Gas-cooled nuclear reactor core shaping using heat exchange intensifiers *

The need to shape reactor cores in terms of coolant flow distributions arises due to the requirements for temperature fields in the core elements (Safety guide No. NS-G-1.12. 2005, IAEA nuclear energy series No. NP-T-2.9. 2014, Specific safety requirements No. SSR-2/1 (Rev.1) 2014). However, any reactor core shaping inevitably leads to an increase in the core pressure drop and power consumption to ensure the primary coolant circulation. This naturally makes it necessary to select a shaping principle (condition) and install heat exchange intensifiers to meet the safety requirements at the lowest power consumption for the coolant pumping. The result of shaping a nuclear reactor core with identical cooling channels can be predicted at a quality level without detailed calculations. Therefore, it is not normally difficult to select a shaping principle in this case, and detailed calculations are required only where local heat exchange intensifiers are installed. The situation is different if a core has cooling channels of different geometries. In this case, it will be unavoidable to make a detailed calculation of the effects of shaping and heat transfer intensifiers on changes in temperature fields. The aim of this paper is to determine changes in the maximum wall temperatures in cooling channels of high-temperature gas-cooled reactors using the combined effects of shaped coolant mass flows and heat exchange intensifiers installed into the channels. Various shaping conditions have been considered. The authors present the calculated dependences and the procedure for determining the thermal coolant parameters and maximum temperatures of heat exchange surface walls in a system of parallel cooling channels. Variant calculations of the GT-MHR core (NRC project No. 716 2002, Vasyaev et al. 2001, Neylan et al. 1994) with cooling channels of different diameters were carried out. Distributions of coolant flows and temperatures in cooling channels under various shaping conditions were determined using local resistances and heat exchange intensifiers. Preferred options were identified that provide the lowest maximum wall temperature of the most heat-stressed channel at the lowest core pressure drop. The calculation procedure was verified by direct comparison of the results calculated by the proposed algorithm with the CFD simulation results (ANSYS Fluent User’s Guide 2016, ANSYS Fluent. Customization Manual 2016, ANSYS Fluent. Theory Guide 2016, Shaw1992, Anderson et al. 2009, Petrila and Trif 2005, Mohammadi and Pironneau 1994).


Introduction
One of the major problems in implementing gas-cooled nuclear reactors is the high core thermal intensity due to the need to achieve a high coolant gas temperature, which, respectively, results in high cooling channel wall temperatures.Because of the uneven heat generation in the core, the maximum wall temperatures in different channels may vary.At given total coolant flow and average core outlet temperature, the maximum wall temperature in the most heat-stressed channel group can be decreased if the coolant flow through the channels of this group is increased by shaping the mass flows.The other option is to install heat exchange intensifiers into the channels of this group while maintaining the coolant flow.It is obvious that, in any option of changing the core hydraulic (aerodynamic) characteristics, the total core pressure drop increases.When implementing measures to reduce the maximum wall temperature in the cooling channels, the best option would be when an acceptable wall temperature in all the channels is achieved with a minimum core pressure drop.
The influence of the coolant mass flow shaping in a gas-cooled reactor on the main gas/temperature parameters of the cooling channel walls was studied in (Podgorny and Kuzevanov 2017).The effect of installed heat exchange intensifiers on changes in the maximum channel wall temperature was analyzed in (Kuzevanov and Podgorny 2017).
The paper focuses on the effect of changes in the maximum wall temperatures from the core coolant mass flow shaping using heat exchange intensifiers.Various shaping conditions were considered with equal (1) heating or increments of enthalpies in the cooling channels, (2) coolant mass flows in the channels, and (3) maximum wall temperatures in the cooling channels.
As in (Podgorny and Kuzevanov 2017, Kuzevanov and Podgorny 2017), the GT-MHR design (NRC project No. 716 2002, Vasyaev et al. 2001, Neylan et al. 1994) is examined.In the theoretical and computational analysis, the following basic system parameters were assumed to be unchanged: the total gas coolant mass flow G 0 , the average coolant temperature at the core outlet Т av , the reactor thermal power Q 0 , and the coolant temperature at the core inlet T 0 .

