TITULO TERCERO

energy balance in problems where a fluid is used for heat transfer and shaft work. They also demonstrate the use- fulness of the enthalpy property. However, as was pointed out, H1 and H2 are seldom both known. Addi-

tional information is frequently provided by the first and second laws of thermodynamics and their consequences. The first and second laws of thermodynamics

The first law of thermodynamics is based on the energy conservation expressed by Equation 6 and, by convention, relates the heat and work quantities of this equation to internally stored energy, u. Strictly speaking, Equation 6 is a complete form of the first law of thermodynamics. However, it is frequently use- ful to use the steady-state formulation provided in Equation 8 and further simplify this for the special case of 1) no change in potential energy due to grav- ity acting on the mass, and 2) no change in kinetic energy of the mass as a whole. In a closed system where only shaft work is permitted, these simplifying assumptions permit the energy Equation 8 for a unit mass to be reduced to:

∆u = q−

(

w J_k/

)

(22a)

or in differential form

du = δq−

(

δw J_k/

)

(22b)

The first law treats heat and work as being inter- changeable, although some qualifications must apply. All forms of energy, including work, can be wholly con- verted to heat, but the converse is not generally true. Given a source of heat coupled with a heat-work cycle, such as heat released by high temperature combus- tion in a steam power plant, only a portion of this heat can be converted to work. The rest must be rejected to an energy sink, such as the atmosphere, at a lower temperature. This is essentially the Kelvin statement of the second law of thermodynamics. It can also be shown that it is equivalent to the Clausius statement wherein heat, in the absence of external assistance, can only flow from a hotter to a colder body.

Concept and definition of entropy

Heat flow is a function of temperature difference. If a quantity of heat is divided by its absolute tem- perature, the quotient can be considered a type of dis- tribution property complementing the intensity fac- tor of temperature. Such a property, proposed and named entropy by Clausius, is widely used in ther- modynamics because of its close relationship to the second law.

Rather than attempt to define entropy (s) in an ab- solute sense, consider the significance of differences in this property given by:

S S S q T 2 1 1 2 − = =

_∫

× ∆

δ _{rev total system mass (23)}

where

∆S = change in entropy, Btu/R (J/K)

qrev = reversible heat flow between thermodynamic

equilibrium states 1 and 2 of the system, Btu/ lb (J/kg)

T = absolute temperature, R (K)

Entropy is an extensive property, i.e., a quantity of entropy, S, is associated with a finite quantity of mass,

m. If the system is closed and the entire mass under-

goes a change from state 1 to 2, an intensive property

s is defined by S/m. The property s is also referred to

as entropy, although it is actually specific entropy. If the system is open as in Fig. 2, the specific entropy is calculated by dividing by the appropriate mass.

Use of the symbol δ instead of the usual differen- tial operator d is a reminder that q depends on the process and is not a property of the system (steam). δq represents only a small quantity, not a differen-

tial. Before Equation 23 can be integrated, qrev must

be expressed in terms of properties, and a reversible path between the prescribed initial and final equilib- rium states of the system must be specified. For ex- ample, when heat flow is reversible and at constant pressure, qrev = cpdT. This may represent heat added reversibly to the system, as in a boiler, or the equiva- lent of internal heat flows due to friction or other irreversibilities. In these two cases, ∆s is always positive.

The same qualifications for δ hold in the case of

thermodynamic work. Small quantities of w similar in magnitude to differentials are expressed as δw. Application of entropy to a reversible process

Reversible thermodynamic processes exist in theory only; however, they serve an important function of de- fining limiting cases for heat flow and work processes. The properties of a system undergoing a reversible pro- cess are constrained to be homogeneous because there are no variations among subregions of the system. More- over, during interchanges of heat or work between a system and its surroundings, only corresponding poten- tial gradients of infinitesimal magnitude may exist.

All actual processes are irreversible. To occur, they must be under the influence of a finite potential dif- ference. A temperature difference supplies this drive and direction for heat flow. The work term, on the other hand, is more complicated, because there are as many different potentials (generalized forces) as there are forms of work. However, the main concern here is expansion work for which the potential is clearly a pressure difference.

