The Scaling Analysis and Numerical Simulation of Single-phase Natural Circulation

In this work, scaling analysis and numerical simulation of the single-phase natural circulation loop with water are presented. First, grid-dependency testing and model validation are performed. Then, according to the assumption of the boundary conditions, the effect of the heat input and structural parameters on the heat transfer characteristic is studied, including the boundary condition and temperature of the steam fluid. The results show that the scaling model can accurately express the flow heat transfer characteristics of the original model, and can reach a steady state under different working conditions. The formation of stable natural cycles is the result of coupling the boundary conditions of the heating and cooling segments under the fixed structure reference. The mass flow rate on the steady state decrease with the increase of the external fluid temperature in the cooling section, and as the heat transfer coefficient of the external convection heat transfer coefficient increases.


Introduction
The single-phase natural circulation loop is a system for conveying heat from a heat source to a heat sink without any extraneous driving forces or devices.The heat sink lies in a higher location than the heat source for improving the performance generally.This system is widely used in heat removal systems due to its design simplicity.The applications of this system are employed in many renewable energy regions [1] [2].
Ishii et al. [3] analyzed the similarity law for the circulation flow system and provided the scaling criteria which were deduced from the continuity, integral momentum, and energy equations in 1D form.
The numerical simulation method can satisfy a mass of assumptions for actualizing real applications.The type of loop fluid, the diameter as well as the height of the loop, and the incline are always researched.Ghorbanali and Talebi [4] numerically studied the efficacy of various loop fluids.It was found that the loop using nanofluid showed more outstanding heat transfer ability than pure Water.In addition, Hashemi-Tilehnoee [5] employed the RELAP5/Mod3.2model to research the loop system and provided the compared result with Fluent.The conclusion indicated the case with high heater power was hard to arrive equilibrium state whereas displayed an excellent heat transfer rate.
Though the numerical simulation method is effective to study the single-phase natural circulation loop, experiment always provides fundamental standard.Bello et al. [6] studied the loop geometry structure, increasing temperature along the heater, and different system operating pressure in the vertical heater and horizontal cooler.Misale et al. [7] researched the parallel single-phase loop combined with four loops concluding the parallel loops abide by the same correlations.
In summary, it could be found that although there is a branch of studies about the natural circulation system, the research always focuses on laminar flow in a loop defining the Nusselt number as a constant value.Of courage, many researchers studied the turbulent flow in a single-phase natural circulation loop selecting the gas as loop fluid.The gas can't be used universally.Literature about the water considering turbulent flow is scarce.This paper employs the constant temperature and constant convective heat transfer coefficient as boundary conditions.In addition, this paper proposes to replace a single pipe with a heat exchanger for excellent heat removal.

Physical model and description
Fig. 1 shows the schematic diagram of the system studied and two heat exchangers and two vertical pipes with four elbows comprise it.The specific geometry parameters considered in this paper are shown in Table 1.

2.2Mathematical model Continuity Equation:
Momentum Equation Energy Equation:

Single-phase scaling laws
The dimensionless criterion number of single-phase natural cyclic convection is obtained by the Momentum and Energy Equations: , , , , .
In addition, geometric similarity and similar flow conditions are also required: axial-dimensionless length: radial-dimensionless area: .
Under the constant pressure drop, height, and fluid conditions, the model modeling scale can be obtained according to the prototype scheme.

The mesh independence
Grid independence is to guarantee the reliability of the simulation results and to reduce a precise gird number for economical and effective results.Fig. 2 exhibits the variation of the Reynold number in a horizontal pipe in the model for reaching a constant value.Therefore, the following numerical simulation results are obtained from 320w grid numbers and more than.

The Validation of the scale and numerical model
It is necessary to confirm the accuracy of the simulated models by comparing them with data from the literature.Fig. 3 shows the comparison results of the relationship of velocity with an increasing temperature difference between the cooler and the heater under single-phase flow.It can be found that the velocity in the loop shows good agreement with the prototype model.The figure testifies that the  In this paper, liquid water at a constant phase-based circulation system is investigated.The secondorder upwind method is adopted by the momentum, energy, and turbulence equations.And the (PRESTO) scheme is adopted to discretize the pressure term.RNG k_ε turbulence model is chosen in this model.To assess the precise properties of the single-phase water, REFPROP is used.

The effect of temperature applied on the cooler and the convective heat transfer coefficient out of the heater
As the figure shows, the system keeps a constant temperature at 423K on the heater and a constant coefficient of heat transfer at 500 (Wm-1K-1) on the cooler.It demonstrates that heat removal ability increases with the ambient temperature outside of the cooler and the convective heat transfer coefficient increasing.The constant temperature is used by the heater causing decreasing temperature deviation when the temperature on the cooling section increases, which leads the fluid to get less buoyancy force.The driving force equals the difference between buoyancy force and friction force, which tells us that the higher temperature on the cooler leads to inferior heat transfer capacity.
As Fig. 5 shows, the mass flow rate reduces as the temperature applied to the cooler increases.the mass flow rate drops from 0.494 kg/s to 0.407 kg/s when the convective heat transfer coefficient keeps invariant while the temperature applied on the cooler varies from 333 K to 363 K. What's more.Interestingly, if the ambient temperature on the cooler keeps constant at 323 K, the rate of growth drops NESP-2023 Journal of Physics: Conference Series 2592 (2023) 012039 IOP Publishing doi:10.1088/1742-6596/2592/1/0120395 from 9.67% to 3.11%.The rate of growth drops from 8.19% to 2.95% dedicating the same tendency at 363 K.It illustrates that the effect of the convective heat transfer coefficient is weakening at the same ambient temperature outside of the cooler.The average temperature of the fluid in the heater increases as the convective heat transfer coefficient develops steadily, which leads to the weakening phenomena.The mass flow rate also appears same condition displaying a tiny difference.

The effect of height between the cooler and heater on a mass flow rate of loop
As we mentioned above, the effective height is defined as the vertical distance between the middle of the heater and the cooler.Fig. 6 shows the variation of the mass flow rate with increasing height, meanwhile, the temperature applied on the cooler and the heater are constant at T h =423 K and T c =363 K.The figure depicts that the mass flow rate increases as the height enhances at a constant step and increases as the convective heat transfer develops.Interestingly, the tendency of the mass flow rate to increase gradually wanes.Importantly, the increase rate of mass flow rate fades from 7.76% to 1.08% with the height shift from 4 m to 14 m.With the height developing, the friction factor due to components such as the elbow keeps constant but the friction factor about flow increases, which leads to the decreasing increase rate.It can be noticed that the rate of increase decreases from 4.13% to 1.39% when the boundary conditions are kept invariant at 4 m height.The above conclusion is proved.Fig. 5 The mass flow rate vary with ambient temperature on the cooler (T h =423 K, hout.c=500Wm-1 K-1).

Conclusion
The present analysis researched the scaling method which was applied to a single-phase circulation loop and explored the effect of heat input, height, incline, and diameter of the riser and downcomer on the performance parameters.Some major conclusions are summarized as follows: 1) The scale model could display the same flow and heat transfer characteristics as the prototype model.The computational results show that the model can reach stable behavior.
2) The mass flow rate show decreasing linear relationship as the ambient temperature applied to the cooler increases; that increases with the enhancement of the convective heat transfer coefficient.
3) The height shows an important effect the mass flow rate of the loop increases as the height develops, although it gradually weakens.
4) This provides a theoretical basis for the design and calculation of the cooling tube and the heat tube of the passive residual heat removal system of nuclear power plant containment.
NESP-2023 Journal of Physics: Conference Series 2592 (2023) 012039 IOP Publishing doi:10.1088/1742-6596/2592/1/0120393 scale model could obtain the same result as the prototype model.What's more, the correlation between Vijayan and the deviation line is plotted as a function at Figure4.Simulation results reveal that the Reynolds number of the present study shows a 10 percent deviation from the value calculated using the Vijayan correlation.

Fig. 1 Fig. 2
Fig.1 The geometry model Table 1 Geometrical parament of this paper Name Parament Height of heat exchanger 5 m Height between heater and cooler 10 m Diameter of hot leg 0.044 m Diameter of cool leg 0.032 m The total area of the heat exchanger 3 m 2 Length of loop 2 m

Fig. 3 Fig. 4
Fig.3The comparison of flow rate with increasing temperature difference