Nuclear Waste Technical Review Board

Thermal-Response Evaluation of Yucca Mountain 

During the Preclosure and Postclosure Phases 

July 2008 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page i of v 

 

Executive Summary 

The purpose of this white paper is to document a systematic evaluation of the thermal response of the proposed 

Yucca Mountain repository for various thermal loadings. The U. S. Nuclear Waste Technical Review Board 

(NWTRB) staff has developed calculation tools that allow performing these calculations rapidly to determine 

what parameters are important to an acceptable thermal response and to identify the bounding conditions both 

for emplacement of the waste and for permanent closure of the repository. The methods used in the document 

have been benchmarked against the U. S. Department of Energy’s (DOE) analytical approaches and show good 

agreement. However, it is important to recognize that these tools represent simplified analyses and models 

whose primary purpose is to gain insights into the repository thermal response and to help identify key 

parameters affecting the thermal response. 

The Yucca Mountain thermal response is an important input to the Total System Performance Analysis (TSPA) 

used by DOE to determine the dose rates to the public from the waste disposed of at the proposed Yucca 

Mountain repository. Until very recently, DOE used an assumption of a single waste stream with a specific 

power and decay function as the basis for determining the thermal response. DOE is now developing an 

integrated thermal management strategy using the Total System Model (TSM) and WPLoad models to evaluate 

several waste acceptance, processing, and emplacement scenarios. In addition, DOE has developed an 

Estimated Limiting Waste Stream (ELWS) that predicts the potentially hottest commercial waste stream to be 

received at the repository and is using the actual assembly powers and decay curves as the basis for the analysis 

rather than a single power and decay rate as has been done in the past. As a result of these efforts, DOE has 

been able to increase the maximum allowable thermal operating requirements at emplacement from 11.8 kW to 

18 kW per waste package and the linear line load from 1.45 kW/meter to 2.0 kW/meter over a seven-package 

segment. The postclosure thermal response to support the License Application (LA) is based on this ELWS, and 

DOE has shown that this waste stream can be emplaced and the repository closed within 100 years from the 

start of waste emplacement. 

DOE has made significant advances both in their understanding of the repository thermal response and in the 

thermal methodology for repository operation. However, the thermal strategy is still based on an estimated 

limiting waste stream rather than on the amount of thermal energy that the mountain can absorb without 

exceeding any established thermal limit. Performing more general evaluations, as presented in this document, to 

determine the maximum thermal energy that can be absorbed by the mountain would provide increased 

flexibility to control when the repository may be closed by changing the ventilation duration and/or the linear 

line load, regardless of the waste emplaced. This is especially true during the preclosure phase, where rather 

arbitrary thermal limits for emplacement have been established on the basis of the predicted waste stream. The 

current 2.0 kW/meter thermal limit provides approximately 90o

C margin to the 200oC drift wall temperature 

limit for the 30-day loss-of-forced-ventilation event. Increasing the allowable thermal limits at emplacement 

would allow the majority of waste received at the repository to be emplaced immediately, thus reducing the 

amount of surface storage. 

Specific findings and recommendations based on the NWTRB staff's analyses are as follow: 

1. The calculations performed by DOE and the NWTRB staff are in reasonable agreement and suggest that 

the DOE technical basis is valid. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page ii of v 

 

2. Establishing the thermal limits for emplacement on the basis of the ELWS appears to be overly 

conservative. DOE should reevaluate the thermal limits for emplacement based on the capacity of the 

mountain to absorb the thermal energy rather than on the predicted waste stream. This approach would 

result in higher thermal limits for emplacement and reduce the amount of surface storage required at the 

repository. 

3. Initiating the loss of forced ventilation during the preclosure period at 30 days after emplacement may 

not be the most limiting case. DOE should reevaluate this event and determine when the loss of forced 

ventilation results in the shortest time to reach the 200oC drift wall thermal limit. 

4. Surface aging of the hotter Commercial Spent Nuclear Fuel (CSNF), as opposed to subsurface aging, 

has limited benefit on the postclosure thermal response. DOE should reevaluate the need to surface-age 

the CSNF at the repository. 

5. The DOE postclosure thermal response to support the LA is based on an ELWS. However, performing 

more general evaluations, as presented in this document, to determine the maximum thermal energy that 

can be absorbed by the mountain would provide increased flexibility to control when the repository may 

be closed by adjusting the ventilation duration and/or the linear line load, regardless of the waste 

emplaced. 

6. The Board agrees with DOE that if the 96o

C mid-pillar temperature limit is required, the requirement 

determines the maximum allowable thermal loading. However, if the 96oC limit were eliminated, the 

drift wall’s maximum temperature of 200o

C would determine the maximum allowable thermal loading, 

and the preclosure ventilation period could be decreased by approximately 25 years. 

7. Thermal conductivity has a significant effect on the peak temperatures. Variations in the thermal 

conductivity used in the calculations to predict the repository thermal response should be taken into 

account on the basis of actual thermal conductivity measurements along with appropriate uncertainties 

and margins. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page iii of v 

 

Preface 

This study and report were prepared by Gene W. Rowe and Bruce Kirstein of the U.S. Nuclear Waste Technical 

Review Board staff in support of the Board's analysis of issues associated with thermal management, an effort 

lead by Professor Andrew C. Kadak.
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page iv of v 

 

Table of Contents

1. Introduction..................................................................................................................................................... 1 

1.1. Background............................................................................................................................................. 1 

1.2. Purpose.................................................................................................................................................... 1 

1.3. Approach................................................................................................................................................. 2 

1.3.1. General............................................................................................................................................ 2 

1.3.2. Age of Waste................................................................................................................................... 2 

1.3.3. Temperature Variations Along Drift Wall...................................................................................... 2 

1.3.4. Estimation of Assembly Cladding Temperature............................................................................. 3 

2. Calculation Models ......................................................................................................................................... 3 

2.1. Assembly Power Representation ............................................................................................................ 3 

2.2. Finite-Length Decaying Line Source Calculation .................................................................................. 4 

2.2.1. Effects of Host-Rock Thermal Conductivity on Temperature........................................................ 6 

2.3. Infinite-Length Cylinder Calculation...................................................................................................... 7 

2.3.1. Effects of Host-Rock Thermal Conductivity on Temperature........................................................ 8 

2.4. Radiant-Heat Transfer Between Two Concentric Cylinders ................................................................ 10 

2.5. Effect of Ventilation ............................................................................................................................. 10 

2.6. Thermal Properties................................................................................................................................ 11 

2.7. Benchmarking of Analytical Solutions................................................................................................. 11 

3. Thermal Limits.............................................................................................................................................. 12 

4. Thermal Evaluations ..................................................................................................................................... 12 

4.1. Preclosure.............................................................................................................................................. 12 

4.1.1. General.......................................................................................................................................... 12 

4.1.2. Comparison of DOE and NWTRB Evaluations ........................................................................... 12 

4.1.3. Preclosure Thermal Response During Normal Operation ............................................................ 13 

4.1.4. Preclosure Thermal Response for Loss of Forced Ventilation ..................................................... 14 

4.1.5. Discussion..................................................................................................................................... 17 

4.2. Postclosure ............................................................................................................................................ 17 

4.2.1. General.......................................................................................................................................... 17 

4.2.2. Comparison of DOE and NWTRB Evaluations ........................................................................... 17 

4.2.3. Mid-pillar Temperature Evaluation .............................................................................................. 21 

4.2.4. Drift Wall Temperature Evaluation .............................................................................................. 22 

4.2.5. Waste Package Surface Temperature............................................................................................ 24 

4.2.6. Discussion..................................................................................................................................... 26 

5. Conclusions................................................................................................................................................... 26 

5.1. General.................................................................................................................................................. 26 

5.2. Comments and Suggestions .................................................................................................................. 26 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page v of v 

 

Tables

Table 1 Average PWR and BWR Assembly Characteristics................................................................................. 4 

Table 2 Summary of DTN: MO0702PASTREAM.001 Host-Rock Thermal Conductivities .............................. 6 

Table 3 Effect of Host-Rock Thermal Conductivity on Peak Temperatures........................................................ 6 

Table 4 Effect of Host-Rock Thermal Conductivity on Drift Wall Temperature.................................................. 9 

Table 5 Comparison of DOE and NWTRB Preclosure Temperatures ............................................................... 13 

Table 6 Loss- of-Forced-Ventilation Example Results ...................................................................................... 15 

Table 7 Loss-of-Forced-Ventilation During Preclosure Period......................................................................... 16 

Table 8 Comparison of DOE and NWTRB WP Power and Peak Mid-pillar Temperatures.............................. 18 

Table 9 Comparison of Linear Line Load and Mid-pillar Temperature over a Seven-Package Segment.......... 19 

Table 10 Comparison of Mid-pillar Temperatures Calculated by WPLoad....................................................... 20 

Table 11 Thermal Conditions to Maintain Mid-pillar Temperature Below 96oC .............................................. 21 

Table 12 Effects of Ventilation Duration on Age of Assemblies at Closure...................................................... 22 

Table 13 Thermal Condition for Maintaining Drift Wall Temperature Below 200o

C ....................................... 23 

Table 14 Peak Temperatures for 96oC Mid-Pillar Temperatures ....................................................................... 25 

Table 15 Peak Temperatures for 200o

C Drift Wall Temperatures ..................................................................... 25 

Figures

Figure 1 Representation of Finite-Length Decaying Line Source........................................................................ 5 

Figure 2 Effect of Host-Rock Thermal Conductivity on Peak Temperatures ...................................................... 7

Figure 3 Representation of Infinite-Length Cylinder ........................................................................................... 8 

Figure 4 Effect of Host-Rock Thermal Conductivity on Drift Wall Temperature ............................................... 9 

Figure 5 Representation of Ventilation Model ................................................................................................... 11

Figure 6 Drift Wall Temperature with Normal Ventilation Flow....................................................................... 14 

Figure 7 Loss-of-Forced-Ventilation Example................................................................................................... 15 

Figure 8 Days Required to Reach 200oC Drift Wall Temperature After Loss of Forced Ventilation................ 16 

Figure 9 Thermal Conditions at Emplacement to Maintain Mid-pillar Temperature Below 96oC .................... 21 

Figure 10 Thermal Conditions at Closure for Maintaining Mid-pillar Temperature Below 96o

C ..................... 22 

Figure 11 Thermal Condition at Emplacement for Maintaining Drift Wall Temperature Below 200o

C ........... 23 

Figure 12 Thermal Condition at Closure for Maintaining Drift Wall Temperature Below 200o

C...................... 24 

Figure 13 Waste Package Temperature for 200o

C Maximum Drift Wall Temperature ..................................... 24 

Appendix

A Development and Basis for Calculation Models 

B Assembly Power Constant Calculation 

C PWR and BWR Decay Constants 

D Comparison of Assembly Power in Origin Tables and Calculated Power 

E Sample Calculation Using Finite-Length Decaying Line Source for Single Waste Package 

F Sample Calculation Using Finite-Length Decaying Line Source for Seven-Waste-Package Segment 

G Sample Calculation Using Infinite-Length Decaying Line Source 

H Sample Calculation Using Radiant-Heat Transfer Between Two Concentric Cylinders 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 1 of 27 

 

1. Introduction 

1.1. Background 

The United States Nuclear Waste Technical Review Board (NWTRB) since its inception has maintained a 

focus on the issue of thermal management. In the Board’s Fifth Report to Congress, it characterized thermal 

management as a pivotal and crosscutting issue, one that requires both an incremental approach and an 

integrated solution and encompasses the waste management system as a whole. For preclosure, the report 

recognized the importance of the relationships among waste acceptance, transportation, surface and 

subsurface facility design, and the concepts of waste handling and waste emplacement. The Board also 

recognized that the performance of the repository is implicitly tied to the thermal management strategy and 

that coupled hydrological, mechanical, and chemical processes are all strongly temperature dependent. The 

Board also concluded that the repository environment would likely produce conditions on the surface of a 

waste package that could lead to corrosion and to degradation of waste package performance. 

The philosophy used by DOE to develop a thermal strategy has evolved over several years and has been 

influenced by several companies and organizations. The approach has been to predict the waste 

characteristics and delivery schedule (referred to as the “waste stream”) of the waste shipped to Yucca 

Mountain and to use this information as the basis for determining the thermal response of the mountain, 

from emplacement of waste through the entire thermal pulse. 

Until very recently, DOE used a single waste stream with a single power and decay function as the basis for 

determining the thermal response. The project recently developed two computer models, the Total System 

Model (TSM) and WPLoad, to create an integrated thermal management strategy that evaluates several 

waste acceptance, repackaging, and emplacement scenarios. The TSM evaluates waste acceptance, 

transportation, and site throughput, and the WPLoad model evaluates the thermal conditions for 

emplacement. As part of the new strategy, DOE developed an ELWS that predicts the potentially hottest 

commercial waste stream to be received at the repository and is using the actual assembly powers and decay 

curves as the basis for the analysis rather than a single power and decay rate as they have done in the past. 

As a result of these efforts, DOE has been able to increase the maximum allowable thermal operating 

requirements at emplacement from 11.8 kW to approximately 18 kW per waste package and the linear line 

load from 1.45 kW/meter to 2.0 kW/meter over a seven-package segment. 

This new approach is much more comprehensive than the original approach and provides a better 

understanding of the thermal response of the repository. However, it is still based on a single predicted 

waste stream and does not evaluate other potential receipt scenarios or determine the maximum amount of 

thermal energy that the repository can absorb without exceeding an established thermal limit. The present 

approach appears to have decreased the amount of surface storage required at Yucca Mountain, but 

significant surface storage is still required to allow sequencing of the waste package emplacement to 

maintain the required 2.0 kW/meter linear line load over a seven-package segment. 

1.2. Purpose 

The purpose of this paper is to document a systematic evaluation of the thermal response of Yucca 

Mountain for various waste stream scenarios. The Board staff has developed calculation tools that allow 

performing multiple calculations quickly to determine what parameters are important to the mountain’s 

acceptable thermal response and to identify the bounding conditions both for emplacement of the waste and 

for permanent closure of the repository. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 2 of 27 

 

The purpose of this white paper is not to evaluate or endorse the validity, the technical basis, or the 

operating requirements for the thermal limits used by DOE as its design basis. 

1.3. Approach 

1.3.1. General 

The thermal response during both the preclosure and the postclosure phases is highly dependent on the 

initial conditions at the start of each phase. The important factors for establishing these initial conditions 

appear to be (1) the characteristics of the waste when it is discharged from the reactor (initial enrichment 

and burnup) and (2) the age since discharge from the reactor of the waste at the start of each phase. 

Because of the acceptance criteria established by the standard contract, 10CFR961, DOE has very little 

influence on the characteristics of the waste that it will receive. However, DOE controls the sequence in 

which the various waste packages will be loaded into the repository. Although sequencing of waste 

packages has some benefits, controlling the age of the waste at the start of the preclosure period (by 

aging the waste on the surface before emplacement) and at the start of the postclosure period (by 

controlling the length of the preclosure ventilation period) has a greater effect on the thermal response. 

Therefore, the approach taken in this evaluation is to determine the appropriate initial conditions (age of 

the waste) for both the preclosure period (thermal operating requirements for emplacement) and the 

postclosure phase (thermal operating requirements for closure) for waste having various characteristics. 

DOE has taken a different approach by establishing one set of thermal operating requirements (for 

emplacement) and assuming a fixed preclosure period to prevent exceeding thermal limits during either 

the preclosure or the postclosure period. Although not technically incorrect, this approach may be overly 

conservative, require additional operations during the preclosure phase, and limit the flexibility to close 

the repository. 

1.3.2. Age of Waste 

Throughout this document, the term “age” generally refers to the number of years since the assembly 

was discharged from the reactor. Age is used to refer to three different events: 

Age at receipt: The number of years between when the assembly was discharged from the 

reactor and is received at the repository. 

Age at emplacement: The number of years between when the assembly was discharged from 

the reactor and is emplaced in the repository. 

Age at closure: The number of years between when the assembly was discharged from the 

reactor and the repository is permanently closed. 

The term “aging” refers to the act of storing the assembly to allow for radionuclide decay in order to 

reduce the assembly power. Aging can be done on the surface before emplacement, referred to as 

“surface aging,” or after emplacement but before repository closure, referred to as “subsurface aging” or 

the “ventilation period”. 

1.3.3. Temperature Variations Along Drift Wall 

The emplacement of waste packages of differing powers will result in varying drift wall temperatures 

opposite those packages. However, the effect of axial radiant-heat transfer in a drift loaded with 

varying-power packages is such that the axial drift wall temperature variations are relatively small, as 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 3 of 27 

 

shown in project document 800-00C-WIS0-00100-000-00B, Repository Twelve Waste Package Segment 

Thermal Calculation. During normal operating conditions, in a segment containing waste packages with 

initial powers varying between approximately 0.5kW and 11.8kW, the axial variation in peak drift wall 

temperature is approximately 9°C during preclosure and approximately 6°C during postclosure. For the 

accident conditions of a loss of forced ventilation during the preclosure period, the variation in drift wall 

temperature is approximately 28°C but quickly returns to normal after forced ventilation is restored. 

Because the maximum drift wall axial temperature variation is small and is much less at longer time 

frames of a few hundred years, modeling individual packages in a drift is not necessary. An average line 

load can be used over the distance spanned by a segment of several waste packages. This allows the 

waste packages in an entire drift to be represented as a series of segments. 

This is the approach that is taken throughout this evaluation. Rather than model each individual waste 

package, an average linear line load for the drift is assumed. The drift can be modeled as a series of 

segments, each segment contains multiple waste packages, and the average linear line load is applied to 

each segment. The mountain thermal response is determined by adding the energy contribution from 

each segment. 

1.3.4. Estimation of Assembly Cladding Temperature 

DOE has adopted the use of a Transportation/Aging/Disposal canister, referred to as a “TAD”, for 

disposing of all Commercial Spent Nuclear Fuel (CSNF). The internal configuration of the TAD has a 

major effect on the ability of the waste package to transfer the energy from the assembly to the outside 

surface of the waste package and therefore the maximum cladding temperature. Because the internal 

design of the TAD has not been completed, calculating the cladding temperature is not possible at this 

time. Therefore, calculation of the cladding temperature was not performed as part of this evaluation. 

The assumption is that the peak cladding temperature will not become a limiting criterion for either the 

preclosure or the postclosure. However, this assumption will need to be verified after the TAD design 

has been finalized. 

2. Calculation Models 

The development and basis for the calculation models used in this evaluation are in Appendix A. They are 

simplified models and are not intended to be the basis for the actual repository design. However, the NWTRB 

believes that the models are accurate enough to allow performing the parametric evaluations necessary to 

understand the thermal response as well as to determine which parameters are important to the thermal 

response. They also provide an indication of the validity of the calculations and results performed by DOE. 

Descriptions of the three models used are in Sections 2.2 through 2.4. 

2.1. Assembly Power Representation 

The power of a decaying radionuclide as a function of time can be represented as a decaying exponential in 

the form of Equation 1. 

 

0 ( ) t Qt Qe−λ = Equation 1 

Because a nuclear fuel assembly is composed of several decaying sources, each with a different decay 

function, a single exponential term cannot be used. A more accurate representation of assembly decay is 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 4 of 27 

 

provided as the sum of three decaying exponentials in the form of Equation 2, where the coefficients are 

watts/meter and the time is years. 

 

t t t Q t Q e Q e Q e 1 2 3

1 2 3 ( ) −λ −λ −λ = + + Equation 2 

For determining the six constants in Equation 2 for an assembly with a particular burnup, six equations were 

written based on the data provided in PWR Source Term Generation and Evaluation, 000-00C-MGR0-

00100-000-00B and BWR Source Term Generation and Evaluation, 000-00C-MGR0-00200-000-00A, at six 

different times, and the constants “Q” and “λ” were determined using MathCad. An example of the 

calculation is in Appendix B, the results for PWR and BWR assemblies are in Appendix C, and a 

comparison of the assembly powers in the PWR Source Term Generation and Evaluation, 000-00C-MGR0-

00100-000-00B, and BWR Source Term Generation and Evaluation, 000-00C-MGR0-00200-000-00A 

tables, and the calculated powers using the methodology in Appendix B is in Appendix D. 

Based on DOE Engineering Study 000-00R-G000-01000-000-000, Total System Model Analysis for 

Repository Postclosure Thermal Envelope Study, Phase2, data file WP_Decay_70K22kw_011707.xls, the 

average PWR and BWR assemblies are as follow: 

Table 1 Average PWR and BWR Assembly Characteristics 

Assembly Type 

Average 

Loading 

(MTHM) 

Average Burnup 

(Gwd/ton) 

Average Age 

at Receipt 

(Years) 

PWR 0.413 47.66 17.95 

BWR 0.175 45.41 14.25 

The assembly powers provided in PWR Source Term Generation and Evaluation, 000-00C-MGR0-00100-

000-00B, and BWR Source Term Generation and Evaluation, 000-00C-MGR0-00200-000-00A, assumed 

the initial loading of a PWR assembly to be 0.475 Metric Tons Heavy Metal (MTHM)/assembly and of a 

BWR assembly to be 0.200 MTHM/assembly. These values are relatively high in comparison to the average 

assembly and result in powers that are unrealistically high. To obtain a more representative assembly power, 

the power coefficients in Appendix B are corrected by multiplying the power coefficients by the ratio of the 

actual MTHM/assembly to the base case MTHM/assembly. In all calculations performed as part of this 

evaluation, the correction factors used are as follow: 

 PWR assembly 0.413/0.475 = 0.869 

 BWR assembly 0.175/0.200 = 0.875 

2.2. Finite-Length Decaying Line Source Calculation 

For the majority of the calculations performed in this evaluation, the thermal response of the mountain was 

determined using a finite-length decaying line source, as shown in Figure 1. 
Thermal Response Evaluation of Yucca Mountain 

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Figure 1 Representation of Finite-Length Decaying Line Source 

The temperature as a function of time for a finite-length decaying line source can be determined using 

Equation 3. 

2 2 ( )

4

0

(, , ,) 8 2 2

t t x y Qe z z L e v x y z t erf erf e d K

λ λθ

κθ θ

π θ κθ κθ

− + + ⎧ ⎫ ⎛ ⎞⎛ ⎞ − −

= − ⎨ ⎬ ⎜ ⎟⎜ ⎟ ⎩ ⎭ ⎝ ⎠⎝ ⎠ ∫ Equation 3 

where x and y represent the radial distance from the center of the waste package, z is the location on the 

segment centerline axis, Q is the lead coefficient in watts/meter that describes the exponential decay of the 

source with a rate of decay of λ reciprocal time, K is the thermal conductivity, L is the source length, and κ

is the thermal diffusivity. For a number of equal length sources, n, lined up on the z axis, the temperature is 

calculated by Equation 4. 

1

( , , ) ( , ( 1) , )

n

i

i

vxzt v xz i Lt == −− ∑ Equation 4

where y = 0 to represent the centerline plane of the waste package and the subscript i is applied to the 

decaying power as Qi and λi. The temperature contribution from neighboring drifts is calculated by Equation 

5, where m equals the number of neighboring drifts. 

1

(, ,) ( _ , ,) ( _ , ,)

m

i i

i

v x z t v m Drift space x z t v m Drift space x z t == × − +× + ∑ Equation 5 

The detailed development of this methodology is presented in Appendix A. Two calculation methods are 

used; one assuming that the repository is loaded with waste packages that all have the same characteristics, 

Attachment E, and one assuming an identical seven-package segment that is replicated throughout the 

repository, Appendix F. 
Thermal Response Evaluation of Yucca Mountain 

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2.2.1. Effects of Host-Rock Thermal Conductivity on Temperature 

Variations in host-rock thermal conductivity will affect the peak postclosure temperatures at both the 

drift wall and the mid-pillar. For determining the magnitude of the temperature variations due to changes 

in host-rock thermal conductivity, the peak temperatures for the drift wall and the mid-pillar were 

calculated for a range of host-rock thermal conductivities using the methodology described in Section 

2.2. The basis for selecting this range of thermal conductivities comes from DTN: 

MO0702PASTREAM.001 and is summarized in Table 2. 

Table 2 Summary of DTN: MO0702PASTREAM.001 Host-Rock Thermal Conductivities 

Upper Lithophysal 

(TSW33) 

Upper Nonlithophysal 

(TSW34) 

Lower Lithophysal 

(TSW35) 

Lower Nonlithophysal 

(TSW36) 

 Dry Wet Dry Wet Dry Wet Dry Wet 

Minimum 0.423 1.171 0.613 1.389 0.570 1.326 0.861 1.573 

10% 0.949 1.550 1.147 1.827 1.071 1.690 1.283 1.944 

50% 1.132 1.749 1.389 2.052 1.240 1.863 1.443 2.102 

90% 1.369 1.82 1.626 2.302 1.414 2.055 1.609 2.274 

Maximum 2.057 2.782 2.511 3.199 1.924 2.585 2.117 2.741 

Because the majority of the repository will be located in the lower lithophysal rock, the conductivities 

for this host rock were selected. The 10-percentile dry represents a reasonable lower bound, and the 90-

percentile wet represents a reasonable upper bound. Therefore, temperatures were calculated using the 

method described in Section 2.2 for conductivities between 1.0 W/m-o

C and 2.0 W/m-oC at 0.1 W/m-oC 

intervals for waste packages containing 48 Gwd/t PWR assemblies that were 18 years old at 

emplacement and were ventilated for 50 and 100 years. The results are shown in Table 3 and Figure 2. 

Table 3 Effect of Host-Rock Thermal Conductivity on Peak Temperatures 

Temperature

(deg-C)

Time of Peak

Temperature

(Years from 

Emplacement)

Temperature

(deg-C)

Time of Peak

Temperature

(Years from 

Emplacement)

Temperature

(deg-C)

Time of Peak

Temperature

(Years from 

Emplacement)

Temperature

(deg-C)

Time of Peak

Temperature

(Years from 

Emplacement)

1.00 313 71 150 447 226 140 131 519

1.10 293 70 145 437 211 141 127 510

1.20 275 70 140 429 199 143 123 501

1.30 260 70 136 421 189 145 119 494

1.40 247 70 132 414 180 148 116 487

1.50 235 69 129 407 172 157 113 480

1.60 224 69 126 400 165 155 111 474

1.70 215 69 123 394 159 158 108 468

1.80 207 69 121 388 153 162 106 464

1.90 199 69 118 383 149 191 104 457

2.00 193 70 116 377 144 171 102 452

50 Years of Ventilation 100 Years of Ventilation

Thermal

Conductivity

Drift Wall

Peak Temperature

Mid-Pillar

Peak Temperature

Drift Wall

Peak Temperature

Mid-Pillar

Peak Temperature
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Figure 2 Effect of Host-Rock Thermal Conductivity on Peak Temperatures 

0

50

100

150

200

250

300

350

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Conductivity - watts/meter-deg C

Peak Temperature - deg C

Drift Wall with 100 Years of Ventilationl Mid-Pillar with 100 Years of Ventilation

Drift Wall with 50 Years of Ventilation Mid-Pillar with 50 Years of Ventilation

For 50 years of ventilation, a variation of thermal conductivity around a nominal value of 1.8 W/(m-o

C) 

of ±0.1 W/(m-o

C) will result in a difference in the peak drift wall temperature of approximately ∓8oC 

and a difference in the mid-pillar temperature of approximately ∓2.5o

C, and for 100 years of ventilation, 

a difference in the peak drift wall temperature of approximately ∓5o

C and a difference in the mid-pillar 

temperature of approximately ∓2o

C. These results suggest that the thermal conductivity has a significant 

effect on the peak drift wall and to a lesser degree on mid-pillar temperatures and is more pronounced 

for higher heat sources. This variation must be accounted for either by measuring actual host-rock 

thermal conductivity and using as-built data in the calculation or by placing sufficient margin on the 

thermal limits so that a variation in rock conductivity will not cause the actual temperatures to exceed 

established limits. 

2.3. Infinite-Length Cylinder Calculation 

As stated in Appendix A, the methodology described in Section 2.2 is valid for drift wall temperatures only 

after approximately 5 years. For the majority of the calculations performed in this evaluation, this is 

acceptable because peak postclosure temperatures usually occur much beyond 5 years after emplacement. 

However, for determining the drift wall temperatures for the shorter preclosure periods, an infinite-length 

cylinder model as shown in Figure 3 was used. 
Thermal Response Evaluation of Yucca Mountain 

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Figure 3 Representation of Infinite-Length Cylinder 

The temperature for the region bounded internally by a cylinder, i.e., a tunnel in the rock, is: 

2

01 0 1

2 22

0 1 1

2 ( )( ) ( )( ) ( ) () ()

t ut

p

Qe e J ur Y ua Y ur J ua v r du

C u J ua Y ua

λ κ

πρ λ κ

∞ − − − − = ⋅ − + ∫ Equation 6 

where “Q” is the energy wall flux, “a” is the cylinder (tunnel) radius, ρ is the bulk density, Cp is the heat 

capacity, κ is the thermal diffusivity, Jν(z) is the Bessel function of the first kind of order ν, and Yν(z) is the 

Bessel function of the second kind of order ν. This equation is two dimensional in r and thus represents a 

drift of infinite length. Equation 7 is applied to account for the heat removed by the ventilation system. 

( , , ) (1 ) ( , , )

n

eff i

i

v r t i Vent v x t i =− × ∑ Equation 7

where Venteff represents the fraction of heat removed by the ventilation system and the subscript i is applied 

to the decaying power as Qi and λi. The detail development of this methodology is shown in Appendix A, 

and a sample calculation is shown in Appendix G. 

2.3.1. Effects of Host-Rock Thermal Conductivity on Temperature 

As shown in Section 2.2.1, variations in rock conductivity have a significant effect on peak postclosure 

temperatures. Although Equation 6 contains the constant “thermal diffusivity” rather than “thermal 

conductivity”, the temperature difference due to variations in thermal diffusivity can be evaluated by 

varying the thermal conductivity because the relationship between thermal conductivity and thermal 

diffusivity is as follows: 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 9 of 27 

 

rock

p

K

κ = ρC Equation 8 

where “к” is thermal diffusivity in m2/day, “Krock” is host rock thermal conductivity in watts/meter-oC, 

“ρ” is rock density = 2097 kg/meter3

, and “Cp” is rock specific heat = 1119 joules/kg-oC. 

Peak drift wall temperatures were calculated using the method described in Section 2.3 for 

conductivities between 1.0 W/m-o

C and 2.0 W/m-oC at 0.1 W/m-oC intervals for waste packages 

containing 48 Gwd/t PWR assemblies that were 16 years old at emplacement. The results are shown in 

Table 4 and Figure 4. 

Table 4 Effect of Host-Rock Thermal Conductivity on Drift Wall Temperature 

Thermal

Conductivity

Time of Peak

Temperature

(Years)

Peak

Temperature

(deg-C)

1.0 15.26 110

1.1 14.99 103

1.2 14.76 98

1.3 14.55 93

1.4 14.36 89

1.5 14.18 86

1.6 14.02 83

1.7 13.87 80

1.8 13.73 77

1.9 13.60 75

2.0 13.47 73

Figure 4 Effect of Host-Rock Thermal Conductivity on Drift Wall Temperature 

40

50

60

70

80

90

100

110

120

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Thermal Conductivity watt/meter-deg C

Temperature - deg-C

Drift Wall Temperature
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 10 of 27 

 

Variation of thermal conductivity around a nominal value of 1.8 W/(m-o

C) of ±0.1 W/(m-oC) will result 

in a difference in the peak drift wall temperature of approximately ∓2.0o

C. This suggests that thermal 

conductivity has less effect on drift wall peak temperature during the preclosure period than during the 

postclosure period. This is probably because the peak temperatures occur much sooner during 

preclosure, so the ability for the rock to dissipate heat is not as important. 

2.4. Radiant-Heat Transfer Between Two Concentric Cylinders 

The waste package surface temperature is calculated from the transfer of radiant heat between two 

concentric cylinders, as described in the open literature. Natural convection and conduction are not taken 

into account. The temperature across an annulus due to radiant-heat transfer is shown in Equation 9. 

( ) 4 4

1 2

12

11 2 2

1 11 1

T T

Q

Ae A e

σ − = ⎡ ⎤ ⎛ ⎞ ⎢ ⎥ + − ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

Equation 9

where σ = 5.67 × 10-8 W m-2 K-4, Q12 is the energy per unit time transported from surface 1 to 2, Ti are the 

absolute surface temperatures, Ai are the surface areas per unit length, and ei are the respective emissivities. 

Given the drift wall temperature and the package power, the package temperature can be calculated. The 

above equation can be applied to the package and the drip shield and to the drip shield and the drift wall. A 

sample calculation is in Appendix H. 

2.5. Effect of Ventilation 

During the preclosure period, forced ventilation through the emplacement drifts removes a large portion of 

the heat generated by the waste packages. The effect of ventilation as used by DOE in its temperature 

analyses is the use of a constant ventilation efficiency for all times and locations in a drift. The ventilation 

efficiency is defined as the energy removed by flowing air divided by the energy of the source for the entire 

drift length for the entire ventilation duration. Therefore, this ventilation efficiency is called an “integrated 

efficiency”. The details of the ventilation model are described in Ventilation Model and Analysis Report, 

ANL-EBS-MD-000030, REV04. The ventilation model is based on heat transfer from the waste package to 

the air by convection and to the rock wall by radiation. The air receives energy by convection from the rock 

wall and the waste package. The bulk rock is heated by the difference in energy received by radiation and 

lost by conduction into the air. The solution for all of the temperatures is time dependent. Typically, 

temperatures are calculated on a one-year time step and the drift is divided into 100-meter lengths, where 

the air exiting one segment is the input to the next segment. This is shown in Figure 5. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 11 of 27 

 

Figure 5 Representation of Ventilation Model 

For the purpose of most of the calculations performed as part of this white paper, the assumption was that 

the ventilation efficiency decreases linearly as the air moves down the drift and is heated. The ventilation 

efficiency was assumed to be 90 percent at the entrance of the drift and 80 percent at the exit of the drift. 

2.6. Thermal Properties 

The set of thermal-physical properties used to calculate temperatures are based on the properties of the 

lower lithophysal unit, which comprises most of the repository footprint. The calculated bulk density is 

2097 kg/m3 and takes into account a water saturation of 90 percent of the porosity, which is 0.13. The heat 

capacity of the bulk rock with water present is calculated as 1119 Joules/(kg-o

C). The thermal conductivity 

of the bulk rock with 90 percent water saturation is a linear interpolation between dry and fully saturated 

and is calculated as 1.83 W/(m⋅

 oC). 

2.7. Benchmarking of Analytical Solutions 

The calculated thermal-physical properties in Section 2.6 were used to calculate the peak drift wall and midpillar temperatures for comparison with those obtained from the Multiscale Thermohydrologic Model, 

(MSTHM), ANL-EBS-MD-000049, REV03, for the lower lithophysal unit. The MSTHM takes into 

account water-vapor movement, hydrology, and stratigraphy. The MSTHM peak drift wall temperature for a 

typical waste package is 136.8°C (refer to MSTHM, Figure 6.3-15a), and the temperature calculated at that 

location using the methods described in this document is 137.5°C. The MSTHM peak mid-pillar 

temperatures for two locations are 87.3°C and 89.6°C, and the temperature calculated using the methods 

described in this document is 87.8°C. These differences in calculated temperatures are considered small, so 

the methods described in this document are of sufficient accuracy for the thermal strategy analyses 

presented in this document. 

 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 12 of 27 

 

3. Thermal Limits 

At the July 31, 2007, fact-finding meeting between DOE and the NWTRB, DOE provided the thermal limits 

listed below as the basis for design. Not clear is whether these limits consider the accuracy of the calculation 

methods or any margin to the thermal limits. They are important considerations that DOE needs to include when 

making its final recommendations for establishing the thermal limits and the thermal operating methodology. 

Although these thermal limits are used throughout this evaluation, the use of these limits does not imply 

acceptance or endorsement of the limits by the NWTRB. 

Maximum CSNF cladding temperature: 

 Before repository closure: 400°C 

 After repository closure: 350°C 

Maximum waste package surface temperature: 300°C 

Maximum drift wall temperature: 200°C 

Maximum temperature halfway between emplacement drifts (“mid-pillar”): 96°C 

4. Thermal Evaluations 

4.1. Preclosure 

4.1.1. General 

During the preclosure phase, a nominal ventilation flow rate of approximately 15m3/sec is provided 

through each emplacement drift by a series of fans located external to the drift. The ventilation system 

removes approximately 85 percent of the waste package heat. This evaluation will calculate the drift 

wall temperature for a uniform drift loading during both normal and loss-of-ventilation conditions using 

the methodology described in Section 2.3. 

The parameters used for this evaluation are as follow: 

WP type – 21 PWR Forced ventilation efficiency – 85% Rock density – 2097 kg/m3

WP length – 5.85 m Natural ventilation efficiency – 0% Heat capacity – 1119 Joules/kg-o

C 

WP spacing – 0.1 m Drift spacing – 81 m Rock thermal conductivity – 1.83W/m- oC 

WP spacing (DOE case) – 0.7 m No. drifts – 1 Initial rock temperature – 22.8o

C 

Drift length – Infinite 

 

4.1.2. Comparison of DOE and NWTRB Evaluations 

DOE document 800-00C-WIS0-00600-000-00B, Temperature in as “As-Loaded” and ThermallyMisloaded Segment, provides preclosure temperatures for both the normal and loss-of-forced-ventilation 

conditions. To evaluate these results, the NWTRB calculated the temperatures using the same linear line 

load of 1.99 kW/meter (waste package spacing was increased to 0.7 meters to achieve the same linear 

line load as used in the DOE calculation), with a loss of forced ventilation for 30 days, 30 days after 

emplacement. The results are compared with the DOE results in Table 5. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 13 of 27 

 

Table 5 Comparison of DOE and NWTRB Preclosure Temperatures 

 Normal Condition Loss-of-Ventilation Condition 

 DOE Results NWTRB Results DOE Results NWTRB Results 

Waste Package Temperature - oC 111.9 115 162.1 148 

Drift Wall Temperature - oC ? 55 113.9 108 

The differences in results could be attributed to the following. 

1. Effect of invert DOE attempts to model the actual properties of the invert; 

the NWTRB model assumes the invert acts as a fixed 

insulator for 25 percent of the surface area 

2. Thermal conductivity The NWTRB model assumes a constant 1.83 watts/meter-o

C 

conductivity; the DOE model has a variable conductivity. 

3. Geometric model The NWTRB model assumes two concentric circles to 

represent the waste package and drift wall; in reality, the 

center of the waste package is offset from the center of the 

drift 

Despite the differences in the two models, the results indicate reasonable agreement. 

4.1.3. Preclosure Thermal Response During Normal Operation 

Several calculations were performed for various assembly powers at emplacement using the 

methodology in Section 2.3. On the basis of these calculations, the determination was that the 

maximum drift wall temperature was 139o

C, or approximately 60oC below the maximum allowable limit 

of 200o

C for all cases with normal ventilation flow. Figure 6 shows the drift wall temperature as a 

function of time for the most severe case of 5-year-old PWR assemblies with a burnup of 70 Gwd/ton 

(which corresponds to a linear line load of 6.51 kW/meter at emplacement). 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 14 of 27 

 

Figure 6 Drift Wall Temperature with Normal Ventilation Flow 

0 1 103 × 2 103 × 3 103 × 4 103 × 5 103 ×

0

30

60

90

120

150

Days from Emplacement

Drift Wall Temperature - Degrees C

These results suggest that under normal preclosure ventilation conditions, any waste that is received at 

the repository can be emplaced immediately with no surface aging without exceeding the maximum 

allowable drift wall temperature of 200o

C. 

4.1.4. Preclosure Thermal Response for Loss of Forced Ventilation 

During the preclosure period, forced ventilation could be lost. DOE considers the loss-of-forcedventilation event the basis for establishing the thermal conditions for emplacement and has set the 

maximum linear line load over a seven-waste package segment at emplacement at less than 2.0 

kW/meter. The results shown in Table 4 make apparent that establishing the linear line load at 2.0 

kW/meter may be overly conservative because there is approximately 60o

C of margin to the drift wall 

maximum temperature of 200o

C. For evaluating the loss-of-ventilation event, several calculations were 

performed using the methodology described in Section 2.3 to determine how long it would take for the 

drift wall temperature to reach the maximum allowable temperature of 200o

C when forced ventilation is 

lost at various times after emplacement. Figure 7 and Table 6 present an example of the calculation for 

5-year-old assemblies with a burnup of 40 Gwd/ton. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 15 of 27 

 

 

Figure 7 Loss-of-Forced-Ventilation Example 

Graph 

0 100 200 300 400 500 0

60

120

180

240

300

Days from Emplacement

Drift Wall TEmperature - deg-C

Table 6 Loss- of-Forced-Ventilation Example 

Results 

Time After 

Emplacement 

when Forced 

Ventilation is 

Lost

(Days)

Time Required After 

Loss of Forced 

Ventilation Until 200 

deg-C Drift Wall 

Temperature is 

Reached

(Days)

30 62

50 61

70 60

90 59

110 59

130 58

150 58

170 58

190 58

210 58

230 58

250 58

270 58

290 59

310 59

This example shows that the shortest time to reach 200o

C after ventilation is lost is 58 days and occurs 

not at 30 days after emplacement but sometime between 110 and 290 days after emplacement. 

For determining the parameters that define the shortest time to reach 200o

C 30 days after forced 

ventilation is lost, several calculations were performed. The calculations were performed for waste 

packages loaded with PWR assemblies with ages of 5, 10, 15, and 20 years out of reactor at 

emplacement and burnup between 35 and 70 Gwd/ton. The results are shown in Table 7 and Figure 8. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 16 of 27 

 

Table 7 Loss-of-Forced-Ventilation During Preclosure Period 

Assembly

Burnup

Age at

Emplacement

Line Load at

Emplacement

Number of 

Days of

Forced 

Ventilation

Minimum

#Days Required

to Reach 200 deg-C

Drift Wall Temperature

35 5 2.76 80 93

40 5 3.19 130 58

45 5 3.68 180 36

50 5 4.17 280 23

35 10 1.83 140 316

40 10 2.12 360 185

45 10 2.45 550 109

50 10 2.79 600 69

55 10 3.17 750 43

60 10 3.55 850 28

65 10 3.99 1350 17

45 15 2.08 800 175

50 15 2.36 1200 109

55 15 2.67 1050 69

60 15 2.99 1000 46

65 15 3.34 1400 29

70 15 3.70 1650 19

45 20 1.86 650 255

50 20 2.11 1000 159

55 20 2.39 1100 101

60 20 2.66 1250 67

65 20 2.97 1500 44

70 20 3.28 1500 30

45 25 1.69 550 365

50 25 1.92 800 227

55 25 2.16 950 145

60 25 2.40 1150 97

65 25 2.67 1350 65

70 25 2.94 1450 45

Figure 8 Days Required to Reach 200o

C Drift Wall Temperature After Loss of Forced Ventilation 

0

10

20

30

40

50

60

70

80

90

100

2.25 2.75 3.25 3.75 4.25

Linear Line Load at Emplacement - kW/meter

Days Required to Reach 200 deg-C

5 years old at emplacement 10 years old at emplacement 15 years old at emplacement

20 years old at emplacement 25 Years old at emplacement
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 17 of 27 

 

These results suggest that an emplacement drift can be loaded to a higher linear line load than is 

presently assumed by DOE, up to approximately 3.25 kW/meter, without exceeding the 200o

C drift wall 

temperature limit, even with a 30-day loss of forced ventilation and no credit taken for natural 

ventilation. 

4.1.5. Discussion 

During the preclosure period, the mid-pillar temperature limit is not important because the peak midpillar temperature occurs hundreds of years after repository closure. Therefore, during the preclosure 

period, the limiting thermal limit appears to be the maximum drift wall temperature of 200o

C. According 

to the above calculations, it appears that during normal operation with forced ventilation, there is no 

restriction on the power of the waste emplaced. However, if forced ventilation is lost for 30 days, the 

drift wall temperature can be exceeded. For preventing exceeding drift wall temperature, an operating 

limit of approximately 3.25 kW/meter linear line load should be maintained. This value is significantly 

higher than the 2.0 kW/meter criteria assumed by the project and corresponds to an average waste 

package power of approximately 19.3 kW. 

4.2. Postclosure 

4.2.1. General 

At some point, the forced ventilation system will be shut down and the repository will be closed. This 

will result in a rise in repository temperatures because none of the heat generated by the waste packages 

will be removed by the ventilation system. The following evaluation will compare waste package power 

and mid-pillar temperatures results from DOE evaluations with NWTRB evaluations. In addition, the 

minimum age of various burnup fuels that result in not exceeding either the maximum mid-pillar 

temperature of 96o

C or the maximum drift wall temperature of 200oC will be calculated. The parameters 

used for this evaluation are as follow: 

Drift length: 583 m Rock density: 2097 kg/m3

No. drifts: 11 (5 on each side) Heat capacity: 1119 Joules/ kg-oC 

Drift spacing: 81 m Rock thermal conductivity: 1.83 W/m-K 

No. segments per drift: 15 Initial rock temperature: 22.8o

C 

No. WP per segment: 7 Initial ventilation efficiency: 90% 

WP spacing: 0.1 m Final ventilation efficiency: 80% 

 

4.2.2. Comparison of DOE and NWTRB Evaluations 

DOE Engineering Study 000-00R-G000-01000-000-000, Total System Model Analysis for Repository 

Postclosure Thermal Envelope Study, Phase2, data file WP_Decay_70K22kw_011707.xls, provides the 

individual average waste package characteristics and the maximum mid-pillar temperature if the entire 

repository were loaded with waste packages with these characteristics. As a check of these results, 

several waste packages were selected, and the waste package power at emplacement and peak mid-pillar 

temperature were calculated. The results are shown in Table 8. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 18 of 27 

 

Table 8 Comparison of DOE and NWTRB WP Power and Peak Mid-pillar Temperatures 

WP ID Shipment

ID

WP

Length

(meters)

Year

Created

Total

MTHM

Average

Burnup

(Gwd/ton)

Age at

Emplacement

(Years OOR)

Age at

Closure

(Years OOR)

Number of

Assemblies

per WP

SNF TYPE DOE

Results

NWTRB

Results

DOE

Results

NWTRB

Results

Power

Difference

(%)

Temperature

Difference

(%)

130958 201706700 5.85 2018 7.476 40 7 103 44 BWR 11,692.6 11,567.7 79 75 1% 5%

131027 201807206 5.85 2019 7.568 38 10 105 44 BWR 9,514.0 9,421.7 76 72 1% 5%

131028 201807207 5.85 2019 7.568 40 11 106 44 BWR 9,654.3 9,590.3 78 75 1% 4%

135949 202908306 5.85 2030 7.591 42 26 110 44 BWR 7,371.3 7,311.5 80 75 1% 6%

135955 202910202 5.85 2030 7.810 51 6 90 44 BWR 17,229.6 17,572.0 94 92 -2% 2%

135962 202908603 5.85 2030 7.717 55 17 101 44 BWR 12,222.5 12,232.1 93 90 0% 3%

141094 204005612 5.85 2040 7.798 46 31 105 44 BWR 7,542.2 7,568.8 85 81 0% 5%

141112 204005703 5.85 2040 7.811 54 5 79 44 BWR 19,129.0 20,879.0 97 99 -9% -2%

141113 204005704 5.85 2040 7.814 62 6 80 44 BWR 21,315.9 22,732.2 106 106 -7% 0%

130915 DTF1WP97 5.85 2017 7.675 24 26 123 21 PWR 4,362.6 4,369.4 61 57 0% 7%

130916 DTF1WP98 5.85 2017 5.705 23 44 141 21 PWR 2,277.8 2,254.4 49 46 1% 6%

130954 DTF1WP108 5.85 2018 7.265 50 12 108 21 PWR 12,722.6 12,814.9 100 97 -1% 3%

135852 202909104 5.85 2030 9.135 47 21 105 21 PWR 12,033.4 12,058.4 115 111 0% 3%

135853 202909400 5.85 2030 9.534 52 8 92 21 PWR 19,938.1 21,587.0 130 130 -8% 0%

135854 202909401 5.85 2030 9.534 54 9 93 21 PWR 20,595.5 21,315.0 134 132 -3% 1%

141090 204007302 5.85 2040 9.040 62 14 88 21 PWR 19,748.8 19,945.1 140 140 -1% 0%

141091 204007401 5.85 2040 9.172 54 8 82 21 PWR 21,317.1 21,839.1 134 133 -2% 1%

141092 204007402 5.85 2040 9.172 50 16 90 21 PWR 16,534.3 14,467.0 124 122 13% 2%

Note: The "Total MTHM" values for WPs DTF1WP97, DTF1WP98, and DTF1WP108 contained in WP_Decay_70K22kw_011707.xls and BatchInfo.mdb were inconsistant.

The value contained in BatchInfo.mdb were used.

Power at Emplacement

(watts)

Peak Post closure

Mid-pillar Temperature

(deg-C)

Comparison of

DOE with NWTRB

Results

Waste Package Characteristics

The NWTRB and DOE results are in reasonable agreement. The difference between the “Power at 

Emplacement” values can be attributed to the following: 

• The NWTRB calculation utilized average waste package characteristics, whereas the 

DOE calculation calculates waste package power on the basis of the individual assembly 

characteristics. 

• The NWTRB calculation rounds assembly burnup values to the nearest integer, whereas 

the DOE calculation uses actual burnup values 

• The NWTRB calculation assumes a constant 4.0 percent BWR and 4.5 percent PWR 

enrichment, whereas the DOE calculation uses actual assembly enrichment. 

The difference in “Peak Postclosure Mid-pillar Temperature” can be attributed to the fact that the 

NTWRB calculation utilized a finite-length drift model, whereas the DOE model utilized an infinitelength drift model. Therefore, the NWTRB results were consistently lower than the DOE results. 

A further check was performed to determine the accuracy of the linear line load and mid-pillar 

temperature for a seven-package segment. Three random seven-package segments were chosen, one 

emplaced in 2021, one emplaced in 2031, and one emplaced in 2050; the results are shown in Table 9. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 19 of 27 

 

Table 9 Comparison of Linear Line Load and Mid-pillar Temperature over a Seven-Package Segment 

DOE

Calculation

(Note 1)

NWTRB

Calculation

(Note 4)

DOE

Calculation

(Note 1 & 5)

NWTRB

Calculation

(Note 4)

DOE

Calculation

(Note 1)

NWTRB

Calculation

(Note 4)

DOE

Calculation

(Note 1 &5)

NWTRB

Calculation

131463 DTF1WP336 WP PWR 0.4616 43.4 2021 20 96 12,008.1 11,738.0 2,003 114 107 85 N/A

131543 DTF1WP372 WP PWR 0.4379 43.7 2021 12 96 13,976.9 13,896.8 2,371 112 107 85 N/A

131642 DTF1WP433 WP PWR 0.4180 36.3 2021 27 96 7,966.8 7,622.6 1,301 92 85 85 N/A

130987 201806105 WPMPC BWR 0.1718 11.9 2021 26 96 1,965.4 1,971.3 336 38 35 85 N/A

131351 202004600 WPMPC BWR 0.1770 52.6 2021 6 96 17,879.3 18,430.6 3,145 93 91 85 N/A

131333 DTF1WP282 WP BWR 0.0762 17.2 2021 45 96 879.6 876.8 150 35 31 85 N/A

131428 202008205 WPMPC PWR 0.3507 57.1 2021 9 96 17,405.8 17,680.9 3,017 111 108 85 N/A

Total 72,082.0 72,216.9 N/A N/A N/A N/A N/A

Average 10,297.4 10,316.7 1,735 1,761 85 81 85 82

DOE

Calculation

(Note 1)

NWTRB

Calculation

(Note 4)

DOE

Calculation

(Note 1 & 5)

NWTRB

Calculation

(Note 4)

DOE

Calculation

(Note 1)

NWTRB

Calculation

(Note 4)

DOE

Calculation

(Note 1 & 5)

NWTRB

Calculation

133759 WP WPCodisposeL 0.4169 2031 0 86 353.0 407.0 78 24 24 82 N/A

134029 202600601 WPMPC PWR 0.4618 56.5 2031 16 86 17694.4 18147.0 3097 138 134 82 N/A

134226 202606302 WPMPC PWR 0.4552 57.1 2031 16 86 17826.4 17887.6 3052 138 133 82 N/A

135834 202908800 WPMPC PWR 0.4566 53.7 2031 14 86 17795.1 17618.3 3007 133 129 83 N/A

133762 WP WPCodisposeL 0.4169 2031 0 86 353.0 407.0 78 24 24 83 N/A

135557 202903302 WPMPC BWR 0.1716 55.1 2031 7 86 16190.7 17313.9 2955 94 93 82 N/A

133765 WP WPCodisposeL 0.4169 2031 0 86 353.0 407.0 78 24 24 82 N/A

Total 70,565.6 72,187.8 N/A N/A N/A N/A N/A

Average 10,080.8 10,312.5 1,779 1,763 82 80 82 81

DOE

Calculation

(Note 1)

NWTRB

Calculation

(Note 4)

DOE

Calculation

(Note 1 & 5)

NWTRB

Calculation

(Note 4)

DOE

Calculation

(Note 1)

NWTRB

Calculation

(Note 4)

DOE

Calculation

(Note 1 & 5)

NWTRB

Calculation

141079 DTF1WP3952 WP PWR 0.4326 26.9 2050 66 67 3256.8 3071.7 516 81 68 81 N/A

141027 204003904 WPMPC BWR 0.1775 44.2 2050 46 67 5566.6 5537.2 931 81 76 81 N/A

132617 202303001 WPMPC BWR 0.1726 44.6 2050 43 67 5739.6 5805.6 976 81 76 81 N/A

140808 203910205 WPMPC BWR 0.1775 44.0 2050 45 67 5627.0 5631.9 947 81 77 81 N/A

137451 203210206 WPMPC BWR 0.1775 42.7 2050 40 67 5940.0 5988.8 1007 81 77 81 N/A

138191 203405610 WPMPC BWR 0.1771 42.6 2050 36 67 6319.3 6413.2 1078 81 78 81 N/A

140044 DTF1WP3604 WP PWR 0.4132 30.9 2050 62 67 3719.2 3635.9 611 81 74 81 N/A

Total 36,168.6 36,084.3 N/A N/A N/A N/A N/A

Average 5,166.9 5,154.9 870 866 81 75 81 75

Notes: 1. Data taken from WP_Empl_Lin_Dana_YFF5_22kw_011707-236_050107a_Mod_R2.xls Sheet Emplaced (85C 4 year 18kw)

2. Data taken from WP_Decay_70k_22kw_011707.xls sheet WP_Decay

3. Data taken from Batch_Info.mdb

4. Calculated value

5. Represents a running average over 7 waste packages

Individual

Mid-pillar

Temperature

(deg-C)

WP Power at

Emplacement

(watts/WP)

Lineal Line Load

at Emplacement

(watt/meter)

Seven Segment Average

Mid-pillar Temperature

(deg-C)

Average

MTHM

(Note 4)

WP

ID

(Note 1)

Shipment

ID

(Note 1 & 2)

WP

Type

(Note (1)

SNF

Type

(Note 1)

Individual

Mid-pillar

Temperature

(deg-C)

WP Power at

Emplacement

(watts/WP)

Lineal Line Load

at Emplacement

(watt/meter) Emplacement

Year

(Note 1)

Average

Age at

Emplacement

(Years)

(Note 4)

Seven Segment Average

Mid-pillar Temperature

(deg-C)

Seven Segment Average

Mid-pillar Temperature

(deg-C)

WP

ID

(Note 1)

Shipment

ID

(Note 1 & 2)

WP

Type

(Note (1)

SNF

Type

(Note 1)

Average

MTHM

(Note 4)

Average

Age at

Emplacement

(Years)

(Note 4)

Ventilation

Duration

(Years)

(Note 4)

Ventilation

Duration

(Years)

(Note 4)

Average

Burnup

(Gwd/ton)

(Note 2)

Average

Burnup

(Gwd/ton)

(Note 2)

Average

Burnup

(Gwd/ton)

(Note 2)

Emplacement

Year

(Note 1)

Emplacement

Year

(Note 1)

WP Power at

Emplacement

(watts/WP)

Average

Age at

Emplacement

(Years)

(Note 4)

Ventilation

Duration

(Years)

(Note 4)

Individual

Mid-pillar

Temperature

(deg-C)

Lineal Line Load

at Emplacement

(watt/meter) WP

ID

(Note 1)

Shipment

ID

(Note 1 & 2)

WP

Type

(Note (1)

SNF

Type

(Note 1)

Average

MTHM

(Note 4)

These results are consistent with the results in Table 8. The waste package powers are in reasonable 

agreement, and the mid-pillar temperatures calculated by the NWTRB are consistently lower because of 

the use of a finite line source rather than an infinite line source. 

DOE also performed an engineering study, and the results were documented in data file 

WPLOAD_OUPUT_case_3b.TXT. A comparison of these results and the NWTRB calculations is 

provided in Table 10. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 20 of 27 

 

Table 10 Comparison of Mid-pillar Temperatures Calculated by WPLoad 

DOE

Calculation

(Note 2)

NWTRB

Calculation

(Note 8)

DOE

Calculation

(Note 5)

NWTRB

Calculation

(Note 8)

DOE

Calculation

(Note 6)

NWTRB

Calculation

(Note 8)

2620 21PWRTAD 5.85 2018 1666 - 1686

1666 - 1670 2149 58 4 1.7143 28.9 1979 39

1670 - 1686 1555 58 17 7.7513 31.5 1978 40

Total 21 9.4656

Average 0.4507 31.0 40 99 5,804 5,569 1.85 0.98 92 78

2641 21PWRTAD 5.85 2018 2214 -2234

2215 - 2221 4342 88 8 3.6568 30.0 1983 35

2222 - 2231 3345 88 13 5.9566 29.8 1981 37

Total 21 9.6134

Average 0.4578 29.9 36 99 5,977 5,832 1.71 0.98 91 78

209 21PWRTAD 5.85 2018 102 - 122

102 - 106 24975 3 & 4 5 1.7297 43.7 2009 9

107 - 115 24563 4, 5, & 6 9 3.1135 47.6 2008 10

116 - 120 24562 6 & 7 5 1.7297 43.7 2008 10

121 - 122 24049 7 & 8 2 0.6919 50.6 2006 12

Total 21 7.2648

Average 0.3459 46.0 10 99 13,011 12,560 1.74 2.11 98 92

2626 44BWRTAD 5.85 2018 1820 - 1863

1820 - 1853 26053 62 34 5.8486 50.6 2012 6

1854- 1863 26052 62 10 1.7202 47.1 2012 6

Total 44 7.5688

Average 0.1720 49.8 6 99 17,983 16,582 1.88 2.79 91 86

2621 21PWRTAD 5.85 2018 1687 - 1707

1687 - 1707 1554 59 21 9.5730 28.4 1978 40

Total 21 9.5730

Average 0.4559 28.4 40 99 5,297 5,037 1.73 0.85 87 73

245 21PWRTAD 5.85 2018 123 - 143

123 - 125 24049 8 3 1.0379 50.6 2006 12

126 - 129 24049 9 4 1.3838 50.6 2006 12

130 - 133 24049 10 4 1.3838 50.6 2006 12

134 - 135 24049 11 2 0.6919 50.6 2006 12

136 - 137 24048 11 2 0.6919 46.8 2006 12

138 - 141 16830 12 4 1.5404 12.2 1996 22

142 16831 12 1 0.3867 12.5 1996 22

143 16821 13 1 0.3852 18.6 1996 22

Total 21 7.5016

Average 0.3572 39.6 15 99 10,450 9,256 1.69 1.56 86 84

2628 21PWRTAD 5.85 2018 1872 - 1892

1872 24976 65 1 0.3459 47.6 2009 9

1873 - 1880 24048 65, 66 & 67 8 2.7675 46.8 2006 12

1881 - 1884 24047 68 4 1.3839 43.0 2006 12

1885 - 1892 23605 69 & 70 8 3.4595 53.2 2005 13

Total 21 7.9568

Average 0.3789 48.6 12 99 12,996 13,700 1.72 2.30 101 102

Average 10,217 9,791 1.72 1.65 92 85

Notes: 1. Taken from WPLOAD_OUPUT_case_3b.TXT Section 1

2. Taken from WPLOAD_OUPUT_case_3b.TXT Section 6 Based on 7 package segment analysis 9,791 1.65 85

3. Taken from WPLOAD_OUPUT_case_3b.TXT Section 8

4. Taken from WPLOAD_OUPUT_case_3b.TXT Section 9

5. Taken from WPLOAD_OUPUT_case_3b.TXT Section 13

6. Taken from WPLOAD_OUPUT_case_3b.TXT Section 14

7. Taken from WASTESTREAM.TXT

8. Calculated

WP

ID

(Note 2 & )

WP

Type

(Note 2)

Shipping

Cask

Number

(Notes 4 & 7)

Emplacement

Year

(Note 2 & 4)

Assembly

Numbers

(Note 3 & )

Batch

Number

(Note 4 & 7)

WP

Length

(Note 1)

Lineal Line Load

at Emplacement

(watt/meter) Number

Assemblies

(Note 7)

Year

Discharged

(Note 4 & 7)

Average

Age at

Emplacement

(Years)

(Note 8)

Ventilation

Duration

(Years)

(Note 8)

Individual

Mid-pillar

Temperature

(deg-C)

WP Power at

Emplacement

(watts/WP)

Average

Burnup

(Gwd/ton)

(Note 7)

Total

MTU

(Note 7)

As with the TSM calculations, the NWTRB and DOE results are reasonably consistent and show the 

same trend; the DOE calculations tend to calculate a higher mid-pillar temperature than the NWTRB 

calculations. 

The result of this evaluation is that the approach and calculation methods used by the DOE appear 

reasonable and consistent with the completely independent calculations performed by the NWTRB staff. 

The “Power at Emplacement” results from the DOE model are potentially more accurate than the 

NWTRB results because the finer detail used in calculating the waste package power and the “Peak 

Postclosure Mid-pillar Temperature” is potentially more conservative because of the use of an infinitelength drift model. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 21 of 27 

 

4.2.3. Mid-pillar Temperature Evaluation 

The approach used by DOE to determine the maximum mid-pillar temperature limit was to predict the 

incoming waste stream and calculate the mid-pillar temperature on the basis of this waste stream. The 

approach taken by the NWTRB was to determine the age of the waste at closure that would prevent 

exceeding the 96o

C mid-pillar temperature for various assembly burnup, age, and ventilation durations. 

Each calculation assumed a similar seven-package segment replicated throughout the repository 

containing one DOE long co-disposal waste package, one DOE short co-disposal waste package, and 

five PWR waste packages with the same characteristics. Several calculations using the methodology 

described in Section 2.2 were performed, and the age of the fuel was adjusted until the peak postclosure 

mid-pillar temperature approached 96o

C without exceeding the 96oC limit. In some cases, the minimum 

age at emplacement of 5 years results in a temperature that does not approach the 96o

C limit. The results 

are shown in Table 11 and Figure 9. 

Table 11 Thermal Conditions to Maintain Mid-pillar Temperature Below 96oC 

Assembly

Burunup

(Gwd/ton)

Ventilation

Duration

(Years)

Age at

Emplacement

(Years from 

reactor 

discharge)

Line Load at

Emplacement

(kW/meter)

Age at 

Closure

(Years from 

reactor 

discharge)

Line Load at

Closure

(kW/meter)

Time of Peak

Drift Wall

Temperature

(Years from 

Emplacement)

Peak

Drift Wall

Temperature

(oC)

Time of Peak

Mid-pillar

Temperature

(Years from 

Emplacement)

Peak

Mid-pillar

Temperature

(oC)

35 25 5 2.2 30 0.93 39 189 302 90

40 25 12 1.58 37 0.96 40 194 331 96

45 25 31 1.25 56 0.82 43 172 360 96

50 25 52 1.01 77 0.7 48 155 372 96

55 25 69 0.89 94 0.64 56 147 365 96

60 25 84 0.81 109 0.6 74 144 356 96

40 50 5 2.53 55 0.71 67 161 392 91

45 50 12 1.82 62 0.74 68 166 383 96

50 50 31 1.4 81 0.65 73 153 394 96

55 50 47 1.2 97 0.6 82 146 386 96

60 50 61 1.07 111 0.58 101 145 378 96

45 75 5 2.9 80 0.58 97 146 420 92

50 75 12 2.05 87 0.6 100 150 412 96

55 75 29 1.62 104 0.55 113 144 406 96

60 75 43 1.41 118 0.53 136 143 397 96

50 100 5 3.28 105 0.49 135 138 440 93

55 100 12 2.32 112 0.52 145 142 422 96

60 100 27 1.86 127 0.49 172 141 422 96

Figure 9 Thermal Conditions at Emplacement to Maintain Mid-pillar Temperature Below 96oC 

0

10

20

30

40

50

60

70

80

90

35 40 45 50 55 60

Assembly Burnup - Gwd/ton

Age of Assemblies at 

Emplacement - Years Out of 

Reactor

25 Years Ventilation 50 Years Ventilation

75 Years Ventilation 100 Years Ventilation

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

3.25

3.5

3.75

4

35 40 45 50 55 60 65

Assembly Burnup - Gwd/ton

Linear Line Load at 

Emplacement- kW/meter

25 Years Ventilation 50 Years Ventilation

75 Years Ventilation 100 Years Ventilation
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 22 of 27 

 

These results indicate that waste with an average burnup of approximately 48 Gwd/ton could be 

emplaced when it is approximately 5 years old out of reactor (which would correspond to an average 

linear line of approximately 3.0 kW/meter) and ventilated for 75 years, and the mid-pillar temperature 

would still be maintained below 96o

C. The same data, but plotted in terms of conditions at closure, are 

presented in Figure 10. 

Figure 10 Thermal Conditions at Closure for Maintaining Mid-pillar Temperature Below 96oC 

0

20

40

60

80

100

120

140

35 40 45 50 55 60

Assembly Burnup - Gwd/ton

Age of Assemblies at 

Closure - Years Out of 

Reactor

25 Years Ventilation 50 Years Ventilation

75 Years Ventilation 100 Years Ventilation

0.4

0.5

0.6

0.7

0.8

0.9

1

35 40 45 50 55 60 65

Assembly Burnup - Gwd/ton

Linear Line Load at ClosurekW/meter

25 Years Ventilation 50 Years Ventilation

75 Years Ventilation 100 Years Ventilation

The data indicate that the duration of the preclosure ventilation phase has less influence on the mid-pillar 

temperature than does the age of the waste at closure. The following can be observed from Table 11: 

Table 12 Effects of Ventilation Duration on Age of Assemblies at Closure 

Assembly Burnup 

(Gwd/ton) 

Ventilation 

Duration 

(Years) 

Age of Assemblies 

Not to Exceed 96o

C Mid-pillar 

Temperature 

(Years Out of Reactor) 

50 50 81 

50 75 87 

Therefore, for 50 Gwd/ton assemblies, ventilating for an additional 25 years reduces the age at closure 

by only 6 years. 

4.2.4. Drift Wall Temperature Evaluation 

The results shown in Table 10 confirm that the mid-pillar thermal limit is reached before the drift wall 

temperature limit. If the mid-pillar temperature limit were eliminated, the thermal power of the 

emplaced waste could be increased. To determine how much the thermal power could be increased, the 

age of the waste at closure was determined that would prevent exceeding the 200o

C drift wall 

temperature for various assembly burnup, age, and ventilation durations. Each calculation assumed a 

similar seven-package segment replicated throughout the repository containing one DOE long codisposal waste package, one DOE short co-disposal waste package, and five PWR waste packages with 

the same characteristics. Several calculations using the method described in Section 2.2 were performed, 

and the age of the fuel was adjusted until the peak postclosure drift wall temperature approaches 200o

C 

without exceeding the 200o

C limit. In some cases, the minimum age at emplacement of 5 years results in 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 23 of 27 

 

a temperature that does not approach the 200oC limit. The results are provided in Table 12 and Figures 

11 and 13. 

Table 13 Thermal Condition for Maintaining Drift Wall Temperature Below 200oC 

Assembly

Burunup

(Gwd/ton)

Ventilation

Duration

(Years)

Age at

Emplacement

(Years from 

reactor 

discharge)

Line Load at

Emplacement

(kW/meter)

Age at 

Closure

(Years from 

reactor 

discharge)

Line Load at

Closure

(kW/meter)

Time of Peak

Drift Wall

Temperature

(Years from 

Emplacement)

Peak

Drift Wall

Temperature

(oC)

Time of Peak

Mid-pillar

Temperature

(Years from 

Emplacement)

Peak

Mid-pillar

Temperature

(oC)

35 25 5 2.20 30 0.93 39 190 302 90

37 25 5 2.33 30 0.99 39 200 296 94

40 25 10 1.71 35 0.99 40 200 322 97

45 25 19 1.54 44 0.98 41 199 330 102

50 25 27 1.50 52 0.97 42 200 334 106

55 25 35 1.46 60 0.96 43 199 328 109

60 25 42 1.43 67 0.95 44 199 322 112

49 50 5 3.20 55 0.89 66 197 352 106

50 50 6 2.93 56 0.90 66 198 352 107

52 50 9 2.46 59 0.90 67 198 349 108

53 50 10 2.39 60 0.90 67 199 347 109

54 50 11 2.35 61 0.91 67 200 345 110

55 50 13 2.25 63 0.90 67 199 346 110

56 50 14 2.24 64 0.90 68 200 344 111

58 50 17 2.17 67 0.90 68 200 343 112

60 50 20 2.12 70 0.89 69 199 342 113

60 75 5 4.11 80 0.78 95 189 364 112

Figure 11 Thermal Condition at Emplacement for Maintaining Drift Wall Temperature Below 200o

C 

 

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

35 40 45 50 55 60 65

Assembly Burnup - Gwd/ton

Linear Line Load at EmplacementkW/meter

25 Years Ventilation 50 Years Ventilation

0

5

10

15

20

25

30

35

40

45

35 40 45 50 55 60

Assembly Burnup - Gwd/ton

Age of Assemblies at 

Emplacement - Years Out of 

Reactor

25 Years Ventilation 50 Years Ventilation
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 24 of 27 

 

Figure 12 Thermal Condition at Closure for Maintaining Drift Wall Temperature Below 200o

C 

0

10

20

30

40

50

60

70

80

35 40 45 50 55 60

Assembly Burnup - Gwd/ton

Age of Assemblies at Closure - Years 

Out of Reactor

25 Years Ventilation 50 Years Ventilation

0.75

0.85

0.95

1.05

1.15

1.25

35 40 45 50 55 60 65

Assembly Burnup - Gwd/ton

Linear Line Load at ClosurekW/m eter

25 Years Ventilation 50 Years Ventilation

These results indicate that for CSNF with an average burnup of approximately 48 Gwd/ton, the 

ventilation duration could be reduced from 75 years to approximately 50 years and drift wall 

temperature would be maintained below 200o

C. 

4.2.5. Waste Package Surface Temperature 

For calculating the waste package surface temperature, the drift wall surface temperatures were first 

calculated using the method described in Section 2.2. On the basis of the drift wall temperature, the drip 

shield and waste package surface temperatures were calculated according to the method described in 

Section 2.4. A plot of drift wall, drip shield, and waste package temperatures is shown in Figure 13 for 

the case of 50 Gwd/ton assemblies emplaced 27 years out of reactor and ventilated for 25 years. 

Figure 13 Waste Package Temperature for 200oC Maximum Drift Wall Temperature 

0 100 200 300

25

50

75

100

125

150

175

200

225

Drift Wall Temperature

Drip Shield Temperature

Waste Package Temperature

Time From Emplacement - Years

Temperature - deg-C

The waste package surface temperature was calculated for the same cases described in Sections 4.2.3 

and 4.2.4. The results are presented in Table 14 and Table 15. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 25 of 27 

 

Table 14 Peak Temperatures for 96o

C Mid-Pillar Temperatures 

Assembly

Burunup

(Gwd/ton)

Ventilation

Duration

(Years)

Age at

Emplacement

(Years from 

reactor 

discharge)

Line Load at

Emplacement

(kW/meter)

Age at 

Closure

(Years from 

reactor 

discharge)

Line Load at

Closure

(kW/meter)

Time of Peak

Drift Wall

Temperature

(Years from 

Emplacement)

Peak

Drift Wall

Temperature

(oC)

Peak

Drip Shield

Temperature

(oC)

Peak

Waste 

Package

Temperature

(oC)

35 25 5 2.2 30 0.93 39 190 196 204

38 25 5 2.4 30 1.02 39 205 211 219

40 25 12 1.58 37 0.96 40 195 201 209

45 25 31 1.25 56 0.82 43 173 179 187

50 25 52 1.01 77 0.7 48 155 161 168

55 25 69 0.89 94 0.64 56 147 153 160

60 25 84 0.81 109 0.6 74 145 149 156

40 50 5 2.53 55 0.71 66 162 168 175

43 50 5 2.75 55 0.77 66 174 179 187

45 50 12 1.82 62 0.74 68 167 173 180

50 50 31 1.4 81 0.65 73 153 159 166

55 50 47 1.2 97 0.6 81 147 152 159

60 50 61 1.07 111 0.58 96 145 149 156

45 75 5 2.9 80 0.58 96 146 151 158

48 75 5 3.13 80 0.62 96 155 160 167

49 75 8 2.42 83 0.61 97 154 159 166

50 75 12 2.05 87 0.6 99 151 156 163

51 75 16 1.89 91 0.58 101 149 154 160

52 75 19 1.82 94 0.58 102 148 152 159

53 75 22 1.76 97 0.57 104 147 151 158

54 75 26 1.67 101 0.56 107 145 150 156

55 75 29 1.62 104 0.55 110 144 149 155

60 75 43 1.41 118 0.53 128 143 147 153

50 100 5 3.28 105 0.49 135 138 143 149

53 100 5 3.53 105 0.52 132 145 150 156

54 100 8 2.74 108 0.52 135 144 149 155

55 100 12 2.32 112 0.52 140 142 147 153

56 100 15 2.18 115 0.51 144 142 146 152

57 100 18 2.08 118 0.5 148 142 146 152

60 100 27 1.86 127 0.49 161 141 145 150

Table 15 Peak Temperatures for 200oC Drift Wall Temperatures 

Assembly

Burunup

(Gwd/ton)

Ventilation

Duration

(Years)

Age at

Emplacement

(Years from 

reactor 

discharge)

Line Load at

Emplaceme

nt

(kW/meter)

Age at 

Closure

(Years from 

reactor 

discharge)

Line Load at

Closure

(kW/meter)

Time of Peak

Drift Wall

Temperature

(Years from 

Emplacement)

Peak

Drift Wall

Temperature

(oC)

Peak

Drip Shield

Temperature

(oC)

Peak

Waste 

Package

Temperature

(oC)

35 25 5 2.2 30 0.93 39 190 196 204

37 25 5 2.33 30 0.99 39 200 206 214

40 25 10 1.71 35 0.99 40 200 206 214

45 25 19 1.54 44 0.98 41 199 205 213

50 25 27 1.5 52 0.97 42 200 206 214

55 25 35 1.46 60 0.96 43 199 205 212

60 25 42 1.43 67 0.95 44 199 205 213

49 50 5 3.2 55 0.89 66 197 203 210

50 50 6 2.93 56 0.90 66 198 204 211

52 50 9 2.46 59 0.90 67 198 204 211

53 50 10 2.39 60 0.90 67 199 205 212

54 50 11 2.35 61 0.91 67 200 206 214

55 50 13 2.25 63 0.90 67 199 205 212

56 50 14 2.24 64 0.90 68 200 206 213

58 50 17 2.17 67 0.90 68 200 205 213

60 50 20 2.12 70 0.89 69 199 205 212

60 75 5 4.11 80 0.78 95 189 194 201

For all cases in Table 14 and Table 15, the peak waste package temperature is well below the maximum 

allowable limit of 300o

C. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 26 of 27 

 

4.2.6. Discussion 

The DOE postclosure thermal response to support the LA is based on an estimated limiting waste 

stream. However, performing more-general evaluations to determine the maximum thermal energy that 

can be absorbed by the mountain would provide increased flexibility to control when the repository may 

be closed by adjusting the ventilation duration and/or the linear line load, regardless of the waste 

emplaced. 

This analysis shows that the postclosure thermal response is more a function of the initial conditions at 

the start of closure than it is of the conditions at waste emplacement. The amount of preheat of the 

mountain during the preclosure period has some influence on the postclosure thermal response, but the 

power of the waste at closure has a much greater influence. This is because the ventilation system 

removes between 80 and 90 percent of the heat during the ventilation phase. Therefore, the use of 

surface aging of the waste has limited benefit in reducing the postclosure mid-pillar peak temperature. 

On the basis of the characteristics of the waste that is expected to be received, the mid-pillar temperature 

limit of 96o

C can be maintained and the repository can be closed in approximately 75 years after 

completion of waste emplacement. If the mid-pillar temperature limit were eliminated, the drift wall 

temperature limit would be controlling and the preclosure ventilation duration could be reduced by 

approximately 25 years. This preclosure period could be decreased further if the linear line load at 

emplacement is reduced by either de-rating the waste packages or increasing the spacing between waste 

packages. 

5. Conclusions 

5.1. General 

DOE has made significant advances in both their understanding of the repository thermal response and in 

the thermal methodology for repository operation. However, their thermal strategy is still based on an 

estimated limiting waste stream rather than on the amount of thermal energy that the mountain can absorb 

without exceeding any established thermal limit. Performing more-general evaluations, as presented in this 

document, to determine the maximum thermal energy that can be absorbed by the mountain would provide 

increased flexibility to control when the repository may be closed by changing the ventilation duration 

and/or the linear line load, regardless of the waste emplaced. This is especially true during the preclosure 

phase, where DOE has established rather arbitrary thermal limits for emplacement that are based on the 

predicted waste stream. The current 2.0 kW/meter thermal limit provides approximately 90o

C margin to the 

200o

C drift wall temperature limit for the 30 day loss of forced ventilation event. Increasing the allowable 

thermal limits at emplacement would allow the majority of waste received at the repository to be emplaced 

immediately, thus reducing the amount of surface storage. 

5.2. Comments and Suggestions 

As a result of this evaluation and review of DOE calculations, the Board offers the following comments and 

suggestions. 

1. The calculations performed by DOE and the NWTRB staff are in reasonable agreement and suggest 

that the DOE technical basis is valid. 

2. Establishing the thermal limits for emplacement based on the estimated limiting waste stream 

appears to be overly conservative. DOE should reevaluate the thermal limits for emplacement 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page 27 of 27 

 

according to the capacity of the mountain to absorb the thermal energy rather than according to the 

predicted waste stream. This would result in higher thermal limits for emplacement and reduce the 

amount of surface storage required at the repository. 

3. Initiating the loss of forced ventilation during the preclosure period at 30 days after emplacement 

may not be the most limiting case. DOE should reevaluate this event and determine when the loss of 

forced ventilation results in the shortest time to reach the 200o

C drift wall thermal limit. 

4. Surface aging of the hotter CSNF, as opposed to subsurface aging, has limited benefit on the 

postclosure thermal response. DOE should reevaluate the need to surface age the CSNF at the 

repository. 

5. The DOE postclosure thermal response for supporting the license application (LA) is based on an 

estimated limiting waste stream. However, performing more-general evaluations, as presented in this 

document, to determine the maximum thermal energy that can be absorbed by the mountain would 

provide increased flexibility to control when the repository may be closed by adjusting the 

ventilation duration and/or the linear line load, regardless of the waste emplaced. 

6. The Board agrees with DOE that if the 96o

C mid-pillar temperature limit is required, this limit 

determines the maximum allowable thermal loading. However, if the 96o

C limit were eliminated, the 

drift wall maximum temperature of 200o

C would determine the maximum allowable thermal loading 

and the preclosure ventilation period could be decreased by approximately 25 years. 

7. Thermal conductivity has a significant effect on the peak temperatures. Variations in the thermal 

conductivity used in the calculations to predict the repository thermal response should be taken into 

account on the basis of actual thermal conductivity measurements along with appropriate 

uncertainties and margins. 
 

Appendix A 

Development and Basis for Calculation Models 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-1 

 

Analysis of Repository Temperatures Using Analytical Solutions 

Introduction: Repository rock temperatures can be calculated using the analytical solution for a finite-length 

decaying line source. Constant-efficiency ventilation can also be simulated with this line-source solution. 

Repository rock temperatures include all temperatures from the drift wall out into the rock mass at any position 

in the repository. The drift wall temperature at the end of a drift where a package is at the end of the drift is an 

upper-bound approximation. In-drift temperatures are calculated after rock temperatures are determined by 

working back into the drift from the drift wall temperature. Waste package powers are approximated as a sum 

of three decaying exponentials for use in the analytical solution. Based on experience any waste package 

power, or combination of waste package powers, can be adequately approximated as a sum of three decaying 

exponentials. 

The principle of superposition is used to calculate the temperature anywhere in the rock mass by repeated use of 

the analytical solution for a finite-length decaying line source. Thus for any location in the rock, the 

temperature is determined from the temperature contribution of all sources, or waste packages, in the same drift 

and in neighboring drifts. The analytical solution for a finite-length decaying line source is simply “placed” at 

all locations of interest by shifting the coordinate frame, a temperature calculated for the location of interest, 

and all the temperatures summed for the final temperature. 

The validation, or verification, of the use of the analytical solution for a finite-length decaying line source for 

temperature predictions has been conducted by comparing predicted temperatures for special cases with those 

published in Analysis/Model Reports. The implementation of the prediction of repository temperatures using 

the line-source solution is accomplished with Mathcad. Details follow here for the line-source analytical 

solution and also discussions of why a single-medium solution can be used instead of taking into account 

stratigraphy, drift wall temperature variation as a function of waste-package power variation, ventilation, and 

neighboring drifts. 

The Finite-Length Decaying Line Source: The derivation of the temperature due to a finite-length decaying 

line source proceeds from the solution for a constant point source as described by H.S. Carslaw and J.C. Jaeger, 

Conduction of Heat in Solids, 1959, Section 10.4, Equation (2). This solution for a constant point source is used 

to derive the solution for a decaying point source through the use of Laplace transforms. This decaying-pointsource solution is then integrated over a finite length to obtain the solution for a finite-length decaying line 

source. The details of this derivation are presented in the Section A-1: Temperatures Due to Decaying Point 

and Line Sources. The temperature due to a finite-length decaying line source is: 

2 2 ( )

4

0

( , , ,) 8 2 2

t t x y Qe z z L e v x y z t erf erf e d K

λ λθ

κθ θ

π θ κθ κθ

− + ⎧ ⎫ ⎛ ⎞⎛ ⎞ − −

= − ⎨ ⎬ ⎜ ⎟⎜ ⎟ ⎩ ⎭ ⎝ ⎠⎝ ⎠ ∫ (1) 

 

where z is the location on the axis and the source is centered at z = +L/2, Q is the lead coefficient in 

power/meter that describes the exponential decay of the source with a rate of decay of λ reciprocal time, K is 

the thermal conductivity, L is the source length, κ is the thermal diffusivity, and x and y are points in the 

medium. For a number of equal-length sources lined up on the z axis, the temperature is: 

 

 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-2 

 

1

( , , , ) ( , , ( 1) , )

n

i

i

vx yzt v x yz i Lt == −− ∑ (2) 

where the subscript i now is applied to the decaying power as Qi and λi. When neighboring sources are to be 

taken into account and the temperature in the plane is of interest, set y = 0 and x equal to the drift spacing, or 

multiples of the drift spacing, and repeat the above equation. 

Decaying Sources: The waste package powers, and for all intents and purposes, can be represented as 

decaying sources. More specifically, the meaning of the word “decaying” is that of a decaying exponential. 

The above equations require that the source be represented as: 

t Q qe−λ = (3) 

The actual use of this representation of a decaying source is implemented as a sum of three such terms as: 

1

i

n t

i

i

Q qe−λ

=

= ∑ (4) 

This summation for three terms is what works; all of the waste package powers, or assembly powers, can be 

represented with this three-term summation. This then requires that the equations for temperature above be 

solved three times and the results added to obtain the final temperature. This is an application of the 

superposition principle. An example of this three-term summation for a 1.42 kW/meter line load at 

emplacement is: 

0.33 0.022 0.0014 101 1052 266 t tt Qe e e − −− =⋅ + ⋅ +⋅ (5) 

where the coefficients are watts/meter and the time is years. This particulate summation describes the line load 

for Enhanced Design Alternative II (EDA-II) from the License Application Design Selection report and is 

essentially the same power decay used in the Total System Performance Analysis to within a scale factor of 

1.021 times the above, which yields 1.45 kW/meter at emplacement (Bechtel SAIC Company 2005a). 

Drift Wall Temperature Predictions: Drift wall temperature predictions are obtained by setting x in Equation 

1 to the drift radius. Since the line-source solution considers only a homogeneous medium, the drift region is 

filled with that medium, i.e., rock. In order to demonstrate that this line solution yields the correct temperatures 

for a drift wall, the solution for the region bounded internally by a cylinder from Carslaw and Jaeger, 

Conduction of Heat in Solids, Section 13.5, Equation 16, is used. This solution is for a constant heat flux 

applied to the wall and is the basis for deriving the solution for a decaying flux applied to the wall. This 

decaying flux is the same form as that given above in Equation 4. The solution for the region bounded 

internally by a cylinder, i.e., a tunnel, is derived in Section A-2: Decaying Cylindrical Source Derivations. The 

temperature for the region bounded internally by a cylinder, i.e., a tunnel in the rock, is: 

 2

01 0 1

2 22

0 1 1

2 ( )( ) ( )( ) ( ) () ()

t ut

p

Qe e J ur Y ua Y ur J ua v r du

C u J ua Y ua

λ κ

πρ λ κ

∞ − − − − = ⋅ − + ∫ (6) 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-3 

 

where Q is the energy wall flux, “a” is the cylinder (tunnel) radius, ρ is the bulk density, Cp is the heat capacity, 

κ is the thermal diffusivity, Jν(z) is the Bessel function of the first kind of order ν, and Yν(z) is the Bessel 

function of the second kind of order ν. This equation is two dimensional in r and θ (or x and y) and thus 

represents a drift of infinite length. The corresponding equation for a line source of infinite length is obtained 

by taking the limit in Equation 1 as the source length becomes infinite, and this limit yields: 

2

4

0

( ) 4

t t r Qe e vr e d

K

λ λθ

κθ θ

π θ

− −

= ⋅ ∫ (7) 

Equation 6 is used to verify that the decaying line-source solution, Equation 7, approximates the drift wall 

temperature by setting r in both equations to the drift radius. The results of this verification show that the 

temperatures at the drift wall are essentially the same for times of interest, longer than 5 years, for physical 

properties pertinent to the medium of interest. 

Ventilation Simulation: The removal of energy by ventilation from a power source in a drift is simulated by 

applying a constant efficiency over the ventilation duration. The ventilation efficiency is defined as the amount 

of energy removed by ventilation air divided by the source energy during the ventilation duration. The 

ventilation efficiency is typically around 86 to 88 percent for the parameters of interest (Bechtel SAIC 

Company 2004). Given that the solution for the temperature of the drift wall, or anywhere in the rock, is 

available as: 

v ft = ( ) (8) 

where f(t) is given by equation (1) or (2), then the drift wall temperature due to a constant ventilation efficiency, 

ε, for duration tv is: 

(1 ) ( ) v v f t for t t =− ≤ ε (9) 

At t = tv, the ventilation ceases and all the source power is delivered to the rock. At this time, the fraction of the 

source power applied as an addition is ε, and the effect of this additional power is to be added to the (1-ε) 

solution. However, the contribution to the drift wall temperature from the application of the ε fraction begins at 

t = tv, and thus the ε fraction solution must have its t = 0 begin at t = tv. As a result, the drift wall temperature, 

or rock temperature, for t > tv is written as: 

(1 ) ( ) ( ) v v v f t f t t for t t = − +− ≥ ε ε (10) 

The solutions f(t) and f(t-tv) are additive because these heat-transfer solutions are linear. 

It is important to recognize that the solution that “starts” at t-tv have its source power also begin with t = 0 at t = 

tv. Each solution above must have its own time scale. To illustrate, suppose that the source power is written as 

equation (3). Equation (3) applies at the start of the problem, applies for all t, and the solution is multiplied by 

(1-ε) as indicated in equation (10). At the end of ventilation the additional power that applies is: 

( ) v v t tt Q qe e for t tv− −− λ λ = > (11) 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-4 

 

Thus the time argument of the exponential begins at zero when t = tv. The temperature solution for this source 

is multiplied by ε and added to the (1-ε) solution. Note that the lead coefficient of this power is “decayed.” 

Stratigraphy, Hydrology and Dryout: Equations (1), (6) and (7) for temperature predictions as presented 

above are based on an infinite homogeneous medium that is described by one set of physical properties. These 

equations are generally referred to as “conduction only” equations. Therefore, there is no stratigraphy, no water 

flow or movement, no latent heat effects due to the vaporization of water, and only a single effective thermal 

conductivity. The purpose of this section is to show that these phenomena are not important for predicting 

temperatures for the situations of interest and that heat transfer by “conduction only” is sufficient. 

The recognition that conduction-only predictions are adequate is documented in the Ventilation Model and 

Analysis Report (Bechtel SAIC Company 2004, Section 6.4.1.3). This document states that conduction heat 

transfer dominates other heat-transfer mechanisms (i.e., convection in fractures and lithophysae, and latent heat) 

in the host rock. This statement originates from Sass et al. 1988 (page 35). This statement is supported by 

conclusions of data and modeling of the Drift Scale Test (Birkholzer and Tsang, 2000, page 1439). The

Ventilation Model and Analysis Report concludes that for the level of confidence required for the ventilation 

model, the assertion that conduction dominates the heat transfer in the host rock is consistent with the validity 

of the conceptual model (for the ventilation model). 

The assertion that “conduction only” is consistent and sufficient for the objective of predicting temperatures in 

Yucca Mountain can be verified by comparing conduction-only temperature predictions with thermohyologybased predictions. This has been done for a drift of infinite length as described by equations (6) and (7) and 

results from the Multiscale Thermohydrologic Model (MSTHM) (Bechtel SAIC Company 2005b). The 

locations chosen from the MSTHM are in the middle of the repository so that there is little or no edge cooling 

effect. These MSTHM predictions are obtained from the two-dimension line-averaged-heat-source, drift-scale, 

thermohydrologic (2-D LDTH) model for the mean thermal conductivity and mean infiltration rate for a BWR 

package that is very close to the line average power of 1.45 kW/meter. The 2-D LDTH peak drift wall 

temperature prediction is 136.8°C. The analytical solution prediction using equation (7) for the peak drift wall 

temperature is 137.5°C. This temperature difference is considered acceptable. The 2-D LDTH peak mid-pillar 

temperatures at two locations in the middle of the repository are 87.3 and 89.6°C. The analytical solution 

prediction using equation (7) for the peak mid-pillar temperature is 87.8°C. Likewise, these temperatures are 

considered acceptable. These analytical results are based on the calculated thermal-physical properties of 

thermal conductivity at 1.83 W/(m K), heat capacity at 1119 Joules/(kg C) and bulk rock density at 2097 kg/m3

. 

The reason these are calculated properties is that they take into account 90 percent saturation of the rock. 

The comparison of peak drift wall and peak mid-pillar temperatures from the two prediction techniques, 

analytical conduction only and thermohydrologic, shows that the temperatures agree to within a few degrees. 

Thus, stratigraphy, hydrology and rock dryout as described in the MSTHM do not have a significant effect on 

temperature predictions. 

