Chapter 3 – Process Variables
Process: to a chemical engineer, the set of tasks or operations that accomplish a chemical
or material transformation to produce a product
Feed or inputs: raw materials and energy that go into a process
Product or output: the desired outcome (e.g. a material) a process is used to make
Process units: hardware used by a process to accomplish specific tasks – for example, a
mixing tank, a heat exchanger, a reactor, an absorption column, etc.
Process streams: the liquid, solid, or gas flows that move material from one process unit
to another
Process variables: the physical and chemical properties of process streams, such as
temperature, pressure, and composition
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Mass and Volume
Density: mass per unit volume of a material (e.g. lbm/ft3); symbol ρ
Specific Volume: volume per unit mass (e.g. ft3/lbm), equals 1/ρ ; symbol Vˆ
For solids and liquids, changes in temperature (T) and pressure (p) have a relatively small
effect on density; for gases, changes in T and p cause large density changes. Solids and
liquids, in this course, will be usually assumed to be incompressible, i.e. ρ = constant.
Specific Gravity: ratio of density (ρ) of a substance to that (ρref) of a reference substance.
The reference substance is often water at 4 oC, whose density is 1.000 g/cm3 = 62.43
lbm/ft3. Symbol: SG. Note that SG is dimensionless.
SG = ρ/ρref
For clarity, the temperatures at which the density and reference density are evaluated
need to be specified. Also, the reference substance must be given.
*Example:
25 o
100 o
The following data are available for a liquid: SG1 = 0.95 o , SG2 = 0.94 o
4
4
The reference substance is water. What is the density of the liquid at 25 oC in AES units?
*Example 3.1-2.
One kg of Hg occupies 7.36 × 10-5 m3 at 0 oC. Given that the volume of a mass of
mercury changes according to
V(T) = V0(1 + 0.18182 × 10-3 T(oC) + 0.0078 × 10-6T 2(oC))
what is the density of mercury at 100 oC? V0 is the volume of mercury at 0 oC.
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Composition
Atomic weight: mass of an atom, measured on a scale on which carbon 12 (12C) has a
mass of exactly 12. If an atom has twice as much mass as 12C, what is its atomic weight?
Molecular weight: the sum of the atomic weights of all the atoms making up a molecule.
Symbol M.
*What is the molecular weight of C6H6? (atomic weight of C = 12.01, atomic weight of H
= 1.01).
gram-mole (g-mole) of a substance: an amount of the substance whose mass, measured
in grams, equals its molecular weight. What is the mass of 1 g-mole of C6H6?
kg-mole of a substance: an amount of the substance whose mass, measured in kg, equals
its molecular weight. What is the mass of 1 kg-mole of C6H6?
lb-moles are similarly defined.
Example:
*How many molecules are in 1 g-mole of O2? Take the molecular weight of O2 to be
32.0. Also, O2 has 16 protons, 16 neutrons, 16 electrons, for a total mass of about 5.32 ×
10-26 kg.
*How many lb-moles are in 150 g of O2? (1 lbm = 453.59 g)
Mass fraction: the fraction of total mass occupied by a component i of a mixture or
solution. Symbol: usually xi or ωi.
Given: 100 lbm of solution of NaCl in water. If the mass of NaCl in the solution is 5 lbm,
what is its mass fraction? What is the mass percent of NaCl present?
Mole (or molar) fraction: the fraction of total moles attributable to a component i of a
mixture or solution. Symbol: usually yi or xi.
Given: 200 g-moles of a solution that contains 20 g-moles of substance A and 180 gmoles of substance B. What are the mole fractions of A and B? What are the mole
percents of A and B?
n
NOTE: For both mass and mole fractions, we must have
∑x
i
=1
i =1
where xi is the mass or mole fraction of species i and there are n species present in the
mixture.
Mass concentration: mass of a species per unit volume of solution (e.g. 0.3 lbm water/ft3
of solution).
Molar concentration: number of moles of a species per unit volume of solution (e.g. 0.2
kg-mole water/m3 of solution). Molarity is molar concentration expressed in units of gmole solute/L of solution. The symbol M is used to indicate units of molarity (e.g. 1 M
solution of NaCl in water means 1 g-mole NaCl/1 L of solution).
