2013/5/22. ( + ) ( ) = = momentum outflow rate. ( x) FPressure. 9.3 Nozzles. δ q= heat added into the fluid per unit mass

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9.3 Nozzles (b) omentum conservation : (i) Governing Equations Consider: nonadiabatic ternal (body) force ists variable flow area continuously varying flows δq f

+d mu ( x) momentum flux Q x Q x dx Q x dq dx dx net force acting in the x-direction ( + ) momentum outflow rate Fressure ( x) Fressure ( x+ dx) + x+ dxd+ f dx df d

+d ( + ) 0 m x dx m x dm x m ( x) + dx m x dx 0 ρ u m constant (mass flow rate) dm x dx 0 ρ udu d +δf +δ u du d f recall for perfect gases: a ρ, ρ a u (c) Energy

1 9.3 Nozzles (b) omentum conservation : (i) Governing Equations Consider: nonadiabatic ternal (body) force ists variable flow area continuously varying flows δq f ternal force per unit volume +d δffdx dx δ q heat added into the fluid per unit mass ( q u) δ ρ c δ c : contact area of CV through which heat is added. +d mu ( x) momentum flux Q x Q x dx Q x dq dx dx net force acting in the x-direction ( + ) momentum outflow rate Fressure ( x) Fressure ( x+ dx) + x+ dxd+ f dx df d dx d dx ( x) dx f dx dx dx dx ressure ( ) dmu d d d d dx dx + dx + dx + f dx dx dx dx dx dx du d m dx dx+ f dx ( ρ u) du d + δf dx dx 3 a) mass conservation : +d ( + ) 0 m x dx m x dm x m ( x) + dx m x dx 0 ρ u m constant (mass flow rate) dm x dx 0 ρ udu d +δf +δ u du d f recall for perfect gases: a ρ, ρ a u (c) Energy conservation : du d δf + u u a ( ρ ) ρ+ ( ρ ) + ( ρ ) d u ud du ud 0 ρu ρu dρ du d ρ u +d E ( x) m h+ u mc + u total energy flux E( x+ dx) E ( x) q δ de dx q δ c c dx d m C + u dx q δc dx 4

2 q δ + δ m c d C u q recall ρr and R C p Cd + udu δq d udu δq + C C R a C p ρr ρ d du δq + ( ) u C for perfect gases p u Look for dρ, d, du, and d : dρ du d (mass) ρ u du d δf + (momentum) u d du δq + ( ) u Cp d dρ d + (equation of state) ρ (energy) du d d d u du d δf du δq du + + ( ) u u C u du δq δf d u C ( ) 5 7 (d) Equation of state Summary : ρr d Rdρ + ρrd dρ du d (mass) d Rdρ ρrd ρ u + ρr ρr du d δf + (momentum) u d dρ d + d ρ du δq + ( ) (energy) u Cp d dρ d + (equation of state) ρ Look for dρ, d, du, and d : ( ) du d δq δf + u C du d δf δq + u C ( ) dρ du d Substitute into (mass): ρ u dρ d δf δq d + ρ C ( ) ~ four algebraic equations for the differentials in terms of u,, ρ,,, f, q, du, dρ, d, d d δf δq + C ( ) ( ) ( ) 6 8

3 Look for dρ, d, du, and d : Substitute into (momentum): d δf du u δf d δf δq + C ( ) d + δf δq + C ( ) ( ) ( ) Substitute into (energy): d δq du ( ) C u p ( ) ( ) d ( ) δf δq + C 9 d Rd ds C ( ) δ δ + R d f q ( ) C d + ( ) δf δq R + ( ) ( ) ( ) C + ( ) R + ( ) δf δq R + ( ) δ ( ) ( ) ( ) q ( ) ( ) δ + R ( ) C f e) Entropy variation st law of thermodynamics : enthalpy h e + C ρ de dw dq d + + ds ρ dh Cd de + d d + ds + d + pd ρ ρ ρ ρ d d ds C d Rd C ρ ( ρr ) s s C ln Rln Remark: du d δf δq + u C ( ) dρ d δf δq + ρ C ( ) ( ) ( ) d d + δf δq + C ( ) ( ) ( ) ( ) ( ) δ ( ) δ ( ) ( ) ( ) d d f q + C δf δq ds R + 0 3

