This Page deals with dimensional analysis,models and similitude,and application of dimensionless parameters.

**DIMENSIONAL ANALYSIS**

**PRE REQUEST DISCUSSION**

This Page deals with dimensional
analysis,models and similitude,and application of dimensionless parameters.

Many important engineering
problems cannot be solved completely by theoretical or mathematical methods.
Problems of this type are especially common in fluid-flow, heat-flow, and
diffusional operations. One method of attacking a problem for which no mathematical
equation can be derived is that of empirical experimentations.

For example, the pressure loss
from friction in a long, round, straight, smooth pipe depends on all these
variables: the length and diameter of the pipe, the flow rate of the liquid,
and the density and viscosity of the liquid. If any one of these variables is
changed, the pressure drop also changes. The empirical method of obtaining an
equation relating these factors to pressure drop requires that the effect of
each separate variable be determined in turn by systematically varying that
variable while keep all others constant. The procedure is laborious, and is
difficult to organize or correlate the results so obtained into a useful
relationship for calculations.

There exists a method intermediate
between formal mathematical development and a completely empirical study. It is
based on the fact that if a theoretical equation does exist among the variables
affecting a physical process, that equation must be dimensionally homogeneous.
Because of this requirement it is possible to group many factors into a smaller
number of dimensionless groups of variables. The groups themselves rather than
the separate factors appear in the final equation.

**Concepts**

Dimensional analysis drastically
simplifies the task of fitting experimental data to design equations where a
completely mathematical treatment is not possible; it is also useful in
checking the consistency of the units in equations, in converting units, and in
the scale-up of data obtained in physical models to predict the performance of
full-scale model. The method is based on the concept of dimension and the use
of *dimensional formulas*.

Dimensional
analysis does not yield a numerical equation, and experiment is required to
complete the solution of the problem. The result of a dimensional analysis is
valuable in pointing a way to correlations of experimental data suitable for
engineering use.

**METHODS OF DIMENSIONAL ANALYSIS**

If the number of variables
involved in a physical phenomenon are known, then the relation among the
variables can be determined by the following two methods.

1.Rayleigh’s method

2. Buckingham’s π theorem

**1****Rayleigh’s method**

This method is used for
determining the expression for a variable which depends upon maximum three or
four variables only. If the number of independent variables becomes more than
four then it is very difficult to find the expression for the dependent
variable.

**2 ****Buckingham’s**** ****π**** theorem.**

If there
are n variables
(independent and dependent
variables) in a
physical

phenomenon and if these variables
contain m fundamental dimensions (M, L, T), then the variables are arranged
into (n-m)
dimensionless numbers. Each term is called Buckingham’s

π theorem.

**Applications**

v It is
used to justify the dependency of one variable with the other.

v Usually
this type of situation occurs in structures and hydraulic machines.

v To solve
this problem efficiently, an excellent tool is identified called dimensional
analysis.

**SMILITUDE ****–****TYPES OF
SIMILARITIES**

Similitude is defined as the
similarity between the model and its prototype in every respect, which means
that the model and prototype are completely similar. Three types of
similarities must exist between the model and prototype.

**Concepts**

Whenever it is necessary to
perform tests on a model to obtain information that cannot be obtained by
analytical means alone, the rules of similitude must be applied. *Similitude *is
the theory and art of predicting prototype performance from model* *observations

**1. Geometric similarity **refers to
linear dimensions. Two vessels of different sizes are** **geometrically
similar if the ratios of the corresponding dimensions on the two scales are the
same. If photographs of two vessels are completely super-impossible, they are
geometrically similar.

**2.Kinematic similarity **refers to
motion and requires geometric similarity and the same** **ratio of
velocities for the corresponding positions in the vessels.

**3.Dynamic similarity **concerns
forces and requires all force ratios for corresponding** **positions to be
equal in kinematically similar vessels.

**SIGNIFICANCE**

The requirement for similitude of
flow between model and prototype is that the significant dimensionless
parameters must be equal for model and prototype

**DIMENSIONLESS PARAMETERS**

Since the inertia force is always
present in a fluid flow, its ratio with each of the other forces provides a
dimensionless number.

1. Reynold’s
number

2. Froud’s
number

3. Euler’s
number

4. Weber’s
number

5. Mach’s
number

**Applications of dimensionless parameters**

1. Reynold’s
model law

2. Froud’s
model law

3. Euler’s
model law

4. Weber’s
model law

5. Mach’s
model law

**Important Dimensionless Numbers
in Fluid Mechanics:**

**MODEL ANALYSIS.**

**PRE REQUEST DISCUSSION**

Present engineering practice
makes use of model tests more frequently than most people realize. For example,
whenever a new airplane is designed, tests are made not only on the general
scale model but also on various components of the plane. Numerous tests are
made on individual wing sections as well as on the engine pods and tail
sections

Models of automobiles and
high-speed trains are also tested in wind tunnels to predict the drag and flow
patterns for the prototype. Information derived from these model studies often
indicates potential problems that can be corrected before prototype is built,
thereby saving considerable time and expense in development of the prototype.

**Concepts**

Much time, mony and energy goes into the design
construction and eradication of hydraulic structures and machines.

To minimize the chances of failure, it is always
desired that the tests to be performed on small size models of the structures
or machines. The model is the small scale replica of the actual structure or
machine. The actual structure or machine is Called prototype.

**Applictions**

1. Civil
engineering structures such as dams, canals etc.

2. Design of
harbor, ships and submarines

3. Aero
planes, rockets and machines.

4. Marine
engineers make extensive tests on model shop hulls to predict the drag of the
ships

**GLOSSARY**

**The three
friction factor problems:**

The
friction factor relates six parameters of the flow:

1. Pipe
diameter

2. Average
velocity

3. Fluid
density

4. Fluid
viscosity

5. Pipe
roughness

6. The
frictional losses per unit mass.

Therefore,
given any five of these, we can use the friction-factor charts to find the
sixth.

Most often, instead of being interested in the average
velocity, we are interested in the volumetric flow rate Q = (p/4)D^{2}V

The three most common types of
problems are the following:

Generally, type 1 can be solved directly, where as types 2 and
3 require simple trial and error.

Three fundamental problems which are commonly encountered in
pipe-flow calculations: Constants: rho, mu, g, L

1. Given D,
and v or Q, compute the pressure drop. (pressure-drop problem)

2. Given D,
delP, compute velocity or flow rate (flow-rate problem)

3. Given Q,
delP, compute the diameter D of the pipe (sizing problem)

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