BASIC ENGINEERING SCIENCES
REVIEWER
(LECTURE)
Engineering Mechanics, Strength of Materials, Fluid Mechanics,
Thermodynamics, and Engineering Economics
Revision 0
2011
Prepared By:
Agerico U. Llovido – PME
CONTENTS
A. ENGINEERING MECHANICS
B. STRENGTH OF MATERIALS
C. FLUID MECHANICS
D. THERMODYNAMICS
E. ENGINEERING ECONOMICS
A - ENGINEERING MECHANICS (LECTURE)
1. Engineering Mechanics
Engineering Mechanics – is the science which deals with the study of forces and motion of rigid bodies.
2. Branches of Engineering Mechanics
2.1
Statics – is a branch of mechanics which studies forces on rigid bodies that remain at rest or bodies
that is stationary.
2.2
Dynamics – is a branch of mechanics which considers the motion of rigid bodies caused by the forces
acting upon them or dynamics deals with bodies not in equilibrium.
3. Branches of Dynamics
3.1
Kinematics – is a branch of dynamics which deals with pure motion or deals only with the motions of
bodies without reference to the forces that causes them to move.
3.2
Kinetics – is a branch of dynamics which relates motion to the applied forces or deals with both the
forces acting on the body and the motion which causes the body to move.
4. Definition of Terms:
4.1
Force – is a push or pull which tends to change the state of rest or of uniform motion, the direction
of motion, or the shape and size of a body. Force is a vector quantity
4.2
Matter – is any substance that occupies space and has mass.
4.3
Inertia – is the property of matter that causes it to resist any change in its state of rest or uniform
motion. There are three kinds of inertia – inertia at rest, inertia of motion and inertia of direction.
The mass of a body is a measure of its inertia.
4.4
Mass – is the quantity of matter contained in a body. The mass of a body remains the same
everywhere. It is a measure of inertia, which means a resistance to a change of motion. The SI unit
of mass is kg.
4.5
Velocity – is the displacement of the body per unit time. It is a vector quantity. The SI units of
velocity are m/s.
4.6
Acceleration – is the rate of change of velocity of a moving body. The SI units of acceleration are
m/s2.
4.7
Rigid Body – is a body in which the relative distance between internal points does not change.
5. Newton’s Laws of Motion for a Particle
5.1
A particle acted upon by a balanced force system has no acceleration.
5.2
A particle acted upon by an unbalance force system has an acceleration in line with and directly
proportional to the resultant of the force system.
5.3
Action and reaction forces between two particles are always equal and oppositely directed.
6. Physical Quantities
6.1
Scalar quantity – has magnitude only. (Ex. mass, speed, volume, time, etc.)
6.2
Vector quantity – has magnitude and direction. (Ex. weight, force, velocity, acceleration).
7. Forces (Static)
7.1
Composition of forces – finding the resultant of two or more forces.
7.2
Resolution of forces – finding the resultant of two or more components of a force.
7.3
Concurrent forces – forces that meet at a common point.
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A - ENGINEERING MECHANICS (LECTURE)
7.4
Non-concurrent forces – forces that are parallel and do not meet at a common point.
7.5
Couple – two forces having equal in magnitude, parallel, and in opposite direction.
7.6
Coplanar – forces that lie in the same plane.
7.7
Non-coplanar – forces that do not lie in the same plane.
7.8
Skewed forces – forces that are non-parallel, non-concurrent, and non-coplanar.
8. Resultant of forces
8.1
Resultant – a single force, which acts on a body to produce the same effect upon a body as two or
more acting together.
8.2
Components of a force – the separate forces which can be so combined.
Fx = F cos θ , Fy = F sin θ
8.3
Resultant by cosine law
R = F12 + F22 − 2 F1 F2 cos(180 − θ )
R = F12 + F22 + 2 F1F2 cos θ
8.4
Resultant by Pythagorean Theorem
R = F12 + F22
tan θ =
8.5
Resultant of three or more concurrent forces
∑F
x
A
F2
F1
= F1 cos θ1 + F2 cos θ 2 + L
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∑F
R=
= F1 sin θ1 + F2 sin θ 2 + L
y
(∑ F ) + (∑ F )
tan θ =
2
x
2
y
∑F
∑F
y
x
9. Moment or Torque of a Force
Moment or Torque – is a measure of the tendency of a force to rotate or twist a rigid body upon which it
acts about a pivot point.
Moment of a Force = Force × Perpendicular Distance
M = F ×d
10. Free Body Diagram
Free body diagram – a diagram of an isolated body which shows only the forces acting on the body.
11. Forces in Equilibrium
∑F =0
∑F = 0
∑M = 0
x
y
12. Parabolic Cables
∑M
A
=0
 L  L 
H (d ) − w   = 0
 2  4 
H=
wL2
8d
Where
d = cable sag
L = cable span
w = pipe weight (kg/m).
  L 
T A = H 2 + w 
  2 
2
Length of Cable = L +
A
8d 2 32d 4
−
3L
5L3
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A - ENGINEERING MECHANICS (LECTURE)
13. Catenary
T A = TB = wy
H = tension at lowest point = wc
y2 = s2 + c2
x = c ln
s+ y
c
L = 2x
14. Friction
Friction – is the force that resists the motion of one surface relative to another with which it is in contact or
a retarding force between two objects that inhibits motion.
F= fN
tan θ = f =
F
N
Where:
F = frictional force
N = normal force (reaction normal to the surface of contact)
f = coefficient of friction. It depends only on the condition of surfaces and the materials in contact.
a. Coefficient of static friction (for bodies that are not moving)
b. Coefficient of kinetic friction (for bodies that are moving)
R = total surface reaction
W = weight of the body
θ = frictional angle = arctan f
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15. Rectilinear Motion
Rectilinear motion – the motion of a body in a straight line.
15.1 Uniform motion
s = vt
15.2
Where:
s = displacement, m
v = displacement over time, m/s
t = time, s
Variable acceleration
ds = vdt or v =
ds
dt
dv = adt or a =
dv
dt
vdv = ads
15.3
Constant acceleration
a=
v 2 − v1
t
s = v1t +
1 2
at
2
v 22 = v12 + 2 as
Where:
a = acceleration or change in velocity over time, m/s2. (+) when accelerating. (-) when decelerating.
v = velocity, m/s
s = distance, m
t = time, s
16. Falling Bodies
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g=
v 2 − v1
t
s = v1t +
1 2
gt
2
v 22 = v12 + 2 gs
Where:
a = acceleration or change in velocity over time, m/s2. (+) when going down. (-) when going up.
17. Rotation or Angular Motion
17.1
Uniform motion
θ = ωt
17.2
Uniform acceleration
ω − ω1
α= 2
t
1
2
θ = ω1t + α t 2
ω 22 = ω12 + 2αθ
Where:
α = angular acceleration, rad/s2 or rev/s2. (+) when accelerating. (-) when decelerating.
ω = angular velocity, rad/s or rev/s.
θ = angular displacement, rad or rev
t = time, s
18. Projectile
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18.1
Velocity
v x = vo cos θ
v y = vo sin θ
18.2
Horizontal displacement
x = (vo cos θ ) t
18.3
Vertical displacement
y = (vo sin θ ) t −
18.4
Equation of path of projectile (parabola)
y = x tan θ −
18.5
gx 2
2vo2 cos 2 θ
Range of projectile (R)
vo2 sin 2θ
g
R=
t=
18.6
1 2
gt
2
2v y
g
Maximum height of projectile, H
H=
t=
vo2 sin 2 θ
2g
vy
g
Where:
vo = initial velocity, m/s
x = horizontal distance, m
y = vertical distance, m
t = time of flight, s.
19. Relation of Angular and Peripheral Motion
s = rθ
v = rω
a = rα
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v2
r
an =
Fc = man =
mv 2
r
aT = an2 + at2
Where:
s = peripheral distance, m
r = radius, m
θ = angle, rad
v = peripheral velocity, m/s
ω = angular velocity, rad/s
a = peripheral acceleration, m/s2
α = angular acceleration, rad/s2
an = normal acceleration, m/s2
Fc = centrifugal force, N
m = mass of body, kg
aT = total acceleration, m/s2
at = tangential acceleration, m/s2
20. Force (Dynamics – Kinetics)
F = ma =
W
a
g
Where:
F = force, N
M = mass of the body, kg
A = acceleration of the body, m/s2
21. D’Alembert’s Principle (Reversed effective force)
W
a = ma = Reversed Effective Force(acceleration force)
g
Apply reverse effective force and treat as statics
∑F
∑F
A
x
=0
y
=0
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A - ENGINEERING MECHANICS (LECTURE)
∑M = 0
22. Work-Energy method
KE1 + PW − NW = KE2
Where:
KE1 = initial kinetic energy =
Wv12
2g
PW = positive work
NW = negative work
KE2 = final kinetic energy =
Wv 22
2g
23. Work, Energy and Power
23.1 Work – is done when a force acting on a body displaces it. Work is a scalar quantity. The SI unit for
work is Joule.
23.2 Energy – is the capacity of a body to do work. Energy is a scalar quantity. The SI unit of energy is
Joule.
23.3 Power - is the rate of doing work. Power is a scalar quantity. The SI unit of power is Watt (1 W = 1
J/s).
Power =
Force × distance
= Force × velocity
time
Efficiency =
Power output Work output
=
Power input
Work input
24. Types of Energy
24.1 Kinetic Energy – is the energy possessed by a body due to its motion.
KE =
24.2
1 2
mv in N-m or Joule
2
Potential Energy – is energy possessed by a body due to its position. There are two types of
potential energies, gravitational and elastic. The potential energy of a body due to its height from
the ground is called its gravitational potential energy. The potential energy of a body due to its
configuration (shape) is called its elastic potential energy.
PE = Wh = mgh in N-m or Joule
Where:
m = mass of the body, kg
v = velocity of the body, m/s
W = mg = weight of the body, N
h = height, m
25. Impulse and Momentum
25.1 Impulse – is equal to the product of the force acting on the body and the time for which it acts.
Impulse = F t
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25.2
Momentum – is defined as the product of its mass and velocity. Momentum is considered to be a
measure of the quantity of motion in a body. Its SI units are kg-m/s.
Momentum = mv
26. Law of Conservation of Momentum
Law of conservation of momentum – the total momentum of a group of interacting bodies remains constant
in the absence of external force.
m1v1 + m2 v2 = m1v1′ + m2 v′2
Types of collision:
Elastic collision – is a collision of the two bodies in which kinetic energy as well as momentum is conserved.
Inelastic collision – is a collision of two bodies in which only the momentum is conserved but not kinetic
energy.
Coefficient of Restitution:
Coefficient of restitution – is the negative ratio of the relative velocity after collision.
v′ − v′
e=− 1 2
v1 − v 2
Note: If e = 1, the collision is perfectly elastic while if e = 0, the collision is completely inelastic.
27. Simple Pendulum
Simple pendulum – is a simple machine based on the effect of gravity. A simple pendulum is a heavy point
mass, suspended by a light inextensible string from a frictionless rigid support.
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v = 2 gh
h=
v2
2g
T = 2π
L
g
Where:
v = final velocity of the pendulum bob, m/s
h = height of the pendulum bob, m
T = time for a double swing, s
L = displacement of the pendulum bob, m
28. Conical Pendulum
tan θ =
Fc ω 2r
=
W
g
29. Centrifugal Force
Centrifugal force – is an apparent outward force on a body following a circular path.
30. Centripetal force
Centripetal force – is the force that holds an object in circular motion, pointed toward the center of the
circle.
Fc =
A
mv 2 Wv 2
=
= mω 2 r
r
gr
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A - ENGINEERING MECHANICS (LECTURE)
Where:
Fc = centrifugal force, N
m = mass of the body rotating, k
v = 2πrN = πDN = velocity of the body rotating, m/s
ω = angular velocity, rev/s or rad/s
31. Kinetic Energy of a Rotating Body
1 2 1
1
mv = mk 2ω 2 = Iω 2
2
2
2
Torque, T = F r = Iα
KE =
Where:
KE = kinetic energy of a rotating body, J
v = peripheral velocity, m/s
I = moment of inertia of the body, kg-m2
I = mk2
ω = angular velocity, rev/s or rad/s
α = angular velocity, rad/s2 or rev/s2
32. Simple Harmonic Motion
Simple Harmonic Motion – is the motion of an object with acceleration proportional to the displacement,
resulting in repetitive motion.
Velocity of B = ω R 2 − r 2 m/s
Acceleration of B = ω 2 x m/s2
t = time of complete oscillation =
2π
ω
seconds
T = general formula for the period of motion = 2π
displacement
acceleration
33. Banking of Highway Curves
θ = ideal angle of banking
Consider a car weight W that makes a horizontal turn on a curve of radius r while travelling at a certain
velocity v. The curve is banked at an angle θ with the horizontal so that there is no tendency to slide up or
down the road.
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tan θ =
v2
gr
With frictional force on a banked curved.
When the car is travelling a banked curve with a velocity greater than the rated speed of the curve and is
about to skid up the plane.
tan µ = f (coefficient of friction)
tan(θ + µ ) =
v2
gr
Where:
µ = frictional angle
θ = banking angle
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34. Unbanked Highway Curves
When the car is travelling on a horizontal, unbanked highway curve of radius r.
W=N
F f = Fc
f =
v2
gr
-
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B - STRENGTH OF MATERIALS (LECTURE)
1. STRENGTH OF MATERIALS
Strength of materials – is the ability of a material to withstand load without failure or deals with the elastic
behaviour of loaded materials. It is also termed as mechanics of materials.
2. LOAD
Load - is defined as any external force acting on a machine part.
Four Types of Load
2.1 Dead or steady load – a load that does not change in magnitude or direction.
2.2 Live or variable load – a load that changes continually.
2.3 Suddenly applied or shock loads – a load that is suddenly applied or removed.
2.4 Impact load – a load that has initial velocity.
3. STRESS AND STRAIN
Stress – is the internal force per unit area set up at various sections of the body.
F
Stress, σ =
A
Where
F = Force or load acting on a body, and
A = Cross-sectional area of the body
Normal stress, σ - is the force per unit of normal area.
Shear stress, τ - is the force per unit of shear or parallel area.
Strain – is the deformation per unit length when a system of forces or loads act on a body,
δL
Strain, ε =
or δ L = ε L
L
Where
δ L = Change in length of the body, and
L = Original length of the body
Dilation – is the sum of the strains in the three coordinate system.
4. HOOKE’S LAW, MODULUS OF ELASTICITY AND MODULUS OF RIGIDITY (SHEAR MODULUS)
Hooke’s Law – is a simple mathematical statement of the relationship between the elastic stress and strain.
Stress is proportional to strain.
For normal stress:
Modulus of Elasticity, E =
B
σ
ε
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B - STRENGTH OF MATERIALS (LECTURE)
For shear stress:
Modulus of rigidity, G =
τ
θ
Relationship of Modulus of Elasticity and Modulus of Rigidity
E
G=
2(1 + µ )
5. STRESS-STRAIN DIAGRAM
Properties of Materials
5.1 Proportional limit (A) – is defined as that stress at which the stress-strain curve begins to deviate from
the straight line.
5.2 Elastic limit (B) - is defined as the stress developed in the material without any permanent set.
5.3 Yield point – (C and D) – is a point where the material yield before the load and there is an appreciable
strain without any increase in stress.
5.4 Ultimate stress (E) – is the largest stress obtained by dividing the largest value of the load reached in a
test to the original cross-sectional area of the test piece.
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5.5 Breaking stress (F) – is the stress corresponding to a point where the specimen breaks area or breaking
force divided by the original cross-sectional area.
5.6 Percentage reduction in area – is the difference between the original cross-sectional area and crosssectional area at the neck (i.e. where the fracture takes place).
5.7 Percentage elongation – is the percentage increase in standard gauge length (i.e. original length)
obtained by measuring the fractured specimen after bringing the broken parts together.
6. WORKING STRESS AND FACTOR OF SAFETY
WORKING STRESS – is a stress lower than the maximum or ultimate stress at which failure of the material
takes place. Also called design stress, safe or allowable stress.
FACTOR OF SAFETY – is the ratio of the maximum stress to the working stress.
Factor of safety =
Maximum stress
Working or design stress
For ductile materials e.g. mild steel,
Yield point stress
Factor of safety =
Working or design stress
For brittle materials e.g. cast iron
Ultimate stress
Factor of safety =
Working or design stress
Notes on design stress and working stress:
Design stress is stress used in determining the size of a member (allowable or less).
Working stress is stress actually occurring under operating conditions.
7. TENSILE STRESS AND STRAIN
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F
A
δL
Tensile Strain:
εt =
L
8. COMPRESSIVE STRESS AND STRAIN
σt =
Tensile Stress:
F
A
δL
Compressive Strain: ε c =
L
9. SHEAR STRESS AND STRAIN
Compressive Stress: σ c =
Shear Stress – is the stress induced when a body is subjected to two equal and opposite forces acting
tangentially across the resisting section tending the body to shear off.
Shear Stress =
Tangential force
Resisting area
Shear Stress, τ =
F
A
Shear Strain – is measured by the angular deformation accompanying the shear stress.
10. BEARING STRESS
Bearing Stress – is a localized compressive stress at the surface of contact between two members of a
machine part that are relatively at rest.
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Application: Riveted Joints, Cotter Joints, Knuckle Joints, etc.
For Riveted Joints:
Bearing Stress:
Where
F
d ⋅t ⋅ n
d = Diameter of the rivet
t = Thickness of the plate
d ⋅ t = Projected area of the rivet, and
n = Number of rivets per pitch length in bearing or crushing
σ b (or σ c ) =
Note:
Bearing Pressure = is the local compression which exist at the surface of contact between two members of a
machine part that are in relative motion.
F
Bearing Pressure: pb =
L⋅d
Where
pb = Average bearing pressure
F = Radial load on the journal
L = Length of the journal in contact, and
d = Diameter of the journal
11. TORSIONAL STRESS AND ANGULAR OF TWIST
Torsion – is the state of a machine member subjected to the action of two equal and opposite couples acting
in parallel planes (or torque or twisting moment.
Torsional Shear Stress – is the stress set up by the torsion. It is zero at the centroidal axis and maximum at
the outer surface.
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Maximum torsional shear stress:
For shaft,
τ
r
=
T G ⋅θ
=
J
l
τ = Torsional shear stress induced at the outer surface of the shaft or maximum shear stress,
Where
r = Radius of the shaft,
T = Torque or twisting moment,
J = Second moment of area of the section about its polar axis or polar moment of inertia,
G = Modulus of rigidity for the shaft material,
l = Length of the shaft, and
θ = Angle of twist in radians on a length l .
For a solid shaft diameter
J=
π
×d4
32
16T
τ= 3
πd
For a hollow shaft with external diameter ( d o ) and internal diameter ( d i ).
J=
τ=
π
[
]
× (d o )4 − (d i )4 and r =
32
16Tdo
(
π do4 − di4
do
2
)
Or
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τ=
16T
πd o3
(1 − k )
4
and k =
di
do
Torsional rigidity of the shaft = G × J
2π ⋅ N ⋅ T
= T ⋅ω
60
Where
T = Torque transmitted, and
ω = Angular speed in rad/s.
English Units:
N ⋅T
P=
, N in rpm and T in in-lb.
63,025
Power transmitted by the shaft = P =
11.1
Shaft in series and parallel
Composite shaft – composed of two shaft of different diameters connected together to form one
shaft.
Shaft in series – where the driving torque is applied at one end and the resisting torque at the
other end.
Shaft in parallel – where the driving torque is applied at the junction of the two shafts and the
resisting torques at the other ends of the shafts.
Shaft in series:
Total angle of twist = θ 1 + θ 2 =
T ⋅ l1 T ⋅ l2
+
G1 J1 G2 J2
For shafts of the same materials, G1 = G2 = G
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Shaft in parallel:
θ1 = θ2
T ⋅ l1 T ⋅ l2
T
l G J
or 1 = 2 × 1 × 1
=
G1 J1 G2 J2
T2 l 1 G2 J2
For shaft of the same materials G1 = G2 = G
T1 l2 J1
= ×
T2 l 1 J2
12. BENDING STRESS OR FLEXURAL STRESS
M σ E
= =
I
y R
Where
M = bending moment acting at the given section.
σ = bending stress.
I = moment of inertia of the cross-section about the neutral axis.
E = Young’s modulus of the material of the beam, and
R = radius of curvature of the beam.
Bending stress:
Mc M
σ=
=
I
Z
Where Z = I c is known as section modulus .
13. THERMAL STRESSES AND THERMAL EXPANSION/CONTRACTION
Thermal stress – is induced in the body if the body is prevented to expand or contract freely with the rise or
fall of the temperature.
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Thermal Expansion/Contraction
∆L = α L∆t
Thermal Strain
∆L
ε th =
= α ∆t
L
Thermal Stress
σ th=ε th⋅E = α E∆t
Where
L = Original length of the body.
∆t = Rise or fall of temperature.
α = Coefficient of thermal expansion.
14. LINEAR AND LATERAL STRAIN
Linear strain – is a strain in the direct stress own direction.
Lateral strain – is a strain at the right angle of the direct stress direction.
15. POISSON’S RATIO
Lateral strain
µ=
Linear strain
Values of Poisson’s ratio range between 0.25 to 0.35 for most materials.
16. IMPACT STRESS
Impact stress - is the stress produced in the member due to the falling load.
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Energy gained by the system = Potential energy lost by weight
1
× P × δ L = W (h + δ L )
2
Then
σi =
W
2hAE
1 + 1 +

