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# Homework Assignment 1 - PowerPoint PPT Presentation

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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• 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)

• Thermodynamics review

• Heat transfer review

• Calculate heat transfer by all three modes

Use total differential to H = U + PV

dH=dU+PdV+VdP , using dH=TdS +VdP →

→ TdS=dU+PdV

Or: dU = TdS - PdV

• 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)

• 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)

PxV = mx Rx∙T

PyV = 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)

• Assume adiabatic mixing and no work done

• What is mixture enthalpy?

• What is mixture specific heat (cp)?

• 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

• Water, water vapor (steam), ice

• Properties of water and steam (pg 675 – 685)

• Alternative - ASHRAE Fundamentals ch. 6

• 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?

• Conduction

• Convection

• Definitions?

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)

Conduction

k - conductivity

of material

TS2

TS1

L

Tair

k - conductivity

of material

• Boundary conditions

• Dirichlet

• Tsurface = Tknown

• Neumann

L

Tair

TS2

h

TS1

x

Dirichlet

Neumann

Unsteady state heat transfer in building walls

External temperature profile

Internal temperature profile

T

time

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)

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

Important Result for Pipes

• Assumptions

• Heat conducts in radial direction

• Thermal conductivity is constant

• No internal heat generation

ri

ro

• 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

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)

Forced Convection

• Transfer of energy by means of large scale fluid motion

• Reynolds number, Re = VD/ν

• Prandtl number, Pr = µcp/k = ν/α

• Nusselt number, Nu = hD/k

• Rayleigh number, Ra = …

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)

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

• Schmidt number, Sc

• Prandtl number, Pr

Pr = ν/α

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)

Forced Convection

• External turbulent flow over a flat plate

• Nu = hmL/k = 0.036 (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 ReD0.5 + 0.06 ReD(2/3) ) (Pr )0.4 (µ∞ /µw )0.25

• Use with care

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)

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

• What temperature does water boil under ideal conditions?

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 = 0.255 kg∙m/kJ, 778 ft∙lbm/Btu

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

• 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

• 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)

• Idealized surface that

• Emits maximum possible energy

• Radiation emitted is independent of direction

• 1) Surface properties are spectral, f(λ)

• Usually: assume integrated properties for two beams:

• 2) Surface properties are directional, f(θ)

• Usually assume diffuse

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

• <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

• α + ρ + τ = 1 α = ε for gray surfaces

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= 0.1713 × 10-8 BTU/hr/ft2/°R4

• Both happen simultaneously on a surface

• Slightly different temperatures

• Often can use h = hc + hr

Tout

Tin

Ri/A

Ro/A

R1/A

R2/A

Tout

Tin

Tin

Tout

l1

l2

k1, A1

k2, A2

(l1/k1)/A1

R1/A1

(l2/k2)/A2

R2/A2

A2 = A1

k3, A3

(l3/k3)/A3

R3/A3

l thickness

k thermal conductivity

R thermal resistance

A area

l3

• Use relationships in text to solve conduction, convection, radiation, phase change, and mixed-mode heat transfer problems