CHAPTER THREE INTEGRAL RELATIONS FOR A CONTROL

VOLUME

Contents • Introduction • Physical Laws of Fluid Mechanics • The Reynolds Transport Theorem • Conservation of Mass Equation • Linear Momentum Equation • Angular Momentum Equation • Energy Equation • Bernoulli’s Equation

Introduction • Fluid Flows are governed by a set of natural laws. And these laws are elegantly

defined by mathematics. We make appropriate choice of the approach we follow for some particular fluid flow problem. We need to call back to the different approaches mentioned in Chapter one: Integral Relations for a Control Volume : In this approach gross analysis of properties of fluid

flow is analyzed for a selected finite control volume. Differential Relations for a Fluid Flow : Unlike the Integral approach, the properties of

interest are evaluated at infinitesimal points and instantaneous time. Dimensional Analysis: This is also another mathematical technique used in fluid flow

analysis which eases a great deal of complications that happen in analytical approaches. It involves extensive experimental investigation.

Numerical or Computational Methods: It utilizes numerical methods based on the basic governing equations. There are many unsolved mathematical relations which can only be approached by this method.

Physical Laws of Fluid Mechanics There are some entities that needs to be well defined before we formulate physical laws in fluid Mechanics. • System : It is an arbitrary quantity of mass with fixed identity. ‘All the laws of

mechanics are written for a system’. • The mass of a system is conserved or

• Surrounding : Anything outside the system • Boundary : It is an imaginary wall that separates the system from the

surrounding• Control volume (CV): is the space occupied by a system at some instant, and

all the laws that apply for the system apply for the CV too, just at the instant.

The Reynolds transport theorem • The Reynolds transport theorem is a model of transport phenomena

of different extensive properties in fluids. • There are three types of CVs:

Fixed CV Moving CV Deformable CV

Arbitrary fixed CV• The figure shows an arbitrary

CV with arbitrary flow pattern. There are extensive properties crossing (entering or exiting) the boundary (Control surface) of the CV.• Part of the extensive properties are accumulated inside the CV.

• The condensed form of the Reynolds transport theorem is given below, with the convention that influx is taken as negative and outflux is positive.

• For fixed CV

• CV moving at constant velocity

• Arbitrarily moving and deformable CV

𝛽=𝑑𝐵𝑑𝑚Intensive property

Extensive property

Conservation of mass Equation • Here, the Reynolds transport theorem is applied to conservation of

mass, where is the extensive property and the intensive property is unity, .

• For fixed CV

A system consists of a fixed quantity of mass, as a result conservation of mass demands .

• For one dimensional inlets and outlets

• For steady flow

• For steady and one dimensional inlet and outlet

• For incompressible flow

• For incompressible flow with one dimensional inlet and outlet

• For steady and one dimensional inlet and outlet

Linear momentum equation • Here, the Reynolds transport theorem is applied to linear momentum

equation , where is the extensive property and the intensive property, .

• For one dimensional momentum flux

Force is equivalent to the rate of change of linear momentum since mass is conserved for a system

Surface forces on a CV• Pressure force • Viscous stress force • Mechanical reaction force (e.g.: - flange connections in pipes)

• Jet exit pressure is normally taken as atmospheric except for supersonic jet.

𝑈𝑛𝑖𝑓𝑜𝑟𝑚𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑓𝑜𝑟𝑐𝑒

The angular momentum equation • Here, the Reynolds transport theorem is applied to angular

momentum equation , where is the extensive property and the intensive property,

• As the rate of angular momentum of the system is equivalent to the net moment about some fixed point O, then

• For fixed CV

• For one dimensional inlet and exit

The Energy Equation • The Reynolds transport theorem also applies for Energy. The

extensive property is , and the intensive property is specific energy, .

• The rate of energy transfer of the system is equal to the rate of work and heat transfer • A system of fluid possesses energy in different forms.

Chemical rxn, Electrostatic, magnetic field etc..

• Work done on a system of fluid is by shaft, by pressure and viscous stresses on surface boundaries. If the surface is part of a machine, the pressure work is accounted in the shaft work.

• Pressure work

• Viscous stress work

• For a fixed CV

• Taking the pressure work term () to the right side, the energy equation takes its most convenient general form for a fixed CV

• One dimensional energy flux

• For steady flow one dimensional inlet and outlet, the energy equation becomes:

• But mass is conserved for a system, therefore:

• The above energy equation is utilized for flow through pipes with no shaft work, negligible viscous work and heat transfer.

Note: Doing work against friction increases internal energy. Therefore, the internal energy term in the energy equation is accounted in the friction head term, as shown in the above equation.

Bernoulli’s Principle • This principle was merely stated by Daniel Bernoulli and later derived

by Leonard Euler. It is the relation b/n pressure, velocity and head in steady, frictionless, incompressible and stream line flow. • We can see Bernoulli’s equation in action in Venturi meter, Nozzles

and diffusers, Flow over airfoil blades (e.g.: - airplane wing), etc…• The equation can be directly derived or simplified from the energy

equation.