Making Correct Parabolic Nozzles

by Richard Garcia, Director of Research, RRS.ORG

I would like to set the record straight on a common mistake I see with designing parabolic rocket nozzles.

The simplest and most common nozzle used in amateur rocketry is the simple cone. Most commonly the cone is a 15° half angle. But the simplicity of the cone comes at the cost of reduced performance and increased length. The next step up in complexity is a parabolic bell nozzle.

The parabolic bell nozzle design is a little more complicated and much more difficult to manufacture. Machining a parabolic shape out of metal on a lathe by hand is doable with a digital read-out (DRO) device but the process would likely be very tedious.  In this case, I would recommend looking into some sort of computer-controlled system like CNC or 3D printing. If you’re using an ablative composite like silica-phenolic then tuning a wood mandrel on a lathe is very doable. Beyond the parabolic nozzle the next step up in complexity and efficiency is using the method of characteristics to design a rocket nozzle. This method was pioneered by Gadicharla V.R. Rao.

.These are sometimes called “Rao” nozzles. The parabolic nozzle is a decent approximation of a Rao nozzle, a method also proposed by Rao. Going further to get the last drop of performance would require a detailed computational fluid dynamics (CFD) based optimization.

I’m writing this report to set the record straight on a common mistake I see with determining the proper geometric and dimensional features of specific parabolic rocket nozzles. Almost all the amateur parabolic nozzles I’ve seen up to this point, including several college teams, and even my own early designs, made this mistake. It’s not really their fault though. The parabolic nozzle is commonly mention in rocket literature but rarely is one particular key fact needed to make them correctly explained. The two of the most prolific textbooks on the topic are, “Rocket Propulsion Elements” by Sutton and Biblarz, and “Modern Engineering for Design of Liquid-Propellant Rocket Engines” by Huzel and Huang. Both give their readers charts to help design the parabola and both also neglect to mention this one important fact.

That there is a problem at all will only become apparent when you try to solve for the exact equation of the parabola. If you read through the common literature (a task left to the reader) and go through the process you will see that the parabola is defined by two points and two angles. The two points are the beginning and the ends of the parabola. The beginning point is where the curvature at the throat, typically a circular arc, transitions into the parabola. The end point is the exit of the nozzle. The two angles, θn at the beginning of the parabola and θe at the exit are the angles between the centerline of the nozzle and lines tangent to the parabola at the two points. These angles are normally found by using graphs given in the literature. These graphs are the angles vs expansion area ratio as shown below. Note that they present the length of the bell nozzle as a percentage, Lf, of the length of an equivalent conical nozzle with the same nozzle expansion area ratio and a 15° half angle.

Graph of theta angles from Rao’s paper

Footnote about the θn and θe vs expansion area ratio graphs.

I’d like to note that the graph I shared is the in the commonly presented form. However, if you look at the original paper on parabolic nozzles “Approximation of Optimum Thrust Nozzle Contour” by G.V.R. Rao in 1960” you will find the information presented differently. Instead of theta versus epsilon at various percentages of nozzle lengths given in terms of the percentage of an equivalent 15° conical nozzle length of the same expansion ratio (epsilon), the axes of graph are in terms of the ratios re/rt versus Ln/rt given at various values of θn and θe. These lines are straight in comparison to the curved lines in the graph above. The more common graphs are supposed to have been derived from the originals. 

However, I have not gone through the math myself, but, I am interested in what the process and results would be. In turn, the originals are supposed to have been derived from full calculations of the Rao method. Again, something I have not tried replicating but it is something I would be interested in seeing the details of. 

Another thing worth mentioning is that I recall reading online that someone had tried to reproduce these results using the method of characteristics. (Unfortunately, I cannot recall exactly where I read this.) The original graphs were based on using a fixed specific heat ratio (gamma) of γ=1.23 and the original paper stated “A different value of γ does not appear to appreciably change the nozzle contour when the area ratio and length are prescribed.” however the testimony I found online stated that they found that the gamma value did significantly affect the angles. I am in doubt over this anonymous internet testimony, but it is something I’m interested in exploring in more detail.  If you ever investigate this yourself, dear reader, be sure to send us what you discover.

In total, you will have six parameters to determine, at point 1 you will have x1, y1, and θn. At point 2, you will have x2, y2, and θe. The obvious route from here would be to take the equation of a parabola (more precisely a square root since the x-axis will be our chamber centerline. But hey, a square root is just a parabola on its side), and these six parameters and try to solve for the unknowns in the equation.

This is where you run into the problem. The parabola is over-defined! You can’t solve for a parabola that meets all six criteria. It is impossible. Five of the defining parameters can be correct, but at least one will always be wrong. This is where most people, after considerable effort and eventually the banging of one’s head against a wall, will give up and let one, or more, of the 6 parameters to be wrong, usually by letting the starting angle of the parabola to be wrong.

When plotting a nozzle contour most put the centerline of the nozzle along the x-axis. The mistake everyone makes, including myself, is to make the natural assumption that this parabola is aligned with the nozzle’s same x and y axis in the same way as all of the parabolas in your algebra classes. That is to say the centerline of the parabola is around the x axis, the same axis of axial symmetry as the nozzle. But what everyone misses, including in the common literature, is that the parabola should be canted! That is to say, it should be rotated by some angle in relation to the x and y axis.

The way I found this fact out is by being a prolific reader. I’ve spent a lot of time gathering papers and books on rockets. I’ve only ever found one source that explicitly states the nature of these parabolasand that is in “Liquid Rocket Engine Nozzles NASA SP-8120” It simply refers the them as a “canted-parabola contour” and offers no further explanation. But that clue was enough for me to figure out what was going on. Adding the rotation of parabola adds a degree of freedom in the math. This allows for the equations to be solved with all the required constraints.

* Footnote on the NASA SP-8000 publication.

By the way, the NASA SP-8000 series of publications is a fantastic resource for the aspiring rocket scientist. Back in the 1970’s, they tried to write a comprehensive review of the state of the art in rockets and spaceflight. I remember reading somewhere that they fell short of their goal, since there was too much know-how to catalog. But what they did manage to publish is great. Whenever diving into an unfamiliar design topic, I’ll read the relevant SP-8000 article on it first as a good overview.  Some topics may be more outdated than others, but much of it is still great and relevant today.

When trying figure this out, I went digging and found the original paper on parabolic nozzles. “Approximation of Optimum Thrust Nozzle Contour, by G.V.R. Rao, 1960” I had hoped it would directly state the parabola should be canted, but it doesn’t. However, it does give enough information to deduce this conclusion.  In the short paper, its one page, Rao describes “a simple geometric construction for the parabolic contour.” It is something that would have been more familiar with the designers and draftsmen of that time. Today, we are more likely use a series of points and connect them with a spline curve, or use mathematical equations in CAD software to draw similar curves. But back then, designers would have to do it by hand on paper. The method suggested by Rao is called the “tangent method” of drawing a parabola. As the name suggests it involves drawing a series of lines that are tangent to the parabola.

Since we are interested in the mathematical equations, I won’t be describing that method any further.  But it is plain to see from figures that the method does not directly depend upon the x, and y axis. The contour is simply plotted on the x-y plane and then the contour is revolved around to x-axis to create the three-dimensional nozzle. Since we cannot fit the desired parabola with the normal parabolic equations, we are left with the question: 

How does one mathematically rotate the equation of a parabola?

Using parametric equations is how we will be able to create aparabola rotated on the x-y plane. Parametric equations also make some of the math easier when solving for circular arcs that are used in the rest of the nozzle contour. 

First let’s start with vertex form a parabola.

Equation 1; vertex form of the parabola equation

To parameterize this, let us set:

Equation 2: parameterization or variable substitution

And solve for x and y.

Equations 3.1 and 3.2: Solving for x and y variables

This pair of equations is now a parametric form of a parabola. But it is just a regular parabola and we need to rotate it. Recall from your study of trigonometry, that the formulas for rotating a point around the origin by angle θ are:

Equations 4.1 and 4.2: Rotating the frame of reference by angle, theta

We can combine equations 3 and 4 by substituting the parametric equations into the place of x0 and y0. Doing this, we get:

Equations 5.1 amd 5.2: Combining prior equations 3 and 4

From this, we can get:

Equations 6.1 and 6.2: further derivation of x and y equations

Note that for our use case, because the rotation angle θ is a constant, h∙cosθ-k∙sinθ and h∙sinθ+k∙cosθ are also constants and can be replaced by single terms, c1 and c2 respectively. But we can name c1 and c2 whatever we like so I shall name them h and k respectively. (Yes, I know this seems like an odd maneuver, but it works fine for us and makes our equations a bit simpler.) With a little rearranging this then gives us the parametric form of a rotated parabola:

Equations 7.1 and 7.2: rearranging equations 6.1 and 6.2

Now we are ready to solve for the parabolic nozzle contour. Before we do this for the above equations, we shall discus the rest of the contour as well.  We will use a piecewise set of equations to describe the entire contour. For each piece we will need to solve for parameters and the split points between the different pieces of the function. We will go through each of them in order.  How to size a thrust chamber is beyond the scope of this paper and is an exercise left to the reader. You should already have calculated the radius of the chamber, throat, and nozzle exit, rc, rt, and re, respectively and the length of the chamber and nozzle, Lc, and Ln respectively.  The figure below shows a typical form of a thrust chamber. You’ll note that I’ve used z and r instead of x and y. This is a notation that I adopted early in my work writing a code for thrust chamber design. I figured z is up and rockets go up, so that’s where I’ll put the axis of symmetry of the chamber. I put the throat at z=0 for convenience. I decided to keep my notation in this report, instead of adopting something more conventional, for my own convenience of not having to rewrite all of my equations.

CAD drawing of nozzle geometry with refernce points and radii, note the parabola axis is canted

The first part of our piecewise function f1 is the chamber section representing by red in the figure. f1 is a simple horizontal line with the equation. 

f1 Chamber Section

Equation 8 descrbing the straight chamber section before the converging section that follows

f2 Concave Curved Converging Section

The exact converging section shape typically does not strongly affect engine performance. So, the designer has a lot of freedom here butthe following geometric convention is common. Three parts are used, f2 (blue in figure) is a concave circular arc, followed by f3(green in the figure) is sloped line and f4 (orange in the figure) is convex circular arc. I’ve found that using the parametric form of a circle makes the math easier. The parametric form of this circular arcin function 2 is the following. 

Equation 9.1 amd 9.2 for a concave-type converging section

Were rf2 is the radius of curvature for function 2. 

f3 Linear Converging Section

The third piece is a straight line and we will use the point slope formof a line.  The designer provides the converging angle and a typical rule of thumb value is 30°. Using a bit of trigonometry, we can see that the slope will be tan(-θc). For the point we will use (z4, r4) which is where f3 intersects f4.

Equation 10: linear converging nozzle

f4 Convex Curved Converging Throat Section

The fourth piece is another circular arc. The radius of curvature is typically given by the designer as a multiple of the throat radius. A typical a rule of thumb value of 1.5 time the throat radius is commonly used. We will call this scalar rc2. Again, we will use the parametric form.

Equation 11: convex curved converging section

About solving for the unknowns upstream of the nozzle throat

Solving the converging section is where we will notice a hiccup (minor problem). Say that you have a large throat and a small contraction ratio, Ac/At. This could mean the point where f4 meets the given converging angle, θc, of f3, could be equal to or larger than the chamber radius. A mathematical solution that uses the given parameters would fail to produce real geometry. A second problem could arise even when a valid solution does exist because the result could be an undesirably sharp radius if the point is too close to the chamber radius.

So, we have two solution paths. For the conventional case, we don’t run into the previously mentioned hiccup. Here, functions f1, f3, and f4 are sufficiently defined by the given parameters mentioned so far. But to define where the function f2 is located, we will need an additional parameter.  In my work, I’ve decided to do this by setting the point r3 to be a percentage of the vertical distance between the chamber radius and r4. Where the concave f2 curve meets the linear converging portion, f3, is the location of r3. Where f3 meets the convex f4 curve is r4. My default value is 50% of the way between r4 and rc. I’ve named that parameter f2vp and its value should be between 0 and 1. The letters used mean: function 2, vertical percentage. (You are probably asking yourself “Why not just define rf2 in the same way as rf4, with a scaler multiple of rt?” Well good question. That’s the way I started doing it. It’s been a while but if I remember correctly, I found that it was hard to pick a default value that worked for a large variety of engines and if picked poorly results were either undesirable or didn’t solve. So, I came up with the above method which I found much more satisfactory.)

For the second solution path an alternat method for defining the converging section must be used. It will be convenient to set the radius of both of the concave, f2, and convex, f4, sections equal. Then we define the curved converging sections, f2 and f4, to each take up a percentage of the radial distance between the throat and the chamber radius. The linear section is the remaining percentage. I’ve named this parameter f2vp_alt. The name means: function 2, vertical percentage alternate method. Since there are 3 functions that take up this vertical distance, and two of them are equal the value for f2vp_alt must be between 0 and 0.5. If it is 0 that means that the linear portion takes up all the vertical distance and the curved portions would disappear. If it is 0.5 then the linear portion would disappear. The default vale I use is 1/3. This means that f2 takes up 1/3 of the distance, and f4 takes up 1/3 of the distance and the linear section, f3, takes up the remaining 1/3. It makes for decent looking converging contours.

This will always be solvable with real geometry, but means that theeither the converging angle or the radius of curvature for the converging portions cannot be pre-defined and is instead solved for. At first, I tried to let the converging angle, θc be derived. However sometimes the solution would be a very shallow angle creating an overly long converging section. Instead, I found better results by keeping the designer given θc and instead solve for the radius of rf2and rf4, which would still be kept equal. I was resistant to this at first because it would remove the ability to use the rule of thumb value for rf4, or any specified value for that matter. This will only work if the assumption that the designer will provide reasonable values for θcis met. To summarize, the primary method lets the radius of function 2, rf2, be an unknown to be solved for, and the converging angle of f3is given and the radius of f4 is given. With the alternate method the converging angle is given along with the vertical space the functions occupy. This leaves the radiuses of curvature rf2 and rf4, which are equal, to be solved fore.

When I have designed thrust chambers, I use the first method and then only switch to the second method if the first method does not work. The remaining question is when to switch to avoid a sharp corner. My solution was to specify that the concave radius, f2, take up a minimum percentage of the vertical distance between the throatand chamber radius. My default value for this is 15% but this was somewhat arbitrary. You can also avoid the second method by increasing the thrust chamber contraction area ratio if possible.

Converging section solution, Method 1

The following is a mathematical solution for the converging section when using the regular method for the converging section described above. From f4 we need to solve for z4 and its corresponding radial point r4. For that we need to find when the circular arc is at the givenconverging angle. This is when its derivative is equal to the slope of f3, the linear section, which is tan(-θc). We will take the derivative ofthe parametric equations of function 4. In case you forgot from your calculus class the derivative of a parametric equation is found by taking the derivative of each equation separately with respect to the parametric independent variable, in this case t, which will give us the equations for and . Then you divide the equation for by the equation for and cancels out and you are left with an equation for . Using this method on the equations 11.1 and 11.2 for function 4 we get

Equation 12:

We solve this at t4 by setting the derivative equal to tan(-θc) which gives us

Equation 13:

Solving for t4 we get

Equation 14:

Note that this solution is periotic so it mathematically solves with any added to it. Through quick trial and error, we find a value n=1will put point (z4, r4) in the correct quadrant on the x-y plane. We then plug t4 into f4 to get the z4 and r4 coordinates. 

Equations 15.1 and 15.2:

Now that we know that value of r4 we can now check if the criteria needed for switching to the alternate method is met. If not, we can carry on to find r3. Recalling our assertion that r3 shall be some given fraction of the vertical distance between rand rc we see that it would be given by the following equation.

Equation 16:

Next, we plug r3 into the f3 function [eq 10] and solve to get z3.

Equation 17:

Now all that remains to be solved is z2 and the radius of curvature, rf2, for the concaved converging function f2. If we take f2 [equation 9] and plug in the known point z3, and r3, which both occur at t3, we get the following:

Equation 18.1 and 18.2:

We can see that the there are three unknowns z2, rf2, and t3, but there are only two equations. So, we need a third equation to solve this system. We know the derivative of this function is the same as for the function f4, [equation12] since they are both the equations of a circle. And we know the value of the derivative at this intersection pointfrom the slope of f3, the linear converging section, which is tan(-θc).Setting equation 12 to tan(-θc) gives us the third equation we need. 

Equation 19:

We then solve this system of three equations, 18.1, 18.2, and 19 for the three unknows z2, rf2, and t3. This gives us:

Equation 20:
Equation 21:
Equation 22:

We now have all the parameters and split points needed for each of the function to fully define the converging section. Equation 20 like in equation 14 also has a periodic portion of the solution, but here n=0.

Converging section solution, Method 2 

With the second method everything is the same as the first method up to calculating t4 [equation 14] then the methods diverge. In method 2, after calculating tthe next things to calculate are r3 and r4. This method defines the percentage of the radial distance between the throat and the chamber that the two curved converging functions, f2and f4, take up. This definition allows us to create, by reasoning, the equations for r3 and r4. They are:

Equation 23:
Equation 24:

Next the radius rf4 can be found by taking f4r [equation 11.2] at t4 when it will equal r4,

Equation 25:

and then solving for rf4 and we get 

Equation 26:

Then we set rf2=rf4. For the remaining unknowns that define the geometry, we can use the same math from the other method. Those remaining unknowns are z4, z3, t3, z2 and are defined by equations 15.1, 17, 20, and 22, respectively.

f5 Diverging Throat Section

For the diverging section function 5, magenta in the figure, is again in the parametric form of a circle.