A
Wl




For sudden load:
h=0
2W
σi =
A
17. STIFFNESS
Stiffness – is the amount of force required to cause a unit of deformation and is often referred to as a spring
constant.
18. TOTAL STRAIN ENERGY
Total Strain Energy – is the energy stored in a loaded member is equal to the work required to deform the
member.
1
F 2L
U = Fδ =
2
2AE
19. COMBINED STRESSES
EQUIVALENT TORQUE = Te = T 2 + M 2
1
EQUIVALENT MOMENT = Me =  M + T 2 + M 2 

2
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19.1
19.2
19.3
Combined axial and bending stress
F Mc
σ= ±
A
I
Maximum Shear induced by combined axial or bending and shear stresses
1
τ max = σ t2 + 4τ 2
2
Maximum Normal Stress induced by combined axial or bending and shear stresses
1
1
σ max = σ t + σ t2 + 4τ 2
2
2
20. THEORIES OF FAILURE UNDER STATIC LOAD
20.1 Maximum principal (or normal) stress theory (also known as Rankine’s theory).
According to this theory, the failure or yielding occurs at a point in a member when the maximum
principal or normal stress in a bi-axial stress system reaches the limiting strength of the material in a
simple tension test.
1
2
σ max = σ t 1 = σ t +
σ max =
σ max =
σy
N
σu
N
1
σ t2 + 4τ 2
2
, for ductile materials
, for brittle materials
Where,
σ y = Yield stress.
σ u = Ultimate stress.
N = Factor of safety.
20.2 Maximum shear stress theory (also known as Guest’s or Tresca’s theory).
According to this theory, the failure or yielding occurs at a point in a member when the maximum shear
stress in a bi-axial stress system reaches a value equal to the shear stress at yield point in a simple
tension test
1
τ max = σ t2 + 4τ 2
2
τ max =
τy
N
Where,
τ max = Maximum shear stress in a bi-axial stress system,
τ y = Shear stress at yield point.
N = Factor of safety.
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B - STRENGTH OF MATERIALS (LECTURE)
Or τ max =
σy
2×N
20.3 Maximum principal (or normal) strain theory (also known as Saint Venant theory).
According to this theory, the failure or yielding occurs at a point in a member when the maximum
principal (or normal) strain in a bi-axial stress system reaches the limiting value of strain (i.e. strain at
yield point) as determined from a simple tensile test.
σ t 1 − µσ t 2 =
σy
N
1
1
σ t 1 = σ t + σ t2 + 4τ 2
2
2
1
1
σ t 2 = σ t − σ t2 + 4τ 2
2
2
20.4 Maximum strain energy theory (also known as Haigh’s theory).
According to this theory, the failure or yielding occurs at a point in a member when the strain energy per
unit volume in a bi-axial stress system reaches the limiting strain energy (i.e. strain energy at the yield
point ) per unit volume as determined from simple tension test.
(σ t1 )
2
+ (σ t 2 ) − 2 µ × σ t1 × σ t 2
2
σ y
= 
 N




2
20.5 Maximum distortion energy theory (also known as Hencky and Von Mises theory).
According to this theory, the failure or yielding occurs at a point in a member when the distortion strain
energy (also called shear strain energy) per unit volume in a bi-axial stress system reaches the limiting
distortion energy (i.e. distortion energy at yield point) per unit volume as determined from a simple
tension test.
(σ t1 )2 + (σ t 2 )2 − 2σ t1 × σ t 2 = 
σ yt 
2

