The Date Palm Trunk Fibers (DPTF) were collected from local farms located in the Al-Agwaar region in the Jordan Valley area of the Balqa governorate (44° 43.574′ N; 122° 41.9052′ W). Fig. 1 shows a sample of the raw fibers.

The available raw fibers were primarily subjected to intensive cleaning by washing them several times using tap water to get rid of all the solid impurities attached to the fibers, such as fine grains of dust, sand, and any foreign things stuck between their fibers. After the cleaning water became clear and free of impurities, a final cleaning process was carried out three times using distilled water to ensure that all impurities were removed and that the fibers were clean and could be used for experiments. Afterward, the DPTF was set aside in an electrical furnace for 15 h at 95°C to produce clean, dry fibers. The DPTF were milled using a multi-purpose grinder. A sieving process was then performed for three hours to classify the sizes and choose the appropriate size to prepare the fixed bed adsorbent. After obtaining the appropriate size, the adsorbent is kept at 105°C to keep it dry and away from moisture. The samples are then cooled and stored in closed glass containers to be ready for preparing the fixed bed adsorbent. The artificial wastewater contaminated with cadmium ions was prepared by dissolving cadmium nitrate in deionized water. A standard solution (Concentration = 1.0 mg/l) was prepared by adding 2.74 g of a high purity grade (p. a.) Cd (NO₃)2.4H₂O to one liter of deionized water. The required concentration of cadmium ions solution was achieved by diluting the standard solution with deionized water.

Fixed bed column experiments were held using the apparatus shown in Fig. 2. The apparatus was designed to measure the breakthrough curves during the continuous adsorption of cadmium ions from an aqueous solution using the DPTF adsorbent. The setup has a 20 L well stirred and closed feed tank to store the prepared artificial wastewater contaminated with cadmium ions at a specific concentration for the intended experiment. The column was made from glass material with a 1.1 cm internal diameter and a length of 50 cm (1 cm² cross-sectional area). Fiberglass wool layers were fixed at the top and bottom of the packed bed adsorbent to keep constant bed height, support the bed, and protect it from escaping the flowing solution. This support was retained using a fixed stainless steel sieve during the whole experiment.

A peristaltic pump (Heidolph, Germany) was used to force the liquid to flow from the feed tank to the top of the column. The flow rate was kept constant during all experiments using the installed valve and the flow meter. The DPTF adsorbent was packed inside the column until the anticipated bed height. The column effluent was collected in a graduated beaker, and samples were taken at steady time intervals to be analyzed for concentration measurements. The residual concentration of cadmium ions in the aqueous solution from the effluents was measured using atomic absorption spectroscopy.

The separation efficiency of cadmium ions from contaminated queues solutions using a fixed-bed adsorption column was investigated by measuring breakthrough curves under different operating conditions. The variation of the effluent concentration with time, the time to reach the breakthrough, and the time to reach bed exhaustion are significant parameters used in the design and operation of the adsorption columns. Continuous fixed-bed column runs were conducted to determine the operating parameters’ influence on the adsorption process of cadmium ions onto the DPTF adsorbent. Several breakthrough curves were measured under controlled experimental conditions. A systematic experimental investigation was performed by keeping the feed concentration and the particle size constant in all experiments and varying the feed flow rate at fixed bed height. Another series of experiments were conducted at different bead heights and fixed flow rates. The experimental conditions are summarized in Table 1.

Various mathematical models and theories have been investigated to predict the breakthrough curves, estimate the design parameters of adsorption columns, and evaluate the dynamic behavior and column performances. Among these models used in this study: are the Thomas model, Adams-Bohart model, Yoon-Nelson model, and Wolborska model.

Thomas model is one of the most commonly applied models used in predicting the breakthrough curves and the dynamic behavior and the performance of the adsorption-backed bed column. It is based on the hypothesis that no axial dispersion with a plug flow pattern in the fixed bed adsorbent. Also, this model assumes that the rate driving force in the adsorption process is governed by second-order reversible reaction kinetics and Langmuir kinetics. It is often used to determine the adsorption capacity of the adsorbent. The linearized form of the Thomas equation is given by the relation [40]:(2) ln⁡(COCt−1)=−COKTHt+MQeKTHF

where (C₀) is the influent concentration of adsorbate, i.e., pollutant (mg/l), (C_t) is the effluent concentration of the pollutant (mg/l), F is the volumetric influent flow rate (L/min), K_Th is Thomas rate parameter (L/min.mg), t is the time (min), Q_e is the maximum (equilibrium) adsorption capacity (mg/g) of the adsorbent. M is the mass of adsorbent in the fixed bed column (g). Hence, the kinetic coefficient K_Th and q can be estimated from the slope and intercept of the straight-line plotting of Ln(COCt−1) vs. t at a given flow rate. Eq. (2) is widely used to fit the measured breakthrough curves obtained from fixed-bed adsorption column experiments. This linearized form relation is a developed form of the Thomas model, which is based on the hypothesis of an irreversible isotherm when developing the differential material balance relation for a fixed bed column [41].

