2.2 A Design study for model paddle wheels

I think that most model paddle steamers have too many blades on the wheels.  I write this to give my reasons.

 

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                   Contents

 

                   Introduction

 

                   The paddle wheel as a mechanical oar

 

                   The analysis of the feathering mechanism

 

                   Appraisal of the mechanism

 

                   Resumé of useful paper on paddle wheels

 

                   Proportions of practical paddle wheels

 

                   Readme

 

 

Comments to ivorbittle@tiscali.co.uk

 


The Paddle Wheel Design Study

 

Introduction

In 2002 I wanted to design a steam-powered boat propelled by paddle wheels. I had no preference for a scale model or a home-brewed design but I did want the result to look attractive and to work well. The paddle wheels were an exercise in mechanical engineering design and one that I fancied because it would all be new to me. So I supposed that I was making a miniature paddle boat with the constraint that the paddle wheels must not look to be wrong to anyone who has seen a paddle steamer. Figure 1 shows the outcome.

 

 

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Fig 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In the event the paddle wheels that I designed and made have been very successful when coupled with my vee-twin, steam engine. They push my boat along at about 2 miles per hour and recently the boat towed a “dumb” tanker that is 10 feet long and weighs at least 300 pounds at a very acceptable speed.

 

I have seen several paddle-powered boat all electrically driven but generally they do not shape up to the speed that would make them look realistic given that real paddle steamers were quite fast. I think that they have used too many blades on their paddle wheels.

 

This section of my website is essentially my design study for the paddle wheels. It has not been altered in sequence. There was a separate design study for the hull and another for the bulbous bow and, of course, another for the engine.

 

The paddle wheel as a mechanical oar

After a lot of thought I have concluded that the paddle wheel is really a mechanical oar using a sequence of blades fitted to a wheel rather than one blade used in a repeated stroke. It makes sense to start with the oar.

 

As an oarsman in a racing boat pulls the blade of his oar through its stroke, water piles up in front of the blade and a hole appears behind it. When the blade is lifted from the water the disturbance is more or less circular in plan and moving in the opposite direction to the boat. This flow pattern is called a puddle. I have made a sketch of what happens in figure 2.

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Fig 2
The important features are the rise of the surface ahead of the blade, the drop behind the blade and the fact that the puddle is more or less circular. The alteration in levels means that the average pressure on the leading face is greater than it would be in still water and vice versa for the trailing face. The net effect of this is that the water produces a force on the blade to resist its motion. The circle means that water is flowing from the raised part in every direction and curling round to join the water that is flowing inwards to fill the hole.

 

What is not evident is the way that the blade is made to enter the water and the way that it leaves. Oarsmen train for a long time to become proficient in handling an oar. They do not just smash the blade into the water at a downward angle. They ease it into the water and increase the force on the oar handle at the same time and create the puddle progressively. At the end of the stroke the blade is flicked out of the water. In this way nothing is done to stop the run of the boat nor is work done to no advantage.

 

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Fig 3
I took some pictures of a Dutch hoe being dragged through the water in my fishpond. They were not good but they were important. I have inserted just one of them It is figure 3. The Dutch hoe represents the blade of the oar. As it moves through the water near to the surface it is clear that water is heaped up ahead of the hoe and a hollow forms behind the hoe. Water flows away from the heap ahead of the hoe in all directions. Some goes over the upper edge, how much depends on the depth of immersion. Water flows forwards towards the hoe from behind the hoe and water flows upwards from under the lower edge to join water from the sides and all of it moves in the direction of the hoe. If the hoe were to be removed suddenly the net effect is a mass of water just like the puddle from an oar moving in the direction of motion of the hoe. I had in mind taking several pictures at various speed and depths and perhaps making a model paddle or even two but there is a message in this single picture. The blade of a paddle wheel must work like an oar.

