- Email: [email protected]
without a porous medium
International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Large-eddy simulations (LES) of temperature fluctuations in a mixing tee with/without a porous medium T. Lu a, P.X. Jiang b,*, Z.J. Guo a, Y.W. Zhang b, H. Li b a b
School of Mechanical and Electrical Engineering, Beijing University of Chemical Technology, Beijing 100029, China Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 30 April 2009 Received in revised form 21 June 2010 Accepted 22 June 2010
Keywords: Large-eddy simulations Temperature fluctuations Mixing Porous media
a b s t r a c t Temperature fluctuations in a mixing tee were simulated with and without a porous media in FLUENT using the LES turbulent flow model with the sub-grid scale (SGS) Smagorinsky–Lilly (SL) model with buoyancy. The normalized mean and fluctuating temperatures are used to describe the time-averaged temperatures and the time-averaged temperature fluctuation intensities. For the tee junction without the porous media, the predicted normalized mean temperature and temperature fluctuations compare well with previous experimental data. Comparison of the numerical results with the porous media with both experimental and numerical data without porous media shows that the porous media significantly reduces the temperature fluctuations. Moreover, analysis of the temperature fluctuations and the power spectrum densities (PSD) at the locations having the strongest temperature fluctuations in the tee junction shows that the porous media significantly reduces the thermal fatigue effects and can be useful in various structures such as tee junctions, elbows, piping systems. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Thermal fatigue is still not fully understood. One of the main obstacles to a full understanding resides in the multi-domain nature of the loading and associated damage, involving three complementary scientific disciplines: thermal-hydraulics, thermo-mechanics and materials science [1]. Prediction of thermal fatigue in mixing tees is a challenging subject that is needed for life cycle management of nuclear power plant piping systems. Thermal striping is one phenomenon that has been identified as a cause of thermal fatigue failures [2,3]. Thermal striping characterizes the phenomenon where hot and cold fluid mixing results in temperature fluctuations of the mixing fluids near a wall. The temperature fluctuations of the mixing fluids may then cause cyclical thermal stresses and subsequent fatigue cracking of the pipe wall. Research on thermal striping was initially carried out for liquidmetal-cooled fast breeder reactors in the 1980’s because of the high thermal conductivity of the liquid metal coolant [4]. Areas susceptible to thermal striping include components in the core outlet region, such as the core upper plenum, flow guide tube, and control rod upper guide tubes. Outside the core region, components where hot and cold fluids come in contact or mix, such as tee junctions, elbows, and leakage from valves may also be affected. The focus of thermal striping studies shifted to light water reactors * Corresponding author. E-mail address: [email protected] (P.X. Jiang). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.07.001
after several incidents of piping failure at some nuclear power plants [5,6]. Following the discovery of a leak due to a longitudinal crack at the outer edge of a bend in the mixing zone of the residual heat removal system of the Civaux nuclear power plant unit in 1998, metallurgical studies showed that the origin of this degradation was thermal fatigue cracking [1,7]. This incident was caused by thermal fatigue associated with temperature fluctuations in the fluid at the mixing tee. The piping system components that are most susceptible to thermal striping fatigue cracking are mixing tees in the residual heat removal systems in both boiling water reactors and pressurized-water reactors. Therefore, the European commission funded the international project ‘‘Thermal Fatigue Evaluation of Piping System Tee-connections” with the main objective being to advance the accuracy and reliability of thermal fatigue load determinations in engineering tools and to formulate research oriented approaches to outline a science based practical methodology for managing thermal fatigue risks [7]. To reach these objectives some of technological and scientific aims are focused on assessments of the fatigue significance of turbulent thermal stratification and mixing effects in the identified piping system tee-connections and determination of the effects of variable mass flow rates in the ‘‘run” and ‘‘branch” pipes by experimental and numerical simulations, e.g. computational fluid dynamics (CFD). The main goal of thermal striping evaluations is to identify the temperature fluctuation magnitudes and frequencies. This can be achieved by mock-up experiments or three-dimensional, unsteady turbulent modeling that resolves both large and small-scale
