FW_Su2013_04

Left, we have the current going through the tungsten electrode and hitting the work piece, and the current travels out the work piece. In a non-transferred arc, the arc current never gets to the workpiece. It goes from the electrode and you have a water-cooled — that's what this little squiggly line is in blue, there's water-cooled — um actually this is air-cooled, um this is air-cooled, I'm sorry. Okay, air-cooled copper anode, and the current goes there, creates a plasma, and the plasma flame hits the surface. But it doesn't have the same heat intensity because it's not bringing the electrons with it. You actually have two airflows. This one will typically be an inert gas like argon. This one could be nitrogen or air or other things, depending on whether you're trying to weld or whether you're trying to cut. You can use a cheaper gas than argon.

In fact for sheet metal work we actually have air-operated plasma torches, plasma cutting torches, where they actually use a hafnium electrode rather than a tungsten electrode, and they actually use compressed air. They don't have to use argon or helium to protect the electrode. They actually form a little skin of hafnium oxide on the surface of the electrode, and the hafnium oxide is fairly stable. You can get about a half an hour's worth of cutting out of these things. And so now you have, for sheet metal workers, these guys just plug into 110 volt wall power, have compressed air from an air compressor or whatever, and they can cut stainless steel or nickel-based alloys or whatever with just a compressed air torch using hafnium electrodes rather than tungsten electrodes.

But in any case, this particular arc is basically a jet burner. And so just like we talked about flames in the very beginning, we said you could have an arc which is at constant pressure, such as this plasma transferred arc. It's constricted because of the cool copper around it, and so you get a higher current density than you do in a gas tungsten arc where the electrode would stick down beneath this. Over here you actually have a plasma jet burner, where you're essentially igniting the gas or heating the gas at constant volume rather than constant pressure, and you get a very strong arc with lots of velocity. And that gives you better conduction, convection, and radiation, but there is no electron heat flow. And so this one — non-transferred arcs — typically give you around 2000 watts per square centimeter, and this will only give you, this will give you, about 10,000 or 20,000.

So it turns out NASA likes to weld the space shuttle main tanks with what they call variable polarity plasma arc. We might talk about why they would use some sort of AC rather than straight DC for welding of aluminum, but in fact on a high conductivity material like aluminum, they like to use the plasma arc rather than gas tungsten arc, because the higher heat intensity with the higher thermal conductivity metal gave better results. So that's why we use plasma arc.

And actually many times in fancier plasma torches, like a big cutting torch for a plate line or something, this copper which is air-cooled here may be water-cooled, and so the torches get fairly complex. Here are some tips, one of the ceramic tips and one that's bonded to copper. These are just replacement tips from a modern plasma cutting torch.

Turns out the whole idea of plasma processing or plasma cutting was sort of discovered serendipitously by a guy named Bob Gage, who was at Union Carboy — Union Carbates — Union Carbide Speedway Labs in Indianapolis, Indiana. In the 1950s he was studying gas tungsten arc torches, and he was just studying the arcs without melting of steel, and he was using water-cooled copper plates. And for whatever reason he decided to drill a hole in the water-cooled copper plate, and what he saw was — he said he made the first plasma torch — and the flame was shooting out past the, through the hole in the copper. When I say plate, probably wasn't much more than a sixteenth or eighth of an inch thick. But the torch, or the flame, plasma flame, was shooting out through the hole, and he thought, that's interesting. And then he started thinking, well could he use it for anything? And then that's where they started looking at constricted gas tungsten arcs.

Again, if you cool the outside of the arc, it heats up the inside. And in fact there are some plasma torches which actually use a swirl of water on the inside to cool things down. Not so much for welding, but when they're trying to generate very high temperatures. Typical temperature in a plasma arc might be 20,000 degrees, but when people want to get to 50,000 degrees, they actually can cool the outside surface of the plasma with a water swirl and get 50,000 degrees in some of these plasma torches. I don't know of a commercial application for that, but sometimes people use it.

So now we want to get into some more pract— well actually this next thing's not so practical. After this one we'll get into things that are a little more practical. If we look at — we talked about this surface boundary layer. I don't know if I have it here quickly. Here it is. So here's my surface boundary layer. And you might say, well in a plasma how thick is that boundary layer? And it depends on what process you're talking about.

So there's a paper back almost 25 years ago in the Journal of Applied Physics by, uh, Emil Pfender [Pfender] — was professor at University of Minnesota who spent his life studying plasmas and electric arcs. And he and a student came up with four different characteristic lengths in an atmospheric plasma like an electric arc. And the first one is the recombination length. They call all these things lambda because it's a length. It's the recombination distance for an electron and an ion going back to forming a neutral atom, and that's on the order of three-tenths to six-tenths of a millimeter. So that's twelve to twenty-four thousandths of an inch. That's pretty thick.

Now the next one is the energy exchange mean free path, the distance the electron travels to thermalize. And that means that the electron is being accelerated in the electric field. Remember the fluorescent lights — the electrons have a hundred thousand degrees temperature because we're at low pressure, and the electrons go long distances along mean free path before they hit anything. That means they're picking up energy much more rapidly than the ions, and that's why you get a two-temperature plasma where the ions and the neutrals are down around 100 or 200 degrees centigrade, and the electrons are at 100,000 degrees centigrade. And those, the electrons, are giving off the light. Well, the electron goes about two-tenths of a millimeter before it has enough collisions that it comes to the same temperature.

In an atmospheric pressure arc, the electrons are effectively the same temperature as the ions and the neutrals. It's local thermodynamic equilibrium, and high pressure is — high pressure is anything over a half an atmosphere. The electron mean free path, the distance the electrons go at atmospheric pressure, is about 10^-3 millimeters. Anybody know anything about random walk processes? If something hits here and then hits there and then goes that way — Brownian motion — Einstein's doctoral thesis on Brownian motion, okay, which is gas molecules hitting little dust particles and they move about randomly. And you can show that they move out as the square root of the number of collisions. The distance they move goes as the square root of the number of collisions. It's a statistical Gaussian normal distribution.

And there's about 10^-3 millimeters. Well if it has to go 0.2 millimeters to get to be thermalized, that means you're going to have about 200 distances of collisions. But you got to square that because they're kind of bouncing — if it hits here, a slot will go off in a different direction and then it's got to go to actually get a certain distance away of two-tenths of a millimeter. You're going to have 200 squared, which is what, uh, 40,000 collisions, right? So you have to have 40,000 collisions to thermalize.

And the last one is the Debye [device] screening length. And this has been known for some time, that the distance over which an ion or an electron can see another charge is equal to the permittivity of free space times Boltzmann constant times temperature, the electric charge squared divided by the number density of particles. At one atmosphere that's about 10^-5 millimeters. Well 10^-5 millimeters is 10 microns, okay, that's not very thick. But in fact, that's the distance over which an electron can even see a positive charged particle.

If you start putting all these together, you can actually rationalize that the real thickness approximately of that gas boundary layer that the electrons have to punch through is the energy exchange mean free path, about two-tenths of a millimeter. You're talking about eight-thousandths of an inch thick is this gas boundary layer, and those electrons punch their way through. And they have a screening distance of 10^-3 millimeter — not a screening distance, but 10^-3 before they collide with something. But in fact they don't even see other charged particles over a distance more than 10^-5. And so they really are just ballistically going through there under a high voltage. Okay, so that's where you get into the details of that physics.

If we now think of something very practical, which is basically, how do I initiate an electric arc? Doesn't do you much good, particularly in robotic welding, if you can't start the arc to start the process of welding. Well, turns out arc ignition can be explained, or has been explained for over a hundred years, by Paschen breakdown — Paschen [position] breakdown. Basically if I bring two metal surfaces very close to each other, within a fraction of a millimeter, people found that the air could become charged and ionize, and you get breakdown of the air and the air is no longer an insulator. And they found that would be thousands of volts. And they call this Paschen breakdown. And so even today, you'll read in welding handbooks that Paschen breakdown is one of the ways of igniting arcs.

In fact then, other people in the 1920s learned about field emission. And I don't know if I have the field emission slide here, but field emission basically — in the 1920s when people started looking at tunneling, where electrons can go through a potential just by quantum mechanical effects, they started finding at 10^8 volts per centimeter, an electric field of 10^8, you could actually start stripping electrons out of a piece of metal. And that's sort of the beginning of vacuum technology — sort of came out of x-ray tubes in the 1910s. By the 1920s people were trying to understand, how did the electrons get out of the cathode in the vacuum tubes that they were generating x-rays, and then they started building electronics out of these vacuum tubes. But as they tried to understand that physics, they learned through quantum mechanics that you could actually strip the electrons out, you could just pull them out, if you have a strong enough electric field.

And somewhere in your notes you actually have a plot that shows — this is sort of the Fermi energy plot, I think I may have put it up once before. This is E equals zero of energy in free space, and you actually have a plot of the energy potential of an electron in free space — this is distance this way. If electron's sitting out here in free space, and this is the Fermi energy — we talked about the Fermi energy before — this is the highest level of the electrons in the conduction band of the solid. And we said this was the work function, voltage between here and here, which is four or five volts essentially. This is the potential of the electron, if this is the surface of the material, and you have an electron out here in space at one micron away, it has zero energy. As it gets closer and closer to the metal, it wants to become bound to the metal down to the Fermi energy.

If I now apply an electric field gradient, essentially I can take this potential and I can bend it over like this. The electric field — if this field gets to 10^8 volts per centimeter — I'm talking about a voltage versus distance, so this is just the slope of this line. I essentially drop this potential, and as I drop the potential, the barrier for an electron to get out of here drops by this amount. If I keep on, getting to 10^8, all of a sudden the electrons can just go out of here, straight out of there by tunneling, quantum mechanical tunneling, okay. The distances here are down in the nanometer, you know 1 to 10 nanometer ranges, so we really are in quantum mechanics. That's field emission.

And people later learned that Paschen breakdown was nothing more than field emission. Because if I get down to a tenth of a millimeter separation and a thousand volts, that's 10^-4 times a thousand, that's 10^7 volts per centimeter — per meter — 10^7 volts per meter, anyway, you're getting something close to this 10^8. And then if you actually start studying surfaces, the surfaces, if you've been watching the videos, are actually very rough, and you'll get an enhancement of the field at the surface peaks of the rough surface that you're trying to do this to. And they found that Paschen breakdown — you could get easier Paschen breakdown if you had rougher surfaces on your materials.

In fact it's most difficult to get Paschen breakdown or field emission on a liquid surface. So liquid mercury is a little bit harder, because the liquids don't have any peaks to them. But in fact when you start getting to these types of voltages, even if it's a liquid surface like this or a curved liquid surface, you find that the electric field can actually create little peaks of liquid. It'll just pull the liquid right off the surface. It starts to defeating the surface tension of the metal, and you actually can get little peaks, but it still takes a greater field to do that.

In any case, Paschen breakdown and field emission are essentially the same physical process. But I have X's beside them, because neither one of them is the way we strike arcs, even though if you go to the welding handbook they'll tell you this is how we strike arcs. Okay. In fact in the 1940s, a guy named Holm — okay, Ragnar Holm — came along. Ragnar was one of these German scientists. Everybody knows the German scientists were the best scientists up through World War II, right? Actually even after World War II. If you look at the MIT curriculum in the 1880s, you were required to take German, because all the good science was done in German, okay, published in Germany.

And the famous names in quantum mechanics and stuff — um, Pauli, well, de Broglie was French, but, uh, Niels Bohr, Albert Einstein was Austrian, but anyway they're all Prussians, right? Anyway, what happened was you had to study in Germany if you wanted to be a great physicist or chemist all the way up through the 1930s. And in fact, the guy who founded Caltech, what's his name, um, he'd been president of MIT — anyway he was the great, the greatest US physical chemist [Arthur Amos Noyes]. He had studied in Germany, okay, around the turn of the century. Anyway, you had to take German if you're an MIT student so you could read the scientific literature in the 1880s and around 1900 and stuff. And after World War II, we brought all those great scientists, if they hadn't already come back because they were Jewish come back earlier — we brought them back, people like Werner von Braun, who was head of their rocket program.

Okay, but Ragnar Holm was an expert on electrical contacts. And Ragnar and his wife both had PhDs. Sort of interesting. [Tom produces a book.] This is the fourth edition of Electric Contacts, Ragnar Holm PhD, in collaboration with Else Holm PhD, both of Saint Marys, Pennsylvania. His wife was Else. They wrote a book but he got all the credit. He went to work for Westinghouse as I remember, on electrical contacts because there are all kinds of little mechanical contacts and things. And he wrote the definitive book, first definitive book on electrical contacts. Such that today the IEEE — International Institute, or Institute for Electrical and Electronic Engineers or something, basically electrical engineering group today — has a Holm conference every year named after Ragnar. And this is his book.

Well in this, Ragnar described the heating that occurs when you have two electrodes initially in contact, and you have a short with zero voltage, and you start to pull one away from the other. Initially the electrical contact resistance is a low value because you have some pressure of these two electrical contacts. But as you pull them apart, the pressure decreases. It's got to go to zero. So as it decreases, the resistance increases.

And so if you solve for a little microscopic asperity contact and idealize, you can show — you can solve Laplace's equation and it tells you the potential field for the voltage. Can't see it, oh sorry uh, that's a problem with this thing okay, um, well let me do it this way. You can solve for the voltage across this thing as you're pulling it apart. And Ragnar goes through, there's V is equal to V0² minus V0² — there's a voltage across this thing. And turns out that's also equal to a temperature field. The temperature field equation is also Laplace's equation. Have you had Laplace's equation yet this summer in one of your math courses? It's basically just looking for potentials. Stress concentrations are Laplace's equation. Voltage isopotentials, thermal isotherms, all come from solving Laplace's equation.

Anyway, you get these nice uniform contours of either temperature or voltage. And it turns out, since mathematically voltage and temperature end up being equivalent, you can end up with a temperature equivalent to a voltage. And so this is what Holm solved for. Here's the pre-exponent. And it turns out, if you look at the typical values for the melting voltage of copper, is 0.41 volts — four-tenths of a volt. The melting voltage of tungsten is 1.1 volts. If you look in the back of Holm's book, he has [a table of] various materials, and he has calculated the softening voltage — U is his notation for voltage — melting voltage and boiling voltage.

For silver, it softens at 90 millivolts equivalent temperature, okay, this is according to his equivalence of temperature. A third of a volt for melting and three quarters of a volt for boiling. Iron — well, it has the melting voltage, it doesn't have the boiling voltage. But even molybdenum, you got a boiling voltage of 1.1 volts. And if you go down here he probably has tungsten — was important contact material — well, he's got tungsten carbide and cobalt anyway. He has these melting and boiling voltages which are all on the order of less than a volt or less than two volts typically. Which means if I pull a metal contact away from another metal surface, even with very small voltages in my power supply, I will boil the material at some of these little asperity contacts as I pull these things apart. And if I boil the material, I'm going to get a bunch of metal ions, and those metal ions will cause breakdown — they'll give up electrons and I'll now strike an arc.

So in fact the way you strike an arc in practice — and when you're welding you kind of always tell the students, you come in and you don't try to wham the surface, because if you do that, anyone's tried to do that, has ever welded, knows that you stick your electrode and you weld the tip of the electrode right to the metal. That's not so good. You come in at an angle, you try to graze the surface, okay, and as you are pulling off, you're going to boil iron, and that's going to cause your arc to strike. So it's touch start.

Student: [inaudible question]

No, it's because, uh, your basis of the process equation is basically the diffusion equation, and you're diffusing heat from that surface.

Student: [inaudible follow-up]

Right, oh yeah okay yes, if you want to think it that way, that's the correct way to think about it. You're getting — you have a very small contact, and your power is not going to zero as fast as your cross-sectional area is going to zero. How about that? Okay, so that means that your power goes to infinity. Power per unit volume essentially goes to near infinity, okay. Now, the way they got this equivalence, if you have to read Holm, there's a relationship between electrical conductivity and thermal conductivity for metals, okay, and he used that. And it works pretty well, okay. I mean, it doesn't work perfectly but for in terms of understanding things, the point is, a very light touch and pulling something away breaking that circuit will always cause an arc.

And in fact that's true when we turn the lights off here, okay. You break that circuit and you always get a little arc in that switch. And the reason you have to pay two or three dollars, or four or five dollars I guess, for your little, you know, light switch is — there are two silver contacts in there, and the silver contacts are silver-cadmium oxide. They have about one percent cadmium oxide. Cadmium — it's not like thorium or cerium oxide in tungsten where you have an easy electron emitter. Although to a certain extent, cadmium is an easier electron emitter than silver or any other metal. But silver has very good thermal conductivity. Cadmium, for reasons that are still not completely understood in terms of the physics of what's going on, cadmium will double the amount of current.

It's not the same thing as thoriated tungsten — thoriated tungsten will double the amount of current — but the physics are a little bit different, okay. But it does have to do with electron emission partly, for both of them, but there are other complicating factors. But you have a little bit of silver-cad oxide. Pure silver contacts are good for about 7 amps. That's a 15 or 20 amp light switch. If you didn't have the cadmium oxide in there, you would get arcs when you turn the light off, and those arcs might not get blown out. And if they don't, you burn down the house.

Now the Swedes proved this. You know the Europeans are very environmentally conscious. And about 30 years ago they realized, oh, there's one percent cadmium oxide in that light switch and cadmium is toxic, and therefore I'm dying every time I turn off the light because I'm being poisoned with cadmium. I mean there's this huge amount, right? No, there's not a huge amount. And in fact the father of toxicology, in 1500 AD, this is Paracelsus, or someone who's in Hungary or whatever — but he's the one who, he's considered the father of toxicology, and he said the dose is — the dose makes the poison. Actually he said it in Hungarian, but nonetheless. You know, he said the dose makes the poison. My father used to always tell me if you eat enough ice cream it will poison you, right? Um, if you have a big enough dose. Well, the Swedes decided to outlaw cadmium in their silver contacts for their light switches, and within three years they were burning down so many houses that they decided it was okay to be poisoned every time you turn off the light switch. It was better to be poisoned than to die in a fire, I guess.

Anyway, but the fourth thing that causes arcs to ignite is high frequency. And if I go all the way down here, I have yes and yes by separation of electrodes initially in contact, and high frequency. These are the two ways that we initiate arcs in welding. Paschen breakdown and field emission, that's what you read in the literature and a lot of welding textbooks — forget it, it's wrong, okay. We initiate it by either touch start or high frequency.

Now if you go into my lab and you look at the power supply — anybody ever welded aluminum? You welded aluminum, you use high frequency, right? Little high flow, higher, right? They basically generate the high frequency by passing the welding current through just a little spark gap. In fact you have to replace these spark gaps on the power supply from time to time because they wear out. But you're generating, over about a half a millimeter, you're putting all the electric current through there, and the arc itself generates a lot of radio frequency.

Lightning generates — which is just a big electric arc — generates radio frequency. And we have 17 stations located around the United States, and they pinpoint every lightning strike that hits the earth, okay, by just picking up the electromagnetic interference. And they can tell you how many amps are in that lightning strike. They can, by triangulating from these 17 stations, they can tell you within three to five kilometers of where the strike hit the ground. It's called the National Lightning Detection Network, okay.

Well in welding we use — we impose a high frequency by jumping a gap in the power supply before it gets out to the work piece, and that high frequency can break down the air, if I — not Paschen breakdown — but if I come within a millimeter of two pieces, or two metals, and I have enough high frequency, I basically — the high frequency can cause oscillations of those electrons on the atoms and cause them to ionize, and once they ionize I'm going to get the conductivity I need across the gap to start the arc.

And so in welding aluminum, where we have a very stable oxide skin, okay, we like to use high frequency to stabilize the arc. Otherwise you're welding aluminum and your arc might go over to one of the areas that's high in oxide, and the arc might extinguish because you basically got an insulated layer there. Or it might extinguish momentarily, like for a fraction of a second, but that's enough to disturb the fluid flow in the weld pool and make a lousy weld bead. With high frequency overlay, you can get a much more stable arc. And so on the power supply you can have high frequency off — if I'm not welding aluminum or magnesium I don't need it — or you can have it on just in the first few seconds, like the first three or four seconds, and then it goes off automatically. Or you can have it on continuously. So this switch has three positions to stabilize the arc.

Which is why if I go back to NASA, I like to use variable polarity for their plasma arc welding of the space shuttle main tanks. And maybe the US Navy, in welding their aluminum ships — you go in a shipyard, you'll probably find that people are using high frequency for the aluminum, but they're also being using variable polarity plasma, okay. And the variable polarity plasma is a form of AC that helps break up these surface oxides. So straight DC welding of aluminum is not the most stable process in the world. People like to use AC for aluminum.

Um, one of the theories is that the argon ions, when you're on the negative half cycle, the argon ions will be thrown against the surface and they will break up the surface oxide. There may be some truth to that, but it's probably a little bit more complex than that. But we don't have to get into all the details of the plasma physics, and cathode spots and things like that. Yep.

Student: So we talked about, you know, high frequency and separation of electrodes, but how does that relate to spark plugs? Spark plugs have an arc across a small gap, right? Really high voltage?

No, they do — no, you have a transformer that brings your spark gap in your automotive engine to 12,000 volts or so. There's a transformer in there. The fancier ones nowadays actually have the transformer right on the plug that goes on top of the spark plug. In the old days, in your distributor cap, you would actually have the transformer in the distributor. This is old technology, right? When I used to do the tune-ups on my car, you'd replace the — you'd actually replace, you didn't know it, but you had a little transformer coil in there, and you were replacing the coil. And the spark on the distributor cap was a high frequency — well, it had the high frequency component, okay. It was already high voltage, high frequency to help get the spark, okay.

But the main thing that they do on spark plugs now is they tip them with platinum, okay, because platinum doesn't oxidize like these other things, and therefore it's easier to start the spark, okay, because it's platinum tip. So they use a little piece of, little platinum ball about a tenth of a millimeter in diameter, maybe two-tenths of a millimeter, and they just weld it to the certain tip of that tungsten, in a little projection weld, a little spherical platinum ball, and they just ram it into the surface. And so platinum tip plugs cost you twice as much, but they last 10 times as long, hopefully, right? But it is high frequency and high voltage.

And you'll get noise — if things, radio frequency noise — if you're welding with a high frequency aluminum power supply and you have radio on, you're just going to get noise in the room. You're generating lots of radio frequency interference. But the welding equipment suppliers have to shield from that, so the Federal Communications Administration, or FCC — Federal Communications Corporation, whatever, whatever the second C stands for, Commission — so they don't get upset with you, for destroying everybody's radio transmission all over. Except now it's all digital, so who cares, okay. Other questions?

So oh, so far as welding, robotic welding, um, you could use high frequency, but high frequency is just another wear component in your system. And one of the problems with plasma arc welding is those little tips, the replaceable tips, last about a half an hour. When they first started coming out, they lasted about five minutes, okay. And so that's one of the reasons plasma didn't just kind of get picked up immediately. But they've improved the cooling of these things and the design, such now they get about a half an hour out of these things. But it's still another expendable, and it's a maintenance item, and as you know, if maintenance is required, it won't get done, and therefore you'll have lousy welds. So people tend not to use plasma unless they have to for the higher heat intensity that you can get, okay.

While we take a break now before I start the next topic — so we'll take five minutes and come back. Here, about the spark plugs on the cars nowadays — they actually have the high voltage transformer right there on top of the spark plug. I mean the wires, your spark plug wires, each one of them has a little transformer in it. And the reason for that is, you lose the highest frequencies of the high frequency the longer the distance of the cable, okay. Essentially they — it just doesn't transmit well, okay. So if they put the high frequency spark right down there next to the electrode gap, they get better sparking, more efficient sparks, as far as that goes.

But in fact they don't even have a distributor cap anymore. Um, I guess they're — what are they doing — they're basically firing low voltage circuits and then that goes into the transformer right there on top of the spark plug. Of course, if you got fuel injection, all this is sort of irrelevant too, right? I mean actually I guess you still have spark plugs, but it works a little differently with fuel injection. I'm not an engine person anymore. I used to know about it when I had to change the oil in my car myself.

Student: [inaudible]

Yeah, you have to worry about timing and everything else. Anyway, okay, so now let's start talking about things that I think are a lot more interesting because they're a lot more practical, and that's plasma jets in welding. And I mentioned them briefly a couple of times, but what causes the plasma jet? We know in a flame coming off the out of a torch, we have the chemical reaction and the heating of the gas which causes a volume expansion. We have the same thing when we heat the gas with an electric arc, we get a volume expansion. And that volume expansion is about a factor of 300 with an electric arc, because the higher power density from the electrons than from the chemical heat of the gas — the fuel gas is reacting with the oxygen.

But in fact we also have something that's trying to constrict this whole thing. If I have a current that's flowing down here, which are these kind of reddish arrows — reddish arrows — so I have a current J, which is current density, it's amps per square centimeter. And with every current, Maxwell's equations tells me I have to have, using the right hand rule, I have to have a B field. I have a circular magnetic field around every current conductor. If you've ever seen conductors running in parallel, if you pulse the current through them, you'll see them spread apart, okay, because they're mutually — they're repelling each other if they're going in opposite directions.

If you run current down — did I get that right or get that backwards? Maybe when they're in opposites repelling. I have to think about when they attract them and when they repel. Because actually we get a J cross B. This is the Lorentz [Lorenz] equation. The force between, um — the force, there's a force generated due to the current density crossed across, vector cross product with magnetic field. In cylindrical coordinates these two are orthogonal, so if I take the absolute value of the force, it's proportional to the current density squared.

Well the current density, if I got 100 or 150 amps going through this whole thing but the arc is narrower up here, I have a higher current density up here than I do down at the bottom. And that means I have a higher force. These blue arrows are supposed to be the force, and I have a higher force at the top than I have at the bottom. And that radially directed inward force, because my cross products are — I got axial current, I got circumferential B field, and the F is the third orthogonal component which is radial in cylindrical coordinates, right? Everybody following that? Remember your vector calculus. Neither do I, okay.

In any case, that means I have a pressure — a high pressure up here and a low pressure down here. If I have a fluid within a pressure gradient, I get flow, okay. So back in 1954 or so a guy named Maecker, in the German handbook of physics, basically explained this for plasma jets and calculated it, and showed that the jet coming off a typical 100 amp or 200 amp arc is on the order of close to the speed of sound, which is on the order of 500 miles an hour, 1000 miles an hour, it depends on what you consider your properties of this rarefied gas in terms of its sonic properties.

So I have a jet shooting off the tip of that electrode, a very strong jet. 500 mile an hour wind, even though it's low density, still pretty strong wind. And it explains a number of things. Why can I weld overhead and have the drops go up against gravity? Because they're in a 500 mile an hour wind. If the drops are small enough, they'll be carried with the flow, okay. So it explains why you can weld overhead and get metal transfer against gravity, as long as the drops aren't too big, okay.

It explains a bunch of other things, we're going to go through. If you have very high frequency, it turns out — these, there's a cathode spot right here and an anode spot down here. The anode spot's larger by about a factor of 10 in area than the cathode spot. But very high frequency — and I mean 100 kilohertz, which is very high frequency, we don't usually weld with anything like that — you will get supposedly a football-shaped arc. You don't get this bell-shaped arc. The bell-shaped arc comes from the plasma jet coming down and spreading out to either side, blowing off to either side, just like the picture I showed of the flame with the boundary layers. The bell shape of the electric arc is due to the plasma being blown off to the side because of the plasma jet.

If you have a high frequency arc — and I guess maybe I have seen some of these but not welding arcs — you get a football-shaped arc, because you don't have time to develop the plasma jets. You just take a one centimeter length and put in 500 miles an hour and find out how long it takes for that a molecule up here at the electrode to reach down here. It's more than 10 microseconds. And therefore you don't get fully developed flow from high frequency arcs that are being turned on and off at very high frequency, okay, or being pulsed at different frequencies.

In fact, we used — well I don't know, we still have it, we may still have one in the basement — but we built about 25 years ago a transistorized, a 20 kilowatt transistor bank, to study higher frequency electric arcs and metal transfer and things. And this thing could pulse — had a pretty good volt-current frequency response out to about five kilohertz, okay. You could get full voltage, full current out to about five kilohertz. Beyond five kilohertz, if you started going higher and higher frequencies, you couldn't get quite the voltage current response that you could get at lower frequencies. But this allowed us to start seeing what would happen if we pulsed the current at frequencies sufficient to try to blow the drops off the tip of the electrode.

And in fact that's what we do many times with pulse current welding. The drops are coming off the tip of that electrode at about 100 hertz, and if we pulse with the right pulsing of current, we can actually use this Lorentz force to squeeze the drops off, just like milking a cow, okay. Pulsing, getting a squirt of molten metal off — well, cows aren't molten metal, but they're molten liquid, right? But it's basically the same type of pulsing action. And you can actually get more stable arc transfer, better behavior, particularly on thin materials.

Now this was not very practical until about 20 years ago when we started to get inverter power supplies, the electrical engineering caught up. Uh, the old power supplies from when I started welding were thousand, two thousand pound machines that could have big copper inductors that might weigh 500 pounds of copper, and a great big transformer to transform the voltage, which had a bunch of steel and copper in it. And so the whole machine weighed, you know, 1200 pounds or whatever. It had to be moved around a shipyard with a crane.

Well nowadays we have inverter power supplies, which basically use solid-state silicon controlled rectifiers at about 15 kilohertz, or 20 kilohertz, some of them are even up to 30 kilohertz switching frequency. And so you take the incoming power at 240 volts or whatever, and you chop it up 15,000 times a second or 20,000 times a second, you put it through a very small transformer. And why can you use small transformers with higher frequency? This is a fundamental principle of magnetics, guys, okay.

If the depth that the magnetic field diffuses into a material is a function of the time for it to diffuse, and the higher the frequency, the less diffusion depth into the magnetic steel, and so with higher frequencies you can use thinner steel laminates for the course of your transformers and not get overheating of your transformer. Of course, we use 60 hertz current for most applications for motors and things. But what do you use on airplanes? 400. Okay. The reason they chose that is because at 400 hertz you can make motors a lot lighter with a lot less magnetic iron in them, because your diffusion time is less at 400 hertz than it is at 60 hertz.

