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
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!
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!).
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!
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