# A Validated Model, Scalability, and Plant Growth Results for an Agrivoltaic Greenhouse

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}concentration [11]. Nearly all thermal models in [12] were locally 1-D transient models, with the discrete components considered as lumped masses. The models differed in regard to the other factors considered. Combinations of RH, CO

_{2}concentration, and heat flow due to certain heat transfer modes were neglected in many of the models. Jouid et al. [5] neglected the evaporation of water from the soil, assumed there was no water in the greenhouse soil, and assumed there were no plants in the greenhouse. Mohammadi et al. [13] assumed that there were no crops in the greenhouse, i.e., no evapotranspiration in the greenhouse, a negligible effect of CO

_{2}concentration on evapotranspiration, and no evaporation from the soil. They also assumed that any water condensed on the inside of the roof or screens was removed from the system. Cooper and Fuller [14] neglected edge losses, and Sethi [15] neglected radiation heat exchange between the walls and the roof, which we note below is not negligible.

_{2}concentration and the effects of the plant canopy inside the greenhouse (the test cell), because CO

_{2}concentrations were beyond the scope of our project, and the fraction of the base shadowed by the canopy would be highly uncertain. For our design, radiation heat exchange among elements inside the cell was found to be larger than free convection. In fact, heat flow rates for radiation were at least double those for free convection. Marucci et al. [6] and Cossu et al. [7] only modeled the solar radiation transmitted from outside the greenhouse to the interior. Most of the effects in our model are included in other models [5,6,7,12,13], but only our model contains all the effects we included. Our model also included transient heat conduction in the ground below the greenhouse, as earth coupling effects are generally important during fall and spring periods.

_{2}concentration.

## 2. Materials and Methods

^{3}, which added thermal mass to the system to attempt to make cell air temperatures more uniform over short times. Three total solar radiation pyranometers, two temperature, and two RH sensors were installed in the test cell, as shown in Figure 2b. The internal sensors were positioned about 1.3 m from the south-facing sidewall and about 65.3 cm from each other and from the east- and west-facing PV panels. A pyranometer and temperature, RH, wind speed, and wind direction sensors were installed outside of the test cell to collect ambient data. Manufacturers, models, and specifications of sensors used are presented in Appendix A.

## 3. Mathematical Modeling

- Heat and mass transfer are lumped in a node, and heat flow is locally 1-D;
- Solar radiation entering the test cell is treated as diffuse over all internal surfaces, except on the polycarbonate sheet;
- No thermal nodes are located in the north- and south-facing side walls (these function only as thermal resistances, coupling the cell temperature with the outside air. The mass of the walls is small due to the small size and the low thermal mass of the wall insulation);
- The thermal diffusivity and thermal conductivity of the soil are assumed to be constant based on the partial water infiltration theory [19];
- The irrigation rate and the rate of water diffusion from the soil are known and constant (Appendix D);
- The infiltration of ambient air and the accompanying moisture and heat transfer occurs by wind-driven infiltration through cracks in the walls and by the stack effect. The cracks are lumped into a single gap area determined during model validation;
- Only beam radiation is considered in the shadowing model; diffuse radiation, generally smaller than beam, is not considered.

_{lex}/dt = (q

_{sol,lex}+ q

_{wind,lex}+ q

_{sky,lex}+ q

_{a,lex}− q

_{rad,lex})/ρ

_{lex}V

_{lex}C

_{plex}

_{sp,out}/dt = (q

_{sol,sp}+ q

_{wind,sp}+ q

_{sky,sp}+ q

_{cond,sp,inner})/ρ

_{sp}V

_{sp}C

_{Psp}

_{sp,in}/dt = (q

_{a,sp}-q

_{rad,sp}+ q

_{cond,sp,outer}+ q

_{ligh,sp})/ρ

_{sp}V

_{sp}C

_{Psp}

_{cell}/dt = (q

_{sp,west,a}+ q

_{sp,east,a}+ q

_{lex,a}+q

_{con,a}+ q

_{sidp,a}-q

_{infi})/ρ

_{a}V

_{a}C

_{Pa}

_{con,up}/dt = (α

_{con}/Δx

^{2}

_{con})(2T

_{con,up+1}− 2T

_{con,up}+ (2Δx

_{con}/k

_{con})[q

_{flux,ligh,con}+ q

_{flux,a,con}+ q

_{flux,evap}+ q

_{flux,rad,con}])

