ch98 - FLOW IN A CONSTANT-AREA DUCT WITH HEAT EXCHANGE...

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Unformatted text preview: FLOW IN A CONSTANT-AREA DUCT WITH HEAT EXCHANGE (Combustion Chambers, Heat Exchangers) Frictionless Flow in a Constant Area Duct with Heat Exchange Quasi-one-dimensional flow affected by: area change, friction, heat transfer , shock R x = 0 Q/dm h 1 , s 1 , h 2 , s 2 , C O N S T A N T A R E A N O F R I C T I O CH 12-4 Governing Equations Cons. of mass Cons. of mom. Cons. of energy 2 nd Law of Thermo. Ideal Gas/Const. c p ,c v p = RT h 2-h 1 = cp(T 2 T 1 ) s = c p ln(T 2 /T 1 )- Rln(p 2 /p 1 ) {1-D, Steady, F Bx =0 only pressure work} H E A T E X C H A N G E Quasi-One-Dimensional, Steady, F Bx = 0, dW s /dt = 0, dW shear /dt = 0, dW/dt other = 0, effects of gravity = 0, ideal gas*, c v , c p is constant Property relations for ideal gas with c v and c p constant Cons. of Momentum Cons. of Energy 2 nd Law of Thermodynamics Cons. Of Mass Constant area , frictionless, heat exchange = Rayleigh Flow No R x Can find: p 2 , 2 , T 2 , s 2 , h 2 , V 2 Q/dm If know: p 1 , 1 , T 1 , s 1 , h 1 , V 1 and Constant area , frictionless, heat exchange = Rayleigh Flow breath C O N S T A N T N O F R I C T I O N TS curve H E A T E X C H A N G E A R E A Frictional, Constant Area, Adiabatic Flow Fanno Line Isentropic Flow T s ? Frictionless, Constant Area with Heat Transfer Isentropic Flow Rayleigh Line dA No Frictional, Changing Area, Adiabatic Flow s 2-s 1 = c p ln(T 1 /T 2 )-Rln(p 2 /p 1 ) Need p 2 /p 1 in terms of T 2 and T 1 After manipulation eqs 12.30a 12.30g T s Rayleigh Line s T x For the same mass flow, each point on the curve corresponds to a different value of q added or taken away. T 1 , s 1 x T o s-s 1 = c v ln(T/T 1 ) + Rln [(T o-T 2 )/(T o-T 1 )] FLASHBACK- FANNO LINE, ADIABATIC & CONSTANT AREA BUT FRICTION h 1 + V 1 2 /2 = h 2 + V 2 2 /2 h O1 = h O2 c p T O1 = c p T O2 T O1 = T O2 RALEIGH LINE, NOT ADIABATIC & CONSTANT AREA BUT NO FRICTION q +h 1 +V 1 2 /2 = h 2 +V 2 2 /2 q = h O2 h O1 q = c p (T O2-T O1 ) The effect of heat addition is to directly change the stagnation ( total ) temperature of the flow C O N S T A N T N O F R I C T I O N TS curve properties H E A T E X C H A N G E A R E A where is sonic ? Rayleigh Line s T A ds/dT = 0 B dT/ds = 0 Properties: at A highest s at B highest T pA (p+ p)A = ( + ) A (V + V) 2- AV 2- p A= V A(V + V) - AV 2- p A= V A V dp/ = -VdV V = ( + )(V + V) Want differential form of governing equations. Momentum : dp/ = -VdV Ideal gas : p = RT dp = Rd(T) + RTd( ) dp/p = dT/T+ d / Continuity : V = constant d / + dV/V = 0 ds/dT = 0 dp/ = -VdV Tds = du + pd v (1.10a) du = d(h-pv) = dh pd v v dp Tds = dh v dp = dh dp/ Ideal gas: dh = c p dT Tds = c p dT dp/ Tds = c p dT + VdV ds/dT = c p /T + (V/T)(dV/dT) ds/dT = c p /T + (V/T)( dV/dT ) Momentum : dp/ = -VdV Ideal gas : dp/p =d / + dT/T Continuity : d / + dV/V = 0- VdV /p = d...
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ch98 - FLOW IN A CONSTANT-AREA DUCT WITH HEAT EXCHANGE...

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