Material and Energy Balances Lecture Notes METU ( Filiz ALSHANABLEH )

Mass and Energy Balances


Table of Content Chapter 1 Dimensions, Units, and Unit Conversion 1 Chapter 2 Introduction to Process Variables and Basic Food Engineering Calculations 21 Chapter 3 Introduction to Material Balances 41 Chapter 5 Material Balances for Multiple Units 87 Chapter 6 Energy Balances 121 Chapter 7 Material & Energy Balances of Some Selected Unit Operations 157 APPENDIX 1 : SYMBOLS, UNITS AND DIMENSIONS 182 APPENDIX 2 : UNITS AND CONVERSION FACTORS 186

Chapter 1: Dimensions, Units, and Unit Conversion

Chapter 1 Dimensions, Units, and Unit Conversion Learning Objectives Upon completing this Chapter, you should be able to: • understand the definitions and physical meanings of dimensions and units • perform operations (i.e. addition, subtraction, multiplication, and division) of numbers accompanied by corresponding units • identify units commonly used in engineering and scientific calculations, including those in cgs, SI, and American Engineering (AE) systems • convert one set of units (or one unit) associated with numbers (or a number) or in an equation into another equivalent set of units (or another unit), using a given conversion factor • explain and utilize the concept of dimensional homogeneity (consistency) of equations to identify units of specific numbers in those equations • appreciate the importance and rationale of dimensionless groups (quantities) James Clark Maxwell, a Scottish Mathematician and Theoretical Physicist (1831– 1879) expressed the definition of unit that “Every physical quantity can be expressed as a product of a pure number and a unit, where the unit is a selected reference quantity in terms of which all quantities of the same kind can be expressed”. Physical Quantities • Fundamental quantities • Derived quantities Fundamental Quantities • Length • Mass • Time • Temperature • Amount of substance • Electric current • Luminous intensity

Examples of Derived Quantities • Area [length × length or (length)2] • Volume [(length)3] • Density [mass/volume or mass/(length)3] • Velocity [length/time] • Acceleration [velocity/time or length/(time)2] • Force [mass × acceleration or (mass × length)/(time)2] Dimension A property that can be measured directly (e.g., length, mass, temperature) or calculated, by multiplying or dividing with other dimensions (e.g., volume, velocity, force) Unit A specific numerical value of dimensions. Systems of Units (commonly used in engineering and scientific calculations) • cgs (centimetre, gram, second) • SI (Le Système International d’ Unitès) • American Engineering (AE) (or fps : foot, pound, second) Units & Dimensions of Fundamental Quantities UNIT SYSTEM QUANTITY DIMENSION cgs SI AE • Length cm m ft L • Mass g kg lb M m • Time s s s t or θ • Temperature oC K o F T • Amount of Substance mol n _ _ • Electric current A A A I • Luminous intensity can* _ _ * stands for “Candela”

Examples of the Dimensions of Derived Quantities • Area (A ) [ A = L×L = L2 ] • Volume ( V ) [ V = L×L×L = L3 ] • Density (ρ ) [ρ = M / V = M / L3 ] • Velocity ( ) [ = L / t ] • Acceleration (a) [ a= V / t = ( L / t) /( t ) = L / t2 ] Example 1.1 Sir Isaac Newton (English physicist, mathematician, astronomer, philosopher, and alchemist: 1643– 1727) established a second law of motion equation that the force (F ) is the product of mass (m) and its acceleration (m), which can be described in the equation form as follows: F=ma What is the dimension of F (force)? From a previous page, the dimensions of • mass (m) ≡ M • acceleration (a) ≡ L / t2 The dimension of force (F ) can, thus, be expressed by those of fundamental quantities as follows F = ma ≡ ( M ) ( L / t2 ) F ≡ ( M )( L ) / ( t2 ) Example From a Physics or Chemistry courses, pressure (P ) is defined as “the amount of force (F ) exerted onto the area (A) perpendicular to the force” What is the dimension of P ? • Dimension of force F ≡ ( M )( L ) / ( t2 ) • Dimension of area (A) ≡ L2 Hence, the dimension of pressure is Dimensions and Units The “dimension” is the property that can be measured experimentally or calculated, and in order to express the physical quantity of a dimension, we use a pure number and its corresponding unit

For example, a ruler has a dimension of “length” (L ), its physical quantity can be expressed as 1 foot (ft) or 12 inches (in) or 30.45 centimetres (cm) Another example, Americans express their normal freezing point of water as 32 o F, while Europeans say that the normal freezing point of water at 1atm is at 0 oC We can see that a physical property [e.g., length (L ) or temperature (T )] with the same dimension may be expressed in different numerical value if it is accompanied with different unit. Units of Derived Quantities and Alternative Units The units of fundamental quantities of different unit systems are summarized on Page 3. What are the units for derived quantities? We can assign the unit (in any unit system) to each individual derived quantity using its dimension For instances, • the unit of area (A ), in SI system, is m2 2 , since its dimension is L • the unit of volume (V ), in AE system, is ft3 Example 1.2. Determine the units of density, in cgs, SI, and AE systems? Since the dimension of density (ρ ) is its corresponding units in • cgs unit system is g / cm3 • SI unit system is kg/ m3 • AE unit system is lb 3 m / ft What are “alternative” units? From our previous example we have learned that the dimension of force (F ) is F ≡ ( M )( L ) / ( t2 )

Hence, its corresponding units in • SI system is (kg)(m) / (s2) • AE system is (lb 2 m)(ft) / (s ) Have you ever heard that the unit of force (F ) is (kg)(m) / (s2)? To honour Sir Isaac Newton (1643-1727), who established the 2nd law of motion, a community of scientists gave the name of the unit of force as “Newton (N)”, which is defined as Newton (N) is an example of an alternative unit. Accordingly, instead of expressing the unit of pressure (P), in SI system, as since the dimension of pressure is we can, alternatively, write the unit of pressure, in SI system, as This comes from the fact that and that the units of force (F) and area (A), in SI system, is N and m2, respectively. However, the unit of pressure in SI system is expressed as “Pascal (Pa)”, which is defined as 1 Pa ≡ 1 (try proving it yourself that 1 Pa ≡ 1 ) In AE system, the unit of pressure is expressed as or psi (note that lb is the unit of force in AE system, not a unit of mass, and that “psi” stands for pound f force per squared inches” 1 N ≡ 0.224809 lb f Example 1.3. Work ( W ) is defined as “force (F ) acting upon an object to cause a displacement (L )”. What are the dimension and the corresponding unit of work, in SI system? 2 • Dimension of force F ≡ ( M )( L ) / ( t )

• Dimension of a displacement = L Hence, the dimension of work is W = FL ≡ ( M )( L 2 2 ) / ( t ) Accordingly, the unit of work, in SI system, is Alternatively, the unit of work, in SI system, can be expressed as W = FL ≡ (N)(m) Commonly, the unit of work, in SI system is expressed as “Joule (J)”, in which 1 J ≡ 1 (N)(m) Units of Work, Energy, and Heat We have just learned that the unit of work, in SI system, is J or (N)(m) or and from Physics courses, we learned that work, energy, and heat are in the same unit Is it true? From the definition of work: “force acting upon an object to cause a displacement” the unit of work can be expressed as J or (N)(m) or ,as mentioned above. How about the unit of “Energy”? Energy: • Potential Energy; Ep =mgL where g is an acceleration (a) caused by gravitational force Thus, in SI system, potential energy has the unit of E =mgL p

• Kinetic Energy; E 2 k = ½(m)(V) 2 Hence, the unit of kinetic energy, in SI system, is E = ½(m)(V) k It is clear that E and E are in the same unit, i.e. P K , and we have already got the fact that 1 J ≡ 1 (N)(m) ≡ Accordingly, we can conclude that work and energy are in the same unit Since, from Physics or Chemistry courses, both work and heat are the form of energy transferring between a system and surroundings, the unit of heat is as same as that of work. Units of Temperature A unit of temperature used in any calculations must be an absolute temperature unit Absolute temperature unit • SI K (Kelvin) • AE R (Rankine) T(K)=T( oC)+273.15* (1.1) T(R)=T( o F)+459.67** (1.2) * For convenience, the value of 273 is used ** For convenience, the value of 460 is used The Conversion of the Temperature Units between o o C and F Principle where Tnb = normal boiling point of water Tnf = normal melting/freezing point of water Hence,

o o T( F)=1.8T( C)+32 (1.3) o Example 1.4. The specific gravity of liquid is normally reported at 60 F in AE system. What is the equivalent temperature in SI system? Employing Eq. 1.3 yields o o T( F)=1.8T( C)+32 o 15.6 C Temperature Difference (ΔT ) Consider the following example: ΔT ( o o o C) = 15 C – 10 C = 5 oC ΔT (K) = (273+15) K – (273+10) K = 5 K Thus, it can be concluded that ΔT ( oC) = ΔT (K) (1.4) When considering in the same manner for o F and R, we shall obtain the fact that ΔT ( o F) = ΔT (R) (1.5)

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