Thermal Hydraulic Characteristics Of Extended Heated Vertical Channels To Enhance Natural Convection In The Core Of A Typical MTR Reactor

Transcription

1 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 Thermal Hydraulic Characteristics Of Extended Heated Vertical Channels To Enhance Natural Convection In The Core Of A Typical MTR Reactor Said M. A. Ibrahim * Department of Mechanical Engineering, Faculty of Engineering, AL-Azhar University, Nasr City, Cairo 37,Egypt * prof.dr.said@hotmail.com Professor of Mechanical Power Engineering & Energy Abstract-- This research deals with natural convection heat transfer from vertical heated cladded plates, which are symmetrically placed in proposed chimneys of variable heights in the core of a typical MTR reactor. The heated plates serve as thermal pumps for pumping fluid of a symmetrical enclosure beneath the chimney. The suggested chimneys are used for increasing the length of the vertical heated channels of the reactor core to give the chimney effect. In the thermal analysis of natural convection in channel chimney systems, the variables that play an important role are heat flux, maximum wall temperatures and geometrical parameters such as the height of the heated channel, the channel spacing and the height and spacing of unheated extensions. A simple numerical procedure to obtain the thermal design charts, a thermal optimization of the system and an uncertainty analysis due to the thermo- physical properties is presented. The present results are obtained from a real domain inside the reactor core data in the following dimensionless parameter ranges: 5 Lh/ b 20; :5 L/Lh 4; 4; 0 2 Ra 0 5. This study results in enhancing the reactor power in the free convection regime from a maximum of 400 kw up to 950 kw of thermal energy. This is quite significant increase in reactor power in the natural convection regime which adds to reactor safety. The results are of importance to reactor operation and safety in the natural convection mode of operation. Keywords-- Thermal hydraulic- Natural convection- Chimney- Vertical heated channel- MTR- Rayleigh number- Nusselt number- Temperature profile- Aspect ratio- Expansion ratio- Extension ratio.. INTRODUCTION Nowadays more recent investigation trends in natural convection heat transfer are oriented towards either seeking of new configuration to enhance the heat transfer parameter or the optimization of standard configurations. Natural convection between heated vertical parallel plates is a physical system frequently employed in technological applications, such as thermal control in electronic equipment, nuclear reactors, solar collectors and chemical vapor deposition reactors and it has been extensively studied both experimentally and numerically ( Gebhart, 988 ), ( Kimm and Lee, 966 ), ( Manca et al, 2000 ). More recent trends in natural convection research are to find new configurations to improve heat transfer parameters or to analyze standard configurations to carry out optimal geometrical parameters for better heat transfer rates ( Manca et al, 2000 ), ( Ledezma, 977 ), ( Bejan et al, 2004 ). Haaland and Sparrow ( 983 ) were the first to show that higher flow rate of fluid through a confined open-ended enclosure can be induced by the chimney effect. They introduced a numerical solution for natural convection flow in a vertical channel with a point heat source or distributed heat source situated at the channel inlet. Oosthuizen ( 984 ) studied numerically the heat transfer enhancement caused by the addition of the straight adiabatic extension at the exit of isothermal parallel-walled channel. He solved the parabolic form of the governing equation by means of a fully implicit forward marching procedure. The results indicated that substantial increase (about 50 %) in the heat transfer rate could be achieved, but very long adiabatic sections were required. Wirtz and Haag ( 985 ) presented experimental results for isothermal symmetrically heated plates with an unheated entry channel portion. Their experiments were carried out over a wide range of the Rayleigh number, from the single-plate limit to the fully developed channel. They found that the flow is quite insensitive to the presence of unheated entry section of large channel spacing, while it is severely affected when the gap spacing is small Asako et al. ( 990 ) examined numerically the heat transfer increment due to an unheated chimney attached to a vertical isothermal tube. The numerical results were obtained by a control volume approach solving the full elliptic form of the governing equation. They evaluated the optimum chimney diameter where the maximum amount of heat is transferred and found that for optimum chimney diameters the heat transfer enhancement was up to 2.5 times for low Rayleigh number and small chimney sizes. Straatman et al. ( 993 ) carried out a numerical and experimental investigation of free convection in vertical isothermal parallel walled channels, with adiabatic extension of various sizes and shapes. They employed a finite element discretization to solve the fully elliptic form of the governing equation with the inlet boundary conditions based on Jeffrey- Hamel flow. The experiments were performed with ambient air, using a Mach-Zender interferometer. The increase in heat transfer rates varied from 25 times at low Rayleigh number to

2 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 2.5 times at high Rayleigh numbers. The authors proposed a single correlation in terms of channel Rayleigh number and all the geometric parameters, e.g. heated length ratio, expansion ratio. Lee ( 994 ) investigated numerically the effect of the unheated exit section for natural convection in vertical channels with isotherm or isoflux walls. The results were obtained by means of the boundary layer approximation. An important finding was that an unheated exit determines larger total heat transfer and flow rate than an unheated entry. Campo et al. ( 999 ) presented a numerical solution to the wall temperature distribution and the thermal and the fluid dynamic fields in a channel with partially isoflux heated parallel plates. They found a reduction in the maximum wall temperature when an insulated extension was placed downstream of the heated part, the larger the Rayleigh number the less relevant the reduction Fisher et al. ( 997 ) developed analytical solution for a vertical parallel plate isothermal heat sink and chimney system whereas Fisher and Torrance ( 998 ) developed an analytical solution for a pin-fin sink and chimney system. In the former investigation a ridge of maximum total heat transfer was observed with respect to the plate spacing and the heat sink height, and the authors showed that smaller heat sinks can be used together with a chimney without compromising the thermal performance and without increasing the system size. In the latter, the chimney effect was shown to enhance local heat transfer rates in such a way that the minimum temperature rise remains approximately constant while the height of the heat sink relative to the total height is reduced. Bianco et al. ( 998 ) studied experimentally the free convection in vertical isothermal parallel walled channels, with adiabatic extension of various sizes and shapes with the heated part at uniform wall heat flux. They presented a limited investigation in terms of geometric parameters and Rayleigh number. Auletta ( 200 ) studied expermintally the effect of adding adiabatic extensions for a vertical isoflux symmetrically heated channel. They offered the best configurations of their system to avoid the maximum wall temperatures around the heated channels. This study was useful for the present investigation. Shahin and Floryan ( 999 ) analyzed numerically the chimney effect in a system of isothermal multiple vertical channels. Each channel had an adiabatic extension. They claimed that the interaction between multiple channels increases the induced flow rate and that the associated chimney effect is stronger than in a single channel with adiabatic extension. Fisher and Torrance ( 999 ) carried out experiments on air natural convection in a finned vertical parallel plate heater, with an adiabatic downstream extension. The effect of fin spacing and the channel length on the total heat transfer was investigated and their results confirmed prior theoretical predictions. The present research is an applied one. It is based on studying how to enhance the natural convection heat transfer around the vertical heated channels in the core of a typical MTR reactor. In doing so, chimneys were introduced to increase the heights of the vertical channels in the reactor core in order to utilize the chimney effect to do the job. The best system configurations are based on a theoretical analysis which includes all possible factors including heat transfer ones that lead to our conclusions This type of applied thermal hydraulic research in a complicated core of a real nuclear research reactor is not readily available and is needed. The subsequent increase in the reactor power in the natural convection mode is rather important. 2. THE REACTOR CORE DATA The reactor core is the main component concerned with the performance of the neutronic and thermal hydraulic calculations. MTR core is an array of aluminium cladded fuel elements, absorber plates inside guide boxes, double wall core chimney and irradiation boxes. Inside the core chimney there are 30 grid positions with 6 x 5 configurations. It is divided by two zones where two guide boxes (for absorber control plate insertion) are placed. As a result, the core grid is divided into a central area of 3x6 and two lateral areas of x 6 each. General data regarding the present MTR core and its fuel elements is given in Table I.

3 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 3 Table I General Data of ETRR-2 Core Reactor Type Open pool Fuel Material U 3O 8-Al Fuel enrichment (w % 235 U) 9.7 % Fuel elements dimension (cm x cm) 8.0 X 8.0 Shape of Fuel Plates Flat Number of Fuel Plates 9 Active length (cm) 80 Fuel Plate dimension: Thickness(cm) 0.50 Width (cm) 7.5 Fuel Meat dimension: Width (cm) 6.4 Thickness (cm) 0.07 Water channel thickness between two fuel plates(cm) Water channel thickness between two fuel elements (cm) Weight of 235 U (g) in fuel elements: Standard fuel element ~404 g Type one fuel element ~46 g Type two fuel element ~209 g Cladding Material Aluminium Absorber Plates Material Ag-In-Cd Moderator Light water Coolant Light water The physical model considered in the present work is a simple design of a vertical channel with symmetrical heat generation according to the fission of the fuel element. The channel domain consists of entrance section, channel bundle section, and exit section, as shown in Fig.. The channel dimensions are 80 mm length, 2.7 mm width and 800 mm in height. Left Wall Outlet Right Wall X g Direction of Flow Y Inlet Fig.. Drawing of the flow channel and its axes.

4 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: PROBLEM FORMULATION The aim of this paper is to present a numerical analysis of natural convection in single phase water in a symmetrically heated vertical channel, considering the presence of two downstream adiabatic extensions to enhance the chimney effect. In the following, the heated part is indicated as channel and the unheated part as chimney. The computational domain for the heated vertical rectangular channels in the cladding of the fuel assembly is depicted in Fig.2. The domain is made up of a vertical channel with two parallel plates, heated at uniform heat flux q; the height of the channel plates is L h whereas the distance between them is b. On top of the channel, there is a chimney made up of two insulated parallel and vertical plates; their height is (L-L h) and the distance between them is B. An enlarged computational domain has been chosen. It is made up of the geometry described previously and of two reservoirs of height L x and width L y, which are placed upstream the channel and downstream the chimney. The reservoirs are important because they simulate the thermal and fluid dynamic behaviors far away from the inflow and outflow regions. Chim ney B Vertical channel L Lh b x Fig. 2. Computational domain of the problem. y 4. NUMERICAL STUDY The numerical calculations were performed for the velocity and temperature fields inside the chimney and the box. The conservation equations were solved numerically. The governing equations solved by FLUENT ( 204 ) are the Navier-Stokes equations combined with the continuity equation, the thermal transport equation, and constitutive property relationships. Continuity Equation ( 204 ) p t + x (ρv x) + r (ρv r) = S m () Navier Stokes Equation ( 204 ) Conservation of momentum in an inertial (non-accelerating) reference frame (ρv ) +. (ρv v ) = p +. (τ ) + ρg + F t (2) where p is the static pressure, τ is the stress tensor (described below), and ρg and F are the gravitational body force and external body forces (e.g., that arises from the interaction with the dispersed phase), respectively. F also contains other model dependent source terms such as porous-media and userdefined sources. The stress tensor τ is given by τ = μ[( v + v T ) 2. v I] (3) 3 Where µ is the molecular viscosity, I is the unit tensor, and the second term on the right hand side is the effect of volume dilation. For two dimensional axisymmetric geometries, the axial and radial momentum conservation equations are given by and (ρv x) + (rρv r x xv x ) + (rρv r r rv x ) = p x + r [rμ (2 v x 2 (. v ))] + x x 3 r [rμ r ( v x r + v r x )] + F x (4)

5 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 5 (ρv t rx) + (rρv r x xv r ) + (rρv r r rv r ) = p Where r + r x [rμ (2 v r r 2 3 (. v ))] 2μ v r r μ 2 ( v ) + ρ v z r r v x r )] + F r (5). v = v x + v r + v z x r r The tangential momentum equation for 2D forced may be written as: [5] + r x [rμ ( v r (6) x + (ρv z) + (rρv r x xv z ) + (rρv r r rv z ) = [rμ v z ] r x x r 2 r [r3 μ r [v z ]] ρ v rv z r r (7) The boundary conditions for the energy equation are based on the natural convection 2D analysis. The heat flux, corresponding to the input power of, for instance, 00 W, has been imposed on the plate. For a constant heat flux, the wall temperature of the plate is uniform. Therefore, the plate was defined in the simulations exactly as if it was built in reality; it had the core which generates heat, and the external layers defined as conducting walls. The thermal conductivity of aluminum was taken as 80 W/m K. For the other boundaries, FLUENT makes it possible to incorporate the heat transfer coefficients of the walls and the outside temperatures in the calculation of the inside temperature field. Thus, the calculations were performed both for adiabatic walls and for walls with heat-transfer coefficients in the real plate of the reactor. The temperature of the surroundings is imposed at the entrance opening. As for the exit opening, FLUENT ( 204 ) adjusts the boundary condition there, extrapolating the temperature values from the interior grid cells adjacent to the exit. 5. RESULTS AND DISCUSSIONS Results for the parametric analysis are carried out for water, Pr = 0.7, in the Rayleigh number ranges from 0 2 to 0 5 and for a channel aspect ratios of L h/b = 5, 0 and 20. The expansion ratio,, is in the range 4 and the extension ratio, L/L h, ranges from.5 to 4. No local flow separation around the entrance corner was found in all considered cases. The analyzed configuration is applied to a nuclear research reactor core chimney cooling. Typical geometrical dimensions are referred to L h = 0.8 m, with L / L h = 3 m, b is in the range of 5-40 mm and, consequently, B changes from 55 to 0 cm. Heat flux ranges between 3 and 500 W/m 2. Two actual limiting cases are b = 50 cm and a heat flux of about 50 W/m 2 and the corresponding Rayleigh number is 00 and b = 60 cm with the same heat flux distribution and Ra = 0 5. The highest considered heat flux, 500 W/m 2 related to Ra = 0 5, is attained for b equal to about 50 cm. Figure 3 illustrates the velocity contour for various chimney designs of the channel. = =.5 =2 =3

6 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 6 L/Lh= L/Lh=2 L/Lh=3 Fig. 3. Velocity contours. Wall temperature profiles for Ra = 0 2 and 0 5 and for L/L h =.5 and 4 are shown in Figs. 4 7 with L h/b = 5, 0, and 20 and for different expansion ratio values. In all cases the highest value of maximum wall temperature is attained for the simple channel configuration. These profiles allow the evaluation of the different thermal behaviors of the channel chimney system in terms of the channel aspect ratio. In all temperature profiles, the maximum wall temperature is not attained at the channel outlet section but at a slightly lower value of the axial coordinate due to the diffusive effects, according to the experimental results given by Haaland and Sparrow ( 983 ). The value X max of the section at which the maximum wall temperature is attained depends on the geometrical parameters and Ra values. In fact, for the simple channel configuration, the point X max is the lowest among the various configurations for the assigned Ra and L h/b, as shown in Figs. 4 7; this effect is more evident for the lowest Ra, as given in Figs The X max value, for the same channel length, increases with increasing L h/b value because of the decreasing diffusive effects toward the external ambient. Moreover, increasing the Rayleigh number, the X max value increases because of the decreasing diffusive effects, as it is noted from comparing Fig. 4 with Fig. 6 and Fig. 5 with Fig. 7. A sharp decrease of wall temperature in the outlet section zone is present due to also the cold inflow inside the chimney, which reaches the outlet section of the channel. For the lowest Rayleigh number, Ra = 0 2, and L/L h =.5, Fig. 4 indicates that wall temperatures decrease with increasing the expansion ratio up to between 2 and 3 and, for higher values wall temperatures increase again, where in Fig. 4 is the dimensionless temperature, and X the dimensionless distance. Moreover, the decrease in the wall temperature at the outlet region for 3 is lower than that for the simple channel. For = 4, this decrease is almost equal to that for the simple channel due to a cold inflow from the outlet section of the chimney. The cold inflow in the chimney was observed by Haaland and Sparrow ( 983 ) and a fluid stream flows down along the adiabatic extensions. It reaches the horizontal wall of the chimney, mixes with the hot plume-jet and goes out of the channel. A consequence of the cold inflow or down flow is a reduction of chimney effect, which gets stronger with increasing the aspect ratio as indicated in Fig.4 (b) and (c). It is possible to estimate the position along the adiabatic wall of the chimney where the vorticity goes to 0. In general, it is observed that the number of configurations with a complete down flow increases with increasing Ra value whereas the number of configurations with a partial separation from the wall decreases. The separation is present for = 2 only when L/L h is equal to.5 at Ra 0 4 whereas, for Ra = 0 5, only a complete down flow is observed. Some possible guide lines to evaluate critical conditions related to the beginning of flow separation and complete down flow will be provided in Figs In fact, after the optimal conditions, thermal and fluid dynamics trends indicate a worsening of the chimney effect. The difference between the maximum values of the wall temperature for the simple channel and for = 2 increases with increasing L h/b, just as the increasing difference between the maximum values of the wall temperature for the simple channel and for = 4. It is possible to affirm that increasing L h/b allows to enhance the channel chimney system heat transfer with respect to the simple channel, particularly for the configurations with >.5. Thus comparing the maximum wall temperatures for the simple channel and the present suggested channel-chimney system allowed to determine the best configuration for better heat transfer and also to minimize the maximum wall temperatures.

7 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 7 Ra=0 2 L/Lh=.5 Lh/b=5 (a) L h/b = 5 Ra=0 2 L/Lh=.5 Lh/b=0 (b) L h/b = 0

8 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 8 Ra=0 2 L/Lh=.5 Lh/b=20 (c) L h/b = 20 Fig. 4. Heated wall temperatures at Ra = 02 and L/Lh =.5 for different channel aspect ratios. For L/L h = 4, Fig. 5 depicts that the absolute differences strongly increase with respect to the previous case (L/L h =.5) and this shows that the chimney effect is remarkably improved. Moreover, these differences increase with increasing the channel aspect ratio, L h/b. The configuration with = 4 gives the lowest wall temperature values, but it has to be underlined that the decrease of the maximum wall temperature is significant even for =.5, whereas the reduction from the configuration with =.5 to the configuration with = 4 is reasonably lower. In fact, the percentage variations of the maximum wall temperature between the configuration with =.5 and the simple channel, in reference to the value pertinent to the configuration with =.5, is almost 60 %, whereas the variation between the configuration with = 4 and that with =.5 is almost 9 % of that for L h/b = 5. Therefore increasing the channel aspect ratio enhances the thermal behavior of the channel chimney system for both low L/L h and large L/L h values, and for low Rayleigh number values. So, the channel aspect ratio is important in upgrading the channel-chimney system effect, for low Rayleigh numbers Ra=0 2 L/Lh=4 Lh/b= X Simple Channel = =.5 =2 =3 =4 (a) L h/b = 5

9 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: Ra=0 2 L/Lh=4 Lh/b= X Simple Channel = =.5 =2 =3 =4 (b) L h/b = Ra=0 2 L/Lh=4 Lh/b= (c) L h/b = 20 Fig. 5. Heated wall temperatures at Ra = 0 2 and L/L h = 4.0 for different channel aspect ratios. X Simple Channel = =.5 =2 =3 =4 For the largest Rayleigh number value considered, Ra = 0 5, Figs. 6 and 7 reveal that the wall temperatures are lower than those for the configurations pertinent to Ra = 0 2. For L/L h=.5, Fig. 6, illustrates that the configuration with =.5 shows the lowest maximum wall temperature values, whereas the configuration with = 4 has wall temperature values similar to the ones pertinent to the simple channel, for all the analyzed L h/b values. In this configuration, the down flow is already present for = 2. This is due to the larger velocity of the hot jet coming out of the channel, which determines the fluid separation from the adiabatic chimney wall. Also in this case, the L h/b increase produces an enhancement of the channel chimney system with respect to the simple channel, as observed in Fig. 6. Here again, for the largest Rayleigh number values considered the aspect ratio is an important factor in showing the superiority of the channel-chimney system over the simple channel system.

10 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: Ra=0 5 L/Lh=.5 Lh/b= X Simple Channel = =.5 =2 =3 =4 (a) L h/b = Ra=0 5 L/Lh=.5 Lh/b= X Simple Channel = =.5 =2 =3 =4 (b) L h/b = 0

11 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: Ra=0 5 L/Lh=.5 Lh/b= (c) L h/b = 20. Fig. 6. Heated wall temperatures at Ra = 0 5 and L/L h =.5 for different channel aspect ratios. X Simple Channel = =.5 =2 =3 =4 For Ra = 0 5 and L/L h = 4, Fig. 7 shows that the lowest wall temperatures are obtained for = 2. This indicates that, by also increasing the chimney height remarkably, the cold inflow will be present, causing a decrease in the chimney effect. In fact, for L h/b = 5, Fig. 7(a), it is observed that the wall temperature decreases up to = 2 and then it increases again for 3.0. For the highest analyzed L h/b values, Figs. 7 (b) and (c), it is observed that the difference between the wall temperature values for = 2 and the ones for the simple channel increases. An increase in the chimney effect, when the channel aspect ratio increases, for the highest Rayleigh number for all the analyzed L/L h values, is also present. For Ra = 0 5 the cold inflow determines optimal configurations for 2 for the highest extension ratio Ra=0 5 L/Lh=4 Lh/b= X Simple Channel = =.5 =2 =3 =4 (a) L h/b = 5

12 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: Ra=0 5 L/Lh=4 Lh/b= X Simple Channel = =.5 =2 =3 =4 (b) L h/b = Ra=0 5 L/Lh=4 Lh/b= (c) L h/b = 20. Fig. 7. Heated wall temperatures at Ra = 0 5 and L/L h = 4 for different channel aspect ratios. To obtain quantitative values and furnish a better analysis of the thermal behavior of the present system, the values of the ratio θ ωmax /θ ωmax0 ( ratio of the the maximum temperature of the channel chimney system and the one of the simple channel ) as a function of the expansion ratio are reported in Figs. 8 and 9, for L/L h from.5 to 4 where θ ωmax /θ ωmax0 is defined as the maximum normalized temperature in the channel walls. For Ra = 0 2, Fig. 8 indicates that the ratio θ ωmax /θ ωmax0 is less than for all the analyzed configurations. In agreement with the wall temperature profiles, the ratio decreases, attaining a minimum value, and then it increases for L/L h =.5 for all L h/b values, whereas for L h/b = 5 the ratio θ ωmax /θ ωmax0 attains a minimum value as well as for the configuration with L/L h = 2, as observed in Fig. X Simple Channel = =.5 =2 =3 =4 8(a). For other analyzed L/L h values the profile of the ratio θ ωmax /θ ωmax0 does not show a minimum or a maximum value in the considered interval. It is interesting to observe that the difference between the ratio θ ωmax /θ ωmax0 for L/L h =.5 and that for L/L h = 4 increases when the expansion ratio increases, for a fixed value. For L h/b = 0, Fig. 8(b) depicts that the values of the ratio θ ωmax /θ ωmax0 are always lower than the ones corresponding to the configuration with L h/b = 5. Moreover, the differences between the values at L/L h =.5 and at L/L h = 4 still increase and for = the value is about 0.05, whereas it is about for = 4. For L h/b = 20 the values are very close to those for L h/b = 0 and the differences are the same.