The effect of infiltration on temperature can be investigated by considering the following simplified problem. 

Consider a slab 600-meters thick with the temperature at each face fixed and a heated plane at the 300-meter 

location. This plane is heated so that the steady-state temperature is about 88°C, i.e., 88°C above ambient. 

Water flows from top to bottom through this slab at a specified rate. Solve for the temperature profile as a 

function of water-flow rate, and compare the temperature profile with the no-flow profile. This problem is 

described by J. Bear in Dynamics of Fluids in Porous Media (1988, page 653). This problem was solved with 

parameters pertinent to Yucca Mountain and water-flow rates of 0, 4 and 10-mm/year. The zero water-flow-
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-5 

 

rate temperature of the heat source is 88°C, the 4-mm/year water-flow-rate temperature is 87.9°C, and the 10-

mm/year water-flow-rate temperature is 87.6°C. Thus, for the anticipated percolation flux at, or through, Yucca 

Mountain, the effect on temperature at the heated plane, and elsewhere, is too small to consider. 

Drift Wall Temperature Variation: The emplacement of waste packages of differing powers will result in 

varying drift wall temperatures opposite the packages. However, the effect of axial radiant-heat transfer in a 

drift loaded with varying-power packages is such that the axial drift wall temperature varies by about ±1.5°C at 

the time of occurrence of peak drift wall temperatures. At times greater than the occurrence of peak drift wall 

temperatures the variation is much less and scales approximately according the power relative to the peak 

power. This statement that the drift wall temperature variation is about ±1.5°C was determined by analyzing a 

simplified radiant-heat transfer problem. This simplified problem is based on a coaxial cylindrical heat source 

inside a cylinder (tunnel). The cylindrical heat source is represented as a string of adjacent one-meter-wide 

rings where the power of each ring can be specified. Five such one-meter-wide rings represent a waste package. 

The drift wall is likewise represented as a string of adjacent one-meter-wide rings but is not a power source. 

Because waste packages in a long drift will not “see” the ends of the drift, external heat loss from the outer rings 

is included. For all intents and purposes, all the source heat from the inner cylinder is transported through the 

outer rings to the adjacent rock medium. The details of the definition of this problem, the derivation, and the 

solution for parameters of interest are presented in Section A-3: Radiant Heat Transfer in a Coaxial 

Cylindrical System.

Because the maximum drift wall axial temperature variation is about 3°C, and much lower at longer time frames 

of a few hundred years, it is not necessary to simulate individual packages in a drift. An average line load can 

be used over the distance spanned by a segment of about seven waste packages, as illustrated in Section A-3. 

This allows the waste packages in an entire drift to be represented as a series of segments if desired, or the 

entire drift can be represented with a single line load. Given that the waste packages are about 5.8 meters in 

length, a seven-package segment is about 40 meters in length. Thus a 600-meter-long drift will contain about 

15 segments, each of which can be represented with equation (1) above. A reason for not using one line load 

for the entire drift is that it may be worthwhile to load segments at the ends of the drift with more power to take 

advantage of the edge-cooling effect. An advantage of using 15 segments, rather than individual packages, is 

that there are only 15 segments to keep track of rather than 105 packages. 

References: 

 

Bear, J., 1988, Dynamics of Fluids in Porous Media, Dover Publications. 

Bechtel SAIC Company, 2004, Ventilation Model and Analysis Report, ANL-EBS-MD-000030 REV04, Las 

Vegas, Nevada: Bechtel SAIC Company. 

Bechtel SAIC Company, 2005a, IED Waste Package Decay Heat Generation [Sheet 1 of 1], 800-IED-WIS0-

00701-000-00A, Las Vegas, Nevada: Bechtel SAIC Company. 

Bechtel SAIC Company, 2005b, Multiscale Thermohydrologic Model, ANL-EBS-MD-000049 REV03, Las 

Vegas, Nevada: Bechtel SAIC Company. 

 

Birkholzer, J.T. and Y.W. Tsang, 2000, Modeling the Thermal-Hydrologic Processes in a Large-Scale 

Underground Heater Test in Partially Saturated Fractured Tuff. Water Resources Research, 36, (6), 1431-

1447. Washington, D.C.: American Geophysical Union. 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-6 

 

Carslaw, H.S., and J.C. Jaeger, 1959, Conduction of Heat in Solids, Oxford University Press. 

Sass, J.H., A.H. Lachenbruch, W.W. Dudley, Jr., S.S. Priest, and R.J. Munroe, 1988, Temperature, Thermal 

Conductivity, and Heat Flow Near Yucca Mountain, Nevada: Some Tectonic and Hydrologic Implications. 

Open-File Report 87-649. Denver, Colorado: U.S. Geological Survey. 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-7 

 

Section A-1: Temperature Due to Decaying Point and Line Sources

Introduction: The derivation for the temperature due to a decaying line source proceeds from the solution for 

the temperature due to a constant point or line source. For the case for a decaying infinite-length line source, 

the starting point is the constant infinite-length line source as given by H.S. Carslaw and J.C. Jaeger, 

Conduction of Heat in Solids, Section 10.4, page 261, equation (5). For the case for a decaying finite-length 

line source, the starting point is the constant point source as given by Carslaw and Jaeger, Section 10.4, page 

261, equation (2). The solution for the constant point source is used as the starting point to derive the 

temperature for a decaying point source, and then this decaying point source is integrated over a finite length to 

yield the solution for a decaying finite-length line source. A decaying source is defined as an exponential 

decay, such as that which describes radioactive decay. The derivations presented here in order are the decaying 

infinite-length line source, the result of which is given in equation (A-1.15); the decaying point source, the 

result of which is given in equation (A-1.26); and the decaying finite-length line source, the result of which is 

given in equation (A-1.55). The Laplace transform is used in these derivations by recognizing that the 

transform solution due to a decaying source is obtained by replacing the transform parameter (denoted as p) 

with the transform parameter plus the rate of decay in the transform for a constant source, as explained below. 

The resulting Laplace transforms are then inverted as described in Carslaw and Jaeger, Chapter XII. 

Decaying Infinite-Length Line Source: The derivation of the temperature due to a decaying line source 

proceeds from the solution for the temperature due to a constant line source. The energy per unit time per unit 

length is defined by a decaying exponential, thus the name “decaying,” as: 

t Q Qeo−λ = (A-1.1) 

The temperature due to constant infinite-length line source is give by Carslaw and Jaeger, Section 10.4, page 

261, equation (5), as: 

2

2

/ 4 4 44

u

r t

Qe Q r v du Ei

u t κ πκ πκ κ

∞ − ⎛ ⎞ = =− −⎜ ⎟ ⎝ ⎠ ∫ A A (A-1.2) 

where the usual notation applies. Heat is liberated at the rate of ρCQl per unit time per unit length so that given 

a linear power as Q watts/meter, Ql = Q/ρC, and Ql has units of (m2 °C)/sec. 

Now consider the Laplace transform of v in equation (A-1.2) by using transform pair (26) in Carslaw and 

Jaeger, Appendix V, page 495. This transform pair is: 

( ) 2

1 1

2 1 1

/4

2 u

x t

p K x p x e u du υ υυ υ

υ

∞ −

⇔ − − −− ∫ (A-1.3) 

Set υ = 0 to obtain: 

( ) 2

0

/4

1

2

u

x t

K xp e du

p u

∞ −

⇔ ∫ (A-1.4) 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-8 

 

The right side above is in the same form as equation (A-1.2). To continue, use the following change of 

variables: 

2

2 r

x

κ

≡ (A-1.5) 

so that the transform pair in equation (A-1.4) becomes: 

( )

2

0

/ 4

1

2

u

r t

K rq e du

p u κ

∞ −

⇔ ∫ (A-1.6) 

where q2 ≡ p/κ as defined in Carslaw and Jaeger, Section 12.4, page 304, equation (2). The Laplace transform 

for a decaying line source becomes: 

( ) 0

2

Q K rq

v

πκ λ p = +

A (A-1.7) 

The reason (p + λ) replaces p in equation (A-1.7) is because the Laplace transform of a constant (boundary 

condition) is 1/p while the Laplace transform of e-λt

 is 1/(p + λ). Therefore, rearrange the above as: 

( ) ( ) 0 2

Q v p pv v K rq λ λ

πκ

+= + = A (A-1.8) 

Term-by-term inversion of the above equation is accomplished using Carslaw and Jaeger, Section 12.2, 

equation (2), page 299, and transform pair (23), Appendix V, page 495. Transform pair (23) written exactly as 

it appears is: 

( ) 2 / 4

0

1

2

x t K qx e

t

⇔ − κ (A-1.9) 

Now let x = r and rewrite the above as: 

( ) 2 /4

0

1

2

r t K qr e

t

⇔ − κ (A-1.10) 

and now invert equation (A-1.8) term by term to obtain: 

2 / 4

4

dv Q r t

v e

dt t

κ λ

πκ

− + = A (A-1.11) 

The integrating factor for this differential equation is e+λt

: 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-9 

 

( ) 2 / 4

4

t t

r t

d ve Q e e

dt t

λ λ

κ

πκ

+

− = A (A-1.12) 

Integrating between 0 and t yields: 

2 / 4

0 4

t

t r Q e ve e d

λθ

λ κθ θ

πκ θ

+

− = ∫ A (A-1.13) 

Clearing the exponential on the left side yields: 

2 /4

0 4

t t Qe e r v ed

λ λθ

κθ θ

πκ θ

− +

− = ∫ A (A-1.14) 

Now eliminate Ql and put in terms of Qo from equation (A-1.1) to obtain the final form of the temperature due 

to a decaying infinite-length decaying line source: 

2 / 4

0 4

t t Q eo e r v ed

K

λ λθ

κθ θ

π θ

− +

− = ∫ (A-1.15) 

Decaying Point Source: The derivation of the temperature due to a decaying point source proceeds from the 

solution for the temperature due to a constant point source. The Laplace transform is used again as described in 

the preceding section by replacing p with (p + λ) for the case of a decaying source as defined in equation (A1.1). 

The temperature due to a constant point source is given by Carslaw and Jaeger, Section 10.4, page 261, equation 

(2) as: 

4 4

Qp r v erfc

πκr κt

⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ (A-1.16) 

Heat is liberated at a rate QpρC. Now consider the Laplace transform of v above by using the transform pair (8) 

in Carslaw and Jaeger, Appendix V, page 494. This transform pair is: 

2

qx e x erfc

p κt

− ⎛ ⎞ ⇔ ⎜ ⎟ ⎝ ⎠ (A-1.17) 

Using this transform pair yields the Laplace transform of v in equation (A-1.16) as: 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-10 

 

4

qr Qp e

v

πκr p

−

= (A-1.18) 

where q2 ≡ p/κ as defined in Carslaw and Jaeger, Section 12.4, page 304, equation (2). The Laplace transform 

for a decaying source is written by replacing p with (p + λ) to obtain: 

4

qr Qp e

v

πκ λ r p

−

= + (A-1.19) 

To invert this equation, rewrite as: 

( ) 4

Qp qr v p pv v e

r

λ λ

πκ

− += + = (A-1.20) 

 

Term-by-term inversion of this equation is accomplished using Carslaw and Jaeger, Section 12.2, equation (2), 

page 299, and transform pair (6), Appendix V, page 494. Transform pair (6) is: 

2

4

3 2

x

qx t x

e e

t

κ

πκ

− − ⇔ (A-1.21) 

Using this information to invert equation (A-1.20) yields: 

( )

2 2

4 4

3/2 3 4 2 8

r r

p p t t dv r Q Q

v ee

dt r t t

κ κ λ

πκ πκ πκ

− −

+= = (A-1.22) 

The integrating factor for this differential equation is e+λt

, using this yields: 

( )

( )

2

4

3/2 8

t t r

p t

d ve Q e

e

dt t

λ λ

κ

πκ

−

= (A-1.23) 

Now integrate between 0 and t to obtain: 

( )

2

4

3/2 3/2 0 8

t r t t Qp e ve e d

λ

λ κθ θ

πκ θ

−

= ∫ (A-1.24) 

or: 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-11 

 

( )

2

4

3/2 3/2 8 0

t t r Q ep e v edλ λθ

κθ θ

πκ θ

− −

= ∫ (A-1.25) 

Now eliminate Qp and put in terms of Qo using the information following equation (A-1.16) to obtain the 

temperature due to a decaying point source: 

( )

2

4

3/2 3/2 8 0

t t r Q eo e v ed

C

λ λθ

κθ θ

ρ πκ θ

− −

= ∫ (A-1.26) 

Decaying Finite-Length Line Source: The derivation of the temperature due to a decaying finite-length line 

source proceeds from the temperature due to a decaying point source. Equation (A-1.25) is chosen for the 

starting point, and the meaning of the “power” coefficient, Qp, will be presented later. Rewrite the “r” term in 

equation (A-1.25) as: 

( ) ( ) ( ) 2 22 2

o oo r xx yy zz =− +− +− (A-1.27) 

and rewrite equation (A-1.25) with this expression for r as: 

( )

{( )( )( ) } 2 22

4

3/2 3/2 8 0

o oo xx yy zz t t t Q ep e ve d

λ λ

κθ θ

πκ θ

− − + − +− −

= ∫ (A-1.28) 

Now integrate with respect to dzo from z1 to z2 parallel to the z axis: 

( )

( ) {( )( ) } 2 2 2

2

1

4 4

3/2 3/2 8 0

o o o xx yy t z t z z

p

o

z

Q e e v e e d dz

λ λθ

κθ κθ θ

πκ θ

− − +− − − − −

= ∫ ∫ (A-1.29) 

and rearrange to: 

( )

( ) {( )( ) } 2 2 2

2

1

4 4

3/2 3/2 8 0

o o o xx yy t z t z z

p

o

z

Q e e v e dz e d

λ λθ

κθ κθ θ

πκ θ

− − +− − ⎧ ⎫ − − ⎪ ⎪ = ⎨ ⎬

⎪ ⎪ ⎩ ⎭

∫ ∫ (A-1.30) 

Now change variables according to: 

( )

2

o z z

u

κθ

− = (A-1.31) 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-12 

 

2

o dz du

κθ = − (A-1.32) 

2 o dz du = − κθ (A-1.33) 

and substitute accordingly so that equation (A-1.30) becomes: 

( ) ( ) {( )( ) } 2 2

2 2

1

4

3/2 3/2

0

2

8

o o o

o

xx yy t t z z

p u

z z

Q e e v e du e d

λ λθ

κθ κθ θ

πκ θ

− − +− − =

−

=

⎧ ⎫ ⎪ ⎪ = − ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ ∫ ∫ (A-1.34) 

 

Simplifying some algebra: 

( )

( )

( ) {( )( ) } 2 2

2 2

1

/ 2

4

3/2

0 / 2

2

8

o o xx yy t t z z

p u

z z

Q e e v e du e d

λ κθ λθ

κθ

κθ

θ

πκ θ

− − − +− −

−

−

⎧ ⎫ ⎪ ⎪ = − ⎨ ⎬

⎪ ⎪ ⎩ ⎭

∫ ∫ (A-1.35) 

Convert the du integral in the above equation to an error function. Let z1 = −L/2 and z2 = +L/2 centered on the 

origin so that the du integral becomes: 

( )

( ) ( )

( ) 2 22

/2 /2 /2 /2 0

/2 /2 /2 /2 0

zL zL

u uu

zL zL

e du e du e du

κθ κθ

κθ κθ

− −

− −−

+ +

= + ∫∫∫ (A-1.36) 

or: 

( ) ( )

( )

( ) 2 22

/2 /2 /2 /2 /2 /2

/2 /2 0 0

zL zL zL

u uu

z L

e du e du e du

κθ κθ κθ

κθ

− +−

− −−

+

=− + ∫ ∫∫ (A-1.37) 

 

Now use the definition of the error function from Carslaw and Jaeger, Appendix II, page 482, equation (1), 

which is: 

( ) 2

0

2 x

erf x e d ξ ξ

π

− ≡ ∫ (A-1.38) 

or: 

( ) 2

0 2

x

erf x e d π ξ ξ − = ∫ (A-1.39) 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-13 

 

and the du integrals in equation (A-1.37) become: 

( ) ( )

( )

( ) 2

/2 /2

/2 /2

/2 /2

2 2 2

z L

u

z L

zL zL

e du erf erf

κθ

κθ

π

κθ κθ

−

−

+

⎧⎪ ⎡ + − ⎤⎡ ⎤⎫⎪ =− − ⎨ ⎢ ⎥⎢ ⎥⎬ ⎩ ⎭ ⎪ ⎣ ⎦⎣ ⎦⎪ ∫ (A-1.40) 

 

Now substitute this result into equation (A-1.35), noting that the (-2) cancels and the square root of π from the 

erf yields π in the denominator leading the integral, and obtain the temperature due to a decaying finite-length 

line source: 

( ) ( ) ( ) 2 2

4

0

/2 /2 ( , , ,) 8 2 2

t x y t Q ep zL zL e v x y z t erf erf e d

λ λθ

κθ θ

πκ θ κθ κθ

− − + ⎧ ⎫ ⎪ ⎪ ⎡ ⎤⎡ ⎤ + − = − ⎨ ⎬ ⎢ ⎥⎢ ⎥ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦⎣ ⎦ ∫ (A-1.41) 

where x0 and yo have been set to the origin. When the integration in equation (A-1.29) was set up, the heat 

release per unit length became ρCQp. To verify that this is true, consider taking the limit of the above equation 

as L → ∞ to obtain the temperature due to a decaying infinite-length line source, which is equation (A-1.15), 

and then deriving the energy in the rock per unit length at time t. The limits as L → ∞ for the error functions 

are erf(∞) = 1, and erf(−∞) = −1, refer to Carslaw and Jaeger, Appendix II, page 482. Substituting these limits 

into the above yields: 

( ) { }

( ) 2 2

2

4

0

,, 2 8

t x y Q ep e v xyt e d

λ λθ

κθ θ

πκ θ

− − +

= ∫ (A-1.42) 

So, there is no z dependence, only x and y (and t), and this becomes: 

( )

( ) 2 2

4

0

, , 4

t x y t Q ep e v xyt e d

λ λθ

κθ θ

πκ θ

− − +

= ∫ (A-1.43) 

This is not quite the same form as equation (A-1.15), because of the way Qp was defined for a point source. 

Proceed now to calculate the energy in the medium (rock) per unit length at time t by integrating over all space 

ρCv(x,y,t): 

( )

2

4

0

, , 4

t t r

p

V V

CQ e e E Cv x y t dV e d dV

λ λθ

κθ ρ ρ θ πκ θ

− −

= = ∫ ∫∫ (A-1.44) 

Use dV = rdrdξ for integrating over all space where dξ is the angle that ranges from 0 to 2π. Integrating dξ

yields 2π so that the above integral becomes: 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-14 

 

( )

2

4

0 0

2

4

t t r CQ ep e E d e rdrλ λθ

κθ ρ π θ

πκ θ

− ∞ −

= ∫ ∫ (A-1.45) 

 

Consider now the dr integral, change variables as: 

2 2

u r = / 4 2udu = 2rdr/4 κθ κθ (A-1.46) 

which yields: 

( )

2

2 2 4

00 0

4 4

r

u u e rdr e udu e udu κθ κθ κθ

∞∞ ∞ −

− − = = ∫∫ ∫ (A-1.47) 

 

Now let: 

2 z u dz udu = = 2 (A-1.48) 

and the left integral in equation (A-1.47) becomes: 

2

0

0 0

4 22 2 u zz κθ κθ κθ κθ e udu e dz e

∞ ∞ ∞ − −− = =−= ⎡ ⎤ ∫ ∫ ⎣ ⎦ (A-1.49) 

Therefore, equation (A-1.45) becomes: 

( )

0 0

2

2

t t t

p t

p

CQ e e E d CQ e e d

λ λθ

λ λθ ρ

κθθρ θ κ θ

−

− = = ∫ ∫ (A-1.50) 

Continuing: 

( ) 0

1 1

t

t t t p

p

CQ e E CQ e e e

λ

λ λθ λ ρ

ρ λ λ

−

− = =− ⎡ ⎤ ⎣ ⎦ (A-1.51) 

or multiplying through the first exponential yields: 

( ) 1 CQp t E e λ ρ

λ

− = − (A-1.52) 

which is the desired result, i.e., heat is liberated at ρCQp per unit time per unit length. When λ = 0 the result is: 

E = ρCQ tp (A-1.53) 
Thermal Response Evaluation of Yucca Mountain 

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Now suppose that equation (A-1.1) describes the heat release per unit time per unit length, which yields Qo = 

ρCQp, so eliminate Qp in equation (A-1.41) to obtain the final result for the temperature due to a decaying 

finite-length line source of length L centered at the origin: 

( ) ( ) ( ) ( ) 2 2

4

0

/2 /2 , ,, 8 2 2

x y t t

o Q e zL zL e v x y z t erf erf e d K

λ λθ

κθ θ

π θ κθ κθ

− + − ⎪ ⎪ ⎧ ⎫ ⎡ ⎤⎡ ⎤ + − = − ⎨ ⎬ ⎢ ⎥⎢ ⎥ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦⎣ ⎦ ∫ (A-1.54) 

This is the solution for the temperature due to a decaying finite-length line source of length L centered at z = 0. 

This location with the center at the origin is sometimes not convenient, and it is more convenient to have the left 

edge of the source at the origin. Therefore, consider shifting this solution by +L/2 to the right. To do this, 

subtract L/2 from the independent variable z. This subtraction shifts the center to z = +L/2. To see this, 

suppose that it is necessary to calculate the temperature due to this decaying finite-length line source at the left 

end of the source at some distance away, and the left end is to be at z = 0. This temperature is the same as that 

calculated from equation (A-1.54) with z = −L/2. Thus subtract L/2 from the independent variable to 

accomplish this shift. In general, to shift by a distant of +L, subtract L from the independent variable. The 

temperature due to a decaying finite-length line source of length L centered at z = +L/2 is: 

( ) ( ) ( ) 2 2

4

0

,,, 8 2 2

x y t t

o Q e z e z L

v x y z t erf erf e d K

λ λθ

κθ θ

π θ κθ κθ

− + − ⎧ ⎫ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ − = − ⎨ ⎬ ⎢ ⎥ ⎢ ⎥ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦ ⎣ ⎦ ∫ (A-1.55) 

Now consider how to calculate the temperature due to series of end-to-end decaying finite-length line sources. 

Do not shift the first source, shift the second source by L, the third source by 2L, and so on. Furthermore, 

suppose that each source has a different decay representation. Then the temperature due to a series of these 

decaying finite-length sources is written as: 

( )( ) ( ) ( ) 12 3 v x yztn v x yzt v x yz Lt v x yz Lt , , ,, , , , , , , , , 2 , = + −+ − + (A-1.56) 

 

or in general: 

( )( )

1

, , , , , , ( 1) ,

n

i

i

v x yztn v x yz i Lt == −− ∑ (A-1.57) 

 

References: 

Carslaw, H.S., and J. C. Jaeger, 1959, Conduction of Heat in Solids, Oxford University Press. 
Thermal Response Evaluation of Yucca Mountain 

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Section A-2: Decaying Cylindrical Source Derivations 

Introduction: The temperature due to a decaying flux applied to the inside wall of a cylinder in an infinite 

medium describes the physical situation of a tunnel in a mountain heated by nuclear-decay heat. The solution to 

this problem is given in this section as equation (A-2.20). The derivation for a decaying cylindrical source in an 

infinite medium proceeds from the solution for a constant-flux cylindrical source. The solution for a constant 

flux applied to the inside wall of a cylinder in an infinite medium is given by H.S. Carslaw and J.C. Jaeger, 

Conduction of Heat in Solids, Second Edition, 1959, Section 13.5, equation (16). In order to fix notation and 

use results from Carslaw and Jaeger, the problem is defined here beginning with the energy balance partial 

differential equation and an exponentially decaying boundary condition. 

Consider a region bounded internally by a cylinder heat with an exponentially decaying heat source. The partial 

differential equation that describes the temperature, v, in the region is: 

2

2

1 1 0 vv v

r rr t κ

∂ ∂∂

+− =

∂ ∂∂ (A-2.1) 

where the thermal diffusivity, κ, is defined as: 

K

C

κ

ρ

≡ (A-2.2) 

and the usual notation applies. The boundary condition at the inside surface of the cylinder is: 

 

t

o

dv K Q e at r a dr

−ξ −= = (A-2.3) 

 

The temperature, v, is to remain finite as r → ∞. The initial condition is taken to be 0, i.e., v(r,0) = 0. 

The Laplace transform of the above is: 

2

2

2

1 0 d v dv

q v dr r dr

+ − = (A-2.4) 

where the bar above v is the Laplace transform of v, and: 

 Qo dv K at r a

dr p ξ −= = + (A-2.5) 

where the transform variable is p, and q is defined as: 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-17 

 

p q

κ

≡ (A-2.6) 

The solution in transform space is: 

1

( ) ( ) ( )()Q K qr o o v p Kq p K qa ξ = + (A-2.7) 

where Kν(z) is the modified Bessel function of the second kind of order ν. 

The above can be inverted using the same technique used to invert the transform solution for the constant-flux 

boundary condition problem described by equation (16), Section 13.5, in Carslaw and Jaeger. To illustrate this 

technique, proceed with equation (16) as noted: 

1

( )

( )

QK qr o v

KpqK qa = (A-2.8) 

This equation is inverted by using the fundamental property: 

o

v L pv v

t

⎧ ⎫ ∂

⎨ ⎬ = − ⎩ ⎭ ∂ (A-2.9) 

where L denotes the Laplace transform operator, and v0 = 0 (the initial condition). 

Therefore, the inversion of equation (A-2.8) is conducted by inverting for the derivative of v with respect to 

time and then the result integrated with respect to time between 0 and some arbitrary time, t. Since the Laplace 

transform of v(r,t) for the problem with an exponentially decaying power source results in replacing p with p + 

ξ, compare equation (A-2.7) and (A-2.8), consider the following: 

( ) { } vp v pv v L L v

t

ξξ ξ ⎧ ⎫ ∂

+ =+= + ⎨ ⎬ ⎩ ⎭ ∂ (A-2.10) 

Use Theorem I, Section 12.2, from Carslaw and Jaeger to write: 

{ } v v L Lv L v

t t

ξ ξ ⎧⎫ ⎧ ⎫ ∂ ∂ ⎨⎬ ⎨ ⎬ +=+ ⎩⎭ ⎩ ⎭ ∂ ∂ (A-2.11) 

Thus, obtain the inverse of equation (A-2.7) by using the inverse of equation (A-2.8), which is equation (17), 

Section 13.5, Carslaw and Jaeger. This is done as follows, start with equation (17) as noted, which is: 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-18 

 

2

2

0

2 (1 ) ( ) u t

c

Q du v e fr

u

κ

πκ

∞

− =− − ∫ (A-2.12) 

where the subscript c on v denotes the constant flux boundary condition. The function f(r) is defined as: 

01 0 1

2 2

1 1

( )( ) ( )( ) ( ) () ()J ur Y ua Y ur J ua f r J ua Y ua− ≡

+ (A-2.13) 

where Jν(z) is the Bessel function of the first kind of order ν, and Yν(z) is the Bessel function of the second kind 

of order ν. Take the partial derivative of equation (A-2.12) to obtain: 

2 2

2

0

2 ( ) vc Q du u ue f r tK u

κ κ

π

∞ ∂ − = − ∂ ∫ (A-2.14) 

Simplifying yields: 

2

0

2 ( ) vc Q u t e f r du t C

κ

πρ

∞ ∂ − = − ∂ ∫ (A-2.15) 

Now use Theorem I as noted above to write the inverse solution for equation (A-2.8) above as: 

2 1 0

0

2 ( ) v Q u t L L v e f r du t C

κ ξ

πρ

∞

− − ⎡ ⎤ ⎧ ⎫ ∂

⎨ ⎬ + =− ⎢ ⎥ ⎣ ⎦ ⎩ ⎭ ∂ ∫ (A-2.16) 

 

Conducting the operations indicated on the left side yields: 

2 0

0

2 ( ) v Q u t v e f r du t C

κ ξ

πρ

∞ ∂ − + =−

∂ ∫ (A-2.17) 

The integrating factor is eξt

 so that: 

( ) 2

0

0

2 ( ) u t t Q e v e f r du t Cξ κ ξ

πρ

∞ ∂ −

⎡ ⎤ = − ⎣ ⎦ ∂ ∫ (A-2.18) 

 

Integrate by changing t to θ as the integration variable on the right side and integrate between the limits of θ = 0 

and θ = t to obtain: 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-19 

 

( )

( ) 2

0

2

0 0

2 1 ( )

t

u t Q e v e f r du C uξκ θ ξ

πρ ξ κ

∞ − ⎡ ⎤ = − ⎢ ⎥ − ⎣ ⎦

∫ (A-2.19) 

Evaluating the limits and clearing eξt

 yields (with f(r) from the definition used) the temperature in the region 

bounded internally by a cylinder with a decaying flux applied at the cylinder wall as: 

( )

2

0 01 0 1

2 2 2

0 1 1

2 ( )( ) ( )( )

() ()

t ut Q J ur Y ua Y ur J ua e e v du

C J ua Y ua u

ξ κ

πρ ξ κ

∞ − − − − = − + ∫ (A-2.20) 

Note that the integration variable u ranges from zero to infinity and thus the quantity (ξ - κu

2

) in the 

denominator can be zero, which implies that there is a singularity. However, this is not the case and can be 

shown by the use of l’Hospital’s rule or a series expansion of the fraction containing the exponentials and this 

quantity. Proceeding to demonstrate that there is no singularity using a series expansion, examine: 

2

2 2 ( , , ) lim

t ut

u

e e f u

u

ξ κ

ξ κ ξ κ ξ κ

− −

→

⎡ − ⎤ = ⎢ ⎥ ⎢⎣ − ⎥⎦

 (A-2.21) 

Expand the exponentials as follows: 

( ) ( ) 2 3

1 . . .

2! 3!

t t t

e t ξ ξ ξ ξ − − − =− + + (A-2.22) 

Bring the negative signs out of the ( ) to obtain: 

( ) ( ) 2 3

1 ...

2! 3!

t t t

e t ξ ξ ξ ξ − =− + − (A-2.23) 

 

Do likewise for the second exponential above to obtain: 

2 ( ) ( ) 2 3 2 2

2 1 ...

2! 3!

u t

ut ut

e u t κ κ κκ − =− + − (A-2.24) 

Now form the difference of these exponentials and group like powers of t to obtain: 

( ) () ( ) ( ) ( ( ) ) 2 2 2 2 22 2 . . 1 . 

2

n n n t ut n t t

e e ut u u .

! n!

