111EQUATION CHAPTER 1 SECTION 1CHAPTER 1

BALANCE EQUATIONS AND LAWS OF THERMODYNAMICS

1.0 SCOPE AND OBJECTIVE

In this chapter, the basic ideas in classical thermodynamics are reviewed. They

include

introduction of basic concepts such as: system and environment; state and state

properties;

definition of thermodynamic variables such as: volume, energy, entropy,

temperature, and pressure; and process variables such as heat and work;

derivation of balance equations of energy and entropy and their relation to First and

Second Laws of Thermodynamics respectively; and

lastly, detailed explanations of the implications of Second Law of Thermodynamics

such as: reversibility and spontaneity, friction and dissipation, efficiency and free

energy.

Students are expected to have a basic understanding of the above concepts and

able to apply energy and entropy balance equations to solve problems.

1.1 BASIC CONCEPTS

1.1.1 SYSTEM, ENVIRONMENT AND UNIVERSE

In all scientific studies, a “system” must first be defined. A “system” is something: a

lump of material, or a boundary of space in which we focus our interests. Everything

outside this system are secondary concerns, they are lumped into the “environment”. In

thermodynamics, the union of the system and environment is often referred to the

“universe”.

Figure 1. 1 System, environment, and universe

1.1.2 OPEN AND CLOSE SYSTEMS

In chemical engineering thermodynamics two types of systems are commonly used:

the “control-volume” system: which is a fixed volume in space with fixed position

and boundary, but variable mass; i.e. it can exchange mass with the surrounding. A

control volume system is also known as an "open" system

the “control-mass” system: which is a fixed mass of material with variable position,

boundary, and volume, i.e.: it can exchange mass with the surrounding. A control

mass system is also known as an “close" system.

It should be pointed out that in classical thermodynamics, the systems of interest

are continuous lumps of materials instead of collection of molecules and atoms.

Figure 1. 2 : Open and close systems

1.1.3 CONTINUOUS MEDIA

In classical thermodynamics, the subjects of interests are lumps of continuous

materials, as opposed to statistical thermodynamics, in which systems are collection of

molecules and atoms. It is postulated that average properties of the system can be

defined and measured, while fluctuations of this average properties due to inter- and

intra- molecular movements can be neglected. To satisfy this assumption, the number of

molecules in a continuous system must be large. The question is how large?

In statistics it is well known that the sampled variance is inversely proportional to

the square root of number of samples. Therefore, for a sample of 1,000,000, the

sampled variance is approximately 1/1000 or 0.1%. Hence we need at least a million

molecules in the system before it can approach a continuous system.

Consider gold at room temperature, the density is 19300kg

m3. There are

19300197

×1000×6.02×1023=3.35×1028 molecules. One million molecules will occupy 1

1.69×10−23m3 of volume. The length of this cube will be approximately 26nm.

Calculations for water and methane also shows that the continuity assumption may

break down as the material become nano-sized.

Table 1. 1 : Limits of continuity for difference materials Gold Water Methane

density [kg/m3] 19300 1000 656mol. wt. 197 18 16

number in cubic meter 5.90E+28 3.35E+28 2.46E+28volume of 1 million [m3] 1.69E-23 2.99E-23 4.06E-23length of volume [nm] 26 31 34

1.1.4 THERMODYNAMIC STATES PROPERTIES AND THERMODYNAMIC STATE

In thermodynamics, we postulates that for a closed system or a fixed mass of

materials, there are certain directly or indirectly measurable characteristics of the

material (X 1 , X2 , X3⋯), known as thermodynamic state properties or variables.

It is also postulated that the “classical” or “macroscopic” thermodynamic state

properties will not change if a subset of these variables (X 1 , X2 , X3⋯ X F) are fixed; F is

the degrees of freedom of the thermodynamic states.

Common thermodynamic state properties include: temperature, pressure,

composition, density, internal energy, enthalpy, entropy, heat capacity etc. Students

should be reminded that transport properties such as thermo-conductivity, viscosity,

diffusion coefficients etc., i.e. properties we learned in transport courses are also

thermodynamic state properties.

Although properties such as thermo-conductivity, viscosity, diffusion coefficients are

properties related to transport of heat, momentum and material; they are nevertheless

properties of the thermodynamic state.

Since thermodynamic states properties depend on the definition of state variables,

they do not depend on how such states are reached.

1.1.5 MACRO- AND MICRO- STATES

The “macroscopic”thermodynamic state of a system (e.g., the condition that a

system has specified values of volume, temperature and number of molecules) provides

only a partial, incomplete description from a molecular point of view.

