by Joe Cole
The Crossflow Turbine
Unfortunately bulletin #25 is not a "step-by-step" manual, much to the disappointment of many. When I first saw in in 1978, I found it fragmented, elusive, overly technical missing a few formulas. It' more like the technical ramblings of someone explaining the concept of time and the theory of how a clock works when all you wanted was to know the time. However with some diligent "head scratching" over a suitable period of time you will eventually sort out the relevant pieces of information contained in the bulletin. There are a few point to keep in mind when reading and working on some of the calculations in the bulletin. Keep in mind that it was originally a German document written in the early 1930s. It is very likely that some meaning was lost in OSC translation of the original document. Also bear in mind that some of the formulas in the document do math conventions that we use today ie. addition, subtraction, multiplication then division. You'll have to "play with" the brackets. However taking the bulletin as a whole it follows the same mechanical & hydraulic principles used today in turbines & pumps. Those are a curved blade's reaction to a jet of water (in turbines) and water's reaction to a moving curved blade.
About half of the math in the bulletin deals with the physical relationships between the mechanical turbine elements (blade geometry and runner inside & outside dimensions) and the other half deals with the forces produced on the blades. The forces are represented as "vector diagrams" Vectors diagrams help one to visualize what is going on inside the runner. If you understand them you can analyze changes made to the turbine and or changes made in operating parameter such as head & load. Next to the calculus used in the bulletin, the vectors are probably the most baffling things in the bulletin. The vectors will be explained shortly however, as for the calculus, "I ain't goin' there."
Don't expect to "re-engineer" the crossflow runner. Banki & Mitchell "did their homework" on it. The proportions of the blades to the runner diameters and angles involved are fairly "fixed" and cannot be arbitrarily changes without adversely affecting the power & speed of the runner because these changes affect the efficiency of the turbine. As alluded to in the paragraph above, the runner diameters and blade dimensions are a compromise of mechanical dimensions & mechanical efficiency. Taking all the above into consideration, this article will not be a step-by-step interpretion of bulletin #24 but will be my personal "practical inturpation & explanation" of it. Before getting started I want to clarify something. Bulletin #25 title "The Banki Crossflow Turbine" is somewhat misleading as it only deals with the "runner" aspect of the turbine. After discussion of the "runner" portion of this article I will expound on the turbine design as a whole as well as some abstract theory about the crossflow turbine.
Before calculating much of anything else we need a little understanding of vector diagrams. It will take several stages to illustrate this in it's entirety go get a cup of coffee and come back. In understanding how a jet of water
Now that we are up to speed on velocity, mass & force, lest look at some vector diagrams along with the blade shapes that produced them. In this diagram the jet is being deflected by 70°or so. In applying these momentum theorems or laws as they should be call to turbines is as follows. If a jet of water strikes a curved blade the water is deflected by the angle ?.A force (F) is imparted to the blade in two directions, x & y. These forces are calculated thusly
Fx = M * (v – v cos ° α) or Fx = M * (v – v cos ° α)
&
Fy = M*v * sin α
In this diagram the two velocities are the same but separated by angle α and the triangle is closed by closed by the line ∆ v as dictated by the laws of cosines
∆v = the square root of v12 + v22 – 2 * v1 * v2 * cos α
∆v = the square root of 2 * v2 – 2 * v2 * cos α
These two forces combined is equal to:
F=M * v * the square root 2 * (1 – cos α)
Here is the general text book vector of a Pelton wheel in motion.
This is the path & vector diagram of a Pelton wheel. It is showing 2 buckets. The bucket on the left is showing the absolute path of the water jet while the right side is showing the relative path if the jet. If the wheel were "locked down", the water path would indeed follow the path as indicated on the left. However in a running machine the wheel is moving in the same direction as V1 . The water jet is trying to follow the inside curvature of the bucket but because the bucket is trying to move away from the jet the path is straighter as indicated in blue then had it mage the near U-turn of the absolute path. The vector is compound, in other words its is showing more then just one part of the blade. The left triangle is the vector for the entry of the jet to the bucket & the right side is of course showing the water exit from the bucket. The inset illustration shows what happens when the wheel (or runner) is in over speed. The path flattens out more and you can see in the inset vector that μ 1 is approaching the same length as V1 Very little power is now being produced and aa a matter of fact the power that is being produces is rather in driving the intended load, is being spent on maintaining a high wheel speed and overcoming windage & friction. Notice that three additional notations are included. V2, v2 * β. In hydraulics the following notation conventions are used.
| 1 st Quad | 2nd Quad | 3ed Quad | 4th Quad | |
| Absolute Velocity | V1 | V21 | V11 | V1 |
| Relative Velocity | v1 | v21 | v11 | v1 |
| Blade Angle | β1 | β21 | β11 | β1 |
| Attack Angle | α1 | α21 | α11 | α1 |
| Runner Velocity | μ 1 | μ 21 | μ 11 | μ 1 |
Before any "design work" is to be done their are a few things that must be known. First you must have a reasonable expectation of the amount of power that might be produced from a given site. For instance, don't expect to supply a full household with electricity produced from a scenic babbling brook running across your back yard. It takes a lot of water to produce electricity. The "head" and "Q" must first be determined. The "head" at least in the US is measured in feet. "Q" is the quantity of water and in "micro-hydro" work it's usually given in CFM (cubic feet per minute) and in larger turbines is given in CFS (cubic feet per second). Be sure when calculation from formulas in other documents you pay attention to & convert units as necessary. In this article I use CFS.
To begin we first determine the power potential of our site. For convenience, (mine) throughout this article I'll be using my own site for the design & evaluation. That is the head ( H ) = 26 feet and the flow ( Q ) = 8 CFS. The formula for determining the potential hydraulic horse power is ( H * Q * 60 ) / 660. This is the raw horse power potential and does not reflect any efficiency or loss's. According to the formula my potential horse power output would be ( 26 * 8 * 60 )/660 or 18.9 HP. Assigning a efficiency figure is difficult. I want to be conservative in this figure so let's use an overall plant efficiency of 75%. Therefore 18.9 * .75 = 14.18 HP. One HP is equivalent to 745.7 watts so 14.18 * 745.7 = 10574 watts or 10.57kW of electrical power. This is enough to "do a house."
Now having that out of the way we can start to design a runner that will accommodate the site. In the bulletin pages 10 through 15 deal with the construction proportions of the runner. The information we need from these pages are: Formula #35 Q=volume of water, Formula #36 L= blade length, Formula #37 ρ=blade radius & Item (E) Central angle on page 15. Once these values ate known you take these figures to a machine shop and have them form the blades from flat stock to conform to “ρ”, machine the blade sections to form the 73.46° arc in item 3, & finely cut the blade to their final length (L). You then pay the man a huge sum of money & prey your calculations were correct. Definitely NOT the way to go. There is a much simpler & cheaper way to arrive at near the same result but first a quick discussion on one aspect of Banki's design. The dimensions and angles in the bulletin represent the "near" optimum dimensions & angles to satisfy mechanical advantage & un-restricted passage of the water. These dimensions are fairly "fixed" and therefore cannot be arbitrary changed without some decrease in efficiency ie. power & speed 
L = 210.6 * Q / D1 * H^.5
L = 210.6 * 8 / 12.27 * 5.099 = 25.5 inches.
Oh by the way, H^.5 is the same thing as "the square root of "H". It took me a while to figure out that one. Anyway with the initial blade length calculated as 25.5 inches, divide the "Jet area" of 28.8 square inches by 25.5 to get the So which in this case = 1.13 inches.
The only thing we need to factor in now is the nozzle efficiency and adjust the length accordingly. If you follow good hydraulic principle and design a good gate/nozzle you should be able to achieve an nozzle coefficient of .98.That’s only a 2% off peak which would be 1.356 which translate to a .223 increase in runner width to compensate. This would bring the runner length to 25.72 inches. If it were me I’d bring the runner on out to 26 to28 inches just for grins and a little more error margin. Now we’re looking at a D:L ratio of 1:2.08 Not terribly bad but! A 28 inch wide small diameter turbine is going to be a machining & welding nightmare. Building the runner itself it not too bad but I would add in 2 extra center support disk for rigidity. Of course you’ll want to extend the runner length again to compensate for the width of the extra center supports. We’ll we’re now out to 28.5 inches. Do I here 30?
My Daddy used to tell me, “You’ve got to use some horse sense”. Although I was never very good with math, I do have to ability to “visualize” how things function & anticipate problem areas. Having some “horse since” also helps. Here comes the first major problem in designing a turbine. Let’s tentatively select 4-inch pipe to make out blade sections from. We might select it because it look good & stout and because it is relatively easy to come by. That will make us a runner 12.27 inches in diameter. At this point my horse is telling me “there aint no way”, you’re getting 8 cubic feet of water a second through a 12 inch runner without major difficulty. It’s not impossible just not practical and here’s why. Anyway fabricating all the flat stock for the gate & nozzle assembly will really be the difficult part. That’s an awfully wide gate assembly. At a 26 foot head you only have a shade over 11 PSI at the lower end of the system but think about it. You have a 28.5 inch wide gate perhaps transitioning back several feet to a round penstock. That might present top panel behind the gate valve of 28.5 x 36 inches. Multiple that times 11.25 PSI and you’ll have in the neighborhood of 11,550 pounds of force acting just on the top panel of the gate. Even if your welds held, the things going to bulge out & distort like a balloon. Personally I’d give it around a 100% failure rate within 10 minutes.