 

If a paddle wheel is to work like a mechanical oar each blade must have enough clear space around it for the full flow pattern to form. This includes the spacing between successive blades. Suppose that a second Dutch hoe had followed the first in my picture above. The outcome would depend on how closely the second followed the first. If the second blade were to be too close to the first the wave ahead of it would fill the depression behind the first. This would interfere with the flow associated with the first blade but, if the second blade were to be  moved backwards relative to the first, the first would become progressively independent of the second. This means that the blades of a paddle wheel should be as far apart as is necessary to preclude interference between the actions of the blades.[1]

The blades actually turn on a wheel so, like the oar the blades must enter cleanly and leave cleanly if the wheel is to be efficient and the only mechanism that has evolved for this is the simple feathering mechanism.

 

In a sense I have laid out the basic design of a paddle wheel. Now I need to understand the feathering mechanism. I do not need to know how many blades will be used to do that so it can wait until later.

 

The feathering mechanism

Every bucket is controlled by a mechanism that is just a simple four-bar chain.[2]

 

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Fig 4
See figure 4. Points O, P, A and B are all moving joints. AB is the link that is extended to carry the float (or blade). Point A is a pivot on the frame of the wheel on which the float lever turns. O is the centre of the shaft on which the wheel turns. Point A clearly rotates around O at radius OA and is the second link of the chain. The third link is OP on the frame of the paddle box and is the offset of the pivot point for the master rod PB.[3] OP is clearly fixed in space and only the other three links PB, BA, and OA move. They complete the four-bar chain. As the wheel rotates A drags B round with it and both A and B follow arcs of circles and AB takes whatever position is needed to link the two points.

 

There are a lot of variables here, too many for me to explore completely, so I shall have to simplify things a bit. For a start I could put AB equal to OP and to put P on the same level as O even though such an arrangement is impractical because the links can all end up in one horizontal line and then the subsequent motion is not predictable and potentially very expensive.

 

This means that I really want diagrams to show the way in which the blades change position as the wheel rotates. In engineering this is called a space diagram and here is just a series of diagrams of the four-bar chain in an orderly set of positions. The starting point must be the case when for OP = AB but it is inevitable that the useful cases will be when PO is less than AB. In order to draw these diagrams, I have to have a ratio for AB/PB. I am sure that others have already ploughed this furrow so it makes sense to see what ratio was used. I have looked at two real wheels and a ratio of about 0.25 looks to be about right.

 

 

One might argue that point P should be above or below the horizontal centre line Text Box:  
Fig 5
but the locus of P is then another circle and, if I draw diagrams for P located in the horizontal on the frame, the whole diagram can be rotated to see what might happen for other positions of P.

 

So it is about drawing space diagrams for various values of OP and a value of 0·25 for AB/PB.

 

I have drawn a first space diagram, figure 5, for OP = AB. The blade is vertical throughout the lower part of its travel, ie through the water. If it is supposed that the mechanism behaves as I have drawn it, the movement of the blade as it goes over the top is quite odd. It involves high accelerations. It arises because I have let the master rod be below the pivot throughout. If the blade flopped downwards and B went up at the left hand end instead of down the diagram would be symmetrical about the horizontal axis. This is not a practical arrangement. We must look at OP less than AB.

 

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Fig 6
Now for a second space diagram, figure 6.

 

I will try OP = 0·8 AB. The result is shown in figure 6. This produces a space diagram in which the blade is angled down at entry and up at exit and vertical in mid position. It looks to be very promising.

 

Now, for this arrangement, there can be no question of the linkage going over centre because PB and AB can no longer get into line. However the high accelerations as the blade goes over the top are still there and look to be unavoidable.

 

The diagram then raises the question of what happens when OP is further reduced to say 0.6 OP.  It is figure 7.

 

This change just increases the angles at entry and at leaving to larger values and the ratio 0·6 must probably be regarded as too small. The motion of the blades over the top is better but this only tells us that we are going to have to accept these high accelerations. In practice they may not matter.[4]

 

This diagram is a little distorted by the two circles being of slightly different diameters. This makes no difference to the message but does show how the difference affects the motion. Making PB shorter increases the angle at entry and decreases it at leaving.

 

 

 

 

 

 

 

 

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Fig 7

 

 

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Fig 8
Now I have to find some way to choose between these diagrams. I have already said that the blades should enter without splash. This means that, for these flat floats I really want to find out how to make them go into the water in a direction that is along the flat face of the float. To an engineer this means that I require a few velocity diagrams and these will tell me something about the exit as well.

 

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Fig 9
I have drawn a velocity diagram for position 3. Start with the blue vector that represents the speed and direction (ie the velocity) of O and therefore of the boat relative to the water. Then we know that, relative to O, the blade or float is moving in a circle and that its velocity is the tangential speed and its direction is at right angles to  OA. The vector representing this is in green. We can combine these two vectors to give the velocity of the float relative to the water. It is in red. What we want is for this red vector to be parallel to the face of the blade at this position. If similar vector diagrams are drawn for the positions in the space diagrams we can see the angles of the float and the direction that it is moving. I have done this for the lower positions in figure 9.

 

In order to do so I had to decide on a suitable value for the ratio of the speed of the float as it rotates on the wheel relative to the boat speed. I already know that the best speed of the boat that I can expect will be about 3 fps and the German paper (see later) gives the ratio of boat speed to linear speed of blades as say 0·75. So the tangential speed will be about 4 fps.

 

This diagram is interesting. The relative velocity at point 1 is not relevant because the blade is above the water. However the direction of the blade at 2 is not far away from being parallel to the face of the blade at 2 in the space diagram for OP = 0·8 AB. The blade would enter cleanly. It seems that it would also leave very well at a position such as 6. This feathering mechanism is good.

 

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Fig 10
But now we have to think. These paddles must affect the water just in the same way as an oar and it follows that the water from which we want to extract the blade cleanly starts to move with the blade and to heap up in front of the blade and form a hollow behind it. The diagram in figure 9 makes no allowance for this and pretends that at the point of extraction the water is still not moving. I made an allowance for this in figure 10. Here I let the blade increase the velocity of the water relative to the boat from 3 fps to 4 fps in equal steps and drew more velocity triangles. This alters the direction of the blade at 6 by a significant angle.

 

This tells us that, for best performance, this paddle wheel will need setting up when on test. The adjustment will be by moving P, probably downwards, to move the whole diagram round the axis of the wheel and. the diagram will become asymmetrical in the way required.

 

I found this to be interesting because I have always been suspicious of P being on the horizontal and this made me feel better.

 

Appraisal of the mechanism

I wrote this at the time. I have chosen to explore this feathering mechanism for just one ratio of OP/AB and for one ratio of AB/OA. Even so I think that I now understand the feathering mechanism for the paddle wheel and how it might be exploited for my model paddle wheel. I think that making one will present no special problem. I will use the 4 to1 ratio between the master rod and the arm of the paddle. Then, if I make provision to adjust the pivot pin of the master rod downwards from the horizontal, the feathering action can be adjusted and, at least, the blades will enter and leave the water properly.[5]

 

Resumé of useful paper on paddle wheels

Ken Fletcher retained an article from Model Boats on paddle wheels. It is of German origin and goes back to 1915.

 

In this article the thrust produced by the paddle wheel is given by an empirical expression:-

                                    Thrust = , where:-

A is the area of the paddle, u is the speed of the paddles at the radius of the centre of pressure, v is the speed of the ship, n is some index and k and m are coefficients.

 

There is nothing unexpected in this because, if n is put equal to 2, it becomes the ordinary expression for a dynamic force albeit with two coefficients. The effective velocity is the difference between the tangential speed of the blades and the speed of the boat.

 

However there is some confusion about m and A. I think that it is intended that m should be the number of paddles immersed. If it is mA is the total area of the immersed paddles. The k is effectively our CD. Then the article says that k varies widely and opts out. I wonder why it varies and with what? Clearly the value of k will vary with say the blade spacing round the wheel, with the setting of the feathering gear if indeed it is not a wheel with fixed blades. Like so much associated with paddle wheels engineering testing is not easy or even possible.

 

A ratio is defined. The bracket  is the speed of the paddles relative to the undisturbed water and a speed ratio that is defined as  and is called the slip of the wheel. I need to understand this. One might argue that the very best that a paddle can do is advance the boat at a speed equal to the tangential speed of the blades. This can then be a reference speed. In use the paddle wheel will produce a speed of a . Of course this speed is that achieved when the thrust produced by the two paddle wheels is equal to the drag on the hull. This speed is therefore the outcome of two relationships; the thrust-speed relationship for the paddle wheels and the drag- speed relationship for the hull. These must vary from hull to hull and wheel to wheel. Inevitably it is most unlikely that there will be some single value for the speed ratio.

 

The author gives figures for this slip of between 0.15 and 0.3 without saying the circumstances of the measurement. Is he saying that these are the figures for this ratio at say maximum efficiency or perhaps for maximum thrust. What sort of ship and wheels were involved. We shall never know. There may be a wide range to the value but at least we have a range. He is saying that the boat speed is between 70% and 85% of the blade speed. This is useful.

 

Now he goes off in another direction. He defines some distance e that he calls the paddle distance. I suppose that this could be the circumferential distance between consecutive paddles. Whatever it is he says that A, the area of one paddle, can be related to e. It seems that A varies from  and . This seems to me to be no more than a statement of the geometry but there is a proviso. He says that if e is large “the water that is displaced by the float can close behind the float and give an increased thrust”. He is really saying that there must be space round the blades especially in front and behind.

 

Now he quotes a few figures. He says that e varies from about 0.7 to 1.2 metres for fixed paddles and goes up to 1.9 metres for feathering paddles but does not give the pitch circle diameter. He gives wheel speeds of 20 to 30 rpm initially and now (1915) 40 to 60 rpm.

 

Then he points out that, if the paddles are big and the spacing large the diameter of the wheel D must be large to get sufficient paddles into the water. How many is “sufficient”?

 

Now he defines the height of a paddle as h (presumably the radial height) and gives typical figures for D in terms of h,

For fast sea-going steamers with feathering paddles                          D = 5.3 to 5.5h

For in-shore steamers                                                                        D = 8.1 to 8.8h

For small river steamers with fixed floats                                           D = 8.2 to 10h

For new river tugs with eccentric between hull and wheel                  D = 4.2 to 5.5h

 

Working back from A the width of the blades b can be related to the height and we get:-

 

h lies between .38 to .48b  for fast sea-going steamers and .17 to .24b for in-shore boats.

 

Proportions of practical paddle wheels

I must record that, in 2002, I knew in the way that engineers do that my wheels needed 5 blades and I went on to design my wheels. Now I have to try to justify that decision.

In 2002 I felt apprehensive about the fact that my analysis came up with a different position for the pivot for the master rod to that commonly used on real paddle wheels. We could not both be correct. Then, I was looking at Hambleton’s book about paddle steamers, when I spotted a page that I reproduce here as figure 11. It is all there. The pivot for the master rod is offset downwards. This gives clean entry and exit. This design was due to William Stroudley who was a railway engineer designing a ferry for the London, Brighton and South Coast Railway. His wheels set new speed records.

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Fig 11

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Stroudley did more than design the wheels, he used engines of his own design as well. This tells us that the state of ship design in 1880 was open to a step change in performance. Ship design has always progressed very conservatively but once the performance of Stroudley’s designs for wheels and engines  was recognised they quickly became the norm.

 

What interests me in this is that it means that, up to the time of Stroudley, design was by very limited cut and try. I am inclined to think that it had stagnated but I am not competent to assert that this was the case. Figure 12 is an extract from a book, but I have forgotten which book, and it tells us that, after William Stroudley, the design of paddle wheels changed and gives us an example of what they changed to. So we need to look to post-1880 designs and probably post-1890.

 

Prior to Stroudley the floats of paddle wheels were mounted in a circular frame as shown in figure 13. The post-Stroudley design is shown in figure 12. The big changes in the mechanical design of the Text Box:  
Fig 12
wheel were the abandonment of the outer rim, the use of a much narrower hub with raked spokes going out to an internal ring to support the blades, and the use of curved wooden floats.

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Fig 13

The wheel design in figure 12 gives us the proportions of the blades in what must be the fully developed wheel and we can use them for our model wheel. We can also scale down the various radii.

 

This leaves us with the thorny problem of the number of blades to use.

 

We can put one constraint on to this immediately; there must be at least two blades in the water at any point of the rotation. This tells us that the design must take account of the depth of submergence and this ties the wheels to the design of the hull. Throughout my building of Canopus I was conscious of the need to get this right. It is not easy to move the wheel shafts once they have been built in. A look at figure 13 shows the position of the water line and that this wheel has two blades in the water.

 

In the article from 1915 I have highlighted a section about the spacing between blades. At the end of it there is the statement that the blade spacing on feathering wheels could be up to 1.9 metres or 6 feet but does not tell us how big the wheels were. If figure 12 there is the astonishing statement that wheels became standardised at 14ft 10in. One must ask “what wheels?” because wheels must have varied with the size of the hull. However this does give us an idea of wheel size and for this diameter and the blade spacing of 6 feet the number of blades would have been 8. This, of course, is typical. Now these blades were about 3 feet “high” so the space between the blades was about twice the height of the blade and the need to avoid interference between the puddles from adjacent blades engineering tells us to regard this as a minimum.

 

In the event I decided to use blades on my model wheel of 1² height and 2.5² wide set on a pitch circle of 3.75² diameter. The minimum spacing should, by the argument above, be at least 2². Using this minimum give the number of blades at 5.9. As either 6 blades or 5 blades would give at least two submerged I settled to have 5 blades and experience has shown that it was the right decision.

 

There remains the question of wheel speed. No paddle-driven boat will plane and we are dealing with a displacement hull. Froude’s critical speed for displacement hulls shows that no model of say 54² length will travel faster than 3 mph. At this speed the bow wave is very large like that in front of model tugs (and real ones). It is more likely that our model will make about 2 mph, that is a slow walking pace. The 1915 paper gives a slip of between 0.15 and 0.3 of the tangential speed of the blades. So the blade speed is about 25% more than the boat speed. If the pitch circle diameter is known it is a simple calculation to find the shaft speed. It is speed in rpm = 430/pitch circle radius in inches. For my wheels it is 230 rpm.

 

My paddle-boat weighs 27 pounds and if, from stationary, the throttle is opened suddenly, the wheels thrash round and spill water through the paddle box louvres. They are going much too fast. They do drive the boat but, as the speed increases, the jumbled water from the boxes gets less as the drive improves and when the correct speed of the boat for the speed of the wheels is reached they operate cleanly. Getting the match right is crucial and, if you make the engine or the gears, plan to be able to change the gear ratio.

 

Going back to the study.

 

In October 2002 I thought that I had reached the stage where I could design a model paddle wheel. I recorded the following.

 

All that has gone before leads me to think that the blades should be 1″ by 2.5″ and operate on a pitch circle diameter of 3.75″ to give an overall diameter of 4.75″. There should be 5 blades that should be curved.[6]

 

Then I have to design a wheel. Looking at real wheels it seems that before William Stroudley wheels had an outer rim and after the designs became rimless. I do not really fancy a rimmed wheel but I suppose that if I made a scale model it would have to have a rim. Either way there is a decision to be made about the mounting of the blades.

 

The blades could fit between two plates, they could overhang two plates, and they could be mounted on a single plate. Before these can be assessed there is another factor to consider. Should the working face of the blade be kept intact or can supports and bearings be cut into the face? I do not know. I suspect that the face should be kept intact unless there is some good reason to do otherwise.

 

Let me look at these separately.

 

The single plate arrangement would require a stiff plate with five arms each curved to clear the blades with tubular bearings set into the arms. The blades would swing on these bearings and the feathering arms would be attached to the ends of the blades. So far as I can see such an arrangement is quite practical. It has no counterpart in full sized wheels but would be ok for a model wheel on a modern design. It is a clean design and it would suit me. However these wheels might have to operate in salt water and this is not good for long bearings.

 

It is clear that the design of the hub has some effect on the overall design. If outside plates with rims are used the hub must either be 2.5″ long or the spokes have to be bent in towards a shorter hub. Stroudley used this design. A long hub adds weight that might be omitted if a short hub can be used. Making bent spokes is not very easy and, in my view, is best avoided.

 

This brings me back to the use of two plates and overhanging blades and this is what I used. See figures 14 and 15. 

 

But in the end I have a decision to take about the ultimate design. Is it to be a modern steamer or a period boat? Could a period boat be fitted with a single plate wheel?[7]

 

I recorded the following in April  2004. I ran the paddler at Herne Bay yesterday. The engine was not at its best but the wheels turned. I was astonished at the grip that the wheels had. The engine seemed to be hardly turning over and the paddles turning very slowly yet the boat moved at a sort of middling pace. Certainly the wheels seemed to perform very well

 

However there is work to be done and we shall see another day whether they really perform.[8]

 

Readme

This section of my website on paddle wheels is here to justify the statement that five blades is enough on a model paddle wheel by giving the fundamental argument that leads to this. The dimensions for the blades came the design in figure 12 and, as it happened these fitted nicely with the emerging design of the hull. The rest comes from any ability that I possess as an engineer.

 

The photographs are sufficient for any competent model engineer to use my ratios and draw up a set of wheels. You will see that I made curved blades. These were milled from brass tubing complete with slots to attach them to their levers.

 

You may think that I should give the drawings but, in fact, I have no formal drawings to hand. I think that I just discarded them as I went along. I would have to draw them up from the wheels and this website is taking up a great deal of time in other directions but I may draw them up some day.

 

Finally my paddle steamer, Canopus, has a bulbous bow that was designed and fitted solely to smooth the water as it approached the wheels. It has a wedge rudder that blends smoothly into the hull. It also has a fin (see front page) that is designed to give a 15 foot turning circle, which it does, and the fin contains 2 pounds of lead to lower the centre of gravity enough to let me have the top works and to have enough stability to spare to stop a paddle digging in. It is driven by a vee-twin of my design having a bore and stroke of 7/16² and piston valves despite the received wisdom in model engineering that they do not work. It is supplied with steam by a boiler made by Martin, Howes and Bayliss but modified by them to permit the fitting of a better superheater. The gas tank is heated electrically with automatic pressure control and a Cheddar burner control system holds the steam pressure at about 50 psi. It is a miniature steam-boat that runs for 25 minutes on one fill.

 

 

 

Ivor Bittle 2008

 

 

 

 

 

 

 

 

 

 

 

 



[1] It is significant that when paddles were proposed for very large boats the paddle wheels were very large with lots of relatively small blades that were not, proportionally, very wide. They were actually fitted to the ss Great Eastern. If this were to be modelled to scale the model blades would be so close as to be useless. Nevertheless it is common for modellers to retain the number of blades when modelling a smaller ship and these do not work either.

[2] I took it for granted that my wheels would feather. I had never seen a wheel with fixed blades working. I have, just recently, and the wheel lifted a great deal of water behind it and used up a lot of power to do it..

[3] When I wrote this I had in mind a conventional wheel with a master rod and slave rods working on it. In the final design each blade had its own mechanism.

[4] My paddle wheels behave as shown in figure 5 but they do not show excessive wear nor have they ever gone over centre and locked.

[5] This is just what I did.

[6] I now know that this decision was correct. The paddle wheels are as good as a screw. The only other choices are 4 blades, which is visually not acceptable, and 6. I think that this would have involved a master rod and probably any potential improvement would have been lost in the distortion of the blade motion due to the use of slave rods.

[7] In the event I did not like the paddle steamers that were used on the Thames. Everything had to be sacrificed to make them fast because they did round trips of 140+ miles in a day. I turned towards continental steamers used on lakes and produced a design that I hoped would look attractive.

[8] When the paddle-boat went afloat for the first time the gear ratio between the engine and the paddles was wrong and I had to cut new gears.