∆P is the pressure difference maintained across the fixed vol- ume. For example, the exponent n in eq. (15.58) is equal to 1 4 in the case of lami- nar flow through stacks of parallel-plate channels, and is comparable to 1 4 in other configurations. The frequency of pulsating flows through microchannels in parallel can be opti- mized for global system performance subject to total volume and void space con- straints. The fundamentals of this optimization opportunity are outlined in Bejan (2000) and Bejan et al. (2004). 15.5 EXTERNAL NATURAL CONVECTION 15.5.1 Vertical Walls In natural convection the body force is due to internal density differences that are induced by heating or cooling effects. The equations of motion reviewed in Section 15.2.2 are complemented by the assumption that density and temperature changes are sufficiently small so that the linear approximation is valid. This is known as the Oberbeck–Boussinesq approximation ρ ≈ ρ 0 [ 1 − β(T −T 0 ) ] (15.60) where β is the thermal expansion coefficient BOOKCOMP, Inc. — John Wiley & Sons / Page 1148 / 2nd Proofs / Heat Transfer Handbook / Bejan 1148 POROUS MEDIA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1148], (18) Lines: 696 to 747 ——— 2.42616pt PgVar ——— Short Page * PgEnds: Eject [1148], (18) β =− 1 ρ ∂P ∂T P (15.61) Note that the gravitational acceleration points in the negative y direction, in other words, the body acceleration vector g appearing in eq. (15.6) and Fig. 15.1a, has (0, −g, 0) as components. Consider the heat and fluid flow in a porous medium adjacent to a heated vertical flat plate on which a thin thermal boundary layer is formed when the Rayleigh number Ra takes large values. Using the method of scale analysis (Bejan, 1984, 1995), the two-dimensional boundary layer equations take the form (Nield and Bejan, 1999) ∂u ∂x + ∂v ∂y = 0 (15.62) v =− K µ ∂P ∂y − ρgβ(T −T ∞ ) (15.63) ∂P ∂x = 0 (15.64) σ ∂T ∂t + u ∂T ∂x + v ∂T ∂y = α m ∂ 2 T ∂x 2 (15.65) Here the subscript ∞ denotes the reference value at a large distance from the heated boundary, and P denotes the difference between the actual static pressure and the lo- cal hydrostatic pressure. It has been assumed that the Oberbeck–Boussinesq approx- imation and Darcy’s law are valid. These equations were solved subject to a variety of boundary conditions, and the existing solutions are cataloged in Nield and Bejan (1999). The most important are the solutions for walls with constant temperature and constant heat flux. The local Nusselt number for a vertical isothermal wall (Fig. 15.3a) is (Cheng and Minkowycz, 1977) Nu y = 0.444Ra 1/2 y (15.66) where Nu y = q y/(T w − T ∞ )k m and the local Rayleigh number is defined as Ra y = Kg βy(T w −T ∞ )/α m ν. Equation (15.66) is valid in the boundary layer regime, Ra 1/2 y 1. The overall Nusselt number is Nu H = 0.888Ra 1/2 H (15.67) where Nu H = q H/ [ k m (T w − T ∞ ) ] and Ra H = Kg βH(T w − T ∞ )/α m ν and where H is the height of the wall. For convection near walls with uniform heat flux, the local Nusselt number (Cheng and Minkowycz, 1977) can be rearranged (Bejan, 1984): BOOKCOMP, Inc. — John Wiley & Sons / Page 1149 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL NATURAL CONVECTION 1149 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1149], (19) Lines: 747 to 747 ——— * 20.927pt PgVar ——— Short Page PgEnds: T E X [1149], (19) Figure 15.3 Natural convection in external flow: (a) impermeable vertical wall; (b) vertical partition embedded in a porous medium; (c) vertical wall separating a porous medium and a fluid reservoir; (d) hot surface facing upward in a porous medium; (e) cold surface facing upward in a porous medium; ( f ) impermeable sphere or cylinder embedded in a porous medium; (g) point heat source, low-Rayleigh-number regime; (h) point heat source, high- Rayleigh-number regime; (i) horizontal line source. BOOKCOMP, Inc. — John Wiley & Sons / Page 1150 / 2nd Proofs / Heat Transfer Handbook / Bejan 1150 POROUS MEDIA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1150], (20) Lines: 747 to 790 ——— 1.88142pt PgVar ——— Normal Page PgEnds: T E X [1150], (20) Nu y = 0.772Ra ∗1/3 y (15.68) where Nu y = q y/k m [ T w (y) − T ∞ ] and Ra ∗ y = Kg βy 2 q /α m νk m . Equation (15.68) holds in the boundary layer regime, Ra ∗1/3 y 1. The corresponding formula for the overall Nusselt number is Nu H = 1.044Ra ∗1/3 H (15.69) where Nu H = q H/k m (T w − T ∞ ) and Ra ∗ H = Kg βH 2 q /α m νk m . Other formula- tions of the boundary layer problem, such as the time-dependent development of the flow near a vertical wall, and the flow near a permeable wall with blowing or suction, are reviewed in Nield and Bejan (1999). When the porous medium of Fig. 15.3a is finite in both directions, the discharge of the heated vertical stream into the rest of the medium leads in time to thermal stratification. This problem was considered in Bejan (1984). As shown in the upper part of Fig. 15.4, the original wall excess temperature is T 0 − T ∞,0 , and the porous medium is stratified according to the positive vertical gradient γ = dT ∞ /dy. The local temperature difference T 0 − T ∞ (y) decreases as y increases, which is why a monotonic decrease in the total heat transfer rate as γ increases should be expected. This trend is confirmed by the lower part of the figure, which shows the integral solu- tion developed for this configuration. The overall Nusselt number, Rayleigh number, and stratification parameter are defined as Nu H = q H k m (T 0 − T ∞,0 ) (15.70) Ra H = Kg βH α m ν (T 0 − T ∞,0 ) (15.71) b = γH T 0 − T ∞,0 (15.72) The accuracy of this integral solution can be assessed by comparing its b = 0 limit, Nu H Ra 1/2 H = 1 (b = 0) (15.73) with the similarity solution for an isothermal wall adjacent to an isothermal porous medium, eq. (15.67). The integral solution overestimates the global heat transfer rate by only 12.6%. Referring again to the single-wall geometry of Fig. 15.3a, the corresponding prob- lem in a porous medium saturated with water near the temperature of maximum density was solved by Ramilison and Gebhart (1980). In place of the Oberbeck– Boussinesq model, this solution was based on ρ = ρ m (1 − α m |T − T m | q ), where ρ m and T m are the maximum density and the temperature of the state of maximum BOOKCOMP, Inc. — John Wiley & Sons / Page 1151 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL NATURAL CONVECTION 1151 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1151], (21) Lines: 790 to 800 ——— 2.23799pt PgVar ——— Normal Page PgEnds: T E X [1151], (21) ␥H y H 0 T ϱ,0 T 0 Wall Porous medium Temperature 0 0.5 1 0.5 1 b= H TT ␥ Ϫ 0,0ϱ b = 0 Similarity solution Integral analysis Nu Ra — HH Ϫ1/2 Figure 15.4 Heat transfer solution for a vertical isothermal wall bordering a linearly stratified porous medium saturated with fluid. (From Bejan, 1984.) density. The parameters ρ m ,T m ,q, and α m (not to be confused with the thermal dif- fusivity α m ) depend on the pressure and salinity, and are reported in Gebhart and Mollendorf (1977). Data on the local Nusselt number are reported in graphical form (Ramilison and Gebhart, 1980); a closed-form analytical substitute for this graphical information for pure water at atmospheric pressure was found by the present author, Nu y = 0.42 ×0.35 − T m − T ∞ T w − T ∞ 0.46 2α m Kgy(T w − T ∞ ) α m ν 1/2 (15.74) Equation (15.74) is accurate within 1% in the range −16<(T m −T ∞ )/(T w −T ∞ )<5. The breakdown of the Darcy flow model in vertical boundary layer natural con- vection was the subject of several studies (Plumb and Huenefeld, 1981; Bejan and Poulikakos, 1984; Nield and Joseph, 1985). Assuming the Forchheimer modification of the Darcy flow model, at local pore Reynolds numbers greater than 10, the local BOOKCOMP, Inc. — John Wiley & Sons / Page 1152 / 2nd Proofs / Heat Transfer Handbook / Bejan 1152 POROUS MEDIA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1152], (22) Lines: 800 to 838 ——— 3.51009pt PgVar ——— Normal Page PgEnds: T E X [1152], (22) Nusselt number for the vertical wall configuration of Fig. 15.3a approaches the fol- lowing limits (Bejan and Poulikakos, 1984): Nu