There’s a lot of energy in the vaporization phase change. In the form of water changing from a liquid into a gas, it has been at the center of most of the world’s energy production systems for the last century. Recently, direct energy conversion has started replacing it with wind and photoelectric power, and hydro-power always did fine without it(1). Although phase change is unlikely to ever power starships, it still has a future in solar collectors and nuclear reactors, most designs for fusion reactors… and it may very well be the best way to cool a starship.
Gaseous water is so common it has its own name; steam. But all materials have a gas phase(2); it’s just that most of them become gases at temperatures that are outside of our usual experience, and that are far beyond the needs of industrial processes.
The following table lists some interesting materials for spaceship cooling using the vaporization phase change. It also highlights the fact that, in most cases, the vaporization phase change transports much more energy than temperature change in the fluid phase.
|
Substance |
Boiling point |
Latent Heat of vaporisation |
Specific heat |
Temperature difference required to equal vaporisation |
for 500 C temp. diff. in fluid |
By phase change |
|
K |
kJ/kg |
kJ/kg°C |
°C |
Tons/s |
Tons/s |
|
|
Helium |
4 |
21 |
5.19 |
4 |
39 |
4762 |
|
Hydrogen |
20 |
449 |
14 |
32 |
14 |
223 |
|
Ammonia |
240 |
1369 |
4.7 |
291 |
43 |
73 |
|
Water |
373 |
2270 |
4.18 |
543 |
48 |
44 |
|
Lithium |
1615 |
21159 |
3.58 |
5910 |
56 |
5 |
|
Indium |
2345 |
2081 |
0.25 |
8324 |
800 |
48 |
|
Silver |
2435 |
2374 |
0.24 |
9891 |
833 |
42 |
|
Gallium |
2673 |
3681 |
0.37 |
9949 |
541 |
27 |
|
Beryllium |
2742 |
32444 |
1.82 |
17827 |
110 |
3 |
|
Aluminium |
2792 |
10500 |
0.897 |
11706 |
223 |
10 |
|
Copper |
2835 |
5060 |
0.385 |
13143 |
519 |
20 |
|
Tin |
2875 |
2475 |
0.227 |
10901 |
881 |
40 |
|
Silicon |
3538 |
13679 |
0.71 |
19266 |
282 |
7 |
Table 1, Mass flow for 100 GW of power
From Table 1 we can see that most elements need much less mass flow to carry 100 GW through vaporization phase change than by heating of the liquid phase. Let’s look at Silver, for example. If it stays liquid it’s a bad performer, requiring 833 tonnes per seconds at 500°C of temperature difference. But by changing from a liquid to a gas, it becomes a very promising coolant, requiring only 42 tonnes per second, or practically 20 times less coolant to move the same amount of energy.
Lithium appears at first glance to be one of the most promising candidates, and is already used in high temperature heat pipes for some industrial cooling purposes(3). So Lithium will be used in our examples from now on in this article.
The phase change equation is extremely simple:
1) Q=m×Ve(4)
Where:
Q= Power (kW)
m= mass flow (kg/s)
Ve= vaporisation energy (kJ/kg)
The same equation is true for the solid to liquid phase change, but the energies in presence are an order of magnitude lower, making the phenomenon less interesting from the starship cooling point of view. Besides, it’s hard to pump a solid.
Although the equation is simple, the physical phenomenon of boiling isn’t. The transition from a liquid, to bubbles, then to a bubbly froth and finally into a gas is accompanied by a large change in volume, and is influenced by the pressures in the boiling area. Fluids are good heat conductors but gases are bad ones, so the vaporisation must happen at the right point, or else the equipment that is supposed to be cooled may melt, despite the presence of the coolant at the correct temperature.
In a phase change system, the radiation shield effectively becomes a boiler, where neutron and x-ray radiation play the role of the combustion gas burner, and the liquid metal the role of water and steam. Depending on the design and the materials used, the radiation may heat the liquid metal directly through neutron capture, or heat an intermediary material that heats the fluid through conduction. Another good analogy is a nuclear reactor core, where neutrons heat fuel elements that then heat a coolant. Some very advanced and compact reactor designs use a circulating liquid nuclear fuel(5), that acts as both the heat source and as a coolant; this is probably the closest existing design to what the radiation shield will be like.
One important point in using the vaporization phase change to transmit the cooling power is that the whole evaporation condensation cycle takes place at practically the same temperature. This means that the whole radiator surface will be approximately at the phase change temperature.
This is a significant gain in radiator efficiency. Radiator designs are limited by the maximum temperature that the pipe walls, as well as the pumps and compressors, can withstand. For example, let’s suppose that temperature is 2000 K. A radiator with a helium coolant and with 500 C of temperature difference would be, at best, at an average temperature of 2000 – 500/2 = 1750K. The phase change radiator using lithium could be at practically 2000K for its whole surface; let’s say 1950K.
From the (simplified) radiation equation:
2) Q=eBT^4, we can simplify further by removing emissivity (e) and the constant (B)
and then, for the fluid cooling radiator: 1750^4 = 9.3e12
For the phase change cooling radiator: 1950^4 = 14.5e12
If we compare the two numbers, 14.5/9.3 = 1.54, or in words the phase change radiator is more than fifty percent more efficient than the temperature change radiator, allowing the ship designer to have radiators 30% smaller for the same power, or to handle 50% more power for the same area, as required by the design.
Metallic gases are ruled by the same gas laws as other gases, and it may be useful to increase the gas pressure to lower the gas density and therefore use smaller piping.
The overall thermal cycle we have been describing here is called the Rankine cycle. Its main particularity is that the coolant is a gas one one side of the system and a liquid on the other side. And that the temperature stays relatively constant throughout the cycle, although this is not an absolute requirement. The liquid side piping is designed following the rules of “Part 1, Liquids in Pipes”. The gas side piping is designed according to “Part II, Gases in Pipes”.
Image 2 Rankine cycle
The third part of the cycle is what happens between the liquid phase and the gas phase, when they both co-exist in what is known as two phase flow, in the shield and the radiator.
Two phase flow has many difficult problems associated with it: the rapid acceleration of fluids, in which entrained droplets can reach very high speed and cause erosion. Water hammer and large pressure surges, that can destroy piping when a liquid phase suddenly vaporizes and stops fluid flow or when a slug of fluid builds up, fills up a pipe and blocks the way of some high velocity gas.
These are elements of detailed design, that will need to wait for actual ship design before they can be worked on. But as a general preliminary design guide, elements are often sized for the gas phase, with the expectation that anything in the liquid phase will necessarily be much smaller.
Pressure
At what pressure level should we expect our system to operate?
Table 1 is based on atmospheric pressure, but that may not the best choice. For lithium, the radiator temperature is perhaps a little low. To increase the temperature and use phase change at this higher temperature, you need to increase the pressure, or else the fluid will boil too early. However, if we operate at higher pressures, the phase change energy will go down. Eventually, at high enough pressures, you don’t get a phase change at all, the metal remains liquid despite the heat added. This is the area of supercritical fluids. A higher boiling point is interesting, since it means the radiators will be operating at higher temperature, and therefore be able to handle more power.
The perfect gas laws can be used to determine the pressure we would like to operate at, in order to get a certain density of gas. Higher pressures will increase the boiling point and reduce the energy in the phase change. Clapeyron’s Equation can be used to determine the new boiling point, and then Watson’s Equation can be used to estimate the phase change energy at these new values. Finding the optimal solution will require an iterative process (that the mathematically inclined are invited to solve for the optimal point), as we choose between a higher pressure that gives us a higher density and smaller piping, and a lower pressure that maximizes the phase change energy.
As a first approximation, we might want to choose to operate our Lithium system at 2000°K, for maximum radiation power.
We can use Clapeyron’s Equation to confirm at what pressure the lithium will vaporize:
3) P2=Pn*e^Ve/R(1/Tn-1/T2)
Where:
P2=new pressure at boiling point (kPa)
Pn=atmospheric pressure = 101 (kPa)
Ve= vaporisation energy at standard pressure = 146 (kJ/mol) for lithium
R=perfect gas constant = 8.3144 (J/°K*mol)
Tn=boiling temperature at standard pressure = 1600 (°K)
T2= New temperature = 2000 (°K)
Filling in values for our example:
P2= 101*e^146,000/8.314*(1/2000- 1/1600 = 900 kPa
This will be the operating pressure of the system.
To determine the density at this pressure, we used the Perfect Gas Law:
4) d=Aw*P / R*T
Where:
d=density (kg/m3)
R=perfect gas constant = 8.314 (J/°K*mol)
Aw=atomic weight = 6.9 g/mol
P=pressure (kPa) = 2000 kPa
T=temperature (°K) = 2000 °K
6.9*900 / 8.314*2000 = 0.37 kg/m3
Finally, using Watson’s Equation we can find the new heat of vaporization:
5) Ve2= Ve *((Tc – T2)/(Tc – Tn))0.38
Where:
Ve2=vaporisation energy at new pressure (kJ/kg)
Ve= standard vaporisation energy (J/kg)= 146,000 (kJ/mol)/6.9 = 21159 kJ/kg
Tn=standard pressure boiling point (°K) = 1600 °K
T2= New temperature (°K) = 2000 °K
Tc= temperature at critical point (°K) = 3220 °K for lithium
Again using the values from our example:
Ve2 =21159*(3220-2000/3220-1600)0.38 = 19,000 kJ/kg
So the phase change at this pressure carries 10% less energy, but the radiators should have a much greater gain in efficiency.
Referring back to Equation 1), for 100 GW (or 100e9 J/s) of power:
100e9 J/s ÷ 19e6 J/kg = 5,300 kg/s.
And 5,300 kg/s ÷ 0.37 kg/m3 = 14,300 m3/s.
This compares very favorably with the 24,000 kg/s and 32,000 m3/s that we calculated for Helium in the ‘Gases in Pipes’ chapter. It’s also better than the 57,000 kg/s that we calculated for liquid Beryllium, but as a gas, the volume flow is more than four hundred times larger than the 35 m3/s of the liquid metal.
These reduced mass flows will allow the designer to use slower velocities in the piping, thereby reducing both the pumping power required, and the wear on the piping.
The main problem with all of these good numbers is that lithium vapor at 2000 °K is highly reactive. It will be a challenge to develop materials able to resist it for the many years of operation required. If it were possible to use Beryllium as a heat transfer medium, the mass flow would go down even further, to about than 3000 kg/s. However, the piping and equipment would then have to withstand temperatures in the order of 2700-2800 K.
Heat pipes
It’s quite encouraging to know that Lithium phase change is in use today as a heat transfer medium in industry and laboratories, as the working gas in the thermodynamic device known as a heat pipe.
Image 4 A very hot heat pipe, showing uniform heat distribution, source: Econotherm
Image 5 Schematic of a heat pipe
Heat pipes are self enclosed vapor and condensate systems, where the return pump is replaced by a wick. In the heat pipe, a liquid evaporates at one end and condenses at the other end, transferring heat through the vaporisation phase change. The tricky part is how the liquid returns back to the hot end against the thermal gradient without any external force applied. This is done using capillarity action and a wicking surface. The fluid is drawn up by the wick through the combined effect of surface tension and intramolecular forces with the wicking material. Even though the term wick brings braided cloth to mind, a tightly grooved surface or a porous media can act just as well to draw up the fluid.
This effect cannot work over long distances or against great pressures, therefore heat pipes are usually quite short, from a few cm to a few meters at most. This limits their applications to certain specific situations, making them a bit of a solution without a problem, although they have recently come into their own as cooling elements for microprocessors.
Heat pipe equations
Capillary pressure in a channel type of wick:
Image 6 Cutaway view through a heat pipe and detail of channel
Let’s take a simple example, a 1m long heat pipe, 150mm in diameter. With lithium as working fluid and with small 5 mm x 5mm channels.
6) Pc=(2σ/Reff)*cos(b)(5)
Where:
Pc=capillary pressure (Pa or N/m2)
σ = Surface tension (N/m) = 0.24 N/m at 1400 oC for lithium (x)
Reff = width of channel (m) = 5mm = 0.005 m
b = angle of fluid to wall = 15 degrees hypothesis (actual data welcome! Not very sensitive to angle anyway, except for quite small angles)
So for lithium in a heat pipe:
Pc= 2*0.32/0.005*cos 15 = 46 Pa.
