## Av

In the following development, only steady-state operation is treated, so m is constant in time and the same at the inlet and exit ports. These are designated by subscripts i and e, respectively. However, the quantities on the right hand side of Eq (4.7) at the inlet may be different from those at the exit port.

Application of the First law to the device illustrated in Fig. 4.9 introduces several features not involved in the First-law treatment of closed systems. These are:

1.The internal energy (U or u) that appears in equations such as (1.4) and (3.2) is supplemented by the kinetic energy of the moving fluid, V2. Changes in gravitational potential energy between the inlet and exit lines should also be added to the total energy, but for simplicity, this component of the energy is not treated here.

2. The internal energy convected through the boundaries of the open system are included in writing the First law.

3. In addition to shaft work, pressure-volume work is inherently involved in pushing the fluid into the inlet and out of the exit line. This work component is called the flow work Wf 1. The total work performed by the system is where Ws is the rate at which the system performs shaft work, and the terms in parentheses are the inlet and outlet flow work rates. The sign convention is that of the First law for closed systems: work is positive if performed by the system and heat is positive if added to the system.

The flow work terms are of the pressure-volume type because they involve force acting over a distance, or, equivalently, pressure displacing volume. Thus the rate of performing flow work is the pressure times the volume flow rate, or where Eq (4.7) has been used to replace the volume flow rate in terms of the mass flow rate.

The First law for the steady-state open system equates the net rate of energy transport across system boundaries at the flow ports to the net rate of energy input in the form of heat and work exchanged with the surroundings:

1 Strictly speaking, the work W with a dot over it is a work rate, or power, but to avoid confusion of terminology, it will be referred to as simply work.

w fi = piAivi = m pivi and W fe = peAeVe = m pevt e

1 Strictly speaking, the work W with a dot over it is a work rate, or power, but to avoid confusion of terminology, it will be referred to as simply work.

Note that the properties of the fluid inside the system do not enter the statement of the First law. The reason for this absence is the restriction to steady-state operation.

An important simplification of the above equation is obtained by combining the internal energy terms with the flow-work terms. The combination u + pv is the enthalpy of the fluid, h, so the First law becomes:

Dividing Eq (4.8) by the mass flow rate converts the heat and shaft work terms to a per-unit-mass basis rather than a per-unit-time basis:

4.5.2 The Second Law for open systems

Only in some cases does the Second law provide an equation that is as universally useful as is the First law in the form of Eq (4.9). The reason is that irreversibilities in the device create entropy that cannot be quantitatively determined from thermodynamics. The Second law can be applied to the open system in two ways. Steady-state flow through the device is assumed so that the entropy and heat can be expressed on a per-unit-mass basis.

Second law in the form of Eq (1.10) relates difference in entropy (per unit mass of the flowing fluid) between the outlet and inlet conditions to the heat added to the system:

In the event that the temperature varies with location in the device, the right hand side of Eq (4.10) is replaced by ^Sqj /Tj, the summation of increments of heat Sqj added at local fluid temperature Tj. However, the single-temperature version is used here for simplicity.

An alternate method of applying the Second law to the open system is via the total entropy change version of Eq (1.13). This involves the entropy changes of the surroundings, Assurr = -q/Tsurr, where the minus sign appears because heat addition to the system (q) is heat removal from the surroundings. Using the equality in Eq (4.10) for As of the system (the moving unit mass of fluid), the total system + surroundings entropy change is:

Astotal = As + Assurr = se - si - q/Tsurr > 0 (4.11)

The distinction between Eqs (4.10) and (4.11) lies in the level of irreversibility present. If the process is internally reversible (Sect. 1.7), the equality in Eq (4.10) applies. However, internal reversibility does not guarantee total reversibility; if the heat q is transferred over a nonzero AT, an external irreversibility is present. This is revealed by substituting (from Eq (4.10)) q/T for se - sI in Eq (4.11):

Only if the process is both internally reversible and externally reversible (T = Tsurr) is total reversibility achieved.

Essentially all common applications of the entropy balance are for devices such as pumps, turbines, nozzles, valves and in-line flow components. A subset of these in-line flow devices operate in a nearly adiabatic manner (q = 0). If they also function reversibly, then Eq (4.8) becomes:

Equations (4.11) and (4.12) are the Second-law analogs of the First law expressed by Eq (4.9). To the extent that the device can be assumed to operate reversibly, the Second law provides another equation for thermodynamic analysis of the open system.

### 4.4.3 Reversible Work of a Flow System

If a flow device operates reversibly, the minimum work needed for the task can be determined by application of the First and Second laws. The First law gives:

where ws>rev is the reversible shaft work. The differential of Eq (1.16a), dh = Tds +vdp, can be integrated to yield:

Because the process is reversible, the Tds integral is the heat added during the process. From the above two equations, the reversible work is:

surr surr

This formula is the open-system analog of the pV work integral in a closed system (Eq (3.1)). The work done by the 100% efficient turbine can be calculated from Eq (4.13) by expressing v as a function of p at constant s using the steam tables. However, if the fluid is a liquid, the specific volume v is approximately independent of pressure and the reversible work simplifies to: 