Quote:
Originally Posted by swamp2
That still incorrect. There is more torque even not counting intertial terms.
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I disagree; I still believe the previous conclusion is accurate where an increased torque arm is offset by a smaller piston area. I thought I could demonstrate this through instantaneous torque, but I was mistaken. The math is more complex and my previous demonstration was oversimplified and inaccurate. I now realize that to complete the demonstration, the full cycle of the engine needs to be considered.
Quote:
Originally Posted by swamp2
Error in that formula. I was also counting too much on a text rather than doing the full derivation manually. The correct expression is (again dropping terms of order (r/l)^2
Tp ≈ P*D/2*(sin(θ) + (S/4*l)*sin(2θ))
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I struggled a bit to reconcile the expression above, maybe due to misinterpretation on my part. By interpreting it the following way, I was able to come decently close to the longer expression I quoted in my previous post:
Tp ≈ P*D/2*(sin(θ) + (S/
(4*l))*sin(2θ))
The latter however remains an approximation. Probably due the omission of the (r/l)^2 term you mention.
Quote:
Originally Posted by swamp2
None of those complex arc functions are required. The exact expression including (r/l)^2 is not too much more complex.
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The expression from my previous post was derived manually and most likely can be simplified to a shorter expression through some algebraic gymnastics; I was just too lazy to do so
. But it remains mathematically valid and exact.
Quote:
Originally Posted by swamp2
However, the inertial terms can be a very significant contributor to the torque…
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I agree that when evaluating instantaneous torque values, the inertial terms do have an important impact. A piston engine relies in “storing” a part of the energy from the compression stroke in momentum for the other three strokes (exhaust, intake and compression). As for the translation momentum of the piston and rods (and other components), if I am not mistaken, the forces used to accelerate them in the first portion of the stroke are recuperated when they slow down in the second portion.
So for an engine operating at steady state, the inertial terms should cancel out through an entire 4 stroke cycle, wouldn’t they
?
Quote:
Originally Posted by swamp2
Thus, "more stroke at a constant displacement does not make more torque" is a pretty decent engineering approximation. However, the inertial terms can be a very significant contributor to the torque, making this good engineering approximation nothing more than a point of mild curiosity (again for those engines with massless components or running at very low rpm).
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I think it is much more than just that.
I now realize that trying to demonstrate this through force equations is way too complex. So I’ll try to present it from a different perspective using the conservation of energy principles instead. This gets rid of all the geometric and inertial factors.
Assume two isentropic engines operating at the same constant RPM for which all parameters are equal (cylinder count, displacement, compression ratio, volumetric efficiency, thermal efficiency, con-rod length, etc...) except for the bore and stroke.
Both engines are sucking-in the same amount of intake charge (same displacement, RPM and VE). So, for a given amount of rotations, the exact same amount of energy (chemical here) is input to the system. The output energy is measured by the torque produced over the given amount of rotations (work). If the engine with the longer stroke generates more torque, where is the other engine loosing its energy to? I know, this is still very simplified, but provides food for thought.
From a design perspective, varying the bore and stroke ratio allows other parameters to be altered such as volumetric efficiency, valve size, blow-by losses, friction losses and many more that will vary the torque output depending on operating conditions (low vs high RPM for example). But IMHO, the conclusion remains, for a given displacement, the mechanical benefit of a longer torque arm is offset by a smaller piston area.