Thermal Response of Materials to a High Energy Radiation He - PowerPoint Presentation

Slide1

Thermal Response of Materials to a High Energy Radiation Heat Source

Dr. Matthew C. Carroll

Texas A & M University at Galveston

2011 Thermal and Fluids Analysis Workshop

Passive Thermal Session #1

Newport News, Virginia, August 2011Slide2

What are we doing?

The Big Picture

Also, many thanks to:

Chris Kostyk

and

Stephen Miller

our Session Chairs.Slide3

Main Objectives

Formulate a volumetric heat generation term that accurately models the energy deposition in a solid structural material resulting from a high-energy radiation heat source.

Develop a solution for the temperature distribution in this material for the cylindrical and spherical geometries resulting from this energy deposition and resulting heat generation.Slide4

Overall Model – A DiagramSlide5

Other PossibilitiesSlide6

Solutions depend on ...

Geometry

Rectangular (Plane Wall)

Cylindrical

Spherical

Toroidal

Heat Generation Modeling

Surface Heating

VEDED (Volumetric Exponentially Decaying

Energy Deposition)Slide7

What solutions are available?

Rectangular

(Plane Wall)

Cylindrical

Concave/Convex)

Spherical

Concave/Convex

Surface

Heating

Textbook

Textbook

Textbook

VEDED

Multi-Group

Straightforward

Analytical

Solution

Finite Difference

Finite DifferenceSlide8

What solutions are proposed today?

Rectangular

(Plane Wall)

Cylindrical

Concave/Convex)

Spherical

Concave/Convex

Surface

Heating

Textbook

Textbook

Textbook

VEDED

Multi-Group

Straightforward

Analytical

Solution

Analytical

Series

Solution

Analytical

Series

SolutionSlide9

Why is multi-group modeling necessary?

Taken from W. H. McMaster, et. al., Lawrence Radiation Laboratory, Livermore, CaliforniaSlide10

What use are series solutions?

Since a convergent series can be evaluated to arbitrary precision, series solutions are essentially exact solutions that can be used to benchmark finite difference solutions.

Series solutions are relatively easy to program and can thus be used in the preliminary analysis that traditionally accompanies concept design.Slide11

Series Solutions

Cylindrical / Spherical Geometries

ProcedureSlide12

Four Cases

Concave Cylindrical Geometry

Inside radius faces radiation source.

Convex Cylindrical Geometry

Outside radius faces radiation source.

Concave Spherical Geometry

Inside radius faces radiation source.

Convex Spherical Geometry

Outside radius faces radiation source.

Derivation is done for concave cylindrical geometry. Other cases are similar.Slide13

(1) Partition Radiation Spectrum

For a spectrally distributed radiation source, a partitioning of the radiation spectrum should be made into N individual groups, with each group in a certain wavelength interval. For groups

i

= 1 through J, the mass attenuation coefficient is small enough to necessitate modeling as a volumetric heat generation term, with

m

i

= mass attenuation coefficient for group

i

P

i

= power fraction: fraction of total power in group

i

For groups J + 1 through N, the attenuation coefficient is large enough that the group can be assumed to cause surface heating.Slide14

Slide15

(2) Develop Governing Equation

General Heat Conduction Equation, Cylindrical Coordinates

Steady State, Constant Thermal Conductivity

Axisymmetric

, Neglecting Axial EffectsSlide16

(3) Formulate Heat Generation Term

Energy flux at point r in the wall due to group

i

:

where

q = total radiation flux (W/m

2

)

P

i

= power fraction group

i

r

1

= inner radius of wall

m

i

= mass attenuation coefficient group

iSlide17

Conduct energy balance on arbitrary cylindrical shell of thickness

dr to obtain heat generation from group i

:

Volumetric heat generation term from all groups:

Key step! Expand exponential into

MacLaurin

series:Slide18

Regroup terms and reverse order of summation to get:

Or in simpler form:Slide19

(4) Establish Boundary Value Problem

The boundary value problem with the heat generation term can then be formulated as:Slide20

Note the Following

A and the a

n are as defined before.

At the inner boundary (facing the radiation) r = r

1

there is a Neumann condition corresponding to the radiation energy groups with high attenuation coefficients.

