Numerical Simulation of Convective Heat Transfer of CO₂ in a Tube Under Supercritical Pressure at Low Reynolds Numbers

Abstract

The supercritical carbon dioxide (S-CO₂) Brayton cycle has the advantages of compact layout, simple structure, high thermal efficiency, clean working quality, its application in lead-cooled fast reactor power conversion system helps the miniaturization and modularization of the whole system. The development of an efficient and compact supercritical CO₂ heat exchanger has important reference significance for improving the thermal efficiency of the system of lead-cooled fast reactor. Supercritical CO₂ can operate in high Reynolds number turbulence and low Reynolds number turbulence in heat exchanger. The convective heat transfer of supercritical pressure CO₂ flowing upward and downward in a vertical circular tube (d = 2 mm) at low inlet Reynolds number (Re_in = 1970) was numerically simulated by different turbulence models to study the effects of variable properties, buoyancy and thermal acceleration on wall temperature and turbulent kinetic energy. The results showed that the heat transfer deterioration and enhancement occurred at the entrance of the heating section during the upward flow, which was mainly attributed to the influence of buoyancy and heat acceleration on the turbulent kinetic energy distribution. The LB turbulence model was used to simulate the heat transfer phenomenon, which was occurred in the downward flow.

1 Introduction

In recent years, researchers at home and abroad have proposed to use supercritical CO₂ Brayton cycle in the power cycle system of the fourth generation reactor (lead-cooled fast reactor) and the energy production of solar energy system [1, 2]. The advantage of this cycle is that the whole cycle will not undergo phase change. Compared with supercritical water, supercritical helium and other circulating refrigerants, supercritical CO₂ has the advantages of high density and large specific heat, which helps to simplify the cycle process and reduce the size of the whole cycle. The development of a high efficiency and compact supercritical CO₂ heat exchanger has important reference significance for improving thermal efficiency of the lead-cooled fast reactor power cycle system.

Supercritical CO₂ can operate in high Reynolds number turbulence and low Reynolds number turbulence in heat exchanger. Domestic and foreign scholars have carried out a lot of experimental research and theoretical analysis on the turbulent flow of supercritical pressure fluid at high Reynolds number [3,4,5,6,7], but there are few studies on the turbulent mixed convection heat transfer in this small structure heat exchanger at low Reynolds number. Therefore, in this paper, the turbulent mixed convective heat transfer of supercritical CO₂ under the condition of low Reynolds number in a tube with an inner diameter of 2 mm is numerically simulated and compared with the experimental results. The effects of variable physical properties, buoyancy and thermal acceleration on flow and heat transfer are analyzed to provide theoretical guidance for the design of supercritical CO₂ heat exchanger.

2 Physical Model and Governing Equation

The physical model and coordinate system are shown in Fig. 1. The origin of the coordinates is the center point of the supercritical pressure CO₂ at inlet section. The axial coordinate x axis is the same as the flow direction, and the radial coordinate r axis is perpendicular to the flow direction and points to the pipe wall. The physical model used for numerical calculation is the same as the material and size of the experimental section in the experiment [8]. The experimental section has an inner diameter of 2.0 mm, an outer diameter of 3.14 mm and a length of 500 mm. In the middle is a heating section that is directly energized, with a length of 290 mm. As the resistivity changes with temperature during the experiment, the local heat flux density is not completely uniform, but its unevenness does not exceed 1%. Therefore, in the process of numerical calculation, it can be regarded as a uniform internal heat source q_v. There are 105 mm long inlet and outlet sections at both ends of the heating section. The material of the experimental section is stainless steel, and the thermal conductivity constant is 16.38 W/m.k. This is a coupled problem of heat conduction and convection. Since the convection heat transfer of supercritical pressure CO₂ in a vertical circular tube is symmetrical, in the process of numerical calculation, half of the circular tube is taken for two-dimensional calculation for convenience. The whole flow is axisymmetric two-dimensional steady flow.

Governing equations in cylindrical coordinates are as follows:

Heat conduction equation:

$$ \frac{1}{r}\frac{\partial }{\partial r}(\lambda r\frac{\partial T}{{\partial r}}) + \frac{1}{x}(\lambda \frac{\partial T}{{\partial x}}) + \dot{\phi } = 0 $$

(1)

Continuity equation:

$$ |\frac{1}{r}\left\{ {\frac{\partial }{\partial x}(prU) + \frac{\partial }{\partial r}(rpV)} \right\} = 0 $$

(2)

Momentum equation in U direction:

$$ \begin{array}{*{20}l} {\frac{1}{r}\left\{ {\frac{\partial }{\partial x}(prU^2 ) + \frac{\partial }{\partial r}(prVU)} \right\} = - \frac{\partial p}{{\partial x}} + pg + } \hfill \\ {\frac{1}{r}\left\{ \begin{gathered} 2\frac{\partial }{\partial x}\left[ {r\mu_e \left( {\frac{\partial U}{{\partial x}}} \right)} \right] + \hfill \\ \frac{\partial }{\partial r}\left[ {r\mu_e \left( {\frac{\partial U}{{\partial r}} + \frac{\partial V}{{\partial x}}} \right)} \right] \hfill \\ \end{gathered} \right\}} \hfill \\ \end{array} $$

(3)

Momentum equation in V direction

$$ \begin{gathered} \frac{1}{r}\left\{ {\frac{\partial }{\partial x}(\rho rUV) + \frac{\partial }{\partial r}(\rho rV^2 )} \right\} = - \frac{\partial p}{{\partial r}} + \hfill \\ \frac{1}{r}\left\{ {\frac{\partial }{\partial x}\left[ {r\mu_e \left( {\frac{\partial U}{{\partial r}} + \frac{\partial V}{{\partial x}}} \right)} \right] + 2\frac{\partial }{\partial r}\left[ {r\mu_e \left( {\frac{\partial V}{{\partial r}}} \right)} \right]} \right\} \hfill \\ - 2\frac{\mu_e V}{{r^2 }} \hfill \\ \end{gathered} $$

(4)

Energy equation:

$$ \frac{1}{r}\left\{ \begin{gathered} \frac{\partial }{\partial x}(\rho C_p rUT) + \hfill \\ \frac{\partial }{\partial r}(\rho C_p rVT) \hfill \\ \end{gathered} \right\} = \frac{1}{r}\left\{ \begin{gathered} \frac{\partial }{\partial x}\left[ {rC_p \left( {\frac{\mu }{\Pr } + \frac{\mu_T }{{\sigma_T }}} \right)\frac{\partial T}{{\partial x}}} \right] + \hfill \\ \frac{\partial }{\partial r}\left[ {rC_p \left( {\frac{\mu }{\Pr } + \frac{\mu_T }{{\sigma_T }}} \right)\frac{\partial T}{{\partial r}}} \right] \hfill \\ \end{gathered} \right\} $$

(5)

The control equations are discretized by the control volume integral method using the ANSYS FLUENT software for numerical calculation. In discretizing the equations, the fluid region and the solid wall region adopt the uniform grid in the axial direction, the non-uniform grid in the radial direction, and the pressure velocity coupling is carried out by the SIMPLEC algorithm. The momentum equation and the energy equation first adopt the first-order upwind scheme, and then change to the second-order upwind scheme after reaching convergence, and then iterate until convergence. For the numerical turbulence models, LB, LS and RNG, standard and realizable turbulence models with enhanced wall function method are used in this paper. Due to the drastic changes in physical properties, relaxation factors in the range of 0.1–0.3 are used for all independent variables. In order to ensure the grid independence, a relatively precise grid is divided near the wall during the calculation process. Through adaptive grid adjustment, the condition y +  < 0.6 is satisfied to obtain an approximate grid independent solution. When all variables meet the following criteria, the numerical solution is considered to be convergent.

$$ \left| {\left( {\phi^{i + 1} - \varphi^i } \right)/\phi^i } \right| \le 10^{ - 6} \quad \phi ,U,V,T,\varepsilon $$

Fig. 1.

