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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 Identity

Use total differential to H = U + PV

dH=dU+PdV+VdP , using dH=TdS +VdP →

→ TdS=dU+PdV

Or: dU = TdS - PdV

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)

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)

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)

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?

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- 1-D steady-state conduction

k - conductivity

of material

TS2

TS1

L

Tair

Unsteady-state conduction

k - conductivity

of material

- Boundary conditions
- Dirichlet
- Tsurface = Tknown
- Neumann

L

Tair

TS2

h

TS1

x

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
- Steady state
- Heat conducts in radial direction
- Thermal conductivity is constant
- No internal heat generation

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

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

Dimensionless Parameters

- Reynolds number, Re = VD/ν
- Prandtl number, Pr = µcp/k = ν/α
- Nusselt number, Nu = hD/k
- Rayleigh number, Ra = …

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)

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

Analogy between mass, heat, and momentum transfer

- 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

Phase Change –Boiling

- 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

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)

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

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

Figure 2.10

- 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

Radiation EquationsCombining Convection and Radiation

- Both happen simultaneously on a surface
- Slightly different temperatures
- Often can use h = hc + hr

Tin

Tout

- Add resistances for series
- Add U-Values for parallel

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

Summary

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

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