Dimensional and Model Analysis MCQ Quiz in বাংলা - Objective Question with Answer for Dimensional and Model Analysis - বিনামূল্যে ডাউনলোড করুন [PDF]

Concept:

Distorted model:

• Distorted models are those which are not geometrically similar to the model. So, we use different scale ratios in different directions.
• Particularly, in the case of shallow rivers, where depth is small compared to the width, without distortion, the depth of flow in the model will be so small that measurements may not be accurate and the surface tension effect will be prominent which will not present in the prototype.
• So for the study of rivers, we have to use a distorted model and use different scales in different directions.

Let LrH and LrV are the horizontal and vertical scale ratios. Thus

\({L_{rH}} = \frac{{{L_p}}}{{{L_m}}} = \frac{{{B_p}}}{{{B_m}}}\)

\({L_{rV}} = \frac{{{h_p}}}{{{h_m}}}\)

Velocity ratio Vr = √LrV

∵ We know that, time = distance/velocity

so, Time scale ratio, \({T_r} = \frac{{{L_{rH}}}}{{{V_r}}} = \frac{{{L_{rH}}}}{{\sqrt {{L_{rV}}} }}\)

⇒ \(\frac{{{T_m}}}{{{T_p}}} = \frac{{{L_{rH}}}}{{\sqrt {{L_{rV}}} }}\)

Where, Tr = Time scale ratio, Tm = Time period of model, Tp = Time period of prototype 

Calculation:

Given: 

LrH = 1/1000, LrV = 1/100, Tp = 12 hour

∵ \(\frac{{{T_m}}}{{{T_p}}} = \frac{{{L_{rH}}}}{{\sqrt {{L_{rV}}} }}\)

\(\frac{{\frac{1}{{1000}}}}{{\sqrt {\frac{1}{{100}}} }} = \frac{{{T_m}}}{{12}}\)

⇒ Tm  = 0.12 Hour

⇒ Tm = 0.12 × 60 min

⇒ Tm = 7.2 min

Advantages of the distorted model:

1) Due to substantial height obtained by distortion, vertical measurement is easier.

2) Hydraulic simulation can be achieved by distortion.

3) Turbulent flow is possible

4) Cost of the model can be reduced, as we can change the dimensions.

Disadvantages of distorted model:​

1) Due to different scales in horizontal and vertical directions, the pressure and velocity distribution are different in the model.

2) Slopes, bends, curves, and cutting in the earth are not correctly represented in the distorted model.

Concept:

The relation between rpm (N), diameter (D), and (H) head is,

 \(\rm \frac{\sqrt H}{DN}\)

Given,

H_m = 20 m, H_p = 20 m

Scale = \(\rm \frac{D_m}{D_p}=\frac{1}{2}\), N_p = 500 rpm,

N_m = ?

Hence

 \(\rm\frac{\sqrt{H_m}}{D_m N_m}=\frac{\sqrt{H_p}}{D_pN_p}\)

⇒ N_m = \(\rm N_p\left(\frac{D_p}{D_m}\right)\sqrt{\frac{H_m}{H_p}}\)

⇒ N_m = 500 × 2 × 1 = 1000 rpm

Explanation:

Euler’s Number is defined as the square root of the ratio of the inertia force of a flowing fluid to the pressure force. Mathematically

Euler number \( = \sqrt {\frac{{Inertia\;force}}{{Pressure\;force}}}= \frac{V}{{\sqrt {P/\rho } }}\)

Other important dimensionless numbers are described in the table below:

Concepts:

Mach number is dimensionless quantity  and it is defined as the ratio of flow velocity past a boundary to the local speed of sound. It is related to compressibility of fluid flow.

Reynolds Number is dimensionless quantity  and it is defined as ratio of inertia force to viscous force in fluid flow. Based on this flow can be considered as either laminar or turbulent.

Froude Number is dimensionless quantity  and it is defined as square root of ratio of inertia force to gravitational force. It governed the fluid flow in open channels.

Euler Number is dimensionless quantity  and it is defined as square root of ratio of inertia force to pressure force. It governs the fluid flow in pipes. 

Concept:

Velocity ratio (V_r):

\(V_r=\sqrt L_r\)

\(V_r={V_m\over V_p}\,, L_r={L_m\over L_p}\)

Where L_r = Length ratio, L_m = length of model, L_p = length of prototype, V_m = velocity of model, and V_p = velocity of prototype.

Solution:

Given

L_r = \(1\over 30\)

V_m = 3 m/sec

\({V_m\over V_p}=\sqrt{L_r}\)

\(V_p={V_m\over \sqrt {L_r}}={3\over \sqrt{1\over 30}}=3\sqrt{30}=16.43\, m/s\)

Velocity of prototype (V_p) = 16.43 m/s

Hence option (1) is correct.

Explanation:

Froude Number:

• The Froude number, Fr, is a dimensionless value that describes different flow regimes of open channel flow.
• The simultaneous motion through two fluids where there is a surface discontinuity. 
• Gravity forces and wave-making effect, as with ship’s hulls, Froude number is significant because in those cases gravity forces are predominant and Froude number is the ratio of inertia force and gravity force given by

\({{\rm{F}}_{\rm{r}}} = \sqrt {\frac{{{\rm{Inertia\;force}}}}{{{\rm{Gravity\;force}}}}} = {\rm{\;}}\frac{{\rm{v}}}{{\sqrt {{\rm{gL}}} }}\)

Froude number has the following applications:

• Used in cases of river flows, open-channel flows, spillways, surface wave motion created by boats
• It can be used for flow classification

Use in open channel design i.e free surface flows

Mach number:

• A dimensionless number that is most significant for supersonics as with projectile and jet propulsion because their elastic forces are predominant.
• Mach number is the ratio of inertia force and the elastic force which is used for compressible flow.

