Show that the pressure loss in a pipe is a function of length (L) and diameter (D) of the pipe, density of the fluid (rho), and the flow velocity (V). Find a relationship for the pressure loss using the Buckingham pi theorem.

Explanation:

Buckingham pi theorem is used to find the relationship between variables in a physical phenomenon.

First, we need to identify the variables involved in the physical problem of pressure loss (ΔP). These are: d, L, p, μ and v

hence, we have

f(ΔP, d, L, p, μ,v) = 0

Counting the number of variables n, (ΔP, d, L, p, μ,v) we have them = 6

The number of fundamental dimensions (Mass, Length, Time), m = 3 (That is, [M], [L], [T])

By Buckingham's pi theorem,

Number of dimensionless groups = number of variables - number of fundamental dimensions.

∴No. of dimensionless groups = n -m = 6-3 = 3

To apply the Buckingham Pi theorem, the reoccurring set must contain three variables that cannot themselves be formed into a dimensionless groups.

For this problem we have the following  variables chosen as the recurring set : d, v and p.

The others were left out for the following reasons:

By dimensional analysis,

The dimensions of these variables are :

d = [1]

v = []

p = []

we can form  dimensionless groups from each of the remaining variables ΔP, L and μ in turn

Dimensions of ΔP

Dimensions of

Hence we have that is a dimensionless quantity

Plugging in the dimensions of each of the variables, we obtain the first Pi group as

Dimensions of L

L has the dimensions of [L]

to make it dimensionless, we will have to multiply it by the inverse

is therefore dimensionless

The second Pi group we have is

Dimensions of μ

μ has the dimensions of

is therefore dimensionless

substituting in the dimensions of each variable, we have  the third dimensionless group as

Thus we have

∴ A relationship for the pressure loss is  

freezing

deposition

condensation

explanation;

melting also called fusion is the process where solid becomes liquid, sublimation is the process where solid becomes gaseous without first changing to liquid, deposition involves change from gaseous to solid state, freezing involves change from liquid to solid, boiling or evaporation involves change of liquid to gaseous state, while condensation is the change from gaseous state to liquid state.

endothermic reactions involves absorption of heat from the surroundings or the environment while exothermic reactions involves loss of heat energy to the surroundings.

in this case, sublimation, evaporation and melting are endothermic reactions (gain energy from the environment) while, freezing, deposition and condensation are exothermic (loss of energy to the surroundings) reactions.

total distance travelled = 60 miles

total time taken =1.5 hours

average speed = total distance / total time

60 / 1.5

= 40

therefore, average speed = 40 m/h

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