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Stability maps for rectangular circulation loops
Applied Thermal Engineering 23 (2003) 965–977 www.elsevier.com/locate/apthermeng
Stability maps for rectangular circulation loops L. Cammarata, A. Fichera *, A. Pagano Dipartimento Ingegneria Industriale e Meccanica, Universita di Catania, Viale Andrea Doria 6, Catania 95125, Italy
Abstract This study aims to define a methodology for the construction of stability maps of rectangular natural circulation loop. The interest for these maps derives from their ability to present in synthetic form the results of stability analysis. In fact, the proposed methodology is based on the linear stability analysis around equilibrium points of a high order nonlinear mathematical model describing the dynamical behaviour of the non-dimensional flow velocity and temperature inside the loop. Imposed heat flux at the loop walls has been assumed at the boundaries; in particular, it has been considered that the fluid is heated and cooled during its passage in the horizontal sections, respectively, at bottom and top of the loop, whereas the vertical legs are supposed adiabatic. Hence, the stability analysis was performed considering the effect of the variation of the modified Grashof number, Grm , for a wide range of loop geometrical configurations, assuming various aspect ratios (ratio of the vertical to the horizontal length of the tube) and inner tube diameters. 2003 Elsevier Science Ltd. All rights reserved. Keywords: Natural circulation loop; Stability map; Design tool
1. Introduction Natural convection is often one of the dominant mechanisms for the transfer of heat from a source to a sink by means of the motion of a fluid, generated by buoyancy. Its study is in general very complex, especially in those cases characterised by high heat transfer rate, which leads to nonlinear dynamical behaviours. Closed loop thermosyphons, also called natural circulation loops, are technical devices specifically designed in order to employ this mechanism for the heat removal from a source, placed at bottom, to a heat sink placed at the topmost section of the loop.
*
Corresponding author. Tel./fax: +39-95-337-994; Tel.: +39-095-7382450; fax: +39-095-330258. E-mail addresses: afi[email protected] (A. Fichera), [email protected] (A. Pagano).
1359-4311/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-4311(03)00027-9
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In this kind of system the fluid motion is channelled in the tube forming the loop, so that the flow is essentially one-dimensional. As no pumping devices are needed, not only the cost of pumping is eliminated but also the heat removal from the heat source is intrinsically safer. In fact, natural is preferred to forced convection in safe systems, such as nuclear power plants and electrical machine rotor cooling [1–3], or when the elimination of the cost of pumping is inviting, such as in geothermal and solar plants [4,5]. Finally, natural convection may represent one of the possible technical solution in those plants in which the pumping system cannot be conveniently positioned, such as cooling systems for internal combustion engines, turbine blade cooling or computer cooling [6,7]. Due to their simplicity, the rectangular [1,8] and the toroidal [9–11] geometrical configurations are indeed the most interesting from a theoretical point of view. In both cases the simplest conditions are those in which the loop lies on a vertical plane, is symmetrical with respect to the vertical axis and is made up of a bottom-placed heat source and a heat sink on the top. The heat source and the heat sink may be connected or not by adiabatic legs. Toroidal loops usually lack of adiabatic legs and simply consist of two semicircular heat exchanging sections directly connected, whereas rectangular loops, as the one schematised in Fig. 1, are usually designed with thermally isolated vertical legs connecting the heat exchanging sections [8]. Due to the relevance of their applications, the stability of natural circulation loops represents a stringent requirement. In fact, the oscillations of the fluid velocity and temperature associated to unstable dynamics are able to compromise the heat removal from the heat source. They are, therefore, extremely dangerous for the plant safety and represent one of the strongest limitations to the diffusion of natural circulation loop application. The system stability strongly depends on both the geometry and the heating conditions at the boundaries. The most common heating conditions considered in theoretical studies are symmetrical with respect to the vertical axis and consist of: imposed wall temperature [9,12]; imposed heat flux [11,13] or mixed conditions [10,14,15]. An accurate model of the dynamical behaviour of natural circulation in closed loops is necessary in order to allow the theoretical study of the system stability. The geometry of the system
L1 cooling section
& cp•∆T m
L
adiabatic legs
x
heating section
q Fig. 1. Rectangular natural circulation loop.
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and the heating conditions at the boundary play a fundamental role in the possibility of defining a reliable model. In particular, for the toroidal geometry, the governing equations describing the flow inside the loop (supposed one-dimensional) have been exactly reduced to a three-dimensional model, both for the case of known wall temperature and for that of known heat flux (but not for the case of mixed boundary conditions). The model reduction has been obtained through Fourier series expansion of the temperature field and of the functions describing the system geometry and the boundary conditions, followed by the truncation of high order terms. These models have been proved to be able to reproduce the chaotic dynamical behaviours experimentally observed in real system [14], which mainly resembles those of the Lorenz system [16]. Only recently [17], the same approach has been adopted to define a higher order model for the case of rectangular loops with adiabatic legs and imposed heat flux at the wall, which has been also experimentally validated on the loop schematised in Fig. 1. A synthetic way to describe the stability of natural circulation loops, without the need of simulating the system dynamics, have been proposed by Vijayan in [1,18], Ferreri in [19] and Chen in [20]. These studies propose stability maps in which the region of stable behaviour is separated from that of unstable dynamics. In [1], such a map is drawn in the space spanned by the modified Grashof and Stanton numbers, namely Grm and Stm , and is obtained through the linear stability analysis of a space-discrete model of a rectangular loop, considering three possible inner diameters of the pipes forming the loop. The map obtained in this way is affected by the severe approximation of the model due to space discretisation. Moreover, the model is studied considering each operating condition in the map defined by independent values of Grm and Stm , which corresponds to admit that the heat power removed in the cooling section is independent on that supplied in the heating section; in technological applications this assumption is not reliable and its validity at last limited to the analysis of transient behaviour. The map reported in [19] is defined, for the case of vertical toroidal loops, in the space spanned by the Grashof number, accounting for the heat power supplied in the heating section, and by the angle of inclination of the loop symmetry axis with respect to the vertical direction. The model considered in this case is obtained through a reliable reduction of the Navier–Stokes equations by means of Fourier series expansion and truncation of all modes higher than the first. Also in this case the map is constructed through the linear stability analysis of the model. In practice, the study aims to analyse the effect of the asymmetric distribution of forces on the stability of the system. Finally, in [20] the stability of the system dynamics has been mapped as a function of the loop aspect ratio. Though interesting from a theoretical point of view the results of this study cannot be properly considered as a useful design tool. The scope of this study is to define a methodology for the construction of stability maps to be used as a tool for both the design and the operations of rectangular natural circulation loops. The model reported in [17] has been rewritten in non-dimensional form and linearised around its equilibrium points. Then, the study of the stability of the equilibrium points has been approached by evaluating the sign of the eigenvalues of the linearised model. Such an analysis has been repeated for a wide range of operating conditions, defined by the modified Grashof number, Grm , and for a wide range of the aspect ratio, defining the system geometry as the ratio between the height of the loop, L, and its width, L1. Such an approach enables to create, for any chosen diameter of the tubes of the loop, a stability map spanning the space ðGrm Þ ðL=L1Þ.
