Applying Computational Methods to Determine the Electric Current Density in a MHD Generator Channel from External Flux Density Measurements V. A. Bokil 1 , N. L. Gibson 1 , D. A. McGregor 1,2 , C. R. Woodside 2 (1) Oregon State University (2)National Energy Technology Laboratory Pacific Coast Carbon Storage / Computational Energy Science Research Closeout Meeting October 29, 2014 Gibson (OSU) ACMDECDMHDGCEFDM NETL 2014 1 / 11
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Applying Computational Methods to Determinethe Electric Current Density in a MHD GeneratorChannel from External Flux Density Measurements
V. A. Bokil1, N. L. Gibson1, D. A. McGregor1,2, C. R. Woodside2
(1) Oregon State University(2)National Energy Technology Laboratory
Pacific Coast Carbon Storage / Computational Energy Science ResearchCloseout Meeting
October 29, 2014
Gibson (OSU) ACMDECDMHDGCEFDM NETL 2014 1 / 11
Finding Arcs in MHD Generators
V. A. Bokil1, N. L. Gibson1, D. A. McGregor1,2, C. R. Woodside2
(1) Oregon State University(2)National Energy Technology Laboratory
Pacific Coast Carbon Storage / Computational Energy Science ResearchCloseout Meeting
Magneto-hydrodynamic (MHD)Power Generation offers significanttheoretical efficiency gains overtraditional turbo-machinery steamcycles when used as a topping cycle
The combustion product is lowconductivity, thermally generatedplasma. Passing this plasma througha strong magnetic field nearelectrodes generates electrical power.
However, MHD Power Generationsuffers from high life-cycle costs due,in part, to high component failurerates inside the channel, potentiallycaused by “arcing”.
Magneto-hydrodynamic (MHD)Power Generation offers significanttheoretical efficiency gains overtraditional turbo-machinery steamcycles when used as a topping cycle
The combustion product is lowconductivity, thermally generatedplasma. Passing this plasma througha strong magnetic field nearelectrodes generates electrical power.
Lorentz Force
F = q(E + u× B)
u = velocity, E,B = electric/magnetic fields
However, MHD Power Generationsuffers from high life-cycle costs due,in part, to high component failurerates inside the channel, potentiallycaused by “arcing”.
Magneto-hydrodynamic (MHD)Power Generation offers significanttheoretical efficiency gains overtraditional turbo-machinery steamcycles when used as a topping cycle
The combustion product is lowconductivity, thermally generatedplasma. Passing this plasma througha strong magnetic field nearelectrodes generates electrical power.
Efficiency gains are primarily due tothe lack of moving parts, whichallows for very high temperatures inthe generator.
However, MHD Power Generationsuffers from high life-cycle costs due,in part, to high component failurerates inside the channel, potentiallycaused by “arcing”.
Magneto-hydrodynamic (MHD)Power Generation offers significanttheoretical efficiency gains overtraditional turbo-machinery steamcycles when used as a topping cycle
The combustion product is lowconductivity, thermally generatedplasma. Passing this plasma througha strong magnetic field nearelectrodes generates electrical power.
Efficiency gains are primarily due tothe lack of moving parts, whichallows for very high temperatures inthe generator.However, MHD Power Generationsuffers from high life-cycle costs due,in part, to high component failurerates inside the channel, potentiallycaused by “arcing”.
Since the wall of the generatorchannel is cool relative to the bulkplasma temperature (for example500◦ K vs. 2500◦ K), a thermalboundary layer forms near the wall ofthe channel.
The plasma is thermally generated sothe conductivity drops dramaticallywith the plasma temperature.
However, the total current across thechannel is constant, so current will“jump the conductivity gap.”
It does so in very high density arcs.These arcs are hot and damage theelectrodes, which must be replaced.
We seek to detect the location ofarcs (areas of particularly largecurrent density) from externalmagnetic flux density measurements.
Since the wall of the generatorchannel is cool relative to the bulkplasma temperature (for example500◦ K vs. 2500◦ K), a thermalboundary layer forms near the wall ofthe channel.
The plasma is thermally generated sothe conductivity drops dramaticallywith the plasma temperature.
However, the total current across thechannel is constant, so current will“jump the conductivity gap.”
It does so in very high density arcs.These arcs are hot and damage theelectrodes, which must be replaced.
We seek to detect the location ofarcs (areas of particularly largecurrent density) from externalmagnetic flux density measurements.
Since the wall of the generatorchannel is cool relative to the bulkplasma temperature (for example500◦ K vs. 2500◦ K), a thermalboundary layer forms near the wall ofthe channel.
The plasma is thermally generated sothe conductivity drops dramaticallywith the plasma temperature.
However, the total current across thechannel is constant, so current will“jump the conductivity gap.”
It does so in very high density arcs.These arcs are hot and damage theelectrodes, which must be replaced.
We seek to detect the location ofarcs (areas of particularly largecurrent density) from externalmagnetic flux density measurements.
Since the wall of the generatorchannel is cool relative to the bulkplasma temperature (for example500◦ K vs. 2500◦ K), a thermalboundary layer forms near the wall ofthe channel.
The plasma is thermally generated sothe conductivity drops dramaticallywith the plasma temperature.
However, the total current across thechannel is constant, so current will“jump the conductivity gap.”
It does so in very high density arcs.These arcs are hot and damage theelectrodes, which must be replaced.
We seek to detect the location ofarcs (areas of particularly largecurrent density) from externalmagnetic flux density measurements.
Since the wall of the generatorchannel is cool relative to the bulkplasma temperature (for example500◦ K vs. 2500◦ K), a thermalboundary layer forms near the wall ofthe channel.
The plasma is thermally generated sothe conductivity drops dramaticallywith the plasma temperature.
However, the total current across thechannel is constant, so current will“jump the conductivity gap.”
It does so in very high density arcs.These arcs are hot and damage theelectrodes, which must be replaced.
We seek to detect the location ofarcs (areas of particularly largecurrent density) from externalmagnetic flux density measurements.
Current densities inside the generator cannot be directly measureddue to high temperatures, magnetic fields, and corrosive gasses
There is, however, a magnetic field that is induced by the internalelectric field, which extends beyond the channel
Measuring internal features by induced magnetic fields has beensuccessfully implemented for Vacuum Arc Remelters
We are interested in Current Reconstruction using externalmeasurements of the induced magnetic flux density
Typically current reconstruction relies on the solution of integralequations, e.g., the Biot-Savart Law [1, 2], which involves specialassumptions of geometry and/or material parameters
Instead, we solve a more flexible differential equations model andperform a simulation-based parameter estimation.
Current densities inside the generator cannot be directly measureddue to high temperatures, magnetic fields, and corrosive gasses
There is, however, a magnetic field that is induced by the internalelectric field, which extends beyond the channel
Measuring internal features by induced magnetic fields has beensuccessfully implemented for Vacuum Arc Remelters
We are interested in Current Reconstruction using externalmeasurements of the induced magnetic flux density
Typically current reconstruction relies on the solution of integralequations, e.g., the Biot-Savart Law [1, 2], which involves specialassumptions of geometry and/or material parameters
Instead, we solve a more flexible differential equations model andperform a simulation-based parameter estimation.
Current densities inside the generator cannot be directly measureddue to high temperatures, magnetic fields, and corrosive gasses
There is, however, a magnetic field that is induced by the internalelectric field, which extends beyond the channel
Measuring internal features by induced magnetic fields has beensuccessfully implemented for Vacuum Arc Remelters
We are interested in Current Reconstruction using externalmeasurements of the induced magnetic flux density
Typically current reconstruction relies on the solution of integralequations, e.g., the Biot-Savart Law [1, 2], which involves specialassumptions of geometry and/or material parameters
Instead, we solve a more flexible differential equations model andperform a simulation-based parameter estimation.
Current densities inside the generator cannot be directly measureddue to high temperatures, magnetic fields, and corrosive gasses
There is, however, a magnetic field that is induced by the internalelectric field, which extends beyond the channel
Measuring internal features by induced magnetic fields has beensuccessfully implemented for Vacuum Arc Remelters
We are interested in Current Reconstruction using externalmeasurements of the induced magnetic flux density
Typically current reconstruction relies on the solution of integralequations, e.g., the Biot-Savart Law [1, 2], which involves specialassumptions of geometry and/or material parameters
Instead, we solve a more flexible differential equations model andperform a simulation-based parameter estimation.
