# Ion Exchange (0D)

The main assumptions of the implemented model are as follows:

1. Model dimensionality is limited to a 0D control volume

2. Single liquid phase only

4. Single solute and single solvent (water) only

5. Plug flow conditions

6. Isothermal conditions

7. Favorable Langmuir or Freundlich isotherm

## Introduction

Ion exchange is the reversible transfer of one or more solutes between a fluid phase and a sorbent. This process is becoming increasingly popular in drinking water treatment applications where it is used for water softening and demineralization. This implementation of the fixed-bed ion exchange model accounts for process equilibrium, kinetics, and hydrodynamics to predict performance, bed and column geometry, and capital/operating costs. The ion exchange process operates as a cycle with four steps:

1. Service

2. Backwashing

3. Regeneration

4. Rinsing

Critical to predicting performance of an ion exchange process is having an estimate for the breakthrough time, or the duration of treatment before the solute begins exiting the column at a concentration unacceptable to the operator. At this time, the mass transfer zone is approaching the end of the ion exchange bed, the resin is nearing exhaustion, and the regeneration cycle can begin. Fundamental to this model is the assumption that the isotherm between the solute and the resin is favorable, and thus the mass transfer zone is shallow.

Note

If using single-use configuration for regenerant, the backwashing, regeneration, and rinsing steps are not modeled and all associated costs for these steps are zero.

### Isotherm Configurations

The model requires the user has either Langmuir or Freundlich isotherm equilibrium parameters for their specific system. Variables in the following equations and their corollary in the WaterTAP model are defined in a following section.

#### Langmuir

For the Langmuir isotherm, the Langmuir parameter $$La$$ (langmuir in the WaterTAP model) is:

$La = \frac{1}{1 + K C_0}$

Where $$K$$ is an equilibrium constant derived from experimental data, and $$C_0$$ is the influent concentration of the target ion. (Note: This equation is not included in the model). $$La$$ is used in the dimensionless form of the Langmuir isotherm:

$La = \frac{X (1 - Y)}{Y (1 - X)}$

$$Y$$ is the ratio of equilibrium to total resin capacity (resin_eq_capacity and resin_max_capacity, respectively in the WaterTAP model). For a favorable isotherm (a core assumption of the model), $$La$$ is less than one.

#### Freundlich

For the Freundlich isotherm, the model assumes the user has fit breakthrough data to the Clark model. The general solution evaluated at 50% breakthrough is:

$\frac{C_b}{C_0} = \frac{1}{\bigg(1 + (2^{n - 1} - 1)\text{exp}\bigg[\frac{k_T Z (n - 1)}{BV_{50} u_{bed}} (BV_{50} - BV)\bigg]\bigg)^{\frac{1}{n-1}}}$

The form often fit to breakthrough data is:

$\frac{C_b}{C_0} = \frac{1}{A \text{exp}\big[\frac{-r Z}{u_{bed}} BV\big]^{\frac{1}{n-1}}}$

The full derivation for both equations is provided in Croll et al. (2023).

## Ports

The model provides three ports (Pyomo notation in parenthesis):

• Inlet port (inlet)

• Outlet port (outlet)

• Regeneration port (regen)

## Sets

The table below outlines example Sets that could be used with the ion exchange model. “Components” is a subset of “Ions” and uses the same symbol j. They can include any ion as long as the ion is configured into the property package. target_ion_set includes the component to be removed via the ion exchange process. The current model implementation is only for a single component, but target_ion_set is included for future development of a multi-component model.

Description

Symbol

Example Indices

Time

$$t$$

[0]

Phases

$$p$$

['Liq']

Components

$$j$$

['H2O', 'Cation_+', 'Anion_-', 'Inert']

Ions

$$j$$

['Cation_+', 'Anion_-']

Target Ion

$$j$$

['Cation_+']

In this example, the influent stream contains H2O (always included), Cation_+, Anion_-, and an uncharged component Inert. The user would specify the concentration of each as part of the property package in the model build. The charged components are included in “Ions”, a subset of “Components”. The model is configured as a cation exchange process since target_ion_set contains a positively charged component, Cation_+.

## Model Components

The ion exchange model includes many variables, parameters, and expressions that are common to both the langmuir and freundlich isotherm configurations. These are provided in the table below.