Physical model and coordinate system

Figure 2 shows the physical properties change of supercritical CO₂ at the pressure of 8.8 MPa. Because the pressure difference between the inlet and outlet of the vertical circular pipe is very small, the change of CO₂ physical properties with pressure has little impact on the results. Therefore, the physical properties of CO₂ in the circular pipe are selected as the corresponding physical properties under different inlet pressures, and the change of physical properties with temperature is processed by piecewise linear interpolation. After verification, the maximum deviation between the processing method and the physical property value calculated by NIST is not more than 1%.

The boundary conditions of numerical calculation are selected according to the boundary conditions of experimental conditions. The boundary conditions at the inlet are the velocity inlet boundary conditions of uniform incoming flow. At the same time, given the inlet temperature, the boundary conditions at the outlet are the pressure outlet boundary conditions. The heating section of the experimental section is a solid wall with internal heat source, and the boundaries of the rest are adiabatic boundary conditions. Coupled solution of heat conduction in solid wall and convection heat transfer in fluid region.

Fig. 2.

CO₂ Physical properties at the pressure of 8.8 MPa

3 Effects of Buoyancy and Thermal Acceleration on Flow and Heat Transfer

In this paper, Bo* number proposed by Jackson hall [5] is used to evaluate the effect of buoyancy on flow and heat transfer. The experimental conditions taken in this paper are calculated as Bo* > 8 × 10⁻⁶, indicating that the buoyancy force has a great influence and are located in the area where the turbulent kinetic energy is enhanced. See literature [9]. The effect of thermal acceleration on flow and heat transfer is evaluated by the k_V number proposed by McElicott [10]. Murphy [11] believes that when Kv > 9.5 × 10⁻⁷, the turbulent kinetic energy is restrained and the heat transfer is deteriorated due to thermal acceleration. Through calculation, the experimental conditions calculated in this paper are greater than this value at the entrance of the heating section, indicating that the thermal acceleration has a great impact.

4 Calculation Results and Analysis

For the convenience of verification, the conditions of numerical calculation are exactly the same as those of experiment. The inlet pressure is 8.8 MPa, the inlet temperature is 25 ℃, the mass flow is 0.77 kg/h, and the inlet Re is about 1970.

Figure 3 shows the comparison between the numerical calculated outside wall temperature and the experimental results for upward flow under three heat flux conditions. Different turbulence models are used in the calculation. It can be seen from the figure that local wall temperature peaks and valleys occur at the inlet of the pipe when flowing upward, that is, local heat transfer deterioration and heat transfer enhancement occur, and this phenomenon becomes more obvious with the increase of heat flux. The heat transfer enhancement and heat transfer deterioration at the inlet of the tube are mainly attributed to the changes of thermal physical properties and turbulent kinetic energy caused by buoyancy and thermal acceleration. The later part of this paper will be explained in detail as shown in Figs. 5 and 6. The maximum deviation between the experimental results and the calculated results is 5.6%, 15%, 10% respectively.

The LB turbulence model with low Reynolds number can simulate this kind of heat transfer phenomenon well. At the same time, the wall temperature calculated by the turbulence model at the inlet of the pipe is in good agreement with the experimental results, indicating that the flow at the inlet has been in a turbulent state.

Figure 4 shows the comparison between the numerical calculated outer wall temperature and the experimental results when flowing downward under similar conditions. Due to the limitation of convergence, RNG, realizable and standard turbulence models are used in the calculation. It can be seen that the calculated results using RNG turbulence model are in good agreement with the experimental results when the heat flux is low. While the calculated results using realizable and standard turbulence models are in good agreement with the experimental results at the rear half of the pipe when the heat flux is high. It may be that the choice of turbulence model is related to heat flux which needs to be further studied. The wall temperature rises continuously along the flow direction, and there is no abnormal distribution phenomenon in the upward flow. The maximum deviation between the experimental results and the calculated results is less then 2%.

Fig. 3.

Comparison of calculated and experimental outside wall temperatures on upward flow. q_w, W/m²: (a)—7 926; (b)—13182; (c)—19933 ■—Experimental values; Solid line—LB; Dashed line—RNG; Dot line—LS

Fig. 4.

Comparison of calculated and experimental outside wall temperatures on downward flow. q_w, W/m²: (a)—7 832; (b)—13065; (c)—19878. ■—Experimental values; Solid line—RNG; Dashed line—Standard; Dot line—Realizable

Figure 5 shows the comparison of turbulent kinetic energy at r/R = 0.9 calculated by LB model with and without buoyancy under the three heat flux conditions on upward flow and downward flow. It can be seen from the figure that: (1) the turbulent kinetic energy of upward flow and downward flow with buoyancy considered is higher than that without buoyancy considered, indicating that buoyancy enhances the turbulent kinetic energy of upward flow and downward flow, which is consistent with the experimental results; (2) The turbulent kinetic energy of downward flow is significantly higher than that of upward flow (except for local positions), indicating that the buoyancy force has a stronger effect on heat transfer enhancement of downward flow than upward flow; (3) When flowing upward, the turbulent kinetic energy begins to decrease to 0 at the inlet of the tube for a short distance, and the heat transfer deteriorates. This may be because the physical property change and thermal acceleration lead to the weakening effect of the turbulent kinetic energy being greater than the enhancement of the turbulent kinetic energy caused by the buoyancy, which will be discussed further below. Later, due to the physical property change and the weakening of the thermal acceleration effect, the buoyancy force changed the flow from laminar flow to turbulent flow again. At a certain position of the pipe, the turbulent kinetic energy increased sharply, and the corresponding wall temperature appeared a valley as shown in Fig. 5, that is, the heat transfer appeared local enhancement; (5) The turbulent kinetic energy in downward flow rises continuously along the path, and there is no abnormal change in upward flow. This may be because the effect of buoyancy on the enhancement of turbulent kinetic energy is stronger than that caused by variable physical properties and thermal acceleration.

Fig. 5.

The change of turbulent kinetic energy under the action of upward, downward and no gravity (R/R = 0.9). q_w, W/m²: (a)—7 832; (b)—13065; (c)—19878 Solid line—LB(upward); Dashed line—LB(g = 0); Dot line—LB(downward)

Figure 6 shows the axial distribution of turbulent kinetic energy at the radial position r/R = 0.9 obtained by considering the density change and all physical parameters without considering the buoyancy force. LB turbulence model is used in the calculation. It can be seen that at the inlet of the pipe, the change trend of turbulent kinetic energy obtained by considering only the change of density and all physical parameters basically coincides, and both decrease to 0, That is, laminar fluidization occurs in turbulence. We know that the thermal acceleration is mainly caused by the axial fluid density difference, which can further explain that the heat transfer deterioration at the pipe inlet in the upward flow is mainly caused by the thermal acceleration, which is also in good agreement with the experimental results.

Fig. 6.

Variation of turbulent kinetic energy without buoyancy (r/R = 0.9) q_w, W/m²: (a)—7926; (b)—13182; (c)—19933 Solid line—Consider only changes in density; Dashed line—Change of all physical parameters

5 Conclusions

• (1) LB turbulence model can better simulate the local wall temperature peaks and valleys of upward flow at low Reynolds number, while RNG turbulence model can better simulate the wall temperature of downward flow.

• (2) The enhancement effect of buoyancy on downward flow heat transfer is greater than that of upward flow.

• (3) The local heat transfer deterioration and enhancement in the upward flow and the heat transfer enhancement in the downward flow are mainly due to the influence of buoyancy and thermal acceleration on the turbulent kinetic energy.

• (4) For the design of supercritical CO₂ heat exchanger of lead-cooled fast reactor, the subsequent research on convective heat transfer under higher pressure and temperature will be carried out in the future.