Scaling factors

[Bierjunge]

In http://www.paddleducks.co.uk/smf/index.php?topic=3260.msg14305 , Derek had written:

To this point, would you consider posting a listing of 'scaling attributes' for vessel modeling? as I am sure many [including myself] are sometimes confused.....like
 
scale speed = the ? [function of the real vessel speed x the ?? function of  scale]
scale displacement =   the ? [function of the real vessel displacement x the ?? function of  scale]
scale revolutions = the ? [function of the real vessel wheel RPM  x the ?? function of  scale]
plus any other scale conversion values you think appropriate to our craft

I am not Ivor, but I had made some own thoughts on this topic of scaling factors, which might hopefully be intersting for others as well:

In the following, I am referring to the model scale as "length scale".
A model in 1:100 scale is 1 m long, if the original boat is 100 m long.
Then we get for all sorts of physical properties:

rotational speed	scale to the power of -0,5
acceleration	scale to the power of 0
velocity	scale to the power of 0,5
length; pressure	scale to the power of 1
area, surface	scale to the power of 2
volume, mass, force	scale to the power of 3
performance	scale to the power of 3,5
torque	scale to the power of 4
momentum of inertia	scale to the power of 5

A few examples:

The displacement is scaled by the cube of the length scale: A battleship of 10.000 metric tons displacement as 1:100-model weighs 10 kg.

The velocity is a bit tricky:
Model railroaders, for example, scale the velocity by the length scale, to achieve the same optic impression (same wheel speeds etc.).But this leads to some conflicts with the time scale. But anyway, it would be far too slow for our model boats:
Instead of 30 km/h of the original, the 1:100 model would travel only 0,3 km/h (yawn), and the wave pattern would be too flat.

To achieve a scale wave pattern, the model speed is scaled according to Froude by the square root of the length scale: Our 1:100 example boat now travels 3 km/h.
By the way, the so called hull speed (maximum speed achievable without planing) is calculated as 4,5 km/h times square root of hull length. This also proves that speed has something to do with square root of length scale.

Velocity leads directly to rotational speed (of scale propellers or paddlewheels), if we assume scale slip and thread of the propeller or wheel.
These turn by the square root of the length scale faster than the prototype: If the propeller of our prototype battleship turns at 100 rpm, then the prop of the 1:100 model should turn (at least) 1.000 rpm.
By the way, the same factor applies to the pitch, yaw and roll motions of the scale
ship.

Dependent on the speed is the ram pressure, which acts on the front of the hull, on the propeller blade or the wheel bucket:
Because the velocity acts square in the ram pressure, the pressure itself is scaled by the length scale.
That's cool, because then the water columns being piled up by the bow or the wheel bucket has scale height. But that's no wonder, because the main motivation of introducing Froude speed was to achieve scale waves.

Ram pressure and area lead to the resistance force: Scale to the power of 3, which perfectly matches the weight force as well.
But also the centrifugal force acting on propeller blade tips or buckets is scaled this way, if scale rotational speed (see above) is kept.
In this case, the tendency of wheels to water clog their houses should be similar to the prototype. Amazing, how well everything fits together!

Weight and length, or even ram pressure, bucket surface and wheel diameter lead to the torque. This is scaled by length to the power of 4.
But only in few cases the torque of the prototype engine will be known, unless you calculate it from power and speed, or cylinder dimensions and steam pressure.

And finally, from resistance force and velocity, or from rotational speed and torque, you get the engine performance: length by the power of 3,5 (which already I.K. Brunel had found...)!
This means: A battleship of 10.000 kW would need as 1:100 scale model only one single tiny watt for scale speed!
Other example: A 1:32 torpedo boat of originally 1.000 kW needs around 5,4 W shaft power.
But this assumes the same efficiency of the drive train, so you practically should keep some power reserves (not only for the case of dog or swan attacks...)

Enough theory for now...
Regards, Moritz

[derekwarner_decoy]

Hi PD's & thanks Moritz for your comments...I will print them out & try to digest & understand them

Ivor has come back off line & advises he will pen a few notes, however similar to your commments...sometimes water & scale can seem to defy the laws of physics

regards

[Bierjunge]

Hi PD's & thanks Moritz for your comments...I will print them out & try to digest & understand them

It is difficult to explain such a complex topic in just a few lines. If you have any specific questions, just feel free to ask! Then I will try to explain it a little better.

Moritz

[sandy_ACS]

 

Hi PD's,

Well,  after a bit of a problem logging in (my fault )... forgot my own log in info, here is something to think about.



Scale paddle speeds?

Whilst I am in total agreement with Moritz regarding the scaling factors he has put forward, one, in particular, requires some further input/study…. Namely rotational speed.

I have no issue with the case of standard propellers (screws), since the results from many hundreds of models clearly show that the typical figures he suggests/calculates are true, however, I am not entirely convinced in the case of paddle wheels.

If we consider that, for the majority of cases, we would consider the rotational speed for model paddle wheels to be between 80 – 150rpm (typical average speeds as stated in many posts on the subject) to be the norm.
If, however, we apply Moritz’s factor to the same typical models, you get a very different, and confusing, range of speeds.

Take a model of PS Waverley at 1/48th scale… gives a model of some 59.75”.

PS Waverley’s engines turn at 57.8 rpm for a given speed of 18.37Knots.

To make life simple, lets say 60 rpm.

Now at 1/48 scale the speed of the paddle wheels would need to be 6.928 (square root of 48) x this, which gives a model paddle wheel speed of 415.68 rpm.
Which is Considerably higher than the highest typical maximum (80 – 150) by a factor of 2.77.

Lets take another case: - i.e. 1/24th scale.
Assuming a similar full size rotation speed this would result in 60 x 4.8989 (square root of 24) = 293.934 RPM, which is almost twice the typical maximum used for models.

 If we now consider the other implications of this: - Engine speed.

Typically we would be thinking of installing a gearbox/pulley system with a ratio of between 2:1 and 3:1 which for a 1/48 scale model would require the engines to be going at: -

For 2:1    = 831.36rpm or
For 2.5:1 = 1039.2rpm or
For 3:1    = 1246rpm.

For a 1/24 scale model engine speed required would be: -

For 2:1    = 587.868rpm or
For 2.5:1 = 734.835rpm or
For 3:1    = 881.802rpm.

For our Derek, with PS DECOY @ 1/24 scale and 5.5:1 ratio geared chain drive the engine speed would need to be a staggering 1616.637rpm.
 

So what! You ask?

Well, on the surface there does not appear to be a problem, just gear your electric motor accordingly……no problem,   but what about steam engines?

The problem here is twofold: -

1.   Steam engines are designed for best output at relatively low rpm.
2.   Steam consumption becomes a major factor at higher speeds.

The typical model steam engine is best suited to speeds of around 300 – 600 rpm (fully loaded) typically.
So for our 1/48 scale model this would imply either direct drive, or a very low gear ratio.

For our 1/24 scale model we could use up to say 2:1(max) gearing.

Even assuming the particular model steam engine could run at speed of up to 1000- 1200 rpm (fully loaded) without placing un-necessary strain on the moving parts, providing the beast with steam (at full pressure) whilst doing so for more than a few minutes, would be a big problem.
This would require a pretty hefty boiler installation.

In our Derek’s case, with his engine going at 1616 odd rpm (fully loaded) he would need to install 2 or 3 more boilers. But I very much doubt that his engine could achieve such speeds, especially at 35-45psi).
Not that I am suggesting his engine is not suitable, just not designed for such high speed use, as indead non of mine are.

 Clearly there is something else at work here, which the scaling factors (as stated) do not take in to account, since the typical RPM figures of 80 – 150rpm paddlewheel speed are at odds with the above.

What this is; is un-clear, at this point in time.

So..... do we need a steam engine which can run at 1600 rpm or more?, if so, then we will certainly have steam generation issues; or one that can produce sufficient torque at say 300 to 400 rpm for direct drive?... which would tend to favour large bores and long strokes, but again these will impact on boiler size..... or should the paddlewheels be going much slower than Moritz's scaling suggests?

I am unable to come up with a suitable answer to this  ... can any of you ?
 
All very interesting and worthy of more study.

Ok that will do for this post.

Keep happy.

Best regards.

Sandy. 

[Bierjunge]

Hello Sandy,

If we consider that, for the majority of cases, we would consider the rotational speed for model paddle wheels to be between 80 – 150rpm (typical average speeds as stated in many posts on the subject) to be the norm.
If, however, we apply Moritz’s factor to the same typical models, you get a very different, and confusing, range of speeds.

Take a model of PS Waverley at 1/48th scale… gives a model of some 59.75”.

PS Waverley’s engines turn at 57.8 rpm for a given speed of 18.37Knots.

To make life simple, lets say 60 rpm.

Now at 1/48 scale the speed of the paddle wheels would need to be 6.928 (square root of 48) x this, which gives a model paddle wheel speed of 415.68 rpm.

You've calculated absolutely correct. But remember that all this is under the assumption that we're aiming for Froude speed. If not, things look different of course.

Anyway, the Waverley seems to have relatively small (and thus high speed) wheels compared to older paddlers. I have no idea what her wheel diameter is, but from your numbers, assuming a typical wheel slip of some 20%, a would guess that the wheel diameter should be around 3,8 meters (please excuse my metric brain). Close enough, I hope?

If we assume that the 1/48 scale model has the same wheel slip (which has basically something to with the ratio of the fluid resistance of the buckets versus the hull), the model wheel MUST roate your 415 rpm in order to reach Froude scale speed of 2,7 knots or 1,37 m/s (instead of the 18,4 kn or 9,45 m/s of the prototype) at 20% wheel slip. Sorry about this!

If the model wheels turn only 150 rpm, the model would attain (again assuming 20% slip) only 0,5 m/s, which is only 36% of Froude speed (but anyhow 250% of "model railway speed"). The drawback of this is the considerably underscale wave pattern (the waves would have only 13% of scale height, because the speed gous in quadratic); the advantage is the much smaller needed power of only 5% (sic!) of the power (resistance times speed) needed for Froude speed...

But don't worry too much abaout power: I've read the Waverley has 2100 hp (being 1565 kW). With an exponent of 3,5 for the power, the 1/48 model would need 2,0 W for Froude scale speed, which should be achievable by a model boiler. That's for the 415 rpm boat, nota bene.
The "slow 150 rpm variant" would need only 5%, being 0,1 W, mechanical power...

1.   Steam engines are designed for best output at relatively low rpm.
2.   Steam consumption becomes a major factor at higher speeds.

The typical model steam engine is best suited to speeds of around 300 – 600 rpm (fully loaded) typically.
So for our 1/48 scale model this would imply either direct drive, or a very low gear ratio.

I tend to agree that for a "typical 2 cyl model engine" of' let's say, somtehing between 1/2" and 2 cm bore and stroke, and the small wheels of a 1/48 Waverley (some 8 cm???), only a low gearing of not much more than 2:1 should be needed.

Clearly there is something else at work here, which the scaling factors (as stated) do not take in to account, since the typical RPM figures of 80 – 150rpm paddlewheel speed are at odds with the above.

What this is; is un-clear, at this point in time.

If we stick to your Waverley example, could you please provide the wheel dimensions (diameter, width and and height of buckets)? I have written a simple Excel calculator, which gives me the estimated wheels speed, shaft power and steam consumption depending on cylinder dimensions, steam pressure and gear ratio. Then we could try to justify the scaling factors.

But again: As soon as you're not heading for Froude (or so called "scale speed"), things might look different.

Best regards, Moritz

[derekwarner_decoy]

Hi PD's....& welcome back Sandy ...I do freely admit I was stumped with my calculations & assumptions of scaling paddle shaft speed for PS Decoy....so applied the recommendation from ANTON as below for his 2CC horizontal Quartz paddle wheel engine

His WEB site quotes a recommendation of 'a reduction of 6:1 to 8:1' ....so considering that my JMC3H has 50% more displacement than the Quartz I considered I could lower the reduction ratio & maintain the preferred odd to even pinion tooth ratio for chain life & adopted the smallest 9 tooth :48 tooth = 5.5:1 ratio

Finding manufacturers that publish model steam engine speeds is difficult  ...however SAITO do publish  engine speeds & these seem to indicate approx the 3000 RPM [unloaded] ....so considering  a 40% reduction for loading this bought me to consider approx 200 RPM paddle shaft speed max for Decoy which had a safety factor of 30% for emergency.....

Interesting subject



Quote from ANTON.....'This engine, in its horizontal version, and is particularly suited to the propulsion of vessels wheel (side or rear). L'installation du moteur se fera avec une réduction d'un rapport de 6 à 8 (courroie cratée ou pignon). The engine installation will be done with a reduction of a report from 6 to 8 (or cratée belt sprocket)'

Ce moteur sera utilisé dans des bateaux de 5à 10kgrs, avec des roues de Ø100 à 150 mm. This engine will be used in boats 5à 10kgrs with wheels Ø100 150 mm.

[Bierjunge]

Finding manufacturers that publish model steam engine speeds is difficult  ...

How could they do this? The engine speed is HIGHLY dependent on the steam pressure, on the output resistance (size of prop or wheel) and an optional output gearing. So engine speed just HAPPENS, depending on what the custumer does to the engine...
All the manufacturer could to is either
- publish a no-load speed (of questionable use for the customer)
- or a typical load speed under normal conditions (e.g. "with a 90 mm prop at 3 Bar")
- or a maximum speed above which mechanical damage to the engine could occur.

If you give full pressure to an engine with no load on the output shaft however, it will flare to silly speeds of several thousand rpm.
Graham for example gives a no load speed of 4000 rpm at 30 psi for their TVR1A engine ("for the youngsters just to see how fast it will go"), but recommends to do this only for a couple of minutes...

Even if a munfacturer gives a typical load speed (e.g. 1000 rpm), this can vary highly depending on the operational conditions:
- Take the double steam pressure (e.g. 3 Bar instead of 1,5 Bar), and you will get 141% of the engine speed (1414 rpm) and 283% of the power consumption.
- Make the gearing to the wheel shaft twice as short (e.g. 8:1 instead of 4:1), and you will get 283% of the engine speed (2828 rpm), again 141% of the wheel shaft speed and again 283% of the power consumption.
- Halve the area of the wheel buckets, and you will get again 141% of the engine- and wheel speed and 141% of the power consumption.
- But if you combine all these three measures, you will get 566% of the engine speed (5655 rpm, which might damage the engine...) and 1131% (!) of the power (which the boiler presumably won't deliver)

You get similar effects for a propeller drives vessel depending highly on the diameter and, even more important, pitch of the propeller.

Maybe we should ask Eddy to transfer this thread into the Live Steam section?

[derekwarner_decoy]

Hi PD's..........Moritz suggests.......maybe we should ask Eddy to transfer this thread into the Live Steam section? .....this is a good idea Eddy.....

More interesting by the day...however...I am not racing out to purchase a 19 or 21 tooth pinion just yet

1) Australian paddlers travel/ed at approx 3 knots in the rivers system...not like the Scottish paddlers of 15> 18+ knots in their trials in the ocean
2) ANTON is aware that I have a JMC3H & I have questioned his recommended 6:1 to 8:1 Quartz speed reduction....... but with little response 
3) I have a photograph of the straight cut spur gear reduction on PS Marion & this must be in the order of 50:1

Even electric driven paddlers seem to hold a certain mystique over prop vessels...you know  wheels thrashing around but going nowhere fast...it is not a SKI boat......so to combine this with a visual of a real steam engine is my plan  

[derekwarner_decoy]

Hi PD's..... ..to attempt another view or opinion...the following is self explanatory to my engine builder Jean Marc Cloup...let's wait & see if I get a response..... or will it be
...


----- Original Message -----
From: Derek L Warner Pty Ltd
To: jmc
Sent: Saturday, February 16, 2008 10:02 AM
Subject: JMC3H


Marc de Jean de salutations... de Derek Warner en Australie
Mes collègues dans notre groupe modèle mondial de navire de palette discutent actuellement le rapport de réduction de vitesse de moteur de vapeur à l'axe de palette Je n'ai toujours pas fini mon modèle, mais enferme un jpg de ma installation de moteur de JMC3h Si vous seriez ainsi sorte pour offrir votre commentaire ou recommendation sur le rapport de réduction de vitesse de moteur de vapeur à l'axe de palette il serait le plus apprécié sincères amitiés Derek
--------------------------------------------------------------------------------

Greetings Jean Marc...from Derek Warner in Australia

My colleagues in our world wide model paddle vessel group are currently discussing steam engine speed reduction ratio to the paddle shaft

I have still not finished my model, but enclose a .jpg of my JMC3h engine installation

If you would be so kind to offer your comment or recommendation on the steam engine speed reduction ratio to the paddle shaft it would be most appreciated

kind regards
Derek

[derekwarner_decoy]

Hi PD's & I understand as Moritz says......how could they do this? The engine speed is HIGHLY dependent on the steam pressure, on the output resistance (size of prop or wheel) and an optional output gearing

But we teach symplistically that steam & electricity are parallel energy sources......one is contained in a pipe, one is contained in a wire...they both have the potential to expend energy & provide work [output]

They also both have a deadly common factor ...like don't  touch or you will get burnt....

So if the manufacturers of electric motors can talk about no load RPM for voltage & resultant current draw & delta T........model steam engine producers have a similar ability

I also recognise the economies of scale or size here with the multi national MABUCHI company producing 50 million minature electric motors PA to Sandy producing a slightly smaller quantity of model steam engines & so cannot fund the $17.5M for the effiency calculations for each engine design 

[Bierjunge]

My colleagues in our world wide model paddle vessel group are currently discussing steam engine speed reduction ratio to the paddle shaft

If you would be so kind to offer your comment or recommendation on the steam engine speed reduction ratio to the paddle shaft it would be most appreciated

Derek, if you tell us
- the number of cylinders, bore and stroke of your engine,
- the boiler's typical operation pressure,
- the diameter of your paddlewheels,
- the size (width and height) of the buckets/floats
- the average number of buckets being simultaneously immersed at one wheel,
- and maybe the scale and length of your model and the speed of the prototype,
I could offer to maxe some rough approximative calculations on which gearing ratios and speeds might be feasible. Not to make own experiments redundant, but it could give some hints narrowing your solution space.

[derekwarner_decoy]

Hi PD's & thankyou Moritz......this is interesting....my answers below in RED

Derek, if you tell us
- the number of cylinders, bore and stroke of your engine, - two cylinders 10 mm bore x 10 stroke..double acting
- the boiler's typical operation pressure, - WP will be set 1.5 Bar typical
- the diameter of your paddlewheels, - 140 mm diameter
- the size (width and height) of the buckets/floats -width = 50 mm wide x 12 mm high
- the average number of buckets being simultaneously immersed at one wheel, - 3 flat floats equally spaced @ 36 degrees
- and maybe the scale and length of your model and the speed of the prototype, - scale of PS Decoy = 1:24, post to post = 1150mm- speed as built was I believe 11 knots....but in the OZ rivers say 3 to 5 knots max...so the latter 3>5 knots is the max desired scale speed

I could offer to maxe some rough approximative calculations on which gearing ratios and speeds might be feasible. Not to make own experiments redundant, but it could give some hints narrowing your solution space.

Moritz....I have drawn the plans for Decoy from two 50 x50 mm postage stamp photographs...& the engine & boiler selection were simply by choice...& not from original specification...but again I thank you  ...as this is an interesting exercise....regards

[Bierjunge]

Hi PD's & thankyou Moritz......this is interesting....my answers below in RED

That was fast! OK, let's go!

What I'm doing:
First, I'm estimating the torque the engine can deliver. With your data, I get piston downforce of 11,8 N. If we assume 50% cylinder filling (cutoff) and 50% mechanical efficiency (half of the piston force being lost by friction), I get 1,9 Ncm as average crankshaft torque.

Now to the paddlewheel: The idea is that the (geared) engine torque is balanced by the wheel's drag torque, caused by the dynamic fluid pressure acting on the floats.
As simplest (and worst) case, I am assuming a standing boat (like being tied to a pole) to calculate the speed of the wheels churning in still water (100% slip). Even worse and simpler, I'm calculating as if the floats (cw=1) didn't help each other by causing a flow; each acts in still water.
The idea behind this worst case: In reality, if the boat is moving, the resistance on the wheels is much less, so either the wheels could turn faster (and the boat could run faster) than the circumferential wheel speed in the "pole-pulling" experiment, or less steam would be needed.

In the following you get the results for different gearing ratios:

gear ratio	3	4	6	8	10	-
engine speed	273	421	773	1191	1664	rpm
wheel speed	91	105	129	149	166	rpm
circumf. speed	0,67	0,77	0,95	1,09	1,22	m/sec
prototype speed	6,4	7,3	9,0	10,4	11,6	knots
steam mass flow	64	99	182	280	392	grams/hour
output power	0,5	0,8	1,5	2,4	3,3	Watt

Nota bene, the "prototype speed" in the table is not the precise vessel's speed, but just the wheel's circumferential speed after applying Froude scaling and converting into knots. And the steam consumption is just a very rough estimate, treating steam as ideal gas (which it isn't).

But anyhow:
So gearing ratios 3:1 and 4:1 seem too weak and too slow.
Ratio 6:1 should deliver healthy and reasonable speeds of wheel and engine without consuming too much steam. By the way, the Decoy's website states a max. wheel speed of 24 rpm, which corresponds to 118 rpm of your model wheels after Froude scaling.
Gearing of 8:1 could still be OK, but 10:1 seems too short, because the engine speed gets very high, and the steam comsumption increases drastically without major speed increase.

So if I were you, I would design the boat for a ratio of roughly 6 (or maybe somewhere between 6 and 8 ), but keep the option to change gear sizes later just in case that the practical experience proves something different.

So much for now. Moritz

[derekwarner_decoy]

Hi PD's...this is amazing.......but my PC broke last year  ...however my calculations & guestimations provided a reduction ratio of 5.5:1 via the 9 tooth to 48 tooth chain drive pinion set for my PS Decoy

Then 2 years later Moritz SCRUNCHES the numbers into his mind/brain & PC  ....& confirms

but anyhow- so gearing ratios 3:1 and 4:1 seem too weak and too slow....Ratio 6:1 should deliver healthy and reasonable speeds of wheel and engine without consuming too much steam

Thank's again Moritz....your calculations suggest my 5.5:1 is a healthy median between 4:1......&.....6:1.....however I do have one scaling trick plan up my OZ sleeve

Irrespective of the final test results....the paddle float height can be varied from 12 mm high to +2 = 14...or - 2 = 10 high which provides a +/- 20% varience in blade surface area

[derekwarner_decoy]

Hi PD's...just  & back checking.... Moritz from my original 2005 assumptions & all seems OK...but I do have one question from this....based upon Decoy @ 1:24 scale & being steam driven ie., ... reversable engine...however I would like the paddles & hence engine to stop prior to engaging the ASTERN engine command  

Rotational mass [RM] =
1) the driven chain sprocket + the shaft + coupling + the chain = 165 gms~~~~
2) paddle set...= 250 gm~~~~
3) engine flywheel + drive sprocket = 50 gm~~~negating crankshaft as effiecency loss]
So RM....10% of the net vessel displacement @ 72 tonnes [which is a good % number for RM] 

We understand that a moving paddle vessel 'without power' will come to rest quickly due to the braking or drag effect of the immersed wheels.....