Well nowadays with these inverter power supplies, I don't even know if they have magnetic steel. They basically are chopping things up so fast, they now can put it through a little transformer, and the little transformer can weigh 10 pounds as opposed to 500 pounds. And you can build a welding power supply that's a size not much bigger than a briefcase and weighs less than 100 pounds with a handle on it so the guy can lift — you don't have to have a crane come by to lift the welding power supply to the new park place in the shipyard or wherever you're going to be working — because of the high frequency. And with that high frequency, you then rectify it to make DC.

Out of that 15 kilohertz, well, if you can rectify it to make DC, you can get any pulsing you want, okay. Any pulsing frequency you want. If you start out with 15 kilohertz, you can rectify it and switch it with silicon controlled rectifiers, solid state electronics, and you can generate any types of waveforms that are interesting for welding, which are basically 500 hertz and less, okay. So actually maybe — let me draw that, because I'm getting some blank stares, okay.

If I have a voltage versus time, which starts out at 15 or 20 kilohertz — I mean it starts out at 60 hertz, but once I go through this piece of silicon, I can basically be switching it on and off at 20 kilohertz, okay, or 15 kilohertz, whatever. I now can take that electronically, and I can convert that to a pulsed waveform that looks like this. Or if I want — I'll do that in a second — I could do it so that it comes down this distance, goes up this distance, and down like this. And this basically has all these high frequency components, okay.

So I can take a bunch of high frequency components, and I can chop up this signal with other silicon controlled rectifiers of this signal to generate signals like this. This would be variable polarity. It's not balanced negative and positive. It has variable polarity, it has a net DC offset. And that's what we use for welding. Fancy welding of aluminum is variable polarity AC. But you can get this all out of these solid-state power supplies that weigh next to nothing, okay. And originally when they came out they were 15 or 20 thousand dollars, now they're 2,000. So they're no more expensive than the old great big hunks of copper and steel that we used to have. So there's been a revolution in power supplies that allows us to do some fancy things.

And if you remember, the plasma jet force varied as the current density squared. [Tom pats his pocket searching for something.] I probably should use that — probably in my pocket, right? Well, where'd I put my, uh — no, whatever. Oh, here it is. Okay. So here's — this goes — the force of this plasma jet goes as the current density squared. If I go over here, and I have high peak currents rather than a lower level DC that's continuous, I can get a net higher force with that arc. I can make the arc stiffer so that when the wind comes along, it doesn't blow it out like blowing out a candle. By pulsing the current, I can pulse it at a current that I can squeeze the drops off the end of that electrode, because that same F, J cross B force — if that was the molten tip of the electrode, I can come in there with a high current pulse and just use the magnetic pressure to squeeze that drop off, and get drops exactly when I want them, not just when nature wants to drop them off by surface tension and everything else, okay.

So there's other things that the plasma jet does. It depresses the weld pool. Okay, well what do we mean by depressing the weld pool? Well, you have to ask, when you start talking about depression — they once, when they before they built the Central Artery, the Big Dig [big tunnel] here in Boston, that cost 17 billion dollars, original estimate of 2 billion. It's done by one of my classmates, an MIT engineer. When I say classmates, he was same class as I was. Um, he was a civil engineer, but anyway, he was the one who proposed depressing the Central Artery. And so the joke in the early 90s was, how do you depress the Central Artery? And one of the answers was, you could have George Keverian sit on it. Well George Keverian was Speaker of the House of Representatives in Massachusetts. He weighed about 450 pounds, so if he sat on it, you could depress it. And the other one was Michael Dukakis was governor, and they used to say, you could have Michael Dukakis talk to it for an hour and that would depress it, okay.

So when we're talking about depressing something, there's different types of depression. But in fact, once I get up to about 200 amperes, the arc force coming off this thing will actually start to gouge the weld pool. And if I go above 200 amperes on a gas tungsten arc, I will start creating defects. And these are what are called humped beads, okay, continuous series of nice uniform defects, okay. So some people say a weld is nothing more than a continuous defect surrounded by parent material, okay. But in any case, you can make continuous defects by having excessive plasma jet pressure.

I told you the story on Friday of the spot welding where they tried to go to 2000 amps. If I square 2000 as opposed to squaring 200, I got tremendous jet pressures, and it just sprayed molten metal all over the room. It's a great cutting action. And in fact, we actually do carbon arc gouging, where we take a tungsten or a hollow carbon electrode and we blow compressed air through it and we strike an arc, and we blow the compressed air to blow the metal away. So we melt the material and we use compressed air to blow it away. You can do that with a regular arc, it's just at 2000 amps you can't control it very well, it'll just go everywhere.

Is there a question over there? No, just stretching, okay. So plasma jets do influence things. I end up solving the plasma jet problem because I put flux around it, and you can go to six, seven hundred amps in submerged arc welding because you cover the outside with this sandy flux, which melts and you create a glass tent, a molten glass tent that collects those drops of metal, and they fall by gravity back into the weld pool where you want them. And that was the submerged arc welding process, originally called Union Melt after Union Carbide.

And I told you there's this letter they have of President Roosevelt writing to Churchill telling him about this wonderful new submerged arc or Union Melt process that allowed us to weld Liberty Ships much faster, okay. But that was because they found a way to deal with the strong plasma jets that, once you get above very high currents, just start blowing metal all over everywhere. If you change your polarity around and start melting off this electrode tip, you can go to three, four, or five hundred amps before you're getting into stability problems at higher currents. But you can't just keep going to higher and higher currents because you run into these plasma jets or other stability problems, okay.

I mentioned it provides arc stiffness. If you have a higher plasma jet force — those old heavy power supplies, they rectified the AC 60 cycle, which gave you 120 cycle peaks, okay. If I take 60 cycle AC and I rectify it, I will get a series of peaks like this, because I'm flipping this one up here and flipping this one up here, and I end up with a bunch of things like that. Well, they didn't filter this very well intentionally in the old power supplies, because they could get an average DC of 100 amps with a peak of 150, and that was giving them 120 hertz peaks, which is almost what you wanted around 100 hertz to not only give a stiff arc but also to help squeeze those drops off the tip of the electrode. They just found that not smoothing out the rectified 60 cycle gave you a power supply that gave you better stability in the arc than if you had a pure DC arc.

A pure DC arc is very quiet. But most welding arcs you've heard them, they kind of crackle. And the welders talk about listening for the crackle, and that's for the old 120 hertz crackle. And now in the transformer rectifier — those are the old transformer rectifiers, but in the new solid state inverter technology, they actually have circuits that help simulate the crackle so the old welders get to hear what they're used to hearing, okay. But actually some of that is real physical stuff, where you're getting pulsing at the right frequency, stiffer arcs, plasma jets.

So there's a little bit of everything in there. It's not just helping the old guys feel comfortable with the crackle. But if you have a — if you run an arc off a DC battery, it's very quiet. So it turns out it's the pulsing of the current which causes an expansion of the gas. And so when we had this 20 kilowatt transistor bank that we could simulate all kinds of power supplies, we did hook it up once as a loudspeaker, and you could talk over the arc, okay.

In fact Professor Leeb in electrical engineering a few years ago patented a process where basically you could use fluorescent light technology as a loudspeaker throughout a building for safety issues and stuff. Because it's an electric arc, it's going at high frequencies, and if you modulate it with a loudspeaker, you can speak over the fluorescent light. You can speak over a welding arc if you want. And in fact there's a German company or Swiss company that will sell you a set of ten thousand dollar speakers that are nothing more than an electric arc, a gas tungsten arc, that has very good high frequency response, because these arcs will respond out to 100 kilohertz.