y = 0.494Ra 1/4 ∞,y for the isothermal wall (15.75) 0.804Ra ∗1/5 ∞,y for the constant heat flux wall (15.76) where Ra ∞,y = gβy 2 (T w − T ∞ )/(bα 2 m ) and Ra ∗ ∞,y = gβy 3 q /(k m bα 2 m ). Equations (15.75) and (15.76) are valid provided that G 1, where G = (ν/K)[bgβ(T w − T ∞ )] −1/2 . In the intermediate range between the Darcy limit and the inertia-dominated limit, that is, in the range where G is of order 1, numerical results (Bejan and Poulikakos, 1984) for a vertical isothermal wall are correlated within 2% by the closed-form expression Nu y = (0.494) n + 0.444G −1/2 n 1/n · Ra 1/4 ∞,y (15.77) The heat transfer results summarized in this section also apply to configurations where the vertical wall is inclined (slightly) to the vertical. In such cases, the grav- itational acceleration that appears in the definition of Rayleigh-type numbers in this section must be replaced by the gravitational acceleration component that acts along the early vertical wall. When a vertical wall divides two porous media and a temperature difference exists between the two systems, there is a pair of conjugate boundary layers, one on each side of the wall, with neither the temperature nor the heat flux specified on the wall but rather, to be found as part of the solution of the problem (Fig. 15.3b) (Bejan and Anderson, 1981). The overall Nusselt number results for this configuration are correlated within 1% by the expression Nu H = 0.382(1 +0.615ω) −0.875 · Ra 1/2 H (15.78) where Nu H = q H/(T ∞,H −T ∞,L )k m and where q is the heat flux averaged over the entire height H . In addition, Ra H = Kg βH(T ∞,H −T ∞,L )/(α m ν); the wall thickness parameter ω is defined as ω = W H k m k w Ra 1/2 H (15.79) In this dimensionless group, W and H are the width and height of the wall cross section, k m and k w are the conductivities of the porous medium and wall material, respectively, and Ra H is the Rayleigh number based on H and the temperature dif- ference between the two systems. In thermal insulation and architectural applications, the porous media on both sides of the vertical partition of Fig. 15.3b may be thermally stratified. If the stratification on both sides is the same and linear (e.g., Fig. 15.4), so that the vertical temperature gradient far enough from the wall is dT /dy = b 1 (T ∞,H − T ∞,L )/H , where b 1 is a constant, and if the partition is thin enough so that ω = 0, it is found that the overall BOOKCOMP, Inc. — John Wiley & Sons / Page 1153 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL NATURAL CONVECTION 1153 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1153], (23) Lines: 838 to 884 ——— 1.11212pt PgVar ——— Normal Page PgEnds: T E X [1153], (23) Nusselt number increases substantially with the degree of stratification (Bejan and Anderson, 1983). In the range 0 <b 1 < 1.5, these findings are summarized by the correlation Nu H = 0.382 1 + 0.662b 1 − 0.073b 2 1 · Ra 1/2 H (15.80) Another configuration of engineering interest is sketched in Fig. 15.3c: a vertical impermeable surface separates a porous medium of temperature T ∞,H from a fluid reservoir of temperature T ∞,L (Bejan and Anderson, 1983). When both sides of the interface are lined by boundary layers, the overall Nusselt number is Nu H = (0.638) −1 + (0.888B) −1 −1 · Ra 1/4 H,f (15.81) where Nu H = q H/(T ∞,H −T ∞,L )k m and B = k m ·Ra 1/2 H /(k f ·Ra 1/4 H,f ). The parameter k f is the fluid-side thermal conductivity, and the fluid-side Rayleigh number Ra H,f = g(β/αν) f H 3 (T ∞,H − T ∞,L ). Equation (15.81) is valid in the regime where both boundary layers are distinct, Ra 1/2 H 1 and Ra 1/4 H,f 1; it is also assumed that the fluid on the right side of the partition in Fig. 15.3c has a Prandtl number of order 1 or greater. Additional solutions for boundary layer convection in the vicinity of vertical partitions in porous media are reviewed in Nield and Bejan (1999). 