Irreversible Effects

We have noted that if there is a significant change in temperature, the thermal energy terms (i.e., q and u) may represent much more energy than the mechanical terms (i.e., pressure, potential and kinetic energy, and work). On the other hand, if the temperature difference between the system and its surroundings is very small, the only source of "heat" (thermal energy) is the internal (irreversible) dissipation of mechanical energy into thermal energy, or "friction." The origin of this "friction loss" is the irreversible work required to overcome intermolecular forces, i.e., the attractive forces between the ''fluid elements,'' under dynamic (nonequilibrium) conditions. This can be quantified as follows.

For a system at equilibrium (i.e., in a reversible or ''static'' state), thermodynamics tells us that du = T ds — P d(1/p) and T ds = Sq (5-28)

That is, the total increase in entropy (which is a measure of ''disorder'') comes from heat transferred across the system boundary (Sq). However, a flowing fluid is in a ''dynamic,'' or irreversible, state. Because entropy is proportional to the degree of departure from the most stable (equilibrium) conditions, this means that the further the system is from equilibrium, the greater the entropy, so for a dynamic (flow) system

where Sef represents the ''irreversible energy'' associated with the departure of the system from equilibrium, which is extracted from mechanical energy and transformed (or ''dissipated'') into thermal energy. The farther from equilibrium (e.g., the faster the motion), the greater this irreversible energy. The origin of this energy (or ''extra entropy'') is the mechanical energy that drives the system and is thus reduced by ef. This energy ultimately appears as an increase in the temperature of the system (Su), heat transferred from the system (Sq), and/or expansion energy [P d(1/p)] (if the fluid is compressible). This mechanism of transfer of useful mechanical energy to low grade (non-useful) thermal energy is referred to as ''energy dissipation.'' Although ef is often referred to as the friction loss, it is evident that this energy is not really lost, but is transformed (dissipated) from useful high level mechanical energy to non-useful low grade thermal energy. It should be clear that ef must always be positive, because energy can be transformed spontaneously only from a higher state (mechanical) to a lower state (thermal) and not in the reverse direction, as a consequence of the second law of thermodynamics.

When Eq. (5-30) is introduced into the definition of enthalpy, we get iP\ dP

Substituting this for the enthalpy in the differential energy balance, Eq. (5-11), gives

This can be integrated along a streamline from the inlet to the outlet of the system to give dp 1

where, from Eq. (5-30), ef = ("o — "i) — q +

Equations (5-33) and (5-34) are simply rearrangements of the steady-state energy balance equation [Eq. (5-10)], but are in much more useful forms. Without the friction loss (ef) term (which includes all of the thermal energy effects), Eq. (5-33) represents a mechanical energy balance, although mechanical energy is not a conserved quantity. Equation (5-33) is referred to as the engineering Bernoulli equation or simply the Bernoulli equation. Along with Eq. (5-34), it accounts for all of the possible thermal and mechanical energy effects and is the form of the energy balance that is most convenient when mechanical energy dominates and thermal effects are minor. It should be stressed that the first three terms in Eq. (5-33) are point functions—they depend only on conditions at the inlet and outlet of the system—whereas the w and ef terms are path functions, which depend on what is happening to the system between the inlet and outlet points (i.e., these are rate-dependent and can be determined from an appropriate rate or transport model).

If the fluid is incompressible (constant density), Eq. (5-33) can be written

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