Fanno Flow
Simple frictional flow ( Fanno Flow Adiabatic frictional flow in a constant-area duct * he Flow of a compressible fluid in a duct is Always accompanied by :- ariation in the cross sectional area of the duct * eat transfer Friction he above parameters contribute to changes the flow properties. * Although, it is difficult in many cases to separate the effects of each of these parameters, yet in order to provide an insight into the effect of friction, adiabatic flow thermally insulated in a constant area duct is analyzed in this chapter.
* he entropy of the system still increase because of friction. - Friction is associated with the turbulence and viscous shear of molecules of the gas. - Irreversibility associated with friction causes a decrease in the stagnation pressure, steady - flow as a perfect gas. Assumptions: - Adiabatic frictional - flow in a constant crosssectional area duct. τ w X ρ X d dx ρ dρ d
omentum equation B 4A A A( d τ wbdx ρa ( d Ad τ wbdx ρad τ w F( ρ ρ hydraulic wetted perimeter ynamic pressure F is the conventional Fanning friction factor ( circular duct F for flow over as the drag coefficient h loss in diameter pipe 4F F L Fanning coefficient of friction g X f ρ. W L τ w g arcy-weisbech coefficient of friction X dx d ρ dρ d
From eq. ( eq. / A Ad τ wbdx ρad B d τ w dx ρd A d 4F F ρ f ( dx ρ d 0 0 Reynold s number, roughness of boundaries d d / d ρ d ρ τ w F ρ 4A B he friction factor for turbulent in smooth ducts is the on-karman Nikurdse formula 4F 0.8 log (Re 4 f 0
ρ R erfect gas law d dρ d ρ m Continuity Eq. ρ A dρ d 0 ρ Energy Eq. ach number h o h Constant Constant dh d 0 5 d d R d nd law of thermodynamics 0 ds 7 3 4 6
4F d d d ρ ρ ρ ρ d dx 0 d d 3 ρ ρ d 0 dh d 0 5 4 omentum equation erfect gas law Continuity Eq. Energy Eq. d d d ds 0 7 4FdX,,, ρ,, S & (τ ariable 6 ach number nd law of thermodynamics 6 equation
From eq. ividing by 4F ρ ρ d d dx 0 d 4F 4F ρ therefore d from _ Eq.5 C d d dx d dx ρ d d 0 R ρ 0 d d 9 d ( C R dh d 0 5 8
d Combining eq.(9 with eq. (6 d d d d d * d d From eq. 3 & 4 (3, From eq. 9 d d C ( d R d d d (9, d d d d dρ d ρ d d 0 0 (6 (4
0 4 ( dx F d d ( d d he d term and the d term in eq. 8 can be replaced to give 0 4 ( ]( [ dx F d 0 (8 4 d dx F d From Eq. 0 (0 * d d
* ( ( 4 d FdX 0 4 ( ]( [ dx F d FdX d 4. ( ( < d ve d -ve >
From continuity eq. ( eq. 4 & from eqs. 0 & d d * (0 d d ( d 4FdX. ( d ( d d ρ ρ 3 4FdX d. ( ( < d ve ρ -ve > d -ve ρ ve
d d From eqs. & 3 ( ( ( From eqs. 3,3& 4 ( ( 4 4FdX. 4FdX. ( ( [ ( ( ( ( ( ( d ] d 4 5
Entropy changes are determined from d d d ds C hen ds R ds 0 d R From eqs. 4 & 5 & ( ( 4FdX ( (. [ ] d (4 ( ( ( 4 ( 4FdX ( d. ( ( ( ( 4FdX R(. ( F ds ( 4FdX. C C Is positive R d (5 6
ds R( ( Also as a result of frictional flow d (6 - For subsonic flow the ach number increases < ( ve ds ve d ve - For supersonic flow the ach number decreases > ds ve ( ve d ve
he stagnation pressure can be calculated from ( 0 d O 0 d / From eqs. & 4 then d d d 0 0 ( ( ( ( ( 4FdX. 4FdX. (, d 4FdX (. ( ( ( [ ( ( ] d (4 d 7
d 0 0 - For subsonic flow 4FdX. ( ( < ( ve d (7 d ve d O ve - For supersonic flow ds ve > ( d ve d O ve ve ds ve
ds d ( d 4FdX. ( 0 d dρ 4FdX. ρ ( d ( ( 4FdX. ( 4 d ( 4FdX. ( R O 4FdX R(. ( 4FdX. ( ( d d d d dρ d d ds d o h o < > ve -ve ve -ve -ve ve -ve ve -ve ve ve ve -ve -ve Constant
Fanno table relations., 4FL*/, /*, /*, /*, ρ/ρ*, o / o*, (S-S * /C p By separating the variables and by establishing the limits at the duct entrance. at distance L. If, then L L *, *, *, *, ρ ρ *, o o*, and S S * ( d 4FdX. ( d ( ( 4FdX. ( d dρ 4FdX. ρ ( 4 d ( 4FdX. ( ds R. 4FdX R( ( d d 4FdX. ( ( O 0 d
he Fanno Line From Continuity eq. (4 * It is defined by the dρ ρ d From Energy eq. (5 C ( 0 he change of entropy for the perfect gas ds d Continuity eq. Energy eq. Eq. of state d( ( d ds Cv d d( O R Cv R ( ds C v O d d( ( C v O O O O R dρ 9 ρ
ds C v d d( ( Eq. 9 is integrated, the change of entropy may be described in terms Of temperature for a given value of 0( constant and m. o / Aconstant. h, h O const. h, h h O, O h O, O const. O < / Fanno Line S S ax O O (9 > h S he Fanno Line equation is explained as a function of ach rove that, this curve is right. S ax
he Fanno Line equation is a function of ach From the st & nd law of thermodynamics dh ds R d dh Continuity dρ d 0, ρ dh dh d dh dρ ρ Eq. of state ( d Cd C ds 0 dh Energy dh d 0 dρ ρ d d C From 0 & dh ds R dh R dh ( d ( dh C R C C d dh d d d d R
dh ds dh ds dh ds R C C RC R C R At < dh/ds -ve ds 0 dh -ve d ve d ve At > dh/ds ve ds 0 dh ve d ve d -ve At C dh/ds h which corresponding to the state of maximum entropy his means that, A Fanno line moving only in the direction from left to right >, or < then o -ve. If m. /A increase S max decease. R ( [ h O, O C h, R RC his equation describe fanno line as a function on ach < > R] R h O const. / h S S ax
It was noted that L*,the maximum length of duct which doesn t cause choking. Flow * L - (L* (L* L - (L* - (L* ultiply the equation by (4F / (4FL - / ((4F L*/ - ((4F L*/
Fanno flow in a constant cross-sectional area duct proceeded by an isentropic nozzle isentropic nozzle Fanno flow ( L d Flow / o 3 e < Back pressure b he flow in nozzle& duct are affected by L d & b X
< the gas will accelerate in the duct owing to friction and pressure decrease e < or depend on b & L d If e (m. /A max choked condition G*(m. /A max o o R [/(] (/((- o increase mass, by decreasing o and/or increasing o at the nozzle inlet. In this case e, but e would be higher.
Changes in duct length do not affect the mass flow rate >, the gas will decelerate in the duct due to friction, if no shock occur, the flow will approach e directly. In this case ( e, m. max is determined by the throat area of the nozzle and also by the area of the duct
Effect of increasing duct length isentropic nozzle ( L d L d Fanno flow h h o ( duct entr < o h o const. 3 4 * isen 3 * 4 * 3 4 /o S (m. /A > (m. /A 3 >(m. /A 4
( duct entr > At certain length, the at exit duct crosssectional area Subsonic conditions can also be attained at exit area, if discontinuity occurs. If [ L d > L d * at e ], a shock appears in the duct If the duct length is increased further, the shock will position itself further upstream. If the duct is very long, the shock will be at the throat of the nozzle. Beyond that length, there will be no shock at all. hen, the flow is subsonic at all point.
Effect of reducing back pressure (ach at duct inlet m, c b back < exit a / o a a b h back / o h o constant a b c X c S
Effect of reducing back pressure (ach at duct inlet > Flow / o Supersonic nozzle > Fanno flow critical length L * L < L * b b > e In the duct L <L*, there are 5 cases:- esign case eb Expansion wave Shock wave stand somewhere N.S.W inside the stand duct at duct exit Oblique shock wave * / o b / o e b b < e he flow in nozzle& duct are affected by L d & b X