I can mention that I have used a linear temp. gradient in the boundary layer to calculate condensation and evaporation of sulphuric acid on rotary heat exchanger heating elements. Works very good.
Yes, the temperature gradient at wall is proportional to thermal boundary layer thickness. But not a linear relation. For more information please refer blasius and Pohlhausen solution.
When I calulated the sulphuric behavior in the Rotary Heat Exchangers I used a linear temp. distribution in the boundary layer, which is not correct if I for example compare with for instance a flat plate. However, the heat transfer elements consists of swirls .e.t.c.. i.e. a lot of different boundary layers. So from practical point of view a linear assumption was used. By doing that one was able to predict the film distribution and the formation of droplets. Maybe I was lucky but it worked very good. Maybe another temp distribution more similar to that on a flat plate also is useful, maybe also better, I do not know.!?
But the original question is about the normal derivative of the temperature at the wall? Of course that implies you are not considering a BC that prescribes the heat flux so that it gives you a prescribed normal derivative.
That's correct. It is the first temp. derivative at the wall compared to the first derivative from the partial pressure as a function of the sulphuric acid (or similar) that determines if it will be film condensation or formation of droplets
I can add to my calculations that this type of Rotary Heat Exchanger (so called Ljungström) is operating under equal conditions at Karlshamn Power Station, Karlshamn, Sweden.
It would be interesting to see how you apply your formulas on a not flat element configuration. As I said before the heat transfer plates are mostly rather complicated and work in the transition between laminar and turbulent flow. I.e. Reynolds number approx. 2000.