1.2 Conservation Principles
3
1.2 Conservation Principles
Conservation laws can be derived by considering a given quantity of matter or
control mass (CM) and its extensive properties, such as mass, momentum and
energy. This approach is used to study the dynamics of solid bodies, where the
CM (sometimes called the system) is easily identified. In fluid flows, however,
it is difficult to follow a parcel of matter. It is more convenient to deal with
the flow within a certain spatial region we call a control volume (CV), rather
than in a parcel of matter which quickly passes through the region of interest.
This method of analysis is called the control volume approach.
We shall be concerned primarily with two extensive properties, mass and
momentum. The conservation equations for these and other properties have
common terms which will be considered first.
The conservation law for an extensive property relates the rate of change
of the amount of that property in a given control mass to externally determined effects. For mass, which is neither created nor destroyed in the flows
of engineering interest, the conservation equation can be written:
On the other hand, momentum can be changed by the action of forces and
its conservation equation is Newton's second law of motion:
where t stands for time, m for mass, v for the velocity, and f for forces acting
on the control mass.
We shall transform these laws into a control volume form that will be used
throughout this book. The fundamental variables will be intensive rather than
extensive properties; the former are properties which are independent of the
amount of matter considered. Examples are density p (mass per unit volume)
and velocity v (momentum per unit mass).
If 4 is any conserved intensive property (for mass conservation, 4 = 1; for
momentum conservation, 4 = v ; for conservation of a scalar, 4 represents the
conserved property per unit mass), then the corresponding extensive property
@ can be expressed as:
where OcM stands for volume occupied by the CM. Using this definition,
the left hand side of each conservation equation for a control volume can be
written:'
This equation is often called control volume equation or the Reynolds' transport
theorem.
3
1.2 Conservation Principles
Conservation laws can be derived by considering a given quantity of matter or
control mass (CM) and its extensive properties, such as mass, momentum and
energy. This approach is used to study the dynamics of solid bodies, where the
CM (sometimes called the system) is easily identified. In fluid flows, however,
it is difficult to follow a parcel of matter. It is more convenient to deal with
the flow within a certain spatial region we call a control volume (CV), rather
than in a parcel of matter which quickly passes through the region of interest.
This method of analysis is called the control volume approach.
We shall be concerned primarily with two extensive properties, mass and
momentum. The conservation equations for these and other properties have
common terms which will be considered first.
The conservation law for an extensive property relates the rate of change
of the amount of that property in a given control mass to externally determined effects. For mass, which is neither created nor destroyed in the flows
of engineering interest, the conservation equation can be written:
On the other hand, momentum can be changed by the action of forces and
its conservation equation is Newton's second law of motion:
where t stands for time, m for mass, v for the velocity, and f for forces acting
on the control mass.
We shall transform these laws into a control volume form that will be used
throughout this book. The fundamental variables will be intensive rather than
extensive properties; the former are properties which are independent of the
amount of matter considered. Examples are density p (mass per unit volume)
and velocity v (momentum per unit mass).
If 4 is any conserved intensive property (for mass conservation, 4 = 1; for
momentum conservation, 4 = v ; for conservation of a scalar, 4 represents the
conserved property per unit mass), then the corresponding extensive property
@ can be expressed as:
where OcM stands for volume occupied by the CM. Using this definition,
the left hand side of each conservation equation for a control volume can be
written:'
This equation is often called control volume equation or the Reynolds' transport
theorem.