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

Abstract

We developed an agrivoltaic greenhouse (a ‘test cell’) that partially trapped waste heat from two photovoltaic (PV) panels. These panels served as parts of the roof of the enclosure to extend the growing season. Relative humidity, internal air temperature, incident solar radiation, wind speed, and wind direction were measured for one year. A locally 1-D transient heat and moisture transport model, as well as a shadowing model, was developed and validated with experimental data. The models were used to investigate the effects of altering various parameters of the greenhouse in a scalability study. The design kept test cell air temperatures generally above ambient throughout the year, with the test cell temperature below freezing for 36% less of the year than ambient. Plant growth experiments showed that kale, Brassica oleraceae, a shade-tolerant plant, can be grown within the test cell throughout the winter. The simulations showed that enlarging the greenhouse will increase cell air temperatures but that powering an electric load from the PV panels will reduce cell air temperatures.

1. Introduction

Local and national governments and private industry continue efforts to expand the deployment of renewable energy sources. A central pillar of renewable energy generation is utility-scale photovoltaic (PV) solar farms. These farms require large amounts of land with high solar irradiance. Growing global populations also require more food, which necessitates more flat land with high solar irradiance for growing crops. The needs of solar energy production have prompted significant debate and even legislative action on the uses of high-quality agricultural land [1,2,3]. Some local communities and governments have already adopted rules limiting the development of solar farms [1,2], but some farmers prefer renting their land to solar companies instead of growing crops, as the income can be more reliable [3]. Agrivoltaics may allow farmers to avoid the conflict by combining PV arrays and agriculture in a way that preserves the yield of both.

Different ideas for combining PV arrays and agriculture have been proposed, but they generally fall into the three categories shown in Figure 1: interspersed PV arrays, greenhouse-mounted PV arrays, and stilt-mounted PV arrays [4,5,6,7,8,9,10]. For interspersed PV arrays, the PV panels are mounted between rows of plants [2,4]. Greenhouse-mounted PV arrays use the framework of a traditional greenhouse to mount the panels on the roof, although the position and orientation of the solar panels may be sub-optimal to accommodate needs of the greenhouse. The greenhouse may also take advantage of primary (electricity) or secondary (heat) products of the panels. Finally, stilt-mounted PV arrays position the panels at a desired height above an otherwise open field, with plants growing under them. The space beneath allows access to the crops by farming machinery.

In [5], Joudi and Farhan replaced a greenhouse roof with a solar air heater panel to partially block light during warmer months and remove heat from the system. It also absorbed light during colder months and redirected heat into the greenhouse. The greenhouse maintained a temperature 16 °C higher than ambient during the winter and 10 °C cooler than ambient during the summer, which showed that the solar air heaters could reduce the cooling load in the summer and support the heating load during the winter.

Marucci et al. [6] placed PV panels in a checkerboard pattern across the roof of a Quonset-type greenhouse. The purpose of their study was to examine the effects of shading by the panels. Another design by Cossu et al. [7] replaced the covering material of a traditional greenhouse with a version of semi-transparent PV panels (STPs). The STPs in this design were interconnected PV cells pressed between two sheets of glass. It was found that the STPs could only partially offset the energy costs of the greenhouse.

There are multiple examples of large commercial greenhouses being converted into agrivoltaic greenhouses by replacing parts of the pitched roofs with PV panels [8,9]. The fraction of roof covered by the PV panels varies, but these designs have the same general limitations. They are quite large, and the shading caused by the PV panels can reduce crop yields inside the structures. Finally, structures that have already been built are difficult or expensive to retrofit.

A stilt-mounted PV array, like the system shown in Figure 1c, was developed by Sekiyama and Nagashima [10]. They compared the effects of a low-density configuration (panel spacing of 1.67-m apart) and a high-density configuration (panel spacing of 71-cm apart) on the growth of corn, a shade-intolerant crop. The panels were mounted at a height of 4 m above the ground, high enough for a tractor to pass under. Sekiyama and Nagashima found improved crop yield for corn grown beneath the low-density configuration and a slightly reduced yield for corn grown beneath the high-density configuration. The configuration of Figure 1c does not extend the growing season due to the lack of a heat-trapping enclosure.

