On the validity of Boussinesq approximation in

International Journal of Thermal Sciences 105 (2016) 224e232

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On the validity of Boussinesq approximation in transient simulation of single-phase natural circulation loops Mayur Krishnani, Dipankar N. Basu* Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 September 2015 Received in revised form 20 February 2016 Accepted 4 March 2016 Available online 22 March 2016

Application of One-dimensional numerical models with Boussinesq approximation is a common approach for stability evaluation of single-phase natural circulation loops. Present work focuses on assessing the viability of such approximation during nonlinear stability appraisal following transient simulation. Accordingly a 3-D computational model of a rectangular loop is developed, with heating at bottom horizontal arm and isothermal sink around the top. Transient conservation equations are solved, performing simulations both with complete variation of all relevant thermophysical properties and the simple Boussinesq model. System exhibits unstable behavior with increase in both heater power and sink temperature, the nature of the temporal response being significantly affected by the coolant-side condition. Boussinesq model predicts instability for substantially lower power levels, along with bidirectional pulsing in flow rate and presence of secondary motion at the corners of the horizontal arm. The system also takes much longer time to initiate the flow, compared to the model with complete property variation, the deviation being larger at enhanced power levels. Hence the Boussinesq approximation provides a highly-conservative estimate of the stability boundary and is not a practicable tool for transient analyses of single-phase loops, particularly at higher powers. © 2016 Elsevier Masson SAS. All rights reserved.

Keywords: Natural circulation loop Stability Transient response Boussinesq approximation

1. Introduction Stability appraisal of single-phase natural circulation loops (NCLs) is of interest both to engineers, due to its practical relevance, and mathematicians, as an excellent non-linear test problem. Its inherent reliability and enhanced passive safety have encouraged application in critical fields such as nuclear reactor cooling [1e4] and ship propulsion [5,6], where ensuring a stable zone of operation is an obligation. However, the fully-coupled nature of momentum and thermal transports and dependence on the prevailing body force field, administers high sensitivity to the operating conditions and susceptibility to instability. As was summarized by Zvirin [7], motion in NCL is initiated by the favourable instability of the second kind, among the four possible types. However the third and fourth kinds of instabilities, namely oscillation growth in steady flow and multiple steady-state solutions, are of major concerns in the practical systems. First reasonable explanation about the emergence of oscillations was proposed by Welander [8],

* Corresponding author. E-mail address: [email protected] (D.N. Basu). http://dx.doi.org/10.1016/j.ijthermalsci.2016.03.004 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.

assuming the fluid to behave like a pendulum, which was subsequently demonstrated experimentally by Creveling et al. [9]. An exact expression for the steady-state velocity was developed by Sen et al. [10] following a 1-D model, which showed the possibility of having zero to three solutions. These early studies made the path for several future endeavour towards exploring the concept of NCL stability, using both experimental and theoretical frameworks. Following the development in computational resources over last two decades, numerical simulation using in-house or commercial codes has emerged as the most common approach for NCL characterization. While a few researchers employed the simplistic Lorenz model [11e13], mostly to identify different possible dynamical flow regimes and nature of bifurcations under unstable condition, the applicability is limited, owing to the assumed nature of velocity and temperature fields. Quite a few recent applications of multidimensional system codes can also be found in literature [14e17], which shows encouraging prospects in reproducing experimental results. However, the huge computational cost required for transient simulation has restricted its use generally to steady-state cases. Therefore adoption of 1-D transient model has traditionally been the preferred option for stability evaluation of single-phase NCLs, along with linear stability analysis. Eigenvalues