That’s not much pressure, but enough to get the job done over a small distance.
How far can the capillary pressure take us? We need to know the pressure drop of the returning liquid in the channel, since the pressure drop will equal the capillary pressure. This is a fairly straightforward application of Bernouilli’s Equation, reduced to laminar flow and with no gravity component. (We use laminar flow since our little finger already told us turbulent flow has too much pressure drop). This simplified condition is covered in the Hagen Poisseuille Equation, that will give us the equivalent flow:
7) Q= pi*Rh4*Pc/8*u*L
Where:
Q= fluid flow (m3/s)
Rh=hydraulic radius (m)
Pc=Capillary pressure (N/m2)(Pa) = 46 Pa
u=viscosity of lithium at 1400 oC = 1.5 Ns/m2
L=equivalent length of the heat pipe (m)
pi=3.1416
We are missing two factors, the hydraulic radius and the length
8) Hydraulic radius equation(7)
Rh = 2*(W*H)/(W+2H)
Where:
W= width = 0.005m
H= height = 0.005m
Rh = 2×.005×.005/.005+.01 = 0.0067m
9) Equivalent length equation(3)
L= 0.5Le +La +0.5Lc
Where:
Le = Evaporation section length
La = Area of adiabatic flow
Lc = Condensation section length
Let’s simplify this and put L at 1m in length.
And putting 7) 8) and 9) together we find:
Q=pi*.0067^4*46/8*1.5*1 = 4 e-8 m3/s
That’s for one channel. For the 40 channels we see in the pipe section, the total flow would be 1.7e-6 m3/s
With the density of lithium, 530 kg/m3, we can find the mass flow: 530 x 1.7e-6 = 0.0009 kg/s.
Doesn’t sound like much, does it? About 1 gram per second.
If we put this flow into equation 1), the total vaporization phase change power for this heat pipe is then :
Q= 0.0009 kg/s x 21,159 kJ/kg = 19 kW.
We can compare this to the heat conduction through a 1m long solid copper pipe, 150mm in diameter:
10) Q=k*(A/x)*dt, the equation for conductivity
Where:
Q=Power (W)
k=Thermal conductivity (W/mK) = 401 for copper
dt=temperature difference (Tin-Tout) (K)
A=Area (m2) = 0.0176 m2
x=Thickness of material (m) =1m
Although we don’t know the temperature difference, since we don’t know how hot the copper will get, we can use the equation to determine the temperature difference required to move 19 kW through the pipe.
dt=Q/k*(A/x) = 19000/401*(.0176/1)= 2126 oC. Since the pipe is only at 1400 oC, it is impossible to remove all the energy. If we worked out all the details, we would find this heat pipe is about 10 times more conductive than the equivalent solid copper pipe. Not bad. In the spreadsheet (see Tools of the Trade at the end of the text), there is an example with silver, that moves about 100 times more energy.
Starship applications of heat pipes
For starship cooling, heat pipes are interesting for extracting heat from massive elements, such as the radiation shields, or as extended radiating surfaces in radiators. They greatly increase the fin effectiveness by increasing the average surface temperature of the fins, and therefore increasing the thermal radiation off into space.
Image 7 A hot gas flowing in a radiator section. White pipe, black radiator fins
For example, in image 6 above, if the fins are solid carbon composite, the heat will need to go through, on average, 0.3 m of fin before it exits as radiation.
If the fins are, instead, composed of a multitude of parallel heat pipes, the heat will need to move through no more than twice the thickness of the heat pipe wall, about 0.001 m , or in other words 300 times less thickness.
In such a case, a well chosen heat pipes will reach an overall temperature very close to the temperature of the hot gas in the pipe, minus the convective losses at the pipe wall.
Limitations of Heat Pipes
Heat pipes are limited in their operation range by four boundaries:
The Sonic Limit: Gases cannot move faster than the speed of sound in the gas. This limits the gas flow in a small diameter heat pipe. Although the speed of sound is very fast for metallic gases.
The Viscous Limit: As the heat load goes down, the heat pipe eventually reaches a point where the pressure at the hot end is not sufficient to drive the hot gas over to the condensing section. This means the designer must ensure that the system can operate in these part load situations.
