This is a basic introduction to engineering calculations in Bioprocess Engineering Principles. The first step in systems quantitative analysis is to express the system properties using mathematical language.

1. Lecture 2: INTRODUCTION TO ENGINEERING
CALCULATIONS
by
Listowel Abugri Anaba
AEN7201 BIO-ENGINEERING

2. Learning Outcomes
• To learn conventions and definitions which form the backbone of
engineering analysis
• To know the nature of physical variables, dimensions and units
• Get to understand dimensionality and be able to convert units
with ease
• How physical and chemical processes are translated into
mathematics

3. Physical Variables, Dimensions and Units
Calculations used in bioprocess engineering require a systematic approach with well-defined methods and rules
The first step in quantitative analysis of systems is to express the system properties using mathematical language

4. 1.1 Physical Variables
• A physical property of a body or substance that can be
quantified by measurement e.g. length, velocity, viscosity etc.
• Seven out of all physical variables are accepted
internationally as basis for measurement
• The base quantities are called dimensions, from which the
dimensions of other physical variables are derived
e.g. velocity is LT-1 , force is LMT-2 etc.

5. Base quantities
Base quantity Dimensional
symbol
Base SI unit Unit symbol American Eng.
Length L metre m foot (ft)
Mass M kilogram kg pound mass (lbm)
Time T second s second
Electric current I ampere A ampere
Temperature Θ kelvin K Rankine (R)
Amount of substance N gram-mole gmol (mol) lbm-mole (lbmmol)
Luminous intensity J candela cd candela
Supplementary fundamental units
Plane angle - radian rad
Solid angle - steradian sr

6. 1.1.1 Substantial variables (1)
• Examples of substantial variables are mass, length, volume,
viscosity, temperature etc.
• Expression of the magnitude of substantial variables requires
a precise physical standard against which measurement is
made
• These standards are called units

7. 1.1.1 Substantial variables (2)
• The magnitude of substantial variables are in two parts: the
number and the unit used for measurement
• The values of two or more substantial variables may be added
or subtracted only if their units are the same
• the values and units of any substantial variables can be
combined by multiplication or division

8. Dimensional quantities (1)
Derived quantity Dimension SI unit
Acceleration LT-2 ms-2
Angular velocity T-1 rads-1
Area L2 m2
Concentration L-3N moldm-3
Conductance (electric) L-2M-1T3I2 m-2kg-1s3A2 (Siemens)
Density L-3M kgm-3
Energy L2MT-2 Nm or J (Joule)
Enthalpy L2MT-2 J
Entropy L2MT-2θ-1 J/K
Force LMT-2 m·kg·s-2 or N (Newton)
Fouling factor M T-3θ-1 Wm-2 K-I
Frequency T-1 s-1 or Hz (Hertz)
Half life T s
Heat L2MT-2 J
Heat flux MT-3 W m-2

9. Dimensional quantities (2)
Derived quantity Dimension SI unit
Heat-transfer coefficient MT-3θ-I Wm-2K-1
Illuminance L-2J Cdm-2 (lux)
Mass flux L-2MT-1 kgm-2s-1
Momentum LMT-1 Kgms-1
Molar mass MN-1 Gmol-1
Osmotic pressure L-1MT-2 Kgm-1s-2
Power L2MT-3 m2kgs-3 or Js-1 or W (Watt)
Pressure/stress L-1MT-2 m-1kgs-2 or Nm-2 or Pa(Pascal)
Specific death constant T-l S-1
Specific growth rate T-l S-1
Specific production rate T-l S-1
Specific volume L3M-1 Kg-1m3
Surface tension MT-2 Nm-l
Viscosity (dynamic) L-1MT-1 Pa.s
Viscosity (kinematic) L2T-1 m2s-1

