Effect of Biot Number in Batch Studies of Adsorption

Effect of Biot Number in Batch Studies of Adsorption

V. Ramakrishna

Professor, Civil Engineering Department, GMR Institute of Technology, Rajam, Srikakulam District-AP- 532127,

Abstract

The Homogeneous Surface Diffusion Model (HSDM) which is based on mass transfer is widely used to analyze the adsorption phenomenon in batch studies. Different investigators used several approaches to find analytical solution using HSDM. They used dimensionless parameters for analyzing HSDM. The Biot number (Bi), a dimensionless parameter which relates external mass transfer coefficient with diffusion coefficient is used by few investigators. The Bi is expressed differently by different investigators based on their objectives of their study. The Bi used by Flora et al. for his successful analytical results is considered in the present study along with Distribution parameter (), another dimensionless parameter used by them. The relation between Bi and is studied and found that the product of these two parameters is a constant. The Bi considered in the study is found to be consisting of three different dimensional groups. The relationship among them is explored by conducting a parametric study using a wide range of values for the dependent parameters. The HSDM is discretized and analyzed in order to substantiate the results. The results obtained are in agreement with the theoretical trends of adsorption.

Introduction

The Homogeneous Surface Diffusion Model (HSDM) which is based on mass transfer is widely used [1, 5, 6, 9] to analyze the adsorption phenomenon in batch reactors. Flora et al. [5] used HSDM along with dimensionless parameters

such as (i) time (ii) adsorbate concentration (iii) quantity adsorbed (iv) particle radius (v) Biot number (Bi) and

(vi) Distribution parameter () in their model. The usage of HSDM for batch studies is limited [11]. Al-Qodah [1] used

HSDM for estimation of adsorption of

dyes on shale oil ash. Hand et al. [6] developed few user-oriented solutions to the HSDM that are applicable to batch studies. They presented correlations and graphical solutions for estimating the diffusivity from the experiments. McKay

[9] analyzed the model incorporating

dimensionless time and dimensionless distance using Crank and Nicholson method. He performed experimental studies for adsorption of dyes onto

activated carbon and applied the developed model. Markovska et al., [13], Slaney and Bhamidimarri [12], Basheer and Najjar [3] used HSDM for column

studies of adsorption.

Flora et al.,[5] have verified their experimental results with those from analytical studies and reported encouraging trends. However,

Ramakrishna [11] reported that the results reported by Flora et al. [5] are erratic and

found that the erratic results that are obtained are primarily based on improper estimation of external mass transfer (kf) and diffusion (D) coefficients. Babu and Ramakrishna [2], Ramakrishna [11] studied the various

models available for estimation of the above two coefficients and observed that four models are available for this purpose:

• Models used by Buzanowski and Yang[4] and Liaw et al.[8] based on Linear Driving Force approach for uptake of adsorbate

• Model used by Kaguei et al. [7] which assumes that the rate of change of

  mass accumulation on the surface of adsorbent is proportional to the

  change of concentration in liquid phase

• Model used by McKay and Allen [10]

  The above four models give an indication of the range of mass transfer coefficients that are best suited for batch adsorption experiments. Babu and Ramakrishna [2] developed a code using

  C language for estimating the mass transfer coefficients obtained from the above models for their use as initial guess values in simulation. They can be adjusted to fit to the actual experimental values. They used the experimental

  values reported by Flora et al., [5] as

  inputs to their code and compared with the values that are reported by Flora et al.[5]. They tested the code with the experimental results reported by Al- Qodah [1]. It is noticed that the results from only two these four models are

  adsorbent characteristics, and adsorption parameters at initial conditions. The Distribution parameter (), the other

  dimensionless parameter used by Flora et al., [5] depends upon ratio of adsorption capacity to amount adsorbed

  at initial conditions. Higher values of indicate higher adsorption under large adsorbent doses. The usage of in adsorption analysis is scarce [11]. Due to the similarity of these dimensionless parameters Bi and in terms of their independent variables, their relative dependency is investigated in the present

  study. The model given by Flora et al [5]

  is taken as the basis for the entire study.

