Proceedings of ICONE14 International Conference on Nuclear Engineering July 17-20, Miami, Florida, USA
ICONE14-89181
NATURAL CONVECTION HEAT TRANSFER IN A RECTANGULAR WATER POOL WITH INTERNAL HEATING AND TOP AND BOTTOM COOLING Jong K. Lee, Seung D. Lee, Kune Y. Suh* Seoul National University San 56-1 Sillim-dong, Gwanak-gu, Seoul, 151-744, Korea *Phone: +82-2-880-8324 Fax: +82-2-889-2688 Email:
[email protected] ABSTRACT During a severe accident, the reactor core may melt and be relocated to the lower plenum to form a hemispherical pool. If there is no effective cooling mechanism, the core debris may heat up and the molten pool run into natural convection. Natural convection heat transfer was examined in SIGMA RP (Simulant Internal Gravitated Material Apparatus Rectangular Pool). The SIGMA RP apparatus comprises a rectangular test section, heat exchanger, cartridge heaters, cooling jackets, thermocouples and a data acquisition system. The internal heater heating method was used to simulate uniform heat source which is related to the modified Rayleigh number Ra′. The test procedure started with water, the working fluid, filling in the test section. There were two boundary conditions: one dealt with both walls being cooled isothermally, while the other had to with only the upper wall being cooled isothermally. The heat exchanger was utilized to maintain the isothermal boundary condition. Four side walls were surrounded by the insulating material to minimize heat loss. Tests were carried out at 1011 < Ra′ < 1013. The SIGMA RP tests with an appropriate cartridge heater arrangement showed excellent uniform heat generation in the pool. The steady state was defined such that the temperature fluctuation stayed within ±0.2 K over a time period of 5,000 s. The conductive heat transfer was dominant below the critical Rayleigh number Ra′c, whereas the convective heat transfer picked up above Ra′c. In the top and bottom boundary cooling condition, the upward Nusselt number Nuup was greater than the downward Nusselt number Nudn. In particular, the discrepancy between Nuup and Nudn widened with Ra′. The Nuup to Nudn ratio was varied from 7.75 to 16.77 given 1.45×1012 < Ra′ < 9.59×1013. On the other hand, Nuup was increased in absence of downward heat transfer for the case of top cooling. The current rectangular pool testing will be extended to include circular and spherical pools.
INTRODUCTION Should a severe accident occur in the absence of effective cooling mechanisms, the reactor core may heat up to the point of eutectic formation. The molten pool may relocate to the lower plenum of the reactor vessel. The high temperature of the molten core material will threaten the thermal and structural integrity of the reactor vessel. The extent and urgency of this threat depend primarily on the intensity of the internal heat sources and the consequent distribution of heat fluxes on the vessel wall in contact with the molten core. The feasibility of external vessel flooding as a severe accident management has received wide attention. The heat transfer inside the molten core material can be characterized by the strong buoyancy-induced flow resulting from internal decay heating of fission products. Thermofluid dynamic characteristics of such flow depend strongly on the thermal boundary conditions. Kulacki and Goldstein [1] reported on natural convection in volumetrically heated layers. The fluid layer was bounded by the isothermal upper and lower plates and adiabatic side walls. The modified Rayleigh number Ra′ varied from 200 to 107. Their test data covered the laminar, transition, and turbulent regimes of natural convection. For Ra′ > 104, the turbulent mixing effect began to play a key role in the overall energy transport process. Any periodicity or near periodicity in the mean temperature fields evident at lower Ra′ tended to disappear. Jahn and Reineke [2] practiced a numerical study on natural convection in internally heated pools contained in a rectangular cavity. They found that the temperature field suggested presence of nonuniform eddies in the upper region of the pool with a stable and calm liquid layer in the lower region. They concluded that heat was transferred more effectively in the upper region versus the lower region in this geometry.
