Alternative equation for sorption data interpretation Patrick van Hemert & K-H.A.A. Wolf
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Transcript of Alternative equation for sorption data interpretation Patrick van Hemert & K-H.A.A. Wolf
Alternative equation for sorption data interpretation
Patrick van Hemert & K-H.A.A. Wolf
May 25, 2006
International Symposium CBM, Tuscaloosa
Technical GeosciencesDelft University of Technology, The Netherlands
Objectives
• For experts:• Introduce alternative equation for sorption
interpretation• Simpler calculation• Easier error analysis
• For non-experts:• Basic understanding of volumetric sorption
isotherms measurements
Technical GeosciencesDelft University of Technology, The Netherlands
Outline
• Relevance • Introduction to volumetric sorption• Alternative formula for interpretation• Comparison to original formula• Conclusion
Technical GeosciencesDelft University of Technology, The Netherlands
Reasons for measuring sorption isotherms
• Determine Gas Storage Capacity• Used for Gas In Place analysis
• Determine fluid-solid interaction• Reservoir models• Science
Technical GeosciencesDelft University of Technology, The Netherlands
Typical laboratory output: The Langmuir
•X-axis is usually pressure in either bar, psia or Mpa.
•Y-axis is usually sorbed gas in mol/kg, or SCF/Ton
( )1ads
n Pn P
bP
Technical GeosciencesDelft University of Technology, The Netherlands
What are volumetric isotherms?
• Isothermal experiments.• Material balance calculations from P,T,V data• Procedure consist of two steps, isolated(1) and connected(2) •Sorbed amount is the difference between the total amount and the equilibrium amount of gas
Technical GeosciencesDelft University of Technology, The Netherlands
- 1 2
Gas supply
Valve 1
Valve 2
sample cell
P
0
vacuum
-
Gas supply
Sorption experimental procedureRepresentation of the two experimental steps in a sorption experiment
From N. SiemonsFrom N. Siemons
Technical GeosciencesDelft University of Technology, The Netherlands
Pressure development during an experiment
Technical GeosciencesDelft University of Technology, The Netherlands
Sorption is a material balance difference calculation
M M MGibbs total void gasn mole n V
1 1 2 2 2 21
[ ] ( , ) ( , ) ( , )M
M i i i i M MGibbs ref void
i
n mole V P T P T V P T
1 1 2 21 1
[ ] 0 ( , ) ( , )M M
M i i i i itotal initial added ref
i i
n mole n n V P T P T
M M Madsorbed total gasn mole n n
1 1 2 2 2 21
1[ ] ( , ) ( , ) ( , )
MM i i i i M MGibbs ref ref
i
molen V P T P T V P T
kg m
1 1 2 2 2 21
[ ] ( , ) ( , ) ( , )M
refM i i i i M MGibbs
i
Vmolen P T P T P T
kg m
M:Measured point
Technical GeosciencesDelft University of Technology, The Netherlands
Determination of volume ratio from material balance conservation
2 2 1 11 1
2 2 2 2
( , ) ( , )
( , ) ( , )
i i i ivoid
i i i iref
V P T P T
V P T P T
•For micro-porous solids needs to be experimentally determined
•Helium sorption experiments with the assumption that helium sorption is negligible
Technical GeosciencesDelft University of Technology, The Netherlands
Alternative formulation for sorption interpretation
simplifies error analysis
1 1 2 2 2 21
( , ) ( , ) ( , )M
refM i i i i M MGibbs
i
Vn P T P T P T
m
Void volume ratio sample and reference cell(defines the dividing surface)
Each parameter can separately be calculated
Technical GeosciencesDelft University of Technology, The Netherlands
High pressure volumetric isotherms are very sensitive
• Changes in the volume ratio have a large effect • The use of an wrong densities has large effects
Technical GeosciencesDelft University of Technology, The Netherlands
Previous formulation is unnecessarily difficult
1 2
1 1 1 1 2 2 2
12 2
1 1 11 2 2 2 2 2 2
, ,
, ,
i iMrefM
Gibbs i i i i i ii
i iMvoid
M M i M M ii
V P Pn
m Z P T T R Z P T T R
V P P
m Z P T T R Z P T T R
•Redundant terms in equilibrium amount•Real gas law formulation unnecessarily complicated•Void volume is not independently measured• This unnecessarily complicates error analysis!
Technical GeosciencesDelft University of Technology, The Netherlands
The alternative equation for volumetric sorption interpretation has the following
advantages:
• Physical interpretation is straightforward• Easy to calculate• Simple error analysis
• Contains only independent parameters
Technical GeosciencesDelft University of Technology, The Netherlands
Questions?
Technical GeosciencesDelft University of Technology, The Netherlands
Alternative formulation for sorption simplifies error analysis
1 1 2 2 2 21
( , ) ( , ) ( , )M
refM i i i i M MGibbs
i
Vn P T P T P T
m
1 2
1 1 1 1 2 2 2
12 2
1 1 11 2 2 2 2 2 2
, ,
, ,
i iMrefM
Gibbs i i i i i ii
i iMvoid
M M i M M ii
V P Pn
m Z P T T R Z P T T R
V P P
m Z P T T R Z P T T R
Technical GeosciencesDelft University of Technology, The Netherlands
Experimental uncertainties from alternative equation
2
1 1
2
1 1
M MMsys M iGibbs Gibbs
Gibbs ref kii kref k
M M MMiGibbs Gibbs Gibbski
i k k
dn dnn V P
dV dP
dn dn dnT m
dT d dm
2 2 2
2 22 2
1 1 1 1
M M MGibbs Gibbs Gibbs
refrefran M
GibbsM MM M
i iGibbs Gibbsk ki i
i k i kk k
n n nV m
V mn
n nP T
P T
Technical GeosciencesDelft University of Technology, The Netherlands
Alternative formulation for sorption interpretation
Volume ratio sample and reference cell
The sorbed amount is calculated from the difference in material balance between the total amount of gas and the equilibrium amount of gas.
Technical GeosciencesDelft University of Technology, The Netherlands
Alternative formulations require unknown parameters
1 1 2 2 2 21
( , ) ( , ) ( , )M
refM i i i i M MAbs
i
Vn P T P T P T
m
2 2
2 2
( , )1
( , )
M M
M Msorb
P T
P T
1 1 2 2 2 21
( , ) ( , ) ( ) ( , )M
refM i i i i M Mswell
i
Vn P T P T P T
m
sample
ref
V
V
Technical GeosciencesDelft University of Technology, The Netherlands
Gibbs sorption is the only rigorous thermodynamic framework for adsorption
equilibrium
• The extent and density of the sorbed phase can not be measured independently or directly.
• Gibbs suggested the mathematical transformation that all fluid-solid interaction is attributed to a dividing surface.
• In micro-porous solids measuring the dividing surface is an experimental challenge.
• This constitutes an ill-defined problem.
Technical GeosciencesDelft University of Technology, The Netherlands
What causes the considerable differences between inter-laboratory sorption
experiment?
• Investigation into the nature of the measurements
• Quantification of the accuracy and precision of the measurements
Technical GeosciencesDelft University of Technology, The Netherlands
Introduce alternative formula for interpretation of volumetric sorption
experiments
•Calculation is simpler•Error analysis is easier•More straightforward interpretation
1 1 2 2 2 21
( , ) ( , ) ( , )M
refM i i i i M MGibbs
i
Vn P T P T P T
m
Technical GeosciencesDelft University of Technology, The Netherlands
Enhanced Coalbed Methane production may be a commercial energy source
• Underground coal contains ,mainly sorbed, methane
• Coalbed Methane projects (CBM) produce methane by pumping gas and water from the reservoir
• Enhanced Coalbed Methane (ECBM) projects may enhance production by injecting gas
• Gas-solid interactions determine production behavior
Technical GeosciencesDelft University of Technology, The Netherlands
Optimizing ECBM requires knowledge of gas-coal interactions
• Equilibrium adsorption data quantify fluid-solid interaction
• Pure component isotherms are the simplest form of adsorption equilibrium data• Methane isotherms on coal are already used
to quantify the amount of methane producible
Technical GeosciencesDelft University of Technology, The Netherlands
Sorption isotherms are measured volumetrically
• Volumetric technique is very widespread• Procedures and apparatus are easily obtainable• Actively used in the field of (E)CBM
• Gravimetric sorption measurements are less common
• More complex equipment and procedures
Technical GeosciencesDelft University of Technology, The Netherlands
Reliability of sorption isotherms is often omitted
• Every measurement has an associated uncertainty
• This uncertainty quantifies the measurements’ significance
• Reliability estimates from error analysis and repeated experiments should concur.
Technical GeosciencesDelft University of Technology, The Netherlands
Inter-laboratory isotherms differ considerably
• Repeated (inter-laboratory) experiments are the best reliability estimators
• The associated experimental errors must be specified.
• The cause of the difference must be investigated
Technical GeosciencesDelft University of Technology, The Netherlands
Sorption is a material balance difference calculation
Gibbs total void gasn mole n V
1 1 2 2 2 21
[ ] ( , ) ( , ) ( , )M
M i i i i M MGibbs ref void
i
n mole V P T P T V P T
1 1 2 21 1
[ ] 0 ( , ) ( , )M M
M i i i i itotal initial added ref
i i
n mole n n V P T P T
adsorbed total gasn mole n n
1 1 2 2 2 21
[ ] ( , ) ( , ) ( , )M
refM i i i i M MGibbs
i
Vmolen P T P T P T
kg m