Energy Equation: General Form http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771...
7.2 Energy Equation: General Form
This section derives the energy equation for a control volume by applying the Reynolds transport theorem to the
system equation. The energy equation for a system is (1, 2):
(7.4)
Equation 7.4, also called the first law of thermodynamics, can be stated in words:
Recall that a system is a body of matter that is under consideration. By definition, a system always contains the
same matter. An imaginary boundary separates the system from all other matter, which is called the environment
(or surroundings).
Equation 7.4 involves sign conventions. Thermal energy is positive when there is an addition of thermal energy
to the system and negative when there is a removal. Work is positive when the system is doing work on the
environment and negative when work is done on the system.
To extend Eq. 7.4 to a control volume, apply the Reynolds transport theorem Eq. (5.21). Let the extensive
property be energy (Bsys = E) and let b = e to obtain
(7.5)
where e is energy per mass in the fluid. Let e = ek + ep + u where ek is the kinetic energy per unit mass, ep is the
gravitational potential energy per unit mass, and u is the thermal energy (or internal energy) per unit mass.
(7.6)
To simplify Eq. 7.6, let*
(7.7)
Similarly, let
(7.8)
where z is the elevation measured relative to a datum. When Eqs. 7.7 and 7.8 are substituted into Eq. 7.6, the
result is
(7.9)
Shaft and Flow Work
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Work is classified into two categories:
Each work term involves force acting over a distance. When this force is associated with a pressure distribution,
then the work is called flow work. Alternatively, shaft work is any work that is not associated with a pressure
force. Shaft work is usually done through a shaft (from which the term originates) and is commonly associated
with a pump or turbine. According to the sign convention for work (see p. 219), pump work is negative.
Similarly, turbine work is positive. Thus,
(7.10)
To derive an equation for flow work, use the idea that work equals force times distance. For example, Fig. 7.3
defines a control volume that is situated inside a converging pipe. At section 7.2, the fluid that is inside the
control volume will push on the fluid that is outside of the control volume. The magnitude of the pushing force
is p2A2. During a time interval "t, the displacement of the fluid at section 7.2 is "x2 = V2"t. Thus, the amount of
work is
(7.11)
Figure 7.3 Sketch for deriving flow work.
Convert the amount of work given by Eq. 7.11 into a rate of work:
(7.12)
This work is positive because the fluid inside the control volume is doing work on the environment. In a similar
manner, the flow work at section 7.1 is negative and is given by
The net flow work for the situation pictured in Fig. 7.3 is
(7.13)
Equation 7.13 can be generalized to a situation involving multiple streams of fluid passing across a control
surface:
(7.14)
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To develop a general form of flow work, use integrals to account for velocity and pressure variation across the
control surface. Also, use the dot product to account for flow direction. The general equation for flow work is
(7.15)
In summary, the work term is the sum of flow work [Eq. 7.15] and shaft work [Eq. 7.10]:
(7.16)
Final Steps in the Derivation of the Energy Equation
Introduce the work term from Eq. 7.16 into Eq. 7.9 and let :
(7.17)
In Eq. 7.17, combine the last term on the left side with the last term on the right side:
(7.18)
Replace p/Á + u by the specific enthalpy, h. The integral form of the energy principle is:
(7.19)
If the flow crossing the control surface occurs through a series of inlet and outlet ports and if the velocity V is
uniformly distributed across each port, then a simplified form of the Reynolds transport theorem, Eq. (5.21), can
be used to derive the following form of the energy equation:
(7.20)
where the subscripts o and i refer to the outlet and inlet ports, respectively.
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