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Ch MAE130B 11 (4) Compressible Flow Changzheng Huang Compressible Flow Non-isentropic adiabatic flow (Fanno flow) Non-isentropic frictionless flow (Rayleigh flow) Ch 11 (4) Changzheng Huang 2 2. Non-isentropic adiabatic flow (Fanno flow) V1 p1A Mass Momentum Energy D w l V pA 1V1 A = VA = m p1 A pA w Dl = m (V V1 ) V 2 V12 m h + + gz h1 + + gz1 = Q + W 2 2 Since flow is adiabatic (Q=0), steady (W=0) and horizontal (gz=0), Mass Energy Ch 11 (4) V = Constant V h+ = h0 = constant 2 Changzheng Huang 2 Note the conditions of validity of this. 3 Temperature-entropy diagram 2. Non-isentropic adiabatic flow (Fanno flow) T M<1 pa V V+dV D w M=1 Ta pA (p+dp)A Fanno line M>1 dx s Second law of thermodynamics states that entropy must be constant or increasing for adiabatic flow. Three typical Fanno flow cases, T 1 M<1 T 2 a M>1 1 s (i) Subsonic Fanno flow Ch 11 (4) T a 2 s (ii) Supersonic Fanno flow Changzheng Huang 2 Shock 1 s (iii) Normal shock Fanno flow 4 a Temperature-entropy diagram 2. Non-isentropic adiabatic flow (Fanno flow) T 1 pa V V+dV D w Ta pA (p+dp)A dx s 0 l l* Question, for a given flow condition (say M), how far can the flow extend before it reaches the chocked condition (l*-l)? d dV Mass d ( V ) = 0 + =0 V Momentum Adp w Ddx = VAdV Energy Entropy Ch 11 (4) V2 d h+ = 0 2 c p dT + VdV = 0 Tds = c p dT Changzheng Huang Tds = dh vdp dp = c p dT 5 dp p p Temperature-entropy diagram 2. Non-isentropic adiabatic flow (Fanno flow) T 1 pa V V+dV D w Ta pA (p+dp)A dx s 0 l l* Momentum Adp w Ddx = VAdV From pipe flow, p 2 2 = =w l r R l V 2 p = f D2 w = f V 2 8 dx 2dp 2dV f = + 2 D V V 6 D2 4 dp f V 2 8 Ddx = V D2 4 dV Ch 11 (4) Changzheng Huang Temperature-entropy diagram 2. Non-isentropic adiabatic flow (Fanno flow) T 1 pa V V+dV D w Ta pA (p+dp)A dx s 0 l l* f dx 2dp 2dV = + 2 D V V dx dp kp 2 2dV d dT 2c 2 2dV f = + = + 2 kV 2 + V D p kV V T 2 dV + dT 2 + 2dV = kM 1 2dV + 2 dT = T kM 2 V kM 2 V kM 2 T V dx kM 2 1 dV 2 2 dT f = + D kM 2 V 2 kM 2 T Ch 11 (4) Changzheng Huang 7 Temperature-entropy diagram 2. Non-isentropic adiabatic flow (Fanno flow) T 1 pa V V+dV D w Ta pA (p+dp)A dx s 0 l l* Momentum Energy dx kM 2 1 dV 2 2 dT f = + D kM 2 V 2 kM 2 T V2 d h+ = 0 2 kR T dT dV 2 + 2 =0 2 k 1 V T 2V 2 22 dV 2 c p dT + =0 2 dT k 1 2 dV 2 M + =0 2 T 2 V 2 Definition V = Mc Ch 11 (4) V = M c = M kRT Changzheng Huang dV 2 dM 2 dT = + 2 2 V M T 8 Temperature-entropy diagram 2. Non-isentropic adiabatic flow (Fanno flow) T 1 pa V V+dV D w Ta pA (p+dp)A dx s 0 l l* Momentum dx kM 2 1 dV 2 2 dT f = + D kM 2 V 2 kM 2 T k 1 dM 2 dT = 2 k 1 2 T M 1+ 2 dV 2 dM 2 1 = 2 k 1 2 M 2 V 1+ M 2 9 Energy dT k 1 2 dV 2 + M =0 2 T V 2 dT dV 2 dM 2 Definition + 2= T V M2 Ch 11 (4) Changzheng Huang Temperature-entropy diagram 2. Non-isentropic adiabatic flow (Fanno flow) T 1 pa V V+dV D w Ta pA (p+dp)A dx s 0 l l* 1 M 2 dM 2 dx kM 4 =f k 1 2 D 1+ M 2 M * =1 M l* 1 M 2 dM 2 dx = f k 1 2 kM 4 l D 1+ M 2 [( k + 1) 2] M 2 = f ( l * l ) 1 1 M 2 k +1 + ln 2 kM 2k 1 + [( k + 1) 2] M 2 D Ch 11 (4) Changzheng Huang 10 Temperature-entropy diagram 2. Non-isentropic adiabatic flow (Fanno flow) T 1 pa V V+dV D w Ta pA (p+dp)A dx s 0 l l* T ( k + 1) 2 = * T 1 + [( k 1) 2] M 2 V [( k + 1) 2] M = = * 2 V 1 + [( k 1) 2] M * 2 1/ 2 1/ 2 p 1 T ( k + 1) 2 = * *= * 2 p M 1 + [( k 1) 2] M T * p0 p0 p p 1 2 k 1 2 M = = 1 + * * * 2 p0 p p p0 M k + 1 Ch 11 (4) Changzheng Huang +1 k 2( k 1) 11 Compressible Flow Non-isentropic adiabatic flow (Fanno flow) Non-isentropic frictionless flow (Rayleigh flow) Ch 11 (4) Changzheng Huang 12 3. Non-isentropic frictionless flow (Rayleigh flow) T V1 V D q pA pA 1 b a s l Mass Momentum Energy Mass Momentum Entropy Ch 11 (4) 1V1 A = VA = m p1 A pA = m (V V1 ) Rayleigh line V 2 V12 m h + + gz h1 + + gz1 = Q + W 2 2 V = 1V1 p + V = p1 + V 2 2 11 ( V ) RT = p + V 2 p+ 1 11 2 p s s1 = c p ln T p R ln T1 p1 13 Changzheng Huang 3. Non-isentropic frictionless flow (Rayleigh flow) T V V+dV D q pA (p+dp)A dx Differential analysis Mass d ( V ) = 0 b a s c p dT + VdV = Tds dp + VdV = 0 d dV + =0 V Momentum dp = VdV Energy Entropy dh + VdV = q Tds = dh vdp Tds = c p dT Ch 11 (4) dp dp d dT =0 p T Changzheng Huang 14 3. Non-isentropic frictionless flow (Rayleigh flow) T V V+dV D q pA (p+dp)A dx 1 dT 1 dV ds +2 = c p dT + VdV = Tds k 1 T M V kR dp + VdV = 0 d dV + =0 V dp 2 dV + kM =0 p V d dV + =0 V b a s dT = T (1 kM 2 ) dp d dT =0 p T Ch 11 (4) dp d dT =0 p T Changzheng Huang 1 M 2 ) ( kM 2 + k 1) kR ( ( k 1) M 2 ds 15 3. Non-isentropic frictionless flow (Rayleigh flow) T V V+dV D q pA (p+dp)A dx dT (1 kM ) ( k 1) M ds = T (1 M ) ( kM + k 1) kR 2 2 2 2 b M<1 M>1 a s ds kR (1 M ) ( kM + k 1) = dT T (1 kM ) ( k 1) M 2 2 2 2 2 2 dT (1 kM ) ( k 1) M T = ds (1 M 2 ) ( kM 2 + k 1) kR Ma =1 1 Mb = k Ch 11 (4) Changzheng Huang 16 3. Non-isentropic frictionless flow (Rayleigh flow) T V V+dV D q pA (p+dp)A dx b M<1 M>1 a s dh + VdV = q Energy Second law of thermodynamics also states that entropy increases with heating and decreases with cooling. T T T b b cooling b heating cooling a a a heating shock heating heating cooling s s s (i) Subsonic (ii) Supersonic (iii) Normal shock Rayleigh flow Rayleigh flow Rayleigh flow Ch 11 (4) Changzheng Huang 17 3. Non-isentropic frictionless flow (Rayleigh flow) T V V+dV D q pA (p+dp)A dx Momentum dp = VdV p + V 2 = pa + aVa2 b M<1 M>1 a s p 1 + kM 1+ k = = pa 1 + kM 1 + kM 2 2 a 2 aVa2 p V 2 p + = 1+ pa p pa pa kV 2 p kVa2 1 + c 2 p = 1 + c 2 a a Ch 11 (4) T p a pV p Mc = = = Ta pa pa Va pa M a ca p kRT M = pa kRTa 1/ 2 T p M = pa Ta 2 1/ 2 T 2 p =M Ta pa Changzheng Huang 18 3. Non-isentropic frictionless flow (Rayleigh flow) T V V+dV D q pA (p+dp)A dx 2 p 1 + kM a 1+ k = = pa 1 + kM 2 1 + kM 2 b M<1 M>1 a s T0 T0 T Ta = T0,a T Ta T0,a T 1+ k =M2 2 Ta 1 + kM 2 T a V Mc == =M Va M a ca Ta Ch 11 (4) 1/ 2 2 T0 2 ( k + 1) M k 1 2 M = 1 + 22 2 T0,a (1 + kM ) p0 T0 p pa = p0,a p pa p0,a p0 1 + k 2 k 1 2 = M 1 + 2 p0,a 1 + kM k + 1 2 Changzheng Huang 19 k k 1 3. Non-isentropic frictionless flow (Rayleigh flow) T V V+dV D q pA (p+dp)A heating dx dh + VdV = q > 0 b M<1 M>1 s a M T0 T cooling p p0 b T a M T0 p p0 b T cooling a b shock heating a heating heating s (i) Subsonic Rayleigh flow Ch 11 (4) heating cooling s s (ii) Supersonic (iii) Normal shock Rayleigh flow Rayleigh flow Changzheng Huang 20 3. Non-isentropic frictionless flow (Rayleigh flow) T V V+dV D q pA (p+dp)A cooling dx dh + VdV = q < 0 b M<1 M>1 s a M T0 T cooling p p0 b T a M T0 p p0 b T cooling a b shock heating a heating heating s (i) Subsonic Rayleigh flow Ch 11 (4) heating cooling s s (ii) Supersonic (iii) Normal shock Rayleigh flow Rayleigh flow Changzheng Huang 21

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