Mass Transport Equation: In support of our Filter Prototype

v: velocity of water through the column. Units: cm.s^-1

D: diffusion coefficient; in the context of a mass transport equation, it quantifies how fast a substance like a metal ion spreads out due to molecular motion from regions of high concentration to low concentration. Units: cm²/s

ρ: Bulk density of beads in the filter; refers to the mass of the beads per unit volume of the entire filter bed, including the void spaces between the beads.
ρ= mass of dry beads / the vol. they occupy Units: g.cm^-3

The following is the plot for concentration of the metal ions in water against different positions on the adsorption filter bed made of sodium alginate beads.

Advection-Diffusion: Mass Tralsport Equation Graph Model Documentation

Parameter Justification

The parameters are chosen based upon literature study. The justification for choosing the parameters are as follows.

Velocity (v = 1×10^-6 m/s)

• Physically realistic for low-permeability groundwater systems
• Falls within typical range: 10^-8 to 10^-5 m/s for groundwater
• Conservative choice for contamination studies where slow transport is critical

Diffusion Coefficient (D = 1×10^-10 m²/s)

Effective diffusion in porous media, accounting for:

• Porosity reduction
• Molecular diffusion in free water

Explanation of the plot

• Blue curve (t = 0): The solute is localized near the inlet.
• Green dashed curve (t = mid): At intermediate time, the solute has moved rightward due to advection while becoming broader and lower in peak height due to diffusion.
• Red dotted curve (t = final): After a long time, the profile has shifted even further right, spreading out more. The peak is smaller because diffusion has smeared out the concentration.

The plot is a dynamic Gaussian distribution, moving rightward due to advection and broadening due to diffusion.

Competitive Langmuir Isotherm: Equilibrium Relationship to model Adsorption

The following is the plot for the amount of adsorbate adsorbed in the sodium alginate beads-filter bedagainst concentration of different metal ions in water.

Competitive Langmuir Adsorption Isotherm fordifferent metal ions Model Documentation

Explanation of Plot

• Hg: The most favorably adsorbed ion among the four, indicated by the steepest and highest curve. It reaches higher saturation faster.
• Cr: Has high adsorption, but lower than Hg.
• Fe: Shows moderate adsorption behavior.
• Al: The least favorably adsorbed, likely due to its lower interaction strength with the adsorbent. It has slower and lower saturation.

The shape of the curves is characteristic of Langmuir isotherms, which assume monolayer adsorption on a finite number of sites or positions on the filter bed.

Breakthrough Curve: To Determine Filter Exhaustion Time

It helps determine when the filter becomes saturated and stops working effectively.

• C: Effluent concentration of the contaminant; concentration of metal ions in the water that is coming out of the filter.
• C₀: Influent concentration of the contaminant; concentration of metal ions in the water that is going into the filter.
• b: First-order breakthrough rate constant.
• t: Time in hours.

The following is the breakthrough curve for different metal ions.

Breakthrough Curves for differentmetal ions Model Documentation

Explanation of the Plot

The plot shows that the adsorbent (sodium alginate beads – filter bed) has the highest affinity for mercury ions and the lowest for aluminium ions.

Adsorption Strength Order

Adsorption strength decreases in the order:

Hg²⁺ > Cr³⁺ > Fe²⁺ > Al³⁺

Facts Regarding Our Math Model

Regarding the Langmuir Isotherm

Q1. Why do we consider Mₘₐₓ_i value in the competitive Langmuir isotherm?

We consider the Mₘₐₓ_i value in the competitive Langmuir isotherm because it defines the maximum adsorption capacity of the adsorbent (sodium alginate beads) for each individual metal ion. This parameter is essential for accurately predicting how much of each ion can be adsorbed in the presence of competition.

• It defines the saturation limit for ion i. Even in a mixture of competing ions, each ion has a theoretical upper limit on how much it could occupy if all available sites were occupied by only that ion.
• It reflects the adsorbent’s physical and chemical capacity.

Mₘₐₓ_i is determined by:

1. Number and nature of binding sites
2. Surface area and porosity
3. Affinity of ion i for those sites

Cᵢ represents the equilibrium concentration of the metal ion i in the solution, that is, the concentration of ion i that remains in the liquid phase after adsorption has occurred and equilibrium is established. It tells us:

• Amount of Ion Not Adsorbed: It tells you how much of metal i remains in the solution after equilibrium.
• At first, adsorption generally increases with increasing Cᵢ because more ions are available to interact with the adsorbent. When the sites are saturated, adsorption will stop at a point when the adsorbent sites can no longer accommodate any adsorbate.

The denominator plays a role in capturing the competitive interactions between multiple metal ions that are adsorbed onto a common surface with a fixed number of adsorption sites.

Each term in the summation Σ_j=1ⁿ K_Mj · C_j represents the tendency of species j to occupy the surface. C_j : concentration of metal ion j: higher concentration means more of that species available to adsorb K_Mj : adsorption constant: higher K_Mj means stronger attraction to the adsorbent.

• The numerator increases with increase in Cᵢ M_maxi , and K_Mj , leading to more adsorption.
• The denominator increases with higher concentrations or higher affinity to the adsorbent of all ions, which reduces the adsorption of a particular ion.
• If many ions with high C_j and K_Mj are present, M_i will decrease due to competitive adsorption.
• Hg²⁺, Al³⁺, Cr³⁺ tend to outcompete Fe²⁺, especially at low pH or when binding sites are limited.

Regarding the Mass Transport Equation

Q2. Why do we use bulk density in the mass transport equation?

The filter bed is made up of many spherical beads. They do not fill the space completely. There is air or liquid in between the beads. Bulk density includes that space between the beads.

Q3. Why do we use ∂M/∂t ?

We are modelling a dynamic time-dependent adsorption process. Metal ions are continuously being adsorbed from the water over time. The term ∂M/∂t tells us how fast the concentration is changing in the water phase. We need the rate of change to properly model mass transfer dynamics over time.

Q4. Why are we subtracting the sink term?

The term represents how fast the metal ions are leaving the water phase and entering the solid phase (beads). So, we subtract it from the equation.

• The left side of the equation tells us how metal ion concentration changes with time and due to water flow.
• The right side tells the effects influencing the change — diffusion, movement of metal ions from liquid to solid phase.
• The concentration of metal ions in water changes due to multiple processes occurring in the system.
• To keep the maths and physics consistent, we must relate all these effects together. That is what the equation does.

To evaluate and optimize our bio-hybrid metal capture system, we modeled the adsorption dynamics within a fixed-bed adsorber, where peptide-functionalized beads are packed into modular mesh cartridges. These cartridges are arranged in alternating concave and convex layers to enhance flow distribution and prevent channeling, ensuring consistent performance under continuous flow conditions.

Filter prototype illustrated in 2D and 3D visualisation. Model Documentation

The performance of this reactor was analyzed using mass-transfer and Langmuir isotherm-based models to predict solute uptake, breakthrough behavior, and overall adsorption efficiency. Parameters such as flow rate, bed height, and bead diameter were systematically varied to determine the optimal contact time and minimize pressure drop across the bed.
By coupling these experimental and theoretical insights, we developed a comprehensive performance model that connects molecular adsorption phenomena to reactor-scale behavior. This modeling framework, supported by real-time monitoring through colorimetric or ion-selective sensors, enables dynamic control of effluent quality and process optimization. Together, these approaches form the foundation of a reproducible, modular, and biodegradable metal sequestration platform that future iGEM teams can extend toward new biosorptive and environmental remediation systems.
Fixed-bed adsorbers are vertical columns packed with adsorbent particles, commonly used in Performance Characteristics of Fixed-Bed Adsorbers commercial adsorption operations because they provide a high adsorption area per unit volume. The liquid containing solute flows through the bed, and solute uptake occurs until the adsorbent is saturated.

Fixed bed adsorber model Model Documentation

Adsorption Dynamics

1. Initial Stage:
   • Adsorbent at the top of the column rapidly takes up solute.
   • Effluent concentration is close to zero.
2. Adsorption Zone / Wave:
   • As top layers saturate, the region where adsorption actively occurs moves down the column, forming the adsorption zone or adsorption wave.
   • This wave moves more slowly than the fluid flow.
3. Breakpoint:
   • When the adsorption zone reaches the bottom, the effluent concentration begins to rise noticeably.
   • Complete saturation occurs when the effluent concentration equals the inlet concentration, CAi.

Breakthrough Curve

• The breakthrough curve plots effluent concentration versus time or volume processed.
• It is a key factor in designing and operating fixed-bed adsorbers:
  • Area under the curve represents solute loss.
  • Stopping adsorption before full saturation reduces solute waste but leaves some unused adsorbent capacity.
• The curve’s shape depends on:
  • Feed rate
  • Solute concentration
  • Adsorption equilibrium
  • Mass transfer rates within and outside the particles

Engineering Considerations

• Fixed-bed adsorption is an unsteady-state process; equations are generally nonlinear.
• Design factors include:
  • Amount of adsorbent
  • Required contact time for a given solute quantity
  • Mass transfer limitations
• Pilot studies are often required, but mathematical analysis and mass balances can help predict breakthrough behavior and optimize operation.

Engineering analysis of biofilter model. Model Documentation

Mass Balance Approach

• To predict the effluent concentration as a function of time, a differential mass balance is applied to small sections of the column:

{Mass in through system boundaries} – {Mass out through system boundaries} + {Mass generated within system} – {Mass consumed within system} = {Mass accumulated within system}

• This allows derivation of the breakthrough curve, which guides operational decisions such as timing of adsorbent replacement or regeneration.

Mass Transfer in Fixed-Bed Adsorbers

The efficiency of adsorption in a fixed-bed system depends not only on the adsorbent capacity but also on the rate of mass transfer from the liquid to the internal surfaces of the adsorbent. There are multiple steps where resistance can occur.

1. Bulk liquid to particle boundary layer: Transfer of solute from the well-mixed bulk liquid to the thin stagnant film around the particle.
2. Diffusion through the boundary layer: Movement of solute across the liquid film surrounding the particle.
3. Transport within particle pores: Diffusion of solute through the internal liquid-filled pores to the adsorption sites.
4. Adsorption at the active site: The actual binding of solute to the internal surfaces of the adsorbent.
5. Surface diffusion: Migration of adsorbed molecules along internal surfaces within the pores.

Key points:

• Adsorption on the outer particle surface is minimal compared to internal pore adsorption.
• Bulk transfer (step 1) is generally fast due to convective flow and can usually be neglected.
• Step 2 (boundary layer diffusion) is often the major external resistance.
• Steps 3 and 5 (internal diffusion and surface migration) represent major internal resistances.
• The overall rate-controlling step can vary depending on system conditions and must often be determined experimentally using pilot-scale columns.