ion_exchange_0D.rst

Ion Exchange (0D)

.. index::
   pair: watertap.unit_models.ion_exchange_0D;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
3. Steady state only
4. Single solute and single solvent (water) only
5. Plug flow conditions
6. Isothermal conditions
7. Constant separation-factor treatment with favorable Langmuir 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, i.e., treatment, (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.

Ports

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

• Inlet port (inlet)
• Outlet port (outlet)
• Regen port (regen)

Sets

Description	Symbol	Indices
Time	t	[0]
Phases	p	['Liq']
Components	j	['H2O', 'Na_+', 'Ca_2+', 'Cl_-', 'Mg2+', 'SO4_2-', 'PFAS_-', 'Hardness_2+']
Ions	j	['Na_+', 'Ca_2+', 'Cl_-', 'Mg2+', 'SO4_2-', 'PFAS_-', 'Hardness_2+']*

*"Ion" is a subset of "Component" and uses the same symbol j. NOTE:The "Components" and "Ions" lists can include any ion as long as the ion is configured into the property package.

Degrees of Freedom

Aside from the inlet feed state variables (i.e., temperature, pressure, component molar flowrate),

the Ion Exchange (0D) model has at least an additional 27 degrees of freedom that the user must specify. The table below gives an outline of these.

Description	Symbol	Variable Name	Index	Units
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}
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}
Service flow rate through resin bed in bed volumes per hour	SFR	service_flow_rate	None	\text{hr}^{-1}
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}
Resin bead diameter	d	resin_diam	None	\text{m}
Resin bulk density	\rho_{b}	resin_bulk_dens	None	\text{kg/L}
Bed porosity	\epsilon	bed_porosity	None	\text{dimensionless}
Dimensionless time	\tau	dimensionless_time	None	\text{dimensionless}
Regenerant dose per volume of resin	C_{regen}	regen_dose	None	\text{kg/}\text{m}^3
Number of cycles before regenerant disposal	N_{regen}	regen_recycle	None	\text{dimensionless}
Regeneration time	t_{regen}	t_regen	None	\text{s}
Backwash time	t_{bw}	t_bw	None	\text{s}
Backwash loading rate	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}
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 eq 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}
Service-to-regeneration flow ratio	R	service_to_regen_flow_ratio	None	\text{dimensionless}

Users must provide values for and 'fix' the following variables to solve the model with DOF=0: 'pressure', 'temperature', 'flow_mol_phase_comp', 'langmuir', 'resin_max_capacity', 'service_flow_rate', 'number_columns', and 'bed_depth'. The other variables can simply be fixed to their default values ('.fix()').

NOTE: Variables for 'temperature', 'pressure', and 'flow_mol_phase_comp' come from the associated property package as state variables and are accessed via {port_name}.{state_var_name}

