The Fuel Shortage and Thermodynamics—The Entropy Crisis*

The purpose of this paper is to bring to the attention of the broad spectrum of specialists currently concerned with the energy question in the United States the necessity for using a yardstick other than energy for assessing the effectiveness of fuel usage.In establishing present patterns of energy consumption, the standard procedure is to find the total amount of energy used in each sector of the economy and, thus, to determine the needs of each sector for different fuels. The term energy in this connection is ambiguous. For example, the heat required in an industrial process may be added to the electrical work required or to the “heating value” of the fuel consumed in producing the electrical work required. None of these quantities represents energy consumed in the process, because it is known from the first law of thermodynamics that energy, rather than being consumed in any process, is always conserved. When opportunities for fuel conservation are to be assessed, it becomes necessary to use a measure other than energy.For example, every engineer knows that a Btu (British thermal unit) of enthalpy in the circulating water of a power plant is less marketable and less valuable than a Btu of enthalpy in a steam main. He realizes also that a cold battery that is charged is more valuable and useful than a discharged battery having the same energy by virtue of being hot.Typical conditions of process steam used in industry are 270 °F and 30 psig. The heat required to change water from ambient conditions (55 °F) into typical process steam conditions is 1150 Btu per lb of steam. Accordingly, since the typical heating value of hydrocarbon fuels is about 20,000 Btu per lb of fuel, it is often concluded that 0.057 lb of fuel per lb of process steam is needed. If this amount of fuel were used then, according to the customary definition, the effectiveness of fuel utilization would be 100%. By virtue of the first and second laws of thermodynamics, on the other hand, it can be shown that the minimum amount of fuel required to accomplish the task just cited is only 0.015 lb of fuel per lb of steam and, therefore, only when this minimum amount is consumed is the process 100% effective. Conversely, if 0.057 lb of fuel per lb of steam is consumed, the effectiveness of fuel utilization is only 0.015/0.057 = 26%.The preceding simple examples illustrate the necessity for using a yardstick other than energy for assessments of fuel needs and of effectiveness of fuel utilization. The laws of thermodynamics indicate that neither energy, nor heat, nor enthalpy, nor Gibbs free energy are in general satisfactory yardsticks. The relevant quantity is a property called available useful work that is in turn uniquely related to another important property called entropy. Hence, the subtitle is chosen to be “The Entropy Crisis” in suggestive distinction to the fashionable characterization of the problem as “The Energy Crisis.”This paper is concerned with the thermodynamic arguments that lead to available useful work or entropy as the objective measure of fuel needs and of effectiveness of fuel utilization.It was observed by Lazar Carnot about the year 1800 that the perfect hydraulic engine would produce enough work to return all the working fluid to its elevated source if that work were used in a perfect pump. Another way of saying this is that the minimum requirement for lifting water from one level to a higher one is that an equal weight must be lowered from the higher level to the lower one. Moreover, more than this minimum is required if any part of the mechanism involved is less than perfect.Such observations are generalized in the laws of thermodynamics from which the following theorem may be proved: for any physical task that is to be performed within an environment that is essentially in a stable equilibrium state, a certain minimum of work is required. The task to be performed may be as simple as the raising of a mass of material from one level to another in the gravity field, or as complex as the conversion of iron ore into steel. The minimum work will be required when the task is performed reversibly, that is, in such a way that all systems involved in the process can be restored to their initial states after the process has occurred, leaving no physical evidence that any process had taken place.It will be observed from the preceding theorem that the work of the reversible process is fixed by the nature of the system and the initial and final states of that system in the process. Even though the number of such possible reversible processes between the same end states may be infinitely large, the work required is identically the same for all of them.Associated with each system in each state will be, then, an amount of work which is the minimum required to create that system in that state out of materials from the atmosphere or in mutual stable equilibrium with the atmosphere. This minimum amount is also equal to the maximum useful work that can be done by the system starting from a given state and ending in a state in mutual stable equilibrium with the atmosphere. It is called the available useful work of the system in that state.1Values of available useful work are, of course, associated with the fuels we use. If a fuel such as the molecular species CH2 were to be formed reversibly from carbon dioxide and water in the atmosphere, the work required would be the available useful work of the fuel. This work could be recovered completely for use on other systems if the fuel were combined with oxygen from the air in a reversible process that restores the carbon dioxide and water as constituents of the gaseous mixture we call air. Clearly, each task that is to be performed on any system within the atmosphere may be carried out by means of the oxidation of fuel without consuming any resource other than the infinite atmosphere, because only work is required and the oxidation of fuel is entirely equivalent in the reversible ideal process to a supply of work.Indeed in the great majority of all physical tasks undertaken by modern American society, whether they be the lifting of a steel beam to the top of a building or the heating of a house, the process drawn upon is the oxidation of fuel. The proliferation of these tasks in an affluent society has increased the magnitude of the basic process, namely oxidation of fuel, to the point where such questions as where is the fuel to come from? how do we get it? and how do we pay for it? have resulted in what is called the energy crisis but which would be more appropriately called the fuel crisis or fuel shortage.It is appropriate, therefore, to give thought to the minimum magnitude of the oxidation process that will permit the desired tasks to be performed—that is, the minimum fuel requirement. This minimum would be attained if the oxidation process and all subsequent operations were to be executed reversibly within the terrestrial (air and water) environment. It can be shown that the annual minimum would consume a tiny fraction of the fuel consumed in the year 1972. That tiny fraction would provide all the lifting jobs, all the metal-forming jobs, and all the industrial and domestic heating to which society is now accustomed or is likely to become accustomed in the foreseeable future.Why then do we consume the massive quantities of fuel that we do? The answer is in part economic; namely, that we use the amount of fuel that minimizes the sum of capital and operating costs. It is also in part technological; namely, that we have not organized our technological skills so as to reduce substantially the fuel requirement.The economic optimization usually performed by either consumer or industry is faulty. It would be valid only if the prices of capital goods and fuel reflected their total social costs. That is, the price of coal would have to include the cost of rehabilitating the landscape after strip mining, and the price of imported oil would have to reflect in some way the effect of the import on the national balance of payments. On the other hand, the price of iron and steel for power plants and other machinery would have to reflect the cost of cleaning up atmospheric and water pollution caused by production of iron and steel.The technological argument is not, of course, entirely independent of the economic one. With proper price allocation the demand for competent engineers and engineering to reduce fuel consumption would rise considerably above present levels. As prices and social pressures for fuel conservation increase, however, engineering efforts will be more and more directed toward devising means for realizing in some degree the vast potential for reducing consumption of fuel. It is to this task that the present paper is directed.The thermodynamic measure of the task-performing value of fuel and air or their products of oxidation is, as stated earlier, the maximum possible work, the available useful work, that could be obtained from them. The amount of this work is dependent on the temperature and other properties of the environment (ambient air and water) to which or from which the fuel-air system may deliver or receive energy in any quantity.In a reversible process the available useful work is conserved. That is, when fuel is oxidized in order to perform a specific task on a specific material, the available useful work is merely transferred from the fuel-air system to the material if the process is executed reversibly. It is informative, therefore, to examine the performance of any particular task so as to trace the losses in available useful work that occur in each step of that task.The available useful work Φ of a system and atmosphere can be shown to be given by the equation(1)Φ=E+p0V-T0S-∑inμi0Niwhere E denotes energy, V volume, and S entropy of the system, Ni for i = 1,2, … ,n the number of moles of molecular component i in the system, p0 the pressure of the atmosphere, T0 the temperature of the atmosphere, and μi0 the total potential of component i in the atmosphere or in mutual stable equilibrium with it.The quantity Φ may be evaluated for any system in any state whether stable equilibrium, nonstable equilibrium, or nonequilibrium. In particular its value is zero when system and atmosphere are in mutual stable equilibrium. That is, Φ is zero when the system is in a stable equilibrium state such that its temperature is T0 its pressure p0 and its total potentials are μi0 for i = 1,2, … ,n.The close relationship between the quantity Φ and the entropy is evident from Equation 1. For given energy, volume, and composition of the system, Φ decreases with increase in entropy of the system. For states of small entropy the available useful work is large, and vice versa. A deficiency in available useful work corresponds to a surplus of entropy. The value of Φ may exceed that of the energy of the system. It cannot, of course, exceed the energy of the system and atmosphere taken together.In general, the available useful work Φ is different from the Gibbs free energy Z and enthalpy H. Unlike Φ, which has a value for any state of the system, equilibrium or nonequilibrium, Z and H are defined (≡) for stable equilibrium states only, as follows:(34.2)Z≡E+pV-TS and H≡E+pVwhere p and T denote, respectively, pressure and temperature which are not in general identifiable for states that are not stable equilibrium states.The available useful work Φ was devised by (Gibbs Ref. [4], p. 77. Equation 54) in 1875. In an earlier paper (Gibbs Ref. [4], p. 58) had introduced the function(34.3)E+p0V-T0Sfor application to a system the constituent substance of which did not enter into or mix with the atmosphere. The maximum possible decrease in the value of this function as the system (body) proceeded toward pressure and temperature equilibrium with the atmosphere (medium) he called the “available energy of the body and medium,” Gouy Ref. [Gouy (1889)] introduced the abbreviated form of Gibbs function(34.4)E-T0Sthe change in which for any change of state is the available work Ω used to introduce the property entropy.A closely related property(34.5)H-T0Swhich is particularly useful for calculating the available work in steady-flow processes, was used by Darricus Ref. [3] for processes in turbines and by Keenan Ref. [6] for analysis of steam power plants and for cost accounting when both process steam and power are produced. It appeared in an engineering textbook by Bosnjakovic Ref. [1] and more recently in European literature [Brennstoff-Warme-Krafl, 1961],2 where it has been given the name exergy.When applied to a hydrocarbon fuel, the quantity Φ is the minimum useful work required to form the fuel in a given state from the water and carbon dioxide in the atmosphere. Since this minimum will be the useful work of a reversible process, the quantity Φ is also the maximum useful work that could be obtained by oxidation of the fuel and return of the products to the atmosphere. Moreover, any change in state of the fuel-air-atmosphere system will produce an amount of useful work less than, or in a reversible change, equal to the corresponding decrease in Φ.The curves of Figure 1 are calculated for 1 pound-mole of a liquid hydrocarbon fuel which may be described as CH2 and which has a so-called heating value of 280,000 Btu (20,000 Btu per pound). This heating value is the decrease in enthalpy when 1 pound-mole of fuel burns in theoretical air to carbon dioxide, water, and nitrogen:(34.6)CH2+32O2+5.65N2→CO2+H2O+5.65N2The available useful work for the reactants of this reaction can be shown to be about 292,000 Btu per pound-mole, namely about 4% greater than the heating value. In order to obtain this work, the following process, as one among many possible ones, could be used: (1) the oxidation is carried out in a reversible fuel cell, at some temperature Tc, which delivers electrical work to the surroundings; (2) the products of the fuel-cell process are cooled from Tc to the temperature of the environment T0 as they provide heat to Carnot engines that produce further work; (3) each of the products CO2,H2O, and N2 is separated from the mixture reversibly by means of a semipermeable membrane and expanded reversibly and iso-thermally at T0 in an engine cylinder until it attains a pressure equal to the partial pressure of that constituent in the atmosphere; (4) each molecular species is introduced reversibly into the atmosphere through a semipermeable membrane. The term semipermeable membrane refers to a device that is impermeable to all molecular species except one. The final state after step (4) corresponds to zero available useful work at atmospheric temperature (55 °F). It is the base state C in Figure 1.Such a fuel-air process would produce the maximum possible work, the available useful work. Moreover, the work so obtained could be used in an inverse process to create the original quantity of fuel from the carbon dioxide and water vapor in the air. Any oxidation process that produced less work than the available useful work would be irreversible, and the loss in available useful work would be a measure of the irreversibility of the process.Work is produced for use in one or more tasks. Because these tasks are almost always performed with a high degree of irreversibility, the available useful work remaining in the material operated on is usually only a small fraction, and more often a tiny fraction, of that available at the beginning of the task. When a steel beam is lifted to the top of a tall building, the available useful work remaining at the end of the task may be of the order of 10% of that in the fuel-air system that was the source of the work. In the manufacture of the steel beam, on the other hand, the available useful work remaining in the beam is a fraction several orders of magnitude less than in the lifting process. The search for potential savings of fuel, therefore, must involve, first, the oxidation process and, the task that the work produced by the fuel for oxidation of a hydrocarbon fuel are not they are in of fuels are almost always in a without production of electrical For CH2 and the reaction cited earlier, the temperature at the end of the process is about the temperature for the irreversible, the process, which is in Figure 1 by the in an increase in entropy and a loss of available useful work. the loss is Btu per pound-mole of our fuel or about of the original value of 292,000 The remaining available useful work is which is the maximum amount of work that could be for example, by heat to Carnot engines and by of the species to the by the environment at process A to in Figure 1 is a process. That is, the for heat in to systems by the process. By virtue of the irreversibility and the associated increase in entropy in the process, available useful work has been at process is the of a perfect through a value from high to entropy In the process available useful work is temperature in the process available useful work is the temperature are processes in which entropy because of with state the available useful work can be in a number of For example, the energy of the products may be transferred in the form of heat to any material at a temperature less than and the transferred energy may be used reversibly to produce work. Because the temperature between the products and the material is the process is and, therefore, the available useful work The in Figure 1 the available useful work in the products that in a material at temperature which has cooled the products to without For this purpose the material is to be of infinite heat because some heat would be transferred to material at less than and the loss in Φ would be That is, the the maximum available useful work after the products to of Figure 1 the heat available from products of in simple steady-flow from temperature to the temperature of the atmosphere (55 °F). It is the of enthalpy with The the available useful work of the products of at temperature It atmospheric temperature at a small value of the corresponding to the work upon reversibly the products with the constituent of the atmosphere. The in between and at any value of is the available useful work from material that has cooled the products from °F to is evident from the that as the temperature of the material is lowered °F the loss in Φ with decrease in a temperature of about the value of Φ is about of that for the fuel or Btu per of typical temperature of the working fluid in a steam power plant is about of the available useful work of the fuel is in the process of energy from the products of to the working fluid a temperature and have been to these temperature and to losses in Φ by reducing an industrial on the other hand, the products of be used to process steam 270 The loss of available useful work is then 292,000 or about of that in the fuel. The between and is, 100% represents the fraction because a steam power plant was not between the products of and the process steam. It can be shown by a simple that for the same amount of process steam an steam power plant may produce Btu of electrical work for an of fuel Btu of electrical work would be obtained at the of 292,000 or Btu of available useful work. A power plant would consume about of CH2 with and Btu of available useful work with to produce the same amount of electrical preceding an of a heating process with a process in order to reduce the loss of available useful work. industrial heating processes, on the other hand, such high that a process would either or Moreover, the to be per of temperature as shown by the of the of Figure is these processes, such as the manufacture of steel or be on available useful work from the material in process and from the products of leaving the heating process. For example, the leaving a heating to as potential to supply heat to other processes as the fuel A of potential may be in the products of leaving the heating process. The opportunities for available useful work through heat to or heating processes are of processes in in order to reduce consumption of fuel and available useful work in a simple industrial process may be by the of to In the modern plants about of are for the amount of fuel that would 1 in a simple The effectiveness of the process in of available useful work produced to that consumed is but it is as and the fuel consumption is as great as in the simple has been that about a of the annual fuel consumption in the United States is to of electrical work in power plants and another to industrial process In of loss of available useful work the power plant is less than the industrial The of the power which is available useful work, for about of the available useful work consumed, the of the industrial process for small of of the available useful work The opportunities for reducing fuel consumption therefore, to be greater in industrial than in power have been of these opportunities the It is a that of industrial heating with power production was more than it is The for this change are of such in of present and of fuel that they be and no the of power production and the high cost of the required for the of a power With more and of fuel the may in the to be more that of than that of the measure of fuel requirement other than the energy or the less enthalpy or heating value is needed. The one is the function Φ, which was devised by It is the maximum useful work that can be obtained from a fuel by it and the products of into the atmosphere. This maximum will be in any reversible process between the initial and final any task to be such as the lifting of a steel beam or the manufacture of steel has a minimum requirement of available useful work when with the maximum from the fuel, the minimum amount of fuel required. this taken from usually to be it the great potential in present processes for these processes, that of the power plant fuel of with power plants will return to and the corresponding of industrial processes may come into as the price of fuel to reflect more the economic and social cost of producing are but examples of what must become a engineering on