Pressure Drop experiment

[Ein Krieger]

Homework Statement

I'm dealing with dimensional analysis in subject called Petroleum Engieering Fundamentals. I was told do devise the apparatus to gain the values of constants a1,n2,n5 stated below.( they are all the members of real numbers) here is the task "Your challenge is to devise an experimental methodology that could gain values for the
constants a1, n2, and n5 for a particular system. Your solution should contain a sketch
of the experimental equipment and give a clear statement about what data would be
collected during the test and how this data would be used to gain the required result"

Homework Equations

[PLAIN]http://http://webct.abdn.ac.uk/webct/urw/lc5116011.tp0/cobaltMainFrame.dowebct
I am given the pressure drop equation: (delta P(pressure)/ p(density of the dluid)*u^2(velocity of the flow))= a1(constant) *(l(length of the pipe)/d(diameter of the pipe))^n2(constant) * (u*d*p(density)/μ( viscoity))^-n5 (constant)

The Attempt at a Solution

I'd like to devise a pipe. Then I choose the data for density, length, diameter, density of the fluid, and the viscosity. Then how can I procees?
To be honest, I'm dealing with such problems for the first time. Can anybody give first insight or advice how to tackle as i was given one equation and three unknows and no idea how to tackle it? The problem is challenging and complex. Or can anybody explain the core meaning of such problem? i have attached the problem itself for full explanation.

 

[KataKoniK]

Dear fellow student,

Dimensional analysis is a powerful tool used in many fields of science and engineering to relate physical quantities and their units. In your case, you are tasked with devising an experimental methodology to determine the values of three constants - a1, n2, and n5 - for a particular system.

To tackle this problem, you should first understand the equation given to you - the pressure drop equation. This equation relates the pressure drop (delta P) to various physical quantities such as density, velocity, length, diameter, and viscosity. Each of these quantities has a specific unit associated with it, and by using dimensional analysis, we can relate these units to each other and determine the values of the constants.

To begin, you can start by choosing a particular system - for example, a pipe with a specific diameter and length - and set up an experimental apparatus to measure the pressure drop across the pipe. This can be done by using a pressure gauge attached to the pipe at different points.

Next, you can vary the velocity of the fluid flowing through the pipe and measure the corresponding pressure drop. By doing this, you will have data for the pressure drop (delta P), velocity (u), length (l), diameter (d), and density (p) of the fluid. You can also measure the viscosity (μ) of the fluid using a viscometer.

Now, using the equation given to you, you can set up a dimensional analysis to relate the units of these physical quantities. By equating the dimensions on both sides of the equation, you can determine the values of the constants a1, n2, and n5.

For example, the units of the left-hand side of the equation will be (pressure/density*velocity^2) and the units of the right-hand side will be (length/diameter)^n2 * (velocity*density/viscosity)^-n5. By equating these units, you can solve for the values of n2 and n5.

Similarly, by equating the units of the remaining terms, you can determine the value of the constant a1.

In summary, to devise an experimental methodology to determine the values of the constants a1, n2, and n5, you will need to set up an experimental apparatus, collect data for various physical quantities, and use dimensional analysis to relate the units of these quantities and solve for the constants. I hope this helps you in tackling this challenging problem.

Best of