TALENTO HUMANO

Any physical variable can be described by certain familiar properties e.g. length, velocity, area, volume, acceleration etc. These are all known as dimensions. Of course value and standardized unit has to be attached to make dimensions useful.

Dimensions are measurable physical quantities without numerical values while units are the standard elements we use to quantify these dimensions. For example, the dimension, length is a

measurable quantity which has units such as microns (µm), centimeters (cm), meters (m), kilometers (km), feet (ft), etc – see Figure 5.1.

Figure 5.1: Illustration of the dimension, unit and magnitude (value) using a ruler

In fluid mechanics, the four primary quantities (also known as fundamental or basic quantities) are usually taken to be mass M, length L, time T, and temperature or an MLTθ system for short. Sometimes one uses an FLTθ system, with force, Freplacing mass. The quantities which are expressed in terms of primary quantities are called derived or secondary quantities (e.g., area, velocity, acceleration, force etc.).

5.3 MODEL AND PROTOTYPE

There are many times when for economic or other reasons (as for precision), it is desirable to determine the performance of a structure or machine by testing another structure or machine.

This type of testing is called MODEL TESTING. The structure or machine being tested is called the MODEL and the structure whose performance is to be pre-determined or predicted is called the PROTOTYPE. A model may be smaller than, the same size or larger than the prototype.

Model experiments of airplanes, rockets, missiles, ships, turbines, pumps and other structures and machines have resulted in savings that more than justified the expenditure of fund for the design, construction and testing of the model.

There are three types of models widely used in engineering:

 Mathematical (Analytically) models = equation

 Analogue (Symbolic) models = computer words (logic)

 Physical (Iconic or Image or Portrait) models

In order to design a model, it is essential to anticipate the forces which predominate in the prototype and ensure that the relevant dimensionless parameters are the same in the model and prototype

5.4DIMENSIONAL ANALYSIS

Dimensions used in mechanics mass M, length L, time T and force F. Corresponding units for these dimensions are kilogramme, metres, second and Newton. Any system in mechanics can be defined by three fundamental dimensions. Two systems are used: the force (FLT) and the mass (MLT). In the force system, mass is a derived quantity and in the mass system, force is a derived quantity.

For example, by Newton’s second law;

Force (F) = Mass (M) × Acceleration (g)

2 2

2

 





 MLT

T ML Time

Length Mass

F

Force and mass are related by Newton’s law viz

F = MLT ^-2 (5.1) M = FL^-1T ² (5.2)

A table of common variables and their units and dimensions are provided in Table 5.1 below Table 5.1 List of dimensions ofsome common physical quantities:

5.5 DIMENSIONALHOMOGENEITY

An equation is a relationship between two or more physical quantities. Hence, any correct equation expressing a physical relationship between quantities must be dimensionally homogeneous and numerically equivalent. The law of dimensional homogeneity states

that every additive term in an equation when reduced to fundamental dimensions must contain identical powers of each of the dimensions. In essence, only quantities having the same dimensions can be added, subtracted or equated in a dimensionally homogeneous equation. Consider, for example, the equation of gauge pressure:

wh gh P_guage  

Dimensions of L.H.S of the equation = ML^1T^2

Dimensions of R.H.S of the equation = ML^2T^2LML^1T^2 The law of the dimensional homogeneity is conserved, thus:

Dimensions of L.H.S of the equation = Dimensions of R.H.S of the equation Equation P_guage ghwhis dimensionally homogeneous.

Dimensional homogeneitycan beuseful for:

1. Checkingunits of equations;

2. Convertingbetween two sets ofunits;

3. Definingdimensionless relationships (seebelow).

WORKED EXAMPLE 5-1

The period of oscillation of a simple pendulum is a function of the length of the string and the acceleration due to gravity. Use the principle of dimensional homogeneity to set up a relationship combining the variables.

Solution:

Let the period of oscillation be “T”, the length of the string be “l” and the acceleration due to gravity be “g”, then

T = ∅ (𝑙, 𝑔)

T is the principal or dependent variable while l and g are independent variables, T = c l^a g^b = c L^a (LT^-2)^b

T – c L^a+b T^-2b

Comparing the powers of the fundamental dimensions, then For T: 1 = -2b => b = -½

For L: 0 = a + b => a = -b = - (-½) = ½

Substituting a and b in formulation equation above, then T = c l^½ g^-½ = = 𝑐^𝑙½

𝑔^½

𝑇 = 𝑐√𝑙 𝑔

WORKED EXAMPLE 5-2

The speed of propagation of sound through a fluid medium is a function of the density of the fluid and the Bulk Modulus of the fluid. Use the principle of dimensional homogeneity to set up a relational equation combining the variables.

Solution:

Let the speed of propagation of sound be “c”, the density of the fluid be “ρ” and the Bulk Modulus be “E”, then

C = φ (ρ, E)

C is the principal or dependent variable while ρ and E are independent variables, C = k ρ^a E^b

LT^-1 = k (ML^-3)^a (ML^-1 T^-2)^b Simplifying, then

LT^-1 = k M^a+b L^-3a-b T^-2b

Comparing the powers of the fundamental dimensions, then For M: 0 = a + b => a = -b

For L: 1 = -3a – b => 1 = 3b – b => 2b = 1 => b =½ and a= -½ For T: -1 = -2b => b = ½

Substituting a and b in formulation equation above, then C = k ρ^-½ E^½ = 𝑘 ^𝐸½

𝜌^½

𝑐 = 𝑘√ 𝐸 𝜌

5.6 METHODS OF DIMENSIONALANALYSIS

In the worked examples solved so far, there were three equations and three fewer unknowns so that equations were solved in each case explicitly. But where there are more unknowns than there are equations, the problems cannot be solved explicitly or completely by the earlier method. Resorting to some special methods would be the likely probable solution. These methods of dimensional analysis allow the development of an equation of a physical quantity in terms of non-dimension parameters and thus reducing the numbers of variables. The methods that can be used to perform dimensional analysis are:

1. Rayleigh’s method, 2. Buckingham’s method, 3. Bridgman’s method,

4. By visual inspection of the variables involved, and 5. Rearrangement of differential equations.

The popular of these are the Rayleigh’s and Buckingham’s methods and will be dealt with.

5.6.1 Rayleigh’s Method

Lord Rayleigh (1842 - 1919) who was born in Essex, England popularized the principle of dynamic similarity by introducing in 1899 a generalization of the principle.

A special form of relationship among the dimensionless groups is given by this method.

However, is inherent weakness is that it does not provide any information regarding the number of dimensionless groups to be obtained as a result of dimensional analysis. This method is no more in use based on its weakness, thus it is obsolete. However, for knowledge sake it can still be learn.

Rayleigh’s method is use to determine the expression for a variable that depends upon maximum of three or four variables only. It will difficult to fine the expression for the dependent variable if the independent variables become more than four. In this method, a functional relationship of some variables is expressed in the form of an exponential equation which must be dimensionally homogeneous. Therefore, if X is a variable which depends on X1, X2, X3, ...Xn; the functional equation can be expressed as:

X = f(X1, X2, X3, ...Xn) (5.3)

In equation (5.1), X is a dependent variable while X1, X2, X3, ...Xn are independent variables. A dependent variable is the one about which information is required while independent variables are those which govern the change of dependent variable.

Equation (5.1) can also be expressed as:

X = C(X1a, X2b, X3c, ... Xnn) (5.4)

where, C is a constant and a, b, c, ...n are the arbitrary powers. The values of a, b, c, ...n are obtained by comparing the powers of the fundamental dimensions on both sides. Therefore, the expression of the dependent variable is obtained.

WORKED EXAMPLE 5-3

Find an expression for the drag force on smooth sphere of diameter D, moving with a uniform velocity V in a fluid density ρ and dynamic viscosity μ..

Solution:

The drag force F, is a function of

(i) Diameter D, (iii) Velocity V

(ii) Fluid density ρ (iv) Dynamic viscosity μ Mathematically, F = φ (D, V, ρ, μ)

= k D^a, V^b, ρ^c, μ^d Where k is a non-dimensionless constant

5.6.2 Buckingham’s π Method/Theorem

The simplest and most popular of the several methods developed to perform dimensional analysis is the method of repeating variables, popularized by Edgar Buckingham (1867–1940).

The method named after Buckingham and often called Buckingham π-method was first published by the Russian scientist Dimitri Riabouchinsky (1882–1962) in 1911. Buckingham theorem is an improvement over Rayleigh’s method in the sense that it gives information, in advance of the analysis, as to the number of non-dimensional group to be expected.

Buckingham used the Greek letter π (Pi) to denote the dimensionless group. The concise six steps to be followed in Buckingham’s theorem are as follows:

Step 1: List all the variables in the problem and count their total number n.

Step 2: List the primary quantities of each of the n parameters.

Step 3: Set the reduction m as the number of primary dimensions. Calculate k, the expected number of II’s,

k = n – m

Step 4: Choose m repeating parameters.

Step 5: Construct the k π’s, and manipulate as necessary.

Step 6: Write the final functional relationship and check your algebra.

Buckingham π-theorem states that:

If n is the total number of variables (dependent and independent variables) in a

dimensionally homogeneous equation and m is the number of fundamental dimensions

(such as L, T, etc.) in the total variables, then the variables are arranged into n-m non-dimensional terms. These non-non-dimensional (dimensionless) terms are called the π-terms.

Mathematically, if a variable X1, is dependent on independent variable,X2, X3, X4... Xn then the functional equation can be express as:

X1 = f( X2, X3, X4... Xn) (5.5)

Of which Eqn. (5.3) can be expressed as:

f1 (X1, X2, X3, X4... Xn) = 0 (5.8) Eqn. (5.4) is dimensionally homogeneous and has a total number of n variables. If there are m fundamental quantities, then according to Buckingham π-theorem, Eqn. (5.4) can be written in terms of number of π-terms (non-dimensional terms) in which number of π-terms is equal to (n-m). Hence Eqn. (5.4) becomes:

f1 (π1, π2, π3, ...πn) = 0 (5.9)

Each dimensionless π-term is formed by combining m variables out of the total n variables with one of the remaining (n-m) variables i.e. each π-terms contains (m+1) variables. Then m

variables which appear repeatedly in each of π-terms are consequently called repeating variables and are chosen from among the variables such that they together involve all the fundamental quantities and they themselves do not form a dimensionless parameter. Let in the case X1, X2, X3 and X4 are the repeating variables if the fundamental quantities m = 3. The norms in fluids is to take m = 3 (corresponding to M, L, T). Then π-each term is express as:

Where, a1, b1 c1; a2, b2, c2 etc. are the constants that are determined by considering dimensional homogeneity. These values are substituted in Eqn. (5.6) and values of π1, π2, π3 ...πn-m are obtained. Afterwards, the values of π’s are substituted in Eqn. (5.5). The final general equation for the phenomenon can then be obtained by writing anyone of the π-terms as a function of the other as:

WORKED EXAMPLE 5-4

The resistance R experienced by a partially submerged body depends upon the velocity V, length of the body l, viscosity of the fluid μ, density of the fluid ρ, and gravitational acceleration g. Obtain a dimensionless expression for R.

Solution:

5.7 LIMITATIONS OF DIMENSIONAL ANALYSIS

5.8 COMMON DIMENSIONLESS PARAMETERS

Some common dimensionless numbers or parameters in fluid flow are due to viscosity, gravity, pressure, surface tension and compressibility.

Ri = Inertia force = 𝑚𝑎 = ^{𝜌𝐿3𝑣3}

𝐿 = 𝜌𝐿²𝑣² (5.10) Rv = Viscous force = 𝜏 𝐴 = 𝜇𝐿^{2 𝑑𝑣}

𝑑𝑦 (5.11) Rg = Gravity force = 𝑚𝑔 = 𝜌𝐿³𝑔 = (𝑔)𝜌𝐿³ (5.12) Rp = Pressure force = (∆𝑃)𝐴 = (∆𝑃)𝐿² (5.13) Rs = Surface tension = σL = (σ)L (5.14) Rc = Compressibility force = kL² = (k)L² (5.15) Note that the quantities in brackets are retained in order that the type of force may be recognize. These forces may be express in a semi-dimensional way in terms of length l, velocity v, density ρ, viscosity μ, gravity g, pressure change ΔP, surface tension σ and compressibility k rather than simply a force F.

