MEEN 461 HEAT TRANSFER Summary of Conduction Fourier’s Law: q Newton’s Law: q q T k ; k in [W/(mK)] A x q h Tsurface T fluid ; h in [W/(m2K)] A 4 Stefan-Boltzmann Law: q T ; emission of an ideal surface (blackbody); = 5.67 10 8 W/(m2K4); T in [K] q AT 4 ; emission of a real surface; 0.0 1.0 q12 A1 (T14 T24 ) ; radiation exchange --- surface 1 in a large enclosure 2 Steady state, 1-D conduction without internal heat generation: resistance Plane wall, T(x): q q T1 T2 T L /( kA) Rth Convection: q Thermal resistance and thermal contact Ts T T 1/(hA) Rth Thermal contact Tinterface Rtc Cylindrical wall, T(r): q Ti To T ln (ro / ri ) Rth 2 kL Spherical wall, T(r): q Ti To 1/ ro 1/ ri 4 k Analogy between conduction/convection and current flow: Ohm’s Law Thermal resistances in series: q qA qB qC ... Toverall Toverall Rth ( Rth, A Rth, B Rth,C ...) 𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅𝑡ℎ,𝐴 + 𝑅𝑡ℎ,𝐵 + 𝑅𝑡ℎ,𝐶 + ⋯ Thermal resistances in parallel: T T 1 T 1 1 q (q1 q2 q3 ...) ... T ... R R th ,1 Rth ,2 Rth ,3 th ,1 Rth ,2 Rth ,3 1 𝑅𝑡𝑜𝑡𝑎𝑙 = 1 𝑅𝑡ℎ,𝐴 + 1 𝑅𝑡ℎ,𝐵 + 1 𝑅𝑡ℎ,𝐶 +⋯ Steady state, 1-D conduction with volumetric internal heat generation: q in [W/m3] Plane wall, T(x): T ( x) qL2 x 2 1 Ts 2k L2 Cylinder, T(r): T (r ) qro2 r 2 1 Ts 4k ro2 Fins: resistance: Fin effectiveness: fin q fin (qbase ) without fin q fin hAc (Tb T ) fin 1.0 T Rth Fin efficiency: fin q fin (q fin ) k q fin fin 1.0 hAs (Tb T ) Finite-difference numerical method: Steady one-dimensional or multi-dimensional conduction Derivation of nodal equations --- based on conservation of energy, qin qout qV 0 T T x x Transient Conduction: Lumped capacitance system, T(t): h V Bi k As hA T (t ) T exp s Ti T cV Time t constant t exp t 0.1 Implicit and explicit finite-difference numerical methods: Transient conduction Derivation of nodal equations --- based on conservation of energy, qin qout qV dE / dt T T t t Solve a set of N equations simultaneously or explicitly for temperatures at N nodes after each t using the computer MEEN 461 --- Heat Transfer Summary of Convection Newton’s Law of Cooling --- Convective heat transfer coefficient in [W/(m2K)] Local heat transfer coefficient, h, and average heat transfer coefficient, h . h h 1 As As qx Tsurface T fluid qconvection qconvection Asurface Tsurface T fluid Tsurface T fluid h dAs Local heat transfer on a surface exposed to a fluid At y 0, u 0, h h(Ts T f ) k T y ; h k (T / y ) y 0 y 0 Ts T f External flow forced convection --- Concepts of hydrodynamic and thermal boundary layers Internal flow forced convection --- Concepts of developing flow (entrance region or length), and hydrodynamically and thermally fully developed flows Laminar flow and turbulent flow in forced convection --- Critical Reynolds number for transition from a laminar boundary layer to a turbulent boundary layer, 5 105, and critical Reynolds number for determining if an internal flow is laminar or turbulent, 2,300. Developing flow (entrance region) and thermally fully developed flow in a channel Thermally fully developed flow q h Tw Tm h qw constant Tw Tm for UHF and UWT Laminar flow (ReDh < 2,300) --- use tables, NuDh = constant for UHF or UWT Turbulent flow (ReDh > 2,300) --- use NuDh (ReDh, Pr) correlations, for a channel of any cross section, and UHF and UWT. qw h ( PL)(Tw Tm ) for UHF(Uniform heat flux) qw h ( PL) Tlm for UWT(Uniform Wall temperature), where Tlm (Tmo Tmi ) / ln[(Tw Tmi )/(Tw Tmo )] Px Ts Tm ( x) exp h mc Ts Tm,i p for constant surface temperature Forced convection Nu Dh constant, Nu L Nu L ( Re L , Pr ) , NuD NuD ( ReD , Pr) , or NuDh NuDh (ReDh , Pr) External flow --- flat plate, cylinder, sphere, a bank of tubes, impinging jet, and … Internal flow --- channels of various cross sections --- round, rectangular, parallel-plate, annular, and … Natural convection Nu L Nu L (GrL , Pr ) , Nu L Nu L ( RaL , Pr ) , Nu L Nu L ( RaL ) , or … External flow --- vertical, horizontal, and inclined plates, long horizontal cylinder, sphere, and … Internal flow --- vertical channel, vertical, horizontal, and inclined rectangular enclosures, concentric cylinders, and … Summary of Radiation 1. Blackbody radiation --Blackbody, Planck’s distribution, Eb ( ) = function of and T; maxT 2,898 μm K; max 1/ T Spectral (or monochromatic) emissive power, Eb (W/m2/m); emissive power, Eb T 4 (W/m2) 2. Real surfaces --Spectral emissive power, E Eb (W/m2/m), and emissive power E Eb (W/m2), where is emissivity (dimensionless; 0 < 1.0). For a blackbody, = 1.0 3. Definitions --- = J – G Irradiation G (W/m2), radiosity J (W/m2), and net radiaiton flux qnet Diffuse radiation; specular radiation; and non-participating medium 4. More Definitions --Absorptivity (), Reflectivity (), and Transmissivity () are fractions of the irradiation absorbed, reflected, and transmitted. For a blackbody, = 1.0, and = = 0. Kirchhoff’s law: 1.0 Similarly, 1.0. For any opaque surface, 1.0. and 1.0. E (T ) (T ) Eb (T ) 0 ( , T ) Eb ( , T ) d T 4 Use of Table 12.2; F(0 ) T 0 Eb d (T ) f (T ) T 5 Intensity (I) is the radiation emitted by a surface per unit projected area of the surface and per unit solid angle in a specific direction. I1 dq12 dq12 (dA1 cos 1 ) d1 2 (dA1 cos 1 ) [(dA2 cos 2 ) / r 2 ] in W/(m2sr) For a blackbody, I b Eb / . For diffuse radiation, I E / . Similarly, Ib Eb / and I E / . (dA cos 1 ) ( dA2 cos 2 ) ( dA cos 1 ) ( dA2 cos 2 ) dq12 I b1 1 Eb1 1 2 , to be used to define view factor r r2 View factor F12 is the fraction of the radiation that leaves a surface 1 arrives at another surface 2. Reciprocity relation (or reciprocity rule): A1F12 A2 F21 Summation rule For the interior surfaces of an enclosure, n Fi j Fi 1 Fi 2 Fi 3 Fi 4 ... 1.0 j 1 Superposition rule Symmetry rule Blackbody Radiation Exchange For a blackbody, 1.0, 0, and J Eb Assumptions: (1) steady state, (2) isothermal surfaces, and (3) non-participating medium separating surfaces Between two black surfaces: qnet ,12 Eb1 Eb 2 (T14 T24 ) Rspace 1/ ( A1 F12 ) where “space” resistance, Rspace 1 A1 F12 Gray body Radiation Exchange “Gray” = radiation properties are not dependent on wavelength, Additional assumptions: (4) gray or black surfaces, (5) opaque surfaces ( = 0), and (6) diffuse radiation Between two surfaces: qnet ,12 J1 J 2 J1 J 2 Rspace 1/ ( A1F12 ) where “space” resistance, Rspace 1 A1 F12 On each of the surfaces (for instance, surface 1) exchanging radiation: qnet ,1 Eb1 J1 Eb1 J1 Rsurface (1 1 ) / (1 A1 ) where “surface” resistance, Rsurface 1 1 1 A1 For three or more surfaces exchanging radiation, first solve a set of equations simultaneously for the values of J. Special cases: A relatively small surface 1 in a large enclosure 2 (or the surroundings); A2 qnet ,12 A1 or A2 Eb1 Eb 2 (T14 T24 ) 1 A1 (T14 T24 ) 1 1 1 1 2 1 1 1 0 1 A1 A1 F12 2 A2 1 A1 A1