_{1}

^{*}

The treatment
s
of raw water using resins embedded with cation and anion w
ere
investigated in this study via a developed mathematical model. The developed mathematical model was used to
predict
the amount of sodium (
Na^{+}
), calcium (
Ca^{2+}
), magnesium (
Mg^{2+}
) and ammonium (NH_{4}^{＋ }) ions removed by the resin embedded with hydrogen ion (
H^{+}
) in the cation bed. The model was also used to investigate the amount of chloride (
Cl^{－}) and sulphate (SO_{4}^{2－ }) ions treated by the resin embedded with hydroxyl ion (
OH^{－}) in the anion bed. The effect of flow rate and superficial velocity were also investigated. The simulation results showed that there was significant reduction of Na^{+}, Ca^{2+}, Mg^{2+}, , Cl^{－} and SO_{4 }^{2－}from their initial concentrations in raw water. This showed that the mathematical model was able to predict the concentrations of cations and anions investigated in this study. The result revealed that the flow rate of water has effect on the treatment of cations and anions in raw water using ion-exchange resins. Thus, operating the resin beds at very high flow rate reduced its performance while at very low flow rate the residence time of wastewater on the bed increased with resultant increase in performance. Similarly, high superficial velocity reduced the amount of ion concentration removed by the resin in both beds. The total final concentrations of cations in the cation bed by the model were 0.0003156, 0.0003452 and 0.0036 mol/m^{3} at 4.0, 4.5 and 6.5 m/min respectively while that from the anion bed were 0.0002597, 0.0002769 and 0.00205 mol/m^{3} at 4.0, 4.5 and 6.5 m/min respectively. The predicted model results, when compared with the maximum allowable limit of total concentration of both cations and anions of a functional industrial company (Notore chemical), showed a maximum percentage deviation ranging from 2.00% to 3.53%. This showed that the developed model has achieved its set objectives.

Boilers are the heartbeat of thermal power plants which require a high purity water to operate in order to protect the internal installations against corrosion and fouling. When mineral salts, acids and bases are in water solution, they break down into ions with either positive or negative electrical charges. Ions with positive charges are called cations, while those with negative charges are called anions. These ions are reactive if allowed into a boiler, therefore detrimental to boiler tubes resulting in poor performance. To solve this problem, plastic beads called ion exchange resins have been developed to trap these unwanted ions from entering into the boiler using a process called Demineralization. The ion exchange resins have high affinity to either cations or anions [

The model was developed by assuming a hypothetical system representing cation and anion bed as shown in

A d z ∂ C A ∂ t = − ∂ ∂ z ( u ¯ C A + N A ) A d z + R A A d z (1)

where C A = concentration of ion in the liquid phase (mol/m^{3}), t = time (min), z = bed height (m), A = bed cross-sectional area (m^{2}), u ¯ = the average fluid flowing velocity in all directions (m/min), N A = molar flux of component A in the mixture (mol/min∙m^{2}), R A = rate of ion-exchange (mol/m^{3}・min).

Dividing all through by A d z yields

∂ C A ∂ t = − ∂ ∂ z ( u ¯ C A + N A ) + R A (2)

Equation (2) is the dynamic model for the cylindrical column embedded with resin to remove ions in wastewater. This model incorporated flow due to diffusion and bulk flow. However, for cylindrical system,

u ¯ = u r + u θ + u z (3)

where u r = velocity in radial direction (m/min). u θ = velocity in axial direction (m/min), u z = velocity in the vertical direction (m/min). The molar diffusion flux of ion concentrations on the bed in the three direction is:

N A = − D e ( ∂ C A ∂ r + ∂ C A ∂ θ + ∂ C A ∂ z ) (4)

Substituting Equation (3) and (4) into Equation (2) and assuming that the effects of fluid flow along the radial and axial directions were negligible, we obtained the model:

∂ C A ∂ t = − ∂ ∂ z [ u z C A − D e ( ∂ C A ∂ z ) ] + R A

∂ C A ∂ t = − ∂ ∂ z ( u z C A ) + D e ∂ 2 C A ∂ z 2 + R A (5)

Equation (5) is the dynamic model for the fixed bed in the vertical direction.

Assumed that the bed operates at steady state, Equation (5) is reduced to

d d z ( u z C A ) = D e d 2 C A d z 2 + R A (6)

Where C_{AO}, C_{A} = initial and final concentration, υ_{o}, υ = initial and final volumetric flow rate, dZ = differential bed height.

From [

D e d C A d z = K L ( C A o − C A ) (7)

where: K L = mass transfer coefficient (m/s), D e = effective diffusion coefficient (m^{2}/s).

Hence, for constant fluid velocity and substituting Equation (7) into Equation (6) yields

u z d C A d z = K L d d z ( C A o − C A ) + R A

u z d C A d z = − K L d C A d z + R A

( u z + K L ) d C A d z = R A (8)

The mass transfer rate of ion exchange, R A in the resin phase has been expressed by [_{ }

R A = d C s d t = k 1 C A ( C s m − C s ) − k 2 C s ( C A o − C A ) (9)

Hence, at equilibrium

R A = d C s d t = k 1 [ C A ( C s m − C s ) − 1 K d C s ( C A o − C A ) ] (10)

where: C s = Concentration of ion in the resin (mol/m^{3}). C A = Concentration of ion in the liquid (mol/m^{3}).

C s m = Maximum concentration of ion in the resin (mol/m^{3}). K d = k 1 k 2 = Equilibrium constant (dimensionless).

Since the amount of ion concentrations removed from raw water are being trapped by the resins, it followed that the rate of ion concentrations in the liquid phase was equivalent to the rate of ion concentration in the resin phase, thus:

R A = d C S d t = − d C A d t (11)

Hence, combining Equation (8), Equation (10) and Equation (11) yields:

∴ − d C A d z = ( k 1 u z + K L ) [ C A ( C S m − C S ) − 1 K d C S ( C A O − C A ) ] (12)

Equation (12) represents the steady state model equation for the ion-exchange bed.

K d was obtained from an expression by [

K d = y A / x A [ ( 1 − y A ) / ( 1 − x A ) ] 2 × C A o C S m (13)

where x A and y A are ionic mole fractions in liquid and resin and they are expressed as:

x A = C A C A o ; y A = C S C S m (14)

In practice, the desired values of the ionic mole fractions in the liquid and resin phases have to be specified [

The residence time, τ m can be expressed as:

τ m = ε r V υ o (15)

where v o = Volumetric flow rate (m^{3}/min), ε r = Bed voidage.

The developed model (Equation (12)) was solved numerically using the 4^{th} order Runge-Kutta algorithm expressed by [

C A ( i + 1 ) = C A ( i ) + [ k 1 + 2 ( k 2 + k 3 ) + k 4 ] h 6 (16)

where:

k 1 = f ( y i , C A ( i ) ) (17)

k 2 = f ( y i + h 2 , C A ( i ) + h 2 k 1 ) (18)

k 3 = f ( y i + h 2 , C A ( i ) + h 2 k 2 ) (19)

k 4 = f ( y i + h , C A ( i ) + h k 3 ) (20)

i = 1 , 2 , 3 , ⋯ , n .

where h = step size.

Using the following initial and boundary conditions. At z = 0 , C A = C A o ; and at maximum exchangeable capacity:

Parameter | Value | Reference |
---|---|---|

Bed height (m) | 5.0 | [ |

Bed radius (m) | 0.6034 | [ |

Voidage (-) | 0.003 | [ |

Mass Transfer Coefficient K_{L} (m/min) | 1.92 × 10^{−6} | [ |

Parameter | Initial Value | Exit value | Reference |
---|---|---|---|

Na^{+} | 0.5862 (37.1 ppm) | 0.0002142 (0.01356 ppm) | [ |

Ca^{2+} | 0.079 (5.0 ppm) | 0.0000293 (0.00185 ppm) | [ |

Mg^{2+} | 0.194 (12.3 ppm) | 0.0000712 (0.00451 ppm) | [ |

NH 4 + | 0.0221 (1.4 ppm) | 0.0000081 (0.00051 ppm) | [ |

Cl^{−} | 0.2228 (14.1 ppm) | 0.0000813 (0.00514 ppm) | [ |

SO 4 2 − | 0.4930 (31.2 ppm) | 0.0001856 (0.01175 ppm) | [ |

ppm = 0.0158 mol/m^{3}.

z > 0 , C A = C ( z )

The model results for cations and anions were simulated and the effects of flow rates and fluid velocity on resin capacity were also studied.

