Two Phase Flows (Section 3) The Basic Model

Transcription

1 Two Phase Flows (Section 3) By: Pro. M. H. Saidi Center o Excellence in Energy Conversion Shari University o Technology

2 Homewor Homewor set#1 Problems 1,,3,4; Chapter 1, Collier and Thome. Due to next Tuesday (Mehr, 14th)

3 Time Averaging B 1 T = T 0 Bdt

4 Volume Averaging Bˆ 1 = V V BdV Volume Averaging { B} 1 = V c V c BdV Phase Averaging Relation between phase and volume averaging or continues phase { B} = α Bˆ c

5 Principal Equation o Two Phase Flow 1. One dimensional low. Steady state 3. Constant physical properties 4. Existence o Thermodynamic equilibrium

6 Conservation o Mass α ρ t Z ( Aα ρ ) + ( Aα ρu ) = Γ Void raction o phase Density o phase Continuity equation o phase For steady state low Γ Mass Generation rate per Γ = 0 unit length or phase t ( Aα ρ ) = 0 For gas (g)- liquid () two phase low d ( A ρ u ) =Γ d ( ρ u ) =Γ g g g g dw g Γ g = Γ = = dw

7 Conservation o Momentum t ( W δz) + ( W u + δz ( W u )) W u z t ( W δz) + δz ( W u ) z rate o creation o momentum o phase rate o inlow o momentum within the control volume W = Aα ρ u Pressure orces on the element Aα p Aα p + δz ( Aα p) p δz ( Aα ) z z Aα ρδzg sinθ τ ρδzg sinθ τ P δz + τ P δz + u Γ w w w nz n 1 Gravity Wall shear Interacial shear n Rate o momentum generation

8 Conservation o Momentum Force= creation o momentum + inlow o momentum within the control volume n p Aα δz τ P δz + τ P δz Aα ρδzg sinθ + u Γ w w nz n z 1 = ( W δz) + δz ( W u ) t z Steady gas- liquid two phase low Adp τ P + τ P A ρ g sinθ + u Γ = W du g gw gw g g g g g g g g Adp τ P + τ P A ρ g sinθ + u Γ = W du w w g g Ι И Momentum conservation at interace τ P + u Γ = τ P + u Γ g g g g g g Ш

9 Conservation o Momentum summation o equation Ι, И & Ш * Adp τ P τ P g sin θ( A ρ + A ρ ) = dw ( u + W u ) gw gw w w g g g g Friction orce or each phase ** dp ( dfg + S) = τgwpgw τg Pg = Ag( gf ) dp ( df S) = τwpw + τg Pg = A ( F ) dp ( dfg + df ) = τgwpgw τwpw = A( F ) Part o total pressure gradient which is need or prevalence o riction

10 Conservation o Momentum substitution equation ** in * yields ( dp ) = ( dp F) + ( dp a) + ( dp z) dp 1 d d x ν g (1 x) ν ( a) = ( W gug + W u ) = G + A α (1 α) dp Ag A ( z ) = g sinθ ρg ρ g sin θ αρg (1 α) ρ + = + A A Total pressure lost

11 Energy Conservation t u u u u [ αρ ( ε + ) Aδz] + W ( ε + ) δz [ W ( ε + ) δz W ( ε + )] z A rate o increase o total energy in the C.V rate o entrance o energy within the control volume ε Internal energy per unit mass Rate o heat entrance to C.V o phase n φ P δz + φ P δz + φ& Aαδz w w n n 1 B Heat low rom channel wall H.V. via the various interaces Internal heat generation within C.V

12 Energy Conservation The wor done by pressure orces Wor done by expansion o phase W p W p W p α δz ( ) W g sinθδz paδz ρ + ρ z ρ t n δzp +Γ + u τ P δ z n n ρ 1 The wor done by body orce C Wor done by pressure and shear orces at the interace with the other phases Mass generation rate per unit length u Γ δz ( ε + ) D

13 A=B+C+D Enthalpy o phase Energy Conservation u u Aα ρ ε W i W g θ φ P t z α u + + +Γ + + ( + ) + ( + ) = sin + w w n n φnpn φ& Aα pa ( i ) u τnpn 1 t 1 i = u + p ρ For steady gas- liquid two phase low in channel with constant area u g d W g( ig + ) + W g g sinθδz = u φwgpwgδz + φg Pgδz + ugτg Pgδz +Γ gδz( ig + ) u d W ( i + ) + W g sinθδz = u φw Pwδz + φgpgδz + uτgpgδz +Γ δz ( i + )

14 Energy Conservation Energy conservation at interace ug u φg Pg + ugτg Pgδz +Γ g( ig + ) = φw Pwδz + φgpgδz + uτgpgδz +Γ δz( i + ) with regard the equations, and d W u [ W i W i ] [ ] ( W W ) g sin Q d g g W u g g g + θ = w1 Heat transer to the luid across the channel wall per unit length Q ( = φ p + φ p ) wl w w wg wg

15 Energy Conservation Total pressure gradient Frictional dissipation dp de Q [ xνg + (1 x) ν ] = W 3 3 d G d xv g (1 x) v + p [ xνg + (1 x) ν ] + + g sinθ + α (1 α) w Acceleration head term Static head term Internal energy per unit mass E = xε + (1 x) ε g

16 Use o the momentum or energy equation to evaluate the pressure gradient Using momentum equation Using void raction to calculate acceleration term rom dp 1 d d x ν g (1 x) ν ( a) = ( W gug + W u ) = G + A α (1 α) or static head term rom dp Ag A ( z ) = g sinθ ρg + ρ = g sin θ αρg + (1 α) ρ A A Then calculating riction pressure term rom correlation equation in terms o independent variables.

17 Use o the momentum or energy equation to evaluate the pressure gradient Using energy equation Calculation o pressure lost arising rom variation o potential energy Calculation o pressure lost arising rom variation o inetic energy Calculate the riction pressure term rom independent variables Note: in two methods we need to the void raction but the degree o importance in each method is not the same.