Modern Engineering Thermodynamics

102 CHAPTER 4: The First Law of Thermodynamics and Energy Transport Mechanisms
or
The energy rate balance is
or
E G = E T (4.1)
_E G = _E T + _E P ðas required by the first lawÞ
_E G = _E T (4.2)
From now on, we frequently use the phrases energy balance and energy rate balance in identifying the proper
equation to use in an analysis. So, for simplicity, we introduce the following abbreviations:
and
EB = energy balance
ERB = energy rate balance
In Chapter 3, we introduce the components of the total system energy E as the internal energy U, the kinetic
energy mV 2 /2g c , and the potential energy mgZ/g c ,or 3
E = U + mV 2
2g c
+ mgZ
g c
(3.9)
In this equation, V is the magnitude of the velocity of the center of mass of the entire system, Z is the height of
the center of mass above a ground (or zero) potential datum, and g c is the dimensional proportionality factor
(see Table 1.2 of Chapter 1). In Chapter 3, we also introduce the abbreviated form of this equation:
E = U + KE + PE (3.10)
and similarly for the specific energy e,
and
e = E m = u + V 2
2g c
+ gZ
g c
(3.12)
e = u + ke + pe (3.13)
In these equations, we continue the practice introduced in Chapter 2 of using uppercase letters to denote extensive
properties and lowercase letters to denote intensive (specific) properties. The energy concepts described in these
equations are illustrated in Figure 4.1.
Center of
mass
Velocity V
In equilibrium thermodynamics, the proper energy balance is given by Eq. (4.1),
where the gain in energy E G is to be interpreted as follows. The system is initially
in some equilibrium state (call it state 1), and after the application of some “process,”
the system ends up in a different equilibrium state (call it state 2). If we
now add a subscript to each symbol to denote the state at which the property is
to be evaluated (E 1 is the total energy of the system in state 1 and so forth), then
we can write the energy gain of the system as
Height = Z
Internal
energy
U
FIGURE 4.1
System energy components.
Z = 0
System (either
open or closed)
System boundary
or
E G = Final total energy − Initial total energy (4.3)
E G = E 2 − E 1 (4.4)
and extending this to Eq. (3.9), we obtain
E G = U 2 − U 1 + m
ðV2 2 2g − V2 1 Þ + mg
c
g
ðZ 2 − Z 1 Þ (4.5)
c
or
E G = m u 2 − u 1 + V 2 2 − V2 1
+ g
2g c
g
ðZ 2 − Z 1 Þ
(4.6)
c
3 In this text, we use the symbol V to represent the magnitude of the average velocity |V|, and the symbol V to represent volume.