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2. Mathematical model A general scheme for the modelling of rectangular circulation loop is reported in Fig. 1. The loop height and width are L and L1 respectively and the inner diameter of all tubes is D. Consider in the following the non-dimensional abscissa s ¼ x=Ltot , with x dimensional abscissa arbitrarily starting at the bottom left corner, (see Fig. 1) and Ltot total loop length. The conservation of mass for the case of uncompressible flow yields the independence of the fluid velocity on the abscissa; therefore, the Reynolds number, Re, is only function of non-dimensional time, s Re ¼ ReðsÞ ¼
qwD l
ð1Þ
where s¼
t U
and U ¼
qDL 2l
ð2Þ
The conservation of momentum for the control volume of length ds, integrated over the whole loop yields I dRe L Grm þ f ðReÞ Re jRej ¼ gðsÞ hðsÞ ds ð3Þ ds D 2 where Grm ¼
q2 gbD2 L2tot q000 l3 cp A
ð4Þ
is the non-dimensional Grashof number, in its modified form directly expressed by means of the heat flux term. This number expresses the ratio between the buoyancy, depending on the thermal boundary conditions, and the viscous stress acting on the fluid element. In the present study, it has been assumed an imposed heat flux per unit area, q000 , for both horizontal section of length L1 . At regime q000 is supplied to and removed from the fluid during its passage through the bottom and the top section, respectively. Moreover, the non-dimensional function 8 1 0 > < 0 L < x < L þ L1 gðsÞ ¼ ð5Þ 1 L þ L1 < x < 2L þ L1 > > : 0 2L þ L1 < x < 2ðL þ L1 Þ describes the H influence of the gravitational field on the various part of the loop (notice that for a close loop gðxÞ ¼ 0), whereas the function f ðReÞ represents the friction law. Under the assumption of smooth tubes b ð6Þ Red with b and d non-dimensional number, whose determination is in general very complex for the kind of system herein considered and is usually done on an experimental basis. The integration of the energy balance for a control volume of length ds over the whole loop can be expressed as f ¼
L. Cammarata et al. / Applied Thermal Engineering 23 (2003) 965–977
oh oh o2 h þ Re ¼ hðsÞ þ Fo 2 os os os
969
ð7Þ
where hðs; sÞ ¼
T ðsÞ T0 DT
ð8Þ
is the non-dimensional temperature, T ðsÞ is the dimensional temperature of the fluid element, T0 is a corresponding reference temperature independent on time and valid over the whole loop, and 2L q000 . DT ¼ ctot pl The non-dimensional function 8 0 >0 q00 ðsÞ < 1 L < x < L þ L1 ð9Þ hðsÞ ¼ 00 ¼ 0 L þ L1 < x < 2L þ L1 > q0 > : 1 2L þ L1 < x < 2ðL þ L1 Þ describes the heating boundary conditions. In Eq. (7) the term depending on the non-dimensional Fourier number Fo ¼
aU ðLtot =2Þ2
ð10Þ
represents the contribution of thermal diffusion along the tube axis [19]. This term is usually neglected in the modelling of toroidal loops but it plays an important role in the damping of thermal oscillations. Its consideration is therefore necessary, as much as that of the friction factor in the momentum balance equation, in order to define a reliable model of the system dynamics. Eqs. (3) and (7) represent the mathematical model of the dynamics of a generic rectangular closed loop thermosyphon. The second of these is a PDE and is therefore infinite dimensional; its reduction to a finite number of one-dimensional equations has been obtained as described in the following section. Rodriguez–Bernal and Van Vleck [21] have described the way to reduce a model formally identical to that expressed by (3) and (7), by means of the following Fourier series expansion of the known functions gðsÞ and hðsÞ and of the variable hðs; tÞ X ak ðsÞ eipks ð11Þ hðs; sÞ ¼ k2Z
hðsÞ ¼
X
bk eipks
ð12Þ
ck eipks
ð13Þ
k2K
f ðsÞ ¼
X k2J
where K; J Z ¼ Z n f0g. Substitution of (11)–(13) in (3) and (7) yields an infinite system of ordinary differential equations; its integration requires the calculation of the coefficients of the Fourier series expansions of the geometry (Eq. (5)) and of the thermal boundary conditions (Eq. (9)), which yields
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ck ¼
1 ½ð1 cos kc þ i sinkcÞð1 cos kpÞ 2pki
ð14Þ
with c¼
pL L þ L1
ð15Þ
for the coefficients of gðsÞ expansion (note that ck ¼ 0 for even k), and bk ¼
1 ð1 cos kc þ i sinkcÞ 2pki
bk ¼ 0
for odd k
for even k
ð16Þ ð17Þ
for the coefficients of hðsÞ expansion. The coefficients of the Fourier series expansion of the non-dimensional temperature are complex variables of the model and it is convenient to perform their separation in their real and imaginary parts ak ðsÞ ¼ ak ðsÞ þ ibk ðsÞ
ð18Þ
Substitution of these coefficients in Eqs. (3) and (7) and application of the method of residues leads to an infinite system of ordinary differential equations. In order to obtain a finite dimensional model, truncation of high order modes is necessary. The number of modes that is convenient to consider mainly depends on the shape of the functions gðsÞ and hðsÞ (the more complex they are the higher number of modes should be taken into account) and on the influence of the friction factor, f ðReÞ, and of the Fourier number, Fo, whose damping effect is stronger on the higher order modes (fast modes) than on the first modes. In the present study the Fourier series expansions of the rectangular geometry and of the heating boundary conditions have been truncated to the third mode (k ¼ 3), which has been assumed a convenient compromise between model complexity and its accuracy. The approximation of model (3), (7) under the previous assumptions reads dRe L b Grm 1 1 RejRej þ a1 sin c b1 ð1 cos cÞ þ a3 sin 3c b3 ð1 cos 3cÞ ¼ ð19Þ p ds D Red 3 3 da1 1 sin c ¼ 4p2 Foa1 þ 2pReb1 þ 2p ds
ð20Þ
db1 1 ¼ 4p2 Fob1 2pRea1 þ ð1 þ cos cÞ 2p ds
ð21Þ
da2 ¼ 4p2 4Foa2 þ 2p 2Reb2 ds
ð22Þ
db2 ¼ 4p2 4Fob2 2p 2Rea2 ds
ð23Þ
L. Cammarata et al. / Applied Thermal Engineering 23 (2003) 965–977
da3 1 sin 3c ¼ 4p2 9Foa3 þ 2p 3Reb3 þ 6p ds
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ð24Þ
db3 1 ð25Þ ¼ 4p2 9Fob3 2p 3Rea3 þ ð1 þ cos cÞ 6p ds Looking at model (19)–(25) it is possible to observe that the first equation comes from the reduction of momentum equation, whereas the others consist of distinct couples of equations, each of which describes just one of the three modes herein considered. Moreover, only the first and third couples, expressed by Eqs. (20), (21), (24) and (25) respectively, are coupled with the first equation. In fact, the second couple (Eqs. (22) and (23)) describes a uncontrollable but stable mode [22], not affecting the velocity field and playing a relatively important role only during the system transient. Notice that the presence of the thermal damping term depending on Fo, usually neglected, is in this case necessary to ensure the damping of the second modes, and of the other truncated odd modes, which are uncontrollable as a consequence of equation (17), which could otherwise diverge towards infinity. Once that the model has been integrated, the reconstruction of the non-dimensional temperature field is possible by means of the following: hðs; sÞ ¼ 2a1 cosps 2b1 sinps þ 2a2 cos 2ps 2b2 sin 2ps þ 2a3 cos 3ps 2b3 sin 3ps
ð26Þ
Experimental validation of model (19)–(25) reported in [17], has allowed to demonstrate its reliable performances in describing the dynamics of rectangular circulation loops.
