Hello Gpsfrontier,
Well I am a mechanical engineer, and I am "impressed with" formulas eheh
You need to consider the max amount of cooling you need in the hottest season, and the max. reservoir size. Your cooler should be able to work at that extreme design point and you can use a thermostat to adjust to more favorable conditions by simply turning the cooler on and off, keeping your NS temp. at the desired level automatically.
You do approximations and put efficiency factors in your formula to simulate real life conditions. You don't need to consider every single physical effect and all the details for that. It will give you a basic idea to start your design. My point is, a vapor compression refrigerator can be 60% efficient tops, so you calculate how many calories you need per second (for the extreme condition) and divide it by 0.6, and you know the least amount of power you need. It is reasonable to expect that your cooler will be even less efficient than that so expect to use more power.
If you calculate this, you will know the minimum cost you can realistically expect and then maybe you will eliminate the option of vapor compression cycles.
The length, wall thickness and thermal conductivity of the tubing has no effect on efficiency. That only effects the heat removal rate from the NS passing through the tubing. Efficiency is all about the percentage of energy you waste. By using a shorter tube you don't waste anything, you only draw heat slower and it will take a longer time to cool down the same amount of solution but you will use the same amount of energy to do that.
Major energy waste is in the compressor and heat absorbtion of the fridge from the surrounding medium, that is air. So what effects your power consumption is mainly how efficient your compressor is and how well your fridge is insulated. It is also important to insulate the tubing between the reservoir and the fridge.