I decided to start on a new motor. So I thought I would share the process I use to design and build a new motor.
Some of the basics:
To start, I knew I was using 2" EMT for the casing. The 2" EMT has an actual inside diameter of 2.08". I just guessed that I wanted the grain length about 15". I also knew a single grain would create a very progressive thrust profile, the motor would build up more thrust the longer it burned. Another problem with a monolithic grain, the Kn would start out very low, then increase dramatically at the end of the burn. That's not really what we want. A strong kick off the pad, then tapering thrust would be better. To compromise, I decided to use three grains, and allow the ends of the grains as well as the core to burn. The more short grains you use, the greater the initial burning surface area you get.
To find the initial surface burning area. Start with the area of the ends of the grains.
3.14xr2=area of circle
3.14x(1.04x1.04)=3.39
The core is .75 inch diameter, so we have to subtract it from the end surface area. The radius of the core is .375 inches.
3.14x(.375x.375)=.44
Subtract .44 from 3.39.
3.39-.44=2.95 square inches
Each end has a burning area of 2.95 square inches, we have 6 ends.
2.95x6=17.7 square inches
So, we know we have 17.7 square inches of burning area on the ends.
Now, what about the burning area of the core.
The core surface area can be found by the following:
Circumference x length= core area
circumference=diameterx3.14
.75x3.14=2.35 inches
2.35(circumference) x15(length)=35.3 square inches
Add the end burning area and the core burning area together, 17.7+35.3=53 square inches.
Now we need to find the burning surface area at the end of the motors burn. The first thing we need to do is find
the length of each grain. Remember, the grains have been burning from the ends, so they are shorter. Assuming the
burn is more or less linear, at the end of the burn the grains will have burned down the same distance as they
burned in. Our motor is 2.08" diameter, subtract the .75" core and you're left with 1.33 inches of propellant,
of course you need to divide by 2 to get the actual web thickness of the grain. So in my case it is .665"
web thickness. So the end of the grains will burn down .665" from each end. We have 6 ends burning, so 6x.665=3.99
inches. So the 3 grains that started out as 15" in overall length, are now effectively 11.01 inches long.
Again, circumferencexlength.
3.14x2.08=6.53
6.53(circumference)x11.01(length)=71.89 square inches
So our surface burn area at the end of the burn is about 71.9 square inches. Our starting burn area was 53 square inches, ideally, they would be the same. As it is, our motor will still have a progressive burn. But, I don't like messing with four or five grains in a motor this size. So let's see how this works out in terms of the Kn.
Let's take a stab at a throat diameter of 5/8", or .625". To find your Kn, divide the surface burning area by the throat area. Our .625" diameter throat has an area of 3.14x(.3125x.3125)=.3066 square inches.
Initial burn surface area is 53 square inches/.3066=172.9, that's our Kn.
Final burn surface area is 71.89 square inches/.3066=Kn234.5.
Above is a graphic with dimensions of one of three identical grains used in the motor.
A Kn of 234.5 should be fine using the 2" EMT, as chamber pressure should not exceed 800 psi, in fact, I would expect the maximum chamber pressure to only be in the 500 to 600 psi range, owing to the fact I am not optimizing the propellant by powdering it first.
Just to make sure, let's take a look at the formula for determining chamber wall minimum thickness.
[(maximum expected pressure)x(chamber diameter)]/[2x(material tensile strength)]
Our material is steel, with an expected ultimate tensile strength of 75,000 psi.
(800x2.08)/(2x75,000)
1,664/150,000=.011 inches minimum
As you can see, our steel EMT is pretty strong stuff. We have a safety margin of about 6:1. Also, steel does not lose strength very quickly when heated, as aluminum would. A few notes on the thermal properties, the 65% KNO3 and 35% sugar propellant has an expected combustion chamber temperature of 2,139 degrees (F) that's at a chamber pressure of 600 psi. Steel has a melting point of 2,750 degrees (F). So no thermal protection is needed.
The c* is 2,749 fps
Isp is 132. (theoretical)
I cast the first grain today. I was going to use tag board (thin cardboard) as an outside inhibitor on the grain. Then, I remembered I had some thin walled 2" diameter PVC pipe. It is actually central vacuum tubing. After checking it out, it was a perfect fit. While it will reduce my grain diameter slightly, its ease of use will outweigh the loss of propellant. I don't think heat will be a problem, as the propellant burns so quickly I don't think the PVC will have time to melt, at least not during the burn. I can imagine it will stick to the chamber wall though.
