Dimensional and Model Analysis MCQ Quiz - Objective Question with Answer for Dimensional and Model Analysis - Download Free PDF

Explanation:

Similarity Between Model and Prototype

• For a model to accurately represent its prototype in engineering analysis, certain similarities must exist between the two. These similarities ensure that the results obtained from studying the model can be extrapolated to predict the behavior or performance of the prototype. The three key types of similarity are geometric, kinematic, and dynamic similarity.

Types of Similarity:

• Geometric Similarity: This refers to the model and the prototype having the same shape and proportional dimensions. All corresponding lengths in the model and prototype should maintain a consistent scale factor. For example, if the prototype is scaled down by a factor of 10, every dimension (length, width, height) of the model should be 1/10th of the corresponding dimension in the prototype.
• Kinematic Similarity: This ensures that the motion of the model and the prototype are similar. The velocities, accelerations, and flow patterns at corresponding points in the model and prototype must maintain a consistent ratio. Kinematic similarity is essential in fluid mechanics and dynamics where motion plays a critical role.
• Dynamic Similarity: This involves the forces acting on the model and the prototype being in proportion. The ratios of corresponding forces, such as inertial forces, viscous forces, pressure forces, and gravitational forces, must be the same. This similarity ensures that the dynamic response of the model accurately predicts the behavior of the prototype.

Importance of All Three Types of Similarity:

• For complete similarity between the model and its prototype, all three types of similarity—geometric, kinematic, and dynamic—must exist simultaneously. This ensures that the model behaves in the same manner as the prototype under similar conditions, allowing engineers to make reliable predictions about the prototype's performance based on model analysis.

Concept:

The Mach angle \( \mu \) is related to Mach number \( M \) as:

\( \sin \mu = \frac{1}{M} \Rightarrow M = \frac{1}{\sin 30^\circ} = 2 \)

Speed of sound:

\( a = \sqrt{\gamma R T} = \sqrt{1.4 \cdot 287 \cdot 300} \approx 347.15~\text{m/s} \)

Velocity of bullet:

\( V = M \cdot a = 2 \cdot 347.15 \approx 694.3~\text{m/s} \approx 280\sqrt{6} \)

 

Explanation:

Froude Model Law

Definition: Froude Model Law is a similarity law used in fluid mechanics for the study and comparison of fluid flow phenomena in models and prototypes. It is primarily applied in cases involving gravitational forces, such as free surface flows, ship modeling, and open channel flows. The law states that the ratio of inertial forces to gravitational forces must be consistent between the model and the prototype to ensure dynamic similarity.

Working Principle: Froude Model Law is based on the principle that the Froude number (Fr), which is the ratio of inertial forces to gravitational forces, must remain the same for both the model and the prototype. The Froude number is given by:

Froude Number \(F_r = \frac{V}{\sqrt{gL}}\)

Where:

• v: Velocity of fluid
• g: Acceleration due to gravity
• L: Characteristic length

Dynamic similarity is achieved when the Froude number for the model and the prototype are equal:

\(\frac{V_m}{\sqrt {L_m g_m}} = \frac{V_p}{\sqrt{L_p g_p}}\)

From this, the relationships between various physical quantities such as velocity, time, force, and power can be derived.

Scale ratio of force = (Scale ratio of length)³

To understand why this is correct, let us analyze the relationship between force and length under Froude Model Law:

Force Relationship:

The force acting in a fluid flow is typically determined by the inertial forces and gravitational forces. According to Froude Model Law, the force scale ratio between the model and the prototype can be derived as follows:

• The inertial force is proportional to mass × acceleration.
• Mass is proportional to the volume, which scales with the cube of the length (l³).
• Acceleration, under Froude similarity, is proportional to g (gravitational acceleration), which does not change.

Therefore, the scale ratio of force (F) is proportional to the scale ratio of length (l) cubed:

Scale Ratio of Force = (Scale Ratio of Length)³

Explanation:

When comparing a model and a prototype where the predominant force is gravity, the Froude Model Law is applied. This law is based on the principle that the ratio of inertial forces to gravitational forces should be the same for both the model and the prototype. The Froude number (Fr) is a dimensionless number that compares the inertial force to the gravitational force and is given by:

Fr = V / √(gL)

where:

• V = velocity
• g = gravitational acceleration
• L = characteristic length (such as depth of water)

The Froude Model Law ensures dynamic similarity between the model and the prototype when gravity is the dominant force. This is particularly important in hydraulic models and fluid dynamics involving free-surface flows, such as in the design of ships, spillways, and river modeling.

Analyzing the Given Options

1. 1) Mach Model Law (Incorrect)

   • The Mach Model Law is used for scenarios where compressibility effects are significant, and the Mach number (ratio of flow velocity to the speed of sound) is used for dynamic similarity. It is not primarily concerned with gravitational forces.

2. 2) Euler Model Law (Incorrect)

   • The Euler Model Law applies to scenarios where pressure forces dominate, and the Euler number (ratio of pressure forces to inertial forces) is used. This law is not suitable for situations where gravity is the predominant force.

3. 3) Weber Model Law (Incorrect)

   • The Weber Model Law is concerned with surface tension forces and is used when studying phenomena like capillary waves and droplet formation. It involves the Weber number, which is the ratio of inertial forces to surface tension forces, and is not applicable for gravity-dominated situations.

4. 4) Froude Model Law (Correct)

   • The Froude Model Law is specifically used for scenarios where gravitational forces are predominant. It ensures dynamic similarity between the model and the prototype by maintaining the same Froude number. This makes it the appropriate choice for the given situation.

Concept:

For open channel flow over a spillway, the head is a length parameter and follows the law of geometric similarity in Froude model laws. So the scale ratio for head (H) is:

\( H_m = \frac{H_p}{\sqrt{L_r}} \)

Where:

• \(H_m\) = head in model
• \(H_p\) = head in prototype = 1.6 m
• \(L_r\) = length scale ratio = 50

Given:

• Prototype head = 1.6 m
• Model scale ratio = 1:50

Calculation:

As per Froude’s model law:

\( H_m = \frac{H_p}{L_r} = \frac{1.6}{50} = 0.032 \, \text{m} \)

 

Concept:

Froude number is the ratio of inertial force to the gravitation force.

\(\text{Froude }\!\!~\!\!\text{ number}=\text{ }\!\!~\!\!\text{ }\sqrt{\frac{\text{Inertia force }\!\!~\!\!\text{ }}{\text{Gravitational force }\!\!~\!\!\text{ }}}\)

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

Important Points

\(\text{Reynold's number}=\frac{\text{Inertia force}}{\text{Viscous force}}\)

\(\text{Euler }\!\!~\!\!\text{ number}=\sqrt{\frac{\text{Inertia force}}{\text{Pressure force}}}\)

\(\text{Weber }\!\!~\!\!\text{ number}=\sqrt{\frac{\text{Inertia force}}{\text{Surface Tension force}}}\)

\(\text{Mach }\!\!~\!\!\text{ number}=\sqrt{\frac{\text{Inertia force}}{\text{Compressibility force}}}\)

Concept:

Unit speed i.e. speed of the turbine under the unit head,

\({{\rm{N}}_{\rm{a}}} = \frac{{\rm{N}}}{{\sqrt {\rm{H}} }}\)

Given: P = 400 kW, H1 = 81 m, N1 = 225 rpm, and H2 = 64 m

To Calculate: N2 = ?

Calculation:

\(\frac{{{{\rm{N}}_1}}}{{\sqrt {{{\rm{H}}_1}} }} = \frac{{{{\rm{N}}_2}}}{{\sqrt {{{\rm{H}}_2}} }} \Rightarrow \frac{{225}}{{\sqrt {81} }} = \frac{{{{\rm{N}}_2}}}{{\sqrt {64} }}\)

\(\Rightarrow {{\rm{N}}_2} = 200 \ rpm\)

Explanation:

Weber number

The Weber number is the ratio of dynamic pressure (i.e. inertia force) to the surface tension force. 

\({W_e} = \sqrt {\frac{{inertia\;force}}{{surface\;tension}}} = \frac{V}{{\sqrt {\sigma /\rho L} }}\)

Additional Information

Other important dimensionless numbers are described in the table below"

Reynold’s number	\(R_e=\frac{{{inertia force}}}{{{viscous force}}} = \frac{{{\bf{\rho VL}}\;}}{{\bf{\mu }}}\)
Froude Number	\({F_r} = \sqrt {\frac{{inertia\;force}}{{gravitation\;force}}} = \frac{V}{{\sqrt {{\bf{Lg}}} }}\)
Euler number	\({E_u} = \sqrt {\frac{{inertia\;force}}{{pressure\;force}}} = \frac{V}{{\sqrt {P/\rho } }}\)
Mach Number	\({M} = \sqrt {\frac{{inertia\;force}}{{Elastic\;force}}} = \frac{V}{{\sqrt {K/\rho } }}=~\frac VC \)

Explanation:

Dimensions:

Density (ρ) = [ML^-3]

Angular velocity (ω) = [T^-1]

Dynamic viscosity (μ) = [ML^-1T^-1]

Characterstic diameter (D) = [L]

By Buckingham π - method:

π = [ρ^a, ω^b, D^c] μ 

Where,

π = dimensionless number

π = [ML^-3]^a [T^-1]^b [L]^c [ML^-1T^-1]

Equating the dimention of both side;

a = -1

b = -1

c = -2

μ/ρωD² is the dimensionless term.

CONCEPT:

• Dimensions: When a derived quantity is expressed in terms of fundamental quantities, it is written as a product of different powers of the fundamental quantities.
• The powers to which fundamental quantities must be raised in order to express the given physical quantity are called its dimensions.
• A quantity without dimension will usually be a ratio of two quantities with similar dimensions and hence, will cancel out. Thus, they will have no units and known as dimensionless quantity.
• Angle: It is defined as the ratio of the length of arc to the radius, i.e.,

​\(Angle\;\left( \theta \right) = \frac{{length\;of\;arc\;\left( l \right)}}{{radius\;\left( r \right)}}\)

\( \Rightarrow \theta = \frac{{\left[ {{M^0}L{T^0}} \right]}}{{\left[ {{M^0}L{T^0}} \right]}} = 1\)

∴ An angle is a dimensionless quantity

• Strain: It is defined as the ratio of change in length to the original length. i.e.,

\(Strain = \frac{{change\;in\;length\;\left( {{\rm{\Delta }}l} \right)}}{{original\;length\;\left( l \right)}}\)

\( \Rightarrow Strain = \frac{{\left[ {{M^0}L{T^0}} \right]}}{{\left[ {{M^0}L{T^0}} \right]}} = 1\)

∴ Strain is a dimensionless quantity

• Specific gravity: It is the ratio of the density of a substance to the density of a given reference material i.e.,

\(Specific\;gravity\;\left( \rho \right) = \frac{{Density\;of\;the\;object\;\left( {{\rho _{object}}} \right)}}{{Density\;of\;water\;\left( {{\rho _{water}}} \right)}}\)

\( \Rightarrow Specific\;gravity = \frac{{\left[ {M{L^3}{T^0}} \right]}}{{\left[ {M{L^3}{T^0}} \right]}} = 1\)

∴ Specific gravity is a dimensionless quantity

EXPLANATION:

As from the above discussion, we can say that:

• An angle is the ratio of the length of arc to the radius. As both have the dimension of length so the angle is a dimensionless quantity. So option 1 follows.
• Strain: It is defined as the ratio of change in length to the original length. As both have the dimension of length so the strain is a dimensionless quantity. So option 2 follows.
• Specific gravity: It is the ratio of the density of a substance to the density of given reference material. As both have the dimension of density so the specific gravity is a dimensionless quantity. So option 3 follows.

As the given three options are correct.

So the correct option is 4.

Explanation:

Buckingham's Pi Theorem:

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

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

f(x1, x2, x3, ……………, xn) = 0

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

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

Concept:

Froude number is the ratio of inertial force to the gravitation force.

\(\text{Froude }\!\!~\!\!\text{ number}=\text{ }\!\!~\!\!\text{ }\sqrt{\frac{\text{Inertia force }\!\!~\!\!\text{ }}{\text{Gravitational force }\!\!~\!\!\text{ }}}\)

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

Important Points

\(\text{Reynold's number}=\frac{\text{Inertia force}}{\text{Viscous force}}\)

\(\text{Euler }\!\!~\!\!\text{ number}=\sqrt{\frac{\text{Inertia force}}{\text{Pressure force}}}\)

\(\text{Weber }\!\!~\!\!\text{ number}=\sqrt{\frac{\text{Inertia force}}{\text{Surface Tension force}}}\)

\(\text{Mach }\!\!~\!\!\text{ number}=\sqrt{\frac{\text{Inertia force}}{\text{Compressibility force}}}\)

Explanation:

Critical flow

It is defined as the flow at which specific energy is minimum or the flow corresponding to critical depth is defined as critical flow.

Relation of critical velocity with critical depth is:

\({V_c} = \sqrt {g \times {h_c}} \)

\(\frac{{{V_c}}}{{\sqrt {g \times {h_c}} }} = Froude\;number = 1\)

Froude number is 1 for critical flow.

Tranquil flow or streaming flow or sub-critical flow

• When the depth of a flow in a channel is greater than critical depth (h_c), the flow is said to be sub-critical flow.
• Froude number is less than 1 for sub-critical flow.

Torrential flow or shooting flow or super-critical flow

• When the depth of a flow in a channel is less than critical depth (h_c), the flow is said to be a super-critical flow.
• Froude number is greater than 1 for super-critical flow.

Concept:

Froude model law:

The gravity force is the only predominant force in addition to the inertia force, which control the motion

(F_r)_model = (F_r)_prototype

Or,

\(\frac{{{V_m}}}{{\sqrt {{g_m}{L_m}} }} = \frac{{{V_p}}}{{\sqrt {{g_p}{L_p}} }}\)

\(\frac{{{V_r}}}{{\sqrt {{g_r}{L_r}} }} = 1\)

\({V_r} = \sqrt {{g_r}{L_r}}\)

Since, in most of the cases g_r = 1, then

V_r = √L_r ………(i)

∵ velocity = distance/time

So, V_r = L_r/T_r

T_r = L_r/V_r = L_r/√L_r = √L_r

Or

\(\frac{{{T_m}}}{{{T_p}}} = \sqrt {\frac{{{L_m}}}{{{L_p}}}}\)

Calculation:

Given,

L_r = 1/36, T_m = 3 hour

∵ We know that, T_r = √L_r

\(\frac{3}{{{T_p}}} = \sqrt {\frac{1}{{36}}}\)

⇒ T_p = 18 hour

Explanation:

Number	Definition	Significance
Reynolds No	\(Re = \frac{{Inertia\;Force}}{{Viscous\;Force}} \Rightarrow \frac{{\rho VL}}{\mu }\)	Flow in closed conduit i.e. flow through pipes.
Froude No	\(Fr = \sqrt {\frac{{Inertia\;Force}}{{Gravity\;Force}}} \Rightarrow \frac{V}{{\sqrt {gL} }}\)	Where a free surface is present and gravity force is predominant. Spillway, Open Channels, waves in the ocean.
Euler No.	\({E_u} = \sqrt {\frac{{Inertia\;Force}}{{Pressure\;Force}}} \Rightarrow \frac{V}{{\sqrt {\frac{p}{\rho }} }}\)	In cavitation studies, where pressure force is predominant.
Mach No.	\(M = \sqrt {\frac{{Inertia\;Force}}{{Elastic\;Force}}} \Rightarrow \frac{V}{C}\)	Where fluid compressibility is important. Launching of rockets, airplanes and projectile moving at supersonic speed.
Weber No.	\({W_e} = \sqrt {\frac{{Inertia\;Force}}{{Surface\;tension\;Force}}} \Rightarrow \frac{V}{{\sqrt {\frac{\sigma }{{\rho L}}} }}\)	In Capillary studies i.e. where Surface tension is predominant.