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You and I both need the input of both Hyundai and experienced automatic transmission shops to know the correct answer. Shops that repair Genesis transmissions learn firsthand !
Here are the issues (which I cannot yet answer).
First, Hyundai may be concerned about flow restrictions, or even OVER COOLING in the winter. Secondly, on adding a cooler after 12,000 miles, you certainly never want to add trash and may even want to add an external inline magnetic transmission filter to catch moving trash and tiny, tiny ground up steel or babbit circulating in the transmission. <Cleanliness is next to Godliness !>
I purchased a Canadian made Dana automatic transmission cooler for use with my ancient 1987 Chevy Astro Van some years back because I thought I was going to tow a trailer. The Canadian DANA model had different diameter holes drilled the length of each cooling tube. One end started with fairly large holes. But as you went to the opposite side, the holes became fairly small.
WHY ? You, of course, ask. The answer is that fluids become very thick at freezing temperatures and may flow hardly at all the first five minutes at 17 degrees BELOW ZERO, especially through an
aftermarket cooler. At freezing temperatures the majority of the transmission fluid went through 1/3 of the transmission cooler easily without restriction avoiding blockage of transmission fluid flow.
It largely BYPASSED the cooling in the cooler when cold. Eight minutes later (December) the transmission fluid was past 120 degrees, slowly rising, and as the fluid temperature increased past 170 degrees, it was viscous enough to then pass through through the whole length of the transmission cooler coil.
What both of us need to know is how Hyundai and 4 -6 good transmission shops feel about this.
I know that one theory is using a unique bimetallic bypass valve to just recirculate transmission fluid until it reaches 170 degrees operating temperature. To use THAT, we have to know that such a valve is 99.7% reliable and cannot remain shut or stuck open.
The advantage of the Dana patent is that is has NO moving parts.