Ancillary Concepts

Many important applications involve single-inlet, single-outlet control vol- umes at steady state. Several concepts related to this class of applications are now presented.

Efficiencies. Consider first the case of an adiabatic control vol- ume: a control volume for which there is no heat transfer. As there is a single inlet and a single outlet, Equation reduces to = Denoting the common mass flow rate by m, Equation gives

(2.24) - - (no heat transfer)

m

Accordingly, when irreversibilities are present within the control volume, the specific entropy increases as mass flows from inlet to outlet. In the ideal case in which no internal irreversibilities are present, mass passes through the control volume with no change in its entropy-that is, isentropically.

Isentropic efficiencies for turbines, compressors, and pumps involve a com- parison between the actual performance of a device and the performance that would be achieved under idealized circumstances for the same inlet state and the same outlet pressure, with heat transfer between the device and its sur- roundings not occurring to any significant extent. The isentropic turbine

compares the actual turbine power to the power that would be developed in an isentropic expansion from the specified inlet state to the specified outlet pressure, ( :

(2.25)

The isentropic compressor compares the actual power input to the power that would be required in an isentropic compression from the spec- ified inlet state to the specified outlet pressure:

(2.26)

2.2 CONTROL VOLUME CONCEPTS 61

An isentropic pump is defined similarly.

Special Cases. Next, consider the control volume of Figure 2.1 where heat transfer occurs only at the temperature Denoting the common mass flow rate by Equations and read, respectively

= Q,, -

+

+

0 = -

+

+

Eliminating the heat transfer term from these expressions, the work developed per unit of mass flowing through the control volume is

= - - - -

+

-

m m

The term in brackets is fixed by the states at the control volume inlet and outlet and the temperature at which heat transfer occurs. Accordingly, an expression for the maximum value of the work developed per unit of mass flowing through the control volume, corresponding to the absence of internal irreversibilities, is obtained when the entropy generation term is set to zero:

Figure 2.1 One-inlet, one-outlet control volume at steady state.

62 THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

-

+ +

int rev

(all heat transfer at temperature This expression is applied in Section 3.3.2.

Using similar reasoning, expressions can be developed for the energy trans- fers by heat and work for a single-inlet, single-outlet control volume at steady state in the absence of internal irreversibilities, but without the restriction that heat transfer occurs only at the temperature For such cases, the heat trans- fer per unit of mass flow is

= T

int rev

(2.28)

Combining Equations 2.28, and (from Section 2.3.1) gives the work developed per unit of mass flow:

(heat transfer according to Equation 2.28)

The integrals of Equations 2.28 and 2.29 are performed from inlet to outlet, denoted I and 2, and the subscripts int rev stress that the expressions apply only to control volumes in which there are no internal irreversibilities. The integral of Equation 2.29 requires a relationship between pressure and specific volume. An example is the expression = const, where the value of is a constant. An internally reversible process described by such an expression is called a polytropic process and n is the polytropic exponent. When the exponent is determined for an actual process by fitting pressure-specific volume data, the use of pv" = const to calculate the integral in Equation 2.29 can yield a plausible approximation of the work per unit of mass flow in the actual process.

Head Loss. Equation 2.29 is a special case of the generalized Bernoulli equation

(2.30)

dp

+

+

2.2 CONTROL VOLUME CONCEPTS 63 where denotes the work developed by the control volume per unit of mass flowing, and accounts for the rate at which mechanical energy is irreversibly converted to internal energy. The term h, is known by various names, including head loss, friction, friction work, and energy dissipation. In the absence of irreversibilities, such as friction, h, vanishes and Equation 2.30 reduces to give Equation 2.29.

The term h, of Equation 2.30 can be obtained experimentally by measuring all the other terms. The accumulated experimental data have been organized to allow estimation of the term for use in designing piping systems. Thus, when h, is known, Equation 2.30 can be used to find some other quantity, such as the work required for pumping a liquid or the volumetric flow rate when the work is known. The term h, is conventionally formulated as the sum of two contributions, one associated with wall friction over the length of the conduit carrying the liquid or gas and the other associated with flow through resistances such as valves, elbows, tees, and pipe entrances and exits. Thus, for a piping system consisting of several resistances and several lengths L of pipe each with diameter D we have denotes viscosity. As discussed further in Section 2.6.2, the function

is available both graphically and analytically. The loss K for resistances of practical interest are also available from the engineering literature (also see Section 2.6.2).

Momentum Equation. Like mass, energy, and entropy, linear momentum is carried into and out of a control volume at the inlet and outlet, respectively.

Such transfers are accounted for by expressions of the form

=

mv

64 THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

1

contained within the control volume

resultant external force control volume linear momentum] acting on the

net rate at which linear momentum is transferred

into the control volume accompanying mass flow

At steady state, the total amount of linear momentum contained in the control volume is constant with time. Accordingly, when applying Newton’s second law of motion of control volumes at steady state, it is necessary to consider only the momentum accompanying the incoming and outgoing streams of matter and the forces acting on the control volume. Newton’s law then states that the resultant external force F acting the control volume equals the difference between the rates of momentum associated with mass flow exiting and entering the control volume:

F - (2.32)

where 1 denotes the inlet and 2 the outlet. The resultant force includes the forces due to pressure acting at the inlet and outlet, forces acting on the portion of the boundary through which there is no mass flow, and the force of gravity. Applications of Equation 2.32 are provided elsewhere 3,