Description

Symbol

Variable Name

Index

Units

Variables

Inlet temperature

$$T$$

temperature

[t]

$$\text{K}$$

Inlet pressure

$$p$$

pressure

[t]

$$\text{Pa}$$

Component molar flow rate

$$N_j$$

flow_mol_phase_comp

[t, 'Liq', 'H2O']

$$\text{mol/s}$$

Control volume mass transfer term

$$\dot{m}_j$$

process_flow.mass_transfer_term

[t, 'Liq', j]

$$\text{mol/s}$$

Service flow rate through resin bed in bed volumes per hour

$$SFR$$

service_flow_rate

None

$$\text{hr}^{-1}$$

Linear velocity through bed

$$u_{bed}$$

vel_bed

None

$$\text{m/s}$$

Interstitial velocity through bed

$$u_{inter}$$

vel_inter

None

$$\text{m/s}$$

Number of operational columns

$$n_{op}$$

number_columns

None

$$\text{dimensionless}$$

Number of redundant columns

$$n_{red}$$

number_columns_redund

None

$$\text{dimensionless}$$

Bed depth

$$Z$$

bed_depth

None

$$\text{m}$$

Column height

$$H_{col}$$

col_height

None

$$\text{m}$$

Column diameter

$$D_{col}$$

col_diam

None

$$\text{m}$$

Column height to diameter ratio

$$R_{HD}$$

col_height_to_diam_ratio

None

$$\text{dimensionless}$$

Total bed volume

$$V_{res, tot}$$

bed_vol_tot

None

$$\text{m}^3$$

$$d$$

resin_diam

None

$$\text{m}$$

Resin bulk density

$$\rho_{b}$$

resin_bulk_dens

None

$$\text{kg/L}$$

Resin surface area per volume

$$a_{s}$$

resin_surf_per_vol

None

$$\text{m}^{-1}$$

Bed porosity

$$\varepsilon$$

bed_porosity

None

$$\text{dimensionless}$$

Number of cycles before regenerant disposal

$$N_{regen}$$

regen_recycle

None

$$\text{dimensionless}$$

Relative breakthrough concentration at breakthrough time

$$X$$

c_norm

target_ion_set

$$\text{dimensionless}$$

Breakthrough time

$$t_{break}$$

t_breakthru

None

$$\text{s}$$

Empty Bed Contact Time (EBCT)

$$EBCT$$

ebct

None

$$\text{s}$$

Reynolds number

$$Re$$

N_Re

None

$$\text{dimensionless}$$

Schmidt number

$$Sc$$

N_Sc

target_ion_set

$$\text{dimensionless}$$

Sherwood number

$$Sh$$

N_Sh

target_ion_set

$$\text{dimensionless}$$

Peclet particle number

$$Pe_{p}$$

N_Pe_particle

None

$$\text{dimensionless}$$

Peclet bed number

$$Pe_{bed}$$

N_Pe_bed

None

$$\text{dimensionless}$$

Parameters

Regeneration time

$$t_{regen}$$

t_regen

None

$$\text{s}$$

Backwash time

$$t_{bw}$$

t_bw

None

$$\text{s}$$

$$u_{bw}$$

bw_rate

None

$$\text{m/hr}$$

Number of bed volumes for rinse step

$$N_{rinse}$$

rinse_bv

None

$$\text{dimensionless}$$

Pump efficiency

$$\eta$$

pump_efficiency

None

$$\text{dimensionless}$$

Service-to-regeneration flow ratio

$$R$$

service_to_regen_flow_ratio

None

$$\text{dimensionless}$$

Pressure drop equation intercept

$$p_{drop,A}$$

p_drop_A

None

$$\text{dimensionless}$$

Pressure drop equation B

$$p_{drop,B}$$

p_drop_B

None

$$\text{dimensionless}$$

Pressure drop equation C

$$p_{drop,C}$$

p_drop_C

None

$$\text{dimensionless}$$

Bed expansion fraction equation intercept

$$H_{expan,A}$$

bed_expansion_frac_A

None

$$\text{dimensionless}$$

Bed expansion fraction equation B parameter

$$H_{expan,B}$$

bed_expansion_frac_B

None

$$\text{dimensionless}$$

Bed expansion fraction equation C parameter

$$H_{expan,C}$$

bed_expansion_frac_C

None

$$\text{dimensionless}$$

Expressions

Fraction of bed depth increase during backwashing

$$X_{expan}$$

bed_expansion_frac

None

$$\text{dimensionless}$$

Additional column sidewall height required for bed expansion

$$H_{expan}$$

bed_expansion_h

None

$$\text{dimensionless}$$

Backwashing volumetric flow rate

$$Q_{bw}$$

bw_flow

None

$$\text{m}^{3}\text{/s}$$

Rinse time

$$t_{rinse}$$

t_rinse

None

$$\text{s}$$

Rinse volumetric flow rate

$$Q_{rinse}$$

rinse_flow

None

$$\text{m}^{3}\text{/s}$$

Regen + Rinse + Backwash time

$$t_{waste}$$

t_waste

None

$$\text{s}$$

Cycle time

$$t_{cycle}$$

t_cycle

None

$$\text{s}$$

Bed volume of one unit

$$V_{res}$$

bed_vol

None

$$\text{m}^{3}$$

Column volume of one unit

$$V_{col}$$

col_vol_per

None

$$\text{m}^{3}$$

Total column volume

$$V_{col, tot}$$

col_vol_tot

None

$$\text{m}^{3}$$

Bed volumes of throughput at breakthrough

$$BV$$

bv_calc

None

$$\text{dimensionless}$$

Regeneration solution tank volume

$$V_{regen}$$

regen_tank_vol

None

$$\text{m}^{3}$$

Pressure drop through resin bed

$$p_{drop}$$

pressure_drop

None

$$\text{psi}$$

Power of main booster pump

$$P_{main}$$

main_pump_power

None

$$\text{kW}$$

Regen pump power

$$P_{regen}$$

regen_pump_power

None

$$\text{kW}$$

Backwash pump power

$$P_{bw}$$

bw_pump_power

None

$$\text{kW}$$

Rinse pump power

$$P_{rinse}$$

rinse_pump_power

None

$$\text{kW}$$

If isotherm is set to langmuir, the model includes the following components:

Description

Symbol

Variable Name

Index

Units

Variables

Langmuir equilibrium parameter for resin/ion system

$$La$$

langmuir

target_ion_set

$$\text{dimensionless}$$

Maximum resin capacity

$$q_{max}$$

resin_max_capacity

None

$$\text{mol/kg}$$

Equilibrium resin capacity

$$q_{eq}$$

resin_eq_capacity

None

$$\text{mol/kg}$$

Unused resin capacity

$$q_{un}$$

resin_unused_capacity

None

$$\text{mol/kg}$$

Sorbed mass of ion

$$M_{out}$$

mass_removed

target_ion_set

$$\text{mol}$$

Number of transfer units

$$N$$

num_transfer_units

None

$$\text{dimensionless}$$

Dimensionless time

$$\tau$$

dimensionless_time

None

$$\text{dimensionless}$$

Partition ratio

$$\Lambda$$

partition_ratio

None

$$\text{dimensionless}$$

Fluid mass transfer coefficient

$$k_{f}$$

fluid_mass_transfer_coeff

target_ion_set

$$\text{m/s}$$

Mass removed during service

$$M_{rem,j}$$

mass_removed

target_ion_set

$$\text{mol}$$

If isotherm is set to freundlich, the model includes the following components:

Description

Symbol

Variable Name

Index

Units

Variables

Freundlich isotherm exponent for resin/ion system

$$n$$

freundlich_n

None

$$\text{dimensionless}$$

Bed volumes at breakthrough

$$BV$$

bv

None

$$\text{dimensionless}$$

Bed volumes at 50% influent conc.

$$BV_{50}$$

bv_50

None

$$\text{dimensionless}$$

Mass transfer coefficient

$$k_T$$

mass_transfer_coeff

None

$$\text{s}^{-1}$$

Average relative breakthrough concentration at breakthrough time

$$X_{avg}$$

c_norm_avg

None

$$\text{dimensionless}$$

Relative breakthrough conc. for trapezoids

$$X_{trap,k}$$

c_traps

k

$$\text{dimensionless}$$

Breakthrough times for trapezoids

$$t_{trap,k}$$

tb_traps

k

$$\text{s}$$

Area of trapezoids

$$A_{trap,k}$$

traps

k

$$\text{dimensionless}$$

## Degrees of Freedom

Aside from the inlet feed state variables (temperature, pressure, component molar flowrate), the user must specify an additional 9 degrees of freedom for both the langmuir and freundlich isotherm model configurations to achieve a fully specified model (i.e., zero degrees of freedom). Depending on the data available to the user and the objectives of the modeling exercise, different combinations of variables can be fixed to achieve zero degrees of freedom.

