Expansion Processes

So far we have seen that a periodic function can be expanded in a discrete basis set of frequencies and a non-periodic function can be expanded in a continuous basis set of frequencies. The expansion process can be viewed as expressing a function in a different basis. These basis sets are the collections of solutions to a differential equation called the wave equation. These sets of solutions are useful because they are complete sets. 

The expansion process consists of three steps creating small discontinuities or cells in a fluid or plastic phase causing these cells to grow to a desired volume and stabilizing this cellular stmcture by physical or chemical means. 

Prom the onset of creaming to the end of the rise during the expansion process, the gas must be retained completely in the form of bubbles, which ultimately result in the closed-ceU stmcture. Addition of surfactants faciUtates the production of very small uniform bubbles necessary for a fine-cell stmcture. 

FIG. 11-113 Claude cycle refrigerator utilizing both expansion processes. 

The Intercooled Regenerative Reheat Cycle The Carnot cycle is the optimum cycle between two temperatures, and all cycles try to approach this optimum. Maximum thermal efficiency is achieved by approaching the isothermal compression and expansion of the Carnot cycle or by intercoohng in compression and reheating in the expansion process. The intercooled regenerative reheat cycle approaches this optimum cycle in a practical fashion. This cycle achieves the maximum efficiency and work output of any of the cycles described to this point. With the insertion of an intercooler in the compressor, the pressure ratio for maximum efficiency moves to a much higher ratio, as indicated in Fig. 29-36. 

Any work developed by the turboexpander is at the expense of the enthalpy of the process stream, and the latter is correspondingly cooleci. A low inlet temperature means a correspondingly lower outlet temperature, and the lower the temperature range, the more effective the expansion process becomes. 

Most ethylene plants operate continuously with the expander functioning at or near design point. However, by using inlet guide vanes, the expander can still provide optimum performance at off-design conditions. Also, the expansion process generates power, which is used by the compressor. The ethylene enters the expanders at approximately 26 bar (377 psia) and exits at approximately 6 bar (87 psia). The expanders generate over 2,000 hp for gas compression. 

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 ... 

A third objective is similarly obvious. If compression and expansion processes can attain more isentropic conditions, then the cycle widening due to irreversibility is decreased, cr moves nearer to unity and the thermal efficiency increases (for a given t). Cycle modifications or innovations are mainly aimed at increasing (by increasing or decreasing a)-... 

We next consider the application of the exergy flux equation to a closed cycle plant based on the Joule-Brayton (JB) cycle (see Fig. 1.4), but with irreversible compression and expansion processes—an irreversible Joule-Brayton (IJB) cycle. The T,.s diagram is as shown in Fig. 2.6. 

Consider again the simplest case of compressor delivery air (mass flow i/>, at T ), mixed at constant pressure with unit mass flow of combustion products (at Tf) to give mass flow (1 + i/>) at Ts (see the T, s diagram of Fig. 4.5). The compression and expansion processes are now irreversible. 

At the instant a pressure vessel ruptures, pressure at the contact surface is given by Eq. (6.3.22). The further development of pressure at the contact surface can only be evaluated numerically. However, the actual p-V process can be adequately approximated by the dashed curve in Figure 6.12. In this process, the constant-pressure segment represents irreversible expansion against an equilibrium counterpressure P3 until the gas reaches a volume V3. This is followed by an isentropic expansion to the end-state pressure Pq. For this process, the point (p, V3) is not on the isentrope which emanates from point (p, V,), since the first phase of the expansion process is irreversible. Adamczyk calculates point (p, V3) from the conservation of energy law and finds... 

Most rigid polyurethane foams have a closed cell structure. Closed cells form when the plastic cell walls remain intact during the expansion process and are not ruptured by the increasing cell pressure. Depending on the blowing process a small fraction (5-10%) of the cells remain open. Closed cell structures provide rigidity and obstruct gaseous or fluid diffusional processes. 

This splitting procedure is now applied reeursively to each of the children nodes just created. The successive expansion process continues until terminal nodes or leaves, over which no further partitions are performed, can be identified. 

If no Mollier diagram is available, it is more difficult to estimate the ideal work in compression or expansion processes. Schultz (1962) gives a method for the calculation of the polytropic work, based on two generalised compressibility functions, X and Y which supplement the familiar compressibility factor Z. 

The calculation of entropy is required for compression and expansion calculations. Isentropic compression and expansion is often used as a reference for real compression and expansion processes. The calculation of entropy might also be required in order to calculate other derived thermodynamic properties. Like enthalpy, entropy can also be calculated from a departure function ... 

The effect of this expansion process is illustrated in Figure 4.4, in which all points in the sample database are drawn from a donut-shaped object. The outline of that object is quickly reproduced, with increasing fidelity as the number of units increases. It is notable that fewer units lie in the interior of the object once the map is complete than was the case when the SOM was trained on a set of points that defined a conical flask. This is a consequence of the flexibility of the geometry of the lattice in which the GCS grows. 

Figure 5.23 SAND can be used to modify amine-containing molecules, and then photo-initiate crosslinking to another amine-containing molecule via a ring-expansion process. The conjugates may be disrupted by reduction of the cross-bridge disulfide with DTT.

Figure 5.31 ASIB can react with sulfhydryl-containing molecules through its iodoacetate group to form thioether linkages. Subsequent exposure to UV light causes a ring-expansion process to occur, creating a highly reactive dehydroazepine intermediate that can couple to amine-containing molecules.

Figure 5.32 APDP can modify sulfhydryl-containing compounds through its pyridyl disulfide group to form disulfide bonds. Its phenyl azide end then can be photolyzed with UV light to couple with nucleophiles via a ring-expansion process. The disulfide group of the crosslink can be selectively cleaved using Dll.

Whilst sodium bicarbonate is the primary blowing agent, it is common compounding practice to use it in conjunction with a proportion of a weak acid, such as stearic or oleic acid, whose function is to trigger the reaction and assist in the uniform decomposition of the bicarbonate. The higher than normal fatty acid level will also act as a process aid, facilitating the bubble expansion process. 

In a hysteresis experiment, the movable barrier would be reversed at a time, designated as t, so that the monolayer comes under an expansion process at the same speed, v. The increase of surface area causes a reduction in the surface pressure. For a reversibly adsorbed monolayer, the desorption of segments may continue during the first period of expansion until the surface pressure is reduced to its equilibrium value. On further expansion, readsorption occurs because the surface pressure is below its equilibrium value. 

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