Working Report 2015-01 Thermal Analysis of KBS-3H Repository Kari Ikonen, Heikki Raiko May 2015 POSIVA OY Olkiluoto FI-27160 EURAJOKI, FINLAND Phone (02) 8372 31 (nat.), (+358-2-) 8372 31 (int.) Fax (02) 8372 3809 (nat.), (+358-2-) 8372 3809 (int.)
Working Report 2015-01 Thermal Analysis of KBS-3H Repository Kari Ikonen, Heikki Raiko VTT May 2015 Working Reports contain information on work in progress or pending completion.
THERMAL ANALYSIS OF KBS-3H REPOSITORY ABSTRACT This report contains the temperature analyses of the KBS-3H type nuclear fuel repository, where the fuel canisters are emplaced at horizontal position in the horizontal drifts in a rectangular geometry according to the preliminary SKB (Swedish Nuclear Fuel and Waste Management Co) and Posiva plan. The analyses are performed for Olkiluoto BWR canisters, Loviisa VVER canisters and Olkiluoto EPR canisters. The temperatures were determined in the initial conditions, with artificially wetted outer gap and in the saturated buffer case. Initial values, related to dimensions with tolerances, are chosen so that temperatures are conservatively overestimated. The maximum temperature on the canister/buffer interface is limited to the design temperature of +100 o C. However, due to uncertainties in thermal analysis parameters (like variation in rock thermal conductivity) the allowable calculated maximum canister temperature is set to 95 o C leaving a safety margin of 5 o C. The allowable temperature can be controlled by adjusting the space between adjacent canisters, adjacent drifts and the distance between separate panels of the repository and the pre-cooling time affecting thermal power of the canisters. In the horizontal disposal the inner air gap is closed underneath the cylinder and heat transfer on the contact area is improved, when compared to corresponding axisymmetric case. The effect of eccentricity was studied by a 2D model in the cross-section of a canister. Sensitivity analyses of the effect of different parameters were made also. Thermal conductivity and capacity of rock, thermal conductivity of bentonite, the emissivity of the copper surface, the gap width on the canister surface, the canister spacing and the precooling time were varied. A comparison to KBS-3V type configuration of a repository was made by using then the same heat loads in canisters, the same spacings between the canister centre points (9.0 m for BWR, 7.2 m for VVER and 10.6 m for EPR) and same average emplacement schedules. The result was that in the initial conditions the temperature on the canister/buffer interface was 2.9 3.7 o C lower than that of the respective KBS-3V configuration. This is caused by that the inner air gap dimension is lower than in KBS-3V (10 mm), outer gap in KBS-3H has better thermal conductivity than pellet filled outer gap in KBS-3V and the eccentricity of the air gap in KBS-3H lowers canister temperature. Although the maximum temperature in KBS-3H is only about 2.9 3.7 C lower than in KBS-3V when using same canister spacing, the minimum canister spacings in KBS-3H (based on the thermal analysis alone) would be for BWR 8.0 m, for VVER 6.4 m and for EPR 9.1 m, when the allowable temperature is 95 C. Keywords: Spent nuclear fuel, repository, decay heat, temperature dimensioning.
KBS-3H LOPPUSIJOITUSTILAN LÄMPÖANALYYSI TIIVISTELMÄ Tutkimusselostuksessa kuvataan korkea-aktiivisten BWR-, VVER- ja EPR-ydinpolttoaineiden loppusijoituksen (LS-tilan) kallioon synnyttämien lämpötilakenttien laskentatulokset, kun kapselit sijoitetaan SKB:n ja Posivan kehittämän KBS-3H ratkaisun mukaisesti vaaka-asentoon pitkiin loppusijoitusreikiin. Lämpötilat määritettiin asennusolosuhteissa, ulkoraon keinokastelun sekä saturoituneen bentoniittipuskurin tapauksessa. Mittatoleransseja sisältävät alkutiedot on valittu siten, että lämpötiloille saadaan konservatiiviset tulokset. Kapselin ja sen ympärillä olevan bentoniittipuskurin maksimilämpötila rajoitetaan suunnitteluarvoon +100 o C. Kuitenkin materiaalien lämpöteknisiin ominaisuuksiin liittyvien epävarmuuksien (kuten kallion lämmönjohtumiskertoimen vaihtelun) takia laskennalliseksi lämpötilaksi sallitaan korkeintaan 95 o C, mikä merkitsee 5 o C:een turvallisuusmarginaalia. Maksimilämpötilaa voidaan säätää muuttamalla kapseliväliä, tunneliväliä, paneeleiden välistä etäisyyttä sekä kapselin lämpötehoa. Kapselin asennon epäkeskisyydestä johtuen kapselin ja bentoniitin välillä on kontakti, mikä vaakaratkaisussa alentaa kapselin lämpötilaa aksisymmetriseen tilaan verrattuna. Asiaa tutkittiin 2D-mallilla kapselin poikkileikkaustasossa. Työssä tehtiin myös herkkyystarkasteluja eri parametrien suhteen varioimalla kallion lämmönjohtavuutta ja lämpökapasiteettia, bentoniitin lämmönjohtavuutta, kuparipinnan emissiviteettia, ilmaraon paksuutta kapselin pinnalla, kapselien keskipisteiden välistä etäisyyttä sekä käytetyn polttoaineen esijäähdytysaikaa. Työssä verrattiin KBS-3H -vaakaratkaisun lämpötiloja KBS-3V pystyratkaisuun käyttämällä samoja kapselin lämpötehoja, samoja kapselivälejä (9.0 m BWR-kapselille, 7.2 m VVER-kapselille ja 10.6 m EPR-kapselille) ja sijoitusnopeuksia. Tulokseksi saatiin, että kapselin ja bentoniitin maksimilämpötila jää 2.9 3.7 o C alhaisemmaksi kuin KBS-3V -ratkaisussa. Tämä aiheutuu vaakaratkaisussa kapselin pienemmästä ilmaraosta (5.1 mm) pystyratkaisun ilmarakoon (10 mm) verrattuna, ulkoraon paremmasta lämmönjohtavuudesta, epäkeskisyyden aiheuttamasta paremmasta lämmönsiirrosta sekä kapselin asennosta. Vaikka maksimilämpötila on KBS-3H vaakaratkaisussa on vain 2.9 3.7 C alhaisempi kuin KBS-3V pystyratkaisussa käytettäessä samoja kapselivälejä, kapselivälin sallittavaksi vähimmäisarvoksi (perustuen ainoastaan termiseen analyysiin) saadaan BWR-kapselille 8.0 m, VVER-kapselille 6.4 m ja EPR-kapselille 9.1 m silloin, kun sallittu maksimilämpötila on 95.0 C. Avainsanat: Käytetty ydinpolttoaine, loppusijoitustila, jälkilämpö, lämpötilamitoitus.
SYMBOL LIST LATIN ALPHABET unit c thermal heat capacity [J/kg/K] c v volumetric thermal heat capacity [J/m 3 /K] erf error function [-] F view factor [-] H actual length of a canister [m] H eff effective length of line heat source [m] k heat flux reduction coefficient [-] P power [W] q thermal density [J/m 3 ] R, r radius [m] r 0 external radius of canister [m] T temperature [ o C or K] T 0 canister surface temperature [ o C] T 0b bentonite surface temperature [ o C] t time [s] V volume [m 3 ] x, y, z cartesian co-coordinates [m] GREEK ALPHABET α heat transfer coefficient [W/m 2 /K] δ air gap width in air-filled gap [m] Δt time increment [s] ε emissivity [-] ε tot total emissivity [-] φ thermal flux [W/m 2 ] Φ volumetric heat generation [W/m 3 ] λ thermal conductivity [W/m/K] ν thermal diffusivity, ν = λ/(ρc) [m 2 /s] ρ rock density [kg/m 3 ] σ Stefan-Bolzmann constant = 5.6697 10-8 [W/(m 2 K 4 )] SPECIAL NOTATION BWR boiling water reactor DAWE Drainage, Artificial Watering and air Evacuation EPR European pressurized water reactor PWR pressurized water reactor VVER Russian type pressurized water reactor LO1-2 Loviisa units 1 and 2 OL1-2 Olkiluoto units 1 and 2 OL3 Olkiluoto unit 3 2D two-dimensional 3D three-dimensional ln natural logarithm SC Supercontainer
1 CONTENTS ABSTRACT TIIVISTELMÄ SYMBOL LIST 1 INTRODUCTION... 3 2 INITIAL DATA... 7 2.1 Geometry of Supercontainer and drift... 7 2.2 Thermal properties of materials... 11 2.3 Emissivity of air gap surfaces... 16 2.4 Decay heat power modelling... 17 2.5 Effective thermal conductivity of perforated container shell and surrounding gaps... 19 3 CALCULATION METHODOLOGY AND ITS VERIFICATION... 23 3.1 Numerical analysis of a single BWR canister in KBS-3H drift... 23 3.2 Effect of artificial wetting of outer gap of a single canister... 30 3.3 Heat flux reduction coefficient... 31 3.4 Analytical solution of line segment heat source... 32 3.5 Extrapolation of canister surface temperature in analytic line segment heat source model... 33 3.6 Effect of the inner air gap eccentricity... 37 4 ANALYSIS AND RESULTS OF A SINGLE PANEL... 43 4.1 Distances in a drift... 43 4.2 Disposal operation and schedule... 45 4.3 Detailed description of BWR canister analyses... 47 4.4 Temperatures in panels containing BWR, VVER and EPR canisters 51 4.4.1 Temperatures in initial conditions... 51 4.4.2 Temperatures with artificially wetted outer gap... 53 4.4.3 Temperatures in saturated buffer case... 53 4.4.4 Summary of maximum temperatures... 56 4.5 Sensitivity to various parameters... 56 4.5.1 Results of variation of parameters... 56 4.5.2 Statistics of heat flux reduction coefficient k values... 62 4.6 Comparison of KBS-3H and KBS-3V temperatures... 63 5 CONCLUDING REMARKS... 65 REFERENCES... 67
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3 1 INTRODUCTION The goal of this study is to determine temperatures in the KBS-3H type spent fuel repository according to the current state of the art and available data of the thermal properties of different materials. This is needed in showing that acceptable temperatures are not exceeded and in the evaluation of proper canister spacings. Initial values, related to dimensions with tolerances, are chosen so that temperatures are conservatively overestimated. Temperatures are determined in three cases, in initial conditions, with artificially wetted outer gap and in saturated buffer case. Initial condition case means the conditions at the disposal moment. Artificial wetting takes place after about two weeks after disposal. Saturated buffer case is reached within some decades. The principal layout of the Posiva repository in Olkiluoto is shown in Figure 1 at the depth of 420 m. In the KBS-3V design the canisters are emplaced at vertical position. In parallel with the KBS-3V concept, the KBS-3H concept, where the canisters are emplaced at horizontal position, has been developed. The objective of this work is to evaluate the thermal behaviour of KBS-3H type spent nuclear fuel repository. The analyses are made for the Finnish BWR and PWR fuel canisters of Olkiluoto and Loviisa nuclear power plants. Figure 1. Principal layout of the Olkiluoto repository (Posiva Oy). In the KBS-3H concept, where the canisters are emplaced in up to 300 m long horizontal deposition drifts, the drift spacing is 25 m. The deposition drift is divided into two compartments both being 150 m long, at maximum, and the compartments are separated