  Model Formulation

  The Biot number (Bi) is mathematically expressed as [5]:

  close to that of the trends reported by

  k f r C0

  Flora et al. [5]. A close examination of

  Bi D q

  —– (1)

  the results from these models revealed [2] that, (i) the specific surface of the adsorbate in the model is important for adsorption of adsorbate (ii) higher the specific surface, higher will be the adsorbate removal.

  The usage of Biot number (Bi) in mass transfer studies as an analogous term to that used in heat transfer studies is available in literature [6,13]. Higher values of Bi indicate domination of kf compared

  0

  where, kf = external mass transfer coefficient, m/s

  D = diffusion coefficient, m2/s

  r = radius of adsorbent particle, m

  = adsorbent particle density, kg/m3

  C0 = initial concentration of adsorbate, kg/m3

  q0 = dimensionless

  = amount adsorbed for C = C0 (from isotherm equation) and is calculated as-

  to that of D. The Bi is defined differently in the literature [5,6,13] based on the

  objectives of the study. Hand et al. [6]

  q abC0 0 1 bC

  0

  for Langmuir

  considered Bi based on surface diffusivity and porosity of the adsorbent.

  Isotherm ———(2)

  The distribution parameter () is mathematically expressed as:

  Markovska et al., [13] considered Bi

  W q

  based on radius of the particle and mass

  0

  ——- (3)

  transfer coefficients. The Bi as defined by Flora et al., [5] depends not only on mass transfer coefficients, but also on

  V C0

  Where, W = mass of adsorbent, kg V = volume of adsorbent, m3

  Combining Eqs. (1) and (3):

  k r W

  Table1: Ranges of parameter values used in the present study

  S.No.	Variable Parameter	Ranges of values -6	Unit
  1	External mass transfer coefficient, kf	0.5×10 ; 2.5×10-6; 5.0×10-6; 10.0×10-6	m/s
  2	Diffusion coefficient, D	1.0×10-10; 2.5×10-10; 5.0×10-10; 10.0×10-10	m2/s
  3	Initial concentration of adsorbate, C0	50; 250; 500	mg/L
  4	Adsorbent dose, W/V	1.0; 0.1	kg/m3
  5	Adsorbent particle density,	500; 841	kg/m3
  6	Partice diameter, d(=2r)	100; 200	mm

  S.No.	Variable Parameter	Ranges of values -6	Unit
  1	External mass transfer coefficient, kf	0.5×10 ; 2.5×10-6; 5.0×10-6; 10.0×10-6	m/s
  2	Diffusion coefficient, D	1.0×10-10; 2.5×10-10; 5.0×10-10; 10.0×10-10	m2/s
  3	Initial concentration of adsorbate, C0	50; 250; 500	mg/L
  4	Adsorbent dose, W/V	1.0; 0.1	kg/m3
  5	Adsorbent particle density,	500; 841	kg/m3
  6	Particle diameter, d(=2r)	100; 200	mm

  Bi f — (4)

  D V

  Eq. (4) is taken as governing equation for the present study. Eq. (4) indicates that,

• Bi is inversely proportional to

• the product of [(Bi)()] is a constant

• there are three terms, which are not dimensionless, on the right hand side of Eq. (4) on which the product of [(Bi)()] is dependant

The present study focused mainly on examining the above relations based on available data.

Results and Discussion

A set of values identified[11] for validating the code developed using Al- Qodah experimental data is considered for studying the relationship between Bi and and the other dependant parameters as per Eq. (4). The range of values are primarily identified (Refer Table-1) based on their ranges used for

The above study showed the inter- dependence of Bi and , the two dimensionless parameters used in the study. However, the Bi for mass transfer

in its conventional form is expressed [13]

as:

successful validation of experimental

k

—– (5)

data of Al-Qodah [1]. Additional ranges of these parameters that are reported in literature [5] are also included in the study.

The values of Bi and are calculated using Eq. (4) and values of these parameters as per Table-1 and are plotted. The slope of the plot (Fig. 1) yielded a slope (i.e., [(Bi)()]) of 0.002, a

constant value, for W/V = 1.0 kg/m3. Similarly, for a W/V value of 0.1 kg/m3

and the resulting slope (i.e., [(Bi)()]) is equal to 0.0002. The values of [(Bi)()] are calculated for a set of values given in Table-1.

