Recap: Conservation Equations & Analysis of Finite Control Volume | Fluid Mechanics for Mechanical Engineering PDF Download

Recap
In this course you have learnt the following

• A control mass or closed system is characterized by a fixed quantity of mass of a given identity, while in an open system or control volume mass may change continuously due to the flow of mass across the system boundary.
• Continuity equation is the equation of conservation of mass in a fluid flow. The general form of the continuity equation for an unsteady compressible flow is given by
  
   
• The concept of stream function is a consequence of continuity. In a two dimensional incompressible flow, the difference in stream functions between two points gives the volume flow rate (per unit width in a direction perpendicular to the plane of flow) across any line joining the points. The value of stream function is constant along a streamline.
• The equation of motion (conservation of momentum) of an inviscid flow is known as Euler’s equation. The general form of Euler’s equation is given by     where  is the body force vector per unit mass and  the velocity vector. Euler’s equation along a streamline, with gravity as the only body force, can be written as
        ( where s represents the coordinate along the streamline.)
   
• Reynolds transport theorem states the relation between equations applied to a system and those applied to a control volume. The statement of the law of conservation of momentum as applied to a control volume is known as momentum theorem. This theorem states that the resultant force (or torque) acting on a control volume is equal to the time rate of increase in linear momentum (or angular momentum) within the control volume plus the rate of net efflux of linear momentum (or angular momentum) from the control volume.
   
• A fluid element in motion possesses intermolecular energy, kinetic energy and potential energy. The work required by a fluid element to move against pressure is known as flow work. It is loosely termed as pressure energy. The shaft work is the work interaction between the control volume and the surrounding that takes place by the action of
  shear force such as the torque exerted on a rotating shaft. The equation for conservation of energy of a steady, in viscid and incompressible flow in a conservative body force field is known as Bernoulli’s equation.
  Bernoulli’s equation in the case of gravity as the only body force field is given by
  
      (The value of C remains constant along a streamline.)
   
• The loss of mechanical energy due to friction in a real fluid is considered in Bernoulli’s equation as

             

where, h_f = frictional work done or loss of mechanical energy due to friction per unit weight of a fluid element while movingfrom station 1 to 2 along a streamline. The term h_f is usually referred to as head loss.