Main calculated dependences
The equations for calculating the coolant mass flow distribution by groups of identical cooling channels were obtained in (Podgorny and Kuzevanov 2017).In this case, channels of the same group have the same diameter and heat load, with a total of m groups of identical channels.
The relation between the total coolant pressure loss ΔP (a component of the pressure difference resulting from a temperature change can be neglected due to its insignifi-cance), when passing from the core inlet to its outlet, and the total coolant mass flow G 0 looks like this: (1) where n i is the number of identical channels in the i-group, 1 ≤ i ≤ m; ξ i and ξ l.i are the friction coefficients and local resistance coefficients reduced to the average coolant thermal parameters, respectively; l is the core height, m; d i are the diameters of round channels, m; ρ i is the average coolant density, kg/m 3 .
The k p coefficients for an arbitrary p-group of channels are defined by the dependence and the friction coefficients for technically smooth round channels are determined by the Blasius dependence ξ p = 0.316 / Re p 0.25 . (3) The coolant mass flow G p in a single channel of an arbitrary group is calculated by the equation With a known flow G p under the conditions of constant specific isobaric heat capacity of helium c p = const, it is easy to obtain a ratio for calculating the coolant temperature at the outlet (the «out» superscript) of a p-group channel: (5) where ν p is the heat load deviation of the p-channel group from the average load.
The thermophysical properties of the coolant gas (helium) necessary for carrying out calculations -the average density ρ av p and dynamic viscosity coefficient μ av p -are found by the ideal gas low equation and Sutherland equation, respectively: where P 0 is the coolant pressure at the core inlet, Pa; ρ 0 is the coolant density at the core inlet; R He is the individual gas constant for helium, J/kg•K; T p is the outlet temperature of an arbitrary p-channel, K; µ 0 * is the control viscosity at temperature Τ 0 * (273 K), Pa•s; S is Sutherland constant or effective gas temperature (reference value), K; µ 0 is the viscosity at the core inlet temperature and pressure, Pa•s.
The accepted shaping condition means specifying the relationship of at least two coolant parameters.For sha-ping under the conditions of the same mass flows in the channels and the same coolant heating in them, the functions of the relationship between the coolant flow and its heating in any channel are trivial.For shaping under the condition of the same maximum wall temperatures in the cooling channels, the following equation for the relationship between the channel coolant flow and its maximum wall temperature Θ wall max was obtained in (Podgorny and Kuzevanov 2017)   (8)   (9) Relation ( 8) is obtained using the dependence for the heat transfer coefficient proposed by B.S. Petukhov (Petukhov and Kirillov 1958); ε(Pr) and k are the parameters of Petukhov's formula for the heat transfer coefficient.In ( 9), χ is the fraction of thermal power released directly in fuel elements; ΔT av is the average balanced value of the coolant heating in the core, K.
It is obvious that, with the same channel coolant flow and the heat load maintained, the maximum channel wall temperature depends on whether the channel is smooth or has an extended area with specific features of the heat exchange surface (i.e., heat exchange intensifier).The dependence for a gaseous coolant, reflecting the effect of a heat exchange intensifier installed into a smooth channel on changes in the maximum wall temperature, was obtained and verified in (Kuzevanov and Podgorny 2017).It is written as is the coolant temperature at the inlet of the area of the installed extended heat exchange intensifier, K; δT h.c = T h.c -T in int ; T h.c is the coolant temperature in cross section with the maximum channel wall temperature, K; the "0" subscript refers to the option of a technically smooth channel.
In Dependence (10), γ reflects the intensifier's effect: where H 0 is the core height equal to the cooling channel length, m; l int is the heat exchange intensifier length, m; ΔP 0 is the pressure difference in a technically smooth channel, Pa; ΔP is the pressure difference in a channel with a heat transfer intensifier at the same channel coolant flow, Pa.