Parts per million (ppm) and parts per billion (ppb): these units are sometimes used
when the concentration of a species is low. One needs to specify whether a molar or mass
concentration is intended. Ppm of a species equals its mass or mole fraction times one
million (1 × 106); ppb of a species equals its mass or mole fraction times one billion (1 ×
109). Thus, if xi is mass or mole fraction of i,
ppmi = xi × 106
ppbi = xi × 109
Example:
A “solution” consists of pure benzene. What are the molar and mass ppm and ppb of
benzene in the “solution”?
1 ng of KOH is present in 1 g of solution. What are the mass ppm and ppb of KOH?
A gas mixture contains 1000 moles total, including 1 mole of HCl. What is the molar
ppm of HCl?
*Example 3.3-3.
A gas mixture possesses following mass fractions of species:
Mass fraction
0.16
0.04
0.17
0.63
O2
CO
CO2
N2
molecular weight (g/g-mol)
32
28
44
28
What is the molar fraction of O2?
Note: the easiest way to start is by assuming a basis of calculation.
Average molecular weight: The average molecular weight M of a solution is the mass
of solution per mole of particles it contains. If we have a solution of n species that
contains molesi of species i, the molecular weight of which is Mi, then:
M
= mass of solution / (moles of particles in solution)
= (M1×moles1 + M2×moles2 + …Mn×molesn) / (moles1 + moles2 + …molesn)
Thus:
n
M
=
∑M
i =1
i
moles i
=
moles total
n
∑M
i =1
i
yi
(1)
Flow Rates
When materials are transported from one location to another, for example between two
process units, the rate at which this transport takes place is quantified by their flow rates.
A flow rate can be expressed in mass, molar, or volumetric units. As with all “rates,”
time must be in the denominator.
Mass flow rate: symbol m& . Example: 0.5 lbm air/s
Molar flow rate: symbol n& . Example: 10 kg-moles toluene/h
Volumetric flow rate: symbol V& . Example: 50 ft3 water/min
In future courses, you will also encounter fluxes of materials, which can also be in mass,
molar, or volumetric units. Fluxes are flow rates per area. For example, a mass flux of 1
kg/m2 ⋅ s means that 1 kg of material passes through an area of 1 m2 each second.
*Given: Fluid flows through a pipe of radius 1 ft. The average volumetric flux is 10
ft3/ft2⋅s (note that volumetric flux has units of speed). What is the volumetric flow rate of
the fluid?
Approximate measurement of liquid flow rates can be accomplished with a bucket and a
timer - just measure how much liquid (expressed in units of mass, moles, or volume)
flows into the bucket within a specified time period. Devices such as rotameters, orifice
meters, turbine flow meters, ultrasonic flow meters, and others are available for more
sophisticated measurement and control of liquid and gas flow rates.
*Example 3.3-5.
A 0.50 molar solution of sulfuric acid (H2SO4) in water flows into a reactor at a rate of
1.25 m3/min. The specific gravity of the solution is 1.03 (relative to water at 4 oC).
What is the total mass flow rate?
What is the mass concentration of H2SO4 in the stream (in kg/m3)? (MH2SO4 = 98 g/mol)
What is the mass flow rate of H2SO4 (in kg/s)?
What is the mass fraction of H2SO4?
What is the molar flow rate of H2SO4 (in g-mole/s)?
PRESSURE
Pressure: pressure is, by definition, force per area. Common units of pressure are: N/m2,
dynes/cm2, lbf/in2. N/m2 is otherwise known as a Pascal (Pa), and lbf/in2 as psi (“pounds
per square inch”).
In a static fluid, no part of the fluid is in motion relative to any other part of the fluid. If
the only body force acting on a static fluid is that of gravity, then the pressure P at a
depth h below the free surface of the fluid is equal to
P = P0 + ρgh
(2)
where P0 is the pressure at the free surface of the fluid (i.e. at a depth of “0”), ρ is the
density of the fluid, and g is gravitational acceleration. The pressure inside a static fluid is
sometimes referred to as hydrostatic pressure. A "body force" is a force that acts
throughout the volume (body) of an object (here, the object is the fluid). An example of a
body force is gravity since gravity "pulls" simultaneously on all parts of a body (as
opposed to, for example, a “surface force” which acts only on the surface of a body).
