Efficiency of Scaled-Up Electric Water Heater Elements

Yep let one slip there.
Now really picky would be if we get into linearity.

Depending on the thermal resistance out of the heating element, a 120°C could give you a boiler instead of a water heater.

Are you suggesting that the thermal conductivity of the scale is higher than water to somehow thermally bridge? I would doubt that - water will already reach fairly efficient equilibrium with the inside tank wall. The temperature drop should all be across the insulation around the tank wall.

I do agree on the first point under the assumption that the thermal conductivity through the scale is lower. Then in that case, it will take longer to equilibrate the water temperature, in which case you may have a slight overshoot. But I think that would still be getting into the noise in terms of overall efficiency. I wouldn't expect a well insulated tank to suffer as a result of this.

-Colin

Maybe not the best way of explaining what I am thinking.

How about this way.
If you start with a tank that holds 40 gallons of water and has a given amount of surface area.
As the amount of scale and sludge builds up the volume of water in the tank goes down but the surface area stays the same.

Actually I would guess the thermal conductivity of water may well be less then that of the scale. Compared to metals water is almost an insulator, otherwise stratified heat storage systems would not work.
This chart gives water a value of 0.58 and concrete a range of 0.9 - 2 .
http://www.engineeringtoolbox.com/thermal-conductivity-d_429.html

But water mainly moves heat by convection and water high specific heat capacity makes it very good for moving heat as long as the water itself can move.

I think the question in my mind is what is more rate limiting - heat transfer within the water/scale or heat transfer across the tank wall to the surroundings. If both water and scale are orders of magnitude more efficient at conducting heat within the tank to the wall as opposed to the conduction out of the tank. I think they are in a halfway decent insulated tank, then the difference between water or scale wouldn't have a material impact on overall heat loss.

Just curious - is the scale described a "wet" slurry on the bottom of the tank or is this a completely impermeable buildup on the heating element? I've actually never owned an electric tank to see how they build up and where the crud exists, but I do hear about people "draining" them.

-Colin