Vanthoor suggested that a comprehensive greenhouse model should consider internal temperature, relative humidity (RH), and CO₂ 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, CO₂ 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 CO₂ 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.

Tiwari et al. [16] considered heat loss from the floor to the ground but handled it as steady-state heat transfer, unlike the rest of the model, which was quasi-steady state. They also considered solar radiation to be partially absorbed by the plants, an effect we neglected by using a model that addresses the overall impact of the solar gain and evapotranspiration. Abdel-Ghani 2011 [17] took an entirely different modeling approach, treating a greenhouse as a solar collector and assuming that elements such as coverings, air, and plant canopies have radiation properties, such as transmittance, absorbance, and reflectance.

There are several differences in our model compared with the above. We chose to neglect CO₂ concentration and the effects of the plant canopy inside the greenhouse (the test cell), because CO₂ 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.

Unlike [16], we modeled transient transfer from floor to ground and included thermal contact conductance between concrete blocks and the ground, as well as contact conductance between possible multi-layers of concrete blocks and modelling transient heat transfer from floor to ground. No individual greenhouse model we examined considered all the heat transfer modes included in this work, although several considered other factors that we neglected, such as CO₂ concentration.

The test cell (Figure 2) has a 20.3-cm (8-in) window of transparent polycarbonate at the top placed between two PV panels, which allows light into the area below and allows plants to grow at the floor level. The PV panels transfer heat by free convection and thermal radiation to the cell inside surfaces when exposed to solar radiation on the outside. The heat is partially trapped in the enclosed space, increasing the cell air temperature. At each instant in time, the value of the cell air temperature results from an energy balance, which includes all cell surfaces and masses, the thermal resistances of the cell walls, and radiation and forced convection on all outer cell surfaces. Due to the traditional “greenhouse” effect (of poor thermal radiation through the polycarbonate from the cell inside surfaces), the energy balance maintains internal temperatures at above the ambient throughout the day, extending the growing season without the use of heaters during periods of cool ambient temperature.

2. Materials and Methods

The test cell used two 0.7 m × 2 m PV panels donated by Solar States LLC [18]. The two panels were positioned facing east and west at 35.5° from the horizontal and were held in position by two wooden frames constructed from 3.81 cm × 8.9 cm (a U.S. “2 × 4”) wooden beams. As shown in Figure 2, the frame has two vertical beams about 78-cm-long, spaced 15.2-cm-apart, and two horizontal beams of about 1.1 m. There is an approximately 20.3-cm gap between the upper edges of the two PV panels covered by a transparent glazing of 1.6-mm-thick transparent polycarbonate.

The north- and south-facing sides of the enclosure are sealed by plywood walls. The north walls are supported by brackets, and the southern walls are held in place by hinges and locks that allow interior access. The test cell is insulated by R-10 foamboard insulation mounted on the inside of the north- and south-facing side walls.

As shown in Figure 2b, the base of the test cell was covered with low-density concrete blocks of density 1920 kg/m³, 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.

The temperature and RH sensors collected data every 5 min, while the pyranometers and wind speed and direction sensors collected data every minute and record five-minute averages. Approximately every 55 days, data were collected, and sensors were reset. The data sets were concatenated to produce a full year data set from 17 January 2020 to 17 January 2021. Gaps in data caused by brief instrumentation failures and brief periods when snow covering rendered the data invalid were filled using linear interpolation.

Three plant growth experiments were carried out in the test cell. In each experiment, two Italian heirloom Toscano Lacinato kale (also known as “dinosaur” kale) seeds (Brassica oleraceae) were planted together six-millimeters-deep in potting soil. Pairs of Toscano kale seeds were about 3.8-cm apart, with six sets of seeds in each planter. Plants within the test cell were watered by a gravity-fed irrigator. Control plants growing adjacent to the test cell and in a conventional greenhouse were watered by hand every few days. Samples of soil were taken for gravimetric water content analysis to assure all plants were receiving sufficient and approximately equal soil moisture.

In the October-2019 experiment, four planters were placed in the test cell, and one planter was placed outside as a control. In the October-2020 growth experiment, four planters were placed in the test cell, four planters were placed outside as a control, and four planters were placed in a nearby conventional greenhouse. Finally, in the March-2021 growth experiment, two planters in the test cell from the October-2020 growth experiment were replaced with new planters, and four new planters were placed outside as controls.

During the October-2020 growth experiment, the total leaf area of sample plants from the test cell and the control was estimated by measuring leaf length and width and assuming an ellipsoid shape for the leaves. The same data were collected for the control plants in the greenhouse; however, those plants were then taken and dried at 62.8 °C (145 °F) in an oven for 24–48 h to estimate dry mass. An allometric function (discussed in Appendix B) correlated total leaf area to plant dry mass. This function was used with the measurements of leaves from the test cell and the adjacent control plants to determine the average dry mass of the plants and thus compare the dry mass of the kale produced in the test cell, greenhouse, and control test.

During the third growth experiment, the same measurements as those in the second growth experiment were taken. These new plants were started in March 2021 in the test cell and in the control area. The allometric function found previously was used to compare the dry mass of kale produced in the test cell and in the control area.

3. Mathematical Modeling

Three models were developed to help understand the relative contributions of the heat and mass transfer processes in the greenhouse and to predict the effects of changes in the test cell design parameters (i.e., scalability) in sensitivity studies. The first was a locally 1-D thermal model with temperature nodes throughout the system, based on an energy balance at each node. The moisture transport model, like the thermal model, had nodes throughout the system and was based on a mass balance at each node. Thermal and moisture transport interact through the latent heat of water, so their equations must be solved simultaneously. The thermal model had a strong effect on the moisture transport model results, but the moisture transport model only weakly affected the thermal model. This interaction is discussed below.

The third model predicted shadowing of the beam and diffuse solar radiation and used the geometry of the test cell and the sun’s transit across the sky to predict which parts of the test cell base were shadowed.

In the thermal model (Figure 3), several processes act at each node. Some nodes experienced incident solar radiation, such as the nodes on the outside of the solar panels. This was found from measured values of surface area and solar radiation flux. Values for the angle of incidence and transmissivity (for the transparent polycarbonate, its absorptivity) were calculated from the instantaneous solar positions and published data. The solar panel nodes also experience radiation heat exchange with the sky, which was found using calculated values of node temperature, sky temperature, and the radiation view factor. Solar radiation transmitted through the transparent polycarbonate was calculated based on the transmissivity of the material and was assumed to be distributed uniformly over the internal surfaces. Forced convection due to external wind occurred at all external surface nodes and was calculated using the measured wind speed and air temperature and the calculated surface temperatures. Free convection occurred at all internal surface nodes. Radiation heat transfer also occurred at all internal nodes and was calculated using the net radiation method (Appendix C).

Air in the test cell is coupled with the ambient air by conduction through the north- and south-facing side walls, as well as through infiltration through gaps in the side walls. Heat flow due to conduction through the side walls was determined using measured ambient and cell air temperatures and a calculated wall thermal conductance. All properties, such as thermal conductivity, were taken from published values. Heat flow due to infiltration was found using the measured ambient air temperature and calculated cell air temperature. The infiltration air flow rate was determined using measured values of wind speed and direction to calculate the stagnation pressure on the outside of the south-facing wall. In addition, infiltration due to the stack effect was calculated from cell air temperature and measured ambient temperature. Conduction in the concrete and soil was modeled by numerically solving the 1D, transient heat conduction equation. The resulting linear algebraic equations were solved implicitly using a matrix method. At the concrete soil interface, the heat flow due to thermal contact conductance was found from calculated temperatures and a thermal contact resistance, the value of which was determined during model validation (by comparison with measured data).

Key assumptions in the models are as follows:

• 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.