M. Krishnani, D.N. Basu / International Journal of Thermal Sciences 105 (2016) 224e232

obtained from the linearized set of conservation equations can only predict the stability threshold, whereas detailed knowledge about the nature of fluctuations and flow reversals can only be identified following the temporal evolution of system variables. Ramos et al. [18] were probably the first to attempt numerical modeling of a loop with variable cross-sectional area. Their approach was subsequently improved by Bernier and Baliga [19], as they coupled 2-D finite volume simulation in the heater and cooler with 1-D loop momentum equation. Development of the pioneering computational model using explicit finite difference method can be credited to Vijayan and Date [20] for a figure-of-eight configuration. Their work, however, was aimed towards steady-state analyses, which was extended towards stability appraisal in their subsequent study [21]. Initial perturbation was provided in two different ways, which resulted in wide disparity between marginal stability curves. That was rectified by Nayak et al. [22], as close agreement was reported between linear and nonlinear stability maps. Similar study was also reported by Mousavian et al. [23], as they discussed about the role of grid spacing on transient prediction and suggested the use of finer meshes. That conclusion was in line with the work of Vijayan et al. [24] employing 1-D system code ATHLET, to facilitate repeated switching between laminar and turbulent frictional correlations. More elaborate discussion about the role of grid spacing, time steps and adopted discretization scheme was presented by Ambrosini and co-workers in a series of studies [25e27]. They suggested the use of explicit upwind scheme and Courant number close to one to circumvent numerical diffusion. The importance of using geometry-specific closure laws was also emphasized [27], supporting the discussion of Vijayan [28]. Such guidelines have since been followed in several subsequent studies [29e31], thereby helping the development of a substantial database on the transient behavior of single-phase NCLs. It is quite evident that numerous aspects of nonlinear stability analysis have received thorough attention. However one common factor among different 1-D models is the adoption of Boussinesq approximation to couple the thermal and momentum fields

(a)

1004

Density

1000

996

992 IAPWS-IF97

988

Boussinesq model

984 275

285

295

305

315

325

Temperature (K)

Dynamic viscosity (mPa-s)

(b)

55 Viscosity

45

Expansion coefficient

1.5

35 1.0

25 Reference values

15

0.5 5 T0=288 K 0.0

Thermal expansion coefficient ×105 (K-1)

2.0

-5 275

285

295

305

Temperature (K)

315

325

Fig. 1. Variation of thermophysical properties of water with temperature at 1.013 bar and comparison with Boussinesq model.

225

Fig. 2. Schematic representation of the rectangular NCL under consideration (drawing not to scale).

Table 1 Details of the mesh systems adopted for numerical simulation.

No. of nodes No. of elements Orthogonal quality Skewness Element quality

Model 1

Model 2

Model 3

143,488 135,900 0.977 0.13 0.24

255,460 240,768 0.979 0.12 0.50

474,496 449,280 0.982 0.11 0.58

[22e24,28,31]. That considerably simplifies the analysis, as density is considered as a linear function of temperature only during the body force evaluation. All other thermophysical properties are taken as constant, thereby eliminating the exclusive inclusion of the equation-of-state in the solution framework. A suitable reference temperature needs to be selected, sink temperature or ambient temperature being the common choices. However, property variation for common fluids generally exhibit substantial nonlinearity with temperature variation. As can be seen from Fig. 1a, density of water estimated using Boussinesq model provides reasonable value only around the selected reference temperature (¼ 15 C in present figure), in comparison to the IAPWS-IF97 standard [32]. Thermal expansion coefficient itself can increase more than 10 times with rise in water temperature from 378 to 388 K (Fig. 1b). Basu et al. [33] suggested the use of loop-averaged fluid temperature as the reference for lessening the error in property estimate. That is also not the greatest option considering the small density differential required in developing the driving force in NCLs and to amplify the transient fluctuations. Therefore it is essential to perform a comprehensive analysis of single-phase NCLs with the most accurate equation-of-state, to ascertain the viability of Boussinesq approximation and its influence on the transient response of the loop. Presence of local buoyancy forces due to the temperature variation across any crosssection can also affect the stability behavior, thereby necessitating multi-dimensional modeling. The same is performed in the present

226


Fig. 3. Pictorial views of the mesh system employed for simulation.

approximation.

10000

2. Computational model and numerical procedure 1000

Ress 100

Turbulent correlation Laminar correlation Present simulation

10 1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

Grm/NG Fig. 4. Comparison of the steady-state simulation results with the correlation of Vijayan [28].