_{i,con}/dt = (α

_{con}/Δx

_{con}

^{2})(T

_{i,con+1}− 2T

_{i,con}+ T

_{i,con-1})

_{con,low}/dt = (q

_{cond,con}+ q

_{tcc,con})/ρ

_{con}A

_{con}[Δx

_{con}/2]C

_{Pcon}

_{soil,up}/dt = (q

_{cond,soil}+ q

_{tcc,soil})/ρ

_{soil}A

_{soil}[Δx

_{soil}/2]C

_{Psoil}

_{i,soil}/dt = (α

_{soil}/Δx

_{soil}

^{2})(T

_{i,soil+1}− 2T

_{i,soil}+ T

_{i,soil-1})

_{soil,low}/dt = (α

_{soil}/Δx

^{2}

_{soil})(2T

_{soil,low-1}− 2T

_{soil,low})

_{a}/dt = Mdot

_{infi,h2o}+ ET

_{plant}+ E

_{liq}

_{a}/dt = Mdot

_{infi,h2o}− M

_{cd}

_{liq}/dt = −E

_{liq}+ Mdot

_{soil}

_{liq}/dt = M

_{cd}(Acon − A

_{plant})/A

_{con}+ Mdot

_{soil}

_{plant}/dt = Mdot

_{irrig}− ET

_{plant}

_{plant}/dt = Mdot

_{irrig}+ Mdot

_{cd}(A

_{plant}/A

_{con})

_{Plant}is evapotranspiration from the plants. It is an adapted form of the Hargraves equation [20] and is discussed in Appendix C.

_{plant}is greater than M

_{plant,max}, Equations (17) and (18) are used. Note that Equation (18) modifies the mass change value predicted by Equations (13) or (14).

_{over}/dt = (M

_{plant}− M

_{plant,max})

_{liq}/dt = dM

_{liq}/dt + dM

_{over}/dt

_{s}) = cos(φ) cos(δ) cos(ω) + sin(φ) sin(δ)

_{s})

_{west}and ω

_{east}, to be sunlit. These angles determine when the west-facing and east-facing solar panels no longer block sunlight from reaching the point under consideration. The angles are found by trigonometric relations based on the horizontal distance, y, and vertical distance, h, to each panel edge from the point under consideration.

_{east}= atan(h

_{1}/y

_{1}) − 90°

_{west}= atan(h

_{2}/y

_{2}) + 90°

_{set}= acos(−tan(φ)tan(δ))

_{rise}= −ω

_{set}

## 4. Results

#### 4.1. Validation

#### 4.1.1. Thermal Model

#### 4.1.2. Moisture Transport Model

_{g}(T)) at the air temperature of the test cell.

_{a}/0.662 p

_{g}(T)

_{g}is nearly an order of magnitude smaller than that at 35 °C (95 °F) [23]. For a given change in the product of the humidity ratio, λ, and the air vapor pressure, p

_{a}, this translates into an order of magnitude greater change in RH at −10 °C (14 °F) compared to that at 35 °C (95 °F). Thus, the sensitivity of RH to λp

_{a}is much greater in the winter than in the summer.

