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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).

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Homework Assignment 1

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Homework assignment 1

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

Objectives

  • Thermodynamics review

  • Heat transfer review

    • Calculate heat transfer by all three modes


Thermodynamic identity

Thermodynamic Identity

Use total differential to H = U + PV

dH=dU+PdV+VdP , using dH=TdS +VdP →

→ TdS=dU+PdV

Or: dU = TdS - PdV


T s diagram

T-s diagram


H s diagram

h-s diagram


P h diagram

p-h diagram


Ideal gas law

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

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

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

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

Properties of water

  • Water, water vapor (steam), ice

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

    • Alternative - ASHRAE Fundamentals ch. 6


Psychrometrics

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

Heat Transfer

  • Conduction

  • Convection

  • Radiation

  • Definitions?


Conduction

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

Unsteady-state conduction

k - conductivity

of material

  • Boundary conditions

    • Dirichlet

    • Tsurface = Tknown

    • Neumann

L

Tair

TS2

h

TS1

x


Boundary conditions

Boundary conditions

Dirichlet

Neumann


Unsteady state heat transfer in building walls

Unsteady state heat transfer in building walls

External temperature profile

Internal temperature profile

T

time


Conduction 3d

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)


Important result for pipes

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

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

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

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

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

Analogy between mass, heat, and momentum transfer

  • Schmidt number, Sc

  • Prandtl number, Pr

    Pr = ν/α


Forced convection1

ReL = Reynolds number based on lengthQ = 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


Natural convection

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 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

Phase Change –Boiling

  • What temperature does water boil under ideal conditions?


Forced convection boiling

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

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

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

Radiation wavelength


Blackbody

Blackbody

  • Idealized surface that

    • Absorbs all incident radiation

    • Emits maximum possible energy

    • Radiation emitted is independent of direction


Surface radiation issues

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

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 & 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

Figure 2.10

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


Radiation1

Radiation


Radiation equations

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 Equations


Combining convection and radiation

Combining Convection and Radiation

  • Both happen simultaneously on a surface

    • Slightly different temperatures

    • Often can use h = hc + hr


Homework assignment 1

Tout

Tin

Ri/A

Ro/A

R1/A

R2/A

Tout

Tin


Homework assignment 1

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


Combining all modes of heat transfer

Combining all modes of heat transfer


Summary

Summary

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


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