And so, you know, I always say you can sell anything to a golfer. Next best thing is selling anything to a music aficionado, okay — someone who loves to pay anything for music. Like these monster cables, where they have grain-oriented copper, single cryo, you know — this stuff is, I mean, it's just a flim-flam job. But they'll, people will pay ten dollars a foot for the cable to hook up their speakers, right? Any of you have that ten dollar a foot cable? You can go over here to Best Buy and they'll sell it to you, okay. I remember the first time I saw it, I thought, what — up anyway. Um, but people will buy it, you tell them it's better. Yeah, it might be better in the fourth decimal place, such that no one could ever hear the difference. But yeah, there's, okay, there's an effect. It's just it's not a real effect.

Okay, so it influences droplet transfer. I can get plasma jets depending on these cathode spots and anode spots that actually will come up from the base and create a plasma jet in the opposite direction and actually get unstable transfer. I told you we didn't use gas metal arc for titanium. This is the reason why. In titanium you get very stable plasma jets coming off the surface. We've done high speed movies of them, and you just blow the drops away. You're not — they're not uniformly going across. You can influence the transfer by getting regular transfer by pulsing, but you can also plasma jets that are not controlled can actually go the other direction and destroy your metal transfer. Any questions? Plasma jets are important, okay. Run into them all the time. Yep.

Student: [inaudible question about the direction of the jet]

The downward jet normally is because of the pressure — period. It's a pressure gradient, but it's due to the high current density at the tip. And if the current density at the bottom is low, then you get a pressure gradient top to bottom. If you get a cathode spot — if you reverse the polarity to DC electrode negative, or electrode positive — now your cathode is down here at the work piece, and the cathode spot's sharper than the anode spot. And in titanium, it can — you can just see it sit there and not move around most the time. On that flat surface, the spot wanders around the surface, time averaged over, you know, 100 hertz. It's a big wide spot, and so you get a nice flow from the electrode down. But in titanium, that spot is stationary. It doesn't like to wander around, okay. And so from the flat surface you'll get a jet shooting up from the surface that interferes with the metal transfer.

So one of my ideas originally was to put a little magnetic field around it to force that anode — that cathode spot — to wander around on titanium. But that was about the time that all the Alpha subs were being benched, okay. What do you call — not in dry dock, they were being laid up, okay. Mothballs — well yeah okay, yeah, basically mothballed, being taken out of duty, okay, and being mothballed. Because they had cracks, and they were noisy as could be, and they're expensive to maintain. And so the great interest in the early 80s subsided by the mid 80s for titanium subs.

However, that didn't mean that — I guess I can tell you part of the story. I don't think I told you the rest of the story after that meeting in the early 80s where the aerospace industry thought heavy section titanium was a quarter inch or three-eighths of an inch, and the Navy thought it was one inch or four inches. It was just complete disconnect. But then, about four or five years later — actually about 85 or 86 — Congress was still concerned about this titanium submarine, even though the word was coming out that they were a problem, they weren't the world's solution for the Soviets. Um, but nonetheless, Congress was still interested in leapfrogging, and they wanted to go to all composite submarines. So they zeroed the SSN-21 budget. Did I tell you about that? Okay, this is 85 or 86. That was the design budget for the SSN-21, which later became the Seawolf, okay.

And the, uh, that was about 100 million dollars a year. Captain Fireball — Millard Firebaugh, graduate of 13A, now 2N — was head of the design team. And all of a sudden Congress said you got no money. And so the Secretary of the Navy and all kinds of people went up on Capitol Hill, and what Congress did is they didn't just give him zero money. They took the 100 million dollars and they gave it to DARPA. Do people know what DARPA is? Defense Advanced Research Projects Agency. And they said, we want you to design the next submarine of the future. And of course for DARPA, that was going to be an all-composite submarine, right? So anyway, the Secretary of Navy and all the other people go up on Capitol Hill, and they argue to get 100 million dollars back. And so after Congress feels like they're listening to them, they give them the 100 million dollars back, but they leave the extra money they gave to DARPA. And so DARPA is now in competition with the US Navy for designing submarines, okay.

This is a different mindset back then. Nowadays everybody just wants to know how can we spend less money. Back then it was sort of, you know, how can we get the job done, who cares what it costs. I mean that was the big huge deficits — it was Ronald Reagan who taught the US government how to overspend, okay, or taught the US government that you can buy votes with tax dollars, okay, by overspending. But anyway, however you want to say it — I guess I have, I don't have any affiliation, but I have some sympathy with the Tea Party idea that we shouldn't overspend everything we have. But nonetheless, they got the money, and Captain Firebaugh ended up designing a submarine. He ended up becoming Chief Engineer of the Navy, and he was Chief Engineer when they had the Seawolf weld cracking problem, but that's another story.

But the story that I wanted to tell you was, so they have this three-day workshop in Reston, Virginia. DARPA has a workshop, and they brought, again, they brought in the aerospace guys, they brought in the titanium guys, they brought in the old steel guys, and they wanted them to get together and decide how to build the next generation submarine. And they didn't care if titanium wasn't the answer. If it was the answer, tell us how to do it. But they brought in a bunch of composite people from the aerospace industry, okay. You gotta remember, the aerospace composites industry folks had still not ever designed an aircraft out of composites, okay. The first aircraft out of composites was the V-22 Osprey, okay. And the only reason they did that is because they couldn't design it any other way. It would just be too heavy out of anything else. They had to make the Osprey out of composites in order for it to fly with any payload.

So they have this three-day workshop, and the first day of the workshop they just talked to all of us about what are the requirements of building a submarine. And they had a guy get up, and he showed the Patrick Henry, which had just finished its life and it was about to be blown up off the coast of Virginia and sunken to the bottom of the Atlantic. And it showed a picture of all the cutouts in the hull that it had — certain that it had — in the previous 30 years, and it looked like a patchwork quilt. I mean, just dozens of holes cutting the sides. And so then they broke us up into the metals people on the second day and the composites people, and we went our separate ways. And on the third day, on the half day, we came back to report.

And we basically reported — I was on the metals group because I was working on titanium — and we reported back. But the composite guys were told that they would have to be able to have ways to get in and repair. That's why they showed us the Patrick Henry with all the repairs that have been done to that hull. And so the composite guys came in and they had certain great big hatch doors in their composite design — conceptual design — and they said, this is where we will bring in the big heavy equipment. And I raised my hand, I said, have you ever seen a nuclear submarine? Do you think there are big hallways that you can just, you know, move the piece of equipment out into the hallway on a big dolly and run it down to the hatch and bring it out? And they said, well sure, you know. They had no clue that there was absolutely no space inside the submarine to bring out anything much bigger than a bread box, right? And the reason you cut the hatches where you cut them is because that's where the thing is that you're going to pull out, okay. They had no concept of what it looks like inside a sub.

And so, but what they were really saying is, you can't cut into composites without destroying the structure, okay. So the reason one of the reasons you no longer have composite submarines, other than little, you know, one or two man vehicles, is because they're unrepairable, okay. You can't cut them out and do repairs. So that was a sort of wasted three days of my life, uh, so far as that goes. But it was sort of interesting to see the great disconnect, okay.

There's another problem of electromagnetic forces in arcs, and I sort of alluded to it a little bit Friday when I talked about the stud welding process, and that is basically arc blow. And there's almost nothing in the literature, even the welding handbook, about arc blow, other than it exists. But what happens if I've got my electrode in my arc, and let's say I've got a piece of material that's long and thin, and I'm going to bring it to ground over here. The current is going to go through, it's going to turn and go down the material, and it's going to go out, right? Now when the current turns 90 degrees, you're going to have a higher density of magnetic field inside, or on the outside — on the inside than on the outside.

