MINIMUM SOLAR RADIATION REQUIREMENTS FOR SOLAR FUELS SYNTHESIS

Pergamon

PII: S0038 – 092X( 98 )00107 – 8

Solar Energy Vol. 66, No. 2, pp. 73–80, 1999  1999 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0038-092X / 99 / $ - see front matter

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MINIMUM SOLAR RADIATION REQUIREMENTS FOR SOLAR FUELS SYNTHESIS ´ CARLOS GOMEZ CAMACHO† Escuela Superior de Ingenieros, Camino de los Descubrimientos, E.S.I., 41092 Seville, Spain Revised version accepted 19 August 1998

Abstract—A methodology that gives the thermodynamic limits to the amount of solar radiation needed to synthesize a pure fuel is proposed. This methodology is based on the concept of ‘‘solar equivalent of a fuel’’, and is related with the reversible reaction of synthesis of the fuel from environmental reactants and solar radiation. Solar equivalent of a fuel is defined as the minimum amount of solar energy needed to synthesize one mole of the pure fuel, at environmental temperature and pressure, from reactants and with the other products contained in the environment, when this one and the Sun are the only energy sources. Through out this paper, the environment is considered as a real mixture of gases, although the methodology can be easily extended to solid or liquid mixtures, or a combination of the different aggregation states. Thermodynamic algorithms depends only on Sun radiation equivalent temperature, thermal equations of state of environment and pure fuel, and standard equilibrium constant of formation for each compound taking part in the chemical reaction. Numerical and graphical results are given for the synthesis of graphite, carbon monoxide, methane, ethane, propane, methanol, ethanol, and dihydrogen, all of them synthesized from atmospheric carbon dioxide and water vapour. The results help to consider under which conditions a fossil fuel should be used when primary energy can be directly solar. Those results give an objective criterium to valorize economically the replenishment costs of fossil fuel in relation to economical costs of solar technologies.  1999 Elsevier Science Ltd. All rights reserved.

defined as the minimum quantity of solar energy necessary for the reaction of synthesis for one mole of pure fuel, at environmental temperature, T, and pressure, p, starting with the reactants and the rest of the products contained in the environment, with this and the Sun being the only sources of energy. Using this design plan, the material replenishments of fuel and the environment are integrally considered. In this paper it is supposed that the environment is a real mixture of gases, although it can be generalized to any other type of aggregation state or mixture of these. Refer to ´ Gomez Camacho (1997) for a complementary thermodynamic deduction which is more systematic than that presented here.

1. INTRODUCTION

The absence of an objective criterion for comparing renewable energy sources and fossil fuels is an obstacle to the development of renewables. Among the most significant failings are that fossil fuels are not explicitly considered: neither the replenishment of the fuels theirselves nor the material replenishment of the environment, which are both aspects which strongly influence the comparison, generally in detriment of renewable energies. Refer to Ruiz de la Torre Ruiz (1996) for the incorporation to energy accounting of the costs of replenishment of fossil fuels, and to Lara Cruz (1995), showing the greater value that should be given to fossil fuels compared to renewable energies. In this paper a methodology for comparing fossil energies and solar energy is proposed, based on the common origin of both: the Sun, as a source of energy, together with the environment as the source and sink of energy and material; along with, in its generation, incorporating and unifying the replenishment of fossil fuels and the environment. The solar equivalent of a fossil fuel, Q SF , is

2. GENERAL APPROACH

Consider the next illustration, (Fig. 1), in which a machine M with a reversible cycle receives heat from the Sun S, receives heat and reactants R from the environment E, for synthesizing one mole of pure fuel F and with the rest of the products P being integrated into the environment. During the course of the reaction it is considered that the temperature, pressure and composition of the environment are constant, as the change of its thermodynamic state as a consequence of the reaction is negligible. The whole reaction takes

†ISES member. E-mail: [email protected] 73

´ C. Gomez Camacho

74

The heat per mole of fuel exchanged between the machine M and the Sun S, Eq. (5), is obtained by eliminating Q EF from Eq. (2) and Eq. (4), followed by canceling Q SF and then applying the definition of Gibbs free energy, G5H2TDS. Q SF ? (1 2 T E /T S ) 5 D r GF (T E , p E , x cE )

