Dimensional Analysis

Question 1

What are the benefits of dimensional analysis?

Answer 1

It allows us to reduce the number of experiments required by allowing us to test variables in groups rather than individually

Can be used to check equations during derivations

Yields some information about the form of the relationships connecting the relevant variables.

Question 2

Define variables in an experiment

Answer 2

Values that can vary

Question 3

Define parameters in an experiment

Answer 3

Values that are fixed

Question 4

What are dimensions?

Answer 4

Dimensions are NOT units. There are 7 primary dimensions but those used in Water Engineering are Length (L), time (T) and Mass (M)

Question 5

Derive the dimensions for Force

Answer 5

We know Force=mass x acceleration

[Force (N)] = [Mass (kg)] x [acceleration (m/s^2)]

= M x L/T^2
= M x L x T^-2
= MLT^-2

Question 6

What will dimensional analysis not do?

Answer 6

Reveal the actual relationships between variables

Reveal analytical solutions

Tell us which parameters are important and which aren’t. We still need to do experiments

Question 7

What is ‘m’ in dimensional analysis?

Answer 7

M is the total number of variables and parameters

Question 8

What is ‘n’ in dimensional analysis?

Answer 8

N is the number of dimensions

Question 9

Briefly describe the steps of the Buckingham Pi method of dimensional analysis

Answer 9

Step 1: Write the functional dependence between variables and parameters

Step 2: Determine m, n and thus the number of dimensionless Pi numbers

Step 3: Choose a set of repeating variables that provide the most pure representation of the variables

Step 4: Write expressions for the Pi numbers

Step 5: Determine the exponents of the repeating variables in each Pi number

Step 6: Write the new relationship in dimensionless form. Pi1=function(Pi2, …, Pi(n)

Question 10

How do you calculate how many experiments are needed before dimensional analysis

Answer 10

Experiments = number of changes^variables

Eg - 9 values for each 5 variables = 9^5 = 59049 experiments

Question 11

How do you calculate how many experiments are needed after dimensional analysis

Answer 11

Experiments=number of changes^pi numbers in function

eg - 9 values and 3 pi numbers in function (4 pi numbers total) = 9^3 = 729 experiments