Thermodynamics

1.1 Introduction

The performance of rocket engines can be calculated from motor geometry and measurements of thermodynamic quantities such as temperature and pressure. The combustion of the fuel-oxidizer mixture, a chemical reaction, increases the pressure and temperature of the gaseous products of the reaction. The gas is accelerated by the converging part of the nozzle until it reaches sonic speed at the throat. At supersonic speeds, the divergent section further accelerates the gas as it expands to atmospheric pressure and temperature. As it expands, its heat content is converted to thrust (kinetic energy).

Combustion thermodynamics explains these processes. Thermodynamics and combustion chemistry (chemical reaction kinetics and equilibrium) combine to provide the theory required to calculate performance behavior of engine designs and to analyze rocket test data. Rocket propulsion is mainly about combustion thermodynamics and is the basis for rocket engine design and analysis. This chapter presents the basic principles of thermodynamics.

1.2 Thermodynamic concepts: the first law and substances

Thermodynamics is the branch of physics involving heat and its conversion to other forms of energy and to work. Rocket engines are heat engines that convert heat to mechanical energy. Energy is neither created nor lost due to engine processes; it is conserved. The initial and final amounts of energy of a process are equal. The first law of thermodynamics is an expression of energy conservation:

E = Q -  W                                                                 (1.2- 1)

where E is energy, Q is heat and W is work.

These quantities are related to either a fixed mass of matter (called a closed system or control mass) or a fixed volume through which mass flows (called an open system or control volume):

bulletfixed mass: control mass (cm) or closed system
bulletfixed volume: control volume (cv) or open system

The mass or volume is chosen for analysis. Control volumes are chosen for flow processes whereas control masses are used when a fixed mass of matter undergoes a change. A fixed mass of gas flowing through the engine, for example, is a control mass. Two cross-sectional surfaces in the engine define the entrance (or inlet) and exit flow boundaries of a control volume. Along with the shape of the chamber, these surfaces define the closed surface of the control volume.

Systems are distinct from their surroundings or environment, which is exterior to the boundary of a system. Heat and work can enter or leave a system and are exchanged with the surroundings. They are signed quantities with sign convention:

bulletHeat into a system is positive.
bulletWork out of (done by) a system is positive.

This convention explains why work is subtracted from heat in (1.2-1).

When a closed system does not exchange heat or work with its surroundings, it is an isolated system. Heat and work are propagated through matter, or substances. They have two kinds of properties:

1. Intensive properties are independent of mass
2. Extensive properties vary with mass.

For example, the volume of a substance is related to its mass through its density whereas density itself is independent of mass. Often, an extensive property, like volume, V, can be expressed as an intensive property by dividing it by its mass. When this is done, it is often indicated by putting the word specific in front of the extensive quantity. Thus, specific volume is:

                                                                 (1.2-2)

It is conventional to use the corresponding lower-case symbol to represent the intensive quantity. Specific volume is the reciprocal of density, or v = 1/r .

1.3 States, processes, and reversibility

The "dynamics" of thermodynamics involves states of matter, defined by thermodynamic properties, and transitions between states, called processes. (Electronics engineers: this is similar to state machines in digital logic circuits.)

The four measureable thermodynamic properties are: pressure, P, temperature, T, volume, V, and mass, m. By combining V and m, we can work with specific volume, v, instead:

Measureable quantities: P, V, T, m or P, v, T, m

Two thermodynamic quantities that are not directly measureable are heat and energy.

U, P, v, and T are properties of a substance, and as such are only state-dependent:

Properties of a substance are state-dependent and process-independent.

Mathematically, they are point properties, are path-independent, and are described by exact differentials, dX. Their values are found by integrating dX:

                                                                     (1.3-1)

A closed thermodynamic cycle has the same initial and final states. Because initial internal energy, Ui, and final internal energy, Uf , are equal, U depends only on state. Then applying the first law to a closed cycle,

D U = Uf - Ui = D (Q - W) = D Q -  D W  = 0 Þ D W = D Q                             (1.3-2)

In other words, the heat exchanged in a closed thermodynamic cycle must equal the work done.