(5)

Q SF , obtained after a reversible cyclic process of M, is obtained canceling in Eq. (5), giving Eq. (6). Fig. 1. General setting for the derivation of the solar equivalent of a fuel F.

place at the pressure and temperature of the environment. The synthesis reaction of the fuel is expressed with the stoichiometric coefficients of the products positive, and those of the reactants negative, as 1?F 1

O n ? B 5 0; B

c [E

(nR , 0; nP . 0; R, P [ E; F [ ⁄ E) The variation of any extensive thermodynamic function, X, in the reaction is defined by D r XF (T E , p E , x Ec )81 ? X *F (T E , p E ) 1

O n ? X (T , p , p ) E

B

E

E c

B

c[E

In this article the heat exchanged is considered positive when it tends to increase the thermodynamic energy of the system from which it is evaluated. The first law of thermodynamics applied to the cyclic machine M gives Eq. (1).

TS Q SF (T S , T E , p E , x cE ) 5 ]]] ? D G (T E , p E , x cE ) TS 2TE r F (6) Q SF , is a lower limit for the heat transferred between M and S, as a consequence of DS FU $0 and of the independence of the function of state of the reversibility or irreversibility of the process. Also, D r GF (T E , p E , x cE ) is the opposite of the exergy of the fuel, as can be deduced indirectly from the development above, or as shown in ´ Gomez Camacho (1996).

3. CALCULATIONS RELATING TO THE STANDARD THERMODYNAMIC STATES

The determination of Q FS depends only on D r GF (T E , p E , x Ec ). This calculation is made through reference to the standard thermodynamic states, which are in turn only functions of the temperature, with pressure, composition and thermodynamic behaviour being predetermined. Following the above, Eq. (7) can be written. D r GF (T E , p E , x Ec ) 5 DG m* (T E ) 1 E 1 [ m *F (T E , p E ) 2 m * F (T )] 1

S E E E E DU M F 5 0 5 Q F 1 Q F 2 D r HF (T , p , x c ) (1)

1

The heat per mole of fuel exchanged between M and E, is Eq. (2). E F

S F

E

E

E c

Q 5 2 Q 1 D r HF (T , p , x )

Q EF Q SF U E E E DS F 5 0 5 2 ] 2] 1 D r SF (T , p , x c )(3) TE TS Canceling for Q FE gives Eq. (4). TE Q EF 5 2 Q SF ? ]S 1 T E ? D r SF (T E , p E , x Ec ) (4) T

E

B

E 2 m* B (T )]

(2)

The application of the second law of thermodynamics to the system of bodies taking part in the reaction, or ‘‘Universe’’ U, gives Eq. (3).

O n ? [m (T , p , x ) 2 B

E

E c

c [E

(7)

The variation of the molar standard Gibbs function for the reaction, DGm * (T ), is related to the standard equilibrium constant, K * (T ), as E * E DG * m (T E )5 2 R?T ?ln K (T ). At the same * time, K (T ) is obtained from those for the * formation of each substance, K f,B (T ), with * K * (T ) 5 F 1c [K f,B (T )] nB . The standard thermodynamic state for a gas is pure, at a pressure of 5 10 Pa, at the considered temperature and with ideal gas behaviour. The difference between chemical potentials is Eq. (8).

P

Minimum solar radiation requirements for solar fuels synthesis pE g B

E

E

E c

m (T , p , x ) 2 m

g* B

E

(T ) 5

E FV (T , p, x ) g B

E

E c

D r GF (T E , p E , x Ec ) 5 2 R ? T E ? ln K * (T E ) 1 pE

0

1

G

p E ? x EB R?TE E 2 ]] ? dp 1 R ? T ? ln ]] p p*

75

E V *(T , p) ? dp 1 O n ? l F

E

B

c[E

p*

(8)

pE

R?T ? dp 1 5E FV (T , p, x ) 2 ]] p G E

The difference between the chemical potential of a pure gas and its standard is Eq. (9).

?

E c

p E ? x EB ]] 1 R ? T ? ln p*

E FV *(T , p) g F

E

0

pE

m gF* (T E , p E ) 2 m gF* (T E ) 5

g B

E

E

J

(12)

0 E

G

R?T pE 2 ]] ? dp 1 R ? T E ? ln ] p p*

(9)

The standard thermodynamic state for a liquid or solid is pure, at a pressure of 10 5 Pa, at the temperature considered and with real behaviour. The difference between chemical potentials is Eq. (10).

4. GENERAL EXPRESSION FOR THE SOLAR EQUIVALENT OF THE FUEL

Q SF of a combustible gas, Eq. (13), is obtained from the substitution of Eq. (11) in Eq. (6). Q SF (T S , T E , p E , x cE )

H

TS 5 ]]] ? 2 R ? T E ? ln K * (T E ) 1 TS 2TE

m lF* (T E , p E ) 2 m lF* (T E ) 5

pE

R?T ? dp 1 5E FV *(T , p) 2 ]] p G p 1 R ? T ? ln ]J 1 O n ? p E

5

E V *(T , p) ? dp l F

g F

1

pE

E

E

0

(10)

E

p*

E

*

D r GF (T E , p E , x cE ) depends on the three: (1) thermal equation of state for the environment, V gm (T E , p E , x Ec ); (2) thermal equation of state of the pure fuel, V F* (T E , p E ); and (3) standard * E equilibrium constants of formation, K f, B (T ), of each of the reacting compounds. For the fuel, reactants and products, which are all gases, D r GF (T E , p E , x Ec ), Eq. (11), is obtained by substituting Eq. (8) and Eq. (9) in Eq. (7): D r GF (T E , p E , x Ec ) 5 2 R ? T E ? ln K * (T E ) 1 pE

1

5E F

E

pE

?

g B

E

G

E c

0

p E ? x BE ? ln ]] p*

66

(13)

S

Q F of a condensed fuel, Eq. (14), is obtained from the substitution of Eq. (12) in Eq. (6). TS ]]] Q (T , T , p , x ) 5 S ? T 2TE

G

0

S

E

E

E c

5

? 2 R ? T E ? ln K * (T E )

O

pE 1 R ? T E ? ln ] 1 nB ? p* c [E

pE

pE

?

5E F

R ?TE ]] V (T , p,x )2 ? dp 1 R ? T E ? p

S F

R?T V gF* (T E , p) 2 ]] ? dp 1 p

J

B

c [E

5E F

E

G

R?T V gB (T E , p, x cE ) 2 ]] ? dp 1 p

1

J

l F

E

B

c[E

p*

0

p E ? x EB 1 R ? T E ? ln ]] p*

E V *(T , p) ? dp 1 O n ? pE

(11)

For liquid or solid fuels, and gaseous reactants and products, D r GF (T E , p E , x cE ), Eq. (12), is obtained by substitution of Eq. (8) and Eq. (10) in Eq. (7).

R?T ? dp 1 R ? T 5E FV (T , p, x ) 2 ]] p G E

?

g B

E

E c

E

?

0

p E ? x BE ]] ? ln p*

66

(14)

´ C. Gomez Camacho

76

is considered as an integral radiator at a temperature of T S 55777 K. The algorithm for the calculation is Eq. (16). The reactions considered are the opposite of those of combustion; in a generic form, the reaction per mole of synthesized fuel is

5. SIMPLIFIED EXPRESSION FOR THE SOLAR EQUIVALENT OF THE FUEL

One significant application is: (1) ideal gas fuel V gF* (T, p)5R?T /p), or liquid or solid; (2) the set of the two, reactants and the rest of the products—the environment—a mixture of ideal gases (V gB (T, p, x c )5R?T /p) and (3) standard pressure E * of the environment, p 5p . In this case, Eq. (11) or Eq. (12) convert into Eq. (15).

E

*

E

E

D r GF (T , p , x c ) 5 R ? T ? ln

P (x ) ]]] c [E *

c ? CO 2 ( g) 1 (h / 2) ? H 2 O( g) 5 C c H h Oo 1 (c 1 h / 4 2 o / 2) ? O 2 ( g) From this expression, the extended product of Eq. (16) is Eq. (17):

E nB B

K (T E )

(15)

P

Q SF , Eq. (16), is obtained from substituting Eq. (15) into Eq. (6), or directly from Eq. (13) or Eq. (14).

P

c [E

(16) 6. SOLAR EQUIVALENT OF FUELS IN NORMAL ATMOSPHERIC CONDITIONS

The solar equivalents are given for eight typical fuels in Table 1, whose synthesis reactions are the opposite to those of combustion, in which only the production of CO 2 and H 2 O are admitted. The environment is considered as atmospheric air, at T E 5298.15 K and p E 5p * 510 5 Pa, with ideal gas mixture behaviour and with the standard composition of atmospheric dry air taken from Maslov (1997), x dO 2 5 0,209476; x dCO 2 5 3,14 ? 10 24 and relative humidity f E 550%. Environmental molar fractions become in this case 24

E

(17)

The values of K * (T ) are obtained from the thermochemical tables of Gurvich et al. (1997). The second row of Table 1 shows the results of Q SF , solar equivalent of the fuel, for graphite, carbon monoxide, methane, ethane, propane, methanol, ethanol, and dihydrogen, under environmental conditions before said. As a practical conclusion, from the above results, and as an example, in order to materially replenish one mole of C 3 H 8 a minimum of 2273 kJ of solar energy would be needed. Also, the material replenishment of C 3 H 8 automatically involves that of the atmosphere, in the process of synthesis analyzed in this article. One different and complementary point of view from that above is to consider that 2273 kJ of solar energy can be directly used, or used to produce a maximum of one mole of C 3 H 8 . As can be seen in Table 1, the solar equivalents of the fuels analyzed varies greatly, from 250 kJ / mol for H 2 to 2273 kJ / mol for C 3 H 8 ; in ascending order they are H 2 , CO, C, CH 3 OH, CH 4 , C 2 H 5 OH, C 2 H 6 , and C 3 H 8 . Values of Q SF are lower for simpler molecules, and higher for the hydrocarbon than for its alcohol.

E nB B

(x ) R ?TE ?TS c [E S S E * F ]]] Q F (T , T , p , x c ) 5 ]]] ? ln TS 2TE K * (T E )

E

h o ] ]

(x EO 2 )sc 1 4 2 2 d (x EB )nB 5 ]]]]] h ] (x ECO 2 )c ? (x EH 2 O )s 2 d

x O 2 5 0,20616; x CO 2 5 3,0903 ? 10 ; E

x H 2 O 5 0,01584 The gaseous fuels behave as ideal gases. The Sun

Table 1. Solar equivalents and some derivatives for eight typical fuels Fuel (standard state) Q SF / (kJ / mol) ≠Q SF / ≠T E / [J /(mol?K)] ≠Q SF / ≠ E / [J /(mol?%)]

C (g)

CO (g)

433

290

139

27

0

0.4

CH 4 (g)

C2H6 (g)

C3H8 (g)

CH 3 OH (g)

C 2 H 5 OH (g)

H2 (g)

879

1581

2273

763

1440

250

241

73

187

22

108

2128

2105

2158

2211

2105

2158

253

Environment is considered as atmospheric air, at 298.15 K and 105 Pa, standard composition of dry air and relative humidity 50%.

Minimum solar radiation requirements for solar fuels synthesis

77

7. DEPENDENCE OF THE SOLAR EQUIVALENT OF THE FUELS WITH ATMOSPHERIC TEMPERATURE AND RELATIVE HUMIDITY

The dependence of Q SF with T E and f E can be directly introduced into the simplified expression, being coherent with the hypothesis used for its deduction. Firstly, it is considered that the margin of variation of T E is sufficiently small as to consider DC * p , m (T ) constant. Integrating the Van’t Hoff equation results in ln K * (T ) 5 ln K * (T 0 ) ? 2 [DH m* (T 0 ) 2 T 0 ? DCp,m (T 0 )] /R ? * ? (1 /T 2 1 /T 0 ) 1 DC p,m (T 0 ) /R ?

? ln(T /T 0 ) In this paper temperatures between 0.018C and 508C have been applied, taking the reference temperature T 0 5258C5298.15 K. The values necessary for DCp , m (T 0 ), DHm (T 0 ) and DGm (T 0 ) for the synthesis of each fuel are taken from Gurvich et al. (1997). E Considering the dependence with f , the molar fraction of water vapour present in the atmosphere is calculated from the expression x H 2 O (T, f ) 5 f ? p f1g (T ) /p * . Water vapour pressure p f1g (T ) is calculated from the Clausius–Clapeyron equation, admitting a linear dependence for the enthalpy of vapourization of pure liquid water with the temperature, in accordance with the expression:

F

GS D

P1 p f1g (T ) T ]]] 5 P2 ? exp ]] ? ] Pa K (T /K)

Fig. 2. Solar equivalent of carbon (graphite).

In the following pages the solar equivalents in kJ / mol of the eight fuels included in Table 1 are shown, represented by contour plots, with T E between 0.018C and 508C on the X-axis and f E between 2% and 100% on the Y-axis. See Figs. 2–9, where can be observed the very different behaviour of the eight fuels considered with environmental temperature and relative humidity.

P0

The coefficients P0 , P1 and P2 are adjusted by means of least squares to the data given by Keenan et al. (1978). The values obtained for the P’s are P0 5 25.718857?10 22 , P1 5 25.270859?10 3 and P2 52.089075?10 11 , with which an excellent adjustment for the range of temperatures considered is obtained. The molar fractions for the rest of the components of atmospheric air are obtained by multiplying the corresponding molar fraction in dry air by [1 2 x H 2 O (T, f )]. The product Eq. (17) then converts into Eq. (18).

P

c [E

h o ] ]

h o ] ]

sc 1 4 2 2 d ? (1 2 x EH O )s 4 2 2 d (x E,d O2 ) 2 E nB (x B ) 5 ]]]]]]]]] h ] E,d 2 E s (x CO 2 ) ? (x H 2 O ) 2 d (18)

Fig. 3. Solar equivalent of carbon oxide.

´ C. Gomez Camacho

78

Fig. 4. Solar equivalent of methane.

Fig. 6. Solar equivalent of propane.

For instance, compare the very vertical isolines of graphite, Fig. 2, or carbon monoxide, Fig. 3, with the rest of figures, or the clearly non- linear behaviour of contour lines of dihydrogen, Fig. 9, with the very linear of the others fuels; or even the changing trend of the slope of isolines for methane or methanol (positives at low relative humidities, negatives at high relative humidities) with the monotonous behaviour of the other fuels. Another important aspect is the variation of Q SF with T E , at constant f E , or the variation with f E ,

and constant T E . Even though it is possible to obtain an analytical formula for the corresponding partial derivatives by expressing Q SF as an explicit function of T E and f E , and later deriving, in this paper only the values corresponding to T5258C, f E 550% with standard composition of dry air and atmospheric pressure of 10 5 Pa will be given. The corresponding values have been obtained numerically and are shown in third and fourth rows of Table 1. C 3 H 8 will be used to explain and, as an

Fig. 5. Solar equivalent of ethane.

Fig. 7. Solar equivalent of methanol.

Minimum solar radiation requirements for solar fuels synthesis

Fig. 8. Solar equivalent of ethanol.

application of, starting with atmospheric conditions T E 258C, p E 10 5 Pa, f E 50% and standard dry air composition. The minimum quantity of solar energy needed to produce 1 mole of C 3 H 8 (its solar equivalent, Q SF ) is 2273 kJ, in agreement with second row of Table 1; if T E increases by 18C, with the rest of the atmospheric variables held constant, Q SF increases by 187 J / mol: from this point of view, a decrease of T E is helpful to the synthesis of C 3 H 8 . However, if f E increases by 1%, with the

79

rest of the atmospheric variables constant, Q SF E decreases by 211 J / mol: an increase of f favours the synthesis of C 3 H 8 . To recapitulate, it can be said that the formation of C 3 H 8 from the atmosphere and the Sun is favoured by cold and humid atmospheres. Repeating this thinking for the rest of the fuels in Table 1, a decrease of T E favors the solar synthesis of C, CO, C 2 H 6 , and C 2 H 5 OH, while an increase favours that of CH 4 , CH 3 OH and H 2 . Also, this dependence can be, in absolute values, very large, such as that for C 3 H 8 , or very small, as in the case of CH 3 O H . With respect to f E , the production of C does not depend on it, its increase lightly favours the synthesis of CO and its decrease is convenient for the production of the rest of the fuels. The independence of Q SF of graphite with f E of the atmosphere is deduced from Eq. (18), taken for this case with c51, h50 and o50. 8. CONCLUSIONS AND RECOMMENDATIONS

As general conclusions for this paper, firstly we have an algorithm which allows the calculation of the minimum solar energy necessary to synthesize a fuel from the environment. Secondly, the results help to give quantitative values for the conditions in which a particular quantity of fossil fuel should be used for a certain energy transformation, when the final energy can be obtained from solar radiation. A third aspect is the basis which is given for putting values to the cost of replenishing both, fossil fuels and environment, in relation to the economic costs of solar technologies. A fourth conclusion is the help that is given for the selection of sites and the operational strategies of producing those solar fuels most adequate for the meteorological conditions. 9. IN MEMORIAM

This paper is devoted to the memory of Jose´ Luis Ruiz de la Torre, a wise gentleman. NOMENCLATURE

Fig. 9. Solar equivalent of dihydrogen.

Subscripts: B F P R c f

generic substance fuel products, except fuel reactants reactants and products, except fuel reaction of formation

´ C. Gomez Camacho

80 m r

per mole chemical reaction of synthesis of the fuel Any extensive quantity with a subscript B or F means partial molar quantity

Superscripts: E environment M transforming machine S Sun U ‘‘Universe’’ d dry air g gas l liquid or solid * standard * pure substance Quantities and units: G Gibbs free energy (J) H enthalpy (J) K * (T ) standard equilibrium constant of the chemical reaction of synthesis of the fuel (dimensionless) Q heat interchanged (J) Q SF solar equivalent of the fuel (J / mol) R molar gas constant [8.31451 J /(mol.K)] S entropy (J / K) T thermodynamic temperature (K) U internal energy (J) V volume (m 3 ) X any extensive quantity p pressure (Pa) x molar fraction (dimensionless) DC * increment of isobaric standard molar heat capacity p,m (T ) of the chemical reaction of synthesis of the fuel [J /(mol?K)] * DG m (T ) increment of standard molar Gibbs free energy of

DH * m (T )

m n f

the chemical reaction of synthesis of the fuel (J / mol) increment of standard molar enthalpy of the chemical reaction of synthesis of the fuel (J / mol) chemical potential (J / mol) stoichiometric coefficient (dimensionless) relative humidity (dimensionless)

REFERENCES ´ ´ Gomez Camacho C. (1996) Analisis exergoambiental de la ´ Aplicacion ´ a las caracterısticas ´ ´ combustion. exergeticas y ´ In Actas ambientales de la biomasa como fuente de energıa. ´ ´ ´ de los III paneles cientıfico-tecnologicos, Cordoba, Uni´ versidad de Cordoba. ´ Gomez Camacho C. (1997) Equivalente solar de combustibles ´ fosiles. In: de Oliveira E., Maldonado E. and Guedes M. ´ (Eds.), Actas del VIII Congresso Iberico de Energia Solar, Porto, pp. 763–768. Gurvich L. V., Iorish V. S., Yungman V. S. and Dorofeeva O. V., (1997) Thermodynamics properties as a function of temperature. In: Lide D.R. (Ed.), CRC Handbook of Chemistry and Physics, CRC Press, 77th ed., pp. 5–61. Keenan J. H., Keyes F. G., Hill P. G. and Moore J. G., (1978) Steam Tables ( SI Units). Wiley-Interscience, New York, p. 8. ´ Lara Cruz A. (1995) In: Gomez C. (Ed.), Conclusions of the Forum European Enterprises Facing Technical and Legal Barriers to Renewable Energy, Seville, Published by the European Commission, Joint Research Center, Institute for Prospective Technological Studies. EUR 16390 EN, p. 12. Maslov I. A. (1997) Physics of the Earth. In: Grigoriev I.S., Meilikhov E.Z. (Eds.), Handbook of Physical Quantities, CRC Press, p. 1472. ´ renovables, Ruiz de la Torre Ruiz J. L. (1996) Energıas ´ de futuro. In Maestrıa ´ sobre Tecnicas ´ ´ energıa de Energıas ´ Sede Renovables, Universidad Internacional de Andalucıa, ´ Iberoamericana de la Rabida.