At the outer boundary (bordering the coolant) r = r

2

there is a

Dirichlet

condition corresponding to the coolant temperature plus a temperature difference due to convective heat transfer. This depends on the coolant configuration.Slide21

(5) Solve Boundary Value Problem

By superposition, the solution T(r) can be considered to be of the form

where

T

1

(

r

) is the solution of

T

2

(

r

) is the solution ofSlide22

and the

φn(r) are solutions of

All three of these differential equations can be easily solved to obtain the following general solution:

Applying boundary conditions to solve for

C

1

and

C

2

and rearranging terms, we obtain the final solution for the concave cylindrical case.Slide23

Final Solutions

Cylindrical Geometry

Final solution for concave case is:

Similarly, final solution for convex case is:Slide24

Final Solutions

Spherical Geometry

Final solution for concave case is where s = -1 and final solution for the convex case is where s = 1, where

sr

2

<

sr

< sr

1

with r

1

the radial coordinate of the surface on the radiation side and r

2

the radial coordinate of the surface on the coolant side:Slide25

Numerical

Implementation

and ResultsSlide26

The equations look worse

than they are ...

“It must be 10 feet tall,” he said,

“And big, and fat, and bad, and red.

And it can bite, and kick, and kill,

and it will do it, yes it will!”

From

A Fly Went By

, Dr.

Suess

But seriously, they are not hard to program.Slide27

For example: this is the code for the subroutine that calculates temperatures for the convex cylindrical case (in FORTRAN 90).

DOUBLE PRECISION FUNCTION TEMP5(R)

DIMENSION F(100), U(100)

DOUBLE PRECISION F,K,Q,R,R1,R2,T0,U,UEFF

DOUBLE PRECISION AN,ARG,FAC,RATIO,RSET,SUM

COMMON F,J,K,NLAST,Q,R1,R2,T0,U,UEFF

SUM = 0.0

FAC = 1.0

RATIO = R/R1

DO 60 N = 1, NLAST

AN = N

FAC = FAC * AN

ARG = -UEFF*R2

RSET =R2**AN*DLOG(RATIO)+R1**AN/AN-R**AN/AN

60 SUM = SUM+QB*UEFF**AN*DEXP(ARG)*RSET/FAC

TEMP4=T0+R2/K*(DLOG(RATIO)*SUM)

RETURN

ENDSlide28

Sample Calculation:

Corrugated Wall

Fusion Plasma Chamber

A typical corrugated wall design consists of

a 1.0-mm

thick corrugated stainless steel panel bonded to a

3.0-mm thick stainless steel backing plate. The material used is usually PCA (“Primary Candidate Alloy”), a type 316 stainless steel with about 70%,

16 – 18% chromium, 10 – 14% nickel, and small amounts of molybdenum, manganese, silicon, and carbon.

A four-group partition is used for the radiation spectrum. The two low-energy groups are modeled as surface heating. The high-energy groups have P

1

= 10% with

m

1

= 3.552 cm

-1

and P

2

= 40% with

m

2

= 43.61 cm

-1

, respectively.Slide29

Total wall power loading is about 77 W/cm

2 and distributed over a large range of wavelengths. There is also about 23 W/cm2

cyclotron and charged particle radiation which is low energy and modeled as surface heating.

Coolant in the corrugated wall channels is at about 2000 psig with 280

o

C and 320

o

C inlet and outlet temperatures. A 50

o

C temperature rise across the coolant-wall interface is a good estimate for most conditions so the boundary condition becomes

T = 370

o

C at the outer surface.

Plane wall, concave, and convex solutions are all applicable for this situation.Slide30

Diagram of Corrugated WallSlide31

Results: Temperature Profiles

Concave, Planar, Convex Wall SectionsSlide32

Conclusions

The series solutions work fine and accurately predict temperatures in materials subjected to high-energy radiation heat sources.

The series solutions are fairly easy to program. It can be somewhat complex to apply the boundary conditions for the various cases.For the alternating series solutions, additional precision may be needed, as the individual terms get very large before getting small and yielding the final temperature.Slide33

Acknowledgements

Work funded by Startup Grant #169310 from Texas

A & M Galveston.

NASA, for their strong encouragement for me to continue these investigations in the midst of several other research projects and areas.

Dr. Vijay Panchang, Head, Department of Marine Systems Engineering and Marine Engineering Technology, for collecting and developing a team of researchers; for encouraging and mentoring them; for transforming the department into one optimally balanced between teaching, research, and service.