Are plants not already capturing the energy embedded in cold LNG for degasification?
(e.g. you could do so by running a traditional nat gas electrical generator, then using the waste heat to preheat the LNG and then using the resulting higher pressure output to drive a turbine.)
This can be done efficiently enough that it's been proposed that liquefying air during times of excess energy then 'discharging' it with the aid of natgas peakers may be a competitive technique for energy storage. The round trip for liquifying air is pretty poor, but the additional energy from the natgas waste heat which is too low grade against traditional heatskinks to usefully extract more energy makes up for it, and the natgas peakers are running at exactly the time you want to discharge the liquid air battery.
So the energy cost of using that process as the required heat-sink for this capture process should be the energy that would have been generated by an alternative process like the above.
(e.g. you could do so by running a traditional nat gas electrical generator, then using the waste heat to preheat the LNG and then using the resulting higher pressure output to drive a turbine.)
This can be done efficiently enough that it's been proposed that liquefying air during times of excess energy then 'discharging' it with the aid of natgas peakers may be a competitive technique for energy storage. The round trip for liquifying air is pretty poor, but the additional energy from the natgas waste heat which is too low grade against traditional heatskinks to usefully extract more energy makes up for it, and the natgas peakers are running at exactly the time you want to discharge the liquid air battery.
So the energy cost of using that process as the required heat-sink for this capture process should be the energy that would have been generated by an alternative process like the above.