Optimum Solar HDH Desalination for Semi-Isolated Communities Using HGP and GA’s

1 Copyright © 2014 by ASME

OPTIMUM SOLAR HDH DESALINATION FOR SEMI-ISOLATED COMMUNITIES

USING HGP AND GA’S

Khalid M. Abd El-Aziz Graduate Teaching & Research Assistant

Dept. of Mechanical Design and Production Cairo University, Cairo, Egypt

[email protected]

Karim Hamza Senior Research Fellow

Dept. of Mechanical Engineering University of Michigan, Ann Arbor, MI 48109-2102

[email protected]

Mohamed El Morsi Associate Professor

Dept. of Mechanical Engineering American University in Cairo, New Cairo, 11835, Egypt

Ain Shams University, Cairo, 1156, Egypt [email protected]

Ashraf O. Nassef Professor

Dept. of Mechanical Engineering American University in Cairo, New Cairo, 11835, Egypt

[email protected]

Sayed M. Metwalli Professor Emeritus

Dept. of Mechanical Design and Production Cairo University, Cairo, Egypt

[email protected]

Kazuhiro Saitou Professor

Dept. of Mechanical Engineering University of Michigan, Ann Arbor, MI 48109-2102

[email protected]

ABSTRACT Modeling and unit-cost optimization of a water-heated

humidification-dehumidification (HDH) desalination system

were presented in previous work of the authors. The system

controlled the saline water flow rate to prevent salts from

precipitating at higher water temperatures. It was then realized

that this scheme had a negative impact on condensation

performance when the controlled flow rate was not sufficiently

high. This work builds on the previous system by disconnecting

the condenser from the saline water cycle and by introducing a

solar air heater to further augment the humidification

performance. In addition, improved models for the condenser

and the humidifier were used to obtain more accurate

productivity estimations. The Heuristic Gradient Projection

(HGP) optimization procedure was also refactored to result in

reduced number of function evaluations to reach the global

optimum compared to Genetic Algorithms (GA’s). A case study

which assumes a desalination plant on the Red Sea near the city

of Hurghada is presented. The unit-cost of produced fresh water

for the new optimum system is $0.5/m3 compared to $5.9/m

3

for the HDH system from previous work and less than the

reported minimum cost of reverse osmoses systems.

NOMENCLATURE 𝐴 surface area [m

2]

𝑎 humidifier specific area [m2/m

3]

𝑏 incidence angle modifier coefficient

𝐶 cost [$]

𝑐 specific heat capacity [J/kg K]

𝑐𝑝 inflation and yearly maintenance cost factor

𝐸𝑂𝑇 equation of time [minute]

𝐹𝑅 heat removal effectiveness factor

ℎ heat transfer coefficient [W/ m2 K]

𝐻ℎ𝑢𝑚 humidifier packing height [m]

𝐼𝐴𝑀 incidence angle modifier

𝐼 irradiance [W/m2]

𝑖 specific enthalpy [kJ/kg dry air, kJ/kg water]

∆𝑖𝑣 latent heat of vaporization [kj/kg]

𝑘 thermal conductivity [W/m K]

𝐾 mass transfer coefficient [kg/ m2s]

𝐿𝐶 longitude correction [hour]

𝐿𝐶𝑇 local clock time [24-hour format]

�̇� mass flow rate [kg/s]

�̇�𝑠 specific fresh water production [kg/m2 day]

𝑚𝑟 mass flow ratio [kg air/kg water]

Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference

IDETC/CIE 2014 August 17-20, 2014, Buffalo, New York, USA

DETC2014-34598

2 Copyright © 2014 by ASME

𝑁 day number [0-365]

�̇� heat transfer [W]

𝑟 tube radius [m]

𝑇 temperature [K]

𝑡𝑝 plant lifetime

𝑡𝑠 solar time [hour]

𝑈 overall heat transfer coefficient [W/m2K]

𝑈𝐿 heat loss coefficient [W/m2 ⁰C]

𝑉 volume [m3]

𝑉𝐷 daily fresh water production [m3/day]

�̇� volume flow rate [m3/s]

�̇�𝑠 specific air flow rate [m3/m

2 s]

𝑌 coordinate along humidifier height [m]

𝛽 collector tilt angle [rad]

𝛿 declination angle [rad]

𝛾 collector azimuth angle [rad]

𝜔 humidity ratio [kg vapor/kg dry air], hour angle [rad]

∅ latitude angle [rad]

𝜌 reflectance of surrounding surface

(𝜏𝛼) effective transmittance-absorptance product

𝜃𝑖 solar angle of incidence [rad]

Subscripts

1 distillate layer

𝑎 air

𝑎𝑚𝑏 ambient

𝐵 brine

𝑏, 𝑛 beam normal

𝑐 collector

𝑐𝑜 coolant

𝑐𝑜𝑛𝑑 condenser

𝐷 distillate

𝑑, ℎ diffuse horizontal

𝑑𝑏 dry bulb

𝑓, 𝑖 fluid at inlet

ℎ𝑢𝑚 humidifier

𝑖 interface

𝑠 saturation

𝑡 transferred heat

𝑡, ℎ total irradiance falling on a horizontal surface

𝑢 useful energy gain

𝑣 vapor

𝑤 water

𝑤𝑏 wet blub

1. INTRODUCTION Although a person requires about 2-5 liters of freshwater

per day for direct consumption, most of human water

requirement is consumed in agriculture and industry. Growth in

these sectors is, therefore, limited by the amount of available

fresh water, which is only 3% of Earth’s water content. This has

long motivated development of seawater desalination systems.

However, conventional desalination technologies are energy

intensive and rely mainly on fossil fuels which are becoming

less available over time.

Desalination using solar energy is friendly to the

environment and relies on a renewable energy source (RES).

The challenge remains for solar water desalination systems to

be cost-effective. Usually, the major cost component in these

systems is the initial investment, since their reliance on RES

allows for low running costs when compared with conventional

systems.

In Egypt, the Nile River is the main source of fresh water.

However, many locations on both the Red Sea and the

Mediterranean Sea are too far from the Nile River to have a

continuous fresh water supply. In this study, we focus on such

remote locations which require a small to medium-scale

desalination plant with a relatively low initial investment.

Several technologies may be utilized to effect water

desalination using solar energy. Reverse Osmosis (RO) is one

of the most widely-used technologies. It depends on pushing

the saline water through a nano-filter (membrane) by a high-

pressure pump, effectively allowing only the water molecules

to pass through the filter leaving salt and most unwanted

constituents behind [1]. RO is known to be a highly productive

process: reported unit-cost of produced fresh water is $2-3/m3

[2]. However, RO desalination systems require expertise in

their installation and maintenance, making them less suitable

for decentralized water production in remote regions. Also, RO

desalination systems are sensitive to inlet water salinity, further

discouraging their use, because the Red Sea is one of the most

saline open water surfaces in the world [3].

HDH desalination is another technology that was studied in

various configurations by a lot of previous work in literature.

The main concept is that air capacity for water vapor increases

logarithmically with temperature [4]. So, if the air is heated to a

modest temperature and passed through a saline water stream, it

will carry water vapor free of salts. The air can then be cooled

so as the water vapor will condense into fresh water.

HDH desalination is usually coupled with solar-thermal

rather than thermal-electric technologies used in case of RO.

Solar-thermal technologies are usually less expensive and

easier to maintain. Also, solar HDH desalination systems

require less expertise in installation and maintenance making

them more suitable for remote regions. In addition, because the

effectiveness of the humidification process is only mildly

affected by inlet water salinity content [5], HDH desalination is

less sensitive to salinity than RO. However, Narayan et al. [6]

reported a unit-cost of $3-7/m3 for produced fresh water using

HDH desalination which is higher than unit-cost of RO. This

work is a continuation of effort to reduce the unit-cost of HDH

desalination in order to take advantage of its versatility.

This paper briefly describes a model of an HDH

desalination system from previous work of the authors [7].

Improvements to that system are then introduced along with

their respective reasons. The model equations for the modified

system are derived, and it is then optimized using HGP. Finally,

comparison of optimum points using HGP and GA’s for both

systems is presented.

3 Copyright © 2014 by ASME

2. RELATED WORK

2.1 Variable Water Flow Rate El-Morsi et al. [8] modeled and optimized a single-stage

water-heated system. Similar to other systems, the condenser is

cooled using inlet saline water in order to also preheat it. A

genetic algorithm (GA) was used for optimizing the system.

The optimum system yielded 4.8 kg fresh water/m2 of solar

collector area per day, and the unit cost of produced fresh water

was $8/m3. The system addressed salt precipitation at high

saline water temperatures by using a solar water heater that will

at most reach the temperature limit for the design weather

conditions. However, in conditions other than maximum, the

water temperature will drop below the limit which had a

negative effect on humidification performance.

To address this issue, Abd El-Aziz et al. [7] modified this

system to control saline water temperature by variating its flow

rate. A closed-loop solar water heater was used to allow the

heat storage medium to reach its full capacity. The modified

system was optimized using a tailored optimization technique

based on HGP. The system yielded 8.3 kg/m2 day. And the

optimum unit-cost of produced fresh water was $5.9/m3. The

temperature control ensured the water will always be at the

temperature limit for maximum humidification performance.

However, when the saline water flow rate was not sufficiently

high, condensation performance was negatively affected. As

will be shown in section 3, this issue is also addressed in this

paper.

2.2 Air-heated Systems Narayan et al. [6] reviewed a wide range of system designs

including water and air-heated systems. The authors noted a

disadvantage of the air-heated cycle: Air temperature drops in

the humidifier as it exchanges heat with saline water.

Eventually, condensation takes place at a low temperature

which has a negative effect on condensation performance.

Farsad and Behzadmehr [9] modeled a hybrid cycle that

heats both air and saline water. This overcomes the

disadvantage of the air-heated cycle because air will leave the

humidifier at a higher temperature.

Orfi et al. [10] also modeled and simulated a system that

heats both air and saline water and reported daily production of

43 kg/m2 of solar collector area per day. It was also concluded

that an open air system is more productive than a closed air

system. However, the system had no mechanism of controlling

saline water temperature. Also, the condenser was being cooled

by the inlet saline water stream similar to many other

configurations [11, 12, 13, 14], but this does not necessarily

imply minimum unit-cost of production. In this paper, we will

model and optimize a similar system that addresses these

issues.

3. SYSTEM MODEL

3.1 Overview The diagram in Fig. 1 shows the system from previous

work of the authors [7]. It is similar to many other systems in

literature but adds a controller to change water and air flow

rates based on temperature reading from state 3 of Fig. 1. This

effectively allows the water temperature to stay within a preset

limit (In this case, 60 °C).

This model was used as basis for design modifications

which are shown in Fig. 2:

Condensation performance is negatively affected when

automatic saline water flow rate is not sufficiently

high. Therefore, the condenser was disconnected from

the inlet water stream to operate on an independent

coolant supply with constant flow rate.

Saline water temperature is limited to 60 °C, air

capacity for water vapor will be less than 0.153 kg

vapor/kg air (if we assume air is saturated at state 5).

Therefore, a solar air heater was added before the

humidifier to allow for higher outlet temperatures and

humidity ratios.

The solar tank of the solar water heater is an expensive

component, and it will not be very useful in

conjunction with the solar air heater because the latter

can’t operate at night. The solar tank was, therefore,

removed.

Fig. 1: Schematic diagram for original system

Fig. 2: Schematic diagram for modified system

4 Copyright © 2014 by ASME

The components of the modified system are as follows:

i. Pump: pumps saline water through the system

ii. Solar water heater: a flat plate solar collector for

heating inlet water (state 1).

iii. Controller: controls water flow rate according to

temperature reading at state 2.

iv. Solar air heater: packed bed solar air heater that heats

inlet air (state 3).

v. Humidifier: uses hot saline water (state 2) to humidify

heated ambient air (state 4) to produce hot humid air

(state 5) and brine (state B).

vi. Condenser: uses a coolant circuit (e.g. water) to cool

humid air (state 5) to produce distillate (state D)

and cool air (state 6).

The following assumptions were made before writing the

system model equations:

1. Heat losses to the atmosphere by the condenser,

humidifier and piping are neglected.

2. Kinetic and potential energy changes for air and water

are neglected.

3. Temperature rise due to pumping and air circulation

are neglected.

4. Thermodynamic properties of seawater are

approximated to that of pure water [5].

Air is saturated at condenser exit.

3.2 Global solar irradiance In order to accurately determine the energy gain from a

solar collector, the global aperture irradiance 𝐼 is calculated

from [15]:

𝐼 = 𝐼𝐴𝑀 {𝐼𝑏,𝑛 cos 𝜃𝑖 + [𝐼𝑑,ℎ (1+cos 𝛽

2) + 𝜌𝐼𝑡,ℎ (

1−cos𝛽

2)]} (1)

Ib,n, Id,h and It,h are calculated from hourly weather data for

the location and 𝜌 may be taken as 0.2 (typical reflectivity for a

concrete ground). 𝛽 is set equal to the latitude angle of the

location for maximum performance. cos 𝜃𝑖 is calculated from

the following equation [15]:

cos 𝜃𝑖 = cos 𝛽 (sin 𝛿 sin ∅ + cos 𝛿 cos ∅ cos𝜔)− cos 𝛿 sin𝜔 sin 𝛽 sin 𝛾+ sin 𝛽 cos 𝛾 (sin 𝛿 cos∅− cos 𝛿 cos𝜔 sin ∅)

(2)

For a collector facing south, 𝛾 = 𝜋, and ∅ depends on the

location. 𝛿 and 𝜔 are given by [15]:

sin 𝛿 = 0.39795 cos[0.98563(𝑁 − 173)] (3)

𝜔 = 15(𝑡𝑠 − 12) (𝑑𝑒𝑔𝑟𝑒𝑒𝑠) (4)

The equation of time (𝐸𝑂𝑇) is given by [15]:

𝐸𝑂𝑇 = 0.258 cos 𝑥 − 7.416 sin 𝑥 + 3.648 cos 2𝑥− 9.228 sin 2𝑥

(5)

Where the angle 𝑥 is defined as a function of the day

number, 𝑁 [15]:

𝑥 =360(𝑁 − 1)

365.242 (𝑑𝑒𝑔𝑟𝑒𝑒𝑠) (6)

𝑡𝑠 can then be calculated from local clock time (𝐿𝐶𝑇) using

[15]:

𝐿𝐶𝑇 = 𝑡𝑠 −𝐸𝑂𝑇

60+ 𝐿𝐶 + 𝐷 (7)

Where longitude correction (𝐿𝐶) is calculated from [15]:

𝐿𝐶 =

(𝑙𝑜𝑐𝑎𝑙 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑒) − (𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑

𝑡𝑖𝑚𝑒 𝑧𝑜𝑛𝑒 𝑚𝑒𝑟𝑖𝑑𝑖𝑎𝑛)

15

(8)

Where 𝐷 = 1 if daylight saving time is in effect, and 0

otherwise. The incidence angle modifier (𝐼𝐴𝑀) can be

calculated from [15]:

𝐼𝐴𝑀 = 1 − 𝑏 (1

cos 𝜃𝑖− 1) (9)

The coefficient b is a property of the solar collector. This

effectively allows the formulation to represent tubular as well

as flat plate collectors.

3.3 Solar Water Heater The energy gain from a flat-plate solar collector is

calculated via the Hottel-Whillier-Bliss equation [16]:

�̇�𝑢 = 𝐹𝑅𝐴𝐶[(𝜏𝛼)𝐼 − 𝑈𝐿(𝑇𝑓,𝑖 − 𝑇𝑎𝑚𝑏)] (10)

𝐹𝑅, (𝜏𝛼) and 𝑈𝐿 are all specific to the solar collector. 𝑇𝑓,𝑖 is

taken as inlet saline water temperature and 𝐼 is calculated as

shown in section 3.2. The energy balance for the saline water

stream across the solar collector is given by:

�̇�𝑢 = �̇�𝑤(𝑖2 − 𝑖1) (11)

For variable flow rate, the controller will constantly change

�̇�𝑤 to satisfy eqn. (10) and (11), i.e.:

�̇�𝑤 =𝐹𝑅𝐴𝐶[(𝜏𝛼)𝐼 − 𝑈𝐿(𝑇𝑓,𝑖 − 𝑇𝑎𝑚𝑏)]

(𝑖2 − 𝑖1) (12)

In the above equation, 𝑖2 is the enthalpy of water at 60 ⁰C

and atmospheric pressure.

3.4 Solar Air Heater The performance parameters of the solar collector, 𝐹𝑅,

(𝜏𝛼) and 𝑈𝐿 are originally affected by fluid flow rate. In case of

liquid heaters, the variation in these parameters can be

neglected. For air heaters, on the other hand, the variation is

significant [17]. Therefore, an experimental correlation for a

packed bed solar air heater [18] was used instead of eqn. (10).

If we define �̇�𝑠 as the volume flow rate of air passing

through the collector per unit area of the collector, the

correlation is:

5 Copyright © 2014 by ASME

(𝑇4 − 𝑇𝑎𝑚𝑏)

𝐼= 𝐶1�̇�𝑠 + 𝐶2 (13)

In the above equation, 𝐼 is calculated as shown in section

3.2. Also, for packed bed air heaters, 𝐶1 = −2.111 and

𝐶2 = 0.07822 [18].

