lect. notes #4.pdf - Flow in Constant-Area Ducts With...

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74 Flow in Constant-Area Ducts With Friction ( Fanno Flow ) Engineering Fields & Applications ? 1. Power plants 2. Air craft propulsion engines 3. Transport of fluids in chemical process plants 4. Transport of natural gas in long pipe lines Now, we’ll consider th e friction alone; i.e. friction of the Duct is the main factor to change fluid properties Ducts SHORT LONG NO heat transfer Heat Transfer Flow is ADIABATIC Flow is ISOTHERMAL
75 Adiabatic Flow Assumptions: 1-D flow steady Q = 0 Ws = 0 P.E. = 0 A = const. Governing Equations Energy Equation: Continuity Equation: State Equation: Energy & Continuity lead to: Then, a Fanno line equations are: For given h 0 & G To draw a Fanno line; choose , get h, then get s o 2 h 2 V h G A m V VA m ) , h ( s s 2 2 0 2 G - h h 2 2 0 2 G - h h ) , h ( s s h or T s . const A m G const. T or h o o M=1 M<1 M>1
76 Fanno lines for different flows per unit are (G's), but for same h o (T o ) Effect of friction ? : increases entropy, s; and as a result, M increases for subsonic flow & decreases for supersonic flow a T s o T Large G Small G 1 M oa p 1 M 1 M * h or T s const. T or h o o M=1 M<1 M>1 *
77 Choking due to Friction When duct length is such that M = 1 at duct exit, duct is said to be choked !.. What happens if L > L* ?? For Subsonic Flow & For Supersonic Flow Before answering: To limit the effect to the friction only, we assume that the back pressure is as low as necessary T s const. T o M=1 M<1 M>1 * 1 1 M=1 * 1 L=L*
78 First: Subsonic Flow M=1 *a 1a (L*)a (a) T o P oi M=1 *b 1b (L*)b (b) M=1 *c 1c (L*)c (c) 1a T s o T (a) (c) 1 M * c a b (b) P oi 1b 1c
79 Remarks: When L>L*, M1 will decrease such that a steady state solution is possible again with exit Mach No. =1. This results in a reduction of mass flow rate; i.e. the flow is choked by friction. Second: Supersonic Flow Remember, P b << P* What happens is: Shock wave inside duct, then choking * 1 L a =L* (a) T o P oi * 1 L b >L a (b) * 1 L c >L b (c) 1 T s o T (a) (c) 1 M * (b ) P oi o
80 Control Volume of Adiabatic Flow in a Constant-Area Duct Objective: To find the variations of flow properties along the constant- area duct ….