Modeling | Bioplus-China

Modeling: A Predictive Tool for Dye Dosage in Textile Coloring

Moving beyond qualitative characterization, we developed a quantitative predictive model based on the Langmuir adsorption isotherm to enhance the practical applicability of our bio-adsorbent system. In this project, we aimed to remove synthetic dyes from wastewater using engineered bacteria. The Langmuir model was employed to quantitatively analyze the adsorption behavior of dyes onto the fabric surface, providing a theoretical framework for describing monolayer adsorption on homogeneous interfaces. This approach allows precise calculation of the dye quantity required to achieve a specific color depth on a given mass of fabric, thereby transforming our biological system into a predictable and scalable tool suitable for industrial applications.

Model Overview: From Adsorption to Prediction

We determined that the adsorption of our bio-derived dye onto target fabric follows the Langmuir adsorption model, which indicates monolayer coverage on a homogeneous surface.

The core Langmuir equation is:

q_{e} = \frac{q_{max} K_{L} C_{e}}{1 + K_{L} C_{e}}

q_e is the adsorption capacity at equilibrium (mg dye / g fabric).
q_max is the maximum adsorption capacity (mg/g).
K_L is the Langmuir constant (L/mg), representing adsorption affinity.
C_e is the equilibrium dye concentration in solution (mg/L).

Parameterization: Fitting the Model to Our Data

We first conducted dyeing experiments using solutions of different initial concentrations (C₀). We measured the absorbance at various time points and found that the absorbance reached equilibrium after 3 hours, as shown in Figure 1. Raw data are shown in Table 1.

Table 1: Dye absorbance (A₄₇₅) values at different times and initial concentrations.
C₀ (mg/L)	A₄₇₅
C₀ (mg/L)
Time	0(min)	10(min)	20(min)	30(min)	60(min)	120(min)	240(min)
200	3.65	3.07	2.72	2.48	2.20	2.07	2.05
180	3.29	2.73	2.39	2.16	1.86	1.74	1.72
160	2.92	2.41	2.10	1.89	1.59	1.46	1.37
150	2.74	2.26	1.96	1.76	1.48	1.35	1.17
140	2.56	2.11	1.84	1.65	1.38	1.25	1.08
100	1.82	1.48	1.26	1.11	0.90	0.80	0.67
70	1.28	0.98	0.79	0.67	0.50	0.43	0.37

Change in dye absorbance over time — Figure 1: Change in dye absorbance (A₄₇₅) over time for different initial concentrations C₀.

We performed batch adsorption experiments with varying initial dye concentrations C₀ and measured the equilibrium concentration C_e via absorbance, as shown on the left of Figure 2. The adsorption capacity q_e was calculated using:

q_{e} = \frac{(C_{0} - C_{e}) V}{m}

where V is the volume of the solution and m is the mass of the fabric (Table 2). The results are shown on the right of Figure 2.

Table 2: Mass of the dyed fabric and calculated C_e and q_e at different initial concentrations.
m (g)	C₀ (mg/L)	C_e (mg/L)	C₀-C_e	q_e (mg/g)
0.0612	200	112.398	87.602	14.31
0.0603	180	94.294	85.706	14.21
0.0624	160	75.237	84.763	13.58
0.0635	150	64.207	85.793	13.51
0.0594	140	59.298	80.702	13.58
0.0521	100	36.497	63.503	12.18
0.0545	70	20.345	49.655	9.11

Measured Ce and calculated qe — Figure 2: Measured values of C_e and calculated q_e for different initial concentrations C₀.

Linearization for Parameter Fitting

To obtain the parameters q_max and K_L, we linearized the Langmuir equation:

\frac{1}{q_{e}} = \frac{1}{q_{max} K_{L} C_{e}} + \frac{1}{q_{max}}

By plotting 1/q_e against 1/C_e from our experimental data (as shown in Figure 3), we performed a linear regression analysis.

Figure 3: Linearized Langmuir isotherm plot for the adsorption of our dye onto fabric.

Langmuir Parameters

The plot shows a strong linear relationship with a high coefficient of determination (R² = 0.9727), confirming the applicability of the Langmuir model to our system. The slope and intercept of the trendline Y = 0.9889X + 0.05901 were used to calculate the adsorption parameters.

Calculation of Parameters:

From the linear equation y = 0.9889x + 0.05901:
q_max = 16.95 mg/g
K_L = 0.0597 L/mg

These parameters are the foundational constants for our predictive tool.

The Predictive Tool: From Design to Recipe

The ultimate goal of our model is to answer the practical question: "How do I configure the dye bath to get the color I want?"

We derived a practical formula to calculate the initial dye concentration C₀ needed:

q_{e} = \frac{(C_{0} - C_{e}) V}{m}

Since q_e is itself a function of C_e (from the Langmuir model), we can combine the equations. For a desired C_e (which determines color depth), and knowing the mass of fabric m and the volume of the dye bath V, the required initial dye amount is calculated by:

C_{0} = C_{e} + \frac{q_{e} m}{V}

Fabric samples dyed with different concentrations — Figure 4: Fabric samples dyed with different initial dye concentrations C₀, showing corresponding color depths.

How to use this tool

Define Desired Outcome: Choose the target equilibrium adsorption capacity q_e. A higher q_e means more dye is adsorbed onto the fabric, resulting in a deeper color.
Define Process Parameters: Set the mass of fabric (m) and the volume of the dye bath (V).
Calculate: Plug the values into the equation above along with our predetermined constants q_max and K_L. The result is the initial dye concentration (C₀) you need to prepare.
Prepare Dye Bath: Dissolve the corresponding mass of dye (C₀×V) in the volume V of solvent.

Conclusion

We have successfully developed a quantitative and predictive mathematical model based on the Langmuir isotherm. The high R2 value from our linear fit confirms the model's validity for our dye-fabric system. This model transcends theoretical description and serves as a practical engineering tool. It allows users to precisamente predict the initial dye dosage required to achieve a specific color outcome, minimizing waste and ensuring reproducibility. This represents a significant step towards the rational and sustainable application of synthetic biology in the textile industry.