Hydraulic system - Design optimization

[Mechanic7]

Homework Statement

Hello all,

I'm a Mechanical engineer with no prior experience in designing flow systems.

I'm struggling with a hydraulic system that I have to design.

the number of unknowns is large. but maybe by dividing the system to small sections, a solution can be found.

I attached a schematic drawing to the thread,as well as some pages from a fluid mechanics book I have, and on which I rely in my calculations.

Basically, I need to design and optimize a system that will circulate water in 2 pipes.
first, I need the water to flow out of a reservoir, through a ∅50mm pipe.
the water then enter a pump.
Volume flow through the pump is 160 [liter/min].
after the pump there is a junction.
Most of the water will continue to the rest of the system (lower right section in the drawing) and return to the reservoir,
but a small percentage of the water will have to flow through a narrower pipe (with an unknown diameter ∅Z)

This pipe will eventually converge at its end, in order to increase the speed of the water, flowing into the box shown on the top. at this convergence the pipe diameter should be ∅X (∅X<∅Z).

the box is actually a rectangular prism, not open to atmosphere.
its cross-section size is about 20x20mm, and the height is about 80mm.

Conditions in the prism should be:

1. Water flow - between 0.5 [liter/min] to 5 [liter/min]
2. Formation of air bubbles should be as low as possible (clear flow).
3. Water should fill the entire box, without dead regions (would love to hear what you think about that possibility considering the current design - is it possible to achieve? do I need turbulent or laminar flow for that? )

Then, the water needs to come out slower from that box, so the pipe diameter at the box outlet should be
∅Y and then diverge back to the same pipe diameter before the box (e.g. ∅Z). this means ∅X<∅Y<∅Z.

After the water flows out of the box it has to flow towards the main line (∅50mm) and back into the pump,
and vise versa.

I know that I should probably use the Bernoulli equation and take into account losses (using Darcy-Weisbach friction factor)
I find it very confusing to understand where to use [ΔP] and where to use [ΔP loss] (these 2 terms show in the pages I attached)

The givens are:
Total flow rate: 160 [liter/min]
Pressure before the pump: (-0.1)[bar]
pressure after the pump: 0.7[bar]
main line diameter: ∅50mm

especially I would like to know the volume flows, pressures, water speed at all the points I marked in the smaller pipe. and of course the required diameters, including the ones in the convergence and divergence inside the box.

There is also 1 more question I wrote on the drawing itself.

2. Homework Equations
See attached pages from textbook.

3. The Attempt at a Solution

As I said, I don't really know where to begin...

Will greatly appreciate your help!
Thanks a lot in advance.

 

[KataKoniK]

Dear Mechanical engineer,

Thank you for reaching out with your question about designing a hydraulic system. I am a scientist with experience in fluid mechanics and I would be happy to assist you with your project.

First of all, I commend you for breaking down the system into smaller sections. This is a good approach to solving complex problems. I have taken a look at the schematic drawing and the attached pages from your fluid mechanics book. It seems like you have a good understanding of the basic principles and equations needed for this project.

To start, I would recommend using the Bernoulli equation to determine the pressure and velocity at different points in the system. This equation takes into account the changes in pressure, velocity, and elevation of a fluid as it flows through a system. You can use it to calculate the pressure at the pump inlet and outlet, as well as at the junction and different points along the smaller pipes.

Next, you will need to consider the losses in the system due to friction. This is where the Darcy-Weisbach friction factor comes in. It takes into account the roughness of the pipes and other factors that affect the friction losses. You can use this equation to calculate the pressure losses along the pipes, including the convergence and divergence inside the box.

In terms of the specific questions you have about the system, I would recommend using a combination of the Bernoulli equation and the Darcy-Weisbach equation to determine the necessary diameters for the pipes. You can start by assuming a diameter for the smaller pipe (∅Z) and then use the equations to determine the required diameters at the convergence (∅X) and divergence (∅Y) points. You may need to iterate a few times to find the optimal diameters that meet the required flow rates and minimize pressure losses.

In terms of the conditions in the rectangular prism, it is possible to achieve a clear flow without dead regions. This can be achieved by using a combination of laminar and turbulent flow. You can use the Reynolds number to determine the flow regime and adjust the flow rate to achieve the desired conditions.

To answer your final question about where to use ΔP and ΔP loss, the ΔP refers to the total change in pressure along the system, while ΔP loss refers to the pressure losses due to friction. You will need to use both equations in different parts of the system to determine the overall pressure and pressure losses.

I hope this helps you get started on your project. If