3. Stability map The aim of the present section is to analyse, on the base of the model defined in the previous section, the dependence of the system stability on both the operating condition and the aspect ratio. The analysis has been performed for different values of the diameter of the tube, which also plays an important role on the system dynamics, especially for its influence on the damping terms (and, hence, on the system stability). The construction of maps represents an interesting tool both for design and operation of the system as it allows, for a given geometry, to predict whether or not an operating condition will be unstable or not. Moreover, it allows to analyse the effects on the stability of the system played by both geometrical parameters and operating conditions. Analyses have been performed through the construction of maps, namely Stability Maps, one for each of the different values of the inner tube diameters considered in this study. Each map is drawn on the space spanned by Grm and L=L1 , whose points are defined by the loop aspect ratio and the operating condition. The approach allows both to separate the regions of stable and unstable behaviour and to associate a sort of stability level to each point of the map. The steps for the construction of the map are: • calculation of the equilibrium points of the model at each point; • linearisation of the model around the equilibrium; • computation of the eigenvalues of the Jacobian of the linearised system and analysis of their sign.
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In order to simplify the calculations, the system order has been reduced by elimination of the couple of equations referred to the second transient mode. This is justified by previous considerations on the stability of this mode, which is damped out after a short transient and does not contribute to the regime behaviour. The equilibrium points are found by solving the algebraic system obtained by setting simultaneously to zero the time derivatives that appear on the lefthand side of Eqs. (19)–(25). Three stationary points are found in this way: • two points are symmetric, as a consequence of the geometrical symmetry of the loop, and correspond to the stationary motion occurring with equal probability either in the clockwise or in the counter-clockwise direction; • the third point corresponds to the absence of motion; this case is possible only when no geometrical or thermal perturbations affect the ideal symmetry of the system, and is stable only for very low values of the heat power. The analysis of the system stability has been performed only for the case of the first two points, which for basic considerations on the system symmetry are dynamically equivalent and lead to identical results. In order to calculate the linearised model around one of these equilibrium points, it is convenient to pose: x ¼ ½ Re
a1
b1
Dx ¼ ½ DRe Da1
a3 Db2
T
b3
Da3
ð27Þ Db3
T
ð28Þ
u ¼ ½Grm
ð29Þ
Du ¼ ½DGrm
ð30Þ
In this way, the original nonlinear model can be written in the following compact form: x_ ¼ f ðx; uÞ
ð31Þ
and the linearised model as D_x ¼ ADx þ BDu
ð32Þ
where A is the Jacobian matrix, i.e. the following squared matrix 5 5 having as generic element Akj the derivative, calculated at the equilibrium point, of the kth equation with respect to the jth variable 2 6 6 6 A¼6 6 4
LTot Gr sin c b½ð1 dÞRed jRej þ Reð1dÞ p D 4p2 Fo 2pb10 2pRe 2pa10 0 6pb30 0 6pa30
Gr Gr sin 3c ð1 cos cÞ p p 3 2pRe 0 4p2 Fo 0 0 36p2 Fo 0 6pRe
3 Gr ð1 cos 3c 7 p 3 7 0 7 7 0 7 5 6pRe 36p2 Fo ð33Þ
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In (32) B is the system gain matrix, which weights the influence of input variations on the linearised system behaviour. In order to analyse the system stability for a given operating condition, which corresponds to fixing B, it is necessary to consider only the eigenvalues of the Jacobian matrix A. In particular, the system is said to be stable, marginally stable or unstable if the real parts of its eigenvalues are all negative, either negative or null, or finally if at least one of them is positive. Therefore, the stability analysis of the points of the map has been performed by looking at the maximum real part of the Jacobian of the linearised model for the operating condition and the geometry defined by each of these points. This approach allows to separate in the map the region of stable behaviour from that of unstable behaviour. Moreover, a family of lines, namely isostability lines, can be drawn on the map; each of these lines corresponds to the same value of the maximum real part of the Jacobian eigenvalues. In practice, the points on these curves are characterised by different geometry and operating conditions but by the same kind of dynamical behaviour.
4. Results and discussion Figs. 2 and 3 report the maps obtained increasing the inner diameter of the pipes forming the loop from 0.010 m to 0.035 m with steps of 0.005 m. The first consideration that can be drawn from these maps, in accordance with several studies on natural circulation loops [9,10], is that in all cases for a given aspect ratio L=L1 the increase of the heat power supplied to the system, i.e. of Grm , leads the system to instability. In the maps this corresponds to the crossing of the isostability line at zero level, which connects the points at which the maximum real part of the eigenvalues of the Jacobian reach zero. Similarly, the values associated to the other isostability lines (denoted by the different colours scaled on the colour-bar on the right of each map) indicate the maximum real part of the eigenvalues of the system Jacobian. The maps defined in this way can be easily used for two main tasks: separation of the stable and unstable operating condition of a loop with given aspect ratio; choice of the best geometrical configuration, if one exists, for the removal of a given heat power. It is possible to notice that the crossing condition between the stable and unstable region is characterised by having a minimum for the aspect ratio L=L1 ¼ 1. This means that the square geometry is the less stable, i.e. those that are less capable of damping out undesired oscillations. Analogous considerations lead to observe that the high aspect ratio loops are unstable for Grm lower than those with lower aspect ratios. To see this, notice that the points of the isostability line at zero having aspect ratios close to L=L1 ¼ 10 are characterised by lower critical Grm than those close to L=L1 ¼ 0:1. Another interesting point is the difference in shape between the top and the bottom of the map that characterises the isostability lines. In particular, the distance between these lines is considerably higher for low values than for high values of the aspect ratio. This means that perturbations of Grm may cause the jump of several isostability lines in high aspect ratio loops, whereas this phenomenon is much weaker in the low aspect ratio loops. In practice, the loops with lower aspect ratio are not only more stable but also less sensitive to variations of Grm . This is probably due to the more uniform and distributed heat supply that can be achieved with this geometry, which allows for thermal oscillations damping directly at the heat source and before they can reach the vertical legs and reflect in oscillations of the buoyancy term.