4 from each other by a titanium compartment plug. The titanium drift plug that is mounted at the drift entrance resembles the compartment plug, but is much sturdier. DAWE (Drainage, Artificial Watering and air Evacuation), selected as the reference design, is based on drainage during installation phase, filling the void spaces with water and air evacuation from the drift. The horizontal deposition drift is slightly inclined upwards and therefore the leakage water flows under the drift components standing on parking feet along the drift bottom down towards the drift entrance until the plug has been mounted. The gap between the drift components and the drift wall is about 42.5 mm and it will be filled by pumping water in the bottom part of the drift via short pipes in the plug lead- throughs. An air evacuation pipe will be placed on supports in the drift wall during the preparation phase of the drift. In the rear section of each compartment a vertical pipe, which leads to the uppermost point of the compartment, where an air pocket will be formed during artificial watering, is attached to the air evacuation pipe. The vertical pipe will be left in the drift when the horizontal air evacuation pipe is removed together with the short wetting pipes from the drift compartment. The penetrations will be sealed and installation operations in the second compartment will be initiated without delay. In the final step, the second compartment and at the same time the whole drift will be sealed with the drift plug. The artificial water filling as well as the pipe removal procedures are carried out also in the latter compartment as described above. The objective of filling the void space in the drift is to accelerate the swelling of bentonite in order to attain counter pressure against the drift wall. The counter pressure can mitigate the impacts of rock spalling or at least mitigate rock spalling induced by rock and thermal stresses. In contrast to the KBS-3V variant, KBS-3H utilises a prefabricated installation package called a Supercontainer. The Supercontainer consists of a perforated protective cylinder made of titanium with a bentonite buffer and copper canister installed inside it. The Supercontainer is assembled before disposal, and installed in the deposition drift. Distance blocks made of compressed bentonite thermally and hydraulically separate the Supercontainers from each other in the long deposition drift. The bentonite inside the Supercontainer and in the distance blocks collectively forms the buffer structure (the bentonite inside the Supercontainers and the bentonite distance blocks are jointly referred to as the buffer). Filling components are used in those drift sections that do not fulfil the positioning criteria for safe disposal (e.g. excessive inflow rate of groundwater into the deposition drift). These drift sections, where the inflow rate is too high, will be post-grouted using colloidal silica aiming at improving the installation conditions and in these drift sections filling blocks are emplaced. The filling blocks are made of highly compacted bentonite and their axial dimensions are dependent on the trace length of the intersecting fracture. This introduction is made according to the description of KBS-3H alternative in (Posiva 2012). In this analysis, the thermal evolution of the canisters and the near field is made in the KBS-3H configuration. Sensitivity analyses of the effect of the material thermal properties are made and, in addition, a comparison to KBS-3V type configuration of a repository. Figure 2 illustrates the structure of a KBS-3H drift.
5 Figure 2. Principal layout of the Posiva KBS-3H drift. The maximum allowable temperature in a canister shall be limited to the design temperature of +100 o C. Due to uncertainties and natural variation in parameters of thermal analysis the allowable calculated maximum canister/buffer interface temperature (reference temperature) is set to 95 o C rendering a safety margin of 5 o C. The work consists of first the adaptation, checking and verification of the calculation process and then the analyses of the repository. This study has been made at Technical Research Centre of Finland (VTT) for Posiva Oy on contract.
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7 2 INITIAL DATA In the following the geometry of canister and Supercontainer, thermo-mechanical properties of materials, decay heat power and emissivity of air gap surfaces are presented. 2.1 Geometry of Supercontainer and drift Figure 3 shows the Supercontainer components in the KBS-3H system. Figure 3. An exploded view of the Supercontainer components in KBS-3H system. Table 1 and Figure 4 show the common dimensions of the fuel canister, disposal Supercontainer and disposal drift used in the analyses. The lengths of the Olkiluoto BWR canister, the Loviisa VVER and Olkiluoto EPR canister are 4752 mm, 3552 mm and 5223 mm, respectively. For all the canister types the external diameter is 1050 mm. At the depth of 420 m the ambient rock temperature in Olkiluoto is +10.5 o C.
8 Table 1. Common dimensions of drift, distance block and around for the BWR, VVER and EPR fuel canisters and Supercontainers. If the dimension used in thermal analysis is not given in table, the nominal dimension has been used. Parameter Dimension used in thermal analysis [mm] Nominal dimension [mm] Drift diameter with tolerances (0/+5) 1855 1850 Diameter of the distance block (bentonite material) 1765 Average gap between the distance block and the rock 45.0 42.5 Gap at top 38.8 *) 33.1 Gap at bottom 51.2 *) 51.8 Outer diameter of solid and ring-shaped bentonite blocks inside SC (Supercontainer) 1738 1740 Length of solid bentonite blocks inside SC 350 Inner diameter of ring-shaped blocks inside the Supercontainer 1058 External diameter of the canisters (copper) 1050 Average width of air gap on copper surface 5.1 4 Gap at top (min 4.8 mm, max 10.2 mm) 10.2 8 Gap at bottom 0 Gap between canister and right end block, (min 1 mm, max 15 mm) 15.0 8 Outer diameter of the Supercontainer 1761 Inner diameter of the Supercontainer 1749 Thickness of cylindrical shell and end plates of SC 6.0 Average width of the outer air gap between the cylindrical Supercontainer and the rock 47.0 44.5 Gap at top 43.3 *) 37.6 Gap at bottom 50.7 *) 51.4 Average width of the inner air gap between buffer and the cylindrical titanium shell of the SC 5.5 4.5 Gap at top (min 6 mm, max 11 mm) 11 9 Gap at bottom 0 Total average width of the outer three layers (air, titanium, air) between buffer and rock = 47.0 mm + 6.0 mm + 5.5 mm 58.5 55 *) The gaps are calculated with the maximum drift diameter 1855 mm in order to get conservatively higher temperatures. The nominal values are calculated with the nominal drift diameter 1850 mm.
9 Figure 4. Dimensions used in thermal analyses (see Table 1). Values of the reference case (BWR canister) are written in red colour. These dimensions are not nominal but conservative values used in the thermal analyses.
10 Figure 5 illustrates the gaps in the cross-section (values presented in Table 1). Thus there are two asymmetric air gaps. The effect of eccentricity of the inner gap between the canister and bentonite ring block is more significant than the outer gap (air+titanium+air) as shown later. Air gap 10.2 (Cu-buffer) Outer super-container gap 43.3 (Ti-rock) Inner super-container gap 11.0 (buffer-ti) Canister 73.5 deg. Bentonite buffer Super-container 6 mm Foot 50 mm Outer super-container gap 50.7 Rock Figure 5. Illustration of the eccentricity of the cylinders. The Supercontainer is supported and centered by small feet (Figure 6) at the position 73.5 deg in the disposal drift. Titanium has quite good thermal conductivity, but the contact area of the feet is small and rock on one side and bentonite on other side has low thermal conductivity. Thus the effect of support feet in thermal conduction is small and it is ignored. Figure 6. Structure of the foot.
11 2.2 Thermal properties of materials The thermal analyses are generally made with theoretically well-founded methods and the thermal properties are selected to reflect the real and expected values of the properties and conditions or the parameters of the system components. Conservatism has been added to the allowable temperatures that are set with a considerable margin. Many of the thermal analyses concerning operational phase can be verified by simple measurements during the pre-operation test phase of the encapsulation plant and repository. As for the large-scale thermal properties of rock, additional data are collected during the construction phase of the repository. Thermal dimensioning of the repository can be updated later, if new data become available and the local canister spacings can be adapted accordingly even during plant operation. It should be noted, however, that in the KBS-3H concept the distance blocks have an additional purpose of separating the Supercontainers hydraulically from each other, and the canister spacings cannot, therefore, be shortened based on thermal factors alone. The thermal conductivity and the heat capacity of Olkiluoto rock are based on laboratory measurements with core drilled samples (Kukkonen et al. 2011). The conductivity of the rock decreases slightly as a function of temperature (Figure 7a) and at the temperatures of 25 C, 60 C and 100 C the thermal conductivity is 2.91±0.51, 2.82 and 2.72 W/m/K, respectively. The temperature varies in the rock around a canister between 10.5 65 C. The constant average value of 2.82 W/m/K (at 60 C) is used for all the Olkiluoto repository analysis in this report in the base cases. The heat capacity of the rock increases slightly as a function of temperature and at the temperatures of 25 C, 60 C and 100 C the capacity is 712, 764 and 824±32 J/kg/K, respectively (Figure 7b). The value of 764 J/kg/K (at 60 C) is used throughout the analysis. With the measured average rock density of 2743 kg/m 3 the applied volumetric (a) (b) Figure 7. Thermal conductivity and heat capacity of Olkiluoto rock (Kukkonen et al. 2011).
