A Dynamic Flow 3-D Solar Energy Cell and Process (techrxiv.org) A Dynamic-flow 3-D Solar Energy Cell and Process

This paper proposes a new type of solar electric cell and method to achieve higher power density compared to conventional solar cells. The proposed apparatus and method comprise photovoltaic liquid fluids (e.g. colloidal quantum dots / perovskite type fluids) in a flow-cell comprised of high porosity 3-dimensional metal substrates. Based on several key factors that will be discussed, it is believed that this new cell and process could enable substantially more power per unit geometric panel area compared to conventional solid-state passive 2-dimensional solar panels. Technical rationale will be analyzed for how this method could de facto overcome current power limitations per unit panel area in conventional solar panels (such as Shockley-Queisser and recombination). Proposed fundamental governing mathematical models and associated variables for this new process are provided as a starting point, with the main goal being to illuminate this concept for other scientists and engineers to add their expertise and ingenuity to make such systems a commercial reality if it is practical to do so. A potential real-world application would be an on-board solar charging panel for electric vehicles, and similar applications where space is limited and maximum power density from a solar panel is desirable.

• The amount of PV fluid flow that can pass through the 3-D cell per unit time • The amount of electrons that the PV fluid can produce per unit time within the cell (via absorption of photon energy to achieve an electron excitation state) • The amount of the excited electrons that can successfully conduct out of the cell to achieve useful work before returning to the fluid/cell to complete the circuit (relative to the excited electrons that recombine to their ground-state and do not conduct out of the cell)

Schematic of the 4DCPV concept
A visual description is that 4DCPV comprises a labyrinth of inter-connected metal strands forming a 3-D volume filled with electron-generating PV fluid flowing throughout. The cell substrate (the highly conductive metal strands inter-connected as a singular object) is porous enough to allow light to penetrate and impact the PV fluid traversing throughout-to create excited electrons-yet also comprise sufficient strand density to be immediately available to conduct the sea of excited electrons generated throughout the cell.

Modeling 4DCPV System Behavior as a Dynamic Chemical Reaction Process
It is useful to derive mathematical models to help understand the potential advantages of the 4DCPV process, as well as help guide predictive system behavior and optimization. The models chosen for this paper are to equate the 4DCPV process steps to chemical reaction / chemical equilibria style processes and associated equations. There are several major reasons for doing this, as shown in the following examples and rationale: Consider an example where we have 100 g of reactant in a fixed solid-state 2-dimensional (2-D) surface area. In a one minute time frame, sufficient activation energy is provided for 20% reaction efficiency-yielding 20 grams of product. Now let's say we have 200 g of the same reactant in a 3-dimensional (3-D) volume, with the identical 2-D surface area. Assuming there is sufficient activation energy to still achieve a 20% reaction efficiency and all else being equal, the system will have generated 40 grams of product in the same 1 minute time frame.
The above example is a static 1-minute comparison. With 4DCPV, there is a continuous flow of PV fluid (at least, whenever there is suitable activation energy to make the process work). We can conceptualize that if say 300 grams of liquid-phase reactant can pass through the 3-D volume (with the same 2-D surface area in the example above) in 1 minute, then a 20% reaction efficiency would yield 60 grams of product. Alternatively, we can consider that a 13% reaction efficiency on 300 grams of liquid-phase reactant would yield 40 grams of product in the 4DCPV process-still 2 times more product compared to the static solid-state 2-D surface area process.

PV fluid
Electron flow out through conductor substrate to capacitor/battery/power a load, then return to complete the circuit Pump

3-D Cell (Porous metal)
Now if we say that the "reaction product" is in fact the electrons (and associated electrical energy output) of a solar energy cell (i.e. the "reactant" is PV fluid that generates electrons upon impact of suitable photon energy, and the "activation energy" comes from the photons provided free from the electromagnetic radiation of the Sun), then the de facto cell efficiency would be 40-60% for the 4DCPV process compared to 20% for the 2-D cell (based on energy output per equivalent 2-D surface area of panel).
It is well worth noting that the 4DCPV examples provided above would generate 2-3X the power output per equivalent square meter of panel area even at 13-20% efficiency (well below the theoretical efficiency of Shockley-Quiesser). In other words, the efficiency of the dynamic 4DCPV process does not violate Shockley-Queisser per unit of photovoltaic material-there is just more photovoltaic material units per time exposed to the sun's irradiance in the 3-D flow cell, so the utilization of the sun's photon energy per 2-D panel surface area is effectively increased. By utilizing two or more types of PV fluid in a 4DCPV, each optimized to specific bands of the Sun's electromagnetic spectrum, it is believed that the 4DCPV process can effectively act as a multi-junction solar cell and exceed the single-junction Shockley-Queisser limitsopening up even more possibilities for cell power density.
The above examples provide a chemical process engineering rationale for 4DCPV. A theoretical physics / thermodynamic approach is also useful and necessary in deriving a mathematical model for predictive system behavior (and how to make the most energetically favorable) 4DCPV process. In deriving these fundamental equations, the electrons are envisioned to follow similar behavior and rules to chemical reaction / chemical equilibria processes. This hypothesis is based on the fact that much of the behavior in chemical equilibria is deeply rooted in the behavior and interactions of the electrons in the associated chemicals, and assumes that the solar PV process can be modeled as a reaction that generates electrons as a product with the Sun's electromagnetic spectrum providing free activation energy to drive it. So called "driving force" process parameters and associated variables govern chemical reactions and equilibria, and should be a reasonable predictive model for interactions in the dynamic 4DCPV as well. This allows for manipulation of the variables and process thermodynamics with techniques such as applying Le Chatelier's Principal.
Both the chemical reaction process engineering and thermodynamic/theoretical physics factors are utilized in deriving the proposed governing equations and selection of optimal process/material parameters for 4DCPV. Ultimately, the goal is to demonstrate that 4DCPV for solar cells can generate more power output per square meter of available surface area, and effectively overcome the current power limitations associated with solid-state passive 2-D solar (such as the Shockley-Quiesser band-gap limitations of single-junction solid-state semiconductors and re-combination effects with solid state p-n junction layers).

Proposed Fundamental Governing Equations and System Parameters
Mathematical modeling of the 4DCPV process starts with three fundamental equations written in chemical equilibria-style format. In addition to assisting in the derivation of associated variables implicit in the 4D process, these equations will provide insights on how to manipulate the process parameters, variables, and materials to achieve the most energetically favorable 4DCPV process.

Equation #1:
ℎ + 0 1 ⇔ * Where: hν = total Planck-Einstein photon energy at one or more incident light frequencies (ν) e 0 = ground state electrons in the PV fluid flowing through the 3D solar cell e* = excited electrons from the PV fluid k1 = "reaction rate factor" that measures the rate of formation of "products" (e* in this case) relative to "reactants" (e 0 in this case) over a given time frame, factoring in any reversion of e* converted back to e 0 Clearly, the selection of PV fluid(s) for 4DCPV would be those that favor the reaction to the right (the "products" side of the chemical equilibria) upon impact by incident light frequencies.
The second chemical-equilibria style equation in 4DCPV involves the interaction between the excited electrons and the material(s) of the solar cell:

Equation #2
: Where: e 0 = ground state electrons in the PV fluid(s) flowing through the 3D solar cell e* = excited electrons from the PV fluid e c = excited electrons from the PV fluid that are conducted into the cell's substrate k2 = "reaction rate factor" that measures the rate of formation of "products" (e c in this case) relative to "reactants" (e* in this case) over a given time frame, factoring in any reversion of e c converted back to e * k1 = the reaction rate factor from Equation #1 Equation #2 is presented as a "tug of war" scenariowhere the excited electron can revert to a reactant (a ground state electron in the PV fluid of Equation #1) or conduct into the cell's substrate. Clearly, an optimal 4DCPV process would favor both reactions to the right, and the selection of cell parameters must be such that it "wins" the tug of war at a high enough propensity to be commercially viable. The reaction rate factor k1 will be affected by Equation #1 as well as the efficiency of excited electrons being conducted into the cell (k2).
Equation #2 provides an excellent opportunity to focus on driving forces and manipulations to achieve favorable results by invoking an electrical version of Le Chatelier's principle. If we assume in Equation #2 that the cell is made of an excellent instantaneous conductor, the chemical equilibria of Equation #2 should be driven very far to the right unless there becomes a surplus of products (e c electrons) in the cell. A highly efficient cell conductor for 4DCPV should accept the excited electrons instantaneously and be able to instantaneously conduct them out of the cell-to continuously "starve" the product's side (e c ) and continuously drive both reactions of Equation  #2 favorably "to the right." Key parameters of the proposed cell to accomplish this will be discussed later in this paper.
The third chemical-equilibria style equation for 4DCPV involves the flow of electrons conducted out of the cell to a staging area (electrical energy storage) until called upon to do electrical work, then returning to the PV fluid to complete the circuit and restore the ground-state electrons for the process of Equations 1 through 3 to repeat.

Equation #3
: Where: e c = excited electrons from the PV fluid that are conducted into the cell's substrate e s = excited electrons that are conducted out of the cell and into a staging/storage system (e.g. capacitors or batteries) before moving on to power an electrical load e 0 = electrons returning to the PV fluid to complete the circuit and restore ground state for the process of Equations #1 through #3 to repeat k3 = "reaction rate factor" that measures the rate of formation of "products" (e s in this case) relative to "reactants" (e c in this case) over a given time frame, factoring in any reversion of e s converted back to e c Again, the obvious goal is to drive this series of reactions as far as possible to the right. By utilizing clever arrangements of Capacitor/Battery Storage in Equation #3, it is believed that the e c of Equation #2 can be maintained in the desired "starved" state favorable to the Le Chatelierstyle process manipulations described above, and in doing so maintain the reverse reactions of Equation #2 and #3 to a minimum.
In conventional 2D solar PV, it is well established that re-combination of excited electrons in the solid-state semiconductor (prior to conducting out of the cell to do electrical work) diminishes the net power output. The 4DCPV process could offer greater options to overcome such limitations; such as with the chemical-equilibria style manipulations described above, a wider variety of process variable manipulations (flow rate, multiple PV fluids, the 3D solar cell parameters…) and by eliminating the need for expensive solid-state multi-junction approaches required to overcome Shockley-Queisser and p-n junction layer issues of passive solid-state 2D PV systems.
These three foundational equations, their driving forces, and their inter-relations set the stage for deriving a governing equation for the 4DCPV process. Since 4DCPV process involves PV fluid flow and time as a variable, the equation that incorporates the net effect of the 3 chemical equilibria-style equations and all related variables must consider system dynamics to predict the performance of the 4DCPV cell. One way to do this is to model the system power for a time frame that represents "one cell