3.5 Condenser The energy and mass balances across the condenser are:

�̇�𝑐𝑜(𝑖𝑐𝑜,𝑜𝑢𝑡 − 𝑖𝑐𝑜,𝑖𝑛) = �̇�𝑎𝑖𝑟(𝑖5 − 𝑖6) − �̇�𝐷𝑖𝐷 (14)

�̇�𝑎𝑖𝑟𝜔5 = �̇�𝑎𝑖𝑟𝜔6 + �̇�𝐷 (15)

In addition, the greatest amount of heat the condenser is

able to transfer is:

�̇�𝑐𝑜𝑛𝑑 = 𝐴𝑐𝑜𝑛𝑑𝑈𝑐𝑜𝑛𝑑(𝑇𝑑𝑏,5 − 𝑇𝑐𝑜,𝑜𝑢𝑡) − (𝑇𝑑𝑏,6 − 𝑇𝑐𝑜,𝑖𝑛)

ln(𝑇𝑑𝑏,5 − 𝑇𝑐𝑜,𝑜𝑢𝑡)

(𝑇𝑑𝑏,6 − 𝑇𝑐𝑜,𝑖𝑛)

(16)

�̇�𝑐𝑜𝑛𝑑 = �̇�𝑐𝑜(𝑖𝑐𝑜,𝑜𝑢𝑡 − 𝑖𝑐𝑜,𝑖𝑛) (17)

In eqn. (16), 𝑈𝑐𝑜𝑛𝑑 is determined by evaluating its three

components: the heat transfer coefficient on the water side, for

the condenser tubes and on the humid air side. For a shell and

tube condenser, it can be calculated from [19]:

1

𝑈𝑐𝑜𝑛𝑑=1

ℎ𝑤+ (

𝑟𝑜𝑘𝑤𝑎𝑙𝑙

) ln (𝑟𝑜𝑟𝑖) + (

𝑟𝑜𝑟𝑖⁄

ℎ𝑎) (18)

Where ℎ𝑤 is the heat transfer coefficient on the water side.

It can be obtained from the Dittus-Boelter correlation [19]:

ℎ𝑤 = 0.023(𝑅𝑒𝑤)0.8(𝑃𝑟𝑤)

0.333(𝑘𝑤𝐷𝑖) (19)

A condensate layer will separate the humid air from the

condenser tubes. Therefore, ℎ𝑎 will have two components: one

for the condensate layer and another for the air stream [19]:

1

ℎ𝑎=1

ℎ1+ (

𝑍𝑔

ℎ𝑔) (20)

Where ℎ1 is the coefficient of the condensate layer which

is given by [19]:

ℎ1 = 0.943 [𝑘1

3𝜌1(𝜌1 − 𝜌𝑔)𝑔∆𝑖𝜗

𝜇1(𝑇𝑠 − 𝑇𝑤)�̇�𝑤

]

14⁄

(21)

The parameter 𝑍𝑔is given by [19]:

𝑍𝑔 = 𝐶𝑝 𝑑𝑇

𝑑𝑖 (22)

Where 𝑇(𝑖) for humid air is given by [4]:

𝑇 =𝑖 − 2501𝜔

1 + 1.805𝜔 (23)

Finally, ℎ𝑔 is given by [19]:

ℎ𝑔 = 𝑁𝑢𝑔 (𝑘𝑔

𝐷𝑡) (24)

Where the air Nusselt number 𝑁𝑢𝑔 is given by [19]:

𝑁𝑢𝑔 = 1.019 × 10−14(𝑅𝑒𝑎)0.4806(𝑃𝑟𝑎)

−95.69 (25)

In the above equations calculation of Reynolds and Prandtl

numbers for air and water require calculation of thermal

conductivity and dynamic viscosity. Correlations from [20]

were used for this purpose. Value of 𝑈𝑐𝑜𝑛𝑑 will vary along the

length of the condenser. Therefore, an average value is used

[19]:

𝑈𝑐𝑜𝑛𝑑 =1

2(𝑈𝑖𝑛𝑙𝑒𝑡 + 𝑈𝑜𝑢𝑡𝑙𝑒𝑡) (26)

3.6 Humidifier To determine the humidity content at state 5 accurately,

consider a volume element (Fig. 3) of the humidifier, and

assume that temperature and humidity gradients are only in the

vertical direction. The mass balance for the control volume of

the whole element becomes:

𝑑�̇�𝑤 = 𝑑�̇�𝑣 = �̇�𝑎𝑑𝜔 (27)

Fig. 3: volume element of humidifier The energy balance for the control volume of the water

side is:

𝑑𝑇𝑤 =ℎ𝑤𝑎

�̇�𝑤𝑐𝑤(𝑇𝑤 − 𝑇𝑖) 𝑑𝑦 (28)

The energy and mass balances for the control volume of

the air side are:

𝑑𝑇𝑎 =ℎ𝑎𝑎

�̇�𝑎(𝑐𝑎 + 𝜔𝑐𝑣)(𝑇𝑖 − 𝑇𝑎) 𝑑𝑦 (29)

𝑑𝜔 =𝐾𝑎

�̇�𝑎

(𝜔𝑖 − 𝜔) 𝑑𝑦 (30)

The interface between air and water is assumed to be

saturated air at temperature 𝑇𝑖 , that is, 𝜔𝑖 can be calculated

knowing 𝑇𝑖 . And the energy balance for the interface is:

6 Copyright © 2014 by ASME

ℎ𝑤𝑎(𝑇𝑤 − 𝑇𝑖)𝑑𝑦 = ℎ𝑎𝑎(𝑇𝑖 − 𝑇𝑎) 𝑑𝑦+ ∆𝑖𝜗𝐾𝑎(𝜔𝑖 − 𝜔) 𝑑𝑦

(31)

For small 𝑑𝑦, equations (27) to (31) can be solved

progressively for 𝑑𝑇𝑤, 𝑑𝑇𝑎 and 𝑑𝜔 along the length of the

humidifier. The boundary conditions are:

𝑇𝑤(0) = 𝑇2, 𝑇𝑎(0) = 𝑇4, 𝜔(0) = 𝜔4 (32)

Heat and mass transfer coefficients are calculated from

[21]:

𝐾 =2.09 �̇�𝑎

0.11515 �̇�𝑤0.45

𝑎

(33)

ℎ𝑤 =5900 �̇�𝑎

0.5894 �̇�𝑤0.169

𝑎 (34)

ℎ𝑎 = 𝑐𝑎𝐾 (35)

Also, a type of humidifier packing was used that has

𝑎 = 157.7 𝑚2 𝑚3⁄ and a cross sectional area of 0.56 𝑚2. The

cost per unit height was obtained from the manufacturer of that

packing type [22].

3.7 Cost Model The following cost functions were obtained by

interpolating actual component prices listed on an online trade

portal [23]:

𝐶𝑐𝑜𝑛𝑑 = 50𝐴𝑐𝑜𝑛𝑑 (36)

𝐶𝑓𝑎𝑛 = 73.4�̇�𝑎 (37)

𝐶𝑝𝑢𝑚𝑝 = 25�̇�𝑤 + 25�̇�𝑐𝑜 (38)

𝐶𝑐𝑜𝑙,𝑤 = 120𝐴𝑐,𝑤 + 1100𝑉𝑡𝑎𝑛𝑘 (39)

The cost for the solar air heater was obtained from [18]:

𝐶𝑐𝑜𝑙,𝑎 = 120𝐴𝑐,𝑎 (40)

The cost for the humidifier was approximated from [22]:

𝐶ℎ𝑢𝑚 = 110𝐻ℎ𝑢𝑚 (41)

4. OPTIMIZATION

4.1 Heuristic Gradient Projection Heuristic Gradient Projection (HGP) is first described by

Metwalli [24, 25]. Abd El-Malek et al. [26] utilized HGP for

3D space frame weight minimization subject to bending stress

constraints [27]. It was later used by [28, 29, 30] to handle

stress constraints of some form because activation of stress

constraints translates to maximum material utilization, hence

the optimum solution.

The method aims at activating the constraints while

maximizing (or minimizing) the objective function value at the

same time. From a general search point, the procedure is to:

1. Find the objective function gradient (𝛻𝑓) at the current

point. To minimize, use the negative of the gradient

(−𝛻𝑓).

2. If there is a violated constraint, project the negative of

its gradient onto 𝛻𝑓:

𝑆 = (−∇𝑔 𝑇∇𝑓)∇𝑓 (42)

3. If there are no violated constraints, but there is an

inactive constraint, project its gradient onto 𝛻𝑓:

𝑆 = (∇𝑔 𝑇∇𝑓)∇𝑓 (43)

4. Project 𝑆 onto the constraint gradient tangents for

active constraints. If we have ‘r’ active constraints,

define:

𝐺𝑟 = [∇𝑔1 ∇𝑔2 … ∇𝑔𝑟] (44)

As per the gradient projection method, the final search

direction becomes [24, 31]:

𝑓𝑝 = {𝐼 − 𝐺𝑟[𝐺𝑟𝑇 𝐺𝑟]

−1𝐺𝑟𝑇}𝑆 (45)

If the active constraint is linear, step 4 will make it remain

active for the rest of the search. If it is convex, step 4 will cause

the search to violate the constraint, and it will be handled in the

next iteration by step 2. If it is concave, step 4 will cause the

search to deactivate the constraint, and it will be handled in the

next iteration by step 3.

4.2 Problem Formulation The objective is to minimize the unit-cost of produced

fresh water ($/𝑚3):

𝑓(𝑋) = (1 + 𝑐𝑝)(𝐶 +𝐶 +𝐶 +𝐶 +𝐶 , +𝐶 , )

𝑉 𝑡 (46)

Where

𝑋 = [𝐴𝑐,𝑎 𝐻ℎ𝑢𝑚 𝐴𝑐𝑜𝑛𝑑 �̇�𝑠 �̇�𝑐𝑜 𝐴𝑐,𝑤]𝑇 (47)

Because �̇�𝑤 is controlled based on 𝐴𝑐,𝑤, system capacity

is strongly related to the value of 𝐴𝑐,𝑤. Therefore, to be able to

have control on the size of the system generated, 𝐴𝑐,𝑤 is

assumed to be a constant input. All other design variables are

taken relative to the input 𝐴𝑐,𝑤 to generate adequate component

sizing.

Also, �̇�𝑠 directly defines air temperature for a given

weather condition. The fact that saline water is kept at 60 ⁰C

suggests existence of an air temperature that provides for

optimum system throughput regardless of all other parameters.

We can, therefore, specify �̇�𝑠 as an input to the algorithm whose

optimum value will be obtained later.

In addition, because the parameters have different units,

they are scaled by an appropriate factor to be in the same order

of magnitude.

𝑋 = [𝐴𝑐,𝑎𝐴𝑐,𝑤

100 𝐻ℎ𝑢𝑚𝐴𝑐,𝑤

𝐴𝑐𝑜𝑛𝑑𝐴𝑐,𝑤

10 �̇�𝑐𝑜

𝐴𝑐,𝑤]

𝑇

(48)

𝑋 = [𝑥1 𝑥2 𝑥3 𝑥4]𝑇 (49)

The cost functions, 𝐶𝑋, are closed-form of the design

variables. However, the yearly production, 𝑉𝑦𝑒𝑎𝑟 , requires

7 Copyright © 2014 by ASME

simulation to be evaluated. A year-long simulation is a time-

consuming task. Therefore, 𝑉𝑦𝑒𝑎𝑟 is approximated as follows:

𝑉𝑦𝑒𝑎𝑟 = 𝑉𝑑𝑎𝑦 × 365 (50)

This approximation is sufficiently accurate for comparative

purposes because the system has no solar tank, which means

that performance in a day is independent of the performance in

the day before it. Also, a day with average weather conditions is

used to evaluate 𝑉𝑑𝑎𝑦 (day 100 in case of Hurghada), so as the

generated parameters will be near-optimum for the whole year.

