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Intro to PLUS by Leta Moser and Kristen Cetin • PLUS accreditation • Peer-Led Undergraduate Studying (PLUS) – assists students enrolled by offering classspecific, 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. – http://www.utexas.edu/ugs/slc/support/plus 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/m2K] Q A h (Twall Tair ) A h (Tw T ) [W] Heat flux area or q h (Tw T ) [W/m2] Specific heat flux Natural convection Forced convection T T Tw Tw L – characteristic length Nusselt number: hL Nu k Convection Conduction h – natural convection k – air conduction L- characteristic length 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 u v 0 x y 2) Momentum u u 2u u v v 2 x y y u, v – velocities n – air viscosity Non-dimensionless momentum equation Using x* x L ; y* y L ; u* u U oo v* v ; U oo L = characteristic length and U0 = arbitrary reference velocity * * 2 * u u 1 u * * u v *2 * * U oo L y x y n ReL Reynolds number Forced convection governing equations Energy equation for boundary layer 2 T T T u v x y y 2 T –temperature, – thermal diffusivity =k/rcp, k-conductivity, r - density, cp –specific cap. Non-dimensionless energy equations T* T 1 T * T u v * * x y Re L . Pr . y *2 * * 2 * T T Tw T Air temperature outside of boundary layer * Reynolds number Re L UL n Inertial force Viscous force Wall temperature Prandtl number Pr n a Momentum diffusivity Thermal diffusivity Simplified Equation for Forced convection General equation Nu f (Re, Pr) For laminar flow: Nu C L Re Pr For turbulent flow: Nu CT Re Pr 1/ 3 Nu hL k Re L 4/5 For air: Pr ≈ 0.7, n = viscosity is constant, k = conductivity is constant Simplified equation: h forced f (U , Ln ) C U m m Or: h forced C ACH m ACH Volume flow rate RoomVolume UL n Natural convection GOVERNING EQUATIONS Natural convection Continuity u v 0 x y • Momentum which includes gravitational force u u 2u u v g T T v 2 x y y • Energy T T 2T u v x y y 2 u, v – velocities , n – air viscosity , g – gravitation, ≈1/T - volumetric thermal expansion T –temperature, T– air temperature out of boundary layer, –temperature conductivity Characteristic Number for Natural Convection Non-dimensionless governing equations Using x* x L ; y * y L ; u * u U ; v* vU ; T* T T Tw T L = characteristic length and U0 = arbitrary reference velocity Tw- wall temperature The momentum equation become * u * g Tw T L * 1 2u * * u u v T *2 * * 2 x y U Re L y Gr * Multiplying by Re2 number Re=UL/n g Tw T L3 * * 2 * u u u 2 u * * v* * (Gr / Re L ) T * (1 / Re L ) *2 x y y n2 Grashof number Characteristic Number for Natural Convection Gr g Tw T L3 n2 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 Nu f (Gr, Pr) Natural convection simplified equations For laminar flow: Nu C L Gr Pr For turbulent flow: Nu CT Gr Pr 1/ 4 1/ 3 For air: Pr ≈ 0.7, n = constant, k= constant, = constant, g=constant Simplified equation: h forced f ((Tw T ) m , Ln ) f (T m , Ln ) Or: Even more simple h forced C T m T∞ - air temperature outside of boundary layer, Ts - surface temperature Forced and/or natural convection In general, Nu = f(Re, Pr, Gr) GrL Re 2L 1 Nu f (Re, Pr, Gr) natural and forced convection GrL Re 2L 1 Nu f (Re, Pr) forced convection GrL Re 2L 1 Nu f (Gr , Pr) natural convection Combined forced and natural convention Churchill and Usagi approach : n hcombined (h1n h2n )1/ n (hnatural hnforced )1/ n 5 h 4 h combined 3 n=2 n=3 2 h n=6 1 1 h 2 T or ACH 0 0 1 2 3 4 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 hnatural (2.12 T 0.33 )3 h forced 1.19 ACH 0.8 hcombinbed (2.12 T n ) (1.19 ACH ) 0.33 3 0.8 3 1/ 3 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 k - conductivity of material • Steady-state q k / L(TS1 TS 2 ) • Unsteady-state TS1 T k T qsource 2 rc p x 2 L • Boundary conditions Tair – Dirichlet – Neumann Tsurface = Tknown T h(Tair Tsurface) x TS2 h Boundary conditions Biot number hL Bi k solid convention conduction Importance of analytical solution 1.0 T h To h h (T-Ts)/(To-Ts) Ts 0 Analytical solution Numerical -3 nodes, =60 min Numerical -7 nodes, =60 min Numerical -7 nodes, =12 min 0.8 0.7 T -L/ 2 0.9 L/2 h omogenous wall L = 0.2 m k = 0. 5 W/ mK c p = 9 20 J/kgK 2 r = 120 0 k g/m 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 1 2 3 4 5 hour 6 7 8 9 10 What will be the daily temperature distribution profile on internal surface for styrofoam wall? External temperature profile T time A. B. What will be the daily temperature distribution profile on internal surface for tin glass? External temperature profile T time A. B. Conduction equation describes accumulation Important numbers Nusselt number Reynolds number Prandtl number Grashof number Biot number hL Nu k Re L Pr Gr Convection Conduction UoL Inertial force n Viscous force n Momentum diffusivity a Thermal diffusivity g Ts T L3 hL Bi k solid n2 Buoyancy forces Viscous forces thermal internal resistance surface film resistance Reference book: Fundamentals of Heat and Mass Transfer, Incropera & DeWitt