 N 
Note:
The maximum distortion energy is the difference between the total strain energy and the strain energy
due to uniform stress.
21. ENDURANCE / FATIGUE LIMIT
Endurance Limit or Fatigue Limit – maximum stress that will nor cause failure when the force is reversed
indefinitely.
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22. RESIDUAL STRESS
Residual Stress (1) – internal, inherent, trapped, locked-up body stress that exists within a material as a
result of things other than external loading such as cold working, heating or cooling, etching, repeated
stressing and electroplating.
Residual Stress (2) – Any internal stresses that exist in a part at uniform temperature and not acted upon by
an external load.
23. VARIABLE STRESSES
Soderberg’s Criterion
1 σm K fσv
=
+
N σy
σe
Goodman’s Criterion (ductile)
1 σm K fσv
=
+
N σu
σe
Where:
N = factor of safety
σψ = yield strength
σu = ultimate strength
σe = endurance strength
σm = mean stress
σv = variable stress
σm =
σv =
B
σ max + σ min
σ max
2
− σ min
2
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24. MOMENTS AND DEFLECTIONS IN BEAMS
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25. PROPERTIES OF SECTIONS
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C - FLUID MECHANICS (LECTURE)
1. FLUID MECHANICS
1.1
Fluid mechanics is a branch of continuous mechanics which deals with a relationship between
forces, motions, and static conditions in continuous material. This study area deals with many and
diversified problems such as surface tension, fluid statics, flow in enclose bodies, or flow round
bodies (solid or otherwise), flow stability, etc.
1.2
Fluid mechanics is the study of fluids (gases and liquids) either in motion (fluid dynamics) or at rest
(fluid statics) and the subsequent effects of the fluid upon the boundaries, which may be either solid
surfaces or interfaces with other fluids.
1.3
Fluid Mechanics encompasses the study of all types of fluids under static, kinematic and dynamic
conditions.
2. FLUID
A fluid is defined as a material which will continue to deform with the application of a shear force.
A fluid is a substance that cannot maintain its own shape but takes the shape of its container. Liquid and
gases are both classified as fluids.
3. IMPORTANT PROPERTIES OF FLUIDS
Viscosity causes resistance to flow.
Surface tension leads to capillary effects.
Bulk modulus is involved in the propagation of disturbances like sound waves in fluids.
Vapour pressure can cause flow disturbances due to evaporation at locations of low pressure.
4. COMPRESSIBLE AND INCOMPRESSIBLE FLUIDS
Compressible fluid - the density of a fluid varies significantly due to moderate changes in pressure or
temperature. Generally gases and vapours under normal conditions can be classified as compressible fluids.
Incompressible fluid - the change in density of a fluid is small due to changes in temperature and or
pressure. liquids are classified under this category.
5. IDEAL AND REAL FLUIDS
Ideal Fluids – are fluids that have no viscosity, incompressible, no resistance to shear, no eddy currents and
no friction between moving surfaces.
Real Fluids – are fluids that are compressible, non-uniform velocity distributions and have friction and
turbulence in flow.
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6. REAL FLUIDS
Newtonian Fluids (1) – is a viscous real fluid and whose shear stresses are a linear function of the fluid strain
(Ex. Air, water, gases, steam, etc.)
Newtonian Fluids (2) - a linear relationship exists between the magnitude of the applied shear stress and the
resulting rate of deformation.
Non-Newtonian Fluids (1) – are real fluids like pastes, gels, electrolyte solutions, slurries, etc.
Non Newtonian Fluids (2) - classified as simple non Newtonian, ideal plastic and shear thinning, shear
thickening and real plastic fluids. In non Newtonian fluids the viscosity will vary with variation in the rate of
deformation.
7. PRIMARY THERMODYNAMIC PROPERTIES OF FLUIDS
Pressure (p) - is the (compression) stress at a point in a static fluid.
patm = 14.7 psia = 101.3 kPaa
Temperature (T) – is a measure of the internal energy level of a fluid.
o
R =o F + 460
o
K = o C + 273
Density (ρ) - is its mass per unit volume.
m
ρ=
V
Specific weight (γ) - is its weight per unit volume.
W mg
γ = ρg = =
V
V
Specific gravity (SG) - is the ratio of a fluid density to a standard reference fluid, water (for liquids), and air
(for gases).
ρgas
SGgas =
ρ air
ρ air = 1.20 kg m3 at 101.3 kPa and 21 C.
SGliquid =
ρ liquid
ρwater
ρwater = 1000 kg m3
8. VISCOSITY
Viscosity (1) - is that property of a real fluid by virtue of which it offers resistance to shear force.
Viscosity (2) – is the fluid resistance to flow or the property of fluid to resist shear deformation.
Newton’s law of viscosity states that the shear force to be applied for a deformation rate of (dV/dy) over an
area A is given by,
F = µ A(dV dy )
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where F is the applied force in N, A is area in m2, dV/dy is the velocity gradient (or rate of deformation), 1/s,
perpendicular to flow direction, here assumed linear, and µ is the proportionality constant defined as the
dynamic or absolute viscosity of the fluid.
Shear Stress = F A = µ (dV dy )
Shear Strain = dV dy = F (µ A)
Viscosity Index – is the rate at which viscosity changes with temperature.
Viscosimeter – an instrument, consisting of standard orifice, used for measuring viscosity (in SSU and SSF).
Absolute (Dynamic) Viscosity – is the viscosity determined by direct measurement of shear resistance (in
Poise or centiPoise.) Units are 1 reyn = 1 lb-sec/in2, 1 Poise = 1 dyne-sec/cm2 = 0.1 Pa-sec. 1 centiPoise (cP) =
0.01 Poise.
Kinematic Viscosity – it the absolute viscosity of a fluid divided by the density (in Stoke or centiStoke.) Units
are ft2/s, m2/s, 1 stoke = 1 cm2/sec. = 0.0001 m2/sec. 1 centiStoke (cSt) = 0.01 Stoke.
9. REYNOLDS NUMBER
Reynolds number – is a dimensionless number which is the ratio of the forces of inertia to viscous forces of
the fluids. It is the primary parameter correlating the viscous behaviour of all newtonian fluids.
Forces of inertia DVρ DV
Re =
=
=
Viscous Forces
µ
v
Where:
Re = Reynolds number, dimensionless
D = inside diameter, m
V = velocity, m/s
ν = kinematic viscosity, m2/s
µ = absolute viscosity, Pa-sec
10. TYPES OF FLOW
Laminar Flow – particles run parallel to each other. Laminar flow occurs if the Reynolds number is less than
2000.
Turbulent Flow – particles run not in same direction. Turbulent flow occurs if the Reynolds number is greater
than 4000. Fully turbulent occurs at very high Reynolds number.
Transitional Flow – also termed as critical flow in which this type of flow occurs if the Reynolds number is
between 2000 to 4000.
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11. SURFACE TENSION AND CAPILLARY ACTION
Surface Tension – is the membrane formed on the free surface of the fluid which is due to cohesive forces.
The reason why insects were able to sit on water is due to surface tension. The amount of surface tension
decreases as the temperature increases.
Capillary Action – this is done through the behaviour of surface tension between the liquid and a vertical
solid surface.
12. COMPRESSIBILITY AND BULK MODULUS
Compressibility, β- the measure of the change in volume of a substance when a pressure is exerted on the
substance.
∆V
−
Vo
β=
∆P
Where:
∆V = change in volume
Vo = original volume
∆P = change in pressure
Bulk modulus, EB - is defined as the ratio of the change in pressure to the rate of change of volume due to
the change in pressure. It is the inverse of compressibility.
13. HYDROSTATIC PRESSURE
Hydrostatic Pressure – is the pressure of fluid exerted on the walls of the container.
Notes:
1. Pressure in a continuously distributed uniform static fluid varies only with vertical distance and is
independent of the shape of the container. The pressure is the same at all points on a given horizontal
plane in the fluid. The pressure increases with depth in the fluid.
2. Any two points at the same elevation in a continuous mass of the same static fluid will be at the same
pressure.
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Pressure = Weight Density x Height
p = γ h = ρ gh
Pressure Head,
p p
h= =
γ ρg
Where:
p = hydrostatic pressure (gage pressure)
h = height of liquid (pressure head)
ρ = Density of liquid
14. MANOMETER
Manometer is a device to measure pressure or mostly difference in pressure using a column of liquid to
balance the pressure.
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pA − pB = (pA − p1 ) + (p1 − p2 ) + (p2 − p3 ) + (p3 − pB )
pA − pB = −γ 1 (z A − z1 ) − γ 2 (z1 − z2 ) − γ 3 (z2 − z3 ) − γ 4 (z3 − zB )
15. BUOYANCY
Buoyancy – the tendency of a body to float when submerged in a fluid.
Two Archimedes Law of Buoyancy
1. A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it
displaces.
2. A floating body displaces its own weight in the fluid in which it floats.
W = FB
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FB = ρV
Where:
FB = buoyant force
W = weight of the body
V = volume of the body submerged or volume of the liquid displaced
ρ = density of the liquid
16. BASIC SCIENTIFIC LAWS USED IN THE ANALYSIS OF FLUID FLOW
1. Law of conservation of mass: This law when applied to a control volume states that the net mass flow
through the volume will equal the mass stored or removed from the volume. Under conditions of steady
flow this will mean that the mass leaving the control volume should be equal to the mass entering the
volume. The determination of flow velocity for a specified mass flow rate and flow area is based on the
continuity equation derived on the basis of this law.
2. Newton’s laws of motion: These are basic to any force analysis under various conditions of flow. The
resultant force is calculated using the condition that it equals the rate of change of momentum. The
reaction on surfaces are calculated on the basis of these laws. Momentum equation for flow is derived
based on these laws.
3. Law of conservation of energy: Considering a control volume the law can be stated as “the energy flow
into the volume will equal the energy flow out of the volume under steady conditions”. This also leads to
the situation that the total energy of a fluid element in a steady flow field is conserved. This is the basis
for the derivation of Euler and Bernoulli equations for fluid flow.
4. Thermodynamic laws: are applied in the study of flow of compressible fluids.
17. FLOW RATES
Volume flow rate (Q) – of a fluid is a measure of the volume flow of fluids passing through a point per unit
time.
Mass flow rate (m) – of a fluid is a measure of the mass flow of fluid passing through a point per unit time.
18. CONTINUITY EQUATION
Continuity Equation - This equation is used to calculate the area, or velocity in one dimensional varying area
flow, like flow in a nozzle or venturi.
m = ρ1Q1 = ρ2Q2
ρ1 A1V1 = ρ2 A2V2
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For incompressible flow. ρ = ρ1 = ρ2
Q = Q1 = Q2
A1V1 = A2V2
19. VELOCITY HEAD
Torricelli’s Theorem:
“The velocity of a liquid which discharge under a head is equal to the velocity of a body which falls in the
same head”.
h=
V2
2g
Where:
h = velocity head
V = velocity of the liquid
20. FRICTION HEAD LOSS IN PIPES
Darcy-Weisbach Equation
hf =
fLV 2
L V2
=f
2gD
D 2g
Where:
hf = velocity head, m or ft
V = velocity of the liquid, m/s or ft/sec
L = length of pipe, m or ft
D = internal diameter, m or ft
f = coefficient of friction, friction factor (Darcy)
g = 9.81 m/s2 or 32.2 ft/sec2
21. COEFFICIENT OF FRICTION, f
1. Coefficient of friction for laminar flow (Re < 2300).
64
f=
Re
2. Coefficient of friction for turbulent flow
Colebrook equation, turbulent flow only (Re > 2300)
 2.51 ε D 
f = −2 log 
+

12
3.7 
 Re f
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Where, ε = nominal roughness of pipe or duct being used
A good approximate equation for the turbulent region of the Moody chart is given by Haaland’s
equation:

 6.9  ε D 1.11  
f = − 1.8 log 
+
 
 Re  3.7   

−2
H. Blasius Equation (4000 < Re < 105)
0.316
f= 4
Re
22. HYDRAULIC DIAMETER
For flow in non-circular ducts or ducts for which the flow does not fill the entire cross-section, we can define
the hydraulic diameter
4A
Dh =
P
where
A = cross-sectional area of actual flow,
P = wetted perimeter, i.e. the perimeter on which viscous shear acts
23. BERNOULLI’S EQUATION
Bernoulli’s equation – is a general energy equation that is used for solving fluid flows. It relates elevation
head, pressure head and velocity head. Some conditions of using Bernoulli’s equation: (1) No fluid friction,
(2) fluid is incompressible, and (3) negligible changes in thermal energy.
Bernoulli’s theorem – “ Neglecting friction, the sum of the pressure head, velocity head and elevation head
of a point is equal to the sum of the pressure head, velocity head and elevation head of another point”.
p1
γ
Where:
+
V12
p V2
+ Z1 = 2 + 2 + Z 2
2g
γ 2g
Pressure head =
Velocity head =
p
γ
V2
2g
Elevation head = Z
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24. VENTURI, NOZZLE AND ORIFICE METERS
Venturi, Nozzle and Orifice meters are the three obstruction type meters commonly used for the
measurement of flow through pipes.
Flow Rate:
A2
Q=
1 − (A2 A1 )2
2gh
This equation needs a modifying coefficient as viscous effects and boundary roughness as well as the
velocity of approach factor that depend on the diameter ratio have been neglected.
The coefficient is defined by,
Qactual = Qtheoretica l × C d
where Cd is the coefficient of discharge. Cd for venturi meters is in the range 0.95 to 0.98. Cd for flow nozzle
is in the range 0.7 to 0.9 depending on diameter ratio and Reynolds number to some extent. For orifice, The
range for coefficient of discharge is 0.6 to 0.65.
25. DISCHARGE MEASUREMENT USING ORIFICES
Actual flow rate:
Qactual = C d A0 2gh
where A0 is the area of orifice and Cd is the coefficient of discharge.
Qactual
Cd =
Qtheoretica l
Coefficient of velocity Cv.
Actual velocity of jet at vena contracta
V
Cv =
=
Theoretical velocity
2gh
The value of Cv varies from 0.95 to 0.99 for different orifices depending on their shape and size.
Coefficient of contraction Cc.
Area of jet at vena contracta Ac
Cc =
=
Area of velocity
A0
The value of coefficient of contraction varies from 0.61 to 0.69 depending on the shape and size of the
orifice.
Coefficient of discharge Cd.
Actual discharge
Actual area
Actual velocity
Cd =
=
×
= C c × Cv
Theoretical discharge Theoretical area Theoretical velocity
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Average value of Cd for orifices is 0.62.
26. PERIPHERAL VELOCITY FACTOR
Cd = φ =
PeripheralVelocity π DN
=
VelocityofJet
2gh
-
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D – THERMODYNAMICS (LECTURE)
1. THERMODYNAMICS
Thermodynamics (1) - is that branch of the physical sciences that treats of various phenomena of energy
and the related properties of matter, especially the laws of transformation of heat into other forms of
energy and vice versa.
Thermodynamics (2) – is the study of heat and work and those properties of substance that bear a relation
to heat and work.
2. THE WORKING SUBSTANCE
Working substance – a substance to which heat can be stored and from which heat can be extracted
Fluid - is a substance that exists, or is regarded as existing, as a continuum characterized by low resistance
to flow and the tendency to assume the shape of its container.
Pure Substance – is one that is homogeneous in composition and homogeneous and invariable in chemical
aggregation. A working substance whose chemical composition remains the same even if there is a change
in phase.
Simple Substance – is one whose state is defined by two independently variable intensive thermodynamic
principles.
Ideal gas - is mathematically defined as one whose thermodynamic equation of state is given by pv=RT,
where p is the absolute pressure, v is the specific volume, R is the gas constant, and T is the absolute
temperature of the gas, respectively. A working substance which remains in gaseous state during its
operating cycle and whose equation of state is pV= mRT
Incompressible substance - is a substance whose specific volume remains nearly constant during a
thermodynamic process. Most liquids and solids can be assumed to be incompressible without much loss in
accuracy.
3. THE SYSTEM
System (1) – is that portion of the universe, an atom, a galaxy, a certain quantity of matter, or a certain
volume in space, that one wishes to study. It is a region enclosed by specified boundaries, which may be
imaginary, either fixed or moving.
System (2) – is nothing more than the collection of matter that is being studied. It is a quantity of matter of
fixed mass and identity upon which attention is focused for study.
Thermodynamic Systems
3.1 Closed System (1) – is one which there is an exchange of matter with the surroundings – mass does not
cross its boundaries. Also called control mass.
Closed System (2) – is one that has no transfer of mass with its surrounding, but may have a transfer of
energy (either heat or work) with its surroundings.)
Isolated System (1) – A special type of closed system that does not interact in any way with its
surroundings.
Isolated System (2) – is one that is completely impervious to its surroundings – neither mass nor energy
cross its boundaries.
Isolated System (3) – is one that is not influences in any way by the surroundings. This means that no
energy in the form of heat or work may cross the boundary of the system. In addition, no mass may
cross the boundary of the system.
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3.2 Open System (1) – is one across whose boundaries there is a flow of mass. Each may have energy
crossing its boundary. Also called control volume.
Open System (2) – is one that may have a transfer of both mass and energy with its surroundings.
Control Volume – is a fixed region in space chosen for the thermodynamic study of mass and energy
balance for the flowing system.
Control Surface – is the boundary of the control volume.
Surrounding or Environment – everything external to the system.
Boundary – distinguishes the system from its surroundings; may be at rest or in motion, usually defined
by a broken or dashed line.
4. THERMODYNAMIC PROPERTIES
Thermodynamic Properties (1) - a quantity which is either an attribute of an entire system or is a function of
position which is continuous and does not vary rapidly over microscopic distances, except possibly for
abrupt changes at boundaries between phases of the system
Thermodynamic Properties (2) – is any characteristic of the system that can be observed or measured;
macroscopic characteristics of a system such as mass, volume, energy, pressure and temperature to which
numerical values can be assigned at a given time even without knowledge of the history of the system.
Classification of Properties
1. Intensive Properties – are independent of mass; for example, temperature, pressure, density, and
voltage.
Specific Properties – are those for a unit mass, are intensive by definition such as specific volume.
2. Extensive Properties – are dependent upon the mass of the system and are total values such as total
volume and total internal energy.
Thermodynamic state - is a set of values of properties of a thermodynamic system that must be specified to
reproduce the system. The individual parameters are known as state variables, state parameters or
thermodynamic variables. Once a sufficient set of thermodynamic variables have been specified, values of
all other properties of the system are uniquely determined. The number of values required to specify the
state depends on the system, and is not always known.
Point Function - A quantity whose value depends on the location of a point in space, such as an electric field,
pressure, temperature, or density.
5. MASS AND WEIGHT
Mass (1) – a property of matter that constitutes one of the fundamental physical measurements or the
amount of matter a body contains. Units of mass are in lbm, slugs, or kg. Symbol m.
Mass (2) - is the absolute quantity of matter in it, an unchanging quantity for a particular mass when the
speed of the mass is small compared to the speed of light (no relativistic effect).
Weight (1) – the force acting on a body in a gravitational field, equal to the product of its mass and the
gravitational acceleration of the field. Units of weight are in lbf or N. Symbol W. Formula W=mg. Where g =
9.81 m/s2 or 32.2 ft/s2.
Weight (2) - is the force exerted by a body when its mass is accelerated in a gravitational field.
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D – THERMODYNAMICS (LECTURE)
6. VOLUME
Volume – the amount of space occupied by, or contained in a body and is measured by the number of cubes
a body contains. Units of volume are in ft3, gallons, liters, cm3, or m3. Symbol is V.
7. SPECIFIC VOLUME AND DENSITY
Density (ρ) of any substance – is its mass (not weight) per unit volume. Units of density are lbm/ft3 or kg/m3.
Symbol ρ.
m
ρ=
V
Specific Volume – is the total volume of a substance divided by the total mass of a substance. Units of
specific volume are in ft3/lbm or m3/kg. Symbol is v.
1
V
or v =
v=
m
ρ
Specific Weight (γ) of any substance – is the force of gravity on unit volume. Units of specific weight are
ft3/lbf or m3/N. Symbol is γ.
1
V
or v =
γ=
W
ρg
Specific Gravity (1) – is a measure of the relative density of a substance as compared to the density of water
at a standard temperature. Symbol is SG.
Specific Gravity (2) – a dimensionless parameter, it is defined as the ratio of the density (or specific weight)
of a substance to some standard density (or specific weight).
For Liquid substances:
SG =
ρ liquid
ρH2 O at std
=
γ liquid
γ H2O at std
For Gaseous substances:
SG =
ρgas
=
γ gas
ρair at std γ air at std
8. STANDARD DENSITIES
Density of water
At approx. 4oC (39.2oF) pure water has it's highest density (weight or mass) = 1000 kg/m3 or 62.4 lb/ft3.
Density of air
At 70 °F (21.1 C) and 14.696 psia (101.325 kPaa), dry air has a density of 0.074887 lbm/ft3 (1.2 kg/m3).
9. PRESSURE
Pressure – is a measure of the force exerted per unit area on the boundaries of a substance (or system). It is
caused by the collisions of the molecules of the substance with the boundaries of the system. Units of
pressure are psi, kg/cm2, kN/m2 or kPa. Symbol is p. Formula p = F/A.
10. ATMOSPHERIC PRESSURE
Atmospheric pressure - is the force per unit area exerted against a surface by the weight of air above that
surface in the Earth's atmosphere.
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The standard atmosphere (symbol: atm) - is a unit of pressure and is defined as being equal to 101.325
kPa.[1] The following units are equivalent, but only to the number of decimal places displayed: 760 mmHg
(torr), 29.92 inHg, 14.696 psi.
Barometric pressure - is often also referred to as atmospheric pressure. Units is normally in Bar. 1 Bar = 100
kPaa.
Air pressure above sea level can be calculated as p = 101325(1 - 2.25577x10-5h)5.25588, where p = air pressure
(Pa) and h = altitude above sea level (m).
Also
p = po
 h
−
h
e  o