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Fig. 2. Stability maps computed for inner diameter of the loop pipe: (a) D ¼ 0:010 m; (b) D ¼ 0:015 m; (c) D ¼ 0:020 m.
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Fig. 3. Stability maps computed for inner diameter of the loop pipe: (a) D ¼ 0:025 m; (b) D ¼ 0:030 m; (c) D ¼ 0:035 m.
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Since now only the general structure common to all the maps has been described. In order to analyse the dependence of the system stability on the inner diameter of the pipe, it is necessary to check out the differences among the various maps. Comparing the various maps reported in Figs. 2 and 3, it is possible to notice that the main effect of the increase of the inner diameter of the loop tube is the progressive translation of the isostability lines towards lower Grm values. To talk of a translation is legitimated by the observation that the position of the isolines changes mainly regardless of the aspect ratio, which corresponds to a reduction of the stable region. In other words, the higher is the diameter of the tubes, the weaker is the system stability. This can be explained as a consequence of the reduction of the overall friction caused by the increase of the tube diameter. In fact, the diminishment of the friction is associated to the weaker damping of the perturbations that leads the system to become unstable. Notice also that the reduction of the system stability is particularly strong for the lower values of the diameter, e.g. when increasing the diameter from 0.010 to 0.015, whereas it becomes almost negligible for the higher diameters, e.g. passing from D ¼ 0:030 to 0.035 m. This is also the reason why further increases of the diameter have not been considered in this study.
5. Conclusions The aim of the present study has been the definition of an analytical methodology for the construction of stability maps for rectangular natural circulation loops. The proposed approach has been based on linearisation and analysis of a high order model for this kind of systems. The analysis of the model has aimed to characterise the system stability through the computation of the complex eigenvalues of the Jacobian matrix of the linearised model and the study of the sign of their real part. For a given diameter of the loop pipes, a map is then created by iterating the approach for various operating condition, defined by the modified Grashof number, Grm , and for different geometrical configuration of the loop, expressed by the ratio between the loop height and width. Several maps have been constructed in this way, each differing from the others for the inner diameter of the loop adopted during the calculation. Reported results show the convenience of adopting loops characterised by low aspect ratios, which ensure stable behaviour for higher Grm and are less sensitive to modification or perturbation of the operating condition. Moreover it has been pointed out the negative effect on stability played by the increase of the diameter of the loop pipes. The proposed maps represent a useful tool for design and operation of natural circulation loops, in fact they account for the dependence of the system stability on both the operating condition and the loop geometry.
References [1] P.K. Vijayan, A.K. Nayak, D.S. Pilkhwal, D. Saha, V. Venkat Raj, Effect of loop diameter on the stability of single-phase natural circulation in rectangular loops, in: Proceedings of NURETH-5, Salt Lake City, 1992, pp. 261–267.
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[2] J. Miettinen, T. Kervinen, H. Tomisto, H. Kantee, Oscillations of single-phase natural circulation during overcooling transient, in: Proceedings of ANS Topical Meeting, Atlanta, 1987, pp. 20–29. [3] R. Greif, Y. Zvirin, A. Mertol, The transient and stability behavior of a natural convention loop, Transaction of the ASME 101 (1979) 684–688. [4] D.B. Kreitlow, G.M. Reistad, C.R. Miles, G.G. Culver, Thermosyphon models for downhole heat exchanger applications in shallow geothermal systems, ASME Journal of Heat Transfer 100 (1978) 713–719. [5] Y. Zvirin, A. Shitzer, A. Bartal-Bornstein, On the stability of the natural circulation solar heater, in: Proceedings of the Sixth International Heat Transfer Conference, Toronto, 20, 1978, pp. 997–999. [6] Y. Zvirin, A review of N.C. loops in PWR and other systems, Nuclear Egineering Design 67 (1981) 203–225. [7] D. Japikse, Advances in thermosyphon technology, in: J.P. Harnett, T.F. Irvine (Eds.), Advances in Heat Transfer, vol. 9, Academic Press, New York, 1973. [8] G. Cammarata, A. Fichera, M. Frogheri M. Misale, M.G. Xibilia, A new modelling methodology of natural circulation loop for stability analysis, in: Proceedings of Eurotherm Seminar No. 63, Single and Two-Phase Natural Circulation, Genova, 1999, pp. 151–159. [9] Y.-Z. Wang, J. Singer, H.H. Bau, Controlling chaos in a thermal convection loop, Journal of Fluid Mechanics 237 (1992) 479–498. [10] J. Singer, Y.-Z. Wang, H.H. Bau, Controlling a chaotic system, Physical Review Letters 66 (9) (1991) 1123–1125. [11] M. Sen, E. Ramos, C. Trevino, The toroidal thermosyphon with known heat flux, International Journal of Heat and Mass Transfer 28 (1) (1985) 219–233. [12] M. Sen, E. Ramos, C. Trevino, O. Salazar, A one-dimensional model of a thermosyphon with known wall temperature, International Journal of Heat and Fluid Flow 8 (3) (1987) 171–181. [13] M. Sen, E. Ramos, C. Trevino, O. Salazar, The effect of axial conduction on a thermosyphon with prescribed heat flux, European Journal of Mechanics, B––Fluids 8 (1) (1989) 57–72. [14] M. Gorman, P.J. Widman, K.A. Robbins, Nonlinear dynamics of a convection loop: a quantitative comparison of experiment with theory, Physica 19-D (1986) 255–267. [15] P.J. Widman, M. Gorman, K.A. Robbins, Nonlinear dynamics of a convection loop II: chaos in laminar and turbulent flows, Physica 36-D (1989) 255–267. [16] E.N. Lorenz, Deterministic nonperiodic flow, Journal of Atmospheric Science 20 (1963) 131–141. [17] L. Cammarata, G. Desrayaud, A. Fichera, A. Pagano, An ordinary differential model for rectangular circulation loops, in: Twelth International Heat Transfer Conference, Grenoble, 2002. [18] P.K. Vijayan, H. Austregesilo, Sealing laws for single-phase natural circulation loops, Nuclear Engineering and Design 152 (1994) 331–347. [19] J.C. Ferreri, Single-phase natural circulation in simple circuits––reflections on its numerical simulation, in: Eurotherm Seminar 63, Single and Two-Phase Natural Circulation, Genova, 1999, pp. 17–30. [20] K. Chen, On the oscillatory instability of closed-loop thermosyphons, Journal of Heat Transfer 107 (1985) 826– 832. [21] A. Rodriguez-Bernal, E. Van Vleck, Diffusion induced chaos in a closed loop thermosyphon, SIAM Journal of Applied Mathematics 58 (4) (1998) 1072–1093. [22] L. Cammarata, G. Desrayaud, A. Fichera, A. Pagano, Experimental validation of a model-based controller for a rectangular natural circulation loop, in: B. Sunden, C.A. Brebbia (Eds.), Advanced Computational Methods in Heat Transfer VII, Wit Press, Southampton, Boston, 2002.
