Integral analysis depends on analyzing finite control volumes (CV): what flows in and out of a control volume and what is generated within the CV. The surface of a control volume is termed the control surface (CS).
Reynold’s Transport Theorem
Reynold’s Transport Theorem states that the rate of change for any extensive fluid flow property in a given system:
\displaystyle\frac{dN}{dt}=\dfrac{\partial}{\partial t}\iiint_{CV}(\rho\eta)d V\llap{—}+\iint_{CS}(\rho \eta\overrightarrow{\bm{V}})\cdot d\overrightarrow{\bm{A}}
An extensive property varies proportional to the size of the system, properties such as energy, mass, and momentum are examples of extensive properties.
The term in the above equation can be thought of as:
- \frac{dN}{dt}: Time rate of change of extensive property N in the system
- \frac{\partial}{\partial t}\iiint_{CV}(\rho\eta)d V\llap{—}: The storage of extensive property N within the control volume.
- \iint_{CS}(\rho \eta\overrightarrow{\bm{V}})\cdot d\overrightarrow{\bm{A}}: Transport of extensive property N in/out of the control volume.
Conservation of Mass
The integral equation for conservation of mass can be obtained by setting N=m and \eta=1.
\displaystyle\frac{dm}{dt}=\dfrac{\partial}{\partial t}\iiint_{CV}(\rho) d V\llap{—}+\iint_{CS}(\rho \overrightarrow{\bm{V}})\cdot d\overrightarrow{\bm{A}}
By definition, the mass of a system is steady and does not change, therefore:
\displaystyle 0=\dfrac{\partial}{\partial t}\iiint_{CV}(\rho) d V\llap{—}+\iint_{CS}(\rho \overrightarrow{\bm{V}})\cdot d\overrightarrow{\bm{A}}
Conservation of Momentum
The integral equation for conservation of momentum can be obtained by setting N=\overrightarrow{\bm{P}} and \eta=\overrightarrow{\bm{V}}.
\displaystyle\frac{d\overrightarrow{\bm{P}}}{dt}=\dfrac{\partial}{\partial t}\iiint_{CV}(\rho\overrightarrow{\bm{V}}) d V\llap{—}+\iint_{CS}(\rho \overrightarrow{\bm{V}})\overrightarrow{\bm{V}}\cdot d\overrightarrow{\bm{A}}
Since the time rate of change of momentum is force we can express it as the sum of surface and body forces acting upon the system.
\displaystyle\overrightarrow{\bm{F}}_B+\overrightarrow{\bm{F}}_S=\dfrac{\partial}{\partial t}\iiint_{CV}(\rho\overrightarrow{\bm{V}}) d V\llap{—}+\iint_{CS}(\rho \overrightarrow{\bm{V}})\overrightarrow{\bm{V}}\cdot d\overrightarrow{\bm{A}}