The Adams-Bohart model is routinely used to describe fixed-bed column breakthrough behavior at its initial stages [42]. The simplified equation of the Adams-Bohart model linearized form is as follows [43,44]:(3) t=NOZCOv−1KabCOLn(COCt−1)

where N₀ is the bed capacity (mg/l), υ is the linear velocity (cm/min), K_ab is Adams-Bohart model rate constant for the (l/mg.min), Z is the bed height of column (cm), t is the breakthrough time, is the breakthrough concentration. Applying Bohart–Adams method requires no less than several individual column tests that need to be performed to obtain the necessary experimental data, which is time-wasting. However, an approach has been developed by Hutchins [45], which involves fewer fixed column runs to obtain the required data. This method, known as the bed depth service time (BDST) model, where the Adams-Bohart equation is rewritten with the following linear equation [45,46]:(4) tb=aZ+b

where(5) a=slope=NOCOv

(6) b=intercept=−1KabCOLn(COCt−1)

where t_b, the breakthrough time is when the normalized concentration (C_t/C₀) reaches the value of 0.01. The BDST model was established by assuming that the adsorption rate is governed by the surface reaction between the adsorbate and the residual capacity of the adsorbent.

The Yoon-Nelson model depends on the principle that the possibility of adsorbate adsorption and breakthrough of adsorbate on the adsorbent is proportionate to the rate of decrees in the adsorption probability for every adsorbate species [47]. The linearized form of the Yoon-Nelson model is [48]:(7) Ln(CtCO−Ct)=KYNt−τKYN

where K_YN and τ, the rate parameter (L/min) and the elapsed time to reach a 50% breakthrough (min), the value of K_YN is influenced by the column operating conditions such as the influent concentration of pollutants, height of the bed and rate of flow. The values of K_YN and τ can be evaluated from the intercepts and slopes of the straight-line Plotting of Ln(CCO−C) vs. t

The model developed by Wolborska is based on the general equations of mass transfer describing diffusion mechanisms in the low concentration range breakthrough curves. The linearized form of this model is expressed by the relation [49]:(8) Ln(CtCO)=βaCONOt−βaZv

Results and Discussion Where βa is the external mass transfer coefficient (1/min). N₀ is the adsorption capacity of the bed per unit volume (mg/L). The values of the parameters of the Wolborska model can be determined from the Plot of Ln(CtCO) vs. time.

In the first stage of removal investigations in the continuous flow fixed-bed column with DPTF adsorbent, the flow rate varied from 5 to 15 ml/min, whereas the inlet Concentration of cadmium ions in the feed water was kept constant at 115 mg/l and the mass of the adsorbent was 11 g corresponds to 10 cm bed height. The plots of experimental measurements of breakthrough curves as normalized cadmium concentration (C_t/C₀) against time (t) at different flow rates are depicted in Fig. 4. Fig. 4 shows a typical S-shaped form at different flow rates. This is substantial evidence that resistance to bulk diffusion mass transfer and the bonding of molecules into the pore surface play a vital role during the adsorption process of cadmium onto DPTF. The shown variation of C_t/C₀ with time in Fig. 4 results from the downwards movement of the interface layer between the mass transfer zone (MTZ) and the saturated zone. This causes decay in the MTZ and growth in the saturation zone with time, thus a continuous increase in the (C/C₀) value. The measured breakthrough data were fitted to the Thomas model using linear regression analysis to determine the Thomas rate constant (K_Th) and maximum sorption capacity (Q_e) at the different feed flow rates. Table 2 shows that increasing the flow rate leads to a decrease in the maximum (equilibrium) adsorption capacity (Q_e) and an increase in the values of the Thomas rate constant (K_Th). The values of R² of the kinetic model alternated from 0.9371 to 0.9598, demonstrating good linearity and that the experimental data can be described by this model to a high level of accuracy, with relatively high (R²) values.