Regardless of whether a process is to be considered reversible or irreversible, it must have specific begin- ning and ending points (limits) in order to be evalu- ated. To apply the first and second laws, the limits must be equilibrium states. Nonequilibrium thermodynam- ics is beyond the scope of this text. Because the limits of real processes are to be equilibrium states, any pro- cess can be approximated by a series of smaller revers- ible processes starting and ending at the same states as the real processes. In this way, only equilibrium conditions are considered and the substitute processes can be defined in terms of the system properties. The

following lists the reversible processes for heat flow and work:

Reversible Heat Flow Reversible Work Constant pressure, dP=0 Constant pressure, dP=0 Constant temperature, dT=0 Constant temperature, dT =0 Constant volume, dv=0 Constant entropy, ds=0

w=0 q=0

The qualification of these processes is that each describes a path that has a continuous functional relationship on coordinate systems of thermodynamic properties.

A combined form of the first and second laws is ob- tained by substituting δ qrev = Tds for δq in Equation

22b, yielding:

du = Tds−δw_k (24)

Because only reversible processes are to be used, δw

should also be selected with this restriction. Revers- ible work for the limited case of expansion work can be written:

δ

(

w_rev

)

= Pdv (25)

In this case, pressure is in complete equilibrium with external forces acting on the system and is related to

v through an equation of state.

Substituting Equation 25 in 24, the combined ex- pression for the first and second law becomes:

du = Tds Pdv− (26)

Equation 26, however, only applies to a system in which the reversible work is entirely shaft work. To modify this expression for an open system in which flow work d(Pv) is also present, the quantity d(Pv) is added to the left side of Equation 25 and added as (Pdv + vdP) on the right side. The result is:

du d Pv+

( )

= Tds Pdv Pdv vdP− + + (27a)

or

dH = Tds vdP+ (27b)

The work term vdP in Equation 27 now represents re- versible shaft work in an open system, expressed on a unit mass basis.

Because Tds in Equation 26 is equivalent to δq, its

value becomes zero under adiabatic or zero heat trans- fer conditions (δq = 0). Because T can not be zero, it

follows that ds = 0 and s is constant. Therefore, the maximum work from stored energy in an open sys- tem during a reversible adiabatic expansion is ∫ vdP

at constant entropy. The work done is equal to the decrease in enthalpy. Likewise for the closed system, the maximum expansion work is –∫ Pdv at constant

entropy and is equal to the decrease in internal en- ergy. These are important cases of an adiabatic isen- tropic expansion.

Irreversible processes

All real processes are irreversible due to factors such as friction, heat transfer through a finite temperature difference, and expansion through a process with a

finite net force on the boundary. Real processes can be solved approximately, however, by substituting a series of reversible processes. An example of such a substitution is illustrated in Fig. 3, which represents the adiabatic expansion of steam in a turbine or any gas expanded from P1 to P2 to produce shaft work. T1, P1 and P2 are known. The value of H1 is fixed by T1

and P1 for a single-phase condition (vapor) at the in-

let. H1 may be found from the Steam Tables, a T-s dia-

gram (Fig. 3) or, more conveniently, from an H-s (Mol- lier) diagram, shown in the chapter frontispiece. From the combined first and second laws, the maximum en- ergy available for work in an adiabatic system is (H1 − H3), as shown in Fig. 3, where H3 is found by the

adiabatic isentropic expansion (expansion at constant entropy) from P1 to P2. A portion of this available en-

ergy, usually about 10 to 15%, represents work lost (wL) due to friction and form loss, limiting ∆H for shaft work to (H1− H2). The two reversible paths used to

arrive at point b in Fig. 3 (path a to c at constant en- tropy, s, and path c to b at constant pressure) yield the following equation:

H₁−H₃ H₂ H₃ H₁ H₂

(

)

−

(

−

)

= − (28)

Point b, identified by solving for H2, now fixes T2; v1

and v2 are available from separate tabulated values

of physical properties.

Note that ∆H 2-3 can be found from:

∆H_{2 3} Tds

3 2

− =

∫

(29)

or, graphically, the area on the T-s diagram (Fig. 3) under the curve P2 from points c to b. Areas bounded

by reversible paths on the T-s diagram in general rep-

resent q (heat flow per unit mass) between the sys- tem and its surroundings. However, the path a to b is irreversible and the area under the curve has no sig- nificance. The area under path c to b, although it has the form of a reversible quantity q, does not represent heat added to the system but rather its equivalent in internal heat flow. A similar situation applies to the re- lationship between work and areas under reversible paths in a pressure-volume equation of state diagram. Because of this important distinction between revers- ible and irreversible paths, care must be exercised in graphically interpreting these areas in cycle analysis. Returning to Fig. 3 and the path a to b, wL was con- sidered to be a percentage of the enthalpy change along the path a to c. In general, the evaluation should be handled in several smaller steps (Fig. 4) for the fol- lowing reason. Point b has a higher entropy than point c and, if expansion to a pressure lower than P2 (Fig.

