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

HW1   1) Using the equations provided in the attached paper sheet and the basic properties of view factors calculate the view factors for internal characteristic surfaces: FSS , FSE , FSI FES , FEE , FEI FIS , FIE , FII 2) Using the geometry of the building, period of the year, and provided data in the excel file calculate: - incident angle of direct solar radiation on all external surfaces for period of 24 hours, - direct (ID) and diffuse (Id) components of solar radiation for period of 24 hours. For the ground surface assume reflectivity of rground = 0.2. 3) Using both a) Swinbank Cole model and b) Berdahl and Martin model (provided in class notes) and data provided in the excel file calculate the equivalent sky temperature for the period of 24 hours.

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

Laminar and Turbulent Flow forced convection

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

Forced convection governing 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 Reynolds number Prandtl number Momentum diffusivity Inertial force Thermal diffusivity Viscous force

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:

Natural convection

GOVERNING EQUATIONS Natural 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

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

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

Natural convection simplified 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

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

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

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

External convective heat flux Presented 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

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 treats every surface with different orientation as separate object. windward

Wind Direction Wind direction: ~225o 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

Conduction

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

Boundary conditions Biot number convention conduction

Importance of analytical solution

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

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

Conduction equation describes accumulation

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

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

HW2 Problem East South Steady State Energy Model 2.5 m Internal surfaces East South Steady State Energy Model

You already defined External Boundaries

Internal Boundaries Window Internal sources Transmitted Solar radiation

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

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