12 heat capacity of the rock material is 2.1 MJ/m 3 /K. The diffusivity ν = λ/(ρc), which appears in the analytic formulas (Equation 8 presented later), is 1.34. 10-6 m 2 /s at temperature 60 C. The thermal conductivity of bentonite buffer depends on the saturation rate (Figure 8). In normal atmospheric humidity conditions the bentonite thermal conductivity is (Börgesson 1994) about 0.75 W/m/K. In dry condition the thermal conductivity is about 0.3 W/m/K and in fully saturated condition 1.3 W/m/K. In the repository condition the effective thermal conductivity of the compacted buffer bentonite is estimated to be 1.0 W/m/K (Hökmark & Fählt 2003; Ageskog & Jansson 1999; Hökmark et al. 2009). This estimate is based on the instrumented in situ demonstration test made earlier in Äspö (Prototype Repository test) and reported in (Hökmark & Kristensson 2010). The thermal conductivity of bentonite may depend on temperature, but its effect is much lower than the effect of water saturation rate. Ageskog and Jansson considered four concentric cylindrical shells to model the ringshaped buffer with conductivities ranging from 0.9 W/m/K for the inner parts to 1.15 W/m/K for the outer parts. This kind of modelling led to λ = 1.0 W/m/K, which corresponds to the effective bentonite conductivity in the whole block. In (Hökmark et al 2009, Appendix 4) bentonite conductivity λ = 1.0 W/m/K is used as a reference value for calculations and values λ = 1.1-1.2 W/m/K are given as the bentonite block thermal conductivity in the unsaturated state at the time of deposition. The experimental data (Figure 8) indicate that the conductivity is not very sensitive to changes in saturation, at least not in the high-saturation range. The saturation must drop below about 65% to bring the conductivity below 1.0 W/m/K. In this analysis the conductivity is conservatively set to 1.0 W/m/K and the saturation is not supposed to take place before the maximum temperature is reached. The effect of DAWE watering is conservatively not taken into account in the bentonite conductivity. Figure 8. Heat conductivity of MX80 bentonite as a function of the degree of saturation. The legend gives the void ratio. (Börgesson 1994).
13 The thermal conductivity of air and humid air (relative humidity 100%) as a function of temperature is presented Figure 9a. Possible humidity in the air is diffused into bentonite. Thus the thermal conductivity for pure air is applied. On the other hand, the possible humidity has only a minor decreasing effect on thermal conductivity. The upper curve in Figure 9a and thermal capacity in Figure 9b for air are approximated by (T in Celsius)!(T) = 0.0243 + 7.07"10 #5 T [W/m/ o C] c v (T) = 1245 # 2.29"T [J/m 3 / o C]. (1) These fittings are used in the analyses for air filled gaps. Very small thermal capacity of air has no effect at all in the calculation process. Figure 9. Thermal conductivity of air (black line) and humid air in atmospheric pressure (left) and thermal capacity (right) (Fletcher 1991). Temperature dependence of the thermal conductivity and volumetric heat capacity of water is plotted in Figure 10 (Fletcher 1991). The curves presented in Figure 10 are approximated by (T in Celsius) and used in the analyses for water!(t) = 0.552 + 2.31"10 #3 T # 1.03"10 #6 T 2 [W/m/ o C] c v (T) = 4.22 # 0.001675"10 #5 T [MJ/m 3 / o C]. (2) Figure 10. Thermal conductivity and volumetric capacity of water (Fletcher 1991).
14 Table 2 summarizes the thermal properties, temperatures and surface emissivities for the BWR, VVER and EPR fuel canisters used in the analyses. Table 2. Thermal properties, initial temperatures and surface emissivities for the BWR, VVER and EPR Supercontainers used in the analyses (reference values). Parameter Value used Unit Reference in the analysis model Canister copper overpack conductivity 390 W/m/K *) Copper volumetric heat capacity 3.45 MJ/m 3 /K *) Bentonite thermal conductivity in installation 1.0 W/m/K Hökmark 2009 condition (not T-dependent) Bentonite thermal conductivity in saturated (wet) conditions 1.3 W/m/K Hökmark 2009 Bentonite (buffer) volumetric capacity 2.4 MJ/m 3 /K Hökmark 2009 Density of rock 2743 kg/m 3 Kukkonen 2011 Rock thermal conductivity at 60 o C 2.82 W/m/K Kukkonen 2011 Rock volumetric heat capacity at 60 o C 2.10 MJ/m 3 /K Kukkonen 2011 Titanium thermal conductivity 19.0 W/m/K *) Titanium density 4500 kg/m 3 **) Titanium volumetric heat capacity 2.43 MJ/m 3 /K **) Emissivity of copper surface 0.30 - ***) Emissivity of bentonite surface 0.80 - ***) Emissivity of rock surface at 50 o C 0.90 - ***) Emissivity of titanium surface 0.5 - ***) Initial canister temperature 50 C Calculated estimate Volumetric heat capacity of outer gap (Supercontainer shell and air gaps) 0.095****) MJ/m 3 /K Calculated Initial undisturbed rock temperature +10.5 C Ikonen 2012 (-400 m) Air thermal conductivity Eq. (1) W/m/K Fletcher 1991 *) http://www.engineeringtoolbox.com/thermal-conductivity-metals-d_858.html **) http://www.etltd.co.uk/titanium-ti/titan-grade-12.html ***) http://www.raytek.com/raytek/en-r0/ireducation/emissivitytablemetals.htm and http://www.omega.com/literature/transactions/volume1/emissivitya.html ****) The effective volumetric heat capacity is determined by weighting the volumetric capacities of air 1140 J/m 3 /K (Fletcher 1991) and of titanium 2.43 MJ/m 3 /K by volumetric fractions 0.93 and 0.07 giving the effective volumetric heat capacity 0.095 MJ/m 3 /K, which is used in the analysis.
15 The emissivities of metallic surfaces depend strongly on the surface quality. Canister outer surface is machined copper. After manufacture the machined surface is clean and not oxidised. However, the canister is exposed to atmospheric oxidation in elevated temperature. The surface temperature is some 50 to 90 C during storage in encapsulation plant and in repository before disposal and even after disposal the canister surface is exposed to the residual air in the drift for some time (until the residual oxygen is used from the air). This makes the canister surface slightly oxidised all over and the surface emissivity is remarkably better than after machining. The copper surface emissivity is assumed to be 0.3 that should be conservative for the condition when the maximum temperature takes place (at 15 years after disposal). Accordingly, the emissivity of the Supercontainer shell made of perforated titanium shell (rolled plate) is lower after manufacture, but it will be covered by bentonite mud after the DAWE-process in the drift immediately after the disposal of each compartment of a drift. Even if the artificial water of the DAWE-process is dismissed due to bentonite absorption, the dried bentonite mud will cover the titanium shell surfaces and thus giving a higher emissivity on the titanium surfaces. Taking this phenomenon into account, the emissivity coefficient 0.5 is used for titanium shell surfaces. The effective thermal conductivity of the Supercontainer and the surrounding inner and outer air gaps is calculated as presented in Chapter 2.5. The effective volumetric heat capacity of 53 mm thick layer is calculated by weighting the volumetric capacity of titanium and air by their relative volumes. Table 3 shows the data of the BWR, VVER and EPR fuel canisters used in the analyses. Table 3. Data of the BWR, VVER and EPR fuel canisters used in the analyses (Ikonen & Raiko 2012). Supercontainer data according to Figure 4. Canister type BWR VVER EPR Average burn-up value [MWd/kgU] 40 40 50 Pre-cooling time [years] 32.95 28.30 50.26 Decay heat when emplaced [W/canister] 1700 1370 1830 Amount of uranium in a canister [tu] 2.11 1.44 2.13 Canister length [mm] 4752 3552 5223 Canister effective heat capacity [MJ/m 3 /K] 2.4 2.5 2.7 Number of canisters in two compartments 15+14*) 19+17*) 12+12*) Average disposal rate [canisters per year] 36 36 50 Distance block length [mm] 3606 3006 4735 Total length of the Supercontainer [mm] 5394 4194 5865 *) The first number is for the inner compartment and the second number for the outer compartment (closer to the central tunnel).