Equations 27.1 and 27.2:

The radius of this function, rf5, is the diverging radius of the nozzle throat, it is given by the designer and is usually a rule of thumb value of 0.382 times the throat radius. But we still need to find the point z5, and r5. To do this we first note that the derivative of f5 is thesame as for f2 and f4 given in equation 12. We know what the angle of the tangent should be at z5, where f5 meets the parabolic portion. It should be the θn mention above and gotten from the θn vs expansion area ratio graph. From the same trigonometry used for the slope of f3we can set the derivative [eq 12] equal to tan(θn) at t5. Then we solve for the parameter twhich gives us:

Equation 28:

Like in equation 14 this also has a periodic portion of the solution, but here n=0. We then plug t5 into f5 [eq 27] to get z5 and r5.

Equation 29:
Equation 30:

f6 Parabolic Nozzle Section

The parabolic section of the nozzle, function 6, is yellow in the figure. As derived above in equation 7 we use a pair of parametric equations for a parabola with a rotation applied. I like to use ψinstead of θ here for the sake of symbolic variety.

Equations 31.1 and 31.2:

Solving for the unknowns downstream of the nozzle throat 

All that’s left is to solve for the remaining six unknowns in function 6. These unknowns are, t5, t6, m, h k and ψ. Note that the parameter variable t is unique for each section, so it’s technically a different t5 than for the t5 for function 5.  We know six key data points on the parabola. We know the starting coordinates z5, and r5 from our prior work. We know the end coordinates z6, r6 from the exit radius and the length of the nozzle provided by the designer. If we substitute these coordinates into function 6, we would have the following four equations:

Equation 32:
Equation 33:
Equation 34:
Equation 35:

But it’s not possible to solve these four equations for six unknowns. We need at least two more equations. We know the slope at the beginning and end of the parabola from θn and θe which we got from the θn and θe vs expansion area ratio graphs. This means we can use the derivative of function 6 to give us the last two equations. To find the derivative function first we take the derivative of each parametric equation with respect to the parameter variable, t.

Equation 36:
Equation 37:

And then we divide by the equation for and we get:

Equation 38:

Taking this function at t5 and t6 and using trigonometry to find the slope of the tangent lines at beginning and end of the parabola, we see that the derivative should be tan(θn) and tan(θe) respectively. This gets us last two equations we need for a system of six equations to solve for our six unknowns. 

Equation 39::
Equation 40:

Now all we have to do is simultaneously solve the system. Here are the six equations fully written out.

Equation 41:
Equation 42:
Equation 43: exit axial position
equation 44: exit radius value
Equation 45: nozzle tangent angle
Equation 46: nozzle exit angle

As far as I can tell this can’t be solved symbolically. I’ve tried several methods including a few computer algebra systems, but always failed. (Give it a try yourself and let us know what you come up with.) 

Instead, I solve the system of equations 41 through 46 numerically for m, h, k, phi, t5 and t6. This can easily be done with a calculator like the Texas Instruments, TI-89. But the way I solve this and the rest of the calculations is with a piece of custom software I wrote. My programing language of choice for doing this has been Octave, but it is possible to do with any Turing-complete programming language.  Once you’ve done this and solved for those unknowns you now have a fully mathematically defined parabolic nozzle contour.  One final note worth mentioning is that I typically use these equations to find the value r at a particular z location. If this z is in one of the parametric sections, then I first solve the equation fz(t)=z for t. Then I take the value for t and use it to solve fr(t)=r for the value of r.

Hopefully, that clears up any confusion and set the record … canted.

List of symbols

Ac​ = Cross sectional area of the thrust chamber

Ac/At​ = Thrust chamber contraction area ratio

Ae​ = Cross sectional area of the nozzle exit

At​ = Cross sectional area of the nozzle throat chamber

c1-c2​ = Numerical constants

f1-f6​ = Parts of the piecewise function that defines the nozzle contour

f2vp​ = Fraction of the vertical distance between r4 and rc occupied by f2

f2vp_alt​ = Fraction of the vertical distance between rt and rc occupied each by f2 and f4

fr1-fr6​ = Radial portion of the of the piecewise functions that defines the nozzle contour

fz1-fz6​ = Axial portion of the of the piecewise functions that defines the nozzle contour

h​ = Horizontal translation of parabola

k​ = Vertical, or radial, translation of parabola

Lc​ = Length of the thrust chamber, measured from the injector to the nozzle throat

Lf​ = Length of a bell nozzle given as a percentage of an equivalent 15° half angle conical nozzle

