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# Is the Volume at the Bottom Dead Center the Same as Displacement Volume?

04/15/2017 3:47 PM

Hi, I'm a new mechanical engineering student in the forum, at the moment I'm studying about the Otto cycle from Thermodynamics from Cengel, 8th edition.

I found a an exercise that has been driving me mad, Exercise 9-2 (1st and 2nd images)

As you can see, they give me a certain total displacement volume, and if we find throughout the exercise the specifics volumes at the TDC and BTC, their difference will be that total displaced volume in terms of mass.

And if we divided the total displacement volume by the difference between the specifics volumes, we should find the total MASS of the content inside the cylinder. In the exercise the don't do that, they divide the total displacement volume only by the specific volume at the BDC, ignoring specific volume at the TDC.

¿Why?

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#1

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/15/2017 3:57 PM

No, the displacement volume is the difference between the volume at bottom dead center and top dead center.

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#2
In reply to #1

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/15/2017 4:21 PM

The compression ratio is the ratio of the Top Dead Center volume and Bottom Dead Center volume. In gasoline engines, it can be as high as 10.5:1. Diesel engines may have a compression ratio over 22:1. So the Bottom Dead Center volume is a fair to good approximation of the displacement volume.

https://en.wikipedia.org/wiki/Compression_ratio

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#4
In reply to #2

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/15/2017 4:26 PM

Ok, but, calculating, the error in the mass with that aproximation is about 12%, a little bit large, is it a good proximation?

Sorry for my childhood english, I'm not native.

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#3
In reply to #1

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/15/2017 4:21 PM

So, is the book wrong?

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#5

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/15/2017 4:34 PM

This might help to understand....sometimes it's best to take a break and come back to something, it takes time to absorb knowledge, time+effort=success....

http://www.ecourses.ou.edu/cgi-bin/ebook.cgi?topic=th&chap_sec=08.1&page=theory

Try not to mix up the terms....

https://www.youtube.com/watch?v=K8RzTmeVWfM

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#10
In reply to #5

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/16/2017 11:28 AM

Ok, I'll try tot take my break

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#6

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/15/2017 5:08 PM

I think I finally understand your question.

Specific volume is the volume occupied by 1 Kg of matter. (It's the inverse of density). So, to determine mass either multiply volume by density or divide volume by specific volume, it's the same thing.

That is all you are doing, calculating the total mass in the cylinders by dividing the volume by the specific volume when the contents of the cylinder are at atmospheric pressure and temperature, which is at BDC. When the pistons compress the gas inside, the mass inside each cylinder does not change.

I hope this explains it.

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#9
In reply to #6

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/16/2017 11:27 AM

I will denote with capital letter the total volume, and with lowercase letter specific volume

I think I'm starting to understand, with the method I thinked was right (m=Vd/(v1-v2)) I only was considering the mass of the displacement volume (That's what v1-v2 represents), but not the rest of the content in the cylinder. , but instead, only v1 does contains the mass of all of the cylinder (what I want to find)

Now, the total volume (wich is divided by the specific volume) shouldn't be only the total displacement volume, but that quantity plus the clearance volume?

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#11
In reply to #9

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/16/2017 12:16 PM

I think you have it. Swept volume or displacement volume represents how much gets pumped through the engine in 2 (four-stroke) or 1 (two-stroke) revolutions. Total volume is BDC volume x 4, which includes some gasses not pumped through. I'm not sure if this agrees with your textbook problem, but it wouldn't be the first time a textbook had an error in it.

One complete cycle of a four-cylinder, four-stroke engine. The volume displaced is marked in orange.

https://en.wikipedia.org/wiki/Engine_displacement

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#12
In reply to #11

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/16/2017 1:04 PM

I think that when they say "total displacement volume", they're talking about the total in four cylinders.

Displacement volume: displaced volume in one cylinder.

Total displacement volume: total displaced in four cylinders.

Anyway, I understood what you wanted to say me, but what I don't quite understand is why in the total volume (no specific) they do not consider clearance volume.

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#13
In reply to #12

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/16/2017 2:39 PM

If you are calculating how much energy is in the fuel mixture, I think you would use the BDC volume mass. The whole thing burns, not just the displacement volume mass.

When ignition occurs, pressure increases, and on the power stroke, the increased pressure and the volume increase (displacement volume) translate to mechanical energy, W, from the engine. W = ∫P(v) dv

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#7

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/16/2017 4:13 AM

This post is intentionally blank...As the question has already been answered.

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#8
In reply to #7

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/16/2017 9:51 AM

Well, it's not quite blank - it does come with a health warning.

Score 1 for Off Topic
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#16
In reply to #7

### Re: Is the volume at the Bottom Dead Center the same as Displacement volume?

04/17/2017 10:15 AM

That should be,"This post is intentionally almost blank as the statement that it is amost blank is not blank." Nuts are irrelevant, though also not blank.

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#14

### Re: Is the Volume at the Bottom Dead Center the Same as Displacement Volume?

04/16/2017 3:38 PM

The displacement volume is the space swept by the piston between BDC and TDC which can be calculated from the area of the piston and the stroke.

Temperature and pressure or mass of gas/fuel do not come into the equation yet.

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#15

### Re: Is the Volume at the Bottom Dead Center the Same as Displacement Volume?

04/17/2017 6:25 AM

Assuming the cylinder is closed, the mass of the contents of the cylinder is the same whatever the position of the piston. The only variables are pressure, temperature and volume.

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#17

### Re: Is the Volume at the Bottom Dead Center the Same as Displacement Volume?

04/17/2017 10:52 AM

Hey you! This is and IDEAL Otto cycle, not a practical one! They give you the compression ratio, and everything else you need to solve the problem.

Use your formula for heat capacity as a function of temperature, and solve for the temperature after combustion (assumption is still at TDC), use the other information they gave you to figure out the average pressure during cycle, etc.

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