So, I'm super imposing these two flows on top of each other and first of all

the, the quaint flow because it's going to

be going in the direction my piston's moving.

I'm going to have a net zero flow rate but the, the

parabolic flow created by the, the poiseuille flow the pressure driven flow.

That will, have a, a net, a net leakage which results in the energy loss.

So let's start modeling these two terms.

First of all, let me make a bunch of assumptions.

First I'm going to assume that I have a concentric piston to cylinder interface.

It actually turns out that if my piston is all the

way offset to one side, I increase my leakage about 150% so.

This assumption does make a big difference.

I'm going to assume I have steady flow, that it's fully developed, because this

gap is so small, my ren, my renolds number is going to be small.

Therefore I have laminar flow, an incompressible fluid, and

all the, the velocity is in the axial direction.

Now with this, I can then start to model the leakage, and I'm going to model

it using the equation typically used for flow

between parallel plates, laminar flow between parallel plates.

And in this equation, we refer to the width of the plates.

Well in this case, I've got a piston cylinder and I can

think of that width as just wrapping around the circumference of each piston.

So in this case, I'm replacing the width.

By the circumference, which is that pi d

that first piece of, of the equation there.

Now you'll notice that there's a cube of the, the clearance here.

So c is the,the radial clearance between the piston and cylinder.

So that clearance is enormously important for how much leakage flow rate I have.

And then in the, in the denominator I've got the

length as well as the dynamic viscosity of the fluid.

And obviously this is also proportional to the.

To the pressure differential.

So if I want to calculate what the energy loss is,

I'm going to integrate the pressure differential times the, the flow rate.

The leakage flow rate with respect to time.

And you'll notice here that I'm doing this over half of a cycle.

Now, the reason I'm doing over half the cycle

is because I only have this pressure graded for

half the cycle, when I am exposing the outlet

here to my high pressure port if you will.

When I'm at tank pressure it's very close to case pressure and, so,

I'm going to say I don't have any leakage during that period of time.

Only when I'm connected to high pressure and therefore half of the, the cycle time.

So here, I'm trying to calculate the energy loss per cycle.

And because I'm only experience pressure half

the cycle, I'm going to use that time.

So, I can say, half the cycle is just going to

be pi divided by the, the angular velocity of my pump.