#### 4.1.3. Shadowing Model

#### 4.2. Experimental Data

#### 4.2.1. Test-Cell Temperature and Ambient Temperature

#### 4.2.2. Plant Growth

#### 4.3. Numerical Results

#### 4.3.1. Parametric Study

- Minimum temperatures in all parametric runs, Figure 19a–n, are only weakly affected by the structural, electrical load (the “load”), and thermal mass changes. Cases 2 (cell twice the length; Figure 19a) and 11 (no blocks; Figure 19j) showed a slightly elevated minimum air temperature, as the cell air received greater heat input from twice the number of PV panels in the absence of the load-leveling feature of energy storage in the concrete block. This suggests that the system was saturated with thermal mass, so additional mass may not have improved heat retention. This conclusion is supported by Figure 19i, in which additional thermal mass was added to the system, with only a slight reduction in daytime maximum temperatures.
- Figure 19a, for which the north–south length of the test cell was doubled (thereby doubling the number of PV panels), shows that the change led to increased heating during the day. This increase in temperature was larger during the summer than during the winter. As expected, the greatest increases in cell temperature were during the highest temperature days.
- Figure 19b shows an increase in daily maximum temperatures due to the widening of the gap area (i.e., increasing the insolation) between the two panels from 20.3 to 40.6 cm. Temperatures much above ambient have an adverse effect on plant growth, so this increase is undesirable. However, the increase in solar gain from the widened glazing is an improvement. The relative benefit of the increased glazing area is addressed in Section 4.3.2 and Section 4.3.4.
- Figure 19c–h shows that powering a load from the PV panels reduces the internal maximum temperatures during the day, especially during the summer, but as noted in comment 1 above, it does not significantly affect the minimum temperatures at night. This is discussed further in Section 4.3.3.
- Figure 19j shows that the removal of the concrete blocks reduces the maximum temperatures slightly during the hottest days of the year and slightly increases the minimum temperatures. The thermal contact conductance between the soil and the concrete blocks, which is removed in this case, reduces heat flow into the soil during the day and reduces heat flow out into the system at night This reduction in daily maximum temperatures (if only slight) can help keep the average cell temperature near the ideal of 20 °C (68 °F). (As thermal diffusivity is the ratio of the thermal conductivity to the product of density and specific heat, the smaller thermal diffusivity material is better at heat storage. The thermal diffusivities of concrete and soil are α
_{con}= 0.45 · 10^{−6}m^{2}/s and α_{soil}= 0.99 · 10^{−6}m^{2}/s, indicating superior heat storage for concrete blocks on a per-mass basis. However, soil was the largest portion of the thermal mass in the system. Note that contact resistance is created by placing the blocks over the soil, which reduces the heat flow to and from the dominant thermal mass.) - Figure 19k,l, shows some increase in temperatures during the spring and fall compared with the base case. This can be explained as follows:
- Figure 19k,l has more planters spread over the base of the test cell than the base case. Due to the increase in planters, less of the test cell base is exposed for evaporation, reducing evaporative cooling during these periods. Reduced evaporative cooling leads to the higher temperatures during the spring and fall as observed.

#### 4.3.2. Instantaneous Photosynthetic Rate (P_{n})

_{n}[24], more information about where kale can grow at the base of the test cell could be determined. P

_{n}> 0 means that a leaf is fixing more CO

_{2}then needed for cellular respiration. P

_{n}= 0 means that a leaf is fixing the necessary amount of CO

_{2}, and P

_{n}< 0 means that a leaf is not fixing enough CO

_{2}. The instantaneous value of P

_{n}can be found using the Mitscherlich equation [24].

_{n}(I) = (1 − e

^{−k(I−I0)}) P

_{max}

_{n}is the instantaneous photosynthetic rate; P

_{max}is a constant (the maximum photosynthetic rate of 20.3–21.0 µmols CO

_{2}fixed/m

^{2}s [24]); I

_{0}is the PAR irradiance at the compensation point where the P

_{n}is equal to zero (which, for kale, is 13 µmols photons/m

^{2}s = 2.85 W/m

^{2}); I is the instantaneous PAR irradiance; and k is Mitscherlich function, reported as 0.0030 [24]. Our shadowing model produces values of instantaneous solar irradiance. These values can be converted into instantaneous PAR (PAR stands for Phototsynthetically Active Radiation, which is solar radiation between 400 and 700 nm) irradiance using a conversion factor of 2.43 µmols photons/J, found using the methods described in [24,25,26] (The conversion factor for PAR in W/m

^{2}to µmol photons/m

^{2}s is about 4.57 µmols photons/J. However, the pyranometers used in this study measure solar radiation intensity between 300 and 1100 nm (visible light spectrum). PAR makes up only a fraction of that energy; as such, the factor of 4.57 must be reduced to 2.43 µmols photons/J when applied to readings from the pyranometers in this work.). If P

_{n}is integrated over a 24-h period, a value for net photosynthetic gain per day can be found. If the net photosynthetic gain is positive valued, then a leaf could grow that day. If the net photosynthetic gain is zero, the plant would maintain its current biomass. Finally, if the net photosynthetic gain is negative, the leaf is likely to lose biomass.

#### 4.3.3. Ideal Temperature Zones

- Plants in ambient conditions are expected to spend more time near the ideal temperature. Compared with the test cell data, Figure 15 reveals that the smaller time near the ideal temperature was mostly due to overheating during summer months. Simple modifications, such as ventilation fans, could reduce this overheating effect substantially and allow for better control of relative humidity. Addressing high temperatures during the summer should result in the test cell performing better than the ambient.
- The test cell spent fewer hours below freezing 0 °C (32 °F) than the ambient (36% fewer hours).
- Doubling the north–south length, as in case 2, reduced the number of hours spent near 20 °C (68 °F).
- Doubling the glazing area, as in case 3, reduced the number of hours spent near 20 °C (68 °F).
- Powering a load from the solar panels, as in cases 4–6 and 7–9, showed a progressive increase in the number of hours spent near 20 °C (68 °F) with increasing load.
- Table 2 shows that changes which increased cell temperature reduced the time spent near 20 °C (68 °F) and that changes that decreased cell temperature increased the time spent near 20 °C (68 °F) throughout the year. This suggests that the test cell overheated during the summer. This is an undesirable characteristic of the design. Keeping the test-cell air near the ideal temperature range would promote better plant growth.

#### 4.3.4. Shadow Model: Transmitted Solar Radiation Compared with a Stilt-Mounted PV Array

## 5. Discussion

## 6. Movement to a Comprehensive Agrivoltaic-Based Plant Growth Model and System Optimization

_{2}history in the greenhouse, which we have not modeled. However, a first-order approximation would be to the consider air infiltration (already included in the model) sufficient to keep the CO

_{2}concentration equal to about 400 ppm, that of the outside ambient air [24], which ignores the CO

_{2}contribution from soil respiration.

## 7. Comment on Scalability

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Variable | Definition |

A | Surface area, m^{2} |

B | Solar radiation transmitted through a transparent surface, W/m^{2} |

C_{p} | Specific heat capacity, J/kg·K |

E_{b} | Emissive power, W/m^{2} |

E | Evaporation rate, kg/s |

ET | Evapotranspiration rate, kg/s |

h | Coefficient of free convection, W/m^{2}·K |

J | Radiosity, W/m^{2} |

k | Thermal conductivity, W/m^{2}·K |

L | Through-thickness length, m |

M | Mass, kg |

Mdot | Mass flow rate, kg/s |

p | Index of the five-minute period per day from 0 to 288 |

p_{a} | Air vapor pressure, Pa |

p_{g} | Saturation air vapor pressure, Pa |

H | Heigh of obstruction (i.e., side panel), m |

q | Heat flow rate, W |

RH | Relative humidity |

R_{a} | Extraterrestrial solar radiation, W/m^{2} |

R_{c} | Thermal contact resistance, m^{2}·K/W |

S | Incident solar radiation, W/m^{2} |

SOF | Sky obstruction factor |

SP | Shadow projection point, m |

T | Temperature, K |

U | Transmitted solar radiation, W/m^{2} |

V | Volume, m^{3} |

y | Horizontal distance from panel edge, m |

α | Thermal diffusivity, m^{2}/s |

β | Angle from horizontal, degrees |

Δx | Material layer thickness, m |

δ | Declination angle, degrees |

ε | Emissivity |

η | Solar altitude angle, degrees |

λ | Humidity ratio |

φ | Latitude, degrees |

ρ | Density, kg/m^{3} |

σ | Stefan-Boltzmann constant, W/m^{2}·K^{4} |

ω | Hour angle, degrees |

Subscript | |

a | Test cell air |

con | Concrete |

cond | Conduction |

cd | Condensation |

east | Eastern facing solar panel |

evap | Evaporative cooling |

cv | Free Convection |

flux | Heat Flux |

i | ith node |

in | Inner |

infi | Infiltration |

irrig | Irrigation |

lex | Polycarbonate (i.e., Lexan) |

ligh | SR transmitted through Lexan |

liq | Liquid water node |

low | Lower thermal node |

Subscript | |

max | Maximum |

out | Outer |

over | Overflow |

plant | Planters |

rad | Radiation heat exchange |

rise | Sunrise |

sidp | Side Panel |

sky | Radiation heat exchange with sky |

soil | Soil |

sol | Incident solar radiation |

sp | Solar Panel |

set | Sunset |

tcc | Thermal Contact Conductance |

up | Upper thermal node |

west | Western facing solar panel |

wind | Free convection with wind |

## Appendix A

^{2}and a resolution of 1.25 W/m

^{2}. The sensor has an uncertainty of ±10 W/m

^{2}or ±5% of the reading (whichever is greater), and an additional ±0.38 W/m

^{2}/°C (±0.21 W/m

^{2}/°F) for a temperature greater or less than 25 °C (77 °F) [33].

## Appendix B

**Figure A1.**Dry mass as a function of total leaf area. Trial curve fits include linear, a power function, and a second-order polynomial. The second-order polynomial showed best agreement with the data.

**Table A1.**Comparison of experimentally derived plant dry mass sampled from the test cell to the value predicted by the allometric correlation of the same plants by a second-order polynomial.

Plant # | Leaf Area (cm^{2}) | Correlation Result (kg) | Experimental Measurement (kg) | % Error |
---|---|---|---|---|

1 | 15.3 | 0.065 | 0.063 | 3.1 |

2 | 26.1 | 0.112 | 0.113 | 1.1 |

3 | 33.3 | 0.161 | 0.112 | 43.7 |

4 | 38 | 0.164 | 0.151 | 8.3 |

## Appendix C

_{cv}= h (T

_{1}− T

_{2}) A

_{sky,lex}= σ ε (T

_{sky}

^{4}− T

^{4}) A

_{sky,sp}= SOF σ ε (T

_{sky}

^{4}− T

^{4}) A (1 + cos β)/2

_{sol}= S A

_{ligh}= U A/A

_{total}

_{rad}= (E

_{b}− J) A / ((1 − ε)/ε)

_{b}is the blackbody emissive power, J is the radiosity, and ε is the emissivity of the participating surface [36].

_{cond}= (k/L) (T

_{i}— T

_{j}) A

_{tcc}= (A/R

_{c}) (T

_{i}— T

_{j})

_{infi}= Mdot

_{a}C

_{pa}(T

_{int}— T

_{amb})

_{int}is the test cell air temperature and T

_{amb}is the ambient temperature.

_{plant}= 0.525 × 0.0022 · R

_{a}· TR

^{0.5}(TC + 17.8)

_{max}− T

_{min}

_{max}− T

_{min})/2

_{infih2o}= Mdot

_{a}(λ

_{amb}− λ

_{cell})

## Appendix D

_{sc}, σ, and R

_{gas}which are the solar constant, the Stefan-Boltzmann constant, and the gas constant, respectively. Other constants, such as all angles and lengths, were measured values from the test cell, some of which were adjusted for model calibration in the parametric simulations. Finally, values such as specific heat capacities, densities, emissivity values, and other material properties were set as general values accepted for those materials. Some values in this appendix were slightly varied during parametric simulations. The values used in the base case are presented here.

${G}_{sc}=1367\frac{\mathrm{W}}{{\mathrm{m}}^{2}}$ | ${\beta}_{west,sp}=0.63\mathrm{rad}$ | ${\beta}_{east,sp}=0.61\mathrm{rad}$ |

${\beta}_{lex}=0\mathrm{rad}$ | $\sigma =5.67\xb7{10}^{-8}\frac{\mathrm{W}}{{\mathrm{m}}^{2}{\mathrm{K}}^{4}}$ | ${R}_{gas}=8.3144\frac{\mathrm{kg}\xb7{\mathrm{m}}^{2}}{\mathrm{K}\xb7\mathrm{mol}\xb7{\mathrm{s}}^{2}}$ |

${\rho}_{lex}=1210\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ | ${A}_{lex}=0.394{\mathrm{m}}^{2}$ | ${\alpha}_{lex}=0.19$ |

$\mathsf{\Delta}{x}_{lex}=0.0024\mathrm{m}$ | ${C}_{{P}_{lex}}=1250\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ | ${\u03f5}_{lex}=0.7$ |

${W}_{SP}=1.4\mathrm{m}$ | ${L}_{SP}=1.97\mathrm{m}$ | ${M}_{SP}=32.3\mathrm{kg}$ |

${k}_{SP,eff}=1.05\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}$ |

**Table A5.**Constant parameters for soil. Also included here are values for the planters which contain soil.

${\rho}_{soil}=1650\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ | ${A}_{plant}=0.48{\mathrm{m}}^{2}$ | ${C}_{{P}_{Soil}}=1000\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ |

${k}_{soil}=1.56\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}$ | ${\alpha}_{soil}=0.99\xb7{10}^{-6}\frac{{\mathrm{m}}^{2}}{\mathrm{s}}$ | ${\u03f5}_{soil}=0.93$ |

${\mathsf{\Delta}\mathrm{x}}_{soil}=0.046\mathrm{m}$ |

${\rho}_{air}=1.2\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ | ${C}_{{P}_{air}}=1005\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ | ${h}_{eff,sp,lex}=2\frac{\mathrm{W}}{{\mathrm{m}}^{2}\xb7\mathrm{K}}$ |

${h}_{eff,con}=3.75\frac{\mathrm{W}}{{\mathrm{m}}^{2}\xb7\mathrm{K}}$ |

${\rho}_{con}=1920\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ | $\mathsf{\Delta}{x}_{con}=0.05\mathrm{m}$ | ${C}_{{P}_{con}}=835\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ |

${\u03f5}_{con}=0.94$ | ${k}_{con}=0.72\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}$ |

**Table A8.**Miscellaneous constant parameters related to thermal contact conductance, liquid water properties, irrigation rates, sky view factors, and the Hargreaves equation scale factor.

${R}_{{c}_{con,con}}=0.04\frac{\mathrm{K}\xb7{\mathrm{m}}^{2}}{\mathrm{W}}$ | ${R}_{{c}_{con,soil}}=0.06\frac{\mathrm{K}\xb7{\mathrm{m}}^{2}}{\mathrm{W}}$ | ${\dot{M}}_{soil}=6.57\xb7{10}^{-6}\frac{\mathrm{kg}}{\mathrm{s}}$ |

$SO{F}_{east}=0.6$ | ${\dot{M}}_{irrig}=1.06\xb7{10}^{-5}\frac{\mathrm{kg}}{\mathrm{s}}$ | |