As depicted in Fig. 4, at the fewest flow rate (5 ml/min), comparatively increased uptake performance was noticed for cadmium ions sorption to DPTF. This is a consequence of the strong adsorption driving force between the cadmium ions on the DPTF and the cadmium ion in the solution; thus, better column performance was observed. Generally sharper breakthrough curves (i.e., shorter mass transfer zone) were measured at elevated rates of flow. The breakpoint time as well as total maximum (equilibrium) adsorption capacity (mg/g) of the adsorbent (see Table 2) also reduced with the rising flow rate. This attitude can be interpreted by the evidence that cadmium adsorption is influenced by the inadequate residence time of the cadmium ions in the column. This inadequate time reduces the bonding potential of the cadmium ions onto DPTF. While shorter MTZ was detected at the higher flow rate, the adsorption capacity and cadmium ions uptake were higher at lower flow rates and thus better column performance. The results presented in Table 2 show that a maximum cadmium adsorption capacity on DTPF was obtained as 51.5 mg/g (favorable) with the longest mass transfer zone of 83 min (unfavorable) at the lowest flow rate at 5 ml/min.

Fig. 5 displays the measured breakthrough curves at different bed heights under the same operating conditions. It can be observed that a reduction in the slope of the breakthrough curve as the bed height is enlarged from 5 to 15 cm. This indicates a rapid mass transfer zone.

In this study, the bed depth service time (BDST) approach will be used to estimate the parameters N₀ and k_ab from the slope and intercept of the linear Plot of t_b vs. Z using the experimentally measured breakthrough curves at different bed heights shown in Fig. 5.

Fig. 6 shows the BDST plot using the experimental data of cadmium ions adsorption onto DPTF at different bed heights (Fig. 5) for a volumetric flow rate of 10 ml/min (linear velocity = 10.53 cm/min). The linear relationship obtained from the BDST plot for an initial concentration of 115 mg/l is expressed in Eq. (9):(9) tb=5.8Z − 28.33

The saturation concentrations (N₀) and the rate constant (k_ab) obtained from BDST is 7022.16 mg/l and 0.0536 l/mg. min, respectively. DTPF has the advantage of a high value of saturation concentration, allowing more solutions to be processed, resulting in a reduced adsorbent exhaustion rate. The linear correlation coefficient is R²= 0.9619, indicating a high degree of linearity and thus an excellent fitting of the experimental data to the BDST model. The estimated BDST model parameters can be useful to scale up to the industrial process and at other different flow rates and concentrations without extra experimental efforts.

The values of K_YN and τ are evaluated from the intercepts and slopes of the straight-line Plot of Ln(CCO−C) vs. t for the laboratory obtained breakthrough data. The evaluated Yoon-Nelson model constants for the studied flow rates are given in Table 3. It is noticeable that operating at a lower flow rate has a more significant (τ) value, which offers the advantage of a long time to reach the 50%-breakthrough, affecting the operation and column design of the adsorption process. The decrease in τ as flow rate decreases shows that as flow rate decreases, the rate at which the adsorbent bed is exhausted is slower, which is industrially and operationally desirable for the adsorption process. However, the value K_YN increases at higher flow rates. Fig. 7 estimates the laboratory breakthrough curves’ fitting degree of the Yoon-Nelson model. It can be predicted that the calculated curves by the Yoon and Nelson model agree precisely with the experimentally measured curves. Moreover, the Yoon-Nelson model presented good approximations of the experimental behavior observed.

The measured breakthrough curves were analyzed applying the Wolborska model (Eq. (8)) to improve the understanding of the role of column operating flow rate on the mass transfer mechanism. This model depicts the diffusion mechanism in adsorption systems of low concentration generated from the mass transfer equation. Fig. 8 shows the linear fitting of the breakthrough data concerning Eq. (8), and the estimated Wolborska model parameters are listed in Table 3.

Regarding the parameters of the Wolborska model given in Table 2, the values of the bed capacity increase with decreasing the flow rate. However, it is noteworthy that the external mass transfer coefficient (β_a) changes considerably by changing the flow rate. The adsorbent has a more significant external mass transfer kinetic coefficient (2.271/min) at a higher flow rate. This result designates an improved external mass transfer rate at a higher flow rate which does not agree with the measured improved performance and high capacity of the fixed bed at a low flow rate, as can be predicted from the breakthrough data. This can be attributed to the fact that the internal mass transfer process mainly influences the mass transfer; the binding to the adsorbent inside the pores is affected by the flow rate and contact time. Predictably, a low flow rate provides more contact time, thus a slight resistance for cadmium ions to be adsorbed to the active sites within the pores of DPTF. This indicates that pores diffusion and binding to the active sites is the controlling mechanism in the adsorption process of cadmium ions onto the DPTF.