Exam 3 Cheat Sheet Heat Transfer

Spring 2019 Exam 3 Cheat Sheet: Heat Transfer: Three Modes: 1. Conduction 2. Convection 3. Radiation Overall Balance for Resistance Models Q= T 1−T 2 Rtot δ Rwall= kA δ=thickness of the wall Convection (fluid: liquid or gas) 1 hA Rcyl= ( rr 21 ) Infinitely Long Fin Temperature Gradient T ( x ) −T inf −xm =e T b−T inf Note: m is not straight mass! Rate of Heat Transfer Adiabatic Fin Tip Temperature Gradient T ( x ) −T inf cosh [ m( L−x )] = cosh [ mL ] T b−T inf 2 πLk r 1=inner radius∧r 2 =outer radius L=the length of thetube Rate of Heat Transfer Q¿ Radiation: (fluids: liquid or gas) 1 Rrad= hradA Lump System Analysis: Where… 2 h∗ p A c∗k Q¿ Cylindrical Wall (Conduction) ln √ Note: p is the perimeter Planar Wall (conduction) Rconv= m= 2 hrad =εσ (T s +Tin f )(Ts+Tinf ) Spherical Wall (Conduction) r 2−r 1 Rsph= 4 πr 1 r 2 k r 1=inner radius∧r 2 =outer radius Summing Resistance: In series: Rtot =R 1+ R 2+ R 3+ … R n If the Bi# < 0.1, then lump system analysis can be used Bi= h Lc Volume where Lc = Surface Area k Lump System Equation: h As T ( t )−T inf −bt =e where b= ρV cp T i −T inf V is volume ( R1 + R1 + … R1 ) −1 In parallel: Rtot = 1 2 n Heat Transfer Through Fins: The purpose of fins is to increase the rate of heat transfer by increasing the surface area The rate of heat transfer between the body and its environment (W) Q¿ The total amount of energy transferred between the body and its surroundings (kJ) Q=m c p ( T (t )−T i ) The maximum amount of possible energy that can be transferred between the body and its surroundings (kJ) Q=m c p ( T inf −T i ) Spring 2019 ℜ= Forced Convection: Questions to ask yourself when determining h 1. What is the geometry 2. Is there a uniform surface temperature of heat flux 3. Am I looking for a local or average value 4. Is the flow laminar or turbulent – Re (watch cut off limits for different geometries) Flat Plates: vx ϑ 5 If ℜ<5∗10 =¿ laminar flow R ex= R e L= vL ϑ **If ℜ<2300=¿ laminar Heat Transfer: Q¿ Uniform Surface Temperature: Heat Transfer: Q¿ ΔT ln=(T i−T e )/ln ¿ Determining exit temperature: −h As T e =T s−( T s−T i) e m ¿ ¿ ¿ Determining h: Convective Heat Transfer: Nu=3.66= Q¿ **q ¿ is the heat flux (W/m^2) Uniform Surface Temperature: Local h x Laminar N ux= v avg D ϑ hD k Uniform Heat Flux: Heat Transfer: Q¿ Determining Surface Temperature: 1 hx x =0.332 R e x0.5 Pr 3 k T e =T i +q ¿¿ ¿ Determining h: Turbulent Nu=4.36= 1 h x N u x = x =0.0296 R ex0.8 Pr 3 k Avg h∨h L Laminar hD k Non-Circular Pipes: For pipes that are not circular, replace D with D h 1 Nu= hL 0.5 =0.664 R e L Pr 3 k Nu= hL 3 =0.037 R e 0.8 L Pr k Where… D h= Turbulent 1 ** determine Nu values from table 19-3 (page 792) Radiation: Combination Laminar and Turbulent Nu= hL 0.8 =(0.037 R e L −871) Pr k Uniform Heat Flux: Laminar 1 h x N u x = x =0.453 R e x0.5 Pr 3 k Turbulent 1 h x N u x = x =0.0308 R e x0.8 Pr 3 k Surface Temperature T s ( x ) =T inf +q ¿ ¿ Circular Pipes: v avg Dh h Dh 4 Ac ℜ= Nu= k p ν 1 3  all objects above 0K emit radiation Blackbody Radiation:    a perfect emitter and absorber blackbody radiation is the maximum amount of radiation a surface can emit emissivity, ε = 1 Emissivity, ε :    how closely a body approximates a blackbody ranges from 0 ≤ ε ≤1 note: by Kirchhoff’s Law ε ( T )=α(T ) where α = absorptivity Emissive power: Spring 2019 4 E (T )=εσ T Wien’s displacement law: ( λT )max power =2897.8 μm K