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

70From the definition of entropyh = u + RT ⇒u − h =− RTEq.B-20• ⎛∂u•W∂u• ⎞ • •W− mW( RWTW) + m W TW P ⎜+W= QW−PWV⎜∂TW∂P⎝⎟W ⎠Eq.B-21• ⎛∂u•W∂u• ⎞ • •WmW( RWTW) − mW TW P ⎜+W= PWV−QW⎜∂TW∂P⎝⎟W ⎠Eq.B-22⎛• • ⎞ • ••∂uW TW ∂uW PW PWV QWmW= mW+ + −⎜∂TW RTW W∂PW RT W WRTW WRT⎜⎝⎟⎠W WEq.B-23Dividing by mW• ⎛• • ⎞ ••mW ∂uW TW ∂uW PW PWV QW= m + + −Eq.B-24W ⎜∂TW RTW W∂PW RT W WmRTW W WmRT⎜⎝⎟⎠W W WFrom Eq.B-9 into Eq.B-1• • • •V mW TW PW= + − Eq.B-25V mW TW PWSubstituting Eq.B-24 into Eq.B-25•• • • • • •V ∂uW TW ∂uW PW PWV QW TW PW= + + − + −Eq.B-26V ∂TW RTW W∂PW RTW WmRTW W WmRTW W WTW PW•• • • • •V QW PWV ∂uW PW PW T ⎛Wu ⎞∂W+ − − + = 1V mRTW W WmRTW W WPW RTW WPW T +WRW TEq.B-27∂⎜⎝∂ ⎟W ⎠However• • •V PWVV ⎛ PWV⎞− = 1V m R T V ⎜− =m R T0 Eq.B-28⎜⎝ ⎟⎠W W W W W WTherefore Eq.B-27 becomes• • •T ⎛Wu ⎞∂WQW PW uW1∂ P •WT + = + −R T ⎝⎜ ∂ ⎟⎠mRT P ∂P RTT TWEq.B-29W W W W W W W W W W• •uW QW PW uWW1 •• ⎛ ⎞∂∂+ WRW T = + −⎝⎜ ∂ ⎟W ⎠ mWRW PW ∂PW RWEq.B-30⎛ • • ⎞ QW PW ∂uWP • WT•+W−mWRW PW ∂PW RWTW= u∂W1+ RW∂T⎜W⎜⎝⎟⎠Eq.B-31