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Optimal design of pressure swing adsorption units for hydrogen recovery under uncertainty

  • Oleg Golubyatnikov ORCID logo EMAIL logo , Evgeny Akulinin und Stanislav Dvoretsky
Veröffentlicht/Copyright: 4. Mai 2023
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Chemical Product and Process Modeling
Aus der Zeitschrift Chemical Product and Process Modeling Band 18 Heft 5

Abstract

The paper proposes an approach to the optimal design of pressure swing adsorption (PSA) units for hydrogen recovery under uncertainty, which provides a reasonable margin of the potential resource of the PSA hydrogen unit and compensates for the negative impact of a random change in uncertain parameters within specified limits. A heuristic iterative algorithm is proposed to solve the design problem with a profit criterion, which is guaranteed to provide the technological requirements for the PSA unit, regardless of the values that take uncertain parameters from the specified intervals of their possible change. An experimental verification of the approach with the root-mean-square error of 19.43 % has been carried out. Optimization problems of searching for a combination of mode and design parameters under uncertainty for a range of 4-bed 4-step VPSA units with a capacity of 100–2000 L/min STP have been solved taking into account the requirements for hydrogen purity of 99.99+ %, gas inlet velocity of 0.2 m/s, and bed pressure drop (no more than 1 atm). It has been established that taking into account uncertainties leads to an increase in energy costs by 8–10 %, a decrease in profit by 10–15 %, and a decrease in hydrogen recovery by 4–5 %, which is a payment for the uninterrupted operation of the PSA unit. The effect of uncertain parameters (percentage composition of the gas mixture; gas temperature; the diameter of adsorbent particles) on the key indicators of the PSA process (recovery, profit, hydrogen purity, unit capacity) has been established and trends in adsorption duration, adsorption and desorption pressure, P/F ratio, valve capacity, bed length, adsorber diameter for design of hydrogen PSA unit, which are necessary for subsequent design and scaling of units.

Keywords: design; hydrogen; optimization; pressure swing adsorption; uncertainty
Corresponding author: Oleg Golubyatnikov, Tambov State Technical University, Ul. Sovetskaya str., 106, Tambov, 392000, Russia, E-mail:

Funding source: Russian Science Foundation

Award Identifier / Grant number: 21-79-00092

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: The study was carried out at the expense of a grant from the Russian Science Foundation (Project No. 21-79-00092, https://rscf.ru/en/project/21-79-00092/).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

APPENDIX

See Tables 4–8.

Table 4:

Mathematical model of the PSA process for hydrogen recovery.

Description Expression
Component-wise material balance in the gas phase flow along the height of the adsorbent ∂ci(x,τ)∂τ+(1−ε)ε∂qi(x,τ)∂τ+∂(νgci(x,τ))∂x=∂∂x(DLc(x,τ)∂yi(x,τ)∂x),i=H2,CO2,CO
Sorption kinetics (Gluckauf formula) ∂qi(x,τ)∂τ=ki(qi*−qi(x,τ))
Heat propagation in the gas mixture flow along the height of the adsorbent Cgρg∂Tg(x,τ)∂τ+Cgρgνg∂Tg(x,τ)∂x−hαε[Ta(x,τ)−Tg(x,τ)]=λg∂2Tg∂x2
Heat propagation in the adsorbent Caρa∂Ta(x,τ)∂τ+α h[Ta(x,τ)−Tg(x,τ)]−∑i∆Hi∂qi(x,τ)∂τ=λa∂2Ta(x,τ)∂x2
Continuity of the flow ∑ici∂νg(x,τ)∂x−νg(∂∑ici∂x)=0
Ergun’s equation ∂P∂x=−(150(1−ε)2dp2ψ2εμgνg+1.75Mgρg(1−ε)dpψε3νg2)
Valve equation Gin(τ)=Kv(Pads−Padsin),Geq(τ)=Kv(Peq−Pppe)
Throttle equation Gdes(τ)=(P/F)Gadsout(τ,L),0≤τ≤τads
Pressure set in adsorbers dPadsindτ=KadsPdesGin(τ)PadsVcolε(Pads−Padsin)
Depressurizing in adsorbers dPdesindτ=KdesPdesGdes(τ)PadsVcolε(Pdes−Pdesin)
Pressurizing pressure equalization dPppedτ=KppePeqGeq(τ)PadsVcolε(Peq−Pppe)
Depressurizing pressure equalization dPdpedτ=−dPppedτ
The Dubinin–Radushkevich–Astakhov equation qi*=W0vi*exp[−(RTglg(Ps,i/Pp,i)βiE)n]
Unit capacity Gout=(Gadsout−Gdes)(Padsout/PSTP)(TSTP/Tg,adsout)
Hydrogen recovery η=cH2outGoutcH2inGin100%
Table 5:

Initial conditions of the PSA model.

