Dimensional analysis is a mathematical technique used to establish relationships between physical quantities in fluid phenomena. It involves considering the dimensions of quantities and grouping dimensionless parameters to better understand flows. Dimensional analysis provides guidance for experimental work by indicating important influencing factors. It is applied to develop equations, convert between units, reduce experimental variables, and enable model studies through similitude. The key concepts are that theoretical equations must be dimensionally homogeneous and empirical equations have limited applications. Dimensional analysis methods include Rayleigh's method of exponential relationships and Buckingham's Π-method of grouping variables into dimensionless terms.
Dimensional analysis offers a method for reducing complex physical problems to the simplest (that is, most economical) form prior to obtaining a quantitative answer.
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms vs. grams) and tracking these dimensions as calculations or comparisons are performed.
Fluid Mechanics Chapter 5. Dimensional Analysis and SimilitudeAddisu Dagne Zegeye
Introduction, Dimensional homogeneity, Buckingham pi theorem, Non dimensionalization of basic equations, Similitude, Significance of non-dimensional numbers in fluid flows
Dimensional analysis offers a method for reducing complex physical problems to the simplest (that is, most economical) form prior to obtaining a quantitative answer.
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms vs. grams) and tracking these dimensions as calculations or comparisons are performed.
Fluid Mechanics Chapter 5. Dimensional Analysis and SimilitudeAddisu Dagne Zegeye
Introduction, Dimensional homogeneity, Buckingham pi theorem, Non dimensionalization of basic equations, Similitude, Significance of non-dimensional numbers in fluid flows
A fluid is a state of matter in which its molecules move freely and do not bear a constant relationship in space to other molecules.
In physics, fluid flow has all kinds of aspects: steady or unsteady, compressible or incompressible, viscous or non-viscous, and rotational or irrotational to name a few. Some of these characteristics reflect properties of the liquid itself, and others focus on how the fluid is moving.
Fluids are :-
Liquid : blood, i.v. infusions)
Gas : O2 , N2O)
Vapour (transition from liquid to gas) : N2O (under compression in cylinder), volatile inhalational agents (halothane, isoflurane, etc)
Sublimate (transition from solid to gas bypassing liquid state) : Dry ice (solid CO2), iodine
Pipe Flow Friction factor in fluid mechanicsUsman Shah
This slide will explain you the chemical engineering terms .Al about the basics of this slide are explain in it. The basics of fluid mechanics, heat transfer, chemical engineering thermodynamics, fluid motions, newtonian fluids, are explain in this process. ,education ,chemical engineerin ,chemical engineering ,fluid mechanics ,heat transfer ,chemical process principles ,macdonald ,kfc ,mazeo ,chemicals ,engineers ,cv formatin ,law ,laptop.
Dimension less numbers in applied fluid mechanicstirath prajapati
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
B.TECH. DEGREE COURSE
SCHEME AND SYLLABUS
(2002-03 admission onwards)
MAHATMA GANDHI UNIVERSITY,mg university, KTU
KOTTAYAM
KERALA
Module 1
Introduction - Proprties of fluids - pressure, force, density, specific weight, compressibility, capillarity, surface tension, dynamic and kinematic viscosity-Pascal’s law-Newtonian and non-Newtonian fluids-fluid statics-measurement of pressure-variation of pressure-manometry-hydrostatic pressure on plane and curved surfaces-centre of pressure-buoyancy-floation-stability of submerged and floating bodies-metacentric height-period of oscillation.
Module 2
Kinematics of fluid motion-Eulerian and Lagrangian approach-classification and representation of fluid flow- path line, stream line and streak line. Basic hydrodynamics-equation for acceleration-continuity equation-rotational and irrotational flow-velocity potential and stream function-circulation and vorticity-vortex flow-energy variation across stream lines-basic field flow such as uniform flow, spiral flow, source, sink, doublet, vortex pair, flow past a cylinder with a circulation, Magnus effect-Joukowski theorem-coefficient of lift.
Module 3
Euler’s momentum equation-Bernoulli’s equation and its limitations-momentum and energy correction factors-pressure variation across uniform conduit and uniform bend-pressure distribution in irrotational flow and in curved boundaries-flow through orifices and mouthpieces, notches and weirs-time of emptying a tank-application of Bernoulli’s theorem-orifice meter, ventury meter, pitot tube, rotameter.
Module 4
Navier-Stoke’s equation-body force-Hagen-Poiseullie equation-boundary layer flow theory-velocity variation- methods of controlling-applications-diffuser-boundary layer separation –wakes, drag force, coefficient of drag, skin friction, pressure, profile and total drag-stream lined body, bluff body-drag force on a rectangular plate-drag coefficient for flow around a cylinder-lift and drag force on an aerofoil-applications of aerofoil- characteristics-work done-aerofoil flow recorder-polar diagram-simple problems.
Module 5
Flow of a real fluid-effect of viscosity on fluid flow-laminar and turbulent flow-boundary layer thickness-displacement, momentum and energy thickness-flow through pipes-laminar and turbulent flow in pipes-critical Reynolds number-Darcy-Weisback equation-hydraulic radius-Moody;s chart-pipes in series and parallel-siphon losses in pipes-power transmission through pipes-water hammer-equivalent pipe-open channel flow-Chezy’s equation-most economical cross section-hydraulic jump.
A fluid is a state of matter in which its molecules move freely and do not bear a constant relationship in space to other molecules.
In physics, fluid flow has all kinds of aspects: steady or unsteady, compressible or incompressible, viscous or non-viscous, and rotational or irrotational to name a few. Some of these characteristics reflect properties of the liquid itself, and others focus on how the fluid is moving.
Fluids are :-
Liquid : blood, i.v. infusions)
Gas : O2 , N2O)
Vapour (transition from liquid to gas) : N2O (under compression in cylinder), volatile inhalational agents (halothane, isoflurane, etc)
Sublimate (transition from solid to gas bypassing liquid state) : Dry ice (solid CO2), iodine
Pipe Flow Friction factor in fluid mechanicsUsman Shah
This slide will explain you the chemical engineering terms .Al about the basics of this slide are explain in it. The basics of fluid mechanics, heat transfer, chemical engineering thermodynamics, fluid motions, newtonian fluids, are explain in this process. ,education ,chemical engineerin ,chemical engineering ,fluid mechanics ,heat transfer ,chemical process principles ,macdonald ,kfc ,mazeo ,chemicals ,engineers ,cv formatin ,law ,laptop.
Dimension less numbers in applied fluid mechanicstirath prajapati
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
B.TECH. DEGREE COURSE
SCHEME AND SYLLABUS
(2002-03 admission onwards)
MAHATMA GANDHI UNIVERSITY,mg university, KTU
KOTTAYAM
KERALA
Module 1
Introduction - Proprties of fluids - pressure, force, density, specific weight, compressibility, capillarity, surface tension, dynamic and kinematic viscosity-Pascal’s law-Newtonian and non-Newtonian fluids-fluid statics-measurement of pressure-variation of pressure-manometry-hydrostatic pressure on plane and curved surfaces-centre of pressure-buoyancy-floation-stability of submerged and floating bodies-metacentric height-period of oscillation.
Module 2
Kinematics of fluid motion-Eulerian and Lagrangian approach-classification and representation of fluid flow- path line, stream line and streak line. Basic hydrodynamics-equation for acceleration-continuity equation-rotational and irrotational flow-velocity potential and stream function-circulation and vorticity-vortex flow-energy variation across stream lines-basic field flow such as uniform flow, spiral flow, source, sink, doublet, vortex pair, flow past a cylinder with a circulation, Magnus effect-Joukowski theorem-coefficient of lift.
Module 3
Euler’s momentum equation-Bernoulli’s equation and its limitations-momentum and energy correction factors-pressure variation across uniform conduit and uniform bend-pressure distribution in irrotational flow and in curved boundaries-flow through orifices and mouthpieces, notches and weirs-time of emptying a tank-application of Bernoulli’s theorem-orifice meter, ventury meter, pitot tube, rotameter.
Module 4
Navier-Stoke’s equation-body force-Hagen-Poiseullie equation-boundary layer flow theory-velocity variation- methods of controlling-applications-diffuser-boundary layer separation –wakes, drag force, coefficient of drag, skin friction, pressure, profile and total drag-stream lined body, bluff body-drag force on a rectangular plate-drag coefficient for flow around a cylinder-lift and drag force on an aerofoil-applications of aerofoil- characteristics-work done-aerofoil flow recorder-polar diagram-simple problems.
Module 5
Flow of a real fluid-effect of viscosity on fluid flow-laminar and turbulent flow-boundary layer thickness-displacement, momentum and energy thickness-flow through pipes-laminar and turbulent flow in pipes-critical Reynolds number-Darcy-Weisback equation-hydraulic radius-Moody;s chart-pipes in series and parallel-siphon losses in pipes-power transmission through pipes-water hammer-equivalent pipe-open channel flow-Chezy’s equation-most economical cross section-hydraulic jump.
Second application of dimensional analysis
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Dimensional analysis means analysis of the dimensions of physical quantities. Dimensional analysis lowers the number of variables in a fluid phenomenon by mixing the some variables to form parameters which have no dimensions.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Online aptitude test management system project report.pdfKamal Acharya
The purpose of on-line aptitude test system is to take online test in an efficient manner and no time wasting for checking the paper. The main objective of on-line aptitude test system is to efficiently evaluate the candidate thoroughly through a fully automated system that not only saves lot of time but also gives fast results. For students they give papers according to their convenience and time and there is no need of using extra thing like paper, pen etc. This can be used in educational institutions as well as in corporate world. Can be used anywhere any time as it is a web based application (user Location doesn’t matter). No restriction that examiner has to be present when the candidate takes the test.
Every time when lecturers/professors need to conduct examinations they have to sit down think about the questions and then create a whole new set of questions for each and every exam. In some cases the professor may want to give an open book online exam that is the student can take the exam any time anywhere, but the student might have to answer the questions in a limited time period. The professor may want to change the sequence of questions for every student. The problem that a student has is whenever a date for the exam is declared the student has to take it and there is no way he can take it at some other time. This project will create an interface for the examiner to create and store questions in a repository. It will also create an interface for the student to take examinations at his convenience and the questions and/or exams may be timed. Thereby creating an application which can be used by examiners and examinee’s simultaneously.
Examination System is very useful for Teachers/Professors. As in the teaching profession, you are responsible for writing question papers. In the conventional method, you write the question paper on paper, keep question papers separate from answers and all this information you have to keep in a locker to avoid unauthorized access. Using the Examination System you can create a question paper and everything will be written to a single exam file in encrypted format. You can set the General and Administrator password to avoid unauthorized access to your question paper. Every time you start the examination, the program shuffles all the questions and selects them randomly from the database, which reduces the chances of memorizing the questions.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
2. DIMENSIONAL ANALYSIS
It is a pure mathematical technique to establish a
relationship between physical quantities involved in a fluid
phenomenon by considering their dimensions.
In dimensional analysis, from a general understanding of
fluid phenomena, we first predict the physical parameters that
will influence the flow, and then we group these parameters
into dimensionless combinations which enable a better
understanding of the flow phenomena. Dimensional analysis is
particularly helpful in experimental work because it provides a
guide to those things that significantly influence the
phenomena; thus it indicates the direction in which
experimental work should go.
3. DIMENSIONAL ANALYSIS
Dimensional Analysis refers to the physical
nature of the quantity (Dimension) and the
type of unit used to specify it.
Distance has dimension L.
Area has dimension L2.
Volume has dimension L3.
Time has dimension T.
Speed has dimension L/T
4. APPLICATION OF DIMENSIONAL ANALYSIS
Development of an equation for fluid phenomenon
Conversion of one system of units to another
Reducing the number of variables required in an
experimental program
Develop principles of hydraulic similitude for model study
5. DIMENSIONAL REASONING &
HOMOGENEITY
Principle of Dimensional Homogeneity
The fundamental dimensions and their respective powers
should be identical on either side of the sign of equality.
Dimensional reasoning is predicated on the proposition that,
for an equation to be true, then both sides of the equation
must be numerically and dimensionally identical.
To take a simple example, the expression x + y = z when x =1,
y =2 and z =3 is clearly numerically true but only if the
dimensions of x, y and z are identical. Thus
1 elephant +2 aero planes =3 days is clearly nonsense but
1 meter +2 meter =3 meter is wholly accurate.
An equation is only dimensionally homogeneous if all the
terms have the same dimensions.
6. FUNDAMENTAL DIMENSIONS
We may express physical quantities in either mass-length-
time (MLT) system or force-length-time (FLT) system.
This is because these two systems are interrelated through
Newton’s second law, which states that force equals mass
times acceleration,
F = ma 2nd Law of motion
F = ML/T2
F = MLT-2
Through this relation, we can covert from one system to the
other. Other than convenience, it makes no difference which
system we use, since the results are the same.
7.
8. DIMENSIONS OF SOME COMMON PHYSICAL
QUANTITIES
[x], Length – L
[m], Mass – M
[t], Time – T
[v], Velocity – LT-1
[a], Acceleration – LT-2
[F], Force – MLT-2
[Q], Discharge – L3T-1
[r], Mass Density – ML-3
[P], Pressure – ML-1T-2
[E], Energy – ML2T-2
9. BASIC CONCEPTS
All theoretical equations that relate physical quantities must be
dimensionally homogeneous. That is, all the terms in an equation
must have the same dimensions. For example
Q = A.V (homogeneous)
L3T-1 = L3T-1
We do, however sometimes use no homogeneous equation, the
best known example in fluid mechanics being the Manning
equation.
Manning's equation is an empirical equation. Generally the use of
such equations is limited to specialized areas.
10. To illustrate the basic principles of dimensional analysis, let us
explore the equation for the speed V with which a pressure wave
travels through a fluid. We must visualize the physical problem to
consider physical factors probably influence the speed. Certainly
the compressibility Ev must be factor; also the density and the
kinematic viscosity of the fluid might be factors. The dimensions of
these quantities, written in square brackets are
V=[LT-1], Ev=[FL-2]=[ML-1T-2], r=[ML-3], ν=[L2T-1]
Here we converted the dimensions of Ev into the MLT system using
F=[MLT-2]. Clearly, adding or subtracting such quantities will not
produce dimensionally homogenous equations. We must therefore
multiply them in such a way that their dimensions balance. So let us
write
V=C Ev
a rb νd
Where C is a dimensionless constant, and let solve for the
exponents a, b, and d substituting the dimensions, we get
11. (LT-1)=(ML-1T-2)a (ML-3)b (L2T-1)d
To satisfy dimensional homogeneity, the exponents of each
dimension must be identical on both sides of this equation. Thus
For M: 0 = a + b
For L: 1 = -a -3b +2d
For T: -1 = -2a – d
Solving these three equations, we get
a=1/2, b=-1/2, d=0
So that V = C √(Ev/r)
This identifies basic form of the relationship, and it also determines
that the wave speed is not effected by the fluid’s kinematic
viscosity, ν.
Dimensional analysis along such lines was developed by Lord
Rayleigh.
13. RAYLEIGH’S METHOD
Functional relationship between variables is
expressed in the form of an exponential relation
which must be dimensionally homogeneous
if “y” is a function of independent variables
x1,x2,x3,…..xn , then
In exponential form as
),.......,,( 321 nxxxxfy
]),.......()(,)(,)[( 321
z
n
cba
xxxxy
14. Procedure :
Write MLT forms of the given data
Write the same equation in exponential form
Select suitable system of fundamental dimensions
Substitute dimensions of the physical quantities
Apply dimensional homogeneity
Equate the powers and compute the values of the
exponents
Substitute the values of exponents
Simplify the expression
Ideal up to three independent variables, can be used for
four.
RAYLEIGH’S METHOD
15. BUCKINGHAM’S ∏ METHOD
A more generalized method of dimension analysis developed by E.
Buckingham and others and is most popular now. This arranges the
variables into a lesser number of dimensionless groups of variables.
Because Buckingham used ∏ (pi) to represent the product of variables
in each groups, we call this method Buckingham pi theorem.
“If ‘n’ is the total number of variables in a dimensionally homogenous
equation containing ‘m’ fundamental dimensions, then they may be
grouped into (n-m) ∏ terms.
f(X1, X2, ……Xn) = 0
then the functional relationship will be written as
Ф (∏1 , ∏2 ,………….∏n-m) = 0
The final equation obtained is in the form of:
∏1= f (∏2,∏3,………….∏n-m) = 0
Suitable where n ≥ 4
Not applicable if (n-m) = 0
16. Procedure :
List all physical variables and note ‘n’ and ‘m’.
n = Total no. of variables
m = No. of fundamental dimensions (That is, [M], [L], [T])
Compute number of ∏-terms by (n-m)
Write the equation in functional form
Write equation in general form
Select repeating variables. Must have all of the ‘m’
fundamental dimensions and should not form a ∏ among
themselves
Solve each ∏-term for the unknown exponents by
dimensional homogeneity.
BUCKINGHAM’S ∏ METHOD