When energy crosses the boundary of a system, the energy in the system changes by an equal amount. Thus a decrease in the energy of a system equals the increase in energy of the surroundings and vice versa.
Energy can cross a system
boundary in the form of heat transfer (to or from the system), in the form of
work (done by or on the system), or by mass entering or leaving. A process
during which no heat is transferred to or from a system is known as an
adiabatic process. The law of conservation of energy states that for a system
of particles
E2 – E1
= Q͂ - Wʹ
Or
dE = d Q͂ - dWʹ
Where:
- E2 – E1: is the change in energy experienced by the system
- Q͂: is the heat added to the system
- Wʹ: is work done by the system
The total energy is a property of the system and includes
internal (U), kinetic (KE), and potential energies (PE):
E = U + KE + PE
If we express this on a per-unit-mass basis, we obtain:
Heat and work are not properties
of the system, but they represent energy transfers moving across the system
boundaries in specific forms.
Remember the general conservation
expression:
If we substitute, N = E and n = e,
we obtain:
Equation 1 is generally known as the first law of thermodynamics and is a statement of the conservation of energy.
The Wʹ term consists of all forms of work crossing the boundary. This includes shaft work, electric and magnetic work, viscous shear work, and flow work. The last of these forms represents work done by the fluid system in pushing mass into or out of the control volume. Thus:
where dWf /dt is flow
work per unit time.
To develop an expression for dWf/dt, we can consider an element of mass dm leaving a control volume through an area dA, as shown below:
The fluid in the control volume must work against the external pressure p to move dm out. Since work = force x distance, we have:
Similarly, for an element of mass entering, work is being done on the system. Therefore, we get:
Combining the previous two equations, we can obtain the flow work as:
If we substitute Equations 3 and 2 into Equation 1 we get:
Combining the integral terms results in:
Enthalpy is a property defined as h = u + p/ρ; the conservation of energy equation now becomes:
For a steady, one-dimensional flow, the energy equation becomes:
For many incompressible flows, Q͂ is usually assumed to be zero, and changes in internal energy are frequently negligible. Thus equation 5 can be simplified into:
For compressible flows, however,
the change in internal energy is significant and cannot be neglected.
