I assume that by "No BigReactors" you really mean "No Big Reactor Steam Turbines" as well: There are other sources of forge compatible steam (RailCraft etc), but this seems counter to the spirit of avoiding the mod and trying out 'something new'.
That said - there is a reason Big Reactors is my - and others - go to mod when large quantities of RF are required: With the other mods you are limited by the output of the engines/dynamos.
So - Listing the mods I know of in direwolf20 that produce RF:
Thermal Expansion dynamos all produce 80RF/t - differing dynamos and fuels merely change how long the fuel burns for. Dynamos do (however) support augments, so you can boost the output of a single dynamo up to 8x with a T3 power output augment. Augments require each tier to be installed to be active, so no multiple T3 augments. So; 640RF/t max = 32 Dynamo's to reach 20kRF/t.
The enderIO generators seem to have power production in the range 10RF/t to 80RF/t - with Octadic capacitor upgrades possible in the Stirling generator so again 640RF/t max per generator, and again, a grid of 32 Dynamo's.
ExtraUtilities introduces a variety of generators, and a 'different' space conserving way of boosting power- recipes can merge 8 of the basic generators to make a x8 Generator, and 8 of those to make a x64 generator. The basic combustion generator yields 40RF/t, so 2560RF/t per generator. This makes 20KRF/t reachable with a mere 8 x64 generators. Some of the other generators produce more that 40RF/t - but so many of them are so silly (a pink generator? Really?) I'll stop at the combustion generator before I feel (more) dirty.
The most powerful engine in Buildcraft, the Combustion generator, yields 60RF/t and has no way to be boosted.
MFR also has some RF generation that I can't find reliable details on.
As a non RF native option:
Assuming you have IC2x and MFR; as the MFR rednet power cables can convert EU to RF at a rate of 1:4 (I think) a 360EU/t Mark I EB reactor will be producing 1440RF/t. MOX reactors can safely produce >5000RF/t, and there are more daring reactor designs that require control computers to not overheat and explode but can produce an order of magnitude more power. So, 14 or less IC2exp nukes will hit your 20kRF/t benchmark.
So, all the non Big Reactors solutions are going to need ~10 to 100 engines to achieve the required RF/t.
And, all things considered - this is why people use Big Reactors for kRF power