# Design and Modelling of Water Chilling Production System by the Combined Effects of Evaporation and Night Sky Radiation

Authors: Ahmed Y. Taha Al-Zubaydi; W. John Dartnall
Journal: Journal of Renewable Energy
 (2014)

---

## Abstract

The design and mathematical modelling of thermal radiator panel to be used primarily to measure night sky radiation wet coated surface is presented in this paper. The panel  consists of an upper dry surface coated aluminium sheet laminated to an ethylene vinyl acetate foam backing block as an insulation. Water is sprayed onto the surface of the panel so that an evaporative cooling effect is gained in addition to the radiation effect; the surface of a panel then is wetted in order to study and measure the night sky radiation from the panel wet surface. In this case, the measuring water is circulated over the upper face of this panel during night time. Initial TRNSYS simulations for the performance of the system are presented and it is planned to use the panel as calibrated instruments for discriminating between the cooling effects of night sky radiation and evaporation.


---

## Body

## 1. Introduction

The demand for energy efficient air conditioning systems to minimize the energy consumption worldwide has been an important issue for researchers since the invention of active air conditioning system. In parallel with the researches conducted to create new conventional air conditioning technologies, researches to find noncompressor systems took a place.Noncompressor cooling systems have been introduced as low electricity consumption alternatives to mechanical vapour compression units, providing a solution that reduces the energy consumption and environmental issues in buildings. Absorption and adsorption chillers, direct and indirect evaporative air coolers, and water passive cooling systems which utilize the night sky radiation effect are among the range of noncompressor systems available. Radiative cooling is a passive cooling process based on the phenomenon of heat transfer between earth-based objects and the sky by the means of long-wave radiation, usually when the effective sky temperature is less than the ground temperature. The transparency of earth’s atmosphere in the infrared range of 8–13μm (atmospheric window) allows a fraction of the thermal radiation to be absorbed by space, especially during the night and early morning where solar short wave radiation is absent. The phenomenon can be applied to cool surfaces and/or fluids, which in turn can be used in air conditioning applications.Cooling with night sky radiation phenomena could be considered as featured technology with capability to be a passive alternative to the conventional air conditioning system. Many researchers have explored the different applications and systems proposed to cool fluids by means of heat rejection by radiation to the sky. Yannas et al. [1] presented the world’s best-known applications of the night sky radiation system with mathematical modelling of a roof pond system. Spanaki [2] presented a comparison between different roof pond systems for cooling. Tiwari et al. [3] presented the analysis of the moving water system in three different cases including roof pond, spray cooling/gunny bags, and moving water over the roof.A commercial system known as “WhiteCap” was introduced in the United States. This system is reported by Bourne and Carew [4, 5]. The WhiteCap system employs the cooling of water by night sky radiation, convection, and evaporation of water in an open water cycle. The cold water is stored for use during the day by direct conduction to the roof (WhiteCap-R), fan coils (WhiteCap-T), or coils entrenched in the floor slab (WhiteCap-F). The WhiteCap system enhances the evaporation heat loss by spraying the water over the roof in the primary stage, and the water droplets evaporate as they travel from the spray outlet to the roof surface. Bourne and Carew [4] reported a 47% reduction in annual use of conventional cooling by using the WhiteCap cooling system on a 2,500-square-meter office building in Los Angeles.The low cooling capacity and the high effectiveness of ambient conditions on the system performance (i.e., cloud cover and relative humidity) are considered practical limitations of the nocturnal cooling by radiation technology applications. However, the system can be applied as a supplemental heat sink system with another HVAC system [6]. Examples of hybrid systems are presented by Heidarinejad et al. [7] and Farmahini Farahani et al. [8]. Both studies investigated the possible combination of nocturnal radiation cooling systems with conventional systems, specifically with an indirect evaporative cooling unit in the first study and a direct evaporative cooling unit in the second.Although the WhiteCap system is available since 1996, no theoretical works available present the system mathematical modelling. In previous work, the author presented a mathematical modelling of three different systems applying the Night sky radiation for water chilling [9]. The system compared three panels to measure the various effects: the first panel had a dry radiative surface with an embedded heat exchange water circuit for the purpose of measuring the radiant heat. The second panel employed a sprayed water flowing over its radiative surface to measure combined effect of (a) heat lost by pre-evaporation during the sprayed water droplet travel through the air, (b) evaporation of water flowed over the radiation surface and (c) radiation from the thin water layer surface. The third panel is like the second, however the water is running via a gutter at the top so that is flows over the radiative surface enables the heat loss by radiation and evaporation from a wetted surface. The comparison between the three systems shows that the second panel performance is the best in term of water chilling.In this work, the author presented the second system presented in the previous work [9] with a detailed mathematical modelling to enable researchers to use it as a design guide for future work. In order to satisfy the above objective, it was decided to construct a panel having similar principles of the WhiteCap system. By using this panel, the separate effects of evaporative cooling and night sky radiative cooling may be deduced from the recorded data. In this work we discussed the design and mathematical modelling of the system (i.e., radiator panel with sprayer). The system under investigation is described in Figure 1. The layout of the system (Figure 1) consists of a radiator panel with a well-insulated back side which is used to simulate the principles of the system similar to the WhiteCap system. Water is sprayed onto the panel flow for a measured distance and then returned to the storage tank and pumped again. The contact of the water with the air causes mass loss by the evaporation through the process; make-up water to the tank is a necessity. The make-up water temperature is different from the water temperature in the storage tank, and this is considered in the calculations. The water temperatures in the tank, spray nozzle inlet, water pond inlet, and water pond outlet are the data collected.Figure 1
Experimental setup for the radiative cooling measurement systems-radiator panel with sprayer.This paper discusses the theoretical work of the project by presenting mathematical modelling and a performance simulation with the Transient System Simulation Tool (TRNSYS) software package [9, 10]. The results are analysed to optimize the design parameters and to analyse the contribution of the three heat transfer phenomena that occur in different applications of cooling water using the night sky radiation effect. We also study the feasibility of producing chilled water for space cooling applications. The simulation results will be analysed and compared with the experimental results in a later stage of the project.