0.06 IAPWS-IF 97 standard Boussinesq approximation

Mass flow rate (kg/s)

0.05

The need of employing multi-dimensional model for NCL simulation was stressed upon by Ambrosini [34], as 1-D codes are unable to predict important phenomenon like flow stratification and secondary flows. The ability of commercial system codes in predicting such intricate details was demonstrated by Ambrosini et al. [27] and Kumar et al. [16]. Subsequently ANSYS-Fluent was employed by a few researchers [14,17,35] with reasonable success. Hence ANSYS-Fluent 14.5 is employed in the present study for numerical simulation of the rectangular loop presented in Fig. 2. Associated conservation equations are summarized below.

vr v þ ruj ¼ 0 vt vxj

(1)

v v vp vtji ðrui Þ þ þ þ rgi ruj ui ¼ vt vxj vxi vxj

(2)

v v v rCp uj T ¼ rCp T þ vt vxj vxj

0.04

* vT

l

vxj

! þ uj tji

þ SE

(3)

Here, 0.03

tji ¼ ðm þ mt Þ 0.02

0.01 0.5

1.0

1.5

Power (kW)

2.0

2.5

Fig. 5. Variation in steady-state mass flow rate with heater power at Tc ¼ 275 K following IAPWS-IF97 standard and with Boussinesq approximation.

study by developing a 3-D model of a rectangular loop with water as the working medium. The loop is subjected to horizontal heating in constant power mode and horizontal cooling through an isothermal sink. Other sections of the loop are insulated. Transient simulations are accomplished adopting both IAPWS-IF97 standard and the Boussinesq model separately, and resultant temporal responses are compared to determine the practicability of such

vuj ¼ vxi

m þ rCm

k2 ε

vuj vxi

l* ¼ l þ

Cp mt Prt

SE is the volumetric energy source term per unit time and its magnitude varies along the length of the loop, being positive in heater section, negative in the cooler and zero for the rest of the loop. The RNG kε turbulence model is employed following Pilkhwal et al. [14] with standard values of the coefficients. Secondorder upwind scheme is used for spatial discretization, whereas the first-order fully-implicit scheme is followed for the temporal terms. Pressure-velocity coupling is resolved using the PISO algorithm, along with PRESTO scheme for discretization of the pressure terms [14]. As mentioned earlier, property variations are modelled in two different ways, namely, following IAPWS-IF97 standards and adopting Boussinesq approximation. In case of the former, NIST real gas model is activated in each Fluent 14.5 session, by dynamically loading REFPROP v7.0 as a shared library. Appropriate grid selection is mandatory for ensuring the correctness of any numerical simulation. Non-uniform staggered mesh


227

Fig. 6. Temporal development of loop flow rate and temperature at source centre for two different power levels with Tc ¼ 275 K.

Fig. 7. Temporal development of loop flow rate at 1.5 kW heater power for two different sink temperatures.

system is employed here, with finer meshes near the wall and also in the connecting elbows, for capturing larger gradients of flow variables. Progressively refined meshes are attempted with, in an effort to get the results with reasonable accuracy, along with a practicable simulation period. Details of three different meshes are summarized in Table 1. Increasing the number of elements from model 2 to model 3, less than 1% change can be observed in average mass flow rate and also in fluid temperatures at different sections of the loop, but at the expense of substantial increase in computational time. Therefore the mesh 2 is followed for all subsequent simulations. Pictorial views of the concerned mesh are shown in Fig. 3. Steady-state predictions from the developed model is compared with the widely-used correlation of Vijayan [28] and the same is presented in Fig. 4. Calculated values are in close agreement with

the relation over the entire range of Reynolds number (Re) considered in the present study. That establishes the steady-state validation of the developed model and allows to proceed for transient analyses. A constant time step of 0.1 s is followed during transient simulation. Computation with smaller time steps are also attempted, without any noticeable deviation in the profiles, and hence this particular value is adhered with. It must also be mentioned that both the mesh systems 2 and 3 (Table 1) produces identical transient predictions and hence the reported transient traces can be considered to be both grid- and time-independent. All the subsequent analyses are performed at 1.013 bar pressure level. 3. Results and discussion In order to ascertain the steady-state behavior, simulations are

228


Fig. 8. Temporal development of loop flow rate and phase portrait at 2.5 kW heater power for three different sink temperatures.

carried out by simultaneously solving the conservation equations (Eqs. (1)e(3)) without the corresponding temporal terms. With increase in the power supply, the temperature differential across the heater increases nearly in proportion. Therefore a larger density difference is available between the vertical arms, leading to enhanced buoyancy effect. Raised temperature level also reduces the average viscosity of water, final outcome being an increase in the loop flow rate. Both the property models employed here, namely the IAPWS-IF97 standard and Boussinesq model, exhibit similar trends, albeit with a substantial difference in concerned magnitudes (Fig. 5). Over the entire range of power considered here, the Boussinesq model predicts significantly higher flow rate, with the percentage deviation gradually reducing with input power. Subsequently water also assumes a lower temperature level.

Transient simulations are performed for five different power levels, starting from 0.5 kW to 2.5 kW in steps of 0.5 kW, and also for three different sink temperatures, namely, 275, 283 and 293 K. The loop is initially assumed to be at quiescent state and 300 K water temperature. On application of power, water in contact with the wall of the bottom horizontal arm gets heated, with the bulk fluid still retaining the initial temperature. That sets up convection current and local motion in the bottom arm. Similarly water in contact with the wall in the cooler section loses energy to the sink. Bulk motion inside the loop is initiated after an interval, when the local convection in both the heater and cooler are sufficiently strong. Present configuration being symmetric, it is not possible to guess the flow direction in priori and hence the counter-clockwise direction is chosen to be positive for the subsequent discussion.


Fig. 9. Effect of grid selection on the transient profile for 1.5 kW heater power and Tc ¼ 275 K following IAPWS-IF97 standard.

Fig. 10. Comparison of temporal development of mass flow rate for 500 W heater power and Tc ¼ 275 K following IAPWS-IF97 standard and with Boussinesq approximation.

Table 2 Comparison of the predicted time required for flow initiation with the two models. Power (kW)

0.5 1.0 1.5 2.0 2.5

Time required for flow initiation (s) with Tc ¼ 275 K IAPWS-IF97 standard

Boussinesq approximation

62.8 48.8 44.3 39.7 36.7

67.1 66.2 65.7 65.1 63.3

Simulations are executed initially adopting IAPWS-IF97 model, thereby allowing complete variation of all thermophysical properties. Corresponding transient developments of loop flow rate for two different power levels are presented in Fig. 6, along with concerned profiles of area-averaged water temperature at the source centre. It is clearly evident that water temperature rapidly increases on application of power, without any bulk motion. When the effect of local convection reaches the vertical arms, a substantial temperature difference is already available between them. Associated large buoyancy force initiates a considerably large flow rate in the loop, which subsequently causes a rapid decline in the water temperature level, prompting an immediate reversal in the flow direction. For lower power levels, frictional force is able to suppress the subsequent oscillations and the system gradually becomes

