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

81Therefore Eq.A-1 and Eq.A-2 become⎛ • ••⎛ ln ln ⎞⎞P Q W ∂ vW ∂ vWP+ − +TWvW⎜ •⎜∂ln∂ln⎜ P m ⎝⎠⎟⎜⎝WRW TW P RW⎟⎠TW=⎛ 1 ⎛ ∂ ln1+ PvW Wv−⎞⎞W⎜c⎜PW∂ ln ⎟⎝ R ⎝ T T ⎠⎠⎟W W W⎛ • ••⎛ ln ln ⎞⎞P Q W ∂ vW ∂ vWP+ − +TWvW⎜ •⎜∂ln∂ln⎜ P m ⎝⎠⎟⎜⎝WRW TW P RW⎟⎠W=⎛ ⎛ ∂ ln ⎞⎞1+ cPWvW−⎜⎜∂ ln⎝ ⎝ R ⎠⎟⎠⎟WTWTEq.C-32Eq.C-33Solving form • b: From Eq.C-17• ⎛∂u•b∂u• ⎞ • •bm ( u − h ) + m T + P ⎜= Q − PV⎜∂⎟b b W b b b b⎝ Tb∂P⎠• • • ⎛∂u•b∂u• ⎞bmb( ub− ( uW + RWTW)) = Qb−PVb− m b Tb+P⎜∂Tb∂P⎝⎠⎟• • • ⎛∂u•b∂u• ⎞bmb( ( ub−uW)− RWTW) = Qb−PVb− m b Tb+P⎜∂Tb∂P⎝⎠⎟From Eq.C-9• • • • •P mb Rb TbV= + + − bP mb Rb TbVbEq.C-34Eq.C-35Eq.C-36Eq.C-9∂ub∂ubHowever if the assumption that = = 0 is made, then∂Tb∂P⎛ • • • ⎞•P mbVbTb= T b− +⎜ P mb V⎜⎝b⎟⎠R • = 0 and thereforebEq.C-37Substituting Eq.C-37 into Eq.C-36 gives⎛ • • • ⎞• • •∂ub P mb Vbumb( ( ub uu)RWTW) Qb PVb mT ∂b− − = − −b b − + −mbP∂T b ⎜ P mb V b⎟∂P⎜⎝⎠Eq.C-38• •• ⎛ u ⎞⎛ ⎞• •bub P Vbm ∂b ( ub uW) RWTW T ∂ bQb PVb mT ∂u •− − − = − − b b + −m bbP⎜ T ⎝ ∂ ⎟b⎠ ∂T b ⎜ P V⎜⎝b⎟∂P⎠Eq.C-39