I'll say this one last time. The bacteria primarily eat CO2, not algae not sunlight. They run on Chemosythesis not Photosynthesis. The main consumables for chemo are compounds that contain carbon, sulfur and phosphorous. They only need the algae to provide a little bit of this and a little of that(think vitamins you don't eat more than a few grams, but you eat several pounds of calories.)
If you run the process perfectly in reverse to you, output O2 and sugars yes it requires energy. They don't run it perfectly in reverse. They also only require heat for a warm enfironment, no other energy sources. I'm really not at all certain why I am even arguing this. Bacteria have been doing this for thousands of years with no sunlight, only heat sulfur and CO2. If you think its impossible then take it up with them. first link
NASA sends volcano-loving bugs into orbit from google.
I don't know how many coal-fired power plants you've been to but the exhaust gases are hot(not from the steam turbine because thats a closed system, but from the actual coal burner). Yes of course you want to get as many BTUs out before you dump it but there are limits.
I've taken thermo, Kelvin and Plank are happy and the first law is still in place. For me it was ME 3210 and it rather clearly points out that heat engines are never 100% efficient. Stick your hand in front of your muffler, its hot too. And what you suggest
is a violation of the Kelvin-Plank statement of the 2nd law of thermodynamics. the law in short says if you are going to do work with heat you will have to reject some heat as non-useable in the process. Thats whats happening out of the exhaust from the burner(not the boiler). Heat that cannot be successfully transferred into the steam or water depending on what kind of plant you are running, is used to preheat incoming air to the burner, which is also used to pre-re-heat liquid in the cycle thats coming out of one of the series of pumps and closed heaters or deaeraters. All said and done its still going to be reasonably hot. Effectively as exactly hot as saturated water(the last regenerative cycle for the exhausts will be at the lowest turbine pressure at saturation temperature and the gasses will be slightly warmer than the fluid approaching the saturation point). Which happens to be the temperature the bacterie exist at in nature.