Multiple approaches for large-scale CO₂ capture by adsorption with 13X zeolite in multi-stage fluidized beds assessment

Abstract

Adsorption is a promising technology for reducing CO₂ emissions from combustion gases. However, some issues must be considered when designing large-scale units with low environmental impact and cost. In this contribution, some of these issues are addressed. Firstly, sugarcane bagasse is selected as a meaningful model system. 13XBF Kostrolith zeolite is used as the adsorbent, and adsorption equilibrium data are determined using a magnetic suspension balance from Rubotherm. 13XBF showed a high CO₂ capacity and CO₂/N₂ selectivity under post-combustion scenario conditions. Secondly, the fluidized bed process was optimized using a 3-stage adsorber and a 3-stage desorber, with heat exchangers that can be used in each stage. The constrained variable in the adsorption column is the solid-to-gas mass feed ratios. It was found that substantial CO₂ recovery with 98% purity can be achieved and that the heat load in the desorption column accounts for the highest energy-optimized cost in the process. Thirdly, a detailed CFD model for the fluid dynamics and heat transfer of fluidization for one stage is presented. Bubbles were defined using a hyperbolic filter as predicted by the model, which was consistent with the semi-empirical model and emphasized the significance of bubble size in gas–solid heat transfer, which is essential to the effectiveness of adsorption/desorption. The bubble portion and its effects on the temperature are estimated. Finally, a semi-empirical approach is applied to understanding the controlling factors in a fluidized bed adsorption stage. Results show that the mass transfer from the bubbles to the emulsion is likely the most important resistance to the fluidized adsorber.

Appendices

Appendix 1

1.1 CFD model

The two-fluid model utilizes governing equations that contain conservative equations and constitutive correlations for solid stress and interfacial drag coefficient. These relations are written as:

• Mass conservative equations for phases $(k=g,\mathrm{s})$ [43]:

  $$\frac{\partial }{\partial t}\left({\varepsilon }_{k}{\rho }_{k}\right)+\nabla \cdot \left({\varepsilon }_{k}{\rho }_{k}{{\varvec{u}}}_{{\varvec{k}}}\right)=0$$

  (1)

  where ${\varepsilon }_{k}$ is the portion of gas or solid, ${\rho }_{k}$ is the density and ${{\varvec{u}}}_{{\varvec{k}}}$ is the velocity.

• Momentum conservative equations for phases $(k=g,s$) [43]:

  $$\frac{\partial }{\partial t}\left( {\varepsilon_{k} \rho_{k} {\varvec{u}}_{{\varvec{k}}} } \right) + \nabla \cdot \left( {\varepsilon_{k} \rho_{k} {\varvec{u}}_{{\varvec{k}}} {\varvec{u}}_{{\varvec{k}}} } \right) = - \varepsilon_{k} \nabla p_{k} + \nabla \cdot \overline{\overline{\tau }}_{k} - D\left( {{\varvec{u}}_{{\varvec{k}}} - {\varvec{u}}_{{\varvec{k}}} } \right) + \varepsilon_{k} \rho_{k} {\varvec{g}}$$

  (2)

  where ${p}_{k}$ is the pressure of gas or solid, $\overline{\overline{\tau }}_{k}$ is the gas or phase stress tensor, ${\varvec{g}}$ is the gravity vector and $D$ is the drag coefficient.

• Energy conservative equations for phases $(k=g,s, j=s,g)$ [43]:

  $$\frac{\partial }{\partial t}\left( {\varepsilon_{k} \rho_{k} C_{pk} T_{k} } \right) + \rho_{k} \nabla \cdot \left( {\varepsilon_{k} {\varvec{u}}_{{\varvec{k}}} C_{pk} T_{k} } \right) = \nabla \cdot \left( {\varepsilon_{k} k_{k} \nabla T_{k} + \overline{\overline{\tau }}_{k} \cdot {\varvec{u}}_{{\varvec{k}}} } \right) + Q_{kj} - \varepsilon_{k} \frac{{\partial p_{k} }}{\partial t}$$

  (3)

  where ${C}_{pk}$ is the specific heat, ${T}_{k}$ is the temperature or phase stress tensor, ${{\varvec{k}}}_{{\varvec{k}}}$ is the thermal conductivity and ${Q}_{kj}$ is the heat exchange between both phases.

