1 R-value = 1 h ft2 F. = m2 K btu. W 1 kw = tons of refrigeration. solar = 1370 W/m2 solar temperature
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1 Quick Reference for Heat Transfer Analysis compiled by Jason Valentine and Greg Walker Please contact ith corrections and suggestions copyleft 28: You may copy, distribute, and modify and redistribute as long as you attribute the original author and distribute ith an equivalent license. Version: October 9, 28 Nomenclature: Conversions: Physical Constants: Assumed values: symbol name units q heat transfer W q heat flux W/m 2 q = g generation W/m 3 k thermal conductivity W/m K E energy rate W R thermal resistance K/W G conductance W/K R thermal resistance m 2 K/W h convection coefficient W/m 2 K U overall heat transfer coefficient W/m 2 K θ temperature difference K (or C) C = ṁc p thermal capacitance W/K ω periodic frequency rad/s R-value = h ft2 F =.76 m2 K btu W kw = 84 tons of refrigeration name parameter Stefan-Boltzmann σ = W/m 2 K 4 Boltzmann k B = J/K blackbody C = 2πhc 2 = W µm 4 /m 2 blackbody C 2 = hc /k B = µm K Wien s C 3 = 2898 µm K speed of light c = m/s Planck s constant h = J s name parameter solar constant q solar = 37 W/m2 solar temperature T solar = 578 K gravity g = 9.8 m/s 2 critical Reynolds for flat plate Re cr = 5 5 critical Reynolds for inside tube Re cr = 23 critical Rayleigh for vertical plate Ra cr = 9 radius of the Sun R Sun = km average distance to the Sun Sun = 5 6 km
2 Conduction. Equivalent resistance Assumptions:. steady state 2. constant properties 3. one-dimensional transport 4. no generation expression cartesian cylindrical spherical d 2 ( T heat equation dx 2 = d r dt ) ( d = r dr dr r 2 r 2 dt ) = dr dr temperature distribution heat flux heat transfer T T x + T ln(r/r 2) ln(r /r 2 ) k T ka T resistance ka critical radius r < r 2 ; T = T k T r ln(r 2 /r ) 2πk T ln(r 2 /r ) ln(r 2 /r ) 2πk k h T T (r /r) (r /r 2 ) k T r 2 (/r /r 2 ) 4πk T /r /r 2 /r /r 2 4πk 2k h.2 Shape Factors escription schematic restrictions shape factor T isothermal sphere in a semi-infinite medium z z > /2 2π /4z T horizontal isothermal cylinder buried in a semi-infinite medium z 2π cosh (2z/) 2
3 T vertical cylinder in a semi-infinite medium 2π ln(4/) to parallel cylinders of length in infinite medium T, 2 2 ( 2π ) cosh circular cylinder of length miday beteen to parallel plates of equal length and infinite idth T z z z /2 z 2π ln(8z/π) cylinder of length centered in a square solid of equal length T > 2π ln(.8/) eccentric circular cylinder of length in a cylinder of equal length d T > d 2π cosh ( 2 +d 2 4z 2 2d ) z conduction through the edge of adjoining alls T >
4 conduction through the corner of three alls ith a temperature difference T 2 across the alls length and idth.5 disk of diameter and temperature T on a semi-infinite medium of temperature T none 2 square channel of length W T2 T W/ <.4 W/ >.4 W 2π.785 ln(w/) 2π.93 ln(w/).5.3 Temperature distribution for fins of uniform cross section tip boundary θ(x)/θ b heat transfer (θ /θ b ) sinh βx + sinh β( x) θ() = θ sinh β dθ cosh β( x) dx = x= cosh β ka c βθ b cosh β θ /θ b sinh β ka c βθ b tanh β θ( ) = e βx ka c βθ b dθ dx = h x= k θ() cosh β( x) + (h/βk) sinh β( x) cosh β + (h/βk) sinh β ka c βθ b sinh β + (h/βk) cosh β cosh β + (h/βk) sinh β β = hp/ka c θ T T θ b = θ() = T () T θ = θ() = T () T 4
5 .4 Fin charts.4. Plate fins (or straight fins) rectangle 2t η f = tanh ξ ξ triangle 2t η f = ξ I (2ξ) I (2ξ) parabola 2t 2 η f = + + 4ξ rectangular fin triangular fin parabolic fin.7.6 η ξ = h/kt 5
