Subject: Maxwell's Daemon and Thermoelectric effect

I had asked this question before, but it was interpretted as an engineering 
problem and the root of my query was ignored.

Maxwell's Daemon tell me that creating a temperature gradient requires 
energy.  With this in mind how is it explained that:

Thermoelectric modules create temperature gradients when electric energy is 
supplied to the module.  The hot side of the module has heat deposited on it 
that was removed from the cold side.  This seems to agree with Maxwell's 
Daemon.

I get confused, however, when I see that the electrical energy used to pump 
the heat, is also deposited on the hot side of the module in the from of more 
heat, further increating the temperature gradient.  

In my head the energy of the system looks like:

-Ecold + Ehot + Eelectric + Egradient

Where E cold is the energy removed form the cold side, Ehot is the same 
quantity dumpe don the hot side, Eelectric is the energy powering the module 
(and also dumped on the hot side) and Egradient is the energy required to form 
the gradient.  Since Ehot and Ecold cancel (is conserved) I get:

Eelectric + Egradient

So im guessing that the heat deposited on the hot side from the electricty is 
NOT precisely given by V*I = W... I hypothesize that is is reduced by some 
amount, namely the energy required to form the gradient.

What is this energy?  How is it calculated?  Am I correct on where this 
energy 'comes from'?

Re: Maxwell's Daemon and Thermoelectric effect