Stanley Hookers Constant of Proportionality. 0.422

[Simon Thomas]

There are no "modern tools" required to see that Hooker's Wc formula with the 0.422 coefficient is only valid for pure air, not for an air/fuel ratio of 93/7 or whatever ratio. Moreover it is obviously only valid for 100% volumetric efficiency and no scavenging.
According to the nomenclature page in the booklet Wc stands for "Rate of mixture consumption - lb/min". c stands for charge to the piston engine.
Not air, that would have been Wa.

Moreover deriving the Wc formula was based on "assuming the specific heat of the charge and exhaust gases tot be equal", which is not really accurate.
Surely in 1941 they were already able to estimate the difference in specific heats and include that into the formula.

I am not going to discuss all the inaccuracies in Hooker's booklet because I don't want to get into endless discussions.
I will say however that Hooker et al, with only a little bit more effort on their slide rules, could have done a much better job.

My books finally arrived! One of them is "Aircraft Propulsion Theory and Performance" by Morley. It expands on Hooker's work.
Seems Dagger made a massive error in his assessment.
In Morley's book, it provides a figure for the Universal Gas Constant G of 2,776 "ft.lb per °K". He also gives an effective molecular weight of air m = 28.9 lb.
He states that this is consistent with the gas constant R of 96 ft.lb per lb.°K.
Obviously mixture strength varies, which will impact the effective weight of the mixture.
Peak EGT is around 34.2 lb. Best power mixture is around 34.8 lb. Typical auto rich will be around 36.7 lb.
This will vary the value of R from 81.2 down to 75.6 96 ft.lb per lb.°K
Using a value of m of 36.15 lb, the R becomes 76.791 ft.lb per lb.°K.
Using this in Hookers formula, gives:

As you can see, Hooker was correct and Dagger was wrong.
Note that Hooker uses "G" where Morley uses "R".

 

[Daggerr]

.....
Seems Dagger made a massive error in his assessment.
.....
As you can see, Hooker was correct and Dagger was wrong.
.....

Dagger was not wrong and certainly did not make a massive error in his assessment.

It is exactly as I wrote before in post #14 above: the 77 in formula (4) is simply 2 times G, expressed in: L·"Hg/(°K·Lb)
For pure air G = 38.4 L·"Hg/(°K·Lb) which would give 76.8 (instead of 77) and therefore an equation constant of 0.422
You would know that that 76.8 in the denominator of the formula is equal to 2*G if you had taken the trouble to derive formula (4) from (2) yourself.
In that case you would also have realised what units of measurement should in this case be used for G, namely L·"Hg/(°K·Lb) in view of the units used for all the other variables. If you were not capable of that you should have politely asked instead of making a fool of yourself by clearly demonstrating that you do not understand the formulas.

What you are doing is completely wrong. If that is copied from Morley's book then that book is also wrong.
You mess around with numbers until you accidentally get a number of about 77 and then assume that that must be the 77 in Hooker's formula (4), but it is not.

The 77 in formula (4) is 2 * 38.4 L·"Hg/(°K·Lb) , not 76.791 ft.lb per lb.°K as you (and Morley ?) seem to think.

For all readers of this topic: there is only one gasconstant in the universe, and that is the universal gas constant.

Gas constant - Wikipedia

en.wikipedia.org

There are however billions and billions of specific gas constants as those depends on the specific gas composition, which gives a specific molar mass, aka molecular weight.
The G in Hooker's booklet is a specific gas constant, so G = R/M in which R is the universal gas constant and M is the molar mass of the gas.

Hooker however does not seem to realise that the G on the left side in equation (2) is not the same as the G on the right side in equation (2) because the molar mass on the left is not equal to the molar mass on the right.
A lot of this kind of mistakes are made all over the process industry. I have seen it too often. Therefore never use a specific gasconstant in an equation, but always use R/M instead so that the impact of M is not hidden and therefore not overlooked.

It is even worse, because Hooker's formula (2) is based on the wrong assumption that there is mixing of 1 volume part of exhaust gas with 5 volume parts of manifold gas. That would only be true if there were no valve overlap and therefore no scavenging. In reality however there is scavenging so the actual mix ratio is much lower than 1 on 5.

 

[Simon Thomas]

Your delusions of understanding are thoroughly entertaining, however you are still profoundly wrong.
I am not going to discuss all the inaccuracies in your posts because I don't want to get into endless discussions.

 

[Daggerr]

I have given Simon Thomas plenty of time now to admit that he blundered, and retract the above insult towards me, but alas.

To avoid that some visitors might get confused by all his nonsense, I will point out his mistakes below.
In my first post here on 5 April I simple ignored his previous posts, which were nonsense, and explained to the topic starter what the basis for the factor 0.422 is. That should have been the end of it, but then, for no good reason whatsoever, Simon starts criticizing and insulting me while posting even worse nonsense.

At the end of this post it will be clear to everyone that Simon is the one with "delusions of understanding" that are "thoroughly entertaining".
He is clearly clueless on this subject and moderators should ban him from this topic to prevent that he continues sabotaging it and spoil it for others.
Best to take all his posts in all topics in the past, and in the future, with a lot of salt, especially if they contain numbers or units of measurement (UOM).

First an introduction and refresher, which most visitors of this topic probably don't need, but just to make sure that we are all on the same page:
There are some 900 (nine hundred) possible UOM's for the Universal gas constant Ru, with some 900 different numerical values. All kinds of strange combinations are possible. I use here Ru instead of R because Hooker already used R for the compression ratio of the supercharger.
There are many examples given here: Universal Gas Constant "R" Values in many units
There are also some 900 possible UOM's for the Specific gas constant (called G by Hooker et al), however with billions of possible numerical values as there are billions of possible gas mixture compositions each with their own molar mass M (aka molecular weight) because G = Ru/M .

However one can't simply select the numerical value that one likes, and treat the associated UOM as mere decoration.
In fact it is the other way around: out of the 900 possibilities one has to select the one with the UOM that is consistent with those of all the other variables in the equation, and the numerical value of that gas constant is then a given, not a choice.

To understand that the UOM's of all variables in an equation have to match (be consistent) seems to be especially a problem for some (not all) of those who grew up with that inconsistent mess of medieval British Imperial units, and express their own weight in "stone".
Moreover anyone who does not understand UOM's cannot understand what a certain formula actually states.

Even Hooker/Reed/Yarker in their green booklet "The Performance of a Supercharged Aero Engine" do not seem to understand what the G they use actually is. On the nomenclature page in their booklet they state that G is the "Gas constant", without indicating that it is "Specific" and without indicating any UOM. They seem to think it is just a general constant with only a numerical value and no UOM, not realizing that it is actually not a constant but a physical property that depends on the molar mass of the gas.
The only correct value for Universal gas constant Ru that is consistent with all the other UOM's in the equations in the green booklet is: 1114 L."Hg/(°K.lbmole) , not any of the 900 or so alternatives.

To obtain the 77 (actually 76.8) in equation (4) requires G to be 38.4 L."Hg/(°K.lb) which means that the 77 , and consequently also the 0.422 in equation (5), are based on a value of M = Ru / G = 1114 L."Hg/(°K.lbmole) / 38.4 L."Hg/(°K.lb) = 29.0 lb/lbmole, which is the molar mass of dry air, not that of a realistic air/fuel mixture, which would always result in a molar mass M that is higher than 29 lb/lbmole.

Surely others must have pointed that out to Hooker already during his life. So later he tried to save face in Appendix IV of his autobiography by claiming that the 0.422 equation is for an air/fuel mixture of 93 % air and 7 % petrol by weight. However in the green booklet there is no mention of any AFR in the derivation of equation (5). That is clearly based on the molar mass and specific gas constant G for pure air to obtain the 77 and 0.422 in his equations (4) and (5), as proven above.
For his claimed AFR of 93/7 = 13.3 the G would be 36.5 instead of 38.4 , the 77 would be 73 and the 0.422 factor would be 0.444 instead, as I already posted here in early April.

Note that equation (5) , the one with the 0.422 factor, has Tc in the denominator, which however cannot be measured. So they came up with equation (7) which supposedly gives Tc as a function of Tci. Actually it is no more than a curve fit of Figure (11) to match desired results with the inaccurate equation (5). Tc is mostly a fudge factor, although they try to give it a plausible scientific explanation but that would only have a small effect on Tc. If they had used 0.444 instead of 0.422 in equation (5) then the curve fit coefficients in equation (7) would simply have gotten different values to again match the same desired results.
Hooker should have combined equations (5) and (7) into one new equation with Tci (which can be measured or calculated) instead of Tc in the denominator, instead of using two separate inaccurate equations, but apparently he did not realise that.
A new equation combining (4) and (7) would still not be perfect though, because equation (6), which is used for Figure (11), is not accurate either.

Also important to note is that the green booklet is thin on thermodynamics, which does not seem to have been their expertise either.
However this, and many other inaccuracies, should not stop anybody from buying and studying their green booklet. It gives a rough indication of the impact of the many variables involved. Far from perfect but usable.
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Simon's first post with nonsense, circled in red:

No, that 0.422 is not the clearance volume. Not even the numerical value would be right, but more importantly: the UOM of 0.422 is °K.lb/"Hg , not Liter, or any other unit of volume.
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His second post with nonsense:

No, although m2/s2.K is one of the 900 possible UOM's for a specific gas constant, it is one of the 899 wrong ones.
They are NOT the dimensions required for 77 as neither m nor s are units in the rest of the equation.
Obviously the units of 77 must match all the units of all the other variables in the equation.
Look what units he came up with for Wc : m2/(s2.°K) . lb/min . Utter nonsense, the units for Wc should simply be: lb/min.
UOM's are a complete mystery to him, so he thinks that Matlab can save him, unaware of the Sh!t-In-Sh!t-Out principle.
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His third post with nonsense:

One of the 900 possible UOM's for Ru is 2782 ft.lb per lbmole.°K, but not 2776 and not ft.lb per °K.
That would however again be merely one of the 899 wrong ones for Hooker's equations.
Molar mass (aka molecular weight) has a UOM of lb/lbmole (or kg/kmol or g/mol), not just lb.
His lack of understanding of UOM's, and consequently sloppiness with their use, also leads to a completely wrong calculation for the molar mass of the air/fuel mixtures that he mentions.
A mixture with an AFR of 93/7 by weight has a molar mass of 30.5 lb/lbmole, not 34.2 or 36.15 or even higher. Even something as simple as that he can't do correctly.
I can guess what mistake was made because it is a classical blunder made by people that don't understand UOM's: they don't realise what it means that molar mass has lbmole in the denominator and simply blend on mass (weight) fractions instead of using mole fractions.

The simplest way to calculate it is: 100 lb of an air/fuel mixture of 93 % air and 7 % petrol by weight (AFR = 13.3) contains 93/29.0 = 3.21 lbmole air,
plus 7/100 = 0.07 lbmole fuel. So mixture molar mass = 100 / (3.21 + 0.07) = 30.5 lb/lbmole.
Or alternatively: mole fraction of air is 3.21 / (3.21 + 0.07) = 0.9787 and mole fraction of fuel is 0.07 / (3.21 + 0.07) = 0.0213
So mixture molar mass = 0.9787 * 29.0 + 0.0213 * 100 = 30.5 lb/lbmole.
Highschool stuff really, but apparently too complicated for certain people.

The Specific gas constant G for this air/fuel mixture would then be:
( 2782 ft.lb per lbmole.°K ) / ( 30.5 lb/lbmole ) = 91.2 ft.lb per lb.°K (which is equivalent to 36.5 L."Hg per °K.lb)
not the 76.791 ft.lb per lb.°K that he fabricated (which would be equivalent to 30.7 L."Hg per °K.lb).
But even if he had used the correct value for M his calculated G would still be wrong because the UOM that he used for G is wrong and therefore its associated numerical value is automatically wrong too.
His statements that I was "wrong" and even made a "massive error" is also wrong. He is the one making "massive errors" all the time.

He does not understand and does not want to hear that G must be expressed in L."Hg/(°K.lb)
I will explain that in more detail below for those of you that are interested in more details.

He does not understand and does not want to hear that the 77 is not G but 2G although I already mentioned that several times here in the past months.
That factor 2 is also a complete mystery to him. To end that discussion I will explain that also in more detail below for those of you that are interested in more details.
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EXPLANATION OF THE FACTOR 2

Hooker's equation (4) in the green booklet (the one with the 77) is derived from his preceding equation (3) as follows:
Hooker's equation (3) gives the weight of charge inhaled per cylinder per cycle:

Merlin has 12 cylinders, so the relation between s (clearance volume per cylinder) and S (swept volume of engine) is:

Inserting that gives that weight of charge inhaled per cylinder per cycle is:

Merlin is a four stroke engine, so it takes 2 crankshaft rotations to complete 1 cycle. Number of cycles per minute is therefore: N / 2
For a Merlin engine with 12 cylinders, running at N rpm, the total engine Rate of mixture consumption (charge consumption) is therefore:

This can be simplified to arrive at Hooker's equation (4):

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UOM ANALYSIS

A UOM analysis (dimensional analysis) can be used to determine what the UOM of G in equation (4) shall be.
Below is equation (4) with all UOM's of all variables indicated. x is dimensionless and has no UOM. In every equation the combination of UOM's on the right side of the equal sign must be the same as on the left side.
In this case both sides have to show only lb/min, which is the UOM of Wc, and any other UOM's must cancel each other out.
The blue UOM's are the ones that shall remain, the red UOM's are the ones that must cancel each other out:

Below only the UOM's are shown without the variable names. G is replaced by ???? because that is the UOM which we want to determine here:

It is evident now that the UOM of G must have Liter and "Hg in the nominator, so that they both end up in the denominator of the equation, and °K and lb must be in the denominator of G, so that they both end up in de nominator of the equation.
As a result the red units in de nominator cancel out the red units in the denominator and only lb/min remains on both sides:

This proves that the UOM of G to be used in this equation is: L."Hg/(°K.lb)
This technique can be applied to any equation in any field of science and engineering.

 

[Snautzer01]

Equation war right there.

 

[Simon Thomas]

How can you change formula 4 in Hooker's paper, and then "prove" he is wrong? You obviously don't understand the step Hooker made. There is no "2" in the denominator. If you were as bright as you attempt to portray, that should be obvious.
You are still wrong.

For those still bored enough to be reading, this figure should have everything you need to know:

This is Hooker's calculation of engine power using thermo and Hawker's calculation of engine power calculated using aerodynamics.
As anyone can see, they are pretty much identical. A logical conclusion would be that using two very separate techniques to calculate the power to be almost identical proves that they are both right (and consequently Dagger is still wrong)
QED

 

[SaparotRob]

You're both wrong. It's all magic and trying to hide it behind "science" is hogwash. You guys aren't fooling me.

 

[Daggerr]

How can you change formula 4 in Hooker's paper, and then "prove" he is wrong? You obviously don't understand the step Hooker made. There is no "2" in the denominator.

I did not "change" formula 4, but merely showed how Hooker/Reed/Yarker derived it from the preceding formula 3 in the green booklet.

The real formula 4 has 2G in the denominator, which Hooker then replaced by 77 (actually 76.8) which is 2 * 38.4 L."Hg/(°K.lb)

The graph you posted is completely irrelevant for the mathematical derivation of formula 4 from formula 3. Apparently only posted it to divert attention from your mistakes.

Even though I clearly outlined above how to go step by step from formula 3 to formula 4 you don't understand it.
Also you don't seem to understand the UOM analysis technique that I outlined above. You are the perfect example of a Briton that does not understand UOM's.
Moreover you are clearly not capable of even the simplest mathematics and physics at high school level.

If anybody else here has questions about my posts feel free to ask and I will gladly answer.
If you prefer to remain anonymous to avoid harassment by a certain person you can send your questions to my mail box.
If your Q and my A could be of interest to others I will post them here but without mentioning your user name, so you will remain anonymous.
Note however that I am not logged in to this forum every day so it can take a while before I notice your message, but you will get an answer.

 

[special ed]

I have read all of these posts and understand none of it, however having made it this far, I feel I am qualified to apply as an aeronautical engineer.
I build and fly models, some more successful than others, and am able to "baffle them with bull shit" as well as most.

 

[Snautzer01]

Daggerr
Simon Thomas

Never in my life i would have thought a fight between aeronautical minds could keep me entertained.

I realley love this stuff. But my talent lie elsewhere.
I do my best to understand what you guys say and as a Neanderthaler try to comprehend.

Stick flies faster because of magic.

So butt heads, but as it should in this wizz kid kind of way.

Please don't be the monkey that throws shit to the other monkey.

So be civil. Your point will be taken more seriously.

Thank you.

 

[Daggerr]

Please don't be the monkey that throws shit to the other monkey.

Simon Thomas started throwing BS around. I merely counteracted by throwing pearls, but it seems that so far they landed before swines.

There is no "2" in the denominator.

Oh yes there is. Unless Morley was a moron there must also be a factor 2 in the denominator somewhere in his book that you mentioned.
And claiming in your every post that I was "wrong" without you being able to provide any proof of that is becoming silly.

I challenge you to prove that all your mistakes that I circled in red in my post #24 are in fact not wrong but strokes of your genius. Show us.

I challenge you to derive Hooker's equation (4) from his equation (3) here yourself. I did it already in my post #24, but apparently you know a trick to come to a different result without that factor 2 in the denominator? Show us.

I challenge you to derive the correct UOM to be used for G in Hooker's equation (4), taking into account that that equation has variables S, Tc and P that are expressed in Liter, °K and "Hg respectively. I proved already in my post #24 that it is L."Hg/(°K.lb), but apparently you know a trick to come to a different result? Show us.

This is not Mission Impossible but merely highschool physics and math.
If you don't accept these challenges then you finally admit that you were wrong in everything you posted here so far.

 

[Snautzer01]

Simon Thomas started throwing BS around. I merely counteracted by throwing pearls, but it seems that so far they landed before swines.

Ah you call me swine. Dearie...

No.

I think you might underestimate a few people here.
You are far from the only one that can throw around math. F.i I am not a stranger to math although this is not my field.

This is not facebook.

For the rest...

MATH FIGHT IN ILSE 3

Relax.

Take a bit of the anger out.

Then it will be a proper fight.

Not dirty.
Like swines do.

Please carry on the debate.
I like this stuff.

 

[SaparotRob]

I'm enjoying this too. I'm usually the pig casting pearls.