Conservation of Energy | Fluid Mechanics for Mechanical Engineering PDF Download

Conservation of Energy
The principle of conservation of energy for a control mass system is described by the first law of thermodynamics

Heat Q added to a control mass system- the work done W by the control mass system = change in its internal energy E

The internal energy depends only upon the initial and final states of the system. It can be written in the form of the equation as
     (13.1a)

Equation (13.1a) can be expressed on the time rate basis as
     (13.1b)

Where δQ and δW are the amount of heat added and work done respectively during a time interval of δt. To develop the analytical statement for the conservation of energy of a control volume, the Eq. (10.10) is used with N = E (the internal energy) and η = e (the internal energy per unit mass) along with the Eq. (13.1b). This gives
    (13.2)

The Eq. (13.2) is known as the general energy equation for a control volume.

The first term on the right hand side of the equation is the time rate of increase in the internal energy within a control volume and the second term is the net rate of energy efflux from the control volume.

Different forms of energy associated with moving fluid elements comprising a control volume are -

1. Potential energy 
The concept of potential energy in a fluid is essentially the same as that of a solid mass. The potential energy of a fluid element arises from its existence in a conservative body force field. This field may be a magnetic, electrical, etc. In the absence of any of such external force field, the earth’s gravitational effect is the only cause of potential energy. If a fluid mass m is stored in a reservoir and its C.G. is at a vertical distance z from an arbitrary horizontal datum plane, then the potential energy is mgz and the potential energy per unit mass is gz. The arbitrary datum does not play a vital role since the difference in potential energy, instead of its absolute value, is encountered in different practical purposes.

2. Kinetic Energy
If a quantity of a fluid of mass m flows with a velocity V, being the same throughout its mass, then the total kinetic energy is mV²/2 and the kinetic energy per unit mass is V²/2. For a stream of real fluid, the velocities at different points will not be the same. If V is the local component of velocity along the direction of flow for a fluid flowing through an open channel or closed conduit of cross-sectional area A, the total kinetic energy at any section is evaluated by summing up the kinetic energy flowing through differential areas as


The average velocity at a cross-section in a flowing stream is defined on the basis of volumetric flow rate as,

The kinetic energy per unit mass of the fluid is usually expressed as    where α is known as the kinetic energy correction factor. 
Therefore, we can write


Hence,  
    (13.3a)

For an incompressible flow,
    (13.3b)

 

3. Intermolecular Energy 
The intermolecular energy of a substance comprises the potential energy and kinetic energy of the molecules. The potential energy arises from intermolecular forces. For an ideal gas, the potential energy is zero and the intermolecular energy is, therefore, due to only the kinetic energy of molecules. The kinetic energy of the molecules of a substance depends on its temperature.

4. Flow Work
Flow work is the work done by a fluid to move against pressure.
For a flowing stream, a layer of fluid at any cross-section has to push the adjacent neighboring layer at its downstream in the direction of flow to make its way through and thus does work on it. The amount of work done can be calculated by considering a small amount of fluid mass A₁ ρ₁ dx to cross the surface AB from left to right (Fig. 13.1). The work done by this mass of fluid then becomes equal to p₁ A₁ dx and thus the flow work per unit mass can be expressed as

  (where p₁ is the pressure at section AB (Fig 13.1)

Fig 13.1   Work done by a fluid to flow against pressure

Therefore the flow work done per unit mass by a fluid element entering the control volume ABCDA (Fig. 13.1) is p₁ /ρ₁ Similarly, the flow work done per unit mass by a fluid element leaving the control volume across the surface CD is p₂/ρ₁

Important- In introducing an amount of fluid inside the control volume, the work done against the frictional force at the wall can be shown to be small as compared to the work done against the pressure force, and hence it is not included in the flow work.

Although ’flow work’ is not an intrinsic form of energy, it is sometimes referred to as ’pressure energy’ from a view point that by virtue of this energy a mass of fluid having a pressure p at any location is capable of doing work on its neighboring fluid mass to push its way through.


Steady Flow Energy Equation
The energy equation for a control volume is given by Eq. (13.2). At steady state, the first term on the right hand side of the equation becomes zero and it becomes
    (13.4)

In consideration of all the energy components including the flow work (or pressure energy) associated with a moving fluid element, one can substitute ’e’ in Eq. (13.4) as


and finally we get
           (13.5)

The Eq. (13.5) is known as steady flow energy equation.


Bernoulli's Equation   
Energy Equation of an ideal Flow along a Streamline

Euler’s equation (the equation of motion of an inviscid fluid) along a stream line for a steady flow with gravity as the only body force can be written as
     (13.6)

Application of a force through a distance ds along the streamline would physically imply work interaction. Therefore an equation for conservation of energy along a streamline can be obtained by integrating the Eq. (13.6) with respect to ds as
     (13.7)

Where C is a constant along a streamline. In case of an incompressible flow, Eq. (13.7) can be written as
     (13.8)

The Eqs (13.7) and (13.8) are based on the assumption that no work or heat interaction between a fluid element and the surrounding takes place. The first term of the Eq. (13.8) represents the flow work per unit mass, the second term represents the kinetic energy per unit mass and the third term represents the potential energy per unit mass. Therefore the sum of three terms in the left hand side of Eq. (13.8) can be considered as the total mechanical energy per unit mass which remains constant along a streamline for a steady inviscid and incompressible flow of fluid. Hence the Eq. (13.8) is also known as Mechanical energy equation.

This equation was developed first by Daniel Bernoulli in 1738 and is therefore referred to as Bernoulli’s equation. Each term in the Eq. (13.8) has the dimension of energy per unit mass. The equation can also be expressed in terms of energy per unit weight as
    (13.9)

 

In a fluid flow, the energy per unit weight is termed as head. Accordingly, equation 13.9 can be interpreted as

      Pressure head + Velocity head + Potential head =Total head (total energy per unit weight).

Bernoulli's Equation with Head Loss
The derivation of mechanical energy equation for a real fluid depends much on the information about the frictional work done by a moving fluid element and is excluded from the scope of the book. However, in many practical situations, problems related to real fluids can be analysed with the help of a modified form of Bernoulli’s equation as
     (13.10)

where, h_f represents the frictional work done (the work done against the fluid friction) per unit weight of a fluid element while moving from a station 1 to 2 along a streamline in the direction of flow. The term h_f is usually referred to as head loss between 1 and 2, since it amounts to the loss in total mechanical energy per unit weight between points 1 and 2 on a streamline due to the effect of fluid friction or viscosity. It physically signifies that the difference in the total mechanical energy between stations 1 and 2 is dissipated into intermolecular or thermal energy and is expressed as loss of head h_f in Eq. (13.10). The term head loss, is conventionally symbolized as h_Linstead of h_f in dealing with practical problems. For an inviscid flow h_L = 0, and the total mechanical energy is constant along a streamline.