Bi f r

D

The values of Bi are calculated based on the range of values given in Table-1. The calculated values are given in Table-2. It can be noted from Table-2 that Bi increases with either increase in kf or decrease in D indicating higher values of Bi reflect the relative dominance of kf over D. This is clearly in agreement with

the definition of Bi [13].

3.0

W/V = 1.0 kg/m3; Slope = 0.002 W/V = 0.1 kg/m3; Slope = 0.0002

W/V = 1.0 kg/m3; Slope = 0.002 W/V = 0.1 kg/m3; Slope = 0.0002

2.5

Distribution parameter

Distribution parameter

2.0

1.5

1.0

Study of the functional groups in Biot number

As discussed above, the Biot number comprises of three dimensional functional groups viz.,

• mass transfer coefficients,

0.5

0.0

200 400 600 800 1000 1200 1400

k with dimensions of (L-1)

f

f

D

D

(1/Biot number)

Figure 1: Plot of Biot number vs.

Distribution parameter

• adsorbent characteristics, r with

dimensions of (L4M-1)

S. No.	Variable Parameter			Biot Number Bi = [(kf)(r)/ (D)]
	kf (m/s)	D (m2/s)	r (mm)
1	0.5×10-6	1.0×10-10	100	500
2	2.5×10-6	1.0×10-10	100	1250
3	0.5×10-6	2.5×10-10	100	200
4	2.5×10-6	2.5×10-10	100	1000

S. No.	Variable Parameter			Biot Number Bi = [(kf)(r)/ (D)]
	kf (m/s)	D (m2/s)	r (mm)
1	0.5×10-6	1.0×10-10	100	500
2	2.5×10-6	1.0×10-10	100	1250
3	0.5×10-6	2.5×10-10	100	200
4	2.5×10-6	2.5×10-10	100	1000

Table 2: Values of Conventional Biot Number

• adsorption parameters C0

q

q

dimensions of (ML-3)

with

0

Comparing Eqs. (1) and (5) it can be observed that, the Bi defined by Flora et al., [5] is having additional terms such as

particle density and adsorption parameters in the equation. These parameters are not used by other[6] researchers. It is hence decided to study the Bi as defined by Flora et al., [5] in detail and examine its applicability in understanding the trends of adsorption phenomenon in batch studies. Further, based on the encouraging results obtained in the study for proving [(Bi)()] as constant, an analysis is

further carried out based on the three functional groups involved in the Bi for their inter-dependence.

The inter-dependence of these three functional groups is studied using a selected range of values (Table-3). Two of the three functional groups are kept constant while the third functional group is varied. The ranges of these values are selected based on the values adopted for initial parametric study (Refer Table-2).

Effect of kf/D: The value of kf is initially kept same (2.5×10-6 m/s) for the ratio of kf/D = 0.25×104 m-1 and 1.0×104 m-1.

The trends of adsorbate removal indicate (Fig. 2) that the effect of reduction in D value for the above ratios is negligible. The kf value is then increased for kf/D = 2.0×104 m-1 onwards (kf = 5.0×10-6, 6×10-6, and 10×10-6 m/s respectively) while there is a marginal reduction of D value for kf/D = 3.0×104 m-1. The trends show increased adsorbate removal for the last three combinations. This could

be largely attributed to the increase of kf value in the kf/D ratios since the effect of D is found negligible in the first two cases. It is evident from the results that, kf is dominant compared to D in this analysis.

aspect, study of W/V values for less than

1.0 1.0 is useful.