For either model configuration, the user can fix the following variables:

• resin_diam

• resin_bulk_dens

• bed_porosity

• service_flow_rate (alternatively, vel_bed)

• bed_depth

• number_columns

### Langmuir DOF

If isotherm is set to langmuir, the additional variables to fix are:

• langmuir

• resin_max_capacity

• dimensionless_time (can be fixed to default value of 1)

### Freundlich DOF

If isotherm is set to freundlich, the additional variables to fix are:

• freundlich_n

• bv

• c_norm

• one of bv_50 or mass_transfer_coeff as determined from Clark model equations

## Solution Component Information

The IonExchange0D model is designed to work with WaterTAP’s Multi-component aqueous solution (MCAS) property package. In addition to providing a list of solute ions, users must provide parameter information for each ion including molecular weight, diffusivity data, and charge data. An example of how this data is used to build a model is provided below.

target_ion = "Ca_2+"
ion_props = {
"solute_list": [target_ion],
"diffusivity_data": {("Liq", target_ion): 9.2e-10},
"mw_data": {"H2O": 0.018, target_ion: 0.04},
"charge": {target_ion: 2},
}
m = ConcreteModel()
m.fs = FlowsheetBlock(dynamic=False)
m.fs.properties = MCASParameterBlock(**ion_props)
ix_config = {
"property_package": m.fs.properties,
"target_ion": target_ion,
}
m.fs.ix = IonExchange0D(**ix_config)


## Equations and Relationships

Description

Equation

Common

Service flow rate

$$SFR = \frac{Q_{p, in}}{V_{res, tot}}$$

Total bed volume

$$V_{res, tot} = V_{bed}n_{op}$$

Flow through bed constraint

$$\frac{Z}{u_{bed}} = \frac{V_{res, tot}}{Q_{p, in}}$$

Total resin volume required

$$V_{res, tot} = Z \pi \frac{D_{col}^2}{4} n_{op}$$

Volume of single column

$$V_{col} = H_{col} \frac{V_{bed}}{Z}$$

Total column volume required

$$V_{col, tot} = n_{op}V_{col}$$

Column height to diameter ratio

$$R_{HD} = \frac{H_{col}}{D_{col}}$$

Column height

$$H_{col} = Z + H_{distributor} + H_{underdrain} + H_{expan}$$

Interstitial velocity

$$u_{inter} = \frac{u_{bed}}{\varepsilon}$$

Contact time

$$t_{contact} = EBCT \varepsilon$$

Empty bed contact time

$$EBCT = \frac{Z}{u_{bed}}$$

Regeneration tank volume

$$V_{regen} = t_{regen} (Q_{p, in} / R)$$

Bed expansion fraction from backwashing (T = 20C)

$$X_{expan} = H_{expan,A} + H_{expan,B}u_{bw} + H_{expan,C}u_{bw}^{2}$$

Bed expansion from backwashing

$$H_{expan} = X_{expan}Z$$

Regen volumetric flow rate

$$Q_{regen} = \frac{Q_{p, in}N_{regen}}{R}$$

Backwashing flow rate

$$Q_{bw} = u_{bw} \frac{V_{bed}}{Z}n_{op}$$

Rinse flow rate

$$Q_{rinse} = u_{bed} \frac{V_{bed}}{Z}n_{op}$$

Main pump power

$$P_{main} = \frac{g \rho_{in} p_{drop}Q_{p, in}}{\eta} \Big( \frac{t_{break}}{t_{cycle}} \Big)$$

Regen pump power

$$P_{regen} = \frac{g \rho_{in} p_{drop}Q_{regen}}{\eta} \Big( \frac{t_{regen}}{t_{cycle}} \Big)$$

Rinse pump power

$$P_{rinse} = \frac{g \rho_{in} p_{drop}Q_{rinse}}{\eta} \Big( \frac{t_{rinse}}{t_{cycle}} \Big)$$

Backwash pump power

$$P_{bw} = \frac{g \rho_{in} p_{drop}Q_{bw}}{\eta} \Big( \frac{t_{bw}}{t_{cycle}} \Big)$$

Pressure drop (T = 20C)

$$p_{drop} = Z(p_{drop,A} + p_{drop,B}u_{bed} + p_{drop,C}u_{bed}^{2})$$

Rinse time

$$t_{rinse} = EBCT N_{rinse}$$

Waste time

$$t_{waste} = t_{regen} + t_{bw} + t_{rinse}$$

Cycle time

$$t_{cycle} = t_{break} + t_{waste}$$

Reynolds number

$$Re = \frac{u_{bed}d}{\mu}$$

Schmidt number

$$Sc = \frac{\mu}{D}$$

Sherwood number

$$Sh = 2.4 \varepsilon^{0.66} Re^{0.34} Sc^{0.33}$$

Bed Peclet number

$$Pe_{bed} = Pe_{p} \frac{Z}{d}$$

Particle Peclet number

$$Pe_{p} = 0.05 Re^{0.48}$$

Resin surface area per vol

$$a_{s} = 6 \frac{1-\varepsilon}{d}$$

Langmuir

Langmuir isotherm

$$\frac{C_{b}}{C_{0}} (1-\frac{q_{eq}}{q_{max}}) = La (1-\frac{C_{b}}{C_{0}})\frac{q_{eq}}{q_{max}}$$

Constant pattern solution for Langmuir isotherm

$$N(\tau - 1) = 1 + \frac{\log{(C_{b}/C_{0})} - La \log{(1 - C_{b}/C_{0})}}{1 - La}$$

Resin capacity mass balance

$$q_{max} = q_{avail} + q_{eq}$$

Partition ratio

$$\Lambda = \frac{q_{eq} \rho_{b}}{C_{0}}$$

Fluid mass transfer coeff

$$k_{f} = \frac{D Sh}{d}$$

Number of mass-transfer units

$$N = \frac{k_{f}a_{s}Z}{u_{bed}}$$

Dimensionless time

$$\tau = (\frac{u_{inter}t_{break} \varepsilon}{Z} - \varepsilon) / \Lambda$$

Height of transfer unit

$$HTU = \frac{u_{bed}}{\rho_{b}k}$$

Rate coefficient

$$k = 6 \frac{(1-\varepsilon)k_{f}}{\rho_{b}d}$$

Mass removed

$$M_{rem,j} = V_{res,tot}q_{eq} \rho_{b}$$

Mass transfer term

$$\dot{m}_j = -M_{rem,j} / t_{break}$$

Freundlich

Breakthrough concentration

$$X = \frac{C_b}{C_0}$$

Bed volumes at breakthrough concentration

$$BV = \frac{t_{break} u_{bed}}{Z}$$

Clark equation with fundamental constants

$$X = \frac{1}{\bigg(1 + (2^{n - 1} - 1)\text{exp}\bigg[\frac{k_T Z (n - 1)}{BV_{50} u_{bed}} (BV_{50} - BV)\bigg]\bigg)^{\frac{1}{n-1}}}$$

Evenly spaced c_norm for trapezoids

$$X_{trap,k} = X_{trap,min} + (k - 1) \frac{X - X_{trap,min}}{n_{trap} - 1}$$

Breakthru time calculation for trapezoids

$$t_{trap,k} = - \log{\frac{X_{trap,k}^{n-1}-1}{A}} / k_T$$

Area of trapezoids

$$A_{trap,k} = \frac{t_{trap,k} - t_{trap,k - 1}}{t_{trap,n_{trap}}} \frac{X_{trap,k} + X_{trap,k - 1}}{2}$$

Average relative effluent concentration

$$X_{avg} = \sum{A_{trap,k}}$$

Mass transfer term

$$\dot{m}_j = -(1 - X_{avg}) N_j$$

## References

LeVan, M. D., Carta, G., & Yon, C. M. (2019).
Section 16: Adsorption and Ion Exchange.
Perry’s Chemical Engineers’ Handbook, 9th Edition.
Crittenden, J. C., Trussell, R. R., Hand, D. W., Howe, K. J., & Tchobanoglous, G. (2012).
Chapter 16: Ion Exchange.
MWH’s Water Treatment (pp. 1263-1334): John Wiley & Sons, Inc.
Inamuddin, & Luqman, M. (2012).
Ion Exchange Technology I: Theory and Materials.
Vassilis J. Inglezakis and Stavros G. Poulopoulos
Adsorption, Ion Exchange and Catalysis: Design of Operations and Environmental Applications (2006).
doi.org/10.1016/B978-0-444-52783-7.X5000-9
Michaud, C.F. (2013)
Hydrodynamic Design, Part 8: Flow Through Ion Exchange Beds
Water Conditioning & Purification Magazine (WC&P)
Clark, R. M. (1987).
Evaluating the cost and performance of field-scale granular activated carbon systems.
Environ Sci Technol, 21(6), 573-580.
doi:10.1021/es00160a008
Croll, H. C., Adelman, M. J., Chow, S. J., Schwab, K. J., Capelle, R., Oppenheimer, J., & Jacangelo, J. G. (2023).
Fundamental kinetic constants for breakthrough of per- and polyfluoroalkyl substances at varying empty bed contact times:
Theoretical analysis and pilot scale demonstration.
Chemical Engineering Journal, 464.
doi:10.1016/j.cej.2023.142587
United States Environmental Protection Agency. (2021). Work Breakdown Structure-Based Cost Models