Chapter 9 – Buckingham Pi Theorem

BUCKINGHAM PI THEOREM

Dimensional Analysis

It is used to determine the equation is right or wrong.

The calculation is depends on the unit or dimensional

conditions of the equations.

For example;

F=ma F=MLT-2

Unit : F=kg.m/s

1

Chapter 9 – Buckingham Pi Theorem

Buckingham Pi Theorem

If an equation involving k variables is dimensionally

homogeneous, it can be reduced to a relationship

among k – r independent dimensionless products,

where r is the minimum number of reference

dimensions required to describe the variables.

For example, the function of G can be written as ;

G(π1, π2, π3, ……., πk-r)=0

Or

π1=G(π2, π3, π4, ……., πk-r)=0

The dimensionless products are frequently referred to

as “pi terms”, and the theorem is called the

Buckingham Pi Theorem.

Buckingham used the symbol П to represent a

dimensionless product, and this notation is commonly

used.

2

Chapter 9 – Buckingham Pi Theorem

To summarize, the steps to be followed in performing

a dimensional analysis using the method of repeating

variables are as follows :

Step 1 List all the variables that are involved in the

problem.

Step2 Express each of the variables in terms of

basic dimensions.

Step 3 Determine the required number of pi terms.

Step 4 Select a number of repeating variables,

where the number required is equal to the

number of reference dimensions. (usually the

same as the number of basic dimensions)

Step 5 Form the pi term by multiplying one of the

nonrepeating variables by the product of

repeating variables each raised to an

exponent that will make the combination

dimensionless.

Step 6 Repeat step 5 for each of the remaining

nonrepeating variables.

Step 7 Check all the resulting pi terms to make sure

they are dimensionless.

Step 8 Express the final form as a relationship

among the pi terms and think about what it

means.

3

Chapter 9 – Buckingham Pi Theorem

Selection of Variables

One of the most important, and difficult, steps in

applying dimensional analysis to any given problem

is the selection of the variables that are involved.

For most engineering problems (including areas

outside fluid mechanics), pertinent variables can be

classified into three groups – geometry, material

properties and external effects.

Geometry :

The geometry characteristics can be usually be

described by a series of lengths and angles.

Example: length [L]

Material properties :

More relates to the kinematic properties of fluid

particles.

Example: velocity [LT-1

]

External effects :

This terminology is used to denote any variable that

produces, or tends to produce, a change in the system.

For fluid mechanics, variables in this class would be

related to pressure, velocities, or gravity.

(combination of geometry and material properties)

Example: force [MLT-2

]

4

Chapter 9 – Buckingham Pi Theorem

Common Dimensionless Groups in Fluid Mechanics

5

Chapter 9 – Buckingham Pi Theorem

items Problems

Reynolds number Flow in pipe.

Froude number Flow of water around ship.

Flow through rivers or open

conduits.

Euler number Pressure problems.

Pressure difference between two

points.

Cauchy number Fluid compressibility.

Mach number Fluid compressibility.

Strouhal number Unsteady, oscillating flow.

Weber number Interface between two fluid.

Surface tension problems.

6

Chapter 9 – Buckingham Pi Theorem

Modeling and Similitude

Models are widely used in fluid mechanics.

Major engineering projects involving structures,

aircraft, ships, rivers, harbor, and so on, frequently

involve the used of models.

A model (engineering model) is a representation of a

physical system that may be used to predict the

behavior of the system in some desired respect.

The physical system for which the predictions are to

be made is called the prototype.

Usually a model is smaller than the prototype.

Therefore, it is more easily handled in the laboratory

and less expensive to construct and operate than a

large prototype.

However, if the prototype is very small, it may be

advantageous to have a model that is larger than a

prototype so that it can be more easily studied.

7

Chapter 9 – Buckingham Pi Theorem

8

Chapter 9 – Buckingham Pi Theorem

9

Chapter 9 – Buckingham Pi Theorem

MODELING AND SIMILITUDE

Models are widely used in fluid mechanics. Major engineering projects involving structures, aircrafts, ships, rivers, harbor, and so on, frequently involve the use of models. A model is a representation of a physical system that may be used to predict the behavior of the system in some desire respect. The physical system for which the predictions are to be made is called the prototype.

1

Chapter 9 – Buckingham Pi Theorem

Theory of Models The theory of models can be readily developed by using the principles of dimensional analysis.

( )nΠΠΠ=Π ,......,, 321 φ If above equation describes the behavior of a particular prototype, a similar relationship can be written for a model of this prototype, that is,

( )nmmmm ΠΠΠ=Π ,......,, 321 φ Pi terms, without a subscript will refer to the prototype. The subscript m will be used to designate the model variables or pi terms. The pi terms can be developed so that Π1 contains the variable that is to be predicted from observations made on the model. Therefore, if the model is designed and operated under the following conditions, Π2m=Π2 , Π3m=Π3 , …, Πnm=Πn …… Eq.(1)

Then with the presumption that the form of φ is the same for model and prototype, it follows that,

Π1=Π1m …… Eq.(2)

2

Chapter 9 – Buckingham Pi Theorem

Equation (2) is the desired prediction equation and indicates that the measured value of Π1m obtained with the model will be equal to the corresponding Π1 for the prototype as long as the other pi terms are equal. The conditions specified by equation (1) provide the model design conditions, also called similarity requirements or modeling laws.

3

Chapter 9 – Buckingham Pi Theorem

Model scales We will take the ratio of the model value to the prototype value as the scale. Length scales are often specified. For example, as 1:10 or as a 10

1 scale model. The meaning of this specification is that the model is one-tenth the size of the prototype, and the tacit assumption is that all relevant lengths are scaled accordingly, so the model is geometrically similar to the prototype. There are, however, other scales such as the ;

Velocity scale, VVm

Density scale, ρρm

Viscosity scale, µµm

And so on.

4

Chapter 9 – Buckingham Pi Theorem

Models for which one or more similarity requirements are not satisfied are called distorted models. Models for which all similarity requirements are met are called true models.

5

Chapter 9 – Buckingham Pi Theorem

TUTORIAL FOR BUCKINGHAM PI THEOREM

EXAMPLE 1 Verify the Reynolds number is dimensionless, using both the FLT system and MLT system for basic dimensions. Determine its value for ethyl-alcohol flowing at a velocity of 3m/s through a 5cm diameter pipe. EXAMPLE 2At a sudden contraction in a pipe the diameter changes from D1 to D2. The pressure drop, ∆P, which develops across the contraction, is a function of D1 and D2, as well as the velocity, V, in the larger pipe, and the fluid density, ρ, and viscosity, µ. Use D1, V and µ as repeating variables to determine a suitable set of dimensionless parameters. Why it be incorrect to include the velocity in the smaller pipe as an additional variables.

1

Chapter 9 – Buckingham Pi Theorem

EXAMPLE 3

Figure 1

Water sloshes back and forth in a tank as shown in Figure 1. The frequency of sloshing, ω, is assumed to be a function of the acceleration of gravity, g, the average depth of the water, h, and the length of the tank, l. Develop a suitable set of dimensionless parameters for this problem using g and l as repeating variables. EXAMPLE 4Assume that the flowrate, Q, of a gas from a smokestack is a function of the density of ambient air, ρa, the density of the gas, ρg, within the stack, the acceleration of gravity, g, and the height and diameter of the stack, h and d, respectively. Use ρa, d and g as repeating variables to develop a set of pi terms that could be used to describe this problem.

2

Chapter 9 – Buckingham Pi Theorem

EXAMPLE 5

Figure 2

a) The water flowrate, Q, in an open rectangular

channel can be measured by placing a plate across the channel as shown in Figure 2. This type of a device is called a weir. The height of the water, H, above the weir crest is referred to as the head and can be used to determine the flowrate through the channel. Assume that Q is a function of the head, H, the channel width, b, and the acceleration of gravity, g. Determine a suitable set of dimensionless variables for this problem.

b) In some laboratory tests, it was determined that if b=0.9m and H=10cm, then Q=0.07m3/s. Based on these limited data, determine a general equation for the flowrate over this type of weir.

3

Chapter 9 – Buckingham Pi Theorem

TUTORIAL FOR BUCKINGHAM PI THEOREM

Question 1 a) The pressure rise, ∆P, generated by a pump

depends on the impeller diameter, D, its rotational speed, N, the fluid density, ρ and viscosity, µ and the rate of discharge, Q. Show that the relationship between these variables may be expressed as :

⎥⎦

⎤⎢⎣

⎡=∆

µρφρ

2

322 , ND

NDQDNP

b) A given pump rotates at a speed of 1000rev/min,

and its duty point it generates a head of 12m when pumping water at a rate of 15 liter per second. Calculate the head generated by a similar pump, twice the size, when operating under dynamically similar conditions and discharging 45 liter per second. The influence of Reynolds number is negligible.

1

Chapter 9 – Buckingham Pi Theorem

Question 2 Kenaikan kapilari, h, untuk suatu cecair dalam tiub berubah menurut diameter tiub, d, pecutan gravity, g, ketumpatan bendalir, ρ, ketegangan permukaan, σ dan sudut sentuh, θ. a) Dengan menggunakan kaedah Teorem Buckingham

Pi, tentukan kumpulan tanpa dimensi Pi yang menghubungkan kesemua parameter yang disebutkan.

b) Dalam ujikaji pertama, kenaikan kapilari ialah

h=3cm. Dalam ujikaji yang lain, diameter tiub dan ketegangan permukaan bendalir adalah separuh daripada ujikaji pertama sementara ketumpatan bendalir pula adalah dua kali ganda. Sudut sentuh untuk kedua-dua ujikaji ini adalah sama. Tentukan nilai h untuk ujikaji kedua.

2

Chapter 9 – Buckingham Pi Theorem

Question 3 a) Kesusutan tekanan, ∆P, untuk aliran likat mantap

dan tidak boleh mampat dalam paip lurus mengufuk dipengaruhi oleh panjang paip, l, halaju purata, U, kelikatan, µ, diameter paip, D, ketumpatan bendalir, ρ dan kekasaran dalaman, ε. Dengan menggunakan kaedah Buckingham Pi, tentukan kumpulan-kumpulan tidak berdimensi yang menghubungkan parameter-parameter ini.

b) Sebatang paip berdiameter 40cm mengalirkan

minyak (s=0.86, µ=10-1Pa.s). Jika keadaan ini diulang dalam makmal dengan menggunakan air (µ=10-3Pa.s) dan paip berdiameter 50mm dari jenis yang serupa, tentukan halaju air yang setara jika minyak mengalir pada halaju 10m/s.

3

Chapter 9 – Buckingham Pi Theorem

TUTORIAL FOR SIMILARITY Question 1 Carbon tetrachloride flows with a velocity of 0.30m/s through a 30mm diameter tube. A model of this system is to be developed using standard air tube as the model fluid. The air velocity is to be 2m/s. What tube diameter is required for the model if dynamic similarity is to be maintained between model and prototype? Question 2 The flowrate over the spillway of a dam is 1000m3/min. Determine the required flowrate for a 1:25 scale model that is operated.

1

Chapter 9 – Buckingham Pi Theorem

Question 3 For a certain fluid flow problem it is known that both the Froude number and Weber number are important dimensionless parameters. If the problem is to be studied by using a 1:15 scale model, determine the required surface tension scale if the density scale is equal to 1. The model and prototype operate in the same gravitational field.

2

Chapter 9 – Buckingham Pi Theorem

Question 4

Figure 1

Water flowing under the obstacle shown in Figure 1 puts a vertical force, Fv, on the obstacle. This force is assumed to be a function of the flowrate, Q, the density of water, ρ, the acceleration of gravity, g, and a length l, that characterized the size of obstacle. A 1/20 scale model is to be used to predict the vertical force on the prototype. a) Perform a dimensional analysis for this problem. b) If the prototype flowrate is 30m3/s, determine the

water flowrate for the model if the flows are to be similar.

c) If the model force is measured as (Fv)m=80N, predict the corresponding force on the prototype.

3

Chapter 9 – Buckingham Pi Theorem

Question 5

Figure 2

The pressure rise, ∆P, across a blast wave, as shown in Figure 2 is assumed to be a function of the amount of energy released in the explosion, E, the air density, ρ, the speed of sound, c, and the distance from the blast, d. a) Put this relationship in dimensionless form. b) Consider two blast; the prototype blast with energy

release E and a model blast with 1/1000 the energy release (Em=0.001E). At what distance from the model blast will the pressure rise be the same as that at a distance of 2 km from the prototype blast?

4

Chapter 9 – Buckingham Pi Theorem

Question 6 Wind blowing past a flag causes it to flutter in the breeze. The frequency of this fluttering, ω, is assumed to be a function of the wind speed, V, the air density, ρ, the acceleration of gravity, g, the length of the flag, L, and the area density, ρA (with dimensions of ML-2) of the flag material. It is desired to predict the flutter frequency of a large L=12m flag in a V=9m/s wind. To do this a model flag with L=1.5m is to be tested in a wind tunnel. a) Determine the required area density of the model

flag material if the large flag has a ρA=0.07kg/m2. b) What wind tunnel velocity is required for testing

the model? c) If the model flag flutters at 6Hz, predict the

frequency for the large flag.

5