The most complete description that is possible about a system is a statement of its

wave function, which describe coordinates of the elementary particles, we have a

specification of the quantum state of the system.

For a macroscopic system (~ 1024 electrons and nuclei), the number of quantum

states, which may be compatible with the same total energy, volume, and composition is

astronomical.

A less detailed description would be the coordinates, velocities and orientations of

molecules, i.e. the classical micro-state. The number of classical microstates consistent

with a macrostate is still astronomical. For example, an ideal gas may be represented by

a box of particles with no interaction with each other, and that linear momentum is

always conserved in collision. The “macroscopic” thermodynamic state is defined since

volume, temperature and the number of molecules, are fixed. However, the particles

can be arranged in different positions and move around with different velocities.

Figure 1. 3 : A box of non-interacting particles

1.1.6 INTENSIVE AND EXTENSIVE VARIABLES

The multitude of thermodynamic properties can be divided into two categories.

Some of them, e.g. temperature, pressure, density, viscosity, thermoconductivity, etc.

are independent of the total mass of material in the system. They are known as

“intensive” variables.

On the other hand, some of them, e.g. the total volume (V ) is homogeneous

function of the total mass of material (N ), i.e: V (kN )=kV (N ). They are known as

“extensive” variables.

1.2 THERMODYNAMIC AND PROCESS VARIABLES

1.2.1 VOLUME, INTERNAL ENERGY AND ENTROPY

It is not possible for us to define all classical thermodynamic variables based on

other classical thermodynamic variables. We must starts with some a priori definitions

based on our conceptualization of microstates.

Let us assume that the extensive thermodynamic state of a pure substance can be

defined by three basic extensive thermodynamic properties. The first two are intuitive

concepts: volume (𝑉), and internal energy (𝑈).

Volume is the space within which the molecules can be found. Internal energy

include

intermolecular kinetic energy due to motions of centers of mass of molecules,

intermolecular potential energy due to attraction and repulsion between

molecules,

intra-molecular kinetic energy due to rotation and vibration of molecules,

intra-molecular electronic energy due to motion of electrons around nuclei,

and

nuclear energy which will remain unchanged in chemical processes.

The third thermodynamic quantity is known as entropy (S), which is a measurement

of uncertainty of how the molecules and atoms were arranged in the system.

Mathematically, entropy S can be expressed as:

S=−kN ∑all microstates

p i ln pi

212\*

MERG

EFOR

MAT

(.)

where pi are probabilities of finding the system in the ith microstate, andk is the

Boltzmann constant, in memory of the physicists Ludwig Eduard Boltzmann (1844-1906),

who laid the groundwork of statistical mechanics by explaining the relation of entropy to

molecular arrangements1.

The fact that entropy is a measurement of uncertainty in microstates can explained

by the following. If there is only one microstate that the system can be found, i.e. we are

absolutely certain about how molecules are arranged, the entropy of the system is zero.

However, this is only a hypothetical situation never found in the real world and can never

be attained according to the Third Law of Thermodynamics2. On the other hand, if the

system can be found in a lot of microstates with equal probabilities, then the above

equation is simplified to

pi=1Ω

313\*

MERG

EFOR

MAT

(.)

S=−kNlnΩ

414\*

MERG

EFOR

MAT

(.)

where Ω is the number of possible micro-states; some times known as “degeneracy” of

the system.

1 https://en.wikipedia.org/wiki/Ludwig_Boltzmann2 https://en.wikipedia.org/wiki/Third_law_of_thermodynamics

Figure 1. 4 : The epitaph of Boltzmann’s grave

1.2.2 TEMPERATURE AND PRESSURE

We have assumed that the extensive macroscopic thermodynamic state can be

defined by the three variables V ,U ,S. However since extensive macroscopic

thermodynamic variables must be homogeneous in nature, the intensive

thermodynamic states can be defined using two variables only. Hence for a closed

system with number of molecules N fixed, the macroscopic thermodynamic state is

defined by two of the above three variables. For example, we can define our

thermodynamic state using S and V only and write

U=U (S ,V )

515\*

MERG

EFOR

MAT

(.)

Given this definition of thermodynamic state of a closed system, two important intensive

properties: temperature and pressure, can be defined as:

T=( ∂U∂S )V

616\*

MERG

EFOR

MAT

(.)

P=−( ∂U∂V )S

717\*

MERG

EFOR

MAT

(.)

Microscopically, temperature is that it is proportional kinetic energy of each

molecule; while pressure is average momentum transfer between the molecules and the

wall of the container.

Furthermore, the above equations results in

dU=TdS−PdV

818\*

MERG

EFOR

MAT

(.)

Note that the above equation is the results of definition of thermodynamic state. It has

nothing to do with the First and Second Law of thermodynamics3.

Given these variables, other thermodynamic potential variables, such as enthalpy4,

Helmholtz free energy5, Gibbs free energy6 can be defined, e.g.:

H=U+PV

919\*

MERG

EFOR

MAT

(.)

A=U−TS

1

0110\

*

MERG

EFOR

MAT

(.)

3 Sometimes, equation 18 was derived in the following way. According to the First Law of thermodynamics for a closed system dE=δQ+δW (see section 1.3.1). If kinetic and potential energy are negligible, we have dE=dU ; furthermore, if the process is reversible and no shaft work, δQ=TdS and δW=−PdV , hence we have dU=TdS−PdV . But since changes of thermodynamic properties only depend on the initial and final thermodynamic states and equation 18 involves only thermodynamic properties, it is valid for all changes. This is a very roundabout way of deriving equation 18. Here the equation is the direct results of the definition of temperature and pressure.4 https://en.wikipedia.org/wiki/Enthalpy 5 https://en.wikipedia.org/wiki/Helmholtz_free_energy 6 https://en.wikipedia.org/wiki/Gibbs_free_energy

G=H−TS

1

1111\

*

MERG

EFOR

MAT

(.)

1.2.3 HEAT

Heat is transfer of energy between systems with different temperatures. Systems

are at the same temperature when there is no heat flow between them.

The Zeroth Law of Thermodynamics stated7

"…if two systems, A and B, are in thermal equilibrium and a third system C is in

thermal equilibrium with system A then systems B and C will also be in thermal

equilibrium (in thermal equilibrium is a transitive relation; moreover, it is an equivalence

relation). … Since A, B, and C are all in thermal equilibrium, it is reasonable to say each of

these systems shares a common value of some property. We call this property

temperature."

Please note that the aforementioned statement of the Zeroth Law has introduced

three quantities: heat, thermo-equilibrium and temperature in a convoluted manner. It

does not really define “temperature”. We do not need this any more if we accept the

definition of temperature in equation 16 based on the statistical definitions of U ,V and

S. Thermo-equilibrium between two systems is then defined as the two having the same

7 https://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamics

temperature, and heat flow is defined as the energy transferred between two system

with different temperatures. The Zeroth law of thermodynamic is no longer needed, or it

may be re-interpreted as:

“There will be energy transfer between two system caused by different

temperatures. The energy transferred induced by different temperatures is known as

heat.”

1.2.4 INTERNAL PRESSURE, EXTERNAL PRESSURE, WORK AND SHAFT-WORK

Any system will sustain external forces acting it by the environment. These forces

will act on the surfaces and induces surface stresses (Figure 1. 5). The external pressure

Pext acting on the system, or hydrodynamic pressure, is defined as the average of the

normal stresses acting on the surface of the system:

Pext=13

(σ xx+σ yy+σzz )

12112\

*

MERGE

FORMA

T (.)

Figure 1. 5 : Surface stresses acting on a system

It should be distinguished from the pressure defined in equation 17 which is sometimes

called internal pressure P∫¿ ¿, or thermodynamic pressure.

The workdone by these external forces will cause the system to change in shape or

volume (Figure 1. 6).

P-V work Shaft work

Figure 1. 6 : PV-work and Shaft-work

The part of work required to change the volume is given by −Pext dV and is known as the

PV-work. The rest of the work is known as shaft work.

δW=−P ext dV+δW shaft

13113\

*

MERGE

FORMA

T (.)

1.2.5 PROCESS AND EXCHANGE

Notice heat is the transfer of energy between two systems, or the system and the

surrounding; and work is the result of external force on the system. Hence they are

exchange quantities which are dependent on the process, i.e. interaction between the

system and its environment that changes the thermodynamic state of the system. They

are not thermodynamic state variables which are characteristic of the system.

As shown in Figure 1. 7, a control mass or closed system can only exchange heat and

work with its environment, while a control volume or open system can exchange mass,

heat and work with its environment.

Figure 1. 7 : Exchanges with the environment of closed and open systems

In thermodynamics, an isolated system is defined as a close system that has no

exchange in heat and work with its environment. Since the universe has no

environment, it cannot exchange anything. An universe is an “isolated system”.

