# BUCKINGHAM PI THEOREM

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1 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

2 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(π 1, π 2, π 3,., π k-r )=0 Or π 1 =G(π 2, π 3, π 4,., π k-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

3 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

4 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

5 Common Dimensionless Groups in Fluid Mechanics 5

6 items Reynolds number Froude number Euler number Cauchy number Mach number Strouhal number Weber number Problems Flow in pipe. Flow of water around ship. Flow through rivers or open conduits. Pressure problems. Pressure difference between two points. Fluid compressibility. Fluid compressibility. Unsteady, oscillating flow. Interface between two fluid. Surface tension problems. 6

7 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

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10 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

11 Theory of Models The theory of models can be readily developed by using the principles of dimensional analysis. Π 1 = φ ( Π2, Π3,..., Π n ) If above equation describes the behavior of a particular prototype, a similar relationship can be written for a model of this prototype, that is, Π = Π Π,..., Π 1m φ ( ) 2m, 3m 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 Π 1 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, Π 2m =Π 2, Π 3m =Π 3,, Π nm =Π n Eq.(1) Then with the presumption that the form of φ is the same for model and prototype, it follows that, Π 1 =Π 1m Eq.(2) nm 2

12 Equation (2) is the desired prediction equation and indicates that the measured value of Π 1m obtained with the model will be equal to the corresponding Π 1 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

13 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, V m V ρ m Density scale, ρ µ m Viscosity scale, µ And so on. 4

14 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

15 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 2 At a sudden contraction in a pipe the diameter changes from D 1 to D 2. The pressure drop, P, which develops across the contraction, is a function of D 1 and D 2, as well as the velocity, V, in the larger pipe, and the fluid density, ρ, and viscosity, µ. Use D 1, 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

16 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 4 Assume that the flowrate, Q, of a gas from a smokestack is a function of the density of ambient air, ρ a, the density of the gas, ρ g, within the stack, the acceleration of gravity, g, and the height and diameter of the stack, h and d, respectively. Use ρ a, d and g as repeating variables to develop a set of pi terms that could be used to describe this problem. 2

17 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=0.07m 3 /s. Based on these limited data, determine a general equation for the flowrate over this type of weir. 3

18 Question 1 TUTORIAL FOR BUCKINGHAM PI THEOREM 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 : P = ρn 2 2 Q D φ ND 3 ρnd, µ 2 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

19 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

20 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=10-1 Pa.s). Jika keadaan ini diulang dalam makmal dengan menggunakan air (µ=10-3 Pa.s) dan paip berdiameter 50mm dari jenis yang serupa, tentukan halaju air yang setara jika minyak mengalir pada halaju 10m/s. 3

21 Question 1 TUTORIAL FOR SIMILARITY 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 1000m 3 /min. Determine the required flowrate for a 1:25 scale model that is operated. 1

22 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

23 Question 4 Figure 1 Water flowing under the obstacle shown in Figure 1 puts a vertical force, F v, 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 30m 3 /s, determine the water flowrate for the model if the flows are to be similar. c) If the model force is measured as (F v ) m =80N, predict the corresponding force on the prototype. 3

24 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 (E m =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

25 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, ρ A (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 ρ A =0.07kg/m 2. 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

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