analysis of water injection into high-temperature mixture of ...

82• ⎛ Tb u ⎞ • • ⎛bmTb bu ⎞ • ⎛bmTb bub mb ubm ∂b ( ub uu) RWT WQb PV ∂b1P∂ ∂ ⎞− − − = − + − +⎜ T bVbP T ⎝ ∂ ⎟⎠ ⎜⎝ ∂ ⎟⎜b ⎠ ⎝ P∂Tb∂P⎟⎠ Eq.C-40• ⎛ u ⎞ • • ⎛bub 1 ⎞ • ⎛ ub1m ∂b ( ub uu)RWTW T bQb PV ∂b1VbP∂ ∂ubm ⎞b− − − = − + − +⎜ ∂T ⎟⎜ ∂T R ⎝ ⎠ ⎝ ⎟⎠ Eq.C-41⎝⎜∂T R ∂P V ⎟⎠b b b b bbFrom Eq.C-2 V = V + VbbWWwe get• • •V = V + VEq.C-42Substituting Eq.C-42 into Eq.C-41• ⎛ u ⎞ • • • ⎛bum ∂⎛ ⎞ ∂b ( ub−uu)−RWTW − T b= Qb−P V− VW1+T⎝ ⎜∂ ⎟⎠ ⎜⎝⎟⎠⎜⎝∂Tb1 ⎞R ⎟⎠• ⎛ ub1 u ⎞b( V VW) P ∂ ∂ mb− − ⎜+⎜∂ Tb Rb∂ P V⎝⎟b ⎠• ⎛ ∂u⎞ • • •bub1mb ( ub uW)RWTW T ⎛ ⎞⎛∂ ⎞− − − b= Qb− PV− PVW1+⎜∂T⎟ ⎜⎝⎟⎠⎜ ∂T⎝ ⎠ ⎝ R ⎟⎠b• ⎛ ub 1 ub m ⎞ • ⎛bub 1 ubm ⎞PV∂ ∂ PV∂ ∂b− + +W+⎜∂T R ∂P V ⎟⎜∂T R ∂P V⎝ ⎠ ⎝ ⎟⎠b b b b b bbbbbbEq.C-43Eq.C-44From Eq.C-25 we can write⎛ • • • ⎞•mWTWPVW= V + −⎜ mW TW P⎜⎝⎟⎠Substituting Eq.C-45 into Eq.C-44⎛• • • ⎞• ⎛ ub ⎞ • ⎛•mW TWP ⎟⎞ ⎟⎛1m ∂b ( ub uW)RWTW T bQ bPV PV∂u⎞bW1− − − = − − + − +⎜∂T ⎟b mW TWP⎟⎟⎜ ∂⎝ ⎠ ⎜⎟⎟⎝⎟⎜⎝⎝ ⎠⎟TbRb⎠⎠• ⎛ ub 1 ub m ⎞ • ⎛bub 1 u ⎞bPV∂ ∂ PV∂ ∂ mb− + +W+⎜∂T R ∂P V ⎟⎜∂T R ∂P V⎝ ⎠ ⎝ ⎟⎠b b b b b bEq.C-45q.C-46