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 2.3, 2.6, /2.10, 2.12, 2.14, 2.20, 2.22• Due on Tuesday 2/3/11 (~2 weeks)
Objectives
• Thermodynamics review
• Heat transfer review• Calculate heat transfer by all three modes
Thermodynamic IdentityUse total differential to H = U + PVdH=dU+PdV+VdP , using dH=TdS +VdP →→ TdS=dU+PdVOr: dU = TdS - PdV
T-s diagram
h-s diagram
p-h diagram
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
Kkg
kJ314.8
R
lbf
lbm
ft1545
MMR
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)
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∙TPy 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 constantT = absolute temperature (K, °R)
Enthalpy of perfect gas mixture
• Assume adiabatic mixing and no work done
• What is mixture enthalpy?
• What is mixture specific heat (cp)?
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
Properties of water
• Water, water vapor (steam), ice
• Properties of water and steam (pg 675 – 685)• Alternative - ASHRAE Fundamentals ch. 6
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?
Heat Transfer
• Conduction
• Convection
• Radiation
• Definitions?
Conduction
• 1-D steady-state conduction
xT
x kAQ dd
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)
∙
L
k - conductivity of material
TS1 TS2
)(/ 21 SSx TTLkAQ
Unsteady-state conduction
• Boundary conditions
• Dirichlet
• Tsurface = Tknown
• Neumann
sourcep qx
T
xk
Tc
)( surfaceair TThx
T
L
Tair
k - conductivity of material
TS1TS2h
x
Boundary conditionsDirichlet Neumann
Unsteady state heat transfer in building walls
External temperature profile
T
time
Internal temperature profile
Conduction (3D)
• 3-D transient (Cartesian)
• 3-D transient (cylindrical)
sourcep qz
Tk
zy
Tk
yx
Tk
x
Tc
sourcep qz
Tk
z
Tk
rr
Tkr
rr
Tc
2
11
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)
∙
Important Result for Pipes
• Assumptions• Steady state• Heat conducts in radial direction• Thermal conductivity is constant• No internal heat generation
o
i
oi
rr
TTk
L
Q
ln
2Q = 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
∙
ri
ro
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
• 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)
ThAQ
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)k = thermal conductivity (W/m/K, Btu/hr/ft/K)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 (ReL
0.8 – 9200 ) (µ∞ /µw )0.25
• External turbulent flow (40 < ReD <105) around a single cylinder• Nu = hmD/k = (0.4 ReD
0.5 + 0.06 ReD(2/3) ) (Pr )0.4 (µ∞ /µw )0.25
• Use with careReL = 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 numberµ∞ = 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)
Pr
TgHTHgRa
/T 2
33
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 numberν = 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/LC = 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)
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
4ATQemited
Short-wave & long-wave radiation
• Short-wave – solar radiation• <3m• Glass is transparent • Does not depend on surface temperature
• Long-wave – surface or temperature radiation• >3m• Glass is not transparent • Depends on surface temperature
Figure 2.10
• α + ρ + τ = 1 α = ε for gray surfaces
Radiation
Radiation Equations
2
2
2
1
211
1
42
411
21111
)(
AA
F
TTAQ
2
2
2
1
211
1
3
2
2
2
1
211
1
21
42
41
111
4
111
)()(
AA
F
T
AA
F
TTTT
havg
r
tAhQ rrad
Q1-2 = Qrad = heat transferred by radiation (W, BTU/hr) F1-2 = shape factorhr = 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
R1/A R2/ARo/A
Tout
Ri/A
Tin
l1k1, A1 k2, A2
l2
l3
k3, A3
A2 = A1
(l1/k1)/A1
R1/A1
ToutTin
(l2/k2)/A2
R2/A2
(l3/k3)/A3
R3/A3
1. Add resistances for series
2. Add U-Values for parallel
l thicknessk thermal conductivityR thermal resistanceA area
Combining all modes of heat transfer
Summary
• Use relationships in text to solve conduction, convection, radiation, phase change, and mixed-mode heat transfer problems