Ismail B Celik Mechanical and Aerospace Engineering Department West Virginia University WVU Morgantown WV 26506 ibcelikmailwvuedu Contributors Zhiyuan Ma Sofiane Benyahia ID: 816013
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Slide1
Discretization Error Estimation for Two-Fluid Models of Gas-Solid Flows
Ismail B. Celik
Mechanical and Aerospace Engineering Department
West Virginia University (
WVU
), Morgantown WV 26506
ibcelik@mail.wvu.edu
Contributors:
Zhiyuan
Ma* Sofiane Benyahia**,
Venkata S.S. Guda*,
Madhava
Syamlal
**
* West Virginia University, WVU
** National Energy Tech. Lab., NETL
Slide2Error Transport Equation (ETE) as an Alternative to Richardson Extrapolation (RE)
Predicted error in U-velocity from ETE
True error in U-velocity
Parsons, D., (2015) “Prediction of Discretization Error Using Error Transport Equation”, Ph.D. Dissertation, Mechanical and Aerospace Engineering, West Virginia University, Morgantown WV (USA
).
Parsons, D., and
Celik
, I. “
Prediction of Discretization Error Using Error Transport Equation
”, Submitted to Journal of Computational Physics, June 2016
Slide3Objective
Develop
a robust discretization error field prediction algorithm for a selected transport variable utilizing an error transport equation preferably on a single mesh or at most two
similar
meshes
Slide4Generalized 1D transient
s
calar
t
ransport e
quation
The corresponding discrete
equations
Modified equation (1) with
Finally, we get
Theory
On a much finer mesh, being the continuous counterpart of
(5)
Slide5Use the modified equation approach to account for time discretization error when ‘
dt
’ is not small
For example, first order backward time discretization
Hence the additional error source term should be added to
rhs
of
Eq
(4)
Theory
Slide6In general we let
and
w
hich can be determined by some calibration procedure
For a
order method
Note
that for small
and
, and have the same sign. However, for coarse meshes can be large negative number which may change the sign of . Approximation for the Ratio Et/Ea
Slide7Transient Model Problem
Solution at time = 10 seconds
Transients of the solution
Slide8Transient Model Problem: Burgers Equation
Transients of discretization error:
nx
=41,
dt= 0.1
Slide9Transient Model Problem: Burgers Equation
Transients of discretization error:
nx
=81,
dt= 0.1
Slide10Transient Model Problem: Burgers Equation
Transients of discretization error:
nx
=41,
dt= 0.01
Slide11Transient Model Problem: Burgers Equation
Transients of discretization error:
nx
=81,
dt= 0.01
Slide12MFIX Equations
Continuity equation
Momentum equation
porous media force
term
gas-solid momentum exchange
where ,
,
Slide13ETE based on Continuity Equation in MFIX
Gas-phase continuity
Subtracting
Eq
(2) from
Eq
(1) gives
Note that
Therefore ,
This equation constitutes the error transport equation for the discretization error
Slide14ETE
Algorithm
Solve
two-phase flow equations on two meshes, fine and coarse, to
obtain
Solve
for
using the ETE
(
Eq
(4)) and calculate
.
Calculate the residual of Eq(5) and adjust until the residual is minimized.The actual error could be taken as an average if need be ETE based on Continuity Equations in MFIX
Slide15ETE based on Momentum Equation in MFIX
T
he
ETE for
,
Where
]
Slide16Results: MFIX Application
References
[1]
Benyahia
, S., Syamlal, M., and O’Brien, T.J., 2007, “Study of the Ability of Multiphase Continuum Models to Predict Core-Annulus Flow,”,
AlchE J., 53(10), pp. 2549-2568.
ETE
for gas-phase volume fraction (Celik et al.,
2016)
:
Gas-velocity error
estimation:
= The case considered is 1D transient ‘core annular flow’ problem presented by Benyahia et al. (2007)
Slide17Transients of MFIX SolutionVariation of
Epg
and
epg-error with time
X = 1.125 cm
Slide18Transients of MFIX SolutionVariation of source terms in the ETE for Gas Vol. fraction,
Epg
X = 1.125 cm
Slide19Transients of MFIX SolutionTransients of ETE solution for Gas Vol. Fraction,
Epg
X = 1.125 cm
Slide20Transients of MFIX Solution
Approximate error in boundary values of
Eps_g
: west boundary (left); east boundary (right); Eps_g_min
= 0.44; Eps_smax = 0.56
Slide21Results: MFIX Application
Error in solid-phase volume fraction
Time averaged and temporal errors in gas phase volume fraction
Blue curves show the error transport equation can predict the time averaged errors accurately. The transients of the error can also be predicted by the ETE with reasonable accuracy.
Slide22Transients of MFIX Solution: gas-phase velocity
Phase-shift: 40cells (+2.6 sec), 60cels (+2.5 sec),
80 cells (0.0 sec)
Gas-phase velocity (Vg) variation with time X = 8.5 cm
Slide23Transients of MFIX Solution: gas-phase velocity
Gas-phase velocity (Vg) error variation with time X = 8.5 cm
Phase-shift: 40cells (+2.6 sec), 60cels (+2.5 sec),
80 cells (0.0 sec)
Slide24Chaotic two-phase flows
Time =6.3 sec
Time = 6.5 sec
(
tke
= turbulent kinetic energy)
Computational Details:
ANSYS-Fluent
Domain
width 16 cm and height 80 cm
Eulerian – Eulerian multiphase model
Gas density = 1.225 kg/m3Solids density = 2000 kg/m3
Uniform inlet gas velocity = 180 cm/sSolids Bed packing 0.1
Remedy: use LES-IQ techniques Celik, I., Klein, M., Janicka, J. (2009). Assessment Measures for Engineering LES Applications. Journal of Fluids Engineering, 131, 031102-1-10.
Slide25A new strategy is proposed to calculate discretization errors in time and space domains
The method is sown to work fro 1D transient Burgers equation
Error transport equations are derived for multi-phase flow two-fluid models
Application to a 1D transient case using the commonly used MFIX code yielded quite encouraging
results for one period corrected for phase shift.
Prediction of the full transients of error transport is challenging !
Application to two-dimensional dense gas-solid flow problems is underway
Conclusions
Slide26Acknowledgements
This work was performed through ORISE/ORAU (Oakridge Associated Universities) Faculty Fellowship Program at NETL (National Energy Technology Laboratory, DOE)
Contributions via discussion and other media by
Madhava
Syamlal of NETL and S. Satish Guda of WVU are acknowledged and greatly appreciated.