System
Definition
(i) System: A quantity of matter in space which is analyzed during a problem.
(ii) Surroundings: Everything external to the system.
(iii) System Boundary: A separation present between system and surrounding.
Classification of the system boundary:-
(i) Real solid boundary
(ii) Imaginary boundary
The system boundary may be further classified as:-
(i) Fixed boundary or Control Mass System
(ii) Moving boundary or Control Volume System
The choice of boundary depends on the problem being analyzed.
Classification of Systems
Types of System
Control Mass System (Closed System)
1. Its a system of fixed mass with fixed identity.
2. This type of system is usually referred to as "closed system".
3. There is no mass transfer across the system boundary.
4. Energy transfer may take place into or out of the system.
Control Volume System (Open System)
1. It’s a system of fixed volume.
2. This type of system is usually referred to as "open system” or a "control volume"
3. Mass transfer can take place across a control volume.
4. Energy transfer may also occur into or out of the system.
5. A control volume can be seen as a fixed region across which mass and energy transfers are studied.
6. Control Surface- Its the boundary of a control volume across which the transfer of both mass and energy takes place.
7. The mass of a control volume (open system) may or may not be fixed.
8. When the net influx of mass across the control surface equals zero then the mass of the system is fixed and vice-versa.
9. The identity of mass in a control volume always changes unlike the case for a control mass system (closed system).
10. Most of the engineering devices, in general, represent an open system or control volume.
Example:-
(i) Heat exchanger - Fluid enters and leaves the system continuously with the transfer of heat across the system boundary.
(ii) Pump - A continuous flow of fluid takes place through the system with a transfer of mechanical energy from the surroundings to the system.
Isolated System
(i) It’s a system of fixed mass with same identity and fixed energy.
(ii) No interaction of mass or energy takes place between the system and the surroundings.
(iii) In more informal words an isolated system is like a closed shop amidst a busy market.
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).
(i) The mathematical expression for the above law is stated as:
∆m/∆t = 0, where m = mass of the system
(ii) For a control volume (Fig.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 (1)
Continuity equation
The above statement expressed analytically in terms of velocity and density field of a flow is known as the equation of continuity.
Continuity Equation - Differential Form
Derivation
(i) The point, at which the continuity equation has to be derived, is enclosed by an elementary control volume.
(ii) The influx, efflux and the rate of accumulation of mass is calculated across each surface within the control volume.
Consider a rectangular parallelepiped in the above figure as the control volume in a rectangular cartesian frame of coordinate axes.
(i) Net efflux of mass along x -axis must be the excess outflow over inflow across faces normal to x -axis.
(ii) 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 
and 
respectively (neglecting the higher order terms in δx).
(iii) Therefore, the rate of mass entering the control volume through face ABCD = ρu dy dz.
(iv) The rate of mass leaving the control volume through face EFGH will be
(v) Similarly influx and efflux take place in all y and z directions also.
(vi) Rate of accumulation for a point in a flow field
(vii) Using, Rate of influx = Rate of Accumulation + Rate of Efflux
(viii) Transferring everything to right side
This is the Equation of Continuity for a compressible fluid in a rectangular Cartesian coordinate system.
Application of the Reynolds Transport Theorem to Conservation of Mass and Momentum
Conservation of mass The constancy of mass is inherent in the definition of a control mass system and therefore we can write
To develop the analytical statement for the conservation of mass of a control volume, the Eq. (1) is used with N = m (mass) and η = 1 along with the Eq. (3a).
This gives
The Eq. (3b) is identical and is the integral form of the continuity equation derived in earlier section. At steady state, the first term on the left hand side of Eq. (3b) is zero. Hence, it becomes
Conservation of Momentum or Momentum Theorem The principle of conservation of momentum as applied to a control volume is usually referred to as the momentum theorem.
Linear momentum The first step in deriving the analytical statement of linear momentum theorem is to write the Eq. (1) for the property N as the linear – momentum 
and accordingly η as the velocity 
. Then it becomes
The velocity defining the linear momentum in Eq. (4) is described in an inertial frame of reference.
Therefore we can substitute the left hand side of Eq. (4) by the external forces 
on the control mass system or on the coinciding control volume by the direct application of Newton’s law of motion. This gives
This Equation is the analytical statement of linear momentum theorem.
In the analysis of finite control volumes pertaining to practical problems, it is convenient to describe all fluid velocities in a frame of coordinates attached to the control volume. Therefore, an equivalent form of Eq.(4) can be obtained, under the situation, by substituting N as and accordingly η as 
, we get
With the help of the Eq. (2) the left hand side of Eq. can be written as
Where 
is the rectilinear acceleration of the control volume (observed in a fixed coordinate system) which may or may not be a function of time. From Newton’s law of motion
The Eq. (6) can be written in consideration of Eq. (7) as
At steady state, it becomes
In case of an inertial control volume (which is either fixed or moving with a constant rectilinear velocity), 
and hence Eqs (8a) and (8b) becomes respectively
and
The Eqs (8c) and (8d) are the useful forms of the linear momentum theorem as applied to an inertial control volume at unsteady and steady state respectively, while the Eqs (8a) and (8b) are the same for a non-inertial control volume having an arbitrary rectilinear acceleration.
In general, the external forces 
in Eqs (4, 8a to 8c) have two components - the body force and the surface force. Therefore we can write
Where 
is the body force per unit volume and 
is the area weighted surface force.
Angular Momentum
The angular momentum or moment of momentum theorem is also derived from Eq.(1) in consideration of the property N as the angular momentum and accordingly η as the angular momentum per unit mass. Thus,
where A Control mass system is the angular momentum of the control mass system. It has to be noted that the origin for the angular momentum is the origin of the position vector 
.
The term on the left hand side of Eq.(9) is the time rate of change of angular momentum of a control mass system, while the first and second terms on the right hand side of the equation are the time rate of increase of angular momentum within a control volume and rate of net efflux of angular momentum across the control surface.
The velocity 
defining the angular momentum in Eq.(9) is described in an inertial frame of reference. Therefore, the term d/dt(ACMS) can be substituted by the net moment ΣM applied to the system or to the coinciding control volume. Hence one can write Eq. (10.19) as
At steady state
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