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

System

Definition

• System: A quantity of matter in space which is analyzed during a problem.
• Surroundings: Everything external to the system.
• System Boundary: A separation present between system and surrounding.

Classification of the system boundary:-

• Real solid boundary
• Imaginary boundary

The system boundary may be further classified as:-

• Fixed boundary or Control Mass System
• Moving boundary or Control Volume System

The choice of boundary depends on the  problem being analyzed.

  Fig 9.1   System and Surroundings

Classification of Systems


 

Types of System

    Control Mass System (Closed System)

• Its a system of fixed mass with fixed identity. 
• This type of system is usually referred to as "closed system".
• There is no mass transfer across the system boundary
• Energy transfer may take place into or out of the system. 

       
Fig 9.2   A Control Mass System or Closed System

   
Control Volume System (Open System)

• Its a system of  fixed volume.

• This type of system is usually referred to as "open system” or a "control volume"

• Mass transfer can take place across a control volume.

• Energy transfer may also occur into or out of the system.

• A control volume can be seen as  a fixed region across which mass and energy transfers are studied. 

• Control Surface- Its the boundary of a control volume across which the transfer of both mass and energy takes place.

• The mass of a control volume (open system) may or may not be fixed.

• When the net influx of mass across the control surface equals  zero then the mass of the system is fixed and vice-versa.

• The identity of mass in a control volume always changes unlike the case for a control mass system (closed system).

• Most of the engineering devices, in general, represent an open system or control volume.
  
  Example:-

• Heat exchanger - Fluid enters and leaves the system continuously with the transfer of heat across the system boundary. 

• Pump - A continuous flow of fluid takes place through the system with a transfer of mechanical energy from the surroundings to the system.

       
Fig 9.3  A Control Volume System or Open System

Isolated System

• Its a system of fixed mass with same identity and fixed energy.

• No interaction of mass or energy takes place between the system and the surroundings.

• In more informal words an isolated system is like a closed shop amidst a busy market.

       
Fig 9.4   An Isolated System
 

Conservation of Mass - The Continuity Equation
 Law of conservation of mass

The law states that mass can neither be created nor be destroyed. Conservation of mass is inherent to a control mass system (closed system)

• The mathematical expression for the above law is stated as:
                                   ∆m/∆t = 0,    where m = mass of the system

• For a control volume (Fig.9.5), the principle of conservation of mass is stated as

Rate at which mass enters = Rate at which mass leaves the region + Rate of accumulation of mass in the region

OR

Rate of accumulation of mass in the control volume 
                              + Net rate of mass efflux from the control volume = 0      (9.1)

Continuity equation
The above statement  expressed analytically in terms of velocity and density field of a flow is known as theequation of continuity.

     Fig 9.5    A Control Volume in a Flow Field


 Continuity Equation - Differential Form
  Derivation

1. The point at which the continuity equation has to be derived, is enclosed by an elementary control volume.

2. The influx, efflux and the rate of accumulation of mass is calculated across each surface within the control volume.

                      
Fig 9.6   A Control Volume Appropriate to a Rectangular Cartesian Coordinate System

Consider a rectangular parallelopiped in the above figure as the control volume in a rectangular cartesian frame of coordinate axes.

• Net efflux of mass along x -axis must be the excess outflow over inflow across faces normal to x -axis.
• Let the fluid enter across one of such faces ABCD with a velocity u and a density ρ.The velocity and density with which the fluid will leave the face EFGH will be    respectively (neglecting the higher order terms in δx).
• Therefore, the rate of mass entering the control volume through face ABCD = ρu dy dz. 
• The rate of mass leaving the control volume through face EFGH will be
  
  
  
    (neglecting the higher order terms in dx)
   
• Similarly influx and efflux take place in all y and z directions also.

• Rate of accumulation for a point in a flow field
  

• Using, Rate of influx = Rate of Accumulation + Rate of Efflux
  

• Transferring everything to right side

                 (9.2)

This is the Equation of Continuity for a compressible fluid in a rectangular cartesian coordinate system.


Continuity Equation - Vector Form

• The continuity equation can be written in a vector form as

       

where     is the velocity of the point

• In case of a steady flow,

• Hence Eq. (9.3) becomes
  
       (9.4)
   
• In a rectangular cartesian coordinate system
                  (9.5)
   

• Equation (9.4) or (9.5) represents the continuity equation for a steady flow.

• In case of an incompressible flow,
  
  ρ = constant

• Hence,
       

•  Moreover
  
  
  
  Therefore, the continuity equation for an incompressible flow becomes
  
              (9.6)
  
            (9.7)

• In cylindrical polar coordinates eq.9.7 reduces to
  
   
• Eq. (9.7) can be written in terms of the strain rate components as

                       (9.8)

Continuity Equation - A Closed System Approach
We know that the conservation of mass is inherent to the definition of a closed system as Dm/Dt = 0 (where m is the mass of the closed system).
However, the general form of continuity can be derived from the basic equation of mass conservation of a system.

Derivation :-
Let us consider an elemental closed system of volume V and density ρ.


Now    (dilation per unit volume)


 

In vector notation we can write this as

The above equations are same as that formulated from Control Volume approach.