The Boiling Limit: As the heat flux rate goes up, the heat pipe can form a film of gas on the hot side, seriously reducing the heat transfer rate.
The Entrainment Limit: The high speed gas moving over the fluid phase creates shear forces, than can eventually lift fluid from the return stream and ‘dry out’ the heat pipe at the evaporator end.
Although we won’t go into the details of all of these, it’s important to remember that you can’t just solve a problem by saying : heat pipe!
___________________________________
Tell me more! Tell me more! (6)
Ok, if you insist. Heat pipes are only 50 years old, they were developed by George Grover at the Los Alamos National Laboratory in 1963. He thought they might be used for cooling small nuclear reactors. Positively a new technology, by thermodynamics standards. So there are new developments. Separate returns and longer runs, for example, as in the image bellow.
Image 8 A heat pipe loop
Image 9 Working fluids
Analogy to Boiling Water Reactors
Image 10 Boiling water reactor, Wikicommons
A very common type of nuclear reactor, the boiling water reactor, can serve as a good analogy to the radiation shield. But there are many differences. We can remove the containment vessel, since coolant leaks will not affect the environment. Instead of a water cooled condenser, we have the radiators that serve the same purpose. As discussed in the ‘Pumps and Turbines’ chapter, we will not be using turbines for large scale power production, they are just too heavy. But it will be possible to run the return pumps from small turbines, either through directly linked shafts or electrical circuits. The shield will likely generate the power of twenty or more power reactors in a fairly wide and short core, rather than the long narrow cores used in boiling water reactors. It’s a good idea to split the shield into multiple circuits, each one perhaps a GW in power. And the actual fuel elements are replaced with radiation absorbing materials, which are a lot less dense than uranium, so the overall density of the shield will be much lower than the density of a nuclear reactor core.
Freezing Fluid Metals
There are some challenges with starting up a liquid metal system. Since Icarus drive will be inactive for almost 70 years during the coast period, the lithium, or any other liquid metal coolant, will freeze solid. Some otherwise promising metals, such as bismuth, expand when they freeze, just as water does. This is bad for piping and worse for pump and valve bodies. We need a substance that shrinks when it freezes.
Unfreezing the metal should be possible by induction heating, using energy from the secondary reactor. As long as the drive is off, we probably won’t be able to turn the metal into vapor though. To keep the heating required within reasonable limits, a good option might be to drastically reduce the efficiency of the radiators, possibly by the simple expedient of adding shutters, or some kind of movable curtain, to reduce the emissivity. A substance with a very wide liquid range, such as tin, might be useful as well since it would be easier to melt.
Tools of the Trade
The equations used and the tables in can be found in the following spreadsheet.
References
(1) In a certain sense, the whole water cycle on Earth depends on phase change and evaporation, so one could argue that hydropower needs phase change too.
(2) Not quite formally true; most materials go directly from solid to gas, or sublimate, at certain pressures and temperatures.
(3) Heat pipes. A nice and clear text, despite slightly approximative english.
http://www.lth.se/fileadmin/ht/Kurser/MVK160/Project_08/Fabian_Korn.pdf
Manufacturer’s website for heat pipes in general:
http://www.thermacore.com/thermal-basics/heat-pipe-technology.aspx
(4) This is a simplified equation. Ve is not a constant but varies with pressure. The complete equation takes ambient pressure into account, and could be written by combining some of the equations that we will be seeing later.
(5) Dreams of nuclear flight. The fifties were a high point of nuclear research.
http://fti.neep.wisc.edu/neep602/SPRING00/TERMPAPERS/stoffel.pdf
(5) This is just one of the many forms of this equation. One equation per type of wick.
(6) Summer nights, Grease. Lyrics by Jacobs Jim, Casey Warren
(7) The hydraulic radius is the radius of a circular pipe that would have the same pressure drop as the open channel in which the condensate fluid flows. Again, a simplified equation was used. The more accurate version for a rectangular channel is:
Rh=(12µL/wh3)*[1 −(192h/π5w)* tanh (πw/2h)]−1
see : MICROFLUIDIC NETWORKING: MODELLING AND ANALYSIS
RELATORE: Ch.mo Prof. Andrea Zanella LAUREANDO: Andrea Biral