229

stable (Fig. 6a), providing a steady clockwise motion. For higher power levels, however, the loop suffers repeated flow reversals (Fig. 6b), characterizing an unstable system. Elevated water temperature results in lower viscosity and hence enhances instability, viscous resistances being the stabilizing factor. As can be seen from Fig. 6, rise in q_ h from 1 kW to 2 kW causes a rise in mean water temperature by about 20 K, which corresponds to about 40% reduction in viscosity (Fig. 1) and that explains the unstable nature. The role of sink temperature is also important in this context. Warmer coolant corresponds to higher temperature levels inside the loop and consequently lower viscosity, lessening the restraining effect inside the loop. That introduces more instability in the system and the same can be demonstrated from Fig. 7. For 1.5 kW heater power, the loop exhibits stable response for a sink temperature of 275 K, but experiences repeated reversals with only 8 K rise in the sink temperature. Even the nature of system response for a typically-unstable system can also deviate significantly with change in Tc, as can be seen from Fig. 8 for 2.5 kW heater power. For both low and high Tc, profiles of mass flow rate exhibit bidirectional pulsing, along with dumbbell-shaped phase portrait. For intermediate sink temperature, however, the response becomes quite chaotic and consists of low-amplitude fluctuations. Accordingly a butterfly shape can be observed on the phase plane. It is important to have a brief discussion on the role of grid selection on the resultant transient traces in the present context. Substantial volume of NCL research is available which discusses the role of truncation error and grid spacing on the transient profiles [36e38]. It is well-established that the intrusion of artificial diffusion through an improper mesh selection can lead to completely wrong prediction and hence it is mandatory to optimize the selected grid. Fig. 9 compares the transient variation of mass flow rate with two different grid systems under the same set of operating conditions. While the model 1 predicts a highly-stable system, repeated flow reversals can be observed with model 2. Any further augmentation in the number of grid elements is found to have insignificant effect on the prediction and hence the model 2 can be viewed as the optimum choice from both steady and transient point of view. In order to evaluate the validity of employing Boussinesq approximation, simulations are attempted setting reference temperature (T0) to 288 K. Accordingly the body force term of Eq. (2) is modified as r0 gi bðT T0 Þ, r0 being the water density at T0, and r¼r0 in all other terms. Other thermophysical properties remain constant at values corresponding to T0. Temporal development of resultant mass flow rate with 0.5 kW heater power is compared with the same estimated following complete property variation in Fig. 10. Boussinesq approximation yields two flow reversals compared to one with the other model and also predicts nearlydouble steady-state flow rate, at the cost of significantly higher settling time. Change in water temperature inside the horizontal arms induces local convective motion. However, density variation predicted with Boussinesq model being relatively smaller than the same with IAPWS-IF97 standard (Fig. 1), developed local buoyancy field is also weaker and hence the loop responds late. Time required to initiate bulk flow for different power inputs are summarized in Table 2, which clearly demonstrate the sluggish response with Boussinesq approximation, with the discrepancy being amplified at higher powers. Largest temperature achieved by the fluid is also relatively higher, with corresponding buoyancy introducing larger instability into the system. Steady-state temperature level predicted for q_ h ¼ 0:5 kW with the total property variation is about 4 K higher than the same for other, owing to the lower mass flow rate. Therefore the nature of both steady-state and transient responses are affected by the adopted simplification. In fact, the entire nature of stability prediction can get reversed

230


Fig. 11. Temporal development of loop flow rate for two different power levels with Tc ¼ 275 K following Boussinesq approximation.

Fig. 12. Temporal development of loop flow rate and phase portrait at 2.5 kW heater power and Tc ¼ 283 K following Boussinesq approximation.

Fig. 13. Comparison of temperature contours at the centre of the left vertical arm for q_ h ¼ 1.5 kW and Tc ¼ 275 K following two different models.

due to such approximations. Fig. 11 presents the transient profiles of loop flow rate for two different cases, which were identified to be stable earlier. However the loop behaves unstable for heater power as low as 1 kW corresponding to the lowest sink temperature considered. The nature of response for unstable systems can also deviate significantly. Loop response on the state-space plot for 2.5 kW heater power and 283 K sink temperature represents period-doubling bifurcation (Fig. 12), instead of the chaotic nature identified earlier. Therefore it is safe to conclude that Boussinesq

approximation can lead to wrong prediction about the stability behavior of NCL. The above conclusion can be reinforced by correlating to the study of Kizildag et al. [39] on natural convection inside a differentially heated cavity for large aspect ratios. For more than 30 C temperature differential, markedly different flow configurations were reported, while the symmetry in isotherms was lost just for 10 C temperature difference. The same can be observed from Fig. 13 for a particular condition at the middle of the vertical riser.


231

Fig. 14. Comparison of velocity vectors at the left corner of the bottom horizontal arm for q_ h ¼ 1.5 kW and Tc ¼ 275 K following two different models.

