Using the Bilgram diagram to calculate the dimensions for the valve of the working cylinder
Fig. 1: Determining the angle C-O-D for a given length of piston travel at cutoff.O-E: Radius of Eccentric CircleE-F: Outside LapE-Z: Inside LapL: Linear Advancea-a: Angular Advance
Many of we hobbyists and buffs like to design a steam engine from the ground up, and one of the principle jobs here is to design the valve for the working cylinder.
There are two approaches to this problem: the cut-and-try method which is best accomplished by constructing a wood or tin working model to some convenient scale of measurements, and then there is the engineering method which follows after a long period of systematic scientific pursuit that usually involves much empirical data and calculation.
This article is concerned with the latter method which is considered to be much more positive and is actually based on the results of much study by the profession over the past 125 years. This leads to direct results through utilization of a limiting drawing-board mockup based upon use of the Bilgram diagram which was most widely used by design engineers during the period when Steam grew from Boyhood to King. Some few engineers worked with a somewhat similar Zeuner diagram; however since this is one of a cut-and-try method, it should likely be consigned to the simple mechanical cut-and-try method mentioned above.
After having decided upon the bore and stroke of our engine for a desired power output and engine size, the very next problem is that of determining the dimensions of the cylinder ports. While steam at, say 150 psi may have an open jet velocity in excess of 3,000 feet per second, this velocity is reduced by the friction and bending of the ports in an ordinary slide valve engine to a value accepted from empirical tables as that near 8,000 feet per minute; and this is further reduced during the opening and closing portions of valve travel to an average value of 6,000 feet per minute.
Now, if we have selected a cylinder size and number of revolutions of crankshaft speed per minute, we can readily calculate the total amount of steam to be passed per minute and consequently the port area of the valve. We must also take into consideration the maximum point of cutoff for our engine, of course, since there will be no steam admitted after this point in each stroke.
So suppose we have selected a 4-inch bore by 5-inch stroke, with maximum cutoff at 3/4 stroke, and speed of 250 rpm. A bit of simple arithmetic reveals that the total amount of steam to and from each end of the cylinder per minute figures out to be: 4 × 4 × 0.7854 × 3/4 × 5 × 250 which, by slide rule, equals about 23,600 cubic inches.
Then, at a mean velocity of 6,000 feet per minute through the ports, empirically, the cross-sectional area of this volume of steam would be: 23,600 divided by (6,000 × 12) which would equal about 0.328 square inches.
Now, the old heads also considered that the width of the cylinder bore for optimum tradeoff against other dimensional problems. If we then adhere to these old standards, our port width will come out to 0.8 × 4.0, or 3.2 inches. Since we have the port area and width, then, the length of the port in the direction of valve travel must be: 0.328 divided by 3.2 which comes out about 0.1025 inches. If this figure is difficult in model scale, it is suggested that this cylinder port be made (1/2 of 3.2 inches) × (2 × 0.1025 inches) or about 1.6 inches × 0.2 inches for the same area. Of course, you can make it as much larger as you may desire, but these are considered sufficient minimums.
In the use of the Bilgram diagram to solve the rest of our valve design problems, it is only necessary to determine the crank angle for the particular maximum point of cutoff which we have selected; in this case 3/4.
Referring to Fig. 1, A-B represents the crosshead (or piston) stroke; F-C equals A-E which is that of the connecting rod which is usually made to be from 2-1/2 to 3 times the piston stroke (mechanically, the longer the better; but there are physical limitations, of course); and E-K-C-D is the crank circle described about center O. From the head end, A-F is laid off as 3/4 of A-B, making it the crosshead position at cutoff. Now, to find the crankpin position at this cutoff, we can swing an arc of radius A-E from point F to intersect the crank circle which will land us at C. The sought-after crank angle at cutoff is consequently that within C-O-D, indicated at “a.” Reversing the process from the crank end of the cylinder results in the angle K-O-E, or “b” which is not equal to “a” because of the “angularity” characteristic of the connecting rod as often referred to and can thus readily be seen. It is equal, in the case of the Scotch Yoke mechanism, of course. This same difference shows up in the eccentric drive also, but is of little consequence ordinarily because it is usually reduced to insignificance by the ratio of eccentric throw to length of eccentric strap. With an infinitely long connecting rod, had we room for it, there would also be no such thing as the problem of angularity, of course. Now, assuming that we have also settled upon the simple matter of linear lead of our valve, we are ready to proceed with layout of the Bilgram diagram.
Referring to Fig. 2, follow these steps:
(1) Draw a horizontal line of convenient length such as K-D. With O as a convenient center, mark off the cutoff angle C-O-D as transferred from Fig. 1.
(2) Draw another horizontal line located above K-D such as M-S, and spaced above K-D by the desired linear lead, usually chosen as about 1/16 inch.
(3) With a radius of O-A equal to the full port opening, describe the Port Opening semicircle. Note: While the dimensional empirical data as discussed was followed closely in larger engine design, it is found in model practice that the width of ports may be reduced to approximately 1/2 the cylinder bore and the length of travel opening doubled accordingly. This makes for somewhat easier machining and dimensioning on small cylinders.
(4) Now for the tricky step; however, it requires only a bit of care and patience to accomplish with a pair of dividers or your compasses. To do this we must find by trial the radius E-F and the resulting center of a circle “E” which shall be tangent under three conditions: the Lead line M-S, the Port Opening Circle, and the Cutoff O-C.
(5) The radius E-F of the above circle may now be designated dimensionally as the Outside Lap.
(6) By dropping a perpendicular from the center E to the horizontal K-D we have actually scaled off the Linear Advance L in inches.
(7) In model work, quite often the Release (opening point of the exhaust port) is taken as full stroke. However, we may be pardoned at this point if we back up a bit and decide that, in accordance with practice upon the bigger boys we wish to establish release at, say, 0.9 stroke, we can find the crank angle at this point similarly to that for cutoff. Then assume that this angle comes out as C'-O-D as transferred similarly from a Fig. 1.
(8) Again, with E as a center, draw a circle tangent to the line O-C'. The lineal distance E-Z, the radius of this circle, is now the lineal dimension for Inside Lap.
(9) A final tangent line O-C' to the Lap Circle yields the crank angle position for Compression C'-O-D, commonly referred to as Cushioning.
(10) To finish up our job, we can now measure off the following dimensions. Note: It should be apparent that, for purposes of easier measurements, the entire diagram may be drawn to two or three times scale of actual valve to be utilized in your engine. Reference Fig. 3.
Length of Valve Face equals Outside Lap plus Width of Steam Port plus Inside Lap, equals A-B. Note: The head end dimension A-B may not be identical to the crank-end dimension C-D because of angularity of the connecting rod; this is further shown if a second Bilgram is drawn for the crank end, proceeding from Fig. 1 as explained above.
The bridge G-H should be at least as wide as the port opening. With the valve drawn as shown in central position, it is thermodynamically desirable to make the dimension of the exhaust port wide as possible, and also the dimension overall from F to F' in order to minimize back pressure and volumetric clearance, both wasteful items; however, unless a balanced piston valve (or poppets) be employed, the load force on the back of an unbalanced slide valve might become prohibitive.
The Seat Limit can of course be found from displacing the valve from its central position to one-half the eccentric throw, and scaling this dimension off on a drawing sketch. In small engines it is usually extended to the full width of the steam chest.
With the above information it is anticipated that some of you ambitious readers may find a bit of pleasure and satisfaction in getting out the old drawing board and instruments and performing a bit of design work for that next engine you are going to start building. As has been mentioned in previous articles by this writer, the old steam engine is quite full of little flaws from wheels to valve gear; but it has certainly provided us with the utmost of pleasure and usefulness.
And I am sure that there is not a soul among us who would not go out and buy a brand new steam traction engine today, were they but available and we could afford one of them. As an old friend of mine (now passed to his happy threshing ground) once said to me, “I must be crazy, but maybe it is just that I can make my own power!” But I also believe that the warmth of the old steamer was the closest thing to the human body ever devised. IMA