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Kulacki and Emara [3] derived upward and downward heat transfer in rectangular geometries under the constraint of relatively small temperature differences. In their study, a planar fluid layer was investigated, where only the top surface was cooled and the bottom surface was insulated to minimize heat loss. For these boundary conditions, heat transfer coefficients were obtained for Ra′ up to 2 × 1012. They also attempted to identify an independent effect of the Prandtl number Pr from 2.75 to 6.86. Kulacki and Nagle [4] investigated natural convection with volumetric heating in a horizontal fluid layer with a rigid, insulated lower boundary and a rigid, isothermal upper boundary for Ra′ from 114 to 1.8 × 106 times the critical value of the linear stability theory. A correlation for the mean Nusselt number Nu was obtained for steady heat transfer. Data were presented on fluctuating temperatures at high Ra′ and on developing temperature distributions when the fluid layer was subjected to step change in power. Hollands et al. [5] reported on an experimental study on the natural convective heat transport through a horizontal layer of air covering Ra′ from subcritical to 4 × 106. Packets of a fluid with a high thermal diffusivity leaving the outer edge of the boundary layer lost heat and took up a temperature equal to the surrounding within a much shorter distance from its starting point than a fluid with a low thermal diffusivity. Correlation of the experimental data on heat transfer from nonboiling, horizontal fluid layers with internal heat generation was cast into a form suitable for analysis of post-accident heat removal in fast reactors by Baker et al [6]. Available data on layers with equal boundary temperatures indicated that the downward heat transfer rate could be treated by conduction alone, while the upward heat transfer rate was largely controlled by convection. Steinberner and Reineke [7] investigated experimentally and numerically the buoyant convection with internal heat sources in a closed rectangular cavity for 107 < Ra′ < 1014. Their investigation of the structure and the dynamic behavior of the turbulent boundary layer at free convection with internal heat sources in rectangular cavities, revealed a number of equal properties in comparison to the turbulent boundary layer at the heated vertical plate. Cheung [8] developed a phenomenological model of eddy heat transfer in natural convection with volumetric heat sources at high Ra′. His model was applied to the problem of thermal convection in a horizontal heated fluid layer with an adiabatic lower boundary and an isothermal upper wall. At high Ra′, the mean temperature was found to be constant throughout the layer except in a sublayer region near the upper wall. The thickness of such a region was observed to be inversely proportional to the mean Nu. Outside the sublayer region the distribution of eddy heat flux was linear. The heat transfer predictions were found to agree well with measurements in the turbulent thermal convection regime. Several dimensionless parameters are related to the natural convection heat transfer phenomena. The spatial and temporal
variation in heat flux on the pool wall boundaries and the pool superheat characteristics relate closely to the natural convection flow pattern inside the molten pool. The natural convection heat transfer phenomena involving internal heat source lead to the strength of the buoyancy force. Natural or free convection phenomena can be scaled in terms of the Grashof number Gr, Prandtl number Pr and additionally, the Dammkoehler number, Da, in the presence of volumetric heat sources. The dimensionless numbers are defined as
Gr =
gβ ΔTL3
ν2
; Pr =
ν QL2 : Da = α kΔT
(1)
The Rayleigh number Ra can be used to characterize the heat transfer in natural or free convection problems, including those involving external heat sources or external heating such as heating from below. This dimensionless number is defined as
Ra = Gr Pr =
gβΔTL3
αν
;α =
k μ ;ν = ρcP ρ
(2)
Ra′ is germane to free or natural convection problems with internal heat sources, and defined as
Ra ' = RaDa = Gr Pr Da =
gβQL5 ανk
(3)