$SO{F}_{west}=0.5$ | $H{G}_{scale,factor}=0.525$ |

ThicknessGlass = 4 mm | ThicknessSi = 0.5 mm | ThicknessCoat = 0.25 mm |

CpSPglass = 700 J/K*kg (quartz glass) | CpSi = 705 J/K*kg (silicon) | |

Cpcoat = 1900 J/K*kg (EVA) | EpsSPb = 0.85 for aluminum (originally 0.77) (Emissivity) | EpsSPt = 0.93 for glass (originally 0.93) (Emissivity) |

Tasp = 0.915 for glass (transmittance–absorbtance product) |

${k}_{q}=0.003$ | ${I}_{0}=13\frac{\mathsf{\mu}\mathrm{mols}\mathrm{photons}}{{\mathrm{m}}^{2}\xb7\mathrm{s}}$ | ${P}_{max}=20.3\frac{{\mathsf{\mu}\mathrm{mols}\mathrm{CO}}_{2}\mathrm{fixed}}{{\mathrm{m}}^{2}\xb7\mathrm{s}}$ |

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**Figure 1.**Categories of agrivoltaic arrays: (

**a**) interspersed PV arrays; (

**b**) greenhouse-mounted PV arrays (PVs shown as horizontal, but they can be appropriately inclined); (

**c**) stilt-mounted PV arrays [10] (open access article distributed under the Creative Commons Attribution License).

**Figure 2.**The test cell: (

**a**) exterior view of the south-facing side; (

**b**) interior view of the north-facing side.

**Figure 4.**Mass transfer nodes and interactions in the computational model. The white bar is the transparent polycarbonate, the brown bars are wood side walls, the tan bars are foam board insulation, the red rectangle represents planters, and the grey rectangle is the concrete block.

**Figure 6.**Solution flow chart. Maximum and minimum daily temperatures are model-predicted maximum and minimum daily temperatures for test cell air. All terms ‘mass’ refer to water mass.

**Figure 9.**Yearlong simulation results and experimental temperature data. “Exp sensor 1” refers to the temperature sensor located on the east side of the test cell base, and “Exp sensor 2” refers to the sensor on the west side. Periods of sensor interruption or snow cover, where data were linearly interpolated, are highlighted.

**Figure 10.**Model-predicted daily maximum temperatures (red asterisks) and minimum temperatures (circles) vs. experimental maximum and minimum temperatures for the same day.

**Figure 11.**Predicted RH and from measurement. Note: over-prediction occurred mostly during the winter and the under-prediction during the summer. Start date of 17 January 2020.

**Figure 12.**Model-predicted daily minimum RH based on the model-predicted cell temperature (diamonds) and the model-predicted minimum RH based on measured temperature (circles) plotted against measured minimum RH for the same day.

**Figure 13.**Example that shows improvement in RH using measured cell air temperature instead of the model predicted. Note: the change in temperature for the two identified points was 6.1 °C (11 °F) from that predicted by the model. 27 February 2020 to 7 March 2020 test period.

**Figure 14.**Predicted (pred) solar radiation on 5 October 2019 and experimental (exp) data for the same day. Sensors 1, 2, 3, and 4 are located west in the test cell, center in the test cell, east in the test cell, and ambient, respectively.

**Figure 16.**Expanded view of test-cell air temperature measurements compared with ambient. In general, wintertime test-cell air temperatures were greater than ambient.

**Figure 17.**Progression of the October-2019 plant-growth experiment: (

**a**) 15 October 2019; (

**b**) 20 November 2019; (

**c**) 19 February 2020; (

**d**) 26 March 2020.

**Figure 18.**Average aboveground plant dry mass in the test cell and control cell for each sampling since the start of the October-2020 growth test. The arrow indicates the point at which control cell plants died.

**Figure 19.**Parity plots of maximum and minimum temperature test-cell air values from the model cases versus the base-case results.

