The Control Volume and Mass Conservation
INTRODUCTION:The Eulerian method focuses on a defined area or location to find the needed information. The use of the Eulerian methods leads to a set differentiation equations that are referred to as Navier–Stokes equations which are commonly used. These differential equations will be used in the later part of this book. Additionally, the Eulerian system leads to integral equations which are the focus of this part of the book. The Eulerian method plays well with the physical intuition of most people. This methods has its limitations and for some cases the Lagrangian is preferred (and sometimes the only possibility). Therefore a limited discussion on the Lagrangian system will be presented (later version).
Lagrangian equations are associated with the system while the Eulerian equation is associated with the control volume. The difference between the system and the control volume is shown in Figure? The green lines in Figure?? Represent the system. The red dotted lines are the control volume. At certain time the system and the control volume are identical location. After a certain time, some of the mass in the system exited the control volume which are marked “an” in Figure? The material that remained in the control volume is marked as “b”. At the same time, the control gains some material which is marked as “c”.
Control Volume: The Eulerian method requires defining a control volume (some time more than one).The control volume is a defined volume that was discussed earlier. The control volumes differentiated into two categories of control volumes, non–deformable and deformable in the case where no mass crosses the boundaries, the control volume is a system. Every control volume is the focus of the certain interest and will be dealt with the basic equations, mass, momentum, energy, entropy etc.
Continuity Equation: the conservation equations will be applied the control volume. In this chapter, the mass conservation will be discussed. The system mass change is
The system mass after some time, according Figure? Is made of msys= mc.v. ma− mc
The change of the system mass is by definition is zero. The change with time (time derivative of equation results in
The first term in equation is the derivative of the mass in the control volume and at any given time is
And is a function of the time. The interface of the control volume can move. The actual velocity of the fluid leaving the control volume is the relative velocity (see Figure). The relative velocity is
Where Uf is the liquid velocity and Ub is the boundary velocity (see Figure). The velocity component that is perpendicular to the surface is
Where ˆn is a unit vector perpendicular to the surface. The convention of direction is taken positive if flow out the control volume and negative if the flow is into the control volume. The mass flow out of the control volume is the system mass that is not included in the control volume. Thus, the flow out is
It has to be emphasized that the density is taken at the surface thus the subscripts. In the same manner, the flow rate in is
It can be noticed that the two equations are similar and can be combined, taking the positive or negative value of Urn with integration of the entire system as
Applying negative value to keep the convention.