13 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: Ra = 0 2 L h /b = max/max L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=3 (a) L h/b = Ra = 0 2 L h /b = 0 max/max L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (b) L h/b = 0

14 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: Ra = 0 2 L h /b = max/max 0 max/max L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (c) L h/b = 20 Fig. 8. Ratio of the maximum wall temperature to the simple channel one vs. for different extension ratio values at Ra = 0 2. At Ra = 0 5, Fig. 9 indicates that the optimal configurations, such as that for which the θ ωmax /θ ωmax0 value is minimum, are those with the expansion ratio value,, between.5 and 2 for all the considered extension ratios. Moreover, for L/L h =.5 and 2, the configuration with = 4 shows a channel thermal behavior equal to the simple channel one for all the analyzed channel aspect ratio values, the θ ωmax /θ ωmax0 ratio being equal to. This is due to the downflow which is present for these configurations. Moreover, the θ ωmax /θ ωmax0 values decrease with increasing L h/b whereas the differences between the values at L/L h =.5 and L/L h = 4 increase. The above results lead to determine the best configurations for the channel-chimney system in order to avoid or rather mitigate the maximum wall temperatures around the heated vertical channels Ra=0 5 L h /b= L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (a) L h/b = 5

15 max/max International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: Ra=0 5 L h /b= L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (b) L h/b =0.05 Ra=0 5 L h /b=20 max/max L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (c) L h/b = 20. Fig. 9. Ratio of the maximum wall temperature to the simple channel one vs. for different extension ratio values at Ra = 0 5. The values of the normalized mass flow rate ratio ψ ψ 0 ( the ratio of mass flow rate of the channel chimney system to that of the simple channel system ), as a function of the expansion ratio are reported in Figs. 0 and, for L/L h ranging from.5 to 4, and for Ra = 0 2 and Ra = 0 5, respectively. The values of the ratio ψ ψ 0 are always greater than, showing that the mass flow rate in the channel chimney system is always greater than that in the simple channel, except for Ra = 0 5 at the lower L/L h values. In fact, for these configurations for = 4, ψ ψ 0 is almost equal to for all L h/b values. For Ra = 0 2, Fig. 0, it is observed that the mass flow rate, pertinent to the channel chimney system, is about two and half times that of the simple channel when 3 for L/L h = 4 and for all values ofl h/b. The differences between the ψ ψ 0 ratios for = for the different analyzed extension ratios are far lower than the same differences for = 4. This means that, for a fixed and low extension ratio, the increase in the expansion ratio produces variations significantly lower than those pertinent to the higher L/L h values. In most configurations with a fixed L/L h value, the maximum values of ψ ψ 0 are present for in the range.5 4.

16 ψ ψ0 ψ ψ International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: Ra=0 5 L h /b= L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (a) L h/b = Ra=0 5 L h /b= L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (b) L h/b = 0

17 ψ ψ0 ψ ψ0 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: Ra=0 5 L h /b= L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (c) L h/b = 20. Fig. 0. Ratio of the dimensionless mass flow rate to the simple channel one vs. for different extension ratio values at Ra = 0 2. For Ra = 0 5,as shown in Fig., the maximum values of the normalized mass flow rate ψ ψ 0 ratio are always obtained for 2 and they depend more significantly on L h/b rather than for the case of Ra = 0 2, especially for L/L h = 3 and 4. These results confirm that the chimney effects are worsened for the channel chimney system when the down flow is present in the chimney and they allow for a quantitative evaluation of the decrease in the mass flow rate. Moreover, comparing the configurations for Ra = 0 2, Fig. 0, with those for Ra = 0 5, Fig., it is observed that, for =, an increase in L/L h leads to a larger increase in ψ ψ 0 ratio for Ra = 0 5 for all the analyzed L h/b values. These results determine the extreme importance of the coolant mass flow rate which must not be allowed to decrease in the reactor core Ra=0 5 L h /b= L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (a) L h/b = 5

18 ψ ψ0 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No: Ra=0 5 L h /b= L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (b) L h/b = 0 ψ ψ Ra=0 5 L h /b=20 L/Lh=.5 L/Lh=2 L/lh=3 L/Lh= (c) L h/b = 20. Fig.. Ratio of the dimensionless mass flow rate to the simple channel one vs. for different extension ratio values at Ra = 0 5. Analogous trends are obtained for the Nu/Nu 0 ratio, where Nu is the average Nusselt number pertinent to the channel chimney system and Nu 0 is the one corresponding to the simple channel. The values of the ratio Nu/Nu 0 as a function of the expansion ratio are reported in Figs. 2 (a) and (b), for L/L h from.5 to 4 and L h/b = 0, for Ra = 0 2 and Ra = 0 5, respectively. The trends and the dependence on L h/b are qualitatively very similar to those shown in Figs. 0 and whereas the differences are more pronounced between the ratios given for Ra = 0 2 as in Fig. 2(a), and that for Ra = 0 5 as in Fig. 2(b). In fact, for Ra = 0 2, Nu/Nu 0 reaches a maximum value of about.8, whereas for Ra = 0 5 the maximum value is slightly higher than.2. This indicates that, for the lowest considered Ra value, the heat transfer enhances more significantly in the channel chimney system, whereas, for the highest considered Ra value, the heat transfer enhancement due to the employment of chimney is larger than 20% with respect to the simple channel. The maximum wall temperature, average Nusselt number and mass flow rate ratio, for smaller L/L h, present their minimum and maximum value, respectively, at =.5 and, for higher value, θ ωmax / θ ωmax0 increases and Nu/Nu 0 and ψ ψ 0 decrease due to

19 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 9 the presence of cold inflow, which determines a decrease of the chimney effect. For L/L h >.5, the cold inflow starts at higher values and, for = 2, θ ωmax /θ ωmax0 attains the minimum value whereas Nu/Nu 0 and ψ ψ 0 present the maximum value..8.6 Ra=0 2 L h /b=0 Nu/Nu L/Lh=.5 L/Lh=2 l/lh=3 L/Lh=4 (a) Ra = Ra=0 5 L h /b=0 Nu/Nu L/Lh=.5 L/Lh=2 L/Lh=3 L/Lh=4 (b) Ra = 0 5. Fig. 2. Ratio of the average Nusselt number to the simple channel one vs. for different extension ratio values and L h/b = 0. This observation is more evident in Fig. 3, where the maximum values of the ratio Nu/Nu0 are founded for different L/L h, Ra and L h/b values. For L h/b = 20, there is always an enhancement of the thermal behavior of the system and the maximum Nu/Nu 0 ratio, (Nu/Nu 0) max, increases when L/L h increases. For Ra = 0 2, with L/L h passing from.5 to 4, the percentage increase of the ratio (Nu/Nu 0) max is about %, whereas for Ra = 0 5 it is only about 2 %.

20 (Nu/Nu 0 ) max International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 20 Ra= L/L h Lh/b=5 lh/b=0 Lh/b=20 (a) Ra = 0 2 (Nu/Nu 0 ) max Ra= L/L h Lh/b=5 Lh/b=0 Lh/b=20 (b) Ra = 0 5. Fig. 3. Maximum values of Nu/Nu 0 vs. L/L h for different L h/b and Ra values. 6. CONCLUSIONS The natural convection flow induced by a localized heat source on the wall of a vertical channel in the core of MTR reactors which uses plate type fuel elements with walls at ambient temperature has been investigated numerically and asymptotically. Numerical solutions have been computed for an infinitely long channel and used to validate the asymptotic scaling for large values of a Rayleigh number based on the channel width. Simplified boundary layer equations have been written on the basis of this scaling. The vertical extent of the flow is found to be finite, and the limiting forms of the solution around the upper and lower ends have been computed. Average Nusselt number, as a function of time, showed minimum and maximum values and oscillations before the steady state according to the temperature profiles. The profiles show that, in terms of Nusselt number, for Ra = 0 2 the worst configuration is B/ b = and the best is for = 4, whereas for Ra = 0 5 the best configuration is = 2 and the worst is for = 4. To conclude increasing the Ra value the optimum value, in terms of Nusselt number, decreases and the worst configuration is obtained at higher value. Temperature wall profiles, as a function of axial coordinate, enables the evaluation of thermal performances of the channel chimney system in terms of maximum wall temperatures for different expansion ratios, as a function of the channel aspect ratio. For the considered Rayleigh number

21 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 2 values, the difference between the highest and the lowest maximum wall temperatures increased with increasing channel aspect ratio. This behavior becomes greater as the extension ratio goes up. These differences decreased significantly for the highest Rayleigh number value. Optimal configurations for assigned L/L h and L h/b were evaluated in terms of corresponding to the minimum value of maximum wall temperatures. The optimal values depend strongly on Ra and L/L h values and slightly on the channel aspect ratio. A more significant increase of maximum average Nusselt number referred to the simple channel value was obtained for the lowest considered Ra value, Ra = 0 2, L h/b = 20 and L/L h = 4 and it was about 80%, whereas for Ra = 0 5 this increase was only about 24% for the same L h/b and L/L h values. This mainly means that the reactor could be operated up to 950 kw, in the free convection regime, instead of only a maximum design value of 400 kw. Increasing the operating power of the reactor in the natural convection mode of operation by about 2.38 times is of extreme importance as far as the reactor safety and operation are concerned in this regime. The present results highlighted the important significant factors to enhance the reactor s free convection heat transfer for the channel-chimney system, such as the Rayleigh and Nusselt numbers, aspect ratio, expansion ratio, and extension ratio. The present results allow to choose the favorite configurations of the suggested channel-chimney system in the core of the typical considered MTR reactor which avoid attaining the maximum wall temperatures around the vertical heated channels in the core and to improve the natural convection heat transfer of the system. Conditions for keeping the coolant mass flow rate in the core within the desired values are lavished. 7. REFERENCES [] Asako Y, Nakamura H, and Faghri M (990). Natural convection in vertical heated tube attached to thermally insulated chimney of a different diameter. ASME J. Heat Transfer. 2: [2] Auletta A, Manca O, Morrone B, Naso V (200). Heat transfer enhancement by the chimney effect in a vertical isoflux channel. Int.. of Heat and Mass Transfer 44: [3] Bejan A, da Silva AK, and Lorente S (2004). Maximal heat transfer density in vertical morphing channels with natural convection. Numer. Heat Transfer. A 45: [4] Bianco N, Manca O, Morrone B, Naso V (998). Experimental analysis of chimney effect for vertical isoflux symmetricaaly heated parallel plates. Proceedings of the Eurotherm Seminar No. 85 on Thermal Management of Electronic Systems. III: [5] Campo A, Manca O, and Morrone B (999). Numerical analysis of partially heated vertical parallel plate in natural convection cooling. Numer. Heat Transfer. Part A 36: [6] Fisher TS, Torrance KE, and Sikka KK (997). Analysis and optimization of a natural draft heat sink system. IEEE Tras. On Component, Packaging Manufacturing Technol. Part A 20: -9. [7] Fisher TS, and Torrance KE (998). Free convection limits for pin fin cooling. ASME J. Heat Transfer. 20: [8] Fisher TS, and Torrance KE (999). Experiments on chimney enhanced free convection. ASME J. Heat Transfer. 2: [9] Gebhart B, Jaluria Y, Mahajan RM, Sammaka B (988). Buoyancy-Induced Flows and Transport. Hemisphere Publ. Corp., New York. [0] Haaland SE, nd Sparrow (983). Solutions for the channel plume and the parallel-walled chimney. Numer. Heat Transfer. 6: [] Kim SJ, nd Lee SW (966). Air Cooling Technology for Electronic Equipment. CRC Press, Boca Raton, FL. [2] Ledezma GA, and Bejan A (977). Optimal geometric arrangement of staggered vertical plates in natural convection. ASME J. Heat Transfer. 9: [3] Lee KT (994). Natural convection in vertical parallel plates with an unheated entry or unheated exit. Numer. Heat Transfer. Part A 25: [4] Manca O, Morrone,B, Nardini S, Naso V (2000). Natural convection in open channels. In Computational Analysis of Convection Heat Transfer, Eds. Suden B, and Comini G, WIT Press, Southampton, UK, pp [5] Oosthuizen PH (984). A numerical study of laminar free convection flow through a vertical open partially heated plane duct. ASME HTD. 32: [6] Shahin GA, and Floryan JM (999). Heat transfer enhancement generated by the chimney effect in systems of vertical channels. ASME J. heat Transfer. 2: [7] Straatman AG, Tarasuk JD, and Floryan JM (993). Heat transfer enhancement from a vertical isothermal channel generated by the chimney effect. ASME J. Heat Transfer. 5: [8] Wirtz RA, and Haag T (985), Effects of an unheated entry on natural convection between heated vertical parallel plates. ASME Paper 85-WA/HT NOMENCLATURE: a thermal diffusivity m 2 /s b channel gap M B chimney gap M g acceleration due to the gravity m/s 2 Gr Grashof number h (x) local convective coefficient W/m 2 k K thermal conductivity W/m 2 k L channel chimney height m L h channel height m L x height of the reservoir m L y width of the reservoir m Nu (x) local Nusselt number Nu average Nusselt number q heat flux w/m 2 Ra Rayleigh number Ra* channel Rayleigh number,

22 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:7 No:0 22 u, v velocity components along x m/s U, V dimensionless components x, y Cartesian coordinates M X, Y dimensionless coordinates, Pr Prandtl number Dimensionless temperature Nu0 Normalized Nusselt number Ψ stream function m 2 /s