Current densities inside the generator cannot be directly measureddue to high temperatures, magnetic fields, and corrosive gasses
There is, however, a magnetic field that is induced by the internalelectric field, which extends beyond the channel
Measuring internal features by induced magnetic fields has beensuccessfully implemented for Vacuum Arc Remelters
We are interested in Current Reconstruction using externalmeasurements of the induced magnetic flux density
Typically current reconstruction relies on the solution of integralequations, e.g., the Biot-Savart Law [1, 2], which involves specialassumptions of geometry and/or material parameters
Instead, we solve a more flexible differential equations model andperform a simulation-based parameter estimation.
Current densities inside the generator cannot be directly measureddue to high temperatures, magnetic fields, and corrosive gasses
There is, however, a magnetic field that is induced by the internalelectric field, which extends beyond the channel
Measuring internal features by induced magnetic fields has beensuccessfully implemented for Vacuum Arc Remelters
We are interested in Current Reconstruction using externalmeasurements of the induced magnetic flux density
Typically current reconstruction relies on the solution of integralequations, e.g., the Biot-Savart Law [1, 2], which involves specialassumptions of geometry and/or material parameters
Instead, we solve a more flexible differential equations model andperform a simulation-based parameter estimation.
We assume a parameterization of the quantity of interest(current density)
Given external field measurements, we find the minimizer of adiscrepancy function involving a simulation(using Newton’s method to explore parameter space)
Requires no special assumptions of geometry or material
In practice, convergence depends on accuracy of initial estimates
Need an accurate model of the essential physics, and an efficientnumerical simulation method
Away from the walls of the generator the plasma flow is modeled with thecompressible Euler equations, while the electromagnetic variables are governed byMaxwell’s equations (with appropriate initial and boundary conditions)
ρ (∂t − u · ∇) u = j× b−∇p Conservation of Momentum
u = velocity e = electric fieldb = magnetic flux density T = thermal energyρ = mass density ρc = charge densityj = current density p = plasma pressureσ = conductivity β = Hall parameterε = (electric) permittivity µ = (magnetic) permeability
Low Magnetic Reynolds NumberAdvection relatively unimportant, magnetic field should relax to adiffusive state.
The system is in equilibriumAll time derivatives are 0. We will lift this assumption after proof ofconcept.
The induced fields are very smallrelative to the applied field, which we denote b0, therefore the plasmaresponds primarily to the applied field.This decouples the induced fields from the fluid flow.
The applied field is constantthroughout generator channel: b0 = [0, 0, b0]T
We choose a heuristic profile for u for now. Eventually a hydrostatic, orhydrodynamic system depending on b0 and j would be solved.
By Helmholtz decomposition, as ∇× e = 0 then e = ∇ψ, where ψ is theelectric (scalar) potential. Using the above consistency condition we havethe following electrostatic problem.
By Helmholtz decomposition, as ∇× e = 0 then e = ∇ψ, where ψ is theelectric (scalar) potential. Using the above consistency condition we havethe following electrostatic problem.
One of the important issues is to numerically maintain the ∇ · B = 0(conservation of magnetic flux) condition, from Maxwell’s equations, toavoid any unphysical effects. Therefore, we discretize with Mimetic FiniteDifferences (MFD) using a technique developed by K. Lipnikov, et al.
MFD are a generalization of Yee-type staggered differences to generalgeometry.
Difference operators are defined in terms of the FundamentalTheorem of Calculus, Divergence Theorem, and Stokes’ Theorem.
MFD describe a compatible discrete calculus which preserves standardrange and kernel theorems.
Range(∇) = Kernel(∇×) Range(∇×) = Kernel(∇·)
Stability is inherited from the existence of a discrete HelmholtzDecomposition.
There are several features of an arc which we wish to be able to detectand quantify. In order to determine the feasibility of a simulation-basedparameter estimation, we first test the sensitivity of the measurementsto
jm: Total current in the system
θ: Tilting of the current due to the Hall effect
s: (Spread of) the distribution of current density.
We assume a parameterized current density profile j which has thesefeatures: