quote; Contact time, residence time or whatever one calls it is not a relevant criteria for heat transfer
That is a ridiculous statement. The concept for "multipass" heat exchangers is to maintain the two fluids in contact as long as possible.
Completely wrong. As I said before this is a common misconception. It is not a reflection on one's intelligence or educational level. Other engineers often make the same mistake because few of them have actually done their own heat exchanger design. In the design groups I've worked with we can tell when someone has never designed an exchanger when they start talking about residence or contact time with respect to the heat transfer. I've designed a lot of exchangers and redesigned or reconfigured a lot of others that were either not meeting requirements or needed to be able to handle greater heat loads and/or throughput. Multipass exchangers are used for a variety of reasons, but not for the reason you claim.
The primary reason for using a multipass is to increase the tubeside velocity, and therefore the heat transfer coefficient. Look it up. (Increasing velocity can also prevent settling/fouling, and even corrosion in some systems...while it can cause erosion in others.) Multiple passes can be used to keep the overall length of an exchanger at a reasonable value or to balance resistance to flow from a common header (most exchangers are not throttled on the cooling water side so if one exchanger is designed with only 1 psi of drop and another with 5 psi of drop, the lower resistance exchanger will get far more than its design flow, robbing others in the network.
Here's the kicker: For a given shell diameter and length increasing the number of tube passes REDUCES the total residence time in the heat exchanger when the flows are fixed (metered.) The reason is simple. Everytime you add another pass you have to add a partition in the exchanger head (or employ U-tubes that have an empty zone within the minimum bend radius of the tubes.) The partition obstructs part of the tube sheet and requires the elimination of some tubes. Fewer tubes = less tube side volume and lower residence time. Again, this comes from experience. Doing tube counts and examining tubesheet layouts is part of the design process.
Another reason I've seen multipass exchangers used was because a refinery had specified that all of the exchangers in a unit have a given tube length when it was built decades ago. They varied the pass count and shell diameter to achieve the desired duties. While I was doing some cooling water distribution analysis and redesign for them I recognized that I could debottleneck their capacity limiting alkyl unit by putting a new head on a small exchanger to reduce the pass count and thereby increase the cooling water flow and duty--a super cheap, easy project. To put it in terms more familiar to you: it had way too much "residence time" for the cooling water and this was restricting the amount of cooling it could accomplish. The tubside film coefficient was high...but so was the exit water temp.
There are various negatives to using multipass tube exchangers: reduction in the LMTD, potential for temperature crosses, increased bypass stream flow on the shell side (empty zones in the tube sheet for partitions or U-tubes), higher pressure drop on the tubeside, complexity/reliability problems of the gasketing of the head partitions, limitations on head side nozzle sizes and orientations, etc.
As for the square tube being the most efficient, a trianglar cross section would be even better, because it only has two unheated sides which are radiating the captured heat into the surrounding area.
An interesting proposal, but it would most likely have the opposite impact because of the geometry. Square tubes against one another will have less external surface area than triangular tubes with one face to the pipe. Essentially the two sides of the square are not exposed to the environment. The flat outside wall is. But for the same triangular tube width, neither of the two outside walls will be touching one another. They will both be losing heat to the environment. So the effective losses for the same width could be as much as twice as high for a triangular tube. However, various effects would tend to suppress the result for the triangular tube and increase that of the square tube somewhat.
Another problem with a triangular shape will be the lower flow. The tube will have half as much open cross section as a square tube, actually even less than that because of wall thickness. So at a given pressure drop the flow rate would be half that of the square tube. This would defeat the purpose of having parallel flow paths. Without running through calcs to verify I suspect the change in the hydraulic radius would improve heat transfer coefficient marginally for a given velocity though.