In contrast, work and heat are process-dependent. They are mathematically represented by inexact differentials, d X, and are path- dependent. That is, to find the total X (heat or work) exchanged in a process, the integral of d X is a line integral and consequently the integration path, C, (representing the process) must be specified:

                                                              (1.3-3)

What this means physically is that a system can change in different ways between two states, and the way it changes affects the value of X. There are many possible processes between two thermodynamic states.

Some important processes are characterized by holding a quantity constant during the process:

Process

Condition

isobaric

D P = 0

isothermal

D T = 0

isentropic

D S = 0

isometric

D v = 0

adiabatic

D Q = 0

A process of special importance is the reversible process: an ideal process that leaves both system and surroundings unaffected in a closed cycle. Real processes can only approach (but not achieve) reversibility. Only reversible processes leave the surroundings in the same state. From the first law, eqn. (1.3-1) can be expressed in line-integral form as:

Þ                                    (1.3-1a)

In addition, for a reversible process,

                                                 (1.3-4)

For closed-cycle, irreversible processes, the system is unchanged but D Q = D W ¹  0. Therefore, the surroundings are changed.

Heat transfer across a finite temperature difference is an irreversible process. So is the conversion of work into heat or unconstrained expansion (no work done) by a gas. In irreversible processes, a disordering takes place in the system that is not recoverable except by affecting its surroundings.

Equilibrium in thermodynamics is static (unchanging) behavior, and a quasiequilibrium process is thus quasistatic. (In electronics, this is low-frequency dynamic behavior.) Reversible processes are quasiequilibrium processes. That is, an infinite number of infinitesimal changes occur so equilibrium is never significantly disturbed. In practice, excessive operating time or product complexity (cost) is required to approach reversibility. But to the extent that processes approach reversibility, thermal efficiency increases.

Thermodynamics involves static states and quasistatic (reversible) processes. ("Thermoquasistatics" would be a more accurate name.) Actual processes are fully dynamic and are described by heat and mass transfer and rate process theory. Thus, the general approach in thermodynamics is to make calculations based on reversible processes, knowing that actual processes are not as ideal. By knowing the causes of irreversibility, we can estimate the extent to which reversibility is approached.

In a reversible isothermal process, reversibility is achieved by an infinitesimal temperature difference between the system and surroundings. The temperature is changed in infinitesimal steps, so that the system maintains temperature equilibrium with its surroundings. The process can similarly be reversed, returning both system and surroundings to the initial state. Therefore,

reversible isothermal process: D T ® 0                                             (1.3-5)

1.4 The second law, thermal efficiency, and entropy

A convenient way of thinking about thermodynamic processes involves constant-temperature heat sources (or sinks) called reservoirs and closed-cycle reversible processes operating between reservoirs at temperatures TH and TL (TH > TL). This can be diagrammed as:

The heat engine exchanges heat with both reservoirs and work with the environment. By the first law,

W = QH - QL

The second law of thermodynamics can be expressed in two equivalent forms, diagrammed below:

The second law states that the processes shown are impossible; heat cannot flow to a higher-temperature source without work input, and work cannot be exchanged with the environment by inputing heat only. That is, work can be exchanged only when the heat engine operates between a temperature difference. The equivalence of the two forms of the second law is illustrated by adding a heat engine to the Clausius form. The result is the Kelvin-Planck form shown below.

The net heat exchanged with the TH reservoir is zero, and QH -  QL of heat is exchanged with the TL reservoir as input to the engine. This results in the Kelvin-Planck form of the second law. The thermal efficiency of a reversible closed-cycle heat engine is:

                                                     (1.4-1)

For an engine, combustion provides heat, QH, which produces work, W, and exhausts heat, QL, at ambient temperature. If the engine is reversible,

                                         (1.4-2)

The smaller QL can be made, the greater is the fraction of QH converted to work, and the higher the efficiency.

It can be shown that h t is the same for all closed, reversible cycles operating between two constant-temperature reservoirs, and is the theoretical maximum efficiency.

From this, the absolute temperature scale with units of degrees Kelvin, K, is defined by the relation:

                                                                     (1.4-3)

and thermal efficiency can be expressed purely in terms of absolute temperatures:

                                                                     (1.4-4)

Carnot Cycle

A reversible, closed cycle of special interest is the Carnot cycle, shown below.

1 ®  2    rev. isothermal processes

3 ®  4
                D T = 0 (reversible)
                D W = 0
                D Q ¹  0

P, V, T, S: point properties
                   state-dependent
                   properties of substance, system

2 ®  3    rev. adiabatic processes
W, Q:         path properties
                   process-dependent

 

4 ®  1
               D Q = 0
               D S = 0 (reversible, isentropic)
               D W ¹  0

The Carnot cycle consists of four processes:

1. reversible isothermal process: QH added to system at TH (+ heat)
2. reversible adiabatic process: work is done by system (+ work)
3. reversible isothermal process: system rejects QL at TL (- heat)
4. reversible adiabatic process: work is done on system (- work)

The Carnot cycle is represented by a heat engine operating between two constant-temperature reservoirs.

A corollary of the second law is the inequality of Clausius:

                                                                 (1.4-5)

The equality holds for reversible, closed cycles. In particular, for the Carnot cycle, applying the definition of absolute temperature, (1.4-3),

                                                 (1.4-6)

For irreversible cycles, Wirrev < Wrev because they are less efficient, and consequently, QL irrev > QL rev. For actual processes, irreversibility is present and d Q/T is always negative.

Entropy, S, is the property that most directly relates to reversibility. For reversible processes, entropy relates heat exchange and temperature. Between two given states,

                                                        (1.4-7)

in which T is in absolute temperature. Entropy is defined differentially according to

                                                                 (1.4-8)

or

                                                             (1.4-8a)

Mathematically, dS is an exact differential and is a point property whereas d Q is a path property. Here, T is an integrating factor, converting d Q to an exact differential for reversible paths.

The change in S of a system undergoing a change in state is

                                                             (1.4-9)

The integration takes place along a reversible path between the initial and final states. For a closed reversible cycle initially at state 1 with intermediate state 2, then from state 1 to state 2, D S1 ® 2 = - D S2 ® 1 and

                                             (1.4-10)

Because entropy is a property of a substance, it is a state property and is path-independent. Consequently, D S is the same for all processes, reversible and irreversible, between two states. Though (1.4-9) provides calculation of D S only along a reversible path, it can be used to determine D S for any path between the same two states because D S depends only on the states, not the path.

In the Carnot cycle, a second reversible process is introduced. It is a reversible adiabatic process and its condition for reversibility is:

reversible adiabatic process Þ D S = 0                                         (1.4-11)

A reversible abiabatic process is thus isentropic.

For all processes (both reversible and irreversible),

                                                     (1.4-12)

The additional heat transfer over that of a reversible process is d Qirrev. This lost energy is added as the irreversible heat term. Eqn. (1.4-12) states the principle of increase of entropy: the entropy of an isolated system can only increase, or, at best remain unchanged.

For a system that interacts with its surroundings, its entropy can increase in two ways:

1. by the addition of heat to the system.
2. by irreversibilities that occur within the system.

1.5 Flow work, enthalpy, ideal gases and heat capacity

Work, like heat, is also process-dependent, and we can develop similar relations for it in terms of pressure and volume. From mechanics, work is defined as

                                                                    (1.5-1)

in which force, F is integrated over a path, C, as a line integral. For a fluid - a liquid or gas - work can be expressed in terms of pressure, P, and volume, V:

                                                     (1.5-2)

This is the work done in displacing a volume of fluid at pressure, P, and is called flow work. An equation similar to (1.4-7) can be written for work:

                                                         (1.5-3)

In (1.4-7), T is a function of S; here, P = P(V). Consequently,

(d W)rev = PdV                                                                (1.5-4)

is an exact differential, just as (d Q)rev = TdS is.

The energy of the fluid in a flow process (open system) is the sum of the internal energy and the flow work across the system boundaries, at inlet and exit. This quantity appears often and is called the enthalpy, h, of the fluid, or

h = u + P× v                                                                 (1.5-5)

Thermodynamic cycles are most easily visualized by two plots, as shown for the Carnot cycle. The P-V and T-S diagrams make it easier to identify particular processes, and heat and work. Because all of the processes are reversible, they are plotted with solid lines. The area within the P-V cycle represents work and within the T-S cycle represents heat. For irreversible processes, the path is often unknown and is shown as a dotted curve. The area under the reversible curve for an irreversible process does not represent heat or work because, depending on the extent of irreversibility, heat or work varies from the reversible value (the area under the curve) to zero for a completely irreversible process.

In engines, the working fluid is the fuel-oxidizer mixture. After combustion, the working fluid is the gaseous products. A very simple thermodynamic engine model approximates the working fluid as a perfect or ideal gas: a gas with specific internal energy, u, dependent only on temperature:

ideal gas: u = u(T)                                                         (1.5-6)

A change in state is therefore a function of T only and is path-independent. The state equation for an ideal gas is:

P× v = R× T                                                                 (1.5-7)

R is a gas constant for the particular gas (in J/kg-K). Another useful thermodynamic quantity is heat capacity or specific heat. Constant-volume heat capacity is defined as:

                                                 (1.5-8)

A real gas approaches ideal as its pressure approaches zero, or

                                                 (1.5-9)

Because P approaches zero and is fixed, for an ideal gas, cv is a function of T only, or

                                                             (1.5-10)

and

du = cv dT                                                         (1.5-10a)

For flow processes, the state equation, (1.5-7), can be substituted into (1.5-5), resulting in:

h = u + P× v = u + R× T                                                 (1.5-11)

and h = h(T) only. Thus, u, h, and T are state-dependent for an ideal gas.

The constant-pressure heat capacity is defined as:

                                                             (1.5-12)

and, for ideal gases,

                                                                 (1.5-13)

or

                                                             (1.5-14)

For a constant-pressure process, Dh = cp× DT. Heat capacities thus relate energy or enthalpy and temperature.

For ideal gases, from (1.5-11):

dh = du + R× dT                                                         (1.5-15)

Substituting (1.5-13) and (1.5-14),

cp× dT = cv× dT + R× dT

or

cp - cv = R                                                             (1.5-16)

The difference between the two heat capacities is the gas constant. For a simple compressible substance such as air, dv ¹  0. For a reversible process, the first law is:

d q = du + d w, reversible process

Because of reversibility, substitutions for d q and d w can be made, yielding:

T× ds = du + P× dv                                                             (1.5-17)

From (1.5-5), the differential enthalpy is:

dh = du + P× dv + v× dP = T× ds + v× dP                                         (1.5-18)

Then (1.5-17) can be expressed as:

T× ds = dh - v× dP                                                         (1.5-19)

Note that these equations all apply to reversible processes. For an ideal gas, the differential entropy follows from the above equations:

                                      (1.5-20)

Integrating ds (an exact differential),

                         (1.5-21)

For a reversible adiabatic process, ds = 0, so T×ds = 0. Then

                                             (1.5-22)

That is,

                                                         (1.5-22a)

Solving,

                                                     (1.5-23)

Again, for an ideal gas and ds = 0, by substituting dT from (1.5-15) and (1.5-18) into (1.5-17),

which reduces to

and

                                                 (1.5-24)

(Sometimes the symbol k is used for g ). Solving the differential equation, we obtain

                                                 (1.5-25)

or

                                                     (1.5-25a)

The specific heats can be expressed in g and R as:

                                             (1.5-26)

The specific heat ratio is assumed constant in the above equations. In practice, it varies inversely with temperature for typical engine working fluids. The general process described by (1.5-25) is a polytropic process:

polytropic process: P× v = constant                                         (1.5-27)

or, in differential form,

d(lnP) + n× d(lnv) = 0                                                 (1.5-27a)

This leads to the expressions:

                                                 (1.5-28)

For an ideal gas, n depends on the particular process, as given in the following table:

Process

n

isobaric (D P = 0)

0

isothermal (D T = 0)

1

isentropic (D S = 0)

g

isometric (D v = 0)

¥

The work performed between states 1 and 2 of a polytropic process is:

                         (1.5-29)

where C is the constant P× vn. For a control volume, Dv = 0 and the flow work is . This follows from the general expression for flow work:

                                  (1.5-30)

For a constant pressure displacement of a system, dP = 0 and (1.5-2) results. For a control volume, dv = 0 and

w = v×dP                                                                 (1.5-31)