Stability maps for rectangular circulation loops L. Cammarata, A. Fichera *, A. Pagano Dipartimento Ingegneria Industriale e Meccanica, Universita di Catania, Viale Andrea Doria 6, Catania 95125, Italy
Abstract This study aims to define a methodology for the construction of stability maps of rectangular natural circulation loop. The interest for these maps derives from their ability to present in synthetic form the results of stability analysis. In fact, the proposed methodology is based on the linear stability analysis around equilibrium points of a high order nonlinear mathematical model describing the dynamical behaviour of the non-dimensional flow velocity and temperature inside the loop. Imposed heat flux at the loop walls has been assumed at the boundaries; in particular, it has been considered that the fluid is heated and cooled during its passage in the horizontal sections, respectively, at bottom and top of the loop, whereas the vertical legs are supposed adiabatic. Hence, the stability analysis was performed considering the effect of the variation of the modified Grashof number, Grm , for a wide range of loop geometrical configurations, assuming various aspect ratios (ratio of the vertical to the horizontal length of the tube) and inner tube diameters. 2003 Elsevier Science Ltd. All rights reserved. Keywords: Natural circulation loop; Stability map; Design tool
1. Introduction Natural convection is often one of the dominant mechanisms for the transfer of heat from a source to a sink by means of the motion of a fluid, generated by buoyancy. Its study is in general very complex, especially in those cases characterised by high heat transfer rate, which leads to nonlinear dynamical behaviours. Closed loop thermosyphons, also called natural circulation loops, are technical devices specifically designed in order to employ this mechanism for the heat removal from a source, placed at bottom, to a heat sink placed at the topmost section of the loop.
*
Corresponding author. Tel./fax: +39-95-337-994; Tel.: +39-095-7382450; fax: +39-095-330258. E-mail addresses: afi[email protected] (A. Fichera), [email protected] (A. Pagano).
1359-4311/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-4311(03)00027-9
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In this kind of system the fluid motion is channelled in the tube forming the loop, so that the flow is essentially one-dimensional. As no pumping devices are needed, not only the cost of pumping is eliminated but also the heat removal from the heat source is intrinsically safer. In fact, natural is preferred to forced convection in safe systems, such as nuclear power plants and electrical machine rotor cooling [1–3], or when the elimination of the cost of pumping is inviting, such as in geothermal and solar plants [4,5]. Finally, natural convection may represent one of the possible technical solution in those plants in which the pumping system cannot be conveniently positioned, such as cooling systems for internal combustion engines, turbine blade cooling or computer cooling [6,7]. Due to their simplicity, the rectangular [1,8] and the toroidal [9–11] geometrical configurations are indeed the most interesting from a theoretical point of view. In both cases the simplest conditions are those in which the loop lies on a vertical plane, is symmetrical with respect to the vertical axis and is made up of a bottom-placed heat source and a heat sink on the top. The heat source and the heat sink may be connected or not by adiabatic legs. Toroidal loops usually lack of adiabatic legs and simply consist of two semicircular heat exchanging sections directly connected, whereas rectangular loops, as the one schematised in Fig. 1, are usually designed with thermally isolated vertical legs connecting the heat exchanging sections [8]. Due to the relevance of their applications, the stability of natural circulation loops represents a stringent requirement. In fact, the oscillations of the fluid velocity and temperature associated to unstable dynamics are able to compromise the heat removal from the heat source. They are, therefore, extremely dangerous for the plant safety and represent one of the strongest limitations to the diffusion of natural circulation loop application. The system stability strongly depends on both the geometry and the heating conditions at the boundaries. The most common heating conditions considered in theoretical studies are symmetrical with respect to the vertical axis and consist of: imposed wall temperature [9,12]; imposed heat flux [11,13] or mixed conditions [10,14,15]. An accurate model of the dynamical behaviour of natural circulation in closed loops is necessary in order to allow the theoretical study of the system stability. The geometry of the system
L1 cooling section
& cp•∆T m
L
adiabatic legs
x
heating section
q Fig. 1. Rectangular natural circulation loop.
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and the heating conditions at the boundary play a fundamental role in the possibility of defining a reliable model. In particular, for the toroidal geometry, the governing equations describing the flow inside the loop (supposed one-dimensional) have been exactly reduced to a three-dimensional model, both for the case of known wall temperature and for that of known heat flux (but not for the case of mixed boundary conditions). The model reduction has been obtained through Fourier series expansion of the temperature field and of the functions describing the system geometry and the boundary conditions, followed by the truncation of high order terms. These models have been proved to be able to reproduce the chaotic dynamical behaviours experimentally observed in real system [14], which mainly resembles those of the Lorenz system [16]. Only recently [17], the same approach has been adopted to define a higher order model for the case of rectangular loops with adiabatic legs and imposed heat flux at the wall, which has been also experimentally validated on the loop schematised in Fig. 1. A synthetic way to describe the stability of natural circulation loops, without the need of simulating the system dynamics, have been proposed by Vijayan in [1,18], Ferreri in [19] and Chen in [20]. These studies propose stability maps in which the region of stable behaviour is separated from that of unstable dynamics. In [1], such a map is drawn in the space spanned by the modified Grashof and Stanton numbers, namely Grm and Stm , and is obtained through the linear stability analysis of a space-discrete model of a rectangular loop, considering three possible inner diameters of the pipes forming the loop. The map obtained in this way is affected by the severe approximation of the model due to space discretisation. Moreover, the model is studied considering each operating condition in the map defined by independent values of Grm and Stm , which corresponds to admit that the heat power removed in the cooling section is independent on that supplied in the heating section; in technological applications this assumption is not reliable and its validity at last limited to the analysis of transient behaviour. The map reported in [19] is defined, for the case of vertical toroidal loops, in the space spanned by the Grashof number, accounting for the heat power supplied in the heating section, and by the angle of inclination of the loop symmetry axis with respect to the vertical direction. The model considered in this case is obtained through a reliable reduction of the Navier–Stokes equations by means of Fourier series expansion and truncation of all modes higher than the first. Also in this case the map is constructed through the linear stability analysis of the model. In practice, the study aims to analyse the effect of the asymmetric distribution of forces on the stability of the system. Finally, in [20] the stability of the system dynamics has been mapped as a function of the loop aspect ratio. Though interesting from a theoretical point of view the results of this study cannot be properly considered as a useful design tool. The scope of this study is to define a methodology for the construction of stability maps to be used as a tool for both the design and the operations of rectangular natural circulation loops. The model reported in [17] has been rewritten in non-dimensional form and linearised around its equilibrium points. Then, the study of the stability of the equilibrium points has been approached by evaluating the sign of the eigenvalues of the linearised model. Such an analysis has been repeated for a wide range of operating conditions, defined by the modified Grashof number, Grm , and for a wide range of the aspect ratio, defining the system geometry as the ratio between the height of the loop, L, and its width, L1. Such an approach enables to create, for any chosen diameter of the tubes of the loop, a stability map spanning the space ðGrm Þ ðL=L1Þ.
