Elsevier

Chemical Engineering Science

Volume 153, 22 October 2016, Pages 411-433
Chemical Engineering Science

The influence of the wall thermal inertia over a single-phase natural convection loop with internally heated fluids

https://doi.org/10.1016/j.ces.2016.06.060Get rights and content

Highlights

  • Analytical/numerical modelling of piping-material influence on natural circulation dynamics.

  • Piping-material properties affect the stability behaviour of a natural circulation loop (NCL).

  • Frequency analysis highlights the effects induced by different piping materials.

  • 1D simulations are suitable to catch the stable/unstable flow regime of NCLs.

  • For 1D systems an all-external heat flux can mime the distributed internal heat generation.

Abstract

This paper deals with the influence of the piping material thermal and geometrical properties on the dynamic stability of single-phase natural circulation loops. To this purpose, a semi-analytical approach is developed by adopting the tools provided by the linear analysis. By considering a generic natural circulation loop configuration with a localized heat flux and a homogenously distributed Internal Heat Generation (IHG), the governing equations (mass, momentum and energy balance) are linearized around a steady-state solution of the system and treated by means of the Fourier transform to obtain dimensionless stability maps. Moreover, in order to verify the linear analysis methodology, a numerical strategy is adopted to solve the nonlinear governing equations and to investigate the natural circulation dynamics in the time domain. In principle, both the developed approaches can be applied to any natural circulation loop configuration. In the present work, the linear and the nonlinear analyses are applied to a specific natural circulation loop geometry, namely the Horizontal Heater Horizontal Cooler (HHHC) one. In this regard, an Object-Oriented (O-O) one-dimensional model of the HHHC loop is developed. For the assessment of the O-O model, the obtained results are compared with RELAP5 and Computational-Fluid-Dynamics (CFD) time-dependent simulations.

Introduction

In presence of density gradients induced by temperature differences, convective motions can arise in a fluid due to the buoyancy force. Systems which are able to adopt this kind of flows in order to transfer heat between a hot source and a cold sink are known as natural circulation loops. Although forced convection can be a more efficient cooling strategy, natural convection does not require any active component and thus it can be used for high reliability engineering applications. In this regard, the high-level safety requirements (which are being even more stringent after the Fukushima accident) needed in the nuclear industry have demanded research on emergency systems based on natural convection. As for the drawbacks of natural circulation systems, they can be subject to dynamic oscillations of both velocity and temperature fields that may compromise their functionality (for example, local temperature increases can occur and provoke damages to the piping materials). The dynamic behaviour of natural circulation has been investigated both from theoretical and experimental point of view. A brief and recent summary can be found in Pini et al. (2016). The first studies on natural circulation were carried out by Keller (1966) and Welander (1967). More recently, Chen (1985), Vijayan et al., 1995, Vijayan et al., 2007, Vijayan (2002) and Swapnalee and Vijayan (2011) have investigated the influence of the loop geometry on natural circulation instabilities, while Pini et al., 2014, Pini et al., 2016 and Ruiz et al. (2015) have studied the effect of a distributed Internal Heat Generation (IHG) in the working fluid on account of the renewed interest in the circulating-fuel Molten Salts Reactors (MSRs),1 in the framework of the Generation IV Project (GIF, 2014, SAMOFAR, 2015). The previously cited works share the limit to consider only the fluid region of natural circulation loops, while they neglect the effects induced by the piping material which can deeply alter the dynamic behaviour of the system during operational transients.

The influence of the pipe-wall material has been studied by Misale et al., 2000, Misale et al., 2005, Misale and Garibaldi (2010), Jiang and Shiji (2002), and Rao et al. (2012), considering either the specific heat or the thermal conductivity.

In Misale et al. (2000), a 2D numerical analysis, which takes into account the piping material specific heat, is applied to a rectangular natural circulation loop, observing that the thermal capacity of the wall influences the mass flux estimation (which is greater if the pipe walls are neglected). From an experimental point of view, the effect of the piping material thermal inertia is investigated in Misale et al. (2005), Misale and Garibaldi (2010). AISI-304 and Plexiglas pipes are tested. In the first case, the analysed loop shows an almost stable behaviour, while in the second one a more unstable dynamics is observed.

In Jiang and Shiji (2002), a theoretical model, compared with experimental data, is developed for toroidal loops considering the effect of the thermal conductivity of the wall. It highlights that in a loop made of better thermal conducting pipe material, the flow is more stable.

In Rao et al. (2012), the effect of the wall thermal inertia of the heat exchanger sections is studied by means of a numerical approach for a rectangular loop. It is observed that piping material thermal capacity widens significantly the stable operating zone of a natural circulation loop.

In this paper, a complete semi-analytical stability analysis, which takes into account the effect of the piping material properties is developed considering a generic natural circulation loop configuration. In particular, the influence of density, specific heat and thermal conductivity (in the radial direction) of the pipes is studied not only for the conventional natural circulation case (i.e., without IHG), but also when an internal and distributed power source is present inside the system (this occurrence being of interest for MSRs). To verify the results of the linear analysis, a nonlinear Object-Oriented (O-O) numerical model is adopted. In addition, RELAP5 and CFD simulations are presented in order to assess the O-O model. All the developed methodologies can be adopted for any rectangular natural circulation loop geometry and for any piping material. In this work, they are applied to the specific Horizontal Heater Horizontal Cooler (HHHC) configuration considering aluminium, copper and AISI-316 pipes. As a matter of fact, the HHHC layout is very interesting from the scientific point of view since it presents large unstable operating regions (Pini et al., 2016). Generally speaking, the presented models highlight that the dynamics of natural circulation is affected by the presence of the piping materials, which cannot be neglected in order to correctly evaluate the system stability behaviour.

The paper is organized as follows. In Section 2, a generic natural circulation loop is described and the governing equations are presented. Section 3 deals with the techniques used for the linear and nonlinear stability analysis. In Section 4, the linear and nonlinear approaches are applied to the HHHC (Horizontal Heater Horizontal Cooler) loop configuration. In Section 5, the HHHC loop O-O model adopted both for the nonlinear and the transient analysis is presented, then RELAP5 and CFD models are introduced for the assessment of the O-O one. In Section 6, the stability maps and the time dependent simulations are presented, and the results of the frequency analysis are discussed as well. In Section 7, the main conclusions are drawn.

Section snippets

System description and governing equations

The thermal-hydraulic system analysed in this work is a vertical rectangular loop with circular pipes (Fig. 1). Two heat sources are taken into account. The first one is a localized external heating zone (hereinafter called heater), and is treated as a heat flux source q″. The second one represents the heat generation inside the fluid (e.g., a chemical reagent or the circulating fuel in MSRs), and is modelled as a homogeneous distributed volumetric heat source q′′′.

A single cooling section

Linear dimensionless governing equations

In this section, the dynamic behaviour of a generic natural circulation loop is evaluated by linearizing the system around a steady-state condition. This method of analysis is general and valid for any loop configuration. The system mass flux (G) and temperatures (Tf,Twi,Two) are perturbed around a stationary solution (G0,Tf,0,Twi,0,Two,0) under the assumptions that the perturbations are small with respect to the stationary values, namely:G(t)G0+δG(t)withδG(t)G0tR+,Tf(s,t)Tf,0(s)+δTf(s,t)

Horizontal Heater Horizontal Cooler configuration (HHHC)

While the previous Sections deal with a generic natural circulation loop geometry, this Section refers to the specific HHHC layout. In light of previous works (Pini et al., 2014, Ruiz et al., 2015, Pini et al., 2016), the HHHC is an interesting configuration to study natural circulation dynamics since its stability map is characterised by very large instability regions. In the HHHC configuration, the heater and the cooler are placed in horizontal position (Fig. 4). As for the dimensions of the

Nonlinear models

As mentioned in Section 3.4, a 1D O-O model is adopted for the computation of the nonlinear stability maps. This choice is justified by the possibility to analyse the entire Stm,0-Re0 plane through the definition of generic fluids. The O-O model is implemented in Dymola (DYMOLA, 2016) using an extended version, hereinafter called ThermoPowerIHG, of the ThermoPower library (Casella and Leva, 2006, ThermoPower, 2016).

As for Modelica, it is “a language for modelling and simulation of complex

Stability maps

Firstly, the stability maps obtained with the developed models are compared with those of Pini et al. (2016). In particular the thermal inertia is neglected, while the B term is taken into account (i.e., the variation of the convective heat transfer coefficient due to fluid flow perturbation). The comparison presented in Fig. 9 points out a very good agreement both for α=1 (conventional natural circulation) and α=0 (homogenously distributed IHG). Fig. 9 is useful also to highlight the IHG

Conclusions

In this work, the influence of piping materials on the dynamics of natural circulation loops with and without internal heat generation (IHG) has been investigated by means of a semi-analytical method and a numerical one. Both the strategies are developed following a generalised approach and hence they can be applied to any natural circulation loop geometry. In the present work, the HHHC (Horizontal Heater Horizontal Cooler) layout has been taken into account in order to apply the developed

Acknowledgments

The authors gratefully acknowledge the FARB (Finanziamenti di Ateneo alla Ricerca di Base) Project (Grant no. DDM2RIST06) for supporting the present activities and for funding future experimental research on this subject. The authors are grateful to Mr. Francesco Fanale (Politecnico di Milano) for the fruitful discussions and his careful review of the manuscript.

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