1 Conduction Heat Transfer

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1 Eng Formula Sheet 3 (December 1, 2015) 1 1 Conduction Heat Transfer 1.1 Cartesian Co-ordinates q x = q xa x = ka x dt dx R th = L ka 2 T x T y T z 2 + q k = 1 T α t T (x) plane wall of thickness 2L, x = 0 at centerline, T s,1 at x = L, T s,2 at x = L, steady state, 1D, uniform source and properties: ( ( ) T (x) = ql2 x 2 ( ) Ts,2 T s,1 x 1 + 2k L) 2 L + T s,1 + T s, Polar-Cylindrical Co-ordinates q r = k2πrl dt dr R th = ln(r 2/r 1 ) 2πkL 2 T r T r r T r 2 θ T z 2 + q k = 1 T α t T (r) in pipe wall (r 2 > r 1 ), T s,1 at r 1, T s,2 at r 2, 1D radial, steady state, no source, uniform properties: T (r) = T ( ) s,1 T s,2 r ln(r 1 /r 2 ) ln + T s,2 r 2 T (r) solid rod of radius r o, T s at r o, 1D radial, steady state, uniform source and properties: 1.3 Spherical Co-ordinates ( 1 r 2 r 2 T ) + r r T (r) = qr2 o 4k q r = k4πr 2 dt dr 1 r 2 sin θ 2 Convection Heat Transfer ( ( ) ) r T s ro ( sin θ T ) + θ θ R th = 1 ( 1 1 ) 4πk r 1 r T r 2 sin 2 θ φ 2 + q k = 1 T α t q = ha surf (T s T ) R th = 1 ha surf 3 Radiation Heat Transfer q rad = F G ɛσa surf (T 4 1 T 4 2 ) h r = ɛσ(t s + T sur )(T 2 s + T 2 sur) R th = 1 h r A surf σ = W/m 2 K 4

2 Eng Formula Sheet 3 (December 1, 2015) 2 4 Fins hp θ x = T x T m = η f = q f = q f ka c q max ha f θ b q f = η f ha f θ b 4.1 Case A: Convection Heat Loss from the Tip of the Fin θ cosh(m(l x)) + (h/mk) sinh(m(l x)) = θ b cosh(ml) + (h/mk) sinh(ml) η f = 1 ml 4.2 Case B: Insulated Tip [ ] sinh(ml) + (h/mk) cosh(ml) cosh(ml) + (h/mk) sinh(ml) q f = hp ka c θ b sinh(ml) + (h/mk) cosh(ml) cosh(ml) + (h/mk) sinh(ml) θ cosh(m(l x)) = θ b cosh(ml) q f = hp ka c θ b tanh(ml) η f = tanh(ml) ml 4.3 Case C: Specified Tip Temperature θ = (θ L/θ b ) sinh(mx) + sinh(m(l x)) θ b sinh(ml) q f = hp ka c θ b cosh(ml) θ L /θ b sinh(ml) 4.4 Case D: Very Long (or Infinite) Fin θ = e mx q f = hp ka c θ b η f = 1 θ b ml 4.5 Case B Approximation of Case A η f = tanh(ml c) L c = L + A tip /P A f = P L c tanh(x) = ex e x ml c e x + e x 4.6 Fin Resistance 5 Contact Resistance q f = η f ha f θ b = θ b R t,f = 1 R t,f η f ha f R t,c = 1 h c A contact = R t,c A contact

3 Eng Formula Sheet 3 (December 1, 2015) 3 6 Forced Convection - Flat Plate 6.1 Parameters Nu x = h xx k Nu L = h LL k Re x = ρu x µ Re L = ρu L µ Re D = ρu D µ P r = µc p k 6.2 Laminar Boundary Layer (Re L ) T s = constant: C f,x = 0.664Re 1/2 x C f = 1.328Re 1/2 L Nu x = 0.332Re 1/2 x P r 1/3 Nu L = 0.664Re 1/2 L P r1/3 P r 0.6 Heating starts ξ from the leading edge: q s = constant: Nu x = 0.332Re 1/2 x P r 1/3 [ ( ] ξ 3/4 1/3 1 x) Nu x = 0.453Re 1/2 x P r 1/3 Nu L = 0.680Re 1/2 L P r1/3 P r Turbulent Boundary Layer T s = constant: q s = 3 2 h L(T s T ) C f,x = Re 1/5 x Nu x = Re 4/5 x P r 1/3 0.6 < P r < 60, Re x < 10 8 Heating starts ξ from the leading edge: Nu x = Re 4/5 x P r 1/3 [1 (ξ/x) 9/10 ] 1/9 0.6 < P r < 60, Re x < 10 8 q s = constant: Nu x = Re 4/5 x P r 1/3 0.6 P r Mixed Boundary Layer Conditions When Re x,c = : Nu L = (0.037Re 4/5 L A)P r1/3 A = 0.037Rex,c 4/ Rex,c 1/2 Nu L = (0.037Re 4/5 L 871)P r1/3 C f = Completely turbulent boundary layer: Re 1/5 L Nu L = 0.037Re 4/5 L P r1/3 C f = 0.074Re 1/5 L 1742 Re L 0.6 < P r < 60, < Re L 10 8

