The Compressibility of a Gas

There are many more design tools for absorbers and distillation columns. These may be found in the text by King (1971) and in Perry s Chemical Engineers Handbook. 

We have applied dimensional analysis to two process units heat exchangers and distillation columns. Dimensional analysis can also be applied to fundamental systems, such as a gas of a pure substance. Given some basic properties of a gas, we would like to be able to predict the molar volume at a given temperature and pressure. 

Dimensional analysis provides the basis for a universal curve for gases. As always, we begin by listing the parameters of a gas. 

You would not have been able to predict that critical temperature and pressure were appropriate for scaling the nonideal behavior of a gas. Until experiments were performed (and you had completed a course in molecular thermodynamics), any other distinguishing features of a gas - boiling point at 1 atm, the triple point, molar volume, etc. - were reasonable choices. And how can we know that the list of parameters in Table 5.12b is sufficient Again, we would not know until we measured and analyzed data for many gases. If all the data lie on the same plot, the list is sufficient. 

We now derive the dimensionless groups. As before we begin by writing a general expression for the dimensionless group, 11  

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The flow of air through the compressor may be regarded as the compression of a gas with properties (Cpa)i2 and (ya)i2 (the double subscript indicates that a mean is taken over the relevant temperature range). The work required to compress the unit mass of air in the compressor is then represented as... 

Space-Filling Models. For most of this century, chemists have tried to answer the size question by using a special set of molecular models known as space-filling or CPK models. The space-filling model of an atom is simply a sphere of fixed radius. A different radius is used for each element, and the radii are chosen to reproduce certain experimental observations, such as the compressibility of a gas, or the spacing between atoms in a crystal. 

Thus, from an investigation of the compressibility of a gas we can deduce the values of its critical constants. We observe that, according to van der Waals theory, liquid and gas are really two distant states on the same isotherm, and having therefore the same characteristic equation. Another theory supposes that each state has its own characteristic equation, with definite constants, which however vary with the temperature, so that both equations continuously coalesce at the critical point. The correlation of the liquid and gaseous states effected by van der Waals theory is, however, rightly regarded as one of the greatest achievements of molecular theory. 

COMPRESSION (Gas). The compressibility of a gas is delined as Ihe rate of volume decrease with increasing pressure, per unit volume of the gas. The compressibility depends not only on the stale of the gas. but also on the conditions under which the compression is uchicved. Thus, if the temperature is kepi constant during compression, the compressibility so defined is called Ihe isothermal compressihilily ft ... 

In order to make magnetic refrigeration practical one must use a continuous cyclic process in which heat must be rejected when the ferromagnet experiences a magnetic field increase (just as the heat is rejected during the compression of a gas in a conventional gas cycle... 

From the first law of thermodynamics, the internal work W needed for the compression of a gas can be calculated from the enthalpies and before and after the... 

This illustration demon.strates that the sum O -I- IVy is the same for a fluid undergoing some change in a continuous process regardless of whether we choose to compute this sum from the closed-system analysis on a mass of gas or from an open-system analysis on a given volume in space. In Illustration 3.2-2 we consider another problem, the compression of a gas by two different processes, the first being a closed-system piston-and-cylinder process and the second being a flow compressor process. Here we will find that the sum Q + W s different in the two processes, but the origin of this difference is easily understood. 

To obtain an equation for the PV work, consider the compression of a gas in a cylinder fitted with a piston whose area is a (Fig. f -2b). Compression is done through the application of an external pressure Pex, which produces a force on the piston, F = Pe a, and causes the piston to move by dx. The amount of work associated with this process is equal to the force on the piston, PexO, multiplied by the displacement dx. Noting that the product adx is equal to the change of volume, -dV, the amount of work is... 

Kinetically one can compare the rubber elastic extension to the compression of a gas, as illustrated in Fig. 5.167. The thermodynamic equations reveal that reversible rabber contraction can just as well drive a heat pump as reversible gas expansion. Raising temperature, increases the pressure of a gas, analogously it takes a greater force to keep a rabber band extended at higher temperature. The two equations for this fact are known as the ideal gas law p = PoTV(/(ToV) with PoV/r = R, the gas... 

P). Note the expression for (C) is also a function of the particle diameter (dp) and includes known thermodynamic and physical properties of the chromatographic system. Consequently, with the aid of a computer, the optimum particle diameter (dp(opt)) can be calculated as that value that will meet the equality defined in equation (18). However, it will be seen in due course that these equations can be simplified. The equation for a flow of liquid though a packed bed will, however, differ for a compressible fluid, i.e., a gas. Due to the compressibility of a gas, the flow rate can not be described by the simple D Arcy law for liquids. From chapter 2, it is seen that... 

Gases may be compressed liquids cannot. Liquid particles are touchingly close to each other. There is no space between them, so they cannot be pushed closer, as in the compression of a gas. 

The compressibility of a gas can be measured, and by doing so at different pressures and temperatures, the resulting set of (P, P, T) data can be used to find the coefficients a and b. In other words, measurement of the isothermal compressibility can be used to determine the equation of state, in this case, for a gas that is well represented by Equation 2.30. 

An amount of energy that can be used to effect any type of desirable change is referred to as useful energy. This change could involve the raising of a weight, the heating of a fluid, the compression of a gas, the production of electricity, etc. 

We must emphasize again that thermodynamic properties are not measurable quantities. They are concepts, mathematical constructs in a strict sense, created for thermodynamics and their values are determined from experimentally measurable quantities through relationships developed by thermodynamics. Yet, these constructs are essential for the solution of everyday chemical engineering problems from the shaft work required for the compression of a gas (entropy and enthalpy), to the design of a distillation tower (fugacity and enthalpy). 

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