I built a wood base, and drilled a centering hole for my coring tool to rest in. The coring tool is a 3/4" steel bar with one end tapered to fit into the centering hole. I then hot glued a 4.5 " section of heavy PVC pipe to the base to act as a brace for the thin PVC casting tube. I inserted a small square of wax paper into the bottom of the support tube, then inserted the thin walled casting tube. The heavy PVC support tube is needed, as the PVC casting tube would deform from the heat of the propellant casting.
Above is a picture of the casting stand, the support tube is glued to the wood base.
Here is the casting stand assembled. The 3/4" steel coring tool is inserted after the melted propellant is scooped into the PVC liner tube.
Here is a picture of the cast grain after it cooled. It has yet to be trimmed. The finished grain size should be 5", I cast the grain 5.25" long so I can trim it up perfectly square.
A little more theory may be in order now. You may wonder where I got chamber pressure from. It wasn't just a guess. I used the same propellant mix in an earlier motor, after static testing of that motor I came up with some useful data that I can now use to help design a new motor.
The other motor I static tested was the T2. It had 2) 5.5" grains with a total propellant weight of 1.18 pounds. The throat was .42" diameter (.1385 square inches). The motor had a similar Kn as the new motor will. The motor burned for 1.2 seconds with a web thickness of .5", meaning the burn rate at pressure was .4166" per second. The C* for sugar propellant is 2,749 fps.
The formula to calculate chamber pressure is:
Chamber Pressure (psi)=[(propellant weight flowrate)*(C star)]/[(throat area)*32.2]
So, our propellant weight flowrate is 1.18(weight of the propellant)/1.2(seconds of burn time)= .983 lbs. per second.
Chamber Pressure=(.983x2749)/(.1385x32.2)
Chamber Pressure=2702.27/4.4597=605.93 psi
Keep in mind. This is an average chamber pressure. The motor was slightly progressive, so the chamber pressure would have been lower at the start of the burn, and higher at the end of the burn.
Nozzle Design
The last step is nozzle design. To make an efficient rocket motor, the nozzle design is crucial. You could just make an opening of the correct size, but the exhaust gases would never accelerate to the super sonic level. A Delavel nozzle performs the task of accelerating the gases through the throat and allowing the exhaust gas to expand at the proper rate as it moves through the exit portion of the nozzle (called the divergent cone).
As a rule of thumb, the entrance cone (called the convergent cone) should be about 60 degrees, or a half angle of 30 degrees. The divergent section of the nozzle should be at about a 15 degree half angle. The divergent cone is the most critical. Too great an angle and the gases expend energy expanding, wasting impulse. Too long of an exit cone and energy is again lost because the gas actually expands to a point where its pressure is below that of atmospheric pressure. Ideally, the gas pressure as it leaves the divergent cone should be the same as atmospheric pressure.
We know our throat diameter is .625", to find the diameter of the exit cone some software comes in handy, or a lot math. I have a reference book with nozzle expansion ratio and thrust coefficient in a tabular form. In my case the nozzle works best with an expansion ratio of 5.8 to 1, yielding a thrust coefficient of 1.49 and an exit cone diameter of 1.505".
You could probably wing it here, without doing a lot of math. Any well shaped divergent cone is going to improve performance. Going a little smaller on the exit diameter is probably better than too big. Just as an example, if I ran the same motor at a chamber pressure of 1,000 psi, the expansion ratio would be 8.5 to 1 and 1.822" in diameter.
I have more information on design and turning a nozzle here. Make sure the area where the throat meets the convergent and divergent cones is polished very smooth, slightly rounded with no rough spots. The actual length of the throat area is not super critical, but I would try to keep it at about 2 throat diameters or less.
Below is a series of pictures taken as I built the motor.
I don't have any decent CAD software, so once I have the basic dimensions of the nozzle, I draw it out 1:1 on graph paper. Sitting on the paper is the 2.125" mild steel bar stock the nozzle will be turned from. The bar stock was cut to length on my 14" abrasive wheel saw.
Here is the bar stock chucked up in the lathe. With a piece this large and no tail stock used, I had trouble getting the stock perfectly aligned. So I moved the tool post into the slowly turning stock. I left the chuck loose, so as I moved the tool post into the stock it bumped the stock and centered it for me. It worked very well.
I always start by drilling the core first. I start out with a small drill, and work up to the bore size of 5/8" in several increments.
Moving ahead a little. The convergent cone is done. The outside diameter around the convergent section has also been turned to size and an o-ring groove cut. The tool is now set to reduce the diameter of the nozzle in the throat area.
The stock has now been turned around and I am cutting the divergent cone.
Once the divergent cone has been cut, it's just a matter of removing outside material for weight reduction.
More weight reduction, and it's starting to look like a nozzle.