Rm = φ1 (ρ, v, g) = characteristics of system = φ2 (ρ, μ, σ, k) = properties of fluid

= φ3 (d, L, M, T) = linear dimension of system (5.16) Then Rm will be a function of the following groups

Rm = φ (Ri, Rv, Rg, Rp, Rs, Rc, ……..) (5.17)

Inertia forces are usually important in any fluid in motion and the ratio of inertia force to each of the other is then expressed as ^𝑅𝑖

𝑅_𝑚

= 𝑓 {

^𝑅𝑖

𝑅_𝑣

,

^𝑅𝑖

𝑅_𝑔

,

^𝑅𝑖

𝑅_𝑝

,

^𝑅𝑖

𝑅_𝑠

,

𝑅_𝑖

𝑅_𝑐

… … … … } (5.18)

Each quotient in the bracket is assigned a number called the DIMENSIONLESS NUMBER.

These are

𝑅

_𝑖

𝑅

_𝑣

= 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒

𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒 = 𝜌𝐿

²

𝑣

²

𝜇𝑣𝐿 = 𝜌𝑣𝐿 𝜇

= 𝑅𝑒 (𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠 𝑛𝑢𝑚𝑏𝑒𝑟) (5.19)

Osborne Reynolds is a British Engineer (1842 - 1912).

𝑅

_𝑖

𝑅

_𝑔

= 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒

𝐺𝑟𝑎𝑣𝑖𝑡𝑦 𝑓𝑜𝑟𝑐𝑒 = 𝜌𝐿

²

𝑣

²

𝜌𝐿

³

𝑔 = 𝑣

²

𝐿𝑔 = 𝑣

√𝐿𝑔

= 𝐹𝑟 (𝐹𝑟𝑜𝑢𝑑𝑒 𝑛𝑢𝑚𝑏𝑒𝑟) (5.20)

Froude is a Naval architect (1810 - 1879)

𝑅

_𝑖

𝑅

_𝑝

= 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒

𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑓𝑜𝑟𝑐𝑒 = 𝜌𝐿

²

𝑣

²

𝛥𝑃𝐿

²

= 𝜌𝑣

²

∆𝑃 = ∆𝑃

½𝜌𝑣

²

= 𝐸𝑢(𝐸𝑢𝑙𝑒𝑟 𝑛𝑢𝑚𝑏𝑒𝑟) (5.21)

The Euler number is also known as the coefficient of pressure.

𝑅_𝑖

𝑅_𝑠 = 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒

𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑡𝑒𝑛𝑠𝑖𝑜𝑛= 𝜌𝐿²𝑣²

𝜎𝐿 =𝜌𝐿𝑣²

𝜎 = 𝑣²

𝜎⁄𝜌𝐿= 𝑣

√𝜎 𝜌𝐿⁄

= 𝑊𝑒 (𝑊𝑒𝑏𝑒𝑟 𝑛𝑢𝑚𝑏𝑒𝑟) (5.22)

Moritz Weber is a Professor of Naval Mechanics at the Polytechnic Institute of Bailey (1871 - 1951).

𝑅

_𝑖

𝑅

_𝑐

= 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒

𝐸𝑙𝑎𝑠𝑡𝑖𝑐 𝑓𝑜𝑟𝑐𝑒 = 𝜌𝐿

²

𝑣

²

𝐸𝐿

²

= 𝜌𝑣

²

𝐸 = 𝑣

²

𝐸 ⁄ 𝜌 = 𝑣

√𝐸 𝜌 ⁄

= 𝑣 𝑐

= 𝑀 (𝑀𝑎𝑐ℎ 𝑛𝑢𝑚𝑏𝑒𝑟) (5.23)

Ernst Mach is an Austrian Physicist and Philosopher (1836 - 1916).

The ratio ^𝑣

2

𝑘⁄𝜌

= 𝑐

is called Cauchy number

Baron Augustin Louis de Cauchy (1789 - 1857) is a French Engineer who later turned to a Mathematician.

𝑅

_𝑖

𝑅

_𝑚

= ∅ (𝑅𝑒, 𝐹𝑟, 𝐸𝑢, 𝑊𝑒, 𝑀)

These numbers are of significant importance in fluid flow system. The importance of Reynolds’ number will be discussed in the next chapter.

However, if the gravitational force influences the pattern of flow over the motion of a ship in rough sea with the result that waves can form about the craft, then Froude number is the most significant parameter.

The Euler number (pressure coefficient) helps us to know the character of the flow at a particular point in a flow field particularly in a flow where the accelerating force is provided by

a change in the intensity of fluid pressure between points in flow field.

The Weber number is an important parameter in liquid atomization. The surface tension of the liquid at the surface of a droplet is responsible for maintaining the shape of the droplet. If a droplet is subjected to an air jet and there is a relative velocity between the droplet and gas, inertia forces due to the relative cause the droplet to deform. If the Weber number is too high, the inertia force overcomes the surface tension force to the point that the droplet shattered into even smaller droplets. Thus, Weber number criterion is useful to predict the droplet size to be expected in liquid atomization.

Compressibility effect are not usually significant unless the Mach number is much more greater than 0.5 (M ≥ 0.5). in fact, compressibility effect are negligible when Mach number is less than 0.2 (M < 0.2). Mach number governs all forms of resistance due to compressibility when a body moves at high speed i.e. speed approaching or exceeding the local velocity of sound.

5.9 MODEL ANALYSIS

The performance of hydraulic structures (e.g. canals, dams, spillways, etc.) or hydraulic machines (e.g. pumps, turbines, compressors, etc.) depends on their models prepared and tested. This enables one to get the basic or required information. The model is a small scale representing and identical to the actual structure and machine which is called the PROTOTYPE.

5.9.1 Advantages of Model testing These include:

1. The model tests are quite economical and convenient because the design, construction and operation of the model may be changed times without number without incurring extra expenditure until the most suitable design is obtained.

2. The performance of the hydraulic structure or machine can be predicted with the use of models in advance.

3. The model testing can be used to detect and rectify the defects of an existing structure which is not functioning properly.

4. The safety and reliability of a structure is possible by means of model testing where analytically and reliable method is not available for the design.

5.9.2 Applications of Model testing

1. It is employed in civil engineering structures such as canals, dams, weirs and spillways.

2. It is used in the design and construction of pumps, turbines and compressors.

3. It is used in flood control, irrigation channels, site investigation, etc.

4. It is basically used to design harbours, ship and sub-marines.

5. The design of aeroplanes, rockets, jets and missiles is by model testing.

6. Tall buildings can be constructed by model testing to predict the wind loads, stability, characteristics and air flow pattern.

5.10 HYDRAULIC SIMILITUDE

When it is necessary to perform test on a model to obtain information that cannot be obtained by analytical method alone, the rule of similitude must be applied. Therefore, SIMILITUDE is an act of predicting prototypes condition from models observation. The theory of similitude involves the application of the dimensionless numbers while the act comes in when the

engineer must make decision about the model design, model construction, performance of test, or analysis of results that are not included in the basic theory.

It is pertinent to mention here that it is essential that models and prototypes be geometrically, kinematically and dynamically similar before certain test can be carried out confidently and reliably on the model.

Model of engineering systems may be either true or distorted models. True models reproduce features of the prototype but at a scale-that is they are geometrically similar.

5.10.1 Geometric similitude

Geometric similitude exists between model and prototype when the ratio of all corresponding dimensions in the model and prototype are equal

All corresponding angles are the same.

Length: ^𝐿𝑚

𝐿𝑝

= 𝐿

_𝑟

(5.24)

Area: ^𝐴𝑚

𝐴_𝑝

= 𝐴

_𝑟

=

^𝐿2𝑚

𝐿²_𝑝

= 𝐿

²_𝑟

(5.25)

Volume: ^𝑉𝑚

𝑉_𝑝

= 𝑉

_𝑟

=

^𝐿3𝑚

𝐿³_𝑝

= 𝐿

³_𝑟

(5.26)

Where m = model, p = prototype and r = ratio

𝐿

_1𝑚

𝐿

_1𝑝

= 𝐿

_2𝑚

𝐿

_2𝑝

(5.27)

Then

(

^𝐿2

𝐿₁

)

𝑝

= (

^𝐿2

𝐿₁

)

𝑚

(5.28)

5.10.2Kinematic similitude

Kinematic similitude is the similarity of time as well as geometry. It exists between model and prototype when their streamlines of homologous moving particles are geometrically similar and the ratios of velocities, accelerations and discharges are the same. It can be simply concluded that kinematic similarity exists if

i. the paths of moving particles are geometrically similar

ii. the ratios of the velocities, acceleration of particles are similar Velocity: ^𝑉𝑚

