Homework Assignment 1

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:…….. • Due on Tuesday 2/7/12 (~2 weeks)

2. Objectives • Thermodynamics review • Heat transfer review • Calculate heat transfer by all three modes

3. Thermodynamic Identity Use total differential to H = U + PV dH=dU+PdV+VdP , using dH=TdS +VdP → → TdS=dU+PdV Or: dU = TdS - PdV

4. T-s diagram

5. h-s diagram

6. p-h diagram

7. 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)

8. 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)

9. Enthalpy of perfect gas mixture • Assume adiabatic mixing and no work done • What is mixture enthalpy? • What is mixture specific heat (cp)?

10. 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

11. Properties of water • Water, water vapor (steam), ice • Properties of water and steam (pg 675 – 685) • Alternative - ASHRAE Fundamentals ch. 6

12. 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?

13. Heat Transfer • Conduction • Convection • Radiation • Definitions?

14. ∙ 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

15. Unsteady-state conduction k - conductivity of material • Boundary conditions • Dirichlet • Tsurface = Tknown • Neumann L Tair TS2 h TS1 x

16. Boundary conditions Dirichlet Neumann

17. Unsteady state heat transfer in building walls External temperature profile Internal temperature profile T time

18. ∙ 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

19. 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

20. 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

21. Dimensionless Parameters • Reynolds number, Re = VD/ν • Prandtl number, Pr = µcp/k = ν/α • Nusselt number, Nu = hD/k • Rayleigh number, Ra = …

22. 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

23. Analogy between mass, heat, and momentum transfer • Schmidt number, Sc • Prandtl number, Pr Pr = ν/α

24. 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

25. 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

26. Phase Change –Boiling • What temperature does water boil under ideal conditions?

27. 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

28. 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

29. 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)

30. Radiation wavelength

31. Blackbody • Idealized surface that • Absorbs all incident radiation • Emits maximum possible energy • Radiation emitted is independent of direction

32. 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

33. 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

34. 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

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

36. Radiation

37. 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

38. Combining Convection and Radiation • Both happen simultaneously on a surface • Slightly different temperatures • Often can use h = hc + hr

39. Tout Tin Ri/A Ro/A R1/A R2/A Tout Tin

40. 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

41. Combining all modes of heat transfer

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