Design and Maintenance of Green House 3(2+1)

Lesson 16.Greenhouse Steady State Energy Balance and Mass Balance Models

16.2 ENERGY BALANCE MODEL

In this study, the energy storage term in the greenhouse was ignored since the heat capacity of internal air and plants was small compared to the existing fluxes (Walker, 1965 and Kindelan, 1980).  The air in each defined zone in the greenhouse was assumed to be well mixed resulting in no spatial variation in temperature.  The greenhouse surface and the surrounding were assumed gray bodies and the sky was at sky temperature.  Also, it was assumed that the greenhouse ground was completely covered with plants.  During the periods of high solar radiation when ventilation was required, the heat loss to ground, the heat of respiration and heat utilized in photosynthesis were assumed to be very small compared to other fluxes and were neglected (Walker, 1965).  The cover was assumed to be double layer polyethylene. Neglecting the rate of change of energy stored in the greenhouse and other small fluxes, the energy balance equation for the greenhouse was written as:

 

                                                                                                          ............................(1)

Where, Q_sr was the amount of direct and diffuse short wave solar radiation in the greenhouse (W), Q_e was the latent heat energy flux due to plant transpiration (W), Q_cd was the conduction heat transfer through the greenhouse covering material (W), Q_v the energy removed by ventilation air (W), and Q_t was the net thermal radiation through the greenhouse covers to the atmosphere (W).  Each term was defined by a relationship. 

The variable Qsr was expressed as:

                                                                     .........................(2)

where τ_c  was the transmissivity of the greenhouse covering materials for solar radiation, S_l was the shading level, I_sr  was the amount of solar radiation energy received per unit area and per unit time on a horizontal surface outside the greenhouse (W/m²), and A_f was the floor area of the greenhouse (m²).

The variable Qe in equation (1) was expressed as:

                                                                           ...............................(3)

Where ET was rate of transpiration (KgH₂O/sec.m²) and Lv was latent heat of vaporization of water (J/Kg H₂O).

The variable Q_cd  in equation (1) was defined by the following relationship:

                                                                        ........................(4)

where U was the heat transfer coefficient (W/m^{2 o}C), A_c  was area of the greenhouse covers (m²), T_i was the inside air temperature (^oC), and T_o was the outside air temperature (^oC).

Since the transmitted thermal radiation loss was considered separately and not included as a part of the U value for conduction heat loss, U was given the value of 2.73 W/m^{2 o}C  (ASHRAE guide and data book fundamentals, 1981)

The variable Q_v in equation (1) was expressed as:

 

                                                                                                                                                                                       ………..(5)

where V_va  was the volumetric ventilation rate (m³/s), v was the specific volume of air (m³/kg), and C_p was the specific heat of air (J/Kg ^oC).   When the evaporative cooling system was used, T_o in equation (5) was the temperature of the air leaving the cooling system.  It was expressed as:

 

                                                                                                                                                                                           ……….(6)

The variable Q_t was the difference between the thermal radiation emitted from the surface and the thermal radiation gained from the atmosphere such that:

                                                                           

                                                                                                                  ………(7)

where σ was the Stefan-Boltzmann's constant (5.67 x 10^-8 W/m².K⁴), τ_tc and τ_os were the transmissivities of the thermal radiation for the double layer polyethylene and the shading material, respectively, ε_i was the average emissivity of the interior surface, and ε_sky was the apparent emissivity of the sky. The emissivity of the sky was evaluated by the following equation (Idso, 1981):

e_sky = 0.70 + 5.95 E -7*e_o* exp (1500/T_o)

                                                                                                                                                                                                   ………(8)

where e(T_o) and T_o were ambient vapor pressure and temperature at a standard height in P_a. and K, respectively.

 The sky temperature was approximated by the Swanbank model (1963) as a function of outside air temperature (T0 in unit of Kelvin):

T_sky= 0.0552 * (To)^1.5

                                                                                                                                                                                                      …………(9)

Substituting equations (2-5) into equation (1), and expressing each term per unit floor area yielded:

                                                                                                                   

                                                                                                                                                                                            ……………..(10)

where V_va  was the volumetric ventilation rate (m³/s.m²  of floor area), a = A_c/A_f, b = (1/v)C_p, T_e = T_o when the natural or fan ventilation was used without evaporative cooling,and I_sri = τ_c S_l I_sr.

Then, an expression for the interior temperature was derived as

T_i = [I_sri - Q_t/A_f  + (aT_o + b V_va T_e) - ET L_v] / (a + b V_av)

                                                                                                                                                                                       ………….(11)

It is clear from equation (11) that the variables affecting inside greenhouse temperature  (T_i)  were  the  inside  solar  radiation  (I_sri),  outside  temperature  (T_o), transpiration rate (ET), and ventilation rate (V_va).

 

16.3 MASS BALANCE MODEL

To evaluate the inside relative humidity for a given ventilation rate and a rate of moisture production, a mass balance calculation was performed.   In this study, it was assumed that moisture loss from the air by condensation on the surfaces was small and was neglected. Also, the plant transpiration was assumed to be the only source of moisture production in naturally and fan ventilated greenhouses.  When the evaporative cooling system was used, additional moisture was added to the greenhouse environment by the cooling system.  The overall mass balance was:

M_v=M_et

                                                                                                                                                            …………………(.12)

Where, M_v was the amount of moisture transferred from the inside air to the outside air via ventilation (Kg H₂O /s), which was expressed as:

M_v=(1/v)V_a (w_i-w_o)

                                                                                                                                                                …………..(13)

Where w_i and w_o were the humidity ratios of the inside and outside air, respectively, in KgH₂O/Kg dry air.

The  variable  M_et   in  equation  (13)  was  the  amount  of  moisture  added  by transpiration (Kg H₂O/s).

The relationship for relative humidity (RH) expressed as a percent was given by the equation:

RH= [e (T)/e*(T)] *100

                                                                                                                                                                  ………….(14)

Where, e(T) was the partial pressure of the water vapor in moist air (P_a) and e*(T) was the saturation vapour pressure (P_a).  The partial pressure of water vapour was defined by:

E(T) = (w+ P_atm)/(w+0.622)

                                                                                                                                                                       ………………….(15)

Where, P_atm was the atmospheric pressure (P_a).

The saturation vapor pressure for a given temperature was computed with the equation given by ASHRAE Handbook of Fundamentals, (1993):

 e*(T) = exp [-5800.2206/T + 1.3914993-0.04860239T + 0.41764768E- 4T²-0.14452093E -7T³ + 6.5459673ln(T)]

                                                                                                                                                                             …………………(16)

Where, T was in K and e(T) was in P_a.

Water use during the mass transfer process between the air and water for the evaporative cooler was evaluated by the steady flow mass balance equation:

M_aw_o+ m_w=m_aw_e

                                                                                                                                                                                                     ……………………(17)

Where m_w was the evaporative cooling system water intake (Kg H₂O/s), m_a was the mass flow rate of the outside air through the cooling system (Kgdry air/s), and w_e was the humidity ratio of the leaving air passed through the cooling system (Kg H₂O/Kg dry air).

Therefore,

M_w= m_a(w_e- w_o)  

                                                                                                                                                                     …………….(18)                    

 Or

Mw= (1/v)V_av (w_e - w_o)

                                                                                                                                                                                ……………….(19)

Equation  (19)  can  be  used  to  evaluate  the  water  evaporation  rate  in  the evaporative cooling system for a given air flow rate through the greenhouse.  However, a general formula for water intake for the cooling system as a function of outside conditions needed to be evaluated.  Cooling load of the evaporative cooling system of a greenhouse was expressed as:

Q_v= Q_sr- Q_e- Q_cd- Q_t

                                                                                                                                                                                    ………………….(20)

The volumetric ventilation rate with an average inside design temperature was defined as:

                                                                                                                                                                                     ……………………(21)

Where ,a and b were described in equation (10)

Since the evaporative cooling process is an adiabatic exchange of heat, the amount of sensible heat removed from the air equals the amount of heat absorbed by the water evaporated as latent heat of vaporization. Therefore, the difference (w_e-w_o) was expressed as:

                                                                                                                                                                      ………………(22)

Substituting equations (22) and (21) in equation (19) yielded

                                                                         .........................(23)

Where, d=1/L_v.

In Equation (23), water intake by the evaporative cooling system was a function of solar radiation, ambient and inside temperatures.

 

 REFERENCES

1. www.faculty.ksu.edu.sa : Lecture note
2. ASHRAE  guide  and  data  book  fundamentals.1981. American  Society of  Heating, Refrigerating and Air-conditioning Engineers . New York
3. ASHRAE guide and data book fundamentals (SI edition).   1993.   American Society ofHeating, Refrigerating and Air-conditioning Engineers . New York
4. Idso, S. B. 1981.  A set of equations for full spectrum and 8-µm to 14-µm and 10.5-µm to12.5-µm thermal radiation from cloudless skies. Water Resources Res. 17: 295-304.
5. Kindelan, M. 1980.  Dynamic modeling of greenhouse environment.  Transactions of theASAE, 23:1232-1239.
6. Walker, J. N. 1965. Predicting temperatures in ventilated greenhouses. Transactions of theASAE 8(3): 445-448.