After the optimum point is found, productivity and cost results

are calculated from a year-long simulation.

4.3 Performance Constraints To assess the nature of the problem, the method of steepest

descent was used to find the optimum. Table 1 shows the

optimization result from two different starting points. Each

starting point converges to a different solution which indicates

existence of multiple local optima.

Table 1: Optimization result from different starting points

Variable Start 1 Result 1 Start 2 Result 2

𝑥1 1 0.7 1.5 0.8

𝑥2 2 7.6 5 1

𝑥3 1 4.3 1 2.6

𝑥4 1 3.9 1.2 0.5

To find the global optimum, performance constraints were

used. These constraints depend on understanding the nature of

the system being optimized and identifying the bottlenecks

which may hinder its performance. The system at hand transfers

vapor, with air being the container that carries it from the

humidifier to the condenser. In a multi-stage dynamic

programming sense [24], optimality in each of the subsequent

stages necessarily implies optimality of the overall system. A

constraint is formulated for each stage as follows.

4.3.1 Vapor capacity constraint It is essential to maximize the capacity of air to carry

water vapor before the humidification stage (i.e. across the

solar air heater). In reference to Fig. 2, a constraint could

be formulated as:

𝑔1 = �̇�𝑤 − �̇�𝑣,4 ≤ 0 (51)

Where �̇�𝑤 = �̇�𝑤(𝐴𝑐,𝑤). Also, �̇�𝑣,4 represents the

maximum capacity of air to carry water vapor at state 4:

�̇�𝑣,4 = �̇�𝑎𝜔𝑠,4 = 𝜌𝑎𝐴𝑐,𝑎�̇�𝑠𝜔𝑠,4(𝑇4(�̇�𝑠)) (52)

Where 𝜔𝑠,4 is the absolute humidity for saturated air at

conditions of state 4. Since �̇�𝑠 and 𝐴𝑐,𝑤 are constant, 𝑔1 is a

linear function of 𝐴𝑐,𝑎 . Also, 𝑔1 requires simulation to be

evaluated. After that simulation is performed, daily average

values for both �̇�𝑤 and �̇�𝑣,4 are used. Since the global

optimum is assumed to activate 𝑔1, ∇𝑔 𝑇∇𝑓 will be

positive if 𝑔1 is inactive, and will be negative if 𝑔1 was

violated. This makes step 2 or 3 of the HGP procedure

reduce to:

S = +𝑘 [𝜕𝑓

𝜕𝑥1

𝜕𝑓

𝜕𝑥2

𝜕𝑓

𝜕𝑥3

𝜕𝑓

𝜕𝑥4]𝑇

(53)

Where 𝑘 is constant. However, for activating 𝑔1 =𝑔1(𝑥1), and since 𝑥1 is independent of all other 𝑥,

activation can be achieved by moving only in 𝑥1:

S = +𝑘[1 0 0 0]𝑇 (54)

To find out 𝑘 that activates 𝑔1, eqn. (51) is used to

quantify the slackness variable 𝑠 as follows:

�̇�𝑤 − �̇�𝑣,4 − 𝑠 = 0 (55)

For activation according to HGP, 𝑠 = 0, and it follows

that:

�̇�𝑤 = �̇�𝑣,4 = 𝜌𝑎𝐴𝑐,𝑎�̇�𝑠𝜔𝑠,4 (56)

Therefore:

𝐴𝑐,𝑎 =�̇�𝑤

𝜌𝑎�̇�𝑠𝜔𝑠,4 (57)

Dividing by 𝐴𝑐,𝑎𝑜𝑙𝑑 to obtain 𝐴𝑐,𝑎𝑛𝑒𝑤 which should

activate 𝑔1:

𝐴𝑐,𝑎𝑛𝑒𝑤𝐴𝑐,𝑎𝑜𝑙𝑑

=𝑥1𝑛𝑒𝑤𝑥1𝑜𝑙𝑑

=�̇�𝑤

�̇�𝑣,4

(58)

Since 𝑔1 is linear in terms of 𝐴𝑐,𝑎, it is activated in a single

evaluation of eqn. (58). This is the clout of HGP.

4.3.2 Humidity content constraint The constraint 𝑔1 ensured the air had enough capacity

to carry all the �̇�𝑤 available. The constraint to ensure the

humidifier is able to fully saturate the air passing through it

could be written as:

𝑔2 = 𝜔5 − 𝜔𝑠,5 ≤ 0 (59)

While:

𝜔𝑠,5 = 𝜔𝑠,5{𝑇4(�̇�𝑠), 𝑇3, �̇�𝑎(𝐴𝑐,𝑎, �̇�𝑠), �̇�𝑤(𝐴𝑐,𝑤), 𝑥2} (60)

and:

𝜔5 = 𝜔5(𝜔𝑠,5, 𝑥2) (61)

𝐴𝑐,𝑎 was determined by 𝑔1 (in the previous stage),

𝐴𝑐,𝑤; �̇�𝑠 are known. Therefore, 𝑔2 = 𝑔2(𝑥2). In a fashion

8 Copyright © 2014 by ASME

similar to 𝑔1, the following relation can be used for

activating 𝑔2:

𝐻ℎ𝑢𝑚𝑛𝑒𝑤

𝐻ℎ𝑢𝑚𝑜𝑙𝑑

=𝑥2𝑛𝑒𝑤𝑥2𝑜𝑙𝑑

=𝜔𝑠,5

2

𝜔52

(62)

However, because 𝑔2 is not quadratic in terms of 𝑥2,

activation will require iterative application of eqn. (62). Each

evaluation will get closer to 𝑔2 until the separation distance is

small enough that 𝑔2(𝑥2) can be approximated quadratically.

