Adiabatic process with work

Thermodynamics and Chemistry, second edibon,version 3 20 by Howard DeVbe. Latest version www.chem.umd.edu/themobook 

Some of the important terms and definitions discussed in this chapter are as follows. 

The derivation of the mathematical statement of the second law shows that during a reversible process of a closed system, the infinitesimal quantity dq/Tb equals the infinitesimal change of a state function called the entropy, S. Here dq is heat transferred at the boundary where the temperature is 7b. 

In each infinitesimal path element of a process of a closed system, d5 is equal to dq/ if the process is reversible, and is greater than dq/Tb if the process is irreversible, as summarized by the relation d5 dq/Tb. 

Consider two particular equilibrium states 1 and 2 of a closed system. The system can change from state 1 to state 2 by either a reversible process, with A 5 equal to the integral / dq/7b), or an irreversible process, with AS greater than / dq/Tb). It is important to keep in mind the point made by Eig. 4.12 because S is a state function, it is the value of the integral that is different in the two cases, and not the value of AS. 

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Therefore, if a simple gas turbine cycle is modified with the compression accomplished in two or more adiabatic processes with intercooling between them, the net work of the cycle is increased with no change in the turbine work. 

During a process with irreversible work, energy dissipation can be either partial or complete. Dissipative work, such as the stirring work and electrical heating described in previous sections, is irreversible work with complete energy dissipation. The final equilibrium state of an adiabatic process with dissipative work can also be reached by a path with positive heat and no work. This is a special case of the minimal work principle. 

There is, however, the additional dimension of temperature in the A-dimensional space. Do the paths for possible reversible adiabatic processes, starting from a common initial point, lie in a volume in the A-dimensional space Or do they fall on a surface described by r as a function of the work coordinates If the paths he in a volume, then every point in a volume element surrounding the initial point must be accessible from the initial point by a reversible adiabatic path. This accessibility is precisely what Caratheodory s principle of adiabatic inaccessibility denies. Therefore, the paths for all possible reversible adiabatic processes with a common initial state must lie on a unique surface. This is an (A — 1)-dimensional hypersurface in the A-dimensional space, or a curve if N is 2. One of these surfaces or curves will be referred to as a reversible adiabatic surface. 

In this case, heat transferred to the surroundings has been turned completely into work done on the system. This does not violate the second law of thermodynamics because the surroundings do not undergo a cyclic process. The final temperature for an irreversible adiabatic process cannot be lower than for a reversible adiabatic process with the same final volume, but it can be higher. 

Flere the subscripts and/refer to the initial and final states of the system and the work is defined as the work perfomied on the system (the opposite sign convention—with as work done by the system on the surroundings—is also in connnon use). Note that a cyclic process (one in which the system is returned to its initial state) is not introduced as will be seen later, a cyclic adiabatic process is possible only if every step is reversible. Equation (A2.1.9), i.e. the mtroduction of t/ as a state fiinction, is an expression of the law of conservation of energy. 

The second law of thermodynamics states that energy exists at various levels and is available for use only if it can move from a higher to a lower level. For example, it is impossible for any device to operate in a cycle and produce work while exchanging heat only with bodies at a single fixed temperature. In thermodynamics, a measure of the unavailability of energy has been devised and is known as entropy. As a measure of unavailability, entropy increases as a system loses heat, but remains constant when there is no gain or loss of heat as in an adiabatic process. It is defined by the following differential equation ... 

Theorem.—A process yields the maximum amount of available energy when it is conducted reversibly.—Proof. If the change is isothermal, this is a consequence of Moutier s theorem, for the system could be brought back to the initial state by a reversible process, and, by the second law, no work must be obtained in the whole cycle. If it is non-isothermal, we may suppose it to be constructed of a very large number of very small isothermal and adiabatic processes, which may be combined with another corresponding set of perfectlyJ reversible isothermal and adiabatic processes, so that a complete cycle is formed out of a very large number of infinitesimal Carnot s cycles (Fig 11). 

When the system goes from state 1 to state 2, the change in internal energy (AC/) is fixed, but Q and W depend on the path taken. If no heat passes between the system and its surroundings during the process, the change is said to be adiabatic. In an adiabatic process, Q = 0 and AC/ = W. The most usual form of work involved in chemical reactions is volume work, that is work associated with a change in volume of a system, in which case... 

Pure hydrogen gas at room pressure and temperature is adiabatically combusted with air. The combustion takes place with an amount of air that is 30% in excess of what is stoichiometrically required. Calculate the adiabatic flame temperature of the process, the work lost, and the thermo-dynamic efficiency of the process. Assume air to consist of a mixture of 79 mol% of N2 and 21 mol% of 02. 

We start by studying the steady-state design and economics of a process with a single adiabatic reactor. The design considers the entire plantwide process reactor, heat exchangers, gas recycle compressor, preheat furnace, condenser, and separator. The economic objective function is total annual cost, which includes annual capital cost (reactor, catalyst, compressor, and heat exchangers) and energy cost (compressor work and furnace fuel). 

Apply Eq. (2.29) to this non-steady-state process, with n replacing m, with the tank as control volume, and with a single inlet stream. Since the process is adiabatic and the only work is shaft work, this equation may be multiplied by dt to give ... 

The shaft work given by Eq. (7.27) is the maximum that can be obtained from an adiabatic turbine with given inlet conditions and given discharge pressure. Actual turbines produce less work, because the actual expansion process is irreversible. We therefore define a turbine efficiency as... 

A design for purifying helium consists of an adiabatic process that splits a helium stream containing 30-mole-percent methane into two product streams, one containing 97-mole-percent helium and the other 90-mole percent methane. The feed enters at lObar and 117°C the methane-rich product leaves at 1 bar and 27 C the helium-rich product leaves at 5Q°C and IS bar. Moreover, work is produced by the process. Assuming helium an ideal gas with Cp 5/2)R and methane an ideal gas with Cp = (9/2)/, calculate the total entropy change of the process on the basis of 1 mol of feed to confirm that the process does not violate the second law. 

Dhar modelled the stretching of a polymer using the stochastic Rouse model, for which distributions of various definitions of the work can be obtained. Two mechanisms for the stretching were considered one where the force on the end of the polymer was constrained and the other where its end was constrained. Dhar commented that the variable selected for the work was only clearly identified as the entropy production in the latter case. In the former case they argue that the average work is non-zero for an adiabatic process, and therefore should not be considered as an entropy production, however we note that the expression is consistent with a product of flux and field as used in linear irreversible thermodynamics. 

Reactions are frequently carried out adiabatically, often with heating or cooling provided upstream or downstream of the reaction vessel. With the exception of processes involving highly viscous materials such as in Problem P8-4, the work done by the stirrer can, usually be neglected. After substituting Equation (8-42) for Q, the energy balance can be written as... 

The present volume involves several alterations in the presentation of thermodynamic topics covered in the previous editions. Obviously, it is not a trivial exercise to present in a novel fashion any material that covers a period of more than 160 years. However, as best as I can determine the treatment of irreversible phenomena in Sections 1.13, 1.14, and 1.20 appears not to be widely known. Following much indecision, and with encouragement by the editors, I have dropped the various exercises requiring numerical evaluation of formulae developed in the text. After much thought I have also relegated the Caratheodory formulation of the Second Law of Thermodynamics (and a derivation of the Debye-Hiickel equation) as a separate chapter to the end of the book. This permitted me to concentrate on a simpler exposition that directly links entropy to the reversible transfer of heat. It also provides a neat parallelism with the First Law that directly connects energy to work performance in an adiabatic process. A more careful discussion of the basic mechanism that forces electrochemical phenomena has been provided. I have also added material on the effects of curved interfaces and self assembly, and presented a more systematic formulation of the basics of irreversible processes. A discussion of critical phenomena is now included as a separate chapter. Lastly, the treatment of binary solutions has been expanded to deal with asymmetric properties of such systems. 

Adiabatic processes. Ratio of the specific heats. If the gas is contained in st vessel, the walls of which are impermeable to heat or adiabatic so that no interchange of heat with the surroundings is possible, the energy of the gas diminishes by the amount of the work done against the external pressure. On the other hand, if the gas is compressed, its energy increases by the amount of the work done in the compression. In the first case there is a fall, in the second a rise in the temperature of the gas. The magnitude of the change in temperature may be calculated from equation (2) as follows ... 

In order to include the compression ratio in the analysis of Curzon and Ahlbom cycle, it is necessary to suppose finite time for the adiabatic processes. Hence, as it is known, with ideal gas as working fluid and using the Newton heat transfer law, the following can be written ... 

In the treatment of adiabatic processes in the text, the heat capacity has been assumed to be independent of temperature. How could allowance be made for the variation of Cp with temperature in equation (10.6) (The gas may be assumed to be ideal in other respects.) Use this method to estimate the work done, in calories, and the final temperature in the reversible, adiabatic compression of 1 mole of oxygen from 10 liters to 1 liter, the initial temperature being 26 C. What would be the result if Cp were taken as having the constant (mean) value of 7 cal. deg. mole ... 

Spin-forbidden reactions are a subset of the broader class of electronically non-adiabatic processes, which involve more than one PES. The fundamental theory of how such processes occur is well understood (7-9), and a very large amount of research is being performed with the aim of elucidating more details in all the areas of nonadiabatic chemistry. It is not possible to present this work here, so I will instead provide an outline of the most important theoretical insights in the... 

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