Single Phase Systems

In (almost) all of the problems that we have dicussed, you were given all the information that you needed. Obviously, in real life, you will be required to get all the information you need by yourself.

This information can come from:

Measured Data from an existing process (experiments!)
Process Specifications/Design
Physical Properties/Laws

(Obviously, you are also limited by physical contraints and must not violate balances of mass or energy.)

Just like we were sometimes given process specifications to be used in our degree-of-freedom analysis, we will also use physical laws as additional relations to lower the number of degrees-of-freedom.

NOTE

We have already seen how to use measurements (calibration curves) to determine the composition and density of a process stream - see problem 4.26

Liquid and Solid Denisties

Most Liquids and Solids -> density decreases with heating, and density increases with pressure.

NOT VERY MUCH -> "incompressible"!

NOTE

Because solids and liquids are essentially incompressible you can use the density at one T and P for almost any other T and P.

Mixtures are difficult, but usually assume either:

DEFINITION

For mixtures of "similar" solid or liquid materials (e.g., ethanol and methanol) the density of a mixture may be approximated by averaging the specific volumes. This is termed assuming volume additivity.

SIMILAR:

DEFINITION

Another method of approximating the density of a solid or liquid mixture is to average the densities.

DEFAULT:

OBJECTIVES:

Determine the density of a mixture of liquids

TEST YOURSELF

(Re)Do homework problem 3.3!

Gas Densities

For gases, the density is obtained by using an equation of state.

DEFINITION

An equation of state relates the molar density (or specific molar volume) of a fluid (so they sometimes work for liquids, too!) to the temperature and pressure of the fluid.

simplest -> ideal gas law (PV=nRT)!

NOTE

The ideal gas law may also be written as for a flowing system or , where is the specific molar volume.

IMPORTANT

The ideal gas law is an approximation (!) that has only limited applicability. It is usually used for diatomic gas when (RT/P)>5 L/mol and for other gases when (RT/P) > 30 L/mol (i.e., at high specific volumes!).

Standard Temperature and Pressure

A technique for dealing with ideal gases that is of dubious utility is to utilize the reference point of standard temperature and pressure.

DEFINITION

STP is an arbitrary reference point chosen to be T = 273K (OC) and P = 1 atm.

OBJECTIVES:

Use the ideal gas law to determine P, V, or T of a single component

TEST YOURSELF

What is the specific volume of an ideal gas at these conditions?

The utility of using this information is that instead of memorizing values of R (the ideal gas constant), you can instead memorize values of the conditions at STP!

Ideal Gas Mixtures

If you have ni moles of each species (for example nA moles of A), you can try to calculate the pressure or volume that that gas alone (i.e., ignoring the other gases that are around) exerts/occupies.

DEFINITION

Partial pressure refers to the pressure that would be exerted by a species (in a mixture) if there were no other species present.

DEFINITION

The pure component volume, vA, refers to the volume that would be occupied by a species (in a mixture) if there were no other species present.

So, in an ideal gas mixture EACH COMPONENT satisfies the ideal gas law provided the partial pressure or pure component volumes are used!

PAV = nART
or
PvA = nART

In this way, the sum of the component pressures (partial pressures) or volumes (pure component volumes) should sum to the total pressure or volume:

PA + PB + ... = P
VA + VA + ... = V

This is easy to see if you divide either the partial pressure equation or the pure component volume equation by the ideal gas law for the total mixture:



Note that RT cancels in both equations and that V cancels in the first and P cancels in the second, also that nA/n = ya. We can then rearrange the result to get:

pA = yAP
vA = yAV

So The volume fraction (or pressure fraction) of an ideal gas is equal to the mol fraction! (vA/V = nA/n)

OBJECTIVES:

Determine the composition of a mixture of ideal gases from their partial pressures or volume fractions

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