4 Eng Formula Sheet 3 (December 1, 2015) 4 7 Forced Convection - Flow over Cylinders and Spheres 7.1 Parameters Nu D = hd k Re D = ρu D µ 7.2 Flow over Cylinders Nu D = CRe m DP r 1/3 P r 0.6 Table 7.2: Constants for Circular Cylinders in Cross Flow Re D C m , , , Table 7.3: Constants for Non-Circular Cylinders in Cross Flow of a Gas 7.3 Flow over Spheres Nu D = 2 + (0.4Re 1/2 D ( ) µ 1/ Re2/3 D )P r0.4 µ s

5 Eng Formula Sheet 3 (December 1, 2015) 5 8 Forced Convection - Internal Flows 8.1 Parameters 8.2 Energy Balance T s = constant: Nu D = hd k Nu D = hd k Re D = ρu md µ q conv = ṁc p (T m,o T m,i ) = UA T lm T lm = (T s T m,o ) (T s T m,i ) ln [(T s T m,o )/(T s T m,i )] q s =const: T m,o = T s (T s T m,i ) exp T m (x) = T m,i + q s P ṁc p x ( UA ) ṁc p UA = 1 ΣR th 8.3 Laminar Flow (Re D < 2300) Fully Developed: Nu D = 4.36 q s = const Nu D = 3.66 T s = const Entry Region (L/D < 0.05Re D P r, constant T s ) Nu D = Turbulent Flow (Re D > 2300) Fully Developed (L/D 10, constant T s or q s ): ( ) ReD P r 1/3 ( ) µ 0.14 L/D µs Nu D = 0.023Re 4/5 D P rn n = 0.4, T s > T m n = 0.3, T s < T m Entry Region (L/D < 10, constant T s or q s ): 8.5 Flows in Noncircular Tubes ( ) Nu D = 0.036Re 4/5 D D P r1/3 L Replace D in all parameters with the hydraulic diameter, D H, where A c is the cross-sectional area of the tube, and P is the wetted perimeter: D H = 4A c P

6 Eng Formula Sheet 3 (December 1, 2015) 6 9 Radiation 9.1 Parameters and Properties 9.2 Shape Factor Algebra α + ρ + τ = 1 α = ɛ E b = σt 4 σ = W/m 2 K 4 A i F ij = A j F ji 9.3 Radiation Resistance n j=1 F ij = 1 F i j,k = F ij + F ik R surface = 1 ɛ ɛa R space = 1 A i F ij 9.4 Radiation Heat Transfer Rates Rate of heat transfer leaving gray surface 1: q 1 = E b1 J 1 R surface R surface = 1 ɛ 1 ɛ 1 A 1 Rate of heat transfer from gray surface 1 to gray surface 2: q 12 = J 1 J 2 R space R space = 1 A 1 F 12 = 1 A 2 F 21 Rate of heat transfer between black bodies 1 and 2: q 12 = E b1 E b2 R space R space = 1 A 1 F 12 = 1 A 2 F 21

7 Eng Formula Sheet 3 (December 1, 2015) Figures and Tables Fin Efficiency Figure 3.18: Efficiency of straight fins (rectangular, triangular, and parabolic profiles). Figure 3.19: Efficiency of annular fins of rectangular profile. 7

8 Eng Formula Sheet 3 (December 1, 2015) Radiation Shape Factors Table 13.1: Shape Factors for Two-Dimensional Geometries

9 Eng Formula Sheet 3 (December 1, 2015) 9 Table 13.1 (cont d.): Shape Factors for Two-Dimensional Geometries

10 Eng Formula Sheet 3 (December 1, 2015) 10 Table 13.2: Shape Factors for Three-Dimensional Geometries

11 Eng Formula Sheet 3 (December 1, 2015) Figure 13.4: Shape Factor for aligned parallel rectangles. Figure 13.5: Shape factor for coaxial parallel disks. 11

Eng6901 - Formula Sheet 3 (December 1, 2015) 12 Figure 13.

12 Eng Formula Sheet 3 (December 1, 2015) 12 Figure 13.6: Shape factor for perpendicular rectangles with a common edge.

1 Conduction Heat Transfer

1 Conduction Heat Transfer Eng690 - Formula Sheet 2 Conduction Heat Transfer. Cartesian Co-ordinates q x xa x A x dt dx R th A 2 T x 2 + 2 T y 2 + 2 T z 2 + q T T x) plane wall of thicness 2, x 0 at centerline, T s, at x, T s,2