Most efficient type of heat pump Can I make a heat pump beat the Carnot efficiency? Why is the Carnot process the most efficient one?
If I have a heatpump that is sphere shaped, and cascaded in layers like a onion can I beat Carnot efficiency? Heat would be transferred and concentrated from the outermost layer that is in contact with ambient air, and brought towards the center to heat a fluid.
 A: No.
The laws of thermodynamics have nothing to do with specific engineering designs - they state properties of the time-evolution of certain observables of physical systems, independent of design. Nor are they empirical laws - they are derived rigorously from first principles that no one doubts.
You can no more beat the Carnot efficiency by clever construction than you can build a time machine if only you had enough gears. Just as no amount of pistons and lasers could ever change the rest mass of the electron, no amount of Peltier cells can undo physics.
Thousands of people have thought along similar lines - "maybe if I just tweak things, my design will work." When they actually do build their physically impossible device and claim it works, the fundamental flaw is always one of two varieties: (1) improper calorimetry misses sources or losses of energy, or (2) the temperature reservoirs are not really at the temperatures the inventors suspect. Without more detailed design specs, I can't say where your reasoning fails, but doing that analysis is beyond the scope of physics anyway.
A: Due to the first and second law of thermodynamics: No.
The Carnot cylce is more efficient for a high temperature gradient of cold and hot reservoir. The efficiency of the Carnot cyle
$\eta = 1 - \frac{T_{cold}}{T_{hot}}$ is the upper limit of efficiency.
E.g.: $T_{cold} = 600\,K$ and $T_{cold} = 300\,K$ yields a high efficency of 0.5. New 2008 Otto Motor techniques (Split cyle engine, german wiki) yield $\eta\approx 0.4$.
The most efficient process would be to increase the hot Temperature und insolate versus the cold reservoir. Cooling $T_{cold}$ to near absolute zero Kelvin is harder and therefore not the way to engineer.
Remark: laws of thermodynamics
First law of thermodynamics: The increase in internal energy of a closed system is equal to the difference of the heat supplied to the system and the work done by it. Using it's mathematic form $dU = \delta Q + \delta W$ allows derivation of carnot efficiency $\eta$.
Second law links the loss of $Q$ to the increased entropy $S$ of the environment.
These laws both underline the carnot efficiency as a limit.
A: 
The first explicit statement of the first law of thermodynamics, by
  Rudolf Clausius in 1850, referred to cyclic thermodynamic processes.



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In all cases in which work is produced by the agency of heat, a
  quantity of heat is consumed which is proportional to the work done;
  and conversely, by the expenditure of an equal quantity of work an
  equal quantity of heat is produced.

Wikipedia
And because a carnot circle is a closed process without any loss in energy or heat, it is the most efficient system imaginable. What you suggest is a perpetual motion machine, which is impossible due to the same law.
