I have this idea for a vastly more effective and efficient heat exchanger.
The basic issue is that heat exchangers have a tradeoff between thermal resistance and fluid resistance. Ideally you want to drive both of these parameters as far as possible toward zero, but thinner tubes or turbulence mean less thermal resistance and more fluid resistance, while thicker tubes or laminar flow mean less fluid resistance and more thermal resistance.
Consider recuperator-type heat exchangers, which use two separate fluids, typically in a countercurrent configuration.
I suspect you may be able to cut this Gordian knot using fractal geometry like that of the lungs. The idea is to perform almost all of the heat exchange along a very convoluted (wrinkly) fractal surface whose Euclidean dimension is 2 but whose Hausdorff-Besicovitch dimension is nearly 3. This surface is full of tiny capillaries, and it separates two volumes from each other, which we can denominate the “arterial” and “venous” volumes. These spaces are not themselves empty; each one contains two separate, non-communicating sets of branching passages which branch down to the capillaries. In one of these sets of passages, one fluid passes from the arterial space to the venous space; in the other, a potentially separate fluid passes from the venous space to the arterial space.
Each of these two spaces looks a lot like a cauliflower, with a branching tree structure (or rather two of them) of passages feeding a very rough surface.
The two sets of passages are separate and do not mix, but they intertwine progressively more intimately until, at the capillary boundary, they are separated only by very thin walls. Nearly all of the heat is transferred during the passage through the capillaries, and nearly all of the fluid resistance is also due to this passage through these thin capillaries; the passages in the arterial and venous spaces are so much wider than the capillaries that they offer relatively little resistance. But, because the surface through which the capillaries pass is so enormously convoluted, the total cross-sectional surface area of the capillaries is immense, allowing the fluids to pass with relatively little resistance.
You could imagine the convoluted membrane filling, say, 80% of a one-liter volume with a 50-μm thickness penetrated by many 10-μm-diameter capillaries some 10 μm apart. Roughly estimating, this gives you 16 m² of membrane pierced by 4 m² of capillary cross-sectional area dived among 40 billion capillaries, 2 m² for each of the two fluids. A flow rate of 500mℓ/s amounts to 250 microns/s through these capillaries; that is, the fluid spends about 200ms in the capillary, during which time it is in somewhat more intimate thermal contact with the other fluid 20μm away through the capillary wall than it is with its own predecessor or successor fluid 50μm away through the capillary.
(Branching to 20 billion capillaries involves some 34 levels of branching if they are binary.)
The cross-section through which the heat must travel by conduction is some twenty times (5× length, 4× four directions if the capillaries are in a checkerboard through the membrane) the cross-section of the capillaries themselves, so 40 m². Aluminum has a thermal conductivity of some 205 W/m/K; multiplying that by 40m²/10μm gives us 820 MW/K, which is a dramatically enormous number.
Some random online pipe pressure drop calculator tells me that, given .000000000025 l/s (25 pℓ/s, 250 microns/s * 10 microns * 10 microns), pipe diameter of 0.010 mm, pipe roughness of 0.001 mm, and pipe length of .000050 m, the flow should be laminar and the pressure drop should be 0.05 millibar if the fluid is water. That’s 5 Pa or a column height of 0.5 mm of water. God only knows if it’s using some approximation or other that isn’t valid at these scales, but it’s somewhat reassuring.
These two figures together suggest that you should be able to pump considerably larger amounts of heat and fluids through this heat exchanger. If we consider a 5 kelvin loss acceptable, then maybe we can deal with 4100 MW in our one-liter heat exchanger. Water can hold perhaps 100kcal/kg; that gives us 9800 kg/s, which is 9800 ℓ/s, which would be 4.9 m/s through the 2m² of capillaries, which amounts to 0.49 microliters per second per capillary. Entering this into the same calculator provides me with an answer of 999.83 millibar, or 10.2 meters of water column height, which is to say, one atmosphere.
If we take this seriously, it would seem that we can probably nanofabricate a 4-gigawatt water-based countercurrent heat exchanger with a 5-kelvin temperature drop, with only one atmosphere of pumping pressure, in a single liter. I find these numbers so outlandish that they are hard to take seriously, but I wonder how close we could really get.
(In some cases you might have a much lower heat capacity per unit volume, for example with air, and desire a much lower delta temperature, like 5 millikelvins. I think this is also achievable.)
