# Minimizing entropy generation in storing heat

I am working on a cogenerative PV (Photovoltaic) array. The idea is to generate electricity from the solar cells while cooling them to both increase solar conversion efficiency and extract waste heat from the array. On a good day, the cells could easily get up to 40°C, and capture some finite (let's say, 100 J, just because it doesn't really matter) amount of energy as heat.

Ultimately, I am trying to figure out if it is better to store those 100 J in a thermal capacitor (aka concrete block) at 40°C, 20°C, or 50°C, which I'll explain below. In any case, 100 J of thermal energy would be stored in a thermal capacitor with an appropriate heat capacity for all 100 J at that temperature.

Also, in order to get storage at 50°C, I would use a heat pump to elevate the temperature. Please ignore the energy required to run the heat pump for this question.

The reason for the three different temperatures is that there are two loads this heat will serve. First, I would like to condition a room at 20°C, meaning there's an HVAC (Heating, Ventilation, and Air Conditioning) load. Second, I would like to heat water to 50°C, meaning a DHW (Domestic Hot Water) load.

Assuming a lossless system, I will have perfect "energy efficiency" since 100 J are produced, 100 J are stored, and 100 J are utilized. What I want to know is how best to optimize this for entropy.

For example, if I try to heat the HVAC load from the thermal storage at 40°C, then I produce entropy, even in this lossless sytem. For example, given an HVAC load of 100 J, I get:

$$dS=\frac{dq}{T}$$ $$dS=\frac{100}{293}-\frac{100}{313}=0.022$$

which means I produce 22 mJ/K of entropy. This would increase if the storage were at 50°C.

On the other hand, storing the energy at 20°C means I need to exchange heat from the PV array at 40°C to the storage at 20°C, so it appears the math works out the same.

I understand that, with the second law of thermodynamics, you can never really win, but is there a way of minimizing entropy generation when storing heat?

EDIT: From the recent migrations and other concerns people have expressed, I'd like to also clarify a few things.

First, the PV array is being considered as a constant temperature source. That is to say, I am considering the heat removal and heat gain to be balanced (in this example) at 40°C.

Second, the heat storage has a large enough heat capacity so that its temperature changes negligibly. To justify this assumption:

$$dq=CdT$$ $$dT=\frac{dq}{C}$$ $$\lim_{C\to\infty}\frac{dq}{C}=dT=0$$

As long as C is an order of magnitude or larger than dq, any dT will be less than 10%, which I'm happy to neglect.

• Since this is primarily about how to design a technology (rather than about explaining/predicting observations), it's really an Engineering question rather than a Physics question. It'd probably be a better fit over at SE.Engineering. – Nat Jan 8 '18 at 20:31
• @Nat I see where you're coming from, and I do actually have a related question there, but I intended this question to understand the theoretical limitations of a system design, not the practical ones. As such, I believe it's a better fit here than on the Engineering SE. Fundamentally, I am asking how to solve a physics problem, which is the minimization of entropy generation in a system given certain constraints. – Hari Ganti Jan 8 '18 at 21:49
• I would recommend Engineering; you're basically asking about waste heat recovery. That said, regardless of where you post the question, a diagram of your system and assumptions might help. It also may help to describe the question in general terms, e.g. how to get the most useful energy out of it, rather than asking for analysis using a particular approach, e.g. "optimizing for entropy". General terms may help, too, e.g. "DHW" is a pretty esoteric acronym that few are liable to recognize. – Nat Jan 8 '18 at 22:07
• Not to come off as aggressive, just trying to point you in the right direction since it'd seem unlikely for this question to receive the answer that you're looking for in current form. – Nat Jan 8 '18 at 22:08
• – Air Jan 11 '18 at 16:40