ML21148A289

International - Response to Request for Additional Information, Holtec Spent Fuel Pool Heat Up Calculation Methodology Topical Report

ML21148A289
Person / Time
Site:	99902086
Issue date:	05/28/2021
From:	Sterdis A Holtec
To:	Document Control Desk, Office of Nuclear Reactor Regulation
Shared Package
ML21174A040	List: ML21148A289
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Download: ML21148A289 (50)

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Holtec Technology Campus, One Holtec Blvd, Camden, NJ 08104 Telephone (856) 797-0900 Fax (856) 797-0909 Page 1 of 2 May 28, 2021 U.S. Nuclear Regulatory Commission ATTN: Document Control Desk Washington, DC 20555-0001 Docket No. 99902086 - HDI Spent Fuel Pool Heatup Calculation Methodology

Subject:

Response to Request for Additional Information - Holtec Spent Fuel Pool Heat Up Calculation Methodology Topical Report

References:

1. Letter from Holtec International to US NRC, Holtec Spent Fuel Pool Heat Up Calculation Methodology Topical Report, September 29, 2020 (ML20280A524)

2. US NRC Electronic Mail Request to Andrea Sterdis (HDI) Formal Transmittal of the US NRC Requests for Additional Information for Holtec Topical Report HI-2200750 Revision 0, Holtec Spent Fuel Pool Heat Up Calculation Methodology, March 31, 2021 (ML21077A102)

Dear Sir or Madam:

In Reference 1, Holtec Decommissioning International, LLC (HDI) submitted a Topical Report providing a methodology for calculating Spent Fuel Pool heat up for NRC review and approval. Holtec believes the methodology will be a large benefit in reducing zirconium fire risks in the spent fuel pool.

In Reference 2, the NRC transmitted a request for additional information (RAI) concerning the Topical Report.

The following Enclosures to this letter provide a response to the NRC RAI.

(submitted separately via the BOX) provides a proprietary version of the RAI response. This enclosure contains information proprietary to Holtec and is therefore supported by an affidavit signed by Holtec which is provided in Enclosure 3.

provides a non-proprietary, redacted version of the RAI response.

If you have any questions, please contact me at 856-797-0900 ext. 3813.

Sincerely, Andrea L. Sterdis VP, Regulatory and Environmental Affairs Holtec Decommissioning International HOL TEC INTERNATIONAL Digitally signed by Andrea Sterdis DN: cn=Andrea Sterdis, c=US, o=Holtec Decommissioning International, ou=HDI, email=a.sterdis@holtec.com Date: 2021.05.28 16:24:25 -04'00' Andrea Sterdis

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HOL TEC INTERNATIONAL Holtec Response to Request for Additional Information concerning Spent Fuel Pool Heat Up Calculation Methodology Topical Report Proprietary Version Withhold Information From Public Disclosure Under 10 CFR 2.390 (19 Pages Submitted Separately)

Holtec Response to Request for Additional Information concerning Spent Fuel Pool Heat Up Calculation Methodology Topical Report Redacted Version (19 Pages Attached)

HOLTEC PROPRIETARY INFORMATION Page l 1

RAI-01

Treatment of near-wall locations

[ from t

Th [PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

e a

.] 4.a, 4.b Regulatory Justification SRP Section 15.0.2, Subsection III.3c Associated Section 3.1.4 Initial and Boundary Conditions Level of Concern 2

Level of Impact 5

Level of Effort 3

Overall Significance Medium Holtec Response

[It is

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

.] 4.a, 4.b Therefore, the methodology proposed by Holtec does improve the safety of the spent fuel storage.

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HOLTEC PROPRIETARY INFORMATION Page l 2

RAI-02

Lumped Analysis vs. Pin by Pin Analysis

[

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b Regulatory Justification SRP Section 15.0.2, Subsection III.3b Associated Section 3.3.1.4 Level of Detail in the Model Level of Concern 1

Level of Impact 3

Level of Effort 1

Overall Significance High Holtec Response

[

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b The methodology in the main part of the report will be expanded as follows:

[

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o I

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HOLTEC PROPRIETARY INFORMATION Page l 3

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b o

The paper shows that the method is validated through measurements,

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b x

The approach is considered conservative due to the following reasons:

[

HOLTEC PROPRIETARY INFORMATION Page l 4

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b This is further discussed in the response to RAI-03.

HOLTEC PROPRIETARY INFORMATION Page l 5 Reference for Response to RAI-02.

[2.1]

Manteufel, R.D. and N.E. Todreas, Effective Thermal Conductivity and Edge Configuration Model for Spent Fuel Assembly, Nuclear Technology, Vol. 105, pp. 421-440, March 1994.

HOLTEC PROPRIETARY INFORMATION Page l 6

RAI-03

Radial and Axial Peaking

[

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b Regulatory Justification SRP Section 15.0.2, Subsection III.3b Associated Section 3.3.1.4 Level of Detail in the Model Level of Concern 2

Level of Impact 2 Level of Effort 1

Overall Significance High Holtec Response

[

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b (see for example the HI-STAR 100 Storage SAR [3.1],

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HOLTEC PROPRIETARY INFORMATION Page l 7 tables 2.1.3 and 2.1.4, and the SER on the initial submittal of this SAR

[3.2], Section 4.3). [

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b For PWR fuel, axial burnups were also extensively analyzed in support of Burnup Credit for spent fuel transportation casks, as documented in NUREG/CR-6801 [3.3], with results documented in Table 5 of that document. [Proprietary Information Withheld Per 10 CFR 2.390.] 4.a, 4.b Note that for lower burnup, the NUREG reports slightly higher values, up to about 1.215 (for burnups between 14 and 18 GWd/mtU). [Proprietary Information Withheld Per 10 CFR 2.390.]

4.a, 4.b For BWR fuel, similar studies were performed and are documented in NUREG/CR-7224 [3.4]. For high burnup assemblies, results are shown in that NUREG in Figure 6.3, with maximum values generally no more than 1.2. [

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

HOLTEC PROPRIETARY INFORMATION Page l 8

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

HOLTEC PROPRIETARY INFORMATION Page l 9

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

HOLTEC PROPRIETARY INFORMATION Page l 10

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b References for Response to RAI-03

[3.1]

HI-STAR 100 Final Safety Analysis Report, Holtec Report HI-2012610, Rev. 0, March 2001

[3.2]

NRC Safety Evaluation Report and CoC, Holtec HI-STAR 100 Cask System, April 1999

[3.3]

Recommendations for Addressing Axial Burnup in PWR Burnup Credit Analyses, NUREG/CR-6801, ORNL/TM-2001/273, Oak Ridge National Laboratory, March 2003.

HOLTEC PROPRIETARY INFORMATION Page l 11

[3.4]

Axial Moderator Density Distributions, Control Blade Usage, and Axial Burnup Distributions for Extended BWR Burnup Credit, NUREG/CR-7224, ORNL/TM-2015/544, Oak Ridge National Laboratory, August 2016.

[3.5]

Horizontal Burnup Gradient Datafile for PWR Assemblies, DOE/RW-0496, Office of Civilian Radioactive Waste Management, May 1997.

[3.6]

Westinghouse Technology Systems Manual, Section 2.2, Power Distribution Limits, USNRC HRTD, Rev. 0508, ML11223A208

HOLTEC PROPRIETARY INFORMATION Page l 12

RAI-04

Time Step Sensitivity

[

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

. ] 4.a, 4.b Regulatory Justification SRP Section 15.0.2, Subsection III.3d, Appendix K to 10 CFR 50, and TMI

{Three Mile Island] action items for PWR Associated Section 3.3.2.1 Numerical Solutions & 3.3.5.4 Sensitivity Studies Level of Concern 3

Level of Impact 3 Level of Effort 4

Overall Significance Low Holtec Response [

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b 4.a, 4.b I

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HOLTEC PROPRIETARY INFORMATION Page l 13

RAI-05

Planar Surface Area

[

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b Regulatory Justification SRP Section 15.0.2, Subsection III.3e Associated Section 3.3.5.1 Important Sources of Uncertainty Level of Concern 3

Level of Impact 3 Level of Effort 4

Overall Significance Low Holtec Response [

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b I

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HOLTEC PROPRIETARY INFORMATION Page l 14

RAI-06

Uncertainty due to emissivity

[

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

]4.a, 4.b Regulatory Justification SRP Section 15.0.2, Subsection III.3e Associated Section 3.3.5.1 Important sources of Uncertainty Level of Concern 3

Level of Impact 3 Level of Effort 3

Overall Significance Low Holtec Response Surface emissivities are significantly affected by surface layers on the cladding (crud usually increases emissivity); therefore, the assumed oxidation layer and any exposed zircaloy surfaces are assumed to have the emissivity resulting from MATPRO Equation 4.1-8 [2] (equal to 0.8 or higher) using the oxidation thicknesses from [1]. Furthermore, Table B-3.II of [2] also shows an emissivity of fuel cladding with crud well over 0.8.

Therefore, use of an emissivity of 0.8 for zircaloy cladding is conservative.

Emissivity of stainless-steel plates that are used for the rack cell walls is 0.587 per ORNL studies [3] and [4]. The variation in emissivity of stainless-steel with temperature is extremely small (~ 0.05) in large temperature range as shown in reference [5].

Moreover, it must be noted that the emissivity values of 0.8 and 0.587 for zircaloy cladding and stainless-steel plates, respectively, have been approved by USNRC in multiple Holtecs dry storage applications (USNRC Docket Nos. 72-1014, 72-1032, 72-1040, 71-9325, 71-9367, 71-9373, 71-9374, etc.). NRC staff further mentions in their SERs (Section 3.2 on Docket Nos. 71-9367, 71-9374) that the material properties and surface emissivities used in these applications are acceptable.

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HOLTEC PROPRIETARY INFORMATION Page l 15 The variances in emissivity can alter the radiation heat transfer characteristics of the surfaces and therefore change the peak cladding temperatures. However, as noted in Section 4.2.7 of [1], the impact of emissivity variations on the peak cladding temperature (PCT) is extremely small. As a defense-in-depth, Holtec also performed sensitivity evaluations [

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

] 4.a, 4.b

References:

[1] Spent Nuclear Fuel Effective Thermal Conductivity Report, US DOE Report BBA000000-01717-5705-00010 REV 0, (July 11, 1996).

[2] Hagrman, Reymann and Mason, MATPRO-Version 11 (Revision 2) A Handbook of Materials Properties for Use in the Analysis of Light Water Reactor Fuel Rod Behavior, NUREG/CR-0497, Tree 1280, Rev. 2, EG&G Idaho, August 1981.

[3] Nuclear Systems Materials Handbook, Vol. 1, Design Data, ORNL TID 26666.

[4] Scoping Design Analyses for Optimized Shipping Casks Containing 1-,

2-, 3-, 5-, 7-, or 10-Year-Old PWR Spent Fuel, ORNL/CSD/TM-149 TTC-0316, (1983).

[5] Process Heat Transfer, D.Q. Kern.

HOLTEC PROPRIETARY INFORMATION Page l 16

RAI-07

Quality Assurance Program In the topical report, Holtec did not discuss the quality assurance program which controlled this analysis. Holtec should confirm that this [Proprietary Information Withheld per 10 CFR 2.390] 4.a, 4.b analysis is kept under a quality assurance program consistent with 10 CFR Part 50 Appendix B that this program contains adequate documentation for design control, document control, software configuration control and testing, and corrective actions, and that the analysis has been independently peer reviewed. Additionally, Holtec should confirm that the important references which the analysis method rely upon have been incorporated into Holtecs quality assurance program.

Regulatory Justification SRP Section 15.0.2, Subsection III.3f Associated Section 3.3.6.1 Appendix B Quality Assurance Program Level of Concern 3

Level of Impact 3 Level of Effort 2

Overall Significance Low Holtec Response The analysis developed for this topical report was developed, reviewed and approved under the Holtec Quality Assurance (QA) Program. The Holtec QA Assurance Program addresses the 10 CFR 50, Appendix B requirements and provides for appropriate design control, document control, software configuration control and testing, and corrective actions. The topical report and the supporting analysis are maintained under the Holtec QA Program.

When the methodology is approved and then is applied to a plant specific spent fuel pool, the site specific calculations will be performed in accordance with the sites Quality Assurance Program.

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HOLTEC PROPRIETARY INFORMATION Page l 17

RAI-08

Comparison to Office of Research (RES) Data

[

] 4.a, 4.b Please provide a plot similar to that given in Figure 7.3 with these comparisons Regulatory Justification SRP Section 15.0.2, Subsection III.3d Associated Section 3.3.3.2 Validation of the Evaluation Model Level of Concern 3

Level of Impact 5 Level of Effort 3

Overall Significance Low Holtec Response Figure 7.3 in the TR has been expanded to show data up to 6 months for BWR fuel assemblies. The revised figure compares data from the method proposed in the TR to data from calculations done by the Office of Research (RES) starting from 6 months of cooling time. The conclusions made in the TR still remain applicable that the proposed method shows conservative results under all configurations for BWR fuel assemblies

[Proprietary Information Withheld per 10 CFR 2.390.] 4.a, 4.b A similar figure has been added to the TR for PWR fuel assemblies.

Response to RAI-08 Revised Figure 7.3

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

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HOLTEC PROPRIETARY INFORMATION Page l 18 4.a, 4.b

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

HOLTEC PROPRIETARY INFORMATION Page l 19

RAI-09

Variation in Heat Capacity

[

] 4.a, 4.b Regulatory Justification SRP Section 15.0.2, Subsection III.3b Associated Section 3.3.1.4 Level of Detail in the Model Level of Concern 3

Level of Impact 3 Level of Effort 4

Overall Significance Low Holtec Response Heat capacity of fuel assemblies is an input to the calculations. [

] 4.a, 4.b

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

[PROPRIETARY INFORMATION WITHHELD PER 10 CFR 2.390]

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Affidavit for Withholding (5 Pages Attached)

U.S. Nuclear Regulatory Commission ATTN: Document Control Clerk Non-Proprietary Enclosure 3 AFFIDAVIT PURSUANT TO 10 CFR 2.390 Page 1 of 5 I, Andrea L. Sterdis, being duly sworn, depose and state as follows:

1) I have reviewed the information provided in the RAI responses provided in which is sought to be withheld, and am authorized to apply for its withholding.

2) The information sought to be withheld is in Enclosure 1 to the May 27, 2021 letter to NRC providing Responses to Request for Additional Information -

Holtec Spent Fuel Pool Heat Up Calculation Methodology Topical Report. The responses contain information that is proprietary to Holtec International.

3) In making this application for withholding of proprietary information of which it is the owner, Holtec International relies upon the exemption from disclosure set forth in the Freedom of Information Act (FOIA), 5 USC Sec. 552(b)(4) and the Trade Secrets Act, 18 USC Sec. 1905, and NRC regulations 10 CFR Part 9.17(a)(4), 2.390(a)(4), and 2.390(b)(1) for trade secrets and commercial or financial information obtained from a person and privileged or confidential (Exemption 4). The material for which exemption from disclosure is here sought is all confidential commercial information, and some portions also qualify under the narrower definition of trade secret, within the meanings assigned to those terms for purposes of FOIA Exemption 4 in, respectively, Critical Mass Energy Project v. Nuclear Regulatory Commission, 975F2d871 (DC Cir. 1992),

and Public Citizen Health Research Group v. FDA, 704F2d1280 (DC Cir. 1983).

U.S. Nuclear Regulatory Commission ATTN: Document Control Clerk Non-Proprietary Enclosure 3 AFFIDAVIT PURSUANT TO 10 CFR 2.390 Page 2 of 5

4) Some examples of categories of information which fit into the definition of proprietary information are:

a. Information that discloses a process, method, or apparatus, including supporting data and analyses, where prevention of its use by Holtecs competitors without license from Holtec International constitutes a competitive economic advantage over other companies;

b. Information which, if used by a competitor, would reduce his expenditure of resources or improve his competitive position in the design, manufacture, shipment, installation, assurance of quality, or licensing of a similar product.

c. Information which reveals cost or price information, production, capacities, budget levels, or commercial strategies of Holtec International, its customers or its suppliers;

d. Information which reveals aspects of past, present, or future Holtec International customer-funded development plans and programs of potential commercial value to Holtec International;

e. Information which discloses patentable subject matter for which it may be desirable to obtain patent protection.

The information sought to be withheld is considered to be proprietary for the reasons set forth in paragraphs 4.a and 4.b, and 4.c above.

5) The information sought to be withheld is being submitted to the NRC in confidence. The information (including that compiled from many sources) is of a sort customarily held in confidence by Holtec International, and is in fact so held. The information sought to be withheld has, to the best of my knowledge and belief, consistently been held in confidence by Holtec International. No public disclosure has been made, and it is not available in public sources. All disclosures to third parties, including any required transmittals to the NRC, have been made, or must be made, pursuant to regulatory provisions or proprietary agreements which provide for maintenance of the information in confidence. Its

U.S. Nuclear Regulatory Commission ATTN: Document Control Clerk Non-Proprietary Enclosure 3 AFFIDAVIT PURSUANT TO 10 CFR 2.390 Page 3 of 5 initial designation as proprietary information, and the subsequent steps taken to prevent its unauthorized disclosure, are as set forth in paragraphs (6) and (7) following.

6) Initial approval of proprietary treatment of a document is made by the manager of the originating component, the person most likely to be acquainted with the value and sensitivity of the information in relation to industry knowledge. Access to such documents within Holtec International is limited on a need to know basis.

7) The procedure for approval of external release of such a document typically requires review by the staff manager, project manager, principal scientist or other equivalent authority, by the manager of the cognizant marketing function (or his designee), and by the Legal Operation, for technical content, competitive effect, and determination of the accuracy of the proprietary designation. Disclosures outside Holtec International are limited to regulatory bodies, customers, and potential customers, and their agents, suppliers, and licensees, and others with a legitimate need for the information, and then only in accordance with appropriate regulatory provisions or proprietary agreements.

8) The information classified as proprietary was developed and compiled by Holtec International at a significant cost to Holtec International. This information is classified as proprietary because it contains detailed descriptions of analytical approaches and methodologies not available elsewhere. This information would provide other parties, including competitors, with information from Holtec Internationals technical database and the results of evaluations performed by Holtec International. A substantial effort has been expended by Holtec International to develop this information. Release of this information would improve a competitors position because it would enable Holtecs competitor to copy our technology and offer it for sale in competition with our company, causing us financial injury.

9) Public disclosure of the information sought to be withheld is likely to cause substantial harm to Holtec Internationals competitive position and foreclose or reduce the availability of profit-making opportunities. The information is part of

U.S. Nuclear Regulatory Commission ATTN: Document Control Clerk Non-Proprietary Enclosure 3 AFFIDAVIT PURSUANT TO 10 CFR 2.390 Page 4 of 5 Holtec Internationals comprehensive decommissioning and spent fuel storage technology base, and its commercial value extends beyond the original development cost. The value of the technology base goes beyond the extensive physical database and analytical methodology, and includes development of the expertise to determine and apply the appropriate evaluation process.

The research, development, engineering, and analytical costs comprise a substantial investment of time and money by Holtec International.

The precise value of the expertise to devise an evaluation process and apply the correct analytical methodology is difficult to quantify, but it clearly is substantial.

Holtec Internationals competitive advantage will be lost if its competitors are able to use the results of the Holtec International experience to normalize or verify their own process or if they are able to claim an equivalent understanding by demonstrating that they can arrive at the same or similar conclusions.

The value of this information to Holtec International would be lost if the information were disclosed to the public. Making such information available to competitors without their having been required to undertake similar expenditure of resources would unfairly provide competitors with a windfall, and deprive Holtec International of the opportunity to exercise its competitive advantage to seek an adequate return on its large investment in developing these very valuable analytical tools.

U.S. Nuclear Regulatory Commission A TfN: Document Control Clerk "lon-Proorietary Enclosure 3 AFFIDAVIT PURSUANT TO 10 CFR 2.390 STATE OF SOUTH CAROLINA )

)

ss::

COUNTY OF RICHLAND J Andrea L. Sterdis, being duly sworn, deposes anci says:

That she has read the foregoing affidavit and the matters stated therein are true and correct to the best of her knowledge, information, and belief.

Executed at Blythewood, South Carolina, thi~ 7 day of May 2021.

~~~

Andrea L. Sterdis Holtec Decommissioning International Holtec International VP, Regulatory & Environmental Affairs Subscribed and sworn before me this ;;J t day of _(h

__ a----j'jr-------

2021

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Reference 2.1 (RAI Response RAI-02)

Effective Thermal Conductivity and Edge Conductance Model for a Spent-Fuel Assembly (20 Pages Attached)

EFFECTIVE THERMAL CONDUCTIVITY AND EDGE CONDUCTANCE MODEL FOR A SPENT-FUEL ASSEMBLY HEAT TRANSFER AND FLUID FLOW KEYWORDS: heet transfer, spent fuel, effective thermal conductivity RANDALL D. MANTEUFEL Southwest Research Institute Center for Nuclear Waste Regulatory Analyses, 6220 Culebra Road San Antonio, Texas 78238-5166 NEILE. TODREAS Massachuseus Institute of Technology Department of Nuclear Engineering, 77 Massachusetts A venue Cambridge, Massachusetts 02139 Received December 7, 1992 Accepted for Publication July 21, 1993 An effective thermal conductivity (keff) and an edge thermal conductance (hedge) model are developed for the interior and edge regions of a spent-/ uel assembly residing in an enclosure. The model includes conduc-tive and radiative modes of hear transfer. Predictions using the proposed ke11l hedge model are compared with five sets of experimental data for validation. The model is compared with predictions generated by the engine maintenance, assembly, and disassembly (E-MAD) and Wooton-Epstein correlations, which represent the state of the art in this field. The model is applied to a typical pressurized water reactor and a typical boiling water re-actor spent-fuel assembly, and a set of both nonlinear and linear formulations of the model are derived. The proposed model is based on rigorous models of the gov-erning heat transfer mechanisms and can be applied to a large range of assembly and enclosure types, enclo-sure temperatures, and assembly decay heat values. The proposed model is more accurate than comparable lumped correlations and is more amenable/or simple, repetitive design applications than other detailed nu-merical models.

I. INTRODUCTION In a typical transportation or storage cask, each spent-fuel assembly resides in a square enclosure that is backfilled with a nonoxidizing gas (usually helium or nitrogen).1-4 For design purposes, it is desirable to have a simple yet accurate method to predict the maximum fuel rod temperature in a spent-fuel assembly in these casks. Regulations pertinent to spent-fuel shipping and storage require that the spent-fuel cladding be protected NUCLEAR TECHNOLOGY VOL. 10S MAR. 1994 from degradation (10CFR7l.43d for transportation casks5 and 10CFR72.122h for storage casks6). The degradation of spent-fuel cladding has been investigated and found to be accelerated by prolonged exposure in an oxidizing environment at elevated temperatures.7*9 Hence, the casks are typically backfilled with a nonox-idizing gas, and the maximum temperature is main-tained below the design limit of -380°C. Frequently, casks are designed to efficiently dissipate the decay heat of the spent fuel so that this design goal is achieved with relatively large margins of safety, especially for spent fuel with long cooling periods.

The U.S. Department of Energy (DOE) is sponsor-ing the development of a new fleet of spent-fuel ship-ping casks. 10,11 One goal of the new fleet is to improve safety, as well as increase shipping payload, for the large number of fuel shipments expected to occur in the 21st century. Designers are encouraged to develop in-novative designs that are based on technically def en-sible engineering methods. Consistent with these goals, analysis methods have continually been reviewed and updated to improve accuracy.

From a recent survey of preliminary shipping cask design reports, l-4 it was found that a substantial frac-tion of the total cask temperature drop (from maximum fuel temperature to ambient environment tempera-ture, i.e., ATc = Tm -

T00 ) is predicted to occur over a single assembly (from maximum fuel temperature to enclosure wall temperature, i.e., AT0 = T,,, - Tw),

The survey is summarized in Table I, where it is noted that the temperature drop for a single assembly ranges from 15 to 310Jo of the total temperature drop (i.e.,

AT0 / ATc)- This is considered a significant fraction of the total cask temperature drop.

In addition, the prediction of AT0 has relatively large uncertainties because the most frequently used method to calculate A T0 is the Wooton-Epstein corre-lation.13 The Wooton-Epstein correlation was originally 421

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY TABLE I A Summary of Parameters and Methods Currently Being Used to Predict the Temperature Drop Due to a Single Assembly (AT0 ) for Current Design Spent-Fuel Shipping Casks*

AT0 Cask Contractor Cask Payload Qa T,n b AT/

AT, d

-t:..T, Cask Type PWR/ BWR Gas (W) coq (OC)

(OC)

(0/o)

Method Babcock & Wilcox Rail 21/52 Helium 576 185 50 160 31 WE*

BR-100 (Ref. I)

Helium 837 234 64 210 31 WE General Atomic GA-4/GA-9 (Ref. 4)

Truck 4/9 N2 552 187 43 160 27 kefl r Westinghouse Titan (Ref. 3)

Truck 3/7 Helium 580 217 27 192 15 E-MADs Nuclear Assurance NAC (Ref. 2)

Rail 26/52 Helium 506 242 41 217 20 WE

• Updated from Ref. 12.

"Q== total assembly decay heat.

bT,,,== calculated maximum assembly temperature.

ct:.. T0== temperature drop associated with an assembly.

dt:,.T,== total temperature drop for cask.

cwE== Wooton-Epstein correlation. 13 rk,.11 = effective conductivity model.

gE-MAD = E-MAD correlation. 14 developed 30 yr ago, assuming only nitrogen backfill (while helium is now predominantly being used), assum-ing turbulent natural convection (where recent experi-mental evidence suggests laminar natural convection),

and for much higher assembly decay heats (8 com-pared with 0.5 kW). Hence, a study of heat transfer within a spent-fuel assembly was conducted and is re-ported herein.

Many investigations have been reported on mod-eling heat transfer in a spent-fuel assembly since the work of Wooton and Epstein. 15-21 However, these models were not used in the recent transportation cask preliminary design reports 1-4 (see Table I for the models used). In addition, a large amount of recent modeling work is available to designers.22* 24 An obvious ques-tion is the following: Why has the 30-yr-old Wooton-Epstein correlation continued to be used in light of more recent work? Also, what will be the unique con-tribution of a new method compared with approaches already available in the literature? The answer involves (a) the inherent uncertainties in the physical problem and (b) the analysis cost associated with a method.

From the discussion of each of these issues, the moti-vation for the development of the proposed model is made apparent.

I.A. Uncertainties A number of significant uncertainties are associated with heat transfer within a spent-fuel assembly: (a) fuel rod emissivity, (b) enclosing wall emissivity, (c) net as-422 sembly decay power, (d) axial decay peaking factor, and (e) backfill gas composition.12 Specifically, the current state of knowledge suggests that each of these quantities have uncertainties ranging from at least I 0 to as much as 25%. For example, the spent-fuel rod emissivity is expected to have a value between 0.6 and 1.0 with a probability of 950Jo (where the mean is 0.8 and one standard deviation is 0. 1), which represents a 25% uncertainty about the typically assumed value of 0.8 (Ref.. 25). The large range of emissivity values ap-pears primarily to be the result of differences in the ex-perimental results from five separate investigations using different states of oxidation (e.g., uniform ox-ide, nodular oxide, or oxide with crud). If radiative heat transfer is the dominant heat transfer mechanism (as it is for nitrogen backfill with T> 200°C), then the pre-diction of tiT0 should be expected to have an ~250Jo uncertainty due to the uncertainty in the emissivity value. This is a significant uncertainty that would over-whelm most approximation errors introduced in the development of more exact theoretical models. lt is pre-sumed that the pragmatic cask designers recognize these inherent uncertainties and see the inappropriateness of performing more exact theoretical analyses than the knowledge of input parameters supports.

1.8. Simple Methods Coupled with uncertainties in data, the task of per-forming analyses and developing designs strongly sug-gests using simple, conservative models coupled with NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY

~

Edge region Fig. 2. Homogeneous idealization of an enclosed rod array that includes both an interior and an edge region.

Edge\\

Wall iiJ p p/2 Fig. 3. Idealization of an enclosed assembly where the as-sembly is smaller than the enclosure and is assumed to reside in the center of the enclosure.

is neglected here, yet it will be discussed briefly in Sec.

III.A. The combined conductive and radiative conduc-tivity yields the total effective conductivity:

k eff = kcond + k,ad *

(1)

The conductive component is discussed in Sec.

II.A. I, and the radiative component is discussed in Sec.

11.A.2.

II.A.I. Stagnant-Gas Conduction The effective conductivity of a composite medium consisting of infinitely long tubes residing in a medium has been considered in the literature.29*30 The effective stagnant-gas conductive conductivity is frequently re-lated to the backfill gas conductivity through a conduc-tion factor:

(2) where kcond = effective conductivity assuming the back-fill gas is stagnant Fcond = conduction factor k gas = backfill gas conductivity.

424 The problem is to calculate Fcond as a function of tube array pattern (square or hexagonal), characteris-tic geometrical data (rod diameter, rod-to-rod pitch, cladding thickness), and conductivities (kgas, kctad, and k1ue1). A set of analytic formulas for the effective con-ductivity has been derived analytically.30 A relatively simple analytic formula has been found to be accurate for pitch-to-diameter (pld) ratios near typical nuclear fuel assembly values (but not for small pld ratios such as found in consolidated applications where more ac-curate formula are required). for illustration purposes, the simple formula is presented here:

F.

I -Jv, cond -

J + f V 1

(3) where f = volume fraction of the tubes (or rods) v1 = coefficient depending on the tube dimensions and conductivities.

For a square array of rods, the volume fraction is given by 11' f = 4(pld) 2 '

(4) which is approximately equal to 0.444 for pld = 1.33.

Representative values for the conductivities are kfuel = 5 W/ (m

• 0C), k c1ad = 15 W/ (m

• 0C), and k gas =

[0.2 W/ (m

• 0 C) for helium and 0.04 W/(m

• 0 C) for ni-trogen]. In practice, a gap exists between the fuel and the clad, and a thermal resistance to heat transfer is typically associated with this gap (i.e., gap thermal con-ductance). The value of the gap conductance is consid-ered to be uncertain; hence, the conductivity of the fuel is frequently neglected (kfuet = 0.0) in the calculation of Fcond* This underestimates the effective conductiv-ity of the medium and leads to a higher estimate of the maximum temperature. The effect of this approxima-tion has been compared to other approximations (i.e.,

