1. Homework Assignment 1 Review material from chapter 2
Mostly thermodynamics and heat transfer
Depends on your memory of thermodynamics and heat transfer
You should be able to do any of problems in Chapter 2
Problems:……..
Due on Tuesday 2/7/12 (~2 weeks)

2. Objectives Thermodynamics review Heat transfer review
Calculate heat transfer by all three modes
Used prometheus

3. Thermodynamic Identity
Use total differential to H = U + PV
dH=dU+PdV+VdP , using dH=TdS +VdP →
→ TdS=dU+PdV
Or: dU = TdS - PdV

4. T-s diagram

5. h-s diagram

6. p-h diagram

7. Ideal gas law Pv = RT or PV = nRT R is a constant for a given fluid
For perfect gasses
Δu = cvΔt
Δh = cpΔt
cp - cv= R
M = molecular weight (g/mol, lbm/mol)
P = pressure (Pa, psi)
V = volume (m3, ft3)
v = specific volume (m3/kg, ft3/lbm)
T = absolute temperature (K, °R)
t = temperature (C, °F)
u = internal energy (J/kg, Btu, lbm)
h = enthalpy (J/kg, Btu/lbm)
n = number of moles (mol)

8. Mixtures of Perfect Gasses
m = mx my
V = Vx Vy
T = Tx Ty
P = Px Py
Assume air is an ideal gas
-70 °C to 80 °C (-100 °F to 180 °F)
Px V = mx Rx∙T
Py V = my Ry∙T
What is ideal gas law for mixture?
m = mass (g, lbm)
P = pressure (Pa, psi)
V = volume (m3, ft3)
R = material specific gas constant
T = absolute temperature (K, °R)

9. Enthalpy of perfect gas mixture
Assume adiabatic mixing and no work done
What is mixture enthalpy?
What is mixture specific heat (cp)?

10. Mass-Weighted Averages
Quality, x, is mg/(mf + mg)
Vapor mass fraction
φ= v or h or s in expressions below
φ = φf + x φfg
φ = (1- x) φf + x φg
s = entropy (J/K/kg, BTU/°R/lbm)
m = mass (g, lbm)
h = enthalpy (J/kg, Btu/lbm)
v = specific volume (m3/kg)
Subscripts f and g refer to saturated liquid and vapor states and fg is the difference between the two

11. Properties of water Water, water vapor (steam), ice
Properties of water and steam (pg 675 – 685)
Alternative - ASHRAE Fundamentals ch. 6

12. Psychrometrics What is relative humidity (RH)?
What is humidity ratio (w)?
What is dewpoint temperature (td)?
What is the wet bulb temperature (t*)?
How do you use a psychrometric chart?
How do you calculate RH?
Why is w used in calculations?
How do you calculate the mixed conditions for two volumes or streams of air?

13. Heat Transfer
Conduction
Convection
Radiation
Definitions?

14. Conduction 1-D steady-state conduction k - conductivity of material ∙
Qx = heat transfer rate (W, Btu/hr)
k = thermal conductivity (W/m/K, Btu/hr/ft/K)
A = area (m2, ft2)
T = temperature (°C, °F) ∙
TS1
TS2 L
Tair

15. Unsteady-state conduction
k - conductivity
of material
Boundary conditions
Dirichlet
Tsurface = Tknown
Neumann L
Tair
TS2 h
TS1 x

16. Boundary conditions
Dirichlet
Neumann

17. Unsteady state heat transfer in building walls
External temperature profile
Internal temperature profile T
time

18. 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 ∙

19. 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

20. Forced 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)

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

22. 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)

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

24. 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)

25. 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)

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

27. 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

28. 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

29. 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)

30. Radiation wavelength

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

32. 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

33. 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

34. 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

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

36. Radiation

37. 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

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

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

40. 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

41. Combining all modes of heat transfer

42. Summary
Use relationships in text to solve conduction, convection, radiation, phase change, and mixed-mode heat transfer problems
Used prometheus