1: k /D = 0.25×104 m-1

f

f

1: k /D = 0.25×104 m-1

2: k /D = 1.0×104 m-1

2: k /D = 1.0×104 m-1

0.8

f

f

4 -1

4 -1

V

3: kf/D = 2.0×10 m

3: kf/D = 2.0×10 m

4: k /D = 3.0×104 m-1

4: k /D = 3.0×104 m-1

f

5: kf/D = 4.0×104 m

f

5: kf/D = 4.0×104 m

-1

-1

C/C0

C/C0

0.6

0.4

0.2

1, 2

4

3

5

q

W

C0

C

——- (6)

0.0

0 20 40 60 80 100 120 140 160 180 200

Contact time, min

1.0

4

0.8 6 5

C/Co

CCo

0.6

1: k = 2.5×10-6 m/s; W/V = 1.0 kg/m3

f

f

2: k = 5.0×10-6 m/s; W/V = 1.0 kg/m3

Figure 2: Effect of contact time on

f

-6 3

3: kf = 10.0×10 m/s; W/V = 1.0 kg/m

0.4 4: k = 2.5×10-6 m/s; W/V = 0.1 kg/m3

f

f

adsorbate removal for various parameter

-6 3

-6 3

1 -6 3

values of kf/D; Operating parameters: C0

0.2 3 2

5: kf = 5.0×10 m/s; W/V = 0.1 kg/m 6: kf = 10.0×10 m/s; W/V = 0.1 kg/m

= 250 mg/L; W/V = 1.0 kg/m3; d = 100 µm;

= 500 kg/m3

0.0

0 20 40 60 80 100 120 140 160 180 200

Contact time, min

Figure 3: Effect of contact time on

To further ascertain the domination of kf over D, the HSDM is discretized and analyzed by developing a code in C language. Variation of kf and D are studied independently and exhaustively

[11] by changing the other independent parameters. The results are discussed below:

Variation of kf with respect to adsorbent dose

The variation of kf with respect to adsorbent dose (i.e., W/V) is studied for a wide range of parameters keeping D, d, , and C0 as constants. It is observed that the trends of adsorbate removal curves are similar for a specific value of C0 and W/V values. The adsorbate removal is increasing with increase of kf value indicating the increased release rate of the adsorbate from the liquid (Refer Fig.3). The amount adsorbed (q) on the adsorbent is increased with decrease of W/V values. This is clear from Eq. 6 given below, which indicates the higher accumulation of adsorbate when the W/V value is reduced. This is important since, the efficiency of adsorbent is judged from its dosage into the system and the corresponding accumulation of adsorbate on adsorbent. In view of this

adsorbate removal for various parameter values of kf and W/V. Operating parameters: D = 5×10-10 m2/s; C0 = 50

mg/L; = 500 kg/m3; d = 100µm

The study showed that, the adsorption system is dependent on W/V value and the accumulation is inversely proportional to the ratio of W/V value. Studies on the effect of kf with respect to particle radius and density indicated that adsorbate removal increased with decrease in particle size (Refer Fig.4). This indicates the availability of higher specific surface of the adsorbent (Eq. 7) when the particle size is reduced.

s d V

s d V

S 6 1 W —— (7)

The adsorbate removal is reduced with decrease of particle density (Refer Fig.4). This also indicates the availability of higher specific surface on the adsorbent (Eq. 7). The lower values of particle diameter and density together resulted in higher adsorbate removal

validating [11] the above assumption.

It may be further noted that, the equilibrium time is reduced with

increase in either C0 or kf (Refer Figs. 3 & 4). This is understandable since, increase of C0 increases amount of adsorbate available for adsorption and increase of kf indicates the rapid release of adsorbate from liquid. This has resulted in achieving equilibrium in shorter durations at higher values of these two parameters. Further, the rapid release of adsorbate from the liquid resulted in a steep slope of the linear

indicates that, external mass transfer (i.e., kf) is the dominant parameter in adsorbate removal of the system considered. Further, it may also be noted from Fig. 5 and 6 that, the accumulation is increasing with decrease in W/V value.

1.0

4, 5, 6

0.8

1: D = 2.5×10-10 m2/s; W/V = 1.0 kg/m3

portion of the curves while the lower values of kf showed a relative flatter slope. Hence, the linear portion of the curve is shifting close to the axis

0.6

C/C0

C/C0

0.4

0.2

0.0

1, 2, 3

0 20 40 60 80 100 120 140 160 180 200

Contact time, min

2: D = 5.0×10-10 m2/s; W/V = 1.0 kg/m3

3: D = 10.0×10-10 m2/s; W/V = 1.0 kg/m3

4: D = 2.5×10-10 m2/s; W/V = 0.1 kg/m3

5: D = 5.0×10-10 m2/s; W/V = 0.1 kg/m3

6: D = 10.0×10-10 m2/s; W/V = 0.1 kg/m3

indicating a steep slope or rapid adsorption while the curve is shifting away from the axis indicating a slower adsorption.

Figure 5: Effect of contact time on adsorbate removal for various parameter values of D and W/V. Operating parameters: kf = 2.5×10-6 m/s; C0 = 50 mg/L; = 500 kg/m3; d = 100µm

1: density = 500 kg/m3; diameter = 100 microns

1.0 2: density = 500 kg/m3; diameter = 200 microns 3: density = 841 kg/m3; diameter = 100 microns

3

1.0

4, 5, 6 1: D = 2.5×10-10 m2/s; W/V = 1.0 kg/m3

0.8

C/C0

C/C0

0.6

0.4

4: density = 841 kg/m ; diameter = 200 microns

4

2

3

1

0.8

C/C0

C/C0

0.6

0.4

0.2

1, 2, 3

2: D = 5.0×10-10 m2/s; W/V = 1.0 kg/m3

3: D = 10.0×10-10 m2/s; W/V = 1.0 kg/m3

4: D = 2.5×10-10 m2/s; W/V = 0.1 kg/m3

5: D = 5.0×10-10 m2/s; W/V = 0.1 kg/m3

6: D = 10.0×10-10 m2/s; W/V = 0.1 kg/m3

0.0

0.2

0 20 40 60 80 100 120 140 160 180 200

Contact time, min

0.0

0 20 40 60 80 100 120 140 160 180 200

Contact time, min

Figure 4: Effect of contact time on adsorbate removal for various parameter values of and d; Operating parameters: kf = 2.5×10-6 m/s; D = 5×10-10 m2/s; C0 =

250 mg/L; W/V = 1.0 kg/m3

Variation of D with respect to adsorbent dose: The variation of D with respect to adsorbent dose (i.e., W/V) is studied for a wide range of parameters keeping kf, d, , and C0 as constants. It is observed that, adsorption is independent with respect to the variations of D. Further the removal is higher for higher values of kf (Refer Fig. 5 and 6). This

Figure 6: Effect of contact time on adsorbate removal for various parameter

values of D and W/V. Operating parameters: kf = 5.0×10-6 m/s; C0 = 250 mg/L; = 500 kg/m3; d = 100µm

Effect of /r: The adsorbent particle density is doubled (500 and 1000 kg/m3) in the first two combinations while the particle radius is reduced 2.5 times (500

and 200 m). This indicates that /r value is increased 5 times. Since the changes made in adsorbent particle density and particle radius are almost same (i.e., 2 and 2.5), a similar trend of adsorbate removal (Fig. 7) is obtained. Further, in the second combination,

particle radius is decreased resulting in availability of higher specific surface of adsorbent. The adsorbate removal is hence increased with decrease in particle size. In the third and fourth combinations, radius of adsorbent is kept constant (50 m) while the particle

density is doubled (500 and 1000 kg/m3). The relatively higher adsorbate accumulation in the third combination

compared to that of the fourth is due to the availability of more specific surface of the adsorbent (Eq. 6) since the density is decreased. The results obtained are in agreement with those discussed earlier. Hence, it is concluded that, lower values of adsorbent particle density () and particle diameter (d) favor adsorbate removal in adsorption.

1: (density/radius) = 1×106 kg/m4 2: (density/radius) = 5×106 kg/m4 3: (density/radius) = 1×107 kg/m4

1.0

4: (density/radius) = 2×107 kg/m4

0.8 1

2

C/C0

C/C0

0.6

4

0.4 3

0.2

0.0

0 20 40 60 80 100 120 140 160 180 200

Contact time, min

Figure 7: Effect of contact time on adsorbate removal for various parameter values of /r, Operating parameters: kf = 2.5×10-6 m/s; D = 5×10-10 m2/s; C0 = 250

mg/L; W/V = 1.0

Effect of q0/C0: This study is conducted with five different adsorbate concentrations viz., 100, 150, 200, 750, and 1000 mg/L respectively. The corresponding values of q0 are calculated using Eq. 2. It should be noted that, the percentage adsorbate removal is increasing with decrease in q0/C0 vaue

(Fig. 8). The value of q0 calculated from Eq. 2 is increasing with increase of C0. It indicates that, the accumulation of adsorbate is increasing with increase of C0. The results obtained are in agreement with this assumption.

