0:00

In our introductory pump video we talked a little bit about

the ideal volumetric flow rate and pressure torque relationships of a pump.

But now let's start exploring some of the inefficiencies of a hydraulic pump and

really we're looking at the energy loss terms of a, of a hydraulic pump.

So.

In a pump, there's lots of different sources of energy loss.

I'm going to pick a certain hydraulic pump architecture,

which happens to be an axial piston pump.

But realize that what we're talking about here is probably applicable to the other

styles of positive displacement pumps that we

talked about in the, in the previous lecture.

So I've got a, a picture of an axial piston pump.

And if you recall how it works.

I've got a swatch plate that's at an angle here.

It is not moving.

And then I'm rotating this, this block with all the pistons relative to that.

So this is drive by the, by the shaft of the pump.

And then as this rotates, these pistons

reciprocate back and forth, causing the pumping action.

So they move nearly [UNKNOWN] in their, in

their pistons, or in their, in their cylinders.

So let's talk about a few sources of the energy loss.

The first that I'm going to focus on here, is in a piston cylinder itself.

And so in that joint we have one leakage of the

high pressure flow past the, the small clearance that we had between

the two of them, so this is purely a clearance we

don't have any other types of seals in this, in this area.

The second is we have viscus friction between these two

between the piston and the cylinder so these moving parts.

1:23

Other sorts of energy loss in this pump we have bearings and they have friction, not

only do our rolling element bearings that are

supporting the shaft have friction, but also we have

a number of hydrostatic bearings in our, in

our system where if you can imagine these

pistons right here, these, the slippers of the

pistons are in contact with that angled swash plate.

Well, they have a high pressure flow path coming from the cylinder.

To the, to the slipper joint.

And then that becomes a hydrostatic bearing creating clearance between the two

of them while there is viscous friction associated with that, with that bearing.

1:56

We have seal friction, we need to be able to seal the, the case

of the pump from, from the outside so we have friction associated with that seal.

We have fluid compressibility, because normally we think about oil

as being incompressible, but there is a small amount of compressibility.

And that adds up to energy loss because every,

every time the low pressure fluid comes in here.

We first of all have to compress it.

Maybe three percent with, with most hydraulic fluids.

And then we push most of the fluid out of the cylinder.

But, there's a small dead volume left.

And that dead volume of fluid is then open to tank pressure and

we decompress it, therefore losing that

energy that went into compressing the fluid.

And then we have valves that connect these cylinders to high and low pressure ports.

And these valves have associated leakage with them as well as viscous friction.

and, and throttling loss as well as we are.

Forcing flow across the, the partially open, open port.

So many sources of energy loss in the pump.

What I want to do is do kind of a

detail dive into one specific form of energy loss, which

is at the, the joint of the, the piston cylinder,

right their interface, so what I've, what I've highlighted there.

And it is this, this pump right here and right between this piston and

the cylinder block, that's the joint that I really want to take a closer look at.

So, I have a diagram of it here just a

single piston of this multi-cylinder pump and it could be

a, a, a axial piston pump like this, it could

be another style of piston pump, it really doesn't matter.

But what I want to focus on, is this interface between these two surfaces.

And, at this interface, we have, first of all, the velocity of the hydraulic or

of the piston itself, which is moving to the left, so I've got a velocity here.

3:38

And I know that I have a zero slip boundary condition between the

fluid and the piston itself, and also between the cylinder and the fluid.

And so, if I draw a velocity profile,

it would look something like this for Newtonian fluid.

Or I've got a velocity gradient along that,

along that gap between the piston and cylinder.

Now I be drastically exaggerated the size of this gap

here, but just to give you a feeling, were talking.

You know, somewhere in the 10 to 30 micron

type of a, a gap region here so fairly small

but still we get a reasonable viscolocity because of

that gap but also a large amount of, of leakage.

So I've got this, this velocity gradient here which

we refer to a quaint flow which is due

to the moving of the piston right here so,

what I've labeled right here this would be quaint flow.

5:08

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.

7:39

And then I substitute that in.

And then I can integrate this term.

And as I integrate it I get a term, or an expression, for

the energy loss, per-cycle of my pump, for the, for the leakage here.

And you'll notice, again, that it's a cube of

the frequency, it's a square of the, the pressure differential.

So as we're increasing pressure, this leakage term is going to go up

with the, with the square of the pressure so this becomes quite important.

8:03

Now let's also take a look at the, the friction.

So our friction, well we're going to be concerned with a forced and a velocity

and so first my friction force I'm just going to use Newton's Law viscosity here,

rearrange it a little bit and I can then say the surface area this is

acting upon, is just the circumference of the piston multiplied by the length of it.

And I'm going to neglect the fact that perhaps the, the length that

this is acting upon might be changing, as I'm moving through the strokes.

I'm going to assume that my, my length is constant here.

And then I have, have this term here where I've

got velocity in the numerator, and the clearance in the denominator.

But we're only linearly equating this to, to clearance.

Not the cube like we had with the, the leakage term.

8:48

Now, to get the frictional energy loss, I'm going to integrate this the, the

force and the velocity with respect to time, again over a over a cycle.

But now, because this viscous loss is occurring both during

the, the high pressure zone and the low pressure intake stroke.

I have to go across the entire cycle with my, my integration limits.

So, I need to get a feeling for what the velocity profile looks like in my piston.

In an axial piston pump this the, the

piston position is very close to sino soil displacement.

Therefore I can take the derivative of that.

And get the velocity with respect to time.

So I've got this term, for velocity, I then substitute that in, and, now I can,

equate, this energy loss and this integral, in

a little bit more, easy to manageable fashion here.

Now, you might be looking at this and saying, I don't quite

remember what the, the integral of the cosine squared term is here.

Well, it turns out to be a, a very simple value

when we're integrating it from 0 to 2 Pi over omega.

This term right here, ends up just being Pi over omega.

9:59

So, when we substitute integrate that and then plug that in,

we get this for the, the frictional energy loss per cycle.

So now that I've got these two different terms dealing

with the piston cylinder interface, I can add these two up.

And then start to look at the, the tradeoffs between them.

And this is one of the classic engineering

trade-offs where I've got so many competing variables.

If you imagine that I, I add these two terms that are boxed here.

In one of them I've got the cube of the clearance in

the numerator, in the other one I have the clearance of the denominator.

In one of them I've got the length of

the numerator and the other one is the denominator.

And again, I've got multiple trade-offs dealing with pressure.

And you know, quite a few number of things.

10:41

I want to talk about how we evaluate a couple of these.

So, I'm going to take a, a little look at what happens

as we trade off the clearance and the length of the piston.

So, what I'm going to do, is I'm going to take

the two equations that I created and I'm just

going to simply code them in MATLAB, and then create

a plot of what the, the energy loss looks like.

The total energy loss for this, this piston cylinder interface.

Now, we're going to talk a lot more about simulation tools.

Later in this class.

But I just want to give you a feeling for how I would plug these

two equations in, to be able to

understand this, this relationship a little bit better.

So I'm going to jump from here right into my MATLAB code.

And I'll give you a little highlight of what I'm doing here in the, in the code.

And what I first of all have is.

As I go down here, I'm going to define my variables, and I've

picked some fairly common values here, 21 Mega Pascals, about 3000 PSI a

diameter of piston of a centimeter, and angular velocity of about 60 Hertz,

which would be about 360 revolutions per minute, a very common pump speed.

I've got a stroke of a centimeter for the piston.

The, the, the movement back and forth, and

a fairly common dynamic viscosity for our hydraulic oil.

11:53

Now, I keep going down here, and now I'm also going to set up

arrays here for the, the clearance c and the length of the piston l.

And so I'm setting these up between reasonable values, and you might say.

How do you know what reasonable values are?

Well, it really kind of comes from

experience having known what most pumps are,

are manufactured at and also saying what

would I reasonably want to try to, to manufacturer.

What would I try, want to try to tolerance these at.

Now can you imagine trying tolerance a piston cylinder interface to have.

Smaller than a 5 micron radio gap, that gets really expensive to manufacture.

So I've defined those, I then just need to put these into a, into

a a grid a, a 2D grid for the clearance and for the leakage.

And then I can write my two energy loss equations, so I've literally copied

on what's on the PowerPoint slide in the back onto my Mat-Lab code here.

For the, the leakage energy loss and for the viscous friction energy loss.

I then I'm going to add these two up.

Create a three-dimensional plot and then just create

some x labels, y labels, the z label.

And then put a put a title on the plot.

So a fairly simple code it's posted on the, the the mook site

so you're welcome to take a look and play around with it yourself.

So what I'm now going to do is I'm going to go ahead and

run this and it's going to create this plot that I mentioned.

And so you can see that on this lower

right axis I've got the clearance in, in microns and

then I've got the piston length on the left axis

and on the z axis I've got the energy loss.

So, again this classic trade-off here.

And you'll notice that where it's driving me, is it's driving me towards

very small clearances, and therefore very small, or short, lengths of the piston.

Now, I was saying I really don't want to

manufacture these very tight tolerances, so quite likely I'm

going to say, all right, maybe 10 microns

is the very smallest gap that I want to have.

And then I can say all right, let me take a cross sectional slice of this plot.

At 10 microns and then I'll find a minimum of the energy

loss which will drive me towards a, a desired length of my piston.

So again classic energy loss trade offs in a pump and in just one

very small example out of the many forms of of energy loss in our pump.

14:16

Now how do you this for a, for a, a pump that you might buy from a manufacturer?

Now, you know, we're, you could try and model all of these but, I mean, you have

to disassemble, you have to, you know, inspect

everything, there's a lot of unknowns going on here.

What more often happens to treat all of these energy

losses, we say, I recognize I have all these losses.

Now let me characterize these experimentally.

So I'm going to run a large number

of experiments and really develop a map of the

efficiency as a function of the, perhaps the pressure,

the speed, perhaps the volumetric displacement of the pump.

And then take all of that and curve fit

it and these some common equations that are used so

you might remember our classic or theoretic equations was

just Q was equal to Omega D over two pi.

Now X is just the, the fractional displacement of the pump, but

whats now in the brackets here, this is just the efficiency term.

And so for our volumetric efficiency, we have first

of all our slip coefficient, this is a leakage term.

And then we have our second term right here, where we have

the bulk module in the denominator, well this deals with the fluid compressibility.

So, these are the two terms that are being added

together to come up with the, the volume metric efficiency.

On a mechanical efficiency, which is

dealing with the torque pressure relationship.

Here we again have two different terms, the first one is a

viscous energy loss term, the second one is a coulomb friction term.

So I've got those two terms they are coming

together to predict a a mechanical efficiency for this pump.

And then we can take that experimental data, create

this model, and then use that for our simulations.

And this is more often what is done instead of trying to model the

nitty gritty details of all these different forms of energy loss in a pump.

15:59

So in summary, we've discussed some of the major forms of energy forms in a pump.

We did a deep dive into one specific example, and I

hope that this just forms an example that you could then use.

For the other forms of energy loss in the pump.

And then we also talked about how we

can take experimental data, fit that to curves.

And then create these, these these maps that

we then use for, for other simulation work.

[BLANK_AUDIO]