where:
p = atmospheric pressure, (measured in bars)
h = height (altitude), km
p0 = is pressure at height h = 0 (surface pressure) = 1.0 Bar (Earth)
h0 = scale height = 7 km (Earth)
Or for every 1,000 feet, there is a corresponding pressure decrease of approximately 1 in Hg.
11. BAROMETER
Barometer - is an instrument used to measure atmospheric pressure. It can measure the pressure exerted
by the atmosphere by using water, air, or mercury.
12. ABSOLUTE AND GAUGE PRESSURES
Absolute pressure, pabs - is measured relative to the absolute zero pressure - the pressure that would occur
at absolute vacuum. All calculation involving the gas laws requires pressure (and temperature) to be in
absolute units. It the sum of the gauge and atmospheric pressure.
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Gauge pressure - the amount by which the total absolute pressure exceeds the ambient atmospheric
pressure.
Formula
pabs = patm + pg
Vacuum pressure (negative gauge pressure) - the amount by which the total absolute pressure is less than
the ambient atmospheric pressure.
Formula
pabs = patm – pv
Pressure gauge - is often used to measure the pressure difference between a system and the surrounding
atmosphere.
13. TEMPERATURE
Temperature – is a measure of the molecular activity of a substance. It is a relative measure of how “hot” or
“cold” a substance is and can be used to predict the direction of heat transfer. It is an intensive property
that is a measure of the intensity of the stored molecular energy in a system.
Temperature Scales:
1. Fahrenheit (F) Scale – 180 units – from 32 F to 212 F.
2. Celsius (C) Scale or Centigrade Scale – 100 units – from 0 C to 100 C.
Relationship:
o
9
F = 32 +   o C
 5
( )
5
C = o F − 32  
9
Absolute zero - is the theoretical temperature at which entropy reaches its minimum value. The laws of
thermodynamics state that absolute zero cannot be reached using only thermodynamic means.
Absolute temperature - is the temperature measured relative to the absolute zero.
Absolute Temperature Scales:
1. Kelvin (K) Scale – the absolute temperature scale that corresponds to the Celsius scale.
2. Rankine (R) Scale – the absolute temperature scale that corresponds to the Fahrenheit scale.
o
(
)
Relationship:
o
R =o F + 460
o
K =o C + 273
* 273 K → 273.15 K
* 460 R → 459.67 R
Temperature Change:
 5
5
∆oC = ∆o K =   ∆oF =   ∆oR
9
9
( )
( )
9
9
∆o F = ∆o R =   ∆oC =   ∆o K
 5
 5
Ice point – the temperature of a mixture of ice and air-saturated water at 1 atm = 0 C or 32 F.
Steam (boiling) point – the temperature pure liquid water in contact with its vapour at 1 atm = 100 C or 212
F.
( )
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D – THERMODYNAMICS (LECTURE)
Triple point - The temperature and pressure at which a substance can exist in equilibrium in the liquid, solid,
and gaseous states. The triple point of pure water is at 0.01 degrees Celsius and 4.58 millimeters of mercury
and is used to calibrate thermometers.
14. ENERGY
Energy - is defined as the capacity of a system to perform work or produce heat.
Stored Energy – otherwise known as possessed energy, it is the energy that is retrieved and stored within
the system; thus, dependent upon the mass flow.
Potential Energy (1) – is defined as the energy of position. Symbol is P.E.
Potential Energy (2) – energy due to the elevation and position of the system
mgz
PE =
gc
Where
m
z
g
= mass (lbm, kg)
= height above some reference level (ft, m)
= acceleration due to gravity (ft/sec2, m/s2)
gc
= gravitational constant.
= 32.17 ft − lbm lbf − sec2
= 1 kg ⋅ m s2 ⋅ N
Kinetic Energy (1) – is the kinetic energy of motion. Symbol is K.E.
Kinetic Energy (2) – energy or stored capacity for performing work; possessed by a moving body, by virtue of
its momentum.
KE =
Where
mv 2
2gc
m
v
g
= mass (lbm, kg)
= velocity (ft/s, m/s)
= acceleration due to gravity (ft/sec2, m/s2)
gc
= gravitational constant.
= 32.17 ft − lbm lbf − sec2
= 1 kg ⋅ m s2 ⋅ N
Joule’s Constant “J” – 778 ft-lbf/ Btu
Internal Energy (1) – is a microscopic forms of energy including those due to the rotation, vibration,
translation, and interactions among the molecules of a substance.
Internal Energy (2) – heat energy due to the movement of the molecules within the substance brought
about its temperature.
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Internal Energy (3) – energy stored within a body or substance by virtue of the activity and configuration of
its molecules and the vibration of the atoms within the molecules.
Specific Internal Energy – is the substance internal energy per unit mass. Unit is Btu/lbm or kJ/kg. Symbol is
u.
P-V Energy – is also called flow energy or flow work.
Specific P-V Energy – is the substance P-V energy per unit mass.
Enthalpy (1) – is the amount of energy possessed by a thermodynamic system for transfer between itself
and its environment. It is equal to H = U + PV .
Enthalpy (2) – the sum of the internal energy of a body and the product of pressure and specific volume.
Unit is Btu/lbm or kJ/kg. Symbol is h.
Specific Enthalpy – is defines as h = u + Pv .
Where
u = specific internal energy
P = pressure
v = specific volume
Chemical Energy – stored energy that is released or absorbed during chemical reactions.
Nuclear Energy – energy due to the cohesive forces of the protons and neutrons within the atoms.
15. HEAT AND WORK
Transitory Energy – otherwise known as energy in transit or in motion; energy that loses its identity once it is
absorbed or rejected within the system; independent of mass flow stream.
Heat (1) - is the transfer of energy that occurs at the molecular level as a result of a temperature difference.
Symbol Q. Unit is Btu, Btu/hr, kJ or kW.
Heat (2) – energy in transition between a system and its surroundings because of a difference in
temperature.
Q is positive (+) when heat is added to the body
Q is negative (-) when heat is rejected by the body
Q
Heat transferred per unit mass = q =
m
Work – is defined for mechanical system as the action of a force on an object through a distance. Symbol is
W. Unit is ft-lb, kJ.
W = Fd
Where
F = force (lbf, N)
d = displacement (ft, m )
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D – THERMODYNAMICS (LECTURE)
•
Work is a process done by or on a system.
Power – is the rate of doing work. Unit is hp or kW.
Work
Power =
Time
16. TYPES OF WORK
Non-flow Work , WNF – work done in a non-flow system with moving boundary. Typical example of which is
the piston-cylinder assembly.
Flow Work, WF – work done in pushing a fluid across a boundary, usually into or out of a system. WF = p∆V =
p(V2-V1)
Steady-flow Work, WSF – additional work done in an open system; used for steady flow systems where there
is neither accumulation nor diminution of mass and energy within the system.
17. SPECIFIC HEAT
Heat Capacity – is the ratio of the heat (Q) added to or removed from a substance to the change in
temperature ( ∆T ).
Specific Heat (1) – is the heat capacity of a substance per unit mass. Unit is Btu/lbm-F or Btu/kg-C.
Specific Heat (2) – is the heat required to raise the temperature of unit mass of a substance by a unit
temperature.
Specific heat at constant pressure – is the change of enthalpy for a unit mass between two equilibrium
states at the same pressure per degree change of temperature.
Q
Cp =
∆T
Q
cp =
m∆T
q
cp =
∆T
Specific heat at constant volume – is the change of internal energy for a unit mass per degree change of
temperature when the end states are equilibrium states of the same volume.
Q
Cv =
∆T
Q
cv =
m∆ T
q
cv =
∆T
Specific heat ratio –
cp ∆h
k= =
cv ∆u
18. ENTROPY
Entropy (1) – is a measure of inability to do work for a given heat transferred.
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Entropy (2) – is a measure of randomness of the molecules of a substance or measures the fraction of the
total energy of a system that is not available for doing work.
Entropy (3) – a property used to measure the state of disorder of a substance; a function of both heat and
temperature.
Entropy production – is the increase in entropy.
∆Q
∆S =
Tabs
∆s =
∆q
Tabs
∆S = the change in entropy of a system during some process (Btu/R or kJ/K).
∆Q = the amount of heat transferred to or from the system during the process (Btu or kJ)
Tabs = the absolute temperature at which the heat was transferred (R or K).
∆s = the change in specific entropy of a system during some process (Btu/lbm-oR or kJ/kg.K).
∆q = the amount of heat transferred to or from the system during the process (Btu/lbm or kJ/kg).
19. LAWS OF THERMODYNAMICS
19.1
ZEROTH LAW OF THERMODYNAMICS – TEMPERATURE (THERMAL EQUILIBRIUM)
-
-
19.2
FIRST LAW OF THERMODYNAMICS – LAW ON CONSERVATION OF ENERGY
-
-
D
states that when each of two systems is in equilibrium with a third, the first two systems
must be in equilibrium with each other. This shared property of equilibrium is the
temperature.
states that when two bodies have equality of temperature with a third body, they in turn
have equality of temperature with each other and the three bodies are said to be in thermal
equilibrium. The third body is usually a thermometer.
states that, because energy cannot be created or destroyed [setting aside the later
ramifications of the equivalence of mass and energy (Nuclear Energy)] – the amount of heat
transferred into a system plus the amount of work done on the system must result in a
corresponding increase of internal energy in the system. Heat and work are mechanisms by
which systems exchange energy with one another.
states that during any cycle a system undergoes, the cyclic integral of heat is proportional to
the cyclic integral of work or for any system, total energy entering = total energy leaving.
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19.3
SECOND LAW OF THERMODYMANICS – ENTROPY
- states that the entropy – that is, the disorder – of an isolated system can never decrease.
Thus, when an isolated system achieved a configuration of maximum entropy, it can no
longer undergo change: It has reached equilibrium.
Significant statements:
- Clausius: It is impossible for a self-acting machine unaided by an external agency to move
heat from one body to another at a higher temperature.
- Kelvin-Planck: It is impossible to construct a heat engine which, while operating in a cycle
produces no effects except to do work and exchange heat with a single reservoir.
- All spontaneous processes result in a more probable state
- The entropy of an isolated system never decreases.
- No actual or ideal heat engine operating in cycles can convert into work all the heat supplied
to the working substance.
- Caratheodory: In the vicinity of any particular state 2 of a system, there exist neighboring
states 1 that are inaccessible via an adiabatic change from state 2.
19.4
THIRD LAW OF THERMODYNAMICS - ABSOLUTE TEMPERATURE
-
states that absolute zero cannot be attained by any procedure in a finite number of steps.
Absolute zero can be approached arbitrarily closely, but it can never be reached.
- The entropy of a substance of absolute temperature is zero.
20. THERMODYNAMIC EQUILIBRIUM
Thermodynamic Equilibrium – is a condition when a system is in equilibrium with regard to all possible
changes in state.
21. CONSERVATION OF MASS
The law of conservation of mass states that the mass is indestructible.
Steady State – is that circumstance in which there is no accumulation of mass or energy within the control
volume, and the properties at any point within the system are independent of time.
Continuity Equation of Steady Flow
Aυ
Aυ
m = ρ1 A1υ1 = ρ2 A2υ2 = 1 1 = 2 2
v1
v2
Usual English units are: υ fps, ρ lb/ft3, ν ft3/lb, A ft2, m lb/sec.
22. HEAT RESERVOIR
Heat reservoir – is a thermodynamic system that generally serves as a heat source or heat sink for another
system.
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23. HEAT OR THERMAL ENGINE
Heat engine or Thermal Engine (1) – is a thermodynamic system that operates continuously with only energy
(heat and work) crossing its boundaries; its boundaries are impervious to the flow of mass.
Heat engine or Thermal Engine (2) – a closed system (no mass crosses its boundaries) that exchanges only
heat and work with its surroundings and that operate in cycles.
24. ADIABATIC SURFACE AND PROCESS
Adiabatic surface – is one that is impervious to heat. It implies perfect insulation.
Adiabatic process – a process that occurs within a system enveloped by an adiabatic surface.
25. AVAILABILITY
Availability of a closed system – is the maximum work that the system could conceivably deliver to
something other than the surroundings as its state changes to the dead state, exchanging heat only with the
environment.
Availability of a steady-flow system – is the maximum work that can conceivably be delivered by, say, a unit
mass of the system to something other than the surroundings as this unit mass changes from its state at the
entrance to the control volume to the dead state at the exit boundary, meanwhile exchanging heat only
with the surroundings.
26. REVERSIBILITY
Reversibility (1) - a characteristic of certain processes (changes of a system from an initial state to a final
state spontaneously or as a result of interactions with other systems) that can be reversed, and the system
restored to its initial state, without leaving net effects in any of the systems involved.
Reversibility (2) - is a process that can be "reversed" by means of infinitesimal changes in some property of
the system without loss or dissipation of energy
27. IRREVERSIBILITY
Irreversibility - a change in the thermodynamic state of a system and all of its surroundings cannot be
precisely restored to its initial state by infinitesimal changes in some property of the system without
expenditure of energy.
27.1
External irreversibility – is some irreversibility external to the system.
27.2
Internal irreversibility – is any irreversibility within the system.
28. AVAILABLE ENERGY
Available energy - is the greatest amount of mechanical work that can be obtained from a system or body,
with a given quantity of substance, in a given initial state, without increasing its total volume or allowing
heat to pass to or from external bodies, except such as at the close of the processes are left in their initial
condition.
29. HELMHOLTZ FUNCTION
Helmholtz function - a thermodynamic property of a system equal to the difference between its internal
energy and the product of its temperature and its entropy. Symbol A. Formula A = u – Ts.
30. GIBBS FUNCTION
Gibbs function - a thermodynamic property of a system equal to the difference between its enthalpy and the
product of its temperature and its entropy. It is usually measured in joules. Symbol G. Formula G = h – Ts.
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31. EQUILIBRIUM
Equilibrium - is the condition of a system in which competing influences are balanced.
Mechanical equilibrium - the state in which the sum of the forces, and torque, on each particle of the system
is zero. Also means that the system is not accelerating.
Thermal equilibrium - a state where an object and its surroundings cease to exchange energy in the form of
heat, i.e. they are at the same temperature.
Chemical equilibrium - the state in which the concentrations of the reactants and products have no net
change over time. Also a system has no tendency to undergo further chemical reaction.
Thermodynamic equilibrium - the state of a thermodynamic system which is in thermal, mechanical, and
chemical equilibrium. Also the system has no tendency toward spontaneous change – meaning a change
without outside influence.
Quasistatic equilibrium - the quasi-balanced state of a thermodynamic system near to equilibrium in some
sense or degree
32. PERPETUAL MOTION MACHINE OF THE SECOND KIND
Perpetual motion machine of the second kind – is a proposed machine that violates the second law of
thermodynamics.
33. BOYLE’S LAW
Boyle’s Law (1) – states that if the temperature of a given quantity of gas is held constant, the volume of gas
varies inversely with the absolute pressure during a quasistatic change of state.
Boyle’s Law (2) – states that the volume of a gas varies inversely with its absolute pressure during change of
state if the temperature is held constant.
p1V1 = p2V2
34. CHARLES’ LAW
Charles’ Law (1) – the volume of a gas varies directly as the absolute temperature during a change of state if
the pressure of the gas is held constant.
V1 V2
=
T1 T2
Charles’ Law (2) – the pressure of a gas varies directly as the absolute temperature during a change of state
if the volume of the gas is held constant in.
p1 p2
=
T1 T2
35. IDEAL GAS LAW
Ideal Gas Law - The equation of state of an ideal gas which is a good approximation to real gases at
sufficiently high temperatures and low pressures; that is, PV = RT, where P is the pressure, V is the volume
per mole of gas, T is the temperature, and R is the gas constant.
pV = mRT
p1V1 p2V2
=
T1
T2
Where p = pressure, V = volume, m = mass, R = ideal gas constant, and T = absolute temperature.
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36. BASIC PROPERTIES OF IDEAL GAS
8.3143
R=
kJ kg − K
or
M
c p − cv = R
cp
cv
R=
1545
ft − lb lb − R
M
=k
kR
k −1
R
cv =
k −1
Where:
R = gas constant
M = molecular weight
cp = specific heat at constant pressure
cv = specific heat at constant volume
k = specific heat ratio
37. PROPERTIES OF AIR
M = 28.97 kg air mole air
cp =
k = 1.4
R = 53.3 ft − lb lb − R = 0.287 kJ kg − K
c p = 0.24 Btu lb − R = 0.24 kcal kg − C = 1.0 kJ kg − C
cv = 0.171 Btu lb − R = 0.171 kcal kg − C = 0.716 kJ kg − C
38. AVOGADRO’S LAW AND NUMBER
Avogadro’s Law – states that “equal volumes of all ideal gases at a particular pressure and temperature
contains the same number of molecules”.
Avogadro’s number, NA – is the number of molecules in a mole of any substance and is equal to 6.0225 x
1023 gmole-1.
39. JOULE’S LAW
Joule’s Law – states that the change of internal energy of an ideal gas is a function of only the temperature
change.
40. DALTON’S LAW OF PARTIAL PRESSURE
Dalton’s Law of Partial Pressure – states that the total pressure pm of a mixture of gases is the sum of the
pressures that each gas would exert were it to occupy the vessel alone at the volume Vm and temperature tm
of the mixture.
pm = pa + pb + pc + L = ∑ pi
i
[Tm = Ta = Tb = Tc
Vm = Va = Vb = Vc ]
pi = X i pm
∑ Xi = 1
i
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41. AMAGAT’S LAW OF PARTIAL VOLUMES
Amagat's Law of Partial Volumes - states that the volume Vm of a gas mixture is equal to the sum of volumes
Vi of the K component gases, if the temperature T and the pressure p remain the same.
Vm = Va + Vb + Vc + L = ∑ Vi
i
[Tm = Ta = Tb = Tc
pm = pa = pb = pc ]
Vi = X iVm
∑ Xi = 1
i
42. THERMODYNAMIC SYSTEMS AND PROCESSES
Thermodynamic Process – is the path of the succession of states through which the system passes.
Cyclic Process or Cycle – is a process where a system in a given initial state goes through a number of
different changes in state (going through various processes) and finally returns to its initial values.
Reversible Process – is defined as a process that, once having taken place, can be reversed, and in so doing
leaves no change in either the system or surroundings.
Irreversible Process – is a process that cannot return both the system and the surroundings to their original
conditions. That is, the system and the surroundings would not return to their original conditions if the
process was reversed.
Adiabatic Process – is one in which there is no heat transfer into or out of the system. The system can be
considered to be perfectly insulated.
Isentropic Process – is one in which the entropy of the fluid remains constant. This will be true if the process
the system goes through is reversible and adiabatic. An isentropic process can also be called a constant
entropy process.
Polytropic Process – is a process when a gas undergoes a reversible process in which there is heat transfer,
the process frequently takes place in such a manner that a plot of the Log P (pressure) vs Log V (volume) is a
straight line. Or stated in equation form PV n = a constant.
Throttling Process – is defined as a process in which there is no change in enthalpy from state one to state
two, h1 = h2 ; no work is done, W = 0 ; and the process is adiabatic, Q = 0 .
Isobaric Process – is an internally reversible (quasistatic, if nonflow) process of a pure substance during
which the pressure remains constant.
Isometric Process (Isochoric Process) – a constant volume process that is internally reversible (quasi-static if
nonflow), involving a pure substance.
Isothermal Process – is an internally reversible (quasistatic, if nonflow) constant temperature process of a
pure substance.
43. EQUATIONS FOR THERMODYNAMIC PROCESSES
dQ
Entropy = ∆S = ∫
T
Non-Flow Equation
Q = ∆U + Wnf
Steady-Flow Equation, ∆K = 0 ∆P = 0
Q = ∆H + Wsf
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Steady-Flow Equation, ∆K ≠ 0 ∆P ≠ 0
Q = ∆H + ∆K + ∆P + Wsf
Where:
∆U = internal energy
∆H = enthalpy
∆K = kinetic energy
∆P = potential energy
Wnf = non-flow work
Wsf = steady flow work
Internal Energy
∆U = mcv ∆T
Enthalpy
∆H = mcp ∆T
Non-Flow Work
Wnf = ∫ pdV
Steady Flow Work, ∆K = 0 ∆P = 0
Wsf = − ∫ Vdp
Steady-Flow Equation, ∆K ≠ 0 ∆P ≠ 0
∆K + ∆P + Wsf = −∫ Vdp
∆H
∆U
44. ISOBARIC PROCESS
Isobaric Process – is an internally reversible (quasistatic, if nonflow) process of a pure substance during
which the pressure remains constant.
Note: k =
p = Constant
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V2 T2
=
V1 T1
∆H = mc p (T2 − T1 )
∆U = mcv (T2 − T1 )
Non-flow: Q = ∆U + Wnf
Wnf = ∫ pdV = p(V2 − V1 )
Q = ∆H
Entropy: ∆S = ∫
∆S = mc p ln
dQ
T
T2
V
= mc p ln 1
T1
V2
Steady Flow: Q = ∆H + Wsf
Wsf = − ∫ Vdp = 0
Q = ∆H
Entropy: ∆S = ∫
∆S = mc p ln
dQ
T
T2
V
= mc p ln 1
T1
V2
Specific Heat Ratio:
∆H Q
k=
=
∆U ∆U
45. ISOMETRIC PROCESS (ISOCHORIC PROCESS)
Isometric Process (Isochoric Process) – a constant volume process that is internally reversible (quasi-static if
nonflow), involving a pure substance.
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V = constant
p2 T2
=
p1 T1
∆H = mc p (T2 − T1 )
∆U = mcv (T2 − T1 )
Non-flow: Q = ∆U + Wnf
Wnf = ∫ pdV = 0
Q = ∆U
Entropy: ∆S = ∫
∆S = mcv ln
dQ
T
T2
p
= mcv ln 2
T1
p1
Steady Flow: Q = ∆H + Wsf
Wsf = − ∫ Vdp = −V (p2 − p1 )
Q = ∆U
Specific Heat Ratio:
∆H ∆H
k=
=
∆U Q
46. ISOTHERMAL PROCESS
Isothermal Process – is an internally reversible (quasistatic, if nonflow) constant temperature process of a
pure substance.
pV = constant, T = constant
∆H = mc p (T2 − T1 )
∆U = mcv (T2 − T1 )
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Non-Flow Equation: Q = ∆U + Wnf
p1V1 = p2V2 = pV = C = mRT
Wnf = pV ln
V2
p
= pV ln 1
V1
p2
Wnf = mRT ln
V2
p
= mRT ln 1
V1
p2
Q = Wnf
Steady Flow: Q = ∆H + Wsf
V2
p
= pV ln 1
V1
p2
Wsf = pV ln
Wsf = mRT ln
V2
p
= mRT ln 1
V1
p2
Q = Wsf
Entropy:
∆S = mR ln
V2
V1
∆S = mR ln
p1
p2
∆S =
pV V2 pV p1
ln =
ln
T V1 T
p2
47. ISENTROPIC PROCESS
Isentropic Process – is a reversible adiabatic process with constant entropy.
pV k = constant, ∆S = 0 , pV = mRT , Q = 0
k
p2  V1   T2   V1 
=   = 
p1  V2   T1   V2 
,
∆H = mc p (T2 − T1 )
D
k −1
T   p 
Or  2  =  2 
 T1   p1 
,
k −1
k
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D – THERMODYNAMICS (LECTURE)
∆U = mcv (T2 − T1 )
Non-flow: Q = ∆U + Wnf
1− k
 p V  p 1−k 
p1V1  V2 
  − 1 = 1 1  1  − 1
Wnf =
1 − k  V1 
 1 − k  p2 





mRT1
Wnf =
1−k
Wnf =
mRT1
1− k
 V 1−k  mRT
1
 2  − 1 =
 V1 
 1−k


 V  k −1  mRT
1
 1  − 1 =
 V2 
 1−k


k −1


 p2  k

− 1
 p 
 1 

 T2  mR(T2 − T1 ) p2V2 − p1V1
=
 − 1 =
1−k
1−k
 T1 
But,
R
k −1
Wnf = −mcv (T2 − T1 ) = −∆U
cv =
Steady Flow: Q = ∆H + Wsf
kp V
Wsf = 1 1
1−k
Wsf =
k −1


 p2  k
 kmRT1
− 1 =
 p 
1−k
 1 

k −1


 p2  k
 kmRT1
− 1 =
 p 
1−k
 1 

 V  k −1  kmRT  T

1 2
 1  − 1 =
− 1 



 V2 
 1 − k  T1



kmR(T2 − T1 )
1−k
But
kR
k −1
Wsf = −mc p (T2 − T1 ) = − ∆H
cp =
Note:
c p ∆H − ∫ Vdp Wsf
k= =
=
=
cv ∆U ∫ pdV Wnf
48. POLYTROPIC PROCESS
Polytropic Process – is an internally reversible process during which pVn = C where n is any constant.
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D – THERMODYNAMICS (LECTURE)
pV n = constant, pV = mRT
T2  V1 
= 
T1  V2 
n−1
T2  p2 
= 
T1  p1 
,
∆H = mc p (T2 − T1 )
n−1
n
∆U = mcv (T2 − T1 )
Non-flow Work: Q = ∆U + Wnf
1 −n
 p V  p 1−n 
p1V1  V2 
  − 1 = 1 1  1  − 1
Wnf =
1 − n  V1 
 1 − n  p2 





mRT1
Wnf =
1−n
Wnf =
mRT1
1−n
 V 1−n  mRT
1
 2  − 1 =


V
n
1
−
 1 



 V  n −1  mRT
1
 1  − 1 =


V
n
1
−
 2 



n −1




p
 2  n

− 1
 p 
 1 

 T2  mR(T2 − T1 ) p2V2 − p1V1
=
 − 1 =
1− n
1−n
 T1 
But,
R
k −1
mR(T2 − T1 ) p2V2 − p1V1
∆U =
=
k −1
k −1
1 
1 
(n − k )(p2V2 − p1V1 )
 1
 1
Q=
+
−
(p2V2 − p1V1 ) = 
(p2V2 − p1V1 ) =
(k − 1)(n − 1)
 k −1 1 − n 
 k −1 n −1 
Steady Flow: Q = ∆H + Wsf
cv =
n−1




np1V1  p2 n
 nmRT1
 
Wsf =
− 1 =
1 − n  p1 
1−n


D
n−1




 p2  n
 nmRT1
− 1 =
 p 
1− n
 1 

 V  n−1  nmRT  T

1 2
 1  − 1 =
− 1 



 V2 
 1 − n  T1



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D – THERMODYNAMICS (LECTURE)
Wsf =
nmR(T2 − T1 )
1−n
But
cp =
kR
k −1
kmR(T2 − T1 ) k (p2V2 − p1V1 )
=
k −1
k −1
1 
n 
 1
 k
Q=
+
−
(p2V2 − p1V1 ) = 
(p2V2 − p1V1 )
 k −1 1− n 
 k −1 n −1 
(n − k )(p2V2 − p1V1 )
Q=
(k − 1)(n − 1)
∆H = mcp (T2 − T1 ) =
Entropy:
 n − k  T2
∆S = mcv 
 ln
 n − 1  T1
Or
V 
∆S = mcv (n − k )ln 1 
 V2 
 n − k   p2 
∆S = mcv 
 ln
 n − 1   p1 
n −1
n
 n − k   p2 
= mcv 
 ln
 n   p1 
Note:
Steady flow work:
Wsf + ∆K + ∆P = − ∫ Vdp
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D – THERMODYNAMICS (LECTURE)
49. CURVES FOR DIFFERENT VALUES OF n.
50. TABLE OF IDEAL GAS FORMULAS
Ideal Gas Formulas
For constant mass systems undergoing internally reversible processes
Process
Isometric
Isobaric
Isothermal
Isentropic
Polytropic
→
V =C
p=C
T =C
S =C
pV n = C
p1V1k = p2V 2k
p1V1n = p2V 2n
p, V, T
relations
T2 p2
=
T1
p1
T2 V2
=
T1 V1
p1V1 = p 2V2
T2  V1 
= 
T1  V2 
T2  p2
=
T1  p1
∫
Q
(n −1) n
V2
V1
p 2V2 − p1V1
1− k
p 2V2 − p1V1
1− n
V ( p 2 − p1 )
0
p1V1 ln
V2
V1
k ( p 2V2 − p1V1 )
1− k
n( p 2V2 − p1V1 )
1− n
∫
∫
m cv dT
m cv dT
mcv (T2 − T1 )
mcv (T2 − T1 )
∫
m cv dT
mcv (T2 − T1 )
D



p1V1 ln
1
U 2 − U1
T2  p2
=
T1  p1
n −1
p(V2 − V1 )
1
2
(k −1) k
T2  V1 
= 
T1  V2 
0
2
∫ pdV
− Vdp



k −1
∫
m c p dT
mc p (T2 − T1 )
0
∫
m cv dT
mcv (T2 − T1 )
mcv (T2 − T1 )
∫
m Tds
V
p1V1 ln 2
V1
∫
m cv dT
0
∫
m c n dT
mcn (T2 − T1 )
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D – THERMODYNAMICS (LECTURE)
n
∞
0
Specific heat, c
cv
cp
m c p dT
m c p dT
mc p (T2 − T1 )
mc p (T2 − T1 )
∫
H 2 − H1
cv dT
T
T
mcv ln 2
T1
m
S 2 − S1
∫
∫
c p dT
T
mc p ln
m
−∞ to + ∞
0
k −n
c n = cv 

 1− n 
(k = C )
m c p dT
m c p dT
mc p (T2 − T1 )
mc p (T2 − T1 )
0
c n dT
T
T
mcn ln 2
T1
∞
∫
m
k
1
∫
T2
T1
∫
0
Q
T
mR ln
∫
m
V2
V1
∫
c p dT
cv dT
V
p
+ mR ln 2 , m
+ mR ln 2
T
V1
T
p1
∫
51. STAGNATION PROPERTIES
Stagnation Properties – are those thermodynamic properties that a moving stream of compressible fluid
would have if it were brought to rest isentropically (no outside work, the kinetic energy brings about the
compression).
Stagnation enthalpy
h0 = h +
v2
2gc J
Stagnation temperature
T0 = T +
v2
2gc Jc p
T0  p0 
= 
T  p 
k −1
k
=1+
v2
2gc Jc pT
52. MACH NUMBER
Mach Number – is the ratio of the actual speed divided by the local speed of sound a in the fluid.
υ
M<1
M>1
M≡
a
[SUBSONIC]
[SUPERSONIC]
Acoustic speed:
1
a = (gc kRT )2
53. PURE SUBSTANCE
Pure Substance – is a working substance that has a homogeneous and invariable chemical composition even
there is a change of phase.
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D – THERMODYNAMICS (LECTURE)
54. PHASES OF PURE SUBSTANCE
Phase (1) - a chemically and physically uniform quantity of matter that can be separated mechanically from a
non-homogeneous mixture. It may consist of a single substance or of a mixture of substances.
Phase (2) – a quantity of matter that is homogeneous throughout in both chemical composition and physical
structure.
Phase (3) - is a distinct and homogeneous state of a system with no visible boundary separating it into parts.
54.1 Solid - has a definite shape and volume.
54.2 Liquid - has a definite volume but it takes the shape of a container.
Saturated Liquid – a liquid substance that has its temperature equal to the saturation temperature
at a given pressure.
Compressed Liquid – liquid whose pressure is higher than the saturation pressure at the given
temperature. If the temperature is held constant , the pressure is increased beyond the saturation
pressure.
Sub-cooled Liquid – liquid whose temperature is lower than the saturation temperature at the given
pressure.
54.3 Gas - fills the entire volume of a container.
Vapor - the state of a substance that exists below its critical temperature and that may be liquefied
by application of sufficient pressure.
Saturated Vapor – a vapour substance that has its temperature equal to the saturation temperature
at a given pressure.
Superheated Vapor – vapor whose temperature is higher than the saturation temperature at the
given pressure.
55. PHASE TRANSITIONS
Phase transition - Conversion between these phases.
55.1 Melting or fusion – is the change of phase from solid to liquid.
55.2 Freezing or solidifying – is the change of phase from liquid to solid.
55.3 Sublimation – is the change of phase from solid to gas.
55.4 Deposition – is the change of phase from gas to solid.
55.5 Condensation – is the change of phase from gas to liquid.
55.6 Vaporization – is the change of phase from liquid to gas.
56. SATURATED TEMPERATURE
Saturation Temperature – the temperature at which vaporization takes place at a given pressure, this
pressure being called the saturation pressure for the given temperature.
57. SATURATED PRESSURE
Saturation Vapor Pressure - the vapor pressure of a thermodynamic system, at a given temperature,
wherein the vapor of a substance is in equilibrium with a plane surface of that substance's pure liquid or
solid phase.
58. DEGREES SUPERHEAT AND DEGREES SUBCOOLING
Degrees Superheat – difference between actual temperature and saturation temperature.
Degrees Subcooling – difference between saturation temperature and actual temperature.
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D – THERMODYNAMICS (LECTURE)
59. SENSIBLE HEAT
Sensible Heat – the heat added to or removed from a substance to produce a change in its temperature
60. LATENT HEAT
Latent Heat – the amount of heat added to or removed from a substance to produce a change in phase.
60.1 Latent heat of fusion, enthalpy of fusion – is the amount of heat added or removed to change phase
between solid and liquid.
60.2 Latent heat of vaporization, enthalpy of evaporation – is the amount of heat added or removed to
change phase between liquid and vapor. It is sometimes called the latent heat of condensation.
61. CRITICAL POINT
Critical Point – is the condition of pressure and temperature at which a liquid and its vapor are
indistinguishable.
62. MOLLIER DIAGRAM
Mollier Diagram (h-s) – is a chart on which enthalpy is the ordinate and entropy the abscissa.
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D – THERMODYNAMICS (LECTURE)
63.
TEMPERATURE-ENTROPY DIAGRAM
64. ph-Chart
66. PROPERTIES
Specific Volume, v – volume per unit mass.
Internal Energy, u – energy stored within a body or substance by virtue of the activity and configuration of
its molecules and the vibration of the atoms within the molecules.
Enthalpy, h – a composite property applicable to all fluids and is defined by the equation: h = u + pv.
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D – THERMODYNAMICS (LECTURE)
Entropy, s – a property used to measure the state of disorder of a substance; a function of both heat and
temperature.
67. MIXTURES
Mixture – substance made up of liquid and vapor portion or two-phase liquid-vapour system.
x= quality or dryness factor or vapour content
y = 1 – x = moisture content or wetness
Properties of mixtures
v = v f + xv fg
u = u f + xu fg
h = h f + xh fg
s = s f + xs fg
68. PROCESSES INVOLVING PURE SUBSTANCES
a. Isobaric or constant pressure process: p1 = p2
b. Isothermal or constant temperature process: T1 = T2
Evaporation and condensation processes occur at constant pressure and constant temperature.
c. Isometric or constant volume process: V1 = V2
For constant mass: v1 = v2
If the final state is a mixture: v1 = (vf + xvfg)2
d. Isentropic or constant entropy process: s1 = s2
Isentropic process is reversible (no friction loss) and adiabatic (no heat loss, that is, completely
insulated system).
e. Throttling or isenthalpic (constant enthalpy) process: h1 = h2
If the final state is a mixture: h1 = (hf + xhfg)2
If the initial state is a mixture, such as in steam calorimeter:
(hf + xhfg)1 = h2
69. ESSENTIAL ELEMENTS OF A THERMODYNAMIC HEAT ENGINE
69.1 Working substance – matter that receives heat, rejects heat, and does work.
69.2 Source of heat (Hot body, Heat reservoir, or just Source) – form which the working substance
receives heat.
69.3 Heat sink (Receiver, Cold body, or just Sink) – to which the working substance can reject heat.
69.4 Engine – wherein the working substance may do work or have work done on it.
70. THE CARNOT CYCLE
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D – THERMODYNAMICS (LECTURE)
T1 − T2 = T4 − T3 and S1 − S4 = S2 − S3
Q A = T1 (S1 − S4 )
QR = T2 (S2 − S3 ) = T2 (S1 − S 4 )
W = Q A − QR = T1 (S1 − S4 ) − T2 (S1 − S4 )
e=
W QA − QR T1 (S1 − S 4 ) − T2 (S1 − S4 )
=
=
QA
QA
T1 (S1 − S 4 )
e=
T1 − T2 TH − TL
=
T1
TH
QA QR
=
TH
TL
Where
e = Carnot cycle efficiency
T1 = TH = highest absolute temperature
T2 = TL = lowest absolute temperature
71. STIRLING AND ERICSSON CYCLE
Ideal Stirling Cycle – is composed of two isothermal and two isometric processes, the regeneration occurring
at constant volume.
Ideal Ericsson Cycle – consists of two isothermal and two isobaric processes, with the regeneration occurring
during constant pressure.
72. BASIC WORKING CYCLES FOR VARIOUS APPLICATIONS
Application
Steam Power Plant
Gasoline Engine (Spark-Ignition)
Diesel Engine (Combustion-Ignition)
Gas Turbine
Refrigeration System
Basic Working Cycle
Rankine Cycle
Otto Cycle
Diesel Cycle
Brayton Cycle
Refrigeration Cycle
-
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E - ENGINEERING ECONOMICS (LECTURE)
1. Engineering Economics
Engineering Economics – is the study of the cost factors involved in engineering projects, and using the
results of such study in employing the most efficient cost-saving techniques without affecting the safety and
soundness of the project
2. Definitions
Investment – is the sum total of first cost (fixed capital) and working capital which is being put up in a
project with the aim of getting a profit.
Fixed Capital – is a part of the investment which is required to acquire or set up the business.
Working Capital – is the amount of money set aside as part of the investment to keep the project or
business continuously operating.
Demand – is the quantity of a certain commodity that is bought at a certain price at a given place and time.
Supply – is the quantity of a certain commodity that is offered for sale at certain price at a given place.
Perfect Competition – is a business condition in which a product or service is supplied by a number of
vendors and there is no restriction against additional vendors entering the market.
Monopoly – is a business condition in which as unique product or service is available from only one supplier
and that supplier can prevent the entry of all others into the market.
Oligopoly – is a condition in which there are so few suppliers of a product or service that action by one will
almost result in similar action by the others.
Law of Supply and Demand - “Under conditions of perfect competition, the price of a product will be such
that supply and demand are equal.”
Law of Diminishing Returns – “When the use of one of the factors of production is limited, either in
increasing cost or by absolute quantity, a point will be reached beyond which an increase in the variable
factors will result in a less than proportionate increase output.”
3. Interest
Interest – is the money paid for the use of borrowed money.
4. Simple Interest
Simple interest – is the interest paid on the principal (money lent) only.
I = Pni
F = P + I = P + Pni = P (1 + ni )
Where:
F = future value
P = present value
n = number of interest period
i = interest rate per period
I = interest
Two Types of Simple Interest
4.1 Ordinary simple interest , 1 year = 360 days
4.2 Exact simple interest, 1 year = 365 or 366 days
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E - ENGINEERING ECONOMICS (LECTURE)
5. Compound Interest
Compound interest – is the interest which is calculated not only on the initial principal but also the
accumulated interest of prior periods.
F = P(1 + i )n
Single payment compound amount factor = (F P , i%, n ) = (1 + i )n
F = P(F P , i%, n )
Single payment present worth factor = (P F , i , n) = (1 + i )− n
P = F (P F , i%, n )
Where:
F = future value
P = present value
n = number of interest period
i = interest rate per period
6. Cash Flow Diagram
Cash flow diagram – is a graphical representation of cash flows drawn on a time scale.
7. Discount
Discount – is the difference between the future worth and the present worth of a unit.
Discount, D = F − P
Rate of discount, d =
F−P
F
8. Nominal and Effective Rate of Interest
Nominal interest rates – is the cost of borrowed money which specifies the rate of interest and the number
of interest periods.
Effective interest rates – is the actual rate of interest on the capital and is equal to the nominal rate if
compounded annually. Effective interest rate is greater than nominal interest rates.
Let in = nominal interest rate or annual percentage rate
m = number of sub periods per year

Effective interest rate = i = 1 +

E
m
in 
 −1
m
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E - ENGINEERING ECONOMICS (LECTURE)
Where:
m = 2 for semi-annually
m = 4 for quarterly
m = 12 for monthly
m = 6 for bi-monthly
m = 360 for daily
9. Continuously Compounding Interest Rate
F = Pe rt
Where:
F = future value
P = principal or present value
r = continuously compounding interest rate
t = number of interest periods
10. Annuity
Annuity – is a series of equal payments occurring at equal intervals of time.
Amortization – is a payment of debt by installment usually by equal amounts and at equal intervals of time.
10.1
10.2
Applications of annuity
10.1.1 Installment purchase.
10.1.2 Amortization of loan.
10.1.3 Depreciation
10.1.4 Payment of insurance premiums.
Types of annuity
10.2.1 Ordinary Annuity – payments occur at the end of each period.
10.2.2 Annuity Due – payments occur at the beginning of each period.
10.2.3 Deferred Annuity – first payment occurs later than at the end of the first period.
10.2.4 Perpetuity – an annuity that continues indefinitely.
11. Ordinary Annuity
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Uniform series compound amount factor = (F A , i%, n ) =
(1 + i )n − 1
i
F = A(F A , i%, n )
Uniform series sinking fund factor = ( A F , i %, n ) =
i
(1 + i )n − 1
A = F ( A F , i%, n )
n
Capital recovery factor = ( A P , i %, n ) =
i (1 + i )
(1 + i )n − 1
A = P( A P , i%, n )
Uniform series present worth factor = (P A , i %, n ) =
(1 + i )n − 1
i (1 + i )n
P = A(P A , i%, n )
Where:
F = future value of the periodic payments at the end of n periods.
P = present value of the periodic payments
A = Annuity or periodic payments
n = number of periodic payments
i = interest rate per period
12. Annuity Due
P = A + A(P A ,i%,4)
F = P(F P ,i%,5)
F = P(1 + i )5
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13. Deferred Annuity
P = A(P A , i%,5)(P F , i%,3)
F = P(F P ,i%,8)
F = P(1 + i )8
14. Perpetuity
P=
A
i
Where:
P = present value of the perpetuity
A = Annuity or periodic payments
i = interest rate per period
15. Arithmetic Gradient
(1 + i )n − 1 − n
i 2 (1 + i )n i (1 + i )n
(1 + i )n − 1 − n
Arithmetic-gradient future worth factor = (F G , i%, n) =
2
Arithmetic-gradient present worth factor = (P G , i%, n) =
i
i
1
i
Arithmetic-gradient uniform-series factor = ( A G , i%, n ) = −
E
n
(1 + i )n − 1
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E - ENGINEERING ECONOMICS (LECTURE)
PG = G (P G , i%, n )
AG = G ( A G , i %, n )
FG = PG (F P , i%, n) = G (F G , i%, n )
Where:
AG = Equivalent annual amount of gradient series.
16. Geometric Gradient
   1 + g n 
 A1 1 − 
 