The present study revealed that:

• HSDM is useful in analyzing adsorption phenomenon in batch studies.

• Dimensionless parameters such as Biot number (Bi) and Distribution parameter () are useful in understanding the relative importance of mass transfer coefficients and adsorption parameters in batch studies of adsorption.

• The product of Bi and is found to be a constant.

• Large values of Bi indicate higher values of external mass transfer coefficient which is proved in the present study.

• Bi as defined by Flora et al.[5] is

  helpful in understanding the trends of adsorption.

• External mass transfer coefficient (kf) is relatively dominant in batch studies of adsorption phenomenon than diffusion coefficient (D). This is ascertained by conducting parametric study with a wide range of values.

• Adsorbate removal efficiency is higher when W/V ratio, particle diameter (d=2r) and density () are low.

• Accumulation of adsorbate on adsorbent is high with increase of Co.

• The percentage adsorbate removal is increasing with decrease in q0/C0 value

• The equilibrium time is reduced with increase in either C0 or kf .

  1: q /C = 0.220 m3/kg

  1: q /C = 0.220 m3/kg

  2: q /C = 0.291 m3/kg

  2: q /C = 0.291 m3/kg

  3

  3

  0 0

  0 0

  0 0

  0 0

  3: q0/C0 = 0.977 m /kg

  3: q0/C0 = 0.977 m /kg

  3

  3

  1.0

  4: q0/C0 = 1.245 m /kg

  4: q0/C0 = 1.245 m /kg

  5: q /C = 1.713 m3/kg

  5: q /C = 1.713 m3/kg

  0 0

  0 0

  0.8 1

  2

  C/C0

  C/C0

  0.6

  0.4

  3

  4

  0.2 5

  0.0

  0 20 40 60 80 100 120 140 160 180 200

  Contact time, min

  Figure 8: Effect of contact time on adsorbate removal for various parameter values of q0/C0 Operating parameters: kf = 2.5×10-6 m/s; D = 5×10-10 m2/s; =500

  kg/m3; d = 100 µm; W/V = 1.0 kg/m3

  Summary and Conclusions

  The effect of Biot number (Bi) on Distribution parameter () in batch studies of adsorption is studied. It is found that Bi is inversely proportional to and the product of [(Bi)()] is a constant. The inter-dependence of the three functional groups (viz., mass transfer coefficients, adsorbent characteristics, and adsorption parameters) in the Biot number is studied exhaustively using a selected range of values and Homogeneous Surface Diffusion Model. The results indicate that,

• kf is dominant compared to D in the analysis.

• lower values of adsorbent particle density () and particle diameter (d) favor adsorbate removal in adsorption.

• accumulation of adsorbate is increasing with increase of C0.

References

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Nomenclature Used

HSDM Homogeneous Surface Diffusion

Model

Bi Biot Number, Dimensionless

Co Initial concentration of adsorbate, mg/L (or) kg/m3

d Particle diameter, m (or) m

D Diffusion coefficient, m2/s

kf External mass transfer coefficient, m/s

q Mass adsorbed on adsorbent, kg/kg

Ss Specific surface area of adsorbate

W/V Adsorbent dose, kg/m3

Distribution Parameter, Dimensionless

Mass density of particle, kg/m3

Table 3 Range of values studied for inter-dependence of functional groups in Biot number

S. No.	Values of the independent functional groups		Values of the dependent functional group
	Functional group	Value	Functional group	Value
1	q0/C0	0.804	kf/D	0.25×104; 1.0×104; 2.0×104; 3.0×104; 4.0×104.
	/r	1×107
2	q0/C0	0.804	/r	1.0×106; 5.0×106; 1.0×107; 2.0×107.
	kf/D	2.0×104
3	/r	1.0×107	q0/C0	0.220; 0.291; 0.977; 1.245; 1.713.