3) is possible, the energy available for this additional expansion is greater than that at point c. In other words, a portion of wL (which has the same effect as heat added to the system) for the first expansion can be recovered in the next expansion or stage. This is the basis of the reheat factor used in analyzing ex- pansions through a multistage turbine. Since the pres- sure curves are divergent on an H-s or T-s diagram, the sum of the individual ∆Hs values (isentropic ∆H) for individual increments of ∆P (or stages in an irre-

versible expansion) is greater than that of the revers- ible ∆Hs between the initial and final pressures (Fig. 4). Therefore the shaft work that can be achieved is greater than that calculated by a simple isentropic expansion between the two pressures.

Principle of entropy increase

Although entropy has been given a quantitative meaning in previous sections, there are qualitative aspects of this property which deserve special empha- sis. An increase in entropy is a measure of that por- tion of process heat which is unavailable for conver- sion to work. For example, consider the constant pres- sure reversible addition of heat to a working fluid with the resulting increase in steam entropy. The minimum portion of this heat flow which is unavailable for shaft work is equal to the entropy increase multiplied by the absolute temperature of the sink to which a part of the heat must be rejected (in accordance with the sec- ond law). However, because a reversible addition of heat is not possible, incremental entropy increases also occur due to internal fluid heating as a result of tem- perature gradients and fluid friction.

Even though the net entropy change of any por- tion of a fluid moving through a cycle of processes is always zero because the cycle requires restoration of all properties to some designated starting point, the sum of all entropy increases has a special significance. These increases in entropy, less any decreases due to recycled heat within a regenerator, multiplied by the appropriate sink absolute temperature (R or K) are equal to the heat flow to the sink. In this case, the net entropy change of the system undergoing the cycle is zero, but there is an entropy increase of the surround- ings. Any thermodynamic change that takes place,

whether it is a stand alone process or cycle of processes, results in a net entropy increase when both the sys- tem and its surroundings are considered.

Cycles

To this point, only thermodynamic processes have been discussed with minor references to the cycle. The next step is to couple processes so heat may be con- verted to work on a continuous basis. This is done by selectively arranging a series of thermodynamic pro- cesses in a cycle forming a closed curve on any sys- tem of thermodynamic coordinates. Because the main interest is steam, the following discussion emphasizes expansion or Pdv work. This relies on the limited dif- ferential expression for internal energy, Equation 26, and enthalpy, Equation 27. However, the subject of thermodynamics recognizes work as energy in tran- sit under any potential other than differential tem- perature and electromagnetic radiation.

Carnot cycle

Sadi Carnot (1796 to 1832) introduced the concept of the cycle and reversible processes. The Carnot cycle is used to define heat engine performance as it con- stitutes a cycle in which all component processes are reversible. This cycle, on a temperature-entropy dia- gram, is shown in Fig. 5a for a gas and in Fig. 5b for a two-phase saturated fluid. Fig. 5c presents this cycle for a nonideal gas, such as superheated steam, on Mollier coordinates (entropy versus enthalpy).

Referring to Fig. 5, the Carnot cycle consists of the following processes:

1. Heat is added to the working medium at constant temperature (dT = 0) resulting in expansion work Fig. 4 Three-stage irreversible expansion – ∆Hs1 + ∆Hs2 + ∆Hs3 > ∆H_sac.

and changes in enthalpy. (For an ideal gas, changes in internal energy and pressure are zero and, therefore, changes in enthalpy are zero.) 2. Adiabatic isentropic expansion (ds = 0) occurs with

expansion work and an equivalent decrease in en- thalpy.

3. Heat is rejected to the surroundings at a constant temperature and is equivalent to the compression work and any changes in enthalpy.

4. Adiabatic isentropic compression occurs back to the starting temperature with compression work and an equivalent increase in enthalpy.