• Gas phase stress:

  $$\overline{\overline{\tau }}_{{\text{g}}} = \varepsilon_{{\text{g}}} \mu_{{\text{g}}} \left\{ {\left[ {\nabla {\varvec{u}}_{{\varvec{g}}} + \left( {\nabla {\varvec{u}}_{{\varvec{g}}} } \right)^{{\text{T}}} } \right] - \frac{2}{3}\left( {\nabla \cdot {\varvec{u}}_{{\varvec{g}}} } \right)\overline{\overline{I}} } \right\}$$

  (4)

• Solid phase stress:

  $$\overline{\overline{\tau }}_{{\text{p}}} = p_{{\text{p}}} \overline{\overline{I}} + \varepsilon_{{\text{p}}} \mu_{{\text{p}}} \left\{ {\left[ {\nabla {\varvec{u}}_{{\varvec{s}}} + \left( {\nabla {\varvec{u}}_{{\varvec{s}}} } \right)^{{\text{T}}} } \right] - \frac{2}{3}\left( {\nabla \cdot {\varvec{u}}_{{\varvec{s}}} } \right)\overline{\overline{I}} } \right\}$$

  (5)

• Solid pressure [76]:

  $$\nabla {p}_{s}={10}^{-8.686{\varepsilon }_{\mathrm{g}}+6.385}\nabla {\varepsilon }_{\mathrm{p}}$$

  (6)

• Solid viscosity [[76]:

  $${\mu }_{\mathrm{p}}=0.5{\varepsilon }_{\mathrm{p}}$$

  (7)

• Drag coefficient [43]:

  If $\left({\varepsilon }_{\mathrm{g}}<0.80\right)$:

  $$\begin{array}{cc}& D=150\frac{{\left(1-{\varepsilon }_{\mathrm{g}}\right)}^{2}{\mu }_{\mathrm{g}}}{{\varepsilon }_{\mathrm{g}}{d}_{\mathrm{p}}^{2}}+1.75\frac{\left(1-{\varepsilon }_{\mathrm{g}}\right){\rho }_{\mathrm{g}}\left|{{\varvec{u}}}_{{\varvec{g}}}-{{\varvec{u}}}_{{\varvec{s}}}\right|}{{d}_{\mathrm{p}}}\end{array}$$

  (8)

  If $\left({\varepsilon }_{\mathrm{g}}\ge 0.80\right)$:

  $$\begin{array}{c}D=\frac{3}{4}\frac{\left(1-{\varepsilon }_{\mathrm{g}}\right){\varepsilon }_{\mathrm{g}}}{{d}_{\mathrm{p}}}{\rho }_{\mathrm{g}}\left|{{\varvec{u}}}_{{\varvec{g}}}-{{\varvec{u}}}_{{\varvec{s}}}\right|{C}_{\mathrm{D}0}{\varepsilon }_{\mathrm{g}}^{-2.7}\end{array}$$

  (9)

  where ${d}_{p}$ is the particle diameter and ${C}_{D}$ is given by [43]:

  $${C}_{D}=\frac{24}{{\varepsilon }_{\mathrm{g}}R{e}_{s}}\left[1+0.15{\left({\varepsilon }_{\mathrm{g}}R{e}_{S}\right)}^{0.687}\right]$$

  (10)

  where ${\mathrm{Re}}_{\mathrm{s}}$ is given by:

  $$R{e}_{S}=\frac{{\varepsilon }_{\mathrm{g}}{\rho }_{\mathrm{g}}{d}_{\mathrm{s}}\left|{{\varvec{u}}}_{{\varvec{s}}}-{{\varvec{u}}}_{{\varvec{g}}}\right|}{{\mu }_{\mathrm{g}}}$$

  (11)