16 2.3 Emissivity of air gap surfaces The radiation heat flux φ rad between two flat parallel surfaces having temperatures of T 1 and T 2 is calculated from! rad = " tot # (T 1 4 $ T 2 4 ), (3) where the total emissivity is calculated from formula (Ikonen 2013)! tot = 1 1 +! 1 =! 1! 2, (4) " 1! 1! 1 +! 2 "! 1! 2 2 where ε 1 and ε 2 are the emissivities of the surfaces. The Stefan-Bolzmann constant is σ = 5.6697 10-8 W/(m 2 K 4 ). Equation (3) is applied in the narrow annular gap between the canister and bentonite. In the initial condition, the most important thermal resistance for canister cooling chain is the 5.1 mm (in average) air gap between the canister and the buffer. The thermal conduction over the gap is the sum of thermal radiation and conduction in the air. Before the maximum temperature is reached after some 15 years, the elevated temperature oxidizes the canister surface from the oxygen in the trapped residual air in the disposal drift, see (Posiva 2015). The increasing oxidation makes the canister surface emissivity better and thus decreases the thermal resistance of the gap and lowers the canister temperature. When the buffer gets water, the thermal resistance is remarkably lowered and the temperature is lowered, too. The canister surface is matte after machining and somewhat oxidized during storage after encapsulation in the canister storage of the encapsulation plant or of the repository in a ventilated room typically for a few weeks. Thus the emissivity coefficient used for dry condition analyses, 0.3, is conservative at least for longer term and, in particular, when the maximum temperature is expected after 10 to 15 years. The emissivity of the copper surface depends strongly on the quality of the surface. Polished surface has an emissivity of about 0.02, clean and machined surface about 0.3 and oxidised surface 0.6. In the analyses the emissivity of the copper surface is assumed to be 0.3. Total emissivity expressed by Equation 3 is applied in a narrow cylindrical gap between the canister and bentonite (view factor = 1). If the gap width increases, view factor is less than one and total emissivity is calculated from Mills (1999) and (Ikonen 2013)! tot = 1 1 "! 1 + 1 + 1 "!, (5) 2! 1 F 12! 2 (r 2 /r 1 ) where r 1 and r 2 are the radii of the internal and external cylinders and view factor F 12 = 1 (other view factors between internal and external cylinders 1 and 2 are F 11 = 0, F 21 = r 1 /r 2 and F 12 = 1 - F 21 ). If for instance r 1 = 0.525 m, gap width is 10 mm, i.e. r 2 = 0.535 m, ε 1 = 0.3 and ε 2 = 0.8, it follows from Equation 4 that ε tot = 0.2794. From
17 Equation 3 follows ε tot = 0.2791. If the gap width is 50 mm, from Equation 4 follows ε tot = 0.2808. Thus Equation 3 gives accurate enough results in case of gap widths in practise. 2.4 Decay heat power modelling The decay heat decreases strongly with time and, for example, after 50 years the decay heat is only a half of the amount it was at disposal. The decay power of the spent fuel was calculated by Anttila (2005) with the ORIGEN-S computer code of the TRITON functional module of the SCALE program package (Oak Ridge National Laboratory 2004). Reasonable decay heat level is reached in 30 to 50 years cooling time depending on burn-up value of the spent fuel. Figure 11 shows decay power of spent fuel according to Anttila (2005). Numerical values of the decay power are shown in Table 4. The average burnup of all the expected spent fuel for Olkiluoto 1 and 2 (OL1, OL2) as well as Loviisa 1 and 2 (LO1, LO2) units is about 40 MWd/kgU and the estimated average burnup from Olkiluoto 3 (OL3) units will be about 45 MWd/kgU. Thus the decay power functions have been selected as 40 MWd/kgU for OL1&2 and LO1&2 canisters and conservatively 50 MWd/kgU for OL3 canisters. Figure 11. Decay heat densities of BWR, VVER and EPR fuels as a function of cooling time in years. Cooling time is measured from the moment, when fuel is removed from a reactor. Decay power between two calculated points is interpolated by a linear fitting on loglog-coordinate system ln P = a + b ln t. (6)
18 By setting the times of the end points t 1 and t 2 and corresponding powers P 1 and P 2 the coefficients a and b of the fitting are solved and the interpolation for the power is ln P P 1 = ln P 1 P 2 ln t 1 t 2 ln t t1. (7) This fitting is accurate enough, if the time increments are short enough, for instance 10 years. Table 4. Decay heat of BWR, VVER and EPR spent fuel (Anttila 2005). Cooling time is measured from the moment, when fuel is removed from a reactor. Decay heat [W/tU] Time [years] BWR 40 MWd/kgU VVER 40 MWd/kgU EPR 50 MWd/kgU 10 1339.0 1416.0 1890.0 20 1036.0 1107.0 1455.0 30 854.4 927.5 1204.0 40 713.8 787.6 1013.0 50 602.7 675.9 862.9 60 514.0 586.6 743.9 70 443.0 514.6 648.8 80 386.1 456.5 572.5 90 340.1 409.2 510.9 100 303.1 370.9 460.8 110 273.1 339.5 419.9 120 248.5 313.6 386.3 130 228.5 292.2 358.4 140 211.8 274.3 335.1 150 198.0 259.2 315.4 160 186.6 246.5 298.5 170 176.8 235.5 284.2 180 168.5 226.1 271.7 200 155.2 210.5 251.2 300 119.5 165.6 191.9 600 77.8 107.0 120.2 1000 51.8 69.5 77.6 3000 22.3 27.7 30.5 6000 17.1 21.3 23.3 10000 13.1 16.8 18.0 30000 4.9 6.8 6.9 60000 2.2 2.9 3.0 100000 1.1 1.4 1.5 300000 0.6 0.7 0.9
19 2.5 Effective thermal conductivity of perforated container shell and surrounding gaps Between bentonite and rock there is a perforated shell of titanium. The wall thickness of the container lids is 6 mm. In the cylindrical part the wall thickness is 6 mm and there are holes of diameter 100 mm covering about 61 % of the cylindrical surface (61 % for BWR and VVER, 62 % for EPR). In the following it is conservatively assumed that the holes and gaps are air-filled. The gap is modelled by a single layer having effective thermal conductivity and capacity. For determining the effective thermal conductivity of perforated container shell and coaxial annular air gaps it is assumed that through a hole heat is transferred by conduction in air and by radiation between bentonite and rock surfaces (right upwards arrow in Figure 12). In position of titanium (left upwards arrow in Figure 12) it is assumed that equal heat flux goes through the internal air gap, through titanium thickness and through the external air gap. Heat flow in pure air-filled area and titanium shell area are summed to determine the effective thermal conductivity. Temperatures T 1, T 2, T 3 and T 4 (Figure 12) are assumed to be constant in axial (horizontal) and circumferential direction. Perforated (61 %) titanium plate of 6 mm Diameter 100 mm Rock T 4 h = 47.0 mm T 3 T 2 T 1 Bentonite 3 h = 6.0 mm 2 h = 5.5 mm 1 Eccentricity here Figure 12. Modelling of effective thermal conductivity of perforated Supercontainer shell and surrounding air-filled gaps, when using average air gap thicknesses. Heat flows are assumed to be parallel (red arrows). For determining the three unknowns T 2, T 3 and λ eff following equations are obtained (neglecting the effect of curvature)! T 1 " T 2 air + # 12 $ (T 4 h 1 " T 4 2 ) =! T 2 " T 3 titanium =! T 3 " T 4 air + # 34 $ (T 4 1 h 2 h 3 " T 4 4 ) 3! T 1 " T 4 eff = 0.38! T 2 " T 3 titanium + 0.62 T [! 1 " T 4 air + # 14 $ (T 4 h 1 + h 2 + h 3 h 1 + h 2 + h 1 " T 4 4 )]. (8) 3 h 2 The thickness values are h 1 = 5.5 mm, h 2 = 6.0 mm and h 3 = 47.0 mm. At 50 o C the thermal conductivity of air is about 0.03 W/m/K and the thermal conductivity of tita-
20 nium was assumed to be 19 W/m/K. The total emissivity (Table 2 and Equation 4) between rock bentonite and titanium surfaces is ε 12 = 1/(1/0.8 + 1/0.5-1) = 0.444, between titanium and rock ε 34 = 1/(1/0.5 + 1/0.9-1) = 0.474 and bentonite and rock ε 14 = 1/(1/0.8 + 1/0.9-1) = 0.735. Figure 13. Effective thermal conductivity of perforated container shell and surrounding gaps. Figure 13 shows three curves with ball symbols. These curves show the effective thermal conductivity λ eff as a function of the average temperature (T 1 + T 4 )/2 in the gap calculated in different ways. The top curve shows the effective thermal conductivity when neglecting the perforated shell and calculating the thermal conductivity from equation! eff =! air + " 14 # (T 1 4 $ T 4 4 ) h 1 + h 2 + h 3 T 1 $ T 4. This overestimates the effective thermal conductivity as shown in Figure 13. The lowermost curve with symbols concerns the case, where the titanium shell exists in the gap, but there are no holes in it. This underestimates the effective thermal conductivity. The applied curve (61 % of the titanium shell area has holes) is located between the curves of the aforementioned extreme cases. A fitting of this curve and the fitting used in the analysis (T in Celsius) is (9)
21! eff = 0.181 + 1.702 " 10 #3 T + 7.007 " 10 #6 T 2. (10) Due to thermal radiation the effective thermal conductivity increases with temperature. According to the applied curve in Figure 13 the effective thermal conductivity of the perforated container is at typical temperature range about 0.3-0.4 W/m/K. When the outer gap is filled with water, the thermal conductivity is better (about 0.6 W/m/K) and the temperature in the canister and in the buffer is decreased (effect is presented in Table 8). In KBS-3V the gap is pellet filled and then the effective thermal conductivity is about 0.17-0.19 W/m/K (Ikonen & Raiko 2012). The study above concerned the case, where the air gaps are axisymmetric. In the actual case they are eccentric (Figure 5). The effect of the eccentricity is studied in the next. The gap thicknesses h 1 and h 2 (Figure 5) change in circumferential direction. The local air gap width is (11)! local =! air " d cos #, where δ air is the average air gap width and d is the vertical displacement of the canister downwards and β is the angle to the counter clockwise direction measured from the bottom. Figure 14 shows the effective thermal conductivity as a function of the angle β at different temperatures. The thermal conductivity is nearly constant except at the bottom area, where the inner air gap is closed. The thermal conductivity is assumed to be constant as a function of the angle β and the improved thermal conductivity underneath the cylinder is conservatively not taken into account. Figure 14. Effective thermal conductivity of the rock/supercontainer interface as a function of the angle β at different temperatures.
22
23 3 CALCULATION METHODOLOGY AND ITS VERIFICATION It is not practical to determine temperatures in the whole repository by numerical models. Large 3D models would be needed and the computing time would become very long. Practical way is to determine temperatures around a single canister by numerical analysis and use information of this analysis in a suitable way for the analysis of the whole repository based on an analytical model. Calculation methodology thus combines numerical and analytical methods. The idea is that for a given canister the effects of all other canisters can be calculated analytically. Numerical calculations are needed only for a single canister. Ultimately also the temperatures at the cross-cutting mid plane due to that single canister can be calculated analytically when a quasi-stationary state has been established. Analytical approach is much more efficient than numerical approach, but canister surface temperature calculation by analytic method needs first information of a numerical solution as described later. With help of that a heat flux reduction coefficient can be calculated allowing to determine e.g. maximum temperatures at canister surfaces quickly. Heat transfer across the inner air-filled inner gap will take place by conduction, radiation and convection. The effect of convection is conservatively neglected, since the gap is small and the surfaces are rough. Furthermore, the contact on the bottom hinders air flow. Also in the outer gap the convection in the gaps is assumed to be small and it is conservatively neglected in heat transfer modelling. 3.1 Numerical analysis of a single BWR canister in KBS-3H drift Numerical thermal heat conduction analyses are performed by control volume method in axisymmetry. For this an in-house computer code named KAPSEL has been developed. It has a lot of special features and is tailored for this purpose. Without going into the details, the equation for a control volume is c!t v!t = V 1 "!T!n S ds + #, (12) where c v is volumetric heat capacity (c v = ρc), T temperature of the control volume centre, t time, V volume of the control volume, λ thermal conductivity, n outward coordinate from the surface, S surface area and Φ volumetric decay heat generation. Fully implicit principle is applied everywhere. Nearly ten different material types are defined in practice. Thermal radiation over air gaps is programmed (Ikonen 2013). Special mesh generators for KBS-3V and KBS-3H are developed. The operation of the code is verified by numerous test cases (Ikonen 2012; Ikonen 2009). Visualization code is also in-house made having features suited for this type of analyses and increasing the reliability in different phases of the calculation process.
