This document provides an introduction to engineering calculations, specifically in the context of bio-engineering. It covers essential topics such as physical variables, units of measurement, dimensional homogeneity, and stoichiometry, all crucial for effective quantitative analysis in engineering. The lecture also emphasizes the importance of understanding both metric and non-metric units, unit conversions, and the significance of various natural and substantial variables in engineering calculations.

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