Introduction. In general, gases are highly compressible and liquids have a very low compressibility. COMPRESSIBLE FLOW
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1 COMRESSIBLE FLOW COMRESSIBLE FLOW Introduction he compressibility of a fluid is, basically, a measure of the change in density that will be produced in the fluid by a specific change in pressure and temperature. In general, gases are highly compressible and liquids have a very low compressibility. art one : Introduction of Compressible Flow
2 COMRESSIBLE FLOW Application ; Aircraft design Gas and steam turbines Reciprocating engines Natural gas transmission lines Combustion chambers Compressibility effect ; Supersonic the flow velocity is relatively high compared to the speed of sound in the gas. Subsonic art one : Introduction of Compressible Flow
3 COMRESSIBLE FLOW Fundamental assumptions. he gas is continuous.. he gas is perfect (obeys the perfect gas law) 3. Gravitational effects on the flow field are negligible. 4. Magnetic and electrical effects are negligible. 5. he effects of viscosity are negligible. Applied principles. Conservation of mass (continuity equation). Conservation of momentum (Newton s law) 3. Conservation of energy (first law of thermodynamics) 4. Equation of state art one : Introduction of Compressible Flow 3
4 COMRESSIBLE FLOW erfect gas law : R : ressure : Density R : Universal gas constant Rair : emperature J 87.04( ) kg K art one : Introduction of Compressible Flow 4
5 COMRESSIBLE FLOW Conservation laws : Conservation of mass Rate of increase of mass of fluid in control volume Rate mass enters control volume _ Rate mass leaves control volume Conservation of momentum : Net force on gas in control volume in direction considered Rate of increase of momentum in direction considered of fluid in control l _ Rate momentum leaves control volume in direction considered Rate momentum leaves control volume in direction considered art one : Introduction of Compressible Flow 5
6 COMRESSIBLE FLOW Conservation of energy : Rate of increase in internal energy and kinetic energy of gas in control volume Rate enthalpy and kinetic energy leave control volume _ Rate enthalpy and kinetic energy enter control volume Rate heat is transferred into control volume _ Rate work is done by gas in control volume art one : Introduction of Compressible Flow 6
7 COMRESSIBLE FLOW Definition : A control volume is a volume in space (geometric entity, independent of mass) through which fluid may flow Enthalpy H, is the sum of internal energy U and the product of pressure and volume appears. H U art one : Introduction of Compressible Flow 7
8 COMRESSIBLE FLOW COMRESSIBLE FLOW Introduction Many of the compressible flows that occur in engineering practice can be adequately modeled as a flow through a duct or streamtube whose cross-sectional area is changing relatively slowly in the flow direction. A duct is a solid walled channel, whereas a streamtube is defined by considering a closed curve drawn in a fluid flow. art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 8
9 COMRESSIBLE FLOW Quasi-one-dimensional flow is flows in which the flow area is changing but in which the flow at any section can be treated as one-dimensional. art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 9
10 COMRESSIBLE FLOW CONINUIY EQUAION he continuity equation is obtained by applying the principle of conservation of mass to flow through a control volume. One-dimensional flow is being considered. here is no mass transfer across the control volume. he only mass transfer occurs through the ends of the control volume. art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 0
11 COMRESSIBLE FLOW Mass enters through the left hand face of the control volume be equal to the rate at which mass leaves through the right hand face of the control volume. m & & m We know that m& A We considered ; A A For the differentially short control volume indicated, art two : he Equation of Steady One-Dimensional Compressible Fluid Flow
12 COMRESSIBLE FLOW above equation gives ; A ( d)( d )( A da) Neglecting higher order terms, we found ; Ad Ad da 0 A Dividing this equation by then gives ; d d da A 0 his equation relates the fractional changes in density, velocity and area over a short length of the control volume. art two : he Equation of Steady One-Dimensional Compressible Fluid Flow
13 COMRESSIBLE FLOW MOMENUM EQUAION (Euler s equation) he flow is steady flow. Gravitational forces are being neglected. he only forces acting on the control volume are the pressure forces and the frictional force exerted on the surface of the control volume. art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 3
14 COMRESSIBLE FLOW he net force on the control volume in the x-direction is ; A ( p d)( A da) [( ( d)][( A da) A] df Note : dx is too small, dda have been neglected. Mean pressure on the curved surface can be taken as the average of the pressures acting on the two end surfaces. dfµ is the frictional force. Rearranging above equation, we found the net force on the control volume in the x-direction is ; Ad df art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 4
15 COMRESSIBLE FLOW Since the rate at which momentum crosses any section of the duct is equal to m&, we found that ; A[( d ) ] Ad he above equation can be written as ; Ad df Ad Frictional force is assumed to be negligible. he Euler s equation for steady flow through a duct becomes; d d art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 5
16 COMRESSIBLE FLOW Integrating Euler s equation ; d C (For compressible) And if density can be assumed constant, Euler s equation become ; C (For incompressible) art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 6
17 COMRESSIBLE FLOW SEADY FLOW ENERGY EQUAION For flow through the type of control volume considered as before, we found ; h h q w h enthalpy per mass velocity q heat transferred into the control volume per unit mass of fluid flowing through it w work done by the fluid per unit mass art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 7
18 COMRESSIBLE FLOW Assumption ; No work is done, w0 erfect gases is considered, h c p Steady flow energy equation ; cp cp q Applying this equation to the flow through the differentially short control volume gives ; c p dq c p ( d ) ( d ) art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 8
19 COMRESSIBLE FLOW Neglecting higher order terms gives ; c p d d dq his equation indicates that in compressible flows, changes in velocity will, in general, induce changes in temperature and that heat addition can cause velocity changes as well as temperature changes. If the flow is adiabatic i.e., if there is no heat transfer to of from the flow, it gives ; c p cp art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 9
20 COMRESSIBLE FLOW Steady flow energy equation for adiabatic flow becomes ; d c p d 0 his equation shows that in adiabatic flow, an increase in velocity is always accompanied by a decrease in temperature. art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 0
21 COMRESSIBLE FLOW EQUAION OF SAE When applied between any two points in the flow ; When applied between the inlet and the exit of a differentially short control volume, this equation becomes ; d ( d)( d ) art two : he Equation of Steady One-Dimensional Compressible Fluid Flow
22 COMRESSIBLE FLOW Higher order terms are neglected and it gives ; d d d d d d 0 his equation shows how the changes in pressure, density and temperature are interrelated in compressible flow. art two : he Equation of Steady One-Dimensional Compressible Fluid Flow
23 COMRESSIBLE FLOW ENROY CONSIDERAIONS In studying compressible flows, another variable, the entropy, s, has to be introduced. he entropy basically places limitations on which flow processes are physically possible and which are physically excluded. he entropy change between any two points in the flow is given by ; s s c p ln R ln () Since R c p c v, this equation can be written; s c p s ln If there is no change in entropy, i.e., if the flow is art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 3
24 COMRESSIBLE FLOW isentropic, this equation requires that : hence, since the perfect gas law gives ; it follows that in isentropic flow : in isentropic flows, then is a constant. If equation () is applied between the inlet and the exit art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 4
25 COMRESSIBLE FLOW of a differentially short control volume, it gives ; ( d d s ds) s c p ln R ln neglecting small value, the above equation gives; ds c d d p R () which can be written as ; ds c p d d lastly, it is noted that in an isentropic flow, equation () gives; c p d R d using the perfect gas law ; art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 5
26 COMRESSIBLE FLOW c p d d (3) but the energy equation for isentropic flow, i.e., for flow with no heat transfer, it gives ; d c p d 0 which using equation (3) gives ; d d 0 art two : he Equation of Steady One-Dimensional Compressible Fluid Flow 6
27 COMRESSIBLE FLOW SOME FUNDAMENAL ASECS OF COMRESSIBLE FLOW Mach number gas velocity mach number, M speed of sound a a R M < : subsonic M : transonic M > : supersonic M >> : hypersonic art three : Mach Number 7