## 2. System Description and Modelling

### 2.1. The Effective Sky Temperature

The heat exchange between the radiator plate and the sky occurs in a wavelength range of 8 to 14μm, or what is known as the atmospheric window. The sky is considered to be a black body of equivalent temperature T
sky or the effective sky temperature. In the literature [11–13], many equations have been introduced to calculate T
sky, as a function of the ambient temperature, cloudiness factor, and the emissivity of the clear sky which in turn is a function of ambient temperature, relative humidity, and atmospheric pressure. Berdahl and Martin [11, 13] introduced an equation to calculate the clear sky emissivity as follows:
(1)
ε
sky
=
0.711
+
0.005
T
dp
+
7.3
×
10
-
5
T
dp
2
+
0.013
cos
⁡
⁡
[
2
π
time
24
]
+
12
×
10
-
5
(
P
atm
-
P
h
)
,
where T
dp is the dew point temperature (°C), P
atm is the atmospheric pressure (atm), P
h is the atmospheric pressure at elevation h (m) above sea level, and the variable (time) is the hour of the day. In the TRNSYS software package, the Type 69 calculates the effective sky temperature T
sky (K) using Martin and Berdahl’s [13] equation as follows:
(2)
T
sky
=
T
amb
(
ε
sky
+
0.8
(
1
-
ε
sky
)
C
clo
)
1
/
4
,
where C
clo is the sky cloudiness factor (0-1), supplied by TRNSYS Type 69.


### 2.2. Thermal Panel Description

A thermal panel for the purpose of cooling water with night sky radiation was designed using basic materials which cost little and make the labour easier. The assembly consisted of an aluminium plate with dimensions of 3000 mm long, 2000 mm wide, and 2 mm thick. The dimensions were chosen to fit the size of a piece of EVA foam supplied directly by the manufacturer. The plate was fixed to a 30 mm thick EVA foam block of closed cell foam made from ethylene vinyl acetate and blended copolymers. In the open cycle water system, water is sprayed over the roof surface and flows onto the low, sloped surface for a distance ofL (m) before being collected and returned to the tank. The operation is repeated during the night (Figure 1). Water exchanges heat with the ambient air by means of evaporation and convection and to the sky by radiation. The relative temperature of the water in the tank will be changed by the end of the operating period. In the following the modelling of each process is presented.


#### 2.2.1. Water Droplet Modelling

In this section, the calculation of the sprayer water droplet final temperature and the quantity of evaporative mass will be discussed. The modelling of water droplet energy balance has been presented earlier by Al-Zubaydi et al. [9]. A numerous number of water droplets formed by the nozzle will travel through the air to the water pond on the panel surface, performing a heat and mass transfer with the surrounding air. The process is assumed to be a quasisteady state that varies with the ambient temperature in the form of Fourier series. The water droplet boundary system energy balance is presented in Figure 2 [14]. The quantity of heat stored in the droplets over the period of time Δ
t is
(3)
Δ
U
˙
=
Q
˙
total
.Figure 2
Water droplet energy balance.The internal energy of the system is calculated by(4)
Δ
U
˙
=
m
d
·
C
p
w
·
Δ
T
d
Δ
t
,
where m
d is the mass of the droplet (kg), C
p
w is the specific heat capacity of the water (J/kg·K), and Δ
T
d is the difference between the droplet final temperature T
d
2 and initial temperature T
d
1 (K).The heat transfer between the droplets, the surroundings, and the ambient air is the sum of all heat transfers by convection, radiation, evaporation, and condensation.Essentially, when evaporation occurs, no condensation takes part in the thermodynamic process and the value of heat transfer by condensation is reset to zero. The total heat transfer from a droplet boundary system is given by(5)
Q
˙
total
=
Q
˙
conv
+
Q
˙
rad
-
Q
˙
evap
,
where
(6)
Q
˙
conv
=
h
conv
·
A
d
·
(
T
amb
-
T
d
)
,
(7)
Q
˙
rad
=
ε
water
·
A
d
·
σ
·
(
T
sky
4
-
T
d
4
)
,
(8)
Q
˙
evap
=
m
˙
evap
·
h
f
g
,
where h
conv is the coefficient of thermal convection (w/m²·K) and h
f
g is the evaporation enthalpy of water vapor (J/kg). A
d is the surface area of the droplet (m²), m
d is the single droplet mass (kg), σ is the Stefan-Boltzmann constant (5.67
×
10
-
8 w/m²·
K
4), and ε
water is the water surface emissivity (value set to 1). By substituting (9)–(13) in (8) and rearranging we get
(9)
T
d
2
=
T
d
1
+
Δ
t
m
d
·
C
p
w
×
[
T
sky
4
h
c
·
A
d
·
(
T
a
-
T
d
)
+
ε
water
·
A
d
·
σ
·
(
T
sky
4
-
T
d
4
)
-
m
˙
evap
·
h
f
g
]
,
where T
d
2 is the water drop final temperature (K) and T
d
1 is the water droplet initial temperature (K). h
c and m
˙
evap values can be calculated with the following equations, the mass evaporation rate m
˙
evap (kg/s):
(10)
m
˙
evap
=
h
m
·
A
d
·
[
ρ
drop
-
ϕ
ρ
atm
]
,
where h
m is the mass transfer convection coefficient (m/s), ρ
d is the specific humidity of saturated air evaluated at the water droplet temperature (kg/m³), ρ
amb is the specific humidity of saturated air evaluated at the ambient temperature (kg/m³), and φ is the relative humidity of the ambient air (%).h
m is derived from the Sherwood number [14]:
(11)
Sh
D
′
=
2
+
0.6
Sc
1
/
3
Re
1
/
2
,
where Re is the Reynolds number of the droplet and is given by
(12)
Re
=
D
d
s
d
v
air
,
where D
d is the droplet diameter (m), s
d is the droplet velocity (m/s), and v
air is the kinematic viscosity for air (m²/s). The variable Sc is the Schmidt number and is given by
(13)
Sc
=
v
air
D
vap
,
where D
vap is the water vapour diffusivity in air (m²/s) [
0.26
E
-
4
m
2
/
s
] [15]. After calculating the Sherwood number, the value of h
m is given by
(14)
h
m
=
Sh
D
′
·
D
vap
D
.The coefficient of thermal convectionh
c can be derived from the Nusselt number equation:
(15)
Nu
=
h
conv
D
d
k
air
,
where k
air is the air thermal conductivity (W/m·k). In forced convection between the droplet and the wind, the Nusselt number can be calculated by the Ranz and Marshal [15] equation:
(16)
Nu
=
2
+
0.6
Pr
1
/
3
Re
1
/
2
.The Prandtl numberPr can be found in the air properties table or calculated as a function of air dynamic viscosity v
a, specific heat of the air C
p
a, and air thermal conductivity k
air, with formula [16]:
(17)
Pr
=
C
p
a
v
a
k
air
.By substituting the outputs of (11) and (16) in (15), the substitute equation (15) output in (14), the value of h
conv will be
(18)
h
conv
=
k
air
[
2
+
0.6
Pr
1
/
3
Re
1
/
2
]
D
d
.With the values ofh
c and m
˙
evap calculated from (16) to (24), the new temperature of the droplet T
d
2 can be estimated; this temperature will be the initial temperature of the second part of the water cooling system, the panel pond. The amount of water entering the water pond boundary system is the initial mass flow rate of the sprayer minus the evaporated losses:
(19)
m
2
s
=
m
1
s
-
(
m
˙
d
·
Δ
t
·
m
1
s
m
d
)
.


#### 2.2.2. Radiative Panel Mathematical Modelling

Similar to the water droplet, the panel pond control volume can be represented as a typical thermodynamic control volume (Figure3). As can be seen, the heat transfer to and from the system boundary is in the form of evaporation, conduction, convection, solar, and radiation, in addition to heat transfer due to water entering and leaving the system.Figure 3
Water pond energy balance.Similar to the water droplet energy equation, the water pond energy equation is(20)
m
w
_
rp
·
C
p
w
·
T
rp
2
-
T
rp
1
Δ
t
=
Q
˙
cond
+
Q
˙
solar
-
Q
˙
conv
-
Q
˙
rad
_
sky
m
w
_
rp
·
C
p
w
·
T
rp
2
-
T
rp
1
Δ
t
=
-
Q
˙
evap
-
Q
˙
back
+
E
˙
wi
-
E
˙
wo
,
where m
w
_
rp is the mass of the water in the water pond (kg), Q
˙
cond is the heat added by condensed water (kJ/s), Q
˙
Solar is the heat absorbed by the panel water pond due to solar radiation (kJ/s), Q
˙
conv is the heat convection between the water pond and ambient air (kJ/s), Q
˙
rad
_
sky is the heat exchange between the water in the roof pond and the sky, by radiation (kJ/s), Q
˙
evap is the evaporation heat exchange between water and the ambient air (kJ/s), Q
˙
back is the heat transfer between the water and the ambient air through the panel layers (kJ/s), and E
˙
wi and E
˙
wo are the energy rates added to the panel pond water by the water added and water leaving the system boundary (kJ/s).As in (3) for the water droplet, the energy equation can be rewritten to calculate the temperature of the water leaving the panel pond:
(21)
T
rp
2
=
T
rp
1
+
Δ
t
m
w
_
rp
·
C
p
w
·
[
Q
˙
cond
+
Q
˙
solar
-
Q
˙
conv
-
Q
˙
rad
_
sky
T
rp
2
0000000000000000000
-
Q
˙
evap
-
Q
˙
back
+
E
˙
wi
-
E
˙
wo
]
.Calculations of heat and energy transfers between the system and the boundary can be derivatives using the basic heat transfer equation; many works presenting the mathematical modelling of the panel pond system can also be referenced [17–22]. The amount of heat dissipated to the sky by radiation is a function of the pond emissivity, the sky temperature, and the pond water temperature; other factors having an effect are the humidity ratio, clouds, and radiation wavelength which have a direct effect on changes to the emissivity and sky temperature values. The Q
˙
rad
_
sky is given by
(22)
Q
˙
rad
_
sky
=
h
rad
_
sky
·
A
rp
·
(
T
w
_
rp
-
T
sky
)
,
where h
rad
_
sky is the sky radiation heat transfer coefficient (W/m²·k), A
rp is the panel surface area (m²), T
w
_
rp is the temperature of the water in the pond (K), and T
sky is the effective sky temperature (K).During the day, solar heat is added to water in the pond area; the amount of heat added depends on the amount of solar irradiation flux(
I
solar
) and the panel pond’s solar absorptivity (
α
solar
). The heat absorbed by water from solar irradiation is given by
(23)
Q
˙
solar
=
α
solar
A
rp
I
solar
.Consider thatQ
˙
conv is the heat transfer from water to the ambient by convection; this heat can be calculated by
(24)
Q
˙
conv
=
h
rp
_
conv
A
rp
(
T
rp
-
T
amb
)
.The convection heat transfer coefficient (h
rp
_
conv) is a function of the type of flow rate, laminar flow, or turbulent flow, and the (h
rp
_
conv) general equation is
(25)
h
rp
_
conv
=
Nu
L
·
K
amb
X
rp
,
where K
amb and X
rp are the thermal conductivity of the ambient air and the distance water flows over the panel, while Nu
L is the Nusselt number, which is dissimilar to the water droplet (10); in the case of panel pond flow, its value depends on the nature of convection (forced or natural) [16]. The water mass evaporation rate from panel pond m
˙
rp
_
evap needs to estimate the value of heat dissipated from water by evaporation. By recalling (4) and replacing the droplet surface area by the panel surface area we get
(26)
m
˙
rp
_
evap
=
h
m
_
rp
·
A
rp
·
[
ρ
w
_
rp
-
ρ
atm
]
,
where ρ
w
_
rp and ρ
atm are the water density at saturation temperature pressure and atmospheric pressure, respectively (kg/m³), h
m
_
rp (m/s), the mass transfer coefficient is calculated from the Sherwood number (Sh
X
′), and the binary mass diffusion coefficient (D
vap) is calculated as
(27)
h
m
_
rp
=
Sh
X
′
D
vap
X
rp
.Under forced convection conditions, the Sherwood number is(28)
Sh
X
′
=
0.644
Re
X
1
/
2
Sc
1
/
3
and the Nusselt number is
(29)
Nu
X
=
0.644
Re
X
1
/
2
Pr
1
/
3
.When the value ofm
˙
rp
_
evap is calculated with (27), value above zero indicates that evaporation has taken place, and the heat transferred by evaporation from the water to the atmosphere is
(30)
Q
˙
evap
=
h
g
·
m
˙
rp
_
evap
.The energy of water entering and exiting from the system boundary (Δ
E
w) is
(31)
Δ
E
w
=
h
f
,
wi
m
˙
wi
-
h
f
,
wo
m
˙
wo
.Theh
f
,
wi and h
f
,
wo are the enthalpies of water entering and exiting the pond, and m
˙
wi
,
m
˙
wo are the mass flow rate of water entering pond. The m
˙
wo is estimated to be the entering water mass flow rate after subtracting the evaporating mass rate.The heat transfer through the back of the panel (Q
˙
back
) is a function of the back material thermal conductivity and is given by
(32)
Q
˙
back
=
h
b
rp
A
rp
(
T
w
_
rp
-
T
b
)
,
where T
b is the back layer temperature. The h
b
_
rp is the heat transfer coefficient between the back layer and the pond water:
(33)
h
b
_
rp
=
K
plate
d
x
plate
+
K
foam
d
x
foam
.Two identical panels with the dimensions of 2400 mm long and 900 mm wide were constructed from aluminum sheets insulated from the bottom to eliminate the heat transfer from the back side. The criteria that determined the panel size included the aim to make a portable prototype and to minimize the labor. As planned, the water will flow over the inclined panels during the night and will be presprayed in one case to evaluate the contribution of water spray over the panel.


## 2.1. The Effective Sky Temperature

The heat exchange between the radiator plate and the sky occurs in a wavelength range of 8 to 14μm, or what is known as the atmospheric window. The sky is considered to be a black body of equivalent temperature T
sky or the effective sky temperature. In the literature [11–13], many equations have been introduced to calculate T
sky, as a function of the ambient temperature, cloudiness factor, and the emissivity of the clear sky which in turn is a function of ambient temperature, relative humidity, and atmospheric pressure. Berdahl and Martin [11, 13] introduced an equation to calculate the clear sky emissivity as follows:
(1)
ε
sky
=
0.711
+
0.005
T
dp
+
7.3
×
10
-
5
T
dp
2
+
0.013
cos
⁡
⁡
[
2
π
time
24
]
+
12
×
10
-
5
(
P
atm
-
P
h
)
,
where T
dp is the dew point temperature (°C), P
atm is the atmospheric pressure (atm), P
h is the atmospheric pressure at elevation h (m) above sea level, and the variable (time) is the hour of the day. In the TRNSYS software package, the Type 69 calculates the effective sky temperature T
sky (K) using Martin and Berdahl’s [13] equation as follows:
(2)
T
sky
=
T
amb
(
ε
sky
+
0.8
(
1
-
ε
sky
)
C
clo
)
1
/
4
,
where C
clo is the sky cloudiness factor (0-1), supplied by TRNSYS Type 69.


## 2.2. Thermal Panel Description

A thermal panel for the purpose of cooling water with night sky radiation was designed using basic materials which cost little and make the labour easier. The assembly consisted of an aluminium plate with dimensions of 3000 mm long, 2000 mm wide, and 2 mm thick. The dimensions were chosen to fit the size of a piece of EVA foam supplied directly by the manufacturer. The plate was fixed to a 30 mm thick EVA foam block of closed cell foam made from ethylene vinyl acetate and blended copolymers. In the open cycle water system, water is sprayed over the roof surface and flows onto the low, sloped surface for a distance ofL (m) before being collected and returned to the tank. The operation is repeated during the night (Figure 1). Water exchanges heat with the ambient air by means of evaporation and convection and to the sky by radiation. The relative temperature of the water in the tank will be changed by the end of the operating period. In the following the modelling of each process is presented.


### 2.2.1. Water Droplet Modelling

In this section, the calculation of the sprayer water droplet final temperature and the quantity of evaporative mass will be discussed. The modelling of water droplet energy balance has been presented earlier by Al-Zubaydi et al. [9]. A numerous number of water droplets formed by the nozzle will travel through the air to the water pond on the panel surface, performing a heat and mass transfer with the surrounding air. The process is assumed to be a quasisteady state that varies with the ambient temperature in the form of Fourier series. The water droplet boundary system energy balance is presented in Figure 2 [14]. The quantity of heat stored in the droplets over the period of time Δ
t is
(3)
Δ
U
˙
=
Q
˙
total
.Figure 2
Water droplet energy balance.The internal energy of the system is calculated by(4)
Δ
U
˙
=
m
d
·
C
p
w
·
Δ
T
d
Δ
t
,
where m
d is the mass of the droplet (kg), C
p
w is the specific heat capacity of the water (J/kg·K), and Δ
T
d is the difference between the droplet final temperature T
d
2 and initial temperature T
d
1 (K).The heat transfer between the droplets, the surroundings, and the ambient air is the sum of all heat transfers by convection, radiation, evaporation, and condensation.Essentially, when evaporation occurs, no condensation takes part in the thermodynamic process and the value of heat transfer by condensation is reset to zero. The total heat transfer from a droplet boundary system is given by(5)
Q
˙
total
=
Q
˙
conv
+
Q
˙
rad
-
Q
˙
evap
,
where
(6)
Q
˙
conv
=
h
conv
·
A
d
·
(
T
amb
-
T
d
)
,
(7)
Q
˙
rad
=
ε
water
·
A
d
·
σ
·
(
T
sky
4
-
T
d
4
)
,
(8)
Q
˙
evap
=
m
˙
evap
·
h
f
g
,
where h
conv is the coefficient of thermal convection (w/m²·K) and h
f
g is the evaporation enthalpy of water vapor (J/kg). A
d is the surface area of the droplet (m²), m
d is the single droplet mass (kg), σ is the Stefan-Boltzmann constant (5.67
×
10
-
8 w/m²·
K
4), and ε
water is the water surface emissivity (value set to 1). By substituting (9)–(13) in (8) and rearranging we get
(9)
T
d
2
=
T
d
1
+
Δ
t
m
d
·
C
p
w
×
[
T
sky
4
h
c
·
A
d
·
(
T
a
-
T
d
)
+
ε
water
·
A
d
·
σ
·
(
T
sky
4
-
T
d
4
)
-
m
˙
evap
·
h
f
g
]
,
where T
d
2 is the water drop final temperature (K) and T
d
1 is the water droplet initial temperature (K). h
c and m
˙
evap values can be calculated with the following equations, the mass evaporation rate m
˙
evap (kg/s):
(10)
m
˙
evap
=
h
m
·
A
d
·
[
ρ
drop
-
ϕ
ρ
atm
]
,
where h
m is the mass transfer convection coefficient (m/s), ρ
d is the specific humidity of saturated air evaluated at the water droplet temperature (kg/m³), ρ
amb is the specific humidity of saturated air evaluated at the ambient temperature (kg/m³), and φ is the relative humidity of the ambient air (%).h
m is derived from the Sherwood number [14]:
(11)
Sh
D
′
=
2
+
0.6
Sc
1
/
3
Re
1
/
2
,
where Re is the Reynolds number of the droplet and is given by
(12)
Re
=
D
d
s
d
v
air
,
where D
d is the droplet diameter (m), s
d is the droplet velocity (m/s), and v
air is the kinematic viscosity for air (m²/s). The variable Sc is the Schmidt number and is given by
(13)
Sc
=
v
air
D
vap
,
where D
vap is the water vapour diffusivity in air (m²/s) [
0.26
E
-
4
m
2
/
s
] [15]. After calculating the Sherwood number, the value of h
m is given by
(14)
h
m
=
Sh
D
′
·
D
vap
D
.The coefficient of thermal convectionh
c can be derived from the Nusselt number equation:
(15)
Nu
=
h
conv
D
d
k
air
,
where k
air is the air thermal conductivity (W/m·k). In forced convection between the droplet and the wind, the Nusselt number can be calculated by the Ranz and Marshal [15] equation:
(16)
Nu
=
2
+
0.6
Pr
1
/
3
Re
1
/
2
.The Prandtl numberPr can be found in the air properties table or calculated as a function of air dynamic viscosity v
a, specific heat of the air C
p
a, and air thermal conductivity k
air, with formula [16]:
(17)
Pr
=
C
p
a
v
a
k
air
.By substituting the outputs of (11) and (16) in (15), the substitute equation (15) output in (14), the value of h
conv will be
(18)
h
conv
=
k
air
[
2
+
0.6
Pr
1
/
3
Re
1
/
2
]
D
d
.With the values ofh
c and m
˙
evap calculated from (16) to (24), the new temperature of the droplet T
d
2 can be estimated; this temperature will be the initial temperature of the second part of the water cooling system, the panel pond. The amount of water entering the water pond boundary system is the initial mass flow rate of the sprayer minus the evaporated losses:
(19)
m
2
s
=
m
1
s
-
(
m
˙
d
·
Δ
t
·
m
1
s
m
d
)
.


### 2.2.2. Radiative Panel Mathematical Modelling

Similar to the water droplet, the panel pond control volume can be represented as a typical thermodynamic control volume (Figure3). As can be seen, the heat transfer to and from the system boundary is in the form of evaporation, conduction, convection, solar, and radiation, in addition to heat transfer due to water entering and leaving the system.Figure 3
Water pond energy balance.Similar to the water droplet energy equation, the water pond energy equation is(20)
m
w
_
rp
·
C
p
w
·
T
rp
2
-
T
rp
1
Δ
t
=
Q
˙
cond
+
Q
˙
solar
-
Q
˙
conv
-
Q
˙
rad
_
sky
m
w
_
rp
·
C
p
w
·
T
rp
2
-
T
rp
1
Δ
t
=
-
Q
˙
evap
-
Q
˙
back
+
E
˙
wi
-
E
˙
wo
,
where m
w
_
rp is the mass of the water in the water pond (kg), Q
˙
cond is the heat added by condensed water (kJ/s), Q
˙
Solar is the heat absorbed by the panel water pond due to solar radiation (kJ/s), Q
˙
conv is the heat convection between the water pond and ambient air (kJ/s), Q
˙
rad
_
sky is the heat exchange between the water in the roof pond and the sky, by radiation (kJ/s), Q
˙
evap is the evaporation heat exchange between water and the ambient air (kJ/s), Q
˙
back is the heat transfer between the water and the ambient air through the panel layers (kJ/s), and E
˙
wi and E
˙
wo are the energy rates added to the panel pond water by the water added and water leaving the system boundary (kJ/s).As in (3) for the water droplet, the energy equation can be rewritten to calculate the temperature of the water leaving the panel pond:
(21)
T
rp
2
=
T
rp
1
+
Δ
t
m
w
_
rp
·
C
p
w
·
[
Q
˙
cond
+
Q
˙
solar
-
Q
˙
conv
-
Q
˙
rad
_
sky
T
rp
2
0000000000000000000
-
Q
˙
evap
-
Q
˙
back
+
E
˙
wi
-
E
˙
wo
]
.Calculations of heat and energy transfers between the system and the boundary can be derivatives using the basic heat transfer equation; many works presenting the mathematical modelling of the panel pond system can also be referenced [17–22]. The amount of heat dissipated to the sky by radiation is a function of the pond emissivity, the sky temperature, and the pond water temperature; other factors having an effect are the humidity ratio, clouds, and radiation wavelength which have a direct effect on changes to the emissivity and sky temperature values. The Q
˙
rad
_
sky is given by
(22)
Q
˙
rad
_
sky
=
h
rad
_
sky
·
A
rp
·
(
T
w
_
rp
-
T
sky
)
,
where h
rad
_
sky is the sky radiation heat transfer coefficient (W/m²·k), A
rp is the panel surface area (m²), T
w
_
rp is the temperature of the water in the pond (K), and T
sky is the effective sky temperature (K).During the day, solar heat is added to water in the pond area; the amount of heat added depends on the amount of solar irradiation flux(
I
solar
) and the panel pond’s solar absorptivity (
α
solar
). The heat absorbed by water from solar irradiation is given by
(23)
Q
˙
solar
=
α
solar
A
rp
I
solar
.Consider thatQ
˙
conv is the heat transfer from water to the ambient by convection; this heat can be calculated by
(24)
Q
˙
conv
=
h
rp
_
conv
A
rp
(
T
rp
-
T
amb
)
.The convection heat transfer coefficient (h
rp
_
conv) is a function of the type of flow rate, laminar flow, or turbulent flow, and the (h
rp
_
conv) general equation is
(25)
h
rp
_
conv
=
Nu
L
·
K
amb
X
rp
,
where K
amb and X
rp are the thermal conductivity of the ambient air and the distance water flows over the panel, while Nu
L is the Nusselt number, which is dissimilar to the water droplet (10); in the case of panel pond flow, its value depends on the nature of convection (forced or natural) [16]. The water mass evaporation rate from panel pond m
˙
rp
_
evap needs to estimate the value of heat dissipated from water by evaporation. By recalling (4) and replacing the droplet surface area by the panel surface area we get
(26)
m
˙
rp
_
evap
=
h
m
_
rp
·
A
rp
·
[
ρ
w
_
rp
-
ρ
atm
]
,
where ρ
w
_
rp and ρ
atm are the water density at saturation temperature pressure and atmospheric pressure, respectively (kg/m³), h
m
_
rp (m/s), the mass transfer coefficient is calculated from the Sherwood number (Sh
X
′), and the binary mass diffusion coefficient (D
vap) is calculated as
(27)
h
m
_
rp
=
Sh
X
′
D
vap
X
rp
.Under forced convection conditions, the Sherwood number is(28)
Sh
X
′
=
0.644
Re
X
1
/
2
Sc
1
/
3
and the Nusselt number is
(29)
Nu
X
=
0.644
Re
X
1
/
2
Pr
1
/
3
.When the value ofm
˙
rp
_
evap is calculated with (27), value above zero indicates that evaporation has taken place, and the heat transferred by evaporation from the water to the atmosphere is
(30)
Q
˙
evap
=
h
g
·
m
˙
rp
_
evap
.The energy of water entering and exiting from the system boundary (Δ
E
w) is
(31)
Δ
E
w
=
h
f
,
wi
m
˙
wi
-
h
f
,
wo
m
˙
wo
.Theh
f
,
wi and h
f
,
wo are the enthalpies of water entering and exiting the pond, and m
˙
wi
,
m
˙
wo are the mass flow rate of water entering pond. The m
˙
wo is estimated to be the entering water mass flow rate after subtracting the evaporating mass rate.The heat transfer through the back of the panel (Q
˙
back
) is a function of the back material thermal conductivity and is given by
(32)
Q
˙
back
=
h
b
rp
A
rp
(
T
w
_
rp
-
T
b
)
,
where T
b is the back layer temperature. The h
b
_
rp is the heat transfer coefficient between the back layer and the pond water:
(33)
h
b
_
rp
=
K
plate
d
x
plate
+
K
foam
d
x
foam
.Two identical panels with the dimensions of 2400 mm long and 900 mm wide were constructed from aluminum sheets insulated from the bottom to eliminate the heat transfer from the back side. The criteria that determined the panel size included the aim to make a portable prototype and to minimize the labor. As planned, the water will flow over the inclined panels during the night and will be presprayed in one case to evaluate the contribution of water spray over the panel.


## 2.2.1. Water Droplet Modelling

In this section, the calculation of the sprayer water droplet final temperature and the quantity of evaporative mass will be discussed. The modelling of water droplet energy balance has been presented earlier by Al-Zubaydi et al. [9]. A numerous number of water droplets formed by the nozzle will travel through the air to the water pond on the panel surface, performing a heat and mass transfer with the surrounding air. The process is assumed to be a quasisteady state that varies with the ambient temperature in the form of Fourier series. The water droplet boundary system energy balance is presented in Figure 2 [14]. The quantity of heat stored in the droplets over the period of time Δ
t is
(3)
Δ
U
˙
=
Q
˙
total
.Figure 2
Water droplet energy balance.The internal energy of the system is calculated by(4)
Δ
U
˙
=
m
d
·
C
p
w
·
Δ
T
d
Δ
t
,
where m
d is the mass of the droplet (kg), C
p
w is the specific heat capacity of the water (J/kg·K), and Δ
T
d is the difference between the droplet final temperature T
d
2 and initial temperature T
d
1 (K).The heat transfer between the droplets, the surroundings, and the ambient air is the sum of all heat transfers by convection, radiation, evaporation, and condensation.Essentially, when evaporation occurs, no condensation takes part in the thermodynamic process and the value of heat transfer by condensation is reset to zero. The total heat transfer from a droplet boundary system is given by(5)
Q
˙
total
=
Q
˙
conv
+
Q
˙
rad
-
Q
˙
evap
,
where
(6)
Q
˙
conv
=
h
conv
·
A
d
·
(
T
amb
-
T
d
)
,
(7)
Q
˙
rad
=
ε
water
·
A
d
·
σ
·
(
T
sky
4
-
T
d
4
)
,
(8)
Q
˙
evap
=
m
˙
evap
·
h
f
g
,
where h
conv is the coefficient of thermal convection (w/m²·K) and h
f
g is the evaporation enthalpy of water vapor (J/kg). A
d is the surface area of the droplet (m²), m
d is the single droplet mass (kg), σ is the Stefan-Boltzmann constant (5.67
×
10
-
8 w/m²·
K
4), and ε
water is the water surface emissivity (value set to 1). By substituting (9)–(13) in (8) and rearranging we get
(9)
T
d
2
=
T
d
1
+
Δ
t
m
d
·
C
p
w
×
[
T
sky
4
h
c
·
A
d
·
(
T
a
-
T
d
)
+
ε
water
·
A
d
·
σ
·
(
T
sky
4
-
T
d
4
)
-
m
˙
evap
·
h
f
g
]
,
where T
d
2 is the water drop final temperature (K) and T
d
1 is the water droplet initial temperature (K). h
c and m
˙
evap values can be calculated with the following equations, the mass evaporation rate m
˙
evap (kg/s):
(10)
m
˙
evap
=
h
m
·
A
d
·
[
ρ
drop
-
ϕ
ρ
atm
]
,
where h
m is the mass transfer convection coefficient (m/s), ρ
d is the specific humidity of saturated air evaluated at the water droplet temperature (kg/m³), ρ
amb is the specific humidity of saturated air evaluated at the ambient temperature (kg/m³), and φ is the relative humidity of the ambient air (%).h
m is derived from the Sherwood number [14]:
(11)
Sh
D
′
=
2
+
0.6
Sc
1
/
3
Re
1
/
2
,
where Re is the Reynolds number of the droplet and is given by
(12)
Re
=
D
d
s
d
v
air
,
where D
d is the droplet diameter (m), s
d is the droplet velocity (m/s), and v
air is the kinematic viscosity for air (m²/s). The variable Sc is the Schmidt number and is given by
(13)
Sc
=
v
air
D
vap
,
where D
vap is the water vapour diffusivity in air (m²/s) [
0.26
E
-
4
m
2
/
s
] [15]. After calculating the Sherwood number, the value of h
m is given by
(14)
h
m
=
Sh
D
′
·
D
vap
D
.The coefficient of thermal convectionh
c can be derived from the Nusselt number equation:
(15)
Nu
=
h
conv
D
d
k
air
,
where k
air is the air thermal conductivity (W/m·k). In forced convection between the droplet and the wind, the Nusselt number can be calculated by the Ranz and Marshal [15] equation:
(16)
Nu
=
2
+
0.6
Pr
1
/
3
Re
1
/
2
.The Prandtl numberPr can be found in the air properties table or calculated as a function of air dynamic viscosity v
a, specific heat of the air C
p
a, and air thermal conductivity k
air, with formula [16]:
(17)
Pr
=
C
p
a
v
a
k
air
.By substituting the outputs of (11) and (16) in (15), the substitute equation (15) output in (14), the value of h
conv will be
(18)
h
conv
=
k
air
[
2
+
0.6
Pr
1
/
3
Re
1
/
2
]
D
d
.With the values ofh
c and m
˙
evap calculated from (16) to (24), the new temperature of the droplet T
d
2 can be estimated; this temperature will be the initial temperature of the second part of the water cooling system, the panel pond. The amount of water entering the water pond boundary system is the initial mass flow rate of the sprayer minus the evaporated losses:
(19)
m
2
s
=
m
1
s
-
(
m
˙
d
·
Δ
t
·
m
1
s
m
d
)
.