assume the fuel has a conductivity equal to the gas, or assume negligible thermal resistance across the gap),

and the differences were found to be insignificant.

The formula for the coefficient, assuming kfuel =

0, is

(,. )2 Ogas-clad+

~

v, =

( '1 )2 '

I + Ogas-clad r o where Ogas-clad= (kgas -

kc1ad) /(kgas + kc1ad) r; = inner radius of the clad wall r 0 = outer radius of the clad wall.

(5)

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY large factors of safety. This appears the most rational path that has been chosen by the designers surveyed in Table I. It is interesting to note that the highest maxi-mum temperature reported in Table I is 242°C, which is significantly below the design goal of 380°C. This represents a significant factor of safety. Although com-puter programs are available to more accurately pre-dict !lT0 (assuming the input parameters are known with precision), there is currently no need to design close to the design limit, which would justify incurring the costs associated with acquiring, porting, learning, and running a complex code. In the future, the maxi-mum temperature in actual cask loadings due to larger payloads or less-aged spent fuel may approach 380°C, and it may then be expected that cask designers would aggressively pursue more accurate thermal analysis methods.

In review, an objective of this work is to improve the state of the art in predicting the maximum temper-ature in a single spent-fuel assembly because of its im-portance to the overall fl T for a cask. The calculation method currently pref erred by cask designers was de-veloped for a range of applications different than cur-rent conditions (especially higher decay power levels and predominantly helium backfill). In addition, al-though newer methods have been developed and pub-lished in the literature, they have not been adopted by cask designers. This is thought to be due to the in-creased complexity associated with the newer methods (1000+ line computer codes compared with a single al-gebraic equation) and the fact that these approaches are not supported by precise knowledge of many input pa-rameters. Hence, this work was motivated to develop a new, theoretically based method for design use that is sufficiently accurate yet acceptably simple.

II. EFFECTIVE CONDUCTIVITY AND EDGE CONDUCTANCE MODEL A typical pressurized water reactor (PWR) or boil-ing water reactor (BWR) fuel assembly has a small cross-sectional dimension (-15 to 21 cm square) in comparison with the axial length (-370 cm). Because of the small width-to-length ratio, the spent-fuel assem-blies are typically modeled as two-dimensional cross sections. A two-dimensional cross section of an 8 x 8 array in a square enclosure is illustrated in Fig. 1. A typ-ical BWR has an 8 x 8 array enclosed in a channel that is part of the fuel assembly, and a typical PWR has a 15 x 15 array but does not have a channel as part of the assembly.

For spent fuel, the spatial distribution of decay heat is generally described as being cross-sectionally uniform and nearly uniform along the long axis. No known sources specifically account for cross-sectionally vary-ing decay heats; however, a number of computer codes appear to have this capability. In comparison, the ax-ial variation in decay heat has been considered more im-NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 Fig. I. Cross-sectional view of an enclosed 8 x 8 rod array.

portant than the cross-section variation. Typically, a two-dimensional model is assumed (as shown in Fig. 1),

a uniform decay heat (which is larger than the average decay heat) is assumed, and axial heat transfer is ne-glected. The maximum axial decay heat is assumed to be larger than the average decay heat by a factor of ei-ther Fpeak = 1.1 (Refs. I, 26, 27, and 28) or Fµeak = 1.2 (Refs. 2 and 3).

A theoretical study of the conductive, convective, and radiative heat transfer mechanisms in an enclosed rod array has been completed, 12 and the pertinent re-sults are summarized here, especially the application of the theory to develop an effective thermal conductiv-ity (keff) and edge conductance (hedge) model for a spent-fuel assembly. Jn this paper, four distinct formu-lations of the ke11lhedge model are discussed: the con-tinuum form (Sec. II.A), the lumped form (Sec. 11.B),

the nonJinear algebraic form (Sec. JV.A), and the linear algebraic form (Sec. IV.B). The forms are progressively simpler, yet each form has additional assumptions or approximations in their derivation. The continuum form is the most general and is discussed next.

II.A. Continuum k.ttlhu,. Model A spent-fuel assembly is modeled as having two pri-mary regions: the interior region and an edge region, as shown in Fig. 2. The interior region is modeled as a homogeneous medium that has macroscopic thermal properties that reflect the net effects of the detailed thermal phenomena. An effective thermal conductiv-ity model is developed for the interior region that ac-counts for both conductive and radiative heat transfer.

An edge conductance model is developed for the edge region that is also based on models of both conductive and radiative heat transfer. The edge region extends from the enclosure wall to one-half the pitch from the outer row of rods, as illustrated in Fig. 3.

Two modes of heat transfer are considered within the interior of an assembly: stagnant-gas conduction and thermal radiation. From experimental evidence re-ported in the literature, heat transfer due to natural convection can be neglected in the storage and trans-port of spent fuel for the vast majority of cases of in-terest. For notational convenience, natural convection 423

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY The ratio of radii can be related to the outer fuel rod diameter and the cladding thickness:

where t

...!.= I 'o d '

t = cladding thickness d = rod outer diameter.

(6)

Representative values for PWR and BWR assem-blies are d = 10.7 mm for PWRs and 12.2 mm for BWRs, and t = 0.615 mm for PWRs and 0.813 mm for BWRs (Refs. 31 and 32). Substitution of these values into Eqs. (6), (5), and (3) yields 2.11 PWR with helium 2.48 PWR with N2 F'c*ond = 2.16 BWR with helium (7) 2.49 BWR with N2 The difference between the values of Fcond due to either the PWR or BWR is considered negligible in comparison to the difference due to either helium or nitrogen backfill. Hence, the distinction between PWR and BWR geometries is frequently neglected, and Fcond is taken simply as a function of the backfill gas (assuming a square array pattern, pld = 1.33, represen-tative cladding thicknesses, representative rod diam-eters, and representative values of the gas and cladding conductivities):

F cond = {

2.1 with helium 2.4 with N2 (8)

These simple results are used in Sec. IV to analyze a typical PWR and a typical BWR assembly. More ac-curate estimates of Fcond are used in the comparison of the model predictions with experimental data in Sec. II I.

11.A.2. Thermal Radiation The radiative conductivity model for a square ar-ray is derived by considering radiative heat transfer in the one-dimensional symmetry section shown in Fig. 4.

The radiative transfer across the surface (q:'ad) is estab-lished as a function of the discrete rod temperatures, rod emissivity, and geometric information (e.g., d and p). Each rod is assumed to be isothermal, which is an approximation that can be justified for spent-fuel rods. 12 The radiative transfer between rod surfaces is related to the radiative absorption factors between col-umns of rods (because of the symmetry surfaces, each rod in Fig. 4 represents an infinite column of rods). The net radiative heat flux through the surface is NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994 Tsurl Symmetry q(ad __i.

C9_@_e_0l~_@_@_(;;)

Symmetry

-.i 14--

I I

p I

p Fig. 4. One-dimensional symmetry section used in the deri-vation of the radiative conductivity in a square array.

ll

+ "

+

ll ll Qrad- * *

• Q(i-2)-(i+I)

Q(i-1)-(i+I) + Q(i-l)- (i+2)

+ Qi-(i+ I) + qf'_(i+2) + Qi-(i+3) +

where (9) q:Od = radiative flux across the imaginary surface q[_i = radiative flux from column i to column j.

The radiative flux between columns of rods is given by where E,1rdG;-/J T4 T4 qf'_j = --~- ( i -

j ) '

p e, = rod emissivity (10)

G;-i = column-to-column radiative absorption factor T; = temperature of rod i.

The column-to-column absorption factors are similar to Hottel's gray-body view factors (also called Hottel's script-F), where E,Gi-i = F;-i* The calculation of the G;-i terms is accomplished by a Monte Carlo ray-tracing algorithm to calculate view factors and a matrix inversion algorithm to calculate absorption factors. 12 The radiative conductivity is related to the total radiative heat flux through a discrete form of the Fou-rier's law (which in effect defines k,ad for this problem):

0 AT Qrad = -krad -Ax (11)

The temperatures for each rod are approximated as varying linearly from the imaginary surface between columns i and i + I:

(12) where n assumes any integer value (e.g., -2, - 1, 0, 1, 2). The fourth power of temperature [as expressed in Eq. (10)] can be approximated using Eq. (12) [where (a - b)4 = a4 - 4ba3 + 6a2 b 2 -

4ab 3 + b4, and if b << a, then (a - b)4== a4 -

4ba3 ]:

425

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY 4

~

4

( 2n - 1) q'/ad 3

T;+n = Tsurf- -- p -

4Tsurf

• 2 k,ad (I 3)

Equation (13) can be substituted into Eqs. (9) and (10),

and through a series of algebraic manipulations, the ra-diative conductivity can be expressed as (14) where C,ad is the radiative coefficient. In Eq. (14), the "surf" subscript on temperature has been dropped since the radiative conductivity is based on the local temper-ature. The radiative coefficient is related to the column-to-column absorption factors [which are introduced in Eq. (10)] as Crad=1:,[G;-(i+1)+22ci-(i2)+... +n2Gi-(i+nd.

(15)

The value C,ad can be calculated numerically or ap-proximated analytically. The computation of Crad is not the primary focus of this paper; hence, the value of interest is simply stated. For a square array of rods, with pld = 1.33 and rod emissivity fr= 0.8, and the ra-diative coefficient has been calculated to be Crad = 0.4 (Ref. 12). For the calculations presented in Sec. lll, Crad was evaluated for different conditions.

ll.A.3. Edge Models Similar to the interior model, stagnant-gas con-duction and thermal radiation models are developed for the edge region. The edge is defined as the region from one-half the pitch from the outer row of rods to the en-closure wall, as illustrated in Fig. 3. A total edge con-ductance can be developed as consisting of a conductive and a radiative component:

(16) where hedge = edge conductance (or edge heat transfer coefficient) hco11d = wall conductive heat transfer coefficient h,ad = wall radiative heat transfer coefficient.

The development of hcond is straightforward alge-braic manipulation following the same principles as outlined for the interior region (Sec. II.A.2) (Ref. 12),

and the result is only summarized here:

h

_ Fcond, wk gas cond -

( I _ J/2) W,

(17) where Fcond, w = conduction factor for the wall 426 f = edge-to-interior heat transfer ratio w = distance from the center of the outer ring of rods to the wall.

The wall conduction factor is defined as

{

Fco11d,

w/p S !

Fcond, w =

Fcond w/p

, 1 J

I I

Wt p > 2 2 + Fco11r1( w/p -

2)

(18)

The wall radiative coefficient is developed with the aid of Fig. 5, where the column-to-wall radiative ab-sorption factors are introduced. Jn summary, the ra-diative coefficient was derived as:

h

_ Crad,w,2<nrd4(7j,w)3 rad -

(1 _ J/2)p (19) where Crad, w,2== second-wall radiative coefficient u = Stefan-Boltzmann constant [ = 5.67 x 10- 8 W/ (m2

• K4)]

T.r."' = wall "film" temperature.

The wall film temperature is T

- Te+ Tw J.w -

2 (20) and the edge-to-interior heat transfer ratio is.

(21) where Crod, w, 1 is the second-wall radiative coefficient.

The extrapolated temperature Te is used in Eq. (20) and represents the temperature of an imaginary surface located at a distance one-half pitch from the center of the outer row of rods. The wall temperature Tw is the temperature of the enclosure. These temperatures are also discussed in Sec. II.B.

The wall radiative coefficients are based on the column-to-wall radiative absorption factors:

Crad,w,I = f,[ G1-w + 3G2-w +... + (2n - I )G,,_wJ (22)

Fig. 5. One-dimensional symmetry section used in the deri-vation of the wall radiative heat transfer coefficient for a square array.

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY and Crad, w.2 = Er[ G1-w + G2-w + * * * + Gn-wJ (23)

The column-to-wall radiative absorption factors are cal-culated by using the same techniques as the column-to-column absorption factors (described in Sec. 11.A.2).

For a square array of rods with pld = 1.33, Er= 0.8, and Ew = 0.2, the coefficients were calculated to be Crad, w, 1 = 0.105 and Crad, w,2 = 0.085. The coefficients are calculated for different conditions in Sec. III.

11.B. Lumped k.nlh,d,. Model A more convenient form of the ke11l hedge model can be developed from the continuum form, in order to quickly estimate the maximum fuel temperature in an array. The heat diffusion equation is solved for the interior region of the assembly in order to develop the lumped formulation of the kef1l hec1ge model. Three distinct temperatures are considered in the lumped keJJlhedge model: the maximum temperature Tm, the extrapolated temperature Te, and the wall temperature Tw (see Fig. 6). The locations of Tm and Tw are consid-ered logical, while Te is located at an imaginary surface that is the extrapolated boundary of the interior region.

The nonlinear conduction equation is solved for the in-terior region assuming the heat generation is spatially uniform and the extrapolated wall temperature is cir-cumferentially uniform. The temperature dependence of the radiative component of the effective conductiv-ity can be solved (without additional approximations) using Kirchoff's transformation.

In general, the nonlinear heat conduction equation can be solved for square, hexagonal, or circular cross-sectional geometries, as shown in Fig. 7. These geom-etries are of interest because the square cross section is a common configuration for PWR and BWR assemblies, the hexagonal cross section is common for liquid-metal reactor assemblies and light water reactor assemblies from Eastern European countries, and the circular cross section may be applicable to consolidated assem-blies stored in a circular canister.

Fig. 6. Locations of the maximum, extrapolated, and wall temperatures.

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994 b

Square Hexagonal Circular Fig. 7. Square, hexagonal, and circular cross-sectional ge-ometries for which a conduction shape factor can be calculated [see Eq. (25)).

The solution of the heat diffusion equation can be expressed in a single equation valid for all three of the geometries:

where Q = total assembly decay power Fpeak = axial decay heat peaking factor (24) l 0 = thermally active axial length of the assembly S = geometry-dependent conduction shape factor.

The only term that depends on the geometry is the shape factor. For each geometry, the shape factor has been calculated 12:

{

13.5738 square S =

12.8365 hexagonal.

4.071' circular (25)

Hence, the lumped ke11lhedge model can be applied to the most probable fuel assembly geometries.

The lumped kef1 l hedge model can be expressed in two coupled algebraic equations where the first equa-tion applies to the assembly interior:

QFpeak _ F.

4 4

SLa condkgas(T,,, -

Te) + Crad<l7rd(Tm -

T e )

(26) and the second equation to the edge region within the enclosing wall:

Fcond, wkgas (T. _ T. )

(1 - f/2)w e

w QFpeak

-- =

C d 2<J7rd

+

ra. w,

( T4 _ T 4 )

(2?)

( I - f/2)p e

w Each of the variables in Eqs. (26) and (27) was previ-ously introduced in Sec. II.A. As presented, Eqs. (26) and (27) represent the lumped formulation of the k e11l hedge model.

427

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY Ill. EXPERIMENTAL VALIDATION The lumped ke11lhedge model predictions have been compared with five sets of experimental data in order to validate the proposed method. The five sets of data are from the engineering, maintenance, and disassem-bly (E-MAD) tests, 14 single assembly heat transfer tests26 (SAHTT), Ridihalgh, Eggers, and Associates (REA) tests,33 Sandia National Laboratories liquid-metal fast breeder reactor (SNL-LMFBR) tests,34 and Massachusetts Institute of Technology (MIT) 8 x 8 tests.35 The comparison of model predictions with ex-perimental data is discussed for each test.

Ill.A. The E*MAD tests A series of experimental tests were conducted at the E-MAD facility at the Nevada Test Site.14*36*37 Exten-sive temperature measurements were recorded for two standard 15 x 15 Westinghouse PWR spent-fuel assem-blies. The assemblies had average decay thermal pow-ers of ~ 1300 and 750 W, respectively. Each assembly was placed withln a cylindrical enclosure (called a can-ister) for a 2-to 3-yr testing period during which back-fill gas and canister temperatures were controlled. The backfill gases were intermittently changed to include he-lium, air (which is thermally equivalent to nitrogen),

and vacuum.

The vacuum pressure was reported to be 13.3 kPa

( ~ I 00 mm Hg) of absolute pressure, which is consid-ered to be a "weak" vacuum and insufficient to elimi-nate conduction in the backfill gas. The low pressure in the vacuum tests was, however, very effective at reducing the importance of natural convection (because the Rayleigh number scales as the total gas pressure squared for an ideal gas). In addition, it was estimated that the helium tests were conducted with ~80% helium and 20% air due to the relatively weak vacuum used to purge the air while changing to helium backfill. 12 The lumped ke11lhedge model [Eqs. (26) and (27))

was used to predict the maximum fuel temperature (ex-pressed as t:,.'I'a = T,11 -

Tw). The input parameters (i.e.,

canister temperature, assembly decay heat, backfill gas, fuel rod emissivity, and canister emissivity) were either obtained directly from the E-MAD report 14 or from other sources that analyzed the test data.27*28 The test data and the model predictions are com-pared in Fig. 8 for tests conducted with helium back-fill and in Fig. 9 for tests conducted with either vacuum or air backfill. As it can be noted, the model slightly overpredicts the maximum temperature, due in part to the conservative approximations introduced in the model, and conservative estimates of model parameters such as conduction factors and radiative coefficients.

The importance of natural convection can be seen in Fig. 9, where the air data yield lower values of t:,.T than the vacuum data, especially at lower canister tem-peratures. The lower values of t:,.Tare assessed to be due 428 140 120 E-MAD 15 x 15 PWR o Helium, Q = 1300 W 100 A Helium, Q = 750 W u 80 0

f..'."

<I 60 40 20 0

100 150 200 250 300 350 Canister temperature (°C)

Fig. 8. Comparison of lumped ke11lheage model predictions (solid lines) with experimental data from E-MAD tests with helium (80% helium and 200Jo air) backfill.

140 120 100 u 80 1-'"

Racri,,

(28)

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY where Ra = Rayleigh number (not defined here, see Ref. 12)

Rac,i, = critical Rayleigh number.

Below Rac,ir, the flow of the gas has a negligible effect on the value of T,111 so that the gas can be mod-eled as being stagnant. Above Racrir, buoyancy-driven gas flows are sufficiently strong to affect the value of T,n. Experimental work has been performed to esti-mate the critical Rayleigh number for a vertical, en-closed heated rod array.20 Based on the previous work, Rae,;, was estimated for the E-MAD geometry and thermal conditions and has been included in the theo-retical predictions. lt was found, however, that good agreement could be achieved by using a Rac,i, one de-cade larger than predicted by the correlation in Ref. 20, as shown in Fig. IO. Based on this value of Rau;1, it was determined that the Rayleigh number did not ex-ceed the critical value for helium backfill and exceeded the critical value for air by only one decade at T.v =

100°c.

As noted in Fig. 8, the difference between the air and vacuum data is distinguishable only at low canis-ter temperatures. This is due primarily to two trends:

(a) the Rayleigh number scales as Ra - 1/T4 for an ideal gas, and (b) the radiative conductivity scales as k,ad - T 3 [see Eq. (14)). Hence, as the bulk tempera-ture of the spent-fuel assembly increases, the Rayleigh number decreases, leading to a decrease in the impor-tance of convective heat transfer. In contrast, as the 140 120 100 G

0

<]

40 20 0

100 o Vacuum, Q = 1300 W

+ Air, Q = 1300 W

6. Vacuum, Q = 750 W x Air, Q = 750 W E-MAD 15 x 15 PWR 150 200 250 300 Canister temperature (°C) 350 Fig. JO. Comparison of lumped ke11lh,dge model predic-tions neglecting natural convection (solid lines) and including natural convection (dashed lines).

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994 bulk temperature increases, the importance of radia-tive heat transfer rapidly increases. The combined ef-fect is that natural convection becomes less important at higher temperatures, which is the area of primary de-sign interest (i.e., near 380°C).

Correlations between tlT0 and the assembly ther-mal power were developed in Ref. 14 for the E-MAD data and were used in the thermal analyses of spent-fuel storage40*41 and shipping3 casks. The correlations are for helium backfill:

!::.Ta= Q(l23.53 - 0.6325Tcan + 0.001202T;an) (29) and for vacuum/ air backfill:

tlTa = Q[172.6 X 10-(0.00119Tcanl ],

(30) where Q = assembly thermal power (kW)

Tcan = canister temperature (°C).

Equations (29) and (30) do not appear to have been de-veloped by using models of conductive or radiative heat transfer. Finally, the vacuum/air equation reported in Ref. 41 is apparently in error (where the constant 0.00179 was reported to be 0.0000179) and has been corrected in Eq. (30).

The E-MAD correlations are compared with the lumped ke11lhedge predictions in Figs. 11 and 12. The E-MAD correlations (as well as the lumped ke11l hedge model predictions) are in good agreement with the ex-perimental data. The only anomaly is that the E-MAD correlation for the helium backfill has a nonphysical increase in tlT0 as Tcan increases above 250°C. This is attributed to the quadratic nature of the correlation [see Eq. (29)). Neither the experimental data nor the basic models of heat transfer (especially k,ad - T 3 ) support this trend in the E-MAD correlation.

The relative importance of conductive, convective, and radiative heat transfer were compared by using the lumped ke11l hedge model. In the array interior, the heat transfer due to conductive transport ranged from 40 to 60% with helium backfill and from 20 to 400/o with air/vacuum backfill. The convective contribution ranged from a maximum of 200/o at Tcan = 100°C to 0% at Tcan = 300°C. At the edge, the heat transfer due to conductive transport was less important than in the array interior primarily because of the large void region created by a square assembly residing in a circular en-closure. At the edge, the heat transfer due to radiative transport ranged from 55 to 85% with helium backfill and from 75 to 90% with air/vacuum backfill. Over-all, the model predictions are considered to be in ex-cellent agreement with the experimental data.

111.B. The SAHTT Tests A series of SAHTT tests26 were conducted by using an electrically simulated, full-scale 15 x 15 PWR 429

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY 140...-........ -..----.---.-----,.-----,,--.-----,-....

120 100 u 80 0

1-"

<I 60 40 20 E-MAD 15 x 15 PWR o Helium, Q = 1300 W

b. Helium, Q = 750 W Lumped k6,,lhec1ge model

• - - E-MAD correlation 0._____._ _________....____._ __________ ___.

100 150 200 250 300 350 Canister temperature (°C)

Fig. 1 I. Comparison of the lumped kef/lhtt11, model predic-tions (solid lines) with the E-MAD correlation (dashed lines) for helium backfill.

140 120 100 u

0

<I 60 40 20 0

100 o Vacuum, Q = 1300 W

+ Air, Q = 1300 W

b. Vacuum, Q = 750 W x Air, Q = 750 W E-MAD 15 x 15 PWR lumped ke,,lhedge model

• • • E-MAD correlation 150 200 250 300 Canister temperature (0C) 350 Fig. 12. Comparison of the lumped ke11lhec18e model predic-tions (solid lines) with the E-MAD correlation (dashed lines) for vacuum and air backfill.

assembly. The assembly was enclosed in a fuel tube such as that found in storage and transportation casks.

Experimental tests were conducted by using three dif-ferent fill gases (helium, air, and vacuum), three axial orientations, and two power levels and with the canis-ter temperature controlled using guard heaters to re-430 1 00,---.---,---r----r----.-,-------.-.......-"""T""----,.........,

80

+ Vacuum o Air SAHTT 15 x 15 u 60 ll. Helium 0

1-'"

<I 40 20 0,___......_____.__..._....___.__...__....... __

0 200 400 600 800 1000 1200 Assembly power (W)

Fig. 13. Comparison of the lumped ke1/hedge model predic-tions (solid lines) with the experimental data from the SAHTT tests.

main at 200°C. As in the E-MAD tests, the lowest reported vacuum pressure was 3.8 kPa (29 mm Hg) of absolute pressure so that natural convection was essen-tially eliminated in the vacuum tests; however, stag-nant-gas conduction was not eliminated. The pertinent modeling parameters were either specified in the SAHTT report26 or taken from other sources that analyzed the data_21,2s The lumped kef1lhedge model predictions (based on a constant tube temperature of 200°C) are compared with the experimental data in Fig. 13. The vacuum data show a slightly larger 6 T0 than the air data, indicating the effects of natural convection. However, the data have enough scatter that the effects of natural convec-tion were assessed to be negligible in the enclosed tube bundle. Overall, there appears to be good agreement between the experimental data and the predictions. The ke11lhedge predictions in Fig. 13 were not "calibrated" to the experimental data (which would have improved the comparison) but are based on the best estimates of input parameters. An alternative explanation for the helium predictions being lower than the data is that the backfill gas was not IOOOJo helium but contained a sig-nificant portion of air.

The relative importance of conductive and radia-tive heat transfer were compared based on the lumped k e11lhedge model. In the array interior, the percentage of heat transfer due to conduction ranged from 20 to 250/o with air/ vacuum backfill and from 55 to 600Jo with helium backfill. At the enclosure wall, the percentage of heat transfer due to conduction was -45% with air/ vacuum and - 750Jo with helium backfill. Overall, the model predictions are in agreement with the SAHTT experimental data.

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY 111.C. The REA Tests A series of tests33 were conducted using an electri-cally simulated 61-cm (2-ft) section of an unconsoli-dated and a consolidated BWR. The consolidated tests were performed with two rod spacings (pld =I.OJ and 1.02). Both the unconsolidated and consolidated arrays were emplaced in a square enclosure. The pertinent ex-perimental details and parameters are taken from the REA report 33 and other sources that analyzed the data.42 The unconsolidated tests were conducted with air and vacuum backfill, and the experimental data and lumped ke11l hedge model predictions are compared in Fig. 14. The lumped k1111l hedge model overpredicts the experimental fl T0, as did other model predictions re-ported in Ref. 42. One possible explanation for the di f-f erence is that axial conduction may be responsible for end heat losses (as suggested in Ref. 42). The previous predictions and those reported herein are considered comparable, so that no further investigation of the dif-ference was pursued.

The lumped ke11l hedge model predicts that nearly 60% of the temperature drop occurred at the wall be-cause of a relatively large wall-to-pitch ratio, wlp =

1.45, and a low wall emissivity, ew = 0.25. Conduction accounts for ~ 20% of the total heat transfer in the in-terior of the assembly and ~25% at the wall. This in-dicates that radiative heat transfer was dominant in both the interior and edge regions (due primarily to a relatively high bulk temperature of 200°C and a low conductivity of the backfill).

The lumped ke11l hedge model predictions are com-pared with the consolidated data in Fig. 15 for both p/d values ( 1.0 J and 1.02), both backfill gases (air and he-120 REA 8 x 8 100 o Air, p/d = 1.33 80 t.. Vacuum, pld = 1.33 G

0

...... 60

<l 40 20 0

0 0

0 200 400 600 800 1000 1200 Equivalent array power (W)

Fig. 14. Comparison of the lumped k,.11l het1g, model predic-tions (solid line) with the experimental data from the unconsolidated REA tests.

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994 lium), and three power levels (100, 400, and 800 W).

For the consolidated tests, the tlT0 are considerably smaller than for the unconsolidated tests (-15 com-pared with ~50°C) for comparable power levels. This indicates that the effective conductivity for the consol-idated tests was - 3. 3 times higher than for the uncon-solidated tests. For air backfill, the model predictions are assessed to slightly overestimate the experimental data, yet be in good agreement. For the helium back-fill, the predictions underestimate the data, possibly be-cause of axial heat transfer or air being present in the helium backfill.

111.D. The SNL-LMFBR Tests Sandia National Laboratories conducted experi-ments using an electrically simulated 217-rod LMFBR fuel assembly.34 The rods in the LMFBR assembly were arranged in a hexagonal array with p/d = 1.24 be-ing maintained by axial wire wrappings around the rods. The rod array was enclosed within a hexagonal tube. Tests were conducted using helium backfill gas at power levels of 1000, 1250, and 1500 W.

The experimental data and the lumped ke11l hedge model predictions are compared in Fig. 16. The predic-tions consistently overpredict the experimental data yet appear to have the same trend of decreasing AT0 with 15 G

0 1-"' 10

<]

5 p/d = 1.02

___ p/d = 1.01 o

Air, pld = 1.02 a Air, pld = 1.01 x

Helium, pfd = 1.02

/::;.

Helium, pld = 1.01 REA consolidated 0 ""---t::,~_.___._

0 200 400 600 800 1000 Equivalent array power (W)

Fig. 15. Comparison of the lumped ke11l he,1ge model predic-tions (lines) with the experimental data from the consolidated REA tests.

431

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY SNL-LMFBR 100 80 u

0 i-" 60

+

+ +

<I 40 o Helium 1500 W 20

o. Helium 1250 W

+ Helium 1000 W 0 100 150 200 250 300 350 Wrapper temperature (°C)

Fig. 16. Comparison of lumped kefrlhedge model predic-tions (solid lines) with experimental data from the SNL-LMFBR tes1s.

increasing wrapper temperature. The overprediction appears to be primarily due to the neglect of the pres-ence of the wire wrapping used to maintain the tight pld in the LMFBR assembly. Overall, the data are prop-erly oriented below the model predictions considering the enhanced conduction effect of the wire, so that these data are useful in validating the lumped kefflhedge model.

111.E. The MIT 8 x 8 Tests A series of electrically heated rod experiments35 simulating a section (61 cm) of an 8 x 8 rod array was conducted at MIT. The rod array was enclosed in an aluminum box, and the backfill medium was intermit-tently changed to include air, nitrogen, and helium gas.

The lumped ke11lhedge model predictions are compared with the experimental data in Figs. 17 and 18 for the air / nitrogen and helium tests, respectively. Best esti-mates of the experimental parameters are used to gen-erate the solid lines (assuming 11 o/o of the input heat was conducted axially and not accounted for; see Ref. 35), and the upper and lower estimates are based on the assessed uncertainties in the experimental param-eters (shown as dash-dotted lines). The experimental data primarily fall within the bounds of predictions.

Two general trends are that the model underpredicts the experimental data at low power values (especially for the helium backfill) and overpredicts the data at high power values. The source of these trends, however, is not apparent, but it is thought to be due to tempera-ture measurement biases. Overall, the predictions are in good agreement with the MIT 8 x 8 experimental data.

432 160 120 u

0 i-" 80

<I 40 MIT 8 x 8 A T(air-11 %)

+ AT(Nr 11%)

A T(prediction)

--- t:.. T(upper)

• • • • • • AT(lower) 0 -t---r----.-....--r----.--,.--~-.----..--...--,---f 0

100 200 300 400 500 600 Power (W)

Fig. 17. Comparison of lumped ke11lhedge model predic-tions with experimental data from the MIT 8 x 8 tests with air and nitrogen backfills.35 160 120 u

0 i-" 80

<I 40 MIT 8 x 8

,,,,..,,,,,,.... P******

0,,J'~_::.:..... -********

J-'...-

o t:..T(helium-11 %)

-~---*****

aT(prediction) o,/f/

--- AT(upper)

• • • • • • t:.. T(lower) 0..--r----.-

..---r-~-....---.---..-....-.....----.---1 0

100 200 300 400 500 600 Power (W)

Fig. 18. Comparison of lumped ke11l hedge model predic-tions with experimental data from MIT 8 x 8 tests with helium backfill.35 IV. APPLICATION TO TYPICAL PWR AND BWR ASSEMBLIES The lumped ke11l hedge model can be applied to a typical PWR and a typical BWR assembly in order to generate simplified equations that can readily be em-ployed in the design of a cask. In particular, typical val-ues are assumed for a PWR and a BWR assembly with either helium or nitrogen backfill. These typical values can be inserted into the lumped kef1lhedge model