Natural convection heat transfer involving the internal heat generation is adequately represented by such dimensionless parameters as Ra′ and Pr, and physical dimensions of the pool. For the natural convection phenomena involved in reactor vessel molten core material flow, the predominant driving force for heat transport process is the internal decay heating, while the conduction effects are relatively small. Heat transfer rates are thus primarily governed by Ra′. Nu has been correlated reasonably well by the following relation, which incorporates the dependence on Ra′
Nu = C ⋅ Ra 'n =
hL
(4)
k
Nu is equal to the dimensionless temperature gradient at the surface providing a measure of the convection heat transfer occurring at the surface. For low values of Ra′ the turbulence intensity is small and so is the turbulent or eddy viscosity. Such flows characterized as laminar would be steady if the internal heat source and boundary conditions vary slowly over the time scale of interest. However, the flow is characterized as turbulent and unsteady for large values of Ra′ at least in domains of vigorous mixing and high turbulent intensity, and the molecular viscosity is small in relation to the eddy viscosity. EXPERIMENTAL APPARATUS A rectangular pool has since been categorized from a plane layer. The aspect ratio L/X of the former is on the order of unity or greater. In contrast, the latter means enclosure for which L/X is much less than unity. The aspect ratio equals the ratio of pool height to an appropriate horizontal length. More study of the
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Table 1. Thermocouple locations in SIGMA RP Objective # Z [mm] X [mm] 3, 50, 100, 150, 200, 250, -150 11 300, 350, 400, 450, 497 Pool 3, 50, 100, 150, 200, 250, temperatur 0 11 300, 350, 400, 450, 497 e 3, 50, 100, 150, 200, 250, 150 11 300, 350, 400, 450, 497 Upper 2 -150, 150 boundary Lower 2 -150, 150 boundary -150, 0, Upper heat 6 49, 50 150 flux -150, 0, Lower heat 6 0, 1 150 flux Total 49 -
rectangular pool was in need than study of the plane layer cases according to Kulacki and Richards [9]. The SIGMA RP (Simulant Internal Gravitated Material Apparatus Rectangular Pool) apparatus was built 500 mm long, 160 mm wide and 500 mm high, as presented in Fig. 1. Forty thin cartridge heaters, 5.0 mm in diameter and 500 mm long, were used to simulate internal heating in the pool. The heater average resistance was 210 Ω within ±5 %. The heaters were uniformly distributed to supply a maximum power of 8 kW to the rectangular pool. Unit: mm
1 2 3
16 17 18
31 32 33
4
19
34
5
20
35
6
21
36
7
22
37
23
38
9
24
39
10
25
40
11
26
41
12
27
13 14 15
28 29 30
50
50
50
5
10
50
58 56 52.5
50
580 500
T/C 1mm
50
8
50
58 56 52.5
50
50
50
50
610
50
Heater
43 44 45
150
=2
Upper copper plate
X 10
10 100
5 =1
Z Y
42 5
3
150
2.5 1 2
100
=1
3
5 =2
2 1 2.5
Fig. 2. Schematic diagram of SIGMA RP test loop
Lower copper plate
EXPERIMENTAL PROCEDURE The DAS bias error was calibrated so as to minimize the measurement error. Once properly calibrated, thermocouples were placed at their designated locations. The water cooling system, or heat exchanger, supplied the well-defined upper and lower boundary conditions. Performance testing of the water cooling system showed the temperature difference of ±0.5 K ranging from 5 °C to 80 °C. After all this procedure to check on proper functioning finished, the pool was started to heat up. Ra′ can be calculated for the present experiment for varying input power Q and the characteristic pool height L. SIGMA RP covered the range of 1011 < Ra′ < 1013 and Pr = 6.5. The aspect ratio of the rectangular pool was maintained at unity for all the cases. After the water leakage test, the thermocouples were calibrated by ISOTECH temperature Reference Unit Model 740. Forty-nine thermocouples were submerged sequentially in a constant temperature pool. The thermocouple outputs were
Fig. 1. Schematic of SIGMA RP heater and thermocouple locations
Table 1 shows the thermocouple location in SIGMA RP. A total of forty-nine T-type thermocouples were used to measure the top and bottom temperatures. Eleven thermocouples were installed at the middle of plane 3, 50, 100, 150, 200, 250, 300, 350, 400, 450, and 497 mm, respectively, from the bottom plate. Other twenty-two thermocouples were installed at the line of symmetry, which is 150 mm off the centerline. In order to measure the exact temperature, three segments of thermocouple guide holes were attached to the upper and lower copper walls, separately. Fig. 2 is a schematic diagram of the SIGMA RP test loop. The apparatus comprises the demineralized water system, test section, heat exchanger, power controller, forty cartridge heaters, forty-nine thermocouples and a data acquisition system (DAS).
