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

84• ⎛1 ⎞ • •∂u ⎛ ⎞ ⎛ ⎞bPVW ∂ ∂( )ub − − − + 1+ = − ubm −b ub uW RWTW Tb Qb P m⎜bVb⎟⎜⎝∂T m ⎜⎝∂T R ⎟⎟ ⎠⎟⎠⎝ ∂P⎠b W b b⎛ • • ⎛ 1 ⎞ • ⎛⎞ ∂u ∂ 1 ∂ ∂b1 ub ub mb ub m+ − + − + − + 1⎞b⎟⎜⎟⎝⎜ mWRW TWPV⎠ PV⎜ ∂ ⎟⎜∂ ∂ ∂⎝ T R ⎠ ⎝ T R P V P V ⎟⎠b b b b b b• ⎛1 ⎞ •∂u ⎛ ⎞bPVW ∂( )ubm − − − + 1+ u u R T T = Q⎜⎝∂T m ⎜⎝∂T R ⎟⎠⎟⎠b b W W W b bb W b b⎛ • • ⎛ 1 ⎞ • ⎛⎞ ∂u ∂ ∂b11 mb ub Vb u ⎞b+ − + − − + + 1⎟⎜⎟⎝⎜ mWRW TWPV⎠ PV⎜ ∂ ⎟⎜ ∂ ∂ ⎟⎝ T R ⎠ ⎝V P V T R ⎠b b b bEq.C-52Eq.C-53From Eq.B-2 V = Vb+VWwe can writeVb VW1 = V+ VEq.C-54Substituting Eq.C-54 into Eq.C-53• ⎛1 ⎞ •∂u ⎛ ⎞bPVW ( )∂ubm − − − + 1+ b ub uW RWTW Tb = Qb⎜⎝∂T ⎝⎜∂ ⎟⎠⎟bmW Tb Rb⎠⎛ • • ⎞⎛ ∂ 1 ⎞ • ⎛u ∂ ∂b1+ − 1+ − mb ub ubV+ +⎟⎜⎟⎜mWRW TWPV⎝ ⎠ PVW⎞⎜ ∂ ⎟⎜ ∂ ∂⎝ T R ⎠ ⎝V P T R V ⎠⎟b b b b• ⎛ • • ⎞⎛ 1 ⎞ • ⎛∂ub 11mb ∂ub ∂ubVQ + − + − + +b ⎟⎜ • ⎜mWRW TWPV⎝ ⎠⎟⎜ ∂ ⎟PVW⎞⎜ ∂ ∂⎝ T ⎠ ⎝⎟bRb V P Tb RbV ⎠b=⎛∂ ⎛ ∂ 1 ⎞⎞ub PVW ( − ) − − + ub1+ub uW RWTW Tb⎜ ⎜ ⎟⎝∂T ⎝ ∂ ⎠⎟bmW Tb Rb⎠mTherefore Eq.A-1 and Eq.A-2 become• ⎛ • • ⎞⎛⎛ Pvb∂ ln v ⎞b1 ⎞Qb+ mWRW TW PV 1c− + Pb−⎝⎜ ⎠⎜ ⎟⎜ Tb lnT ⎝ ⎝⎜ ∂b⎠⎟ Rb⎠⎟= ⎛ Pvb ln vb PVW Pvb ln vb1 ⎞⎛ ∂ ⎞ ⎛ ⎛ ∂ ⎞ ⎞( ub−uW)−RWTW −Tb cPb 1 c− + + Pb−⎜ Tb lnT bm WTb lnT ⎝⎜⎝ ∂ ⎠⎟⎜⎝⎝⎜∂b⎠⎟R⎟b⎠⎟⎠• ⎛mb∂ ln vbln vb Pvb ln vb 1 V ⎞VP⎛ ⎞ ⎛ ⎞v∂ ∂Wbc+ + Pb− +⎜ V ∂ lnTb ln P Tb lnT ⎝⎜⎝ ∂ ⎟⎠ ⎜⎝ ∂b⎠⎟RbV ⎟⎠−⎛⎛ Pvb ∂ln v ⎞bPV ⎛ ⎛WPvb ∂ln v ⎞b1 ⎞⎞( ub−uW)−RWTW −Tb cPb 1 c− + + Pb−⎜ Tb lnT bm WTb lnT ⎝⎜⎝ ∂ ⎠⎟⎜⎝⎝⎜∂ ⎟b⎠R⎟b⎠⎟⎠Eq.C-55Eq.C-56Eq.C-57