Governing equations for the thermal model are given below. The heat flow rate equations for Equations (1)–(10) are discussed in Appendix C, along with mass flow rates from Equations (11)–(18).

dT_lex/dt = (q_sol,lex + q_wind,lex + q_sky,lex + q_a,lex − q_rad,lex)/ρ_lexV_lexC_plex

(1)

dT_sp,out/dt = (q_sol,sp + q_wind,sp + q_sky,sp + q_{cond,sp,inner})/ρ_spV_spC_Psp

(2)

dT_sp,in/dt = (q_a,sp-q_rad,sp + q_{cond,sp,outer} + q_ligh,sp)/ρ_spV_spC_Psp

(3)

dT_cell/dt = (q_sp,west,a + q_sp,east,a + q_lex,a+q_con,a + q_sidp,a-q_infi)/ρ_aV_aC_Pa

(4)

dT_con,up/dt = (α_con/Δx²_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])

(5)

dT_i,con/dt = (α_con/Δx_con²)(T_i,con+1 − 2T_i,con + T_i,con-1)

(6)

dT_con,low/dt = (q_cond,con + q_tcc,con)/ρ_conA_con[Δx_con/2]C_Pcon

(7)

dT_soil,up/dt = (q_cond,soil + q_tcc,soil)/ρ_soilA_soil[Δx_soil/2]C_Psoil

(8)

dT_i,soil/dt = (α_soil/Δx_soil²)(T_i,soil+1 − 2T_i,soil + T_i,soil-1)

(9)

dT_soil,low/dt = (α_soil/Δx²_soil)(2T_soil,low-1 − 2T_soil,low)

(10)

In Equations (7) and (8), we assumed there is no heat storage in the interface. Equation (10) is a variant of Equation (9) and predicts temperature changes in the lower soil surface layer, which is insulated.

It is assumed that liquid water can condense and collects on the concrete base and that this liquid water can subsequently evaporate. Three water mass nodes are considered, one each for the air, the accumulated liquid water, and the planter (Figure 4). In the moisture transport model, a mass balance is written for each node, just as is done for heat in an energy balance.

Condensation begins when the humidity ratio of air reaches the saturation humidity ratio. When the humidity ratio is less than the saturation humidity ratio, evaporation occurs.

The equations used in the moisture transport model are as follows:

dM_a/dt = Mdot_infi,h2o + ET_plant + E_liq

(11)

dM_a/dt = Mdot_infi,h2o − M_cd

(12)

dM_liq/dt = −E_liq + Mdot_soil

(13)

dM_liq/dt = M_cd(Acon − A_plant)/A_con + Mdot_soil

(14)

dM_plant/dt = Mdot_irrig − ET_plant

(15)

dM_plant/dt = Mdot_irrig + Mdot_cd(A_plant/A_con)

(16)

Equations (11)–(15) are used when evaporation occurs, and Equations (12)–(16) are for condensation. ET_Plant is evapotranspiration from the plants. It is an adapted form of the Hargraves equation [20] and is discussed in Appendix C.

A separate set of equationsgoverns the overflow of water from the planters when the water in that node exceeds the maximum capacity of the soil. If M_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).

dM_over/dt = (M_plant − M_plant,max)

(17)

dM_liq/dt = dM_liq/dt + dM_over/dt

(18)

The thermal and moisture transport models are solved using the algorithm below in Matlab (Mathworks, Natick, MA, USA) with the integration package ODE15s. See Figure 5 and Figure 6.

The equations for the shadowing model are presented below [21]. The hour angle (ω = 0 is assumed to occur at midnight) and cell-based surface azimuthal angle are defined by

ω = −180° + 1.25°p

(19)

sin(α_s) = cos(φ) cos(δ) cos(ω) + sin(φ) sin(δ)

(20)

The shadow projection point SP is the distance in the north–south direction, measured from the base of the south-facing side wall (see Figure 7). It determines the length of the shadow projected by the sidewall. The value of cos(ω) was found from [22]. A point must be at a distance greater than SP from the south-facing side wall to be sunlit.

SP = cos(ω) H/tan(α_s)

(21)

A point on the base must also be between two different hour angles, defined as ω_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.

ω_east = atan(h₁/y₁) − 90°

(22)

ω_west = atan(h₂/y₂) + 90°

(23)

The sunrise and sunset angles are

ω_set = acos(−tan(φ)tan(δ))

(24)

ω_rise = −ω_set

(25)

Once the values calculated in Equations (19)–(25) are known, the algorithm to determine if a point at the base of the test cell is shaded is as follows. First, a check is made to determine if the current hour angle in the simulation is before sunrise or after sunset (if so, no points are illuminated).

For daytime, a check is performed to determine whether a point on the base is within the projected shadow of the side panel. If so, then the point is not illuminated.

Finally, if the point is not within the projected shadow, a check is made to see if the current hour angle is between the east and west blocking angles for that point. If it is, then the point is irradiated (if not, then it is shaded). Note that the shadowing model only considers beam radiation. The model is illustrated in Figure 7 and Figure 8. Values of constants used in the above models are presented in Appendix D.

4. Results

The models were validated by comparing the calculated minimum and maximum test cell air temperatures with measurements and, by trial-and-error, adjusting key parameters that were identified as most uncertain. These included the area of the air infiltration gap, irrigation flow rate, ground-source moisture infiltration rate, Hargreaves Equation scaling factor, the thermal contact resistances (between the soil, first concrete layer, and second concrete layer), the coefficients of free convection (between the cell air and the concrete, as well as the cell air and every other internal surface), the SOF factors for both solar panels, and the transmittance-absorptance for the solar panels. Once validated, measured data were used as inputs to the models in a parametric study to predict performance of the test cell subject to a variety of changes in dimensions, configurations, and other operational parameters.

4.1. Validation

4.1.1. Thermal Model

Figure 9 and Figure 10 show that the predicted cell temperatures were in good agreement with experimental data. Predicted and measured daily maximum temperatures differed by an average of 1.9 °C (3.5 °F), and predicted and measured daily minimum temperatures differed by an average of 1.5 °C (2.7 °F).

4.1.2. Moisture Transport Model

When comparing the moisture transport model with measurements, the predicted relative humidity (RH), seen in Figure 11 and Figure 12, showed a larger disagreement than that in Figure 9 and Figure 10. The predicted RH was noticeably greater during the winter than the observed RH, and opposite was the case during the summer. Cell air temperatures have a more pronounced effect on the RH values during the winter. From psychrometrics in classical thermodynamics, RH is inversely proportional to the saturation vapor pressure (p_g(T)) at the air temperature of the test cell.

RH = λp_a/0.662 p_g(T)

(26)

Consider the case of air temperature at −10 °C (14 °F), where p_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.

Figure 13 compares model-predicted RH values calculated using both the model predicted and measured temperatures. The RH calculated using measured temperatures was significantly closer to the measured RH. The high sensitivity of RH to temperature, as seen in Figure 13, is due, at least in part, to the high RH at low temperatures, as noted above.

Another possible reason for the differences in Figure 13 is the use of a constant irrigation rate in Equations (15) and (16), while the actual irrigation rate probably fluctuated due to interruptions caused by winter freezing (We selected a constant value for the irrigation rate by first finding a piecewise irrigation rate function that produced good agreement like that shown in Figure 11, and then finding the average value of this function over the course of a year.). We also assumed constant rate of water infiltration to the test cell from the moist soil below concrete blocks, although this value was uncertain and likely varied with time.

4.1.3. Shadowing Model

The shadowing model was validated by comparing the solar radiation data collected on a near completely clear day with that predicted by the shadowing model at the locations of the sensors. Figure 14 shows excellent agreement between the model and the experimental data. Late in the day, the model over-predicted the incident radiation, possibly due to light cloud cover or atmospheric haze near the horizon.

4.2. Experimental Data

4.2.1. Test-Cell Temperature and Ambient Temperature

Figure 15 shows that daytime temperatures inside the test cell were greater than the ambient temperature for most of the year. Furthermore, as shown in Figure 16, even near day 350 and approaching the winter solstice, the minimum temperatures observed at night were typically higher than ambient. This occurred throughout nearly a year of observations.

4.2.2. Plant Growth

The October-2019 plant growth experiment is shown in photographs taken weekly over the course of the experiment.

Figure 17 shows the progression of plant growth for kale in the test cell. The kale was planted in early October 2019 and grew slowly over the course of the next few months. In February 2020, plant growth accelerated, presumably due to improving conditions, and continued in March 2020.

Allometric measurements were used to find the average dry mass of the plants in the test cell and control cell throughout the October-2020 growth experiment. The allometric function is discussed in Appendix B.

Figure 18 shows the same progression as Figure 17. Plant growth was slow over the course of the winter, and during this time, the control cell plant’s average dry mass was greater. At the end of the winter, the plants in the test cell caught up with those of the control cell plants and resumed growth. The control cell plants, which were outside the test cell and unprotected, died on 21 January 2021 (six weeks after data collection started) due to exposure to snow and prolonged freezing temperatures.

4.3. Numerical Results

4.3.1. Parametric Study

A parametric study using the above models was undertaken to evaluate test cell scalability, where we interpreted scalability as the performance of the cell under conditions different than those of the base case. The base-case parameter values are defined in Appendix D, and variations of the parameter values from those of the base case appear in

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	-

. The base-case parity plot appears in Figure 10.

The results of the parametric study are presented in Figure 19. The maximum and minimum temperatures predicted by the model for each day of the year and those predicted by the base case are compared in each parity plot.

Based on the results in Figure 19, the following conclusions could be reached regarding the test-cell scalability.

• 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 m²/s and α_soil = 0.99 · 10⁻⁶ m²/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.
  • Summertime temperatures in Figure 19k,l are similar to those in Figure 19a, because in all three cases, all of the water contained in the liquid water node evaporated. Thus, with no water left in the liquid water node, evaporative cooling was reduced in all three cases.
• Figure 19m,n show similar behavior to that in Figure 19k,l, respectively. We suspect this is because the planters have a much higher potential water storage, and thus, less water spills into the liquid water node during the test period.

4.3.2. Instantaneous Photosynthetic Rate (P_n)

Our plant growth experiments confirmed that kale can be grown in the test cell. Using the shadowing model and the equation to calculate P_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 CO₂ then needed for cellular respiration. P_n = 0 means that a leaf is fixing the necessary amount of CO₂, and P_n < 0 means that a leaf is not fixing enough CO₂. The instantaneous value of P_n can be found using the Mitscherlich equation [24].

P_n(I) = (1 − e^{−k(I−I₀)}) P_max

(27)

where P_n is the instantaneous photosynthetic rate; P_max is a constant (the maximum photosynthetic rate of 20.3–21.0 µmols CO₂ fixed/m² s [24]); I₀ is the PAR irradiance at the compensation point where the P_n is equal to zero (which, for kale, is 13 µmols photons/m² s = 2.85 W/m²); 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/m² to µmol photons/m² 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.

The net photosynthetic gain per day is shown for the irradiance predicted by our shadowing model (Figure 20) and the measured irradiance (Figure 21).

Figure 20 shows that the only viable location for plant growth in the test cell is the area near the centerline between the east and west side of the test cell (dashed green line). If plants are placed close to the east (dashed blue line) or the west side (dashed red line), they would not be able to grow, because the net photosynthetic gain there is always negative. Clearly, the net photosynthetic gain inside the test cell was found to be substantially lower than outside, as expected.

The results based on solar radiation measurements (Figure 21) show that throughout most of the year, leaves would be able to grow in the center of the test cell (solid green line), as well as the east (solid blue line) and western sides (solid red line) of the test cell. The net photosynthetic gain values were higher than those observed in Figure 20 for the interior and exterior points. The solar radiation measurements included diffuse radiation, while the model-predicted solar radiation included only beam radiation.

4.3.3. Ideal Temperature Zones

Kale grows optimally at a temperature in small band centered around 20 °C (68 °F) [25]. Temperatures above freezing are desirable for nearly all plants. An analysis was conducted to determine how much of the year our test cell spent within three temperature bands surrounding the ideal temperature, based on parametric simulations and the ambient conditions (

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

). Another analysis was conducted to determine how much time the test cell, the ambient conditions, and our parametric simulations spent below the freezing point (Table 2).

A brief examination of Table 2 shows a few trends discussed here.

• 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.

The reason that the predicted temperatures were below freezing for more of the year than the measured temperatures is likely because the model underpredicted nighttime temperatures by an average of 1.5 °C (2.7 °F). As a result, the predicted temperatures may have fallen just below freezing at times when the experimental values were higher than 0 °C (32 °F). For example, if the measured temperature was 1.1 °C (34 °F), the model may have predicted −0.4 °C (31.3 °F).

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

Using the shadowing model developed in this work, the solar radiation transmitted to the cell base could be determined and compared with that in other agrivoltaic designs. A stilt-mounted PV array was selected (refer to Figure 1c), and the shadowing model was used to determine the fraction of solar beam radiation transmitted to the base beneath each array as compared to an area equal to the floor of the test cell.

Two types of stilt-mounted PV arrays with geometries from [10] were considered. The low-density stilt-mounted PV array in [10] is composed of 4 PV panels placed 1.67-m apart. The high-density array had 8 PV panels, positioned 0.71-m apart. The calculation was for the summer solstice. The low-density panel configuration of the stilt-mounted PV array allowed 82% of radiation to reach the base beneath the panels, compared to an unshaded area. The high-density configuration allowed 65% to reach the base beneath the panels. By comparison, the test cell allowed 5.2% to reach the base beneath the solar panels (Figure 22).

5. Discussion

The present work focused primarily on evaluating the viability of our test cell, developing a computer model of the system, and on identifying potential improvements and scalability through a parametric study. The project has produced promising results and highlighted several areas where the design or models could be improved.

Based on the results presented in Section 3, we were able to grow kale in the test cell. However, only plants located directly under the glazing grew well. Those located more east and west in the test cell did not grow as well. However, our test cell also kept kale alive throughout the winter and allowed it to resume growth with the arrival of spring. Temperatures within the test cell were greater than those outside (ambient) for almost the entire year, which confirms our hypothesis that the heat produced by the PV panels could be used to augment the normal greenhouse effect and extend the growing season of crops. Based on the net photosynthetic gain (Appendix B) analysis of the interior of the test cell, it is clear that kale leaves can grow within the test cell, despite the lower light compared to other agrivoltaic designs. The test cell may be suitable for other shade-tolerant plants, such as arugula, lettuce, carrots, or potatoes. Trials with multiple plant species should be carried out. Furthermore, if the glazing were removed and the panels completely sealed against solar radiation, then crops such as mushrooms may grow quite well.

The proposed design would be difficult to use with traditional farming techniques. Possible modifications include increasing the height of the test cell to allow more comfortable access to the interior and the integration of automated crop-tending systems.

One option for greater sunlight transmission is to widen the glazing. However, as demonstrated in Section 4.3.3, this would increase cell air temperatures. Based on results from Section 4.3.3, drawing electric power from the PV panels decreases internal temperatures. If the glazing area was widened, then more light would be transmitted to the interior of the test cell, and if the PV panels had at least an efficiency of 14%, then no significant increase in test-cell air temperature would occur. An efficiency of 21% would lead to a reduction in daytime temperatures throughout the summer. The drop in test-cell temperatures during the day would keep the test-cell air closer to the ideal temperature of 20 °C (68 °F).

Several improvements to the computer models have been identified. Including temperature-difference-dependent free convection heat transfer coefficients would improve the fidelity of the heat transport model (constant values are currently in place based on assumed average thermal conditions). Improving the heat transport model would also improve the moisture transport model by increasing the accuracy of the RH calculation (see Section 4.1.2 and Equation (26)). The addition of a flow meter to the irrigation system would produce data that could further improve the moisture transport model. Adding diffuse radiation [27] would improve the accuracy of the shadowing model, especially in overcast climates. The effect of thermal earth coupling should be improved by modeling the ground as a two-dimensional (depth direction and radial outward direction normal to this), transient thermal conductor.

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

One of the next steps in this work is to produce a model that incorporates the thermal and humidity transport models above with a model for plant growth rate. The most comprehensive of the latter is STICS [28,29,30,31]. Once completed, the agrivolatic greenhouse can be optimized by maximizing the monetary value of the two outputs, namely the value of the electricity produced as predicted by the solar input and PV efficiency from above and the value of the wet mass of the agricultural product [24,32]. These two outputs are predicted by the plant growth from STICS subject to changes in greenhouse geometry, construction materials, plant types, numbers and locations in the greenhouse, and environmental conditions. Clearly, STICS requires values for many input parameters that we have not considered, since growth-rate models were beyond the scope of the present study. One influencing growth-rate factor is the CO₂ 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 CO₂ concentration equal to about 400 ppm, that of the outside ambient air [24], which ignores the CO₂ contribution from soil respiration.

7. Comment on Scalability

Scalability, the extent to which the models may be accurately applied to agrivoltaic greenhouse designs of different sizes, dimensions, and materials enters the problem in several ways. For the thermal and moisture transport models, constant values of the convective heat transfer coefficients (h), radiation surface properties, and radiation view factors were used based on fundamentals from the heat transfer literature and were adjusted slightly during model calibration as described above. Good scalability may be assured if the designer refers to these fundamentals in carrying out the design. For example, free convection (in addition to radiation) heat transfer occurs between the inside of PV panels and the concrete block on the test-cell floor. Correlations for h exist in the literature for the geometry of the test cell and the intensity of the convection (i.e., laminar or turbulent flow). These should be sought out and used in the design. The same holds for radiation view factors and the need for accurate radiation surface properties for test-cell materials. The equations for the shadowing model and the model for heat transfer in the concrete block and soil mass are both general (no correlations are used) and will scale without any restrictions. Scalability of the relative humidity model is more challenging as we used the Hargreaves equation for evapotranspiration of water from the planters with a correction factor (0.525) determined by comparing with measured humidity data. We have no data to suggest the general nature of this correction factor. See Equation (A10).

8. Conclusions

Our test cell has shown promising results, and further work on developing its design is already underway. Low-light-tolerant plants, such as kale, can be grown in the test cell during extended time periods outside of the normal growing season. Light levels should be increased to improve the instantaneous photosynthetic rate. A design change such as widening the glazing area between the panels would have such an effect. The test cell reduces the number of hours the cell air spends below 0 °C (32 °F) by 36.6%, as compared to ambient. Powering an electrical load from the PV panels will generally improve growing performance by keeping temperatures near 20 °C (68 °F) for more of the year. More importantly, the load tends to reduce daily maximum temperatures, especially during the summer, but has a negligible effect on daily minimum temperatures. Days with lower maximum temperatures, like those during the winter, are also affected less.

Stilt-mounted PV arrays, as seen in [10], block much less sunlight than the panels in our test cell. However, stilt-mounted PV arrays do not extend the growing season, as the plants grow in their normal ambient environment. One of our major objectives was to extend the growing season, which stilt-mounted PV arrays cannot do.

Based on the net photosynthetic gain each day per unit area shown in Section 4.3.2, it is possible to grow kale in the test cell, because plants inside it receive sufficient PAR irradiance to produce a net gain in energy and biomass on most days. When both beam and diffuse radiation are considered, it is clear that most of the base area can be productively used for plant growth. The best growing area is along a line running north–south, which is centered between the east and west sides of the test cell.

Increasing the panel area results in an increase in cell temperatures during periods of intense solar radiation over those seen in the base case. This increase in temperatures can be counteracted, in full or in part, by powering a load from the panels. The same behavior is expected if the glazing area between the panels is doubled. If the system is scaled up for use in agricultural land, and the glazing area is increased, it is expected that the system will function as intended and that temperatures in the test cell will remain above ambient throughout the year.