_{1}

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A steady state optimization model used to define the optimum salt to carnallite ponds area ratio in a solar pond system was developed. The model is based on material balance analysis using a cascade of complete-mix reactors model (cascade of CFSTR, continuous-flow stirred-tank reactor) prepared for the solar pond system. The basic material balance model shall use the basic phase chemistry relations and physical parameters of the solar pond system under optimization. The Arab Potash solar pond system data was used to examine the developed model where the Arab potash solar system was used as a Case Study. In the course of the model development, calibration and validation of the model is performed. Using this steady state model the optimum salt pond to carnallite pond area ratio is deduced. This optimum ratio is defined as the optimum area ratio that maximizes the carnallite production per the total pond system area. This term, which could be expressed as tons per km^{2}, presents the best pond system efficiency. The results show that a 1.88 ratio of salt to carnallite ponds area is the optimum ratio.

Solar ponds are simple pools of saltwater where it acts as a large scale solar thermal energy collector [

Solar Ponds System considered under this article is a series of evaporation ponds that utilize the sea water or brine as a raw material under the effect of the solar energy to precipitate carnallite salt (MgCl_{2}KCl·6H_{2}O) as a product of the process. The system consists of two main types of ponds: 1) salt ponds where sodium chloride NaCl is precipitated and 2) carnallite ponds where carnallite is precipitated. The last portion of the salt ponds is a control pond where the brine in this pond is monitored carefully to define the carnallite point and to control the feed to the carnallite ponds.

The materials balance is a quantitative description of all materials that enter, leave, generate and accumulate in a system with defined boundaries. A materials balance is based on the law of conservation of mass (i.e., mass is neither created nor destroyed). The general word statement [5,6] of the materials balance is

The mass rate of generation can be positive or negative where in most cases it has negative value. These material balance equations were first proposed for chemical substances in ponds and lakes in early 1960s [

In applying material balance two operation states can be considered; steady state and unsteady (transient) state. The primary requirement for steady state is that there is no accumulation within the system [5,6]. In other words the beginning and ending brine inventory within the ponds are essentially the same (same concentration and depth). This condition in natural system is applicable in case the system is subjected to a constant load for a long enough time [

The ponds and lakes behave like a continuously flow stirred tank reactor (CFSTR) (complete mix reactor) [7, 8]. In CFSTR reactions are instantaneous with no concentration gradient within the pond this is to say concentration are uniform throughout the pond [5-8]. The result is that the concentration of any material leaving the pond is exactly the same as the concentration at any point in the pond. As the pond system is composed of several ponds in series, cascade of CFSTR model can be used. A general computational framework for modeling such systems was developed by [

Solar pond system is more efficient if the ponds are divided into several ponds [2,10] this is called the sequential pond theory. Sequential pond theory analysis [

The depositing sequence of salts crystallized from the evaporation of sea water is well established from studies on solar ponds bitterns [_{3}, Calcite, followed by CaSO_{4}, Gypsum, and then NaCl, halite, [

In sea water solar ponds epsomite (MgSO_{4}·7H_{2}O) is the next salt to crystallize after halite, and with cold weather it can happen early and massively, [_{4}·2.75H_{2}O) crystallizes next with epsomite and halite. Under favorable conditions these salts can contain as much as 60% kanite, 25% epsomite and 15% halite.

Carnallite (KCl·MgCl_{2}·6H_{2}O) and halite are usually the last salts to crystallize and deposit with small amount of MgSO_{4}·6H_{2}O, this normally represents the end point of sea water evaporation.

The above mentioned sequence of crystallization is very dependent on the brine temperature. This in turn is determined by the concentration of the sea water bitterns, the ambient day-night temperature cycle, wind conditions, bitterns depth and the evaporation rate. Pan evaporation provides the basis for the mass balance calculation of evaporation [

The objective of this study is to develop a tool for sizing of salt and carnallit ponds with the aim of increasing the production of carnallite salt deposits. Through this study, a steady state material balance model using cascade complete-mix reactor model is developed. The model utilizes the basic material balance relations for both the salt and carnallite ponds. The model was further used to estimate the optimum salt to carnallite pond area ratio for the Solar Pond System developed at the Arab Potash Company.

The intent of this section is to discuss the fundamentals of the material balance in the solar pond system.

These basic streams are:

1) The feed brine, which is expressed in tons.

2) Evaporation from the pond area, which is a function of the evaporation rate, and the pond surface area available for evaporation. The evaporation is expressed in mm/day.

3) Salts which represent the deposited salts in the pond, which is expressed in tons.

4) Entrainment which represents the brine that is lost within the salt deposits. As the salt crystals grow or accumulate on the pond floor, there are voids created and some brine is trapped therein. The quantity of entrainment is a function of the quantity and type of salt deposited. The entrainment is generally expressed as a weight percent of the salt deposited.

5) Leakage is the brine lost through seepage towards groundwater. The quantity of leakage is usually described in mm/day and is a function of the pond area.

