Forgive me if anyone finds this to be off topic. I would like to write an essay about how pressure vessels relate to calculus and focus on steam locomotives. How does one calculate steam locomotive boiler pressure? Is it possible to determine the rate at which a boilers temperature increases over time?

This is a great question and one that you could go far with. There are many here on RYPN who focus a great deal of their professional time doing nothing but crunching numbers for the sake of boiler safety. In simple answer to your question; yes, you can calculate how a boiler will heat up and there are many factors in coming to a reasonable answer to that question.

May I offer you a friendly suggestion that should help with your essay and will also promote a healthy discussion from the "who's who" of RYPN.

If you want to do an essay, start by picking one specific boiler. You may want to pick one specific locomotive. The smaller the locomotive the easier the calculations will become. For example, doing simple calculations on a 4-4-0 American would be much easier than doing full calculations on say a 2-6-6-6 Allegheny which had a very complex boiler that included superheaters, syphons and a large combustion chamber.

Once you have established a locomotive or boiler to work with, establish a few basic questions that you can gather mathematical answers to. For example: On a 4-4-0 American, burning coal, with a total heating surface area of _______ what would be the estimated time it would take to raise the boiler pressure to _______ p.s.i. You can come up with hundreds, if not thousands of questions that can be solved through math all related to boilers alone.

Finally, you asked what calculations or formulas we use to determine boiler pressure. You can refer to two basic documents. The ASME Code of New Construction is one. It contains many volumes, is very expensive and unless you have specific questions, you would drown yourself in a lot of information that you may never want or need to know. There is an Engineering text book out (sorry I do not have a copy on hand, my engineer does though) entitled something like....."Understanding the ASME Code of New Construction for Power Boilers". I am told it is a very good book. Finally, you could ring up Strasburg's Kelly Anderson who hangs out here, as Does G.Mark Ray of the TVRM (both very smart when it comes to these types of things) and ask either of them (there are many others that hang out here that are related to the ESC as well) if they can help you get a copy of the Engineering Standards Committee Compendium. The compendium is a compilation of specific formulas related to Locomotive type boilers. This is the basic formula book for preparing a FRA Form 4. I have copies here at WRC, but I am not authorized to make additional copies.

Once you have all of that put together, bring your specific questions back to the group and I promise, you will insight some healthy and even flattering discussion among the boiler giants of the steam world!

Best of luck with your project. If you want to chat off line, feel free to contact me, or page people who can help you. There are many here willing to help.

I'd say, while maybe not explicitly "preservation of the machine", it is preservation of understanding about the machine; it's an entirely valid topic and we would all do well to better understand these concepts and help others to do likewise. :-)



I'm guessing, and this is an inference based the way you've worded things, you're not a mechanical engineer. Having said that, what you're looking into is within the discipline of mechanical engineering, specifically focusing on thermodynamics. I'll do what I can to assist, and I'm sure there are other mechanical engineers who can help broaden yours and others understanding of this discipline in engineering.



You're asking a broad question which can go several different directions, I'll try to elaborate a little (starting with the most elementary):
1. What is steam pressure?
2. Where does steam pressure come from?
3. How does one determine at what pressure and temperature a boiler will operate under given conditions?
4. What is the most pressure a boiler can have?

1: Fundamentally, pressure of any sort is a measure of the relation of several factors: the type and number of molecules involved, how much energy is contained within them (i.e. how fast/how much are they moving around) [temperature], and how much space is there between these molecules? for more information, see: http://en.wikipedia.org/wiki/Pressure

Steam pressure (specifically, gauge pressure) is the phenomenon that occurs when vaporized water is retained in a closed (or mostly closed) container or vessel with a density higher or lower than it would be if it were not retained. For the purposes of our discussion, we'll ignore the case where steam pressure is lower than the ambient pressure.

2: Steam pressure "comes" from increasing the energy contained within the water/vaporized water contained in the pressure vessel. When the water and water vapor reach the saturation point for the given temperature, the production of steam basically ceases and the system is in equilibrium. Any change in temperature, volume, or number of molecules in this system will upset the equilibrium and change the pressure.

3: To determine the pressure and temperature at any given moment of the conditions inside the boiler (temperatures and pressures within the boiler are never perfectly uniform), you must consider all of the factors influencing those conditions. For starters, in locomotive boilers, there is always heat (energy) flowing out (unless the boiler has only water at ambient temperature) and usually flowing in (unless the fire has gone out).

First, we'll assume the fire is out but there is still pressure in the boiler since this is a bit simpler: Heat flows out of the boiler primarily in two ways. First, conduction through the insulation and more so radiation from and convection over uninsulated areas: firebox sheets, tubes, tube sheets, pipes, steam dome, whistle, etc. Convection through firebox, tubes, & smokebox increase the flow of heat out of an unfired boiler when the stack is not capped. Second, steam (or water) leaving the boiler carry heat out of the boiler. Finally I need to mention adding water to the boiler, while not really removing heat (except in the case of a pump or an injector (involves lots of calculus and differential equations all on its own!) which each do remove some heat by transforming it into motion and by conduction, radiation, and convection, as mentioned above), but dilutes the heat energy in the boiler (analogous to increasing the denominator in a fraction).

Now, assuming the locomotive has a fire in the firebox, we are constantly in a state of adding heat to the boiler. All of the unfired conditions still apply, even convection through the tubes, but the latter only in the case where enough "cold" (ambient temperature) air is able to squeeze past the fire and actually come into contact with the sheet(s) and/or tube(s) and actually remove some heat (i.e. in a coal fired locomotive, where the grates are not completely covered with ignited coal). From this point on, we are both constantly removing heat from and adding heat to the boiler.

When the rate of adding heat (calculus! dQ/dt) exceeds the combined rate of heat removal and dilution by adding water, the steam pressure rises. Inversely, when the heat combined rate of heat removal and dilution exceed the rate heat is being added, steam pressure will fall. To simplify, this is a question of the rate of BTUs in vs. BTUs out. What is the rate fuel is being added to the fire? Subtract the rate the BTUs going up the stack (directly from the fire). Assuming that fuel is not being added at such an extreme rate as to consume all of the heat being produced (which is possible: think about throwing LOTS of "green" coal over a bed of coals at a rate fast enough that you extinguish the fire), this is "Q" in, the actual heat being added to the boiler (by radiation in the firebox and combustion chamber, if it has one and convection & conduction on the sheets and tubes). If the locomotive has superheaters, they enter into our equation in one of two ways: If they are after the throttle valve, then heat from the burned fuel into the superheaters is subtracted from the total like the heat going up the stack. Alternatively, if the superheaters are connected to the boiler BEFORE the throttle valve, then this become a little more interesting and depends on how far the throttle is open. Heat from the superheaters would only ENTER the boiler for a short time in the period when both the throttle is closed and there is steam in the superheaters below the flue gas temperature. If the throttle valve is open, or superheated steam is being consumed in any other way, there is flow of some rate out of the boiler and this becomes a non-issue.

Now a bit about temperature. Temperatures in the boiler vary quite a bit depending primarily on both the particular location within the boiler, the conditions of steam consumption (heat out), and the condition of the fire (heat in). The most uniform temperature distribution within the boiler occurs when, as I mentioned earlier, there is no fire and the stack is capped. This is when there is the least circulation (none) and convection of water within the boiler. In this state, as time goes on, the lowest parts of the boiler will be the coolest. Convection thus causes the cooler water to move down and warmer water to move up. The next likely most even distribution of temperature is when the locomotive is working. As steam (and heat) flow out and water in pumped/injected in and fuel is added, a steady balance of these will yield a fairly even rate of circulation within the boiler. The coldest parts are likely to be first at the front of the boiler, near the tube sheet, where the "cold" water is added. Some of this will convect downward to the bottom of the boiler barrel and then further flow backward toward the firebox toward the mud ring. This water will mix somewhat with water in the boiler and also be warmed by the lowest tubes in the boiler. Also, the area above the mudring but below the fire rarely has insulation on the outside and is constantly being subjected to a "cold" breeze flowing over it on the inside (of the firebox). This area, while usually relatively small is also a cold spot in the boiler. Remember that is you're developing a mathematical model that this location inside is not part of the heat flowing out of the boiler particularly because it is basically pre-heating the air that flows over it and into the fire.

Circulation in the boiler is a product of at least two constituents: convection and the generation of steam, with the latter having much more influence. When steam is generated, a small volume of water changes phase a gas and occupies significantly larger volume, temporarily displacing the adjacent water molecules. When the steam bubble release from the heating surface, the water around it is stirred and agitated as it moves up toward the surface of the water in the boiler. As a side note, the continuous nature of the generation of steam in a boiler while steam is being used causes the water level in the glass water gauge (water glass) to rise while the steam bubbles remain underwater.

It should be noted that while the vast majority of steam bubbles are generated adjacent to the heating surface, steam bubbles can form elsewhere, if the water molecules are sufficiently heated. An example would be if you [super]heated a sealed container of water (to more than boiling), removed it from the heat source, then opened it. Whether all at once or a small amount at a time, some of that superheated water would flash into steam.

When the throttle is closed, the generation of steam ceases and most of the steam bubbles rise to the surface due to their buoyancy & density. Assuming that the consumption of steam is now relatively insignificant, the water will, as in a boiler with the fire extinguished, but less so, begin to stratify. Convection will be the primary rule instead of circulation. As a rule, the warmest water will rise toward the top and the coolest water will make its way toward the mudring (assuming the boiler is relatively level).

Another important factor to keep in mind when considering the generation of steam is pressure. The higher the pressure within the boiler, the lesser the need to open the throttle in a given condition. When boiler pressure is high, the steam and water are both warmer and thus contain more energy and the steam, which is more dense, can do more work. This translates to a lower rate of steam generation, meaning the water level is less volatile than with lower pressures and the potential for water to carry over into the dry pipe is significantly reduced. An additional benefit here is also that less water (albeit very slight) is required to be added to the boiler, thus slightly lowering the rate the "cold" feedwater dilutes the heat energy present in the boiler. This phenomenon will, in most cases, be imperceptible to the engine crew as other factors of operation will more greatly influence the need for water to be added.

One final general thought about pressure is that pressure in the boiler is never completely uniform. Since water and even steam have mass and thus weight, just like in earth's atmosphere and with the oceans, the lowest widespread pressure should theoretically be at the highest point. the highest pressure in the steam would be at or near the water line, and the highest pressure in the entire boiler is in the lowest part of the water legs above the mudring. Granted these differences are miniscule when compared to the overall pressure difference with the atmosphere, but they are there. There will be a localized pressure drop, though slight, anywhere steam is moving out of the boiler and the immediate area near the opening functions not unlike a vacuum cleaner's opening. If the water level is too high this can induce the dangerous condition of carryover (also called priming). As I mentioned earlier higher boiler pressure reduces the effective velocity of steam within the boiler (and up to the throttle valve, or of that's wide open, up to the slide or piston valves) needed to get the job done and a lower boiler pressure conversely responds like a more powerful vacuum with a higher velocity. The lower pressure's natural tendency to be more volatile coupled with the higher velocity of the steam within the boiler greatly increase the likelyhood of carryover.

4. The "maximum" pressure at which a boiler can operate mostly due to physics and a little bit due to law. Simply put, the pressure within a boiler is always pushing outward on the shell, sheets, and other parts of the boiler. Most parts are under tension and some are under compression. A simple review of forces shows that the simplest pressure vessel is a sphere and requires neither stiffeners nor reinforcements, like a ball. Next would be a tube, cylinder, or pipe which also needs no additional supports, provided there is enough strength in the wall. The greater the ratio of length to wall thickness, the more support will be required. The weakest shape is a flat surface. With the relatively even load that pressure applies to the inside of the flat surface, the pressure tends to try to shape the flat surface to be more like a sphere. Take for example, a locomotive boiler with no staybolts or other reinforcements to is firebox and backhead. To further illustrate my point with this theoretical "boiler", we'll remove the firebox door, the tubes & flues, and and bracing. Now we have a pressure vessel with relatively large flat surfaces that are unsupported against the forces from within. As the pressure begins to rise, what happens? the flat surfaces begin to bulge. Further increase of pressure would would eventually try to round the external corners of the firebox. The front "tubesheet" would also bulge and start to more closely resemble half of a sphere. Finally with enough pressure, assuming nothing failed, the entire firebox itself would be turned inside-out. If everything were somehow sufficiently stretchable, the resulting mess would then look something like a tubular boomerang, then straighten to be more like a pipe, and finally a sphere.

Knowing that the end of the above is in practice, absurd, let's take a brief assessment of the forces involved in a relatively conventional locomotive boiler: relatively thin flat surfaces always need some sort of support like braces, staybolts, crown bars, etc. The forces on these devices is a function of the effective force due to pressure applied over the areas adjacent to the support. Staybolts and braces work in much the same way as ropes or chains support the seat of a swing on a swing set. The load born by each reinforcement is inversely related to the distance to the next support. When a staybolt breaks or a brace fails, its load is instantly transferred to the surrounding supports. This can ultimately have a cascading effect if too many fail in too close proximity. Other considerations in terms of supporting surfaces are corners and the concept of stress-risers. When corners on boilers are relatively sharp, the forces due to pressure and uneven thermal expansion have a tendency to have a more negative effect than flanged corners with decent radii. Remember that the load from between the corner and the last row of staybolts is partially transferred to stretching the sheet around the corner in addition to that sheet's own load, and vice versa. Having said this, the physical and legal "Maximum Allowable Working Pressure" boils down to allowable stress in each material and is a function of at least the following: Material (steel, stainless, copper, etc.), thickness (including wastage due to corrosion), construction method (weld vs. rivet) and seam style (butt vs. lap), support spacing, hole locations, and also applicable laws. Federal rules do differ from state rules so it is important to these in mind. The ASME Code and the FRA form 4 is a good place to see where engineering & physics meet law and work together for the safety of everyone involved.

I think I've temporarily exhausted my brain; given the hour and that I've been writing this for something north of 4 hours, I may very well have missed something. Feel free to ask more questions, but I'll not likely be as elaborate as this. As far as the paper you're wanting to write, pick up or borrow an Mechanical Engineering book on Thermodynamics to give you more specific direction on the calculus formulas you'll need. Q is the engineering symbol for heat/energy. Qin - Work = Qout
Work / Qin = % Thermal Efficiency
Q and Work are both units of energy and can be converted between BTU, kWh, Joule, ft-lb, N-m, HP-hr, etc. (force x distance)
Q/t is power. BTU/hr, kW, Joule/sec, ft-lb/sec, N-m/s, HP. (force x distance / time)

dQ/dt is the calculus you seek.

For those not knowing calculus but curious, the 'd' is simply an abbreviation for the Greek letter "delta" which means change; making the above mean "the change in energy over [the change in] time".

Adam Wright
(wondering if I should consider being a teacher at some point)

It seems to me that the specific questions at hand were:
1) How do pressure vessels relate to calculus?
2) How does one calculate steam locomotive boiler pressure?
3) Is it possible to calculate the rate at which the temperature of the boiler increases as a function of time?

While it has been a few years since engineering school, I would suggest the following:
1) While the equations to calculate pressure affects on the flat surfaces of the boiler are rather straightforward (and can be arrived at using calculus), engineers really needed calculus to derive the simple algebraic formula for stress in the boiler barrel (a cylinder). The derivation starts by looking at the forces on an extremely small section of the "thin" boiler shell. By smart selection of coordinate system and making certain simplifying assumptions, one can use integral calculus to sum the forces on an infinite number of those small sections of the boiler such that a very simple algebraic relationship between pressure, boiler wall thickness, and boiler radius can be used to determine the stress in the metal of the boiler. Engineers then check that against the known measured strength of the material to determine if it will hold pressure. A safety factor is used to decrease the measured strength of the material to help protect against failures. These types of calculations are at best an approximation of something real, so one has to allow for imperfections in the real thing. However, the calculation of stress in the boiler barrel is probably the cleanest/simplest use of calculus in this group of questions.
2) Water is an interesting substance in that it can exist as a solid, liquid or vapor. To make it more interesting, water can exist as liquid and vapor at the same time! In practice, we don't generally "calculate" per se the pressure in that the relationship between temperature and pressure for water is, as you might guess, quite complex. Instead, tables of measured data are used relating temperature, pressure, state (solid/liquid/vapor), energy and other properties. To design a boiler, one would pick an operating pressure and then use the empirical data tables (the steam tables) to determine the rate of energy input needed to boil water. This is where the calculus part comes in: the laws of thermodynamics can be used to determine the rate of energy input (heat in to boil the water plus work done to pump the water up to pressure less heat losses and other inefficiencies equals work done. The rates can be integrated (calculus) to determine the total amount of energy expended. I'm intentionally being a little vague here as you will want to study definitions of work, etc. on your own...just trying to connect the dots to calculus
3) For this one, we need to add the engineering field of "heat & mass transfer" to the thermodynamics previously discussed. I should also mention that defining the "control volume" you are studying is very important. Calculus can be used in a fairly straightforward manner to determine the change in temperature over time of the boiler metal given known heat inputs and outputs. The temperature change of water in the boiler as it moves from water to saturated steam to superheated steam can be tracked using calculus, but is much more complicated. In fact, I would solve such a problem using three "sets" of equations: one as the water was heated to the boiling point, a second for when it changed from all water to all steam (which takes place at a constant temperature even though you are continuing to add heat) and then a third as the steam begins to heat up in the superheaters.

I hope you are seeing a pattern that one typically needs equations based on measured data to make the calculus work as the natural phenomena involved are very complicated and have yet to be adequately modeled using basic physical relationships (such as the laws of thermodynamics).

One can spend ridiculous amounts of time trying to model (create an equation to describe the behavior of something) the many aspects of physics that come into play in a boiler (combustion, boiling, heat transfer, strength of materials, fluid dynamics, etc.), the trick is to make simplifying assumptions along the way combining mathematics (including calculus) and empirical relationships (measured) to get to the answer you need.

I hope that helps...

My experience is that John does not respond to emails or phone calls! I would not waste your time!

Absolutely facinating read! Thanks to the posters.

Also makes me realize I made the right choice in becoming a Civil Engineer instead of Mechanical! LOL

Lot easier to design and build the track you run on.

CV the civil E in NJ

"Is it possible to determine the rate at which a boilers temperature increases over time?"

Yes. The time rate of change for temperature (which has a direct correlation to temperature) is simple data to collect using thermocouples and a clock however I am not sure that it will be of much use as the time rate of change has many variables which will influence the curve. For example the firing rate or pounds of fuel burnt per time interval is subject to draft, grate area (cross section of damper), completeness of combustion, the person actually firing, and other factors. Hard firing will cause the curve to steepen.

To make sense of dT/dt you will have to establish boundary conditions and make assumptions about the process which are then taken as constants for your example. You will want to establish BTUs per pound of fuel, pounds of fuel
consumed, temperature and humidity of combustion air, temperature of combustion, stack temperature, byproducts of combustion as any un-burnt products represent energy lost. You will also want to address heat loss due to less than perfect insulation. With this you will want to account for parasitic energy uses.

For any given boiler there are some things we cannot change, for example heating surface, thickness and composition of boiler materials. We cannot influence the shape of any given boiler for a given test. Your question asks about change in temperature however that isn't overly significant in boiler calculations. Because the purpose of a boiler is to make steam, what would be germane is the capacity of the boiler in an approximately steady state of operation. That comes down to how many lbs of steam can be generated in relation to lbs of fuel consumed at a steady temperature and pressure for an extended period of time.

Fortunately thermodynamics is well understood for carbon fueled power plants and has been studied for well over a hundred years. We are fortunate that many of the formulas used for this have been distilled down to algebraic math from rigorous differential calculus. There are many books which address boiler trials.

For a locomotive I suggest that an older text might address these issues more appropriately as later texts look at modern fixed installations such as power plants.

I highly recommend that you look at “The Steam Locomotive, Its theory, Operation, and Economics” by Ralph Johnson, Simmons-Boardman, 1942, second printing 1981. The first few chapters address fuels, combustion, water, evaporation, superheat, and steam utilization to name but a few. It is an excellent treatise covering the engineering and design issues as they relate to modern locomotives.