Download - CIS/ME 794Y A Case Study in Computational Science & Engineering 1 A Case Study in Computational Science & Engineering: Supersonic flow of ionized gas through

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CIS/ME 794Y A Case Study in Computational Science & Engineering

A Case Study in Computational Science & Engineering:

Supersonic flow of ionized gas through a nozzle

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Course Objectives• To provide insight and understanding of issues and

difficulties in computational modeling through a quarter-long case-study.

• Evaluate relative merits of various methods using programs you have written and pre-developed software modules.

• To understand the differences in performance when computations are done serially versus in parallel.

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Introduction to Case Study• Many engineering applications and processes involve reacting

and plasma flows.

• Some important examples:

– plasma processes in manufacturing of integrated circuits (etching, deposition)

– manufacturing processes (welding, coatings, synthesis of novel materials)

– space propulsion (positioning and station-keeping of satellites)

– Gas lasers, wind-tunnel test facilities, nozzles/shock tubes for studying chemistry

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Manufacturing of Semiconductor Devices

A trench 0.2 m wide by 4 m deep in single-crystal Si, produced by plasma etching (from Lieberman & Lichtenberg)

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Plasma welding

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Diamond deposition using a plasma arcjet

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Diamond growth on silicon using an oxy-acetylene flame

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• There are numerous examples of reacting flows in industrial applications:

– Energy generation & conversion: combustion processes

– Automotive engines

– Gas Turbine engines

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Motivation for modeling

• Gives detailed insight and basic understanding into the problem

• Helpful for design, control and optimization; can identify improved geometries for reactors, scale-up, etc.

• Availability of detailed experimental measurements enable in-depth understanding of cause-effect relationships (important for process control).

• Helpful in interpreting system or sensor response (e.g.. Ionization probe), and experimental data

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Existing modeling tools• “Canned” programs exist:

– Fluent, Fidap, StarCD, Chemkin, etc.

• Why write one’s own code?– greater flexibility

– speed (canned codes trade off speed for user-friendliness), and most importantly

– ability to model additional phenomena

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• This quarter’s case study will focus on an illustrative example involving a supersonic flow in a nozzle with ionization & recombination processes.

• This case study is intended to help bring out issues related computational modeling of a prototypical engineering problem, using high-performance computing methods.

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OSU supersonic afterglow wind-tunnel

Supersonic afterglow of Nitrogen over a wedge

Supersonic afterglow of Helium over a wedge

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Case Study Problem• Argon gas flows through a converging-diverging channel of

known cross sectional area

• Given:

– upstream total pressure, total temperature, and channel geometry

– desire supersonic flow in the diverging portion of the channel

• Find:

– distributions of velocity, density, pressure, temperature, Mach number, electron density, and ionization fraction throughout the channel, at steady state/transient state.

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Case Study Problem

Argon gas flow

2A

A

throat

inlet 4

A

A

throat

exit

Po=1 atmTo=300 K

L = 1 m

Adiabatic walls, i.e. no heat flow

CONVERGING-DIVERGINGOR CD NOZZLE Zone of heat

addition

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Background• Flow of a gas at high speeds such as in CD

nozzles, is characterized by changing density, .

• the mass density, , of a gas can change due to temperature changes or pressure changes. When changes because of pressure changes, the flow is called a compressible flow.

• To illustrate some of the basic characteristics of such a flow through a varying area channel, we begin by with a quasi one-dimensional (quasi 1-D) model of steady flow

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Quasi 1-D steady flow• Quasi 1-D that flow varies in the streamwise,

i.e. flow direction only, and transverse variations are ignored.

• Steady , i.e., no time variation.

• Governing equations, i.e. rules that govern such a flow are conservation of mass (or continuity), conservation of linear momentum, conservation of energy and species number density.

0t

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Governing equations for quasi 1-D flow

dx

(x)

ni(x)

P(x)

T(x)

u(x)

A(x)

(x+dx)

ni(x+dx)

P(x+dx)

T(x+dx)

u(x+dx)

A(x+dx)x x+dx

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Conservation of mass

0uAxA

1

t

or

Adxt

dxuAx

uA

Adxt

uAuA

or

mmm

x

dxxx

storedoutin

At steady state, we have ttanconsmuA

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Conservation of momentum

x

PAu

xA

1u

t2

Conservation of energy

QuAP2

uT

1

R

xA

1

2

uT

1

R

t

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Species conservation (electrons) 3

erAefeee nknnknuAn

xA

1

t

n

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Equation of State

RTP

Definition of Density AnenAmAnAmeneminim

Unknowns:, u, P, T, ne, and nA

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Governing equations for quasi 1-D steady flow

ttanconsmuA

0dx

duu

dx

dP

3eAefee nrknnknuAn

dxd

A1

RTP

AnenAm

Q2

uT

1

R

xu

2

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Governing equations for quasi 1-D, steady, adiabatic, frictionless, compositionally frozen

flow

ttancons2

u1

RT 2

ttanconsmuA

0dx

duu

dx

dP

0uAndxd

e

RTP

AnenAm

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Choking condition• The foregoing equations can be combined to yield:

where M=u/(RT)1/2 is the Mach number based on the isentropic speed of sound

• Note that when M=1, dA/dx must be zero in order for there to be smooth acceleration through M=1.

• Further, for M<1, du/dx>0 for dA/dx<0• Similarly, for M>1, du/dx>0 for dA/dx>0

1M

dx

dA

A

1

dx

du

u

12

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Implications of the choking condition

• Subsonic nozzles are supersonic diffusers:

• Subsonic diffusers are supersonic nozzles:

M < 1Flow

accelerates M > 1Flow

decelerates

M < 1 Flowdecelerates

M > 1Flow

accelerates

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Two-dimensional flow• Conservation of mass

• Conservation of momentum

• Conservation of energy

• Species conservation

Navier-Stokes Equations

x

y

Download - CIS/ME 794Y A Case Study in Computational Science & Engineering 1 A Case Study in Computational Science & Engineering: Supersonic flow of ionized gas through

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