!f t? * 0, then u0.m and ho,m are different. Most calculations involve changes in enthalpy or internal energy, and if the composition during a process is invariant the values of ho,m or u0.m will cancel.

However, if the composition changes during a process it is necessary to know the difference between the values of u0,m for the different species at the reference temperature. This is discussed below.

Obviously u0 ,m and ho.m are consequences of the ideal gas assumption and the equation

hm = hm(T) + ho,m

contains the assumption that the ideal gas law applies down to T0 • If T0 = 0, or a value outside the superheat region for the gas being considered, then the gas ceases to be ideal, often becoming either liquid or solid. To allow for this it is necessary to include latent heats. This will not be dealt with here, but the published data do include these corrections.

9.2.3 ENTROPY OF AN IDEAL GAS - THIRD LAW OF THERMODYNAMICS

The change in entropy during a process is defined as

ds = d~ - 9tdp m (9.15)

T p

If the functional relationship hm = hm(T) is known then eqn (9.15) may be evaluated, giving

s -s =JT d~ -9tin.£.. m 0,m Tc

o T Po (9.16)

where So.m is the value of Sm at T0 and p0 •

It is convenient to take the reference temperature T0 as absolute zero. It was previously shown that

and hence d~ = cp.m(T) dT

(9.17) T T

Consider

If cp.m(T0 ) ~ 0 then the expression is either infinite or indeterminate. However, before reaching absolute zero the substance will cease to be an ideal gas and will become a solid. It can be shown by the Debye theory and experiment that the specific heat of a solid is given by the law cP = aT3

• Hence

( ) (

3 ) d~ aT dT 2 lim - = lim -- = lim (aT dT)~O T -> 0 T T->0 T T-> 0

(9.18)

To integrate from absolute zero to T it is necessary to include the latent heats, but still it is possible to evaluate f dhm / T.

If sm(T) is defined as

J T dh,.,

Srr,(T) = -dT To T

then entropy

Sm= Sm(T) - 9t In !!_ + So.m Po

(9.19)

(9.20)

The term So,m is the constant of integration and this can be compared with u0.m and ho,m· If the composition is invariant then the value of so,m will cancel out when evaluating changes in s and it is not necessary to know its value. If the composition varies then it is necessary to know, at least, the difference between s0 values for the substances involved. It is not possible to obtain any information about s0 from classical (macroscopic) thermodynamics but statistical thermodynamics shows that 'for an isothermal process involving only phases in internal equilibrium the change in entropy approaches zero at absolute zero'. This means that for substances that exist in crystalline or liquid form at low temperatures it is possible to evaluate entropy changes by assuming that differences in so.m are zero.

It should also be noted that the pressure term, p, in eqn (9.20) is the partial pressure of the gas ifit is contained in a mixture, and Po is a datum pressure (often chosen as 1 bar or, in the past, 1 atmosphere).

9.2.4 THE GIBBS ENERGY (FUNCTION) OF AN IDEAL GAS

The Gibbs energy will now be derived for use later when considering the equilibrium composition of mixtures (dissociation). By definition,

and

hm = hm(T) + ho,m

If the pressure ratio, pf p 0 , is denoted by the symbol p" i.e. Pr= pf p0 , then

Sm= Sm(T) - 9t In Pr+ So,m

and

g = h(T) + h0 - T(s(T)- 9t In Pr+ s0)

= (h0 - Ts0 ) + (h(T)- Ts(T)) + 9tT In Pr

If the terms at T0 are combined to give

go,m = ho,m - Tso,m

and the temperature-dependent terms are combined to give

then

(9.21)

(9.22)

(9.23)

(9.24)

(9.25)

(9.26)

(9.27)

Often gm(T) and g0 ,m are combined to give g~ which is the pressure-independent portion of the Gibbs energy, i.e. g~ = gm(T) + g0 ,m, and then

(9.28)

g~ is the value of the molar Gibbs energy at a temperature T and a pressure of p0 , and it is a function of temperature alone. The datum pressure, p 0 , is usually chosen as 1 bar nowadays (although much data are published based on a datum pressure of 1 atm; the difference is not usually significant in engineering problems, but it is possible to convert some of the data, e.g. equilibrium constants).

Gibbs energy presents the same difficulty when dealing with mixtures of varying composition as u0 , h0 and s0 • If the composition is invariant, changes in Gibbs energy are easily calculated because the g0 terms cancel. For mixtures of varying composition g0 must be known. It can be seen from eqn (9.26) that if the reference temperature, T0 , equals zero then

(9.29)

9.3 Tables of u( T) and h( T) against T

These tables are based on polynomial equations defining the enthalpy of the gas. The number of terms can vary depending on the required accuracy and the temperature range to be covered. This section will limit the number of coefficients to six, based on Benson (1977). The equation used is of the form

~(T) hm - ho,m 2 3 4 -- = =a1 +aiT+~T +a4T +asT

9\T 9\T (9.30)

The values of the coefficients for various gases are listed in Table 9.3, for the range 500 to 3000 K. If used outside these ranges the accuracy of the calculation will diminish.

Table 9.3 Enthalpy coefficients for selected gases found in combustion processes (based on kJ / kmol, with the temperature in K)

ho= ao Substance a, a. a, a2 a, a6 kJ/kmol

H, 0.0000 - l.44392e-11 9.66990e-8 -8.18100e-6 3.43328 -3.84470 0.0000

co 0.0000 -2.19450e-12 -3.22080e-8 3.76970e-4 3.317000 4.63284 -l.13882e5

N, 0.0000 -6.57470e-12 l.95300e-9 2.94260e-4 3.34435 3.75863 0.0000

NO 0.0000 -4.90360e-12 -9.58800e-9 2.99380e-4 3.50174 5.11346 8.99147e4

co, 0.0000 8.66002e-l 1 -7.88542e-7 2.73114e-3 3.09590 6.58393 -3.93405e5

o, 0.0000 l.53897e-ll - l.49524e-7 6.52350e-4 3.25304 5.71243 0.0000

H20 0.0000 -l.81802e-ll 4.95240e-8 5.65590e-4 3.74292 9.65140e-l -2.39082e5

CH., -8.586lle-15 l.62497e-10 - l.24402e-6 4.96462e-3 1.93529 8.15300 -6.6930e4

0 0.0000 - l.38670e-l l l.00187e-7 -2.51427e-4 2.76403 3.73309 2.46923e5

Hence the enthalpy, internal energy, entropy and Gibbs energy can be evaluated as follows.

Enthalpy

hm(T) = 9tT(a1 + a2T + a3T 2 + a4T 3 + a5T 4)

= 9t(a 1T + a2T 2 + a3T 3 + a4T 4 + a5T 5)

5

=9tLa;T; i= I

Internal energy

um(T) = 9tT((a 1 - 1) + a2T + a 3T 2 + a4 T 3 + a5T 4)

= 9t((a1 - l)T + a2T 2 + a3T 3 + a4T 4 + a5T 5)

5

= 9t L a;T; - 9tT = h(T) - 9tT i= I

Entropy

This is defined by eqn (9.15) as

ds = d~ -9t dp m T p

Integrating eqn (9 .15) gives

Sm(T) =Sm - So,m = J~ d~ - 9t ln(:)

Equation (9.20) provides some problems in solution. The first is that

Jr d~ =Jr cp,mdT

r,, T 70 T

(9.31)

(9.32)

and this results in ln(O) when T = T0 • Fortunately these problems can be overcome by use of the Van't Hoff equation, which will be derived in Chapter 12, where dissociation is introduced. This states that

h = -T - (µ':JT) 2( d 0 ) m dT

(9.33)

where, for T0 = 0 K

µ~=chemical potential= gm(T) + g0.m = gm(T) + ho,m (9.34)

and

hm = hm(T) + ho,m

(Note: the term ~. is similar to g~, which was introduced in eqn (9.28). At this stage it is sufficient to note that the chemical potential has the same numerical value as the specific Gibbs energy. The chemical potential will be defined in Section 12.2.)

By definition, in eqn (9.31),

hu,(T) hu, - ho.m 2 3 4 -- = = al + ai.T + ll]T + ~T + asT 9tT 9tT

(9.35)

giving

hu,(T) a1 2 3 -- =-+ai+llJT+~T +asT

9tT T (9.36)

Substituting eqns (9.34), (9.13) and (9.36) into eqn (9.33) and rearranging gives

(9.37)

which can be integrated to

- a ln T + a2T + _3_ + _4_ + _s_ - ~ +A = m "0,m l a T2

a T3

a T4

z. I g, (T) + z.

I 2 3 4 9tT 9tT (9.38)

It is conventional to let

(9.39)

(9.40)

The value of entropy can then be obtained from

s. (T) = hu,(T) - 8m(T) m T (9.41)

(9.42)

Gibbs energy

The Gibbs energy can be evaluated from eqn (9.40) or eqn (9.26):

8m(T) = hm(T) - Tsm(T)

The values obtained from this approach, using eqns (9.31) to (9.42) are given in Tables 9.4 for oxygen, nitrogen, hydrogen, water, carbon monoxide, carbon dioxide, nitric oxide and methane. The tables have been evaluated up to 3500 k, slightly beyond the range stated in relation to Table 9.3, but the error is not significant. This enables most combustion problems to be solved using this data. Other commonly used tables are those of JANAF (1971).

9.3.1 TABLES OF MEAN SPECIFIC HEAT

Sometimes data are given in terms of mean specific heat rather than the actual specific heat at a particular temperature. This approach will now be described.

Consider the change of enthalpy, h, between temperatures T1 and T2 • This is

I 1i h2 - h, = Cp(T) dT = h(1;) - h(T..)

7j

Now, this can be written

(9.43)

(9.44)

where cP is the mean specific heat between T1 and T2• Normally cP is given at a particular temperature T, and it is then defined as the mean between the temperature T and a reference temperature Trer• i.e.

_ h(T) - h(Trer) (cph=-----

T-Trer (9.45)

It is also possible to write the mean specific heat as a polynomial function of temperature, in which case

(9.46)

where a, band care tabulated coefficients. If it is required to calculate the enthalpy difference between two temperatures T1 and T2 ,

then

(9.47)

Having calculated the change in enthalpy, h2 - h1, then the change in internal energy, u2 - u1, may be evaluated as

(9.48a)

·>.-::. .·E.~tha_l_PY at T, .. ·· .. ·· ... · .

T0 = 0 T, Temperature, T

Fig. 9.1 Mean specific heat at constant pressure

or, in molar terms, by

um,2 - um.1 = hm.2 - ~T2 - (hm.1 - ~Ti)= hm,2 - hm.1 - ~(T2 - Ti) (9.48b)

If the change of internal energy, u2 - u 1, is known, then the change of enthalpy, h2 - h 1,

may be calculated in a similar way. The use of mean specific heat values enables more accurate evaluation of temperature changes because the mean specific heats enable the energy equation to be solved with an allowance for the variation of specific heats. The mean specific heats approach is depicted in Fig 9.1.

9.4 Mixtures of ideal gases

Many problems encountered in engineering involve mixtures of gases - air itself is a mixture of many gases although it can be considered to be oxygen and atmospheric nitrogen (which can be assumed to contain the nitrogen in the air and the other 'inert' gases such as argon and carbon dioxide). If the gases in a mixture are at a temperature well above their critical temperature and a pressure below the critical pressure, they act as ideal gases. Expressions relating the properties of mixtures of ideal gases will now be formulated. These expressions are a direct consequence of the Gibbs-Dalton laws.

9.4.1 DALTON PRINCIPLE

Dalton stated that

any gas is as a vacuum to any gas mixed with it.

This statement was expanded and clarified by Gibbs to give the Gibbs-Dalton laws.

9.4.2 GIBBS-DALTON LAW

(a) A gas mixture as a whole obeys the equation of state (eqn (9.2))

pV=n~T

where n is the total amount of substance in the mixture. (b) The total pressure exerted by a mixture is the sum of the pressures exerted by the

individual components as each occupies the whole volume of the mixture at the same temperature.

( c) The internal energy, enthalpy and entropy of the mixture are, respectively, equal to the sums of the internal energy, enthalpy and entropy of the various components as each occupies the whole volume at the temperature of the mixture.

