The resisting force R of a supersonic plane depends on its length l, velocity V, air viscosity μ, air density ρ, and bulk modulus of air k. Using Buckingham's π-theorem with repeating variables l, V, and ρ, the relationship can be written as three dimensionless terms: π1 = R/lVρ, π2 = μ/lV2ρ, π3 = k/lV2ρ. Equating the powers of fundamental dimensions gives the relationship between the resisting force R and the variables it depends on.

1. PROCESS CALCULATIONS & THERMODYNAMICS
(CHC 331
Dr. Bimal Das

2. PROCESS CALCULATIONS & THERMODYNAMICS
(CHC 331
Module - I (10 hrs)
• Significance of Units and Dimensions: Conversion of Equations, Systems of Units, Dimensional
Homogeneity and Dimensionless Quantities, Buckingham Pi-theorem for Dimensional Analysis
Mathematical Requisites: Use of log-log and semi-log graph paper, Triangular Diagram.
• Introduction to Chemical Engineering Calculations: Basis, Mole Fraction and Mole Percent,
Mass Fraction and Mass Percent, Concentration of different forms, Conversion from one form to
another.
• Ideal gas laws and its significance, Molar concept, Concept of partial pressure & partial volume,
Dalton’s law and Amagat’s law and Numerical problems on their applications.
• Fundamental concept of vapor pressure & boiling point, Clausius-Clapeyron equation, Antoine
equation and numerical problems on their applications.
• Ideal & non-ideal solutions, Raoult’s law, Henry’s law and their applications in numerical
problems.

3. Module – II (10 hrs)
 Material Balances with and without chemical reaction: Material balances in crystallizers,
gas - liquid absorbers, evaporators, distillation plant. Systems with recycle,drying,
extraction.
 Energy Balance: Enthalpy calculation for systems without Chemical Reaction, Estimation
of Heat Capacities of solids, liquids and gases. Heat of fusion and vaporization
 Enthalpy calculation for systems with Chemical Reaction, Thermo-chemistry,
Calculations of heat of reaction, heat of combustions, heat of formation and heat of
neutralization, Effect of Temperature and Pressure on Heat of Reaction, Hess’s Law,
Adiabatic Flame Temperature, Theoretical Flame Temperature.

4. Module – III (10hrs)
• Scope of thermodynamics, Terminology and fundamental concepts. Microscopic and
macroscopic view. State and path functions, thermodynamics processes,
• Zeroth and First law of thermodynamics: Applications of first law to close and open system.
Limitations of first law, Heat pump, heat engine,
• Second law of thermodynamics: Reversibility and irreversibility, Carnot cycle, concept and
estimation of entropy, third law of thermodynamics, Clausius inequality, Gibb’s and Helmholtz
free energy.

5. Module – IV (10 hrs)
• PVT behavior of pure substance, Equations of state for ideal and real gases, cubic and virial
equation of state, problems, Compressibility factor, thermodynamic properties of pure
substances.
• Refrigeration of gases: Refrigerator, Co-efficient of performance, capacity of refrigerator,
Vapor compression cycle, Choice of refrigerants.

6. Text Books:
1. Unit Operations–Chemical Process Principles – Part-I - Haugen, Wartson & Ragatz (CBS)
2. Basic Principles and Calculations in Chemical Engineering – Himmelblau ((Prentice Hall of
India)
3. Stoichiometry, Bhatt and Vora, Tata McGraw Hill Companies.
4. Chemical Engineering Thermodynamics – J. M. Smith & H. C. Van Ness and M. M. Abbott
(Tata McGraw Hill)
5. Chemical & Engineering Thermodynamics – S. I. Sandler (Wiley)

7. Learning Objectives
1. Introduction to Dimensions & Units
2. Use of DimensionalAnalysis
3. Dimensional Homogeneity
4. Methods of DimensionalAnalysis
5. Rayleigh’s Method
6. Buckingham’s Method
Units and Dimensions

8. Dimensions
Any Physical quantity that can be counted or measured using standard
size defined by custom or law
Dimensions are basic concepts of physical measurements such as:
– Length = [L]
– Time = [T]
– Mass = [M]
– Temperature = [θ]
 Every measurement or quantitative statement requires a unit
 Units are terms that precede and describe the dimensions.
 Unit is any measure or amount used as a standard for measurement.
 A physical quantity is always express as the product of a number and unit

9. Dimensions and Units
In dimensional analysis we are only concerned with the nature of the dimension
i.e. its quality not its quantity.
 Dimensions are properties which can be measured.
Ex.: Mass, Length, Time etc.,
 Units are the standard elements we use to quantify these dimensions.
Ex.: Kg, Metre, Seconds etc.,

10. Units
• All physical quantities are measured w.r.t. standard magnitude of the same
physical quantity and these standards are called UNITS. eg. second, meter,
kilogram, etc.
So the four basic properties of units are:—
• 1. They must be well defined.
• 2. They should be easily available and reproducible.
• 3. They should be invariable e.g. step as a unit of length is not invariable.
• 4. They should be accepted to all.

11. Classification of dimensions
Fundamental or basic dimension
It consists of four quantities, length, mass, time and temperature. These are called dimensions or
base units and are represented by the symbols L, M, q and T respectively. The fundamental
quantities are represented by a system of units according to the system of measurement. Basically,
the physical
system representing the base unit differs in different systems of units.
– dimensions that are measured independently and enough to express essential physical
quantities
The following are the Fundamental Dimensions (MLT)
 Length = [L]
 Time = [T]
 Mass = [M]
 Temperature = [θ]
 Amount of substance (mole)
 Current (Ampere)
 Luminous intensity (Candela)

13. Secondary or Derived Dimensions
Secondary dimensions are those quantities which posses more than one fundamental
dimensions.
1. Geometric
a) Area
b) Volume
m2
m3
L2
L3
2. Kinematic
a) Velocity
b) Acceleration
3. Dynamic
m/s
m/s2
L/T
L/T2
L.T-1
L.T-2
a) Force N ML/T M.L.T-1
b) Density kg/m3 M/L3 M.L-3
Derived dimensions – dimensions that are products or quotients of fundamental dimensions
It consists of quantities which derived from the fundamental quantities, such as area,
force, pressure, energy etc. It follows, therefore, that derived quantities are represented
algebraically in terms of base units by means of the mathematical symbols of multiplication and
division.

15. Problems
Find Dimensions for the following:
1. Stress / Pressure
2. Work
3. Power
4. Kinetic Energy
5. Dynamic Viscosity
6. Kinematic Viscosity
7. Surface Tension
8. Angular Velocity
9.Momentum
10.Torque

16.  Surface tension (σs), has dimensions of force per unit length. The dimensions of
surface tension in terms of primary dimensions is
 Force has the same dimensions as mass times acceleration (by Newton’s second
law). Thus, in terms of primary dimensions,

21. Dimensional Homogeneity
•Dimensional homogeneity means the dimensions of each terms in a equation on both side
are same.
• Thus if dimension of each terms on both side of the equation are same , the equation is
known as dimensionally homogeneous
• The power of fundamental dimensions (L,M,T) on both side of the equation will be
identical for dimensionally homogeneous equation.
• Such equations are independent of the system of units
Dimensional homogeneity suggests that the dimensions of each term in an equation
on both sides will be equal.
Dimension of LHS = Dimension of RHS

22. Dimensional Homogeneity means the dimensions in each equation on
both sides equal.
Dimensional Homogeneity

27. Many practical real flow problems in fluid mechanics can be solved by using
equations and analytical procedures. However, solutions of some real flow problems
depend heavily on experimental data.
Sometimes, the experimental work in the laboratory is not only time-consuming, but also
expensive. So, the main goal is to extract maximum information from fewest experiments.
In this regard, dimensional analysis is an important tool that helps in correlating
analytical results with experimental data and to predict the prototype behavior from the
measurements on the model.
DimensionalAnalysis

28. Use of DimensionalAnalysis
1. Conversion from one dimensional unit to another
2. Checking units of equations (Dimensional Homogeneity)
3. Defining dimensionless relationship using
a) Rayleigh’s Method
b) Buckingham’s π-Theorem
4. ModelAnalysis
Dimensional analysis is a method to determine relationships among the physical
quantities, such as velocity, density and viscosity by using their fundamental
properties length L, mass M, time T and temperature q. It is a simple mathematical
technique to determine the expression for dependent variable. It is used in research
work for conducting model tests. Dimensional analysis can be carried out by the
following methods:
1. Rayleigh’s method
2. Buckingham’s π-method.

29. Rayeligh’s Method
To define relationship among variables
This method is used for determining the expression for a variable which depends upon
maximum three variables only.
Rayleigh’s method of dimensional analysis is generally used when the expression contains
maximum of three independent variables. If the number of independent variables is three or less,
then it is very easy to determine the expression for the dependent variable. If the number of
independent variable is four or more, then it is very difficult to determine the expression for the
dependent variable.
In this method, the expression can be written as

30. Methodology:
Let X is a variable, which depends on X1 ,X2, and X3 independent variables
So, X is a function of X1 ,X2, X3 and mathematically it can be written as
X = f(X1, X2, X3)
This can be also written as
X = K (X1
a , X2
b ,X3
c ) where K is constant and a, b and c are arbitrarily powers
The values of a, b and c are obtained by comparing the powers of the fundamental
dimension on both sides. Thus the expression is obtained for the dependent variable.
Rayeligh’s Method

31. Rayeligh’s Method
Problem: Find the expression for Discharge Q in a open channel flow when Q is
depends on Area A and VelocityV.
Solution:
Q = K.Aa.Vb …… (1)
where K is a Non-dimensional constant
Substitute the dimensions on both sides of equation (1)
M0 L3 T-1 = K.(L2)a.(LT-1)b
Equating powers of M, L, T on both sides,
Power of T,
Power of L,
-1 = -b → b=1
3= 2a+b → 2a = 2-b = 2-1 = 1
Substituting values of a, b, and c in Equation 1m Q = K.A1. V1 = V.A

34. Buckingham’s π-Theorem:
Since Rayleigh’s Method becomes laborious if variables are more than fundamental
dimensions (MLT), so the difficulty is overcome by Buckingham’s π-Theorem which states that
“If there are n variables (Independent and Dependent) in a physical phenomenon and if these
variables contain m fundamental dimensions then the variables are arranged into (n-m)
dimensionless terms which are called π-terms.”
Let X1, X2, X3,…,X4, Xn are the variables involved in a physical problem. Let X1 be the
dependent variable and X2, X3, X4,…,Xn are the independent variables on which X1 depends.
Mathematically it can be written as
X1=f(X2 ,X3 ,X4 ,Xn) which can be rewritten as f(X1X2 ,X3 ,X4,Xn)=0
Above equation is dimensionally homogenous.
It contain n variables and if there are m fundamental dimensions then it can be written in terms of
dimensions groups called π-terms which are equal to (n-m)
Hence; f(π1, π2 π3,…πn-m)=0

35. Buckingham’s π-Theorem
• Let n numbers of variables among these X2, X3 and X4 are repeating variables, if the
fundamental dimension m (M, L, T) = 3 . Then each π term is written as
•
• ……………………..
• ………………………..
• Each equation is solved by the principle of dimensional homogeneity and the values of a1 , b1 , c1
etc. are obtained. These values are substituted in those equation and values of 𝜋1, 𝜋1……….
𝜋(𝑛−𝑚) are obtained. These π terms are substitute in
• f (π1, π2, π3, ……… πn - m) = 0
• The final equation for the physical phenomenon is obtained by expressing any term of
the π term is function of others as
• π1 = f (π2, π3, ……… πn - m)
• π2 = f (π1, π3, ……… πn - m)
𝜋1 = 𝑋2
𝑎1
. 𝑋3
𝑏1
. 𝑋4
𝑐1
. 𝑋1
𝜋(𝑛−𝑚) = 𝑋2
𝑎(𝑛−𝑚)
. 𝑋3
𝑏(𝑛−𝑚)
. 𝑋4
𝑐(𝑛−𝑚)
. 𝑋𝑛
𝜋2 = 𝑋2
𝑎2
. 𝑋3
𝑏2
. 𝑋4
𝑐2
. 𝑋5

36. Properties of π-terms:
• Each π-term is dimensionless and is independent of system of units.
• Division or multiplication by a constant does not change the character of the π-
terms.
• Each π-term contains m+1 variables, where m is the number of fundamental dimensions
and also called repeating variable.
Each equation is solved by principle of dimensionless homogeneity and values of a1,
b1 & c1 etc are obtained. Final result is in the form of
Π1=ϕ’(Π2, Π3, Π4 ,…, Π(n-m))
Π2=ϕ”(Π1, Π3, Π4 ,…, Π(n-m))

37. METHODS OF SELECTING REPEATINGVARIABLES
The number of repeating variables are equal to number of fundamental dimensions of
the problem. The choice of repeating variables is governed by following considerations;
• As far as possible, dependent variable should not be selected as repeating variable
• The repeating variables should be chosen in such a way that one variable contains
geometric property, other contains flow property and third contains fluid property
• The repeating variables selected should not form a dimensionless group
• The repeating variables together must contain all three fundamental dimension i.e.,
MLT
• No two repeating variables should have the same dimensions.
Note: In most of fluid mechanics problems, the choice of repeating variables may be (i)
d, v, ρ, (ii) l, v, ρ or (iii) d, v, µ.

38. Selecting Repeating Variables:
1. As far as possible, the dependent variable should not be selected as repeating variable.
2. Repeating variables put together should not form dimensionless group.
3. No two repeating variables should have same dimensions.
4. Repeating variables can be selected from each of the following
properties.
 Geometric property → Length, height, width, area
 Flow property → Velocity,Acceleration, Discharge
 Fluid property → Mass density, Viscosity, Surface tension
Buckingham’s π-Theorem

42. Q 1. The resisting force R of a supersonic plane during flight can be considered as dependent
upon the length of the aircraft l, velocity V, air viscosity µ, air density ρ, and bulk modulus of air k.
Express the functional relationship between the variables and the resisting force.
R  f (l,V,, , K)  f (R,l,V,, , K)  0
Total number of variables, n= 6
No. of fundamental dimension, m=3 No. of dimensionless  -terms, n-
m=3 Thus: f (1, 2 ,3 ) 0
No. Repeating variables =m=3 Repeating variables =l,V , 
Thus π-terms are written as
1  l V  R
a1 b1 c1
 la2
Vb2
c2

2
a3 b3 c3
3  l V  K

43. Now each Pi-term is solved by the principle
homogeneity
of dimensional
Power of M:
Power of L:
Power of T:
1  term  M L T  L (LT ) (ML ) MLT
o o o a1 1 b1 3 c1 2
Equating the powers of MLT on both sides, we get
0=c1+1 c1 =-1
0=a1 +b1-3c1+1 a1  2
0=-b1-2  b1=-2
   l-2
V -2
-2
R   
1 1
L2
V 2
 2  term M L T  L (LT ) (ML ) ML T
o o o a2 1 b2 3 c2 1 1
Equating the powers of MLT on both sides, we get
Power of M:
Power of L:
Power of T:
0  c2 1 c 2  -1
0  a2  b2 - 3c2 -1 a2  1
0  -b2 -1 b2  -1
R
2 2
  l-1
V -1
-1
   

lV


44.  La3
(LT 1
)b3
(ML3
)c3
ML1
T2
3  term  M L T
o o o
Equating the powers of MLT on both sides, we get
Power of M:
Power of L:
Power of T:
0  c3 1 c 3  -1
0  a3  b3- 3c3 -1 a3  0
0  -b3 - 2  b3  -2
 3 2
  l0
V -2
-1
K   
V2
1 2 3 2 2 2
R   K    K 
  ,  R  l2
V2
 ,
l2
V2 lV  V 
f (   )  f , ,  0 or
K
Hence
l V lV  V 
 R  K 
lV  V 
2  2 
 
 
   

45. Q 2. A thin rectangular plate having a width, w, and height, h, is located so that it is normal to a moving
stream of fluid. Assume the drag D, that the fluid exerts on the plate is a function of w and h, the fluid
viscosity and density µ, and ρ, respectively, and velocity V of the fluid approaching the plate. Determine
a suitable set of pi terms to study this problem experimentally.
From the statement of problem, we can write:
D = f (w,h,, ,V )
Total number of variables, n 6
No.of fundamental dimension,m  3
No.of pi terms, n - m  6-3  3
Repeating variables
w,,V
w  L, h  L
The dimension of variables using MLT system are
D  MLT-2
,
  ML1
T1
V  LT 1

46. a2 b2 c2 a3 b3 c3
 2  hw V  ,  3  w V 
Now the pi terms can be written as
w2
V2

1
 
D
For M:
For L :
For T:
0 1 c1
0 1 a1  b1 -3c1
0  -2-b1
a1 b1 c1
1  Dw V  ,
For 1 :
a1 b1 c1
1  Dw V 
Therefore, a1  -2,b1  -2 and c1  -1
2 2 1
1  Dw V 
M 0
L0
T0
 MLT 2
La1
LT 1
b1
ML3
c1

47. a2 b2 c2 a3 b3 c3
 2  hw V  ,  3  w V 
Now the pi terms can be written as
M 0
L0
T0
 LLa2
LT 1
b2
ML3
c2

h
w
0  c2
0 1 a2  b2 -3c2
0  -b2
2
For M:
For L :
For T:
Therefore,a2  -1,b2  0 and c2  0
1 0 0
 2  hw V 
a1 b1 c1
1  Dw V  ,
For 2 :
a2 b2 c2
 2  hw V 


48. a3 b3 c3
 3  w V 
Now the pi terms can be written as
 hwa2
Vb2
c2
,
 ML1
T1
La3
LT 1
b3
ML3
c3
0 1 c3
0  1 a3  b3 -3c3
0  -1-b3
a1 b1 c1
1  Dw V  , 2
For 3 :
a3 b3 c3
 3  w V 
 

w V 
Therefore, a3  1,b3  1and c3  -1
1 1 1
 3  w V 
M 0
L0
T0
For M:
For L :
For T:
3
Finally the results of dimensional analysiscan be
represented in theform
1,2 ,3,4 ,...,nm  0
1  2 ,3,4 ,...,nm 


 wV 
 h
  ,
D  w  
w2
V2


49. Types of Forces Acting on MovingFluid
1. Inertia Force, Fi
It is the product of mass and acceleration of the flowing fluid and acts in the
direction opposite to the direction of acceleration.
 It always exists in the fluid flow problems
2. Viscous Force, Fv
It is equal to the product of shear stress due to viscosity and surface area of the flow.
3. Gravity Force, Fg
 It is equal to the product of mass and acceleration due to gravity of the flowing fluid.

50. 4. Pressure Force, Fp
 It is equal to the product of pressure intensity and cross sectional area of flowing fluid
Types of Forces Acting on MovingFluid
5. Surface Tension Force, Fs
 It is equal to the product of surface tension and length of surface of the flowing
6. Elastic Force, Fe
 It is equal to the product of elastic stress and area of the flowing fluid

51. Dimensionless Numbers
V
Lg
InertiaForce
Gravity Force

Dimensionless numbers are obtained by dividing the inertia force by viscous
force or gravity force or pressure force or surface tension forceor elastic force.
1. Reynold’s number, Re =
2. Froude’s number, Fe =
3. Euler’s number, Eu =
e
4. Weber’s number, W =
5. Mach’s number, M =
V
p/ 
InertiaForce
PressureForce

InertiaForce V
SurfaceTensionForce  / L

 
Viscous Force
Inertia Force

VL
or
VD
Inertia Force

V
Elastic Force C
Dimensionless numbers reduce the number of variables that describe a system, thereby
reducing the amount of experimental data required to make correlations of physical
phenomena to scalable systems.

56. Some common established nondimensional parameters

60. Application of Different Types of Graph
In science and technology, the different types of graph generally used are:
1. Ordinary graph
2. Semi-log graph
3. Log-log graph
4. Triangular graph.
Ordinary Graph
In ordinary graph, data are plotted between X-axis
and Y-axis to obtain a straight line or curve. This is
generally used when both the variables have the
restricted range of values. The data can be plotted
on first, second, third and fourth quadrant of the
graph. It is the simplest among all kinds of graph. A
typical ordinary graph is shown in Figure.

61. Semi-log Graph
Semi-log graph consists of one axis ordinary (generally Y-axis) and other axis logarithmic
(generally X-axis). Semi-log paper (3 cycles) is written at the top of the graph. While plotting the
data, it is to be remember to hold the graph in such a way that semi-log paper (3 cycles) should
represent the top right hand side corner of the graph.
It is also observed that in horizontal axis, first cycle vertical line started with 0.1, second cycle
started with 1.0 and third cycle started with 10 and end with 100. But, the line of basic numerical
values does not change. The placement of decimal point is allowed to change and they always
differ by one decimal point per cycle.
Semi-log graph is useful when one of the variable being plotted to cover a large range of values
and the other has only a restricted range. For example, in control engineering, it is useful in
plotting phase margin diagram where wt has large range of values and f has restricted range of
values

68. Log-Log Graph
In log-log graph both x-axis and y-axis consist of logarithmic scale. Log-log paper (3 cycles ×
4 cycles) is written at the top of the graph. While plotting the data, it is to be remembered to
hold the graph in such a way that log-log paper (3 cycles × 4 cycles) should represent the top
right hand side corner of the graph.
It is also observed that in horizontal axis, first cycle vertical line is started with 0.1, second cycle is
started with 1.0, third with 10 and fourth with 100, and end with 1000. Whereas, in vertical axis,
the first cycle horizontal line is started with 0.01, second with 0.1, third with 1.0 and end with 10.
But the line of basic numerical values in case of vertical axis as well as horizontal axis does not
change. The placement of decimal point is allowed to change and they always differ by one
decimal point per cycle.
Log-log graph is useful when both of the variables being plotted to cover a large range of values.
For example, in control engineering, it is useful in plotting phase margin diagram for the
determination of stability of the system.

70. Triangular Graph
A triangular graph is of the form of equilateral triangle. It has three sides AB, BC and CA of
equal length, and all of its angles A, B and C are of same measure of 60°.
Why would we use a triangular graph?
Though not all data categories neatly fall into three
and only three sub-categories, for those that do,
triangular graphs offer a spatial method of seeing the
relative abundance and position of such data. They
are easy for the researcher to read and create; they
can utilise colour to show further subdivisions in the
data.
In the above example, the point shows:
A = 59% B = 34% C = 7%

71. It has three axes, A, B and C. These axes move in either clockwise or anticlockwise directions. It
is observed from the graph that the number moves from 0 to 100 for a axis and again it starts
from ‘0’ and so on. The data represents the axis in the form of percentage. A triangular graph is
generally used for plotting the data of liquid-liquid extraction system. For example, consider the
liquid-liquid extraction system of benzene, acetic acid and water. If A = 30, B = 10 and C = 60, a
graph can be plotted on the triangular graph.
It is a useful method for examining the varying proportion of three related set of data. Also, it is
the only graph which enables three variables to be plotted. It has one disadvantage that it cannot
be used to represent the absolute values.