The Mo Dynamics

8/12/2019 The Mo Dynamics

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Volumetric Properties of Pure Fluids

The three-dimensional p

v

T surface is useful for bringing out thegeneral relationships among the three phases of matter normally

under consideration. However, it is often more convenient to work

with two-dimensional projections of the surface.

pvT sur face and projections for a substance that expands on f reezing

Three-dimensional view.


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PT Diagram for a pure Substance

If the pvT surface is projected onto the pressuretemperature plane, a

property diagram known as a phase diagram resul ts. As shown in F ig., when the

surface is projected in this way, the two-phase regions reduce to lines. A point on

any of these lines represents all two-phase mixtures at that particular temperatureand pressure.

The triple line of the three-dimensional pvT surface projects onto a point on

the phase diagram. This is called the tr iple point.

The line representing the two-phase solidliquid region on the phase diagram

slopes to the left for substances that expand on freezing and to the right for those

that contract.

The term saturation temperature designates the

temperature at which a phase change takes place at

a given pressure, and this pressure is called the

saturation pressurefor the given temperature. It isapparent from the phase diagrams that for each

saturation pressure there is a unique saturation

temperature, and conversely.


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Phase diagram for water

The temperature assigned to the triple point of water is 273.16 K.

The measured pressure at the triple point of water is 0.6113 kPa.


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PV Diagram

On PV diagram the boundaries of PT diagram are areas i.e., regions where twophases, solid/ liquid, solid/ Vapour, and liquid/vapour co exist in equilibrium.

The triple point here becomes a horizontal line, where the three phases co-exist

at a single a single temperature and pressure.

Fig shows a PV diagram, with 3 isotherms

super imposed. the line labeled T


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The horizontal segments of the isotherms in the 2-phase region become

progressively shorter at higher temperatures, being ultimately reduced to apoint at C, thus the critical isotherm labeled Tc exhibits a horizontal inflection

at the critical point C at the top of the dome. Here the liquid and vapour phases

cannot be distinguished from each other, because their properties are the same.

At the top of the dome, where the saturated liquid and saturated vapor lines

meet, is the cri tical point. The critical temperature Tc of a pure substance is

the maximum temperature at which liquid and vapor phases can coexist inequilibrium. The pressure at the critical point is called the critical pressure, pc.

The specific volume at this state is the critical specific volume.


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Sketch of a temperaturespecif ic volume diagram for water

showing the liquid, two-phase liquidvapor, and vapor regions

TemperatureSpecific volume diagram


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SATURATION TABLES

The saturation tables are a list property values for the saturated liquid and

vapor states. The property values at these states are denoted by the subscripts fand g, respectively. (ref tables)

The specific volume of a two-phase liquidvapor mixture can be

determined by using the saturation tables and the definition of quality given by,

The total volume of the mixture is the sum of the volumes of the liquid

and vapor phases

V= Vliq+ V vap

Dividing by the total mass of the mixture, m, an average specific

volume for the mixture is obtained


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Since the liquid phase is a saturated liquid and the vapor phase isa saturated vapor, Vliq =mliq vfand V vap = mvapvg, so

by the definition of quality, x= mvap /m, and noting that

mliq/m =1- x, the above expression becomes


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Generalized Compressibility Chart

UNIVERSAL GAS CONSTANT, R

Let a gas be confined in a cylinder by a piston and

the entire assembly held at a constant temperature. Thepiston can be moved to various positions so that a series of

equilibrium states at constant temperature can be visited.

Suppose the pressure and specific volume are measured at

each state and the value of the ratio is volume per mole

determined. These ratios can then be plotted versus pressureat constant temperature. The results for several temperatures

are sketched in Fig. When the ratios are extrapolated to zero

pressure, precisely the same limiting value is obtained for

each curve. That is,Sketch of PV/T versus pressur e

for a gas at several specif ied

values of temperatur e.

If this procedure were repeated for other gases, it would be found in every instance that

the limit of the ratio as p tends to zero at fixed temperature is the same, namely Since the

same limiting value is exhibited by all gases, is called the universal gas constant. I ts

value as determined experimentally is

R 8.314 kJ/kmol . K


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COMPRESSIBILITY FACTOR, Z

The dimensionless ratio is called the compressibil i ty factor and is denoted by Z.

That is,Z=pv/RT

Vari ation of the compressibil ity factor of hydrogen

with pressur e at constant temperature.


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EQUATIONS OF STATE

Considering the curves of two fig. (refer slide 12 & 13), it is reasonable to think

that the variation with pressure and temperature of the compressibility factor forgases might be expressible as an equation, at least for certain intervals ofp and

T. Hencce we can write Two expressions.

1. One gives the compressibility factor as an infinite series expansion in

pressure:

where the coefficients depend on temperature only. The dots in Eq. represent

higher-order terms. The other is a series form entirely analogous to Eq. 3.29 but

expressed in terms of Volume V instead ofP

These Equations are known as vir ial equations of state, and the coeff icients

andB, C, D, . . . are called virial coefficients.


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Equations of state can be classified by the number of adjustable constants they

include. Let us consider some of the more commonly used equations of state in

order of increasing complexity

1. Two-Constant Equations of State

a. VAN DER WAALS EQUATION

An improvement over the ideal gas equation of state based on elementary

molecular arguments was suggested in 1873 by van der Waals, who noted that

gas molecules actually occupy more than the negligibly small volumepresumed by the ideal gas model and also exert long-range attractive forces on

one another.

Based on these elementary molecular arguments, the van der Waals equation

of state is

The constant b is intended to account for the finite volume occupied by the

molecules, the term accounts for the forces of attraction between molecules,

and is the universal gas constant. Note than when a and b are set to zero, the

ideal gas equation of state results.


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b. REDLICHKWONG EQUATION

The RedlichKwong equation, considered by many to be the best of the two-

constant equations of state, is


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Internal Energy, Enthalpy, and Specific Heats of Ideal Gases

For a gas obeying the ideal gas model, specific internal energy depends only ontemperature. Hence, the specific heat cv is also a function of temperature alone.

That is,

This is expressed as an ordinary derivative because u depends only on T.

By separating variables

On integration


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For a gas obeying the ideal gas model, the specific enthalpy depends only on

temperature, so the specific heat cp, is a function of temperature alone.That is

Separating variables

On integration


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An important relationship between the ideal gas specific heats can be developed

by differentiating Eq. with respect to temperature

and introducing Eqs. for dh& duto obtain

On a molar basis, this is written as

For an ideal gas, the specific heat ratio, k, is also a function of temperature only

and

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