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Dr. Matthew C. Carroll Texas A & M University at Galveston 2011 Thermal and Fluids Analysis Workshop Passive Thermal Session #1 Newport News, Virginia, August 2011

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What are we doing? The Big Picture Also, many thanks to: Chris Kostyk and Stephen Miller our Session Chairs.

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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.

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Overall Model – A Diagram

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Other Possibilities

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Solutions depend on... Geometry Rectangular (Plane Wall) Cylindrical Spherical Toroidal Heat Generation Modeling Surface Heating VEDED (Volumetric Exponentially Decaying Energy Deposition)

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What solutions are available? Rectangular (Plane Wall) Cylindrical Concave/Convex) Spherical Concave/Convex Surface Heating Textbook VEDED Multi-Group Straightforward Analytical Solution Finite Difference

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What solutions are proposed today? Rectangular (Plane Wall) Cylindrical Concave/Convex) Spherical Concave/Convex Surface Heating Textbook VEDED Multi-Group Straightforward Analytical Solution Analytical Series Solution

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Why is multi-group modeling necessary? Taken from W. H. McMaster, et. al., Lawrence Radiation Laboratory, Livermore, California

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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.

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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.

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(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 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.

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(2) Develop Governing Equation General Heat Conduction Equation, Cylindrical Coordinates Steady State, Constant Thermal Conductivity Axisymmetric, Neglecting Axial Effects

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(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 i = mass attenuation coefficient group i

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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:

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Regroup terms and reverse order of summation to get: Or in simpler form:

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(4) Establish Boundary Value Problem The boundary value problem with the heat generation term can then be formulated as:

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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.

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(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 of

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and the φ n (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.

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Final Solutions Cylindrical Geometry Final solution for concave case is: Similarly, final solution for convex case is:

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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:

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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.

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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 60SUM = SUM+QB*UEFF**AN*DEXP(ARG)*RSET/FAC TEMP4=T0+R2/K*(DLOG(RATIO)*SUM) RETURN END

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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 1 = 3.552 cm -1 and P 2 = 40% with 2 = 43.61 cm -1, respectively.

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Total wall power loading is about 77 W/cm 2 and distributed over a large range of wavelengths. There is also about 23 W/cm 2 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.

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Diagram of Corrugated Wall

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Results: Temperature Profiles Concave, Planar, Convex Wall Sections

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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.

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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.

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Chapter 2 HEAT CONDUCTION EQUATION

Chapter 2 HEAT CONDUCTION EQUATION

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