The profile in ^{3} to a final concentration of 0.0000283 mol/m^{3} after been in contact with the resin. This showed that 0.07897 mol/m^{3} of calcium ion concentration had been removed by the resin in the cation bed indicating that about 99.96% of calcium ion were removed by the resin. However, the maximum concentration of calcium ion in the effluent water from the plant cation bed was 0.0000293 mol/m^{3} which is greater than the final concentration as obtained from the model equation with deviation of 3.53%.

The profile in ^{3} to final concentration of 0.00021 mol/m^{3} indicating that 0.58599 mol/m^{3} of sodium ion concentration had been removed by the resin in the cation bed accounting for about 99.96% removal with a deviation of 2%.

Component | Model Prediction (mol/m^{3}) | Plant Data (mol/m^{3}) | % Deviation |
---|---|---|---|

Cation Bed | |||

Na^{+} | 0.00021000 | 0.0002142 | 2.00 |

Ca^{2+} | 0.00002830 | 0.0000293 | 3.53 |

Mg^{2+} | 0.00006940 | 0.0000712 | 2.59 |

NH 4 + | 0.00000791 | 0.0000081 | 2.40 |

Anion Bed | |||

Cl^{−} | 0.0000797 | 0.0000813 | 2.00 |

SO 4 2 − | 0.0001800 | 0.0001856 | 3.11 |

The magnesium ion concentration in ^{3} to 0.0000694 mol/m^{3} accounting for 0.19393 mol/m^{3} of magnesium ion concentration removal. However, the comparison of model predictions (0.0000694 mol/m^{3}) with plant data (0.0000718 mol/m^{3}) for magnesium ion removal in the effluent water showed a deviation of 2.59%.

Similarly, ^{3} to an 0.00000791 mol/m^{3} with

0.022092 mol/m^{3} of ammonium ion concentration been removed by the resin. The comparison of model predictions (0.0000081 mol/m^{3}) showed a deviation of 2.40%.

^{3}). However, the level of ion concentrations in the exit stream from the cation bed for the four metal ions investigated showed that the efficacy of the exchange capacity in the resin was suitable as predicted by the model developed.

The chloride ion concentration in effluent water from the cation bed flowing into the anion bed at velocity of 4.0 m/min as presented in ^{3}) with 0.22272 mol/m^{3} of the chloride ion concentration removed by the resin in the anion bed before flowing into the mixed bed. The comparison of model predictions (0.0000797 mol/m^{3}) with plant data (0.0000831 mol/m^{3}) for chloride ion removal showed a deviation of 2.00%.

Similarly, sulfate ion concentration reduction is presented in

flowing into the anion bed at velocity of 4.0 m/min decreased (from initial concentration of 0.4930 to exit concentration of 0.00018 mol/m^{3}), with 0.49282 mol/m^{3} of the sulfate ion concentration removed by the resin in the anion bed before flowing into the mixed bed. The comparison of model prediction (0.000018 mol/m^{3}) with plant data (0.00001856 mol/m^{3}) for sulfate ion removal showed a deviation of 3.11%.

The effect of fluid velocity as a factor that could improve or mar the removal of chloride ion was investigated as shown in

Like the anion bed, the effect of raw water inlet velocity in the cation bed on the removal of sodium ion by the resin as shown in

The effect of raw water inlet velocity on the removal of magnesium ion as shown in

6.5 m/min), For sodium, ammonium and calcium ions resulted to decrease in the amount of ions along the bed height.

The effect of flow rate on the residence time of cations removal was investigated as shown in

Similarly, in the cation bed, increasing the flow rate of water decreased the residence time. However, the residence time of contaminants increased with the resins increased in bed height as shown in

A model for an industrial ion exchange facility for demineralization of boiler feed water has been developed using the principle of conservation of mass. The model developed was integrated numerically using Runge-Kuttaalgorithm imbedded in MATLAB computer programming software. Results obtained were compared with industrial plant data and it agreed reasonably well with a percentage deviation ranging from 2.0% to 3.53% in the cation bed and 2.0% to 3.11% in the anion bed respectively, indicating that the developed model was adequate.

Simulation was performed on functional parameters to predict optional variables for best performance of the ion exchange facility.

The author declares no conflicts of interest regarding the publication of this paper.

Dagde, K.K. (2018) Development of Performance Models for Boiler Feed Water Treatment Ion Exchange Facility. Advances in Chemical Engineering and Science, 8, 280-297. https://doi.org/10.4236/aces.2018.84020