The math thread.

[Indy]

I had an idea last night. It was that it would be nice to have a definitive thread with all sorts of automotive and performance-related mathematical equations in one spot, where questions could also be answered (because I realize these can be found on other sites on the net). I'm looking for everything from the basics, to advanced as people want to get. Engines, suspension, flow dynamics, whatever. Just post it up, say what it's for, and try to be clear about variables and such (duh).

Have at it.
Nate

[groovieguy]

How Do I Calculate Compression Ratio?
As you can see from the formula below, to calculate compression ratio we need to know two things: swept cylinder volume and combustion chamber volume.


Swept Cylinder Volume + Combustion Chamber Volume
Compression Ratio =
--------------------------------------------------------------------------------

Combustion Chamber Volume



Swept Cylinder Volume
Swept cylinder volume is the area of the cylinder covered by the "sweep" of the piston as it moves from BDC to TDC, indicated by the blue region in Figure 3. Crankshaft stroke and the size of the cylinder bore influence swept cylinder volume, as seen in the formula below: Piston at BDC





Swept Cylinder Volume = (pi / 4) x Cylinder Bore2 x Crankshaft Stroke



As an example let's take a typical 350 small-block Chevy V8. The factory stock cylinder bore is 4 inches and the crankshaft stroke is 3.48 inches. Applying these values to the formula for swept cylinder volume we get:


Swept Cylinder Volume = 0.7854 x 16 x 3.48 = 43.731072 cubic inches



That's it for swept cylinder volume. Now let's take a look at the combustion chamber.

Combustion Chamber Volume
Combustion chamber volume is the total volume above the piston when it's at TDC, indicated by the yellow region in Figure 4. To calculate combustion chamber volume we need to consider four things:


Cylinder head chamber volume
Compressed head gasket volume
Piston-to-deck clearance
Piston dish or dome
Piston at TDC



Cylinder Head Chamber Volume
Cylinder head chamber volume refers to the volume of the head's combustion chamber area, measured with the valves closed. Unless the cylinder head is brand new it might have been milled, ported, or otherwise modified. Thus for maximum accuracy it is best to measure cylinder head chamber volume using a process known as "cc'ing" (so named because head chamber volume is commonly expressed in cubic centimeters, or cc's.) For our example engine let's assume we have unmodified, original equipment heads with 76cc chambers.

Compressed Head Gasket Volume
The lowly head gasket contributes to compression ratio as well, both through its compressed thickness and its bore (gasket bore is typically larger than cylinder bore.) Gasket bore and compressed gasket thickness figures are available from the head gasket manufacturer.


Compressed Head
Gasket Volume = (pi / 4) x Gasket Bore2 x Compressed Gasket Thickness



Using a 0.042 inch thick gasket with a 4.125 inch bore the formula yields:


Compressed Head
Gasket Volume = 0.7854 x 17.015625 x 0.042 = 0.561291 cubic inches



Piston-to-Deck Clearance
Piston-to-deck clearance refers to the position of the top of the piston relative to the block's deck surface when the piston is at TDC. The piston might be even with the deck surface (often referred to as "zero deck"), above the deck surface, or below it. If the piston is below deck then volume is added to the combustion chamber, as shown by the green region in Figure 5a. If the piston is above deck then volume is subtracted from the combustion chamber, as shown in Figure 5b. Pistons at zero deck do not contribute to piston-to-deck clearance volume.


Here's the formula to calculate piston-to-deck clearance volume:


Piston-to-Deck
Clearance Volume = (pi / 4) x Cylinder Bore2 x Piston-to-Deck Clearance



Note: For pistons above deck use a negative number for "Piston-to-Deck Clearance". For zero deck applications use 0.

If the piston in our small block Chevy is 0.015 inches below deck then the clearance volume will be:


Piston-to-Deck
Clearance Volume = 0.7854 x 16 x 0.015 = 0.188496 cubic inches



Piston Dish or Dome
Not all pistons are totally flat across their top. Some have "popups" or domes, some have a recessed area or dish, and others have valve reliefs or have been fly-cut for valve clearance. These differences in piston design affect combustion chamber volume. In most cases the required data can be obtained from the piston manufacturer. For our example let's assume we have pistons with 4cc valve reliefs. Note: Use a negative number for domed or popup pistons. For flat-top pistons use 0.

Putting It All Together: Combustion Chamber Volume
Now we finally have enough information to calculate combustion chamber volume:


Combustion Chamber Volume = Cylinder Head Chamber Volume + Compressed Head Gasket Volume + Piston-to-Deck Clearance Volume + Piston Dish or Dome



But wait—before plugging in the numbers there's one important detail we need to address. You might have noticed that we calculated compressed head gasket volume and piston-to-deck clearance volume in cubic inches, but our cylinder head chamber volume and piston dish/dome were expressed as cubic centimeters. We need to convert all our figures to the same unit of measure. 1 cubic centimeter = 0.0610237441 cubic inches, so our 76cc cylinder heads equal 4.637805 cubic inches and our 4cc piston becomes 0.244095 cubic inches. Using cubic inches as the common unit of measure, total combustion chamber volume is:


Combustion Chamber Volume = 4.637805 + 0.561291 + 0.188496 + 0.244095 = 5.631687



Putting It All Together: Compression Ratio

Whew! All that's left is to plug our swept cylinder volume and combustion chamber volume into the formula for compression ratio. Note that both volumes are expressed as cubic inches (Figure 6): Piston at BDC
Combustion Chamber Volume
5.631687 in3
Swept Cylinder Volume
43.731072 in3
Figure 6


Once again the formula for compression ratio is:


Swept Cylinder Volume + Combustion Chamber Volume
Compression Ratio =
--------------------------------------------------------------------------------

Combustion Chamber Volume



Doing the math and rounding to three decimal places we get:


Compression Ratio = 43.731072 + 5.631687 / 5.631687 = 8.765



...so our engine's compression ratio is 8.765:1.




The pi symbol I forget how to do it on the computer

[p8ntman442]

Swept Volume = bore x bore x stroke x 12.8704

I dont follow this, that would be a square bore. the volume of a cyl is pi*r^2*L or
3.14*4.00*3.48 = 43.71

please correct me if Im wrong, and if so delete this post so this thread (which is a good idea) can stay clean.

[groovieguy]

I corrected it, we were both wrong.

[Shaun41178(2)]

E=MC2

[The Dark Side of Will]

Memorizing formulas is just an excuse for not understanding how things really work.
Once understanding is attained, memorizing formulas is not necessary.

[eHoward]

V=IR

[Pyrthian]

Memorizing formulas is just an excuse for not understanding how things really work.
Once understanding is attained, memorizing formulas is not necessary.

lol - so so true

[Doug Chase]

Here's a handy number:

.7854

It's 1/4 of pi.

Now when you're calculating area you don't have to divide diameter in half then square it.

Just do d * d * .7854