• In supersonic case Ma > 1 and in subsonic case Ma < 1.

Mach number is given by Ma = \(\sqrt {\frac{{{\rm{Inertia\;Force}}}}{{{\rm{Elastic\;Force}}}}} \) = \(\frac{{\rm{V}}}{{\sqrt {\frac{{\rm{K}}}{{\rm{\rho }}}} }}\) = \(\frac{{\rm{V}}}{{\rm{C}}}\) 

where V = velocity of an object in the fluid, K = elastic stress and ρ = density of the fluid medium, C = Velocity of sound in the fluid medium

Darcy friction factor:

It is a dimensionless quantity. It is given by-

\({\rm{f}} = \frac{{64}}{{{\rm{Re}}}}{\rm{\;where}},{\rm{\;Re}} = {\rm{Reynold's\;no}}.{\rm{\;}}\)

\(Re ={\rho VD\over \mu}={VD\over \nu}\)

where, ρ = density of fluid, V = velocity of fluid, D = Diameter of pipe,

v = kinematic viscosity

If Re > 4000 then the flow becomes a turbulent flow.

If Re < 2000 then the flow becomes a laminar flow.

Explanation:

• Forces encountered in flowing fluids include those due to inertia, viscosity, pressure, gravity, surface tension and compressibility.

These forces can be written as follows:

Reynolds number (Re): 

• It is defined as the ratio of inertia force to viscous force.
• \({\rm{Re = }}\frac{{{\rm{\rho Vl}}}}{{\rm{\mu }}}\)

Froude number (Fr): 

• It is defined as the ratio of inertia force to gravity force.         
• \({\rm{Fr = }}\frac{{\rm{V}}}{{\sqrt {{\rm{gL}}} }}\)

Weber number (We): 

• It is defined as the ratio of the inertia force to surface tension force.
• \({\rm{We = }}\frac{{{\rm{\rho }}{{\rm{V}}^{\rm{2}}}{\rm{l}}}}{{\rm{\sigma }}}\)

Mach number (M): 

• It is defined as the ratio of inertia force to velocity of sound.
• \({\rm{M = }}\frac{{\rm{V}}}{{\rm{c}}}{\rm{ = }}\frac{{\rm{V}}}{{\sqrt {\frac{{{\rm{dP}}}}{{{\rm{d\rho }}}}} }}\)

Buckingham's Pi Theorem:

Assume, a physical phenomenon is described by m number of independent variables like x1, x2, x3, ..., xm

The phenomenon may be expressed analytically by an implicit functional relationship of the controlling variables as:

f (x1, x2, x3, ……………, xm) = 0

Now if n be the number of fundamental dimensions like mass, length, time, temperature, etc., involved in these m variables, then according to Buckingham's Pi theorem -

The phenomenon can be described in terms of (m - n) independent dimensionless groups like π1, π2, ..., πm-n, where p terms, represent the dimensionless parameters and consist of different combinations of a number of dimensional variables out of them independent variables defining the problem.

Total no of π terms = m - n

Here, m = total parameter = 5

n = Fundamental dimensions (M, L, T) = 3

Explanation:

Mach number

Mach number is defined as the ratio of inertia force to elastic force.

\(M = \sqrt {\frac{{Inertia\;force}}{{Elastic\;force}}} = \sqrt {\frac{{\rho A{V^2}}}{{KA}}} = \sqrt {\frac{{{V^2}}}{{\frac{K}{\rho }}}} = \frac{V}{{\sqrt {\frac{K}{\rho }} }} = \frac{V}{C}\;\;\;\;\left\{ {\sqrt {\frac{K}{\rho }} = C = Velocity\;of\;sound} \right\}\)

\(M = \frac{{Velocity\;of\;body\;moving\;in\;fluid}}{{velocity\;of\;sound\;in\;fluid}}\)

For the compressible fluid flow, Mach number is an important dimensionless parameter. On the basis of the Mach number, the flow is defined.

Other important dimensionless numbers are described in the table below

Explanation:

Cauchy Number:

Cauchy number is defined as the ratio of inertia force to elastic force.

Mach number:

Mach number is defined as the ratio of inertia force to Compressibility force.

\(M = \sqrt {\frac{{Inertia\;force}}{{Compressibility\;force}}} = \sqrt {\frac{{\rho A{V^2}}}{{KA}}} = \sqrt {\frac{{{V^2}}}{{\frac{K}{\rho }}}} = \frac{V}{{\sqrt {\frac{K}{\rho }} }} = \frac{V}{C}\;\;\;\;\left\{ {\sqrt {\frac{K}{\rho }} = C = Velocity\;of\;sound} \right\}\)

\(M = \frac{{Velocity\;of\;body\;moving\;in\;fluid}}{{velocity\;of\;sound\;in\;fluid}}\)

For the compressible fluid flow, Mach number is an important dimensionless parameter. On the basis of the Mach number, the flow is defined.

Other important dimensionless numbers are described in the table below: