Objectives Review: Heat Transfer Fluid Dynamics

Heat Transfer Conduction Convection Radiation Definitions?

Conduction (3D) 3-D transient (Cartesian) 3-D transient (cylindrical) Q’ = internal heat generation (W/m3, Btu/hr/ft3) k = thermal conductivity (W/m/K, Btu/hr/ft/K) T= temperature (°C, °F) τ = time (s) cp = specific heat (kJ/kg/degC.,Btu/lbm/°F) ρ = density (kg/m3, lbm/ft3) ∙ Conduction (3D) 3-D transient (Cartesian) 3-D transient (cylindrical)

Important Result for Pipes Assumptions Steady state Heat conducts in radial direction Thermal conductivity is constant No internal heat generation ri ro Q = heat transfer rate (W, Btu/hr) k = thermal conductivity (W/m/K, Btu/hr/ft/K) L = length (m, ft) t = temperature (°C, °F) subscript i for inner and o for outer ∙

Convection and Radiation Similarity Both are surface phenomena Therefore, can often be combined Difference Convection requires a fluid, radiation does not Radiation tends to be very important for large temperature differences Convection tends to be important for fluid flow

Convection Transfer of energy by means of large scale fluid motion V = velocity (m/s, ft/min) Q = heat transfer rate (W, Btu/hr) ν = kinematic viscosity = µ/ρ (m2/s, ft2/min) A = area (m2, ft2) D = tube diameter (m, ft) T = temperature (°C, °F) µ = dynamic viscosity ( kg/m/s, lbm/ft/min) α = thermal diffusivity (m2/s, ft2/min) cp = specific heat (J/kg/°C, Btu/lbm/°F) k = thermal conductivity (W/m/K, Btu/hr/ft/K) h = hc = convection heat transfer coefficient (W/m2/K, Btu/hr/ft2/F)

Dimensionless Parameters Reynolds number, Re = VD/ν Prandtl number, Pr = µcp/k = ν/α Nusselt number, Nu = hD/k Rayleigh number, Ra = …

What is the difference between thermal conductivity and thermal diffusivity? Thermal conductivity, k, is the constant of proportionality between temperature difference and conduction heat transfer per unit area Thermal diffusivity, α, is the ratio of how much heat is conducted in a material to how much heat is stored α = k/(ρcp) Pr = µcp/k = ν/α k = thermal conductivity (W/m/K, Btu/hr/ft/K) ν = kinematic viscosity = µ/ρ (m2/s, ft2/min) α = thermal diffusivity (m2/s, ft2/min) µ = dynamic viscosity ( kg/m/s, lbm/ft/min) cp = specific heat (J/kg/°C, Btu/lbm/°F) α = thermal diffusivity (m2/s)

Analogy between mass, heat, and momentum transfer Schmidt number, Sc Prandtl number, Pr Pr = ν/α

Forced Convection External turbulent flow over a flat plate Nu = hmL/k = 0.036 (Pr )0.43 (ReL0.8 – 9200 ) (µ∞ /µw )0.25 External turbulent flow (40 < ReD <105) around a single cylinder Nu = hmD/k = (0.4 ReD0.5 + 0.06 ReD(2/3) ) (Pr )0.4 (µ∞ /µw )0.25 Use with care ReL = Reynolds number based on length Q = heat transfer rate (W, Btu/hr) ReD = Reynolds number based on tube diameter A = area (m2, ft2) L = tube length (m, ft) t = temperature (°C, °F) k = thermal conductivity (W/m/K, Btu/hr/ft/K) Pr = Prandtl number µ∞ = dynamic viscosity in free stream( kg/m/s, lbm/ft/min) µ∞ = dynamic viscosity at wall temperature ( kg/m/s, lbm/ft/min) hm = mean convection heat transfer coefficient (W/m2/K, Btu/hr/ft2/F)

Natural Convection Common regime when buoyancy is dominant Dimensionless parameter Rayleigh number Ratio of diffusive to advective time scales Book has empirical relations for Vertical flat plates (eqns. 2.55, 2.56) Horizontal cylinder (eqns. 2.57, 2.58) Spheres (eqns. 2.59) Cavities (eqns. 2.60) For an ideal gas H = plate height (m, ft) T = temperature (°C, °F) Q = heat transfer rate (W, Btu/hr) g = acceleration due to gravity (m/s2, ft/min2) T = absolute temperature (K, °R) Pr = Prandtl number ν = kinematic viscosity = µ/ρ (m2/s, ft2/min) α = thermal diffusivity (m2/s)

Phase Change –Boiling What temperature does water boil under ideal conditions?

Forced Convection Boiling Example: refrigerant in a tube Heat transfer is function of: Surface roughness Tube diameter Fluid velocity Quality Fluid properties Heat-flux rate hm for halocarbon refrigerants is 100-800 Btu/hr/°F/ft2 (500-4500 W/m2/°C) Nu = hmDi/kℓ=0.0082(Reℓ2K)0.4 Reℓ = GDi/µℓ G = mass velocity = Vρ (kg/s/m2, lbm/min/ft2) k = thermal conductivity (W/m/K, Btu/hr/ft/K) Di = inner diameter of tube( m, ft) K = CΔxhfg/L C = 0.255 kg∙m/kJ, 778 ft∙lbm/Btu

Condensation Film condensation Correlations On refrigerant tube surfaces Water vapor on cooling coils Correlations Eqn. 2.62 on the outside of horizontal tubes Eqn. 2.63 on the inside of horizontal tubes

Radiation Transfer of energy by electromagnetic radiation Does not require matter (only requires that the bodies can “see” each other) 100 – 10,000 nm (mostly IR)

Radiation wavelength

Blackbody Idealized surface that Absorbs all incident radiation Emits maximum possible energy Radiation emitted is independent of direction

Surface Radiation Issues 1) Surface properties are spectral, f(λ) Usually: assume integrated properties for two beams: Short-wave and Long-wave radiation 2) Surface properties are directional, f(θ) Usually assume diffuse

Radiation emission The total energy emitted by a body, regardless of the wavelengths, is given by: Temperature always in K ! - absolute temperatures – emissivity of surface ε= 1 for blackbody – Stefan-Boltzmann constant A - area

Short-wave & long-wave radiation Short-wave – solar radiation <3mm Glass is transparent Does not depend on surface temperature Long-wave – surface or temperature radiation >3mm Glass is not transparent Depends on surface temperature

Figure 2.10 α + ρ + τ = 1 α = ε for gray surfaces

Radiation

Radiation Equations Q1-2 = Qrad = heat transferred by radiation (W, BTU/hr) F1-2 = shape factor hr = radiation heat transfer coefficient (W/m2/K, Btu/hr/ft2/F) A = area (ft2, m2) T,t = absolute temperature (°R , K) , temperature (°F, °C) ε = emissivity (surface property) σ = Stephan-Boltzman constant = 5.67 × 10-8 W/m2/K4 = 0.1713 × 10-8 BTU/hr/ft2/°R4

Combining Convection and Radiation Both happen simultaneously on a surface Slightly different temperatures Often can use h = hc + hr

Tout Tin Ro/A R1/A R2/A Ri/A Tout Tin

Add resistances for series Add U-Values for parallel l1 l2 k1, A1 A2 = A1 Tout Tin k3, A3 (l3/k3)/A3 R3/A3 l3 l thickness k thermal conductivity R thermal resistance A area

Combining all modes of heat transfer

Fluid Flow in Pipes [ft] [Pa] Analogy to steady-flow energy equation Consider incompressible, isothermal flow What is friction loss? [ft] [Pa]

Pitot Tubes

Summary Use relationships in text to solve conduction, convection, radiation, phase change, and mixed-mode heat transfer problems Next class Analyze heat exchangers Apply Bernoulli equation to flow in a duct Answer all of your questions on review material Used prometheus