9.4.3 MIXTURE RELATIONSHIPS

The total mass, m, of a mixture can be related to the mass of constituents by n

m=m0 +mb+mc+ ··· = I:lnj ; ~ 1

From the ideal gas law

m pV= mRT= - mwRT= n~T

mw

(9.49)

and hence for the individual constituents

pVa = maRaT = n/Jl.T

pVb = m~bT = nb9\T

pVc = mficT = n/J\.T, etc

(9.50)

In eqn (9.50), Va= volume of constituent a at the pressure and temperature of the mixture, and Vb and Ve are similar volumes for constituents b and c. But the total volume, V, is given by

n

V=V+V.b+v+ .. ·= '\"'v a c L...ii i= 1

and therefore

Let

n

n=na+nb+nc+ ··· = Ini i= 1

n ~ = xa = mole fraction of a in the mixture n

Similarly, nb/ n = xb, and nJ n = x 0 etc, with nJ n = xi. Therefore

n

Xa+Xb+Xc+···=IA;=l i= I

(9.51)

(9.52)

(9.53)

(9.54)

It is possible to develop the term for the partial pressure of each constituent from statement (b) of the Gibbs-Dalton laws. Then, for constituent a occupying the total volume of the mixture at the pressure and temperature of the mixture,

PaV= n/Jl.T

But, for the mixture as a whole (eqn (9.2))

pV=n9\T

Hence, by dividing eqn (9.50) by eqn (9.2)

giving

Pa= xap, and in general Pi= X;P

and, also

n n n

LP;= Ix;p=p Ix;=p i= 1 i= I i= I

(9.55)

(9.56)

(9.57)

Often mixtures are analysed on a volumetric basis, and from the volumetric results it is possible to obtain the partial pressures and mole fractions of the components. Normally

volumetric analyses are performed at constant pressure and temperature. Then, considering the ith component,

n/Jl.T V;=--

p

and the total volume of the mixture, V, is

n~T V=-­

p

Hence, from eqns (9.58) and (9.59)

V; '1j - = - =X; V n

(9.58)

(9.59)

(9.60)

If the gas composition is given in volume percentage of the mixture, the mole fraction is

V;(%) X;=lQO

and the partial pressure is

V;(%) p;=--p

100

(9.61)

(9.62)

The terms relating to the energy of a mixture can be evaluated from statement ( c) of the Gibbs-Dalton laws. If em= molar internal energy, and hm =molar enthalpy, then for the mixture

and

n

E = nem =I n;em,i i= I

n

H=n~= In;~,; i= I

If the molar internal energy and molar enthalpy of the mixture are required then

and

1 n n

i:, = - "'\'"' n.l. . ="'\'"'xi:, . "m L.,. 11'm,1 L.,. i''m,1

n i=I i=l

(9.63)

(9.64)

(9.65)

(9.66)

Neglecting motion, gravity, electricity, magnetism and capillary effects, then em = um, and hence

(9.67)

The definition of enthalpy for an ideal gas is

hm= um+ !RT

Thus, for the mixture

hm = L(Xi(um + ~T);) = L X;Um,i + L X;~T = L X;Um,i +~TL X;

= L X;Um,i + ~T

9.4.4 SPECIFIC HEATS OF MIXTURES

Statement (c) of the Gibbs-Dalton law and the above expressions show that

and

hm = L X;hm,i

By definition, the specific heats are

Thus

du c =-

v dT

dh and c =-.

P dT

Cv m = ( dum) = L X;( dum) = L X;(Cv m}i . dT dT ; .

and, similarly

Cp,m = L X;(Cp,m);

9.5 Entropy of mixtures

(9.68)

(9.69)

(9.70)

(9.71)

Consider a mixture of two ideal gases, a and b. The entropy of the mixture is equal to the sum of the entropies which each component of the mixture would have if it alone occupied the whole volume of the mixture at the same temperature. This concept can present some difficulty because it means that when gases are separate but all at the same temperature and pressure they have a lower entropy than when they are mixed together in a volume equal to the sums of their previous volumes; the situation can be envisaged from Fig 9.2. This can be analysed by considering the mixing process in the following way:

(i) Gases a and b, at the same pressure, are contained in a control volume but prevented from mixing by an impermeable membrane.

(ii) The membrane is then removed, and the components mix due to diffusion, i.e. due to the concentration gradient there will be a net migration of molecules from their original volumes. This means that the probability of finding a particular molecule at any particular point in the volume is decreased, and the system is in a less ordered state. This change in probability can be related to an increase in entropy by statistical thermodynamics. Hence, qualitatively, it can be expected that mixing will give rise to an increase in entropy.

,' Gas 'a' , , · Pa ,'

T T l;_ . ..,, ____ ..ir.. _ _. ___ ._..._ .......... __ Wll

Fig. 9.2 Two gases at the same pressure, p, contained in an insulated container and separated by a membrane; p 0 =Pb= p

Considering the mixing process from a macroscopic viewpoint, when the membrane is broken the pressure is unaffected but the partial pressures of the individual components are decreased.

The expression for the entropy of a gas is (eqn (9.20))

where sm(T) is a function of T alone. Hence, considering the pressure term, which is actually the partial pressure of a component, a decrease in the partial pressure will cause an increase in the entropy of the gas. Equation (9.20) can be simplified by writing the pressure ratio as pr = p I Po' giving

Sm= sm(T) - 9\ In Pr+ So,m

Now consider the change from an analytical viewpoint. Consider the two gases a and b. The total amount of substance in the gases is

The entropies of the gases before mixing are

Sm. = nasma = naCsm.(T) - 9\ ln Pra + So,mJ

and

smb = nbsmb = nb[smb(T) - 9\ In p,b + So,mbl

giving the sum of the entropies before mixing (i.e. separated) as

(9.72)

(9.73)

(9.74)

(SmT)scp = (nTsmT)SCp = nT[xasm,(T) + xbsm/T) + XaSO,ma + xbso,mb - 9\(xa ln Pra + xb ln p,b)]

(9.75)

Since the gases are separated and both are at the pressure, p,, then

(Sm T)sep = nT[xasm.(T) + XbSmb(T) + XaSo,ma + xbso,mb - 9\ ln p)

Hence, the molar entropy, when the gases are separate, is

(smr)sep = X0 Sm 0(T) + XbSm 0(T) + X0 So,ma + XbSo,mb -9\ ln p

= L X;Sm; + L X;So,m ; - 9\ In p

(9.76)

(9.77)

The entropy of the gases after mixing is

(9.78)

where sma and smb are the molar entropies if each component occupied the whole volume. Thus, substituting for sm from eqn (9.20a) gives

(nTsmT)mix = nasma(T) + nbsmb(T) - 9\(na ln Pra + nb ln Prb) + n.so,ma + nbsO,mb (9.79)

(smT)mix = XaSma(T) + xbsmb(T) - 9\(xa ln Pra + xb ln Prb) + xasO,ma + xbsO,mb

where Pr;= pj p 0 , and p; =partial pressure of constituent i. Equation (9.80) may be written

(9.80)

(smT)mix = L X;Sm/T) + L X;So,m; - 9\ L X; ln Pr; (9.81)

Now, consider the final term in eqn (9.81), based on two constituents to simplify the mathematics

Hence

Xa ln Pra + xb ln Prb = Xa ln(XaPr) + xb ln(xbpr)

= Xa ln Xa + Xa ln Pr+ xb ln xb + xb ln Pr

= I X; lo X; + In Pr I X;

1 = I X; ln X; + In Pr = ln Pr - I X; ln -

X;

(sm)mix =I X;sm,(T) +I X;So,m, + 9\(I X; In ~; - In Pr)

Considering the terms on the right-hand side of eqn (9.83):

1st and 2nd terms summation of entropies before mixing, (SmT)sep

(9.82)

(9.83)

3rd term change of entropy due to mixing; this is due to a change of partial pressures when mixed

4th term pressure term

The change of entropy due to mixing is given by

J}.S = (SmT)mix - (SmT)sep

which is the difference between eqns (9.83) and (9.77). This gives

1 J}.S = 9\ I X; ln -

X;

The numerical value of J}.S must be positive because xi is always less than unity.

(9.84)

(9.85)

Equation (9.85) shows that there is an entropy increase due to mixing, and this is caused by the reduction in the order of the molecules. Before mixing it is possible to go into one side of the container and guarantee taking a particular molecule, because only molecules of a are in the left-hand container, and only molecules of b are in the right-hand container. After mixing it is not possible to know whether the molecule obtained will be of a or b: the order of the system has been reduced.

What happens if both a and bin Fig 9.2 are the same gas?