So this is the way you draw an arrow — is a circle for the arrow point, and then the tail is an X, or actually it's usually a circle with a dot, is the way electrical engineers like to do it. So I've got a magnetic field — maybe I got it backwards, but anyway, you got a magnetic field, you've got J, a current coming down through here, and you've got a B field. But the B field is more concentrated on the inside of the radius than it is on the outside of the radius. And electric arcs don't like magnetic fields, even the ones they produce themselves. And so the arc will actually be blown by electromagnetic forces in this direction, okay.

Or it can be blown by magnetic steels. 50 gauss — and the earth's magnetic field is about a half a gauss — but if you want to screw up a weld, just bring a 10 pound electromagnet and stick it down a few inches from where the weld's going to be, and when the arc comes along and gets close to that thing, it'll just blow off away from the magnet, okay. It doesn't — the arcs don't like magnetic fields.

Well, if you start thinking about a complex structure and where you ground it, you're going to have all kinds of arc blow problems. Because at one or two hundred amps you can be generating 25 gauss. And there have been plenty of cases where welders cannot make a good weld, period, unless you go in and demagnetize the steel. And so those who've been in shipyards have seen demagnetizing coils around the steel — just great big copper coils. They wrap around great big plates, and they run an AC current — actually, they put a 60 cycle AC current through it, and they bring the current down. So they magnetize everything, and then with AC they droop that power down. And on the 60 hertz, you're magnetizing one direction in the opposite direction, until you get down to near nothing hopefully.

Just having a big long hunk of steel sitting in the earth's magnetic field will cause it to magnetize. So many times they have to go and demagnetize the steel before they can continue welding. Because doesn't matter what the welder is doing, what his skill is, he can't control the arc if magnetic fields blowing it where he doesn't want it, okay.

So magnetic arc blow — you can use AC if possible, you can use high frequency. Again you get the high current peaks, which gives a stiffer arc, can stabilize it. You can change the ground position. You can have symmetric grounds. You could have another ground over here, and sometimes that will solve the welder's problem, by just changing the ground location or adding another ground, better grounding. Or you can actually add magnetic fields, if it's a short weld, on the side where it's blowing, so that it blows it back essentially, so far as that goes.

So another problem with all of this — oh yeah, it'll just kind of want to wander off to one side, and he may get lack of fusion on one side. If it's not really bad — if it's really bad, he can see it, just like someone came and blew out a candle. But mostly it's usually a fairly continuous process. And so it's just, he's trying to guide his electrode over to the right, and it keeps on wanting to hang over on the left. And he can't get good fusion. He's making defective welds on the right, lack of fusion, because there's a big magnetic field, or a bigger magnetic field, on that side, okay.

We'll get to that also when we do high-power electron beam, because there are some problems that cannot be solved because of magnetic arc blow on dissimilar metals in electron beam. But the basic physics is, um, arcs don't like big electric field or big magnetic fields, okay. And so you have to do something to stabilize it, but there's very little in the open literature that really talks about it. Sort of like residual stresses — it's one of these things we don't really want to talk about. Any questions?

I want to start on heat flow. And again, we're not meeting Tuesday, Wednesday, and Thursday this week, so you can figure out what you're doing. We will meet on Friday. Okay, so there's a little bit of history to heat flow in welding that involves MIT, sort of, in a number of different times. You have this handwritten thing in your notes so you don't have to copy all of this. But if we have heat flow during welding, we can think of it — and you can go to books on heat flow — that have a one dimensional planar solution.

So there's a heat per unit area, for a one-dimensional problem this would be something like the friction stir well — not friction stir weld, but, well, it could be a friction stir weld, but this is a friction weld — and it's a planar heat source, okay. So I want to see how the heat affected zone grows in one direction, for example. And the solution to the heat flow, Fourier's first law, is T minus some average initial temperature is equal to the heat per unit area that I'm going to impose on this surface, which I could do by friction welding, some area welding process. 4π α, thermal diffusivity of the material, t is time, to the one-half power, one over the density times the heat capacity, exponent. And you always get an exponent which has the form of x²/αt.

And in fact x²/4αt is often called the Fourier number. It's a dimensionless group. α is centimeters squared per second times time, which is centimeter squared, x² is centimeters, so it's dimensionless. And in fact everything that goes in an exponent like this has to be dimensionless. You can't take the exponent of something that has length or mass — it has to be a pure number with no dimensions. So the Fourier number you can prove comes out of the diffusion equation. And in fact there was a guy who looked at all this math, guy named Albert Einstein, years ago. And sometimes this x² proportional to t is called the Einstein equation. But in this case for heat flow, it's called the Fourier number. In the case of mass diffusion it's called the Fick number. Um, there's, uh, viscosity diffusion, and basically it's part of the Navier-Stokes equation, but anyway.

In any case, the heat is generated and follows basically a decaying exponential as you get further and further away, with longer and longer times. But if you impose the heat initially at some t-zero, it will start to diffuse away as a simple Gaussian distribution. And these are just, t1 is a little bit later, t2 is a lot later, and the heat's diffusing away to the side. You also can do this in a two-dimensional line source instead of a planar source. And the line source would be similar to an electron beam hole drilling or a laser hole drilling — you just have a linear heat source punching a hole through a plate.

Here instead of the one-half power, it's T minus T₀, Q over δ, and everything here is the same, but instead of the one-half power, it's not an area, it's a length. This is a heat per unit length of the heat source, and this is two halves to make all this have dimensions of temperature. One over ρ c sub p Q is basically temperature. And this is all the same except it's a radius from the line, so it's a cylindrical coordinate system. r is √(x² + y²). So deep penetration laser, electron beam — you can go to a third dimension. Once a mathematician has solved one, he's just so excited, he's got two more solutions.

Three dimensional — you can have a point heat source buried beneath a plate at some origin zero. And now it's just the total heat at that point source. There's no dimension that you have to divide by, but this goes up to three halves rather than two halves. And r now is spherical coordinates, so square root of x² + y² + z². And Q is the buried strength of the heat source. And if you plot a temperature versus time, it just follows the same type of Gaussian versus r, okay. And if you slice the whole thing in two and do a semi-infinite heat source, you basically have a point heat source on the surface of a plate. And that's fine if you're doing a stud weld or something like that we discussed on Friday.

But in fact, we'd really like to know this as this point heat source travels along the plate, what does the heat look like? And that's where Daniel Rosenthal came along. Daniel Rosenthal was one of these — he was Jewish. I don't remember if he's originally from Switzerland or, he did go through the Netherlands in his flight from Germany in 1939. And he ended up in Morocco for a year, and by 1940 he was here in — over here in, I don't know, building three or five, but he was over here in mechanical engineering at MIT.

And while he was traveling through, to get out of Germany through the Netherlands and Morocco and finally to MIT, he was solving the Rosenthal equation for a traveling distributed heat source. He took this solution for a point heat source that was in the literature, and he said okay well let's give it some velocity. And he solved that to come up with what was known as the Rosenthal equation. And now it's a velocity times a radius r just like we had up above, and a distance x where v is the velocity in the x direction, and I'm traveling at velocity v in the x direction. And so you end up with the Rosenthal traveling point heat source solution. He published about three papers in the ASME Journal.

And he ended up somewhere after World War II as a professor at UCLA in material science. But so Rosenthal had an MIT connection. When he published these things, if you look at the original paper, it'll say Department of Mechanical Engineering, MIT, as the [affiliation]. And then it turns out in the mid-60s — and you actually have a copy of this paper in your handouts — there's a guy, Nils Christensen, from Norway, who came along. And Professor Christensen basically extended the Rosenthal formula to — I don't know if I can find it — here we go — to dimensionless form. So here's Nils Christensen, Distribution of Temperatures in Arc Welding. He was at the Royal Institute in Trondheim, Norway at the time. He had some accident, was laid up in a hospital bed for a year.