Variables

Description	Symbol	Variable Name	Index	Units
Resin Variables
Maximum resin capacity	q_{max}	resin_max_capacity	None	\text{mol/kg}
Usable resin capacity at equilibrium	q_{eq}	resin_eq_capacity	None	\text{mol/kg}
Available resin capacity at equilibrium	q_{avail}	resin_unused_capacity	None	\text{dimensionless}
Resin bead diameter	d	resin_diam	None	\text{m}
Resin bulk density	\rho_{b}	resin_bulk_dens	None	\text{kg/L}
Resin particle density	\rho_{p}	resin_particle_dens	None	\text{dimensionless}
Separation factor	\alpha	separation_factor	target_ion_set	\text{dimensionless}
Resin surface area per volume	a_{s}	resin_surf_per_vol	None	\text{m}^{-1}
Langmuir equilibrium parameter for resin/ion system	La	langmuir	target_ion_set	\text{dimensionless}
Bed/Column Variables
Ratio of bed depth to column diameter	X	bed_depth_to_diam_ratio	None	\text{dimensionless}
Bed volume of one unit	V_{bed}	bed_vol	None	\text{m}^{3}
Total bed volume	V_{tot}	bed_vol_tot	None	\text{m}^{3}
Bed depth	Z	bed_depth	None	\text{m}
Bed porosity	\epsilon	bed_porosity	None	\text{dimensionless}
Column height	H	col_height	None	\text{m}
Column diameter	D_{col}	col_diam	None	\text{m}
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}
Number of operational columns	n_{op}	number_columns	None	\text{dimensionless}
Number of redundant columns	n_{red}	number_columns_redund	None	\text{dimensionless}
Underdrain height	H_{underdrain}	underdrain_h	None	\text{m}
Distributor height	H_{distributor}	distributor_h	None	\text{m}
Kinetic Variables
Partition ratio	\Lambda	partition_ratio	None	\text{dimensionless}
Fluid mass transfer coefficient	k_{f}	fluid_mass_transfer_coeff	target_ion_set	\text{m/s}
Rate coefficient based on fluid-phase concentration driving force	k	rate_coeff	target_ion_set	\text{m}^{3}\text{kg*s}
Number of transfer units	N	num_transfer_units	None	\text{dimensionless}
Height of a transfer unit	HTU	HTU	target_ion_set	\text{m}
Position of breakthrough on constant-pattern wave	lh	lh	None	\text{dimensionless}
Influent mass of ion	M_{in}	mass_in	target_ion_set	\text{mol}
Sorbed mass of ion	M_{out}	mass_removed	target_ion_set	\text{mol}
Effluent mass of ion	M_{rem}	mass_out	target_ion_set	\text{mol}
Hydrodynamic Variables
Service flow rate through resin bed in bed volumes per hour	SFR	service_flow_rate	None	\text{hr}^{-1}
Velocity through resin bed	u_{bed}	vel_bed	None	\text{m/s}
Interstitial velocity	u_{inter}	vel_inter	None	\text{m/s}
Holdup percent	holdup	holdup	None	\text{dimensionless}
Pressure drop through resin bed	P_{drop}	pressure_drop	None	\text{psi}
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}
Time Variables
Rinse time	t_{rinse}	t_rinse	None	\text{s}
Dimensionless time	\tau	dimensionless_time	None	\text{dimensionless}
Breakthrough time	t_{breakthru}	t_breakthru	None	\text{s}
Cycle time	t_{cycle}	t_cycle	None	\text{s}
Contact time	t_{contact}	t_contact	None	\text{s}
Regen + Rinse + Backwash time	t_{waste}	t_waste	None	\text{s}
Regeneration time	t_{regen}	t_regen	None	\text{s}
Backwash time	t_{bw}	t_bw	None	\text{s}
Dimensionless Variables
Reynolds number	Re	Re	None	\text{dimensionless}
Schmidt number	Sc	Sc	target_ion_set	\text{dimensionless}
Sherwood number	Sh	Sh	target_ion_set	\text{dimensionless}
Peclet particle number	Pe_{p}	Pe_p	None	\text{dimensionless}
Peclet bed number	Pe_{bed}	Pe_bed	None	\text{dimensionless}
Ratio of breakthrough concentration to influent concentration	C_{b}/C_{0}	c_norm	target_ion_set	\text{dimensionless}
Regeneration Variables
Service-to-regeneration flow ratio	R	service_to_regen_flow_ratio	None	\text{dimensionless}
Number of cycles before regenerant disposal	N_{regen}	regen_recycle	None	\text{dimensionless}
Regenerant dose per volume of resin	C_{regen}	regen_dose	None	\text{kg/}\text{m}^3
Backwashing Variables
Backwashing volumetric flow rate	Q_{bw}	bw_flow	None	\text{m}^{3}\text{/s}
Backwash loading rate	u_{bw}	bw_rate	None	\text{m/hr}
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}
Bed expansion fraction eq 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}
Rinsing Variables
Rinse volumetric flow rate	Q_{rinse}	rinse_flow	None	\text{m}^{3}\text{/s}
Number of bed volumes for rinse step	N_{rinse}	rinse_bv	None	\text{dimensionless}
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}
Assumed efficiency for all pumps	\eta	pump_efficiency	None	\text{dimensionless}

Solution Component Information

In addition to providing a list of solute ions, the users will need to provide parameter information for each ion including molecular weight, diffusivity data, and charge data.

To provide this information to the unit model, users must add dictionaries to the initialization of the unit model. These dictionaries must have the following format.

def get_ix_in(ions):
 diff_data = {
     "Na_+": 1.33e-9,
     "Ca_2+": 9.2e-10,
     "Cl_-": 2.03e-9,
     "Mg_2+": 0.706e-9,
     "SO4_2-": 1.06e-9,
     "PFAS_-": 0.49e-9,
     "Hardness_2+": 0.706e-9,
 }
 mw_data = {
     "Na_+": 23e-3,
     "Ca_2+": 40e-3,
     "Cl_-": 35e-3,
     "Mg_2+": 24e-3,
     "SO4_2-": 96e-3,
     "PFAS_-": 414.1e-3,
     "Hardness_2+": 100.0869e-3,
 }
 charge_data = {
     "Na_+": 1,
     "Ca_2+": 2,
     "Cl_-": -1,
     "Mg_2+": 2,
     "SO4_2-": -2,
     "PFAS_-": -1,
     "Hardness_2+": 2,
 }
 ix_in = {
     "solute_list": [],
     "diffusivity_data": {},
     "mw_data": {"H2O": 18e-3},
     "charge": {},
 }
 for ion in ions:
     ix_in["solute_list"].append(ion)
     ix_in["diffusivity_data"][("Liq", ion)] = diff_data[ion]
     ix_in["mw_data"][ion] = mw_data[ion]
     ix_in["charge"][ion] = charge_data[ion]
 return ix_in

NOTE: 'ions' is an ion_set, which is a configuration argument of the property package as shown below

ions = m.fs.unit.config.property_package.ion_set

NOTE: The above example assumes you have already constructed a pyomo model named 'm' and attached an IDAES flowsheet named 'fs' to it.