Appendix 2

2.1 B–E model

The mass balance for component i in the gas phase of the bubble region at stage j is given by:

$$\frac{d{F}_{i,j}^{(b)}}{dV}={f}_{b,j}{{K}_{be}}_{i,j}\left({y}_{i,j}^{\left(e\right)}-{y}_{i,j}^{\left(b\right)}\right)$$

(1)

$$\sum_{i=1}^{m}{y}_{i,j}^{(b)}=1$$

(2)

Using the following initial condition:

$${F}_{i,j}^{\left(b\right)}{|}_{\left(V=0\right)}={\delta }_{b,j}{F}_{j-1}{y}_{i,j-1}$$

(3)

The mass balance for component i in the gas phase of the emulsion region at stage j is given by:

$$\frac{{\delta }_{e,j}{F}_{j-1}{y}_{i,j-1}-{F}_{j}^{\left(e\right)}{y}_{i,j}^{\left(e\right)}}{{V}_{j}}={\int }_{0}^{{V}_{j}}{f}_{b,j}{{K}_{be}}_{i,j}\left({y}_{i,j}^{\left(e\right)}-{y}_{i,j}^{\left(b\right)}\right)dV+{f}_{e,j}\left(1-{\varepsilon }_{e,j}\right){\rho }_{p}{{k}_{LDF}}_{i,j}\left({q}_{i,j}^{*}-{q}_{i,j}\right)$$

(4)

$$\sum_{i=1}^{m}{y}_{i,j}^{(e)}=1$$

(5)

The mass balance for component i adsorbed on the solid phase at stage j is given by:

$$\frac{{\dot{m}}_{s}\left({q}_{i,j+1}-{q}_{i,j}\right)}{{V}_{j}}=-{f}_{e,j}\left(1-{\varepsilon }_{e,j}\right){\rho }_{p}{{k}_{LDF}}_{i,j}\left({q}_{i,j}^{*}-{q}_{i,j}\right)$$

(6)

The molar flow rate and composition of the outlet gas stream of stage j is determined using the following Equations, valid for 1 ≤ j ≤ N (j = 0 represents the system feed and N is the column number of stages):

$${F}_{j}={F}_{j}^{\left(e\right)}+{F}_{j}^{\left(b\right)}{|}_{\left(V={V}_{j}\right)}$$

(7)

$${y}_{i,j}=\frac{{y}_{i,j}^{\left(e\right)}{F}_{j}^{\left(e\right)}+\left({{y}_{i,j}^{\left(b\right)}F}_{j}^{\left(b\right)}\right){|}_{\left(V={V}_{j}\right)}}{{F}_{j}^{\left(e\right)}+{F}_{j}^{\left(b\right)}{|}_{\left(V={V}_{j}\right)}}$$

(8)

The energy balance for the gas phase of the bubble region at stage j is given by Equation:

$${f}_{b,j}\frac{d}{dV}\left({F}_{j}^{(b)}{H}_{g}^{(b)}\right)={f}_{b,j}{H}_{be,j}\left({T}_{g,j}^{\left(e\right)}-{T}_{g,j}^{\left(b\right)}\right)+-\sum_{i=1}^{m}{{J}_{be}^{{\prime}{\prime}{\prime}}}_{i,j}\left({\varphi }_{{be}_{i,j}}{H}_{i,j}^{\left(e\right)}+(1-{\varphi }_{{be}_{i,j}}){H}_{i,j}^{\left(b\right)}\right)-{\alpha }_{w,j}{h}_{wb,j}\left({T}_{g,j}^{\left(b\right)}-{T}_{w,j}\right)$$

(9)

Using the following initial condition:

$$\left({F}_{j}^{(b)}{H}_{g,j}^{\left(b\right)}\right){|}_{\left(V=0\right)}={\delta }_{b,j}{F}_{j-1}{H}_{g,j-1}$$

(10)

where φ_be takes on values of 0 or 1 as a function of the difference in the mole fractions of i in the bubble gas and the emulsion, according to the relationship shown below:

$${\varphi }_{{be}_{i}}=\left\{\begin{array}{c}1 if {y}_{i}^{(e)}\ge {y}_{i}^{(b)}\\ 0 if {y}_{i}^{(e)}<{y}_{i}^{(b)}\end{array}\right.$$

(11)

The energy balance for the gas phase of the emulsion region at stage j is given by:

$$\frac{{\delta }_{e,j}{F}_{j-1}{H}_{g,j-1}-{F}_{j}^{\left(e\right)}{H}_{g,j}^{\left(e\right)}}{{V}_{j}}={\int }_{0}^{{V}_{j}}{f}_{b,j}{H}_{be,j}\left({T}_{g,j}^{\left(e\right)}-{T}_{g,j}^{\left(b\right)}\right)dV+-\sum_{i=1}^{m}{\int }_{0}^{{V}_{j}}{{J}_{be}^{{\prime}{\prime}{\prime}}}_{i,j}\left({\varphi }_{{be}_{i,j}}{H}_{i,j}^{\left(e\right)}+(1-{\varphi }_{{be}_{i,j}}){H}_{i,j}^{\left(b\right)}\right)dV+\sum_{i=1}^{m}{\dot{q}}_{i,j}^{{\prime}{\prime}{\prime}}\left({\varphi }_{{es}_{i,j}}{H}_{i,j}^{\left(e\right)}+(1-{\varphi }_{{es}_{i,j}}){H}_{i,j}^{\left(s\right)}\right)+-{f}_{e,j}\left(1-{\varepsilon }_{e,j}\right)\frac{6{h}_{p,j}}{{d}_{p}}\left({T}_{s,j}-{T}_{g,j}^{\left(e\right)}\right)+{\alpha }_{w,j}{h}_{we,j}{\left({T}_{g,j}^{\left(e\right)}-{T}_{w,j}\right)}_{lm}$$

(12)

where φ_es takes on values of 0 or 1 as a function of the difference in the equilibrium adsorbed amount and the actual adsorbed amount of i in the solids, according to the relationship shown below:

$${\varphi }_{{es}_{i,j}}=\left\{\begin{array}{c}1 if {q}_{i,j}^{*}\ge {q}_{i,j}\\ 0 if {q}_{i,j}^{*}<{q}_{i,j}\end{array}\right.$$

(13)

The molar flux of component i between the gas phase in the bubbles and the gas phase in the emulsion per unit bed volume at stage j, the molar flux of component i between the gas phase in the emulsion and the solids per unit bed volume at stage j, and the heat exchanger area per unit bed volume are obtained by Equations:

$${{J}_{be}^{{\prime}{\prime}{\prime}}}_{i,j}=-{f}_{b,j}{{K}_{be}}_{i,j}\left({y}_{i,j}^{\left(e\right)}-{y}_{i,j}^{\left(b\right)}\right)$$

(14)

$${\dot{q}}_{i,j}^{\mathrm{^{\prime}}\mathrm{^{\prime}}\mathrm{^{\prime}}}={f}_{e,j}\left(1-{\varepsilon }_{e,j}\right){\rho }_{p}{{k}_{LDF}}_{i,j}\left({q}_{i,j}^{*}-{q}_{i,j}\right)$$

(15)

$${\alpha }_{w,j}=\frac{{A}_{wext,j}}{{V}_{j}}$$

(16)

The energy balance for the solid phase at stage j is given by Equation:

$$\frac{{\dot{m}}_{s}{C}_{p,s}\left({T}_{s,j+1}-{T}_{s,j}\right)}{{V}_{j}}={f}_{e,j}\left(1-{\varepsilon }_{e,j}\right)\frac{6{h}_{p,j}}{{d}_{p}}\left({T}_{s,j}-{T}_{g,j}^{\left(e\right)}\right)+-\sum_{i=1}^{m}{\dot{q}}_{i,j}^{\mathrm{^{\prime}}\mathrm{^{\prime}}\mathrm{^{\prime}}}\left({\varphi }_{{es}_{i,j}}{H}_{i,j}^{\left(e\right)}+(1-{\varphi }_{{es}_{i,j}}){H}_{i,j}^{\left(s\right)}+(-\Delta {H}_{{ads}_{i,j}})\right)$$