Adsorption, first cycle Depressurizing pressure equalization Desorption Pressurizing pressure equalization Adsorption, subsequent cycles
ci(x,0)=ci0 ci(x,τ)=ciads(x,τads) ci(x,τ)=cidpe(x,τdpe) ci(x,τ)=cides(x,τdes) ci(x,τ)=cippe(x,τppe)
qi(x,0)=0 qi(x,τ)=qiads(x,τads) qi(x,τ)=qidpe(x,τdpe) qi(x,τ)=qides(x,τdes) qi(x,τ)=qippe(x,τppe)
Tg(x,0)=Tg0 Tg(x,τ)=Tgads(x,τads) Tg(x,τ)=Tgdpe(x,τdpe) Tg(x,τ)=Tgdes(x,τdes) Tg(x,τ)=Tgppe(x,τppe)
Ta(x,0)=Ta0 Ta(x,τ)=Taads(x,τads) Ta(x,τ)=Tadpe(x,τdpe) Ta(x,τ)=Tades(x,τdes) Ta(x,τ)=Tappe(x,τppe)
νg(x,0)=νg0 νg(x,τ)=νgads(x,τads) νg(x,τ)=νgdpe(x,τdpe) νg(x,τ)=νgdes(x,τdes) νg(x,τ)=νgppe(x,τppe)
P(x,0)=P0 P(x,0)=Pads(x,τads) P(x,0)=Pdpe(x,τdpe) P(x,0)=Pdes(x,τdes) P(x,0)=Pppe(x,τppe)
Padsin(0)=P0 Pdpe(0)=Pads(L,τads) Pdesin(0)=Pdpe(L,τdpe) Pppe(0)=Pdes(L,τdes) Padsin(0)=Pppe(L,τppe)
Table 6:

Boundary conditions of the PSA model.

Adsorption x = 0 Adsorption x = L DPEa x = 0 DPEa x = L Desorption x = 0 Desorption x = L PPEa x = 0 PPEa x = L
ci(0,τ)=ciin(τ) ∂ci∂x(L,τ)=0 ∂ci∂x(0,τ)=0 ci(L,τ)=ciads(L,τads) ∂ci∂x(0,τ)=0 ci(L,τ)=ciads(L,τ) ci(0,τ)=cidpe(L,τ) ∂ci∂x(L,τ)=0
Tg(0,τ)=Tgin(τ) ∂Tg∂x(L,τ)=0 ∂Tg∂x(0,τ)=0 Tg(L,τ)=Tgads(L,τads) ∂Tg∂x(0,τ)=0 Tg(L,τ)=Tgads(τ) Tg(0,τ)=Tgdpe(L,τ) ∂Tg∂x(L,τ)=0
νg(0,τ)=Gin(τ)εS ∂νg∂x(L,τ)=0 ∂νg∂x(0,τ)=0 νg(L,τ)=Geq(τ)εS ∂νg∂x(0,τ)=0 νg(L,τ)=Gdes(τ)εS νg(0,τ)=Geq(τ)εS ∂νg∂x(L,τ)=0
P(0,τ)=Padsin(τ) ∂P∂x(L,τ)=0 ∂P∂x(0,τ)=0 P(L,τ)=Pdpe(τ) ∂P∂x(0,τ)=0 P(L,τ)=Pdesin(τ) P(0,τ)=Pppe(τ) ∂P∂x(L,τ)=0

  1. aDepressurizing pressure equalization.

  2. bPressurizing pressure equalization.

Table 7:

Values of coefficients and adsorbent characteristics of the PSA model.

Description Expression
Specific heat capacity, Cg (J/mol K) 9971
Thermal conductivity coefficient, λg (W/m K) 0.129
Dynamic viscosity, μ g (10−5 Pa s) 1.07
Molar mass, Mg (10−3 kg/mol) 15.45
Diffusion coefficient in the gas phase, D L (10−5 m2/s) 7.46
Heat transfer coefficient, h (W/m2 K) 130
Zeolite adsorbent 13X
Density of the adsorbent, ρ a (kg/m3) 2140
Adsorbent porosity coefficient, ε (m3/m3) 0.39
Thermal conductivity coefficient of the adsorbent, λa (W/m K) 0.139
Sphericity coefficient of adsorbent particles, ψ 1
Input temperature in the adsorbent, Tain (K) 298
Characteristic energy of adsorption, E (J/mol) 12,902
Limiting adsorption volume, W0 (cm3/g) 0.262
Table 8:

Values of coefficients PSA model for gas mixture.

Description H2 CO2 CO
Component concentration, c (vol%) 48–68 27–47 5
Heat of sorption, ΔH (kJ/mol) 7.49 31.56 24.76
Mass transfer coefficient, k (1/s) 0.7 0.015 0.065
Affinity coefficient, β 0.11 2.20 1.08
Exponent DRA-equation, n 0.81 2 2
Critical molar volume, v* (cm3/mmol) 0.007 0.04 0.05
Saturation pressure, Ps (atm) 620.78 212.49 343.99

Nomenclature

Greek symbols

α

specific surface of adsorbent granules (1/m)

β

affinity coefficient

γ

adiabatic exponent

ΔH

heat of sorption (J/mol)

ΔP

pressure drop (atm)

ε

adsorbent porosity coefficient (m3/m3)

η

hydrogen recovery (%)

λ

thermal conductivity coefficient (W/m K)

μ g

dynamic viscosity (Pa s)

ν

iteration number

ν g

gas velocity (m/s)

Ξ

uncertainty region

ξ

uncertain variables

ρ a

adsorbent density (kg/m3)

ρ g

adsorbent molar density (mol/m3)

τ

time, duration (s)

ϕ

profit (K$)

ψ

sphericity coefficient of adsorbent particles

ω

weight coefficient

Latin symbols

C a

specific heat of the adsorbent (J/kg K)

C g

specific heat capacity (J/mol K)

c i

concentration of the ith component in the gas mixture (mol/m3)

c

total mole concentration (mol/m3)

D

inner diameter of the adsorber (m)

D L

diffusion coefficient in the gas phase (m2/s)

d

vector of design parameters

d p

diameter of adsorbent particles (mm)

E

characteristic adsorption energy (J/mol)

EC

energy cost ($/J)

F

function of the mathematical model

G

flow rate (L/min STP)

G out

unit capacity (L/min STP)

g

constraint function

h

heat transfer coefficient (W/m2 K)

I

capital costs ($)

K

kinetic pressure coefficient (1/s)

K v

valve capacity (l atm/min)

k

mass transfer coefficient (1/s)

L

height of the adsorbent bed (m)

LT

life time (year)

M g

molar mass (kg/mol)

N

quantity

n

exponent of Dubinin–Radushkevich–Astakhov equation

OC

operating costs ($/year)

P

pressure (atm)

P s

saturation pressure (atm)

P p

partial pressure (atm)

P/F

purge-feed ratio

PC

product cost ($/year)

p

product price ($/m3)

Q

optimization criterion

q i

concentration in the adsorbent (mol/m3)

qi*

equilibrium concentration in the adsorbent (mol/m3)

R

ideal gas constant (J/mol K)

S 1

set of approximation points

S 2

set of critical points

T

temperature (K)

u

vector of mode parameters

v *

critical molar volume (cm3/mmol)

W

theoretical volume of work (J/m3)

W 0

limiting adsorption volume (cm3/g)

x

spatial coordinate along the bed length (m)

ycss

output model variables in cyclic steady state

yi

mole fraction of the ith component in the bulk

Z

cost ($)

z

price ($)

Subscripts

a

adsorbent

ads

adsorption

col

adsorber

com

compressor

def

defined value

des

desorption

dpe

depressurizing pressure equalization

eq

equalization

g

gas mixture

i,j,k

serial number

in

inlet

out

oulet

ppe

pressurizing pressure equalization

vp

vacuum pump

Abbreviations

CSS

cyclic steady state

PSA

pressure swing adsorption

RMS

root-mean-square error

STP

standard temperature and pressure

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Received: 2022-12-12
Accepted: 2023-04-20
Published Online: 2023-05-04

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https://doi.org/10.1515/cppm-2022-0081
Schlagwörter für diesen Artikel
design; hydrogen; optimization; pressure swing adsorption; uncertainty

Artikel in diesem Heft

  1. Frontmatter
  2. Review
  3. A review of frictional pressure drop characteristics of single phase microchannels having different shapes of cross sections
  4. Research Articles
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Heruntergeladen am 23.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/cppm-2022-0081/html?lang=de
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