C12_Time_Dependent___2D_Nonisothermal_Flow

# C12_Time_Dependent___2D_Nonisothermal_Flow - Time-Dependent...

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Time-Dependent & 2D Temperature Distribution Energy equation in terms of temperature, neglecting the viscous heating 1. Ideal gases 2 ˆ p D TD P Ck T S D tD t ρρ =∇ + + 3. Constant-density fluids (or small variation of density) 2 ˆ p DT T S Dt ρ 2. For stationary fluids or solids 2 ˆ p T T S t 4. Boussinesq approximation for small density variation ( ) TT β −− 2 D p µ = ∇+ +∇ u gu Forced convection 2 D ρµ =∇ − u ug 1 p T = ⎛⎞ =− ⎜⎟ ⎝⎠ for forced convection 0 gTT ρβ ⎡⎤ −≈ ⎣⎦ for natural convection p +≈ g Free convection

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Time-Dependent & 2D Temperature Distribution x y 22 0 01 0 0 with initial and boundary condictions ˆ p ttt yyy TTTT Ck ty ty TT T T ρα = == + ∂∂ ∂∂ =⇔ = == = 1. Assumptions: constant C p 2. Non-zero components: 3. Equation of energy /; / Tt Ty ∂∂ ∂ ( Introducing dimensionless temperature: ) ( ) 0 / TTT TT =− ± 2 2 ty α ∂∂ = ±± 0 0 0 1 0 TTT = + = ±±± Heat transfer problems with a sudden change in boundary temperature: semi-finite boundary
Time-Dependent & 2D Temperature Distribution Heat transfer problems with a sudden change in boundary temperature: semi-finite boundary 2 2 xx uu ty ν ∂∂ = Recall equation of motion for a simple shear flow between 2 parallel plates with similar initial and boundary conditions, having the solution 0 1 4 x u y erf u t =− Thus 0 10 1 4 TT y t α ⇒= Heat flux at the heated wall () 0 0 y y y Tk qk T T y t πα = = =

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Time-Dependent & 2D Temperature Distribution Heat transfer problems with a sudden change in boundary temperature: finite boundary Introducing dimensionless variables ( ) ( ) 01 0 2 / / / TTT TT yy b t tb α =− == ± ± ± 2b +b -b y x 2 0 2 1 with boundary conditions 1 0 tt TT ty = ∂∂ ⇒= = = ±± ± ± Solution () [] 2 0 2 0 10 ( 1/2) 1/2 (1 ) 2e x p c o s n n nt n y n b b π απ = ⎛⎞ ++ ⎜⎟ −+ ⎝⎠
Time-Dependent & 2D Temperature Distribution

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C12_Time_Dependent___2D_Nonisothermal_Flow - Time-Dependent...

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