Dimensional Analysis

DIMENSIONAL ANALYSIS

• Velocity potential = [LT–1]
Stream function = [L2 T–1]
Acceleration = [LT–2]
Vorticity = [T–1] 
• Total no. of variables influencing the problem is equal to the no. of independent variables plus one, one being the no. of dependent variable. 
• Buckingham π theorem states that if all the n-variable are described by m fundamental dimensions, they may be grouped into (n - m) dimensions p terms. 
• Selection of 3 repeating variables from the geometry of flow, fluid properties and fluid motion. 
• Geometric similarity - similarity of shape
Kinematic similarity - similarity of motion
Dynamic similarity - similarity of forces
 Number                                 Equation                                       Significance Reynolds No.                                         Flow in closed conduit pipeFroude No.                                        where a free surface is present, structureEulers No.                                           In cavitation studies.Mach No.                                               where fluid compressibility is important.Weber No.                                      In capillary studies.

 Reynolds Model Law :

(i) Velocity ratio

(ii) Time ratio

(iii) Acceleration ratio,

(iv) Force ratio

(v) Power ratio

(vi) Discharge ratio

Applications of Reynold’s Model Law :-

• Flow through small sized pipes 
• Low velocity motion around automobiles and aeroplane. 
• Submarines completely under water. 
• Flow through low speed trubo machines. 

Froude’s Model law :

(i) (Fr)prototype  =  (Fr)model
Vp / √ gL= Vm / √ gm Lp
It the place of model and prototype is same, then gm = gp
V =  √Lr
(ii) Time scale ratio
Tr  =  √ Lr
(iii) Acceleration scale ratio
ar  = 1
(iv) Discharge scale ratio
Qr  = Lr5/2
(v) Force scale ratio
F = ρp / ρm x (L/ Lm)x (Vp / Vm)2
If the fluid used in model and prototype is same, then
Fr  = Lr3
(vi) Pressure scale ratio
Pr  =  Lr

Applications :

• Open channels 
• Notches & weirs 
• Spill ways & dams 
• Liquid jets from orifice 
• Ship partially submerged in rough & turbulent sea
The document Dimensional Analysis | Mechanical Engineering SSC JE (Technical) is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).
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## FAQs on Dimensional Analysis - Mechanical Engineering SSC JE (Technical)

 1. What is dimensional analysis in mechanical engineering?
Ans. Dimensional analysis is a method used in mechanical engineering to analyze and understand the relationships between various physical quantities. It involves studying the dimensions and units of different variables involved in a problem to derive meaningful insights and solve engineering problems.
 2. How is dimensional analysis applied in mechanical engineering?
Ans. Dimensional analysis is applied in mechanical engineering to determine the functional relationship between variables and to develop dimensionless parameters. It helps in scaling up or down physical models, predicting the behavior of systems, and validating experimental results. It also aids in simplifying complex equations and improving the efficiency of engineering designs.
 3. Can dimensional analysis be used to solve complex mechanical engineering problems?
Ans. Yes, dimensional analysis can be used to solve complex mechanical engineering problems. By identifying the relevant variables and their dimensions, engineers can establish the relationships between these variables and determine the impact of each variable on the overall system. This approach can help in simplifying the problem and providing insights into the behavior of the system without the need for extensive experimentation or complex mathematical calculations.
 4. What are the advantages of using dimensional analysis in mechanical engineering?
Ans. The advantages of using dimensional analysis in mechanical engineering include: 1. Simplification of complex problems: Dimensional analysis helps in simplifying complex problems by reducing the number of variables and identifying the most significant factors affecting the system. 2. Scaling and modeling: It facilitates scaling up or down physical models, allowing engineers to test and analyze the behavior of systems in a controlled and cost-effective manner. 3. Prediction and optimization: Dimensional analysis enables engineers to predict the behavior of systems based on known relationships between variables. It aids in optimizing designs and improving the efficiency of engineering solutions. 4. Experimental validation: Dimensional analysis helps in validating experimental results by identifying dimensionless parameters that should remain consistent across different scales or systems.
 5. Are there any limitations or assumptions associated with dimensional analysis in mechanical engineering?
Ans. Yes, there are certain limitations and assumptions associated with dimensional analysis in mechanical engineering: 1. Linearity assumption: Dimensional analysis assumes that the relationships between variables are linear, which may not always be the case in complex systems. 2. Neglecting small effects: Dimensional analysis may neglect small effects or interactions between variables that could be significant in some cases. 3. Limited applicability: Dimensional analysis may not be suitable for all types of problems, especially those involving highly nonlinear or turbulent phenomena. 4. Dependence on accurate measurements: Dimensional analysis relies on accurate measurements of variables to establish meaningful relationships. Errors in measurements can lead to inaccurate results. Overall, while dimensional analysis is a valuable tool in mechanical engineering, it should be used in conjunction with other analytical and experimental methods to ensure accurate and reliable results.

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