Objectives Calculate heat transfer by all three modes Phase change Next class Apply Bernoulli equation to flow in a duct

Heat Transfer Conduction Convection Radiation Definitions?

Conduction 1-D steady conduction Q x = heat transfer rate (W, Btu/hr) k = thermal conductivity (W/m/K, Btu/hr/ft/K) A = area (m 2, ft 2 ) t = temperature (°C, °F) ∙

Conduction (2) 3-D transient (Cartesian) 3-D transient (cylindrical) Q’ = internal heat generation (W/m 3, Btu/hr/ft 3 ) k = thermal conductivity (W/m/K, Btu/hr/ft/K) t = temperature (°C, °F) τ = time (s) c p = specific heat (kJ/kg/degC.,Btu/lbm/°F) ρ = density (kg/m 3, lbm/ft 3 ) ∙

Important Result for Pipes Assumptions Steady state Heat conducts radially only Thermal conductivity is constant No internal heat generation 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 ∙ riri roro

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

Forced Convection (1) 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 (m 2 /s, ft 2 /min) A = area (m 2, ft 2 ) D = tube diameter (m, ft)t = temperature (°C, °F) µ = dynamic viscosity ( kg/m/s, lbm/ft/min)α = thermal diffusivity (m 2 /s, ft 2 /min) c p = specific heat (J/kg/°C, Btu/lbm/°F) k = thermal conductivity (W/m/K, Btu/hr/ft/K) h = h c = convection heat transfer coefficient (W/m 2 /K, Btu/hr/ft 2 /F)

Dimensionless Parameters Reynolds number, Re = VD/ν Prandtl number, Pr = µc p /k = ν/α Nusselt number, Nu = hD/k

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/(ρc p ) Pr = µc p /k = ν/α k = thermal conductivity (W/m/K, Btu/hr/ft/K) ν = kinematic viscosity = µ/ρ (m 2 /s, ft 2 /min) α = thermal diffusivity (m 2 /s, ft 2 /min) µ = dynamic viscosity ( kg/m/s, lbm/ft/min) c p = specific heat (J/kg/°C, Btu/lbm/°F) k = thermal conductivity (W/m/K, Btu/hr/ft/K) α = thermal diffusivity (m 2 /s)

Forced Convection (2) External turbulent flow over a flat plate Nu = h m L/k = (Pr ) 0.43 (Re L 0.8 – 9200 ) (µ ∞ /µ w ) 0.25 External turbulent flow (40 < Re D <10 5 ) around a single cylinder Nu = h m D/k = (0.4 Re D Re D (2/3) ) (Pr ) 0.4 (µ ∞ /µ w ) 0.25 Better than nothing, but use with care Re L = Reynolds number based on lengthQ = heat transfer rate (W, Btu/hr) Re D = Reynolds number based on tube diameter A = area (m 2, ft 2 ) 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) h m = mean convection heat transfer coefficient (W/m 2 /K, Btu/hr/ft 2 /F)

Natural Convection (1) 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/s 2, ft/min 2 ) T = absolute temperature (K, °R) Pr = Prandtl number ν = kinematic viscosity = µ/ρ (m 2 /s, ft 2 /min) α = thermal diffusivity (m 2 /s)

Phase Change – Pool 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 h m for halocarbon refrigerants is Btu/hr/°F/ft 2 ( W/m 2 /°C) Nu = h m D i /k ℓ =0.0082(Re ℓ 2 K) 0.4 Re ℓ = GD i /µ ℓ G = mass velocity = Vρ (kg/s/m 2, lbm/min/ft 2 ) k = thermal conductivity (W/m/K, Btu/hr/ft/K) D i = inner diameter of tube( m, ft) K = CΔxh fg /L C = kg∙m/kJ, 778 ft∙lbm/Btu

Condensation Film condensation On refrigerant tube surfaces Water vapor on cooling coils Correlations Eqn on the outside of horizontal tubes Eqn 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) Issues Surface properties are spectral, f(λ) Assume integrated properties Surface properties are directional, f(θ) Usually assume diffuse Assume “total properties”

Blackbody Idealized surface that Absorbs all incident radiation Emits maximum possible energy Equation 2.66 Radiation emitted is independent of direction

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

Radiation

Radiation Equations Q 1-2 = Q rad = heat transferred by radiation (W, BTU/hr) F 1-2 = shape factor h r = radiation heat transfer coefficient (W/m 2 /K, Btu/hr/ft 2 /F) A = area (ft 2, m 2 ) T,t = absolute temperature (°R, K), temperature (°F, °C) ε = emissivity (surface property) σ = Stephan-Boltzman constant = 5.67 × W/m 2 /K 4 = × BTU/hr/ft 2 /°R 4

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

T out T in R1/AR1/A R2/AR2/A Ro/ARo/A T out Ri/ARi/A T in

l1l1 k 1, A 1 k 2, A 2 l2l2 l3l3 k 3, A 3 A 2 = A 1 (l 1 /k 1 )/A 1 R 1 /A 1 T out T in (l 2 /k 2 )/A 2 R 2 /A 2 (l 3 /k 3 )/A 3 R 3 /A 3 1.Add resistances for series 2.Add U-Values for parallel l thickness k thermal conductivity R thermal resistance A area

R1/A1R1/A1 T out T in R2/A2R2/A2 R3/A3R3/A3 1.R 1 /A 1 + R 2 /A 2 = (R 1 + R 2 ) /A 1 = R 12 /A 1 =1/(U 12 A 1 ) 2.R 3 /A 3 =1/(U 3 A 3 ) 3.U 3 A 3 + U 12 A 1 4.q = (U 3 A 3 + U 12 A 1 )ΔT A 1 =A 2

Combining all modes of heat transfer

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