For a reversible steady-state, steady-flow process through a control volume with inlet state, i, and exit state, e:

             (1.5-32)

The state equation for an ideal gas, (1.5-7), is a polytropic process with n = 1. For a reversible isothermal process,

                                 (1.5-33)

For this kind of process, Dh = Du = 0 and q1® 2 = w1® 2. For a reversible, isothermal, steady-state, steady-flow process,

                                             (1.5-34)

1.6 Reversible work, Gibbs function, and Maxwell relations

The maximum work possible between two states is the work of a reversible process, Dwrev = Da; a is the Helmholtz function:

a = u - T× s                                                                 (1.6-1)

The availability is the maximum reversible work minus the work done on the surroundings. The final state is in equilibrium with the surroundings. It is the net work available or maximum work transfer to the environment:

                             (1.6-2)

The A subscript refers to the surroundings. Availability can also be expressed in terms of the Gibbs function or free energy,

g = h - T×s                                                         (1.6-3)

and g = a + P×v. Then the availability is:

availability = g - gA                                                 (1.6-4)

or (Dg)A. Four exact differentials can now be expressed:

du = T× ds -  P× dv                                                  (1.6-5)

dh = T× ds + v× dP                                                  (1 .6-6)

da = - P× dv - s× dT                                               (1 .6-7)

dg = v× dP - s× dT                                                   (1.6-8)

These equations are all of the form:

                                     (1.6-9)

The partial derivatives are taken with the other independent variable held constant. Taking partial derivatives of M and N,

                                         (1.6-10)

Applying this relation to (1.6-5) through (1.6-8), we obtain the Maxwell relations. For example, from (1.6-7),

                                                             (1.6-11)

in which v and T are constant while differentiating with respect to the other variable. A relation for functions x(y, z), y(x, z), and z(x, y) is:

                                                  (1.6-12)

1.7 Multiple-phase systems

When two phases of a pure substance, such as water, are in equilibrium, P and T are constant. The fraction of mass that is vapor is the quality, x. (A vapor is a gas in equilibrium with a liquid.) Then quantities such as v or s can be calculated from the general formula,

                                                         (1.7-1)

in which z is the multiphase quantity, and zl and zg are the single-phase (liquid, l, and vapor, g) quantities. Both liquid and vapor are in a saturated state. By defining

                                                                 (1.7-2)

then (1.7-1) can be written as

                                                             (1.7-1a)

In the saturation region, T and P are independent of v, and (1.6-11) becomes:

                                                             (1.7-3)

Because this is a constant-temperature process,

                                                         (1.7-4)

From the first law, heat added in the constant-pressure vaporization process is:

                                             (1.7-5)

Solving for s and substituting into (1.7-4), the Clapeyron equation results:

                                                      (1.7-6)

With an empirical relation for P(T), and from separately determined values for vl and vg, hlg and hg can be determined using (1.7-6).

Finally, we derive a general relation for the enthalpy of a pure substance. In general, h = h(T, P). Beginning with (1.5-19), notice that

                                                     (1.7-7)

and, applying (1.5-17), that

                                                     (1.7-8)

Then the differential, dh(T, P), is:

                                             (1.7-9)

We need expressions for the partial derivatives. From (1.5-19), it follows that

                                                 (1.7-10)

From (1.6-8), the Maxwell relation can be derived:

                                                     (1.7-11)

Substituting this into (1.6-21), and then (1.6-21) and (1.6-18) into (1.6-20),

(1.7-12)

This differential can be integrated along T and P to give the Dh associated with a change of state. A similar derivation for u = u(T, v) results in

                                     (1.7-13)

and for s(T, P) and s(T, v), respectively,

                         (1.7-14)

Rearranging the last two expressions of (1.6-25) and simplifying,

                                             (1.7-15)

From this, we note that (dv/dT)P is small for liquids; then cp »  cv, and cp = cv when (dP/dv)T = 0, as at the temperature in which the density of water is maximum. Also, cp ®  cv as T ®  0, and the specific heats are equal at absolute zero. We also know that for an ideal gas, (1.7-15) is R, the gas constant. Finally, cp > cv because the signs of the derivatives are as shown for all known substances.

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