Ln​ = Length of the nozzle, measured from the nozzle throat to the nozzle exit

m​ = Scalar that is the stretch factor of a parabola 

n​ = scalar for periodic solution, n can mean any whole number

r​ = Radial coordinate

r1 – r6​ = Radial coordinate of points 1 through 6, see figure for location of points

rc​ = Radius of the cylindrical portion of the thrust chamber

rc1​ = Scalier for the concave converging radius rf2 = rc1 ∙ rt

rc2​ = Scalier for the convex converging radius rf4 = rc2 ∙ rt rule of thumb value of 1.5

rd​ = Scalier for the diverging arc radius rf5 = rd ∙ rt rule of thumb value is 0.382

re​ = Radius of the nozzle exit

rf2​ = Radius of curvature for function 2 

rf4​ = Radius of curvature for function 4

rf5​ = Radius of curvature for function 5

rt​ = Radius of the nozzle throat

t​ = Parametric independent variable, it is unique for each pair of parametric equations

t1 – t6​ = Parametric independent variable at points 1 through 6, and unique for each parametric function

x​ = Horizontal coordinate

y​ = Vertical coordinate

z​ = Coordinate perpendicular to the x-y plane

z1 – z6​ = Axial coordinate of points 1 through 6, see figure for location of points

ε​ = Nozzle expansion area ratio, At/Ae

θ​ = Rotation angle of a parametric parabola

θc​ = Converging angle of the linear converging section, function 3

θe​ = Angle between the nozzle centerline and parabola tangent line at the the nozzle exit

θn​ = Angle between the nozzle centerline and parabola tangent line at the beginning of the parabola

ψ​ = Rotation angle of the parabola of the nozzle contour

For questions and comments:

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80th Anniversary of the Reaction Research Society

by Dave Nordling, President, RRS.ORG

Today, January 6, 2023 marks the 80th anniversary of the Reaction Research Society.

Founded in 1943 by George James, this small club of young boys gathered in Glendale to study and conduct experiments in reaction propulsion. It all began with a child’s chemistry set that included zinc and sulfur powder as a demonstration of exothermic reactions. This would lead to rockets and newer propellants and ever increasingly complicated experiments in the remote northern areas of Los Angeles from Mint Canyon to Rosamond to our present location east of Koehn Dry Lake. Many members of the Reaction Research Society have pursued professional aerospace careers making important contributions to the field.

In the war years of the 1940’s, rocketry was more the subject of science fiction and speculation than the robust science it is today in the early 21st century. Our society has seen eight decades of growth as a collective group of enthusiasts and colleagues supporting research and offering shared testing spaces while remaining dedicated to responsible projects and scientific progress. It is with pride that we share this milestone with the public on a bold new year of launch, facility improvement and looking forward to the future.

Photo credit: Mani Suri

Our month meetings are held at 7:30pm on the second Friday of each month by teleconference and in person at the front office of the Compton/Woodley Airport on Alondra Blvd. in Compton. For details, contact the RRS secretary.

Burst discs in liquid rocketry

by Dave Nordling, Secretary, Reaction Research Society

I was recently asked for advice on the installation of a burst disk in a university liquid rocket project. As any pressure relief device is an important subject to consider carefully, I wanted to present a summary of my thoughts to our broader readership.

The Reaction Research Society (RRS) is happy to offer advice, but my first recommendation to any university team would be to talk with your university professors, professional advisers and mentors that are involved with your project. A burst disk is an important component and its function can be critical for safety and preserving your vehicle in any over-pressurization scenario. The subject of your rocket system pressurization, venting and relief devices is extremely important to study well and thoroughly understand before proceeding with any component selection or testing.  Your university is the best place to start.

For those who are doing a liquid rocket project outside of a university program, I would also recommend to consult with experts and reputable manufacturers and distributors of pressure relief devices including burst disks.

Burst discs (the spelling “disk” or “disc” is interchangeable) are one simple form of a pressure relief device or valve that is designed to prevent over-pressurization of a pressure vessel and potential catastrophe.  Burst disks are also sometimes called “rupture disks” which clearly describe their function.

In liquid rocket system designs, burst disks are often placed not only at the pressurant bottle to protect the higher pressure part of the system, but also at the lower pressure end of the regulator which protects the propellant tanks being pressurized. In the event of pressure regulator failure, the burst disk can protect the propellant tank.

Burst disks are usually in the form of a dead-ended pressure fitting that is adapted to directly connect into the pressure vessel either directly into the pressure vessel volume boundary itself or by a tube connection that is also directly connected into the pressure vessel volume boundary. These fittings have a frangible or breakable membrane that is designed to fail when the pressure reaches a specific design point.

An illustration of the burst disk fitting concept

A burst disk is a “one-time use” device and can not be reset after they have “actuated”. As a pressure relief device, the burst disk is often chosen for its compact size and simplicity. They are in common usage in many industries and can fulfill their relief function very well if they are sized and located properly.

