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Figure 4.12: Problem formulation and boundary conditions employed for model vali- dation Here, for the purpose of the model validation, the numerical simulation of stepwise density stratification in a fluid layer with lateral heating is performed. The results are compared to those reported in [21]. Here, for the purpose of the model validation, the numerical simulation of stepwise — Figure 4 12: Problem formulation and boundary conditions employed for model vali- dation Here, for the purpose of the model validation, the numerical simulation of stepwise density stratification in a fluid layer with lateral heating is performed. The results are compared to those reported in [21]. Here, for the purpose of the model validation, the numerical simulation of stepwise

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Figure 1.1: Schematic of the IVMR.

The conservation equations for fluid 1 are Continuity of temperature, velocity, heat flux and shear stresses must be satisfied at the fluid-to-fluid interface. Continuity of heat flux gives the dimensionless group K4Q >= ky /ko. Here, Gry (or Ray) is defined on the reference values of fluid 2 and on the tem- perature difference (7 — T,). So the investigation of the two-layer problem becomes significantly more complicated.

Table 2.1: Correlations for Rayleigh-Bénard convection.

Table 2.2: Correlations for internally-heated horizontal fluid layers with an adiabatic lower wall and an isothermal upper boundary, with L being the height and X being the length of the fluid layer. The first heat transfer measurements for turbulent thermal convection in a fluid bounded by two rigid isothermal planes of equal temperature were reported by Kulacki and Goldstein [65]. In their study, the volumetric energy source was provided by Joule heating. The fluid height was varied from 1.27 to 6.35 cm, and the length and depth of the channel were both 25.4 cm. Horizontally-averaged temperature profiles were de- termined optically. Heat transfer correlations for the upper and lower boundaries were obtained from 67 observations. From the empirical correlation, the critical Rayleigh

Table 2.3: Correlations for internally heated horizontal fluid layers with isothermal lower and upper boundaries. Experimental studies in a similar configuration were reported by Mayinger [74], Jahn and Reineke [54], Kulacki and Emara [64]. Table 2.3 summarizes the heat transfer correlations obtained from these investigations.

Circular and spherical cavities. Mayinger et al. [74] were the first to report ex- perimental and computational results for thermal convection in volumetrically-heated semicircular pools. In the last decade, a number of research programs have appeared world-wide: USA (UCLA facility, [1], ACOPO at UCSB, [108]), Russia (OECD RAS- PLAV project, [2]), France (BALI, [11]), Finland (COPO, [70]), and Sweden (FOR- EVER, [94], SIMECO, [93]). The main objective of these tests is to obtain the heat transfer characteristics applicable to prototypical configurations and simulants. The heat transfer correlations for different geometries (Figs. 2.2) are summarized in Tables 2.4-2.5 below.

Table 2.4: Correlations for internally-heated semicircular and spherical pools heat transfer correlations for different geometries (Figs. 2.2) are summarized in Tables

Table 2.5: Correlations for internally-heated semicircular and spherical pools (contd). Numerical studies. The earliest two-dimensional (2D) simulations of thermal con- vection in rectangular enclosures with internal heat generation were performed in the early 1970s by Thirlby [104] Peckover and Hutchinson [82] and Jahn and Reineke [54]. Mayinger e¢ al. [74] predicted natural convection heat transfer in a hemispherical cavity.

Table 3.1: The experimental Nusselt number correlations of Kulacki and Nguen [68] Table 3.2: Physical properties (taken at T’ = 30°C’)

Chapter 3. Analytical Study Figure 3.1: Semiempirical (dashed lines) and experimental (solid lines) Nusselt num. ber as a function of the Rayleigh number for Heptane- Water.

Figure 3.5: Nusselt number dependence on kinematic viscosity ratio (£12 = 0.1). Figure 3.4: Nusselt number dependence on conductivity ratio (L12 = 0.1). Figure 3.3: Nusselt number dependence on height ratio (kg = 0.2). Here, we plot Nu as a function of Ly9 for fixed values of the other parameters. It can be seen in Fig. 3.3 that the effect of depth ratio is weak for 0.01 < Lig < 0.1. Nu becomes greater as [Lj increases for L;2 > 0.1. The correlation is similar to the behaviour observed experimentally for Nu as Lyj9 is varied. The analysis cannot be applied for Ly2 < 1, when capillary forces and other effects must be taken into account. the behaviour observed experimentally for Nu as Ly is varied. The analysis cannot

Thus, we have obtained the equation for 7 as a function of the following six di- mensionless parameters: Rag, Prj, 12, Lig, 12,12, 12. AS a result, investigation of this problem becomes substantially more complicated than for the one-fluid case. The equation 3.20 can be solved numerically. The values of 7 as a function of Rag are plotted in Fig. 3.8. It can be seen that more heat is being transferred downwards in a double-layer configuration as the upper layer imposes additional thermal resistance. As Rag increases, the difference in 7 between one- and two-layer cases decreases.

