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)
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Homework Assignment 1 Review material from chapter 2 Mostly thermodynamics and heat transfer Depends on your memory of thermodynamics and heat transfer.
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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