They must be securely and directly connected into the volume of the pressure vessel and have no valves or other hardware which would isolate, block, impinge or constrain the relief function in any way. The American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel (B&PV) code does have some general advice on this subject and this is a good place to start your study.

These devices are simple to understand but fairly complex to size properly. Beyond the design of the burst disk, you must also consider where these devices will physically fit on your vehicle, where are they located and what is the environment doing around your relief device

The burst disk body and membrane can be subject to corrosion or physical damage that could reduce it’s effective bursting pressure. It’s important to consider the material compatibility of all body, seals and membrane materials that are exposed or “wetted” to the gases inside. Also, its important to avoid getting gouges, nicks or marks on the membrane that would form stress concentrations and weaken the membrane. Even when being cautious, don’t leave your burst disk covered when it needs to be ready to perform. Careful handling is good advice at all points in the project.

There are three things to consider when locating and installing a burst disk:
(1) relief (set) pressure, (2) minimum flow rate required, (3) where is your burst disk pointed?

(1) Set pressure of the relief device

Any relief device must be set to actuate (or in the case of a burst disk, to rupture) at a pressure above all of your nominal conditions, but also adequately below any and all failure modes.  In some pressure vessel or relief device codes, there are rules of thumb about the set pressure must be a specific percentage (%) above the maximum expected operating pressure (MEOP) or maximum allowable working pressure (MAWP). The thorough examination of all operating conditions and hardware limitations is essential of finding the right set pressure for the relief device. 

ASME also has codes for sizing relief valves in process piping, but the rocket industry doesn’t have a particular specification. The aerospace industry does often draft their own specifications and requirements which follow good industrial practices and always include careful design and testing as part of proving the designs to be sufficient.

Another consideration beyond the static pressure in your pressure vessel is the temperature environment of the gases inside. Beyond the fact that higher temperatures from a thermodynamic standpoint create higher pressures, a burst disk relies on the material strength of the membrane and the yield and ultimate strength can weaken under higher temperatures. Some materials (examples are low quality steels) can also become weaker under cold temperatures. Always consider the full range of temperature environments in every application. It’s important to size each burst disk individually and resist the temptation to assume that one device will suit all environments.

There’s a big tolerance on a burst disk set pressure, so be aware of that imprecision. Burst disks are compact but getting a membrane to burst at an exact pressure is not really practical and thus these devices are not very precise.  Ask the manufacturer about the expected tolerance on any relief device. It’s also wise to test a few of these devices to measure the actual burst pressure. Make sure you are recording data because failure happens suddenly and you are unlikely to visually see the last pressure reading before burst. If you blink, you can miss the most important data point. Therefore, use a data acquisition system when testing your pressure relief devices.

(2) Minimum flow rate required

Any pressure relief device when activated must be able to drop pressure fast enough to avoid over-pressurizing and failing the pressure vessel. This is a less commonly evaluated situation but its equally important to recognize any scenarios where the transient pressure rise would challenge the relief flow rate needed to keep the pressure below a safe level at all times. Steam pressure systems have this problem and so do cryogenic vessels.  Most designers just choose a fitting similar in size to the lines being used, but this isn’t always accurate. 

Relief devices are nearly always sized relative to their flow rate afforded.  This is sometimes called the “capacity” of the relief valve or burst disk. You’ll need to know your gas and upstream conditions. With this, you’ll need to know the open area when the valve is opened and make this is the smallest restriction in the entire flow path. The open area can be expressed as either the discharge area (Cd A) or the valve coefficient “Cv” value. With each device in each specific location, you must select a burst disk capable of venting enough flow to cover the whole range of expected conditions. This is crucial to finding the right burst disk or relief valve. A device that does not have a large enough capacity will not protect your fluid system.

Another consideration for your relief device is if you have any flow path that is smaller than the area of your relief device. One example of poor design is having your pressure relief device located at the end of a long skinny tube. Even if the open area of the tubing is larger than the pressure relief valve opening, the length of the line can accumulate enough flow friction in the tubing that can unintentionally add up enough pressure drop to pose a significant restriction to your relief flow. This is to say nothing of someone accidentally denting or kinking the tubing which would create a severe blockage of the relief flow. It’s always smart to have your pressure relief device very closely coupled to the pressure vessel volume that you are protecting. This means keeping the distance as short as possible. Always know all of your flow path areas and line lengths!