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The thermophysical properties of the fluids needed to compute the dimensionless numbers were determined at the mean temperature Tm between the maximum temperature Tmax and boundary temperature Tb, respectively. All data were recorded when steady state conditions had been established. The steady state was defined as the state when the temperature fluctuation stayed within ±0.2 K over a period of 5,000 s. The working fluid and test matrix of the SIGMA RP experiments are presented in Tables 2 and 3 pursuant to boundary conditions.
checked with temperature reading from a separate thermometer. The calibration curves were drawn for each thermocouple. The relative error in reading from a T-type thermocouple was determined from these curves. ERROR ESTIMATES The power input per unit volume by the voltmeter readings produced an experimental uncertainty of 6 %. The uncertainty in measurement of temperature related to the material properties of the working fluid produced an experimental uncertainty of 2 %. The uncertainty in the geometrical factor L5 was 2 %. A reasonable uncertainty in Ra′ would be on the order of 10 %. For the duration of 14 hr, the variation in the boundary water temperature was neglected since the differences stayed within ±0.5 K. The uncertainty in measurement of the heat flux was 4 % and that of the heat transfer coefficient was 6 %. The total experimental uncertainty from the computed values of Nu was found to be 10 %. The uncertainty in the thermal conductivity was 1 %. The maximum uncertainty in Nu was 12 %.
Table 2. Test matrix for top cooling of water
RESULTS The SIGMA RP tests covered the range of 1011 < Ra′ < 13 10 , in which the thermal convection flow was in the turbulent heat transfer regime. Uniform heat generation is a prerequisite to natural convection experiments. SIGMA RP has adopted the internal heater heating method to heat up the rectangular pool. Fig. 3 illustrates the temperature rise rate at different locations in the rectangular pool. The average temperature increase rate was 2.61×10-5 K/s with a standard deviation of 1.04×10-6 K/s. The rate of temperature rise at any location in the pool did not differ by more than ±4 % from the mean value. The results demonstrated the feasibility of using the internal heater heating method to simulate uniform volumetric heat generation in SIGMA RP.
-5
Average ise ate = 2.60x10 [K] -6 Standard eviation = 1.04x10 [K]
Temperature [K]
285.0
284.5
283.5
283.0 0
10000
20000
30000
40000
Qup [W]
Qdn [W]
Ra'
TC-1 TC-2 TC--3 TC-4 TC-5 TC-6 TC-7 TC-8 TC-9 TC-10 TC-11
42.1 80.7 171.8 304.0 484.8 691.5 937.3 1213.0 1563.6 1901.0 2330.0
42 79.2 165.6 286.0 461.4 659.3 890.4 1132.8 1511.0 1776.8 2150.0
-
7.23×1011 1.39×1012 2.95×1012 5.22×1012 8.33×1012 1.19×1012 1.61×1013 2.08×1013 2.69×1013 3.27×1013 4.00×1013
Case
Qin [W]
Qup [W]
Qdn [W]
Ra'
AC-1 AC-2 AC-3 AC-4 AC-5 AC-6 AC-7 AC-8 AC-9 AC-10 AC-11 AC-12
42.3 81.1 174.4 300.2 484.8 683.4 936.0 1205.0 1554.0 1888.0 2377.0 2789.0
37.2 72.7 158.4 265.8 439.6 581.8 816.9 1037.5 1338.0 1672.6 2123.4 2411.9
4.8 6.5 15.4 23.4 35.6 46.1 59.0 74.3 92.1 106.7 128.0 143.8
1.45×1012 2.72×1012 5.99×1012 1.03×1013 1.67×1013 2.35×1013 3.22×1013 4.14×1013 5.34×1013 6.49×1013 8.17×1013 9.59×1013
Fig. 4 presents the schematic of dimensionless temperature profile in the volumetrically heated rectangular pool. Experimental results indicated that one could divide the rectangular pool into two distinct regions: the upper boundary layer and the turbulent mixing core. Within the upper boundary layer, conduction was dominating. Above the conduction region, the fluid was in the state of turbulent mixing for high Ra′ natural convection. Nearly all the heat generated in the rectangular pool was transported from the core to the upper surface because the lower wall was insulated. Thus, the local heat flux near the lower wall was negligible. The thermal boundary layer was developed at the upper wall surface for large Ra′ expressed as
TC20 TC21 TC22 TC23 TC24 TC25 TC26
284.0
Qin [W]
Table 3. Test matrix for top and bottom cooling of water
286.0
285.5
Case
50000
Time [s]
Fig. 3. Temperature rise rate at different locations
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Ra' =
gβ mQL5 2α mν m k m
(5) 1.0
Dimensionless elevation [z/L]
11
Ra'=7.23x10 12 Ra'=1.39x10 12 Ra'=2.95x10 12 Ra'=5.22x10 12 Ra'=8.33x10 13 Ra'=1.19x10 13 Ra'=1.61x10 13 Ra'=2.08x10 13 Ra'=2.69x10 13 Ra'=3.27x10 13 Ra'=4.00x10