ξ κ ξκ ξ κ ξ κ − − − =− − + − + − − (A-2.25) 

 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-20 

 

Clearly, dividing each right-hand term by (ξ - κu

2

) yields the lead term as t, but what about the other terms? To 

proceed, note that the difference of two like powers can be factored as: 

( )( ) 12 21 . . nn n n n n a b a b a a b ab b −− −− −=− + + + (A-2.26) 

from S.M. Selby, Standard Mathematical Tables, 23-th edition, 1975, page 101. Therefore, each term above 

factors and yields a coefficient of tn

. The n-th coefficient of tn is of the form: 

( ) ( ) 1 2 n-1 1

. . . + b

!

n

n n a ab

n

− − − + (A-2.27) 

where there are n terms in the ( ). Now examine ξ → κu

2

 (or κu

2 → ξ), or a → b, which yields: 

( ) ( ) ( ) ( )

( )

12 1 1 1 1 11

lim . . 

! ! 1!

n nn

nn n n n

a b a a b b na a

n nn

−− − − −

→

− −− + ++ = = − (A-2.28) 

Factoring each term in equation (A-2.25), canceling the denominator, and using the results of equation (A-2.28) 

yields: 

( ) ( )

2

2

1

2

1

lim 1

1 !

t ut n n n

u n

e e

t

u n

ξ κ

ξ κ

ξ

ξ κ

− −

→ =

⎡ ⎤ − ⎢ ⎥ = − ⎢ ⎥ − − ⎣ ⎦ ∑ (A-2.29) 

Continuing with some algebra on the right side: 

( ) ( )

( )

1

1

1

1 !

n

n

n

t

t

n

ξ −

=

= − − ∑ (A-2.30) 

( ) ( )

( )

1

1

1

1

1 !

n

n

n

t

t

n

ξ −

−

=

=− − − ∑ (A-2.31) 

( )

( )

1

1 1 !

n

n

t

t

n

ξ −

=

− = − − ∑ (A-2.32) 

 

But note that: 

( ) ( )

( )

1 2 3

0 1

1 . . 1

2! 3! ! 1 !

n n n x

n n

xx x x

e x

n n

−

−

= =

− =− + − + = − = − ∑ ∑ (A-2.33) 
Thermal Response Evaluation of Yucca Mountain 

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so that using equation (A-2.33), equation (A-2.29) becomes: 

( )

( )

2

2

1

2

1

lim

1 !

n t ut

t

u n

e e t

t te

u n

ξ κ

ξ

ξ κ

ξ

ξ κ

− − −

−

→ =

⎡ ⎤ − − ⎢ ⎥ =− =− ⎢ ⎥ − − ⎣ ⎦ ∑ (A-2.34) 

and this limit proves that the above fraction does not yield a singularity in equation (A-2.20). 

References: 

Carslaw, H.S., and J.C. Jaeger, 1959, Conduction of Heat in Solids, Oxford University Press. 

Selby, S.M., 1975, Standard Mathematical Tables, 23rd edition, CRC Press, Inc. 
Thermal Response Evaluation of Yucca Mountain 

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Section A-3: Radiant Heat Transfer in a Coaxial Cylindrical System 

Introduction: The purpose of this appendix is to demonstrate that it is reasonable to calculate drift wall 

temperatures using a single linear power in an entire drift rather than using individual package powers. This 

demonstration is conducted by analyzing the drift wall temperature due to individual packages within a drift 

where only axial radiant heat transfer occurs. The conclusion derived in this appendix is that the drift wall 

temperature variation due to varying powers is small for the parameters of interest, on the order of 3°C, and a 

variation such as this is not consider large enough to justify the computation of temperatures to an accuracy 

better than this. 

 

Radiant-heat transfer is known to spread, or smear, energy transport. Consequently, maintaining large 

temperature variations over short distances without power sources is difficult, if not impossible. In the Yucca 

Mountain Project’s analysis for predicting in-drift natural convection and condensation, package-to-package 

power variation was not used to calculate drift wall temperatures; instead, a single linear power for the entire 

drift was used (Bechtel SAIC Company 2004, Section 6.3.5.1.1). By virtue of using a single linear power and 

not the actual package-to-package power variation, the resulting drift wall temperature is smooth with respect to 

variation over the distance scale of a package length. Thus, the implicit assumption was that the package-topackage power variation would not have a significant effect on the drift wall temperature opposite these 

packages. This implicit assumption can be tested by analyzing a simplified problem as described here that 

considers only axial radiant heat transfer in the drift between packages of varying power and the drift wall. The 

simplified radiant-heat-transfer-only problem analyzed here calculates the drift wall temperature variation 

opposite a seven-package segment with appropriate individual package powers that approximates that used by 

the Project (Bechtel SAIC Company 2005a, Figure 6.2-2). This particular package segment embeds low-power 

packages between high-power packages. The package powers use for the source powers for the problem 

analyzed here are the powers at approximately the time of peak postclosure temperature, about 75 years after 

emplacement, and are derived from Bechtel SAIC Company 2005b. 

Consider several heat-generating cylinders, axially positioned end to end, with the entire group emplaced 

coaxially within a single, larger, outer cylinder, with ends open to the environment. Radiant-heat transfer is the 

only mode of heat transfer within the system. If the thermal power of each heat-generating cylinder varies from 

one to the next, an axial temperature variation will be imparted on the outer cylinder. In some applications, 

such as an underground nuclear waste repository, this axial temperature variation on the outer cylinder can be 

important. This appendix analyzes the axial temperature variation on the outer cylinder. 

The problem can be simplified by replacing the smaller heat-generating cylinders with a single heat-generating 

cylinder, equal in length to the outer cylinder. The reason the heat-generating center cylinder runs the entire 

length of the outer cylinder is because of the tedium involved in calculating the outer cylinder view factors with 

a finite-length center cylinder present. Typical dimensions to be considered are 5.5 meters for the outercylinder diameter and 2.5 meters for the inner-cylinder diameter. The cylinder length of interest is 100 meters. 

The effects of individual heat-generating cylinders are accounted for by subdividing the inner cylinder into 100 

equal sections along its length (one meter wide each), with each section capable of having a different thermal 

power. The power emitted by the inner cylinder is specified for a unit length; for example, 100 watts/meter. 

Every unit length shall have a specified power, and the entire length of the inner cylinder emits power. 

In order to solve a realistic problem, a convective boundary condition is used to transfer energy from the outer 

cylinder to the outside medium (rock). The reason for doing this is because it is not possible to transport all the 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-23 

 

source power out the ends of the cylinder, the cylinder of interest is too long, and the view factors from virtually 

any position to the ends are very small. The temperature of the outside medium is specified along with the 

temperature of the environment at each end of this cylindrical system. Because the outside medium (rock) 

temperature in the situation of interest changes very slowly, energy transfer within the system is quasi-steadystate. The derivation of the heat-transfer equations and solution technique follow. 

Derivation: A cylindrical source is placed coaxially in a larger, open-ended cylinder. The outer cylinder 

wall is divided into a number of equal-width constant-temperature rings, as is the cylindrical source. The inner 

and outer rings are the same width and are aligned (i.e., an outer ring above an inner ring). 

For each outer ring, j, the energy balance for this system is written as: 

( ) , , () ()

1 1

0

n n

ij j left j right j e kj j j r

i k

i j

Q E E T T P hA T T

= =

≠

∑ ∑ − + −+ − − = (A-3.1) 

In equation (A-3.1), the first term is the energy transported between outer rings. The second term is the energy 

transported from the outer rings to the ends, where, the Ej terms are the end loss terms from outer ring, j, Tj is the 

temperature of outer ring, j, and Te is the temperature at the ends. The third term is the energy transported from 

the inner rings to the outer rings. The fourth term is the energy transported out to the rock, where Aj is the area 

of outer ring, j, h is a heat transfer coefficient, and Tr is the temperature of the rock. 

In the first term in equation (A-3.1), Qij is expressed as the radiant-heat transfer from the outer ring, i, to outer 

ring, j, with an emissivity of unity for all rings (R.B. Bird, W.E. Stewart, and E.N. Lightfoot, equation 14.4-9, 

page 440): 

4 4 ( ) Q AF T T ij i ij i j = − σ (A-3.2) 

Equation (A-3.2) can be written in linearized form as: 

( ) Q BT T ij ij i j = − (A-3.3) 

where: 

32 23 ( ) Bij i ij i i j i j j ≡ +++ σ AF T T T TT T (A-3.4) 

and Fij is the view factor from outer ring, i to outer ring, j. 

 

In the second term in equation (A-3.1), each end-loss term (Ej) for outer ring, j, is linearized, typically, as: 

32 23 ( ) E AF T T T TT T j j je j j e j e e ≡ ++ + σ (A-3.5) 

where Fje is the view factor from outer ring, j, to the end. 

 
Thermal Response Evaluation of Yucca Mountain 

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In the third term in equation (A-3.1), Pkj is expressed as the radiant-heat transfer from inner ring, k, to outer 

ring, j, with an emissivity of unity for all rings: 

4 4 ( ) P AF V T kj k kj k j = − σ (A-3.6) 

where Vk is the temperature of the inner ring, k. Equation (A-3.6) can be written in linearized form as: 

( ) P CV T kj kj k j = − (A-3.7) 

where: 

32 23 ( ) C AF V V T V T T kj k kj k k j k j j ≡ ++ + σ (A-3.8) 

Fkj is the view factor from inner ring, k, to outer ring, j. 

The temperature of inner ring, k, is not specified. However, the source energy (power) for each inner ring, k, is. 

An energy balance for each inner ring, k, relates the source energy and the end losses for inner ring, k, and the 

temperature of inner ring, k, to each outer ring, i: 

, ,

1

( ) ( )( )

n

ki k i k k left k right k e

i

CV T Q E E V T

=

∑ −= − + − (A-3.9) 

where Qk is the specified source energy for inner ring, k, and the Ek terms are the end losses from inner ring, k, 

which also are linearized, typically, as: 

32 23 ( ) Ek k ke k k e k e e ≡ ++ + σ AF V V T V T T (A-3.10) 

where Fke is the view factor from inner ring, k, to the end. 

Solving equation (A-3.9) for Vk yields: 

, ,

1

, ,

1

( )

( )

n

k k left k right e ki i

i

k n

k left k right ki

i

Q E E T CT

V

EE C

=

=

++ +

=

+ +

∑

∑

 (A-3.11) 

Thus, the temperature of inner-ring, k, depends on the temperatures of all the outer rings, i, and the open-end 

temperature. For convenience, the sum in the denominator is rewritten as: 

1 1

n n

ki km

i m

C C

= =

∑ ∑= (A-3.12) 

With this notational change, equation (A-3.11) can be condensed to: 
Thermal Response Evaluation of Yucca Mountain 

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1

n

k k ki i

i

V T α β== + ∑ (A-3.13) 

so that α k and β ki are defined as: 

, ,

, ,

1

( )

( )

k k left k right e

k n

k left k right km

m

QE E T

EE C

α

=

+ + ≡

+ + ∑

 (A-3.14) 

and: 

, ,

1

( )

ki

ki n

k left k right km

m

C

EE C

β

=

≡

+ + ∑

 (A-3.15) 

Substitution of equation (A-3.13) into equation (A-3.7), followed by substitution of equation (A-3.7) into equation 

(A-3.1), as well as substitution of equation (A-3.3) into equation (A-3.1), and collecting the coefficients of Ti yields: 

1 11

( )

n nn

ij i j kj k ki i j

i ki

i j

B TT C TT α β = ==

≠

⎧ ⎫

−+ + − ⎨ ⎬ ⎩ ⎭ ∑ ∑∑ , , ( )( ) ( ) 0 − E E T T hA T T j left j right j e j j r + −− − = (A-3.16) 

Note that: 

1 1 11

n n nn

kj ki i i kj ki

k i ik

C T TC β β = = ==

∑ ∑ ∑∑ = (A-3.17) 

Rearranging equation (A-3.16), collecting the Tj, yields: 

, ,

1 11 1 1

( )

n nn n n

ij i i kj ki ij kj j left j right j j

i ik i k

ij ij

B T T C B C E E hA T β = == = =

≠ ≠

⎧ ⎫ ⎪ ⎪ + − + ++ + ⎨ ⎬

⎪ ⎪ ⎩ ⎭

∑ ∑∑ ∑ ∑ 

, ,

1

( )

n

kj k j left j right e j r

k

C E E T hA T α ==− − + − ∑

(A-3.18) 

Tj appears once in the double sum in equation (A-3.18) above. Placing this occurrence of Tj with the {}Tj term, and 

regrouping the Ti yields: 

, ,

1 1 111

( )

n n nn n

ij kj ki i ij kj kj ki j left j right j j

i k ikk

ij ij

B C T B C C E E hA T β β = = ===

≠ ≠

⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎨ ⎬⎨ ⎬ + − + − ++ +

⎩ ⎭ ⎪ ⎪ ⎩ ⎭

∑ ∑ ∑∑∑

, ,

1

( )

n

kj k j left j right e j r

k

C E E T hA T α ==− − + − ∑ 

(A-3.19) 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-26 

 

This is the working equation for Tj and is the j-th row in the matrix equation, 

 [[A]] ⋅ [T] = [B] (A-3.20) 

with the index j running from 1 to n outer-ring temperatures. 

View Factors: The view factors used in the calculation are defined according to the following equation (R.B. 

Bird, W.E. Stewart, and E.N. Lightfoot; 1960, Transport Phenomena, page 440): 

1 2

12 1 2 2

1 12

1 cos cos F dA dA

A r

θ θ

π = ⌠ ⌠ ⎮ ⎮ ⌡ ⌡ (A-3.21) 

 

where 12 r is the distance between surfaces 1 and 2, A1 is the area of surface 1, A2 is the area of surface 2, θ1 is the 

angle between 12 r and the normal of surface 1, and θ 2 is the angle between 12 r and the normal of surface 2. 

In equation (A-3.4), Fij is the view factor from outer ring, i, to an outer ring, j, with an obscuring, coaxial cylinder 

present, and is: 

( ) 1 1 2 2cos

4

2 2

2 2

2

2

0

cos

8 2

4cos

2

r r

ij

y F d

r L

r

θ

θ

π θ

−

⎛ ⎞ ⎜ ⎟ Δ ⎝ ⎠ = ⎡ ⎤ ⎛ ⎞ ⎢ ⎥ + ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

⌠

⎮

⎮

⎮

⎮

⌡

(A-3.22) 

where L is the i-to-j separation. 

In equation (A-3.8), Fkj is the view factor from inner ring, k, to outer ring, j, and is: 

2

2 12

2

2 cos cos

o

kj

r x F d

R

π

ξ

θ θ ξ

π = ⌠⎮

⌡ (A-3.23) 

where r1 = radius of the inner cylinder, r2 = radius of the outer cylinder, L = k-to-j ring separation, x = ring 

width, and: 

1 1

0

2

sin r

r

ξ − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ (A-3.24) 

2 22 2

2 1 12 R = +− − + ⎡ ⎤ r r rr L 2 cos( / 2 ) π ξ ⎣ ⎦ (A-3.25) 

1

2 2 2

1 2 1 12 R r r rr cos 2 cos( / 2 ) θ πξ = +− − ⎡ ⎤ ⎣ ⎦ (A-3.26) 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-27 

 

2222

2 1

2

2

cos

2

R r Lr

Rr

θ + − − = (A-3.27) 

In equation (A-3.5), Fje is the view factor from outer ring, j, to an end plate with an obscuring coaxial cylinder 

present, and is: 

2

min

max

0

2

2 cos cos r

f b

je

r

F r dr d

R

θ

θ θ

θ

π

⎧ ⎫ ⎪ ⎪ = ⎨ ⎬ ⎪ ⎪ ⎩ ⎭⌠

⎮ ⌠

⎮ ⎮ ⌡ ⌡

(A-3.28) 

where S is the separation of the ring from the end, and: 

2 22 2

2 2 R = r r rr S +− + 2 cosθ (A-3.29) 

1

max 1 2 θ α2 2cos ( / ) r r − = = (A-3.30) 

For θ max < < θ α , the minimum value of r is obtained from β =θ α− , and: 

1

min

cos

r

r

β = (A-3.31) 

and for α < < θ 0 : 

min 1 r r = (A-3.32) 

In equation (A-3.10), Fke is the view factor from inner ring, k, to an end plate, and is: 

2

min

0

2

2 cos cos

r

a b

ke

r

F r dr d

R

α

θ θ θ

π

⎧ ⎫ ⎪ ⎪ = ⎨ ⎬ ⎪ ⎪ ⎩ ⎭⌠

⎮ ⌠

⎮ ⎮ ⌡ ⌡

(A-3.33) 

where S is the separation of the ring from the end, and: 

1

min

cos

r

r

θ = (A-3.34) 

1 1

2

cos

r

r

α − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ (A-3.35) 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-28 

 

1

1

cos

cos

r r

R

θ θ − = (A-3.36) 

2 cos cos b

S

R

θ θ = = (A-3.37) 

 

The view factors given in equations (A-3.22), (A-3.23), (A-3.28), and (A-3.33) were compared with the open 

literature (i.e., W. M. Rohsenow, J. P. Hartnett, and Y. I. Cho, 1998, Handbook of Heat Transfer, and John R. 

Howell, 1982, A Catalog of Radiation Configuration Factors) and found to be satisfactory. 

Solution Technique: The view factors given in equations (A-3.22), (A-3.23), (A-3.28), and (A-3.33) were 

integrated numerically using a Simpson’s integration scheme. The cubic temperature functions written in equations 

(A-3.4), (A-3.5), (A-3.8) and (A-3.10) are the result of the linearization of radiant-heat transfer and thus require an 

iterative solution technique. The solution proceeds from an initial temperature guess (e.g., the environment 

temperature beyond the ends of the cylinder) and then solving equation (A-3.20) for all the outer-ring temperatures. 

These calculated temperatures are then substituted into the cubic temperature functions, and the solution repeated. 

At each outer-ring temperature iteration equation (A-3.20) is solved using Gaussian elimination with back 

substitution. The inner-ring temperatures are then calculated from equation (A-3.13). The iteration proceeds until 

the temperature for each outer ring from iteration to iteration changes very little. The temperature-convergence 

criterion used is the sum of the absolute values of each temperature difference from iteration to iteration. 

Convergence is considered obtained when this sum is less than 0.01. When the solution converges, the energy 

balance, or power balance, for each ring (inner and outer) and a total energy balance for the system are calculated 

using the solution temperatures and the classical 4 T difference. In any such iterative calculation for temperature, 

the energy balances are the objective functions on which to base the conclusion that the convergence criterion is 

acceptable. The energy balances should always “close” to within a few tenths of a percent. The words “energy 

balance” and “power balance” are used interchangeably here, and the use of “energy balance” pertains to a specific 

unit of time, say one second, so that watts becomes joules. 

 

As stated earlier, in order to solve a realistic problem, a convective boundary condition is used to transfer 

energy from the outer cylinder to the outside medium (rock). The boundary conditions are as follows: The 

temperature of the outside medium (rock) behind the outer cylinder was set to 140°C, which is about the bulk 

rock temperature based on a similar transient calculation. The heat-transfer coefficient used for the outside 

medium was 4 W/m2

-K. The temperature of the environment at the open ends of the cylindrical system was 

maintained at 140°C. The power output of the inner cylinder was specified as shown in Table A-3.1, and the 

values used here represent the base-case package powers at the time of peak postclosure temperature at 75 years 

after emplacement with 50 years of ventilation. Each package is five meters in length and thus the ring power 

over five-meter segments is the same. 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-29 

 

Table A-3.1. Thermal Power of Each Inner Cylinder One-Meter-Wide Ring, these powers are based on the base-case package 

powers (1.45 kW/meter) at the time of peak postclosure temperature 

Ring # 

(meters) 

Power 

(W) Ring Power (W) Ring Power (W) Ring Power (W) 

1 1 26 1 51 440 76 1 

2 1 27 1 52 440 77 1 

3 1 28 1 53 440 78 1 

4 1 29 1 54 448 79 1 

5 1 30 1 55 448 80 1 

6 1 31 1 56 448 81 1 

7 1 32 1 57 448 82 1 

8 1 33 1 58 448 83 1 

9 1 34 720 59 104.4 84 1 

10 1 35 720 60 104.4 85 1 

11 1 36 720 61 104.4 86 1 

12 1 37 720 62 104.4 87 1 

13 1 38 720 63 104.4 88 1 

14 1 39 33.8 64 720 89 1 

15 1 40 33.8 65 720 90 1 

16 1 41 33.8 66 720 91 1 

17 1 42 33.8 67 720 92 1 

18 1 43 33.8 68 720 93 1 

19 1 44 736 69 440 94 1 

20 1 45 736 70 440 95 1 

21 1 46 736 71 440 96 1 

22 1 47 736 72 440 97 1 

23 1 48 736 73 440 98 1 

24 1 49 440 74 1 99 1 

25 1 50 440 75 1 100 1 

Results and Conclusions: The energy (or power for one second) transported out of the system calculated from 

the temperatures determined from the defining equations above is given in Table A.3-2. 

Table A.3-2. Calculated Power Losses 

Power radiated out both ends from the outer rings (W) 5.820E+01 

Power radiated out both ends from the inner rings (W) 1.192E+01 

Power transferred through the outer-ring walls (W) 1.820E+04 

Total power transported out of the system (W) 1.827E+04 

The total inner-ring specified source power (calculated as the sum of values in Table A.3-1) is 1.827E+04 W. 

Comparing this value with the total power transported out (Table A.3-2) yields a power-balance error of -

0.02%. Thus, the total energy balance is satisfied, and the convergence criterion specified earlier is appropriate. 

The temperature distribution along the outer cylinder, e.g., the drift wall, is presented in Figure A.3-1. There is 

a package every five meters at the locations specified in Table A-3.1, and the no-package region runs from 0 

meters to 33 meters, and from 74 meters to 100 meters. The solution yields a temperature variation along the 

drift wall of approximately 3°C, from about 144.5°C (at about 40 meters) to slightly less than 147°C (at about 

47 meters), opposite the packages in the region of 34 meters to 73 meters. Because of the end effects, the 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-30 

 

temperatures at each end of the package segment are not used here (in reality, there will be more than 100 

packages in a drift). 

Figure A-3.1 Temperature Distribution Along the Drift Wall (outer cylinder wall) 

139

140

141

142

143

144

145

146

147

148

0 20 40 60 80 100 120

Meters

Degrees C

These results indicate that the drift wall temperature variation opposite packages of varying power is not large 

at the time of peak postclosure temperature, less than 3°C. Thus, conclude that it is justified to use a linear 

power for the entire drift to calculate drift wall temperatures. Since the problem analyzed here does not 

consider any axial conduction or natural convection, these results are bounding; the actual drift wall temperature 

variation will be less. The implicit assumption of a smooth drift wall temperature that does not feel package 

power variation in the in-drift natural convection and condensation analysis cited earlier is justified. 

At the times of interest in the in-drift natural convection and condensation analyses at 300 years and longer, the 

drift wall temperature variation will be much less. The expectation can be that the drift wall temperature 

variation will be decreased approximately by the decrease in package powers, which between 75 and 300 years 

will be a factor of 0.36. Thus the drift wall temperature variation at 300 years will be approximately 1°C, and at 

other times of interest longer than 1,000 years, the temperature variation will be a small fraction of a degree. 

The analyses presented here are based on an emissivity of unity. Clearly, this will not be the case in a 

repository environment, but the emissivities of the surfaces in a repository environment will be close to unity 

due to metal oxidation and the presence of dust on the metal surfaces. This dust will most likely determine the 

emissivities, and thus the emissivities are expected to be in the range of 0.9 and greater. Emissivities in this 

range will not significantly change the results presented here. 
Thermal Response Evaluation of Yucca Mountain 

July 2006 Page A-31 

 

References: 

Bechtel SAIC Company, 2004, In-Drift Natural Convection and Condensation, MDL-EBS-MD-000001 

REV00, Las Vegas, Nevada: Bechtel SAIC Company. 

Bechtel SAIC Company, 2005a, Multiscale Thermohydrologic Model, ANL-EBS-MD-000049 REV03, Las 

Vegas, Nevada: Bechtel SAIC Company. 

Bechtel SAIC Company, 2005b, IED Waste Package Decay Heat Generation [Sheet 1 of 1], 800-IED-WIS0-

00701-000-00A, Las Vegas, Nevada: Bechtel SAIC Company. 

Bird, R.B.; W.E. Stewart; and E.N. Lightfoot, 1960, Transport Phenomena, New York, New York: John Wiley 

& Sons. 

Howell, John R., 1982, A Catalog of Radiation Configuration Factors, New York, New York: McGraw-Hill. 

Rohsenow, W.M.; J. P. Hartnett; and Y. I. Cho, 1998, Handbook of Heat Transfer, Third Edition, New York, 

New York: McGraw-Hill. 
 

Appendix B 

Assembly Power Constant Calculation 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page B-1 

 

4% Enriched BWR Assemblies

The following assembly powers were taken from the tables generated with Origin. 

Low_BU 60 := High_BU 70 :=

5 years OOR Low_5 1714.0800 := High_5 2120.4600 :=

10 years OOR Low_10 1156.7100 := High_10 1440.6400 :=

25 years OOR Low_25 783.5390 := High_25 959.3630 :=

50 years OOR Low_50 500.2433 := High_50 598.2805 :=

100 years OOR Low_100 259.6294 := High_100 300.4353 :=

300 years OOR Low_300 100.6072 := High_300 109.8012 :=

Most of the Origan tables provide data at 10 Gwd/ton intervals. The following equations perform a linear interpulation to 

calculate the assembly powers at 1 Gwd/ton intervals. The variable Cal_BU is the desired burnup value for the calculation

Calc_BU 61 := Burnup value for calculation 

Assembly Power 

for burnup Calc_BU 

at age 5, 10, 25, 50, 

 100, and 300 yrs OOR. 

Interperlation for assembly burnup Cal_BU 

between assembly burnups Low_BU and High_BU. 

Cal_5 Low_5 High_5 Low_5 ( ) − Calc_BU Low_BU −

High_BU Low_BU − := + ⋅ Cal_5 1754.718000 =

Cal_10 Low_10 High_10 Low_10 ( ) − Calc_BU Low_BU −

High_BU Low_BU − := + ⋅ Cal_10 1185.103000 =

Cal_25 Low_25 High_25 Low_25 ( ) − Calc_BU Low_BU −

High_BU Low_BU − := + ⋅ Cal_25 801.121400 =

Cal_50 Low_50 High_50 Low_50 ( ) − Calc_BU Low_BU −

High_BU Low_BU − := + ⋅ Cal_50 510.047020 =

Cal_100 Low_100 High_100 Low_100 ( ) − Calc_BU Low_BU −

High_BU Low_BU − := + ⋅ Cal_100 263.709990 =

Cal_300 Low_300 High_300 Low_300 ( ) − Calc_BU Low_BU −

High_BU Low_BU − := + ⋅ Cal_300 101.526600 =
Thermal Response Evaluation of Yucca Mountain 

July 2008 Page B-2 

 

The following six "Guess values" are required by MathCad to solve the six equations in six unknowns provided 

below. These six equations represent the assembly power as the summation of three expedentional terms at 

assambly ages of 5, 10, 25, 50, 100, and 300 years out of reactor.. 

P1 1600 := P2 130 := P3 15 := λ1 .5 := λ2 .02 := λ3 .0011 :=

Given

P1 e− λ 5⋅ 1 ⋅ P2 e− λ 5⋅ 2+ ⋅ P3 e− λ 5⋅ 3+ ⋅ Cal_5

P1 e− λ 10⋅ 1 ⋅ P2 e− λ 10⋅ 2+ ⋅ P3 e− λ 10⋅ 3+ ⋅ Cal_10

P1 e− λ 25⋅ 1 ⋅ P2 e− λ 25⋅ 2+ ⋅ P3 e− λ 25⋅ 3+ ⋅ Cal_25

P1 e− λ 50⋅ 1 ⋅ P2 e− λ 50⋅ 2+ ⋅ P3 e− λ 50⋅ 3+ ⋅ Cal_50

P1 e− λ 100⋅ 1 ⋅ P2 e− λ 100⋅ 2+ ⋅ P3 e− λ 100⋅ 3+ ⋅ Cal_100

P1 e− λ 300⋅ 1 ⋅ P2 e− λ 300⋅ 2+ ⋅ P3 e− λ 300⋅ 3+ ⋅ Cal_300

P1_Value

P2_Value

P3_Value

λ1_Value

λ2_Value

λ3_Value

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

:= Find P1 P2 ( ) , , λ P3 , λ 1 , λ 2, 3

MathCad calculated the following values for the six constants: 

Calc_BU 61 = P1_Value 3002.31 = λ1_Value 0.340099 =

P2_Value 1098.44 = λ2_Value 0.026099 =

P3_Value 246.08 = λ3_Value 0.002966 =

Which results in the following half lives: 

Half_Life1 .693

λ1_Value := Half_Life2 .693λ2_Value := Half_Life3 .693λ3_Value :=

Half_Life1 2.04 = Half_Life2 26.55 = Half_Life3 233.69 =
Thermal Response Evaluation of Yucca Mountain 

 

 

Appendix C 

PWR and BWR Decay Constants 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix C- 1 

 

PWR Decay Constants

4.5 % Enriched

Burnup Decay 1 Decay 2 Decay 3 Power 1 Power 2 Power 3 Half Life 1 Half Life 2 Half Life 3

10 0.406173 0.024076 0.001286 616.78 171.11 15.47 1.71 28.78 538.88

11 0.404778 0.024028 0.001380 672.46 186.22 18.96 1.71 28.84 502.17

12 0.403622 0.023988 0.001447 728.21 201.32 22.47 1.72 28.89 478.92

13 0.402648 0.023954 0.001498 784.00 216.43 25.97 1.72 28.93 462.62

14 0.401815 0.023925 0.001537 839.83 231.53 29.48 1.72 28.97 450.88

15 0.401096 0.023899 0.001568 895.68 246.63 32.99 1.73 29.00 441.96

16 0.400468 0.023877 0.001593 951.56 261.73 36.51 1.73 29.02 435.03

17 0.399914 0.023857 0.001615 1007.45 276.82 40.02 1.73 29.05 429.10

18 0.399423 0.023839 0.001633 1063.36 291.92 43.53 1.74 29.07 424.37

19 0.398985 0.023823 0.001648 1119.28 307.02 47.05 1.74 29.09 420.51

20 0.398590 0.023808 0.001662 1175.21 322.12 50.56 1.74 29.11 416.97

21 0.394035 0.023725 0.001680 1213.11 337.02 54.46 1.76 29.21 412.50

22 0.390021 0.023650 0.001696 1252.10 351.92 58.35 1.78 29.30 408.61

23 0.386458 0.023580 0.001710 1291.97 366.83 62.23 1.79 29.39 405.26

24 0.383273 0.023515 0.001722 1332.57 381.74 66.12 1.81 29.47 402.44

25 0.380409 0.023456 0.001732 1373.76 396.65 70.00 1.82 29.54 400.12

26 0.377818 0.023400 0.001741 1415.45 411.57 73.87 1.83 29.62 398.05

27 0.375464 0.023348 0.001750 1457.56 426.48 77.75 1.85 29.68 396.00

28 0.373314 0.023300 0.001757 1500.03 441.41 81.62 1.86 29.74 394.42

29 0.371345 0.023255 0.001764 1542.81 456.33 85.49 1.87 29.80 392.86

30 0.369533 0.023213 0.001770 1585.87 471.25 89.35 1.88 29.85 391.53

31 0.368735 0.023316 0.001826 1636.24 487.72 94.21 1.88 29.72 379.52

32 0.367997 0.023413 0.001878 1686.69 504.18 99.08 1.88 29.60 369.01

33 0.367314 0.023505 0.001926 1737.20 520.65 103.94 1.89 29.48 359.81

34 0.366678 0.023591 0.001970 1787.77 537.11 108.81 1.89 29.38 351.78

35 0.366087 0.023672 0.002011 1838.38 553.58 113.68 1.89 29.28 344.60

36 0.365534 0.023749 0.002049 1889.05 570.04 118.55 1.90 29.18 338.21

37 0.365017 0.023822 0.002084 1939.75 586.51 123.43 1.90 29.09 332.53

38 0.364531 0.023891 0.002118 1990.49 602.97 128.31 1.90 29.01 327.20

39 0.364075 0.023957 0.002149 2041.26 619.44 133.18 1.90 28.93 322.48

40 0.363646 0.024019 0.002178 2092.06 635.91 138.06 1.91 28.85 318.18

41 0.361731 0.024100 0.002222 2133.48 655.36 143.00 1.92 28.76 311.88

42 0.359940 0.024177 0.002263 2175.26 674.82 147.94 1.93 28.66 306.23

43 0.358262 0.024249 0.002303 2217.38 694.27 152.88 1.93 28.58 300.91

44 0.356688 0.024318 0.002340 2259.79 713.72 157.83 1.94 28.50 296.15

45 0.355205 0.024383 0.002376 2302.47 733.17 162.78 1.95 28.42 291.67

46 0.353809 0.024445 0.002410 2345.39 752.61 167.74 1.96 28.35 287.55

47 0.352490 0.024505 0.002442 2388.53 772.06 172.70 1.97 28.28 283.78

48 0.351243 0.024561 0.002473 2431.86 791.50 177.66 1.97 28.22 280.23

49 0.350062 0.024615 0.002502 2475.38 810.93 182.63 1.98 28.15 276.98

50 0.348941 0.024667 0.002531 2519.06 830.37 187.60 1.99 28.09 273.80

51 0.347982 0.024829 0.002579 2562.38 854.18 193.04 1.99 27.91 268.71

52 0.347075 0.024983 0.002626 2605.86 878.01 198.46 2.00 27.74 263.90

53 0.346217 0.025129 0.002670 2649.50 901.88 203.86 2.00 27.58 259.55

54 0.345403 0.025268 0.002712 2693.27 925.78 209.24 2.01 27.43 255.53

55 0.344631 0.025401 0.002752 2737.18 949.70 214.62 2.01 27.28 251.82

56 0.343897 0.025527 0.002791 2781.20 973.64 219.98 2.02 27.15 248.30

57 0.343199 0.025647 0.002828 2825.33 997.61 225.33 2.02 27.02 245.05

58 0.342533 0.025763 0.002864 2869.57 1021.59 230.67 2.02 26.90 241.97

59 0.341898 0.025873 0.002898 2913.90 1045.59 236.00 2.03 26.78 239.13

60 0.341292 0.025978 0.002931 2958.31 1069.60 241.32 2.03 26.68 236.44

61 0.340099 0.026099 0.002966 3002.31 1098.44 246.08 2.04 26.55 233.65

62 0.338965 0.026214 0.002999 3046.54 1127.30 250.83 2.04 26.44 231.08

63 0.337885 0.026324 0.003031 3090.99 1156.19 255.56 2.05 26.33 228.64

64 0.336855 0.026429 0.003063 3135.63 1185.08 260.29 2.06 26.22 226.25

65 0.335872 0.026529 0.003093 3180.46 1214.00 265.01 2.06 26.12 224.05

66 0.334933 0.026624 0.003123 3225.45 1242.93 269.72 2.07 26.03 221.90

67 0.334034 0.026715 0.003151 3270.59 1271.87 274.43 2.07 25.94 219.93

68 0.333174 0.026803 0.003179 3315.88 1300.82 279.13 2.08 25.86 217.99

69 0.332350 0.026886 0.003206 3361.30 1329.78 283.82 2.09 25.78 216.16

70 0.331559 0.026967 0.003233 3406.86 1358.76 288.52 2.09 25.70 214.35
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix C- 2 

 

BWR Decay Constants

4% Enriched

Burnup Decay 1 Decay 2 Decay 3 Power 1 Power 2 Power 3 Half Life 1 Half Life 2 Half Life 3

10 0.443891 0.023654 0.001124 240.96 69.33 5.84 1.56 29.30 616.55

11 0.438826 0.023609 0.001222 254.28 75.49 7.01 1.58 29.35 567.10

12 0.434581 0.023571 0.001294 267.89 81.64 8.19 1.59 29.40 535.55

13 0.430966 0.023539 0.001350 281.69 87.80 9.36 1.61 29.44 513.33

14 0.427852 0.023511 0.001395 295.65 93.95 10.54 1.62 29.48 496.77

15 0.425141 0.023487 0.001431 309.74 100.10 11.72 1.63 29.51 484.28

16 0.422759 0.023465 0.001461 323.93 106.25 12.90 1.64 29.53 474.33

17 0.420650 0.023447 0.001486 338.20 112.40 14.08 1.65 29.56 466.35

18 0.418768 0.023430 0.001507 352.53 118.55 15.26 1.65 29.58 459.85

19 0.417079 0.023414 0.001526 366.93 124.70 16.44 1.66 29.60 454.13

20 0.415555 0.023401 0.001542 381.36 130.85 17.62 1.67 29.61 449.42

21 0.412691 0.023427 0.001597 394.24 136.98 19.06 1.68 29.58 433.94

22 0.410117 0.023452 0.001646 407.27 143.11 20.50 1.69 29.55 421.02

23 0.407790 0.023475 0.001689 420.41 149.23 21.94 1.70 29.52 410.30

24 0.405677 0.023497 0.001727 433.67 155.36 23.39 1.71 29.49 401.27

25 0.403749 0.023516 0.001762 447.00 161.48 24.84 1.72 29.47 393.30

26 0.401982 0.023535 0.001793 460.42 167.60 26.29 1.72 29.45 386.50

27 0.400358 0.023552 0.001821 473.90 173.73 27.74 1.73 29.42 380.56

28 0.398859 0.023569 0.001847 487.44 179.85 29.19 1.74 29.40 375.20

29 0.397471 0.023584 0.001871 501.03 185.96 30.64 1.74 29.38 370.39

30 0.396182 0.023598 0.001892 514.66 192.08 32.10 1.75 29.37 366.28

31 0.393343 0.023633 0.001936 526.51 198.54 33.58 1.76 29.32 357.95

32 0.390741 0.023665 0.001977 538.56 205.00 35.06 1.77 29.28 350.53

33 0.388341 0.023695 0.002016 550.77 211.45 36.55 1.78 29.25 343.75

34 0.386124 0.023724 0.002052 563.13 217.90 38.04 1.79 29.21 337.72

35 0.384069 0.023752 0.002085 575.61 224.35 39.53 1.80 29.18 332.37

36 0.382159 0.023778 0.002117 588.21 230.79 41.02 1.81 29.14 327.35

37 0.380379 0.023802 0.002147 600.90 237.24 42.52 1.82 29.12 322.78

38 0.378716 0.023826 0.002175 613.68 243.68 44.01 1.83 29.09 318.62

39 0.377159 0.023848 0.002202 626.54 250.12 45.51 1.84 29.06 314.71

40 0.375698 0.023870 0.002227 639.48 256.56 47.01 1.84 29.03 311.18

41 0.372588 0.023975 0.002276 648.60 264.20 48.59 1.86 28.91 304.48

42 0.369701 0.024074 0.002323 658.02 271.84 50.16 1.87 28.79 298.32

43 0.367013 0.024168 0.002368 667.69 279.49 51.74 1.89 28.67 292.65

44 0.364505 0.024258 0.002410 677.60 287.13 53.32 1.90 28.57 287.55

45 0.362158 0.024343 0.002450 687.70 294.78 54.89 1.91 28.47 282.86

46 0.359957 0.024424 0.002489 697.98 302.42 56.47 1.93 28.37 278.43

47 0.357888 0.024501 0.002526 708.43 310.07 58.05 1.94 28.28 274.35

48 0.355941 0.024574 0.002561 719.02 317.72 59.62 1.95 28.20 270.60

49 0.354104 0.024644 0.002595 729.73 325.36 61.20 1.96 28.12 267.05

50 0.352369 0.024712 0.002628 740.57 333.01 62.78 1.97 28.04 263.70

51 0.352843 0.024955 0.002702 758.20 342.49 64.89 1.96 27.77 256.48

52 0.353326 0.025187 0.002772 775.91 352.00 66.98 1.96 27.51 250.00

53 0.353816 0.025407 0.002838 793.68 361.54 69.06 1.96 27.28 244.19

54 0.354310 0.025617 0.002901 811.52 371.11 71.12 1.96 27.05 238.88

55 0.354807 0.025817 0.002961 829.41 380.70 73.16 1.95 26.84 234.04

56 0.355307 0.026008 0.003019 847.37 390.32 75.19 1.95 26.65 229.55

57 0.355807 0.026190 0.003073 865.37 399.95 77.21 1.95 26.46 225.51

58 0.356306 0.026365 0.003126 883.43 409.61 79.22 1.94 26.28 221.69

59 0.356805 0.026532 0.003176 901.55 419.28 81.21 1.94 26.12 218.20

60 0.357302 0.026692 0.003224 919.71 428.97 83.20 1.94 25.96 214.95

61 0.352606 0.026709 0.003228 921.16 440.21 83.90 1.97 25.95 214.68

62 0.348225 0.026726 0.003233 923.53 451.45 84.59 1.99 25.93 214.35

63 0.344126 0.026742 0.003237 926.68 462.68 85.28 2.01 25.91 214.09

64 0.340282 0.026756 0.003241 930.52 473.91 85.97 2.04 25.90 213.82

65 0.336670 0.026770 0.003245 934.98 485.14 86.66 2.06 25.89 213.56

66 0.333267 0.026783 0.003249 939.98 496.37 87.35 2.08 25.87 213.30

67 0.330056 0.026795 0.003253 945.47 507.59 88.04 2.10 25.86 213.03

68 0.327020 0.026806 0.003257 951.39 518.82 88.72 2.12 25.85 212.77

69 0.324145 0.026816 0.003261 957.70 530.04 89.40 2.14 25.84 212.51

70 0.321418 0.026826 0.003264 964.36 541.26 90.08 2.16 25.83 212.32
 

Appendix D 

Comparison of Assembly Power in Project Documents 

with Calculated Power 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix D- 1 

 

Comparison of Project and NWTRB Assembly Powers 

Decay 1 0.396182 0.375698 0.352369 0.369533 0.363646 0.348941

Decay 2 0.023598 0.02387 0.024712 0.023213 0.024019 0.024667

Decay 3 0.001892 0.002227 0.002628 0.00177 0.002178 0.002531

Power 1 514.66 639.48 740.57 1585.87 2092.06 2519.06

Power 2 192.08 256.56 333.01 471.25 635.91 830.37

Power 3 32.1 47.01 62.78 89.35 138.06 187.6

Origan Power Calculated Power Origan Power Calculated Power Origan Power Calculated Power Origan Power Calculated Power Origan Power Calculated Power Origan Power Calculated Power

5 273.5 273.5 371.9 371.9 483.4 483.4 758.1 758.1 1040.1 1040.1 1359.3 1359.3

6 242.0 246.2 331.8 335.8 433.0 438.3 655.5 671.1 904.4 922.9 1191.2 1211.3

7 222.6 226.7 306.0 309.5 399.1 404.6 593.4 608.2 820.7 837.6 1081.0 1102.0

8 210.4 212.3 287.2 289.8 375.2 378.9 551.8 562.0 763.6 774.5 1006.8 1020.0

9 200.2 201.4 273.5 274.8 357.6 359.0 522.7 527.3 723.2 727.0 951.4 957.4

10 193.0 193.0 263.0 263.0 343.1 343.1 500.8 500.8 690.3 690.3 908.6 908.6

11 186.9 186.2 254.5 253.4 330.7 330.1 482.4 479.9 663.9 661.4 873.5 869.7

12 180.9 180.5 247.0 245.5 320.4 319.2 467.1 463.0 642.0 637.8 843.8 837.9

13 175.7 175.6 239.7 238.6 311.1 309.8 453.1 448.8 623.4 618.1 817.7 811.1

14 171.7 171.3 234.3 232.6 302.9 301.5 441.2 436.6 606.1 601.1 794.8 788.0

15 167.8 167.4 227.9 227.1 295.8 294.0 430.5 425.9 591.1 586.1 773.4 767.6

16 163.7 163.7 222.7 222.0 288.8 287.1 420.0 416.2 577.4 572.6 755.3 749.2

17 160.7 160.3 218.4 217.3 281.8 280.7 410.6 407.3 563.9 560.1 737.4 732.3

18 156.8 157.0 213.2 212.9 275.9 274.6 401.2 398.9 551.5 548.5 720.7 716.6

19 153.8 153.9 209.0 208.6 269.0 268.9 393.0 391.0 540.4 537.5 705.3 701.8

20 150.9 150.9 204.7 204.5 264.2 263.4 384.8 383.4 528.3 527.0 690.0 687.7

21 147.9 148.0 200.5 200.5 258.4 258.1 377.6 376.2 518.4 516.9 674.9 674.2

22 145.0 145.2 196.3 196.7 252.6 252.9 370.5 369.2 507.6 507.2 661.9 661.2

23 142.0 142.4 193.2 192.9 247.9 248.0 362.5 362.4 497.9 497.8 649.1 648.7

24 139.0 139.7 189.0 189.3 243.1 243.1 356.4 355.8 488.3 488.7 636.3 636.5

25 137.1 137.1 185.8 185.8 238.4 238.4 349.4 349.4 479.8 479.8 624.7 624.7

26 134.2 134.6 181.6 182.3 233.8 233.9 342.5 343.2 470.3 471.2 612.1 613.2

27 132.1 132.1 178.4 179.0 229.1 229.4 336.4 337.1 461.9 462.8 600.6 602.0

28 129.3 129.7 175.3 175.7 224.5 225.1 330.5 331.1 453.6 454.5 589.2 591.1

29 126.9 127.3 172.1 172.5 219.9 220.8 324.5 325.3 446.3 446.5 578.8 580.5

30 124.7 125.0 169.0 169.3 216.3 216.7 318.6 319.6 438.0 438.7 568.4 570.1

31 122.3 122.7 165.8 166.3 211.7 212.7 313.6 314.1 430.7 431.1 558.1 560.0

32 120.2 120.5 162.7 163.3 208.1 208.7 307.7 308.6 423.5 423.6 547.9 550.1

33 118.0 118.3 160.6 160.4 204.6 204.9 302.7 303.4 415.3 416.3 538.7 540.5

34 116.0 116.2 157.4 157.5 201.1 201.2 297.8 298.2 408.2 409.2 529.5 531.1

35 114.0 114.1 154.2 154.8 196.5 197.5 291.8 293.1 402.0 402.3 520.3 521.9

36 111.9 112.1 152.1 152.0 193.0 193.9 287.8 288.2 394.9 395.5 510.1 512.9

37 109.9 110.2 149.0 149.4 189.5 190.4 282.8 283.3 387.8 388.9 502.0 504.2

38 108.0 108.2 146.5 146.8 187.1 187.0 277.8 278.6 381.7 382.4 492.9 495.6

39 106.2 106.3 144.0 144.2 183.5 183.7 272.9 274.0 375.6 376.0 484.8 487.3

40 104.4 104.5 141.6 141.8 180.1 180.4 268.8 269.5 369.6 369.8 476.7 479.1

41 102.6 102.7 139.1 139.3 176.6 177.3 263.9 265.0 363.5 363.8 469.6 471.1

42 100.9 100.9 136.9 137.0 174.2 174.2 259.8 260.7 357.4 357.9 461.5 463.4

43 99.1 99.2 134.5 134.6 170.7 171.1 255.8 256.5 352.4 352.1 454.5 455.7

44 97.5 97.5 132.3 132.4 168.3 168.2 251.8 252.4 346.3 346.5 446.4 448.3

45 95.8 95.9 130.1 130.2 165.1 165.3 247.7 248.3 340.3 340.9 439.4 441.1

46 94.2 94.3 128.0 128.0 162.4 162.5 243.7 244.4 336.3 335.5 432.3 434.0

47 92.7 92.7 125.8 125.9 159.6 159.7 239.7 240.5 331.2 330.3 425.3 427.0

48 91.1 91.2 123.8 123.8 156.9 157.0 236.6 236.7 325.2 325.1 419.3 420.3

49 89.7 89.7 121.7 121.8 154.3 154.4 232.5 233.0 320.2 320.1 412.2 413.7

50 88.2 88.2 119.8 119.8 151.8 151.8 229.4 229.4 315.2 315.2 407.2 407.2

51 86.9 86.8 117.9 117.9 149.3 149.3 225.3 225.9 311.1 310.4 400.2 400.9

52 85.4 85.4 116.0 116.0 146.8 146.9 222.3 222.4 306.1 305.7 394.2 394.7

53 84.1 84.0 114.2 114.2 144.4 144.5 219.2 219.1 301.1 301.1 388.1 388.7

54 82.7 82.7 112.4 112.4 142.2 142.2 215.1 215.7 297.1 296.6 382.1 382.8

55 81.5 81.4 110.6 110.6 139.9 139.9 212.0 212.5 293.1 292.2 377.1 377.1

56 80.1 80.1 109.0 108.9 137.7 137.6 208.9 209.4 288.1 287.9 371.1 371.4

57 78.9 78.9 107.3 107.2 135.5 135.5 205.8 206.3 284.1 283.7 365.1 365.9

58 77.8 77.6 105.7 105.6 133.4 133.3 203.7 203.2 281.1 279.6 360.1 360.6

59 76.5 76.4 104.0 104.0 131.4 131.3 200.5 200.3 276.1 275.6 356.1 355.3

60 75.4 75.3 102.5 102.4 129.4 129.2 197.3 197.4 273.1 271.6 350.1 350.2

61 74.2 74.1 101.0 100.9 127.4 127.2 194.2 194.6 268.1 267.8 345.1 345.2

62 73.1 73.0 99.5 99.4 125.5 125.3 192.1 191.8 265.1 264.1 340.1 340.3

63 72.1 71.9 98.0 97.9 123.6 123.4 188.9 189.1 262.0 260.4 335.1 335.5

64 71.0 70.9 96.6 96.4 121.8 121.5 186.8 186.5 257.0 256.8 330.1 330.8

65 70.0 69.8 95.2 95.0 120.0 119.7 183.7 183.9 254.0 253.3 327.1 326.2

66 68.9 68.8 93.8 93.7 118.2 118.0 181.3 181.3 251.0 249.9 322.0 321.8

67 68.0 67.8 92.5 92.3 116.5 116.2 178.9 178.9 247.0 246.5 317.0 317.4

68 67.0 66.8 91.2 91.0 114.8 114.5 176.4 176.4 244.0 243.2 313.0 313.1

69 66.0 65.9 89.9 89.7 113.2 112.9 174.1 174.1 241.0 240.0 309.0 308.9

70 65.1 64.9 88.7 88.5 111.6 111.3 171.8 171.7 238.0 236.9 305.0 304.8

71 64.1 64.0 87.4 87.2 110.0 109.7 169.5 169.5 235.0 233.8 301.0 300.8

72 63.3 63.1 86.2 86.0 108.4 108.2 167.3 167.3 231.0 230.8 297.0 296.9

73 62.4 62.3 85.1 84.9 106.9 106.6 165.1 165.1 229.0 227.9 294.0 293.1

74 61.5 61.4 83.9 83.7 105.4 105.2 163.1 163.0 226.0 225.0 290.0 289.4

75 60.7 60.6 82.8 82.6 104.0 103.7 161.0 160.9 223.0 222.2 286.0 285.7

76 59.9 59.8 81.6 81.5 102.6 102.3 158.9 158.8 220.6 219.5 283.0 282.2

77 59.0 59.0 80.6 80.4 101.2 100.9 156.9 156.9 217.3 216.8 279.0 278.7

78 58.3 58.2 79.5 79.4 99.9 99.6 155.0 154.9 215.0 214.2 275.0 275.2

79 57.5 57.4 78.5 78.4 98.5 98.3 153.1 153.0 212.8 211.6 272.0 271.9

80 56.7 56.7 77.5 77.3 97.3 97.0 151.2 151.1 209.6 209.1 269.0 268.6

81 56.1 55.9 76.5 76.4 96.0 95.7 149.4 149.3 207.5 206.6 265.0 265.4

82 55.3 55.2 75.5 75.4 94.8 94.5 147.6 147.5 205.4 204.2 262.0 262.3

83 54.6 54.5 74.6 74.5 93.5 93.3 145.8 145.8 202.4 201.8 260.0 259.2

84 54.0 53.8 73.6 73.5 92.4 92.1 144.1 144.1 200.4 199.5 256.2 256.2

85 53.2 53.2 72.8 72.6 91.2 91.0 142.5 142.4 198.5 197.3 253.9 253.3

86 52.7 52.5 71.8 71.8 90.1 89.8 140.8 140.7 195.6 195.1 250.6 250.4

87 52.0 51.9 71.0 70.9 88.9 88.7 139.2 139.1 193.8 192.9 247.4 247.6

88 51.3 51.3 70.2 70.0 87.8 87.7 137.6 137.6 191.0 190.8 245.3 244.9

89 50.8 50.6 69.3 69.2 86.8 86.6 136.0 136.0 189.3 188.7 242.2 242.2

90 50.1 50.0 68.5 68.4 85.8 85.6 134.6 134.5 187.6 186.7 239.1 239.6

91 49.5 49.5 67.7 67.6 84.7 84.6 133.1 133.1 184.9 184.7 237.1 237.0

92 49.0 48.9 67.0 66.8 83.7 83.6 131.6 131.6 183.3 182.8 234.1 234.5

93 48.4 48.3 66.1 66.1 82.8 82.6 130.2 130.2 181.7 180.9 232.2 232.0

94 47.8 47.8 65.4 65.3 81.8 81.7 128.8 128.8 179.2 179.0 229.4 229.6

95 47.3 47.2 64.7 64.6 80.9 80.7 127.5 127.5 177.6 177.2 226.5 227.2

96 46.7 46.7 63.9 63.9 79.9 79.8 126.2 126.1 176.2 175.4 224.8 224.9

97 46.2 46.2 63.2 63.2 79.0 79.0 124.8 124.8 173.7 173.6 222.0 222.6

98 45.7 45.7 62.6 62.5 78.1 78.1 123.5 123.6 172.3 171.9 220.3 220.4

99 45.2 45.2 61.9 61.9 77.3 77.2 122.3 122.3 170.9 170.3 217.7 218.2

100 44.7 44.7 61.2 61.2 76.4 76.4 121.1 121.1 168.6 168.6 216.1 216.1

110 40.3 40.4 55.3 55.4 68.9 69.0 110.1 110.2 153.8 153.9 195.9 197.1

120 36.8 36.9 50.4 50.6 62.6 63.0 101.1 101.3 141.5 141.9 180.7 181.5

130 33.9 34.0 46.4 46.7 57.5 58.0 93.7 94.0 131.2 132.0 166.8 168.6

140 31.5 31.7 43.1 43.5 53.3 53.9 87.6 88.0 122.5 123.8 155.9 157.9

150 29.4 29.7 40.4 40.8 49.8 50.5 82.6 83.0 115.6 116.9 146.4 148.9

160 27.8 28.1 38.1 38.5 46.8 47.6 78.3 78.8 109.6 111.1 138.1 141.2

170 26.4 26.7 36.1 36.6 44.3 45.1 74.7 75.2 104.4 106.1 131.7 134.5

180 25.2 25.6 34.4 35.0 42.1 43.0 71.7 72.2 100.0 101.7 126.0 128.7

190 24.3 24.6 32.9 33.5 40.1 41.1 69.0 69.6 96.2 97.9 120.9 123.6

200 23.3 23.7 31.7 32.3 38.5 39.5 66.7 67.3 92.8 94.5 116.3 119.1

250 20.3 20.5 27.2 27.6 32.6 33.2 58.4 58.8 80.4 81.7 99.3 101.4

300 18.4 18.4 24.3 24.3 28.7 28.7 53.0 53.0 72.3 72.3 88.3 88.3

350 16.9 16.6 22.2 21.6 26.0 25.1 48.8 48.2 66.1 64.6 80.0 77.5

400 15.7 15.1 20.4 19.3 23.7 22.0 45.3 44.1 61.0 57.8 73.4 68.2

450 14.7 13.7 19.0 17.3 22.0 19.2 42.4 40.3 56.6 51.8 67.8 60.1

500 13.8 12.5 17.7 15.4 20.4 16.9 39.7 36.9 52.8 46.5 63.0 52.9

550 13.0 11.3 16.6 13.8 19.1 14.8 37.4 33.8 49.5 41.7 58.9 46.6

600 12.2 10.3 15.7 12.4 17.9 13.0 35.2 30.9 46.5 37.4 55.2 41.1

650 11.6 9.4 14.8 11.1 16.9 11.4 33.3 28.3 43.8 33.5 51.9 36.2

700 11.0 8.5 14.0 9.9 16.0 10.0 31.5 25.9 41.3 30.1 48.9 31.9

750 10.4 7.8 13.2 8.8 15.1 8.7 29.9 23.7 39.1 27.0 46.2 28.1

800 9.9 7.1 12.6 7.9 14.3 7.7 28.4 21.7 37.0 24.2 43.7 24.8

850 9.5 6.4 12.0 7.1 13.6 6.7 27.0 19.8 35.2 21.7 41.4 21.8

900 9.0 5.8 11.4 6.3 13.0 5.9 25.7 18.2 33.4 19.4 39.3 19.2

950 8.6 5.3 10.9 5.7 12.4 5.2 24.5 16.6 31.8 17.4 37.4 16.9

1000 8.3 4.8 10.4 5.1 11.8 4.5 23.5 15.2 30.3 15.6 35.7 14.9

Time from 

Discharge

(years)

40 Gwd/t 4.5% 50 Gwd/t 4.5%

BWR Assembly PWR Assembly

30 Gwd/t 4% 40 Gwd/t 4% 50 Gwd/t 4% 30 Gwd/t 4.5%
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix D- 2 

 

Comparison of PWR Origin Power with Calculated Power

0

250

500

750

1000

1250

1500

0 100 200 300 400 500 600 700 800 900 1000

Time from Discharge - Years

Assembly Power - watts

30 Gwd/t Origan 30 Gwd/t Calculated 40 Gwd/t Origan 40 Gwd/t Calulated 50 Gwd/t Origan 50 Gwd/t Calculated

Comparison of BWR Origin Power to Calculated Power

0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

450.0

500.0

0 100 200 300 400 500 600 700 800 900 1000

Time from Discharge - Years

Assembly Power - watts

30 Gwd/t Origan 30 Gwd/t Calculated 40 Gwd/t Origan 40 Gwd/t Calculated 50 Gwd/t Origan 50 Gwd/t Calculated
 

Appendix E 

Sample Calculation Using 

Finite-Length Decaying Line Source Single Waste Package 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix E- 1 

 

Finite Length Decaying Line Source, Single Package

This file calculates temperatures at locations in the repository plane based on a fixed number of neighboring drifts. 

Each drift is similar and contains a series of end-to-end finite-length decaying line sources, called segments. Each 

segment consists of the same number of WPs and each WP has the same length and power characteristics. The WP 

power is determined from the assembly burnup and age out of reactor and is represented by an equation with three 

decaying exponential terms. During the preclosure phase, forced ventilation provides heat removal with the ventilation 

efficiency decreasing linearly as the flow moves down the drift. 

Independent Variables

The following variables define the conditions for the calculation. The burnup should be an integer between 10 and 70 

Gwd/ton. 

Initial Rock Temperature tzero 22.8 := oC 

Est_Drift_Length 60 := 0 meters 

Ventilation Efficiency at Start of Drift Eff_1 0.90 := % 

Drift_Space 8 := 1 meters 

Ventilation Efficiency at End of Drift Eff_2 0.80 := % 

WP_Length 5.8 := 5 meters 

Age at Emplacement A1 := 5 Years Out of Reactor 

WP_Spacing .1 := meters 

Ventilation Duration Vent 75 := Years 

WP_Per_Seg 1 := 2 WP/segment 

Age at Closure A2 A1 Ven := + t A2 80 = Years Out of Reactor 

Burnup 45 := Gwd/ton 

Number of Neighboring Drifts nbors := 5 Drifts 

Constants

The following constants are required for the calculation. 

Krock 1.83 := watt/meter - oC ρ := 2097 kg/meter3 Cp 1119.0 := joules/kg - oC 

κ Krock 3600 ⋅ ⋅24⋅365

ρ⋅Cp := meter2/year 

Dependent Variables

The following calculation determines the exact length of the drift and the number of waste packages per drift 

based on the assumptions provided above. 

No_WP_Est round Est_Drift_Length

WP_Length WP_Spacing +

⎛

⎜

⎝

⎞

⎟

⎠ := No_Seg round No_WP_Est

WP_Per_Seg

⎛

⎜

⎝

⎞

⎟

⎠ :=

No_WP No_Seg WP_Per_Seg := ⋅ Drift_Length No_WP WP_Spacing WP_Length := ⋅( ) + − WP_Spacing

Seg_L Drift_Length

No_Seg :=

Actual Drift Length Drift_Length 571.10 = meters Number of WP per Drift No_WP 96 = WP per Drift 

Number of Segments per Drift No_Seg 8 = Segments Segment Length Seg_L 71.4 = meters 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix E- 2 

 

Decay Variables Calculation

The following files contain the six constants that define the assembly power and decay as a function of burnup for BWR 

and PWR assemblies. The power values are calculated based on a PWR assembly with an initial loading of 0.475 

MTHM/Assembly and a BWR assembly with an initial loading of 0.200 MTHM/Assembly. 

BWR := PWR :=

The six constants are selected from the above files based on the burnup defined above and whether BWR or PWR 

assemblies are loaded into the waste package. To select the type of assembly, change the first term inside the () below 

to either "BWR" or "PWR" and define the number of assemblies contained in the WP, normally 44 for BWR WPs and 21 

for PWR WP's in the "No_Ass" variable. In addition, a correction for the initial assembly MTHM contained in each 

assembly needs to be performed by defining the base loading, 0.475 for PWR assemblies and 0.200 for BWR 

assemblies in the "MTHM_Base" variable, and the actual assembly MTHM in the "MTHM_Actual" variable. 

No_Ass 2 := 1 MTHM_Act .4130 := MTHM_Base 0.475 := 0

Decay submatrix PWR Burnup 1 := ( ) , − , Burnup 1 − , 1, 3 Power_Base submatrix PWR Burnup 1 := ( ) , − ,Burnup 1 − , 4, 6

Power_Base 2302.47 733.17 162.78 = ( )

Decay 0.355206 0.024383 0.002376 = ( )

Power_Correction MTHM_Act

MTHM_Base :=

Half_Life .693

Decay :=

Power Power_Correction Power_Base := ⋅

Half_Life 2.0 28.4 291.7 = ( ) Years Power 2001.94 637.47 141.53 = ( ) watts 

Arrays

The following arrays are necessary in order to perform the linear line load and temperature calculations. 

Q

Qj Power0 j , ⋅No_Ass WP_Per_Seg

Seg_L ← ⋅

for j 02 ∈ ..

return Q

:= λ

λj Decay0 j, ←

for j 02 ∈ ..

return λ

:= t 0 1000 := ..

watts/meter 

Eff

x Seg_L

2 ← + i Seg_L ⋅

Effi

( ) Eff_2 Eff_1 −

Drift_Length ← ⋅x + Eff_1

for i 0 No_Seg 1 ∈ .. −

return Eff

:= Line_Load

Line_Loadt ← 0

Line_Loadt Line_Loadt Qj

e

− λj⋅t

1000 ← +

for j 02 ∈ ..

for t 0 A2 ∈ ..

return Line_Load

:=

kW/meter 

WP_Power

WP_Powert ← 0

WP_Powert WP_Powert Power0 j , ⋅No_Ass e

− λj⋅t ← + ⋅

for j 02 ∈ ..

for t 0 A2 ∈ ..

WP_Power

1000

return

:=

kW/WP 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix E- 3 

 

Power

The waste package power and linear line load at emplacement, A1, and closure, A2, calculated from array "WP_Power" 

and "Line_Load" are: 

WP power at emplacement WP_Power A1 = 21.91 kW Line load at Emplacement Line_LoadA1 = 3.68 kW/meter 

WP power at closure WP_Power A2 = 4.36 kW Line load at closure Line_LoadA2 = 0.73 kW/meter 

Temperature Calculation

The equation for calculating the temperature due to a single finite-length decaying line source (or segment) is: 

vzx ( ) , , t, i, j

e

− λj⋅( ) A1 t + Qj

8⋅π ⋅Krock

0

t

erf θ z

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ erf z Seg_L −

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ − ⎛

⎜

⎝

⎞

⎟

⎠

e

λj⋅θ

θ ⋅ e

x

2 −

4⋅ θ κ⋅ ⋅

⌠

⎮

⎮

⎮

⎮

⌡

:= ⋅ d

The temperature contribution after ventilation shutdown is as follows, the A2 in the leading exponential re-zeros Qij for 

ventilation shutdown. 

va z x ( ) , , t, i, j

e

− λj⋅( ) A2 t + Qj ⋅

8⋅π ⋅Krock

0

t

erf θ z

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ erf z Seg_L −

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ − ⎛

⎜

⎝

⎞

⎟

⎠

e

λj⋅θ

θ ⋅ e

x

2 −

4⋅ θ κ⋅ ⋅

⌠

⎮

⎮

⎮

⎮

⌡

:= ⋅ d

The segment temperature contribution for the i-th segment for the three decaying components is: 

WPtemp z x ( ) tsum 0.0 , , t, i ←

tsum tsum 1. Effi − ( ) ← + ⋅v z Seg_L i ( ) − ⋅ , x, t, i, j

tsum tsum Effi ← + ⋅va z Seg_L i ( ) − ⋅ , x, t Vent − , i, j if t Vent >

for j 02 ∈ ..

return tsum

:=

The temperature contribution from all segments, "No_Seg", and neighbors, "nbors", is: 

WPSum z x ( ) tsum 0. , , t ←

tsum tsum WPtemp z x ← + ( ) , , t, i

for i 0 No_Seg 1 ∈ .. −

x1Loc n Drift_Space ← ⋅ − x

x2Loc n Drift_Space ← ⋅ + x

tsum tsum WPtemp z x1Loc ← + ( ) , , t, i

tsum tsum WPtemp z x2Loc ← + ( ) , , t, i

for i 0 No_Seg 1 ∈ .. −

for n 1 nbors ∈ ..

return tzero tsum +

:=
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix E- 4 

 

years after emplacement 

The following variables define the radial, "X_Drift" and "X_Pillar", and axial, "Z-Drift" and "Z_Pillar", locations used to 

calculate the drift wall and mid-pillar maximum temperatures. The location of the drift wall peak temperature is at the midpoint of the drift. The location of the mid-pillar peak temperature is near the end of the drift, based on a series of 

sensitivity calculations the peak temperature normally occurs at approximately 80% down the length of the drift. 

X_Drift 2.7 := 5 X_Pillar Drift_Space

2 := Z_Pillar Drift_Length2 := Z_Drift Drift_Length 0. := ⋅ 8

Radial Distance from Drift Centerline Axial Distance from Start of Drift 

X_Drift 2.75 = X_Pillar 40.50 = Z_Pillar 285.55 = Z_Drift 456.88 =

The following loops are used to calculate the drift wall and mid-pillar maximum temperatures and time of maximum 

temperatures at the locations defined above. 

Drift_Wall Drift_Max_Time 0 ←

Drift_Max_Temp 0 ←

Drift_Wall 0 ←

Drift_Wallt ← WPSum Z_Drift X_Drift ( ) , , t

Drift_Max_Time t ← Drift_Wallt if > Drift_Max_Temp

Drift_Max_Temp Drift_Wallt ← Drift_Wallt if > Drift_Max_Temp

for t Vent 200 ∈ ..

Drift_Max_Time

Drift_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠ return

:=

Drift_Max_Time

Drift_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠

:= Drift_Wall

Peak Post Closure Drift Wall Temperature Drift_Max_Temp 180 = oC 

Time of Peak Temperature Drift_Max_Time 97 = years after emplacement 

Pillar Pillar_Max_Time 0 ←

Pillar_Max_Temp 0 ←

Pillar 0 ←

Pillart ← WPSum Z_Pillar X_Pillar ( ) , , t

Pillar_Max_Time t ← Pillart if > Pillar_Max_Temp

Pillar_Max_Temp Pillart ← Pillart if > Pillar_Max_Temp

for t 350 520 ∈ ..

Pillar_Max_Time

Pillar_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠ return

:=

Pillar_Max_Time

Pillar_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠

:= Pillar

Peak Post Closure Mid-pillar Temperature Pillar_Max_Temp 112 = oC 

Time of Peak Temperature Pillar_Max_Time 422 =
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix E- 5 

 

Plot of Temperatures as a Function of Time

The plot below represents the temperature response as a function of time at a location on the drift wall at location 

"X_Drift" and "Z_Drift" (red) and at the mid-pillar "X_Pillar" and "Z-Pillar" (blue). The following variables determine the 

time duration of the plot, t in years, the maximum allowable drift wall temperature y1 and x1, and the maximum 

allowable mid-pillar temperature, y2 and x2. 

y1

200

200

⎛

⎜

⎝

⎞

⎟

⎠ := x1

0

550

⎛

⎜

⎝

⎞

⎟

⎠ := y2

96

96

⎛

⎜

⎝

⎞

⎟

⎠ := x2

0

550

⎛

⎜

⎝

⎞

⎟

⎠ :=

0 50 100 150 200 250 300 350 400 450 500 550

0

50

100

150

200

250

300

Drift Wall Temperature

Maximum Allowable Drift Wall Temperature

Mid-Pillar Temperature

Maximum Allowable Mid-pillar Temperature

Time from Emplacement (years)

Temperature (Deg-C)

A1 5 = Line_LoadA1 = 3.68 Burnup 45 = Drift_Length 571 = No_WP 96 =

A2 80 = Line_LoadA2 = 0.73 MTHM_Act 0.4130 = Drift_Space 81 = WP_Length 5.85 =

Vent 75 = No_Seg 8 = WP_Spacing 0.10 =

WP_Per_Seg 12 = Eff_1 0.90 = Z_Drift 456.88 = X_Drift 2.75 =

Eff_2 0.80 = Z_Pillar 285.55 = X_Pillar 40.50 =
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix E- 6 

 

Plot of Temperatures as a Function of Axial Drift Location

The plot below represents the temperature response as a function of axial location along the drift at drift wall location 

"X_Drift" at time of peak temperature "Drift_Max_Time" (red) and at the mid-pillar location "X_Pillar" at the times of 

peak temperature "Pillar_Max_Time" (blue). The following variables determine the axial extent of the plot, z in meters, 

the location where the first waste package begins, y1 and the location where the last waste package ends, y2. 

y1

300

0

⎛

⎜

⎝

⎞

⎟

⎠ := x1

0

0

⎛

⎜

⎝

⎞

⎟

⎠ := y2

300

0

⎛

⎜

⎝

⎞

⎟

⎠ := x2 Drift_Length

Drift_Length

⎛

⎜

⎝

⎞

⎟

⎠ :=

z 100 := − , −90.. 700

− 100 0 100 200 300 400 500 600 700

0

50

100

150

200

250

300

Drift Wall Temperature

Start of Drift

Mid-Pillar Temperature

End of Drift

Distance Along Drift (meters)

Temperature (Deg-C)

A1 5 =

A2 80 =

Line_LoadA1 = 3.68 Burnup 45 = Drift_Length 571 = No_WP 96 = Vent 75 =

Line_LoadA2 = 0.73 MTHM_Act 0.4130 = Drift_Space 81 = WP_Length 5.85 = Eff_1 0.90 =

No_Seg 8 = WP_Spacing 0.10 = Eff_2 0.80 =

WP_Per_Seg 12 =

Time of drift wall temperature plot 

(years from emplacement) 

Time of mid-pillar temperature plot 

(years from emplacement) 

Drift_Max_Time 97 = Pillar_Max_Time 422 =
Appendix F 

Sample Calculation Using 

Finite-Length Decaying Line Source Seven-Waste-Package Segment 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix F- 1 

 

Finite Length Decaying Line Source, 7 package Segment

This file calculates temperatures at locations in the repository plane based on a fixed number of neighboring drifts. 

Each drift is similar and contains a series of identical end-to-end finite-length decaying line sources, called segments. 

Each segment contains 7 WPs and each WP has unique characteristics. The WP power is determined from the 

assembly burnup and age out of reactor and is represented by an equation with three decaying exponential terms. The 

total WP power of a segment is calculated as a function of time and the three decaying exponential terms associated 

with the total segment power are determined. During the preclosure phase, forced ventilation provides heat removal 

with the ventilation efficiency decreasing linearly as the flow moves down the drift. 

Independent Variables

The following variables define the conditions for the calculation. The burnup should be an integer between 10 and 70 

Gwd/ton. 

Initial Rock Temperature tzero 22.8 := oC 

Est_Drift_Length 60 := 0 meters 

Ventilation Efficiency at Start of Drift Eff_1 0.90 := % 

Drift_Space 8 := 1 meters 

Ventilation Efficiency at End of Drift Eff_2 0.80 := % 

WP_Spacing .1 := meters 

Ventilation Duration Vent 75 := Years 

Number of Neighboring Drifts nbors 5 := Drifts 

Decay Variables Calculation

The following files contain the six constants that define the assembly power and decay as a function of burnup for BWR 

and PWR assemblies. The power values are calculated based on a PWR assembly with an initial loading of 0.475 

MTHM/Assembly and a BWR assembly with an initial loading of 0.200 MTHM/Assembly. 

BWR := PWR :=

The following files contain the six constants that define the canister power and decay. 

DOELong := DOEShort :=

Age 5 := BU 45 :=
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix F- 2 

 

The following matrix defines the segment characteristics. The first column defines the type of waste package PWR, BWR, 

DOE Long, or DOE short. The next three columns determine the characteristics of the commercial waste packages (the 

DOE waste packages are assumed to have the same power and decay characteristics independent of age), The second 

column is the average burnup, the third column is the average age, and the forth column is the average MTHM. For DOE 

waste packages the value in the second column should be 10, the third column should be 0, and the forth columns should 

be 1. 

Seg

PWR

DOELong

PWR

PWR

DOEShort

PWR

PWR

BU

10

BU

BU

10

BU

BU

Age

0

Age

Age

0

Age

Age

.4130

1

.4130

.4130

1

.4130

.4130

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

:=

The six constants defining the power and decay are selected from the above files based on the type of waste package and, for 

the commercial waste packages, the assembly burnup defined in matrix "Seg" . 

λ0 submatrix Seg0 0, Seg0 1, , − 1 Seg0 1, , − 1, 1, 3 ( ) := P0 submatrix Seg0 0, Seg0 1, , − 1 Seg0 1, , − 1, 4, 6 ( ) :=

λ1 submatrix Seg1 0, Seg1 1, , − 1 Seg1 1, , − 1, 1, 3 ( ) := P1 submatrix Seg1 0, Seg1 1, , − 1 Seg1 1, , − 1, 4, 6 ( ) :=

λ2 submatrix Seg2 0, Seg2 1, , − 1 Seg2 1, , − 1, 1, 3 ( ) := P2 submatrix Seg2 0, Seg2 1, , − 1 Seg2 1, , − 1, 4, 6 ( ) :=

λ3 submatrix Seg3 0, Seg3 1, , − 1 Seg3 1, , − 1, 1, 3 ( ) := P3 submatrix Seg3 0, Seg3 1, , − 1 Seg3 1, , − 1, 4, 6 ( ) :=

λ4 submatrix Seg4 0, Seg4 1, , − 1 Seg4 1, , − 1, 1, 3 ( ) := P4 submatrix Seg4 0, Seg4 1, , − 1 Seg4 1, , − 1, 4, 6 ( ) :=

λ5 submatrix Seg5 0, Seg5 1, , − 1 Seg5 1, , − 1, 1, 3 ( ) := P5 submatrix Seg5 0, Seg5 1, , − 1 Seg5 1, , − 1, 4, 6 ( ) :=

λ6 submatrix Seg6 0, Seg6 1, , − 1 Seg6 1, , − 1, 1, 3 ( ) := P6 submatrix Seg6 0, Seg6 1, , − 1 Seg6 1, , − 1, 4, 6 ( ) :=

λ := stack ( ) λ0 , λ λ1 , λ 2 , λ 3 , λ 4 , λ 5 , 6 P stack P0 P1 := ( ) , ,P2, P3,P4, P5, P6

λ

0.355206

0.540572

0.355206

0.355206

0.196721

0.355206

0.355206

0.024383

0.023333

0.024383

0.024383

0.020202

0.024383

0.024383

0.002376

0.000000

0.002376

0.002376

0.000000

0.002376

0.002376

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

= P

2302.5

5.6

2302.5

2302.5

541.5

2302.5

2302.5

733.2

401.4

733.2

733.2

2386.2

733.2

733.2

162.8

0.0

162.8

162.8

0.0

162.8

162.8

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

=
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix F- 3 

 

The next loop defines the base case MTHM per assembly for commercial waste packages, the number of assemblies per 

commercial waste package, and the length of the waste package. 

WP_Data WP_Data 0 ←

WP_Data i 0, ← 0.475 Segi 0, if PWR

WP_Data i 1, ← 21 Segi 0, if PWR

WP_Data i 2, ← 5.85 Segi 0, if PWR

WP_Data i 0, ← 0.200 Segi 0, if BWR

WP_Data i 1, ← 44 Segi 0, if BWR

WP_Data i 2, ← 5.85 Segi 0, if BWR

WP_Data i 0, ← 1 Segi 0, if DOEShort

WP_Data i 1, ← 1 Segi 0, if DOEShort

WP_Data i 2, ← 3.697 Segi 0, if DOEShort

WP_Data i 0, ← 1 Segi 0, if DOELong

WP_Data i 1, ← 1 Segi 0, if DOELong

WP_Data i 2, ← 5.220 Segi 0, if DOELong

for i 06 ∈ ..

return WP_Data

:=

WP_Data

0.4750

1.0000

0.4750

0.4750

1.0000

0.4750

0.4750

21.0000

1.0000

21.0000

21.0000

1.0000

21.0000

21.0000

5.8500

5.2200

5.8500

5.8500

3.6970

5.8500

5.8500

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

=

Individual waste package power at time of emplacement, "WP_Individual" in watts/WP, is calculated in the next loop. 

WP_Individual

WP_Individual i ← 0

WP_Individual i WP_Individual i WP_Data i 1, Pi j, 

Segi 3, 

WP_Data i 0, 

⋅ e

− λi j, Seg ( ) i 2, ⋅ ← + ⋅

for j 02 ∈ ..

for i 06 ∈ ..

return WP_Individual

:=

WP_Individual

21905.4

407.0

21905.4

21905.4

2927.7

21905.4

21905.4

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

= watts 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix F- 4 

 

Segment Power and Decay Constants

The total segment power, "Power" in watts, is calculated in the next loops where "t" represents the time from 

emplacement 

t 0 100 := .. 0

Power

Powert ← 0

Powert Powert WP_Data i 1, P

i j, 

Segi 3, 

WP_Data i 0, 

⋅ e

− λi j, Segi 2, ⋅( ) +t

← + ⋅

for j 02 ∈ ..

for i 06 ∈ ..

for t 0 500 ∈ ..

return Power

:=

The loop "Power" calculates the total segment power as a function of time. The following section determines the six 

power and decay constants that define the total segment power. The six "Guess values" are required by MathCad to 

solve the six equations in six unknowns provided below. These six equations represent the segment power as the 

summation of three exponential terms at assembly ages of 0, 25, 50, 100, 200 and 300 years from emplacement. 

Guess values: 

P1 700 := 0 P2 6500 := 0 P3 1700 := 0 λ1 .1 := 4 λ2 .0 := 2 λ3 .00 := 2

Given

P1 e− λ 0⋅ 1 ⋅ P2 e− λ 0⋅ 2+ ⋅ P3 e− λ 0⋅ 3+ ⋅ Power0 

P1 e− λ 25⋅ 1 ⋅ P2 e− λ 25⋅ 2+ ⋅ P3 e− λ 25⋅ 3+ ⋅ Power25

P1 e− λ 50⋅ 1 ⋅ P2 e− λ 50⋅ 2+ ⋅ P3 e− λ 50⋅ 3+ ⋅ Power50

P1 e− λ 100⋅ 1 ⋅ P2 e− λ 100⋅ 2+ ⋅ P3 e− λ 100⋅ 3+ ⋅ Power100

P1 e− λ 200⋅ 1 ⋅ P2 e− λ 200⋅ 2+ ⋅ P3 e− λ 200⋅ 3+ ⋅ Power200

P1 e− λ 500⋅ 1 ⋅ P2 e− λ 500⋅ 2+ ⋅ P3 e− λ 500⋅ 3+ ⋅ Power500

P1_Value

P2_Value

P3_Value

λ1_Value

λ2_Value

λ3_Value

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

:= Find P1 P2 ( ) , , λ P3, λ 1 , λ 2 , 3
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix F- 5 

 

MathCad calculated the following values for the six constants: 

λ1_Value 0.2910 = P1_Value 36217.34 =

λ2_Value 0.0242 = P2_Value 61943.63 =

λ3_Value 0.0024 = P3_Value 14700.5535 =

λ

λ1_Value

λ2_Value

λ3_Value

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

:= P

P1_Value

P2_Value

P3_Value

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

:= Half_Life .693

λ :=

Half_Life

2.4

28.7

291.4

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

= λ

0.2910

0.0242

0.0024

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

= P

36217.3449

61943.6302

14700.5535

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

=

Emplacement Drift Parameters

The following calculation determines the exact length of the drift and the number of waste packages per drift 

based on the assumptions provided above. 

Seg_L Seg_L 0 ←

Seg_L Seg_L WP_Data i 2, ← + + WP_Spacing

for i 06 ∈ ..

:=

No_Seg round Est_Drift_Length

Seg_L

⎛

⎜

⎝

⎞

⎟

⎠ := Drift_Length No_Seg Seg_ := ⋅ L No_WP No_Seg := ⋅7

Actual Drift Length Drift_Length 583.01 = meters Number of WP per Drift No_WP 105 = WP per Drift 

Number of Segments per Drift No_Seg 15 = Segments Segment Length Seg_L 38.87 = meters 

The next expression converts the power in watts to linear power in watts/meter 

Q

P

Seg_L :=

Q

931.8276

1593.7333

378.2271

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

= wats/meter 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix F- 6 

 

Ventilation Efficiency

The following loop calculates the ventilation efficiency for each segment based on the efficiency at the start of the drift 

of Eff_1 and the efficiency at the end of the drift Eff_2. 

Eff_1 0.9000 = Eff_2 0.8000 =

Eff

x Seg_L

2 ← + n Seg_L ⋅

Effn

( ) Eff_2 Eff_1 −

Drift_Length ← ⋅x + Eff_1

for n 0 No_Seg 1 ∈ .. −

return Eff

:=

Power

The next loops calculates the average waste package power, "WP_Power", and linear line load, "Line_Load", as a function 

of time from emplacement. 

Line_Load

Line_Loadt ← 0

Line_Loadt Line_Loadt Qj

e

− λj⋅t

1000 ← + ⋅

for j 02 ∈ ..

for t 0 Vent 100 ∈ .. +

return Line_Load

:=

WP_Power

WP_Powert ← 0

WP_Powert WP_Powert P

j e

− λj⋅t ← + ⋅

for j 02 ∈ ..

for t 0 200 ∈ ..

WP_Power

7

return

:=

The Average waste package power and linear line load at emplacement, time = 0, and closure, time = vent, calculated from 

array "WP_Power" and "Line_Load" are: 

WP power at emplacement WP_Power 0 = 16123.1 Watts Line load at Emplacement Line_Load0 = 2.90 kW/meter 

WP power at closure WP_Power Vent = 3200.1 Watts Line load at closure Line_LoadVent = 0.58 kW/meter 

Constants

The following constants are required for the calculation. 

Krock 1.8 := 3 watt/meter - oC ρ := 2097 kg/meter3 Cp 1119. := 0 joules/kg - oC 

κ Krock 3600 ⋅ ⋅24⋅365

ρ⋅Cp := meter2/year 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix F- 7 

 

WPSum z x ( ) tsum 0. , , t ←

tsum tsum WPtemp z x ← + ( ) , , t, n

for n 0 No_Seg 1 ∈ .. −

x1Loc m Drift_Space ← ⋅ − x

x2Loc m Drift_Space ← ⋅ + x

tsum tsum WPtemp z x1Loc ← + ( ) , , t, n

tsum tsum WPtemp z x2Loc ← + ( ) , , t, n

for n 0 No_Seg 1 ∈ .. −

for m 1 nbors ∈ ..

return tzero tsum +

:=

Temperature Calculation

The equation for calculating the temperature due to a single finite-length decaying line source (or segment) is: 

vzx ( ) , , t, j

e

− λj⋅t

Qj ⋅

8⋅π ⋅Krock

0

t

erf θ z

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ erf z Seg_L −

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ − ⎛

⎜

⎝

⎞

⎟

⎠

e

λj⋅θ

θ ⋅ e

x

2 −

4⋅ θ κ⋅ ⋅

⌠

⎮

⎮

⎮

⎮

⌡

:= ⋅ d

The temperature contribution after ventilation shutdown is as follows, the A2 in the leading exponential re-zeros Qij for 

ventilation shutdown. 

va z x ( ) , , t, j

e

− λj⋅( ) Vent t + Qj ⋅

8⋅π ⋅Krock

0

t

erf θ z

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ erf z Seg_L −

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ − ⎛

⎜

⎝

⎞

⎟

⎠

e

λj⋅θ

θ ⋅ e

x

2 −

4⋅ θ κ⋅ ⋅

⌠

⎮

⎮

⎮

⎮

⌡

:= ⋅ d

The segment temperature contribution for the n-th segment for the three decaying components is: 

WPtemp z x ( ) tsum 0.0 , , t, n ←

tsum tsum 1. Effn − ( ) ← + ⋅v z Seg_L n ( ) − ⋅ , x, t, j

tsum tsum Effn ← + ⋅va z Seg_L n ( ) − ⋅ , x, t Vent − , j if t Vent >

for j 02 ∈ ..

return tsum

:=

The temperature contribution from all segments, "No_Seg", and neighbors, "nbors", is: 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix F- 8 

 

The following variables define the radial, "X_Drift" and "X_Pillar", and axial, "Z-Drift" and "Z_Pillar", locations used to 

calculate the drift wall and mid-pillar maximum temperatures. The location of the drift wall peak temperature is at the midpoint of the drift. The location of the mid-pillar peak temperature is near the end of the drift, based on a series of 

sensitivity calculations the peak temperature normally occurs at approximately 80% down the length of the drift. 

X_Drift 2.7 := 5 X_Pillar Drift_Space

2 := Z_Pillar Drift_Length2 := Z_Drift Drift_Length 0. := ⋅ 8

Radial Distance from Drift Centerline Axial Distance from Start of Drift 

X_Drift 2.75 = X_Pillar 40.50 = Z_Drift 466.40 = Z_Pillar 291.50 =

The following loops are used to calculate the drift wall and mid-pillar maximum temperatures and time of maximum 

temperatures at the locations defined above. 

Pillar Pillar_Max_Time 0 ←

Pillar_Max_Temp 0 ←

Pillar 0 ←

Pillart ← WPSum Z_Pillar X_Pillar ( ) , , t

Pillar_Max_Time t ← Pillart if > Pillar_Max_Temp

Pillar_Max_Temp Pillart ← Pillart if > Pillar_Max_Temp

for t 350 450 ∈ ..

Pillar_Max_Time

Pillar_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠ return

:=

Pillar_Max_Time

Pillar_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠

:= Pillar

Peak Post Closure Mid-pillar Temperature Pillar_Max_Temp 92 = oC 

Time of Peak Temperature Pillar_Max_Time 420 = years after emplacement 

Drift_Wall Drift_Max_Time 0 ←

Drift_Max_Temp 0 ←

Drift_Wall 0 ←

Drift_Wallt ← WPSum Z_Drift X_Drift ( ) , , t

Drift_Max_Time t ← Drift_Wallt if > Drift_Max_Temp

Drift_Max_Temp Drift_Wallt ← Drift_Wallt if > Drift_Max_Temp

for t Vent 225 ∈ ..

Drift_Max_Time

Drift_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠ return

:=

Drift_Max_Time

Drift_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠

:= Drift_Wall

Peak Post Closure Drift Wall Temperature Drift_Max_Temp 146 = oC 

Time of Peak Temperature Drift_Max_Time 96 = years after emplacement 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix F- 9 

 

Plot of Temperatures as a Function of Time

The plot below represents the temperature response as a function of time at a location on the drift wall at location 

"X_Drift" and "Z_Drift" (red) and at the mid-pillar "X_Pillar" and "Z-Pillar" (blue). The following variables determine the 

time duration of the plot, t in years, the maximum allowable drift wall temperature y1 and x1, and the maximum 

allowable mid-pillar temperature, y2 and x2. 

y1

200

200

⎛

⎜

⎝

⎞

⎟

⎠ := x1

0

550

⎛

⎜

⎝

⎞

⎟

⎠ := y2

96

96

⎛

⎜

⎝

⎞

⎟

⎠ := x2

0

550

⎛

⎜

⎝

⎞

⎟

⎠ := t 02 := , .. 550

0 50 100 150 200 250 300 350 400 450 500 550

0

50

100

150

200

250

300

Drift Wall Temperature

Maximum Allowable Drift Wall Temperature

Mid-Pillar Temperature

Maximum Allowable Mid-pillar Temperature

Time from Emplacement (years)

Temperature (Deg-C)

Line_Load0 = 2.90 Z_Drift 466.40 = Drift_Length 583 =

Line_LoadVent = 0.58 Z_Pillar 291.50 = Drift_Space 81 =

WP_Spacing 0.10 = Vent 75 = X_Drift 2.75 = No_WP 105 = Eff_1 0.90 = X_Pillar 40.50 = No_Seg 15 = Eff_2 0.80 =
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix F- 10 

 

Plot of Temperatures as a Function of Axial Drift Location

The plot below represents the temperature response as a function of axial location along the drift at drift wall location 

"X_Drift" at time of peak temperature "Drift_Max_Time" (red) and at the mid-pillar location "X_Pillar" at the times of 

peak temperature "Pillar_Max_Time" (blue). The following variables determine the axial extent of the plot, z in meters, 

the location where the first waste package begins, y1 and the location where the last waste package ends, y2. 

y1

300

0

⎛

⎜

⎝

⎞

⎟

⎠ := x1

0

0

⎛

⎜

⎝

⎞

⎟

⎠ := y2

300

0

⎛

⎜

⎝

⎞

⎟

⎠ := x2 Drift_Length

Drift_Length

⎛

⎜

⎝

⎞

⎟

⎠ :=

z 100 := − , −90.. 700

− 100 0 100 200 300 400 500 600 700

0

50

100

150

200

250

300

Drift Wall Temperature

Start of Drift

Mid-Pillar Temperature

End of Drift

Distance Along Drift (meters)

Temperature (Deg-C)

Line_Load0 = 2.90 Z_Drift 466.40 = Drift_Length 583 =

Line_LoadVent = 0.58 Z_Pillar 291.50 = Drift_Space 81 =

WP_Spacing 0.10 = Vent 75 = X_Drift 2.75 = No_WP 105 = Eff_1 0.90 = X_Pillar 40.50 = No_Seg 15 = Eff_2 0.80 =

Time of drift wall temperature plot 

(years from emplacement) 

Time of mid-pillar temperature plot 

(years from emplacement) 

Drift_Max_Time 96 = Pillar_Max_Time 420 =
Appendix G 

Sample Calculation using 

Infinite Length Decaying Line Source 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix G- 1 

 

Infinite Length Cylinder Calculation

This file calculates the drift-wall temperature due to an average line load for time frames less than approximately 15 years. 