1.3 BALANCE EQUATIONS AND LAWS OF THERMODYNAMICS

1.3.1 ENERGY BALANCE AND FIRST OF THERMODYNAMICS

The total energy in a system include internal energy U , kinetic energy KE and

potential energy PE.

E=U+KE+PE

14114\

*

MERGE

FORMA

T (.)

The kinetic and potential energies are related to the velocity and position of the center

of mass of the system. The kinetic and potential energies of movement of molecules

relative to the center of mass were included in the internal energy U .

The accumulation of total energy in a close system with variable volume V is

therefore given by:

dEdt

=∑j

Q j+∑k

W kshaft−Pext dV

dt

15115\

*

MERGE

FORMA

T (.)

where Q j, and W kshaft are various sources of heat or shaft-work exchanged between the

system and the environment. We have adopted the convention that input to the system

is positive.

The corresponding accumulation of total energy in an open system with fixed

volume and boundary is therefore given by:

dEdt

=∑i

ni(U i+KEi+PEiMW i

+PiextV i)+∑

j

Q j+∑k

W kshaft

16116\

*

MERGE

FORMA

T (.)

ni are the molar flow rates of various streams of material exchange with the

environment. U i, MW i are the molar internal energy and molecular weight of these

streams. KEi and PEi are the kinetic and potential energy of these streams. The terms

PiextV i represents the work required pushing these streams into the control-volume

system.

The above balance equations simply stated the fact that:

energy accumulation=energy input−energy output

Since there is no source or sink, we are implying that total energy is conserved, which

constitute the First Law of Thermodynamics8.

1.3.2 ENTROPY BALANCE AND THE SECOND LAW OF THERMODYNAMICS

The entropy balance of a closed system is shown here.

8 If there is only one heat source and work source, and by omitting the time derivative; we can recover the more common but limited form of the First Law of Thermodynamics (for a closed system) that is found in many undergraduate textbooks dE=δQ+δW , where δW=−P ext dV+δW shaft.

dSdt

=∑j

Q j

T js +d Sgendt

17117\

*

MERGE

FORMA

T (.)

The term on the left is the accumulation of entropy in the system, while the first term on

the right is the entropy brought into and out of the system by heat exchange with the

surrounding. T js are the temperatures of the surrounding sources or sinks at the points

of exchange. The second term d Sgendt

is the entropy generated during the process.

The corresponding entropy balance equation for an open system is given by:

dSdt

=∑i

niS i+∑j

Q j

T js +d Sgendt

18118\

*

MERGE

FORMA

T (.)

The first term on the right hand side represents the entropy brought into/out-of the

system by the material input/output streams.

Note that the above equations are not the Second of Thermodynamics, they are

only an accounting equation of entropy change. However, the inclusion of the term

d Sgendt

expressed the important concept that unlike energy, entropy is not a conserved

quantity. The Second Law of Thermodynamics states that entropy can be generated but

not destroyed:

d Sgendt

≥0

19119\

*

MERGE

FORMA

T (.)

It should be pointed out that the entropy of a system can increase or decrease, it is

only the entropy generated in the process that is positive. There are many ways of

expressing the Second Law of Thermodynamics9,10, but equation 119 offers the most

concise way of stating it with the least qualifications.

1.3.3 REVERSIBILITY

Entropy can only be generated but not destroyed in a process, a process in which no

entropy is generated is known as a “reversible” process:

d Sgendt

=0⇒Process is reversible

20120\

*

MERGE

FORMA

T (.)

9 If there is only one heat exchange δQ and the system has an uniform temperatureT , equations 117 and 119 reduce to the more common form of the Second Law of Thermodynamics for a closed system dSdt≥δQT

.

10 It is very common to give the Second Law of Thermodynamic as “The entropy of the universe cannot decrease.” While this is indeed true, a more precise way to put it is “The entropy of an isolated system cannot decrease.” It is certainly not true that “The entropy of a system cannot decrease.” Increase in entropy in an open system, “self-organization” is very important phenomenon in biological and social systems.

To illustrate the concept of reversibility, we shall use the adiabatic expansion of a mass of

gas against and external pressure Pext with no shaft work as an example (Figure 1. 8).

Under what condition is the expansion reversible?

Figure 1. 8 : The adiabatic expansion of a mass of gas against and external pressure Pext with no shaft work

According to 117, since there is no heat exchange with the surrounding and the

process is reversible, we have

dSdt

=∑j

Q j

T js +d Sgendt

=0

21121\

*

MERGE

FORMA

T (.)