6) Exit brine is the brine leaving the pond either to the following pond or the point out of the system.

Generally, the brine concentration throughout the pond is uniform and equal to the exit brine concentration. This is a key assumption for the CFSTR model used.

From the above we can develop the balance equations, where these equations include the brine flow balance and material balance equations of the brine constituents.

The salt pond system consists of a series of pond system rather than a single pond system.

Following the basic solar ponds streams shown in

Mass:

Magnesium:

Sodium:

where:

• X is the input feed in tons.

• Mg_{x} is the weight fraction of Mg in the input feed.

• Na_{x} is the weight fraction of Na in the input feed.

• Y is the exist brine in tons.

• Mg_{y} is the weight fraction of Mg in the exist brine.

• Na_{y} is the weight fraction of Na in the exist brine.

• S is the NaCl salt deposit in tons.

• E_{f} is the entrainment factor as percent of the salt deposit.

• 0.3934 is the Na percent weight in NaCl.

• L_{k} is the seepage rate in m/yr.

• A is the pond area.

• D is the brine density.

• F_{1} is the evaporation scale factor.

• E_{R} is the evaporation rate in m/y.

For the first pond the following are the known parameters:

• The area A.

• The Mg and Na concentrations at the feed brine X and the exist brine Y.

• The evaporation rate E_{R}.

• The leakage rate L_{k}.

• The entrainment factor E_{f}.

• The pond brine density D.

• The evaporation factor F_{1}.

This implies that the input feed X, the pond exit brine Y and the deposited salt S as the three unknown parameters. The above three relations are regrouped to lead to the following relations:

Mass:

Magnesium:

Sodium:

Solving the above three relations, the three unknowns X, Y, and S are calculated.

The exit brine from the first pond is considered as the input feed to the second pond, this implies that the input feed, X, to the second pond is known, and accordingly the balance is changed to evaluate how big the area should be to concentrate the input brine to a set Mg concentration.

The following are the known parameters in the balance relations for the second salt pond:

• The input feed brine X (equal to the exist feed of the first pond).

• The Mg and Na concentrations at the feed brine X and the exist brine Y.

• The evaporation rate E_{R}.

• The leakage rate L_{k}.

• The entrainment factor E_{f}.

• The pond brine density D.

• The evaporation factor F_{1}.

While the unknown parameters are the pond area A, the exit brine, Y, and the NaCl salt deposited, S. The main material balance relations could be regrouped again to show the following relations:

Mass:

Magnesium:

Sodium:

Solving the above three relations, the three unknowns A, Y and S are calculated.

The same procedure presented for the second salt pond is used for ponds 3 to 6 till the pre-carnallite pond (PC). Calculation for the PC pond yields the value of exit brine, Y, which is the tonnage of brine available to enter the first carnallite pond.

As previously shown the NaCl is only deposited in the salt ponds, however in the carnallite ponds potassium and magnesium (as carnallite) are deposited along with sodium. Accordingly, a new unknown appears in the material balance relations, this new unknown require defining a new material balance relation. This new relation tracks the carnallite deposited in the carnallite ponds.

Following the balance streams shown in

Mass:

Magnesium:

Sodium:

Potassium:

Where• K_{x} is the weight fraction of K in the input feed.

• K_{y} is the weight fraction of K in the exist brine.

• 0.0875 is the Mg percent weight in carnallite (MgCl_{2}KCl·6H_{2}O).

• 0.1407 is the K percent weight in carnallite (MgCl_{2}KCl·6H_{2}O).

The calculations of the last salt pond (pre-carnallite) leads to the quantity of brine leaving PC pond and entering as an input feed to the first carnallite pond.

The known parameters for the material balance of the carnallite ponds are:

• The input feed brine X (equal to the exist feed of the PC pond).

• The Mg, Na and K concentrations at the feed brine X and the exist brine Y.

• The evaporation rate E_{R}.

• The leakage rate L_{k}.

• The pond brine density D.

• The evaporation factor F_{1}.

While the unknown parameters are; the pond area A, the exist brine Y, the NaCl salt deposited, S, and the carnallite salt deposited, C. The main material carnallite ponds balance relations could be regrouped to show the following relations:

Mass:

Magnesium:

Potassium:

Solving the above four relations, the four unknowns A, Y, S and C are obtained.

The same procedure presented above for the first carnallite pond is used for ponds 2 to 6 till the last pond from where the exit brine is discharged to the waste channel.

The above material balance equation developed shows that the salt and carnallite pond is a function of pond area, evaporation rate, different flow streams and brine chemistry. Experience has shown that the behavior of any pond system brine chemistry can best be illustrated on a concentration path diagram. Through the solar pond system studied, only potassium K and sodium Na are precipitated in the pond system while calcium Ca and Magnesium Mg do not precipitate. In this sense all concentration paths is deduced as a function of Mg or Mg and Ca.

These relationships are deduced using field data collected for the specific solar pond system. For new systems the basic relations should be estimated based on experiments to be performed prior the solar pond system design. The relations should cover density, evaporation, sodium, potassium, and evaporation as a function of Mg or Mg + Ca.

Other major inputs as area, entrainment factor, seepage leakage factor and evaporation factor are physical parameters that should be estimated based on field measurements. Entrainment factor depends on the type of salt deposited as it is directly related to the voids and porosity in the deposited salt. Seepage factor and evaporation factor are two parameters that are difficult to estimate as they require extensive measurements and field investigations. Both parameters are deduced based on material balance model calibration.

Evaporation rate is temperature sensitive and a single relation is difficult to be defined accordingly averages for the evaporation rate over the year should be considered while developing the steady state model.

Also Potassium is a temperature sensitive and a single relation for the concentration path of potassium is difficult to define. Through the steady state model development the data for the months of summer months should be used. This is due to the fact that the most evaporation and carnallite production are during these months. It is important to note here that it is not possible to define an optimum ratio of the salt to carnallite pond area that is applicable during all the year. Thus the study uses the period of the year where most of the carnallite is produced in the development of the Potassium concentration path diagram.

Estimating these phase chemistry relations and physical relations the optimization model is developed.

The solar pond system consists of two main sections with different operating objectives. The first section, the salt ponds, is operated to provide the maximum amount of carnallite point brine to the first carnallite pond. The second section, the carnallite ponds, is fed by the flow of the carnallite point brine produced by the salt ponds. There are three critical brine concentrations that need to be known. One is the feed brine concentration, the second is the Carnallite Point concentration, and the third is the concentration at the point where brine will be discarded. The optimization model requires the definition of these three locations chemical analysis.

The optimization model was developed based on the material balance relations presented previously and the deduced phase chemistry and physical parameters relations. The model has 6 salt ponds and 6 carnallite ponds. This selection of ponds system is arbitrary. Enough ponds must be chosen to satisfy the sequential pond theory. The sequential pond theory proves that as we increase the number of ponds the performance of the ponds improves. Usually this is satisfied with about 4 ponds.

To start the model calculation, the area of the first salt pond will arbitrarily be set at 10,000 square meters (1 hectare) and the Magnesium concentration in the brine in each pond will be divided equally from the concentration at the Feed brine to the concentration at the carnallite point. For example, the carnallite point occurs at say 6.1% magnesium and the Feed brine is at say 2%. Each pond will have a successive jump of (6.1 − 2)/6 or 0.683%. Based on this assumption the Magnesium concentrations in the brine entering the pond (Mg_{x}) and leaving the pond Mg_{y} are known values.

Using the developed relations for the density, evaporation, sodium and magnesium, the values of evaporation, density, Na_{x}, Na_{y}, K_{x} and K_{y} are evaluated.

A simple spreadsheet is developed to solve the model relations and calculate the unknown parameters. However it is important to note here that the model results depend on the input parameters and the calibration of the different parameters involved in the calculations.

The best indicator of pond efficiency is the tons of carnallite per square kilometer of total pond area where this value is maximized for the optimum salt ponds area to carnallite ponds area ratio to be reached.

The model should be run for different salt to carnallite ponds area cases and a relation between the area ratio and the tons of carnallite per unit area is developed. From this relation the optimum area ratio is deduced.

The Arab Potash solar pond system is located to the south of the Dead Sea the lowest point at the surface of the earth with an area of about 112 km^{2} [_{2}·6H_{2}O) which is the raw material for the potash production. The brine used in the solar ponds is pumped from the Dead Sea. This brine is rich in potassium chloride. The Arab Potash solar pond system is composed of two parts the salt ponds and the carnallite ponds.

The model developed in the previous section was used to optimize the Arab Potash Solar Pond System (APC). The data available for a stretch of 6 years during the period 1997 to 2002 were used for this case study [14-19]. This optimization for the Arab Potash Solar Pond System was appreciated by the Company as the Arab Potash Company was studying the expansion of the solar ponds system and as such would like to define the optimum salt to carnallite ponds areas.

The following sub-sections present the basic phase chemistry relations and the physical parameters relations. This is followed by model calibration and validation. The optimum salt to carnallite ponds area ratio is then deduced.

The following sub-sections define the composition of the pond brines in term of concentration paths. Other parameters necessary to the APC solar pond model development are also presented.

The relation for the density (ton/m^{3}) as a function of magnesium is developed based on the high carnallite production period data for the years 1997 to 2002. There are three distinct groupings of points that are fitted by a polynomial as presented in

It is needless to mention that the density is needed in material balance calculations to convert from volume to mass.

To develop a relation between the evaporation rate and the magnesium two steps are performed. First a relation between evaporation rate and density is developed, then magnesium is substituted for the density based on the density relation developed above.

If we substitute the density versus magnesium relation presented in