1855 1765 1050 1060.2 1738 1761 24 A checking point in calculation process after every time increment is the energy balance requirement: total energy penetrating the outer surfaces + change of the enthalpy of the model should equal the energy generated inside the model. Figure 15 presents the dimensions for the numerical model. An annular air gap of 5.1 mm is modelled to exist between the canister and the bentonite buffer. The cylindrical shell of the 6 mm titanium Supercontainer has D100 mm diameter holes covering 61 % of the cylindrical area. An annular air gap of 5.5 mm is assumed to exist between the buffer and the Supercontainer shell and an annular air gap of 47.0 mm between the Supercontainer shell and the rock. The cylindrical Supercontainer with surrounding air gaps is modelled by a single homogeneous annular layer (5.5 mm + 6 mm + 47.0 mm = 58.5 mm), whose thermal conductivity is described by the fitting (10) and the effective volumetric thermal heat capacity is 0.095 MJ/m 3 /K (Table 2). The wall thickness of the Supercontainer lids is 6 mm. No axial gaps in the system are assumed in the horizontal direction except 15 mm air gap between the right-hand-side end of the canister and the rigid bentonite block. The geometry of the right end of the canister is simplified by assuming for it the same planar shape as for the left end. Air gap, 5.1 Air gap 5.5 Air gap 47 Distance block 350 Canister Air gap15 Bentonite Titanium end plate of 6 mm 4752 5394 Perforated (61 %) titanium plate of 6 mm 58.5 mm (titanium+gaps) Figure 15. Dimensions for axisymmetric numerical model. Figure 16 (upper) shows the model of three adjacent canisters illustrating material types. The thermal properties of the distance block (see Figure 33) are equal to the properties
25 Figure 16. Upper figure shows axi-symmetric model of BWR canister illustrating grid and material types. Lower figure shows temperature distribution, when maximum temperature on canister surface is reached. Three adjacent canisters are modelled with heat production only in the middle one. BWR canister spacing is 9 m. The result is used for calculation of heat flux reduction coefficient. of the bentonite buffer. Identical meshes are duplicated to the left and to the right side of the central canister. In the model there are about 9 600 axisymmetric control volumes. Thermal heat power is generated only in the central canister (Figure 16 lower). The model is thus separated from the infinitely long canister queue by two vertical planes passing through the middle point of the distance blocks. In these vertical planes temperatures are determined from the analytic solution presented later by Equation (17). A small inaccuracy is caused by the fact that in the vertical end planes on the drift level there is a distance block of bentonite, which has smaller thermal conductivity than rock. In the vertical (radial) direction the outer radius is 20 m and there also temperatures from the analytic solution are applied. An alternate way would be to extend the outer radius very far from the canister to avoid heat pulse reflection from the model edge. A separate subroutine performs the operations related to boundary conditions in practice.
26 Thermal radiation is modelled in the inner canister/buffer air gap. The canister is assumed to be homogeneous with uniform power generation over its volume and the contents of it is not modelled in detail, since the copper shell has very high thermal conductivity (390 W/m/K) causing nearly uniform temperature distribution on the external surface of the canister. The decay heat power of BWR canister is 1700 W at the beginning of the analysis. The effect of the grid density was investigated by varying the mesh density. The grid presented in Figures 16 and 17, was found to be adequate. The applied time increment is 600 s. Figure 17 illustrates more in detail the temperature distribution of a single canister after 1.3 years, when the maximum temperature 81.1 o C on canister surface is reached when heat is produced only in the single canister. Because there is 15 mm air gap on the right-side end of the canister, the temperature distribution deviated from that on the left end area. On the middle of the canister the temperature distribution is quite symmetric with respect to a vertical plane. The effect of the air gap 15 mm on the top end of the canister was studied by changing the material in this gap from air to bentonite. This decreased the canister maximum surface temperature by 1.1 C to 80.0 o C. Figure 17. Temperature distribution of BWR canister after 1.3 years, when maximum temperature 81.1 o C on canister surface is reached. Three adjacent canisters are modelled with heat production only in one canister. On black isotherm the temperature is 30 o C and on white 20 o C.
27 On the middle of the canister and close to the canister, the temperature field is ellipsoid shaped, whereas with greater distances the temperature field becomes gradually more spherical as Figure 17 illustrates. Due to symmetry the longitudinal thermal gradient equals zero in the middle of the canister and the heat flow in longitudinal direction is zero. The history of the maximum temperature on the canister surface, on the drift surface and deeper in the rock is presented Figure 18. Figure 18. History of the maximum temperature on the canister surface, on drift surface and deeper in the rock (1.75 m from the axis of the canister in radial direction). These curves are obtained by three adjacent canister model (Figure 16) in case of BWR fuel.
28 Figure 19 shows the radial temperature profile after 1.3 years at the middle of the canister. Temperature drop over the air gap surrounding the canister is 10.1 C, over the buffer ring 22.4 C and in the outer gap 10.1 C. Figure 19. Radial temperature profile in the middle of the single canister after 1.30 years, when the highest temperature of 81.1 C is encountered. Figure 20 shows with ball symbols the heat flux distribution along canister external surface. The distribution is not symmetric because of the 15 mm air gap at the righthand-side. The highest heat flux is reached in the corner of the canister, where the angle of the heat flow lines to the bentonite direction is largest and the temperature gradient is highest in outwards direction. The heat flux is 86.4 W/m 2 in the middle of the canister. After 1.3 years the heat power is 1660 W. By dividing this by the outer surface 17.4 m 2 of the canister, the average heat flux 95.4 W/m 2 is obtained. The ratio is 86.4/95.4 = 0.906, which means that in the middle the heat flux is about 10 % lower than the average value due to the end effect. After 20 years the heat power is 1209 W and the ratio is 0.902.
29 Figure 20. Heat flux distribution along canister external surface between centres of left and right hand side lids after 1.3 years. Figure 21. Heat flux distribution along canister external surface between centres of left and right hand side lids after 20 years.
30 Figure 22. Temperature distribution on copper surface and on the internal surface of bentonite/buffer after 1.3 years. Figure 22 presents the temperature distribution on the copper surface and on the internal surface of bentonite. On the left end of the canister the temperatures are equal because of the assumption of straight contact. On the right-hand-side lid, due to 15 mm air gap, the temperature difference is about 15 o C. On the cylindrical area of the canister the temperature is nearly constant. In the middle of the cylinder the temperature is 81.4 C and in the corners 80.5 C. The difference is only 0.9 C. The differences are thus very small. This is caused by the very high thermal conductivity of the copper overpack. The conclusion of above is that the internal parts of the canister are not needed to be modelled in detail, although the maximum temperature in individual fuel pellets is about 200 C Ikonen (2006). The canister can be treated as a homogeneous cylinder having a uniform thermal conductivity, effective thermal capacity and internal heat generation inside the 50 mm copper layer. 3.2 Effect of artificial wetting of outer gap of a single canister According to curves in Figure 23 the artificial wetting of the outer gap with titanium shell and two air gaps after 14 days decreases the canister temperature about 5 o C. This result is calculated by using the model shown in Figure 16.
31 Figure 23. Effect of artificial wetting of the outer gap. k = 0.893 in initial conditions and in artificially wetted outer gap case with k = 0.905. The upper curve is the same as upper curve in Figure 18. 3.3 Heat flux reduction coefficient The dependence of k in the initial conditions on the canister spacing is shown in Figure 24. The dependence is very weak for all canister types. For BWR and EPR k 0.89 and for VVER k 0.88. Lower value for VVER is caused by the shorter canister length. Consequently, due to end effects of the canister the heat flux in the middle of the canister is 11-12 % lower than in the infinite long (line source) case. Figure 24. Heat flux reduction coefficient k vs. canister spacing in the initial conditions.
32 In the vertical concept KBS-3V the ends of separate canisters are not close to each other like in the horizontal concept. In addition, in the horizontal concept between canisters there is a bentonite distance block, which has lower conductivity than rock. And further there is a radial air gap between the distance block and the rock. For these reasons k is greater in the horizontal concept than in the vertical concept, where it is about 0.84. In the vertical concept, due to end effects the heat flux in the middle of the canister is about 16 % lower than the average heat flux. 3.4 Analytical solution of line segment heat source The analytic models described below are practical in calculation of temperatures in a panel having lot of canisters. The superposition principle is applied to determine the interaction of great number of fuel canisters. The idea to use the analytic line segment heat source model to predict the canister surface temperature was proposed by (Hautojärvi et al. 1987). The analytic line segment heat source gives temperatures outside of the canister area, but it cannot describe actual temperatures in the vicinity of the canister. However, as described later, it is possible to determine temperatures also in the vicinity of the canister and especially the maximum temperature on the canister surface by using information of a numerical solution of a singe canister. The starting point for the analytic solution is a case, in which an amount of energy (Q) is instantaneously released at a certain point in an infinite space. The spherically symmetric temperature at distance r from the point heat source at time t is originally according to Adolf Fick 1855 (Carslaw & Jaeger 1959) $ Q T(r, t) =! c (4 " # t) e 4 r2 # t, (13) 3/2 where ν = λ/(ρc) is the thermal diffusivity of the material (λ, ρ and c are thermal conductivity, density and thermal capacity of the material, respectively, and they are constants in the whole space). If energy is released continuously, then the temperature is T(r, t max ) = 0 t max $ r 2 P(t)! c [4 " # (t max $ t)] e 4 # (t max $ t) dt, 3/2 (14) where P is the power, t max is the time of observation, and t is the time of heat emission from the canister. This solution can be extended to a finite line heat source by integration over the length H of the line heat source. The canister has end plates with radius of r 0 enhancing heat transfer compared to a finite line source of the same length. This is taken into account by increasing the actual length H to the canister effective value H eff = H + r 0 of the corresponding line source (presented in more detail in Ikonen & Raiko 2012). This increases the accuracy in describing near-field temperatures. Then the temperature at a point in the rock is obtained from the formula
33 T(x, y, z, t max ) = 1/H eff! c 4" # 0 t max $ P(t) t max $ t e x 2 + y 2 4 # (t max $ t) % 1 2 {erf [ 1 2 # (t max $ t) ( H eff 2 + z)] + erf [ 1 2 # (t max $ t) ( H eff 2 $ z)]} dt, (15) where z axis of the rectangular xyz co-ordinate system is aligned parallel to the longitudinal axis of the canister and the origo is in the centre of the canister. The assumptions concerning Equation (15) are thus that the material (in this case the surrounding rock) is homogeneous, extends to infinity, has constant thermal diffusivity and the line segment heat source has uniform power generation. The line segment heat source described by Equation (15) can be applied only for rock temperatures and is accurate there and reasonably accurate even at the rock wall. 3.5 Extrapolation of canister surface temperature in analytic line segment heat source model Due to high thermal conductivity of copper the overpack temperature is practically constant. In the analytic solution this temperature can be determined in the middle of the canister. The temperature gradient equals zero in the horizontal direction from symmetry reasons (Figure 17). The temperatures are changing all the time, but so slowly in time scale of weeks that the situation can be considered as quasi-stationary heat conduction in a circular vertical plane. Figure 25. Gaps on canister cylinder surface and between bentonite buffer and rock.
34 In case of a single canister the analytic solution gives the correct temperature field to quite near the Supercontainer. In the quasi-stationary case the canister surface temperature can be calculated by starting from the temperature T fix in some point in the rock (Figure 25) in the plane through the midpoint of the canister and perpendicular to its axis. Temperature can be and determined by the line segment model from Equation (15), and summing the temperature differences over different material layers from Equation (17). Due to the canister end and corner effects and other differences between the numerical and analytical models the heat flux φ 0 in the middle of the canister is less than the average heat flux φ mean, which equals the power divided by the external area of the canister. The reduction of the radial heat flux φ 0 in the middle of the canister can be presented by introducing a heat flux reducing coefficient k of the radial heat flux relative to the average heat flux from the canister! 0 = k! mean = k P 2"r 0 (H + r 0 ). (16) By summing the logarithmic temperature distributions in the different layers for the canister surface temperature T 0 the formula is obtained T 0 = T fix + k P 2!r 0 (H + r 0 ) [ r 0 ln r fix "rock r + r 0 r ln rock rock " SC r rock # $ SC + r 0 "ben ln r rock # $ SC r 0 + $air + 1 ]. (17) " air r 0 ln (1 + $ air /r + % tot & (T 0 + T 0b ) (T 2 0 + T 2 0b ) 0 ) In the Supercontainer layer the effective thermal conductivity λ SC is determined from the fitting (10), which is plotted also in Figure 13. The applied temperature in the fitting is the current temperature in the Supercontainer. The transient is so slow that the assumption of steady state is valid (thermal capacity multiplied by the partial derivative of the temperature with respect to time can be ignored in this consideration). The logarithmic temperature distributions (well known in literature) are obtained when thermal conductivity in each layer has a constant value. If this were not valid, the thickness can be divided to so many layers that the thermal conductivity can be considered to be constant inside a separate layer. Thermal radiation over the gap depends strongly on the temperature. The total emissivity in Equation (17) is calculated from Equation (4). In the last term related to radiation the temperatures are expressed in Kelvin scale. Heat transfer by thermal radiation can also be described by the effective thermal conductivity in transverse direction as follows! eff =! air + " air # tot $ (T 0 + T 0b ) (T 2 0 + T 2 0b )!! air + 4 " air # tot $ T 3 mean, (18)
35 where T mean is the mean temperature in the gap (the average of the surface temperatures in Kelvin). In longitudinal direction only air thermal conductivity is applied. The radiant part of the effective thermal conductivity decreases proportionally to the air gap width δ air and increases proportionally to the third power of T mean. The thermal conductivity of air at 80 C is 0.030 W/(m K) (Figure 9). If for instance, the gap width is 10 mm, the mean temperature 80 C and the total emissivity 0.5, the effective thermal conductivity is λ eff = 0.050 W/(m K). Figure 26. Maximum temperature history on the middle heated canister surface and on rock wall obtained by three adjacent canisters model. Line with green symbols shows temperature in the rock at distance of 1.75 m. Canister spacing 9 m in the model shown in Figure 16. Figure 26 presents the temperature evolution on the canister surface and on the rock wall. According to the numerical solution the maximum temperature on the central canister surface is 81.1 C after 1.3 years. Due to differences between the models the analytic solution without correction (k = 1) gives 11.8 C higher temperature on the canister surface than the numerical one. In the rock point, in this case 1.75 m (arbitrarily chosen distance) from the canister centre, the analytical solution of an ideal finite line heat source gives about 0.4 C higher temperature than the numerical solution. The shape of the curves calculated numerically and analytically are very similar. The upper curves related to the copper surface coincide, if the mean heat flux in the analytic Equation (15) is lowered by the coefficient k = 0.893 in this example. The coefficient is determined by changing k so that the maximum temperatures are close to each other. The coefficient could be determined also from Equation (16), but in the calculation of the
36 heat flux the temperature gradient need to be estimated and this may cause inaccuracy. Thus the maximum temperature is only applied in the fitting. The curves in Figure 26 were calculated also longer, to 100 years, and as good agreement between numerical and analytical results was obtained. Temperature history on the canister surface as shown in Figure 26 was calculated also to 200 years and a good agreement between the analytic and numerical solution at any time was obtained, when the coefficient k = 0.893 was used. Figure 27 shows the detailed radial temperature profile at 1.3 years. The numerical and analytical profiles correspond each other well in the canister overpack, bentonite and rock. Figure 27. Numerically and analytically calculated radial temperature profiles of single canister after 1.30 years, when the highest temperature of 81.1 C (numerical analysis) is encountered. Figure 26 shows that the canister surface temperatures determined by the numerical and analytical analysis are very close to each other up to 20 years. Also in the drift surface, the temperatures are very close to each other (k is not needed in calculation of the rock point temperature by the analytical method). Figure 27 shows that the temperature profiles coincide at 1.3 years. This is apparent also at other times, since the canister surface and the rock point temperatures are equal according to the curve in Figure 26 and the intermediate points in the analytic solution are calculated from Equation (17), which can be applied at all times in a slow quasi-stationary temperature evolution. Thus the ana-
37 lytic solution (Equation 17) gives quite precise results close to the canister, when only the constant k is first determined from the numerical near-field analysis. The location of the point in the rock has insignificant influence on the canister surface temperature determined from Equation (17) and that point can even be on the surface of the drift. Small differences are caused by canister ends compared to the line segment end effects. Similar results as above are obtained also in case, when the bentonite buffer is saturated and then there is no gap between the canister and the buffer. Due to direct contact and improved heat transfer the temperature on the canister surface is lowered. As a matter of fact, the only quantity, which is needed from the numerical analysis for the analytic solution, is the heat flux reduction coefficient k. From the point of view of the analytic analysis the only purpose of the numerical analysis is to produce the heat flux reduction coefficient k. After this all rock temperatures can be obtained by analytic models for a single canister and for the whole repository. The calculation of k takes place iteratively so that the maximum temperature on the canister surface calculated by the numerical and analytical models becomes equal. Figure 26 demonstrates this. Without correction the analytic solution (k = 1) gives 11.8 o C higher temperature than the numerical solution on the canister surface. The analytic solution is calibrated to give equal canister temperature history as the numerical solution, when a value of k = 0.893 is chosen for the parameter. The line heat source model estimates successfully the temperature in the middle of the canister, where the highest temperature is encountered. The drift wall temperature (not depending on k) is correctly evaluated, when the effective length of the canister is used. In the vertical concept KBS-3V the ends of separate canisters are not close to each other like in the horizontal concept. A separate numerical modelling is needed to calculate the thermal conditions between the canister ends in the horizontal concept KBS-3H. 3.6 Effect of the inner air gap eccentricity In the horizontal disposal of canisters the inner air gap is closed underneath the cylinder due to gravity and heat transfer on the contact area is improved. As shown in Figure 14, the effect of eccentricity of the outer gap (air+titanium+air) is small and the outer gap can be assumed to be axisymmetric and thermal conductivity in it is shown by the curve in Figure 13. In the next the effect of the inner air gap eccentricity is studied by a 2D model in the cross-section of a canister. Figure 28 illustrates the grid and the temperature distribution after 1.3 years in the axisymmetric and asymmetric case. The canister temperature is higher in the axisymmetric case than in the asymmetric case. The outer radius of the model is R = 20 m and in Figure 28 only layers near the canister are drawn. Tempera-
38 ture on every time increment on the outer radius is calculated from the analytic Equation (21). Figure 28. Temperature distribution after 1.3 years of an infinite long 2D model, when the air gap 5.1 mm is axisymmetric (left) and when there is asymmetric air gap, where the bentonite surface is in contact at the bottom side (right). Thermal radiation in a narrow gap is assumed to take place only in the radial direction and view factor is assumed to be equal to one. The effective thermal conductivity in the air gap is calculated from! eff!! air + 4 " local # tot $ T3 mean, (19) where δ local is the local air gap width and T mean is the mean temperature in the gap (the average of the surface temperatures). The local air gap width δ local is calculated from! local =! air " d cos #, (20) where δ air is the average air gap width and d is the vertical displacement of the canister downwards and β is the angle to the counter clockwise direction measured from the bottom. In a fully displaced case and in contact underneath the cylinder d = δ air = R r (R is the radius of bentonite surface and r is the radius of the canister). The modelling of the contact area can be done in different ways. First, the original axisymmetric mesh is compressed, but the gap is not fully closed since zero area elements are then formed. Other alternate is to close the gap fully and remove zero area elements. Third way, which is now applied, is that, after compression on the bottom and contact, the thin air gap elements are changed to bentonite material. This means a straight contact between the canister and bentonite. This is done up to ±18 degrees measured from the bottom.
39 The boundary condition on the edge of the model can to be done by two alternatives. First, the radius of the numerical model could be extended far from the canister to avoid heat pulse reflection from the edges. However, for long-term analysis the radius should be very far. The other alternative is to apply analytically calculated temperature on the edge. When the segment length, H eff, in Equation (15) increases the error function terms in the lower line approach the value of one and for the temperature increase at the outer radius R is obtained T(R, t max ) = 1! c 4" # 0 t max $ p(t) t max $ t e R 2 4 # (t max $ t) dt, (21) where p(t) is the thermal power per heat source line segment element. This equation is applied in numerical analyses for the boundary temperature on the outer edge. Figure 29 illustrates the temperature distribution in the circumferential direction on the copper surface, bentonite surfaces and rock surface in an axisymmetric case and in an asymmetric case, when the average air gap is 5.1 mm (at top 10.2 mm and at bottom 0 mm). Figure 29. Temperature distribution on the copper surface, bentonite surfaces and rock surface after 1.3 years, when the average air gap around canister is 5.1 mm and on the bottom the copper surface and bentonite surface are in contact (see Figure 5). Figure 30 shows the temperature difference between the axisymmetric and asymmetric cases after 1.3 years. On the contact area the temperature of bentonite is about 5 C higher than in the axisymmetric case and above the canister it is about 8 C lower. On
40 the horizontal line passing through the centre of the canister the temperature difference is nearly constant. With greater radius the temperature in the circumferential direction is nearly constant showing that the eccentricity effect is quite local. Conclusion from the analyses (Figure 30) is that the effect of asymmetry of dipolar nature and it is local so that if the distance (radius) is more than 5 m, there is practically no asymmetry in the circumferential direction and the temperature field is axisymmetric. Temperatures obtained by both models are also equal far enough from the canister. Figure 30. Temperature difference after 1.3 years, when average inner air gap is 5.1 mm is axisymmetric or in contact at the bottom line. The temperature difference between the axisymmetric and the corresponding eccentric case was determined up to 100 years. The outer gap around the Supercontainer was assumed to be axisymmetric. The heat flux through the gap was integrated over the circumference and the integral corresponded the current decay heat power of the canister.
41 Figure 31. Temperature difference between axisymmetric and corresponding eccentric case, when the average air gap is 5.1 mm. The time is measured from the moment, when the canister is emplaced. Due to thermal power decrease in the canister, the temperature difference shown in Figure 31 decreases with time. An exponential function fitting to the curve in Figure 31 is!t(t) = 1.21 e " 0.0115 t oc, (22) where time t is expressed as years. This fitting is applied in later analyses to model the effect of the asymmetry of the inner gap. The difference in the canister temperature is decreased according to this curve.
42
43 4 ANALYSIS AND RESULTS OF A SINGLE PANEL The temperature of the whole panel is calculated with an analytical model taking into account a great number of canisters. For this an in-house computer code named OSASTO has been developed. The different canister spacings caused by transition blocks around the plugs will be taken into consideration. The result will be the highest surface temperature of a canister, which is located in the middle of the compartment area. The deposition schedule and order of the canisters will be taken into account in the calculation process. At first some conservative scoping calculations by omitting the artificial watering will be performed. 4.1 Distances in a drift Figures 32 to 35 show the lengths of the components in the drift. Minimum distances are conservatively chosen for distances for all the fuel types. Figure 32. Filling component at the drift bottom (Posiva 2012). Figure 33. Filling components adjacent to a compartment plug (Posiva 2012).
44 6310 mm 1300 mm Figure 34. Design of filling adjacent to the drift plug (Posiva 2012). Figure 35. Principle of filling one drift. Values of the reference case (BWR canisters) are written in red colour. Only the Supercontainer, distance blocks and near-field rock are modeled in the axisymmetric numerical model. The effect of all other canisters in the panel on the temperature of the considered canister is determined by the analytic model by superposing principle. The effect of transition blocks, pellet fillings, plugs and central tunnels are not modeled. The reason for this is that analytic model is based on assumption of homogeneous and isotropic properties of rock.
45 Based on Figures 35-38 for the length L of the drift can be written L = 0.5 m + n. l + 2. (l l SC ) + 15.22 m + 18.19 m, (23) where l is the spacing between canisters and l SC the length of a Supercontainer. Values and results are presented in Table 5. The 100 % utilization degree of potential canister locations is assumed. Table 5. Data related to the BWR, VVER and EPR fuel canisters. Maximum nominal length of the drift is 300 m. Drift spacing 25 m. Canister type BWR VVER EPR Number of compartments in one drift 2 2 2 Number of canisters in two compartments 15+14*) 19+17*) 12+12*) Number of canisters n in one drift 29 36 24 Spacing between canisters l 9 m 7.2 m 10.6 m Length of a Supercontainer l SC 5.394 m 4.194 m 5.865 m Number of canisters in one panel with 30 drifts 870 1080 720 Length of drift calculated from Equation (23) 292.9 m 298.5 m 297.5 m *) The first number is for the inner compartment and the second number for the outer compartment (closer the central tunnel). 4.2 Disposal operation and schedule The deposition of the Supercontainers and other drift component is performed with the deposition machine. The Supercontainer is pushed forward with the deposition machine using water cushion technique. The lift pallet moves on the top of a thin water film between the slide plate and the lift pallet. All this operation description is according to (Posiva 2012). A favourable initial state of saturation is obtained by using artificial water filling so that the buffer should swell uniformly and fill all the open space. The bentonite should seal the drift and reduce the flow inside the drift that can cause piping and transport of bentonite. The artificial watering system is required to fill all void spaces in the drift compartment at approximately the same time, thus ensuring that the bentonite will swell and seal the drift uniformly. This minimises the risk of bentonite piping and erosion, as well as water pressure displacement of distance blocks and/or Supercontainers during the saturation phase after sealing of the drift. The water filling of the compartment is carried out with three pipes that extend through the plug (compartment plug or drift plug) into the transition zone, more specifically to the gap between the drift wall and the transition block that is located between the pellet
46 filling section and the first distance block. The wetting starts after the plug has been mounted in place. The air evacuation pipe will be installed at the lower part of the drift, but in order to function in a proper way the end part at the rear section of the compartment needs to be turned upwards to the top of the drift, where the air will accumulate. This will be achieved by a separate pipe, which will be fixed on the end face of the compartment. When removing the long air evacuation pipe the coupling will allow it to be released from this bottom pipe, which will be left inside the drift. The water filling will accelerate the swelling of bentonite and the nearly simultaneous water filling and buffer swelling in the whole drift is beneficial regarding piping and erosion. As a result all void spaces in the drift compartment are filled at approximately the same time, thus ensuring that the bentonite will swell and seal the compartment uniformly. The distance blocks and filling blocks, together with the compartment and drift plugs, have the important design function of keeping the engineered components in the drift in place, and not allowing any significant loss or redistribution of buffer mass by piping and erosion. The distance blocks and filling blocks have a low hydraulic and thermal conductivities at saturation and will develop swelling pressure against the drift wall, such that friction will resist buffer displacement. Furthermore, each compartment plug is designed to stay in place under the applied loads until the next compartment is filled and a further compartment plug or drift plug installed. The disposal operation of a drift is made in two steps; first the compartment deeper in the drift is filled with distance blocks, Supercontainers and filling components until about half of the drift length is assembled, and then a compartment plug is installed with its auxiliary systems in the drift. Thereafter, the assembled compartment is artificially wetted with the DAWE-system and the air removal piping is removed. The second compartment is then assembled in same way, in principle and the entire drift is finally plugged with drift plug and air evacuation pipe near the central tunnel end of the drift. The time sequence of the drift assembly operation is given in following Table 6, where the estimated time needed for various steps is given. Drifts are then assembled in sequence in the whole repository panel during tens of years in a pre-planned average rate. Table 6. Time to deposit one compartment including plug for the BWR, VVER and EPR fuel canisters used in the analyses. Canister type BWR VVER EPR Time to deposit inner compartment t 1 =16 days t 1 =19 days t 1 =14 days Time to deposit outer compartment t 2 =14 days t 2 =17 days t 2 =13 days
47 Deposition time of the panel is Time = N drifts. ( t 1 + t 2 ) + (N drifts 1). t 3, (24) where N drifts is the number of drifts in a panel and t 3 is the time to go to next drift. The average deposition target rate per year (365 days) is calculated from Target rate = 365 / [Time / (N. drifts n )]. (25) Then t 3 can be solved from Equation (24) and (25) and results are shown in Table 7. Table 7. Emplacement rates with the BWR, VVER and EPR fuel canisters. Canister type BWR VVER EPR Target emplacement rate [canisters/year] 36 36 50 Time to go to next drift [calendar days] t 2 = 263 t 3 = 360 t 3 = 168 Time to emplace the whole panel [years] 24.2 30.0 14.4 4.3 Detailed description of BWR canister analyses In the following one panel containing BWR canisters of the repository is analysed. Tables 1 to 4 present the initial values in the analysis. The k = 0.893 was determined by the numeric model as presented earlier. The analyses here are performed by applying the analytic line segment heat source model (15) and superposing the effect of all the canisters in a considered point. This point is not in the centre of a canister, but in the rock in the plane passing the centre of a canister and being perpendicular to the axis of the canister in some direction. Other canisters do not cause axisymmetric temperature distribution in this perpendicular plane. The effect of the direction was studied by displacing the point to the direction of the drifts emplaced with canisters earlier. The maximum temperature change was 0.05 o C. The point was displaced to the opposite direction and the maximum temperature change was 0.16 o C. Thus the effect is small and it was ignored. When temperature T fix on the rock near the canister is determined and the heat flux φ 0 in the middle of the canister is calculated from Equation (16), the canister surface temperature is calculated from Equation (17). Figure 36 shows emplacement order of canisters in drifts starting from the lower left corner. Two compartments can be seen in the figure. Tables 6 and 7 show the deposition times of the different phases. The disposing of all the canisters in this panel takes 24.2 years. Figure 36 shows also temperature distribution after 25 years.
48 Inner compartment Outer compartment 25 m Curve in Figure 38 from 20. drift and 8. canister BWR fuel canisters First emplaced canister Figure 36. Temperature distribution of one panel containing BWR canisters after 25 years operation in the horizontal plane passing through the axis of drifts and canisters. Canister spacing is 9 m and drift spacing 25 m. The highest temperature of 91.0 o C is encountered in 20th drift from the bottom and in 8th canister row from left in KBS-3H. On white isotherm the temperature is 11.0 o C and on pink isotherm 15.0 o C. At the depth of 420 m the ambient rock temperature is +10.5 o C.
49 Figure 37 shows the temperature distribution after 25 years in the vertical plane passing through the middle of canisters on the 8 th vertical row from the left in Figure 36. First filled drift is on the left side. Asymmetry in vertical direction is caused by the initial undisturbed temperature gradient in vertical direction. Due to the low thermal diffusivity of the rock the heated area spreads rather slowly. Figure 37. Temperature distribution of one panel after 25 years operation in the vertical plane passing through the canisters on the 8th row from the left in Figure 36. On white isotherm the temperature is 11.0 o C and on pink isotherm 15.0 o C. The vertical distance of the pink curves on the left (first emplaced drifts) is about 100 m. At the depth of 420 m the ambient rock temperature is +10.5 o C. Figure 38 illustrates the temperature evolution in the canister position marked in Figure 36 in initial condition case. The highest buffer temperature equals the highest canister surface temperature 91.1 C in areas where the buffer is in contact with the canister.
50 Figure 38. Temperature evolution of a BWR canister locating in a KBS-3H panel with BWR canisters in the canister position marked in Figure 36 in initial condition case. Canister spacing is 9.00 m. The greater canister spacing in the middle of the drift due to the compartment plug decreases the maximum temperature only by 0.12 C. If all the canisters were emplaced fictitiously simultaneously, the highest temperature would be only about 1.0 o C higher than in case of actual emplacement rate. This means that the effect of the disposal rate, if being faster than in the actual case, is not significant.
51 4.4 Temperatures in panels containing BWR, VVER and EPR canisters 4.4.1 Temperatures in initial condition case Figure 39 shows the maximum temperature evolution for BWR, VVER 400 and EPR canister panels. The eccentricity effect is taken into account by lowering the canister temperature according to the curve in Figure 31. First the canister having maximum temperature in the panel is searched and then the temperature evolution for that canister is determined and presented as curves in Figure 39. The temperature evolution of the rock surface related to the same canister is also plotted. The time in Figure 39 is measured from the moment, when the canister having maximum temperature is emplaced. With greater times (over 100 years) the greater difference between the temperature curves of the different canister types is caused by the different decay heat powers (Table 4). Figure 39. Maximum temperature evolution of a BWR, VVER and EPR canisters in initial condition case. Figure 40 presents the temperature profiles at the moment, when the maximum temperature on the canister surface is reached. VVER canister has higher temperature than BWR and EPR canisters. The reason for this is mainly the canister spacing 7.2 m, which is not exactly the calculated reference value from KBS-3V analysis, 7.5 m. This slightly different spacing causes higher temperature for the VVER case in all the analyses. The old spacing for VVER canister 7.2 m was updated to 7.5 m for KBS-3V.
52 In Figure 40 the temperatures and temperature differences are at highest after about 15 years. Large number of canisters on the central areas has nearly same temperature profile as in Figure 40. With longer times temperatures and temperature differences decrease due canister thermal power decrease. Figure 40. Temperature profiles of a BWR, VVER and EPR canisters in initial condition case. Figure 41 shows the maximum temperatures on the canister surface, in the buffer, in the Supercontainer and in the rock both in case of a single BWR canister and in the hottest repository canister in initial condition.
53 Figure 41. Temperature profiles of a BWR Supercontainer and near-field rock in initial condition of the hottest canister in the panel and, for comparison only, for a single Supercontainer according to Figure 19 (average air gap). 4.4.2 Temperatures with artificially wetted outer gap This case is like an initial condition case having the inner air filled gap, but in the outer gap air is replaced by water having an improved thermal conductivity (Figure 13). These results are presented in Table 8. When compared to initial condition case, the artificially wetted outer gap lowers the canister surface temperature about 4 C. The heat flux reduction coefficients are in this case k BWR = 0.905, k VVER = 0.897 and k EPR = 0.904. Artificial wetting was not assumed to cause permanent humidity changes in the bentonite buffer. 4.4.3 Temperatures in saturated buffer case In the saturated buffer case the air space in the inner and outer gaps are filled with saturated bentonite, whose thermal conductivity is set to 1.3 W/m/K. The air gap 15 mm on the top end (right) of the canister is also filled with bentonite. The maximum temperature evolution for BWR, VVER and EPR canister panels is presented in Figure 42. The time is measured from the moment, when the canister having maximum temperature is emplaced. The heat flux reduction coefficients are k BWR = 0.882, k VVER = 0.881 and k EPR = 0.880.
54 Figure 42. Maximum temperature evolution of a BWR, VVER and EPR canisters in saturated buffer case. The time is measured from the moment, when the canister having maximum temperature is emplaced. Figure 43 shows the temperature profiles from the moment, when the maximum temperature on the canister surface is reached. Large number of canisters on the central areas has nearly same temperature profile as in Figure 43. VVER canister has about 2.0 C higher temperature than BWR and EPR canisters as in the initial condition case.
55 Figure 43. Temperature profiles of a BWR, VVER and EPR canisters in saturated buffer case. Figure 44. Temperature profiles of a BWR canister in saturated case and for the hottest canister in the panel, and for comparison, only for a single canister.
56 Figure 44 shows the maximum temperatures on the canister surface, in the buffer, in the Supercontainer and rock in case of a single BWR canister and in the repository canister in saturated buffer conditions. 4.4.4 Summary of maximum temperatures Table 8 summarizes the maximum canister surface temperatures in the whole canister panel. Table 8. Maximum temperature for the BWR, VVER and EPR fuel canisters. Drift spacing 25 m. Canister type BWR VVER EPR Canister spacing 9 m 7.2 m 10.6 m Max. canister temperature in initial condition 91.1 C 93.0 C 90.7 C Max. temperature with artificially wetted outer gap 87.1 C 89.0 C 86.2 C Maximum canister temperature in saturated case 74.8 C 76.6 C 74.3 C The time to the buffer saturation cannot be known precisely beforehand. That is why the maximum canister temperatures can be somewhere between the initial condition case (Figure 39) and the saturated buffer case (Figure 42). 4.5 Sensitivity to various parameters Sensitivity analyses are performed to determine the impact of different parameters. The sensitivity to various parameters was studied earlier in Ikonen (2009) for KBS-3V. 4.5.1 Results of variation of parameters In the next sensitivity analyses of the effect of different parameters are performed. Thermal diffusivity and capacity of rock, thermal conductivity of bentonite, the emissivity of the copper surface, the gap width on the canister surface, the canister spacing and the pre-cooling time are varied.
57 Figure 45. Effect of canister spacings on the maximum canister surface temperature in initial condition case, when the drift spacing is 25 m. Initial values are listed in Tables 1 to 4. Figure 45 shows the effect of the canister spacing on the maximum canister temperature for BWR, VVER 400 and EPR canisters. Based on Figure 45 the canister spacings could be reduced to 6.8 m (VVER), 7.9 m (BWR) and 9.0 m (EPR), if only the temperature is taken into account. In 3H, however, the canister spacing (distance block length) is affected by hydraulic conditions too, and these will be studied in performance assessment. Thermal diffusivity ν describing for instance the spreading of a heat pulse is calculated from! =! ["(T), c v (T)] = " (T) c v (T). (26) In an orthotropic case thermal conductivity depends also on the orientation. Figure 46 presents the effect of the variation of the rock thermal diffusivity on the maximum canister surface temperature change. In these analyses the heat capacity is 2.1 MJ/m 3 /K. The diffusivities in the three points in Figure 46 are 1.19. 10-6 m 2 /s, 1.34. 10-6 m 2 /s and 1.43. 10-6 m 2 /s and the corresponding conductivities 2.5 W/m/K, 2.82 W/m/K and 3.0 W/m/K, respectively. The sensitivity was studied so that first the single canister case was calculated by the numerical model as shown in Figure 16 in order to determine the heat flux reduction coefficient k. Then the whole panel was calculated by the analytic model. Practically same results were obtained for all canister types.
58 Figure 46. Effect of rock thermal diffusivity on the maximum canister surface temperature change in reference case (reference rock thermal diffusivity 1.34. 10-6 m 2 /s). Volumetric heat capacity is kept constant. Figure 47 presents the effect of the variation of the rock volumetric heat capacity on the maximum canister surface temperature change. The analyses were performed in an analogous way as above for the diffusivity. Practically same results were obtained for all canister types. Figure 47. Effect of rock volumetric heat capacity on the maximum canister surface temperature change in reference case (reference rock volumetric heat capacity is 2.1 MJ/m 3 /K in Table 2). Diffusivity is kept constant.
59 Figure 48 presents the effect of the bentonite thermal conductivity on the maximum canister surface temperature. These curves were calculated by a numerical model (Figure 16) for a single canister, since the changes are local and the change of the bentonite conductivity in remote canisters has no influence. According to Figure 48 the decrease of bentonite thermal conductivity increases significantly canister temperature. For selection of the effective conductivity of the buffer, see Section 2.2. Figure 48. Effect of bentonite thermal conductivity on the maximum canister surface temperature in reference case (reference bentonite thermal conductivity 1 W/m/K in Table 2). It was studied also a case, where the bentonite buffer was divided to radial layers having different thermal conductivity, lowest value near the canister surface. The result was that a relatively thin layer with lower conductivity is dominant and mostly determines the canister temperature. The artificial wetting of the compartments is advantageous in increasing the saturation rate and the thermal conductivity of bentonite buffer. The recent test results of bentonite conductivity have large variation. Bentonite conductivity seems to be a sensitive function of the water content, the saturation rate and the chemical content of bentonite. Thus it is worth remarking that the thermal properties of bentonite shall be followed when making the buffer development.
60 Figure 49. Effect of copper surface emissivity on the maximum canister surface temperature in reference case (reference emissivity 0.3 in Table 2). Figure 49 presents the effect of the copper surface emissivity on the maximum temperature on the canister surface. These curves were calculated by a numerical model for Figure 50. Effect of the inner air gap width on the maximum canister surface temperature in reference case (reference gap 5.1 mm in Table 1, BWR fuel).