1. Objectives
Review:
Heat Transfer
Fluid Dynamics

2. Heat Transfer
Conduction
Convection
Radiation
Definitions?

3. 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)

4. 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 ∙

5. 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

6. 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)

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

8. 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)

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

10. Forced Convection External turbulent flow over a flat plate
Nu = hmL/k = (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 ReD 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)

11. 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)

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

13. 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 Btu/hr/°F/ft2 ( 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 = kg∙m/kJ, 778 ft∙lbm/Btu

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

15. 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)

16. Radiation wavelength

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

18. 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

19. 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

20. 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

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

22. Radiation

23. 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 = × 10-8 BTU/hr/ft2/°R4

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

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

26. 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

27. Combining all modes of heat transfer

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

29. Pitot Tubes

30. 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