Intro to PLUS by Leta Moser and Kristen Cetin PLUS accreditation Peer-Led Undergraduate Studying (PLUS) –assists students enrolled by offering class- specific, weekly study groups. –Students can attend any study group at any point in the semester to review for an exam, discuss confusing concepts, or work through practice problems. –

Lecture Objectives: Review - Heat transfer –Convection –Conduction –Radiation Analysis of a practical problem

Example Problem –radiant barrier in attic

Example Problem –heat transfer in window construction

Convection

Convection coefficient – h [W/m 2 K] Conduction Convection Natural convectionForced convection L – characteristic length h – natural convection k – air conduction L- characteristic length or Nusselt number: area Specific heat flux Heat flux

Which surface in this classroom has the largest forced convection A. Window B. Ceiling C. Walls D. Floor Which surface has the largest natural convection

How to calculate h ? What are the parametrs 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 – air viscosity Non-dimensionless momentum equation Using L = characteristic length and U 0 = arbitrary reference velocity Re L Reynolds number

Forced convection governing equations Energy equation for boundary layer Non-dimensionless energy equations T –temperature,  – thermal diffusivity  =k/  c p, k-conductivity,  - density, c p –specific cap. Wall temperature Air temperature outside of boundary layer Inertial force Viscous force Momentum diffusivity Thermal diffusivity Reynolds number Prandtl number 

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

Natural convection

GOVERNING EQUATIONS Natural convection Continuity Momentum which includes gravitational force Energy u, v – velocities, – air viscosity, g – gravitation,  ≈1/T - volumetric thermal expansion T –temperature, – air temperature out of boundary layer,  –temperature conductivity

Characteristic Number for Natural Convection Non-dimensionless governing equations Using L = characteristic length and U 0 = arbitrary reference velocity T w - wall temperature The momentum equation become Multiplying by R e 2 number R e =U  L/ Gr

Grashof number Characteristic Number for Natural Convection 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. Buoyancy forces Viscous forces General equation

Even more simple Natural convection simplified equations For laminar flow: For turbulent flow: For air: Pr ≈ 0.7,  = constant, k= constant,  = constant, g=constant Simplified equation: Or: T ∞ - air temperature outside of boundary layer, T s - 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 (h 1 or h 2 ), and exponent coefficient ‘n’ determines the value for h combined when both terms have the same order of value

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

What kind of flow is the most common for indoor surfaces A. Laminar B. Turbulent C. Transitional D. Laminar, transitional, and turbulent What about outdoor surfaces?

Conduction

Conductive heat transfer Steady-state Unsteady-state Boundary conditions –Dirichlet T surface = T known –Neumann L T air k - conductivity of material T S1 T S2 h

Boundary conditions Biot number convention conduction

Importance of analytical solution

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

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

Conduction equation describes accumulation

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