Heat Transfer Heat transfer rate by conduction is related to the temperature gradient by Fourier s law. For the one-dimensional heat transfer problem in Fig. 1.8, in which temperature varies in the y- direction only, the heat transfer rate is obtained by Fourier s law q = y dt k dy (1.66) 1
Figure 1.8 One-dimensional conduction. 2
For heat conduction in a multidimensional system, eq. (1.66) can be rewritten in the following generalized form q = k where both the heat flux and the temperature gradient are vectors, i.e., T (1.67) q = iq + jq + kq x y z (1.68) 3
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell Unlike the isotropic materials whose thermal conductivity is a scalar, the thermal conductivity of the anisotropic material is a tensor of the second order: kxx kxy k xz k = k yx k yy k yz (1.69) kzx kzy k zz and eq. (1.67) will become q = k T (1.70) Transport Phenomena in Multiphase Systems with Phase Change 4
In a multicomponent system, mass transfer can also contribute to the heat flux N N N x x D J q = k T + hi Ji + crut i= 1 i= 1 j = 1( j i) ρ D ρ ρ T i j i Ji j i ij i j (1.71) where the second term on the right-hand side represents the interdiffusional convection term, and the third term is the contribution of concentration gradient to the heat flux (the diffusion-thermo or Dufour effect). 5
For a binary system ω and ω D = x x D 1 2 12 12 ω 1ω 2 (1.72) where 1 are 2 mass fraction of component 1 and 2 respectively. x1 x2 D12D33 D13D 23 D12 = ω ω D + D D D (1.73) 1 2 12 33 13 23 Transport Phenomena in Multiphase Systems with Phase Change 6
Figure 1.9 Forced convective heat transfer. q = h( Tw T ) Nu = hl k (1.74) (1.75) 7
T a b l e 1. 5 Typical values of mean convective heat transfer coefficients Mode Geometry 2 h (W/m -K) Forced convection Air flows at 2 m/s over a 0.2 m square plate 12 Air at 2 atm flowing in a 2.5 cm-diameter tube with a velocity of 10 m/s 65 Water flowing in a 2.5 cm-diameter tube with a mass flow rate of 0.5 kg/s 3500 Airflow across 5 cm-diameter cylinder with velocity of 50 m/s 180 Free convection Vertical plate 0.3 m high in air 4.5 o ( T = 20 C) ) Horizontal cylinder with a diameter of 2 cm in water 890 Evaporation Falling film on a heated wall 6000-27000 Condensation of water at 1 Vertical surface 4000-11300 atm Outside horizontal tube 9500-25000 Boiling of water at 1 atm Natural convectioncontrolled melting and solidification Pool 2500-3500 Forced convection 5000-100000 Melting in a rectangular enclosure 500-1500 Solidification around a horizontal tube in a superheated liquid phase change material 1000-1500 8
The third mode of heat transfer is radiation. When matter is heated, some of its molecules or atoms are excited to a higher energy level. Thermal radiation occurs when these excited molecules or atoms return to lower energy states. Although thermal radiation can result from changes of the energy states of electrons, as well as vibrational and rotational energy of molecules or atoms, all of these radiant energies travel at the speed of light. The wavelength is related to the frequency by An electromagnetic wave can also be viewed as a particle a photon with energy of ε λ ν = = hν c (1.76) (1.77) Transport Phenomena in Multiphase Systems with Phase Change 9
For a blackbody, the spectral emissive power can be obtained by Planck s law c E 1 b, λ c /( T ) 2 2 λ ( e 1) The emissive power for a blackbody, is = Substituting eq. (1.78) into eq. (1.79), Stefan-Boltzmann s law is obtained For a real surface, the emissive power is obtained by λ Eb = Eb, λ d λ E b = 0 σ (1.78) (1.79) (1.80) (1.81) If the temperature of the surroundings is T sur, the heat transfer rate per unit area from the small object is obtained by SB T E = ε E b q = ε σ T T 4 4 4 SB ( w sur ) (1.82) Transport Phenomena in Multiphase Systems with Phase Change 10
T w Figure 1.10 Radiation heat transfer between a small surface and its surroundings. 11