   1 + i  
Pg = 
i−g

n

A1
1+ i

g ≠i
g =i
Fg = Pg (F P , i %, n )
17. Depreciation
Depreciation – is the decrease in value of a physical property due to the passage of time. Depreciation of a
property is an example of capitalization.
17.1
Types of Depreciation
17.1.1 Physical depreciation – is a type of depreciation caused by the lessening of the physical
ability of the property to produce results, such as physical damage, wear and tear.
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17.2
17.3
17.1.2 Functional depreciation – is a type of depreciation caused by the lessening in the
demand for which the property is designed to render, such as obsolescence and
inadequacy.
Purposes of Depreciation
17.2.1 To provide for the recovery of capital invested in the property.
17.2.2 To enable the cost of depreciation to be charged to the cost of producing the products
turned out by the property.
Depreciation Terms
First Cost (FC) – is the total amount invested on the property until the property is put into
operation.
Economic Life (n) – is the length of time at which a property can be operated at a profit.
Valuation (Appraisal) – is the process of determining the value or worth of a physical property
for specific reasons.
Value – is the present worth of all the future profits that are to be received through ownership
of the property.
17.4
Classification of Values
Market value – is the price that will be paid by a willing buyer to a willing seller for a property
where each has equal advantage and is under no compulsion to buy or sell.
Book value (BV) – is the worth of a property as shown in the accounting records of an
enterprise.
Salvage or resale value (SV) – is the price of a property when sold second-hand; also called
trade-in value.
Scrap value (SV) – is the price of a property when sold for junk.
Fair value – is the worth of a property as determined by a disinterested party which is fair to
both seller and buyer.
Use value – is the worth of property as an operating unit.
Face or par value (F) – is the amount that appears on the bond which is the price at which the
bond is first bought.
18. Methods of Computing Depreciation
18.1
Straight Line Method
Straight line method – is a method of computing depreciation in which the depreciation has the
same value each year.
Annual Depreciation = d =
FC − SV
n
Book Value for m years.
 FC − SV 
BV = FC − m

n


Depreciation rate (d) – is the annual rate for reducing the value of a property using depreciation
method.
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Depreciation rate =
18.2
Annual Depreciation
First Cost
Sinking Fund Method
Sinking fund method – is a method of computing depreciation in which the initial depreciation is
low.
Annual Depreciation = d =
FC − SV
FC − SV
=
(F A , i%, n) (1 + i )n − 1
i
Where:
i = interest rate or worth of money
Book Value for m years.
 (1 + i )n − 1 
BV = FC − d 

i


18.3
Sum-of-the-Years-Digits (SOYD) Methods
SOYD – is a method of computing depreciation in which the digits from year 1 to year n are
added and the depreciation in a certain year decreases by a constant amount each year.
T = SOYD =
n(n + 1)
2
 n − k +1 
dk = 
(FC − SV )
 T

n
d1 =  (FC − SV )
T 
 n −1 
d2 = 
(FC − SV )
 T 
n−2
d3 = 
(FC − SV )
 T 
Etc.
Book Value for m years.
BV = FC − (d1 + d 2 + d 3 + L d m )
18.4
Declining Balance Method
(Also called Diminishing Balance Method, Matheson Method, Constant-Percentage or Constant
Ratio Method).
SV = FC (1 − r )n
d k = r (FC )(1 − r )k −1
d1 = r (FC )
d 2 = r (FC )(1 − r )
d 3 = r (FC )(1 − r )2
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Etc.
Book Value for m years.
BV = FC (1 − r )m
18.5
Double Declining Balance Method
Double declining balance method – is a method of computing depreciation in which the
depreciation of salvage and the book value never stops decreasing. Double declining balance is
dependent on accumulated depreciation.
 2
SV = FC 1 − 
 L
L
dk =
2
(FC )1 − 2 
L
 L
d1 =
2
(FC )
L
d2 =
2
(FC )1 − 2 
L
 L
2
 2
d 3 = (FC )1 − 
L
 L
k −1
2
Etc.
Book Value for m years.
 2
BV = FC 1 − 
 L
18.6
m
Service Output or Production Units Method
Service output method – is a method of computing depreciation in which depreciation is
calculated based on the total production produced per year.
Depreciation per unit, d =
18.7
FC − SV
No. of units capacity
Working Hours or Machine Hours Method
Depreciation per hr, d =
FC − SV
No. of hours capacity
19. Depletion - is the decrease in value of property due to the gradual extraction of its contents, such as mining
properties, oil wells, timber lands and other consumable resources.
Two Methods of Depletion
19.1
Cost Depletion – is based on the level of activity or usage, not time, as in depreciation. It may be
applied to most types of natural resources.
Cost depletion factor for year t is the ratio of the first cost of the resource to the estimated
number of units recoverable.
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pt =
first cost
resource capacity
The annual depletion charge is pt times the year’s usage or volume. The total cost depletion
cannot exceed the first cost of the resource. If the capacity of the property is reestimated some
year in the future, a new cost depletion factor is determined based upon the undepleted
amount and the new capacity estimate.
19.2
Percentage Depletion – is a special consideration given for natural resources. A constant, stated
percentage of the resource’s gross income may be depleted each year provided it does not
exceed 50% of the company’s taxable income. For oil and gas property, the limit is 100% of
taxable income. The annual depletion amount is calculates as
Percentage depletion amount = percentage x gross income from property
Using percentage depletion, total depletion charges may exceed first cost with no
limitation.
20. Capital Recovery (Factors of Annual Cost)
20.1
Using Sinking Fund Method
Capital Recovery = Annual Depreciation + Interest on Capital
Annual Depreciation =
FC − SV
(1 + i )n − 1
i
Interest on Capital = i(FC )
20.2
Using Straight Line Method
Capital Recovery = Annual Depreciation + Average Interest
Annual Depreciation =
Interest on Capital =
FC − SV
n
i  n +1 

(FC − SV ) + i(SV )
2 n 
21. Capitalized Cost
Capitalized Cost – is the sum of the first cost and the present worth of all cost of replacement, operation,
and maintenance for a long time.
Methods of Computing Capitalized Cost
21.1
For perpetual life.
Capitalized Cost = FC +
21.2
For life n.
Capitalized Cost = FC +
E
OM
i
OM FC − SV
+
i
(1 + i )n − 1
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Where:
OM = annual operation and maintenance cost.
22. Break-Even Analysis
Break-even analysis – is a method of determining the income to equal the expenses or the costs of two
alternatives are equal.
Break-even point – is the value of a certain variable for which the costs of two alternatives are equal.
Income = Expenses
P(x ) = M ( x ) + L( x ) + V ( x ) + FC
Where:
x = no. of units produced and sold
P = selling price per unit
M = material cost per unit
L = labor cost per unit
V = variable cost per unit
FC = first cost
23. Business Organizations
Types of Business Organizations (Forms of Business or Company Ownership)
23.1
Individual Ownership or Single Proprietorship - Is also termed as sole proprietorship and is the
type of ownership in business where individual exercises and enjoy rights in their own interest.
The owner has the total control of the business and makes all decisions.
23.2
Partnership – is also termed as general partnership and is an association of two or more
individuals for the purpose of operating a business as co-owners of a profit.
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23.3
Corporation – is an artificial being created by operation of law, having the right of succession
and the powers, attributes, and properties expressly authorized by law or incident to its
existence. It is an association of not less than five but not more than 15, all of legal age.
Private corporation – are those formed for some private purpose or benefits.
Public corporation – are those formed or organized for the government.
Semi-public corporation – are those formed that is partly government and partly a private
individual.
Quasi-public corporation – are those formed for public utilities and contracts, involving public
duties but which are organized for profit.
Non-profit organization – are those formed for community service and religious activities, but
organized for non-profit.
Four Classes of Persons Composing a Corporation
23.3.1 Corporators – are those who compose the corporation.
23.3.2 Incorporators – are those corporators originally (5-15) forming and composing the
corporation.
23.3.3 Stockholders – are owners of shares of stock.
23.3.4 Members – are corporators of corporation who has no capital stock.
Stock – a certificate of owners corporation
a. Common stock – a residual owners of a corporation.
b. Preferred stock – which entitles the holder thereof to certain preferences over the holder of
common stock.
24. Contract
Contract – is a legally binding agreement to exchange services.
24.1
Four Basic Requirements in a Contract
24.1.1 There must be a clear, specific and definite offer with no room for misunderstanding.
24.1.2 There must be some form of conditional future payments.
24.1.3 There must be an acceptance of the contract and the agreement must be voluntary.
24.1.4 Both parties must have legal capacity and the purpose must be legal.
24.2
Breach of contract – it occurs when one party fails to satisfy all obligations of the contract.
24.3
Negligence – is an action, whether willful or unwillful, which is taken without proper care for
safety, resulting to property damages or injury to persons.
24.4
Torts – a civil wrong committed by one person causing damage to another person or his
property, emotional well-being, or reputation.
25. Bond
Bond – a certificate of indebtedness of a corporation usually for a period of not less than 10 years and
guaranteed by a mortgage on certain assets of the corporation or the subsidiaries.
25.1
Types of bonds
25.1.1 Mortgage bonds – a type of bond in which the security behind are the assets of the
corporation.
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25.2
25.1.2 Collateral bonds – a type of bond in which the security behind are the assets of a well
known subsidiary.
25.1.3 Debenture bonds – a type of bond in which there is no security behind except a promise
to pay.
Bond Value
P = Fr (P A , i %, n) + R(P F , i%, n)
Where:
P = value of bond n periods before maturity
F = face or par value of the bond
Fr = periodic dividend
n = number of periods
R = redeemable value (usually equal to face or par value)
i = investment rate
26. Basic Investment Studies
26.1
Rate of return – usually stated in percent per year and is an effective annual interest rate.
Rate of Return =
26.2
Net Profit
Total Investment
Payout period – is the length of time the investment can be recovered.
Payout Period =
Total Investment − Salvage Value
Net Annual Cash Flow
27. Selection of Alternatives
27.1
Present Economy
This involves selection of alternatives in which interest or time value of money is not a factor.
Studies usually involve the selection between alternative designs, materials, or methods.
27.2
Rate of Return
Rate of return – is usually stated in percent per year and is an effective annual interest rate. The
alternative which gives a higher rate of return on investment is then the favorable choice.
27.3
Payout Period
Payout period – this usually expressed in years and is the length of time for the net annual profit
to equal the initial investment.
27.4
Annual Cost
Annual Cost = Depreciation + Interest on Capital + Operation and Maintenance + Other out-ofpocket Expenses
The alternative with a lower annual cost is then the more economical alternative.
27.5
Present Worth
Present Worth - is applicable when the alternatives involve future expenses whose present
value can be easily determined.
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27.6
Future Worth
Future Worth – is applicable when the alternatives involve expenses whose future worth is
more suitable basis of comparison.
28. Replacement Studies
Replacement studies – is an application of selection of alternatives in which the alternatives are to replace
the old equipment with a new one or to continue using the old equipment.
28.1
Rate of Return
Rate of Return =
28.2
Savings Incurred By Replacement
Additional Capital Required
Annual Cost
Annual Cost = Depreciation + Interest on Capital + Operation and Maintenance + Other out-ofpocket Expenses
In computing depreciation and interest of the old equipment in either method, actual present
realizable values and not historical values should be used.
29. Benefit-to-Cost Ratio in Public Projects
Benefit-to-Cost Ratio – is commonly used in public project evaluations where present worth of all benefits is
divided by the present worth of all costs.
Where:
FC = first cost
SV = salvage value at the end of life
n = useful life
OM = annual operation and maintenance cost
i = interest rate or worth of money
B = annual benefits, that is, the annual worth of benefits incurred because of the existence of the project.
C = annual equivalent of the cost
C = FC ( A P , i%, n ) − SV ( A F , i%, n)
Benefit-to-Cost Ratio
B C=
B − OM
C
B/C should be greater than 1 for the project to be justifiable.
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30. Economic Order Quantity
Economic order quantity (E O Q) – is the order quantity which minimized the inventory cost per unit time.
An assumption of the basic (E O Q) with no shortages: “There is no upper bound on the quantity ordered.”
E OQ =
2ak
h
Where:
a = the constant depletion rate (items per unit time)
k = the fixed cost per order, Pesos
h = the inventory storage cost (Pesos per item per unit time)
31. Principles of Accounting
31.1
Bookkeeping System
Bookkeeping system – is used to record all financial transactions of the company. All
transactions are recorded in a journal then categorized and posted in a ledger. A ledger is
classified into asset, liability and owner’s equity.
31.2
Balancing System
Balancing system – where balancing the book means maintaining the equality of the basic
accounting equation:
Assets = Liability + Owner’s Equity
31.3
31.4
E
Double entry bookkeeping system - is a balancing system by maintaining the equality by
entering each transaction into two ledger accounts. All transactions are either debits or credits.
For liability and owner’s equity, credit increases the account and debt decreases the account.
Cash System
Cash system – is the simplest form of bookkeeping system and transactions recorded into the
journals are the present cash and expenses.
Financial Statements
Financial statements – the process of determining the success or failure of the company.
Financial statements are usually evaluated by accountants, business management and
stockholders.
31.4.1 Two Types of Financial Statements
31.4.1.1
Balance sheet – is the presentation of the basic accounting equation.
31.4.1.2
Profit and loss statement – is the presentation of income source and
expenses. This is also known as statement of income and retained
earnings. Examples of income or revenue are sales, interest, etc. and for
expenses are salaries, supplies, utilities, etc.
31.4.2 Terms used in the Balance Sheets
31.4.2.1
Current assets – also known as liquid assets and defined as cash and
other assets that can be converted quickly into cash such as accounts
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31.4.2.2
Fixed assets – are properties that will not be converted into cash or
difficult to convert into cash such as buildings, land, machinery,
equipment and fixtures.
31.4.2.3
Current liabilities – are liabilities which will due within a short period,
usually a year such as accounts payable and accrued expenses.
31.4.2.4
Long-term liabilities – are liabilities that are not payable within a short
period of time such as notes payable and mortgage.
31.4.3 Terms used in the Financial Statements
31.4.3.1
Current ratio – an index of short term paying ability.
Current Ratio =
31.4.3.2
Current Assets
Current Liabilities
Acid-test ratio – also known as quick ratio and defined as measure of
short-term paying ability.
Acid − Test Ratio =
31.4.3.3
Quick Assets
Current Liabilities
Receivable turn-over – measures the speed of collections of accounts
receivables.
Net Sales on Credit
Average Receivables
Receivable Turnover =
31.4.3.4
Gross margin – gross profit as a percentage of sales.
Gross Margin =
31.4.3.5
Gross Profit
Net Sales
Profit margin ratio – percentage of sales that is net income.
Profit Margin Ratio =
31.4.3.6
Net Income Before Tax
Net Sales
Return on investment ratio – percent return on the investment.
Return on Investment =
Net Income
Owner ' s Equity
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