15.5.2 Horizontal Walls With reference to Fig. 15.3d, boundary layers form in the vicinity of a heated hori- zontal surface that faces upward (Cheng and Chang, 1976). Measuring x horizontally away from the vertical plane of symmetry of the flow, the local Nusselt number for an isothermal wall is Nu x = 0.42Ra 1/3 x (15.82) where Nu x = q x/k m (T w −T ∞ ) and Ra x = Kg βx(T w −T ∞ )/α m ν. The local Nusselt number for a horizontal wall heated with uniform flux is Nu x = 0.859Ra ∗1/4 x (15.83) where Ra ∗ x = Kg βx 2 q /k m α m ν. Equations (15.82) and (15.83) are valid in the boundary layer regime, Ra 1/3 x 1 and Ra ∗1/4 x 1, respectively. They also apply to porous media bounded from above by a cold surface; this new configuration is obtained by rotating Fig. 15.3d by 180°. The transient heat transfer associated with suddenly changing the temperature of the horizontal wall is documented in Pop and Cheng (1983). The other horizontal wall configuration, the upward-facing cold plate of Fig. 15.3e, was studied in Kimura et al. (1985). The overall Nusselt number in this configura- tion is Nu = 1.47Ra 1/3 L (15.84) BOOKCOMP, Inc. — John Wiley & Sons / Page 1154 / 2nd Proofs / Heat Transfer Handbook / Bejan 1154 POROUS MEDIA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1154], (24) Lines: 884 to 916 ——— 7.9953pt PgVar ——— Normal Page * PgEnds: Eject [1154], (24) where Nu = q / [ k m (T ∞ − T w ) ] and Ra L = Kg βL(T ∞ −T w )/α m ν and where q is the overall heat transfer rate through the upward-facing cold plate of length L. Equation (15.84) holds if Ra L 1 and applies equally to hot horizontal plates facing downward in an isothermal porous medium. Note the exponent 1 3 , which is in contrast to the exponent 1 2 for the vertical wall in eq. (15.67). Other solutions and flow instabilities in natural convection above or under horizontal walls are reviewed in Nield and Bejan (1999). 15.5.3 Sphere and Horizontal Cylinder With reference to the coordinate system shown in the circular cross section sketched in Fig. 15.3f, the local Nusselt numbers for boundary layer convection around an impermeable sphere or a horizontal cylinder embedded in an infinite porous medium are, in order, Nu θ = 0.444Ra 1/2 θ 3 2 θ 1/2 sin 2 θ 1 3 cos 3 θ − cos θ + 2 3 −1/2 (15.85) 0.444Ra 1/2 θ (2θ) 1/2 sin θ(1 −cos θ) −1/2 (15.86) where Nu θ = q r 0 θ/ [ k m (T w − T ∞ ) ] and Ra θ = Kgβθr 0 (T w − T ∞ )/α m ν. These steady-state results have been reported in Cheng (1982); they are valid provided that the boundary layer region is slender enough, that is, if Nu θ 1. The overall Nusselt numbers for the sphere and horizontal cylinder are, respectively (Nield and Bejan, 1999), Nu D = 0.362Ra 1/2 D (15.87) 0.565Ra 1/2 D (15.88) where Nu D = q D/ [ k m (T w − T ∞ ) ] and Ra D = Kg βD(T w − T ∞ )/α m ν. The tran- sient flow of and heat transfer near a horizontal cylinder are described in Ingham et al. (1983). Solutions for convection at low and intemediate Rayleigh numbers are summarized in Nield and Bejan (1999). 15.5.4 Concentrated Heat Sources The convection generated in a porous medium by a concentrated heat source has been studied in two limits: first, the low-Rayleigh-number regime, where the temperature distribution is primarily due to thermal diffusion, and second, the high-Rayleigh- number regime, where the flow driven by the source is a slender vertical plane (this second regime may be called the boundary layer regime). In the low-Rayleigh-number regime, the transient flow and temperature fields around a constant-strength heat source q that starts to generate heat at t = 0 is (Bejan, 1978) BOOKCOMP, Inc. — John Wiley & Sons / Page 1155 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL NATURAL CONVECTION 1155 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1155], (25) Lines: 916 to 966 ——— 8.33742pt PgVar ——— Normal Page * PgEnds: Eject [1155], (25) ψ = α m K 1/2 · Ra q τ 1/2 8π sin 2 φ 2η erfc η + 1 η erf η − 2 π 1/2 exp −η 2 (15.89) T −T ∞ = q k m K 1/2 1 4πR erfc R 2τ 1/2 + Ra q cos φ 64π 2 τ 1/2 × 1 η − 4 3π 1/2 + 6 5π 1/2 η 2 − 16 45π η 3 − 152 315π 1/2 η 4 + 64 315π η 5 + 517 3780π 1/2 η 6 − 992 14,175π η 7 − 2039 69,300π 1/2 η 8 + 2591 155,929π η 9 +··· (15.90) where η = R 2τ 1/2 R = r K 1/2 τ = α m t σK (15.91) and Ra q = Kg βq α m νk m (15.92) is the Rayleigh number based on point-source strength. The stream function ψ is defined in the axisymmetric spherical coordinates of Fig. 15.3g via v r = 1 r 2 sin φ ∂ψ ∂φ v φ =− 1 r sin φ ∂ψ ∂r (15.93) The time-dependent solution of eqs. (15.89)–(15.93) is valid in the range 0 < η < 1. The steady-state flow and temperature fields around a constant point source q in the low-Rayleigh-number regime is (Bejan, 1978) ψ = αr 8π Ra q sin 2 φ + Ra 2 q 24π sin φ sin 2φ − 5Ra 3 q 18,432π 2 8 cos 4 φ − 3 +··· (15.94) T −T ∞ = q 4πkr 1 + Ra q 8π cosφ + 5Ra 2 q 768π 2 cos 2φ + Ra 3 q 55,296π 3 cosφ 47 cos 2 φ − 30 −··· (15.95) BOOKCOMP, Inc. — John Wiley & Sons / Page 1156 / 2nd Proofs / Heat Transfer Handbook / Bejan 1156 POROUS MEDIA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1156], (26) Lines: 966 to 1004 ——— -1.69669pt PgVar ——— Short Page * PgEnds: PageBreak [1156], (26) The series solution of eqs. (15.94) and (15.95) is sufficiently accurate if Ra q is on the order of 20 or less. Numerical solutions for the point-source steady-state problem at Rayleigh numbers in the range 10 −1 to 10 2 are reported in Hickox and Watts (1980). The same authors reported analytical results for the limit Ra q → 0 and numerical results in the Ra q range 10 −1 to 10 2 for steady-state flow near a point source located at the base of a semi-infinite porous medium bounded from below by an impermeable and insulated surface. A subsequent paper (Hickox, 1981) shows that the transient and steady-state solutions for the point source in the strict Ra → 0 limit can be superimposed to predict the flow and temperature fields around buried objects of more complicated geometries. In the high-Rayleigh-number regime, the point source generates a vertical plume flow the thickness of which scales as y ·Ra −1/2 q , where y is the vertical position along the plume axis (Fig. 15.3h). The analytical solution for the flow and temperature field was constructed in Bejan (1984), T −T ∞ q/(k m y) = v (α m /y)Ra q = 2C 2 1 1 + (C 1 η/2) 2 (15.96) where η = (r/y)Ra 1/2 q and C 1 = 0.141. The solution holds provided the plume region is slender (i.e., when Ra 1/2 q 1). The two-dimensional frame of Fig. 15.3i shows a plume above a horizontal line heat source. The temperature and flow fields generated in a porous medium at high Rayleigh numbers by a horizontal line source of strength q (W/m) are T −T ∞ (q /k m )Ra −1/3 q = v (α m /y)Ra 2/3 q = C 2 2 /6 cosh 2 (C 2 η/6) (15.97) where C 2 = 1.651, η = (x/y)Ra 1/3 q , and Ra q = Kg βyq /α m νk m is the Rayleigh number based on line source strength. The derivation of this solution can be found in Wooding (1963) and is a special case of vertical boundary layer convection (Cheng and Minkowycz, 1977). The boundary layer solution in eq. (15.97) is valid at suf- ficiently high Rayleigh numbers Ra 1/3 q 1. The low-Rayleigh-number regime for convection near a horizontal line source (in an infinite medium or near a vertical insulated and impermeable surface) is described in Nield and Bejan (1999). 15.6 INTERNAL NATURAL CONVECTION 15.6.1 Enclosures Heated from the Side The most basic configuration of a porous layer heated in the horizontal direction is sketched in Fig. 15.5a. In Darcy flow, the heat and fluid flow driven by buoyancy depends on two parameters: the geometric aspect ratio H/L, and the Rayleigh number based on height, Ra H = Kg βH(T h −T c )/α m ν. There exist four heat transfer regimes . fields around a constant-strength heat source q that starts to generate heat at t = 0 is (Bejan, 1978) BOOKCOMP, Inc. — John Wiley & Sons / Page 1155 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL. uniform heat flux, the local Nusselt number (Cheng and Minkowycz, 1977) can be rearranged (Bejan, 1984): BOOKCOMP, Inc. — John Wiley & Sons / Page 1149 / 2nd Proofs / Heat Transfer Handbook. a constant, and if the partition is thin enough so that ω = 0, it is found that the overall BOOKCOMP, Inc. — John Wiley & Sons / Page 1153 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL