Constant-pressure work

The system expands as the temperature increases at constant pressure. Work is needed against the molecules and the atmosphere as they move apart. In the critical region the coefficient of thermal expansion at constant pressure is large and thus the effect on Cp is also large. 

EXAMPLE 7.7 Constant-pressure work. When the externally applied pressure pext is constant, the total work in a quasi-static process of expansion from volume Va to Vb is... 

Figure III-l depicts a hypothetical system consisting of some liquid that fills a box having a sliding cover the material of the cover is such that the interfacial tension between it and the liquid is zero. If the cover is slid back so as to uncover an amount of surface dJl, the work required to do so will he ydSl. This is reversible work at constant pressure and temperature and thus gives the increase in free energy of the system (see Section XVII-12 for a more detailed discussion of the thermodynamics of surfaces).

This is the working equation for a constant volume calorimeter. Alternatively, a calorimeter can be maintained at constant pressure p equal to the external pressure p in which case... 

Combustion. The primary reaction carried out in the gas turbine combustion chamber is oxidation of a fuel to release its heat content at constant pressure. Atomized fuel mixed with enough air to form a close-to-stoichiometric mixture is continuously fed into a primary zone. There its heat of formation is released at flame temperatures deterruined by the pressure. The heat content of the fuel is therefore a primary measure of the attainable efficiency of the overall system in terms of fuel consumed per unit of work output. Table 6 fists the net heat content of a number of typical gas turbine fuels. Net rather than gross heat content is a more significant measure because heat of vaporization of the water formed in combustion cannot be recovered in aircraft exhaust. The most desirable gas turbine fuels for use in aircraft, after hydrogen, are hydrocarbons. Fuels that are liquid at normal atmospheric pressure and temperature are the most practical and widely used aircraft fuels kerosene, with a distillation range from 150 to 300 °C, is the best compromise to combine maximum mass —heat content with other desirable properties. For ground turbines, a wide variety of gaseous and heavy fuels are acceptable. 

With the above as an introduction, we now consider the important operational case of filtration performed under constant pressure. In practice, all the parameters defined above are nearly constant under steady state conditions except V and r, which are varied by the operator. We may therefore integrate the working expression for filtration over the limits of volume from 0 to V, and for residence time over the limits of 0 to x ... 

Specific heat The amount of heat (or mechanical work) required to raise the temperature of a unit mass of a substance one degree Celsius. In the case of gases there are two specific heats, according as to whether the heating takes place at constant pressure or at constant volume. 

Consider next a similar recuperative cycle, but one in which the compression process approximates to isothermal rather than isentropic, with the work input equal to the heat rejected (this may be achieved in a series of small compressions of polytropic efficiency Tjp, followed by a series of constant pressure heat rejections). It may then be shown that the thermal efficiency of this cycle is given by... 

If the definition of work is limited to mechanical work, an interesting simplification is possible. In this case, AE is merely the heat exchanged at constant volume. This is so because if the volume is constant, no mechanical work can be done on or by the system. Then AE = q. Thus AE is a very useful quantity in constant volume processes. However, chemical and especially biochemical processes and reactions are much more likely to be carried out at constant pressure. In constant pressure processes, AE is not necessarily equal to the heat transferred. For this reason, chemists and biochemists have defined a function that is especially suitable for constant pressure processes. It is called the enthalpy, H, and it is defined as... 

For points 2-3, there is constant entropy (S) compression for a one pound of air from Pg to P3. From points 3-5 the air cools at constant pressure, and gives up heat, Q, to the intercooler. From points 5-6 the air is compressed at constant S to the final pressure Pg. Note that point Tj = point Tg for constant temperature. For minimum work Tg = T3. Then the heat, Q, equals the Work, Wl of Figure 12-36B. Figure 12-38 is convenient for estimating the moisture condensed from an airstream, as well as establishing the remaining water vapor in the gas-air. 

A = work function (Helmholtz free energy), Btu/lb or Btu C = heat capacity, Btu/lb °R Cp = heat capacity at constant pressure = heat capacity at constant volume F= (Gibbs) free energy, Btu/lb or Btu g = acceleration due to gravity = 32.174 ft/s ... 

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 -... 

If a change from state A to state B occurs in a system at constant pressure (isobaric) so that only p V work is done, then w = pA V and... 

As noted earlier, for a reaction at constant pressure, such as that taking place in an open coffee-cup calorimeter, the heat flow is equal to the change in enthalpy. If a reaction is carried out at constant volume (as is the case in a sealed bomb calorimeter) and there is no mechanical or electrical work involved, no work is done. Under these conditions, with w = 0, the heat flow is equal to the change in energy, AE. Hence we have... 

The heat absorbed in the change at constant pressure is this plus the external work ... 

A convenient unit for gaseous expansion is obtained by measuring p in standard atmospheres and v in litres the work done by expansion through a volume of 1 litre under a constant pressure of one atmosphere is called a litre-atmosphere (1. atm,). Its value in ergs or other units may be calculated as follows ... 

To complete the cycle, we must get the gas out of solution, and restore it to its initial state, by an osmotic process., Raise a and 7 simultaneously through the spaces r and Y respectively, so that the solution maintains a constant composition throughout, and the gas and osmotic pressures are constant. The work done is pQv — BY, and since the whole work in the cycle vanishes ... 

All the impervious diaphragms are now removed, and the two gases slowly compressed into the box under the constant pressures, the steam produced being removed as fast as it is formed through the third semipermeable diaphragm, so that the equilibrium mixture remains unchanged. The amounts of work done are — 2RT, — RT and + 2RT for the 2H2,02, and 2H20. respectively. 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

Thus, in a reversible process that is both isothermal and isobaric, dG equals the work other than pressure-volume work that occurs in the process." Equation (3.96) is important in chemistry, since chemical processes such as chemical reactions or phase changes, occur at constant temperature and constant pressure. Equation (3.96) enables one to calculate work, other than pressure-volume work, for these processes. Conversely, it provides a method for incorporating the variables used to calculate these forms of work into the thermodynamic equations. 

Line 4-1 represents the introduction of fresh gas into the cylinder at constant pressure Ps. The work done on the gas during each stage of the cycle is as follows. 

What Are the Key Ideas Heat and work are equivalent ways of transferring energy between a system and its surroundings. The total energy of an isolated system is constant. The enthalpy change for a process is equal to the heat released at constant pressure. 

We can relate pressure to the work of expansion against a constant pressure by using the fact that pressure is the force divided by the area to which it is applied P = PI A (Section 4.2). Therefore, the force opposing expansion is the product of the pressure acting on the outside of the piston, Pex, and the area of the piston, A (P = P(XA). The work needed to drive the piston out through a distance d is therefore... 

A piston confines 0.100 mol Ar(g) in a volume of 1.00 L at 25°C. Two experiments are performed. In one, the piston is allowed to expand through 1.00 L against a constant pressure of 1.00 atm. In the second, it is allowed to expand reversibly and isothermally to the same final volume. Which process does more work ... 

The work done by any system on its surroundings during expansion against a constant pressure is calculated from Eq. 3 for a reversible, isothermal expansion of an ideal gas, the work is calculated from Eq. 4. A reversible process is a process that can be reversed by an infinitesimal change in a variable. 

Self-Test 6.7A In an exothermic reaction at constant pressure, 50. kj of energy left the system as heat and 20. kj of energy left the system as expansion work. What are the values of (a) AH and (b) At/ for this process ... 

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