10. 1.1.2 Natural Variables (1)
• Dimensionless variables, dimensionless groups or dimensionless
numbers
• No unit(s) or any standard of measurement is required for their
magnitudes
e.g. the aspect ratio of a cylinder
• Other natural variables involve combinations of substantial
variables that do not have the same dimensions
• Engineers make frequent use of dimensionless numbers for
succinct representation of physical phenomena
e.g. 𝐑𝐞 =
𝐃𝐮𝛒
𝛍

11. 1.1.2 Natural Variables (2)
• Other dimensionless variables relevant to bioprocess engineering are the
Schmidt number, Prandtl number, Sherwood number, Peclet number,
Nusselt number, Grashof number, power number etc.
• Rotational phenomena;
𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒓𝒂𝒅𝒊𝒂𝒏𝒔 =
𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒂𝒓𝒄
𝒓𝒂𝒅𝒊𝒖𝒔
𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒓𝒆𝒗𝒐𝒍𝒖𝒕𝒊𝒐𝒏𝒔 =
𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒂𝒓𝒄
𝒄𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆
• Degrees, which are subdivisions of a revolution, are converted into
revolutions or radians before application in engineering calculations

12. 1.1.3 Dimensional Homogeneity in Equations (1)
• Equations representing relationships between physical variables must
be dimensionally homogeneous
Margules equation for evaluating fluid viscosity:
𝝁 =
𝑴
𝟒𝝅𝒉𝜴
𝟏
𝑹 𝒐
𝟐
−
𝟏
𝑹𝒊
𝟐
• The argument of any transcendental function, such as a logarithmic,
trigonometric, exponential function, must be dimensionless ;
e.g. cell growth is: 𝐥𝐧
𝒙
𝒙 𝒐
= 𝝃𝒕
where x = cell concentration at time t, xo = initial cell concentration, and 𝝃 = specific growth
rate

13. 1.1.3 Dimensional Homogeneity in Equations (2)
• The displacement y due to action of a progressive wave with
Amplitude A, frequency ω/2π and velocity v is given by the equation:
𝒚 = 𝑨 𝐬𝐢𝐧 𝝎 𝒕 −
𝒙
𝒗
• The relationship between α the mutation rate of Escherichia coli and
temperature T, can be described using an Arrhenius-type equation:
𝜶 = 𝜶 𝒐 𝒆
𝑬
𝑹𝑻
• Integration and differentiation of terms affect dimensionality

14. 1.1.4 Equations Without Dimensional Homogeneity
• Equations in numeric or empirical equations
• Equations derived from observation rather than from
theoretical principles
• Richards' correlation for the dimensionless gas hold-up ϵ in a
stirred fermenter
𝐏
𝐕
𝟎.𝟒
𝐮
𝟏
𝟐 = 𝟑𝟎𝛜 + 𝟏. 𝟑𝟑
P (hp)
V = ungassed liquid volume(ft3)
u = linear gas velocity(ft/s)
ϵ = fractional gas hold-up (dimensionless)

15. 1.2 Units (1)
• Unit names and their abbreviations have been standardised
according to SI convention
• SI convention - unit abbreviations are the same for both
singular and plural and are not followed by a period
• SI prefixes are used to indicate multiples and sub-multiples
of units
• No single system of units has universal application

16. 1.2 Units (2)
• Base Units - units for base quantities
• Multiple units - multiples or fraction of base unit e.g. minutes,
hours, milliseconds or all in term of base unit second
• Derived units - obtained in one of two ways;
Multiplying and dividing base units (m2, ft/min, kgm/s2)
Defined as equivalents of compound units ( 1 erg = 1 g. cm/s2, 1 lbf = 32. 1
74 lbm. ft/s2)

17. 1.2 Units (3)
• Familiarity with both metric and non-metric units is necessary
• In calculations it is often necessary to convert units
• Units are changed using conversion factors
1 in = 2.54 cm ; 2.20 lb = 1 kg ; 1 slug = 14.5939kg
• Unit conversions are not only necessary to convert imperial units to
metric; some physical variables have several metric units in common
use e.g. (centipoise, kgh-1m-1), (Pa, atm, mmHg), (km/h, m/s, cm/s)
• Unity bracket e.g. 1lb = 453.6g
𝟏 =
𝟏𝒍𝒃
𝟒𝟓𝟑.𝟔𝒈
=
𝟒𝟓𝟑.𝟔𝒈
𝟏𝒍𝒃
; 𝟏 =
𝟏𝒇𝒕
𝟑𝟎.𝟒𝟖𝒄𝒎
=
𝟑𝟎.𝟒𝟖𝒄𝒎
𝟏𝒇𝒕

18. 1.2.1 SI Prefixes
Factor Prefix Symbol Factor Prefix Symbol
1024 yotta Y 10-1 deci d
1021 zetta Z 10-2 centi c
1018 exa E 10-3 milli m
1015 Peta P 10-6 micro μ
1012 Tera T 10-9 nano n
109 Giga G 10-12 pico p
106 Mega M 10-15 femto f
103 Kilo k 10-18 atto a
102 Hecto h 10-21 zepto z
101 Deca da 10-24 yocto y

19. 1.2.2 UNIT CONVERSION DEVICES
THE CALCULATOR

20. 1.3 Force and Weight
• In the British or imperial system, pound-force (lbf) = (1 lb mass) x
(gravitational acceleration at sea level and 45o latitude)
Units N, kgms-2, gcms-2, lbfts-2 ; 1N = 1kgms-2, 1lbf = 32.174lbmfts-2
• Dimensionless unity−bracket,𝒈 𝒄 = 𝟏 =
𝟏𝑵
𝟏𝒌𝒈𝒎𝒔−𝟐 =
𝟏𝒍𝒃 𝒇
𝟑𝟐.𝟏𝟕𝟒𝒍𝒃 𝒎 𝒇𝒕𝒔−𝟐
Calculate the kinetic energy of 250 Ibm liquid flowing through a pipe at 35 ft s-I.
Express your answer in units of ft-lbf
• Weight changes according to the value of the gravitational acceleration

21. MEASUREMENT CONVENTIONS

22. 1.4 Density, Specific Weight and Specific Volume
• Densities of solids and liquids vary slightly with temperature
• Specific gravity a dimensionless variable also known as
relative density
• Specific volume is the inverse of density
• The density of solutions is a function of both concentration
and temperature
• Gas densities are highly dependent on temperature and
pressure

23. 1.5 Mole
• Amount of a substance containing the same number of
atoms, molecules, or ions as the number of atoms in 12
grams of 12C
• There are 6.022 × 1023 (Avogadro’s Constant) atoms of
carbon in 12 grams of 12C
• 𝑵𝒖𝒎𝒃𝒆𝒓𝒐𝒇 𝒎𝒐𝒍𝒆𝒔 =
𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒑𝒂𝒓𝒕𝒊𝒄𝒍𝒆𝒔
𝟔.𝟎𝟐𝟐𝒙𝟏𝟎 𝟐𝟑 =
𝒎𝒂𝒔𝒔 (𝒈)
𝒎𝒐𝒍𝒂𝒓 𝒎𝒂𝒔𝒔(𝒈𝒎𝒐𝒍−𝟏)

24. 1.5.1 Molar mass
• It is the mass of one mole of substance, and has dimensions MN-l
• Unit: g/mol
• Examples
H2 hydrogen 2.02 g/mol
He helium 4.0 g/mol
N2 nitrogen 28.0 g/mol
O2 oxygen 32.0 g/mol
CO2 carbon dioxide 44.0 g/mol
• Molar mass also referred to us molecular weight
e.g. How many atoms of Cu are present in 35.4 g of Cu? (Cu = 63.5)

25. 1.6 Chemical compositions
• Mole fraction
• Mass fraction
• Mass percent e.g. sucrose solution with a concentration of
40% w/w
• Volume fraction
• Volume percent e.g. H2SO4(aq) mixture of 30% (v/v) solution
• Molarity
• Molality
• Normality

