Analysis of Finite Control Volumes - 2 | Fluid Mechanics for Mechanical Engineering PDF Download

Dynamic Forces on Curve Surfaces due to the Impingement of Liquid Jets

The principle of fluid machines is based on the utilization of useful work due to the force exerted by a fluid jet striking and moving over a series of curved vanes in the periphery of a wheel rotating about its axis. The force analysis on a moving curved vane is understood clearly from the study of the inlet and outlet velocity triangles as shown in Fig. 11.6.

The fluid jet with an absolute velocity V₁ strikes the blade at the inlet. The relative velocity of the jet V_r1 at the inlet is obtained by subtracting vectorially the velocity u of the vane from V₁. The jet strikes the blade without shock if β₁ (Fig. 11.6) coincides with the inlet angle at the tip of the blade. If friction is neglected and pressure remains constant, then the relative velocity at the outlet is equal to that at the inlet  (V_r2 = V_r1).

Fig  11.6    Flow of Fluid along a Moving Curved Plane

The control volume as shown in Fig. 11.6 is moving with a uniform velocity u of the vane.Therefore we have to use Eq.(10.18d) as the momentum theorem of the control volume at its steady state. Let F_c be the force applied on the control volume by the vane.Therefore we can write

To keep the vane translating at uniform velocity, u in the direction as shown. the force F has to act opposite to F_cTherefore,
    (11.14)

From the outlet velocity triangle, it can be written
    (11.15a)

Similarly from the inlet velocity triangle. it is possible to write
     (11.15b)

Addition of Eqs (11.15a) and (11.15b) gives

Power developed is given by
   (11.16)

The efficiency of the vane in developing power is given by
   (11.17)

Propulsion of a Ship

Jet propulsion of ship is found to be less efficient than propulsion by screw propeller due to the large amount of frictional losses in the pipeline and the pump, and therefore, it is used rarely. Jet propulsion may be of some advantage in propelling a ship in a very shallow water to avoid damage of a propeller.
Consider a jet propelled ship, moving with a velocity V, scoops water at the bow and discharges astern as a jet having a velocity Vr relative to the ship.The control volume is taken fixed to the ship as shown in Fig. 11.7.

Fig  11.7    A control volume for a moving ship

Following the momentum theorem as applied to the control volume shown. We can write

Where F_c is the external force on the control volume in the direction of the ship’s motion. The forward propulsive thrust F on the ship is given by
  (11.18)

Propulsive power is given by
   (11.19)

Jet Engine

A jet engine is a mechanism in which air is scooped from the front of the engine and is then compressed and used in burning of the fuel carried by the engine to produce a jet for propulsion. The usual types of jet engines are turbojet, ramjet and pulsejet.

              Fig 11.8    A Turbojet Engine


Fig 11.9  An Appropriate Control Volume Comprising the Stream of Fluid Flowing through the Engine

A turbojet engine consists essentially (Fig. 11.8) of -

• a compressor,
• a combustion chamber,
• a gas turbine and
• a nozzle.

A portion of the thermal energy of the product of combustion is used to run the gas turbine to drive the compressor. The remaining part of thermal energy is converted into kinetic energy of the jet by a nozzle. At high speed fiight, jet engines are advantageous since a propeller has to rotate at high speed to create a large thrust. This will result in excessive blade stress and a decrease in the efficiency for blade tip speeds near and above sonic level. In a jet propelled aircraft, the spent gases are ejected to the surroundings at high velocity usually equal to or greater than the velocity of sound in the fluid at that state.

In many cases, depending upon the range of fight speeds, the jet is discharged with a velocity equal to sonic velocity in the medium and the pressure at discharge does not fall immediately to the ambient pressure. In these cases, the discharge pressure p₂ at the nozzle exit becomes higher than the ambient pressure p_atm. Under the situation of uniform velocity of the aircraft, we have to use Eg. (10.18d) as the momentum theorem for the control volume as shown in Fig. 11.9 and can write


where, F_x is the force acting on the control volume along the direction of the coordinate axis ”OX” fixed to the control volume, V is the velocity of the aircraft, u is the relative velocity of the exit jet with respect to the aircraft,    and  are the mass flow rate of air, and mass burning rate of fuel respectively. Usually    is very less compared to    usually varies from 0.01 to 0.02 in practice).

The propulsive thrust on the aircraft can be written as
   (11.20)

The terms in the bracket are always positive. Hence, the negative sign in F_T represents that it acts in a direction opposite to ox, i.e. in the direction of the motion of the jet engine. The propulsive power is given by

     (11.21)

Non-inertial Control Volume

Rocket engine

Rocket engine works on the principle of jet propulsion.

• The gases constituting the jet are produced by the combustion of a fuel and appropriate oxidant carried by the engine. Therefore, no air is required from outside and a rocket can operate satisfactorily in a vacuum. 
   
• A large quantity of oxidant has to be carried by the rocket for its operation to be independent of the atmosphere.
   
• At the start of journey, the fuel and oxidant together form a large portion of the total load carried by the rocket.
   
• Work done in raising the fuel and oxidant to a great height before they are burnt is wasted. 
   
• Therefore, to achieve the efficient use of the materials, the rocket is accelerated to a high velocity in a short distance at the start. This period of rocket acceleration is of practical interest.
   
  Fig 11.10   A Control Volume for a Rocket Engine

  Let   be the rate at which spent gases are discharged from the rocket with a velocity u relative to the rocket (Fig. 11.10) Both  and u are assumed to be constant.

  Let M and V be the instantaneous mass and velocity (in the upward direction) of the rocket. The control volume as shown in Fig. 11.10 is an accelerating one. Therefore we have to apply Eq. (10.18b) as the momentum theorem of the control volume. This gives
     (11.22)
  
  where ΣF is the sum of the external forces on the control volume in a direction vertically upward. If p_e and p_a be the nozzle exhaust plane gas pressure and ambient pressure respectively and D is the drag force to the motion of the rocket, then one can write
       (11.23)
  
  Where, A_e is outlet area of the propelling nozzle. Then Eq. (11.22) can be written as
  

• In absence of gravity and drag, Eq (11.23) become