Compressible flows ideal adiabatic flow

In this example we describe the calculation of the minimum work for ideal compressible adiabatic flow using two different optimization techniques, (a) analytical, and (b) numerical. Most real flows lie somewhere between adiabatic and isothermal flow. For adiabatic flow, the case examined here, you cannot establish a priori the relationship between pressure and density of the gas because the temperature is unknown as a function of pressure or density, hence the relation between pressure and... 

The differential energy balances of Eqs. (6.10) and (6.15) with the friction term of Eq. (6.18) can be integrated for compressible fluid flow under certain restrictions. Three cases of particular importance are of isentropic or isothermal or adiabatic flows. Equations will be developed for them for ideal gases, and the procedure for nonidcal gases also will be indicated. 

Adiabatic flow of a perfect gas without friction. An ideal frictionless process is represented where a gas is compressed, the area indicating the flow work, which is equal to the change in kinetic energy if no mechanical work in a machine is involved. Thus ... 

Strictly speaking, most of the equations that are presented in the preceding part of this chapter apply only to incompressible fluids but practically, they may be used for all liquids and even for gases and vapors where the pressure differential is small relative to the total pressure. As in the case of incompressible fluids, equations may be derived for ideal frictionless flow and then a coefficient introduced to obtain a correct result. The ideal conditions that will be imposed for a compressible fluid are that it is frictionless and that there is to be no transfer of heat that is, the flow is adiabatic. This last is practically true for metering devices, as the time for the fluid to pass through is so short that very little heat transfer can take place. Because of the variation in density with both pressure and temperature, it is necessary to express rate of discharge in terms of weight rather than volume. Also, the continuity equation must now be... 

The gas is cooled before it enters the compressor to a temperature such that the exit gas from the compressor will be at 450°F and 360 psia assuming ideal adiabatic compression. To get the compressor entering-gas flow rate, assume that pxv - p2v. Neglect pressure drop due to the cooler. 

Fig. 3.53 Temperature profiles in compressible flow of ideal gases. Adiabatic wall...

Moving on to compressible flow, it is first of all necessary to explain the physics of flow through an ideal, frictionless nozzle. Chapter S shows how the behaviour of such a nozzle may be derived from the differential form of the equation for energy conservation under a variety of constraint conditions constant specific volume, isothermal, isentropic and polytropic. The conditions for sonic flow are introduced, and the various flow formulae are compared. Chapter 6 uses the results of the previous chapter in deriving the equations for frictionally resisted, steady-state, compressible flow through a pipe under adiabatic conditions, physically the most likely case on... 

Example 4.2 Dissipated energy in an adiabatic compression In an adiabatic compression operation, air is compressed from 20°C and 101.32kPa to 520 kPa with an efficiency of 0.7. The air flow rate is 22 mol/ s. Assume that the air remains ideal gas during the compression. The surroundings are at 298.15 K. Detemiine the thermodynamic... 

Using the formula for one-dimensional compressible flow presented in Section IV.B, we calculated the pressure, temperature, and velocity profiles describing the subsonic adiabatic expansion of pure carbon dioxide inside the orifice and the capillary up to the nozzle exit (i.e., point 2 in Figures 3 and 5). Both the Bender (38) and Camahan-Starling-van der Waals (39) equations of state were used to calculate the necessary PvT properties for CO2, and results using either of the two equations were essentially identical. Downstream of the nozzle exit, we calculated the pressures and temperatures on the upstream and downstream sides of the Mach disk by using the formulas of Ashkenas and Sherman (36) (see Section V). These formulas assume an ideal gas with y = 1.286, close to the value of CO2 at ambient conditions. We should remember, however. 

A stream of air is compressed in an adiabatic, steady-state flow process at 50 mol/s. The inlet is at 300 K and 1 bar. The outlet is at 10 bar. Estimate the minimum power that the compressor uses. You may assume air behaves as an ideal gas. 

Two-dimensional compressible momentum and energy equations were solved by Asako and Toriyama (2005) to obtain the heat transfer characteristics of gaseous flows in parallel-plate micro-channels. The problem is modeled as a parallel-plate channel, as shown in Fig. 4.19, with a chamber at the stagnation temperature Tstg and the stagnation pressure T stg attached to its upstream section. The flow is assumed to be steady, two-dimensional, and laminar. The fluid is assumed to be an ideal gas. The computations were performed to obtain the adiabatic wall temperature and also to obtain the total temperature of channels with the isothermal walls. The governing equations can be expressed as... 

To calculate the change in entropy in this irreversible flow, it is necessary to consider a corresponding reversible process. One process would be to allow an ideal gas to absorb reversibly the quantity of heat Q at the temperature T2. The gas then can be expanded adiabatically and reversibly (therefore with no change in entropy) until it reaches the temperature Ti. At Ti the gas is compressed reversibly and evolves the quantity of heat Q. During this reversible process, the reservoir at T2 loses heat and undergoes the entropy change... 

For a steady flow of an incompressible fluid in a uniform pipe, the only property that varies along the pipe is pressure. However, for a compressible fluid when the pressure varies (i.e., drops), the density also drops, which means that the velocity must increase for a given mass flow. The kinetic energy thus increases, which results in a decrease in the internal energy and the temperature. This process is usually described as adiabatic, or locally isentropic, with the effect of friction loss included separately. A limiting case is the isothermal condition, although special means are usually required to achieve constant temperature. Under isothermal conditions for an ideal gas. 

A compressor is a gas pumping device that takes in gas at low pressure and discharges it at a higher pre.ssure. Since this proce.ss occurs quickly compared with heat transfer, it is usually assumed to be adiabatic that is, there is no heat transfer to or from the gas during its compression. Assuming that the inlet to the compressor is air [which we will take to be an ideal gas with Cp = 29.3 J/(mol K)] at 1 bar and 290 K and that the discharge is at a pre.ssure of 10 bar, estimate the temperature of the exit gas and the rate at which work is done on the gas (i.e.. the power requirement) for a gas flow of 2.5 mol/s. 

We shall see, as an example, a simple counterflow cold exchanger connecting the cold and warm ends between (B) and (D) is the nearest realistic equivalent to a Carnot cycle. The idealized constant-mass flow system in a perfect counterflow heat regenerator operating with an idealized gas is thermodynamically equivalent to the adiabatic expansion paired with the adiabatic compression in the Carnot cycle, since the following intrinsic energy transfer is fulfilled in terms of a reciprocal isobaric transformation. (See Fig. 2)... 

In contrast to Eq. (8.8), Eq. (8.11) is termed the theoretical adiabatic equation for the mass rate of flow of an ideal compressible fluid across section A 2 in terms of the initial pressure or the pressure difference and the density The value of in Eq. (8.11) should be computed with the general equation of state for the actual gas being metered. 

One mole of oxygen is compressed in a flow process from state 1(1 atm and 25 C) to state 2 (6 atm). The compression is adiabatic but not reversible, with an efficiency of 0.75. Assuming oxygen to behave as an ideal gas under these... 

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