An efficient solar water desalination system using natural vacuum pressure

3.1. Vacuum parameters

As discussed earlier, after system startup, empty space is created in the top surface of seawater inside the evaporator. This empty space will be filled with vapor that evaporated from seawater, and then, pressure ($P_{sat}$) corresponding to the seawater temperature is created. The value of $P_{sat}$ is estimated from the follow equation [13]:

where $P_{sat}$ is the saturated pressure corresponding to the seawater saturated temperature ($T_{sat}$) at the top surface; $P_{atm}$ is the atmospheric pressure; $\rho$ is the density of seawater; $g$ is the acceleration of gravity; and $h$ is the height of a seawater column that is required to create $P_{sat}$. The height ($h$) is given by:

We see that $h$ is a function of seawater temperature and salinity, since $P_{sat}$ changes with seawater temperature, also $\rho$ changes with seawater temperature and salinity and was calculated from the correlations given by El-Dessouky and Ettouney [14] as:

where:

where, $\rho$ is the seawater density in kg/m³, $S$ is the salinity of seawater in ppm (parts per million), and $T$ is the temperature of seawater in ℃. The red sea is one of the most saline seas in the world. The salinity is about 41,000 ppm (41 g/kg). Also, the red sea is warm with temperature ranges from 20 ℃ to 31 ℃ [15]. In our calculations, we will consider the temperature of seawater entering the system to be $T_1=$ 20℃. El-Dessouky and Ettouney [14] suggested an equation to calculate $T_{sat}$ corresponding to $P_{sat}$:

where $P_{sat}$ is in kPa and $T_{sat}$ will be in ℃.

3.2. Evaporation calculations

The equation of mass conservation of seawater in the evaporation chamber is given by:

where ${\dot{m}}_i$ is the mass flow rate of injected seawater (kg/s); ${\dot{m}}_e$ is the rate of evaporated seawater (also called evaporation rate kg/s); ${\dot{m}}_b$ is the mass flow rate of rejected seawater (also called brine kg/s). The parameters used in the equations of this section are shown in Fig. 2. The evaporating rate of seawater (${\dot{m}}_e$) is estimated from the excessive thermal energy (${\dot{Q}}_{exc}$) contained in seawater. In other words, when the temperature of heated seawater ($T_{hot}$) become higher than the saturated temperature ($T_{sat}$) by ${∆T}_{sat}$, the seawater contains an excessive thermal energy. This excessive thermal energy is given by [16]:

where $c_{p,i@Thot}$ is the specific heat of seawater (evaluated at $T_{hot}$). Therefore, the amount of evaporated seawater, also called evaporation rate, is calculated as:

where $h_{fg}$ is the specific enthalpy (kJ/kg) required to evaporate 1 kg of seawater at saturated conditions ($P_{sat}$ and $T_{sat}$). The value of $h_{fg}$ is determined using steam tables, or using the following equation suggested by [16, 17]:

Fig. 2Schematic diagram of the first cycle

The calculation of the specific heat of seawater depends on its temperature and salinity. In order to calculate it, we will use the following correlation suggested by El-Dessouky and Ettouney [14] as follow:

where:

where, $S$ is the salinity of seawater in (g/kg) or in (part per thousand). The amount of thermal energy (${\dot{Q}}_{heating}$) supplied by the FPC is calculated as:

where $c_{p,@Tp4}$ is the specific heat of pure water evaluated at $T_{P4}$. Due to this amount of heat, the seawater (inside the evaporation chamber) will acquire sensible and latent heats. The temperature of seawater will rise from $T_4$ to $T_5$ (sensible heat). Therefore, ${\dot{Q}}_{heating}$ is given by:

where ${\dot{Q}}_{exc}$ (= ${\dot{m}}_e{h}_{fg}$) is the latent heat of evaporated seawater. Assuming that the evaporation chamber is well-insulated, the rate of heat losses is negligible. The surface area of the evaporator (for heat transfer) is given by:

where $U_{eva}$ is the overall heat transfer coefficient in the evaporator; and ${\left(LMTD\right)}_{eva}$ is the Logarithmic Mean Temperature Difference and evaluated by considering the counter-flow. $U_{eva}$ and ${\left(LMTD\right)}_{eva}$ are calculated as given in Eq. (13) and Eq. (14), respectively [17]:

The effectiveness of the evaporator tube is given in Eq. (15):

LMTD method is appropriate when the inlet and outlet temperatures are known. Therefore, this method is suitable to determine the size (surface area) of heat exchanger. When the outlet temperatures are unknown, it is difficult to use LMTD method. Therefore, $\epsilon$-NTU method is preferred. The ε-NTU approach is used to study and analyze the performance of heat exchangers and also to predict the outlet temperatures of the working fluids [18]. $\epsilon$-NTU approach is based on the effectiveness of heat exchanger. For our system, the effectiveness of the evaporator is given by:

Fig. 3Energy balance on the evaporator

Another method used to estimate the temperature difference in heat exchangers is Arithmetic Mean Temperature Difference (AMTD). This method is preferred to avoid the complexity of LMTD method especially when the temperatures are unknown [18]. Eq. (12) can be written as:

where ${\left(\mathrm{A}MTD\right)}_{eva}$ is calculated as [51]:

The brine has a salinity which is higher than that injected seawater. The salinity of brine ($S_b$) is calculated using Eq. (20). Since:

Therefore, salinity of brine is:

3.3. Condensation calculations

The pure water (${\dot{m}}_p$), that is used to condense the evaporated seawater in the condenser, is preheated before entering the FPC. This pure water is preheated by absorbing the thermal energy of evaporated seawater (${\dot{Q}}_{cond}$). This energy is the same value of excessive thermal energy estimated from Eq. (6). In other words, ${\dot{Q}}_{exc}$= ${\dot{Q}}_{cond}$, assuming that the evaporated steam is condensed completely. Therefore, the temperature of cooling pure water leaving the condenser (entering to the FPC) is calculated as:

where $c_{p@Tp2}$ is the specific heat of pure water and we will evaluate it at $T_{P2}$. Because it is not important to use the exact value of $c_p$, and to simplify the calculations, we will evaluate $c_p$ at $T_{P2}$.

The surface area of the condenser tube (for heat transfer) is given by:

where $U_{cond}$ is the overall heat transfer coefficient in the condenser; and ${\left(LMTD\right)}_{cond}$ is the Logarithmic Mean Temperature Difference. $U_{cond}$ and ${\left(LMTD\right)}_{cond}$ are calculated as given in Eq. (23) and Eq. (24), respectively [17]:

Fig. 4Schematic diagram of the second cycle

The effectiveness of the condenser tube is given in Eq. (25):

As mentioned in the previous section, we use $\epsilon$-NTU method when the temperatures are unknown. For condensation calculations, Equ. (22) can be written as:

where ${\left(\mathrm{A}MTD\right)}_{cond}$ is calculated as:

The effectiveness of the condenser is calculated using the ε-NTU method as following [18]:

where $NTU$ is the number of transfer units and we use Eq (29) to estimate $NTU$:

3.4. Tanks and pipes calculations

The pure water leaving the evaporation chamber (with $T_{P5}$) is mixed with the distilled water according to Eq. (30) and Eq. (31). After that, this fresh water is used to preheat the inlet seawater according to Eq. (32):

where ${\dot{Q}}_{fresh}$ is the rate of heat transfer in the fresh water tank. This amount of heat is transferred from the fresh water tank to preheat the seawater. Eq. (31) and Eq. (32) are then used to calculate the temperatures of fresh water ($T_9$) and seawater ($T_2$), respectively. The tanks and pipes are treated as double-pipe heat exchangers with counter flow. The surface area of heat transfer in these is calculated as following:

where $U_{fresh}$ is the overall heat transfer coefficient in the fresh water tank which is calculated using Eq. (13) at $T_1$; and ${\left(LMTD\right)}_{fresh}$ is the Logarithmic Mean Temperature Difference and is given by:

Arithmetic Mean Temperature Difference (AMTD) method may be used instead of LMTD. Then Eq. (33) can be written as:

where ${\left(AMTD\right)}_{fresh}$ is calculated using Eq. (36) and the effectiveness of the fresh water tank is given in Eq. (37):

The inlet seawater inters the second tank with a temperature of $T_2$ and preheated again in this tank and the riser pipe. The amount of heat used for preheating the seawater in this stage (${\dot{Q}}_{tank}$ plus ${\dot{Q}}_{piping}$) is transferred from the brine that leaves the evaporator with a high temperature. The temperature of seawater increases from $T_2$ to $T_3$ in the seawater tank, then from $T_3$ to $T_4$ in the riser pipe. In the other hands, the temperature of the brine decreases to $T_7$ at the outlet of brine tank. The governing equations of this stage are the same manner of the fresh water tank. These equations are given below:

The values of $U_{tank}$ and $U_{piping}$ are determined using Eq. (13) at $T_2$ and $T_3$, respectively. Arithmetic Mean Temperature Difference (AMTD) method may be used instead of LMTD. Therefore, Eq. (40) and Eq. (41) can be written as Eq. (44) and Eq. (45), respectively:

where ${\left(AMTD\right)}_{tank}$ and ${\left(AMTD\right)}_{piping}$ are given by Eq. (46) and Eq. (47), respectively. The effectivenesses of the seawater tank and pipes are given by Eq. (48) and Eq. (49), respectively:

3.5. Solar collector

In this study, FPC is used. The performance of solar collectors varies by the amount of solar radiation, the time and season and other factors such as wind velocity and the ambient temperature and. In order to evaluate the temperature of water exiting from the collector, we should use Eq. (52). The efficiency of FPC is the ratio of a thermal energy (${\dot{Q}}_{gain}$) gained by water (the working fluid) to the solar energy incident on the solar collector [19] and given by:

where ${\dot{m}}_P$ is the mass flow rate of the pure water flowing in the collector; $c_{p@Tp3}$ is the specific heat at constant pressure of the pure water flowing in the collector; $T_{P4}$ is the outlet temperature of water from the collector; $T_{P3}$ is the inlet temperature of water to the collector; $A_{col}$ is the effective area of an FPC; and $G$ is the solar irradiation per unit area. The efficiency of FPC varies by the temperature difference ($∆T$) between the working fluid and the ambient air (i.e., $∆T=T_{P3}-T_{amb}$). According to the review study by S. A. Kalogirou [20], the efficiency of FPC is inversely proportional to $∆T$ (i.e., ${\eta }_{col}$ increases linearly as $∆T$ decreases) as shown in Table 2. To find the efficiency of FPC at any value of $G$ and $∆T$, we will use the following relation [19]:

For a given solar radiation, FPC area, water flow rate and inlet temperature, and ambient temperature, we can calculate the outlet temperature of water as:

3.6. Evaporation ratio (ER)

The evaporation ratio (ER) is the ratio of the distilled water production. In other words, ER is the amount of distilled water to the amount of injected seawater. ER is given by:

3.7. Gain output ratio (GOR)

The gain output ratio (GOR) is the ratio between the latent heat of the evaporated seawater (${\dot{m}}_e{h}_{fg}$) and the supplied heat (${\dot{Q}}_{heating}$). GOR is given by [21]:

3.8. Specific productivity (SP)

The specific productivity (SP) is the ratio of the distilled water in kg/h (kilograms per hour) to the heat supplied by the FPC in kW. SP is given by:

where ${\dot{m}}_d$ is the rate of water produced (distilled water in kg/s) which is equal to the evaporation rate (${\dot{m}}_e$) as following:

3.9. Thermal efficiency

The thermal efficiency of the system, which is defined as the ratio of useful energy (${\dot{Q}}_{exc}$) to the solar energy input (${\dot{Q}}_{th,in}$), is calculated using Eq. (57). ${\dot{Q}}_{th,in}$ is the solar energy absorbed by the FPC, and given by Eq. (58):

3.10. Basic considerations for calculations

In order to initiate the calculations and get the results, we should consider some assumptions. These assumptions are summarized as follow:

1) Seawater properties: the temperature and salinity of seawater play important roles when designing the SNVD system. Seawater with high salinity has high density, and then lower height for the chambers is required. Also, the density and specific heat of seawater depends on the temperature and salinity. In our calculations, we considered the average properties of the Red Sea (i.e., $S=$41,000 ppm and $T_1=$ 20 °C).

2) Solar irradiation ($G$): the operation of the proposed SNVD system depends on the solar energy and weather conditions. The solar radiation incident on the FPC and the ambient temperature vary along the day, therefore, the production of distilled water will be variable. In order to estimate the area of FPC required to heat the working fluid (pure water), we will consider average values for solar irradiation and ambient temperature (in 21th December) as 409 W/m² and $T_{amb}=$ 27.5 °C, respectively (Uqu weather station).

3) All the evaporated seawater (${\dot{m}}_e$) is completely condensed and then converted into distilled water (${\dot{m}}_d$). These considerations mentioned above are shown in Table 3.

3.11. Calculations methodology

The calculations required for designing the proposed SNVD system are done utilizing Engineering Equation Solver (EES) software. These calculations are done in two steps. The first step is the initial calculations, from which we want to determine the appropriate area of the FPC ($A_{col}$), surface area of the evaporator and condenser ($A_{eva}$ and $A_{cond}$), the height of evaporation and condensation chambers ($h$), and the rate of pure water (${\dot{m}}_p$). For this purpose, we, initially, assumed some conditions, for example, the seawater inside the evaporator is to be heated 15 °C above the saturation temperature (i.e., ${∆T}_{sat}{=T}_{hot}-T_{sat}=$ 15 °C). The other initial conditions assumed are listed in Table 4.