(17)

The change in the enthalpy of adsorption for component i is described by the Equation:

$$\Delta {H}_{{ads}_{i,j}}={\overline{\lambda }}_{{ads}_{i,j}}+{H}_{i,j}^{\left(s\right)}-({\varphi }_{{es}_{i,j}}{H}_{i,j}^{\left(e\right)}+(1-{\varphi }_{{es}_{i,j}}){H}_{i,j}^{\left(s\right)})$$

(18)

The average isosteric heat of adsorption of the inlet and outlet solid adsorbed concentrations of component i at stage j is obtained from Equation:

$${\overline{\lambda }}_{{ads}_{i,j}}=\frac{{\int }_{{q}_{i,j+1}}^{{q}_{i,j}}{\lambda }_{ads,i}dq}{{q}_{i,j}-{q}_{i,j+1}}$$

(19)

The specific enthalpy of the j-stage outlet gas stream formed by mixing the emulsion and bubble outlet gas streams is calculated using Equation:

$${H}_{g,j}=\frac{{F}_{j}^{(e)}\sum_{i=1}^{m}{y}_{i,j}^{(e)}{H}_{i,j}^{(e)}+\left({F}_{j}^{(b)}\sum_{i=1}^{m}{y}_{i,j}^{(b)}{H}_{i,j}^{(b)}\right){|}_{\left(V={V}_{j}\right)}}{{F}_{j}^{(e)}+{F}_{j}^{(b)}{|}_{\left(V={V}_{j}\right)}}$$

(20)

The volume fraction of emulsion and bubbles in a Geldart B particle bed at stage j is obtained from the following correlation [77], assuming stages of equivalent cross-sectional area.

$${f}_{e,j}=\mathrm{0,466}+\mathrm{0,534} exp\left(-\frac{({u}_{0,j-1}-{u}_{mf,j})}{\mathrm{0,413}}\right)$$

(21)

$${f}_{e,j}+{f}_{b,j}=1$$

(22)

The minimum fluidization porosity in the j-stage bed can be estimated using Broadhurst and Becker’s [78] correlation for minimum bubbling porosity, which can be approximated to the minimum fluidization porosity in Geldart B particle beds:

$${\varepsilon }_{mf}\cong {\varepsilon }_{mb}=\mathrm{0,586}\cdot {\phi }_{s}^{-\mathrm{0,72}}{\left[\frac{{\mu }_{g}^{ 2}}{{\rho }_{g}g\left({\rho }_{p}-{\rho }_{g}\right){\left({d}_{p}/{\phi }_{s}\right)}^{3}}\right]}^{\mathrm{0,029}}{\cdot \left(\frac{{\rho }_{g}}{{\rho }_{p}}\right)}^{\mathrm{0,021}}$$

(23)

The pressure drop in fixed bed is calculated from Ergun’s Equation and that in fluidized bed is approximately equal to the apparent weight of the bed [79], according to:

$$\frac{\Delta {P}_{m}}{{L}_{m}}=150\frac{{(1-{\varepsilon }_{m})}^{2}}{{{\varepsilon }_{m}}^{3}}\frac{{\mu }_{g}{u}_{0}}{{({\phi }_{s}{d}_{p})}^{2}}+\mathrm{1,75}\frac{(1-{\varepsilon }_{m})}{{{\varepsilon }_{m}}^{3}}\frac{{\rho }_{g}{u}_{0}^{ 2}}{{\phi }_{s}{d}_{p}}$$

(24)

$$\frac{\Delta {P}_{f}}{{L}_{f}}=(1-{\varepsilon }_{f})\left[({\rho }_{p}-{\rho }_{g})g\right]$$

(25)

Equalizing the pressure drop terms from the two Equations above for the imminent fluidization condition (u₀ = u_mf) gives:

$$150\frac{{(1-{\varepsilon }_{mf})}^{2}}{{\varepsilon }_{mf}^{ 3}}\frac{{\mu }_{g}{u}_{mf}}{{({\phi }_{s}{d}_{p})}^{2}}+\mathrm{1,75}\frac{(1-{\varepsilon }_{mf})}{{\varepsilon }_{mf}^{ 3}}\frac{{\rho }_{g}{u}_{mf}^{ 2}}{{\phi }_{s}{d}_{p}}=(1-{\varepsilon }_{mf})\left[({\rho }_{p}-{\rho }_{g})g\right]$$

(26)

The pressure in stage j is given by the following equation:

$${P}_{j}={P}_{j-1}-\Delta {P}_{f,j}$$

(27)

The gas flow velocity in the bubble region of stage j is obtained from the following relation [79]:

$${u}_{0,j}={f}_{b,j}{u}_{b,j}+{f}_{e,j}{u}_{e,j}$$

(28)

The average fluidized bed porosity and the fluidized bed height relative to the initial fixed bed state are calculated by Equations [79]:

$${\varepsilon }_{f,j}={f}_{b,j}{\varepsilon }_{b,j}+{f}_{e,j}{\varepsilon }_{e,j}$$

(29)

$$\frac{{L}_{f,j}}{{L}_{m,j}}=\frac{\left(1-{\varepsilon }_{m,j}\right)}{(1-{\varepsilon }_{f,j})}$$

(30)

The superficial velocity of the outlet gas stream determines the volumetric outlet gas flow rate of stage j, and the volumetric outlet gas flow rate is related to the molar flow rate of the outlet gas of stage j using the ideal gas mixture model, according to Equations:

$${\dot{v}}_{g,j}={u}_{0,j}{A}_{t,bed}$$

(31)

$${P}_{j}{\dot{v}}_{j}={F}_{j}{R}_{g}{T}_{g,j}$$

(32)

The fractions of the j-stage gas feed flow rate that are directed to the emulsion and bubbles are calculated from Equations:

$${\delta }_{e,j}=\frac{{f}_{e,j}{u}_{e,j}}{{u}_{0,j}}$$

(33)

$${\delta }_{b,j}=\frac{{f}_{b,j}{u}_{b,j}}{{u}_{0,j}}$$

(34)

According to Yates, Ruiz-Martinez, and Cheesman [80], the equation of Darton et al. [81] for the volumetric equivalent diameter of spherical bubbles in fluidized beds in the absence of immersed tubes can be modified to determine the equivalent bubble diameter in a fluidized bed that contains rows of immersed horizontal tubes. The bubble growth in the fluidized bed is given by the following Equation:

$${d}_{b}(z)=\mathrm{0,54}\cdot {\left[\frac{{u}_{0}-{u}_{mf}}{{a}_{tw}}\right]}^{\mathrm{0,4}}\cdot {\left(z+4\sqrt{{A}_{cd}}\right)}^{\mathrm{0,8}}\cdot {g}^{-\mathrm{0,2}}$$

(35)

The breakdown of bubbles when passing through a row of horizontal tubes is represented by the following correlation [80]:

$$\frac{{t}_{w}}{{d}_{b,spl}}=\mathrm{0,4}{\left(\frac{{d}_{w,ext}}{{d}_{b,imp}}\right)}^{2}+\left[\mathrm{1,38}-\mathrm{0,65}\left(\frac{{d}_{w,ext}}{{d}_{b,imp}}\right)\right]\left(\frac{{t}_{w}}{{d}_{b,imp}}\right)$$

(36)

The fraction of the cross-sectional area of the bed free for bubble flow in the region of the tube bank is given by Equation:

$${a}_{tw}=1-\frac{{A}_{pr,w}}{{A}_{t,bed}}$$

(37)

where A(pr,w) is the projected area of a row of tubes on the horizontal plane of the bed which, for cylindrical horizontal tubes, can be obtained by Equation:

$${A}_{pr,w}={n}_{t}{L}_{w}{d}_{w,ext}$$

(38)

Thus, it is possible to calculate the average equivalent bubble diameter along the bed by the following Equation:

$${d}_{b,m}=\frac{{\int }_{0}^{{L}_{f}}{d}_{b}(z)dz}{{L}_{f}}$$

(39)

The gas-phase mass exchange coefficients of component i in the bubble–cloud (bc) and cloud-emulsion (ce) paths can be calculated using the following equations [79]:

$${K}_{bc,i}=\mathrm{4,5}\left(\frac{{u}_{mf}}{{d}_{b,m}}\right)+\mathrm{5,85}\frac{{\left({D}_{i,m}\right)}^{1/2}{g}^{1/4}}{{\left({d}_{b,m}\right)}^{5/4}}$$

(40)

$${K}_{ce,i}=\mathrm{6,77}{\left[\frac{{D}_{i,m}{\varepsilon }_{mf}{u}_{br}}{{\left({d}_{b,m}\right)}^{3}}\right]}^{1/2}$$

(41)

The natural rise velocity of a gas bubble in a fluidized bed can be estimated by the following correlation:

$${u}_{br}=\mathrm{0,711}{(g{d}_{b,eq})}^{1/2}$$

(42)

The overall mass transfer coefficient between the bubble and the is obtained by Equation:

$${K}_{be,i}={{C}_{g,be}\left(\frac{1}{{K}_{bc,i}}+\frac{1}{{K}_{ce,i}}\right)}^{-1}$$

(43)

Using an analogy between heat and mass transfer, the heat transfer coefficients from bubble to cloud and cloud to emulsion, considering emulsion packet properties for the cloud-emulsion heat transfer coefficient, can be calculated according to the equations:

$${H}_{bc}=\mathrm{4,5}\left(\frac{{u}_{mf}{\rho }_{g}{C}_{p,g}}{{d}_{b,m}}\right)+\mathrm{5,85}\frac{{\left({k}_{g}{\rho }_{g}{C}_{p,g}\right)}^{1/2}{g}^{1/4}}{{\left({d}_{b,m}\right)}^{5/4}}$$

(44)

$${H}_{ce}=\mathrm{6,77}{\left[\frac{{k}_{ce}{\rho }_{ce}{C}_{p,ce}{\varepsilon }_{mf}{u}_{br}}{{\left({d}_{b,m}\right)}^{3}}\right]}^{1/2}$$

(45)

$${H}_{be}={\left(\frac{1}{{H}_{bc}}+\frac{1}{{H}_{ce}}\right)}^{-1}$$

(46)

The dynamic viscosity and thermal conductivity of the gas mixture are determined by Wilke’s Mixing Rule [82], according to the Equations below.

$${\phi }_{i,j}=\frac{{\left[1+{\left(\frac{{\mu }_{i}}{{\mu }_{j}}\right)}^{1/2}{\left(\frac{{M}_{w,j}}{{M}_{w,i}}\right)}^{1/4}\right]}^{2}}{\sqrt{8}{\left[1+\left(\frac{{M}_{w,i}}{{M}_{w,j}}\right)\right]}^{1/2}}$$

(47)

$${\mu }_{g}=\sum_{i=1}^{m}\frac{{\mu }_{i}}{1+\frac{1}{{y}_{i}}\sum_{j=1,j\ne i}^{m}{y}_{j}{\phi }_{i,j}}$$

(48)

$${k}_{g}=\sum_{i=1}^{m}\frac{{k}_{i}}{1+\frac{1}{{y}_{i}}\sum_{j=1,j\ne i}^{m}{y}_{j}{\phi }_{i,j}}$$

(49)

The properties of the cloud-emulsion film packets are obtained by an analogy with the particle packet model from Kim et al. [83] using the Equations:

$${\rho }_{ce}=\left(1-{\varepsilon }_{ce}\right){\rho }_{p}+{\varepsilon }_{ce}{\rho }_{g}^{(ec)}$$

(50)

$${C}_{p,ec}=\left(1-{\varepsilon }_{ec}\right){C}_{p,s}+{\varepsilon }_{e}{C}_{p,g}^{(ec)}$$

(51)

$${k}_{ec}={\varepsilon }_{ec}{k}_{g}^{(ec)}+(1-{\varepsilon }_{ec}){k}_{s}\left(\frac{1}{{\varphi }_{e}({k}_{s}/{k}_{g}^{(ec)})+2/3}\right)$$

(52)

where ε_ce represents the porosity of the cloud-emulsion film and is approximated as equal to the emulsion porosity and φ_e is the equivalent thickness of the gas film on the contact point between particles. For the system in this work, a value of 0.13 for φ_e was adopted [79].

The heat transfer coefficient between the gas in the emulsion and the particles in fluidized beds can be estimated by means of the following correlations for the particle Nusselt Number over different ranges of the particle Reynolds Number (Re_p) [84].

$${Nu}_{p}=2+\mathrm{0,6}{\varepsilon }_{e}^{ \mathrm{3,5}}{Re}_{p}^{ 1/2}{Pr}^{1/3} , {Re}_{p}<200$$

(53)

$${Nu}_{p}=2+\mathrm{0,5}{\varepsilon }_{e}^{\mathrm{ 3,5}}{Re}_{p}^\frac{1}{2}{Pr}^\frac{1}{3}+\mathrm{0,02}{\varepsilon }_{e}^{\mathrm{ 3,5}}{Re}_{p}^{\mathrm{ 0,8}}{Pr}^\frac{1}{3}, 200<{Re}_{p}<1500$$

(54)

$${Nu}_{p}=2+\left(\mathrm{4,5}{\cdot 10}^{-5}\right){\varepsilon }_{e}^{ \mathrm{3,5}}{Re}_{p}^{ \mathrm{1,8}}, {Re}_{p}>1500$$

(55)

where the dimensionless numbers Re_p and Nu_p are given by the following expressions:

$${Re}_{p}=\frac{{\rho }_{g}{u}_{e}{d}_{p}}{{\mu }_{g}}$$

(56)

$${Nu}_{p}=\frac{{h}_{p}{d}_{p}}{{k}_{g}}$$

(57)

The mass transfer of component i between the gas in the emulsion and the solids through the adsorptive process is described by a Linear Driving Force model using the global kinetic coefficient for adsorption of i (${\text{k}}_{{\text{LDF,i}}}$), whose value is calculated using the mass transport resistances in the outer film, of the diffusion in the macropores and the diffusion in the micropores of the particle, according to the expression [85]:

$$\frac{1}{{k}_{LDF,i}}=\frac{{r}_{p}{q}_{i,0}}{3{k}_{f,i}{C}_{i,0}}+\frac{{r}_{p}^{ 2}{q}_{i,0}}{15{\varepsilon }_{p}{D}_{p,i}{C}_{i,0}}+\frac{{r}_{c}^{ 2}}{15{D}_{c,i}}$$

(58)

The coefficient k_f,i was calculated from the correlation of the Sherwood Number of particle in fluidized beds given by Scala [86], according to Equations:

$${Sh}_{p,i}=\mathrm{2,0}{\varepsilon }_{mf}+\mathrm{0,70}{\left({Re}_{p,mf}/{\varepsilon }_{mf}\right)}^{1/2}{{Sc}_{i}}^{1/3}$$

(59)

$${Sh}_{p,i}=\frac{{k}_{f,i}{d}_{p}}{{D}_{m,i}}$$

(60)

where Re_p,mf is the particle Reynolds Number at the minimum bed fluidization conditions and Sc_i represents the Schmidt Number, given by Equation:

$${Sc}_{i}=\frac{{\mu }_{g}}{{\rho }_{g}{D}_{m,i}}$$

(61)

The diffusivity of i in the particle macropores was calculated using the Bosanquet Equation [82]:

$$\frac{1}{{D}_{p,i}}={\tau }_{p}\left(\frac{1}{{D}_{m,i}}+\frac{1}{{D}_{k,i}}\right)$$

(62)

The Knudsen diffusivity of i is given by Equation (Tables 8 and 9):

$${D}_{k,i}=9700{r}_{mp}\sqrt{\frac{T}{{M}_{w,i}}}$$

(63)

Table 8 Properties of the 13X zeolite adsorbent particles applied to the model

Table 9 Bed geomety parameters