Specific features of calculated dependences
The coolant flows are usually redistributed through the cooling channels by individual local resistances installed at the inlet to the core channels.This shaping arrangement was considered in (Podgorny and Kuzevanov 2017).
The use of heat exchange intensifiers with uniquely shaped coolant mass flows in the core at given local resistances ξ м.i for each identical i-group channel implies implementing the following obvious relations: where ξ int,i is the aerodynamic drag coefficient of a single i-group channel if a heat exchange intensifier is installed in it; ξ * l.i are the modified local resistances for a single i-group channel with a heat exchange intensifier.
The uniquely shaped mass flows are determined by the condition or by the condition ΔT p = ΔT av = idem, when These flows do not depend on whether there are heat exchange intensifiers in the cooling channels.
For the mass flow shaping options, it is possible to calculate a decrease in the maximum wall temperatures in different cooling channel groups using Equation (10) if the aerodynamic characteristics of heat exchange intensifiers used for shaping are known.These characteristics can be represented as the γ parameter from Equation ( 11).
The GT-MHR core consists of fuel assemblies (FA) made of grade H-451graphite (Engle 1977, Engle andJohnson 1976) in the form of perforated prismatic hexagonal elements.The core height and each cooling channel length correspond to the height of ten fuel assemblies.It is obvious that heat exchange intensifiers should be installed in fuel assemblies with the maximum cooling channel wall temperature.For the GT-MHR core, it is the eighth assembly along the coolant flow.It is also possible to install a heat exchange intensifier of greater length in several series-connected fuel assemblies, however, as shown in (Kuzevanov and Podgorny 2017), an increase in the intensifier length over 0.1H 0 has little effect on the maximum channel wall temperature caused by this intensifier, but leads to an increase in the pressure difference.That is why, in analyzing the heat exchange intensifier effect, when the shaping condition is formulated as ΔT p °= idem, G p = idem or Θ wallp max = idem, we consider the option of installing intensifiers in channels only along the length of assemblies with the maximum channel wall temperatures.
In contrast to the shaping conditions G p = idem and ΔT p = idem, the local coolant mass flow distribution under the condition that Θ wallp max = idem depends on whether or not there are heat exchange intensifiers in the channels.In this case, the basic calculational equation ( 8) for this condition of shaping the coolant mass flows will be where Note that it does not specify the features of the heat exchange intensifier, in particular, its design.There are numerous options.We assume that the integral characteristics of the intensifier used, from which the γ parameter can be calculated, are known.We also take into account the fact that the range of γ values is limited: The results presented in (Kuzevanov and Podgorny 2017) make it possible to estimate the maximum value γ max ≈ 2.5 from the condition of the technical characteristics of the intensifiers, which determine their aerodynamic characteristics.All graphs are also constructed for the values γ < γ max , i.e., for all possible γ values.In this case, the γ limitation is due only to the minimum temperature difference between the maximum wall temperature and the coolant temperature.

Calculation results
The calculations of the GT-MHR core took into account the following design features.
The cylindrical ring-shaped core consists of perforated graphite hexagonal blocks (fuel assemblies) with cooling and fuel channels.The core thickness corresponds to the size of three rows of fuel assemblies; therefore, two peripheral (outer and inner) and central areas of the core are distinguished, each of which consists of one row of fuel assemblies located circumferentially and having their own specific heat load.The cooling channels are represented by two groups: (1) with an inner diameter of 15.88 mm and (2) with a diameter of 12.7 mm.The length of each channel is equal to the core height H 0 = 7.93 m.The core thermal power is 600 MW.The total helium mass flow is G 0 = 320 kg/s, its temperature at the core inlet T 0 = 491 °C (NRC project No. 716 2002, Vasyaev et al. 2001, Neylan et al. 1994).As in (Podgorny and Kuzevanov 2017), the cosine power shape was adopted in the core height and thickness.
With the above core specification, there will be four identical channel groups in the reactor: two groups of channels in the central area with diameters of 15.88 and 12.7 mm, respectively, and two groups of analogous channels in the peripheral area.
The analysis of the influence of heat exchange intensifiers involved in shaping the coolant mass flows on changes in the maximum channel wall temperature is made by comparison with the results obtained, when the shaping conditions were implemented by installing local resistances at the inlet of channels, and presented in (Podgorny and Kuzevanov 2017).
Table 1 shows the maximum wall temperatures in groups of identical channels at the nominal reactor power with cores that differ only in the coolant mass flow distribution through the channels.
The arrangement of shaping the coolant mass flows for any shaping condition leads to an increased core pressure drop.Table 2 shows the "price" for the result achieved under different shaping conditions, presented as the ratio ΔP/ΔP 0 , where ΔP 0 is the core pressure drop without shaping, and ΔP is the core pressure drop with arranged coolant mass flow shaping.
Figures 1-3 show the calculated changes in the maximum wall temperature of different cooling channel groups, depending on the integral characteristic of heat exchange intensifiers installed in them.
Figure 1 shows the temperature changes in the reactor core under study, where the coolant mass flow is shaped under the condition that ΔT i = idem and heat exchange intensifiers are installed in the channels of three groups with corresponding reductions in the local resistances at the inlet of these channels.
Figure 2 shows the changes in the maximum wall temperatures in different cooling channels groups when the heat exchange intensifiers partially replace the local resistances installed for shaping the coolant mass flow in the core under the condition that G i = idem.
Figure 3 continues to show the changes in the maximum cooling channel wall temperatures when heat exchange intensifiers are used, which together with local resistances ensure that the shaping condition is met.Curves 1 and 2 in Fig. 3 show that installing a heat exchange intensifier into the core shaped under the condition that Θ walli max = idem is more efficient than installing intensifiers into the core channels shaped under the condition that ΔT i = idem only to Boundary 3 (or γ°= 2.247).In addition, after Boundary 4 (or γ = 2.446), the core pressure drop shaped under the condition that Θ walli max = idem becomes greater than the core pressure drop shaped under the condition that ΔT i = idem.
In order to verify the adequacy of the method for analyzing changes in the temperature regime of the reactor core with the combined use of local resistances and heat exchange intensifiers to ensure shaping the coolant mass flow through the core cooling channels, the calculation results were compared with the CFD simulation results (Fig. 4).
Simulations were made for single cooling channels of all the groups with given (calculated) coolant flow/heat load values.The intensifier used corresponded to the design presented in (Kuzevanov and Podgorny 2017).

Conclusion
The paper has demonstrated the capabilities of the method for calculating temperature changes in the coolant and cooling channel walls during the channel-by-channel coolant mass flow shaping using heat exchange intensi-fiers in the reactor core.For the investigated GT-MHR core, there are two options that can reduce the maximum cooling channel wall temperature.The first option, when there is a strict limitation on the core pressure drop, is to shape the coolant mass flows under the condition that Θ walli max = idem.In this case, the use of heat exchange intensifiers slightly changes the maximum channel wall temperature and is impractical at γ ≥ 2.247.The second option, when there are no principal obstacles to increasing the core pressure drop, is shaping under the condition that ΔT i = idem with the combined use of local resistances and heat exchange intensifiers.Moreover, the second option is definitely preferable for the integral characteristic of heat exchange intensifiers (γ°>°2.247).
The calculation procedure was verified by direct comparison of the results calculated by the proposed algorithm with the results of a detailed gas flow simulation in heated channels with a specific heat exchange intensifier design.

Figure 1 .
Figure 1.Dependence Θ wall max = f (γ) for identical channels of different groups with ΔT i = idem: 1 -channels with ϕ = 15.88 mm, of the peripheral core area; 2 -channels with ϕ = 12.7 mm, of the peripheral core area; 3 -channels with ϕ = 15.88 mm, of the central core area; 4 -the maximum temperature of the channel walls with ϕ = 12.7 mm, of the central core area.

Figure 2 .
Figure 2. Dependence Θ wall max = f (γ) for identical channels of different groups with G i = idem: 1 channels with ϕ = 15.88 mm, of the peripheral core area; 2 -channels with ϕ = 12.7 mm, of the peripheral core area; 3 -channels with ϕ = 15.88 mm, of the central core area; 4 -the maximum temperature of the channel walls with ϕ = 12.7 mm, of the central core area.

Figure 3 .
Figure 3. Dependence Θ wall max = f(γ)for the core channel with the maximum Θ walli max : 1 -the maximum wall temperature of identical channels with ϕ = 15.88 mm, of the central core area, when shaped under the condition that ΔT i = idem; 2 -the maximum wall temperature for any channel when shaped under the condition that Θ walli max = idem; 3, 4 -efficient shaping condition boundaries.

Table 1 .
Maximum cooling channel wall temperatures depending on the coolant mass flow distribution.

Table 2 .
The ΔP/ΔP 0 value for the mass flow shaping options.