From the initial calculations, we can estimate the values of $h$, $A_{col}$, $A_{eva}$, $A_{cond}$, ${\dot{m}}_i$ and ${\dot{m}}_p$. The second step is to use these values as additional inputs to determine all the parameters of our system. From these calculations we can analyze and study the performance of our system during the year. The performance of the system varies with time depending on the change in the solar irradiation ($G$) incident on the tilted solar collector. We should note that the basic considerations listed in Table 3 are used for both initial and analysis calculations.

4.1. Summary of system sizing

The sizing of the proposed SNVD system is obtained by executing the initial calculations as discussed in Section 3.11 and based on the basic considerations and conditions in Table 3 and Table 4. The sizing of the system is summarized below, in Table 5, including the size of the evaporator, condenser, tanks, pipes, and solar collector, in addition, the amount of pure water (${\dot{m}}_p$) required for condensing and heating. The results of simulation are based on a constant injected seawater of 0.1 kg/s.

4.2. Annual operation and performance simulation

The performance of the system is simulated hourly throughout the year. The distilled water production rate, ${\dot{m}}_d$, in kilogram per hour is shown in Fig. 5. The outlet temperature from the solar collector, $T_{p4}$, and the temperature of seawater inside the evaporator, $T_{hot}$, are shown in Fig. 6 and Fig. 7, respectively.

Fig. 5Distilled water throughout the year (kg/hour)

Fig. 6Outlet temperature from solar collector

Fig. 7Temperature of seawater inside the evaporator

Fig. 8 represents the production rate of water per day throughout the year. The maximum production of distilled water is achieved during Spring season, especially in March and April. The production of our proposed system in that duration reaches 92.88 kg per day, which is equivalent to 2.13 % of the seawater injected. In addition, in summer season, the system achieves maximum gain output ratio ($GOR$), specific productivity ($SP$), and thermal efficiency (${\eta }_{th}$) of 0.98, 1.45 kg/kWh, and 39.6 %, respectively.

Fig. 8Daily distilled water and evaporation rate

The accumulated amount of distilled water per month exceeds 2500 kg in April, May and July. It reaches 2625 kg, as a maximum value of production, and is achieved in July, while in January the system produces minimum distilled water of 1861 kg. The production of water per month is shown in Table 6 and Fig. 9. Also, the monthly average evaporation ratio ($ER$), monthly average specific productivity ($SP$), and monthly average $SP$ are represented in Table 6 and illustrated in Fig. 10 and Fig. 11.

Fig. 9Accumulated distilled water per month

Fig. 10Monthly average evaporation ratio (ER) and monthly average specific productivity (SP)

Fig. 11Monthly average GOR

4.3. Characteristics of the SNVD system

In order to analyze the performance of the SNVD system evidently, we executed a simulation for the system operation throughout a single day (21st December). All the parameters required for analyzing the performance are discussed in this section. These parameters comprise the temperatures, heat flow, efficiencies, evaporation rate and ER.

Fig. 12Temperatures in the first cycle

The temperature distribution along the first and second cycles are shown in Fig. 12 and Fig. 13, respectively. These figures show that the temperatures along the system increase with time as the irradiation increase until midday at which the solar radiation is peak value, then decreasing gradually with time. We can note, from these two figures, that all temperatures show the same trend, however, the outlet temperature from solar collector ($T_{p4}$) shows a slightly different profile. This is because $T_{p4}$ is affected strongly by solar radiation.

Fig. 13Temperatures in the second cycle

Fig. 14Heat flow and solar irradiation

Fig. 15Accumulated distilled water in 21th December

The useful heat (${\dot{Q}}_{heating}$) and the input heat (${\dot{Q}}_{th,in}$) are represented in Fig. 14. The useful heat increases with time until peak value of 16.3 kJ/s at 1:00 PM, then decreases with time. Note that the useful heat (${\dot{Q}}_{heating}$) and the input heat (${\dot{Q}}_{th,in}$) increase rapidly, with solar radiation, from 9:00 AM to 11:00 AM. The system produces maximum amount of distilled water at midday (12:00 AM-2:00 PM). The temperature of brine leaving the system ($T_7$) is relatively high compared to the temperature of injected seawater ($T_1$). In addition, the temperature of pure water exiting the evaporation chamber ($T_{p5}$) is relatively high compared to $T_{p1}$. Fig. 15 represents accumulated distilled water in 21st December and the instantaneous rate of water production.