MASS, MOMENTUM , AND ENERGY EQUATIONS
INTRODUCTION: The mass equation is an expression of the conservation of mass principle. The Bernoulli equation is concerned with the conservation of kinetic, potential, and flow energies of a fluid stream and their conversion to each other in regions of flow where net viscous forces are negligible and where other restrictive conditions apply. The energy equation is a statement of the conservation of energy principle.
In fluid mechanics, it is found convenient to separate mechanical energy from thermal energy and to consider the conversion of mechanical energy to thermal energy as a result of frictional effects as mechanical energy loss. Then the energy equation becomes the mechanical energy balance.
Conservation of Mass: The conservation of mass relation for a closed system undergoing a change is expressed as msys = constant or dmsys /dt= 0, which is a statement of the obvious that the mass of the system remains constant during a process.
For a control volume (CV) or open system, mass balance is expressed in the rate form as
Where minand moutare the total rates of mass flow into and out of the control volume, respectively, and dmCV/dt is the rate of change of mass within the control volume boundaries.
In fluid mechanics, the conservation of mass relation written for a differential control volume is usually called the continuity equation.
Conservation of Mass Principle: The conservation of mass principle for a control volume can be expressed as: The net mass transfer to or from a control volume during a time interval t is equal to the net change (increase or decrease) in the total mass within the control volume during t. That is,
Where ∆mCV = mfinal – minitialis the change in the mass of the control volume during the process. It can also be expressed in rate form as
Equations above are often referred to as the mass balance and are applicable to any control volume undergoing any kind of process.
Mass Balance for Steady-Flow Processes: During a steady-flow process, the total amount of mass contained within a control volume does not change with time (mCV= constant). Then the conservation of mass principle requires that the total amount of mass entering a control volume equal the total amount of mass leaving it.
When dealing with steady-flow processes, we are not interested in the amount of mass that flows in or out of a device over time; instead, we are interested in the amount of mass flowing per unit time, that is, the mass flow rate It states that the total rate of mass entering a control volume is equal to the total rate of mass leaving it.
Many engineering devices such as nozzles, diffusers, turbines, compressors, and pumps involve a single stream (only one inlet and one outlet).
For these cases, we denote the inlet state by the subscript 1 and the outlet state by the subscript 2, and drop the summation signs.