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

75⎛• •• • ⎛ ln ⎞ ⎛ ln ln ln ⎞ ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎟1 c Pb∂ vb c ∂ ∂ ∂Pbvb vb vb − + − − − + + V TWQbPbV 1+ −⎜•⎜ ∂ln ⎟ PV⎟⎜⎜ ∂ln ⎟ ⎜∂ln ∂ln⎟⎝ ⎝R ⎠⎠ ⎝⎝ ⎠ ⎝ ⎠⎟⎠ bTb Rb Tb Tb Pb ⎜⎝⎜ V TW⎟⎠b=Eq.B-58⎜⎜⎛⎛ ∂ ln ⎞( − ) − − v ⎛bu −⎛ ln ⎞ ⎛ ln ln ⎞⎞⎟buW RWTW c PbTb Pv⎜⎜cPb ∂ vb ∂ vb ∂ v ⎞b ⎟⎟b⎜ ∂ ln+ Pv ⎜⎜ − + + ⎟W⎝T ⎟⎜b ⎠ ⎜⎜ ∂ln ⎟ ⎜∂ln ∂ln⎜⎝⎝⎝R ⎠ ⎝⎟⎟⎟ ⎠⎟⎠ bTb Tb Pb⎠⎜⎝⎟ ⎠mSolving for P • b:• •− mW= mb• • •Substituting Eq.B-3, into Eq.B-45 gives⎛⎞•−mbTWVPW= P + −⎜ mW TW V⎜⎝⎟⎠Eq.B-59• • •From Eq.B-42, P= Pb+PWwe can write⎛ • • • ⎞• •−mbTWVP− Pb= P + −Eq.B-60⎜ mW TW V⎜⎝⎟⎠⎛ • • • ⎞• mbTWV•Pb= P − + + PEq.B-61⎜mWTWV⎜⎝⎟⎠The equation for the calculation of the burned gas temperature but one was derivedusing a similar methodology as above. Again, starting with the energy equation forthe burned gas Eq.B-17 and rearranging• ⎛∂u•b∂u• ⎞ • •bmb( ub− hW) + m b Tb P ⎜+b= Qb−PbV⎜∂Tb∂P⎝⎟b ⎠Eq.B-62⎛ u•bu• ⎞ • •bm ∂ ∂b Tb P ⎜+b= mb( hW − ub)+ Qb−PbTbPV •⎜∂ ⎝ ∂ ⎟b ⎠Eq.B-63∂u• • • •b∂u•bmb Tb = mb( hW − ub)+ Qb−PbV−mbPb∂T∂PEq.B-64bb⎛ • • •∂u• ⎞bmb( hW ub) Qb PbV mb P•− + − − b∂PbTb=⎜⎜u∂bmb⎜∂ T⎝⎠⎟bEq.B-65Therefore Eq.A-1 and Eq.A-2 become