Mass and Energy Balances
PHE1014/FTE1013
Lecture 1
Content
• Units and Dimensions
• Force and Weight
• Pressure
• Chemical Composition
• Data Representation and Analysis
Chapters 2 and 3
Mass and Energy Balances: Lecture 1
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Units and Dimensions
• Base SI units: length (m), mass (kg), moles (mol), time (s),
temperature (K), electric current (A), light intensity (cd, candela)
• Derived SI units: volume (L), force (N, 1 kg∙m2/s2), pressure (Pa, 1
N/m2), energy/work (J, 1 N∙m), power (W, 1 J/s)
• Work with 4 significant figures
• 0.02562 g∙in/min2
• 4.529 × 10-5 ton∙miles/wk2
• Convert 1 cm/s2 to km/yr2
• Tera (T) = ____ ; Giga (G) = ____ ; Mega (M) = ____ ; kilo (k) = ____ ;
centi (c) = ____ ; milli (m) = ____ ; micro (μ) = ____ ; nano (n) = ____ .
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Force and Weight
• Force
• SI system: 1 N = 1 kg∙m/s2
• American engineering system: 1 lbf = 32.174 lbm∙ft/s2
• Weight
• Force exerted on object by gravity
• W=mg
where m = mass ;
g = gravitational acceleration = 9.8066 m/s2
Mass and Energy Balances: Lecture 1
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Force and Weight
• E.g. Water has density of 62.4 lbm/ft3. How much does 2.0 ft3 of
water weigh?
• At sea level (g = 32.174 ft/s2)
• On a mountain (g = 32.139 ft/s2)
Mass and Energy Balances: Lecture 1
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Mass and Volume
• Density: mass per unit volume of the substance (kg/m3, g/cm3,
lbm/ft3 )
• Specific volume: volume occupied by unit mass of substance;
inverse of density
• Density of pure solids and liquids independent of pressure and
vary slightly with pressure
• Specific gravity: ratio of the density of the substance to density of
reference substance (SG = ρ/ρref) at specific conditions, usually
water at 4°C (ρwater, 4°C = 1g/cm3 = 62.43 lbm/ft3)
Mass and Energy Balances: Lecture 1
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Mass and Volume
• Given specific gravity of mercury at 20°C as 13.546,
• Calculate density of mercury in lbm/ft3
• Calculate volume in ft3 occupied by 215 kg of mercury
Mass and Energy Balances: Lecture 1
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Mass and Volume
• The effect of temperature on the volume of mercury is given as
V(T) = V0 ( 1 + 0.18182 × 10-3 T + 0.0078 × 10-6 T2 )
where V is in ft3 and T is in °C.
• What volume (in ft3) would the mercury occupy at 100°C?
Mass and Energy Balances: Lecture 1
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Mass and Volumetric Flow Rate
• Most processes involve the movement of material from one point
to another
• Rate at which the material is transported is the flow rate
• Flow rate is expressed as mass flow rate or volumetric flow rate
𝑚(𝑘𝑔
ሶ
𝑓𝑙𝑢𝑖𝑑/𝑠)
ሶ 3 𝑓𝑙𝑢𝑖𝑑/𝑠)
𝑉(𝑚
• Suppose a fluid (gas or liquid) flows in the cylindrical pipe above,
where the shaded area represents a section perpendicular to the
direction of the flow
• At every second, m kilogram of the fluid pass through the cross
section
• At every second, V cubic meters of the fluid pass through the cross
section
Mass and Energy Balances: Lecture 1
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Mass and Volumetric Flow Rate
• What is 𝑉ሶ 𝐴 (ft3/s) of a stream of liquid acetone (C3H6O) whose molar
flow rate is 2.17 × 103 kmol/h? SGacetone = 0.791
Mass and Energy Balances: Lecture 1
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Mass and Volumetric Flow Rate
• Non-additivity of liquid volumes. Suppose we mix two miscible
liquids
• mT = mA + mB
• nT = nA + nB
• VT ≠ VA + VB
(law of conservation of mass)
(provided that A and B do not react)
(volumes of liquids are NOT additive, but close)
Mass and Energy Balances: Lecture 1
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Pressure
• Pressure = ratio of a force to the area on which the force acts
(N/m2 = Pa, lbf/in2 = psi)
• A typical value of the atmospheric pressure at sea level, 760 mm
Hg, has been designated as a standard pressure of 1 atmosphere.
• The earth’s atmosphere can be considered a very tall column of
fluid (air)
• The pressure at the bottom of that column is atmospheric pressure
• Pabsolute = Pgauge + Patmospheric
• Pgauge = given by pressure-measuring devices, pressure relative to
atmospheric pressure
Mass and Energy Balances: Lecture 1
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Pressure
• Consider a container of a fluid of density ρ (kg/m3), with height h
(m) and cross-sectional area A (m2). A uniform pressure Po (N/m2)
is exerted on the upper surface of the fluid
• Find the weight of the fluid, Wf
• Given the forces exerted on the top
and bottom surfaces as Fo and F, what
is the pressure exerted by the fluid
on the bottom surface of the container?
Mass and Energy Balances: Lecture 1
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Pressure
• Pressure can also be expressed as a height of mercury. Given SGHg
= 13.6, express the pressure of 14.7 lbf/in2 in the unit mm Hg.
Mass and Energy Balances: Lecture 1
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Temperature
• A degree is both a temperature and a temperature interval
• The conversion factors above refer to the temperature intervals,
not temperatures
• To find the number of Celcius degrees between 32°F and 212°F:
• To find the Celcius temperature corresponding to 32°F:
Mass and Energy Balances: Lecture 1
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Chemical Composition
• Atomic weight: mass of an atom on a scale that assigns 12C a
mass of exactly 12
• Molecular weight: sum of the atomic weights of the atoms
• E.g. MWCO2 = 12 + 2(16) = 48
• Gram-mole (mol in SI unit): amount of a species that the mass in
grams is numerically equal to the molecular weight
• If the molecular weight of a substance is M, then there are M
kg/kmol, M g/mol, and M lbm/lb-mole of this substance
• One gram-mole of any species contains approximately 6.02 × 1023
(Avogrado’s number) molecules of that species
Mass and Energy Balances: Lecture 1
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Chemical Composition
• How many of each of the following are contained in 100.0 g of
CO2 (MW = 44.01)?
• mol CO2
• mol O2
• lb-moles CO2
• gO
• mol C
• g O2
• mol O
• Molecules of CO2
Mass and Energy Balances: Lecture 1
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Chemical Composition
• Given:
Compound
m (g)
Mol. Wt.
n (mol)
ρ (kg/L)
Methanol
79
32.04
0.792
Water
100
18.016
1.00
Mixture
179
V (mL)
195
• Mass fraction, xM =
• Mole fraction, yM =
• Concentration, CM =
• Average molecular weight =
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Data Representation and Analysis
• The operation of a pharmaceutical plant is based on the
measurement of process variables – temperatures, pressures,
flow rates, concentrations, etc.
• Sometimes possible to measure directly, but more often we must
relate one variable to another that is easier to measure.
• E.g. instead of measuring concentration of salt A solution directly,
the electrical conductivity of the solution is measured
• We might do some kind of calibration experiment from which we
can develop an equation relating one variable to another
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Data Representation and Analysis
• Possible calibration curves:
• (a) or (b):
• (c): nonlinear equation – extrapolation is risky
• Interpolation
Extrapolation
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Data Interpolation
• What is the partial vapor pressure of SO2 over water at 0.2 wt%
SO2 and 28°C?
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Linear Fitting of Non-linear Data
x
0
1
2
3
y (x)
1
2
9
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x
0
1
2
3
y (x)
1
2
9
28
x2
0
1
4
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Mass and Energy Balances: Lecture 1
x
0
1
2
3
y (x)
1
2
9
28
x3
0
1
8
27
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Linear Fitting of Non-linear Data
y
• What is the equation to relate x and y?
x2-2
• How would you plot (x,y) data to get a straight line, and how
would you determine a and b for
y=𝑎 𝑥+𝑏
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Logarithmic Coordinates
• Given a non-linear plot
• Data has an exponential relationship 𝑦 = 𝑎𝑒𝑏𝑥 or ln 𝑦 = ln 𝑎 + 𝑏𝑥
• Two approaches to produce linear fits
(i) change the axis
(ii) change the data
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