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

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 = c v Δt Δh = c p Δt c p - c v = R M = molecular weight (g/mol, lbm/mol) P = pressure (Pa, psi) V = volume (m 3, ft 3 ) v = specific volume (m 3 /kg, ft 3 /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 = m x m y V = V x V y T = T x T y P = P x P y Assume air is an ideal gas -70 °C to 80 °C (-100 °F to 180 °F) P x V = m x R x ∙ T P y V = m y R y ∙ T What is ideal gas law for mixture? m = mass (g, lbm) P = pressure (Pa, psi) V = volume (m 3, ft 3 ) 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 (c p )?

Mass-Weighted Averages Quality, x, is m g /(m f + m g ) Vapor mass fraction φ= v or h or s in expressions below φ = φ f + x φ fg φ = (1- x) φ f + x φ g s = entropy (J/K/kg, BTU/°R/lbm) m = mass (g, lbm) h = enthalpy (J/kg, Btu/lbm) v = specific volume (m 3 /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 (t d )? 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 Q x = heat transfer rate (W, Btu/hr) k = thermal conductivity (W/m/K, Btu/hr/ft/K) A = area (m 2, ft 2 ) T = temperature (°C, °F) ∙ L T air k - conductivity of material T S1 T S2

Unsteady-state conduction Boundary conditions Dirichlet T surface = T known Neumann L T air k - conductivity of material T S1 T S2 h x

Boundary conditions Dirichlet 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) Q’ = internal heat generation (W/m 3, Btu/hr/ft 3 ) k = thermal conductivity (W/m/K, Btu/hr/ft/K) T= temperature (°C, °F) τ = time (s) c p = specific heat (kJ/kg/degC.,Btu/lbm/°F) ρ = density (kg/m 3, lbm/ft 3 ) ∙

Important Result for Pipes Assumptions Steady state Heat conducts in radial direction Thermal conductivity is constant No internal heat generation 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 ∙ riri roro

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 (m 2 /s, ft 2 /min) A = area (m 2, ft 2 ) D = tube diameter (m, ft)T = temperature (°C, °F) µ = dynamic viscosity ( kg/m/s, lbm/ft/min)α = thermal diffusivity (m 2 /s, ft 2 /min) c p = specific heat (J/kg/°C, Btu/lbm/°F) k = thermal conductivity (W/m/K, Btu/hr/ft/K) h = h c = convection heat transfer coefficient (W/m 2 /K, Btu/hr/ft 2 /F)

Dimensionless Parameters Reynolds number, Re = VD/ν Prandtl number, Pr = µc p /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/(ρc p ) Pr = µc p /k = ν/α k = thermal conductivity (W/m/K, Btu/hr/ft/K) ν = kinematic viscosity = µ/ρ (m 2 /s, ft 2 /min) α = thermal diffusivity (m 2 /s, ft 2 /min) µ = dynamic viscosity ( kg/m/s, lbm/ft/min) c p = specific heat (J/kg/°C, Btu/lbm/°F) k = thermal conductivity (W/m/K, Btu/hr/ft/K) α = thermal diffusivity (m 2 /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 = h m L/k = (Pr ) 0.43 (Re L 0.8 – 9200 ) (µ ∞ /µ w ) 0.25 External turbulent flow (40 < Re D <10 5 ) around a single cylinder Nu = h m D/k = (0.4 Re D Re D (2/3) ) (Pr ) 0.4 (µ ∞ /µ w ) 0.25 Use with care Re L = Reynolds number based on lengthQ = heat transfer rate (W, Btu/hr) Re D = Reynolds number based on tube diameter A = area (m 2, ft 2 ) 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) h m = mean convection heat transfer coefficient (W/m 2 /K, Btu/hr/ft 2 /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) 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/s 2, ft/min 2 ) T = absolute temperature (K, °R) Pr = Prandtl number ν = kinematic viscosity = µ/ρ (m 2 /s, ft 2 /min) α = thermal diffusivity (m 2 /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 h m for halocarbon refrigerants is Btu/hr/°F/ft 2 ( W/m 2 /°C) Nu = h m D i /k ℓ =0.0082(Re ℓ 2 K) 0.4 Re ℓ = GD i /µ ℓ G = mass velocity = Vρ (kg/s/m 2, lbm/min/ft 2 ) k = thermal conductivity (W/m/K, Btu/hr/ft/K) D i = inner diameter of tube( m, ft) K = CΔxh fg /L C = kg∙m/kJ, 778 ft∙lbm/Btu

Condensation Film condensation On refrigerant tube surfaces Water vapor on cooling coils Correlations Eqn on the outside of horizontal tubes Eqn 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

Short-wave & long-wave radiation Short-wave – solar radiation <3  m Glass is transparent Does not depend on surface temperature Long-wave – surface or temperature radiation >3  m Glass is not transparent Depends on surface temperature

Figure 2.10 α + ρ + τ = 1 α = ε for gray surfaces

Radiation

Radiation Equations Q 1-2 = Q rad = heat transferred by radiation (W, BTU/hr) F 1-2 = shape factor h r = radiation heat transfer coefficient (W/m 2 /K, Btu/hr/ft 2 /F) A = area (ft 2, m 2 ) T,t = absolute temperature (°R, K), temperature (°F, °C) ε = emissivity (surface property) σ = Stephan-Boltzman constant = 5.67 × W/m 2 /K 4 = × BTU/hr/ft 2 /°R 4

Combining Convection and Radiation Both happen simultaneously on a surface Slightly different temperatures Often can use h = h c + h r

T out T in R1/AR1/A R2/AR2/A Ro/ARo/A T out Ri/ARi/A T in

l1l1 k 1, A 1 k 2, A 2 l2l2 l3l3 k 3, A 3 A 2 = A 1 (l 1 /k 1 )/A 1 R 1 /A 1 T out T in (l 2 /k 2 )/A 2 R 2 /A 2 (l 3 /k 3 )/A 3 R 3 /A 3 1.Add resistances for series 2.Add U-Values for parallel l thickness k thermal conductivity R thermal resistance A 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