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