In other words, the system’s entropy remain constant, the process is “isentropic”. In

addition, if we assume that the changes in kinetic and potential energy are negligible

equation 115 reduces to

dU+KE+PEdt

=∑j

Q j+∑k

W kshaft−Pext dV

dt

22122\

*

MERGE

FORMA

T (.)

But according to equation 18, we have

dUdt

=T dSdt

−P∫¿ dV

dt¿

23123\

*

MERGE

FORMA

T (.)

Hence

dUdt

=−P∫ ¿ dV

dt=−Pext dV

dt⇒ P∫ ¿=P ext ¿ ¿

24124\

*

MERGE

FORMA

T (.)

For an expansion to occur adiabatically the internal pressure is equal to external

pressure. However, if the internal pressure is equal to external pressure, how can there

be any expansion?

The above questions can be answered if we examine an irreversible adiabatic

expansion with no shaft work, we have

dSdt

=∑j

Q j

T js +d Sgendt

25125\

*

MERGE

FORMA

T (.)

The energy balance in 122 have not changed, but equation 123 becomes

dUdt

=TdSdt

−P∫¿ dV

dt=T

d Sgendt

−P∫ ¿ dV

dt¿¿

26126\

*

MERGE

FORMA

T (.)

Td Sgendt

=¿

27127\

*

MERGE

FORMA

T (.)

Hence the Second Law of Thermodynamics stipulated that when internal pressure is

greater than the external pressure, the system must expand:

P∫¿ ≥ Pext⇒ dV

dt≥0¿ 28128\

*

MERGE

FORMA

T (.)

when internal pressure is less than the external pressure, the system must contract:

P∫¿ ≤ Pext⇒ dV

dt≤0¿

29129\

*

MERGE

FORMA

T (.)

Hence the reversible expansion is the hypothetical limiting case when the internal

pressure is infinitesimally greater than external pressure and the expansion rate is

infinitesimally small.

We can extend the above observation to all other reversible processes. All

reversible processes are hypothetical limiting cases when there are only infinitesimally

small gradient, be it temperature, pressure, concentration or other, that drives the

process and the process rate is infinitesimally small.

1.3.4 EXAMPLE—TANK EMPTYING

So far, we have only introduce concepts. In this example, we shall see how the

balance equations and laws of thermodynamics can be used to solve problems. The

example (Figure 1. 9) is as follows:

Figure 1. 9 : Adiabatic emptying of a tank

An ideal gas is withdrawn from a tank V with initial pressure Pi and initial

temperature T i until the pressure inside Pf equals the outside pressure Po through a

well-insulated valve. The evacuation process is so fast that it can be regarded as

adiabatic. We would like to know

How much gas is left inside the cylinder?

What is the final temperature of the cylinder at this time?

If we draw a control volume around the tank as shown in Figure 1. 10

Figure 1. 10 : Control volume around the tank

There is no heat or work exchange with the surrounding. The only exchange is the gas

flowing out of the tank. Hence we have:

dEdt

=d (U+KE+PE )

dt=dnUdt

=U dndt

+n dUdt

=dndt (U+

KEi+PEiMW i

+PV )+∑j

Q j+∑k

W kshaft

30130\

*

MERGE

FORMA

T (.)

ndUdt

=PV dndt

31131\

*

MERGE

FORMA

T (.)

For an ideal gas, the internal energy is only dependent on temperature11

11 We shall introduce the ideal gas model and other theory of states in the next chapter.

ndUdt

=nCVdTdt

=PV dndt

=RT dndt

32132\

*

MERGE

FORMA

T (.)

CVRT

dTdt

=1ndndt

33133\

*

MERGE

FORMA

T (.)

ln ( nfn i )=CVRln(T fT i )

34134\

*

MERGE

FORMA

T (.)

(P f VRT fPiVRT i

)=(T fT i )CV

R ⇒( PfPi )=(T fT i )CV

R+1=(T fT i )

CP

R

35135\

*

MERGE

FORMA

T (.)

By now one must realize that the temperature-pressure relation obtained using control

volume approach is the same as an adiabatic reversible expansion of a fixed mass of gas.

However, in our derivation, the Second Law of Thermodynamics was never used! This is

possible because we have already specified the final pressure and constrained the

process to be adiabatic and no shaft work. But the question is which mass of gas has

under undergone an adiabatic reversible expansion? This is left as exercise for the

readers.