Figure 3.10: Dependence of the fraction of heat transfer downwards on k 19. Figure 3.10: Dependence of the fraction Figure 3.8: Dependence of the fraction of heat transfer downwards on Rag.

Figure 3.9: Dependence of the fraction of heat transfer downwards on Ly9.

Figure 3.11: Dependence of the fraction of heat transfer downwards on 119.

The simulations were performed with the individual layer Rayleigh number ranging from 104 to 10’. The computed Nusselt numbers are presented in the reduced form given by Eq. 2.18, and compared to the correlation Eq. 2.20 suggested by Haberstroh and Reinders [46]. It can be seen that the computed results agree well with the predicted ones.

Figure 4.2: Computed isotherms.

Figure 4.3: Mean dimensionless temperature distribution along the height of the do- main

Figure 4.4: Schematic of computational domain adjacent to the interface. The fixed interface two-fluid model is implemented into the code by means of the code’s user FORTRAN routines. The flow domain is divided in two sub-domains cou- pled through the boundary conditions at the interface. At each time step n+ 1 and each iteration, temperature and velocity at ¢-node at fluid-fluid interface (the upper boundary for fluid 2 and the lower boundary of fluid 1 as it is depicted in Fig. 4.4) are approxi- mated from the continuity of shear stress and heat flux as:

Figure 4.5: a) Computed velocity field. b) Velocity distribution along y-axis. is equal to the analytical value of Ujnz = <qeee = 4/6 m/s. In the same way, heat transfer across the interface was checked for the conduction problem of two fluids having different conductivities.

Figure 4.9: Temperature field (Raj = 10°, Lig = 6 : 26). Figure 4.8: Velocity field (paraffin oil- water).

Figure 4.10: Computed isotherms for Rag = 2-10°, Lig = 8: 18, Qu, = Quy. 1.3 Computations with a double-diffusion model

Figure 4.14: The local Nusselt number distribution at For = 0.1

Figure 4.15: Velocity field, crust and mushy zone front (t = 1000)

Figure 4.16: Average Nusselt number Nw, variation with T* &C' (left) and Ra, (right).

The internally and uniformly heated fluid is contained in a square cavity with adi- abatic top and bottom walls. The walls are kept at constant temperature equal to the solidus temperature. Such a configuration provides simplicity in analyzing computed results and is able to capture major features of convection heat transfer in the near- wall region of an internally heated pool. The main parameter of interest in this case is the energy split, i.e. the fraction of heat generated within the layer that is transferred downward. Calculations are first performed for a case of without phase change and the results in terms of energy split, 7, are found to agree well with the correlation of Baket et al. [4]. The calculations are performed for different values of T, Q based on reference value of volumetric heat generation and a fixed value of Ra; equal to 8.8 x10°. Di- mensionless parameter 16 is defined as (Thig — Tso1)/QyH 2. The value of C is taker equal to 10°.

Table 4.3: Calculated results against the predictions of a semiempirical correlation by ~ 3 ». FF

Figure 4.18: Temperature field (left) and mushy zone front (right) for TS = 0.058 with the correlation of Baker. The results are given in Table 4.3 below. The computed temperature field and mushy zone front are shown in Fig. 4.18.

Figure 4.19: Visual observation of the pool melting at steady state (SIMECO test SNEu-03. Ref. [601). Figure 4.20: Velocity field, crust and mushy zone front for Ra; = 1.4- 10°.

Figure 4.21: Isotherms and mushy layer front for Ra; = 1.4 - 10°: no phase-change (left) and solidification (right).

Figure 4.22: Temperature field (Ra = 2- 10°).

Figure 4.23: Velocity field (Ra = 2- 10°).

Figure 4.24: Dimensionless mean eddy heat transport. Figure 4.25: Production of thermal vari- ance 7”.

Figure 4.26: Dimensionless mean temperature distribution.

Table 4.4: Computed vs. experimental average Nusselt numbers.

Figure 4.29: Dimensionless mean eddy heat transport. Figure 4.30: Production of thermal vari- ance 7”.

on Pr agrees with the trend given by the Globe & Dropkin [40] correlation. It should be noted that the values given by different correlations might have as great as nearly 40% difference for the Rayleigh number under consideration. At high Rayleigh numbers, the difference can be of a factor two or even larger. Mean temperature profiles for different values of Pr are plotted in Fig. 4.52. The values in the core are similar, and the difference near the boundaries is due to the differences in the thickness of the thermal boundary layer.

Figure 4.34: Temperature fluctuations iT field (Ra = 4- 10°, Pr = 0.6).

Figure 4.35: Temperature fluctuations 7 field (Ra = 4- 10°, Pr =6).

Figure 4.38: Dimensionless production and dissipation of turbulent kinetic energy far differant racac (Pn —9 AV INT) Figure 4.36: Dimensionless T;,,,, (Ra = Fig 2.4 x 10°). 2.4

Figure 4.37: (Dimensionless v;,; Ra = 2.4 x 10°).

Figure 4.39: Production of thermal vari- ance 7"? for different cases (Ra = 2.4 x 10°).

Figure 4.42: Turbulent viscosity (Ra = 2.4 x 10", Pr = 6).

‘igure 4.40: Dimensionless mean tem-

(Ra = 2.4 x 10°).

Figure 4.43: The flux Richardson number (Ra = 2.4 x 10’, Pr = 6).

Figure 4.45: Vertical profile of the horizontally-averaged temperature (Ra = 6.3 x 10°. Pr = 0.71).

Figure 4.44: Computed temperature field at z=0.5 (Ra = 6.3 x 10°, Pr = 0.7).

Figure 4.46: Variation of reduced mean temperature profiles for different Prandtl numbers (Ra = 63 x 10°).

Figure 4.49: Vertical profile of dimen- sionless Upm; (Ra = 6.3 x 10°,Pr = 0.71). Figure 4.49: Vertical profile of dimen-

Figure 4.50: Vertical profiles of terms in the thermal variance equation (Ra = 6.3 x 41N5 N.. nwt.

Figure 4.48: Vertical profile of dimen- sionless Upms (Ra = 6.3 x 10°,Pr = 0.71).

Figure 4.51: Vertical profile of Nus- selt numbers for different Prandtl numbers (Ra = 6.3 x 10°).

Figure 4.52: Vertical profile of dimen- sionless centerline temperatures for differ- segs Wess ce ee) eeeeedccres fT . & BW ce TASS

Figure 5.1: Schematic of the SIMECO experimental facility. Chapter 5. Experimental Study

Figure 5.2: Main dimensions.

Figure 5.3: Temperature field taken by infra-red camera.

Figure 5.4: Post-test section. Figure 5.4 shows the post-test view of the solidified pool. The crust region can be distinguished from the molten region by its darker shade. The contact between he top heat exchanger and the pool was found to contain no gaps. However, due to shrinkage of the salt during its sudden solidification, after the top heat exchanger had been placed, a small air-gap can form, although it is unlikely to affect the thermal processes in the pool, since it is assumed that the boundary temperature for the molten pool is the liquidus temperature, T),;,, of the eutectic salt.

Figure 5.5: The mean upward Nusselt number.

similar tests with different stimulant materials, but without crust formation. Figure 5.6: Computed temperature field for a) Qin = 2KW, b) Qin = 2.8kW. 5.4 Low Temperature Tests

Figure 5.7: Local side wall heat flux dis- tribution for Qj, = 2kW. Figure 5.8: Local side wall heat flux dis- tribution for Qj, = 2.8kW.

Table 5.1: Physical properties (taken at T’ = 250° C for NaNO3 — K.NOs3 and T = 80° C for Cerrobend). In this section, a series of experiments employing an eutectic salt and Cerrobend is described. Cerrobend is a low melting point metal alloy composed of bismuth, lead, tin and cadmium. The objective of these tests is to investigate the ’focusing effect” when a metallic layer is employed as the top layer in an integral manner. The properties of the materials employed are summarized in Table 5.1.

Chapter 5. Experimental Study of the paraffin oil increases the interfacial boundary layer resistance. Also, the higher viscosity of the oil somewhat reduces the intensity of the motion above the boundary layer.

Figure 5.10: Local downward heat flux distribution.

Figure 5.12: Readings from thermocou- ples inside the side wall heat exchanger (test no.2). Figure 5.11: Energy balance (test no.2). Chapter 5. Experimental Study

Figure 5.13: Readings from thermocouples inside the brass side wall (test no.2).

Table 5.2: Test Matrix (VaNO3 — K NO3/Cerrobend) In the adiabatic case, due to higher temperature in the metallic layer, the crust becomes thinner and nearly uniform, and completely disappears on the upper side. Qualitatively, MVITA predicts the major trends in the molten pool. [In the adiabatic case, due to higher temperature in the metallic layer, the crust becomes

Figures 5.29-5.30 plot time average centerline temperature distributions for Dyj2 = 3: 27 and Lig = 6 : 27, respectively. In contrast to salt-Cerrobend tests, the heat transfer in the two pools is coupled through the interface. Thus, any variation of con- ditions in the upper layer leads to changes in the temperature field in the lower pool. It can be seen that in the uniform pool the turbulent mixed region (i.e. the part of the pool with nearly uniform temperature distribution along centerline) is substantially larger than for pools with a metallic layer. A non-heat-generating upper layer imposes addi- tional thermal resistance. As a result, higher temperatures are reached in the core of the stratified pools than in the core of the uniform one.

Figure 5.14: Centerline temperature dis- tributions for Qin = 2 kW, Ly2 = 3: 27. Figure 5.15: Centerline temperature dis- tributions for Qin = 3 kW, Ly2 = 3 : 27. Figure 5.14: Centerline temperature dis- Table 5.3: Test Matrix (Glycerol/Cerrobend)

Table 5.4: Heat Transfer in the Metallic Layer (Glycerol-Cerrobend) the metallic region are caused by a small driving temperature difference between the bulk temperature of the metal layer and the side boundary temperature of the brass wall. In fact, temperature of the relatively thick (2.3 cm) side brass wall increase with angle ® and the difference between the coolant and the brass wall temperature can reach considerable values (up to 20% higher) at the region around the interface. Thus, the side boundary temperature for the metallic layer is higher than the coolant inlet temperature and the boundary temperature, strictly speaking, is not constant. In the salt-Cerrobend tests, when the crust is formed in the metallic layer, due to low coolant temperatures, the side boundary temperature is the melting point of Cerrobend and the inhomogeneous brass wall temperature distribution affects only crust thickness, and not thermal processes in the metallic layer.

Figure 5.16: Centerline temperature dis- tributions for Qin = 2 kW, Lio = 6: 27.

Figure 5.17: Centerline temperature dis- tributions for Qin = 2 kW, Ly2 = 6 : 27. Chapter 5. Experimental Study

Figure 5.19: Local heat flux distributions for Qin =2 kW, Dy = 6: 27. Figure 5.18: Local heat flux distributions for Qin =2 kW, Ly =3: 27.

Figure 5.21: Local heat flux distributions for Qin = 3kW, Ly = 6: 27. Figure 5.20: Local heat flux distributions for On = 3kW, Ly = 3:27. Chapter 5. Experimental Study

Figure 5.22: Local dimensionless heat flux distributions for Qj, = 2 kW, Lig = 3:27.

Figure 5.24: Local dimensionless heat flux distributions for Qj, = 3 kW, Lig = 3: 27.

Figure 5.23: Local dimensionless heat flux distributions for Qj, = 2 kW, Lig = &.97

Figure 5.26: Computational mesh for a) [£12 = 3:27; b) Ly = 6:27. Chapter 5. Experimental Study

Figure 5.29: Centerline temperature dis- tributions for Qin = 1.5 kW, Ly. = 3 : 27. Figure 5.28: Computed temperature field for L12 = 6:27, Qin = 2kKW: a) isothermal top boundary; b) adiabatic top boundary

Figure 5.30: Centerline temperature dis- tributions for Qin = 1.5 kW, Ly. = 6: 27.

Figure 5.31: Local heat flux distributions for Ly = 3:27.

Figure 5.33: Dimensionless local heat flux distributions for [45 = 3 : 27.

Chapter 5. Experimental Study

Figure 5.37: Heat capacity as a function of temperature.

Figure 5.35: Viscosity as a function of temperature.

Figure 5.36: Thermal conductivity as a function of temperature.