4.3.3 Distillation mass rate constraint The final stage is actual distillation of fresh water in

the condenser. The condenser has to be large enough for

distillation of all the vapor content in the humid air (state

5):

𝑔3 = (�̇�𝐷 + �̇�𝑎𝜔𝑠,𝑇 , ) − �̇�𝑎𝜔5 ≤ 0 (63)

Where 𝜔𝑠,𝑇 , is absolute humidity at saturation at

inlet water temperature. 𝑔3 is a function of 𝑥3 and 𝑥4 and

the following relations could be used to achieve activation

(similar to 𝑔2):

𝑥3𝑛𝑒𝑤𝑥3𝑜𝑙𝑑

=𝑥4𝑛𝑒𝑤𝑥4𝑜𝑙𝑑

=(�̇�𝑎𝜔5)

2

(�̇�𝐷 + �̇�𝑎𝜔𝑠,𝑇 , )2 (64)

This relation scales 𝑥3 and 𝑥4 by the same factor. However,

their optimum values could be in different scales relative to

each other. This will be remedied as will be shown in the next

section. Fig. 4 shows the HGP activation procedure for 𝑔1, 𝑔2

and 𝑔3.

4.4 Constrained Search Now that all the performance constraints are active, a

search is performed to find the constrained optimum. This step

corrects the scale between 𝑥3 and 𝑥4. To proceed, ∇𝑓 and 𝐺𝑟

[32] are numerically calculated to obtain the direction 𝑓𝑝 from

eqn. (45) of the HGP procedure using 𝑆 = −∇𝑓. Small steps are

taken in 𝑓𝑝 until:

𝑓 starts to increase: 𝑓𝑝 is recalculated to step

again.

Any of the constraints deactivates: the constraint

is reactivated as before, then 𝑓𝑝 is recalculated to

step again.

The process is repeated until an 𝑓𝑝 calculation fails to take

a single minimizing step. Fig. 5 shows the stepping procedure.

4.5 Constraint Relaxation By activating the performance constraints and performing

the constrained minimization, the search is localized to the

closest vicinity of the global optimum. The assumption that the

global optimum exactly activates all of the performance

constraints can now be relaxed, and the global optimum can be

found by using the procedure illustrated in Fig. 5 by setting

𝑓𝑝 = 𝑆 directly, and by removing the check and the procedure

for constraint activation (i.e. deactivate all the constraints).

Fig. 4: Constraint activation procedure

Fig. 5: Stepping procedure

4.6 Optimum Specific Air Flow Rate For a given value of �̇�𝑠, optimum values for the rest of the

parameters can now be determined. This allows one-

dimensional optimization (e.g. Fibonacci search) on top of the

optimization procedure discussed before. That is, minimize:

𝑓2(�̇�𝑠) = 𝑓(�̇�𝑠 , [𝐴𝑐,𝑎 𝐻ℎ𝑢𝑚 𝐴𝑐𝑜𝑛𝑑 �̇�𝑐𝑜 𝐴𝑐,𝑤]𝑇) (65)

For different values of 𝐴𝑐,𝑤, �̇�𝑠 always converged to 0.011

𝑚3 𝑚2𝑠⁄ . This value was later used to obtain all results. This

process is an implementation of ridge path optimization. That

𝑆 = −𝛻𝑓

𝑋 = 𝑋 + 𝜖𝑓

𝑓 increased No

𝑓𝑝 from eqn. (45)

𝑋0 = 𝑋

𝑋 = 𝑋0 No

𝑋 = 𝑋 − 𝜖𝑓

𝑔

inactive?

No

Activation procedure

𝑥1𝑛𝑒𝑤 from eqn. (58)

𝑔1 active ?

𝑥2𝑛𝑒𝑤 from eqn. (62)

No

𝑔2 active ? No

Eqn. (64)

𝑔3 active ? No

9 Copyright © 2014 by ASME

is, optimum values of parameters given a certain value of �̇�𝑠, are used to determine the optimum value of �̇�𝑠 [33, 34].

4.7 Genetic Algorithm The optimization problem has been solved by “Optimera”

[35] which is a GA optimization package for C# [36]. The

package was activated from the developed program with 0.9

crossover rate and 0.2 mutation rate. The number of generations

was 40, each containing 20 individuals. The termination

threshold was 0.001 $/𝑚3. Several runs were carried out to

make sure that the global optimum has been reached. Several

very close outputs were attained with the best being reported.

That is why the number of function evaluations was varying

between 1000 and 1600. This was monitored to compare the

GA performance to the HGP as indicated in the next results and

discussion section.

5. DEVELOPED PROGRAM A program was written by the authors in C# [36] to

simulate and optimize the system. It has a simulation as well as

optimization user interfaces.

The simulation interface allows the user to input custom

values for all the system parameters and then simulates the

system behavior in the chosen period of the year. After the

simulation completes, an output chart draws variation of the

different state measures of the system along the simulation

period. Accumulative system output (e.g. average daily

production, unit cost of produced fresh water, total production)

can also be displayed [37].

Finally, the optimization parameter values are obtained

from the input to the simulation interface. The optimization

interface gives log messages for each step the optimizer

performs. When the optimum point is found, the input fields are

updated to reflect its value so the yearly simulation can be run

right away [37].

The following libraries were written by the authors to serve

the purpose of the user interface:

Thermodynamic property calculator for water

which implements the IAPWS industrial

formulation [38].

Thermodynamic property calculator for humid air

which implements the ASHRAE formulation [4,

39].

