Adiabatic processes constant pressure

E3.7 A block of copper weighing 50 g is placed in 100 g of HiO for a short time. The copper is then removed from the liquid, with no adhering drops of water, and separated from it adiabatically. Temperature equilibrium is then established in both the copper and water. The entire process is carried out adiabatically at constant pressure. The initial temperature of the copper was 373 K and that of the water was 298 K. The final temperature of the copper block was 323 K. Consider the water and the block of copper as an isolated system and assume that the only transfer of heat was between the copper and the water. The specific heat of copper at constant pressure is 0.389 JK. g l and that of water is 4.18 J-K 1-g 1. Calculate the entropy change in the isolated system. 

Henceforth we concentrate on the use of Eqs. (l.lS.lf), (1.13.2f), (1.13.3f), (1.13.4e) as the fundamental building blocks (as applied to equilibrium processes) for all subsequent thermodynamic operations. The enormous advantage accruing to their use is that by the First Law all of these functions depend solely on the difference between the initial and the final equilibrium state. We no longer rely on the use of quantities such as heat and work that are individually path dependent. As will be shown shortly and in much of what is to follow, these functions of state may be manipulated to obtain useful information for characterizing experimental observations. One should note that the choice of the functions E, H, A, or G depends on the experimental conditions. For example, in processes where temperature and pressure are under experimental control one would select the Gibbs free energy as the appropriate function of state. Processes carried out under adiabatic and constant pressure conditions are best characterized by the enthalpy state function. 

In the gas turbine (Brayton cycle), the compression and expansion processes are adiabatic and isentropic processes. Thus, for an isentropic adiabatic process 7 = where Cp and c are the specific heats of the gas at constant pressure and volume respectively and can be written as ... 

To calculate the heat duty it must be remembered that the pressure drop through the choke is instantaneous. That is, no heat is absorbed or lost, but there is a temperature change. This is an adiabatic expansion of the gas w ith no change in enthalpy. Flow through the coils is a constant pressure process, except for the small amount of pressure drop due to friction. Thus, the change in enthalpy of the gas is equal to the heat absorbed. 

Adiabatic Reaction Temperature (T ). The concept of adiabatic or theoretical reaction temperature (T j) plays an important role in the design of chemical reactors, gas furnaces, and other process equipment to handle highly exothermic reactions such as combustion. T is defined as the final temperature attained by the reaction mixture at the completion of a chemical reaction carried out under adiabatic conditions in a closed system at constant pressure. Theoretically, this is the maximum temperature achieved by the products when stoichiometric quantities of reactants are completely converted into products in an adiabatic reactor. In general, T is a function of the initial temperature (T) of the reactants and their relative amounts as well as the presence of any nonreactive (inert) materials. T is also dependent on the extent of completion of the reaction. In actual experiments, it is very unlikely that the theoretical maximum values of T can be realized, but the calculated results do provide an idealized basis for comparison of the thermal effects resulting from exothermic reactions. Lower feed temperatures (T), presence of inerts and excess reactants, and incomplete conversion tend to reduce the value of T. The term theoretical or adiabatic flame temperature (T,, ) is preferred over T in dealing exclusively with the combustion of fuels. 

Figure 15.5 shows the ideal open cycle for the gas turbine that is based on the Brayton Cycle. By assuming that the chemical energy released on combustion is equivalent to a transfer of heat at constant pressure to a working fluid of constant specific heat, this simplified approach allows the actual process to be compared with the ideal, and is represented in Figure 15.5 by a broken line. The processes for compression 1-2 and expansion 3-4 are irreversible adiabatic and differ, as shown from the ideal isentropic processes between the same pressures P and P2 -... 

The conditions existing during the adiabatic flow in a pipe may be calculated using the approximate expression Pi/ = a constant to give the relation between the pressure and the specific volume of the fluid. In general, however, the value of the index k may not be known for an irreversible adiabatic process. An alternative approach to the problem is therefore desirable.(2,3)... 

Besides the reversible and irreversible processes, there are other processes. Changes implemented at constant pressure are called isobaric process, while those occurring at constant temperature are known as isothermal processes. When a process is carried out under such conditions that heat can neither leave the system nor enter it, one has what is called an adiabatic process. A vacuum flask provides an excellent example a practical adiabatic wall. When a system, after going through a number of changes, reverts to its initial state, it is said to have passed through a cyclic process. 

The adiabatic flame temperature is defined as the maximum possible temperature achieved by the reaction in a constant pressure process. It is usually based on the reactants initially at the standard state of 25 °C and 1 atm. From Equation (2.20), the adiabatic temperature (7 i[Pg.30]

It is common to equate the strength of interaction of an acid and a base with the enthalpy of reaction. In some cases this enthalpy may be measured by direct calorimetry AH q for an adiabatic process at constant pressure. 

CARNOT CYCLE. An ideal cycle or four reversible changes in the physical condition of a substance, useful in thermodynamic theory. Starting with specified values of die variable temperature, specific volume, and pressure, the substance undergoes, in succession, an isothermal (constant temperature) expansion, an adiabatic expansion (see also Adiabatic Process), and an isothermal compression to such a point that a further adiabatic compression will return the substance to its original condition. These changes are represented on the volume-pressure diagram respectively by ub. he. ctl. and da in Fig. I. Or the cycle may he reversed ad c h a. 

The inequalities of the previous paragraph are extremely important, but they are of little direct use to experimenters because there is no convenient way to hold U and S constant except in isolated systems and adiabatic processes. In both of these inequalities, the independent variables (the properties that are held constant) are all extensive variables. There is just one way to define thermodynamic properties that provide criteria of spontaneous change and equilibrium when intensive variables are held constant, and that is by the use of Legendre transforms. That can be illustrated here with equation 2.2-1, but a more complete discussion of Legendre transforms is given in Section 2.5. Since laboratory experiments are usually carried out at constant pressure, rather than constant volume, a new thermodynamic potential, the enthalpy H, can be defined by... 

The process may be assumed to occur adiabatically and at constant pressure. It is therefore isenthalpic, and may for calculational purposes be considered to occur in two steps ... 

For an adiabatic combustion process at constant pressure, the enthalpy stays constant. Figure 2 shows the exergy of the reactants, e, and the exergy of the reaction products, e , after this reaction has taken place. The loss in exergy is ... 

The equation of state-does not include all the experimental information which we must have about a system or substance. Ve need to tnow also its heat, capacity or specific heat, as a function of temperature. Suppose, for instance, that we know the specific heat at constant pressure Cp as a function of temperature at a particular pressure. Then we can find the difference of internal energy, or of entropy, between any two states. From the first state, we can go adiabatically to the pressure at which we know Cp, In this process, since no heat is absorbed, the change of internal energy equals the work done, which we can compute from the equation of state. Then we absorb heat at constant pressure, until we reach the point from which another adiabatic process will carry us to the desired end point. The change of internal energy can be found for the process at constant pressure, since there we know CP) from which we can... 

This is the compressibility usually employed sometimes, as in considering sound waves, we require the adiabatic compressibility, the fractional decrease of volume per unit increase of pressure, when no heat flows in or out. If there is no heat flow, the entropy is unchanged, in a reversible process, so that an adiabatic process is one at constant entropy. Then we have... 

This equation relates temperature and volume for a mechanically reversible adiabatic process involving an ideal gas with constant heat capacities. The analogous relationships between temperature and pressure and between pressure and volume can be obtained from Eq. (3.22) and the ideal-gas equation. Since P, V,/ T, = P2V2j T2, we may eliminate V,/ V2 from Eq. (3.22), obtaining ... 

The idealization of the gas-turbine cycle (based on air, and called the Bray cycle) is shown on a PV diagram in Fig. 8.12. The compression step AB represented by an adiabatic, reversible (isentropic) path in which the press increases from PA (atmospheric pressure) to PB. The combustion process replaced by the constant-pressure addition of an amount of heat QBC. Work produced in the turbine as the result of isentropic expansion of the air to press... 

Let the superscripts a and b denote two initial binary solutions, consistin of n and nb moles respectively. Let superscript c denote the final solutio obtained by simple mixing of solutions a and b in an adiabatic process. Th process may be batch mixing at constant pressure or a steady-flow pro involving no shaft work or change in potential or kinetic energy. In either c... 

A piston of cross section A moving horizontally in a hollow tube (adiabatically insulated) is propelled by a force Fa against a gas held at constant pressure Pg < Fa/A. What is the initial acceleration of the piston How far is the energy of the gas raised after the piston has been moved a distance d How much is the energy of the walls increased in this process What can you say about the work transfer after the piston has been stopped and the entire system is allowed to equilibrate ... 

A similar argument is used to deal with s this is based on the empirical observation that in an adiabatic process involving a noble gas at low pressures, the product PV7 is virtually constant. Here 7 is a fixed quantity (which will later turn out to be the ratio of molar heat capacities at constant pressure and volume) whose exact significance is irrelevant at this stage near room temperature and for monatomic gases 7 has a value close to 5/3. We therefore use the product PV7 as a measure of the empirical entropy through the simple relation... 

A thermodynamic quantity of considerable importance in many combustion problems is the adiabatic flame temperature. If a given combustible mixture (a closed system) at a specified initial T and p is allowed to approach chemical equilibrium by means of an isobaric, adiabatic process, then the final temperature attained by the system is the adiabatic flame temperature T. Clearly depends on the pressure, the initial temperature and the initial composition of the system. The equations governing the process are p = constant (isobaric), H = constant (adiabatic, isobaric) and the atom-conservation equations combining these with the chemical-equilibrium equations (at p, T ) determines all final conditions (and therefore, in particular, Tj). Detailed procedures for solving the governing equations to obtain Tj> are described in [17], [19], [27], and [30], for example. Essentially, a value of Tf is assumed, the atom-conservation equations and equilibrium equations are solved as indicated at the end of Section A.3, the final enthalpy is computed and compared with the initial enthalpy, and the entire process is repeated for other values of until the initial and final enthalpies agree. 

If the whole process is carried out at constant pressure, then all the heat generated goes into increasing the enthalpy of the products. This internally generated heat is designated as Q, where Q = n AH (heat generated by the reaction at standard state conditions), and Q = n Ai/[products] (heat absorbed by the products of the reaction, at adiabatic conditions). 

Determining the final proiduct temperature,, based upon the above is exemplified by the following example What is the adiabatic temperature of the product gases from the detonation of PETN The gases are allowed to expand freely at one atmosphere (a constant-pressure process), but adiabatic conditions are maintained. 

Consider now an irreversible process in a closed system wherein no heat transfer occurs. Such a process is represented on the P V diagram of Fig. 5.6, which shows an irreversible, adiabatic expansion of 1 mol of fluid from an initial equilibrium state at point A to a final equilibrium state at pointB. Now suppose the fluid is restored to its initial state by a reversible process consisting of two steps first, the reversible, adiabatic (constant-entropy) compression of tile fluid to tile initial pressure, and second, a reversible, constant-pressure step that restores tile initial volume. If tlie initial process results in an entropy change of tlie fluid, tlien tliere must be heat transfer during tlie reversible, constant-P second step such tliat ... 

A single gas stream enters a process at conditions T, P, and leaves at pressure P2. The process is adiabatic. Prove that the outlet temperahire T2 for the actual (irreversible) adiabatic process is greater than that for a reversible adiabatic process. Assume the gas is ideal with constant heat capacities. 

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