[Eqs. (26) and (27)] to generate two coupled nonlinear algebraic equations, which constitute the nonlinear al-gebraic form of the lumped kef1lhedge model (see Sec.

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY IV.A). For additional simplicity, the nonlinear equa-tions can be linearized to generate the linear algebraic form of the lumped k e11lhedge model (see Sec. IV.B).

Hence, two simplified forms (nonlinear and linear) of the lumped ke11lhedge model are presented in this section.

IV.A. Nonlinear Algebraic Form of the Lumped k,,,!h.,,,_ Model In practice, thermal analyses are performed by using characteristics for both a typical PWR and a typical BWR assembly.1*4 For the current problem, Eqs. (26) and (27) can be applied to either of these assemblies, with either nitrogen or helium backfill, by using values from a comprehensive survey of spent-fuel characteristics.31 Typical values of input parameters for spent-fuel assemblies include for geometric parameters: La =

3.66 m, p/d = 1.33, d = 0.0107 m for a PWR and 0.0122 m for a BWR, p = ( pld)d, Le= 0.854 m for a PWR and 0.556 m for a BWR, w/p = 1.0, w = ( w/p)p.

Typical values of input parameters for spent-fuel as-semblies include four model parameters: Fpeak = 1.2, Grad= 0.40 (for a square array, pld = 1.33 and Er=

0.8), Fcond = 2.1 for helium and 2.4 for N2, kgas = 0.2 W/ (m

• 0 C) for helium and 0.04 W/ (m

• 0 C) for N2, Crad, w.2 = 0.085, Fcond, w = 1.355 for helium and 1.412 for N2 assuming w/p = 1, and f = 0.45 for helium and 0.33 for N2

• These parameter values have been in-serted into Eqs. (26) and (27), leading to an interior equation:

and an edge equation:

Q = C3(Te - Tw) + C4(T: - T!).

(32)

The coefficients (C1, C2, C3, and C4 ) in Eqs. (31) and (32) have been calculated and are presented in Ta-ble II. Typically, the total assembly thermal power Q and the average enclosure waJI temperature Tw are specified as inputs, and the extrapolated wall temper-ature Te and maximum temperature Tm are to be cal-cu lated. For illustration purposes, Eqs. (31) and (32) are solved by using the nonlinear, algebraic equation solver available in a commercially available mathemat-ics program43 for the typical PWR with either nitrogen (Fig. 19) or helium backfill (Fig. 20).

In Fig. 19, the predictions for a typical PWR with nitrogen backfill are plotted for three wall temperatures (Tw = 100, 200, and 300°C) and for a range of assem-bly decay powers ( 100 < Q < 1500 W). Note that in-creasing the wall temperature decreases t:,.T0 because of the increased effectiveness of radiative heat transfer at higher temperatures. The curves for constant wall tem-perature are approximately linear at low assembly de-cay heats and are concave downward at the higher wall temperatures (this linearity is used in Sec. IV.B). For reference, the range of application for the current trans-portation cask designs (see Table I) is near Tw = 200°C and Q= 500 W, which yields tlTa = 40°C from Fig. 19.

Typical PWR 200.--............... --.....---r----r-~---r--.--,--.--,

180 160 140 cr 120 0

f-"' 100

<l 80 60 40 20 Square enclosure E, = 0.8 Ew = 0.2 d ::; 1.07 cm p/d = 1.33 w/p = 1.0 L = 3.66 m Fposk = 1.2 N2 backfill 0 L..._.__,L__.,__..i,__.__... __

0 250 500 750 1000 1250 1500 Assembly decay heat (W)

Fig. 19. Lumped ke1flhtt1,, model predictions of t:.T0 for a typical PWR assembly with nitrogen backfill.

TABLE U Coefficients in the Nonlinear Form of the Lumped kefflhet1ge Model* for a Typical PWR and a Typical BWR Spent-Fuel Assembly in Either Helium or Nitrogen Backfill

c.

C2 C3 C4 (W/K)

(10- 8 X W/ K4)

(WIK)

(10- 8 x W/ K4)

PWR with helium 17.38 3.16 64.0 3.83 PWR with N2 3.97 3.16 12.38 3.55 BWR with helium 17.38 3.60 36.54 2.49 BWR with N2 3.97 3.60 7.07 2.31

• Equations (31) and (32).

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994 433

Mantcufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY For higher fuel burnups and shorter cooling times, the assembly thermal power may exceed 1000 W; hence, the solutions are presented up to 1500 W.

Similarly, the results for helium backfill (compared with nitrogen backfill) are shown in Fig. 20. When com-paring Figs. 19 and 20, it is noticeable that helium significantly reduces the dT0 (or equivalently the max-imum temperature) because the thermal conductivity of helium is approximately five times larger than that of nitrogen.

The relative importance of conduction and radia-tion in both the interior region and the wall region of a typical PWR assembly are compared in Figs. 21 and

22. A general trend is that the radiation heat transfer becomes increasingly more important with increasing wall temperature. For the interior and edge regions of a typical PWR assembly, the general trends are as follows:

I. Conduction is more important when helium (rather than nitrogen) gas is used as the backfill.

2. The importance of conduction decreases as the wall temperature increases (likewise indicating an in-creasing importance of radiative heat transfer).

3. Conduction is more important at the edge of the array, compared with the interior of the array (based primarily on the input parameters: rod emissiv-ity Er= 0.8, enclosing wall emissivity Ew = 0.2, pitch-to-diameter ratio pld = 1.33, wall-to-pitch ratio wlp = 1.0).

4. The importance of conduction is not strongly in-fluenced by the assembly decay heat.

200 180 160 140 u 120

~

..,:' 100

<I 80 60 40 20 Typical PWR Square enclosure t,== 0.8 tw = 0.2 d = 1.07 cm p/d== 1.33 wlp== 1.0 L = 3.66 m Fpeak== 1.2 Helium backfill 0 L..-=~-~.....___.__.__...._~_..._.....___.___,

0 250 500 750 1000 1250 1500 Assembly decay heat (W)

Fig. 20. Lumped kef1l het1ge model predictions of 1l T0 for a typical PWR assembly with helium backfill.

434

5. Conduction accounts for 10 to 300/o of the to-tal heat transfer in the interior and 30 to 550/o at the edge of an enclosed PWR assembly with nitrogen backfill.

6. Conduction accounts for 55 to 7511/o of the to-tal heat transfer in the interior and 70 to 900Jo at the Typical PWR 1.0.----.---,----,.-....--.----r-..----.---,----,

l 0.8 Q = 500, 1000, 1500 W Radiation with helium

~ 0.6

~

Radiation with N2 Helium

-g 8

~ 0.4 Nitrogen Conduction with helium 0.2 Q== 500, 1000, Conduction 1500 W with N2 0.0 100 150 200 250 300 350 Wall temperature (°C)

Fig. 21. Relative importance of conduction and radiation heat transfer in the interior region of a typical PWR assembly with either helium or nitrogen backfill.

Typical PWR 1.0.----.---.----,.---.--.----.--

+ Radiation f with helium 0.8 0.2 Q = 500, 1000, 1500 W Radiation with N2 Nitrogen Conduction with N2

~Helium Conduction with helium Q = 500, 1000, 1500 W 0.0._____.__...____.,_......,__......_.....__..__......... _ _.___,

100 150 200 250 300 350 Wall temperature (°C)

Fig. 22. Relative importance of conduction and radiation heat transfer at the edge region of a typical PWR assembly with either helium or nitrogen backfill.

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY edge of an enclosed PWR assembly with helium backfill.

The temperature drop at the edge (Te - Tw) is com-pared with the total assembly temperature drop (T,n -

7;.,) in Fig. 23 for a typical PWR assembly. The gen-eral trends are as follows:

I. A significant fraction (25 to 50%) of the total temperature drop (T,11 -

T...,) is associated with the edge region (Te - T.,.,) for both helium and nitrogen backfill.

2. Nitrogen backfill leads to a larger fraction

(-45%) of the total temperature drop being associated with the edge region.

3. The fraction of the total temperature drop asso-ciated with the edge region is not significantly influ-enced by the value of the wall temperature.

4. Forty to fifty percent of the total assembly tem-perature drop is associated with the edge region for ni-trogen backfill.

5. Twenty-five to thirty-five percent of the total as-sembly temperature drop is associated with the edge re-gion for helium backfill.

The predictions generated by the lumped ke11lhedge model are compared with the Wooton-Epstein corre-lation 13 because the correlation has been and remains widely used in the industry (see Table I). The Wooton-Epstein correlation is similar to the lumped ke11lhedge equations in that it is a nonlinear algebraic equation.

For a typical 15 x 15 PWR assembly, the Wooton-Epstein correlation is I

I

~ 1-E -

1.0 Typical PWR 0.8 Interior t:.T with N2 Interior t:.T with helium 0.6 Nitrogen Q = 500, 1000, 4

1500 W 0.4

' I 0.2 Q = 500, 1000, 1500 W Wallt:.T Helium Wall t:. T with N2 with helium 0.0 100 150

~

200 250 Wall temperature (°C) 300 350 Fig. 23. Relative importance of the temperature drop asso-ciated with either the interior or wall region for a typical PWR assembly with either helium or nitro-gen backfill.

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994 (33) where q.. = heat flux at the enclosure surface (W/ m2)

C 1 = 0.234 (dimensionless)

C2 = 0.815 [W/ (m2 *K413)].

The Wooton-Epstein correlation is based on models of thermal radiation and turbulent convection (instead of stagnant-gas conduction) heat transfer. A concentric ring model for radiative heat transfer is used to relate the maximum temperature to the wall temperature. The convective coefficient [C2 in Eq. (33)) was determined from experimental data by using an electrically simu-lated 17 x 17 rod array with air backfill. 13 The 4/3 ex-ponent for (Tm -

Tw) in Eq. (31) indicates that the natural convection was assumed to be turbulent.

The lumped ke11lhedge predictions (solid lines) are compared with the predictions generated by using the Wooton-Epstein correlation (dashed lines) in Fig. 24.

As illustrated, the lumped ke11lhedge predictions agree with the correlation for the case of Tw = 200°C, which is approximately the range of experimental data used to generate the Wooton-Epstein correlation. 13 The so-lutions, however, increasingly differ for different wall u

0 1-'"

<l Typical PWR 160-~~---;..._---,,--~---.--.---.-----.----,

140 120 100 80 60 40 20 Square enclosure fr= 0.8 Ew = 0.2 d = 1.07 cm pld = 1.33 wlp = 1.0 L = 3.66 m Fpeak = 1.2 N2 backfill Lumped k8,1lhedge model

- - - Wooton-Epstein correlation o.._....____.____. ___......__ _ __,_....__.____._...___.___,

0 250 500 750 1000 1250 1500 Assembly decay heat (W)

Fig. 24. Comparison of the lumped k~11l hMg, model predic-tions (solid lines) and the Wooton-Epstein correla-tion (dashed lines) for a typical PWR assembly with nitrogen backfill.

435

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY temperatures. At Tw = 300°C, the Wooton-Epstein correlation overpredicts !J.T0, and at Tw = 100°C, it underpredicts !J.T0

• Finally, the Wooton-Epstein cor-relation [Eq. (33)] and the plot of its predictions (dashed lines in Fig. 24) were presented with errors in Ref. 12, and the errors have been corrected in Refs. 44 and 45 and in this paper.

In comparison, the lumped ke11lhedge method is based on a more rigorous model of the governing heat transfer mechanisms and is proposed to yield more ac-curate predictions. In addition, the lumped ke11l hedge model has a much broader range of application in that it can accommodate different backfill gases, different array cross-sectional geometries, both square and hex-agonal rod patterns, different rod diameters, different pitch-to-diameter ratios, and different wall-to-pitch ratios.

IV.B. Linear Algebraic Form of the Lumped **"Jh.,,,. Model Equations (26) and (27) may be simplified into the form of a single linear algebraic equation so that cal-culations may then be performed quickly without re-liance on a computer. The linear solution will be shown to be conservative by overestimating !J.T0 (hence over-estimating the maximum temperature). The linear equa-tions are generated by linearizing the difference in the fourth powers of temperature in the radiative transfer term:

4 4

(T;+ 0) 3 T; - Tj = 4 2

(T; - 1j),

(34) where the subscripts i and j are used for generality.

One objective of linearization is to make Eqs. (26) and (27) appear as simple heat transfer equations that contain an effective conductivity term and an edge con-ductance term. In particular, Eq. (26) can be cast in the following form:

QFpeak k

( T.

T. )

~

= eff m -

e a

(35) where the effective conductivity is defined in Eqs. (1),

(2), and (14). Equation (35) can be manipulated to yield (36) with R * = Fpeak tnl SL k a eff (37) where R ;m is the thermal resistance in the interior re-gion of an assembly. The effective thermal conductiv-ity is evaluated at the wall temperature, so that R;111 can be evaluated to a constant value. This is considered an approximation, which leads to an underestimate of keff (because the wall temperature is always less than either the maximum or extrapolated temperatures), hence 436 overprediction of Tm. Alternatively, Eq. (37) can be derived from Eq. (31) [instead of Eq. (26)], where it can be shown that R;111 = (C1 + C24T3 ) - 1*

Similarly, the edge model can be cast in the linear form as QFpeak = hedge(Te - Tw) '

LaLc (38) where the edge conductance is defined in Eqs. (16), (17),

and (19). Equation (38) can be manipulated to yield (39) with R

Fpeak edge -

L L h a c edge (40) where Redge is the thermal resistance at the edge region of an enclosed assembly. The edge conductance is eval-uated at the wall temperature, so that Redge can be evaluated to a constant value. Equation (40) can also be derived from Eq. (32) [instead of Eq. (27)], where it can be shown that R edge = (C3 + C44T3) - 1*

Equations (36) and (39) can be combined to elimi-nate Te and yield one equation:

(41) with R101 = R;111 + Redge,

(42) where Rw, is the total thermal resistance for the en-closed spent-fuel assembly.

The interior, edge, and total thermal resistances have been calculated for a typical PWR and a typical BWR assembly, for both helium and nitrogen backfiU and for three enclosure wall temperatures (100, 200, and 300°C). The results are summarized in Table Jll.

Equation (41) overestimates !J.T0 compared with the solutions of Eqs. (31) and (32), and a comparison is shown in Fig. 25 for the PWR with nitrogen backfill.

The linear solutions [from Eq. (41)) are straight lines that are coincident with the nonlinear solutions at the low assembly decay heat values and overshoot the non-linear solutions at higher assembly decay heat values.

The usefulness of the linear formulation of the ke11lhedge model is that the model can be used quickly to assess the tJ. T0 without reliance on a computer.

V. CONCLUSIONS The objective of this work was to develop a simple, accurate, defensible method to predict the maximum temperature of a spent-fuel assembly residing in an en-closure as encountered in shipping and transportation casks. An effective thermal conductivity and edge con-ductance model have been developed for both an inte-rior and an edge region of the enclosed assembly. The NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY TABLE III Coefficients in the Linear Form of the Lumped k e11l htt1ge Model* for a Typical PWR and a Typical BWR Spent-Fuel Assembly in Either Helium or Nitrogen Backfill with Enclosure Wall Temperature of Either 100, 200, or 300°C R;n1 R edge R 101

(°C/kW)

(°C/kW)

(°C/kW)

Typical PWR

• with helium backfill T.., = 100°C 41.8

13. I 54.9 T.., = 200°C 32.5 12.2 44.7 T..., = 300°C 24.3 11.0 35.3 Typical PWR with N2 backfill T.., = 100°C 95.0 49.0 144.0 T.., = 200°C 57.7 36.3 94.0 T.., = 300°C 36.0 26.0 62.0 Typical BWRb with helium backfill T.., = 100°C 40.3 22.7 63.0 T., = 200°C 30.7 21.0 51.7 T., = 300°C 22.5 18.5 41.0 Typical BWR with N2 backfill T.., = 100°C 87.4 82.0 169.4 Tw = 200°C 52.1 59.3 111.4 r.. = 300°c 32.2 41.5 73.7

• Equations (35) through (42).

8A typical PWR has an average design burnup of ~ 36 GWd/ tonne U and ~0.46 tonne U so that the assembly decay heat after 10 yr cooling is -0.562 kW. The maximum design burnup is ~ 50 GWd/tonne U (up to -60 GWd/ tonne U) so that the assembly decay heat after 10 yr cooling is ~0.856 kW (up to - I.I I kW).

bA typical BWR has an average design burnup of ~30 GWd/ tonne U and ~0.20 tonne U so that the assembly decay heat after 10 yr cooling is -0.195 kW. The maximum design burnup is ~40 GWd/ tonne U so that the assembly decay heat af-ter 10 yr cooling is -0.275 kW.

Typical PWR 200 ~~~-----------------.-------

180 160 140 u 120 0

1--'" 100

<I 80 60 40 20 0,.

0 250 500 750 1000 1250 1500 Assembly decay heat (W}

Fig. 25. Comparison of the linear lumped keftfhedge model predictions [dashed lines, Eq. (41)] with the nonlin-ear lumped ke11l htdge model predictions [solid lines, Eqs. (3 1) and (32)) for a typical PWR assem-bly with nitrogen backfill.

NUCLEAR TECHNOLOGY VOL. 105 MAR. 1994 models include conductive and radiative heat transfer.

Convection heat transfer has a negligible effect on the value of the maximum temperature within an enclosed spent-fuel assembly for the vast majority of shipping and storage casks; hence, it is frequently neglected in the model. T he lumped formulation of the ke11lhedge model was derived by solving the nonlinear heat diff u-sion equation within the interior region of an array. The lumped ke11lheage model generates two coupled nonlin-ear algebraic equations relating the maximum, extrap-olated, and wall temperatures.

The lumped ke11lheage model has been validated by using data from five separate experiments. The model has been applied to a typical PWR and a typical BWR spent-fuel assembly with helium or nitrogen backfill where nonlinear algebraic equations are generated. The model predicts that the heat transfer due to conduction in an array interior ranges from 10 to 30% with nitro-gen backfill and from 45 to 75% with helium backfill.

The heat transfer due to conduction in the edge region ranged from 30 to 55% for nitrogen and 70 to 90% for helium. It was also noted that -30% of the total as-sembly temperature drop is associated with the edge re-gion for helium backfill, and 45% for nitrogen backfill.

437

Manteufel and Todreas EFFECTIVE THERMAL CONDUCTIVITY The lumped kefflhedge model was compared with the E-MAD correlation and the Wooton-Epstein correla-tion. These correlations have recently been used in the industry and represent the state of the art. Overall good agreement was reported for a large range of wall tem-perature and assembly decay heat values. A simplified linear form of the kefflhedge model has been developed so that calculations may be performed quickly without reliance on a computer.

The benefits of the proposed kettlhedge model is that it is just as easy (if not easier) to use as the previ-ous correlations, and the proposed model is based on rigorous formulations of the governing heat transfer mechanisms. As such, the model is proposed to yield more accurate predictions for a larger range of assem-bly types, enclosure temperatures, and assembly decay powers.

ACKNOWLEDGMENTS This work was originally performed at MIT with sup-port by DOE through SNL under contract DE-AC04-76DP00789. This paper was prepared at the Center for Nu-clear Waste Regulatory Analyses with support by the U.S.

Nuclear Regulatory Commission (NRC) under contract NRC-02-88-005. This paper does not reflect the views or po-sitions of either the DOE or NRC.

REFERENCES I. " Preliminary Design Report B&W BR-100100-Ton Rail/Barge Spent Fuel Shipping Cask," DOE/ 10/ 12701, U.S. Department of Energy (1990).

2. "Preliminary Design Report for NAC Combined Trans-port Cask," DOE/ lD/ 12702, U.S. Department of Energy (1990).

3. "TITAN Legal Weight Truck Cask Preliminary Design Report," DOE/ ID/ 12699, U.S. Department of Energy (1990).

4. "GA-4/GA-9 Legal Weight Truck from Reactor Spent Fuel Shipping Casks," DOE/ID/12698, U.S. Department of Energy (1990).

5. "Packaging and Transportation of Radioactive Mate-rial," Code of Federal Regulations, Tille 10, Part 7 I (I 992).

6. "Licensing Requirements for the Independent Storage of Spent Nuclear Fuel and High-Level Radioactive Waste,"

Code of Federal ReguJations, Title JO, Part 72 (1992).

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TASOOJI, J.C. WOOK, J. R. KELM, B. A. SURETTE, and C.R. FROST, "Estimates of Zircaloy Integrity During Dry Storage of Spent Nuclear Fuel," EPRI-NP-6387, Elec-tric Power Research Institute (1989).

JO. W. H. LAKE, "Developing a New Generation of Spent Fuel Casks," Proc. 9th Int. Symp. Packaging and Transpor-tation of Radioactive Materials, Washington, D.C., June 11-16, 1989, CONF-890631, Oak Ridge National Laboratory (1989).

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16. S. R. FIELDS, "STAFF-5, A Two-Dimensional Com-puter Model for Simulation of the Performance of Spent Fuel in Storage/ Disposal," HEDL-TC-1601, Hanford En-gineering Development Laboratory (1981).

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32. N. E. TODREAS and M. S. KAZIMI, Nuclear Systems I: Thermal Hydraulic Fundamentals, Hemisphere, New York (1990).

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38. M. A. GOTOVSKY, E. D. FEDOROVICH, V. N.

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( 1989).

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ment of Nuclear Engineering, Massachusetts Institute of Technology (1991).

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Soc., 65, 65 (1992).

440 RaodalJ D. Maoteufel [BS, engineering science, University of Texas at Aus-tin, 1984; MS, mechanical engineering, University of Texas at Austin, 1987; PhD, mechanical engineering, Massachusetts Institute of Technology (MIT),

1991] is a research engineer at the Center for Nuclear Waste Regulatory Analy-ses at Southwest Research Institute. His main interests are in heat and mass transfer with application to radioactive waste management. He is currently ac-tive in the areas of performance assessment, thermohydrology, and coupled processes in the support of the U.S. high-level radioactive waste program.

Neil E. Todreas (BS and MS, mechanical engineering, Cornell University, 1958; ScD, nuclear engineering, MIT, I 966) is the KEPCO Professor of Nu-clear Engineering and a professor of mechanical engineering at MIT. His tech-nical activities are focused on thermal-hydraulic aspects of nuclear system performance under steady-state and accident conditions.

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