GAS CYCLIC STEADY STATE MODELING

Introduction

Aspen Adsim 2004.1 presents an innovative new modeling approach to maximize profitability in the design, simulation, and optimization of periodic adsorption processes for gas separation, processes, such as Pressure Swing Adsorption (PSA), Thermal Swing Adsorption (TSA), Vacuum Swing

Adsorption (VSA), etc.

Direct determination of the cyclic steady state, without carrying out a dynamic simulation over a large number of cycles, is now available using Aspen Adsim 2004.1.

This powerful tool - Cyclic Steady State (CSS) modeling (the result of

complete discretization of both time and space) presents a periodic adsorption process as a steady state problem.

The Aspen Adsim 2004.1 CSS models offer an extremely efficient design tool that can be more readily used as an optimization package to determine optimal design and operating conditions for an adsorption process.

The following sections outline CSS modeling tasks and include instructions on using CSS models for your engineering business:

• What is CSS Modeling…?

• Discretization Techniques for Time and Space

• Connectivity Between CSS Models

• Bed Model Details

• Material Balance

• Momentum Balance

• Kinetic Model

• Energy Balance

• Adsorption Equilibrium Models

• User Guidelines

• How to Create a CSS Simulation Flowsheet

• How to Create a Dynamic Simulation Flowsheet using CSS Models

• How to Convert a CSS Flowsheet to a Dynamic Flowsheet

• How to a Convert Dynamic Flowsheet to a CSS Flowsheet

• Developer’s Tips to Get Better Convergence Property in CSS Simulation

What is CSS Modeling…?

A periodic adsorption process operates on sequential steps (for example, continuously repeated steps of Feed, Purge, Pressure equalization, Blow down, Production, etc.) with multiple adsorbers packed with single or multiple adsorbent layers. Although the operation of each bed is batchwise, the whole system is continuous because of the use of multi-beds that are ultimately operated in a cyclic steady state within a confined cycle time. Cyclic Steady State (CSS), which is the nature of periodic adsorption processes, implies a steady state in which the conditions at the end of each cycle are identical to those at the beginning.

The traditional approach for CSS determination is to carry out a dynamic simulation of the system, beginning with a specified set of initial condition, over a large number of cycles until a CSS is eventually confirmed from a defined criteria, e.g., the cycle initial state at t0 must be identical to the cycle end state at tN, as illustrated in Figure 1.

Spatial

Domain Time

Domain

Cycle end state(tN)

Step1

Step2

StepN

t1

t2

t^N t^N-1

t⁰ dynamic simulation

Cycle in itial sta

te(t0)

Figure 1 Illustration for traditional dynamic simulation of a periodic adsorption process

t⁰ t^N

t¹ t2

t^N-1

Periodic Boundary State(tN) = State(t0) i.e. Cyclic Steady State

Time domain (t)

Spa tial d

omain (x)

Figure 2 Illustration for the concept of CSS modeling system in Aspen Adsim

From a mathematical point of view, the criterion for CSS is considered a unique characteristic of a periodic adsorption process, and has brought ideas to explore a better numerical method toward CSS in terms of cost-effective process simulation. The existence of periodic time boundary inspires to replace the initial condition by a periodicity condition requiring that the system state at the end of each cycle is identical to that at its beginning.

As illustrated in Figure 2, the forced reformulation also constrains the system within a specified time domain length, from the starting point (t₀) to the ending point (t_N). This suggests a steady state simulation is feasible by complete discretization of space and time within a confined time length (i.e., cycle time).

Based on the above concept, the CSS models in Aspen Adsim 2004.1 have been developed to determine CSS from purely steady state simulation. Direct determination of CSS will effectively save the costs for the optimization of periodic adsorption process since the technique could offer an extremely efficient design tool that can be more readily used as an optimization package to determine optimal design and operating conditions.

Further benefits come from the fact that the graphic user interface of the freshly released CSS models from Aspen Adsim 2004.1 is the same as those of existing Aspen Adsim dynamic models. Therefore, existing Adsim users should find it easy to use this new feature.

The high-level functionalities of CSS bed model (gCSS_Adsorber) in Aspen Adsim 2004.1 are listed in Table 1, compared with the original Aspen Adsim dynamic bed model (gas_bed).

Table 1. Functional comparison of CSS and dynamic bed models in Aspen Adsim 2004.1

Discretization Techniques for Time and Space

Spatial derivatives of CSS bed model (gCSS_Adsorber) are discretized by one of the following numerical methods:

• CFD4 – 4^th Order Central Finite Difference, equivalent to CDS2 in gas_bed

• OCFE2 – 2^nd Order Orthogonal Collocation on Finite Elements

• OCFE4 – 4 ^th Order Orthogonal Collocation on Finite Elements

Time derivatives of CSS models are explained using 1^st Order Backward Finite Difference approximation:

( )

t x t u x t u t

x t

u _{n j n j n j}

∆

≈ −

∂

∂ , ( , ) ( ₋₁, )

Connectivity between CSS Models

CSS models contain at least an input and an output port (gCSS_Port). Each port has associated variables that correspond to the material connection stream (gCSS_Material_Connection) that allows reversible flow.

These are the available connections for CSS models:

Bed Model Details

Material Balance

The CSS bed model (gCSS_Adsorber) uses the following material balance for the bulk gas adsorption:

( ) 0

The physical meanings of each term are:

2

ρ ∂

Adsorbed phase accumulation³

The following continuity equation is required to complete the material balance around the system,:

g

i

C

i

= ρ

∑

Notation

C

i Gas phase concentration for component i, kmol/m³

D

Li Axial dispersion coefficient for component i, m²/s

t

Time, s

Q

i Amount adsorbed for component i, kmol/kg-adsorbent

v

g Superficial gas velocity, m/s

x

Axial distance coordinate, m

ε

b Bed (interparticle) voidage

ε

p Intraparticle voidage

ρ

g Gas density, kmol/m³

ρ

b Bed packing density, kg/m³

ρ

p Particle density (solid density, true density), kg/m³ References

1 If a concentration gradient exists in a packed bed, a diffusive mass flux will occur. In addition, eddy (turbulent) diffusion due to the flow also contributes to the mass flux. The resultant flux is referred to as mass dispersion, which may be expressed mathematically in terms of Fick’s law, where the proportionality constant is called dispersion coefficient.

Dispersion occurs in both radial and axial directions in the bed. The axial dispersed mixing often occurs when a fluid flows through a packed bed and may cause unfavorable separation efficiency as the separation factor is becoming smaller. In general, flow through a packed bed may be adequately represented with inclusion of the axial dispersed plug flow consideration.

2 Here,

ε

_tis the total bed voidage, which is the combined interparticle and intraparticle voidages calculated from

(

_b

)

Momentum Balance

Gas flow through a packed bed can be described by a relevant pressure drop correlation. Within the CSS adsorber model (gCSS_Adsorber), one of the following pressure drop correlations may be chosen as the one. Note that there is no other option to assume an ideal flow regime, such as Constant Pressure and Velocity and Constant Pressure with Variable Velocity since the CSS models has been developed fundamentally for cyclic process for gas separation.

(1) Darcy’s Law:

(2) Blake-Kozeny:

( )

(3) Burke-Plummer:

K

p Darcy Coefficient, bar.s/m²

M

w Molecular weight of gaseous mixture, kg/kmol

P

Gas pressure, bar

r

p Particle radius, m

v

g Superficial gas velocity, m/s

x

Axial distance coordinate, m

ε

b Bed voidage (void fraction)

µ

g Gas mixture viscosity, cP

ρ

g Gas density, kmol/m³

ψ

Particle shape factor

Kinetic Model

Rigorous simulation of an adsorption process requires a reliable representation of the adsorption kinetics for the adsorbent used. In adsorption, the mass transfer mechanism consists of four steps:

• Fluid film transfer

• Pore diffusion

• Adhesion on surface

• Surface diffusion

Because the surface adhesion rate approximates the order of the collision

pores of the particle, adsorbed on the active sites and then diffused along the surface. While fluid film transfer and pore diffusion are treated as sequential steps, pore diffusion and surface diffusion generally occur in parallel. Any combination of the three steps can constitute the rate-controlling mechanism.

This mechanism definitely depends on the adsorption system and can vary with the operating conditions of the process. Typically, a film adjacent to the surface confines the mass transfer rate between solid and fluid phases and this external film mass transfer resistance may be determined by the hydrodynamic condition. It is in fact more convenient to depict film transfer rate in terms of an effective transfer coefficient or a lumped resistance coefficient rather than to use a diffusion equation to represent adsorption kinetics in a rigorous manner.

The CSS adsorber model (gCSS_Adsorber) within Aspen Adsim 2004.1 limits two types of lumped kinetic models for application. They are: Linear Driving Force Approximation and Quadratic Driving Force Approximation. Both

approximations have a lumped resistance coefficient that may be determined at either fluid or solid film where the mass transfer occurs:

(1) Linear Driving Force Approximation (LDFA):

(

i i^*

)

(2) Quadratic Driving Force Approximation (QDFA):

( ) ( )

constant or by a certain relationship according to the dynamic conditions of adsorption system. The CSS adsorber model (gCSS_Adsorber) provides the following choices in determining the lumped mass transfer coefficient from the empirical assessment by Aspen Adsim users:

• Constant

• Effective Diffusivity

2

15

p ei

i r

k = D Linear Driving Force Approximation

2 2

p i ei

r k ₌ π D

Quadratic Driving Force Approximation

• Pressure Dependent

P k

_i

= k

^Pi

• Pressure Dependent Arrhenius



 



−

=

s i i Pi

RT E P

k k⁰ exp

Notation

C

i Gas phase concentration for component i, kmol/m³

*

C

i Equilibrium gas phase concentration for component i, kmol/m³

D

ei Effective diffusivity for component i, m²/s

E

i Activation energy for component i, MJ/kmol

k

i Mass transfer coefficient (fluid or solid) for component i, 1/s

LDFi

k

Mass transfer coefficient as a constant for component i, 1/s

k

Pi Pressure dependent mass transfer coefficient for component i, bar/s

k

₀i Pre-exponent for component i, 1/s

k

₀Pi Pre-exponent for component i, bar/s

k

Fi Fluid film mass transfer coefficient for component i, 1/s

k

Si Solid film mass transfer coefficient for component i, 1/s

P

Gas pressure, bar

Q

i Amount adsorbed for component i, kmol/kg-adsorbent

t

Time, s

T

s Solid temperature, K

R

Gas constant (8.31451e-3), MJ/kmol/K

ρ

b Bed packing density, kg/m³

Energy Balance

The CSS adsorber model (gCSS_Adsorber) uses the following energy balances to represent the heat transportations of non-isothermal system with

compressible flow:

(1) In Fluid Phase:

(2) In Solid Phase:

− ∂

Axial thermal conduction

t C

_{Ps b}

T

^s

∂

ρ ∂

Thermal accumulation in solid phase

∑ ^ _ ^{ ∆ ∂} _∂ ^ _ ^

− ∂

Axial thermal conduction along the wall

t

H A −

Heat transfer between gas and wall

(

_{w amb}

)

a

p Particle external surface area to particle volume ratio (=3/rp), m

A

Hi Internal wall heat transfer area, m

A

External wall heat transfer area, m

C

Pw Adsorber material (e.g., stainless steel) specific heat

capacity, MJ/kg/K

H

s Fluid/solid heat transfer coefficient, MW/m²/K

H

w Fluid/wall heat transfer coefficient, MW/m²/K

H

amb Wall/environment heat transfer coefficient, MW/m²/K

k

g Gas mixture thermal conductivity, MW/m/K

k

s Solid phase thermal conductivity, MW/m/K

k

w Wall phase thermal conductivity, MW/m/K

P

Gas pressure, bar

Q

i Amount adsorbed for component i, kmol/kg-adsorbent

t

Time, s

T

g Gas temperature, K

T

s Solid temperature, K

T

w Wall temperature, K

v

g Superficial gas velocity, m/s

V

Hi Internal wall element volume for heat transfer, m²

V

Ho External wall element volume for heat transfer, m²

x

Axial distance coordinate, m

H

i

∆

Enthalpy of adsorption for component i (i.e., heat of adsorption), MJ/kmol

ε

b Bed voidage (void fraction)

ε

t Total voidage

ρ

g Gas density, kmol/m³

ρ

b Bed packing density, kg/m³

ρ

w Wall material density, kg/m³

Adsorption Equilibrium Models

Introduction

Adsorption equilibrium established after the adsorptive has been in with the adsorbed surface for a long time, and can be represented in general form:

0 ) , ,

( Q T =

f

_i

ρ

_i (Eqn 1)

In this equation, Q_i is the concentration for component i on adsorbed phase, i.e., amount adsorbed, ρ_ι is the density for component i in fluid phase, and T is the temperature. For an isothermal condition, the Eqn1 can be represented by the adsorption isotherm:

T i

i

f

Q = ( ρ )

and

ρ

_i

= f ( Q

_i

)

_T (Eqn 2)

Eqn 2, which is commonly referred to as adsorption equilibrium isotherm, is most frequently used in researches including adsorption process simulation.

For pure component adsorption, an equilibrium relationship could simply be represented by mathematical equation such as the Langmuir, the Freundlich, the Sips, the Toth, and so on. Eqn 1 can also take the following form and is called the adsorption isostere(see Ref. 1):

Qi

i

= f ( T )

ρ

(Eqn 3)

However, the adsorption isostere cannot be measured directly because it is impractical to hold

Q

_iconstant.

For multi-component system, the explanation of adsorption equilibrium relationship often causes considerable attention due to a unique and complex mixing rule that governing an adsorption system of interest. For many

decades, numerous researchers have considered multi-component adsorption equilibria from thermodynamic perspective and developed a number of theories or models based on various assumptions concerning the nature of adsorbed phase.

The CSS model in Aspen Adsim offers the following types of adsorption equilibrium models for multi-component system. Please note all equilibrium models only require pure equilibrium information in order to predict mixture equilibrium:

References

1 D. M. Young and A. D. Crowell, Physical Adsorption of Gases, Butterworths, London (1962).

Mathematical Equation Form for Extended Langmuir 1

{ }

(Pressure dependent equilibrium)

{ }

(Concentration dependent equilibrium)

i i

IP

IP

₁

,

₂ Isotherm parameters for component i

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e., amount adsorbed) for component I

Mathematical Equation Form for Extended Langmuir 2

[ ]

(Pressure dependent equilibrium)

[ ]

(Concentration dependent equilibrium)

i i

i

IP IP

IP

₁

,

₂

,

₃ Isotherm parameters for component i

T

s Adsorbent particle temperature in Kelvin

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e., amount adsorbed) for component I

Mathematical Equation Form for Extended Langmuir 3

( ) ( [ ] )

(Pressure dependent equilibrium)

( ) ( [ ] )

(Concentration dependent equilibrium)

i i i

i

IP IP IP

IP

₁

,

₂

,

₃

,

₄ Isotherm parameters for component i

T

s Adsorbent particle temperature in Kelvin

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e.,

amount adsorbed) for component I

Mathematical Equation Form for Extended Langmuir 4

( ) ( [ ] )

(Pressure dependent equilibrium)

( ) ( [ ] )

(Concentration dependent equilibrium)

i i i

i

IP IP IP

IP

₁

,

₂

,

₃

,

₄ Isotherm parameters for component i

T

s Adsorbent particle temperature in Kelvin

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration

(i.e., amount adsorbed) for component I

Mathematical Equation Form for Extended Langmuir 5

[ ]

(Pressure dependent equilibrium)

[ ]

(Concentration dependent equilibrium)

i i i

i

IP IP IP

IP

₁

,

₂

,

₃

,

₄ Isotherm parameters for component i

T

s Adsorbent particle temperature in Kelvin

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e., amount adsorbed) for component I

Mathematical Equation Form for Loading Ratio Correlation 1

( ) ( )

(Pressure dependent equilibrium)

{ }

(Concentration dependent equilibrium)

IP

_?i Isotherm parameters for component i

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e., amount adsorbed) for component I

Mathematical Equation Form for Loading Ratio Correlation 2

[ ]

(Pressure dependent equilibrium)

[ ]

(Concentration dependent equilibrium)

IP

_?i Isotherm parameters for component i

T

s Adsorbent particle temperature in Kelvin

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e., amount adsorbed) for component I

Mathematical Equation Form for Loading Ratio Correlation 3

( ) ( [ ] )( )

(Pressure dependent equilibrium)

( ) ( [ ] )

(Concentration dependent equilibrium)

IP

_?i Isotherm parameters for component i

T

s Adsorbent particle temperature in Kelvin

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e., amount adsorbed) for component I

Mathematical Equation Form for Loading Ratio Correlation 4

( ) ( [ ] )( )

(Pressure dependent equilibrium)

( ) ( [ ] )

(Concentration dependent equilibrium)

IP

_?i Isotherm parameters for component i

T

s Adsorbent particle temperature in Kelvin

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e., amount adsorbed) for component I

Mathematical Equation Form for Loading Ratio Correlation 5

[ ]

(Pressure dependent equilibrium)

[ ]

(Concentration dependent equilibrium)

IP

_?i Isotherm parameters for component i

T

s Adsorbent particle temperature in Kelvin

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e., amount adsorbed) for component I

Mathematical Equation Form for Extended Dual-Site Langmuir 1

{ } ∑ { }

(Pressure dependent equilibrium)

{ } ∑ { }

(Concentration dependent equilibrium)

IP

_?i Isotherm parameters for component i

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e., amount adsorbed) for component I

Mathematical Equation Form for Extended Dual-Site Langmuir 2

[ ]

(Pressure dependent equilibrium)

[ ]

(Concentration dependent equilibrium)

IP

_?i Isotherm parameters for component i

T

s Adsorbent particle temperature in Kelvin

P

Total gas pressure

y

i Gas phase mole fraction for component i

C

i Fluid phase concentration for component i

Q

i Adsorbed phase concentration (i.e., amount adsorbed) for component I

I.A.S.T. (Ideal Adsorbed Solution

In document Aspen Adsim (Page 99-123)