1.4 SECOND LAW REVISITED

1.4.1 IRREVERSIBILITY AND SPONTANEITY

Consider an adiabatic chamber partitioned into two equal parts each containing the

same gas A. If we remove the partition the gas will mix. If we put the partition back in

place, we will recover the original state with two chambers of gas A again. We note that

since the initial and final conditions are the same, and that the process is adiabatic we

have

∆ S=∆ Sgen=0

In other words, the process is reversible

Figure 1. 11 : Partition and mixing of two chambers of the same gas

However, if the two parts contain two different gases A and B, the results are quite

different. If we remove the partition the gases will mix. If we put the partition back in

place, we will have two chambers containing mixtures of A and B, instead of the original

state with one chamber of pure A and one chamber of pure B again.

A B A,B A,B A,B

ln ln 0A A B BS R n x n x 0S

Figure 1. 12 : Partition and mixing of two chambers of two different gas

Again, using theory of ideal gas, we have

∆ S=∆ Sgen=−R (nA ln xA+nB ln x B )>0

The process is irreversible.

Whether the process is irreversible or not depends on how easily can we return the

original state. In this case, the irreversibility is caused by the spontaneous mixing of the

two gases. Thus we can generalized that a spontaneous process must be irreversible.

1.4.2 IRREVERSIBILITY AND FRICTION AND DISSIPATION

Let us consider a simple example of pushing a block from one end of a table and

back (Figure 1. 13).

(a) Initial

(b) Intermediate

(c) FinalFigure 1. 13 : Moving a block on a table

We know that how easily it can be done, i.e. how reversible is the process depends on

the friction between the block and the table. Since the initial and final state were the

same, we have

∆U=0=δQ+δW Friction

36136\

*

MERGE

FORMA

T (.)

∆ S=0= δQ

T env+∆Sgen

37137\

*

MERGE

FORMA

T (.)

∆ Sgen=δW Friction

T env≥0

38138\

*

MERGE

FORMA

T (.)

The entropy generated which is index of irreversibility is proportional to the workdone

against friction. In general, friction dissipated as heat causes irreversibility.

The relation between frictional forces and irreversibility can also be demonstrated

by steady state, adiabatic fluid flow in a pipe (Figure 1. 14).

Figure 1. 14 : Fluid flow in a pipe

The energy balance is given by

dEdt

=0=∑i

ni(U i+KEi+PEiMW i

+PiextV i)+∑

j

Q j+∑k

W kshaft

39139\

*

MERGE

FORMA

T (.)

0=−∆ H−∆ KE−∆PE+ Wshaft

n=−∆H−∆ KE−∆ PE+W shaft

40140\

*

MERGE

FORMA

T (.)

The ∆ is defined as the difference between outlet and inlet. W shaft is the total amount of

work per unit flow that is required in the system. The negative signs are there because

we define originally input to the system as positive. The entropy balance is given by:

dSdt

=0=∑i

n iSi+∑j

Q j

T js +d Sgendt

41141\

*

MERGE

FORMA

T (.)

0=−∆ S+ 1n

d Sgendt

=−∆S+Sgen

42142\

*

MERGE

FORMA

T (.)

Sgenis the entropy generated in the process per unit flow. According to the definition of

enthalpy, we have

dH=TdS+VdP⇒∆ H=T ∆ S+V ∆P=T ∆S+ 1ρ∆ P=T Sgen+

1ρ∆P 43143\

*

MERGE

FORMA

T (.)

W shaft=∆ KE+∆ PE+ 1ρ∆ P+T Sgen

44144\

*

MERGE

FORMA

T (.)

The above equations can be compared to the Bernoulli equation that include the viscous

loss12 which is obtained by momentum balance:

∆W shaft=∆ KE+∆ PE+ 1ρ∆P+Φvis

45145\

*

MERGE

FORMA

T (.)

The last term is known as the viscous loss of the system which is caused by friction

between the flow fluid of different velocities and the friction with the pipe wall.

Combining 144 and 145, we can see that the entropy generation, or irreversibility, is

caused by the viscous loss:

12 The Bernoulli equation between two section of the pipe is often given in the following form

P1ρ

+v12

2+gz1=

P2ρ

+v22

2+g z2. The last terms on both side of the equations represent the kinetic and

potential energies per unit mass, which correspond to KE and PE in equations 144 and 145. However this applied only to inviscid flow, i.e. the effect of viscosity can neglected and there is no pumping work added to the system. A more general form should include both viscous loss and pump, for example, equation 4.74 of “Unit Operation of Chemical Engineering”, 7th Edition, McCabe, Smith and Harriott, McGraw-Hill 2005.