Another classic mistake in fluid system design is putting a valve or any other restriction device in-between the pressure boundary volume and the pressure relief device that is protecting it. Careful consideration of all valve placements and their positions in all operating modes and under all possible operating scenarios. Put simply: “Do NOT EVER create a situation where the pressure relief device can be isolated or impeded in its operation at any time for any reason, even temporarily. Some piping codes absolutely forbid this. Careful peer-review of your pressure and instrument diagrams (P&ID’s) must look for this situation and avoid it. More than reviewing the paper schematics, one should physically trace all flow paths to be sure the builder hasn’t made such a mistake. The physical hardware must always match the P&ID.

(3) Watch where your burst disk is pointed! 

When your burst disk goes off, any foreign object debris (FOD) near the discharging outlet can be thrown out at high speed causing injury or damage to nearby hardware and structures. Even without particulates or FOD, the impinging high-speed sonic jet of gas is very dangerous.  No one should be standing near a fluid system while any part of it is pressurized anyway, but you should always consider what might happen when your burst disk goes off. You won’t always know when the device will go off. Be prepared at all times.

Make sure all hardware is also secure enough to take the sudden thrust from the burst disk relieving itself. This can be a sudden and powerful force that breaks hardware or knocks things over. The rocket thrust equation also applies in this case. To calculate this thrust value, you do this in two parts: (1) You consider the choked flow pressure differential multiplied by the discharge area and (2) add in the product of the mass flow rate of the gas escaping multiplied by the sonic velocity of the upstream gas conditions.

Calculation of the thrust load from a discharging relief device such as a burst disk

As a design note, for nearly all gases, if the upstream pressure is more than double that of the downstream pressure, the flow velocity through any flow path restriction(s) or “orifice area” is sonic or at the speed of sound as computed by the upstream gas pressure and temperature conditions. This is called “choked” flow.

One potential fix to the jet thrust problem out of relief device is to divert and diffuse the discharging outlet flow in opposing or evenly distributed directions as long as the combined discharge flow areas are sufficiently large and balanced.

An illustration of a burst disk device with balanced venting

Another consideration to be made with a burst disk or pressure relief device is to consider the downstream environment where your burst disk is discharging.

Is the relieving gas or gas mixture going to create a flammable or toxic environment? If so, you need to consider how and where you are diverting the hazardous gases being relieved. Some burst disk fittings have threaded ends on both ends which allows the discharging flow to be routed to a safe location, if this is a necessary feature.

Screw-type burst disk fittings made by Zook in three basic types

Are you creating a dangerous environment (reduced oxygen) within a confined space? The subject of confined space safety is very important and worthy of a separate article in itself. Most testing will be done outdoors and in a very well ventilated environment, but the rocket business is full of horror stories of people who have become injured or asphyxiated simply from improper consideration of confined space safety.

A less often considered scenario is whether the space where the burst disk or relief valve is discharging into is fully open to the environment or not. It is possible to overly restrict or “back up” a burst disk or relief valve if the interstage volume in your rocket isn’t very large or isn’t adequately vented to the outside. Sometimes your discharge space simply isn’t big enough. It is very important to know your vehicle hardware geometry very well, measure your volumes and consider all flow areas out of all assemblies.

Find a reputable burst disk manufacturer and distributor

There are a few reputable manufacturers of burst disks. Fike is one that comes to mind, but they tend to be for very large piping sizes used in facility plants. Fike has been providing reliable products for many years to many industries including oil/gas and the aerospace industry. Swagelok has access to a lot of fluid component manufacturers which may be more suitable.

Zook is another manufacturer of burst disk fittings. These in-line devices come as a holder fitting and replaceable disk. The screw-type fittings are two-piece assemblies and have standard pipe thread ends. The disks come in a range of nominal set pressures.

Screw-type burst disk fittings by Zook

There are certainly other manufacturers and all of them should be able to provide you with good advice or transfer you to a distributor company to help you with selecting an appropriate device. Before you call or email, you must have already taken the time to understand your pressure environment, capacity and design requirements first. A good component distributor is one that is willing to work with you to find the right part for your application and educate you in making the best choice. Literature is easy to find online and always consider more than one manufacturer to get a good price.

A few last words of caution

Burst disk devices can be manufactured from scratch and other amateur rocketry hobbyists have attempted to do so. This is not a good idea. There are a lot of considerations to make in building a reliable burst disk from scratch not to mention the time and materials to adequately prove the design. To make a burst disk from scratch would become every bit as expensive as simply going to a reputable manufacturer and using their product.

As much as your group may want to save money, pressure relief devices are a critical part of your fluid system to which lives may be at stake.  Don’t be cheap. Find a quality manufacturer, select the right product and test them.  Ebay is not the place to find quality products.

If anyone has anything to add to this subject, please contact the RRS secretary or the RRS director of research.