Here's the finished nozzle next to the drawing.
The next step was the motor casing tube. I cut it to size on my abrasive saw, then drilled and tapped a total of 12) 10-32 holes around the circumference of the tube. I wanted a smooth surface on the top end of the motor, so I would weld a top bulkhead in place. I turned my bar stock down to the needed 2.08" to fit snugly into the casing. The problem was, my bar stock was too short to cut with my abrasive saw. So I tapped the bar stock into the motor casing, then held the casing in the saw vice to cut off the .3" thick wafer I needed as a forward closure.
I also wanted to get chamber pressure readings during static testing. So before I installed the forward bulkhead I drilled and tapped it for a 1/4" NPT fitting. I then installed the forward bulkhead and welded it in place. I inserted the bulkhead about 1/2" from the top into the casing. I wanted room to weld, and also room to install a plug in the pressure port when the motor was used in a rocket.
Here is the motor casing with the forward bulkhead welded in place. It's not cleaned up real well, but you can see the threaded port in the center.
On top is the completed motor (less screws), below is the T-2 motor for comparison.
On the left is the 2" nozzle, on the right the 1.5" T-2 nozzle.
Here are some numbers on the 2" motor.
The expected Isp should be in the 120 to 130 range. That would put the motor in the high J or low K class (1200-1300 Ns).
Update: Feb. 21, 2004
I performed the first static test firing of the new motor today. The video can be seen by clicking here. I had to perform the test a little close the my garage. It was the only place close enough to my PC that wasn't snow covered. So I video taped the test through a window, so the sound level is a little off too.
Here is the motor at full thrust.
I wanted data from the test, but the motor diameter was too large for my test stand. So I decided to just measure chamber pressure, and convert the chamber pressure to thrust through math. I did use my test stand, but I simply strapped the motor to the stand using hose clamps. I used the same pressure transducer I have been using on my strand burner, it was set up and calibrated, making installation easy.
After taking a short break to find my dog (he's smart enough to leave home when I do static tests).When he sees me setting up a motor he runs!
It seemed kind of odd, not watch the burn. I was at my computer in the house, and had my launch controller next to me, so I never saw the test until I watched the video. I started the data acquisition software recording, did a last continuity test, and pushed the fire button. The motor lit quickly, came to full thrust with a roar as I watched the chamber pressure climb, then tail off.
I went outside to do a quick inspection of the motor. All seemed well, there was no evidence of blow by or pressure leakage. So I went back inside and watched the video.
I then started the process of evaluating the data from the test. I transferred the pressure/time trace to Excel, and totaled the psi readings. I had captured the data at 60 Hz, that is a time frame of one sample per .01666 seconds. So I multiplied the total psi (33,054) by the time increment of .01666 seconds. That gives a total average pressure of 550.88 psi. The 550.88 psi is divided by the burn time of 1.417 seconds, leaving us with an average pressure of 388.87 psi. Thrust=(chamber pressure)*(throat area)* (nozzle coefficient) So, 388.87*.3066*1.49=177.65 lb seconds average thrust. Take that times the burn time of 1.417 seconds leaves us with 251.7 lb seconds total thrust.
251.7 lb seconds of total thrust divided by our fuel weight of 2.19 lbs leaves us with an Isp of 115.
So I crunched some more numbers. Working from another approach. There were 85 samples in the test, divide the 85 into the total of 33,054 and again, you get 388.87 psi average pressure. It works the same.
Here is a chart of the Pressure/Time trace.
It is a rather odd curve. It would have been expected that the curve would go up initially, then flaten out to a modest increase before dropping off sharply at the end of the burn. That brings up another question. How did I get 1,199.2 psi chamber pressure? that's much higher than planned. I believe the answer is in how I loaded the motor. I didn't use any spacers between the grains, and I think the grains only burned from the core out. I other words, the ends never burned. I was a little concerned about that, as the grains were a tight fit in the casing. That would lead to a very progressive burn, and very high maximum chamber pressure. In the future, I will use spacers between the grains.
Above is a photo after the test.
As you can see in the picture, the motor survived the test with flying colors, even going to 150% of maximum expected chamber pressure. The PVC vacuum tubing worked well as a casting tube and inhibitor. The liner was even easy to remove, they pretty much just slid out. The chamber walls remained cool enough they were not even heat darkened as is usual. The nozzle suffered no adverse effects other than the usual heat discoloration.
I will use grain spacers on the next test. And I need to make a mount on my test stand to hold a motor this size. I'll post more results after the next test.
Update: The motor has been test fired again. With even stranger results. Check out the Static Test 26 page for all the information.