Rectangular Fin Type Heat Sink

Your use of the term "design" is incorrect.

"Guess" is the correct term in this case.

The safest thing to do is go to a catalog and take a guess at which one looks like it will "maximize heat transfer".

I'd guess you'd want to think about total surface area of the heat sink, and maybe even the material of construction. Diamond would be best, but silver will work.

Google "heat sink design".

Few types of fins are smooth, wavy and louvered. Each has its own heat transfer characteristics. Any heat transfer or chemical engineering design hand book will provide required mathematical equations and the theory.

Use 1/8 inch minimum copper plate to spread the heat across the heatsink base . Mount fins vertically. Have the fin surfaces slightly rough to encourage turbulence, and get rid of (insulating) static laminar layer close to the surface.

Assuming natural heat flow, unassisted:

The fins closest to the heat source should be the longer and thicker, the remote fins shorter and thinner (economics). Taper the fin thickness as you get further from the heat source. Assuming heat is applied at the centre of the base, the overall shape of the heatsink assembly will be hemispherical for max heat removal with least material. If heat applied along a straight strip, overall shape will be half-cylinder.

For maximum number of (thin) fins use milling techniques to manufacture the heatsink (not economic). Heatsinks are usually drawn (work-hardening can reduce heat conduction) or moulded (needs finishing).

We cannot get an exact solution,

yes but can be optimize the heat sink in between cost and heat sink efficiency with help of manufacturer of heat exchanger 's catalog or tables.

Where does the heat go? The types shown are all air cooled so a small local fan drawing fresh air along those fins will help alot. It's a simple bit of thermo engineering to determine how much heat the air can take away given room temp is 20degC and exhaust must stay below 60deg C to avoid setting light to adjacent materials.

Since you want to maximize heat transfer, you need an infinite number of fins. Since that is impossible, make as many as you can with the material you choose.

Note: If generator burns up, try adding a fan next time. If that doesn't work, re-evaluate your plan.

Peltier junction with a fan.

That gets even deeper into the theoretical muck. Peltier junctions require more heat to be dissipated by the passive heat sink. The advantage a Peltier provides is that a it raises the temperature of the passive heatsink. This can then increase the amount of heat transfered to the cooling media. However, the total heat load now includes the heat the Peltier drew from the power supply. You cannot get something for nothing.

much will depend on how you plan on the air to flow. laminar flow is poison to heat transfer, you want turbulent flow as much as possible and you don't want one fin "shading" the next and impinging warmer air on it. so a lot of CFD analysis will be required in order to make sure one fin isn't merely heating the next one and the vortices are mixing the heated air with cooler ambient air before they encounter the next fin.

I'd be happy to help here. Can you describe your thermal load, your desired result, maximum operating condition, your containment, whether its passive or active cooling and if you have flexibility in your real estate (mostly height). We have tons of wild custom solutions in stock (one off need) or can build you a solution as needed up to 30:1 aspect ratios, although its about the balance of the transfer-to-dissipation (base thickness and configuration to fins/surface area) that really matters.
Dan

Your problem is apparently very complicated but in fact an easy one. You have two ways and have to make a choice:

1- you look over the net to proposed fin heat sinks from different sources and see which one has the dimensions you want to have. You can be sure that the designers already tried to get the maximal transfer. In most cases they are extruded aluminum in few cases cooper. The material does not have a great influence since the thermal limitation is at the convection level between fin and air.

2- It requires a lot more work but do not be afraid. In almost all heat transfer books (look at Lienhard which is free to download) there are examples how a fin has to be designed and in general presented in a simple and affordable way. Basically you should know how you want to manufacture the heat sink since the form of fins depends on your facility or choice. You would have to do a series of trials till you maximize the capability but with help of a spread sheet you can do it quite simple. I hope you consider the steady state of heat transfer since if you want to maximize the transients then you should look at what Red Fred wrote. You can make at start a reverse engineering effort analyzing several heat sinks available on the market and determine the convection coefficient they are based on. This could help to make a good first step.