Page 235 - The Mechatronics Handbook

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where v denotes the specific volume (the reciprocal of density) and the subscript n has been dropped
from velocity for simplicity.

Energy Balance

Energy is a fundamental concept of thermodynamics and one of the most significant aspects of engi-
neering analysis. Energy can be stored within systems in various macroscopic forms: kinetic energy,
gravitational potential energy, and internal energy. Energy also can be transformed from one form to
another and transferred between systems. Energy can be transferred by work, by heat transfer, and by
flowing matter. The total amount of energy is conserved in all transformations and transfers. The extensive
property balance expressing the conservation of energy principle takes the form

2
2
(
dU + KE + PE) Q W + ∑ m ˙ h i + ---- + gz i ∑ m ˙ h e + ---- + 



--------------------------------------- =
˙
v i
˙
v e
dt – i  2  – e  2 gz e  (12.7a)
i e
where U, KE, and PE denote, respectively, the internal energy, kinetic energy, and gravitational potential
energy of the overall control volume.
The right side of Eq. (12.7a) accounts for transfers of energy across the boundary of the control volume.
Energy can enter and exit control volumes by work. Because work is done on or by a control volume
when matter flows across the boundary, it is convenient to separate the work rate (or power) into two
contributions. One contribution is the work rate associated with the force of the fluid pressure as mass
is introduced at the inlet and removed at the exit. Commonly referred to as flow work, this contribution
is accounted for by m ˙ (p i v i ) and m ˙ (p e v e ) , respectively, where p denotes pressure and v denotes specific
i
e
volume. The other contribution, denoted by W ˙ in Eq. (12.7a), includes all other work effects, such as
˙
those associated with rotating shafts, displacement of the boundary, and electrical effects. W is considered
positive for energy transfer from the control volume.
Energy also can enter and exit control volumes with flowing streams of matter. On a one-dimensional
flow basis, the rate at which energy enters with matter at inlet i is m ˙ (u i + v i /2 + gz i ) , where the three
2
i
terms in parentheses account, respectively, for the specific internal energy, specific kinetic energy, and
specific gravitational potential energy of the substance flowing through port i. In writing Eq. (12.7a) the
sum of the specific internal energy and specific flow work at each inlet and exit is expressed in terms of
the specific enthalpy h(=u + pv). Finally, Q ˙ accounts for the rate of energy transfer by heat and is
considered positive for energy transfer to the control volume.
By dropping the terms of Eq. (12.7a) involving mass flow rates an energy rate balance for closed
systems is obtained. In principle the closed system energy rate balance can be integrated for a process
between two states to give the closed system energy balance:
( U 2 – U 1 ) + ( KE 2 – KE 1 ) + ( PE 2 – PE 1 ) = QW
–
(12.7b)
(closed systems)
where 1 and 2 denote the end states. Q and W denote the amounts of energy transferred by heat and
work during the process, respectively.

Entropy Balance
Contemporary applications of engineering thermodynamics express the second law, alternatively, as an
entropy balance or an exergy balance. The entropy balance is considered here.
Like mass and energy, entropy can be stored within systems and transferred across system boundaries.
However, unlike mass and energy, entropy is not conserved, but generated (or produced) by irreversibilities

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