4 (ii) diabatic flows without ternal forces Design of an isentropic nozzles (diabatic flows without ternal forces) < throat du d < 0 and < > 0 when u d > 0 and > a) Stagnation properties Recall energy equation Cd + uduδ q 0 chocking du d u d ρ d ρ ( ) ds 0 (isentropic flow) > ( ) d d ( ) d d ( ) 3 or d h+ u 0 h0 h+ u u stagnation enthalpy the enthalpy if the fluid were decreased to be rest isentropically constant everywhere in an isentropic nozzle C 0 (stagnation temperature) 5 (ii) diabatic flows without ternal forces < > Convergent part: d < 0 subsonic du > 0 dρ < 0 d < 0 d < 0 supersonic du < 0 dρ > 0 d > 0 d > 0 4 a) Stagnation properties isentropic relations: C 0 C + u u ( ) 0 0 ρ0 u 0 + ρ C R a u Recall C (ideal gases) ( ) ( ) ( ) u ( ) C 0 stagnation temperature: stagnation pressure: + 0 ( ) + ρ0 stagnation density: + ρ 6 4

b) hroat properties at throat,,temperature, pressure, density ρ 0 ( ) + + + m ρ 0 0 R 0 + + +

Operation of a convergent nozzle b : back pressure ass conservation m ρ u ρ u ρ R ρ a ρ R b ρ

5 b) hroat properties at throat,,temperature, pressure, density ρ 0 ( ) m ρ 0 0 R ρ0 0 ρ ρ c) throat area with a given mass flow rate m d) flow area of the nozzle with a given variation of the ach number. < > m ρ u constant ρ u ρ a ( ) ρ R R ρ0 0 + ( + ) 7 < > 9 d) flow area of the nozzle with a given variation of the ach number. Operation of a convergent nozzle b : back pressure ass conservation m ρ u ρ u ρ R ρ a ρ R b ρ ρ ρ ρ0 0 ρ ρ isentropic + + ( + ) ( ) 8 Converging nozzle operating at various back pressures

6 Operation of a convergent nozzle (i) he valve is dosed : 0 throughout (ii) b is slightly less than 0 : subsonic at it b e (iii) b further decreases but larger than : Operation of a convergent-divergent nozzle subsonic with as b and b e ( ) (iv) sonic at it b 0 + (v) b < : the nozzle is choked. Reduction in b below has no effect on the flow condition in the nozzle (neither pressure distribution nor mass flow rate.) 3D complicated pansion waves occur at it. x ressure distributions for isentropic flow in a converging-diverging nozzle. 3 Design of convergent nozzles: Specify the inlet flow conditions (including the mass flow rate ) and the it pressure. Evaluate the stagnation properties from the inlet conditions. ( ) u 0 0 ρ0 0 + and C ρ Calculate the required throat area. + ρ 0 0 m R + Calculate the it ach number from stagnation properties and the it pressure. 0 + b Calculate the it area of the nozzle. ( + ) + + Operation of a convergent-divergent nozzle ressure distributions for isentropic flow in a converging-diverging nozzle. (i) b is slightly less than 0 : flow accelerates before throat and decelerates after throat; subsonic and isentropic everywhere in the nozzle. (ii) 0 > b > cr : higher ach Number at throat but still subsonic and isentropic everywhere. x 4 6

7 Operation of a convergent-divergent nozzle Operation of a convergent-divergent nozzle ressure distributions for isentropic flow in a converging-diverging nozzle. (iii) b cr : flow accelerates and reached at throat; it then decelerateing and becomes subsonic after the throat; isentropic everywhere. (iv) b cr3 : flow accelerates and reached at throat; it continues accelerating xand becomes supersonic after the throat; isentropic everywhere. ressure distributions for isentropic flow in a converging-diverging nozzle. (v) cr < b < cr ~ normal shock appears downstream from the throat. < > < before throat at throat after throat and before normal shock after normal shock ~ s b decreases, the normal xshock wave moves downstream further until it appears at the it of the nozzle. 5 7 Operation of a convergent-divergent nozzle Operation of a convergent-divergent nozzle b cr or cr3 Given and, there are two possible it ach numbers, one subsonic ( ) and the other supersonic ( 3 ). (vi) cr3 < b < cr ~ series of 3D oblique compression shock waves forms outside the nozzle. 0 cr 0 cr ( ) ( ) 3 ( + ) ( ) 3 > > < cr3 cr ressure distributions for isentropic flow in a converging-diverging nozzle. (vii) b < cr3 he nozzle is choked. 3D Expansion waves at it. x 6 8 7

8 Design of a convergent-divergent nozzle Specify the inlet flow conditions (including the mass flow rate ) and the it pressure. Evaluate the stagnation properties from the inlet conditions. ( ) u 0 0 ρ0 0 + and C ρ Calculate the required throat area. + ρ 0 0 m R + Calculate the it ach number from stagnation properties and the it pressure. 0 + b Calculate the it area of the nozzle. ( + ) Design of a convergent-divergent nozzle (i)~ iv (vi) and (vii) b < cr x x x isentropic everywhere inside the nozzle. ( x) ( x + ) ( ) 0 ( x) ( x) ( x) ( x) ( + ) ( ) ( x) 3 Design of a convergent-divergent nozzle Design of a convergent-divergent nozzle given the geometry of the nozzle (x): Find the critical back pressures: th th + +, cr ( + ) ( ) (v) cr < b < cr Where is the normal shock? How strong is the normal shock? Normal shock relations: s s s s s + + ( ) s + x equations 4 unknowns,,, s s s s

isentropic from the inlet to including the throat : s s s th + s + 0 s + s ( ρ ) stagnation state:,, sonic throat 0 0 0 th ( + ) ( ) equations additional unknowns s 33 s s + s s 0 s s s 0 s + + s s x

9 isentropic from the inlet to including the throat : s s s th + s + 0 s + s ( ρ ) stagnation state:,, sonic throat th ( + ) ( ) equations additional unknowns s 33 s s + s s 0 s s s 0 s + + s s x s s s th th + s + + ( ) s + s s + s + ( + ) ( ) ( + ) ( ) 35 isentropic from to the it: ( ρ ) stagnation state:,, ( + ) ( ) s s + s + 0 s sonic throat s + + ( + ) ( ) ( + ) ( ) 4 equations 3 additional unknowns, 0, obtain ( ) s 36 9

045 37 39 isentropic from to the it: s s ρ s s + + s s ρs + ( + ) s s 0 0 + 0

10 isentropic from the inlet to including the throat : x th x 0 x ( x) ( x + ) ( ) 0 0 s + ( ) ( x) s ( x) ( + ) ( ) x x x 0 x isentropic from to the it: s s ρ s s + + s s ρs + ( + ) s s x x x 0 x s s ( ) x + ( ) 0 ( x) x x ( x) ( x + ) ( x) ( + ) ( )

11 x x x 0 x Nozzles (i) Governing Equations (variable area, friction, heating) (ii) diabatic flows without ternal forces (variable area) du d δf δq + u C ( ) ρ δ δ + ρ C d d f q ( ) ( ) ( ) d d + δf δq + C ( ) ( ) ( ) ( ) ( ) δ ( ) δ ( ) ( ) ( ) d d f q + C δf δq ds R Nozzles (i) Governing Equations (variable area, friction, heating) (ii) diabatic flows without ternal forces (variable area) d( u) mass ρ 0 momentum energy entropy equation of state ρ u du d+δf Cd+ uduδq d d ds C R ρr Look for dρ, d, du, d, and ds 9.3 Nozzles (iii) Fanno line ~ adiabatic flows in a constant-area channel (friction) mass ρ u ρ u m momentum mu u + Fx energy C ( ) + ( u u ) 0 entropy s s C ln Rln equation of state ρ R and ρ R ( ρ u m s ) ( ρ u s F ) Given:,,,,, Want:,,,,, x 5 equations with 6 unknowns! solutions! nd law: s >s

12 (iii) Fanno line ~ adiabatic flows in a constant-area channel (friction) Example: 0 96 K, 0 0 ka, 98.5 ka, 87K D 7.6mm 0 0 ( ) ( ) + 94K ρ ρ.7 R kg m a R a m s u a u 65.3 m s πd m 5 3 (iii) Fanno line ~ adiabatic flows in a constant-area channel (friction) friction δ f < 0 subsonic branch: < du u dρ ρ ( ) ( ) δf δ f ( ) d + ( ) δf ( ) d ( ) δf ( ) ( ) ( ) δf ds R ds R du > 0 dρ < 0 d < 0 d < 0 ds > 0 supersonic branch: > du < 0 dρ > 0 d > 0 d > 0 ds > 0 ds R (iii) Fanno line ~ adiabatic flows in a constant-area channel (friction) Example: 0 96 K, 0 0 ka, 98.5 ka, 87K D 7.6mm K 0 ( ) a R a 339.6m s u a u 34.5 m s ρ u ρ u ρ kg m 0.57 ρr 47ka ( ) 0 0 ρ u u u + F x + 5.4ka 3 Fx.86N ~ acting on the fluid by the channel walls (iii) Fanno line ~ adiabatic flows in a constant-area channel (friction) Fanno line ~ the location of all possible downstream states ~ flow states must always move to the right because s > s R C u u ( ) + + constant 0 + ( ) ( ) ( ) 0 ( ) d ds ( ) R R0 ( )

13 Fanno line d ( ) ds ( ) 0 R ( ) 0 ( + ) ( ) ( ) d ds 0 R 0 ( + ) ( ) ( ) ds d 0 R 0 X X X X X X ( ) ( ) X X X + X ( ) Fanno line ( ) 0 ( + ) ds dx + X X ( ) ( X) R Fanno line s s ln ( X) + ln X ( ) R X X ln ln s + s 0 ( ) R 0 < >, Nozzles (iii) Rayleigh line ~ frictionless flow in a constant-area channel (heating) 5 equations mass ρ u ρ u m with 6 unknowns! momentum m ( u u) ( ) solutions! energy mc ( ) + m( u u ) mδ q Q entropy s s C ln Rln nd law: s >s equation of state ρ R and ρ R ρ Want: ( ρ, u,,, s, Q ) ( u m s ) Given:,,,,, 3

14 Rayleigh line Q mc + m u u ( ) ( ) m C + u m C+ u mh mh 0 0 Qm Δ q h h δ q dh0 ds 0 0 Rayleigh line ( ) ( ) d ds C du ds u ( ) C R R R R u R u u u u du u d d d du d ( ) du u ( ) ( ) ds d C ds C d Rayleigh line du δq u ( ) C dρ δq ρ C ( ) d q 0 ( ) ( ) ( ) δ C ( )( + ) d δq δq C ( )( + ) C dh δ q ds heating δ q > 0 : d > 0 for < d < 0 for < < d > 0 for > heating δ q > 0 : subsonic ( < ) du > 0 supersonic ( > ) dρ< 0 du < 0 d < 0 dρ > 0 d > 0 Rayleigh line d ( + ) ( ) C ( ) d + ( + ) ( + ) ds ds C s s + + ln ln C + d ( ) ( ) ds C + + ( ) ( + ) d d 4

15 Rayleigh line Rayleigh line (heating) < : 0, 0 : 0, 0 : 0, 0 Rayleigh line 5

Introduction to Fluid Mechanics. Chapter 13 Compressible Flow. Fox, Pritchard, & McDonald

Introduction to Fluid Mechanics Chapter 13 Compressible Flow Main Topics Basic Equations for One-Dimensional Compressible Flow Isentropic Flow of an Ideal Gas Area Variation Flow in a Constant Area Duct