control volume analysis of mass momentum &energy notes

Control-Volume Analysis of Mass, Momentum and Energy is an important topic of Fluid mechanics that deals with topics such as control mass, control volume, momentum equation, continuity equation and Impact of Jets on planes and vanes. 

Control Mass

• A fixed mass of a fluid element in the flow-field is identified and conservation equations for properties such as momentum, energy, or concentration are written.
• The identified mass moves around in the flow-field.
• Its property corresponds to the same contents of the identified fluid element may change from one location to another.

Control Volume

• This approach is popular and widely applied in the analysis.
• An arbitrarily fixed volume located at a certain place in the flow-field is identified and the conservation equations are written.
• The property under consideration or analysis may change with time.

The Momentum Equation

It is based on the law of conservation of momentum or on the momentum principle, which states that the net force acting on a fluid mass is equal to the change in momentum of flow per unit time in that direction.

F.dt = d(mv)                  ………… (1)

which is known as the impulse-momentum equation and states that impulse of a force F acting on a fluid of mass m in a short interval of time dt is equal to the change of momentum d(mv) in the direction of force.

•   Force exerted by a flowing fluid on a pipe bend

The impulse-momentum equation (1) is used to determine the resultant force exerted by a flowing fluid on a pipe bend.

Fig.: Forces on the bend in the pipe flow

Let v₁, p₁ and A₁ are the velocity, pressure and area at the section 1 of the pipe.

and v₂, p₂, A₂ = corresponding values of velocity, pressure and area at section (2).

Let F_x = horizontal component of force exerted by fluid element on the bend in x-direction.

F_y = vertical component of force exerted by fluid element on the bend in y-direction

The momentum equation in the x-direction is given by,

P₁A₁ - P₂A₂ cosθ - F_x = ρQ (V₂Cosθ- V₁)                       ……………… (2)

Similarly, the momentum equation in y-direction gives

 P₂A₂ sinθ  - F_y = ρQ (V₂ sinθ - 0)                                  ………………. (3)

F_y = ρQ (- V₂ sinθ) - P₂A₂sinθ

Now the resultant force (F_R) acting on the bend ,

  

And the angle made by the resultant force with horizontal direction is given by:

Moment Of Momentum Equation,

Moment of momentum equation is derived from moment of momentum principle which states that the resulting torque acting on a rotating fluid is equal to the rate of change of moment of momentum.

According to moment of momentum principle,

Resultant torque = rate of change of moment of momentum

T = ρQ [V₂r₂ - V₁r₁]  

Vortex Flow,

Vortex flow is defined as the flow of fluid along a curved path or the flow of a rotating mass of fluid is known as vortex flow.

• Forced vortex flow is defined as that type of vortex flow, in which some external torque is required to rotate the fluid mass
•  Free vortex flows are the plane circular vortex flows where the total mechanical energy remains constant in the entire flow field i.e. does not vary from streamline to streamline

For free vortex flows,

mvr = Constant

So, vr = Constant,

For forced vortex flow:

v = ω × r

where ω = Angular velocity = Constant.

Fig.: Figure showing the points of the free surface of a paraboloid

If the point 1 lies on the axis of rotation, then v₁ = ω× r₁ = ω × 0 = 0

The above equation becomes as

Let Z₂ = Z₁ = Z, then,

 

FLOW THROUGH JET

Impact of Jet

• The liquid comes out in the form of a jet from the outlet of a nozzle which is fitted to a pipe through which the liquid is flowing under pressure. Jets can travel long distances without dissipating the energy.

 Force Exerted by the Jet on a stationary vertical plate, 

Consider a jet of Water coming out from the nozzle, strikes a flat vertical plate,

 

Fig.: Force exerted by jet on vertical plate

F_x = ρaV[V – 0]

F_x = ρaV²

Force exerted by the jet on curved plate(vane) when the plate/vane is moving in the direction of jet,

Fig.: Force exerted by jet on a moving curved plate

Force Exerted by a Jet on a Hinged Plate,

Consider a jet of water striking a vertical plate at the center which is hinged at O. Due to the force exerted by the jet on the plate, the plate will swing through some angle about the hinge as shown in Fig.      

                  

Fig. : Force on a hinged plate