𝑉𝑝 =

𝐿_𝑚 𝑇𝑚

⁄ 𝐿_𝑝

𝑇𝑝

⁄ = (^𝐿𝑚

𝐿𝑝

) (^𝑇𝑝

𝑇𝑚

) = ^𝐿𝑟

𝑇𝑟 (5.29)

Acceleration: ^𝑎𝑚

𝑎𝑝 =

𝐿𝑚 𝑇_𝑚²

⁄ 𝐿𝑝

𝑇_𝑝²

⁄

= (^𝐿𝑚

𝐿𝑝

) (^𝑇𝑝

𝑇_𝑚)² = ^𝐿𝑟

𝑇_𝑟² (5.30)

Discharges:^𝑄𝑚

𝑄𝑝 =

𝐿³_𝑚 𝑇𝑚

⁄ 𝐿_𝑝³

𝑇𝑝

⁄

= (^𝐿𝑚

𝐿_𝑝)

3

(^𝑇𝑝

𝑇𝑚

) = ^𝐿3𝑟

𝑇𝑟 (5.31)

2

5.10.3 Dynamic similitude

Dynamic similitude exists between geometrically and kinematically similar systems if the ratios of all forces in the model and prototype are the same.

This occurs when the controlling dimensionless group on the right hand side of the defining equation is the same for model and prototype. The basic requirement for dynamic similarity Type equation here.is that the homologous forces that act on corresponding masses in the model and prototype be the same ratio. i.e.

𝐹

_𝑚

𝐹

_𝑝

= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (5.32)

must be constant throughout the entire flow field. The homologous forces may be one or a combination of the forces earlier discussed under dimensionless number. Vectorial summation of all the forces is the inertia force i.e.

𝑅_𝑖

⃗⃗⃗ = 𝑅⃗⃗⃗⃗ + 𝑅_𝑣 ⃗⃗⃗⃗ + 𝑅_𝑔 ⃗⃗⃗⃗ + 𝑅_𝑝 ⃗⃗⃗⃗ + 𝑅_𝑠 ⃗⃗⃗⃗ (5.33) _𝑐 and according to Newton’s second law of motion, then

∑ 𝐹_𝑥= 𝑚𝑎_𝑥

This implies that Ri = ma (5.34) and so the inertia ratio of model to prototype is

(𝑅

_𝑖

)𝑚

(𝑅

_𝑖

)𝑝 = 𝑅

_𝑟

= 𝑚

_𝑚

𝑎

_𝑚

𝑚

_𝑝

𝑎

_𝑝

=

𝜌

_𝑚

𝐿

³_𝑚

( 𝐿

_𝑚

𝑇

_𝑚²

⁄ )

𝜌

_𝑝

𝐿

³_𝑝

( 𝐿

_𝑝

𝑇

_𝑝²

⁄ )

= ( 𝜌

_𝑚

𝜌

_𝑝

) ( 𝐿

_𝑚

𝐿

_𝑝

)

3

( 𝐿

_𝑚

𝐿

_𝑝

) ( 𝑇

_𝑝²

𝑇

_𝑚²

)

= (𝜌_𝑚 𝜌_𝑝) (𝐿_𝑚

𝐿_𝑝)

2

(𝐿_𝑚 𝐿_𝑝)

2

(𝑇_𝑝 𝑇_𝑚)

2

𝑅

_𝑟

= 𝜌

_𝑟

𝐴

_𝑟

𝑉

_𝑟²

(5.35)

Where ρr = density ratio Ar = area ratio

Vr = velocity ratio

From equation (5.33), then

1 = 𝑅 ⃗⃗⃗

_𝑖

𝑅

_𝑣

⃗⃗⃗⃗ + 𝑅 ⃗⃗⃗

_𝑖

𝑅

_𝑔

⃗⃗⃗⃗ + 𝑅 ⃗⃗⃗

_𝑖

𝑅

_𝑝

⃗⃗⃗⃗ + 𝑅 ⃗⃗⃗

_𝑖

𝑅

_𝑠

⃗⃗⃗⃗ + 𝑅 ⃗⃗⃗

_𝑖

𝑅

_𝑐

⃗⃗⃗⃗ (5.36)

which yields the parameters earlier discussed.

If there are two or more forces acting together or influencing the behaviour of fluid flow, the ratio of the number for the model or prototype must be equal.

CHAPTER SIX

6.0 FLOW IN PIPES AND DUCTS

6.1 INTRODUCTION

The flow of fluids in pipes depends on the prevailing condition(s) during the flow. The properties of the fluid in the pipe are taken into consideration to know the prevailing condition for the fluid flow under review. The type of flow depends on certain criterion which determines whether the flow is smooth or not.

Fluid flow in pipes could either be for compressible fluid or incompressible fluid. However, whichever may be the case, the properties governing the condition of flow will have to be considered. Fluid flowing through pipes could be either Newtonian or non-Newtonian. Fluids that are Newtonian must obey the Newton’s law of viscosity i.e

𝜏 = 𝜇 𝑑𝑣

𝑑𝑦 (6.1)

6.2 CLASSIFICATION OF FLOW

The flow of Newtonian fluid can be classified as (i) Laminar flow

(ii) Turbulent flow

A laminar flow occurs when the paths taken by the individual particle do not cross one another and move along well defined paths. It is also called STREAMLINE FLOW or VISCOUS FLOW. Examples are flows through capillary tube and flow of blood in veins and arteries. On the other hand, a turbulent flow occurs when the paths taken by the individual particle cross one another and move along the flow channel without any defined pattern. Example is high velocity flow in a conduit of large size.

The characteristics of Laminar flow are

 There is no slip at the boundary;

 The flow is irrotational;

 There is shear stress in fluid layers due to viscosity;

 There is continuous dissipation of energy as a result of viscous shear;

 The loss of energy in flow depends on velocity and viscosity;

 There is no mixing between different fluid layers.

The characteristics of Turbulent flow are

 Random, irregular and haphazard movement of fluid particles;

 The velocity gradient near the boundary is quite large resulting in more shear;

 The pressure distribution fluctuates with time about a mean value.

6.3 CRITERION OF FLOW

The Reynolds’ experiment using the Reynolds’ apparatus gives an insight into the two categories of fluid flow which are the laminar and turbulent flow

The classified flow is characterized on the basis of Reynolds’ number. This serves as the criterion for a typical flow. Based on dimensional analysis, the Reynolds’ number is defined as

𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠

^′

𝑛𝑢𝑚𝑏𝑒𝑟, 𝑅𝑒 =

𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒 𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒

=

^𝐹𝑖

𝐹_𝑣

=

^{𝜌𝑙2𝑉2}

𝑙𝜇𝑉

=

^𝜌𝑉𝑙

𝜇 (6.2) If l = d, then

𝑅𝑒 =

^𝜌𝑉𝑑

𝜇

=

^𝑉𝑑

𝑣 (6.3) Where ρ = fluid density

V = velocity of flow

d = characteristic diameter of pipe μ = dynamic viscosity

l = length of pipe

ν = kinematic viscosity

Reynolds’ number, Re, from the relation above (Refer equation 6.3) depends on the following 1. the diameter of the pipe;

2. the density of the fluid;

3. the viscosity of the fluid;

4. the velocity of fluid flow.

6.3.1 Classification of flow based on Reynolds’ number

The foregoing equation (6.3) determines whether a flow is laminar or turbulent. The following points on this classified flow should be noted:

For Laminar flow-

 It occurs when the Reynolds’ number is low;

 Experimental result shows that the Reynolds’ number, Re < 2000;

 The viscous force predominates;

 The pressure (head) loss varies directly with the velocity (linear relationship) For Turbulent flow-

 It occurs when the Reynolds’ number is high;

 Experimental result shows that the Reynolds’ number, Re > 4000;

 The inertia force predominates;

 The pressure (head) loss varies directly with the velocity to index n where 1.75 < n < 2.00 (exponential relationship).

For Re between 2000 and 4000, the flow is unpredictable. Within this range, the flow is at the Transition stage which could be either laminar or turbulent.

The graph of pressure head, hf with the velocity of fluid flow is shown below

In document Análisis y estudio para la implementación de un plan piloto de un Círculo de Control de Calidad en el área de producción de NAJÓMEZ MERCANTIL J.C.H.G (página 55-61)