CEE-233

Advanced Air Pollution Control and Engineering

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Lecture 7: Electrical Properties

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A charged particle makes an electric field at every location in space except its own location

(a) At any of the eight marked spots around a positive point charge +q, a positive test charge would experience a repulsive force directed radially outward. (b) The electric field lines are directed radially outward from a positive point charge +q.

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Millikan Oil drop experiment

Aerosol electrometer

Transmission Electron Microscope particle samplers

Simple electrical mobility analyzer

Electrostatic precipitator

Differential Mobility Analyzer (DMA)

Unipolar aerosol charger

Charging depends on the ion product

Bipolar charger

• Bipolar diffusion charging (exposing aerosol particles to both positive and negative ions) is used to bring aerosols to stationary state charge distributions independent of their initial charge state
• Different ionization methods have been used to generate bipolar ions for aerosol charging, e.g., corona ionization, radioactive sources, and soft X-ray.
• Bipolar corona neutralizers often suffer from the difficulty of maintaining a balance of positive and negative ions; formation of contaminant particles and ozone was reported for corona neutralizers
• Radioactive sources (e.g., ²¹⁰Po, ⁸⁵Kr, and ²⁴¹Am) are most commonly used to produce bipolar ions because of their simplicity and effectiveness
• Safety regulations on using and transporting radioactive sources promoted the search for alternative ionization methods, such as soft X-ray

Equilibrium charge distribution after bipolar charging

Differences in chargers

• For 210Po neutralizers with an initial radioactivity of 18.5 MBq (0.5 mCi), stationary state charge distributions are achieved when the source is less than 3.25 years old (residual activity no less than 0.0527 MBq).
• Stationary state was achieved for 85Kr neutralizers having residual radioactivity greater than 70 MBq.
• Aerosol charge fractions remain reasonably invariant over a wide range of flow rates.
• The positive charge fractions achieved by the soft X-ray neutralizer are higher than those by the other five neutralizers using radioactive sources while negative charge fractions for all neutralizers studied are all in a similar range

https://www.tandfonline.com/doi/full/10.1080/02786826.2014.976333

Static triangular DMA transfer function (Knutson and Whitby, 1975)

Correction for particle diffusion

Particle loss inside a DMA

Standard DMA setup

Size distribution measurement

SMPS Transfer Function

Generalized Problem: Fredholm Integral Equation

where K(q,s) is the kernel. The inverse problem is to find f(s) for a given K(q,s) and g(q).

Second Example: Aerosol size distribution measurement with a scanning mobility particle sizer

V = step or scan

CPC

Neutralizer

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DMA

V = constant (sets z^s)

Neutralizer

Ambient Aerosol Size Distribution

Mobility Classified Distribution

DMA transfer function

Charge distribution

Size distribution

DMA

V = constant (sets z^s)

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Neutralizer

CPC

The CPC measures the integral counts of the transmitted distribution

DMA

V = scanning (sets z^s)

Neutralizer

Ambient Aerosol Size Distribution

Response function

CPC

Generalized Problem: Fredholm Integral Equation

where K(q,s) is the kernel. The inverse problem is to find f(s) for a given K(q,s) and g(q).

Can we infer the aerosol size distribution from measurements of R(z^s)?

Solving the Fredholm Integral Equation by discretization into matrix form (e.g. by quadrature)

where �x = [x₁, …, x_i] = f(s_i) is a discrete vector representing f(s),�b = [b₁, ..., b_j] = g(q_j) is a vector representing g(q) and �A is the i×j design matrix.

Where N is a vector of number concentration in discrete size bins

Where the matrix A subsumes

• Charge distribution
• DMA Flow Rates
• DMA Thermodynamic State
• DMA Geometry
• DMA Transfer Efficiency

Forward model in matrix form

Forward model in matrix form + random error

Inverting the Fredholm Integral Equation through the matrix inverse (regular least squares solution)

where �x = [x₁, …, x_i] = f(s_i) is a discrete vector representing f(s),�b = [b₁, ..., b_j] = g(q_j) is a vector representing g(q) and �A is the i×j design matrix.

This inverse model suffers from noise amplification. Sometimes the simple inverse is good enough.

Inverting the Fredholm Integral Equation through Tikhonov Regularization

L₂ Regularization

• L is a smoothing matrix
• λ is an optimized parameter
• x0 is an optional a-priori estimate

Inverting the Fredholm Integral Equation through Tikhonov Regularization

L₂ Regularization

• L is a smoothing matrix
• λ is an optimized parameter
• x0 is an optional a-priori estimate

Analytical Solution!

Repeat calculation to get an ensemble of solution. Finding the optimal λ through semi-empirical metrics.

L-curve Method�

• The optimal λ occurs at the corner of the resulting L-curve.
• The corner can be found through algorithmic search.

Using the regularized inverse with optimal λ filters the noise