## 2.2.2. Radiative Panel Mathematical Modelling

Similar to the water droplet, the panel pond control volume can be represented as a typical thermodynamic control volume (Figure3). As can be seen, the heat transfer to and from the system boundary is in the form of evaporation, conduction, convection, solar, and radiation, in addition to heat transfer due to water entering and leaving the system.Figure 3
Water pond energy balance.Similar to the water droplet energy equation, the water pond energy equation is(20)
m
w
_
rp
·
C
p
w
·
T
rp
2
-
T
rp
1
Δ
t
=
Q
˙
cond
+
Q
˙
solar
-
Q
˙
conv
-
Q
˙
rad
_
sky
m
w
_
rp
·
C
p
w
·
T
rp
2
-
T
rp
1
Δ
t
=
-
Q
˙
evap
-
Q
˙
back
+
E
˙
wi
-
E
˙
wo
,
where m
w
_
rp is the mass of the water in the water pond (kg), Q
˙
cond is the heat added by condensed water (kJ/s), Q
˙
Solar is the heat absorbed by the panel water pond due to solar radiation (kJ/s), Q
˙
conv is the heat convection between the water pond and ambient air (kJ/s), Q
˙
rad
_
sky is the heat exchange between the water in the roof pond and the sky, by radiation (kJ/s), Q
˙
evap is the evaporation heat exchange between water and the ambient air (kJ/s), Q
˙
back is the heat transfer between the water and the ambient air through the panel layers (kJ/s), and E
˙
wi and E
˙
wo are the energy rates added to the panel pond water by the water added and water leaving the system boundary (kJ/s).As in (3) for the water droplet, the energy equation can be rewritten to calculate the temperature of the water leaving the panel pond:
(21)
T
rp
2
=
T
rp
1
+
Δ
t
m
w
_
rp
·
C
p
w
·
[
Q
˙
cond
+
Q
˙
solar
-
Q
˙
conv
-
Q
˙
rad
_
sky
T
rp
2
0000000000000000000
-
Q
˙
evap
-
Q
˙
back
+
E
˙
wi
-
E
˙
wo
]
.Calculations of heat and energy transfers between the system and the boundary can be derivatives using the basic heat transfer equation; many works presenting the mathematical modelling of the panel pond system can also be referenced [17–22]. The amount of heat dissipated to the sky by radiation is a function of the pond emissivity, the sky temperature, and the pond water temperature; other factors having an effect are the humidity ratio, clouds, and radiation wavelength which have a direct effect on changes to the emissivity and sky temperature values. The Q
˙
rad
_
sky is given by
(22)
Q
˙
rad
_
sky
=
h
rad
_
sky
·
A
rp
·
(
T
w
_
rp
-
T
sky
)
,
where h
rad
_
sky is the sky radiation heat transfer coefficient (W/m²·k), A
rp is the panel surface area (m²), T
w
_
rp is the temperature of the water in the pond (K), and T
sky is the effective sky temperature (K).During the day, solar heat is added to water in the pond area; the amount of heat added depends on the amount of solar irradiation flux(
I
solar
) and the panel pond’s solar absorptivity (
α
solar
). The heat absorbed by water from solar irradiation is given by
(23)
Q
˙
solar
=
α
solar
A
rp
I
solar
.Consider thatQ
˙
conv is the heat transfer from water to the ambient by convection; this heat can be calculated by
(24)
Q
˙
conv
=
h
rp
_
conv
A
rp
(
T
rp
-
T
amb
)
.The convection heat transfer coefficient (h
rp
_
conv) is a function of the type of flow rate, laminar flow, or turbulent flow, and the (h
rp
_
conv) general equation is
(25)
h
rp
_
conv
=
Nu
L
·
K
amb
X
rp
,
where K
amb and X
rp are the thermal conductivity of the ambient air and the distance water flows over the panel, while Nu
L is the Nusselt number, which is dissimilar to the water droplet (10); in the case of panel pond flow, its value depends on the nature of convection (forced or natural) [16]. The water mass evaporation rate from panel pond m
˙
rp
_
evap needs to estimate the value of heat dissipated from water by evaporation. By recalling (4) and replacing the droplet surface area by the panel surface area we get
(26)
m
˙
rp
_
evap
=
h
m
_
rp
·
A
rp
·
[
ρ
w
_
rp
-
ρ
atm
]
,
where ρ
w
_
rp and ρ
atm are the water density at saturation temperature pressure and atmospheric pressure, respectively (kg/m³), h
m
_
rp (m/s), the mass transfer coefficient is calculated from the Sherwood number (Sh
X
′), and the binary mass diffusion coefficient (D
vap) is calculated as
(27)
h
m
_
rp
=
Sh
X
′
D
vap
X
rp
.Under forced convection conditions, the Sherwood number is(28)
Sh
X
′
=
0.644
Re
X
1
/
2
Sc
1
/
3
and the Nusselt number is
(29)
Nu
X
=
0.644
Re
X
1
/
2
Pr
1
/
3
.When the value ofm
˙
rp
_
evap is calculated with (27), value above zero indicates that evaporation has taken place, and the heat transferred by evaporation from the water to the atmosphere is
(30)
Q
˙
evap
=
h
g
·
m
˙
rp
_
evap
.The energy of water entering and exiting from the system boundary (Δ
E
w) is
(31)
Δ
E
w
=
h
f
,
wi
m
˙
wi
-
h
f
,
wo
m
˙
wo
.Theh
f
,
wi and h
f
,
wo are the enthalpies of water entering and exiting the pond, and m
˙
wi
,
m
˙
wo are the mass flow rate of water entering pond. The m
˙
wo is estimated to be the entering water mass flow rate after subtracting the evaporating mass rate.The heat transfer through the back of the panel (Q
˙
back
) is a function of the back material thermal conductivity and is given by
(32)
Q
˙
back
=
h
b
rp
A
rp
(
T
w
_
rp
-
T
b
)
,
where T
b is the back layer temperature. The h
b
_
rp is the heat transfer coefficient between the back layer and the pond water:
(33)
h
b
_
rp
=
K
plate
d
x
plate
+
K
foam
d
x
foam
.Two identical panels with the dimensions of 2400 mm long and 900 mm wide were constructed from aluminum sheets insulated from the bottom to eliminate the heat transfer from the back side. The criteria that determined the panel size included the aim to make a portable prototype and to minimize the labor. As planned, the water will flow over the inclined panels during the night and will be presprayed in one case to evaluate the contribution of water spray over the panel.


## 3. Results and Discussion

Results presented in the following section are based on the simulation performance of the described system with TRNSYS software package, the standard weather data file TMY2, based on Australian weather data in summer (28/29 January) and on the following input data for the systems:(i)
total panel area for wet surfaceAws
=
6.00
m
2. Water Tank Volume VT
=
250 L.In the simulation, the panel is treated as a roof sloped by 30° facing south. The simulation performed in two different locations in Australia, that is, Sydney (humid weather) and Alice Spring (hot and dry area).The system cold water temperature variation during the assigned simulation period from 21:00 hours to 6:00 hours in the following morning is plotted in Figure4. The pump water flow rate varied in several simulation runs to optimize the system performance; the results show that the temperature in the tank dropped to 11.8°C by the end of the period in the clear night arid conditions of Alice Springs, using the open cycle system with sprayer. However, the high relative humidity rates in Sydney compared to Alice Springs (Figure 6) resulted in a less cooling rate in the water as shown in Figure 5. That is mainly related to the reduction of the heat transfer by evaporation (Q evap) from both drop and water layer.Figure 4
Water cooling with nocturnal cooling effect in Alice Springs using the system described in this work.Figure 5
Water cooling with nocturnal cooling effect in Sydney using the system described in this work.Figure 6
Relative humidity rate in the two locations.The convective heat transfer coefficient value from (6) and (24) is proportional to the wind velocity. Analysis of wind velocity data (Figure 7) against the storage tank water temperature in Figures 4 and 5 indicates that the convection effect added heat to the water, thus minimizing the cooling process by radiation. The cloudiness factor (C
clo) for the selected day did not exceed 0.19 on average in Sydney and 0.02 in Alice Springs; thus the negative effect of clouds was eliminated. The analysis of the simulation results, built on a simulation time step of 10 seconds, indicates that the system recorded higher net sky radiation value in Sydney than that in Alice Spring (Figure 8). This is strongly related to the absence of the high evaporation rate of water where the water temperature has dropped to a lower value and reduced the temperature difference between the water and the sky temperature (used in (7) and (22)).Figure 7
Wind velocity in the four locations.Figure 8
Net sky radiation in two different geographical locations.The net long wave radiations from the radiator panel in the 2 selected locations are 59.2 W/m² in Sydney and around 47 W/m² in Alice Springs (Figure8) calculated from (7) and (23). Although the relative humidity in Alice Springs is lower than in Sydney, the effect of the higher ambient temperature in Alice Springs reduced the rate of the net sky radiation rate.The storage tank water temperature in both cases was set to a start temperature of 25°C. Lower start-up temperature resulted in lower storage tank final temperature. For example, in Alice Spring weather, a start temperature of 22°C resulted in final temperatures that were 2.5°C lower than this for a start temperature of 25°C. But the results show that, even under unfavourable conditions, the systems have the capacity to cover a fraction of the cooling demand. Generally, the simulations indicate that sufficient radiative cooling is obtained in periods with modest or above air humidity and ambient night temperatures, cooling below 20°C in most cases.


## 4. Conclusion

In this research, a brief literature review of several researches covering the phenomenon of long wave radiation to the sky was presented. The thermal behaviour of a system utilizing this technology was described by a detailed mathematical modelling. These models simulated different climatic conditions. The simulation results specify that the net long wave radiation to a clear sky on an Australian summer night is sufficient to cool the water in a storage tank with a total volume of 260 litres to low temperature (as low as to 13 degrees lower than the starting temperature of 25°C). The mathematical modelling and simulation results will be used in future research to set up experimental prototype systems to validate the simulation results.



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*Source: 624502-2014-03-02.xml*