Superficially it might be imagined that there will still be an increase of entropy on mixing. However, since a and b are now the same there is no change in partial pressure due to mixing. This means that the molar fraction is unaltered and

LiS = 0 (9.86)

9.6 Concluding remarks

A detailed study of ideal gases and ideal gas mixtures has been undertaken in preparation for later chapters. Equations have been developed for all properties, and enthalpy coefficients have been introduced for nine commonly encountered gases. Tables of gas properties have been presented for most gases occurring in combustion calculations.

Equations for gas mixtures have been developed and the effects of mixing on entropy and Gibbs energy have been shown.

Combustion is an oxidation process and is usually exothermic (i.e. releases the chemical (or bond) energy contained in a fuel as thermal energy). The most common combustion processes encountered in engineering are those which convert a hydrocarbon fuel (which might range from pure hydrogen to almost pure carbon, e.g. coal) into carbon dioxide and water. This combustion is usually performed using air because it is freely available, although other oxidants can be used in special circumstances, e.g. rocket motors. The theory that will be developed here will be applicable to any mixture of fuel and oxidant and any ratio of components in the products; however, it will be described in terms of commonly available hydrocarbon fuels of the type used in combustion engines or boilers.

The simplest description of combustion is of a process that converts the reactants available at the beginning of combustion into products at the end of the process. This model presupposes that combustion is a process that can take place in only one direction and it ignores the true statistical nature of chemical change. Combustion is the combination of various atoms and molecules, and takes place when they are close enough to interact, but there is also the possibility of atoms which have previously joined together to make a product molecule separating to form reactants again. The whole mixture is really taking part in a molecular 'barn dance' and the tempo of the dance is controlled by the temperature of the mixture. The process of molecular breakdown is referred to as dissociation; this will be introduced in Chapter 12. In reality a true combustion process is even more complex than this because the actual rate at which the reactions can occur is finite (even if extremely fast). This rate is the basic cause of some of the pollutants produced by engines, particularly NOx. In fact, in most combustion processes the situation is even more complex because there is an additional factor affecting combustion, which is related to the rate at which the fuel and air can mix. These ideas will be introduced in Chapter 15. Hence, the approach to combustion in this chapter is a simplified one but, in reality, it gives a reasonable assessment of what would be expected under good combustion conditions. It cannot really be used to assess emissions levels but it can be extended to this simply by the introduction of additional equations: the basic approach is still valid.

The manner in which combustion takes place is governed by the detailed design of the combustion system. The various different types of combustion process are listed in Table 10.1, and some examples are given of where the processes might be found. There is an interdependence between thermodynamics and fluid mechanics in combustion, and this interaction is the subject of current research. This book will concentrate on the thermo-

dynamics of combustion, both in equilibrium and non-equilibrium states. The first part of the treatment of combustion will be based on equilibrium thermodynamics, and will cover combustion processes both with and without dissociation. It will be found that equilibrium thermodynamics enables a large number of calculations to be performed but, even with dissociation included, it does not allow the calculation of pollutants, the production of which are controlled both by mixing rates (fluid mechanics) and reaction rates (thermodynamics).

Table 10.1 Factors affecting combustion processes

Conditions of combustion

Time dependence

Spatial dependence

Mixing of initial reactants

Flow

Phase of reactants

Reaction sites

Reaction rate

Convection conditions

Compressibility

Speed of combustion

Classification

steady unsteady

zero-dimensional

one-dimensional

two-dimensional three-dimensional

premixed non-premixed

laminar turbulent

single

multiphase

homogeneous heterogeneous

equilibrium chemistry (infinite rate)

finite rate

natural

forced

incompressible compressible

deflagration detonation

Examples

gas turbine combustion chamber, boilers petrol engine, diesel engine

only used for modelling purposes, welJ­stirred reactors approximated in pipe flows, flat flame burners axisymmetric flames, e.g. Bunsen burner general combustion

petrol , or spark ignition, engine diesel engine, gas turbine combustion chamber

special cases for measuring flame speed most real engine cases, boilers

spark-ignited gas engines, petrol engines with fuel completely evaporated; gas-fired boilers diesel engines, gas turbines, coal- and oil­fired boilers

spark-ignition engines diesel engines, gas turbines, coal fired boilers

approached by some processes in which the combustion period is long compared with the reaction rate all real processes: causes many pollutant emissions

Bunsen flame, gas cooker, central heating boiler gas turbine combustion chamber, large boilers

free flames engine flames

most nonnal combustion processes 'knock' in spark ignition engines, explosions