26. 1.7 Temperature
• Two most common temperature scales are defined using the freezing point (Tf )
and boiling point (Tb ) of water at 1 atm.
Celsius(or centigrade) scale
 Tf = 0oC and Tb = 100oC
 Absolute zero on this scale falls at -273.15oC
Fahrenheit scale
 Tf = 32oF and Tb = 212oF
 Absolute zero on this scale falls at - 459.67oF
The Kelvin and Rankin scale are defined at absolute value of Celsius and
Fahrenheit;
 T(K) = T(oC) + 273.15
 T(oR) = T(oF) + 459.67
 T(oR) = 1.8 T(K)
 T(oF) = 1.8T (oC) + 32

27. 1.8 Pressure
• Units - psi, mmHg, atm, bar, Nm-2 etc.
• Absolute pressure is pressure relative to a complete vacuum
• It is independent of location, temperature and weather,
absolute pressure is a precise and invariant quantity
• Pressure-measuring devices give relative pressure, also called
gauge pressure
Absolute pressure = gauge pressure + atmospheric pressure

28. 1.9 Standard Conditions and Ideal Gases
• Ideal gas - a hypothetical gas that obeys the gas laws perfectly at all
temperatures and pressures
• A standard state of temperature and pressure is used when specifying
properties of gases, particularly molar volumes
• Volume of a gas depends on the quantity present, temperature and
pressure
• 1gmol of a gas at standard conditions of 1atm and 0°C occupies a
volume of 22.4 litres

29. Boyle’s law
At fixed n and T,
PV = constant or
P
1
V 
n = number of moles of gas molecules
1.9.1 Ideal gas equation(1)

30. 1.9.1 Ideal gas equation (2)
At fixed n and P,
Charles’ law
TV 
T is the absolute temperature in Kelvin, K

31. 1.9.1 Ideal gas equation (3)
PV = nRT
P
RnT
V 
R is the same for all gases
R is known as the universal gas constant
nV  Avogadro’s law
P
1
V  Boyle’s law
Charles’ lawTV 
Ideal gas equation

32. 1.10 Chemical Equation and Stoichiometry
• What can we learn from a chemical equation?
C7H16 + 11O2 7CO2 + 8H2O
1. What information can we get from this equation?
2. What is the first thing we need to check when using a chemical
equation?
3. What do you call the number that precedes each chemical formula?
4. How do we interpret those numbers?

33. 1.11 Stoichiometry
• It’s concerned with measuring the proportions of elements that
combine during chemical reactions
• Atoms and molecules rearrange to form new groups in chemical or
biochemical reactions
C6H12O6 2C2H5OH + 2CO2
• Total mass is conserved
• Number of atoms of each element remains the same
• Moles of reactants ≠ moles of products

34. C7H16 + 11O2 7CO2 + 8H2O
If 10 kg of C7H16 react completely with the stoichiometric
quantity, how many kg of CO2 will be produced? = 30.8 kg
Example

35. 1.11.1 Stoichiometry Terminologies (1)
• Limiting reactant is the reactant present in the smallest stoichiometric
amount. It is the compound that will be consumed first if the reaction
proceeds to completion
• Excess reactant is a reactant present in an amount in excess of that
required to combine with all of the limiting reactant
• % 𝐄𝐱𝐜𝐞𝐬𝐬 =
𝐀𝐜𝐭𝐮𝐚𝐥 𝐚𝐦𝐨𝐮𝐧𝐭 𝐩𝐫𝐞𝐬𝐞𝐧𝐭−𝐓𝐡𝐞𝐨𝐫𝐞𝐭𝐢𝐜𝐚𝐥 𝐚𝐦𝐨𝐮𝐧𝐭 𝐧𝐞𝐞𝐝𝐞𝐝
𝐓𝐡𝐞𝐨𝐫𝐞𝐭𝐢𝐜𝐚𝐥 𝐚𝐦𝐨𝐮𝐧𝐭 𝐧𝐞𝐞𝐝𝐞𝐝
𝒙𝟏𝟎𝟎

36. 1.11.1 Stoichiometry Terminologies (2)
• Limiting and Excess Reactants
Consider a balanced chemical reaction: aA +bB cC +dD
• Suppose x moles of A and y moles of B are present and they react
according to the above reaction,
 𝑰𝒇
𝒙
𝒚
<
𝒂
𝒃
, ⇒ 𝑨 = 𝒍𝒊𝒎𝒊𝒕𝒊𝒏𝒈
 𝑰𝒇
𝒙
𝒚
>
𝒂
𝒃
, ⇒ 𝑩 = 𝒍𝒊𝒎𝒊𝒕𝒊𝒏𝒈

37. 1.11.1 Stoichiometry Terminologies (3)
• Conversion is the fraction or percentage of a reactant converted into
products
% 𝑪𝒐𝒏𝒗𝒆𝒓𝒔𝒊𝒐𝒏 =
𝑨𝒎𝒐𝒖𝒏𝒕 𝒐𝒇 𝒓𝒆𝒂𝒄𝒕𝒂𝒏𝒕 𝒄𝒐𝒏𝒗𝒆𝒓𝒕𝒆𝒅
𝑨𝒎𝒐𝒖𝒏𝒕 𝒐𝒇 𝒓𝒆𝒂𝒄𝒕𝒂𝒏𝒕 𝒔𝒖𝒑𝒑𝒍𝒊𝒆𝒅
𝒙𝟏𝟎𝟎
• Degree of completion is usually the fraction or percentage of the
limiting reactant converted into products
𝑫𝒆𝒈𝒓𝒆𝒆 𝒐𝒇 𝒄𝒐𝒎𝒑𝒍𝒆𝒕𝒊𝒐𝒏 =
𝑨𝒎𝒐𝒖𝒏𝒕 𝒐𝒇 𝒍𝒊𝒎𝒊𝒕𝒊𝒏𝒈 𝒓𝒆𝒂𝒄𝒕𝒂𝒏𝒕 𝒄𝒐𝒏𝒗𝒆𝒓𝒕𝒆𝒅
𝑨𝒎𝒐𝒖𝒏𝒕 𝒐𝒇 𝒍𝒊𝒎𝒊𝒕𝒊𝒏𝒈 𝒓𝒆𝒂𝒄𝒕𝒂𝒏𝒕 𝒔𝒖𝒑𝒑𝒍𝒊𝒆𝒅

38. 1.11.1 Stoichiometry Terminologies (4)
• Selectivity is the ratio of the moles of the desired product produced to
the moles of undesired product (by-product)
𝑺𝒆𝒍𝒆𝒄𝒕𝒊𝒗𝒊𝒕𝒚 =
moles of desired product formed
moles of undesired product formed
• Yield is the ratio of mass or moles of product formed to the mass or
moles of reactant consumed
𝒀𝒊𝒆𝒍𝒅 =
actual amount of product formed
theoretical amount of product expected

39. 2CH3OH ⇌ C2H4 + 2H2O
3CH3OH ⇌ C3H6 + 3H2O
If the desired product is ethylene, then the selectivity is
𝑺𝒆𝒍𝒆𝒄𝒕𝒊𝒗𝒊𝒕𝒚 =
𝑴𝒐𝒍𝒆𝒔 𝒐𝒇 𝒆𝒕𝒉𝒚𝒍𝒆𝒏𝒆 𝒇𝒐𝒓𝒎𝒆𝒅
𝑴𝒐𝒍𝒆𝒔 𝒐𝒇 𝒑𝒓𝒐𝒑𝒚𝒍𝒆𝒏𝒆 𝒇𝒐𝒓𝒎𝒆𝒅
Example

40. THANK YOU VERY MUCH