Piston, differential

The counterplaten is connected to the cylinder-platen by three columns, using split nuts and lock nuts. The return motion of the plunger crosshead is performed by differential pistons sealed by stuffing boxes, which may easily be retightened. 

In the example of the previous section, the release of the stop always leads to the motion of the piston in one direction, to a final state in which the pressures are equal, never in the other direction. This obvious experimental observation turns out to be related to a mathematical problem, the integrability of differentials in themiodynamics. The differential Dq, even is inexact, but in mathematics many such expressions can be converted into exact differentials with the aid of an integrating factor. 

When a differential amount of a pure liquid in eqmlibrium with its vapor in a piston-and-cyhnder arrangement evaporates at constant temperature T and vapor pressure Eq. (4-16) apphed to the process reduces to d njG = 0, whence... 

In biphase systems velocity of the steam is often 10 times the velocity of the liquid. If condensate waves rise and fill a pipe, a seal is formed with the pressure of the steam behind it (Fig. 2). Since the steam cannot flow through the condensate seal, pressure drops on the downstream side. The condensate seal now becomes a piston accelerated downstream by this pressure differential. As it is driven downstream it picks up more liquid, which adds to the existing mass of the slug, and the velocity increases. 

It is desirable to have additional axial-load control on the multistage compressor. A balance piston, also refeired to as the balance drum, can be located at the discharge end (see Figure 5-46). The balance piston consists of a rotating element that has a specified diameter and an extended rim for sealing. The area adjacent to the balance piston (opposite the last stage location) is vented, normally to suction pressure. The differential pressure across the balance piston acts on the balance piston area to develop a thrust force opposite that generated by the impellers. The pressure on the... 

A typical pilot-operated PR valve is illustrated in Figure 5. Under normal operating conditions, the vessel pressure acts on the main valve seat at the bottom of the free-floating differential area piston, and by means of the pilot supply line is also applied to the top of the piston and under the pilot valve disc. Since the top area of the piston is larger than the nozzle area at the lower end of the piston, there is a large net load holding the piston down on the nozzle. Under static... 

The flow field in front of an expanding piston is characterized by a leading gas-dynamic discontinuity, namely, a shock followed by a monotonic increase in gas-dynamic variables toward the piston. If both shock and piston are regarded as boundary conditions, the intermediate flow field may be treated as isentropic. Therefore, the gas dynamics can be described by only two dependent variables. Moreover, the assumption of similarity reduces the number of independent variables to one, which makes it possible to recast the conservation equations for mass and momentum into a set of two simultaneous ordinary differential equations ... 

Mechanical efficiencies of steam pumps vary with the types of pump, stroke and the pressure differential. Some representative values are 55 to 80 percent for piston pumps with strokes of 3 inches and 24 inches respeedvely, and pressure differential up to 300 psi. For the same strokes a plunger design varies from 50 to 78 percent, and at over 300 psi differendal the efficiencies are 41 to 67 percent [9]. Steam required is approximately 120 Ibs/hour per BHP. 

Balance piston labyrinths in straight-through designs are required to withstand differential pressures as high as 5,000 psi). 

When the piston is assumed to be measured from the LGSs (top eurve), the corresponding mode is singular because one cannot measure the contribution of each layer to the total piston. This case is not realistic, since no wavefront sensor measures the piston. When the tilts are measured from the LGSs (case of the polychromatic LGS), the odd piston is not measured again. The even piston is no longer available. And the two odd tilt modes are not also, because whereas the tilt is measured, the differential tilt between the two DMs is not one does not know where the tilt forms. Thus there are 4 zero eigenvalues. 

A differential balance written for a vanishingly small control volume, within which t A is approximately constant, is needed to analyze a piston flow reactor. See Figure 1.4. The differential volume element has volume AV, cross-sectional area A and length Az. The general component balance now gives... 

FIGURE 1.4 Differential element in a piston flow reactor. 

FIGURE 3.1 Differential volume elements in piston flow reactors (a) variable cross section (b) constant cross section. 

The dAc/dz term is usually zero since tubular reactors with constant diameter are by far the most important application of Equation (3.7). For the exceptional case, we suppose that Afz) is known, say from the design drawings of the reactor. It must be a smooth (meaning differentiable) and slowly varying function of z or else the assumption of piston flow will run into hydrodynamic as well as mathematical difficulties. Abrupt changes in A. will create secondary flows that invalidate the assumptions of piston flow. 

Perfectly mixed stirred tank reactors have no spatial variations in composition or physical properties within the reactor or in the exit from it. Everything inside the system is uniform except at the very entrance. Molecules experience a step change in environment immediately upon entering. A perfectly mixed CSTR has only two environments one at the inlet and one inside the reactor and at the outlet. These environments are specifled by a set of compositions and operating conditions that have only two values either bi ,..., Ti or Uout, bout, , Pout, Tout- When the reactor is at a steady state, the inlet and outlet properties are related by algebraic equations. The piston flow reactors and real flow reactors show a more gradual change from inlet to outlet, and the inlet and outlet properties are related by differential equations. 

Steady-state temperatures along the length of a piston flow reactor are governed by an ordinary differential equation. Consider the differential reactor element shown in Figure 5.3. The energy balance is the same as Equation (5.14) except... 

In the general case of a piston flow reactor, one must solve a fairly small set of simultaneous, ordinary differential equations. The minimum set (of one) arises for a single, isothermal reaction. In principle, one extra equation must be added for each additional reaction. In practice, numerical solutions are somewhat easier to implement if a separate equation is written for each reactive component. This ensures that the stoichiometry is correct and keeps the physics and chemistry of the problem rather more transparent than when the reaction coordinate method is used to obtain the smallest possible set of differential... 

Unlike stirred tanks, piston flow reactors are distributed systems with one-dimensional gradients in composition and physical properties. Steady-state performance is governed by ordinary differential equations, and dynamic performance is governed by partial differential equations, albeit simple, first-order PDEs. Figure 14.6 illustrates a component balance for a differential volume element. 

FIGURE 14.6 Differential volume element in an unsteady piston flow reactor. 

---