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

85• ⎛ • • ⎞⎛⎛ ln ⎞⎞1cPb∂ vbQ b+ •⎜mWRW TW− PV⎟⎟⎜⎜ + −⎝ ⎠⎜ ⎜∂ ln⎝ ⎝R⎟⎠⎟bTb⎠b= ⎛ ⎛ ∂ln⎞ ⎛ ⎛ ∂ln⎞⎞( − ) − − vb− + 1+ cPbv ⎞b−⎟⎟ub uW RWTW c PbTb RTb b⎜⎜ ∂lnPvW⎟ ⎜⎜ ln⎝⎝ T ⎠ ⎝ ⎝ ⎟⎟bRb∂ Tb⎠⎠⎟⎠m• ⎛⎛∂⎜ ln vb∂lnvbln ⎞VP⎜ +⎞ ⎛ ⎞cPb ∂ vb VW⎜⎜ + − +⎜⎜ ⎝∂lnTb∂lnPln⎝⎜⎜⎟⎠ ⎜⎝R ∂ ⎟⎠⎟bTbV ⎠−⎛⎛ ∂ln⎞ ⎛ ⎛ ∂ln⎞⎞( − ) − − vb− + 1+ cPbv ⎞b−⎟ub uW RWTW c PbTb RTb b⎜⎜ ∂ln⎟ Pv⎟W⎜⎜ ∂ln⎟⎟⎝⎝ T ⎠ ⎝ ⎝⎠bRbTb⎠⎟⎠Eq.C-58Solving for V • b:• •WmbSubstituting Eq.C-3, − m = into Eq.C-45 gives⎛ • • • ⎞•−mbTWPVW= VW+ −⎜ mbTWP⎜⎝⎠⎟Eq.C-59• • •From Eq.C-42, V = Vb+VWwe can write⎛ • • • ⎞• •−mbTWPV− Vb= VW+ −⎜ mbTWP⎜⎝⎠⎟⎛ • • • ⎞• −mbTWP•Vb= V W+ − + V⎜ mbTWP⎜⎝⎠⎟Eq.C-60Eq.C-61The equation for the calculation of the burned gas temperature but one was derived usinga similar methodology as above. Again, starting with the energy equation for the burnedgas Eq.C-17 and rearranging• ⎛∂u•b∂u• ⎞ • • •bmub b+ m b Tb P ⎜+ = mhb W+ Qb−PVb⎜∂T∂PEq.C-62⎝⎟⎠b⎛∂u∂u⎞m T + P⎟= m h − u + Q − PV • Eq.C-63• • • •bbb ⎜b b( W b)b⎜∂TbP⎝ ∂ ⎟⎠b⎛∂ubm T⎟⎞ ⎛ ⎞m h u Q PV m∂u•P ⎜ ⎝⎜∂P⎠⎟• • • •bb b⎟= b( W−b)+b− b−bT⎝⎜∂ ⎟b ⎠Eq.C-64