This cycle has no counterpart in practice. The only way to carry out the constant temperature processes in a one-phase system would be to approximate them through a series of isentropic expansions and constant pressure reheats for heat addition, and isentropic com- pressions with a series of intercoolers for heat rejections. Another serious disadvantage of a Carnot gas engine would be the small ratio of net work to gross work (net work referring to the difference between the expansion work and the compression work, and gross work being expansion work). Even a two-phase cycle, such as Fig. 5b, would be subject to the practical mechanical difficul- ties of wet compression and, to a lesser extent, wet ex- pansion where a vapor-liquid mixture exists.

Nevertheless, the Carnot cycle illustrates the basic principles of thermodynamics and, because the processes are reversible, the Carnot cycle offers the maximum thermodynamic efficiency attainable between any given temperatures of heat source and sink. The efficiency of the cycle is defined as the ratio of the net work output to the total heat input. Various texts refer to this ratio as either thermodynamic efficiency, thermal efficiency, or simply efficiency. Using the T-s diagram for the Carnot cycle shown in Fig. 5a, the thermodynamic efficiency depends solely on the temperatures at which heat addi- tion and rejection occur:

η = T T1_T− 2 = −T_T 1 2 1 1 ₍₃₀₎ where

η =thermodynamic efficiency of the conversion from heat into work

T1 =absolute temperature of heat source, R (K) T2 =absolute temperature of heat sink, R (K)

The efficiency statement of Equation 30 can be ex- tended to cover all reversible cycles where T1 and T2

are defined as mean temperatures found by dividing the heat added and rejected reversibly by ∆s. For this

reason, all reversible cycles have the same efficiencies when considered between the same mean tempera- ture limits of heat source and heat sink.

Rankine cycle

Early thermodynamic developments were centered around the performance of the steam engine and, for comparison purposes, it was natural to select a revers- ible cycle which approximated the processes related to its operation. The Rankine cycle shown in Fig. 6, proposed independently by Rankine and Clausius, Fig. 5 Carnot cycles.

meets this objective. All steps are specified for the sys- tem only (working medium) and are carried out re- versibly as the fluid cycles among liquid, two-phase and vapor states. Liquid is compressed isentropically from points a to b. From points b to c, heat is added reversibly in the compressed liquid, two-phase and finally superheat states. Isentropic expansion with shaft work output takes place from points c to d and unavailable heat is rejected to the atmospheric sink from points d to a.

The main feature of the Rankine cycle is that com- pression (pumping) is confined to the liquid phase, avoiding the high compression work and mechanical problems of a corresponding Carnot cycle with two- phase compression. This part of the cycle, from points a to b in Fig. 6, is greatly exaggerated, because the difference between the saturated liquid line and point b (where reversible heat addition begins) is too small to show in proper scale. For example, the temperature rise with isentropic compression of water from a satu- ration temperature of 212F (100C) and one atmo- sphere to 1000 psi (6.89 MPa) is less than 1F (0.6C). If the Rankine cycle is closed in the sense that the fluid repeatedly executes the various processes, it is termed a condensing cycle. Although the closed, con- densing Rankine cycle was developed to improve steam engine efficiency, a closed cycle is essential for any toxic or hazardous working fluid. Steam has the important advantage of being inherently safe. How- ever, the close control of water chemistry required in high pressure, high temperature power cycles also favors using a minimum of makeup water. (Makeup is the water added to the steam cycle to replace leak- age and other withdrawals.) Open steam cycles are still found in small units, some special processes, and heating load applications coupled with power. The con- densate from process and heating loads is usually re- turned to the power cycle for economic reasons.

The higher efficiency of the condensing steam cycle is a result of the pressure-temperature relationship between water and its vapor state, steam. The lowest temperature at which an open, or noncondensing,

steam cycle may reject heat is approximately 212F (100C), the saturation temperature corresponding to atmospheric pressure of 14.7 psi (101.35 kPa). The pressure of the condensing fluid can be set at or be- low atmospheric pressure in a closed cycle. This takes advantage of the much lower sink temperature avail- able for heat rejection in natural bodies of water and the atmosphere. Therefore, the condensing tempera- ture in the closed cycle can be 100F (38C) or lower.

Fig. 7 illustrates the difference between an open and closed Rankine cycle. Both cycles are shown with nonideal expansion processes. Liquid compression

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