Alright, as promised, I'm throwing a monkey wrench into the works. The problem comes when calculating how long the runner needs to be. Notice in the calculations & illustrations in the bulletin all the math used an infinitely thin blade. If this is not realized it will cause you to calculate the runner too short. You might not notice this until you go to full load and “it just aint makin’ it”. Refer to page 9 of the bulletin to figure 5 for the spacing used. Using our 12.27 turbine as an example, if we multiply it’s diameter by pi we have a circumference of 38.52 inches. This gives a blade spacing( t )in the outer periphery of the runner of 2.14 inches. The illustration to the right shows the problem very well using a exaggerated blade thickness in blue. 
Specific Speed
Specific speeds is a dimensionless number. In broadcast engineering they call this term "normalizing", if any of you are familiar with Smith Charts. The term is used to “level the playing field” if you will, so that all types of runners can be evaluated under the same conditions .As a result the term via it’s number define the shape of the runner. I remember from a long time ago one hydraulic document described it this way. If a model of any given turbine were build with a 1 foot diameter and operated with a 1 foot head, then the specific speed is the speed that the runner would turn to produce 1 horsepower. I guess that about sums it up for a level playing field.
What it all means is that a turbine with a high specific speed will while running a full load, be turning faster then one with a low specific speed. An extreme example are the Pelton wheel which has a very low specific speed. It is usually thought of as a high-head machine. However it can be very efficient a low heads. It’s just that it turns so slow at low heads that the cost the equipment needed to increase the shaft to something usable by a alternator may cause the whole project scrapped or re-dome with a turbine of a higher specific speed. Also a low specific speed is also thought of as a low volume unit. This really makes it an ideal selection for mountainous terrains where large quantities may not be available.
Nozzles
I believe that bigger is better up to a point. In the case of selecting a runner diameter, using a larger & therefore narrower runner not only saves money and add durability but does offer up a few extra chances at increasing the efficiency of the crossflow. Take a second look at the Horse power formulas #2 & #6 on page 7 of the bulletin. Remember the lows of cosines? The 16 ° α1 is usually chosen as a compromise between hydraulic efficiency and mechanical clearances in adjusting So. Therefore if a1 is reduced the efficiency & power output will increase. With a large diameter runner this is much easier then in a small turbine. You could lay that angle down to maybe as low as 8 °.Of course you would want to lengthen the runner to compensate for the smaller So.
General Layout of Flow In Nozzle
The nozzle diagram above is meant to show some general proportions. For maximum efficiency the runner should be designed for single blade operation. However in the interest of construction difficulties in building a wider runner, a double nozzle - blade arrangement may be used at some loss in efficiency. The proportions are general. For instance I chose the radius of the nozzle curvature arbitrarily at 2 times the runner diameter. The exact radius in not important so long at it gives a nice long gentle sweep into the blades. The arc of the nozzle is also an arbitrary figure. I placed this one at 73 degrees “just because”. That long sweep & mechanical clearance is all that matters. You could go “straight in” as the folks a OSC did when they built their turbine using the freshly translated document from Banki’s original papers. By the way, does anyone know how to get or has a copy of the “original Banki papers? However they had some pretty horrible efficiency numbers with their turbine. They may have been “Jim Dandy” mathematicians but would have made a few changes on their nozzle design & transition assembly. Probably the thing that hurt them the most was the nozzle. It was a sliding gate that opens & closed “laterally”, that is across the runner face. What happened when you pot your thumb over the end of a garden hose? Using their arrangement that’s exactly what happened in their turbine. My guess is that at small gate openings the water might have been disbursed by 10-20 degrees. Another thing that hurt a little was not having a smooth transition between their supply pipe and the nozzle. It was a blunt sharp edge transition. That hurt them more at full power then anything else. I’m not trying to belittle any of the people involved I’m just trying to make you aware of “design flaws in engineering.” Left click on the illustration above to save it to your computer. It's actual size is about twice what you are seeing here.
What is important is the angle the water hits the blade at. This is generally taken at 16 deg. However, that is "relative: to the blade angle B, which itself is relative to the periphery of the runner. Through out the bulletin you see constant reference to a1. This is an extremely important angle, for it more then any other factor, determines the power output of the runner. However I’d have to say that 16 degrees is the maximum angle that one should use. Us it as a design figure then see if you can go smaller. Getting small requires “laying the nozzle down” closes to the runner. If you use a large enough radius and a long enough arc for the nozzle, you could get a1 on down to the 8 to 12 degree range. Any smaller though, and you’ll have to start thing about lengthen the runner. Going to excess on this could get you a nozzle with a low coefficient because of excess friction because of excess length.
commercial turbines that's a very good idea because it saves manufacturing cost and makes an extremely ridged turbine assembly. This method is a little impractical for us little guys because not too many of us have casting facilities in our back yards or want to shell out major bucks for some "real machine shop work" Besides if the nozzle is cast in with the housing we can't adjusts the attack angle now can we? I took my design from one of the old Ossburger design. In stead of the water following the runner housing after it leaves the gate it follows a curved guide which is the top of the nozzle assembly. In mine I'll use a nylon or duron spring loaded back seal on the gate shaft. The side seals and shaft bearing are not show in my illustration but they are mounted externally on the nozzle housing. The sealing method on edges of the gate plate are not shown but they
Something Radical
Using just one blade set water leaves the blade between A & B. It re-enters the runner for it's 3ed quadrant of operation at D through E. At full gate operation using 2 blade sets, water leaves quadrant 3 from A to C & then re-inters at D through F. In the illustration shown here the water path for single blade operation is the blue lines. Adding a 2nd blade is shown by the green lines. This illustrated the spread of the water upon entering the 3edred. The pink area shows the water path for single blade operation. The light blue shows the path should 2 blades be used. In single blade operation point E is about as far back to your right as the jet will reach at normal speeds.. What I've attempted to do here is re-direct the water back open more blade set and inter at 16 degrees at that point. A similar situation occurs when the 2nd blade section is added. The water in confined to a course that does not vary with speed and it is always forces to enter quadrant 3 at 16 deg. quadrant of operation. I plotted the water path graphically and came up with the curve necessary for an internal guide shown in
APPENDIX I
Notes on the "Efficiency Formula" in Bulletin #25
The efficient formula is every bit a complicated as it looks. I really have very little though on how it works. Due to my lack of understanding or motion mechanics I’m forced to take Banki’s word on this one. However I can tell you what is going on in the formula. In the formula “C” is the nozzle coefficient. He’s accounting for that in the first part of the formula in dealing with ? 1 & V1.In the middle of the formula the term y is the factor describing the loss of energy caused by the separate jets crossing each other between the 3ed & 4th quadrants. I believe that the loss of power is also represented in ? due to “shock” loss. Shock loss is when the relative velocities v21 & v11are not parallel (in the vector diagram). This can be seen on page 11 of the bulletin on the left side of fig 7.The relative velocity of v21 suggest that the inside diameter of the runner at point “B” in the drawing above is turning at a different speed to the corresponding point “C”These actually turning at different speeds id clearly impossible since they the same physical surface. The last part of the formula is the velocity difference between V1 & ? 1to extract power.
APPENDIX II
Notes on the "Horse Power Formula" in Bulletin #25
The Horse power formula is not as complicated as it might seem. The formula can be divided into parts.
(W * Q * ? 1 / g)the momentum part.g=gravity constant at 32.3 In checking my Excel and Q-Basic programs that calculate this I found a discrepancy withW.As stated earlier W the weight of water per cubic foot of at sea level is 62.2 ft. However to make the formula produce the correct answer in Excel a value of .13 has to be used. At the moment I don’t have time to fix it so I use the correction multiplier of .00291 in the horse power equations. In Q-Basic the value of 62.6 works just fine. I think they mean "mass" rather then weight.
- ((V1 * cos(a1) – u1) takes account of the laws of cosines & subtracts the wheel peripheral speed from V1 to that power is produced.
- (1+y) * (cos b1 / cos b2) the lump sum factor or runner coefficients taking into account the cosine blade angles of the 1st quadrant and the 4th quadrant. Their ratio would always be 1:1 because they are physically the same piece of steel.
This page will be an ongoing document. It will be up-dates and expanded as I have time. Eventually I hope to cover every aspect of building a crossflow turbine. I welcome comments on this page. Please let me know of any mistakes you find or clarifications that nee to be made. I will also entertain contributions to this page. Just email me and let talk about it.
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