Simulation with complete property variation predicts asymmetric temperature contours, with the highest temperature zone shifted towards the inner surface due to the local buoyancy effects. Boussinesq model, however, predicts a symmetric profile, with the lowest temperature fluid located centrally. Here the variation in water temperature at any cross-section is about 2 C, which can produce significant alteration in property values. The symmetric variation with Boussinesq model allows the possibility of bidirectional flow, as can be found from Fig. 14. For the same operating condition, smooth unidirectional flow is observed with IAPWS-IF97 standard. But the simplified model exhibit significant presence of secondary flow at the same location, which eventually lead to the unstable behavior. Hence the adoption of Boussinesq approximation predicts instability for much smaller powers and accordingly leads to a highly-conservative estimate of the stability threshold. Before drawing the final conclusions, one important factor to consider is the computational cost involved in adopting an exhaustive property code like IAPWS-IF97 in a computational framework. The Boussinesq model being very simplistic, with linear density variation and constant magnitude of other thermophysical properties, it is invariably able to perform a simulation within much shorter time span. The complete model, on the contrary, estimates all the required properties as functions of local pressure and temperature at all the nodes and also at every time step, thereby imposing substantial computational resource requirement and also significant simulation time. That is probably the main reason behind the popularity of Boussinesq model during stability appraisal of single-phase NCLs, particularly in 1-D models. As such codes ignore several important features like the flow stratification, wall functions, secondary flow etc., use of an approximate property model can be justified. However, as has already been observed earlier, the Boussinesq approximation in a 3D system consistently lead to over-prediction under steady-state and occasionally wrong nature of system stability. Still considering the significant computational cost associated with IAPWSIF97, probably a guideline can be proposed. Following the argument of Kizildag et al. [39] and also observing present trends, it is proposed to use the Boussinesq model as long as the temperature variation across the heater is limited to 10 C and also the average water temperature is close to T0. The later constraint is important to maintain a reasonable representation of fluid density throughout the loop. However, the average temperature is not known in priori and is only a part of the steady-state solution. Therefore a bit of trial-and-error may become necessary. Simulations can also be repeated with different choices of the reference temperature. It must be noted that, even under such conditions, the Boussinesq

model is likely to over-predict the steady-state flow rate and may provide a different transient trace. But the conclusion about the nature of system stability is expected to be unaffected. The concerned limit of heater power with the present system is about 800 W with 275 K sink temperature, up to which both the models predict qualitatively similar transient profiles. 4. Conclusions For systematic appraisal of the viability of employing Boussinesq approximation during transient simulation of single-phase NCLs, a 3-D computational model of a rectangular loop is developed. Grid- and time-independent nature of the solution is ensured and steady-state results are satisfactorily compared with existing correlation from literature. Transient simulations are performed employing the most-updated property relations for water, as well as, the simplified Boussinesq model. The major observations can be identified as followings. (1) System exhibits unstable behavior with increase in both heater power and sink temperature due to the reduction in the magnitude of frictional resistances. The nature of unstable response can drastically vary with the sink temperature. (2) Simulation with Boussinesq approximation takes much longer time to capture flow initiation in the loop and the concerned difference with the other model increases with heater power. (3) A larger steady-state mass flow rate and hence a lower temperature level is predicted with Boussinesq model. (4) The simplified model predicts instability for much lower power levels, characterized by bidirectional pulsing in loop flow rate and presence of secondary motion at the corners. (5) Temperature contour predicted with Boussinesq model at any particular cross-section is symmetric, thereby minimizing the effect of local convection. Therefore it can definitely be concluded that the Boussinesq approximation is not valid for nonlinear stability evaluation of single-phase NCLs, particularly at higher power levels. Such simplification only provides a highly-conservative estimate of the stability boundary. It is therefore essential to incorporate more accurate property relation in the solution framework. The Boussinesq model can be employed to get a rough estimate at lower powers, as long as the fluid temperature variation is limited to 10 C and average fluid temperature is around the reference temperature.

232


Acknowledgment Financial support provided by the Department of Science and Technology, India under SERB Fast track project for young scientists vide sanction no. SERB/F/5300/2012-2013 dated 19-12-2012, is gratefully acknowledged. Nomenclature A d G Grm H L p q_ h SE T x Cp g k NG Pr Re t u

[12] [13] [14]

[15] 2

Cross-sectional area (m ) Diameter (m) Mass flux (kg m2 s1) Modified Grashof number ð¼ gbd3 r20 q_ h H=ACp m3 Þ Loop height (m) Loop length (m) Pressure (N m2) Heater power (W) Volumetric energy source (W m3) Temperature (K) Coordinate direction Specific heat (J kg1 K1) Gravitational acceleration (m s2) Turbulent kinetic energy (m2 s2) Geometric parameter (¼L/d) Prandtl number Reynolds number (¼Gd/m) Time (s) Velocity (m s1)

Greek symbols Volumetric expansion coefficient (K1) Thermal conductivity (W m1 K1) Density (kg m3) ε Turbulent dissipation rate (m2 s3) m Dynamic viscosity (kg m1 s1) t Shear stress (N m2)

b l r

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