1DIMENSIONAL ANALYSIS

Methods of Dimensional AnalysisThere are two methods of dimensional analysis.

1. Rayleighs method

2. Buckinghams ( theorem) method

1. Rayleighs methodRayleighs method of analysis is adopted when number of parameters or

variables are less (3 or 4 or 5).

MethodologyX1 is a function ofX2, x3, X4, , Xn then it can be written asX1 = f(X2, x3, X4, , Xn)X1 = K (X2a, x3b, X4c, )

Taking dimensions for all the quantities[X1] = [X2]a [X3]b [X4]c

Dimensions for quantities on left hand side as well as on the right hand sideare written and using the concept of Dimensional Homogeneity a, b, c . can bedetermined.

Then,

X1 = K X2a X3b X4c

Problems 1: Velocity of sound in air varies as bulk modulus of electricity K,

Mass density . Derive an expression for velocity in form C =K

Solution:C = f (K, )C = M Ka b

M Constant of proportionality

2[C] = [K]a []b

[LMoT-1] = [L-1MT-2]a [L3MTo]b

[LMoT-1] = [L-a+(-36) Ma+b T-2a]- a 3b = 1

a + b = 0- 2b = 1

b =21

a =21

C = MK1/2 -1/2

C = MK

If, M = 1, C =K

Problem 2: Find the equation for the power developed by a pump if it depends onbead H discharge Q and specific weight of the fluid.

Solution:P = f (H, Q, )P = K Ha Qb c

[P] = [H]a [Q]b []c

[L2MT-3] = [LMoTo]a [L-2MT-2]b [L-2MT-2]c

2 = a + 3b 2c1 = c- 3 = - b - 2- 3 = - b 2b = - 2 + 3b = 1

2 = a + 3 2a = 1

C Velocity LMoT-1

K Bulk modulus L-1MT-2

- Mass density L-3MTo

Power = L2MT-3

Head = LMoTo

Discharge = L3MoT-1Specific Weight = L-2MT-2

3P = K H1 Q1 1

P = K H Q When, K = 1

P = H Q

Problem 3: Find an expression for drag force R on a smooth sphere of diameter Dmoving with uniform velocity V in a fluid of density and dynamic viscosity ..

Solution:R = f (D, V, , )R = K Da Vb c, d

[R] = [D]a [V]b []c []d

[LMT-2] = [LMoTo]a [LMoT-1]b [L-3MTo]c [L-1MT-1]d

c + d = 1c = 1 d

b d = 2b = 2 d

1 = a + b 3c d1 = a + 2 d 3 (1 d) d1 = a + 2 d 3 + 3d da = 2 d

R = K D2-d V2-d 1-d, d

R = K ddd2

d

2

VV

DD

R = K V2 D2d

VD

R = V2 D2

VD

Force = LMT-2

Diameter = LMoTo

Velocity = LMoT-1

Mass density = L3MTo

Dynamic Viscosity = L-1MT-1

4R = V2 D2

VD

R = V2 D2 [NRe]

5 Problem 4: The efficiency of a fan depends on the density dynamic viscosity ,angular velocity , diameter D, discharge Q. Express efficiency in terms ofdimensionless parameters using Rayleighs Method.

Solution: = f (, , , D, Q) = k a b c Dd Qe

[] = []a []b []c [D]d [Q]e

[LoMoTo] = [L-3MTo]a [L-1MT-1]b [LoMoT-1]c [LMoTo]d [L3MoT-1]-1 [LoMoTo] [L-3a-b+d+3e Ma+b T-b-c-e]a + b = 0a = b

b c e = 0c = b e

3a b + d + 3e = o+ 3b b + d + 3e = 0d = 2b 3e

= K -b b -b-e D-2b-3e Qe

= K 2

e3b2ebb

b QDD111

= Ke

3

b

2 DQ

D

= Q

32 D

Q,

D

- LoMoTo

- L-3MTo

- L-1MT-1

- LoMoT-1

D - LMoTo

Q - L3MoT-1

6 Problem 5: The capillary rise H of a fluid in a tube depends on its specific weight

and surface tension and radius of the tube R prove that

2RRH

.

Solution:H = f (, , R)H = K a b Rc

[H] = []a [] b [R]c

[LMoTo] = [L-2MT-2]a [LoMT-2]b [LMoTo]c

[LMoTo] = [L-2a+c Ma+b T -2a-2b]2a + c = 1a + b = 02a 2b = 0

a =

21c

0b2

1c

b =

2c1

H = K c2c1

21c

R

H = K c2

c

21

21

2c

R

RH

= K RR c

2c

21

21

2c

RH

= K 1R2

c

21

21

21

H = K -b b R1-2b

H = K b2bb

RR

H - LMoTo

- L-2MT-2

- LoMT-2

R - LMoTo

7H = Kb

2R

2RRH

.

2. Buckinghams MethodThis method of analysis is used when number of variables are more.

Buckinghams TheoremIf there are n variables in a physical phenomenon and those n-variables

contain m dimensions, then the variables can be arranged into (n-m) dimensionlessgroups called terms.

Explanation:If f (X1, X2, X3, Xn) = 0 and variables can be expressed using m

dimensions then.f (1, 2, 3, n - m) = 0

Where, 1, 2, 3, are dimensionless groups.

Each term contains (m + 1) variables out of which m are of repeating typeand one is of non-repeating type.

Each term being dimensionless, the dimensional homogeneity can be usedto get each term.

Selecting Repeating Variables1. Avoid taking the quantity required as the repeating variable.2. Repeating variables put together should not form dimensionless group.3. No two repeating variables should have same dimensions.4. Repeating variables can be selected from each of the following properties.

a. Geometric property Length, height, width, areab. Flow property Velocity, Acceleration, Dischargec. Fluid property Mass density, Viscosity, Surface tension