28 COMRESSIBLE FLOW Isentropic flow in a streamtube In order to illustrate the importance of the Mach number in determining the conditions under which compressibility must be taken in account, isentropic flow, i.e., frictionless adiabatic flow, through a streamtube will be first considered. From previous chapter, we know that ; d d and a the above equation can be written as : d d M a d () his equation shows that the magnitude of the fractional pressure change, induced by a given fractional velocity change, depends on the square of Mach number. art three : Mach Number 8
29 COMRESSIBLE FLOW Next, consider the energy equation. Since adiabatic flow is being considered ; d c p d R c p M d Since; R c v p c and R c p Above equation can be written as ; d ( ) M d () Lastly, consider the equation of state; d d d combining above equation with eq.() and eq.() art three : Mach Number 9
30 COMRESSIBLE FLOW d M ( ) M d d his equation indicates that: d M d (negative sign means, density decrease when velocity increased) at M0., at M0.33, d d d d % % At low mach number, density changes will be insignificant. art three : Mach Number 30
31 COMRESSIBLE FLOW Normally at M<0.3, the fluid is assumed incompressible. It should also be noted that above equation can we written as ; d d ( ) M Similarly, the temperature difference is neglected at lower value of Mach number. art three : Mach Number 3
32 COMRESSIBLE FLOW Mach waves Disturbances tend to propagated ahead of the body in motion to warn the gas of the approach of the body. his is due to pressure at the surface is higher than surrounding gas and pressure waves spread out from the body. he pressure waves spread out at the of sound Effect of the velocity of the body relative to the speed of sound (pressure wave velocity) on the flow field. art three : Mach Number 3
33 COMRESSIBLE FLOW Consider for subsonic flow M<, figure (). Speed of the body u and speed of sound a, where u<a. Body position at a, b, c and d at time interval t. Waves generated at time 0, t, t and 3t. Since u<a, a body moves slower than the waves and therefore a body will never overtake it. art three : Mach Number 33
34 COMRESSIBLE FLOW If u>a, then M>, the flow is supersonic, a body moves faster than the waves and will overtake it, (figure ()). he waves lie within a cone which has its vertex at the body at the time considered. On gas within this cone aware of the presence of the body. ertex angle α is called Mach angle, where ; a sin α u M he cone is therefore termed a conical Mach wave. art three : Mach Number 34
35 COMRESSIBLE FLOW art three : Mach Number 35
36 COMRESSIBLE FLOW art three : Mach Number 36
37 COMRESSIBLE FLOW ONE-DIMENSIONAL ISENROIC FLOW INRODUCION An adiabatic flow (a flow in which there is no heat exchange) in which viscous losses are negligible, i.e., it is an adiabatic frictionless flow. Although no real flow is entirely isentropic, there are many flows of great practical importance in which the major portion of the flow can be assumed to be isentropic. art four : One-Dimensional Isentropic Flow 37
38 COMRESSIBLE FLOW For example, in internal duct flows there are many important cases where the effects of viscosity and heat transfer are restricted to thin layers adjacent to the walls, i.e., are only important in the wall boundary layers, and the rest of the flow can be assumed to be isentropic. Even when non-isentropic effects become important, it is often possible to calculate the flow by assuming it to be isentropic and to then apply an empirical correction factor to the solution so obtained to account for the non-isentropic effect, for example, in the design nozzle. art four : One-Dimensional Isentropic Flow 38
39 COMRESSIBLE FLOW GOERNING EQUAION By definition, the entropy remains constant in an isentropic flow. c (c:constant) (4.) From equation (4.) art four : One-Dimensional Isentropic Flow 39
40 COMRESSIBLE FLOW Hence, since the general equation of state gives ; or It follows that in isentropic flow ; Recalling that R a, that ; a a eq.(4.5) he steady flow adiabatic energy equation is next applied between the point and point. his gives ; c c p p art four : One-Dimensional Isentropic Flow 40
41 COMRESSIBLE FLOW It can be written as ; ) ( ) ( c p c p From ; M c R R c p p So, it follows that ; ) ( M ) ( M eq.(4.6) his equation applies in adiabatic flow. If friction effects are also negligible, i.e., if the flow is isentropic, eq.(4.6) cam be used in conjunction with the isentropic state relations given in eq.(4.5) to obtain ; art four : One-Dimensional Isentropic Flow 4
42 COMRESSIBLE FLOW ) ( ) ( M M and ) ( ) ( M M Lastly, it is called that the continuity equation gives ; A A which can be rearranged to give ; A A art four : One-Dimensional Isentropic Flow 4
43 COMRESSIBLE FLOW SAGNAION CONDIIONS Stagnation conditions are those that would exist if the flow at any point in fluid stream was isentropically brought to rest. If the entire flow is essentially isentropic and if the velocity is essentially zero at some point in the flow, then the stagnation conditions will be those existing at the zero velocity point. art four : One-Dimensional Isentropic Flow 43
44 COMRESSIBLE FLOW However, even when the flow is non-isentropic, the concept of the stagnation conditions is still useful, the stagnation conditions at a point the being the conditions that would exist if the local flow were brought to rest isentropically. If the equations derived in the previous section are applied between a point in the flow where the pressure, density, temperature and Mach number are,,, M respectively, then if the stagnation conditions are denoted by the subscript 0, the stagnation pressure, density and temperature will, since the Mach number is zero at the point where the stagnation conditions exist, be given by ; art four : One-Dimensional Isentropic Flow 44
45 COMRESSIBLE FLOW 0 M 0 M 0 M ( for the particular case of 4. ) art four : One-Dimensional Isentropic Flow 45
46 COMRESSIBLE FLOW CRIICAL CONDIIONS he critical conditions are those that would exist if the flow was isentropically accelerated or decelerated until the Mach number was unity, (M ) hese critical conditions are usually denoted by an asterisk. By setting M, we found ; * M * M a a * M art four : One-Dimensional Isentropic Flow 46
47 COMRESSIBLE FLOW * M By setting M 0, we found ; 0 * 0 * a a 0 * 0 * For the case of air flow ; *, *, * art four : One-Dimensional Isentropic Flow 47
48 COMRESSIBLE FLOW MAXIMUM DISCHARGE ELOCIY Also known as maximum escape velocity, is the velocity that would be generated if a gas was adiabatically expanded until its temperature has dropped to absolute zero. Using the adiabatic energy equation gives the maximum discharge velocity as : 0 ˆ c c p his can be rearranged to give ; 0 ) ( ˆ c c p ) ( ˆ 0 a a art four : One-Dimensional Isentropic Flow 48
49 ONE-DIMENSIONAL ISENROIC FLOW SUMMARY OF MAJOR EQUAIONS ISENROIC FLOW RELAIONS a a ) ( ) ( M M ) ( ) ( M M ) ( ) ( M M ) ( ) ( ) ( M M M M A A A A
50 SAGNAION CONDIIONS 0 ) ( M 0 ) ( M 0 ) ( M CRIICAL CONDIIONS * M * M * M
51 RELAIONSHI BEWEEN CRIICAL AND SAGNAION CONDIIONS 0 * 0 * 0 * 0 * a a
52 able Some typical values for the speed of sound at 0 ºC Source: Compressible Fluid Flow, he McGraw-Hill Companies, Inc. atrick, H.O. and William, E.C. Gas Molar mass Speed of sound at 0 ºC (m/s) Air Argon (Ar) Carbon dioxide (CO ) Freon (CCl F ) Helium (He) Hydrogen (H ) Xenon (Xe)
53 able Isentropic flow tables for air with.4 Source: Compressible Fluid Flow, he McGraw-Hill Companies, Inc. atrick, H.O. and William, E.C. M o / o / o / a o / a A / A* θ
54 M o / o / o / a o / a A / A* θ
55 M o / o / o / a o / a A / A* θ
56 able 3 Approximate properties of the standard atmosphere Source: Compressible Fluid Flow, he McGraw-Hill Companies, Inc. atrick, H.O. and William, E.C. H a v ( m ) ( K ) ( a 0 5 ) ( kg/m 3 ) ( m/s ) ( m /s 0-5 )
57 H a v ( m ) ( K ) ( a 0 5 ) ( kg/m 3 ) ( m/s ) ( m /s 0-5 )
58 utorial for compressible flow. An air stream enters a variable area channel at a velocity of 30m/s with a pressure of 0ka and a temperature of 0ºC. At a certain point in the channel, the velocity is found to be 50m/s. Using Bernoulli s equation (i.e., p /constant), which assunmes incompressible flow, find the pressure at this point. In this calculation use the density evaluated at the inlet conditions. If the temperature of the air is assumed to remain constant, evaluate the air density at the point in the flow where the velocity is 50m/s. Compare this density with the density a the inlet to the channel. On the basis of this comparison, do you think that the use of Bernoulli s equation is justified.. wo kilograms of air at an initial temperature and pressure of 30ºC and 00ka undergoes an isentropic process, the final temperature attained being 850ºC. Find the final pressure, the initial and final densities and the initial and final volumes. 3. wo air streams are mixed in a chamber. One stream enters the chamber through a 5cm diameter pipe at velocity of 00m/s with a pressure of 50ka and a temperature of 30ºC. he other stream enters the chamber through a.5cm diameter pipe at a velocity of 50m/s with a pressure of 75ka and a temperature of 30ºC. he air leaves the chamber through a 9cm diameter pipe at a pressure of 90ka and a temperature of 30ºC. Assuming that the flow is steady, find the velocity in the exit pipe. 4. he jet engine fitted to a small aircraft uses 35kg/s of air when the aircraft is flying at a speed of 800km/h. he jet efflux velocity is 590m/s. If the pressure on the engine discharge plane is assumed to be equal to the ambient pressure and if effects of the mass of the fuel used are ignored, find the thrust developed by the engine. 5. he engine of a small jet aircraft develops a thrust of 8kN when the aircraft is flying at a speed of 900km/h at an altitude where the ambient pressure is 50ka. he air flow rate through the engine is 75kg/s and the engine uses fuel at a rate of 3kg/s. he pressure on the engine discharge plane is 55ka and the area of the engine exit is 0.m. Find the jet efflux velocity.
59 6. A solid fuelled rocket is fitted with a convergent-divergent nozzle with an exit plane diameter of 30cm. he pressure and velocity on this nozzle exit plane are 75Kpa and 750m/s respectively and the mass flow rate through the nozzle is 350kg/s. Find the thrust developed by this engine when the ambient pressure is (a) 00ka and (b) 0ka. 7. In a hydrogen powered rocket, hydrogen enters a nozzle at a very low velocity with a temperature and pressure of 000ºC and 6.8Mpa respectively. he pressure on the exit plane of the nozzle is equal to the ambient pressure which is 0ka. If the required thrust is 0MN, what hydrogen mass flow rate required? he flow though the nozzle can be assumed to be isentropic and the specific heat ratio of the hydrogen can be assumed to be Carbon dioxide flows through a constant area duct. At inlet to the duct, the velocity is 0m/s and the temperature and pressure are 00ºC and 700ka respectively. Heat is added to the flow in the duct and at the exit of the duct the velocity is 40m/s and the temperature is 450ºC. Find the amount of heat being added to the carbon dioxide per unit mass of gas and the mass flow rate through the duct per unit cross-sectional area of the duct. Assume that for carbon dioxide, Air enters a heat exchanger with a velocity of 0m/s and a temperature and pressure of 5ºC and.5mpa. Heat is removed from the air in the heat exchanger and the air leaves with a velocity 30m/s at a temperature and pressure 80ºC and.45ma. Find the heat removed per kilogram of air flowing through the heat exchanger and the density of the air at the inlet and the exit to the heat exchanger.
60 utorial for compressible flow. Air enters a tank at a velocity of 00m/s and leaves the tank at a velocity of 00m/s. lf the flow is adiabatic find the difference between the temperature of the air at exit and the temperature of the air at inlet. Air at a temperature of 5 C is flowing at a velocity of 500m/s. A shock wave occurs in the flow reducing the velocity to 300m/s. Assuming the flow through the shock wave to be adiabatic, find the temperature of the air behind the shock wave. 3. Air being released from a tire through the valve is found to have a temperature of 5 C. Assuming that the air in the tire is at the ambient temperature of 30 C, find the velocity of the air at the exit of the valve. he process can be assumed to be adiabatic. 4. Gas with a molecular weight of 4 and a specific heat ratio of.67 flows through a variable area duct. At some point in the now the velocity is 80m/s and the temperature is 0 C. At some other point in the flow the temperature is minus 0 C. Find the velocity at this point in the flow assuming that the flow is adiabatic. 5. At a section of a circular duct through which air is flowing the pressure is 50ka, the temperature is 35 C, the velocity is 50m/s, and the diameter is 0.m. If, at this section, the duct diameter is increasing at a rate of 0.m/m, find dp/dx, d/dx and d/dx 6. Consider an isothermal air flow through a duct. At a certain section of the duct the velocity, temperature,and pressure are 00m/s, 5 C,and 0ka respectively. lf the velocity is decreasing at this section at a rate of 30 percent per meter, find dp/dx, ds/dx and d/dx. 7. Consider adiabatic air flow through a variable area duct, At a certain section of the duct the flow area is 0.m, the pressure is 0ka, the temperature is 5 C and the duct area is changing at a rate of 0.m /m. lot the variations of dp/dx, d/dx and d/dx with the velocity at the section for velocities between 50m/s and 300m/s.