6 .4.2 Spines (or pin fins) rectangle 2t η f = tanh 2ξ 2ξ triangle 2t η f = 2 2ξ I 2 (2 2ξ) I (2 2ξ) parabola 2t η f = ξ2.9.8 rectanglar fin trianglar fin parabolic fin.7.6 η ξ = h/kt 6
7 .4.3 Annular fins (or circular fins) r b r e 2t η.5.3. r e /r b = r e /r b = 2 r e /r b = 3 r e /r b = ξ = h/kt.5 Semi-infinite slab solutions penetration depth (9%): δ p = 2.3 αt.5. Constant surface temperature T (, t) = T s T (x, t) T s T i T s ( ) x = erf 4αt q s (t) = k(t s T i ) παt.5.2 Constant surface heat flux q (, t) = q o [ ) ( ) T (x, t) T i = q o 4αt ( ] k π exp x2 x x erfc 4αt 4αt.5.3 Surface convection k dt/dx x= = h[t T (, t)] T (x, t) T i T T i ( ) ( ) ( x hx = erfc exp 4αt k + h2 αt x k 2 erfc + h αt 4αt k ) 7
8 .5.4 Surface pulse of energy E T (x, t) T i = ) E ( ρc παt exp x2 4αt.5.5 Steady-periodic T (x, t) T i T penetration depth (9%): δ p = 4 α/ω NB: ω has units of rad/s, not Hz. ( = exp x ) ( ω/2α sin ωt x ) ω/2α.5.6 To semi-infinite slabs in contact T s = ( ρck) A T A,i + ( ρck) B T B,i ( ρck) A + ( ρck) B 8
9 2 Convection 2. Forced/external: plate and sphere Geometry Correlation Conditions flat plate Nu x =.332 Re /2 x Pr /3 laminar, local, T f, Pr >.6 flat plate Nu =.664 Re /2 Pr/3 laminar, average, T f, Pr >.6 flat plate Nu x =.296 Re 4/5 x Pr /3 turbulent, local, T f, Re x < 8,.6 < Pr < 6 flat plate Nu = (.37 Re 4/5 87) Pr/3 mixed, average, T f, Re cr = 5 5, Re < 8,.6 < Pr < 6 sphere Nu = 2 + ( Re /2 +.6 Re2/3 ) average, T, 3.5 < Re < 7.6 4, Pr/3.7 < Pr < 38 falling drop Nu = Re /2 Pr/3 average, T Re x = U x ν Nu x = hx k f 2.2 Forced/external: cylinder in a cross-flo Nu = h k = C Rem Pr /3 Re = U ν Re C m , , 4, Nu air ater oil Re 2.3 Forced/external: non-circular bars Nu = h k = C Rem Pr /3 Geometry Re C m square V 6 6, V 5 6, thin plate V front, 5, back 7 8, is the height of the object Nu /Pr/3 2 diamond square plate front plate back 4 5 Re 9
10 2.4 Forced/internal: turbulent flo in circular tubes Geometry Correlation Conditions circular tube Nu =.23 Re 4/5 Prn turbulent, fully developed,.6 < Pr < 6, n = for T s > T m, n =.3 for T s < T m Re = 4ṁ πµ h = 4Ac P Re cr = Forced/internal: fully developed laminar flo in tubes section uniform q s uniform T s circular rectangular see chart belo infinite plates (b ) infinite plates (one sided) Nusselt number (Nu ) a b aspect ratio (b/a) uniform q s uniform T s
11 2.6 Natural/external Geometry Correlation Conditions vertical plate Nu =.59Ra /4 laminar ( 4 < Ra < 9 ) vertical plate Nu =.Ra /3 turbulent ( 9 < Ra < 3 ) horizontal plate Nu =.54Ra /4 horizontal plate Nu =.5Ra /3 horizontal plate Nu =.52Ra /5 horizontal cylinder Nu = { sphere Nu = 2 + Ra x = gβ(ts T )x3 να.6 + }.387Ra /6 2 Ra [ + (.559/Pr) 9/6 ] 8/27 < 2 upper surface of hot plate or loer surface of cold plate; 4 < Ra < 7 ; Pr >.7 upper surface of hot plate or loer surface of cold plate; 7 < Ra < loer surface of hot plate or upper surface of cold plate; 4 < Ra < 9, Pr >.7.589Ra /4 [ + (69/Pr) 9/6 ] 4/9 Ra < ; Pr >.7 For < θ < 6, here θ is the angle measured from vertical, use the vertical plate expression ith g replaced by g cos θ. 2.7 Natural/internal Geometry Correlation Conditions horizontal cavity Nu =.69Ra /3 heated from belo; Pr < Ra < 7 9 horizontal cavity Nu = heated from above ( ) 8 ( ) /4 vertical cavity Pr H 2 < H/ < ; Pr < 5 Nu = 2 + Pr Ra ; 3 < Ra < ( ) 9 vertical cavity Pr < H/ < 2; 3 Nu =.8 + Pr Ra < Pr < 5 ; 3 < Ra Pr +Pr vertical cavity Nu = 2Ra /4 Pr.2 (H/).3 < H/ < 4; < Pr < 2 4 ; 4 < Ra < 7 H is cavity height; is distance beteen heated sides.
