Stanley Hookers Constant of Proportionality. 0.422

[Steveb01]

Hi,
I've read Stanly Hooker's autobiography "Not much of an Engineer" and tried to follow the maths in Appendix IV.
On the second page he introduces a constant of proportionality 0.422 in the calculation for Charge Consumption.

He says he uses and lists the Merlin dimensions to arrive at the equation and the constant.

The equation was developed to account for the discrepancy between predicted aircraft performance and the inferior performance actually measured.

I'm wondering if anyone clever here has worked out how he arrived at 0.422 as his constant of proportionality on page 236 in Appendix IV.

I've had several goes at this and the nearest I've come is 0.42
Close, but am I missing something?

In the attached photos Equ 1 is quite easily derived.
Equ 2 introduces the constant.

I've google searched, but although 0.422 is mentioned there has been no explanation.
Thank you if you can clarify.

 

[mikewint]

It's all in the fancy name you give it. If you don't want to impress the on-lookers it is simply put a "FUDGE-FACTOR". You have a bunch of data that you are mathematically fooling with, trying to relate them in some constant manner BUT the results on the left of the "=" sign don't match. Bring on the "FUGE-FACTOR" Multiply the left-side by some number until it equals the right-side. Now to add some validity the number you choose must turn out to be a constant and can't vary much from data set to data set.
Einstein knew that Mass (m) and Energy (E) were two sides of the same coin and were interchangeable BUT the two values were not equal. Enter the "Fudge-Factor" (c^2). Now the two sides are equal E = m c^2
General Relativity: Early on Einstein's equations of General Relativity only worked one of two ways. The Universe was either expanding or contracting. Einstein's fundamental belief was a STATIC universe. Enter the Fudge-Factor (Cosmological Constant). Introduced into General Relativity we have a static universe.
But the wheel turns - Todays discovery of an accelerating expanding universe re-invoke the Cosmological Constant rechristened: DARK ENERGY

 

[SaparotRob]

But the wheel turns - Todays discovery of an accelerating expanding universe re-invoke the Cosmological Constant rechristened: DARK ENERGY

What color is it?

 

[blueskies]

1650 cu inches of displacement goes through 825 cuin of air per rev. This translates to 0.477 cuft per rev. A value of 0.422 would imply a volumetric efficiency of ~88% which seems reasonable.

Per this - AC Eng Perf Analysis at R-R - it would be 88% at a Pe/Pi (pressure exhaust vs pressure intake) ratio of 1.2

 

[Simon Thomas]

Looking at the text, 0.422 = S = clearance volume.

 

[bf109xxl]

 

[special ed]

This stuff is why I do not have a college degree.

 

[Steveb01]

View attachment 825810
View attachment 825811

Thank you,
That's helped, I was aware of the 1/77 constant and can now see the relation to it and the 0.422.

At 6:1 compression ratio and 27 litres.
From EQU 4
1/77 ( 6/(6-1))*27=0.421

Half way between my calculation 0.42 and Sir Stanley's 0.422
It would still be nice to see how 0.422 was derived.
It was not just plucked out of the air.
Thanks again,
Steve.

 

[Steveb01]

View attachment 825810
View attachment 825811

Hi, Sorry, I'm still learning how to navigate around the forum.

Here is my calculation, it's based on 3000rpm; 30"Hg for boost and ambient pressure; and 273degK ambient temperature.

Swept Volume = (5.4/2)**2 *3.142*6 =137.431 IN Cubed per cylinder.
SW*12 and divided by 2 for a 4-Stroke engine = 824.5 IN Cubed per rev.
824.5 Cubic IN = 0.4771 Ft Cubed per rev.

At 3000rpm 0.4771 * 3000= 1431.3 Ft Cubed per minute.
1 Cu Ft of air weighs 0.0807 Lbs.
Wc = 1431.4 * 0.0807=115.514 Lbs per minute = Charge Consumption.

From the given equation Wc = K( N/Tc)( Pc - Pe/6)

K= ((Wc * Tc) / N(Pc - Pe/6)

K= ( 115.514 * 273) / 3000( 30 - 30/6 )

K= 31,531.5 / 3000 * 25

K= 31,531.5 / 75000 = 0.42

It's close, but I'm concerned that it's only at 0 psi boost pressure and at 0 degrees C.

Sir Stanley used the equation to predict further supercharger performance by endeavouring to maximise Wc by increasing Pc and decreasing Tc.

The 1/77 constant is interesting, EQU 4 is a the generic equation that reduces 1/77 to 0.421 for the Merlin engine at 27 litres and 6:1 CR.
I'm still wondering how he calculated them from first principles,
Thank you again,
Steve.

 

[Simon Thomas]

Looking at the units, the gas constant x the mol/lb of fuel has dimensions of m2/s2.K

Putting eqn (4) into mathcad with the provided units, this shows that m2/s2.K are the dimensions required for the constant 77.

[ignore the numerical results, I just wanted the units to drop out. 14.91 mol/lb is AI's guess for a F/A mixture.]

 

[Steveb01]

Looking at the units, the gas constant x the mol/lb of fuel has dimensions of m2/s2.K

Putting eqn (4) into mathcad with the provided units, this shows that m2/s2.K are the dimensions required for the constant 77.

[ignore the numerical results, I just wanted the units to drop out. 14.91 mol/lb is AI's guess for a F/A mixture.]

View attachment 825886

Thank you, I would have expected Wc just to be lbs/min.

 

[z42]

My head hurts from the mixture of SI and imperial units.

 

[Simon Thomas]

Agreed. I blame Hooker, as most of those are what he used.

I suspect the conversion factors are where the 77 comes from , but I hope someone else can work that out. I would like to, but all my books are still on their way from Canada to Australia. My wife is still miffed that I packed 34 boxes of books.

 

[Daggerr]

Just some remarks, without going in too much detail:

1650 cu inches of displacement goes through 825 cuin of air per rev. This translates to 0.477 cuft per rev. A value of 0.422 would imply a volumetric efficiency of ~88% which seems reasonable.

Per this - AC Eng Perf Analysis at R-R - it would be 88% at a Pe/Pi (pressure exhaust vs pressure intake) ratio of 1.2

Hooker does not include a volumetric efficiency in his approach. In other words: he assumes 100 % volumetric efficiency.

Here is my calculation, it's based on 3000rpm; 30"Hg for boost and ambient pressure; and 273degK ambient temperature.

...............................
1 Cu Ft of air weighs 0.0807 Lbs.


...............................

It's close, but I'm concerned that it's only at 0 psi boost pressure and at 0 degrees C.

..........................

The 1/77 constant is interesting,

1) Note that the manifold gas density not only depends on Tc and Pc but also on its molecular weight, which depends on the air/fuel ratio.

2) Tc and Pc are both in Hooker's equation so you need not worry as long as you use the same Tc and Pc in your calculation of gas density as you use in your calculation of K.

3) The 77 is not something magical, but simply 2 times G (gasconstant), 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 instead of 0.421 or 0.42
However for an air/fuel mix of 93/7 as mentioned by Hooker on the second page you posted, G is only about 36.5 L·"Hg/(°K·Lb) so that gives a factor of 73 instead of 77, which would result in an equation constant of 0.444 instead of 0.422 . A difference of about 5 %.

Hooker was a bit sloppy. He ignored volumetric efficiency as well as scavenging.
He mentioned 93/7 air/fuel ratio but actually produced an equation that is only valid for pure air, not for an air/fuel mixture that has a 5 % higher density than pure air.

His booklet 'The Performance of a Supercharged Aero Engine' is much more elaborate than Appendix IV of his biography, but unfortunately contains multiple inaccuracies (simplifications, wrong assumptions, mistakes). Nevertheless it is worth buying as there is nothing else that comes even close.

 

[bf109xxl]

His booklet 'The Performance of a Supercharged Aero Engine' is much more elaborate than Appendix IV of his biography, but unfortunately contains multiple inaccuracies (simplifications, wrong assumptions, mistakes). Nevertheless it is worth buying as there is nothing else that comes even close.

For fun and to practice my LaTeX skills, I translated the scan of this booklet into LaTeX format. I was too lazy to convert the graphics into vector form (although I have all the tools required for that), so I left them as raster images. I can post TeX/DVI/PDF files if it doesn't violate copyright - I'm not quite sure if it's legal.

 

[MIflyer]

Here's a little article I found interesting.

 

[Steveb01]

Here's a little article I found interesting.

Hi, Thank you, that article is indeed very interesting.

 

[Simon Thomas]

I am not sure I would agree that Hooker's work was sloppy. The overall purpose of the text was to find a better way to determine power at altitude. Hawker's separately performed the same task. They both determined a technique that successfully measured power at altitude within the sort of tolerance expected with engines and airframes.
It is easy to look back now with modern tools and find the simplifications too coarse, but back then with slide rules and grunt work it would have taken half a day to do what can be done now in seconds.
It would be interesting to go through each of the mistakes and find out if they are genuine errors, rounding, transposing numbers or what-not.

 

[Daggerr]

It is easy to look back now with modern tools and find the simplifications too coarse, but back then with slide rules and grunt work ............

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.

 

[Simon Thomas]

Would you agree that sinø = ø for small angles of ø?