**Figure 20.**Net photosynthetic flux of CO

_{2}per day per unit area as found by using solar irradiance from the shadowing model. (

**a**) Shows the net photosynthetic flux within the test cell at the points, and (

**b**) shows the ambient. The zero line delineates a value of no net gain or loss.

**Figure 21.**Net photosynthetic gain of CO

_{2}per day per unit area as found using solar irradiance from measured solar radiation. This was found using an assumed clearness index of 1 for the whole year. The zero line delineates a value of no net gain or loss.

**Figure 22.**Distribution of solar radiation at the base of the test cell over a day (the summer solstice). The vertical axis is the ratio of the actual beam radiation incident on each discrete node surface to an unshaded node surface.

**Table 1.**Cases for the parametric study in which the calculations converged (only case 16 did not). Note: Case 14 had the same number of planters as in the base case. For this case the planter height was doubled.

Case | Change from Base Case | Figure 19 |
---|---|---|

1 | Base Case | - |

2 | Twice the length in the N–S dir. | a |

3 | Twice the glazing width in the E–W dir. | b |

4 | Case 2 with 7% load | c |

5 | Case 2 with 14% load | d |

6 | Case 2 with 21% load | e |

7 | Case 3 with 3% load | f |

8 | Case 3 with 14% load | g |

9 | Case 3 with 21% load | h |

10 | Two layers of concrete blocks | i |

11 | No concrete blocks | j |

12 | Twice the planters in the test cell | k |

13 | Four times the number of planters in the test cell | l |

14 | Twice the soil mass in the planters | m |

15 | Four times the soil mass in the planters | n |

16 | Twice the test-cell height | - |

**Table 2.**Percentage of the year each parametric run, the data from the test cell, and the ambient conditions spent within bands of ±1.1, ±2.2, and ±3.3 °C in relation to 20 °C (68 °F).

Case | % of Year in ±1.1 Band | % of Year in ±2.2 Band | % of Year in ±3.3 Band | % of Year below Freezing |
---|---|---|---|---|

Test Cell | 4.83 | 10.19 | 15.83 | 5.70 |

Ambient | 7.61 | 15.26 | 21.98 | 9.01 |

Base | 4.97 | 10.00 | 15.34 | 9.22 |

2 | 4.14 | 8.77 | 13.79 | 8.61 |

3 | 4.44 | 9.03 | 14.31 | 8.99 |

4 | 4.67 | 9.42 | 14.82 | 9.41 |

5 | 4.93 | 9.93 | 15.26 | 9.73 |

6 | 4.83 | 9.77 | 15.13 | 8.92 |

7 | 5.10 | 10.02 | 15.53 | 9.21 |

8 | 5.19 | 10.43 | 15.82 | 9.45 |

9 | 5.31 | 10.89 | 16.81 | 9.71 |

10 | 4.88 | 9.96 | 15.27 | 8.92 |

11 | 4.26 | 8.85 | 14.09 | 9.13 |

12 | 5.04 | 10.01 | 15.42 | 9.14 |

13 | 5.01 | 10.02 | 15.47 | 9.05 |

14 | 5.06 | 10.04 | 15.43 | 9.19 |

15 | 5.05 | 10.04 | 15.47 | 9.18 |

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**MDPI and ACS Style**

Evans, M.E.; Langley, J.A.; Shapiro, F.R.; Jones, G.F.
A Validated Model, Scalability, and Plant Growth Results for an Agrivoltaic Greenhouse. *Sustainability* **2022**, *14*, 6154.
https://doi.org/10.3390/su14106154

**AMA Style**

Evans ME, Langley JA, Shapiro FR, Jones GF.
A Validated Model, Scalability, and Plant Growth Results for an Agrivoltaic Greenhouse. *Sustainability*. 2022; 14(10):6154.
https://doi.org/10.3390/su14106154

**Chicago/Turabian Style**

Evans, Michael E., J. Adam Langley, Finley R. Shapiro, and Gerard F. Jones.
2022. "A Validated Model, Scalability, and Plant Growth Results for an Agrivoltaic Greenhouse" *Sustainability* 14, no. 10: 6154.
https://doi.org/10.3390/su14106154