Exam 3 Equation Sheet

# Exam 3 Equation Sheet - FORMULA SHEET Conservation Laws...

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FORMULA SHEET Conservation Laws Control Volume Energy Balance: .. . . in out gen st E EE E  ; . st p dT Em C dt ; gen E qV Surface Energy Balance: 0 in out  Conduction Fourier’s Law: " , cond x T qk x  ; " , cond n T n ; " cond cond qq A Heat Flux Vector: "" " " xy z TT T iq jq k k i j k x yz     Heat Diffusion Equation: Rectangular Coordinates: . p TTT T kkk q C x xyyzz t      Cylindrical Coordinates: . 2 11 p T T kr k k q C rr r r z z t   Spherical Coordinates: . 2 22 1 sin sin sin p T T kr k k q C r t    Thermal Resistance Concepts: Conduction Resistance: , planewall tcond L R kA ;   , ln 2 cylinder oi t cond R lk ;     , 4 sphere io R k Convection Resistance: , 1 planewall t conv conv R hA ; , 1 2 cylinder t conv conv R rlh ; , 2 1 4 sphere tconv conv R rh Radiation Resistance: , 1 planewall trad rad R ; , 1 2 cylinder rad R rlh ; , 2 1 4 sphere rad R Combined Convection and Radiation Surface: ,, 1 conv rad t conv t rad RR R Contact Resistance: , 1 tcon tac t contact contact R Thermal Energy Generation:  . 2 1 2 plane wall s qL x Tx T kL ; . 2 2 2 1 4 cylinder o s o qr r Tr T kr

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Extended Surfaces: Convective Tip:        cosh sinh cosh sinh b mL x hm k x mL h mk mL          1/2 sinh cosh cosh sinh fin c b mL h mk mL qh P k A mL h mk mL Adiabatic Tip: cosh cosh b x mL ;     tanh fin c b P k A m L Prescribed Tip Temperature:       sinh sinh sinh Lb b mx m L x x mL       cosh sinh fin c b mL P k A mL Infinitely Long Fin: mx b x e ;   fin c b P k A 2 c hP m kA ; bb TT  ; ,, fin conv finsurface conv tip qq q ; , conv tip c L A Fin Effectiveness: , fin fin cb b q hA ; , , t conv base fin tcond f in R R Fin Efficiency: fin fin fin b q hA ;   tanh adiabatic fin mL mL ; c c A LL P  ; tanh c fin c mL mL 11 fin total o fin total b total NA q hA A   ; , 1 t cond fin fin fin R hA ; , 1 t cond finarray ot o t a l R hA Two Dimensional Steady Conduction: Conduction Shape Factor: 2 , 1 D tcond R Sk Finite Difference Method: 1, 1, , 1 , 1 , 4 i j i j ij TTTT T  Transient Conduction: Lumped System Analysis: t cond conv c t conv solid R hL Bi Rk ; exp ii t t    ; 2 c t Fo L   2 exp exp conv c is o l i d c t B iF o kL   ; p t t solid t conv conv s VC CR hA ; Analytical Solutions: ; x x L ; o r r
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Exam 3 Equation Sheet - FORMULA SHEET Conservation Laws...

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