0.8
0.6
0.4
0.2
0.0 0.000
0.005
0.010
0.015
0.020
0.025
0.030
2
Dimensionless mean temperature [(Tmax-Tb)/(QL /2k)] Fig. 5. Dimensionless mean temperature profile with top surface cooled isothermally
Fig. 4. Schematic of temperature profile with top surface cooled isothermally
Fig. 5 illustrates the dimensionless horizontal mean temperature distribution. A steady state was reached for 14 hr. The temperature profile flattened as Ra′ was increased. The temperature gradient across the lower wall was negligible, which signified that there was no heat transfer in the lower wall region. The temperature gradient was dominant in the region of the upper boundary layer, however. The buoyancy affected the central isothermal core of the layer. As a result, the upper boundary layer became thinner and heat transfer was improved. This means that the heat transfer coefficient at the upper surface was relatively large. At the upper wall, because convection played a dominant role in the total heat transfer, the upward heat transfer Nuup depended strongly on Ra′. During the process of heat transport, the turbulent thermal energy which had been produced in the upper wall gradually diffused into the turbulent mixing core. Simultaneously, a portion of the turbulent energy had been dissipated by the mixing process until the steady state was reached. The overall balance was such that the turbulent energy distribution was maintained constant in the mixing core. Fig. 6 demonstrates the dimensionless temperature profile in the rectangular pool. There were three distinct regions: the upper boundary layer, turbulent mixing core, and the lower boundary layer. The onset of convection was characterized in the upper half layer and conduction was dominant in the lower layer. The core region expanded both upwards and downwards as Ra' was increased. Also, a thin thermal boundary layer was rapidly developed at the upper surface as Ra' was increased. Nu demonstrated a strong dependency on Ra'. On the other hand, the lower region of the layer was practically dominated by conduction at all Ra' expressed as
Ra'=
gβm QL5 αm νm κ m
Fig. 6. Schematic of temperature profile with top and bottom surfaces cooled isothermally
Fig. 7 depicts dimensionless horizontal mean temperature distributions. Differences in dimensionless mean temperatures were mainly caused by differing volumetric heat sources. The temperature profile was flattened as Ra' was increased. Note also that the thermal boundary layer thickness at the upper surface was less than that at the lower surface. The buoyancy affected the central isothermal core of the layer. As a result, the upper boundary layer was thinned. Note that the heat transfer coefficient at the upper surface was relatively large. Figs. 8 and 9 present the upward heat transfer data. During the process of heat transfer, the turbulent thermal energy which had been produced in the upper wall region was being diffused gradually into the turbulent mixing core. Simultaneously, a portion of the turbulent energy was being dissipated by the mixing process until a steady state was reached. The overall balance was such that the turbulent energy distribution was maintained constant in the mixing core.
(6)
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CONCLUSION Natural convection tests were conducted in a rectangular pool heated by discrete cartridge heater used to simulate uniform volumetric heating at high Ra′. The internal heating method using the cartridge heaters was adopted to simulate volumetric heat sources. Smaller diameter cartridge heaters are being adopted to minimize interruption of heaters for further investigation on the subject of natural convection thermo fluid dynamics. The SIGMA CP (Circular Pool) and SIGMA SP (Spherical Pool) test sections will be utilized to correlate the present SIGMA RP results with natural convection in circular and hemispherical pools.
100
Nudn
48.4%
1 5 10
1.0
Dimensionless elevation [z/L]
12
1.45x10 12 2.72x10 12 5.99x10 13 1.03x10 13 1.67x10 13 2.35x10 13 3.22x10 13 4.14x10 13 5.34x10 13 6.49x10 13 8.17x10 13 9.59x10
0.8
0.6
0.4
0.2
0.005
0.010
0.015
0.020
Fig. 7. Dimensionless mean temperature profile with top and bottom surfaces cooled isothermally
1000
Greek α β δ μ ν Δ
33.7%
Nuup 1 5 10
Kulacki & Goldstein (1972) Jahn & Reineke (1974) Mayinger (1976) Steinberner & Reineke (1978) Emara & Kulacki (1980) Water (4