The line source strength, watts/meter, is for an infinite-length drift. Ventilation is taken into account with two ventilation

efficiencies, the first one for normal operation, "Eff_F", and the second one for natural ventilation after forced ventilation 

failure, "Eff_N". The second ventilation efficiency can be set to 0. There are no neighboring drifts. This short-term 

approximation is valid because for these short time frames a neighbor will not be "felt." The mathematics are based on the 

temperature due to a decaying flux on the wall in the infinite region bounded internally by a cylinder. 

Independent Variables

The following variables define the conditions for the calculation. The burnup should be an integer between 10 and 70 

Gwd/ton. 

Initial Rock Temperature tzero 22.8 := oC WP_Length 5.85 := Meters 

Ventilation efficiency with forced ventilation Eff_F 0.9 := % WP_Spacing 0.7 := Meters 

Ventilation efficiency with natural ventilation Eff_N 0.0 := % Drift_Rad 2.75 := Meters 

Age at Emplacement A1 16 := Years Out of Reactor Burnup 48 := GWd/ton 

Constants

The following constants are required for the calculation. 

Krock 1.83 := watt/meter - oC ρ := 2097 kg/meter3 Cp 1119.0 := joules/kg - oC 

κ Krock 3600 ⋅ ⋅24

ρ⋅Cp := meter2/day 

κ = 0.0674

Decay Variables Calculation

The following files contain the six constants that define the assembly power and decay as a function of burnup for BWR 

and PWR assemblies. The power values are calculated based on a PWR assembly with an initial loading of 0.475 

MTHM/Assembly and a BWR assembly with an initial loading of 0.200 MTHM/Assembly. 

BWR := PWR :=

The six constants are selected from the above files based on the burnup defined above and whether BWR or PWR 

assemblies are loaded into the waste package. To select the type of assembly, change the first term inside the () below 

to either "BWR" or "PWR" and define the number of assemblies contained in the WP, normally 44 for BWR WPs and 21 

for PWR WP's in the "No_Ass" variable. In addition, a correction for the initial assembly MTHM contained in each 

assembly needs to be performed by defining the base loading, 0.475 for PWR assemblies and 0.200 for BWR 

assemblies in the "MTHM_Base" variable, and the actual assembly MTHM in the "MTHM_Actual" variable. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix G- 2 

 

MTHM_Base 0.475 := MTHM_Act 0.413 := No_Ass 2 := 1

Decay submatrix PWR Burnup 1 := ( ) , − , Burnup 1 − , 1, 3 Power_Base submatrix PWR Burnup 1 := ( ) , − , Burnup 1 − , 4, 6

Decay 0.351243 0.024561 0.002473 = ( )Years-1 Power_Base 2431.86 791.50 177.66 = ( ) watts 

Power_Correction MTHM_Act

MTHM_Base :=

Half_Life .693

Decay :=

Power Power_Correction Power_Base := ⋅

Half_Life 2.0 28.2 280.2 = ( ) Years Power 2114.44 688.19 154.47 = ( ) watts 

Arrays

The following arrays are necessary in order to perform the linear line load and temperature calculations. 

T_Max 500 := 0

t 0 T_Ma := .. x

λ

λj Decay0 j , ←

for j 02 ∈ ..

return λ

:= Q

Qj Power0 j , 

No_Ass e− λj⋅A1 ⋅

WP_Length WP_Spacing +

← ⋅

for j 02 ∈ ..

return Q

:=

Line_Load_Emplace Line_Load_Emplace 0. ←

Line_Load_Emplace Line_Load_Emplace Qj ← +

for j 02 ∈ ..

Line_Load_Emplace

1000

return

:=

Line load at Emplacement Line_Load_Emplace 1.99 = kW/meter 

Q and λ need to be corrected for unit consistency. 

λ λ

365 := Q

3600. 24. ⋅ ⋅Q

2⋅π ⋅Drift_Rad :=

λ

0.00096231

0.00006729

0.000006775

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

= Days-1 Q

122881.88

7447649.47

2380351.6

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

= joules/meter2-day 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix G- 3 

 

oC 

The temperature for the infinite region bounded internally by a cylinder of radius "Drift_Rad" with a decaying wall flux is: 

vr t ( ) , , j

2. Qj ⋅

π ρ⋅ ⋅Cp

0

∞

u e

− λj⋅t e

− κ u

2 ⋅ ⋅t −

λj κ u

2 − ⋅

J0 u r ( ) Y1 u Drift_Rad ⋅ ( ) Y0 u r ⋅ − ( ) J1 u Drift_Rad ⋅ ⋅ ( ) ⋅

J1 u Drift_Rad ( ) ⋅ 2 Y1 u Drift_Rad ( ) ⋅ 2 +

⎛

⎜

⎝

⎞

⎟

⎠

⋅

⌠

⎮

⎮

⎮

⎮

⌡

:= ⋅ d

Units in front of the integral are Joules/(meter2-day) divided by (kg/m3)*(Joules/kg-oC) which yields (meter oC)/day. The 

units for the integral are day/m from the in the denominator and du. Therefore, the units are degrees C. The "time" of this 

equation is determined by the units of, which in this case is days. Calculate the temperature as a function of time, t, with a

forced ventilation efficiency "Eff_F" as follows. 

Wall_Temp

Wall_Tempt ← tzero

Wall_Tempt Wall_Tempt ← + ( ) v Drift_Rad t 1 Eff_F − ⋅ ( ) , , j

for j 02 ∈ ..

for t 0 T_Max ∈ ..

return Wall_Temp

:=

Wall t( ) Wall_Temp tzero ←

Wall_Temp Wall_Temp 1 Eff_F ← + ( ) v Drift_Rad t − ⋅ ( ) , , j

for j 02 ∈ ..

return Wall_Temp

:=

Max_Wall Wall_Max_Time 0 ←

Wall_Max_Temp 0 ←

Wall_Max_Time t ← Wall_Tempt if > Wall_Max_Temp

Wall_Max_Temp Wall_Tempt ← Wall_Tempt if > Wall_Max_Temp

for t 0 T_Max ∈ ..

Wall_Max_Time

Wall_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠ return

:=

Wall_Max_Time

Wall_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠

:= Max_Wall

Wall_Max_Years Wall_Max_Time

365 :=

Wall_Max_Time 4996 = Days from Emplacement 

Wall_Max_Years 13.69 = Years from Emplacement Wall_Max_Temp 55.33 =
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix G- 4 

 

Below is a plot of drift wall temperature as a function of time for the normal case. 

0 1 103 × 2 103 × 3 103 × 4 103 × 5 103 ×

0

20

40

60

80

100

Drift Wall Temperature with Normal Forced Ventilation

Days from Emplacement

Drift Wall Temperature - Degrees C

WP_Spacing 0.7 =

Age at emplacement (years) A1 16 = WP_Length 5.85 =

Line load at emplacement (Kw/meter) Line_Load_Emplace 1.99 = Drift_Rad 2.75 =

Burnup 48 = Forced ventilation efficiency Eff_F 0.9 =
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix G- 5 

 

0 5 10 15 20 25 30 35 40 45 50 55 60

0

30

60

90

120

150

Drift Wall Temperature When Forced Ventilation is Lost 30 Days After Emplacement

Days from Emplacement

Drift Wall Temperature - Degrees C

The temperature after ventilation is lost has the leading Q rezeroed according to the ventilation duration, "vt",: 

va r vt ( ) , , t, j

2. Qj ⋅ e

− λj⋅vt ⋅

π ρ⋅ ⋅Cp

0

∞

u e

− λj⋅t e

− κ u

2 ⋅ ⋅t −

λj κ u

2 − ⋅

J0 u r ( ) Y1 u Drift_Rad ⋅ ( ) Y0 u r ⋅ − ( ) J1 u Drift_Rad ⋅ ⋅ ( ) ⋅

J1 u Drift_Rad ( ) ⋅ 2 Y1 u Drift_Rad ( ) ⋅ 2 +

⎛

⎜

⎝

⎞

⎟

⎠

⋅

⌠

⎮

⎮

⎮

⎮

⌡

:= ⋅ d

The calculation of temperature as a function of time for a loss of ventilation event at rock location "Drift_Rad", with a specified 

preclosure forced ventilation duration," vt", forced ventilation efficiency "Eff_F", natural ventilation efficiency, "Eff_N",is as 

follows. 

Twall Drift_Rad vt ( ) sum 0. , , effcy, t ←

sum sum 1 Eff_F ← + ( ) v Drift_Rad t − ⋅ ( ) , , j

sum sum Eff_F Eff_N ← + ( ) va Drift_Rad vt − ⋅ ( ) , , t vt − , j if t vt >

for j 02 ∈ ..

return sum tzero +

:=

Drift wall temperature 30 days after loss of ventilation Twall Drift_Rad 30 ( ) 108.11 , ,Eff_F, 60 = oC 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix G- 6 

 

The following function calculates the number of years required to reach 200 oC drift wall temperature for several times 

times of ventilation loss. The matrix "Max_Time" provides the results of this calculation. The first column is the case 

number, the second column is the number of days from emplacement that the forced ventilation is lost, and the third 

column is the number of days required to reach 200 oC. 

Time_to_200 Max_Time 0. ←

Drift_Max_Temp 200 ←

Drift_Wall 0 ←

Number 0 ←

Drift_Wall Twall Drift_Rad vt ← ( ) , , Eff_N, t

Time_to_200Number 0, ← Number

Time_to_200Number 1, ← vt if Drift_Wall Drift_Max_Temp >

Time_to_200Number 2, ← t vt − if Drift_Wall Drift_Max_Temp >

Number Number 1 ← + if Drift_Wall Drift_Max_Temp >

break Drift_Wall Drift_Max_Temp if >

for t 0 T_Max ∈ ..

for vt 100 200 ∈ , .. 1400

return Time_to_200

:=

Time_to_200

0

1

2

3

4

5

6

7

8

9

10

11

12

13

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

229

226

225

224

224

224

225

226

228

229

231

233

235

238

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

=

From the "Time_to_200" matrix it can be seen that the most limiting case (i.e. the shortest time to reach 200 oC) is between 

300 and 400 days from emplacement. 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix G- 7 

 

The graph below shows the drift wall temperature as a function of time for the above scenarios. 

x1

0

5000

⎛

⎜

⎝

⎞

⎟

⎠ := y1

200

200

⎛

⎜

⎝

⎞

⎟

⎠ :=

0 1 103 × 2 103 × 3 103 × 4 103 × 5 103 ×

0

80

160

240

320

400

Drift Wall Temperature When Forced Ventilation is Lost at Various Times After Emplacement

Days from Emplacement

Drift Wall Temperature - Degrees C
Appendix H 

Sample Calculation Using 

Radiant-Heat Transfer Between Two Concentric Cylinders 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix H-1 

 

Finite Length Decaying Line Source, 7 package Segment

This file calculates temperatures at locations in the repository plane based on a fixed number of neighboring drifts. 

Each drift is similar and contains a series of identical end-to-end finite-length decaying line sources, called segments. 

Each segment contains 7 WPs and each WP has unique characteristics. The WP power is determined from the 

assembly burnup and age out of reactor and is represented by an equation with three decaying exponential terms. The 

total WP power of a segment is calculated as a function of time and the three decaying exponential terms associated 

with the total segment power are determined. During the preclosure phase, forced ventilation provides heat removal 

with the ventilation efficiency decreasing linearly as the flow moves down the drift. 

Independent Variables

The following variables define the conditions for the calculation. The burnup should be an integer between 10 and 70 

Gwd/ton. 

Initial Rock Temperature tzero 22. := 8 oC 

Est_Drift_Length 60 := 0 meters 

Ventilation Efficiency at Start of Drift Eff_1 0.90 := % 

Drift_Space 81 := meters 

Ventilation Efficiency at End of Drift Eff_2 0.80 := % 

WP_Spacing .1 := meters 

Ventilation Duration Vent 2 := 5 Years 

WP_Dia 1.882 := meters 

Number of Neighboring Drifts nbors := 5 Drifts 

Decay Variables Calculation

The following files contain the six constants that define the assembly power and decay as a function of burnup for BWR 

and PWR assemblies. The power values are calculated based on a PWR assembly with an initial loading of 0.475 

MTHM/Assembly and a BWR assembly with an initial loading of 0.200 MTHM/Assembly. 

BWR := PWR :=

The following files contain the six constants that define the canister power and decay. 

DOELong := DOEShort :=

Age 27 := BU 50 :=
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix H-2 

 

The following matrix defines the segment characteristics. The first column defines the type of waste package PWR, BWR, 

DOE Long, or DOE short. The next three columns determine the characteristics of the commercial waste packages (the 

DOE waste packages are assumed to have the same power and decay characteristics independent of age), The second 

column is the average burnup, the third column is the average age, and the forth column is the average MTHM. For DOE 

waste packages the value in the second column should be 10, the third column should be 0, and the forth columns should 

be 1. 

Seg

PWR

DOELong

PWR

PWR

DOEShort

PWR

PWR

BU

10

BU

BU

10

BU

BU

Age

0

Age

Age

0

Age

Age

.4130

1

.4130

.4130

1

.4130

.4130

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

:=

The six constants defining the power and decay are selected from the above files based on the type of waste package and, for 

the commercial waste packages, the assembly burnup defined in matrix "Seg" . 

λ0 submatrix Seg0 0, Seg0 1, , − 1 Seg0 1, , − 1, 1, 3 ( ) := P0 submatrix Seg0 0, Seg0 1, , − 1 Seg0 1, , − 1, 4, 6 ( ) :=

λ1 submatrix Seg1 0, Seg1 1, , − 1 Seg1 1, , − 1, 1, 3 ( ) := P1 submatrix Seg1 0, Seg1 1, , − 1 Seg1 1, , − 1, 4, 6 ( ) :=

λ2 submatrix Seg2 0, Seg2 1, , − 1 Seg2 1, , − 1, 1, 3 ( ) := P2 submatrix Seg2 0, Seg2 1, , − 1 Seg2 1, , − 1, 4, 6 ( ) :=

λ3 submatrix Seg3 0, Seg3 1, , − 1 Seg3 1, , − 1, 1, 3 ( ) := P3 submatrix Seg3 0, Seg3 1, , − 1 Seg3 1, , − 1, 4, 6 ( ) :=

λ4 submatrix Seg4 0, Seg4 1, , − 1 Seg4 1, , − 1, 1, 3 ( ) := P4 submatrix Seg4 0, Seg4 1, , − 1 Seg4 1, , − 1, 4, 6 ( ) :=

λ5 submatrix Seg5 0, Seg5 1, , − 1 Seg5 1, , − 1, 1, 3 ( ) := P5 submatrix Seg5 0, Seg5 1, , − 1 Seg5 1, , − 1, 4, 6 ( ) :=

λ6 submatrix Seg6 0, Seg6 1, , − 1 Seg6 1, , − 1, 1, 3 ( ) := P6 submatrix Seg6 0, Seg6 1, , − 1 Seg6 1, , − 1, 4, 6 ( ) :=

λ := stack ( ) λ0 , λ λ1 , λ 2 , λ 3 , λ 4 , 5 ,λ6 P stack P0 P1 := ( ) , ,P2, P3, P4, P5, P6

λ

0.348941

0.540572

0.348941

0.348941

0.196721

0.348941

0.348941

0.024667

0.023333

0.024667

0.024667

0.020202

0.024667

0.024667

0.002531

0.000000

0.002531

0.002531

0.000000

0.002531

0.002531

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

= P

2519.1

5.6

2519.1

2519.1

541.5

2519.1

2519.1

830.4

401.4

830.4

830.4

2386.2

830.4

830.4

187.6

0.0

187.6

187.6

0.0

187.6

187.6

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

=
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix H-3 

 

The next loop defines the base case MTHM per assembly for commercial waste packages, the number of assemblies per 

commercial waste package, and the length of the waste package. 

WP_Data WP_Data 0 ←

WP_Data i 0, ← 0.475 Segi 0, if PWR

WP_Data i 1, ← 21 Segi 0, if PWR

WP_Data i 2, ← 5.85 Segi 0, if PWR

WP_Data i 0, ← 0.200 Segi 0, if BWR

WP_Data i 1, ← 44 Segi 0, if BWR

WP_Data i 2, ← 5.85 Segi 0, if BWR

WP_Data i 0, ← 1 Segi 0, if DOEShort

WP_Data i 1, ← 1 Segi 0, if DOEShort

WP_Data i 2, ← 3.697 Segi 0, if DOEShort

WP_Data i 0, ← 1 Segi 0, if DOELong

WP_Data i 1, ← 1 Segi 0, if DOELong

WP_Data i 2, ← 5.220 Segi 0, if DOELong

for i 06 ∈ ..

return WP_Data

:=

WP_Data

0.4750

1.0000

0.4750

0.4750

1.0000

0.4750

0.4750

21.0000

1.0000

21.0000

21.0000

1.0000

21.0000

21.0000

5.8500

5.2200

5.8500

5.8500

3.6970

5.8500

5.8500

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

=

Individual waste package power at time of emplacement, "WP_Individual" in watts/WP, is calculated in the next loop. 

WP_Individual

WP_Individual i ← 0

WP_Individual i WP_Individual i WP_Data i 1, P

i j , 

Segi 3, 

WP_Data i 0, 

⋅ e

− λi j, Seg ( ) i 2, ⋅ ← + ⋅

for j 02 ∈ ..

for i 06 ∈ ..

return WP_Individual

:=

WP_Individual

10992.2

407.0

10992.2

10992.2

2927.7

10992.2

10992.2

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

= watts 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix H-4 

 

Segment Power and Decay Constants

The total segment power, "Power" in watts, is calculated in the next loops where "t" represents the time from 

emplacement 

t

t

k ← k

for k 0 1000 ∈ ..

return t

:=

Power

Powert ← 0

Powert Powert WP_Data i 1, P

i j , 

Segi 3, 

WP_Data i 0, 

⋅ e

− λi j, Segi 2, ⋅( ) +t

← + ⋅

for j 02 ∈ ..

for i 06 ∈ ..

for t 0 500 ∈ ..

return Power

:=

The loop "Power" calculates the total segment power as a function of time. The following section determines the six 

power and decay constants that define the total segment power. The six "Guess values" are required by MathCad to 

solve the six equations in six unknowns provided below. These six equations represent the segment power as the 

summation of three exponential terms at assembly ages of 0, 25, 50, 100, 200 and 300 years from emplacement. 

Guess values: 

P1 700 := 0 P2 6500 := 0 P3 1700 := 0 λ1 .1 := 4 λ2 .0 := 2 λ3 .00 := 2

Given

P1 e− λ 0⋅ 1 ⋅ P2 e− λ 0⋅ 2+ ⋅ P3 e− λ 0⋅ 3+ ⋅ Power0 

P1 e− λ 25⋅ 1 ⋅ P2 e− λ 25⋅ 2+ ⋅ P3 e− λ 25⋅ 3+ ⋅ Power25

P1 e− λ 50⋅ 1 ⋅ P2 e− λ 50⋅ 2+ ⋅ P3 e− λ 50⋅ 3+ ⋅ Power50

P1 e− λ 100⋅ 1 ⋅ P2 e− λ 100⋅ 2+ ⋅ P3 e− λ 100⋅ 3+ ⋅ Power100

P1 e− λ 200⋅ 1 ⋅ P2 e− λ 200⋅ 2+ ⋅ P3 e− λ 200⋅ 3+ ⋅ Power200

P1 e− λ 500⋅ 1 ⋅ P2 e− λ 500⋅ 2+ ⋅ P3 e− λ 500⋅ 3+ ⋅ Power500

P1_Value

P2_Value

P3_Value

λ1_Value

λ2_Value

λ3_Value

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

:= Find P1 P2 ( ) , , λ P3, λ 1 , λ 2 , 3
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix H-5 

 

MathCad calculated the following values for the six constants: 

λ1_Value 0.1318 = P1_Value 663.03 =

λ2_Value 0.0243 = P2_Value 41620.43 =

λ3_Value 0.0025 = P3_Value 16012.3844 =

λ

λ1_Value

λ2_Value

λ3_Value

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

:= P

P1_Value

P2_Value

P3_Value

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

:= Half_Life .693

λ :=

Half_Life

5.3

28.5

273.6

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

= λ

0.1318

0.0243

0.0025

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

= P

663.0348

41620.4314

16012.3844

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

=

Emplacement Drift Parameters

The following calculation determines the exact length of the drift and the number of waste packages per drift 

based on the assumptions provided above. 

Seg_L Seg_L 0 ←

Seg_L Seg_L WP_Data i 2, ← + + WP_Spacing

for i 06 ∈ ..

:=

No_Seg round Est_Drift_Length

Seg_L

⎛

⎜

⎝

⎞

⎟

⎠ := Drift_Length No_Seg Seg_ := ⋅ L No_WP No_Seg := ⋅7

Actual Drift Length Drift_Length 583.01 = meters Number of WP per Drift No_WP 105 = WP per Drift 

Number of Segments per Drift No_Seg 15 = Segments Segment Length Seg_L 38.87 = meters 

The next expression converts the power in watts to linear power in watts/meter 

Q

P

Seg_L :=

Q

17.0591

1070.8424

411.9789

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

= watts/meter 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix H-6 

 

Ventilation Efficiency

The following loop calculates the ventilation efficiency for each segment based on the efficiency at the start of the drift 

of Eff_1 and the efficiency at the end of the drift Eff_2. 

Eff_1 0.9000 = Eff_2 0.8000 =

Eff

x Seg_L

2 ← + n Seg_L ⋅

Effn

( ) Eff_2 Eff_1 −

Drift_Length ← ⋅x + Eff_1

for n 0 No_Seg 1 ∈ .. −

return Eff

:=

Power

The next loops calculate the average waste package power, "WP_Power", and linear line load, "Line_Load", as a function 

of time from emplacement. 

Line_Load

Line_Loadt ← 0

Line_Loadt Line_Loadt Qj

e

− λj⋅t

1000 ← + ⋅

for j 02 ∈ ..

for t 0 Vent 100 ∈ .. +

return Line_Load

:=

WP_Power

WP_Powert ← 0

WP_Powert WP_Powert Pj e

− λj⋅t ← + ⋅

for j 02 ∈ ..

for t 0 200 ∈ ..

WP_Power

7

return

:=

The Average waste package power and linear line load at emplacement, time = 0, and closure, time = vent, calculated from 

array "WP_Power" and "Line_Load" are: 

WP power at emplacement WP_Power 0 = 8328.0 Watts Line load at Emplacement Line_Load0 = 1.50 kW/meter 

WP power at closure WP_Power Vent = 5386.3 Watts Line load at closure Line_LoadVent = 0.97 kW/meter 

Constants

The following constants are required for the calculation. 

Krock 1.83 := watt/meter - oC ρ := 2097 kg/meter3 Cp 1119.0 := joules/kg - oC 

κ Krock 3600 ⋅ ⋅24⋅365

ρ⋅Cp := meter2/year Emissivities are: 

e_Drift 0.9 := e_Drip 0.9 := e_WP 0.9 := σ 5.67 10− 8 := ⋅ watts/meter2-oC4
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix H-7 

 

Temperature Calculation

The equation for calculating the temperature due to a single finite-length decaying line source (or segment) is: 

vzx ( ) , , t, j

e

− λj⋅t

Qj ⋅

8⋅π ⋅Krock

0

t

erf θ z

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ erf z Seg_L −

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ − ⎛

⎜

⎝

⎞

⎟

⎠

e

λj⋅θ

θ ⋅ e

x

2 −

4⋅ θ κ⋅ ⋅

⌠

⎮

⎮

⎮

⎮

⌡

:= ⋅ d

The temperature contribution after ventilation shutdown is as follows, the A2 in the leading exponential re-zeros Qij for 

ventilation shutdown. 

va z x ( ) , , t, j

e

− λj⋅( ) Vent t + Qj ⋅

8⋅π ⋅Krock

0

t

erf θ z

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ erf z Seg_L −

2⋅ κ θ⋅

⎛

⎜

⎝

⎞

⎟

⎠ − ⎛

⎜

⎝

⎞

⎟

⎠

e

λj⋅θ

θ ⋅ e

x

2 −

4⋅ θ κ⋅ ⋅

⌠

⎮

⎮

⎮

⎮

⌡

:= ⋅ d

The segment temperature contribution for the n-th segment for the three decaying components is: 

WPtemp z x ( ) tsum 0.0 , , t, n ←

tsum tsum 1. Effn − ( ) ← + ⋅v z Seg_L n ( ) − ⋅ , x, t, j

tsum tsum Effn ← + ⋅va z Seg_L n ( ) − ⋅ , x, t Vent − , j if t Vent >

for j 02 ∈ ..

return tsum

:=

The temperature contribution from all segments, "No_Seg", and neighbors, "nbors", is: 

WPSum z x ( ) tsum 0. , , t ←

tsum tsum WPtemp z x ← + ( ) , , t, n

for n 0 No_Seg 1 ∈ .. −

x1Loc m Drift_Space ← ⋅ − x

x2Loc m Drift_Space ← ⋅ + x

tsum tsum WPtemp z x1Loc ← + ( ) , , t, n

tsum tsum WPtemp z x2Loc ← + ( ) , , t, n

for n 0 No_Seg 1 ∈ .. −

for m 1 nbors ∈ ..

return tzero tsum +

:=
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix H-8 

 

The following variables define the radial, "X_Drift", "X_Drip", and "X_WP" and axial, "Z_Drift" locations used to calculate 

the maximum temperature. The location of the Drift Wall peak temperature is near the end of the drift, based on a series 

of sensitivity calculations the peak temperature normally occurs at approximately 80% down the length of the drift. 

X_Drift 2.7 := 5 X_Drip 1. := 3 X_WP WP_Dia

2 := Z_Drift Drift_Length 0. := ⋅ 8

The following loop are used to calculate the drift wall maximum temperature and time of maximum temperature at the locations 

defined above. 

Drift_Wall Drift_Max_Time 0 ←

Drift_Max_Temp 0 ←

Drift_Wall 0 ←

Drift_Wallt ← WPSum Z_Drift X_Drift ( ) , , t

Drift_Max_Time t ← Drift_Wallt if > Drift_Max_Temp

Drift_Max_Temp Drift_Wallt ← Drift_Wallt if > Drift_Max_Temp

for t Vent 225 ∈ ..

Drift_Max_Time

Drift_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠ return

:=

Drift_Max_Time

Drift_Max_Temp

⎛

⎜

⎝

⎞

⎟

⎠

:= Drift_Wall

Peak Post Closure Drift Wall Temperature Drift_Max_Temp 200 = oC 

Time of Peak Temperature Drift_Max_Time 42 = years after emplacement 

The invert is assumed to be an insulator so the surface areas "A_Drift", "A_Drip", and "A_WP" used to calculate the drip shield

and waste package surface temperatures are reduced to 75% of their actual area. 

A_Drift 2 := ( ) 0.7 ⋅π ⋅X_Drift ⋅ 5 A_Drip 2 := ( ) 0.7 ⋅π ⋅X_Drip ⋅ 5 A_WP 2 := ( ) 0.7 π ⋅X_WP ⋅ 5

The energy transported between two surfaces, "Q1",can be calculated as follows: 

Q1

Q1t ← 0

Q1t Q1t Qj

e

− λj⋅t ← +

for j 02 ∈ ..

for t 0 300 ∈ ..

return Q1

:=

Q1Drift_Max_Time = 755.76 joules/meter-sec 
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix H-9 

 

The following expressions are used to calculate the drip shield and waste package temperatures using the radiant heat 

transfer between two concentric cylinders at the time of maximum drift wall temperature. 

Drip shield temperature

Drip_Max_Temp

Q1Drift_Max_Time

σ

1.

A_Drip e_Drip ⋅

1

A_Drift

1

e_Drift − 1 ⎛

⎜

⎝

⎞

⎟

⎠ + ⎡

⎢

⎣

⎤

⎥

⎦ ⋅ ( ) 273 Drift_Max_Temp +

4

+

⎡

⎢

⎣

⎤

⎥

⎦

0.25

:= − 273

Maximum drip shield temperature Drip_Max_Temp 206 = oC 

Waste package surface temperature

WP_Max_Temp

Q1Drift_Max_Time

σ

1.

A_WP e_WP ⋅

1

A_Drip

1

e_Drip − 1 ⎛

⎜

⎝

⎞

⎟

⎠ + ⎡

⎢

⎣

⎤

⎥

⎦ ⋅ ( ) 273 Drip_Max_Temp +

4

+

⎡

⎢

⎣

⎤

⎥

⎦

0.25

:= − 273

Maximum waste package temperature WP_Max_Temp 214 = oC 

The temperature profile of the waste package, drip shield, and drift wall as a function of time are calculated from the three 

loops below. 

Drift Drift 0 ←

Driftt ← WPSum Z_Drift X_Drift ( ) , , t

for t 0 300 ∈ ..

return Drift

:=

Drip

Dript

Q1t

σ

1.

A_Drip e_Drip ⋅

1

A_Drift

1

e_Drift − 1 ⎛

⎜

⎝

⎞

⎟

⎠ + ⎡

⎢

⎣

⎤

⎥

⎦ ⋅ 273 Driftt + ( )4

+

⎡

⎢

⎣

⎤

⎥

⎦

0.25

← − 273

for t 0 300 ∈ ..

return Drip

:=

WP

WPt

Q1t

σ

1.

A_WP e_WP ⋅

1

A_Drip

1

e_Drip − 1 ⎛

⎜

⎝

⎞

⎟

⎠ + ⎡

⎢

⎣

⎤

⎥

⎦ ⋅ 273 Dript + ( )4

+

⎡

⎢

⎣

⎤

⎥

⎦

0.25

← − 273

for t 0 300 ∈ ..

return WP

:=
Thermal Response Evaluation of Yucca Mountain 

July 2008 Appendix H-10 

 

The graph of waste package, drip shield, and drift wall temperature as a function of time is provided below. 

0 100 200 300

25

50

75

100

125

150

175

200

225

Drift Wall Temperature

Drip Shield Temperature

Waste Package Temperature

Time From Emplacement - Years

Temperature - deg-C

Drift_Max_Temp 200 = oC Drip_Max_Temp 206 = oC WP_Max_Temp 214 = oC 

Age := 27 BU := 60

Seg

PWR

DOELong

PWR

PWR

DOEShort

PWR

PWR

BU

10

BU

BU

10

BU

BU

Age

0

Age

Age

0

Age

Age

.4130

1

.4130

.4130

1

.4130

.4130

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

:= Ventilation Efficiency at Start of Drift Eff_1 0.90 = % 

Ventilation Efficiency at End of Drift Eff_2 0.80 = % 

Ventilation Duration Vent 25 = Years 

Line load at Emplacement Line_Load0 = 1.50 watts/meter 

Line load at closure Line_LoadVent = 0.97 watts/meter