Heat Transfer: Understanding Conduction, Convection, Radiation, and Phase Change - Prof. J, Study notes of Architecture

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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

Qx = 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) cp = specific heat (kJ/kg/degC.,Btu/lbm/°F) ρ = density (kg/m^3 , lbm/ft^3 )

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) 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/m^2 /K, Btu/hr/ft^2 /F)

Dimensionless Parameters

• Reynolds number, Re = V D/ν

• Prandtl number,^ Pr =^ μcp/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 /( ρcp)

• Pr =^ μcp/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) cp = 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

• External turbulent flow over a flat plate

• Nu = hmL/k = 0.036 (Pr )0.43^ (ReL0.8^ – 9200 ) (μ∞ /μw )0.

• External turbulent flow (40 <^ ReD <10^5 )^ around

a single cylinder

• Nu = hmD/k = (0.4 ReD0.5^ + 0.06 ReD(2/3)^ ) (Pr )0.4^ (μ∞ /μw )0.

• Better than nothing, but 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 (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) hm = mean convection heat transfer coefficient (W/m^2 /K, Btu/hr/ft^2 /F)

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/ft^2

(500-4500 W/m^2 /°C)

Nu = hmDi/kℓ=0.0082(Reℓ^2 K)0. Reℓ = GDi/μℓ G = mass velocity = V ρ (kg/s/m^2 , lbm/min/ ft^2 ) 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

• 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)

• 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.

• Radiation emitted is independent of direction

Radiation Equations

Q1-2 = Qrad = heat transferred by radiation (W, BTU/hr) F1-2 = shape factor hr = 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 × 10 -8^ W/m^2 /K^4 = 0.1713 × 10 -8^ BTU/hr/ft^2 /°R^4

Combining Convection and Radiation

• Both happen

simultaneously on a

surface

• Slightly different temperatures
• Often can use^ h = hc + hr

Tout Tin Ro / A R 1 / A (^) R 2 / A Tout Ri / A Tin l 1 k1, A 1 k 2 , A 2 l 2 l 3 k3, A 3 A 2 = A 1 (l 1 /k 1 ) / A 1 R 1 / A 1 Tout Tin (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

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