61 utorial 3 for compressible flow. he velocity of an air flow changes by one percent. Assuming that the flow is isentropic,plot the percentage changes in pressure, temperature, and density induced by this change in velocity with flow Mach number for Mach numbers between 0. and.. Calculate the speed of sound at 88 K in hydrogen, helium and nitrogen. Under what conditions will the speed of sound in hydrogen be equal to that in helium? 3. Find the speed of sound in carbon dioxide at temperatures of 0ºC and 600ºC. 4. A very weak pressure wave, i.e, a sound wave, across which the pressure rise is 30a moves through air which has a temperature of 30 C and a pressure of 0ka. Find the density change, the temperature change, and the velocity change across this wave. 5. An airplane can fly at a speed or800km/h at sea-level where the temperature is 5 C. lf the airplane flies at the same Mach number at an altitude where the temperature is -44 C, find the speed at which the airplane is flying at this altitude. 6. he test section of a supersonic wind tunnel is square in cross-section with a side length of.m. he Mach number in the test section is 3.5, the temperature is - 00 C, and the pressure is 0ka. Find the mass flow rate of air through the test section. 7. A gas with a molar mass of 44 and a specific heat ratio.67 flows through a channel at supersonic speed. he temperature of the gas in the channel is 0 C. A photograph of the flow reveals weak waves originating at imperfections in the wall running across the flow at an angle to 45 to the flow direction. Find the Mach number and the velocity in the flow. 8. An observer at sea level does not hear an aircraft that is flying at an altitude of 7000m until it is a distance of 3km from the observer. Estimate the Mach number at which the aircraft is flying. In arriving at the answer, assume that the average temperature of the air between sea level and 7000m is -0 C. 9. An observer on the ground finds that an airplane flying horizontally at an altitude of 500m has traveled 6 km from the overhead position before the sound of the airplane is first heard. Assuming that, overall, the aircraft creates a small disturbance, estimate the speed at which the airplane is flying. he average air temperature between the ground and the altitude at which the airplane is flying is 0 C. Explain the assumptions you have made in arriving at the answer.
62 utorial 4 for compressible flow. A gas with a molar mass of 4 and a specific heat ratio of.67 flows through a variable area duct. At some point in the flow the velocity is 00m/s and the temperature is 0ºC. Find the Mach number at this point in the flow. At some other point in the flow the temperature is -0ºC. Find the velocity and Mach number at this point in the flow assuming that the flow is isentropic.. he exhaust gases from a rocket engine have a molar mass of 4. hey can be assumed to behave as a perfect gas with a specific heat ratio of.5. hese gases are accelerated through a nozzle. At some point in the nozzle where the crosssectional area of the nozzle is 0.7m, the pressure is 000ka, the temperature is 500 C and the velocity is 00 m/s, Find the mass flow rate through the nozzle and the stagnation pressure and temperature. Also find the highest velocity that could be generated by expanding this flow. lf the pressure at some other point in the nozzle is 00ka, find the temperature and velocity at this point in the flow assuming the flow to be one-dimensional and isentropic. 3. lf Concorde is flying at a Mach number of., at an altitude of 0000m in the standard atmosphere, find the stagnation pressure and temperature for the flow over the aircraft. 4. If a gas is flowing at 300m/s and has a pressure and temperature of 90ka and 0 C, find the maximum possible velocity that could be generated by expansion of this gas if the gas is air and if it is helium.
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