Lecture 2: INTRODUCTION TO ENGINEERING CALCULATIONS

by

Listowel Abugri Anaba

AEN7201 BIO-ENGINEERING

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

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

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.

Base quantitiesBase quantity Dimensional

symbolBase SI unit Unit symbol American Eng.

Length L metre m foot (ft)Mass M kilogram kg pound mass (lbm)Time T second s secondElectric current I ampere A ampereTemperature 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 radSolid angle - steradian sr

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

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

Dimensional quantities (1)Derived quantity Dimension SI unitAcceleration 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 JEntropy L2MT-2θ-1 J/KForce 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 sHeat L2MT-2 JHeat flux MT-3 W m-2

Dimensional quantities (2)Derived quantity Dimension SI unitHeat-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-1Molar 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.sViscosity (kinematic) L2T-1 m2s-1

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.

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;

• Degrees, which are subdivisions of a revolution, are converted into revolutions or radians before application in engineering calculations

1.1.3 Dimensional Homogeneity in Equations (1)• Equations representing relationships between physical variables must

be dimensionally homogeneous

Margules equation for evaluating fluid viscosity:

• The argument of any transcendental function, such as a logarithmic, trigonometric, exponential function, must be dimensionless ;

e.g. cell growth is: where x = cell concentration at time t, xo = initial cell concentration, and = specific growth rate

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:

• The relationship between α the mutation rate of Escherichia coli and temperature T, can be described using an Arrhenius-type equation:

• Integration and differentiation of terms affect dimensionality

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 (hp)V = ungassed liquid volume(ft3)u = linear gas velocity(ft/s)ϵ = fractional gas hold-up (dimensionless)

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

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)

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

;

1.2.1 SI PrefixesFactor Prefix Symbol Factor Prefix Symbol1024 yotta Y 10-1 deci d1021 zetta Z 10-2 centi c1018 exa E 10-3 milli m1015 Peta P 10-6 micro μ1012 Tera T 10-9 nano n109 Giga G 10-12 pico p106 Mega M 10-15 femto f103 Kilo k 10-18 atto a102 Hecto h 10-21 zepto z101 Deca da 10-24 yocto y

1.2.2 UNIT CONVERSION DEVICES

THE CALCULATOR

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

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

MEASUREMENT CONVENTIONS

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

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

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 weighte.g. How many atoms of Cu are present in 35.4 g of Cu? (Cu = 63.5)

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

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

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

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

Boyle’s lawAt fixed n and T,

PV = constant or

P1V

n = number of moles of gas molecules

1.9.1 Ideal gas equation(1)

1.9.1 Ideal gas equation (2)

At fixed n and P,

Charles’ law

TVT is the absolute temperature in Kelvin, K

1.9.1 Ideal gas equation (3)

PV = nRT

PRnTV R is the same for all gases

R is known as the universal gas constant

nV Avogadro’s law

P1V Boyle’s law

Charles’ lawTV

Ideal gas equation

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?

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

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

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

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,

1.11.1 Stoichiometry Terminologies (3)• Conversion is the fraction or percentage of a reactant converted into

products

• Degree of completion is usually the fraction or percentage of the limiting reactant converted into products

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)

• Yield is the ratio of mass or moles of product formed to the mass or moles of reactant consumed

2CH3OH C2H4 + 2H2O

3CH3OH C3H6 + 3H2O

If the desired product is ethylene, then the selectivity is

Example

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