How does pressure arise? If we think of a surface immersed in a fluid, the particles
(molecules, atoms) of the fluid will push against and therefore exert a force on the
surface. This force, divided by the area of the surface, is pressure. In the context of
equation (2), the force is due to the weight of the particles plus the force exerted on the
fluid particles at the free surface. With this “hint,” how do we derive equation 2?*
Pressure is sometimes expressed as a head Ph of a reference fluid. The “head” is the
height h of the reference fluid that would be needed to exert the pressure, according to
equation (2), if P0 is taken as zero. Thus,
Ph = P/ρg
where ρ is the density of the reference fluid.
(3)
*Example:
Express atmospheric pressure (1.013 × 105 Pa) as a pressure head of water at 4 oC, in m.
*Example 3.4-2: What is the pressure 30.0 m below the surface of a 4 oC lake, assuming
the pressure at the free surface is 1 atm (1.013 × 105 Pa)?
A few more definitions:
Absolute pressure: this is total pressure, that is, total force acting on a surface divided by
the area of the surface.
Gauge pressure: this is pressure relative to atmospheric pressure. Thus:
if the absolute pressure is 1.1 atm, and atmospheric pressure is 1.1 atm, gauge pressure =
if the absolute pressure is 2 atm, and atmospheric pressure is 0.9 atm, gauge pressure =
if the absolute pressure is 0, and atmospheric pressure is 0.95 atm, gauge pressure =
Gauge pressure is sometimes reported in units of psig, “pounds force per square inch
gauge.”
Pgauge = Pabsolute – Patmospheric
(4)
*Example: Derive an expression for the gauge pressure being measured by the
manometer in the below figure at the point indicated.
Standard pressure: Chemical engineers often use various reference states for reporting
material properties or for performing calculations. The so called standard pressure (often
used in calculations involving gases – Chapter 5) is, by convention, chosen to be 1 atm.
TEMPERATURE
Temperature is a measure of the random kinetic motion of the particles (atoms,
molecules) of a substance. Aside: how is this different from the kinetic energy of a
moving train?
Material properties are a function of the random kinetic motion of the material’s
molecules. For example, if you think of the molecules of a liquid, if their degree of
agitation (center of mass motion, vibrations of the molecular bonds) increases, you might
suspect that they will not pack as well as at lower degrees of agitation, because of the
increased force and frequency of collisions between the molecules. Thus, you may
forecast that the density of the liquid will decrease with an increase in temperature (note
that the actual trend in density with temperature is more complex to anticipate than the
above simple argument suggests – for example, see ρ(T) of water at temperatures near 0
o
C). Since we can measure physical properties of materials, such as density, we can use
such measurements to also specify the temperature (e.g. as in a mercury thermometer).
The state of random thermal agitation of matter (temperature) is expressed in terms of
temperature scales which, in turn, are most often defined relative to occurrence of phase
transitions of certain materials. For example, in the Celsius scale, the temperature Tf at
which water freezes, under a pressure of 1 atm, is assigned the value zero degrees Celsius
(0 oC) while the temperature Tb at which water boils, under a pressure of 1 atm, is
assigned the value of 100 oC. On the Celsius scale there are thus 100 temperature
intervals, or degrees, between Tf and Tb of water.
Note that the Celsius scale is not an absolute temperature scale. An absolute temperature
scale is one in which a temperature of zero degrees (0 o) is assigned to the lowest possible
temperature achievable in nature, so called absolute zero. The four common temperature
scales, relative to the values of Tf and Tb of water, are:
1). Celsius scale:
Tf = 0 oC
Tb = 100 oC
Absolute zero = -273.15 oC
2). Fahrenheit scale:
Tf = 32 oF
Tb = 212 oF
Absolute zero = -459.67 oF
3). Kelvin scale:
Tf = 273.15 oK
Tb = 373.15 oK
Absolute zero = 0 oK
4). Rankine scale:
Tf = 491.67 oR
Tb = 671.67 oR
Absolute zero = 0 oR
Note that “degrees” can be used to specify both temperature intervals as well as
temperature magnitudes. For example, “the temperature increased by 10 degrees Celsius”
specifies a temperature interval. On the other hand, “the temperature is 30 degrees
Celsius” specifies a temperature. See *Example 3.5-3 in the text.