12 3 Radiation 3. Blackbody Emissive Poer (band emission) nλt F nλt nλt F nλt (µm K) (µm K) F -nλt emissive poer (W/m 2 µm) raper point.936 nλt (µm K) -.. avelength (µm) 3 K 77 K 33 K 798 K 578 K Wien s la 2
13 3.2 Vie factors configurations (infinite into page) Configuration h ➋ α ➋ h r s ➋ ➊ ➊ ➋ ➊ ➊ r Relation H = h/ F 2 = F 2 = + H 2 H ( α ) F 2 = F 2 = sin 2 H = h/ F 2 = ( + H ) + H 2 2 X = + s 2r F 2 = F 2 = π ( X2 + sin ) X X ➊ ➋ r c F 2 = r a b a b ( tan b c a ) tan c 3
14 configurations Configuration Relation. 2. ➊. h F 2. y ➋ y/h=. x.. x/h ➊ r.6.5 h ➋ r 2 F r 2 /h=.3. h/r.5.5. h ➋ ➊ l F /l=. h/l 4
15 4 Heat exchanger relations Single stream (C r = ): ε = exp( NT U) NT U = UA C min ; ε = q q max ; C r = C min C max parallel counterflo ε ε C r =. C r = 5 C r =.5 C r =.75 C r = NTU C r =. C r = 5 C r =.5 C r =.75 C r = NTU shell and tube -pass shell and tube 2-pass ε ε C r =. C r = 5 C r =.5 C r =.75 C r = NTU C r =. C r = 5 C r =.5 C r =.75 C r = NTU crossflo (unmixed) crossflo (mixed) ε ε C r =. C r = 5 C r =.5 C r =.75 C r = NTU mixed C r =. dashed C max C r = 5 solid C C r =.5 min C r =.75 C r = NTU 5
16 5 Material Properties 5. Solids substance k (W/m K) c p (J/kg K) ρ (kg/m 3 ) α (m 2 /s) aluminum brick cork copper diamond fiberglass germanium glass gold silicon stainless steel styrofoam tungsten Fluids substance k (W/m K) c p (J/kg K) ρ (kg/m 3 ) ν (m 2 /s) α (m 2 /s) Pr air freon mercury oil ater Phase change heat of fusion for ater: h sf = kj/kg heat of vaporization for ater: h fg = 2257 kj/kg 6
17 6 Non-dimensional quantities name expression description Biot Fourier Bi = h k s Fo = αt 2 Grashof Gr = gβ(t s T ) 3 ν 2 Jakob Ja = c p(t s T sat ) h fg Nusselt Prandtl Nu = h k f Pr = ν α Rayleigh Ra = gβ(t s T ) 3 Reynolds να Re = U ν ratio of internal thermal resistance to the external thermal resistance dimensionless time ratio of buoyancy forces to viscous forces here β = /T ratio of sensible heat to latent energy ratio of convection to conduction in a fluid ratio of momentum b.l thickness to thermal b.l. thickness ratio of thermally derived buoyancy to thermal dissipation (Ra = GrPr, here β = /T ) ratio of inertial and viscous forces 7 Math 7. Solutions to OEs OE dy dx + λy = α solution y = C exp( λx) + α λ d 2 y dx 2 λ2 y = α y = C exp( λx) + C 2 exp(λx) α λ 2 y = C 3 cosh(λx) + C 4 sinh(λx) α λ 2 d 2 y dx 2 + λ2 y = α y = C cos(λx) + C 2 sin(λx) + α λ 2 7
18 7.2 Special functions 7.2. Error functions.8 f(x).6 erf erfc 2 x Hyperbolic functions f(x) sinh cosh tanh x 8
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