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In our introductory pump video we talked a little bit about

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

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

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

Â So.

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

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

Â which happens to be an axial piston pump.

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

Â styles of positive displacement pumps that we

Â talked about in the, in the previous lecture.

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

Â And if you recall how it works.

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

Â It is not moving.

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

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

Â And then as this rotates, these pistons

Â reciprocate back and forth, causing the pumping action.

Â So they move nearly [UNKNOWN] in their, in

Â their pistons, or in their, in their cylinders.

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

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

Â And so in that joint we have one leakage of the

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

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

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

Â The second is we have viscus friction between these two

Â between the piston and the cylinder so these moving parts.

Â 1:23

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

Â only do our rolling element bearings that are

Â supporting the shaft have friction, but also we have

Â a number of hydrostatic bearings in our, in

Â our system where if you can imagine these

Â pistons right here, these, the slippers of the

Â pistons are in contact with that angled swash plate.

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

Â To the, to the slipper joint.

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

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

Â 1:56

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

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

Â We have fluid compressibility, because normally we think about oil

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

Â And that adds up to energy loss because every,

Â every time the low pressure fluid comes in here.

Â We first of all have to compress it.

Â Maybe three percent with, with most hydraulic fluids.

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

Â But, there's a small dead volume left.

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

Â we decompress it, therefore losing that

Â energy that went into compressing the fluid.

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

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

Â and, and throttling loss as well as we are.

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

Â So many sources of energy loss in the pump.

Â What I want to do is do kind of a

Â detail dive into one specific form of energy loss, which

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

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

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

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

Â So, I have a diagram of it here just a

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

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

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

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

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

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

Â 3:38

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

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

Â And so, if I draw a velocity profile,

Â it would look something like this for Newtonian fluid.

Â Or I've got a velocity gradient along that,

Â along that gap between the piston and cylinder.

Â Now I be drastically exaggerated the size of this gap

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

Â You know, somewhere in the 10 to 30 micron

Â type of a, a gap region here so fairly small

Â but still we get a reasonable viscolocity because of

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

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

Â we refer to a quaint flow which is due

Â to the moving of the piston right here so,

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

Â 5:08

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

Â the, the quaint flow because it's going to

Â be going in the direction my piston's moving.

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

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

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

Â So let's start modeling these two terms.

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

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

Â It actually turns out that if my piston is all the

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

Â This assumption does make a big difference.

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

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

Â Therefore I have laminar flow, an incompressible fluid, and

Â all the, the velocity is in the axial direction.

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

Â it using the equation typically used for flow

Â between parallel plates, laminar flow between parallel plates.

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

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

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

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

Â By the circumference, which is that pi d

Â that first piece of, of the equation there.

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

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

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

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

Â length as well as the dynamic viscosity of the fluid.

Â And obviously this is also proportional to the.

Â To the pressure differential.

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

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

Â The leakage flow rate with respect to time.

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

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

Â is because I only have this pressure graded for

Â half the cycle, when I am exposing the outlet

Â here to my high pressure port if you will.

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

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

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

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

Â And because I'm only experience pressure half

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

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

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

Â 7:39

And then I substitute that in.

Â And then I can integrate this term.

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

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

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

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

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

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

Â 8:03

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

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

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

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

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

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

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

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

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

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

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

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

Â 8:48

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

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

Â But now, because this viscous loss is occurring both during

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

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

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

Â In an axial piston pump this the, the

Â piston position is very close to sino soil displacement.

Â Therefore I can take the derivative of that.

Â And get the velocity with respect to time.

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

Â equate, this energy loss and this integral, in

Â a little bit more, easy to manageable fashion here.

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

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

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

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

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

Â 9:59

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

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

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

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

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

Â And this is one of the classic engineering

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

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

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

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

Â In one of them I've got the length of

Â the numerator and the other one is the denominator.

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

Â And you know, quite a few number of things.

Â 10:41

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

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

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

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

Â the two equations that I created and I'm just

Â going to simply code them in MATLAB, and then create

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

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

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

Â Later in this class.

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

Â two equations in, to be able to

Â understand this, this relationship a little bit better.

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

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

Â And what I first of all have is.

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

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

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

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

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

Â The, the, the movement back and forth, and

Â a fairly common dynamic viscosity for our hydraulic oil.

Â 11:53

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

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

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

Â How do you know what reasonable values are?

Â Well, it really kind of comes from

Â experience having known what most pumps are,

Â are manufactured at and also saying what

Â would I reasonably want to try to, to manufacturer.

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

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

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

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

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

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

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

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

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

Â Create a three-dimensional plot and then just create

Â some x labels, y labels, the z label.

Â And then put a put a title on the plot.

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

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

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

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

Â And so you can see that on this lower

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

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

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

Â So, again this classic trade-off here.

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

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

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

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

Â going to say, all right, maybe 10 microns

Â is the very smallest gap that I want to have.

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

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

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

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

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

Â 14:16

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

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

Â to disassemble, you have to, you know, inspect

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

Â What more often happens to treat all of these energy

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

Â Now let me characterize these experimentally.

Â So I'm going to run a large number

Â of experiments and really develop a map of the

Â efficiency as a function of the, perhaps the pressure,

Â the speed, perhaps the volumetric displacement of the pump.

Â And then take all of that and curve fit

Â it and these some common equations that are used so

Â you might remember our classic or theoretic equations was

Â just Q was equal to Omega D over two pi.

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

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

Â And so for our volumetric efficiency, we have first

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

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

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

Â So, these are the two terms that are being added

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

Â On a mechanical efficiency, which is

Â dealing with the torque pressure relationship.

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

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

Â So I've got those two terms they are coming

Â together to predict a a mechanical efficiency for this pump.

Â And then we can take that experimental data, create

Â this model, and then use that for our simulations.

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

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

Â 15:59

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

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

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

Â For the other forms of energy loss in the pump.

Â And then we also talked about how we

Â can take experimental data, fit that to curves.

Â And then create these, these these maps that

Â we then use for, for other simulation work.

Â [BLANK_AUDIO]

Â