T Sgen=Φvis

46146\

*

MERGE

FORMA

T (.)

1.4.3 IRREVERSIBILIT AND CARNOT EFFICIENCY

The development of the Second Law of Thermodynamics started with the study of

heat engines. The French engineer/physicist Carnot13 was the first to theorize that when

heat is converted into work in an engine cycle, there is a limit in the efficiency η, i.e.

amount of work obtained per unit of heat input. Some heat must be rejected as waste

heat. Moreover, Carnot derived that this limit is determined by the temperature of the

heat source from which we receive useful heat T H, and the temperature of the heat sink

to which we output waste heat T L.

η=(1−T L

T H )

47147\

*

MERGE

FORMA

T (.)

Consider the following cyclic process, as shown in Figure 1. 15, which takes in QH

amount of heat from a heat source at T H , generate W amount of work and release QL

amount of heat from a heat sink at T L.

13 https://en.wikipedia.org/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot

QH

QLTL

TH

W

Figure 1. 15 : A General Heat Engine Cycle

Since the engines run in cycles, there is no change in thermodynamic state in the

working fluid. The energy balance and entropy balance are given by:

ΔU=QH−W−QL=0⇒QH=W+QL

48148\

*

MERGE

FORMA

T (.)

Δ S=QH

T H−QL

T L+Δ Sgen=0⇒QL=

T LT HQH+T L ΔSgen

49149\

*

MERGE

FORMA

T (.)

W=(1− T LT H

)QH−T LΔ Sgen

50150\

*

MERGE

FORMA

T (.)

Since the entropy generation is non-negative, we have

T L ΔSgen≥0⟹W ≤(1− T LT H

)QH

51151\

*

MERGE

FORMA

T (.)

ηmax=Wmax

QH(1− T L

T H)

52152\

*

MERGE

FORMA

T (.)

1.4.4 IRREVERSIBILITY AND FREE ENERGY/EXERGY

The idea of the maximum amount of work that can be extracted, or minimum

amount of work that is required, can be extended to other systems. For example,

consider a closed system held at constant volume, and isothermal at the temperature T

by a surrounding reservoir also at temperature T env=T , we can simplify the energy and

entropy balance equations as:

dU+KE+PEdt

=∑j

Q j+∑k

W kshaft−Pext dV

dt⇒ dUdt

=Q+W shaft 53153\

*

MERGE

FORMA

T (.)

dSdt

=∑j

Q j

T js +d Sgendt

⇒ dSdt

=QT

+d Sgendt

54154\

*

MERGE

FORMA

T (.)

According to the definition of Helmholtz free energy in equation 110, its rate of change is

given by:

dAdt

=dUdt

−T dSdt

−S dTdt

=Q+W shaft−Q−Td Sgendt

=W shaft−Td Sgendt

55155\

*

MERGE

FORMA

T (.)

We have

W shaft=dAdt

+Td Sgendt

56156\

*

MERGE

FORMA

T (.)

Since d Sgendt

≥0; we can derive that if the system increases in Helmholtz free energy, the

minimum work required is the increase in Helmholtz free energy:

dAdt

>0 ;d Sgendt

≥0⇒Wshaft

≥dAdt

+Td Sgendt

⇒

Wminshaft=dA

dt

57157\

*

MERGE

FORMA

T (.)

Conversely, if the system decreases in free energy, the maximum amount of work that

can be extracted from the process is the decrease in Helmholtz free energy:

dAdt

<0 ;d Sgendt

≥0⇒ ˙−Wshaft

≤−dAdt

−Td Sgendt

⇒

−W maxshaft=−dA

dt

58158\

*

MERGE

FORMA

T (.)

For a closed system held at constant pressure, and isothermal at the temperature T

by a surrounding reservoir also at temperature T env=T , we have

|dGdt |={ Wminshaft dG

dt>0

−W maxshaft dG

dt<0

59159\

*

MERGE

FORMA

T (.)

For an open system, we can define a term known as exergy or available energy14

14 https://en.wikipedia.org/wiki/Exergy

B=H−Te S

60160\

*

MERGE

FORMA

T (.)

where T e is an arbitrarily defined reference temperature of the environment, e.g. 300 K

is commonly used. If the system is at steady state, the changes in potential and kinetic

energy are neglected, and the external pressures are equal to the system pressure at the

inlet and outlet conditions, we have

dEdt

=∑i

ni(U i+KEi+PEiMW i

+PiextV i)+∑

j

Q j+∑k

W kshaft⇒ 0=∑

i

niH i+∑j

Q j+∑k

W kshaft

61161\

*

MERGE

FORMA

T (.)

dSdt

=∑i

niS i+∑j

Q j

T js +d Sgendt

⇒

0=∑i

niS i+∑j

Q j

T js +d Sgendt

62162\

*

MERGE

FORMA

T (.)

Multiply equation 162 by T e and subtract from 161, we have

0=∑i

ni (H i−Te S i)+∑

j (1−Te

T js )Q

j

+∑k

W kshaft−Te

d Sgendt

63163\

*

⇒−∑i

n iBi+Te d Sgendt

=∑j (1−T

e

T js )Q

j

+∑k

W kshaft

MERGE

FORMA

T (.)

The term (1−Te

T js ) in front of Q j looks exactly like the Carnot efficiency with the

temperature of the environment as heat sink. Hence we can define the total equivalent

work ∆W tot as:

∆W tot≡∑j (1−T

e

T js )Q

j

+∑k

W kshaft

64164\

*

MERGE

FORMA

T (.)

If we define the exergy input-output change as ∆ B:

∆ B≡−∑i

ni Bi

65165\

*

MERGE

FORMA

T (.)

we have

∆ B+T ed Sgendt

=∆W tot 66166\

*

MERGE

FORMA

T (.)

Then the concepts of maximum work obtained or minimum work required can also be

related to the absolute change of exergy:

|∆B|={ ∆Wmintot ∆B>0

−∆W maxtot ∆B<0

67167\

*

MERGE

FORMA

T (.)

Furthermore the thermodynamic efficiency η can be defined as:

η={∆ B

∆W tot=∆B

∆B+T ed Sgendt

∆B>0

−∆W tot

−∆ B=

−∆B−T ed Sgendt

−∆B∆B<0

68168\

*

MERGE

FORMA

T (.)

1.4.5 EXAMPLE—EXERGY ANALYSIS AND THERMODYNAMIC EFFICIENCY OF A DISTILLATION COLUMN

The following example is used to show how a process can be assessed using exergy

analysis and thermodynamic efficiency. Methanol and water are separated by a

distillation columns. Results of a specific design simulated using ASPEN Plus are shown

in Figure 1. 16.

Figure 1. 16 : Methanol+Water column and Aspen Plus simulation results

The enthalpies and entropies of the input and output streams are given. Assuming an

environmental reference temperature of 300 K, the increase in exergy ∆ B can be

calculated

∆ B=0.014 kmols (−2.80e6 kJ

kmol+300K ∙147 kJ

kmol ∙K )+0.014 kmols (−2.35e6 kJkmol

+300K ∙227 kJkmol ∙K )−0.028 kmols (−2.58e6 kJ

kmol+300K ∙186 kJ

kmol ∙K )=16.5kW

69169\

*

MERGE

FORMA

T (.)

The reboiler output is 371 K, to supply this duty, low pressure steam (~4 barG, 425 K) is

usually used as the heat source. The condenser operates around 337 K, normal cooling

water at 300 K can be used as the heat sink. Therefore the total equivalent work can be

calculated as:

∆W tot=(1−300K425K )1002kW−(1−300K300K )975kW=293kW 70170\

*

MERGE

FORMA

T (.)

Hence the efficiency of the process is merely:

η=16.5kW293kW

×100%=5.6 %

71171\

*

MERGE

FORMA

T (.)

If we examine many chemical processes, the thermodynamic efficiencies can be quite

low. It is our duty as chemical engineers to find ways to improve these processes.

Exergy and thermodynamic efficiency analysis do not in themselves tell us how to

improve the process. But they provide an evaluation criterion so that different options

can be compared and a target can be set15. Furthermore, we note that poor efficiency is

caused by large irreversibility. Hence, we should look for waste generated by

unnecessary mixing, failure to recover high temperature heat, failure to use sub-room-

temperature streams as cooling medium and equipments that generate large frictional

dissipation.

1.5 SUMMARY

Important concepts introduced in this chapter include

Open and close systems

15 With current technology, a process with a thermodynamic efficiency of 50% is considered very good and difficult to improve.

State and state function

Balance equations

Laws of Thermodynamics

Implications of Second Law of Thermodynamics

Reversibility and spontaneity

Friction and Dissipation

Efficiency and Free Energy