Fibonacci optimizer.

Sun angle and irradiance calculator which

implements the equations in [15].

Numerical function gradient calculator [32].

The program makes use of the following libraries and

codes:

ZedGraph: charting library.

Math .NET: linear algebra package.

CSVReader: reads comma separated value files

(csv) for reading weather data.

Optiemera: a genetic algorithm optimization

library to optimize the system at hand.

6 RESULTS AND DISCUSSION The study considers a small desalination plant operating

near the city of Hurghada. Weather data for the location were

obtained from [40]. The study assumes a small-scale

desalination plant with required productivity between 5 and 10

m3 per day. Inflation and yearly maintenance cost factor (cp)

was assumed to be 10% and plant lifetime was assumed to be

30 years, which are typical values for the type of equipment

considered [41].

Table 2 shows a comparison between the optimum system

from this work and from previous work by the authors [7, 8].

For the new systems, the objective function value between

brackets is based on the previous cost function from [7]. The

unit-cost of produced fresh water relative to previous work was

reduced by about 90% if we use the new cost function and by

83% if we use the original cost function. The much smaller

condenser and the absence of a solar tank considerably reduced

the system’s overall cost as well as unit-cost of produced fresh

water.

Table 2 also lists the optimization and simulation result for

a system with larger input 𝐴𝑐,𝑤 (1000 𝑚2) and smaller input

𝐴𝑐,𝑤 (100 𝑚2). The value of 𝑓 is only slightly affected by the

system size.

The last two columns in Table 2 list the results of

optimization runs that use a larger (𝐴𝑐,𝑤 = 1000 𝑚2)

The condenser area in this system is much smaller than

previously reported [7] because the coolant flow rate is now

constant at a high value and also because the simulation

calculates 𝑈𝑐𝑜𝑛𝑑 during each iteration based on fluid properties

rather than using a single average value.

The values of 𝑔1, 𝑔2 and 𝑔3 at the optimum points are

always close to 0 which means that the final search converged

in their vicinity. This supports the concept of using performance

constraints to guide the search for the global optimum using

HGP. To further verify the optimization procedure, a genetic

algorithm (GA) optimization was performed for a system with

𝐴𝑐,𝑤 = 500 𝑚2. Table 2 shows no significant difference in unit-

cost between HGP and GA results. However, there’s a small

difference in specific fresh water production in favor of the

HGP-optimized system. In terms of function evaluations,

however, the HGP optimization converges in about 10% the

number of function evaluations required by the GA

optimization.

10 Copyright © 2014 by ASME

Table 2: Optimization Results and Comparison

[8]

(GA)

[7]

(HGP)

This work

(HGP)

This work

(GA)

This work

(larger)

This work

(smaller)

𝐴𝑐,𝑤[𝑚2] 1434.7 1247.9 500 500 1000 100

𝐴𝑐,𝑎[𝑚2] - - 518.1 525.5 1036.2 103.3

𝐴𝑐𝑜𝑛𝑑[𝑚2] 494.0 972.7 922 303.5 586.9 77.6

𝐴ℎ𝑢𝑚[𝑚2] 1051.7 1072.9 19.262 1373.4 1963.5 662.1

𝐻ℎ𝑢𝑚[m] - - 17.8 15.6 22.3 7.5

𝑉𝑡𝑎𝑛𝑘[𝑚3] 69.3 158.8 - - - -

𝐴ℎ𝑒[𝑚2] - 8.9 - - - -

�̇�𝑠[𝑚3 𝑚2. 𝑠⁄ ] - - 0.011 0.011 0.011 0.011

�̇�𝑐𝑜[𝑘𝑔 s⁄ ] - - 2.67 100.9 215.8 9.9

𝑉𝐷[𝑚3] 6.9 10.3 26.2 27 54.3 5.1

𝑉𝑦𝑒𝑎𝑟[𝑚3] 2518.5 3773.2 9891.4 9836.4 19803.1 1872.5

�̇�𝑠[𝑘𝑔 𝑚2. 𝑑𝑎𝑦⁄ ] 4.8 8.3 26.4 24.3 26.7 25.2

Function evaluations 940 6204 99 1000-1600 101 103

Running time [minutes] - 90-120 2 10 2 2

𝑔1[𝑘𝑔/𝑠] - - 0.047 0.038 0.093 0.01

𝑔2[𝑘𝑔/𝑘𝑔] - - -0.003 -0.004 -0.003 -0.006

𝑔3[𝑘𝑔/𝑠] - - -0.002 -0.008 -0.01 -0.006

𝑓[$ 𝑚3⁄ ] 8.0 5.9 0.5 (1.2) 0.5 (1.2) 0.5 (1.1) 0.6 (1.8)

7. CONCLUSION This work modified a previously optimized system by

disconnecting the condenser from the saline water cycle and by

introducing a solar air heater to further augment the

humidification performance. Using further improved models

for the condenser and the humidifier obtained more accurate

productivity estimations. The Heuristic Gradient Projection

(HGP) optimization procedure reduced the number of function

evaluations to reach the global optimum compared to Genetic

Algorithms (GA’s). The developed procedure can simulate and

optimize the new model based on selected weather data file,

among various other inputs in any other prospective locations.

A case study, however, which assumes a desalination plant on

the Red Sea near the city of Hurghada is presented. The unit-

cost of produced fresh water for the new optimum system is

$0.5/𝑚3 compared to $5.9/𝑚3 for the HDH system from

previous work and less than the cost of other reported reverse

osmoses systems. The HGP optimization procedure proved to

be robust and efficient. It depends, however, on the total

understanding of the system and its constraints and their effect

on the design parameters and performance.

ACKNOWLEDGMENTS This research was supported by the U.S. Department of

Agriculture Grant # 58-3148-1-162 and Egypt Science and

Technology Development Fund, STDF Project # 3832.

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