Equations and Relationships

Description	Equation
Separation factor	\alpha = \frac{1}{La}
Langmuir isotherm	\alpha \frac{C_{b}}{C_{0}} (1-\frac{q_{eq}}{q_{max}}) = (1-\frac{C_{b}}{C_{0}})\frac{q_{eq}}{q_{max}}
Reynolds number	Re = \frac{u_{bed}d}{\mu}
Schmidt number	Sc = \frac{\mu}{D}
Sherwood number	Sh = \frac{1.09}{\epsilon}Re^{0.33}Sc^{0.33}
Bed Peclet number	Pe_{bed} = Pe_{p} \frac{Z}{d}
Particle Peclet number	Pe_{p} = 0.05 Re^{0.48}
Resin capacity mass balance	q_{max} = q_{avail} + q_{eq}
Interstitial velocity	u_{inter} = \frac{u_{bed}}{\epsilon}
Resin surface area per vol	a_{s} = 6 \frac{1-\epsilon}{d}
Contact time	t_{contact} = \frac{Z}{u_{inter}}
Service flow rate	SFR = \frac{Q_{p, in}}{V_{tot}}
Flow through bed constraint	\frac{Z \epsilon}{u_{bed}} = \frac{V_{bed} \epsilon}{Q_{p, in} / n_{op}}
Total bed volume	V_{tot} = V_{bed}n_{op}
Column height	H = Z + H_{distributor} + H_{underdrain} + H_{expan}
Column volume calculated from bed volume	V_{col} = H \frac{V_{bed}}{Z}
Column volume calculated from column diameter	V_{col} = \pi (\frac{D_{col}}{2})^{2} H
Column diameter calculation	(\frac{D_{col}}{2})^{2} = (\frac{H}{2X})^{2}
Fluid mass transfer coeff	k_{f} = \frac{D Sh}{d}
Rate coefficient	k = 6 \frac{(1-\epsilon)k_{f}}{\rho_{b}d}
Height of transfer unit	HTU = \frac{u_{bed}}{\rho_{b}k}
Partition ratio	\Lambda = \frac{q_{eq} \rho_{b}}{ñ_{in}}
Left hand side of constant pattern solution	lh = N(\tau - 1)
Right hand side of constant pattern solution	lh = 1 + \frac{\log{(C_{b}/C_{0})} - La \log{(1 - C_{b}/C_{0})}}{1 - La}
Dimensionless time	\tau = (\frac{u_{inter}t_{breakthru} \epsilon}{Z} - \epsilon) / \Lambda
Number of mass-transfer units	N = \frac{k_{f}a_{s}Z}{u_{bed}}
Flow conservation	Q_{p, in} - \frac{Q_{bw}t_{bw} + Q_{rinse}t_{rinse}}{t_{cycle}} = Q_{p, out} - \frac{Q_{regen}t_{regen}}{t_{cycle}}
Influent total mass of ion	M_{in} = Q_{p, in}t_{breakthru}ñ_{in}
Removed total mass of ion	M_{rem} = V_{bed}q_{eq}n_{op} \rho_{b}
Mass of ion in effluent	M_{out} = M_{in} - M_{rem}
Steady-state effluent concentration (for target ion)	ñ_{out} = \frac{M_{out}}{Q_{p, in}t_{breakthru}}
Steady-state effluent concentration	ñ_{out} = ñ_{in}
Steady-state regen concentration (for target ion)	ñ_{regen} = \frac{M_{rem}N_{regen}}{Q_{p, regen}t_{regen}}
Steady-state regen concentration	ñ_{regen} = 0
Cycle time	t_{cycle} = t_{breakthru} + t_{waste}
Waste time	t_{waste} = t_{regen} + t_{bw} + t_{rinse}
Regen volumetric flow rate	Q_{p, regen} = \frac{Q_{p, in}N_{regen}}{R}
Regen pump power	P_{regen} = \frac{9.81 \rho_{in} 0.70325P_{drop}Q_{p, regen}}{\eta}
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
Backwashing flow rate	Q_{bw} = u_{bw} \frac{V_{bed}}{Z}n_{op}
Backwash pump power	P_{bw} = \frac{9.81 \rho_{in} 0.70325P_{drop}Q_{bw}}{\eta}
Rinse time	t_{rinse} t_{contact} + N_{rinse}
Rinse flow rate	Q_{rinse} = u_{bed} \frac{V_{bed}}{Z}n_{op}
Rinse pump power	P_{rinse} = \frac{9.81 \rho_{in} 0.70325P_{drop}Q_{rinse}}{\eta}
Main pump power	P_{main} = \frac{9.81 \rho_{in} 0.70325P_{drop}Q_{p, in}}{\eta}
Pressure drop (T = 20C)	P_{drop} = Z(P_{drop,A} + P_{drop,B}u_{bed} + P_{drop,C}u_{bed}^{2})
Total column volume required	V_{col, tot} = n_{op}V_{col}

References

Hand, D. W., Crittenden, J. C., & Thacker, W. E. (1984). Simplified models for design of fixed-bed adsorption systems. Journal of Environmental Engineering, 110(2), 440-456.

Crittenden, J., Rhodes, R., Hand, D., Howe, K., & Tchobanoglous, G. (2012). MWHs Water Treatment. Principles and Design. EditorialJohn Wiley & Sons.

LeVan, M. D., Carta, G., & Yon, C. M. (2019). Section 16: Adsorption and Ion Exchange. Perry's Chemical Engineers' Handbook, 9th Edition.

Inamuddin, & Luqman, M. (2012). Ion Exchange Technology I: Theory and Materials.