Apparently Lingai Luo wrote a book on this in 2013, “Heat and Mass Transfer Intensification and Shape Optimization”; they proposed doing this in 2001. In 2002, Yongping Chen and Ping Cheng wrote a paper, “Heat transfer and pressure drop in fractal tree-like microchannel nets”, surveying the existing work and proposing a two-dimensional coolant duct structure for cooling semiconductor chips. They have 275 citations in Google Scholar, and since then there has been a lot of work in convection in porous and complex structures, including a 2004 book by Bejan, Lorente, and others. It contains this remarkable passage:
Tree-shaped flows in balanced counterflow are a prevailing flow structure in subskin vascularized tissues (Weinbaum and Jiji, 1985; Bejan, 2001). The purpose of the intimate thermal contact between the streams in counterflow is to minimize the leakage of heat (an enthalpy current) along the counterflow, from the warm end to the cold end. The counterflow provides thermal insulation in the flow direction: this insulation effect has its origin in the minimization of thermal resistance in the direction perpendicular to the streams (Bejan, 1979b, 1982). This special feature, and the fact that the streamwise leakage of heat vanishes as the thermal contact between streams becomes perfect, is the reason why the balanced counterflow is the best arrangement from the point of view of minimizing heat transfer irreversibilities.
Unfortunately, although he arranges his flows dendritically, Bejan’s work does not seem to contemplate distributing the capillaries themselves over a fractal surface, although he alludes to lungs at some point.
He also describes our lungs:
The alveoli act as the primary gas exchange units of the lungs. It is estimated that an adult has approximately 600 million alveoli with a surface area for gas exchange of about 75 m², which are perfused by more than 2000 km of capillaries (see Section 6.2). ... In order to optimize its function (see Section 4.11), the airway tree exhibits 23 levels of bifurcations after the trachea (Weibel, 1963).
Note that this 75 m² in a few liters is comparable to the exchange area I described above for the heat exchanger.
That topology-optimization guy in Scandinavia has been using topology optimization to design heatsinks using CFD simulation of convection. Understandably, they come out dendritic.
Luo’s book is about “process intensification”, which she defined in 2001 as “enhancement of the density of flux transferred between two phases through an interface”, which covers the heat-exchanger thing above as well as many other possibilities. Its chapter 4 is all about heat exchangers, and it says:
Microchannel heat exchangers usually have a surface area density above about 10,000 m²·m⁻³ (Shah 1991). One typical example in nature is the human lungs, as a very high performance ultra compact heat and mass transfer system which have a surface area density of about 17,500 m²·m⁻³.
The numbers I postulated above work out to about 80,000, so it isn’t too far from what people were already achieving in 1991. A Karlsruhe project got 18000 MW/m³ by these techniques, with a pressure drop of 4 bar (400 kPa), a 10-kelvin temperature jump, and a residence time of about 2ms, while my calculations above suggested you should be able to get 4 terawatts/m³ (i.e. 4,000,000 MW/m³), which suggests that my calculations are perhaps a bit optimistic but not entirely out of the ballpark. The Karlsruhe device, however, did not use a fractal geometry, and it used crossflow rather than counterflow.
Luo also points out that for microchannel heat exchangers you probably don’t actually want to use a highly conductive material, because it conducts heat longitudinally in the wrong direction. You’d be better off with a highly insulating material, something silly like polyethylene! And this is far more important for the kinds of geometries I'm considering.
Luo does at some point start using fractal designs for her multi-scale distributors, but never for the capillary-bearing surface itself. She proposes a multi-scale branched tree structure, but don’t give performance figures. She does cite da Silva et al. 2004 (“Constructal multi-scalar tree-shaped heat exchangers”) and Zimparov et al. 2006 (“Constructal tree-shaped parallel flow heat exchangers”). The designs in Zimparov’s and da Silva’s papers are basically the same as the Chen and Cheng 2002 paper: essentially planar trees.
Luo’s Chapter 7 shows the optimization of the distributor/collector network from chapter 3 (which is basically the same as Chen and Cheng’s) using cellular automata to improve flow, but never leaving two dimensions.
Sun, Huang, and Zhang did a paper in 2015 where they did a CFD analysis of the Chen and Cheng planar fractal heat exchanger.
Poltorak applied for a US patent in 2012 (publication 20120285660 A1, application US 13/106,640) which is more about branching heatsinks fractally to keep them from whistling. He’s now formed a patent troll company to extort money with his patent, called Fractal Heatsinks Inc. But this is not really relevant.
As mentioned in Recuperator heat storage, Galen discovered heat exchangers with this structure 1800 years ago dissecting animals, and in anatomy they are known as retia mirabilia. They are used both for heat transfer (for example, to prevent heat loss in the legs of sheep) and for mass transfer (for example, in all mammalian kidneys).
Although heat exchangers themselves are very important — a perfect heat exchanger makes compressed-air energy storage 100% efficient, for example — there are other possible uses of this geometry.
Luo’s book mentions a number of applications of “process intensification” beyond just heat exchangers, and Luo’s and Bejan’s books both go into some detail on mass transfer in lungs and kidneys. But in fact this kind of broccoli-like design can be used for a variety of other things as well: