Electronic Appendix of the Manuscript Entitled:
Solar Greenhouses Can be Promising Candidate for CO2 Capture and Utilization: Mathematical Modeling
Mohammed B. Effat*, Hamdy M. Shafey, A. M. Nassib
Mechanical Engineering Department, Assiut University, 71516, Assiut, Egypt
Mobile number: +2-010-01099663
This appendix includes the details of all terms included in the governing equations of the proposed model of the present study. An overview of the model, its basic assumptions, and its governing equations are provided here at first for consistency. The appendix illustrates the expressions of the energy storage term of the greenhouse inside air and convective heat transfer and mass transfer specific rates existing in the mass species and energy balance equations of the model. Then the appendix illustrates in detail the treatment of solar radiation in the greenhouse cover, canopy and soil, and the thermal radiation exchange between the different components of the greenhouse. It presents in detail the mechanistic photosynthesis biochemical sub model considered in the present study to simulate the CO2 capture and utilization process. This sub model is applicable to all plant types of the C3 species giving power to the main model to simulate the capturing performance of greenhouses considering any plant type. The appendix also includes the details of the CO2 enrichment strategy developed in the present study to keep the CO2 concentration inside the greenhouse air nearly constant around the required high value within small specified margin. The appendix also includes the details of the strategies developed in the present study to control the greenhouse air temperature and relative humidity to the required favorable level. Finally, the method of solving the model equations is presented and the values of model inputs and parameters used for obtaining the results of the present study are provided in a table at the end of the appendix.Nomenclature
Ac,Aj,Aph / Carboxylation, Light saturation, and sucrose transport limits of photosynthesis, respectively, mol CO2m-2 s-1
An / Net assimilation specific rate, mol CO2m-2 s-1
Ax / Surface area of greenhouse component x, m2
A1 / Projected area where solar radiation enters the greenhouse but will not pass out from the another side of the cover, m2
A2 / Projected area where radiation enters and will pass out from the another side of the cover, m2
b / Value of stomatal conductance when photosynthesis flux of leaf is equal to zero, mol m-2 s-1
C, Ci / CO2 concentration inside the greenhouse and in intercellular air spacemol /mol air, respectively
Cd, Cw / Drag coefficient andwind coefficient, respectively.
Cp,air,Cv,air / Specific heat of dry air at constant pressure and constant volumeJ/kg.K, respectively
Cx / Specific heat of greenhouse component x other than air, J/kg.K
Eb-x / Black body emissive power of the greenhouse component x, W/m2
EB, EQ / Activation energies for Kc, Ko, , and Vcmax0, Jmax0 J/mole, respectively
esat / Saturation water vapor pressure, kPa
Fi-j / View factor from surface i to surface j.
f / Fraction of absorbed PPFD unavailable for photosynthesis.
G / CO2 injection specific rate for enrichment, mol CO2 m-2 s-1
g / Gravitational acceleration, m/s2
gs / Stomata conductance for CO2 transfer, mol air/m2.s
gs,wv / Stomata conductance for water vapor transfer, mol air/m2.s
H / Greenhouse height, m
Hd / Deactivation energy, J/mole
h / Enthalpy of the humid air per unit mass of dry air, kJ/kg
h` / Distance between sidewalls and roof openings, m
hsolid-fluid / Convective heat transfer coefficient between a soild surface and a fluid, W/m2.K
/ Leaf boundary layer conductance for CO2 transfer , m/s
/ Canopy boundary layer conductance for water vapor transfer, m/s
/ Canopy conductance for water vapor transfer, m/s
/ Total canopy conductance for water vapor transfer, m/s
hfg / Latent heat of vaporization, J/kg
hg / Enthalpy of saturated water vapor, J/kg
hmass / Convective mass transfer coefficient between the humid air and the inner surface of the cover, m/s
hw,bl / Boundary layer conductance for water vapor at canopy level, m/s
hw,c / Canopy conductance for water vapor transfer, m/s
hwv / Total conductance of water vapor transfer, m/s
I / The incident amount of photons in the visible spectrum of solar radiation, mol photonm-2 s-1
J / Electron flux potential, mol electronm-2 s-1
Ji / Radiosity of surface i, W/m2
Jmax / Maximum (biochemical capacity limited) potential of the electron transport flux, mol electronm-2 s-1
Kc,Ko / Michaelis-Menten constants for carboxylation and oxygenation, respectively, mol/mol
k / Thermal conductivity, W/m.K
L / Greenhouse length, m
Le / Lewis Number
LAI / Leaf area Index
l / Depth of the greenhouse soil, m
Mair / Molecular weight of the greenhouse air, kg/mole
m / Optical air mass
mwv / Coefficient that depends on the plant type and ranges from 8-16.
m`` / Mass transfer specific rate, kg/m2.s
/ Molar specific rateaccounting for the loss of CO2 by ventilation, mol m-2 s-1
O2 / Oxygen concentration in the intercellular air space inside the leaf, mol O2/mol air
Pair-dry / Dry air pressure, kPa
P0 / Atmospheric air pressure at sea level, kPa
q``solid-fluid / Convective heat specific rate between the humid air and the cover inner surface, W/m2
q``cool / Energy specific rate accounting for cooling of the greenhouse by ventilation or other alternative cooling method, W/m2
R / Universal gas constant, J/kg.K
Rair / Specific gas constant of dry air, J/kg.K
Rbl / Boundary layer resistance for CO2 transfer, m2.s /mol air
Rd / Dark respiration flux,mol CO2 m-2 s-1
Rd,vis/NIR / Direct visible and near infrared solar radiation fluxes, respectively, W/m2
Rdf,vis/NIR / Diffuse visible and near infrared solar radiation fluxes, respectively, W/m2
Ri / Surface resistances, m-2
Ri-j / Geometric resistances between surface i and j, m-2
Rs / Stomatal resistance for CO2 transfer, m2.s /mol air
Rs-x / Solar radiation specific rate absorbed by the greenhouse component x, W/m2
Rx-y / Net thermal radiation energy specific rate exchanged between surface x and surface y of the greenhouse, W/m2
R0,vis/NIR / Extraterrestrial visible and near infrared solar radiation fluxes, respectively, W/m2
RH / Relative humidity.
rhs / Relative humidity at the leaf surface.
Tx / Temperature of the greenhouse x component, K
t / Time, s
u, ug / Specific internal energy of the greenhouse air and dry saturated water vapor J/kg, respectively
Vcmax / Maximum rate of Rubisco activity for carboxylation,mol m-2 s-1
W / Greenhouse width, m
Xh / Fraction of the area of holes of the perforated plate to the plate area
j / Tilt angle of sub surfaces of the greenhouse cover, rad.
/ Thickness, m
` / Operator that has a value of 1 for amphistomatous leaf and 2 for hypostomatous leaf.
, p / Solar zenith angle and solar profile angle, respectively, rad.
/ Material density, kg/m3
d,df/vis/nir / Reflectivity of the greenhouse component to direct or diffuse, visible or near infra red spectrum.
/ Efficiency of energy conversion for electron transport.
/ Ventilation specific rate , m3/m2.s
/ Humidity ratio of the greenhouse air, kg H2O/kg air
/ CO2 compensation point, ppm
t / Time interval, s
air / Greenhouse air
atm / atmosphere
avg / Average
bl / Boundary layer
base / Base of the greenhouse soil
cov / Greenhouse cover
can / Canopy
cond / condensation
cool / Cooling
d / direct
df / diffuse
dehumid / dehumidification
floor / Greenhouse floor
inj / injection
leaf / leaf
lower / Lower limit
NIR / Near infra red radiation
opt / optimal
P / Perforated plate
rf / Roof vent
sky / Sky
soil / Soil
side / Side wall vent
tran / Transpiration
upper / Upper limit
vis / Visible radiation
wv / Water vapor
1MODEL OVERVIEW AND ASSUMPTIONS
Fig. 1 shows the important features of the greenhouse conceptual model. The greenhouse dimensions are Width, W, Height, H, and a length, L perpendicular to the plane of the figure. The greenhouse components are the greenhouse cover, inside air, plant canopy and the soil. The figure shows the energy and mass species specific rates exchanged between the greenhouse components. It also shows that greenhouse can be cooled by natural ventilation represented by the openings of the greenhouse cover, or by alternative method when the greenhouse is closed represented by the cooling load arrow. There is also a storage tank for CO2 which stores the CO2 that will be injected to the plant inside the greenhouse to capture and utilize it through the photosynthesis process. The following assumptions are taken into consideration in the formulation of the basic equations of the mathematical model:
- Green leaves of the plant canopy are treated considering the big leaf approach. According to this approach, the overall plant canopy is assumed as a lumped system with uniform temperature, CO2 concentration, and humidity ratio. The greenhouse inside air is considered as well mixed with no spatial distribution of the corresponding microclimatic variables (temperature, CO2 concentration and humidity ratio). The greenhouse cover (side walls and roof) is thin enough to be considered as one lump in heat transfer analysis. The soil is a semi-infinite medium that extends in the direction of the Z-coordinate and is treated as a thick slab of thickness l in the numerical thermal analysis.
- The greenhouse soil is covered with thinplastic sheet to prevent any mass transfer of water that may evaporate from the soil to the greenhouse air.
- The greenhouse cover is considered to be blocking (reflecting) to the ultra violet (UV) spectrum of solar radiation.
- The greenhouse is oriented in the east-west direction.
- Greenhouse cover is tightly sealed against infiltration. The only exchange of the greenhouse air with the atmospheric air is through ventilation (if used).
- Any reflected solar radiation from the canopy will escape through the cover directly to outside due to the high cover transmittance to solar radiation.
- The greenhouse inside air is not participating medium in radiation analysis.
- The surfaces of the greenhouse components involved in thermal radiation exchange are considered diffuse, gray, and opaque.
- The directional values of radiative properties for canopy, cover, and floor associated with solar radiation transport are assumed equal to the hemispherical values.
2MODEL GOVERNING EQUATIONS
The mass species balance equation governing the CO2 concentration C (t) in the inside air is expresses as:
= G – An –(1)
where(kg/m3)and (kg/mol) are the mass density and molecular weight of the dry air, respectively. The source term G (mol/m2.s) accounts for the CO2 enrichment by injection. The source term -An(mol/m2.s)accounts for the net CO2 assimilation due to photosynthesis and respiration by the plant canopy. The source term (mol/m2.s) accounts for theloss of CO2 by ventilation (if exits).
The mass species balance equation governing the humidity ratio (t) in the inside air is expresses as:
air H = - – m"dehumid - (2)
The source term (kg H2O/m2.s) accounts for transpiration from plant canopy. The source term -(kg H2O/m2.s) accounts forthe condensation of water vapor on the inner surface of the greenhouse cover. The source term -m"dehumid(kg H2O/m2.s) accounts for of the removal of water vapor from the humid air to control the relative humidity of the inside air (if needed). The source term (kg H2O/m2.s) accounts for the exchange of water vapor due to ventilation (it exits).
The energy balance equation governing the inside air temperature Tair (t) is expressed as:
air H =++ + hg ()- hg () -(3)
The heat transfer terms, , andaccount respectively forthe convection from canopy, from floor, and to cover (W/m2). The terms hg () and hg ()account for the energy in associated with transpiration from the plant canopy and the energy out associated with condensation on the cover surface (W/m2), respectively.The water vapor that enters the greenhouse inside air from canopy through transpiration, leaves the greenhouse inside air to cover through condensation, and that exits inside the greenhouse is in superheated condition, however, it can be reasonably approximated as saturated water vapor, thus the use of hg in transpiration and condensation terms . The term accounts for the cooling load removed from the greenhouse air, when needed.
The energy balance equation governing the greenhouse cover temperature Tcov (t) is expressed as:
covcov Ccov = Rs_cov + Rfloor-cov + Rcan-cov– Rcov-sky-– + hfg(Tcov) (5)
where Afloor=W L is the total floor area (m2), and cov (kg/m3), Acov (m2), cov (m), and Ccov (J/kg.K) are the density,the total surface area, the thickness and the specific heat of the greenhouse cover, respectively. The radiative heat transfer terms Rs_cov, Rfloor-cov, Rcan-cov,and Rcov-skyaccounts respectively for absorbed solar radiation, the net floor-cover and the net canopy-cover radiation exchanges,and cover radiation to sky (W/m2). The heat transfer terms andhfg(Tcov)account respectively forthe convection to ambient and theenergy in to cover by condensation of water vapor(W/m2), respectively.
The energy balance equation governing the plant canopy temperature Tcan (t) is expressed as:
leaf LAIleaf Cleaf= Rs-can+ Rfloor-can – Rcan-cov - - hfg() (6)
where leaf(kg/m3),leaf (m), and Cleaf (J/kg.K) are the density, thickness, and specific heat of the plant leaf, respectively. The radiative heat transfer terms Rs_can and Rfloor-can accounts respectively for absorbed solar radiation and the net floor-canopy radiation exchange (W/m2). The term hfg() accounts for the energy out associated with transpiration (W/m2).
The use of hfg with transpiration and condensation terms in equations (5 & 6) instead of hg as in equation (3) is due to the following. Regarding transpiration, water content of the leaf that is in liquid state hf and needs energy to be converted to water vapor state hg that enters the greenhouse. This energy equals the difference between hf and hgwhich is the latent heat of vaporization hfg. Also, water vapor inside the greenhouse and close to the greenhouse cover inner surface is in the hg state, when it is converted to condensate in the hf state, it will leave its latent heat of vaporization hfg to the cover.
The energy balance equation for the non-steady one dimensional conductiongoverning the soil temperature Tsoil (z,t) is given by:
The associated boundary conditions are:
At the floor (z=0.0): -Ksoil = Rs_floor - Rfloor- can - Rfloor- cov - (8)
At z=: Tsoil=Tbase (9)
where(kg/m3), ksoil(W/m.K) and Csoil(J/kg.K) are the density, thermal conductivity, the and the specific heat of the soil, respectively. The base temperature is defined as the temperature at depth l at which the temperature is constant and is not affected by environmental conditions. In this study, Tbase is taken as 15 at l=1.0 m .
3EXPRESSIONS OF TERMS INCLUDED IN THE GOVERNING EQUATIONS
3.1Estimation of the Energy Storage Term of the Humid Air
The humid air at atmospheric pressure and a dry bulb temperature Tair typical to the greenhouse inside air temperature is treated according to the psychometric principles. In this treatment, the humid airis considered as a mixture of ideal gases namely, dry air and superheated water vapor. Consequently, the humid air enthalpy h, can be estimated as the sum of both dry air and water vapor enthalpy contributions (sensible and latent). Thus, the enthalpy h, of the humid air per unit mass of dry air can be expressed as:
h= (hair,0+) + (hg,0+) (10)
Considering that the reference specific enthalpy hair,0 is defined to be zero at a reference temperature T0=273 K, and assuming that Cp,air and are constant at their averaged values within the temperature range of interest, the expression for h can be reduced to:
h= Cp,air(Tair-T0) + [hg,0 + (Tair-T0)] (11)
Where Cp,air and are the specific heats at constant pressure of dry air and water vapor, respectively. The constant value hg,0 is the specific enthalpy of the dry saturated water vapor at the reference temperature T0.
The specific internal energy, u of the humid air is derived from the specific enthalpy, h and can be expressed as:
u= Cv,air(Tair-T0) -RairT0 + [ug,0+ (Tair-T0)] (12)
Where Cv,air and are the specific heats at constant volume of dry air and water vapor, respectively. The constant values Rair and ug,0 are the specific gas constant of dry air and the specific internal energy of the dry saturated water vapor at the reference temperature T0, respectively.
The time derivative of the storage term of equation (3) can be expressed based on equation (12) as:
= (Cv,air+ ) + [ug,0+ (Tair-T0)] (13)
3.2Estimation of Convective Heat Specific Rates and Condensation Mass Specific Rate
In this section, specific rates of convective heattransfer and the condensation mass specific rate included in the energy and mass species balance equations are developed. The following correlations are used for the air inside the greenhouse (Pr 0.7).
3.2.1Convective heat specific rate between the greenhouse floor and greenhouse air
The convection heat specific rate between the greenhouse floor and the greenhouse air is estimated by:
= hfloor-air (Tsoil(0,t)-Tair) (14)
Where hfloor-air is the heat transfer coefficient between the greenhouse floor and the greenhouse air. This coefficient depends on the mode of convection between the floor and the greenhouse air.
When the greenhouse is not opened (e.g.: ventilation is not acting), convection between the floor and air is purely free convection. As the floor is a horizontal surface, correlations of free convection for horizontal surface is used as :
where Nu is the Nusslet number,Gr is the Grashof number, Ra is the Rayleigh number, Pr is the Prandtl number, kair is air thermal conductivity and Lc_floor is the characteristic length of the greenhouse floor.
When the greenhouse is ventilated,forced convection is not considered to be acting purely due to the relatively low flow rates and associated air velocities allowable in the greenhouse for the plant to withstand. Thus, the mixed mode of convection (forced and free) is suitable for the ventilated case of the greenhouse. For this condition, the following correlation is used :