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2. Mathematical model A general scheme for the modelling of rectangular circulation loop is reported in Fig. 1. The loop height and width are L and L1 respectively and the inner diameter of all tubes is D. Consider in the following the non-dimensional abscissa s ¼ x=Ltot , with x dimensional abscissa arbitrarily starting at the bottom left corner, (see Fig. 1) and Ltot total loop length. The conservation of mass for the case of uncompressible flow yields the independence of the fluid velocity on the abscissa; therefore, the Reynolds number, Re, is only function of non-dimensional time, s Re ¼ ReðsÞ ¼
qwD l
ð1Þ
where s¼
t U
and U ¼
qDL 2l
ð2Þ
The conservation of momentum for the control volume of length ds, integrated over the whole loop yields I dRe L Grm þ f ðReÞ Re jRej ¼ gðsÞ hðsÞ ds ð3Þ ds D 2 where Grm ¼
q2 gbD2 L2tot q000 l3 cp A
ð4Þ
is the non-dimensional Grashof number, in its modified form directly expressed by means of the heat flux term. This number expresses the ratio between the buoyancy, depending on the thermal boundary conditions, and the viscous stress acting on the fluid element. In the present study, it has been assumed an imposed heat flux per unit area, q000 , for both horizontal section of length L1 . At regime q000 is supplied to and removed from the fluid during its passage through the bottom and the top section, respectively. Moreover, the non-dimensional function 8 1 0
L. Cammarata et al. / Applied Thermal Engineering 23 (2003) 965–977
oh oh o2 h þ Re ¼ hðsÞ þ Fo 2 os os os
969
ð7Þ
where hðs; sÞ ¼
T ðsÞ T0 DT
ð8Þ
is the non-dimensional temperature, T ðsÞ is the dimensional temperature of the fluid element, T0 is a corresponding reference temperature independent on time and valid over the whole loop, and 2L q000 . DT ¼ ctot pl The non-dimensional function 8 0
aU ðLtot =2Þ2
ð10Þ
represents the contribution of thermal diffusion along the tube axis [19]. This term is usually neglected in the modelling of toroidal loops but it plays an important role in the damping of thermal oscillations. Its consideration is therefore necessary, as much as that of the friction factor in the momentum balance equation, in order to define a reliable model of the system dynamics. Eqs. (3) and (7) represent the mathematical model of the dynamics of a generic rectangular closed loop thermosyphon. The second of these is a PDE and is therefore infinite dimensional; its reduction to a finite number of one-dimensional equations has been obtained as described in the following section. Rodriguez–Bernal and Van Vleck [21] have described the way to reduce a model formally identical to that expressed by (3) and (7), by means of the following Fourier series expansion of the known functions gðsÞ and hðsÞ and of the variable hðs; tÞ X ak ðsÞ eipks ð11Þ hðs; sÞ ¼ k2Z
hðsÞ ¼
X
bk eipks
ð12Þ
ck eipks
ð13Þ
k2K
f ðsÞ ¼
X k2J
where K; J Z ¼ Z n f0g. Substitution of (11)–(13) in (3) and (7) yields an infinite system of ordinary differential equations; its integration requires the calculation of the coefficients of the Fourier series expansions of the geometry (Eq. (5)) and of the thermal boundary conditions (Eq. (9)), which yields
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ck ¼
1 ½ð1 cos kc þ i sinkcÞð1 cos kpÞ 2pki
ð14Þ
with c¼
pL L þ L1
ð15Þ
for the coefficients of gðsÞ expansion (note that ck ¼ 0 for even k), and bk ¼
1 ð1 cos kc þ i sinkcÞ 2pki
bk ¼ 0
for odd k
for even k
ð16Þ ð17Þ
for the coefficients of hðsÞ expansion. The coefficients of the Fourier series expansion of the non-dimensional temperature are complex variables of the model and it is convenient to perform their separation in their real and imaginary parts ak ðsÞ ¼ ak ðsÞ þ ibk ðsÞ
ð18Þ
Substitution of these coefficients in Eqs. (3) and (7) and application of the method of residues leads to an infinite system of ordinary differential equations. In order to obtain a finite dimensional model, truncation of high order modes is necessary. The number of modes that is convenient to consider mainly depends on the shape of the functions gðsÞ and hðsÞ (the more complex they are the higher number of modes should be taken into account) and on the influence of the friction factor, f ðReÞ, and of the Fourier number, Fo, whose damping effect is stronger on the higher order modes (fast modes) than on the first modes. In the present study the Fourier series expansions of the rectangular geometry and of the heating boundary conditions have been truncated to the third mode (k ¼ 3), which has been assumed a convenient compromise between model complexity and its accuracy. The approximation of model (3), (7) under the previous assumptions reads dRe L b Grm 1 1 RejRej þ a1 sin c b1 ð1 cos cÞ þ a3 sin 3c b3 ð1 cos 3cÞ ¼ ð19Þ p ds D Red 3 3 da1 1 sin c ¼ 4p2 Foa1 þ 2pReb1 þ 2p ds
ð20Þ
db1 1 ¼ 4p2 Fob1 2pRea1 þ ð1 þ cos cÞ 2p ds
ð21Þ
da2 ¼ 4p2 4Foa2 þ 2p 2Reb2 ds
ð22Þ
db2 ¼ 4p2 4Fob2 2p 2Rea2 ds
ð23Þ
L. Cammarata et al. / Applied Thermal Engineering 23 (2003) 965–977
da3 1 sin 3c ¼ 4p2 9Foa3 þ 2p 3Reb3 þ 6p ds
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ð24Þ
db3 1 ð25Þ ¼ 4p2 9Fob3 2p 3Rea3 þ ð1 þ cos cÞ 6p ds Looking at model (19)–(25) it is possible to observe that the first equation comes from the reduction of momentum equation, whereas the others consist of distinct couples of equations, each of which describes just one of the three modes herein considered. Moreover, only the first and third couples, expressed by Eqs. (20), (21), (24) and (25) respectively, are coupled with the first equation. In fact, the second couple (Eqs. (22) and (23)) describes a uncontrollable but stable mode [22], not affecting the velocity field and playing a relatively important role only during the system transient. Notice that the presence of the thermal damping term depending on Fo, usually neglected, is in this case necessary to ensure the damping of the second modes, and of the other truncated odd modes, which are uncontrollable as a consequence of equation (17), which could otherwise diverge towards infinity. Once that the model has been integrated, the reconstruction of the non-dimensional temperature field is possible by means of the following: hðs; sÞ ¼ 2a1 cosps 2b1 sinps þ 2a2 cos 2ps 2b2 sin 2ps þ 2a3 cos 3ps 2b3 sin 3ps
ð26Þ
Experimental validation of model (19)–(25) reported in [17], has allowed to demonstrate its reliable performances in describing the dynamics of rectangular circulation loops.
3. Stability map The aim of the present section is to analyse, on the base of the model defined in the previous section, the dependence of the system stability on both the operating condition and the aspect ratio. The analysis has been performed for different values of the diameter of the tube, which also plays an important role on the system dynamics, especially for its influence on the damping terms (and, hence, on the system stability). The construction of maps represents an interesting tool both for design and operation of the system as it allows, for a given geometry, to predict whether or not an operating condition will be unstable or not. Moreover, it allows to analyse the effects on the stability of the system played by both geometrical parameters and operating conditions. Analyses have been performed through the construction of maps, namely Stability Maps, one for each of the different values of the inner tube diameters considered in this study. Each map is drawn on the space spanned by Grm and L=L1 , whose points are defined by the loop aspect ratio and the operating condition. The approach allows both to separate the regions of stable and unstable behaviour and to associate a sort of stability level to each point of the map. The steps for the construction of the map are: • calculation of the equilibrium points of the model at each point; • linearisation of the model around the equilibrium; • computation of the eigenvalues of the Jacobian of the linearised system and analysis of their sign.
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In order to simplify the calculations, the system order has been reduced by elimination of the couple of equations referred to the second transient mode. This is justified by previous considerations on the stability of this mode, which is damped out after a short transient and does not contribute to the regime behaviour. The equilibrium points are found by solving the algebraic system obtained by setting simultaneously to zero the time derivatives that appear on the lefthand side of Eqs. (19)–(25). Three stationary points are found in this way: • two points are symmetric, as a consequence of the geometrical symmetry of the loop, and correspond to the stationary motion occurring with equal probability either in the clockwise or in the counter-clockwise direction; • the third point corresponds to the absence of motion; this case is possible only when no geometrical or thermal perturbations affect the ideal symmetry of the system, and is stable only for very low values of the heat power. The analysis of the system stability has been performed only for the case of the first two points, which for basic considerations on the system symmetry are dynamically equivalent and lead to identical results. In order to calculate the linearised model around one of these equilibrium points, it is convenient to pose: x ¼ ½ Re
a1
b1
Dx ¼ ½ DRe Da1
a3 Db2
T
b3
Da3
ð27Þ Db3
T
ð28Þ
u ¼ ½Grm
ð29Þ
Du ¼ ½DGrm
ð30Þ
In this way, the original nonlinear model can be written in the following compact form: x_ ¼ f ðx; uÞ
ð31Þ
and the linearised model as D_x ¼ ADx þ BDu
ð32Þ
where A is the Jacobian matrix, i.e. the following squared matrix 5 5 having as generic element Akj the derivative, calculated at the equilibrium point, of the kth equation with respect to the jth variable 2 6 6 6 A¼6 6 4
LTot Gr sin c b½ð1 dÞRed jRej þ Reð1dÞ p D 4p2 Fo 2pb10 2pRe 2pa10 0 6pb30 0 6pa30
Gr Gr sin 3c ð1 cos cÞ p p 3 2pRe 0 4p2 Fo 0 0 36p2 Fo 0 6pRe
3 Gr ð1 cos 3c 7 p 3 7 0 7 7 0 7 5 6pRe 36p2 Fo ð33Þ
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In (32) B is the system gain matrix, which weights the influence of input variations on the linearised system behaviour. In order to analyse the system stability for a given operating condition, which corresponds to fixing B, it is necessary to consider only the eigenvalues of the Jacobian matrix A. In particular, the system is said to be stable, marginally stable or unstable if the real parts of its eigenvalues are all negative, either negative or null, or finally if at least one of them is positive. Therefore, the stability analysis of the points of the map has been performed by looking at the maximum real part of the Jacobian of the linearised model for the operating condition and the geometry defined by each of these points. This approach allows to separate in the map the region of stable behaviour from that of unstable behaviour. Moreover, a family of lines, namely isostability lines, can be drawn on the map; each of these lines corresponds to the same value of the maximum real part of the Jacobian eigenvalues. In practice, the points on these curves are characterised by different geometry and operating conditions but by the same kind of dynamical behaviour.
4. Results and discussion Figs. 2 and 3 report the maps obtained increasing the inner diameter of the pipes forming the loop from 0.010 m to 0.035 m with steps of 0.005 m. The first consideration that can be drawn from these maps, in accordance with several studies on natural circulation loops [9,10], is that in all cases for a given aspect ratio L=L1 the increase of the heat power supplied to the system, i.e. of Grm , leads the system to instability. In the maps this corresponds to the crossing of the isostability line at zero level, which connects the points at which the maximum real part of the eigenvalues of the Jacobian reach zero. Similarly, the values associated to the other isostability lines (denoted by the different colours scaled on the colour-bar on the right of each map) indicate the maximum real part of the eigenvalues of the system Jacobian. The maps defined in this way can be easily used for two main tasks: separation of the stable and unstable operating condition of a loop with given aspect ratio; choice of the best geometrical configuration, if one exists, for the removal of a given heat power. It is possible to notice that the crossing condition between the stable and unstable region is characterised by having a minimum for the aspect ratio L=L1 ¼ 1. This means that the square geometry is the less stable, i.e. those that are less capable of damping out undesired oscillations. Analogous considerations lead to observe that the high aspect ratio loops are unstable for Grm lower than those with lower aspect ratios. To see this, notice that the points of the isostability line at zero having aspect ratios close to L=L1 ¼ 10 are characterised by lower critical Grm than those close to L=L1 ¼ 0:1. Another interesting point is the difference in shape between the top and the bottom of the map that characterises the isostability lines. In particular, the distance between these lines is considerably higher for low values than for high values of the aspect ratio. This means that perturbations of Grm may cause the jump of several isostability lines in high aspect ratio loops, whereas this phenomenon is much weaker in the low aspect ratio loops. In practice, the loops with lower aspect ratio are not only more stable but also less sensitive to variations of Grm . This is probably due to the more uniform and distributed heat supply that can be achieved with this geometry, which allows for thermal oscillations damping directly at the heat source and before they can reach the vertical legs and reflect in oscillations of the buoyancy term.
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Fig. 2. Stability maps computed for inner diameter of the loop pipe: (a) D ¼ 0:010 m; (b) D ¼ 0:015 m; (c) D ¼ 0:020 m.
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Fig. 3. Stability maps computed for inner diameter of the loop pipe: (a) D ¼ 0:025 m; (b) D ¼ 0:030 m; (c) D ¼ 0:035 m.
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Since now only the general structure common to all the maps has been described. In order to analyse the dependence of the system stability on the inner diameter of the pipe, it is necessary to check out the differences among the various maps. Comparing the various maps reported in Figs. 2 and 3, it is possible to notice that the main effect of the increase of the inner diameter of the loop tube is the progressive translation of the isostability lines towards lower Grm values. To talk of a translation is legitimated by the observation that the position of the isolines changes mainly regardless of the aspect ratio, which corresponds to a reduction of the stable region. In other words, the higher is the diameter of the tubes, the weaker is the system stability. This can be explained as a consequence of the reduction of the overall friction caused by the increase of the tube diameter. In fact, the diminishment of the friction is associated to the weaker damping of the perturbations that leads the system to become unstable. Notice also that the reduction of the system stability is particularly strong for the lower values of the diameter, e.g. when increasing the diameter from 0.010 to 0.015, whereas it becomes almost negligible for the higher diameters, e.g. passing from D ¼ 0:030 to 0.035 m. This is also the reason why further increases of the diameter have not been considered in this study.
5. Conclusions The aim of the present study has been the definition of an analytical methodology for the construction of stability maps for rectangular natural circulation loops. The proposed approach has been based on linearisation and analysis of a high order model for this kind of systems. The analysis of the model has aimed to characterise the system stability through the computation of the complex eigenvalues of the Jacobian matrix of the linearised model and the study of the sign of their real part. For a given diameter of the loop pipes, a map is then created by iterating the approach for various operating condition, defined by the modified Grashof number, Grm , and for different geometrical configuration of the loop, expressed by the ratio between the loop height and width. Several maps have been constructed in this way, each differing from the others for the inner diameter of the loop adopted during the calculation. Reported results show the convenience of adopting loops characterised by low aspect ratios, which ensure stable behaviour for higher Grm and are less sensitive to modification or perturbation of the operating condition. Moreover it has been pointed out the negative effect on stability played by the increase of the diameter of the loop pipes. The proposed maps represent a useful tool for design and operation of natural circulation loops, in fact they account for the dependence of the system stability on both the operating condition and the loop geometry.
References [1] P.K. Vijayan, A.K. Nayak, D.S. Pilkhwal, D. Saha, V. Venkat Raj, Effect of loop diameter on the stability of single-phase natural circulation in rectangular loops, in: Proceedings of NURETH-5, Salt Lake City, 1992, pp. 261–267.
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[2] J. Miettinen, T. Kervinen, H. Tomisto, H. Kantee, Oscillations of single-phase natural circulation during overcooling transient, in: Proceedings of ANS Topical Meeting, Atlanta, 1987, pp. 20–29. [3] R. Greif, Y. Zvirin, A. Mertol, The transient and stability behavior of a natural convention loop, Transaction of the ASME 101 (1979) 684–688. [4] D.B. Kreitlow, G.M. Reistad, C.R. Miles, G.G. Culver, Thermosyphon models for downhole heat exchanger applications in shallow geothermal systems, ASME Journal of Heat Transfer 100 (1978) 713–719. [5] Y. Zvirin, A. Shitzer, A. Bartal-Bornstein, On the stability of the natural circulation solar heater, in: Proceedings of the Sixth International Heat Transfer Conference, Toronto, 20, 1978, pp. 997–999. [6] Y. Zvirin, A review of N.C. loops in PWR and other systems, Nuclear Egineering Design 67 (1981) 203–225. [7] D. Japikse, Advances in thermosyphon technology, in: J.P. Harnett, T.F. Irvine (Eds.), Advances in Heat Transfer, vol. 9, Academic Press, New York, 1973. [8] G. Cammarata, A. Fichera, M. Frogheri M. Misale, M.G. Xibilia, A new modelling methodology of natural circulation loop for stability analysis, in: Proceedings of Eurotherm Seminar No. 63, Single and Two-Phase Natural Circulation, Genova, 1999, pp. 151–159. [9] Y.-Z. Wang, J. Singer, H.H. Bau, Controlling chaos in a thermal convection loop, Journal of Fluid Mechanics 237 (1992) 479–498. [10] J. Singer, Y.-Z. Wang, H.H. Bau, Controlling a chaotic system, Physical Review Letters 66 (9) (1991) 1123–1125. [11] M. Sen, E. Ramos, C. Trevino, The toroidal thermosyphon with known heat flux, International Journal of Heat and Mass Transfer 28 (1) (1985) 219–233. [12] M. Sen, E. Ramos, C. Trevino, O. Salazar, A one-dimensional model of a thermosyphon with known wall temperature, International Journal of Heat and Fluid Flow 8 (3) (1987) 171–181. [13] M. Sen, E. Ramos, C. Trevino, O. Salazar, The effect of axial conduction on a thermosyphon with prescribed heat flux, European Journal of Mechanics, B––Fluids 8 (1) (1989) 57–72. [14] M. Gorman, P.J. Widman, K.A. Robbins, Nonlinear dynamics of a convection loop: a quantitative comparison of experiment with theory, Physica 19-D (1986) 255–267. [15] P.J. Widman, M. Gorman, K.A. Robbins, Nonlinear dynamics of a convection loop II: chaos in laminar and turbulent flows, Physica 36-D (1989) 255–267. [16] E.N. Lorenz, Deterministic nonperiodic flow, Journal of Atmospheric Science 20 (1963) 131–141. [17] L. Cammarata, G. Desrayaud, A. Fichera, A. Pagano, An ordinary differential model for rectangular circulation loops, in: Twelth International Heat Transfer Conference, Grenoble, 2002. [18] P.K. Vijayan, H. Austregesilo, Sealing laws for single-phase natural circulation loops, Nuclear Engineering and Design 152 (1994) 331–347. [19] J.C. Ferreri, Single-phase natural circulation in simple circuits––reflections on its numerical simulation, in: Eurotherm Seminar 63, Single and Two-Phase Natural Circulation, Genova, 1999, pp. 17–30. [20] K. Chen, On the oscillatory instability of closed-loop thermosyphons, Journal of Heat Transfer 107 (1985) 826– 832. [21] A. Rodriguez-Bernal, E. Van Vleck, Diffusion induced chaos in a closed loop thermosyphon, SIAM Journal of Applied Mathematics 58 (4) (1998) 1072–1093. [22] L. Cammarata, G. Desrayaud, A. Fichera, A. Pagano, Experimental validation of a model-based controller for a rectangular natural circulation loop, in: B. Sunden, C.A. Brebbia (Eds.), Advanced Computational Methods in Heat Transfer VII, Wit Press, Southampton, Boston, 2002.