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

69Dividing by PV givesdV dP mR mT RT+ = dT + dR + dmEq.B-7V P PV PV PVUsing Eq.C-4 givesdV dP dT dR dm+ = + + Eq.B-8V P T R mWriting in reduced notation and applying to water and burned zone gives• • • • • • • • •V mb Rb Tb Pb mW RW TWP= + + − = + + − W Eq.B-9V mb Rb Tb Pb mW RW TWPWWriting the energy equation for the water zone gives2m u = m u −Δ mh + Q −∫ P dVEq.B-10W2 W2 W1 W1W W W12m u − m u = −Δ mh + Q −∫ P dVEq.B-11W2 W2 W1 W1W W W1•Writing in differential form which defines to mW2uW2− mW1uW1=mWuW• • •mWuW = mW hW + QW− PWdVEq.B-12• • • •mW uW + mW uW = mW hW + QW− PWVEq.B-13Writing the energy equation for the burned zone gives2m u = m u −Δ mh + Q −∫ PdVEq.B-14b2 b2 b1 b1b b b12m u − m u = −Δ mh + Q −∫ PdVEq.B-15b2 b2 b1 b1b b b1•Writing in differential form which defines to mb2ub2− mb 1ub1=mbub• • •mub b= mbhb + Qb− PdVbEq.B-16• • • •mu+ mu= mh+ Q− PV• Eq.B-17b b b b b b b bSolving for T• WFrom Eq.B-13 we can write• ⎛∂u•W∂u• ⎞ • •WmW( uW − hW) + mW TW P ⎜+W= QW−PWV⎜∂TW∂P⎝⎟⎠But from Eq.B-1 this simplifies to• ⎛∂u•W∂u• ⎞ • •WmW( uW − hW) + mW TW P ⎜+W= QW−PWV⎜∂T∂P⎝⎟⎠WWEq.B-18Eq.B-19