Stack design is generally carried out using software, and vendors are usually more than happy to run the selections. Sizing for diameter is basically a balance between the duct friction loss and the stack effect.
You will need to provide your operational criteria (discharge sp, exhaust temp) from the generator manufacturer and ambient summer and winter conditions. Stack height is usually determined by code, site and discharge requirements.
I have found generator manufacturers to be extremely unhelpful on this matter. You would think otherwise, but I never have any luck. Most don't know what an inch of water column is.
You should be able to determine the external static pressure that the unit can deliver. I think you just size the pipe like you would ductwork, but I also am no expert.
some of this is what the other responders to this post have said, but for the sake of completeness...
Take a look at or try to obtain the cut sheets with operating data on the generator. There should be a section on the exhaust where it lists flow rate a rated output (CFM), exhaust temperature at rated output, exhaust outlet size, and maximum recommended back pressure (kPa, "Hg, etc.). This last one is kind of like the ESP the unit can deliver as stated by others on this post.
So to size the stack, you want a stack-to-wind-velocity ratio of 1.5:1. Check the wind data (e.g., 96th percentile) in your area. 1:1 is marginal and very poor flow occurs at 1:2. In most cases, 3,000 to 4,000 fpm will provide good performance. Then make sure you don't exceed the back pressure limitations given a diameter that gives you the required velocity.
If, for example, your exhaust outlet is an 8" ANSI flange, start your iterations by seeing how an 8" pipe will do.
I'm not sure exactly what else you need, but I'll try to clarify further.
I obtained the stack-to-wind-velocity ratio rules of thumb from SMACNA's "Guide for Steel Stack Construction", 2nd Ed. Generally speaking, the purpose of having a higher stack discharge velocity than the prevailing wind speed is to prevent downwash. This is similar to the airflow over buildings, so the ASHRAE Fundamentals Handbook may have some info on this as well. This Fundamentals book also has the "extreme annual windspeed" data for various locations which you'll also need. If you don't have this resource, let me know the location and I can look it up.
For example, Chicago has extreme annual windspeeds tabulated at between 19.2 and and 24.8 mph (5% thru 1% data, respectively). So to be conservative (in this example), you might just want to use 25 mph.
Therefore, at 1.5:1, your stack velocity needs to be 3300 fpm (minimum) to avoid downwash.
Now you still need to obtain the generator data I mentioned before regarding the exhaust: airflow rate, back pressure, temperature, connection size. You should be able to find some tables (or engineering toolbox type thing) where it will tell you the pressure drop (usually in psi/100' of say schedule 40 steel pipe) through various size pipes at certain flow rates. You will have to apply some factors since you're not dealing with standard air. Temperature will be a key factor since your generator exhaust might be over 1000 deg F.
Anyway, assuming you can come up with a number from the tables adjusted for the temperature/pressure, etc. For an example, let's say you come up with 80" W.C. pressure drop/100' of sch 40 pipe given 2000 CFM and an 8" pipe. If you're exhaust connection size is 8", you probably don't want to go smaller than that (manufacturer probably wouldn't allow it anyway). If the generator exhaust airflow is greater than ~1200 CFM, you'll satisfy the min velocity (3300 FPM) we calculated above, so 2000 CFM will work. Now let's say the max recommended back pressure is 1.5" Hg (20" W.C). If your proposed stack height is 25' or less, you're in good shape because you will have a pressure drop of 20" in 25' of 8" stack (given the 80"/100' we determined above). If, however, you need to put in a 50' stack, then you'll exceed the max recommended back pressure and you'll have to go with a larger stack size. Try 10" for the next iteration, and make sure your velocity is still adequate.
If you're adding any additional items to this stack you'll have to account for their pressure drop as well (e.g., muffler, turns, stack cap, etc.).
CountOlaf has provided almost everything you need. As an Electrical Engineer that specifies generators I would also caution you to obtain specific back pressure data from the silencer manufacturer. I am currently working on a site where I am specifying several 2MW units. These units have dual exhaust ports using steel flanges. These are collected by flanged steel piping to a "critical" level silencer which in turn has a single larger exhaust. Due to the generator locations I am being forced to extend the stack above an adjacent portion of the building to avoid fouling the face of the building with soot. My mechanical engineer had to take all these fittings into aocount on top of the other issues.
Also be careful. Depending on when this is going to be constructed and its location there are new exhaust requirements in some areas. Here in California an engine set permitted after 2011 will require a Tier 4 diesel exhaust system which is an entirely different animal utilizing exhaust catylist tanks, controls, piping, seperators, etc....
