Lecture Objectives:

1. Lecture Objectives: • Answer questions related to HW 1 • Learn about Internal and External Surface Convection • Learn about conduction

2. Convection How to calculate h ?What are the parameters that affect h ?What is the boundary layer ?

3. Laminar and Turbulent Flowforced convection

4. Forced convection governing equations 1) Continuity 2) Momentum u, v – velocities n – air viscosity Non-dimensionless momentum equation Using L = characteristic length and U0 = arbitrary reference velocity ReL Reynolds number

5. Forced convectiongoverning equations Energy equation for boundary layer  T –temperature,a – thermal diffusivity a=k/rcp, k-conductivity,r- density, cp –specific cap. Non-dimensionless energy equations Air temperature outside of boundary layer Wall temperature Prandtl number Reynolds number Momentum diffusivity Inertial force Thermal diffusivity Viscous force

6. Simplified Equation for Forced convection General equation For laminar flow: For turbulent flow: For air: Pr ≈ 0.7, n = viscosity is constant, k = conductivity is constant Simplified equation: Or:

7. Natural convection

8. GOVERNING EQUATIONSNatural convection Continuity • Momentum which includes gravitational force • Energy u, v – velocities , n – air viscosity , g – gravitation, b≈1/T - volumetric thermal expansion T –temperature, – air temperature out of boundary layer, a –temperature conductivity

9. Characteristic Number for Natural Convection Non-dimensionless governing equations Using L = characteristic length and U0 = arbitrary reference velocity Tw- wall temperature The momentum equation become Gr Multiplying by Re2 number Re=UL/n

10. Grashof number Characteristic Number for Natural Convection Buoyancy forces Viscous forces The Grashof number has a similar significance for natural convection as the Reynolds number has for forced convection, i.e. it represents a ratio of buoyancy to viscous forces. General equation

11. Natural convectionsimplified equations For laminar flow: For turbulent flow: For air: Pr ≈ 0.7, n = constant, k= constant, b= constant, g=constant Simplified equation: Even more simple Or: T∞ - air temperature outside of boundary layer, Ts - surface temperature

12. Forced and/or natural convection In general, Nu = f(Re, Pr, Gr) natural and forced convection forced convection natural convection

13. Combined forced and natural convention Churchill and Usagi approach : This equation favors a dominant term (h1 or h2), and exponent coefficient ‘n’ determines the value for hcombined when both terms have the same order of value

14. Example of general forced and natural convection Equation for convection at cooled ceiling surfaces n

15. External convective heat fluxPresented model is based on experimental data, Ito (1972) Primarily forced convection (wind): Velocity at surfaces that are windward: Velocity at surfaces that are leeward: U -wind velocity Convection coefficient: u surface u windward leeward

16. Boundary Conditions at External Surfaces 1. External convective heat flux Required parameters: - wind velocity • wind direction • surface orientation N leeward Consequence: U Energy Simulation (ES) program treatsevery surface with different orientation as separate object. windward

17. Wind Direction Wind direction is defined in TMY database: “Value: 0 – 360o Wind direction in degrees at the hou indicated. ( N = 0 or 360, E = 90,   S = 180,W = 270 ). For calm winds, wind direction equals zero.” N http://rredc.nrel.gov/solar/pubs/tmy2/ http://rredc.nrel.gov/solar/pubs/tmy2/tab3-2.html leeward U windward Wind direction: ~225o

18. Conduction

19. Conductive heat transfer k - conductivity of material • Steady-state • Unsteady-state • Boundary conditions • Dirichlet Tsurface = Tknown • Neumann TS1 TS2 L h Tair

20. Boundary conditions Biot number convention conduction

21. Importance of analytical solution

22. What will be the daily temperature distribution profile on internal surface for styrofoam wall? A. B. External temperature profile T time

23. What will be the daily temperature distribution profile on internal surface for tin glass? A. B. External temperature profile T time

24. Conduction equation describes accumulation

25. Important numbers Convection Nusselt number Conduction Inertial force Reynolds number Viscous force Momentum diffusivity Prandtl number Thermal diffusivity Grashof number Buoyancy forces Viscous forces thermal internal resistance Biot number surface film resistance Reference book: Fundamentals of Heat and Mass Transfer, Incropera & DeWitt

26. HW2 • Writhe Energy Balance Equations for the 3 elements of your room from HW1 • Conduction • Convection • Radiation • Solar and • Long vawe

27. 2.5 m Internal surfaces 8 m 8 m HW2 Problem South West Steady State Energy Model

28. You already defined External Boundaries

29. Internal Boundaries Internal sources Window Transmitted Solar radiation

30. Surface Energy Balance Energy coming in = Energy going out Direction does not matter except for the Solar energy

31. Air balance - Convection on internal surfaces + Ventilation + Infiltration Uniform temperature Assumption Affect the air temperature - h, and Q as many as surfaces - maircp.airDTair= Qconvective+ Qventilation Tsupply Qconvective= ΣAihi(TSi-Tair) Ts1 mi Qventilation= Σmicp,i(Tsupply-Tair) Q2 Q1 Tair h1 h2