But his connection to MIT was actually — when his first connection, back in World War II — was he was a Norwegian freedom fighter, skiing across the mountains from Switzerland — not Switzerland, long ski — to Norway. And, but from Sweden to Norway. Sweden was neutral in World War II, he was a Norwegian, and he was basically fighting the Germans as a terrorist, I guess is what we call him today. Um, but Christensen, after World War II — wouldn't we — hurry — terrorist, oh okay, new definition of terrorists, okay. I mean, you know, one definition of terrorists — all the guys who fought on the good side and were the revolution were terrorists, right? I mean, freedom fighters. Oh, freedom fighter if you win, okay. Revolutionary — well I don't know, revolutionaries got negative anyway.

But in any case, so Christensen actually got a scholarship right after World War II to come to MIT and worked in this department with a guy named John Chipman, working on welding fluxes, and did his doctoral thesis on welding fluxes. And it's one of the earliest Welding Research Council bulletins, is publication of his doctoral thesis. Actually it was his second doctoral thesis, because his first thesis was all written, ready to turn in, and it was lost. And Professor Christensen told me once that the only thing they could figure is that some Soviet spies were running around MIT, and they just stole it out of his office, because they didn't have good copying machines right then. So anyway, he had to redo all his experiments, but fortunately he knew what the outcome would be, but it took him an extra three quarters of a year to graduate.

Anyway, so Nils Christensen was laid up in a hospital bed, and what he did is he took the Rosenthal equation and he put it into dimensionless form, okay. And I don't know that I have it written down in dimensionless form, but I do have some of the plots out of that book or out of his paper. And here is one of the plots of the shape of things that you will get. Whether this is in dimensionless form — actually I did, should have it on that other thing — or no, I didn't, oh no, I did. Okay, so that's the shape of the temperature profile in dimensionless form.

It becomes θ, which is T minus T₀ over T₀ minus some standard, or something — it's a dimensionless temperature divided by an operating parameter which is a complex thing in the paper. 1 over the density, exponential minus ρ plus λ, which is the dimensionless distance in the x direction. ρ in this case is not the same ρ as before, it's not density, it's actually square root of x² + y² + z². But in any case, you take the Gaussian distribution and you skew it in that direction because you're traveling in that direction. And you now have temperature profiles behind the weld. And in fact, if we look at all this, we see steep temperature gradients, and these are the things that lead to residual stresses and things like that in the welding operation. So heat flow in welding is pretty important.

Christensen later, in his later years, came to MIT for a sabbatical, came back in the early 80s and spent three months here, just for old times' sake, I guess. He's passed away now, but anyway. If you look at some of those plots in two dimensions, here's the temperature — if this would be the lambda, the x direction, transverse to that you can see your Gaussian distributions. These are all just Gaussians at different times. And then if you're looking down from the top, it can be a series of circles.

So if you look — actually this is looking down all the way from the top — and you can see you get all these ellipses, and you can think of one or two of these ellipses can represent some isotherm, like the melting isotherm. And this dashed line would be the locus of maximum temperatures. That's basically where this thing runs horizontal. The maximum diffusion distance for that particular temperature. And so you actually can start seeing it's like a boat going through the water and leaving a wake of maximum temperatures along here. This is the locus of temperature maximums, and some temperature max we might say defines the heat affected zone.

So you can define the heat affected zone. If you look at this at different travel speeds — and again these are just right out of Christensen's book, and that book is — he wrote his paper in the British Welding Journal in '66. This is very slow travel speed, or maybe zero, near zero travel speed. Just looking down from the top, it looks like a bunch of circles. If you go a little bit faster like typical arc welding, you're moving — but here's your locus of maximum temperatures. And you can sort of see the heat affected zone grows while this weld pool is going to be solidifying. If you go faster, you get the one we showed before. You go faster yet, like lasers and electron beams, this locus of maximum temperature is actually — you can see — the heat affect zone may actually form after you've passed the weld pool and it's solidified, which is some of the stuff I mentioned to you before.

Student: Yeah, when you say locus of maximum temperature, you say choose the temperature of concern?

That dotted line shows part of the metal that will reach — it'll go from the melting point at the beginning of that dotted line to all the way to room temperature, and you march down that dotted line in different temperatures. And if you say a thousand degrees is my heat affected zone, you can find where a thousand degrees would be on that line, okay. And that will give you the width of the heat affected zone right there, okay. This is the locus of maxima going from the highest temperature to the lowest temperature. But the maximum temperatures always fall along that line, okay. There's a bunch of these isotherms at any given point in time, but the locus of the maximum, where the peak was, falls along that dashed line. And so the dashed line is not a maximum single temperature, it's a locus of maxima, okay. But it can tell you width of the heat affected zone.

So this is all interesting, but as we got into the age of digital computers, you can take a single point source solution like this — and you probably haven't gotten Green's functions yet in your math classes, right? But Green's functions basically — in math, a Green's function basically can take a point source solution and stack them up, a bunch of point source solutions, and just add them up. There's something called in heat flow, or in math, the principle of superposition. So I can take a solution — well, if my point heat source solution says that my distribution of heat on the surface — this is the surface — is a point heat source, a quantity Q, then I can do all these semi-circles from that point.

But in fact my heat source is not a single valued point-located source, it's a series of heats. And it has a distribution. There's a distribution to the heat source. And the Green's function can take all of these solutions and put them into an integral. And so now you have not a point heat source algebraic solution, you have an integral solution, and you use an integral calculus. And if you want to see this — I don't think I gave you the paper, maybe I did, but it's a paper that was written by myself and Nunsen Chai back in the early 80s, using a PDP 11/23 computer.

Which had 16 — it was a 16-bit computer. And it had — I remember, the hard drives were 10 megabytes and they were about 20 inches in diameter, okay, and they cost about a thousand dollars for 10 megabyte hard drives. This is in the early days of PCs, but PCs weren't as powerful as this computer. But Nunsen would run these things over the weekend to get one solution. Nowadays on your laptop you can probably run it in five seconds. Yeah, they're awesome, yeah. So basically, but 30 years ago this was state of the art. And so now you could use distributed heat sources, and rather than always having a semi-circular shape weld pool, you could have different depth-width ratios, and so we calculated that.

So it turns out, in heat flow in welding, from Daniel Rosenthal to Nils Christensen to Nunsen Chai — Nunsen Chai is now number two person at Taiwan Semiconductor Manufacturing. So he didn't become the CEO of the world's largest silicon foundry, but he did a thesis looking at intensity of heat gas tungsten arcs, and extending Rosenthal's dimensionless thing to a distributed heat source. And I guess there's another guy, Vivek Dave, in the early 90s under some DARPA sponsorship, basically took it and said, okay well now rather than just having a distributed heat source on the surface, I can have surface depression and other things. What happens if I allow for heat distribution through the surface, okay, down beneath the surface?

So you can keep extending these things, but in fact computers are powerful enough no one does this anymore. You know, 20 years later, you basically just take a finite element heat flow equation and you just put whatever you want in and you calculate it. But that's where we came over the first 60 years to get there. And tomorrow — or not tomorrow, but Friday — we will go through heat flow in general and the Fourier number. And I will teach you how to solve any — well, how to solve 90 percent of all heat flow problems in five to ten minutes, okay. To get the important part of the solution you need, as opposed to all this stuff they tell you on solving differential equations and stuff. Most the time you don't need to do that, okay. But I'll teach you how to use the Fourier number to get an estimate of what you're looking for very quickly, okay.