As the combustion chamber necks down and flares out, the 1-D cross section past which which fuel must pass as it cools the wall of the chamber will change. If there are lots of tubes, they will bunch together at the neck. Is there a standard way this is handled? Is it just a matter of making the channels wider in the radial direction near the neck, so the total 2-D cross section is the same? If there is not a standard way, perhaps the F-1 can be used as a specific example of a way this issue can be handled. (I now realize this is more about the nozzle than the combustion chamber, but I'm leaving the title alone just for historical consistency.)

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    $\begingroup$ Comment rather than answer because I don't know much about fluid dynamics, but I think kerosene isn't very compressible, so you primarily get a changing flow rate as the cross section changes, rather than a changing pressure. $\endgroup$ – Russell Borogove Sep 9 '18 at 21:47
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    $\begingroup$ Yeah, I revised it to have fewer presuppositions. It does seem like varying flow rate might be the real problem I'm asking about, since it would certainly change cooling. $\endgroup$ – Mark Foskey Sep 9 '18 at 21:58
  • $\begingroup$ As you may see in this picture the radii of the combustion chamber of the F-1 are quite uniform. But the diameters of the nozzle are nonuniform. That is why the bifurcation joints are at the nozzle (at the 3:1 expansion plane) and not at the chamber. $\endgroup$ – Uwe Sep 10 '18 at 8:41
  • $\begingroup$ The pipes will bunch at the neck, where the combustion gases are hottest? $\endgroup$ – SF. Sep 10 '18 at 11:13
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    $\begingroup$ There is a NASA paper about this theme. A PDF scan with 130 pages about the state of art from 1972. A master thesis paper of 100 pages. $\endgroup$ – Uwe Sep 10 '18 at 14:59

Two different methodes were used, modifying the shape of the cooling tubes from a flat rectangle (1) to a circle (2) and increasing the number of tubes using a bifurcation joint (3).

The flat rectangle tube shape was used at the smaller diameter of the combustion chamber and the circular shape at the wider diameter. For a smaller expansion ratio of the nozzle only this method was used.

Another method was the use of tapered tubes with varying diameters.

For a bigger expansion ratio of the nozzle, the number of tubes were doubled by using bifurcation joints (4).

An initial design of the F-1 engine had two bifurcation joints for a fully liquid cooled nozzle, the flown version had only one bifurcation and a nozzle extension cooled with the gas from the turbine exhaust. The joints connected 178 primary tubes to 358 secondary tubes. 3000 feet of welding joints were needed between the parallel tubes.

For more information about cooling of the F-1 engine see this excellent page. All links to pictures from this page.

There are better ways to build a combustion chamber with cooling channels now than to weld it from a lot of tubes. The channels may be milled into the inner wall (made of copper) of the combustion chamber, then these channels are filled with electric conductive wax and the outer wall is build by electrolytic deposition of nickel. After that the wax is removed by heat and solvents. Another method is 3D printing. Very complex and difficult welding of many tubes as used for the F-1 engine is not necessary anymore.

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    $\begingroup$ Thanks also for linking that "Heroic Relics" page - looks as if there's plenty of bedtime reading across that site $\endgroup$ – Tom Goodfellow Sep 11 '18 at 6:28

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