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
turbulent motions. This requires CFD simulations using either direct numerical simulations (DNS) [8] or large-eddy simulations (LES) [9]. Thermal striping experiments have been performed using flow visualizations, temperature measurements, and velocity measurements [8,10]. In experiments where fluid and pipe wall temperatures were measured, both the frequencies and the magnitudes of the temperature fluctuations were obtained at selected locations in the pipes. There are few studies using analytical approaches or computer modeling of thermal striping due to the extensive computational capacity required for unsteady, three-dimensional turbulent flow simulations. Thermal striping was modeled previously using LES and DNS [4,8,11], however, these studies were performed for only a small volume around the mixing zone due to the computational constraints. The DNS method requires extreme grid resolution and computational expense that is not viable for realistic geometries and flow conditions. However, there have been many recent develops in LES methods for turbulent flows. The present work simulates mixing flows with relatively high Reynolds numbers and low Richardson numbers both in the main duct and the branch duct using the FLUENT LES model with the sub-grid scale (SGS) Smagorinsky–Lilly (SL) model. The normalized mean temperatures and the normalized temperature fluctuations are used to describe time-averaged temperatures and time-averaged temperature fluctuation intensities. The numerical results for the normalized mean temperature and the normalized temperature fluctuation are compared with experimental data published in the literature [8]. Wakamatsu et al. [12] reported that the temperature oscillations causing thermal striping have relatively high frequencies on the order of several Hz, but there is little information about the power spectrum densities (PSD) of temperature fluctuations in the literature [13]. The PSD of temperature fluctuations, which statistically represent how power densities change with the temperature fluctuation frequency, is one of the key parameters for characterizing thermal fatigue. Therefore, the present work presents the PSD of the normalized temperatures versus frequency using Fast Fourier Transforms (FFT) of the time series results to analyze the energy density. In many fields such as for piping systems of nuclear power plants and in the cooling of turbine blades, engineers try to design the pipe or blades so that they do not experience large temperature fluctuations resulting in thermal fatigue. The designers then seek to reduce the temperature fluctuations to improve the structural integrity. There have been many theoretical and numerical investigations [14–25] of fluid flow and heat transfer in channels fully or partially packed with porous media in recent years due to the interest in a wide range of engineering applications such as heat exchangers, heat pipes, electronic cooling, thermal insulation, geothermal energy systems, enhanced oil recovery, and drying. The porous material improves the convective heat transfer due to the high surface area to volume ratio in the system and the enhanced flow mixing caused by the tortuous path through the porous matrix which improves the thermal dispersion. Studies of the flow and heat transfer in porous media have resulted in great progress in recent decades. Most mathematical models are based on the volume-averaging theory popularized by Whitaker [26]. The Darcy, Darcy-Brinkman or Darcy–Brinkman– Forchheimer models are often applied to describe the fluid momentum conservation in a porous media [27–31]. There are two basic energy conservation models used for porous media analyses. One is the local thermal equilibrium model which uses one energy balance equation for both the fluid and the solid assuming thermal equilibrium between the solid matrix and the fluid [32–34], while the other is the local non-thermal equilibrium model which uses separate energy balance equations for the solid and fluid phases with different temperatures [35–37]. The local thermal equilibrium model is no valid when the particles or pores are not very small,
4459
when the thermal properties differ widely, when convective transport is not important, or when there is significant heat generation. The local non-thermal equilibrium model using two temperature equations should then be used to model the heat transfer between the solid phase and the fluid phase. However, the local non-thermal equilibrium model requires additional information on the interfacial heat transfer coefficient between the solid phase and the fluid phase [35,38], which limits application of the local non-thermal equilibrium model. Various investigations have studied the heat transfer enhancement in tubes or ducts with porous media. Numerical simulations by Yang and Hwang [29] investigated the turbulent heat transfer enhancement in a pipe filled with a porous media using the k–e turbulence model to calculate the fluid flow and heat transfer characteristics in a pipe filled with porous media. The flow resistance and heat transfer characteristics of the air flow from laminar to fully turbulent ranges of Reynolds numbers in a tube inserted with porous media were investigated experimentally and numerically by Huang et al. [30]. The porous media is also expected to reduce the temperature fluctuations in a tee by improving the flow mixing in the tee. In the present work, large-eddy simulations were used to investigate for temperature fluctuations during mixing of hot and cold fluids in a mixing tee partially packed with porous media in the mixing zone of the main duct. Comparisons of the numerical results for the porous media with numerical and experimental results without porous media show how the porous media reduces the temperature fluctuations. 2. Governing equations 2.1. Fluid flow region without porous media Fig. 1 shows a schematic of the mixing of hot and cold fluids in a tee junction as numerically simulated on the FLUENT platform. The tee junction is constructed of square ducts [8]. The main duct is 100 mm by 100 mm while the branch duct is 50 mm by 50 mm. Hot water at a temperature of 343.48 K and velocity of 0.15 m/s enters the main duct in the horizontal direction while cold water at a temperature of 296.78 K and velocity of 0.3 m/s enters the branch duct in the vertical direction. The filtered LES equations for isothermal incompressible flows with a passive scalar h transport are [9,39–41]:
i @ q @ qu ¼0 þ @t @xi i @ qu i u j @ qu @p @ þ ¼ q0 bðT T 0 Þg þ 2lSij sij @t @xj @xi @xj j @ h @ hu @ @h þ ¼ a qj @t @xj @xj @xj
ð1Þ ð2Þ ð3Þ
i are the velocity components, p is the pressure, q is the where u density, b is the thermal expansion coefficient, and q0 is the reference density at the reference temperature, T0. The molecular dynamic viscosity is denoted by l. a is diffusivity. The fluid mechanics are combined with the energy transport in the energy equation [42]
@T @ðui TÞ 1 @ 2 T @qi þ ¼ @t @xi RePr @x2i @xi
ð4Þ
where Re represents the Reynolds number and Pr is the molecular Prandtl number. qi represents sub-grid scale heat flux, which is modeled by the SGS model. As shown in Fig. 1, no-slip and adiabatic conditions were imposed at the walls as the velocity and thermal boundary conditions. The inlet velocity boundary conditions for both the main duct and the
4460
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
Fig. 1. Schematic of hot and cold fluids mixing in a tee junction.
branch duct are specified as the inlet flow boundary conditions with the out flow boundary condition used at the main duct outlet. The operating pressure was ambient pressure, 101,325 Pa. There, three governing equations for the flow and thermal boundary conditions were solved for the simulations. Eq. (1) is the conservation of mass. Eq. (2) is the Navier–Stokes equations which represent the momentum conservation. Eq. (4) represents the energy conservation. The LES approach was then used to simulate the fluid mixing mechanics. The details of SGS model are given in Appendix A. 2.2. Fluid flow region with porous media The center of the mixing zone is assumed to be the coordinate system origin. The copper porous media is then packed in part of the main duct from x/db = 2.5 to x/db = 2.5. With the assumption of constant porosity, /, the continuity equation is given by
/
p;i @ q @ qu ¼0 þ @t @xi
ð5Þ
Assuming local thermal equilibrium, which means that the solid and fluid temperatures are the same at every location, the governing energy equation can be written as: o @ðq cp;f u p;i TÞ @ n @ f ½/qf cp;f þ ð1 /Þqs cp;s T þ ¼ @t @xi @xi
keff ¼ /kf þ ð1 /Þks
gbðT cold T hot Þdb u2m
p;i @ qu p;i u p;j @ qu @p @ ¼ q0 bðT T 0 Þg þ ð2lSij sij Þ þ @xi @xj @t @xj
3. Numerical results
The first part of the last term on the right side (Darcy’s term) accounts for the microscopic viscous drag while the second part accounts for the form drag due to inertial effects (direction changes) inside the pores and to turbulent dissipation. For the particular case of stacked spheres, Ergun [43] suggested the following expressions for the constants C1 and C2
C1 ¼ C2 ¼
150ð1 /Þ
2
/3 dp 1:75ð1 /Þ /3 dp
ð7Þ
@qi @xi
ð9Þ
ð10Þ
ð11Þ
The water properties such as the dynamic viscosity, the specific heat at constant pressure, and the thermal diffusivity were assumed to be constants in the simulations.
The numerical procedure is described in Appendix B. The normalized mean temperatures and normalized temperature fluctuations are used to describe the time-averaged temperatures and time-averaged temperature fluctuation intensities [11,44]. The normalized temperature at a given location is defined as
T i ¼
T i T cold T hot T cold
ð12Þ
where T is the instantaneous temperature at a given location, Tcold is the cold fluid inlet temperature, and Thot is the hot fluid inlet temperature. The time-averaged normalized mean temperature at a given location is
ð8Þ
where dp is the equivalent diameter of the packed spheres. Here, we assumed dp = 28 mm and a porosity of about 30%.
The Smagorinsky–Lilly SGS model is used to simulate the turbulence in the porous media. The buoyancy force due to the temperature difference is incorporated by adding a body force term to the Navier–Stokes equations based on the Boussinesq approximation since the Reynolds numbers of 36, 300 in the main duct and 16, 300 in the branch duct are both relatively high in the turbulent flow range. The effects of buoyancy are related to the Richardson number defined as
Ri ¼
ð6Þ
!
where the effective thermal conductivity in the porous region, keff, is simplify computed as the volume average of the fluid conductivity, kf, and the solid conductivity, ks
p;i ¼ /u in;i , has been where the Dupuit–Forchheimer relationship, u in;i is the intrinsic (liquid) average of the local velocity used and u vector. For a simple homogeneous porous media, the momentum equations for the porous media are modeled by the addition of a momentum source term to the standard fluid flow model in Eq. (2). The momentum equation is then
p;i ju p;i Þ p;i þ C 2 qju ðC 1 lu
keff
@T @xi
T ¼
N 1 X T N n¼1 i
where N is the total number of sampling points.
ð13Þ
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
4461
Fig. 2. Combined temperature distribution and velocity vectors in the y = 0 mm plane at 6 s. (a) Without porous media; (b) with porous media.
The normalized temperature fluctuation is defined as the rootmean square (RMS) of the instantaneous temperature at a given location
T RMS
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 2 u1 X ¼t T i T N i¼1
ð14Þ
The temperature and velocity fields both with and without the porous media were calculated using the LES simulations with the SL SGS model. The combined temperature and velocity distributions on the y = 0 mm plane are shown in Figs. 2 and 3 for both cases at two different times. The temperature and velocity distributions on the plane of y = 0 mm at 6 s are quite different from those at 10 s
Fig. 3. Combined temperature distribution and velocity vectors in the y = 0 mm plane at 10 s (a) without porous media; (b) with porous media.
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
a
1.0 0.8 0.6 0.4 0.2
z/db
due to the unsteady turbulent flow. The results show that the buoyancy influences the mixing of the hot and cold fluids in the tee junction, with the hot fluid in the main duct rising upwards into the branch duct and mixing with the cold fluid in the branch duct. Moreover, the cause of the thermal striping phenomena can be seen with the fluid temperatures varying with time in the downstream section of the main duct after the hot and cold fluids mix. The temperature distributions correspond to the flow field shown by the velocity vectors in Figs. 2 and 3. Many vortices occur in both ducts downstream of the mixing zone and even upstream of the mixing tee, which improve the temperature distributions and affect the influence of the buoyancy on the mixing in the tee junction. However, the back flow up into the branch duct with the porous media is more serious than without the porous media due to the higher flow resistance in the main duct packed with the porous media. The mixing region from x/db = 0 to x/ db = 2.5 with the porous media is much narrower than without the porous media. Because the porous media restricts the flow, and results in more uniform temperatures and velocities with the porous media than without the porous media. The instantaneous temperatures and velocities at the intersections between the plane of y/db = 0.3 and the planes of x/db = 1 and 2 along the z direction were abstracted from the temperature and velocity fields in the tee junction to compare with experimental data without the porous media [8]. The normalized mean temperatures, T , and the normalized temperature fluctuations, T RMS , were obtained using Eqs. (13) and (14). The predicted normalized mean temperatures and normalized temperature fluctuations along these three lines are compared with the experimental data in Figs. 4 and 5. Firstly, the numerical results without the porous media in Fig. 4 for the normalized mean temperatures at different intersections agree well with the experimental data, which verifies the validity of the LES model for predicting the mixing of hot and cold fluids in a tee junction. For x/db = 1 (Fig. 4(a)), the fluid near the top wall (1 < z/db < 0.6) has a relatively high normalized mean temperature, which means that hot fluid is pushed up along the top wall as shown in Figs. 2 and 3 due to the downward flow of the cold fluid from the branch duct and the effect of buoyancy. The fluid in the middle height (0.6 < z/db < 0.2) has relatively low normalized mean temperatures, which means that the hot and cold fluids are well mixed. However, the normalized mean temperature of the fluid in the range of 0.6 < z/db < 1.0 is equal to 1, which indicates that the hot fluid from the main duct does not mix with the cold fluid from the branch duct in this region. The normalized mean temperature distribution at the intersection for x/db = 1 has a large range, which suggests that thermal striping will occur downstream of the mixing zone in the main duct. With further mixing of the hot and cold fluids in the main duct, the normalized mean temperature distributions along the lines of intersection for x/db = 2 (Fig. 4(b)) become smoother and the range of the normalized mean temperatures is narrower. The variations of the normalized mean temperatures with the porous media along the lines of intersection for x/db = 1 and 2 are much smaller than those without the porous media because the porous media restricts the turbulence and enhances the heat transfer. Although the normalized mean temperatures can describe how the fluid temperature varies at a location with time, they cannot explain the fluid temperature fluctuations. The distributions of the normalized temperature fluctuations along the same lines of intersection are shown in Fig. 5. The numerical results for the normalized temperature fluctuations without the porous media agree reasonably well with the experimental data. The normalized temperature fluctuations are most intensive for x/db = 1 (Fig. 5(a)) than in the other locations for both cases which implies that the hot and cold fluids are mixing strongly here with the strongest velocity fluctuations. The largest normalized temperature fluctuations at each location near the middle height are reduced as the flow
0.0 -0.2 -0.4
y/db=0.3 x/db=1 Exp.[8] Without Porous With Porous
-0.6 -0.8 -1.0 0.0
b
0.2
0.4
T
*
T
*
0.6
0.8
1.0
0.6
0.8
1.0
1.0 0.8 0.6 0.4 0.2
z/db
4462
0.0 -0.2 -0.4
y/db=0.3 x/db=2
-0.6
Exp.[8] Without Porous With Porous
-0.8 -1.0 0.0
0.2
0.4
Fig. 4. Normalized mean temperature distributions in different x planes at y/db = 0.3 (a) x/db = 1; (b) x/db = 2.
moves downward, especially with the porous media. The normalized temperature fluctuations decrease due to mixing of the hot and cold fluids. The normalized temperature fluctuations with the porous media are much less than those without the porous media, which again shows that the porous media strongly reduces the temperature fluctuations to improve the thermal conditions in the tee junction. The distributions of the normalized mean temperatures and the normalized temperature fluctuations for the planes of y/db = 0.1 are shown in Fig. 6. The y/db = 0.1 plane is closer to the middle of the main duct than the plane of y/db = 0.3. For the case the normalized mean temperature distributions at y/db = 0.1 (Fig. 6) are quite different from those at y/db = 0.3 (Fig. 4). The normalized mean temperatures with the porous media are not always lower than those without the porous media. As shown in Fig. 6, the normalized mean temperatures with the porous media for x/db = 1 and x/db = 2 are higher in the middle than without the porous media. The corresponding normalized temperature fluctuations for y/db = 0.1 are shown in Fig. 7. The temperature fluctuations with the porous media are less in all three planes than without the porous media which shows that the porous media in the tee junction effectively reduces the temperature fluctuations for hot and cold mixing in a tee junction. Fig. 8 shows the area averaged pressure drops in the x direction for both cases. The pressure drop with the porous media in the
4463
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
a
1.0
1.0 y/db=0.3 x/db=1
0.8
0.8
Exp.[8] Without Porous With Porous
0.6 0.4
0.6 0.4 0.2
z/db
z/db
0.2 0.0
0.0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1.0 0.0
0.1
0.2
y/db=0.1 without Porous with Porous x/db=1 x/db=1
-1.0 0.0
0.3
0.1
0.2
x/db=2
x/db=4
x/db=4
0.4
0.5
* TRMS
* RMS
T
b
0.3
x/db=2
Fig. 7. Normalized temperature fluctuations in different x planes at y/db = 0.1.
1.0 y/db=0.3 x/db=2
0.8
Exp.[8] Without Porous With Porous
0.6 0.4
0
z/db
0.2 -500
0.0 -0.2 Δ p/Pa
-1000
-0.4 -0.6 -0.8
-1500 -2000
without Porous with Porous
-1.0 0.0
0.1
0.2
0.3
-2500
* TRMS
-3000
Fig. 5. Normalized temperature fluctuations in different x planes at y/db = 0.3 (a) x/db = 1; (b) x/db = 2.
-4
-3
-2
-1
0
1
2
3
4
x/db Fig. 8. Area averaged pressure drops in x direction.
1.0 0.8 0.6
without Porous x/db=2,y/db=0.9,z/db=1
on top wall
y/db=0.1
on bottom wall
without Porous with Porous x/db=1 x/db=1
x/db=2
x/db=2
x/db=4
x/db=4
in tee
with Porous x/db=2,y/db=0.7,z/db=1
x/db=2,y/db=0.3,z/db=-1
x/db=4,y/db=0.7,z/db=-1
x/db=1,y/db=0.1,z/db=-0.1
x/db=2,y/db=0.3,z/db=-0.5
1.0
0.8
0.4
0.0
T*
z/db
0.2 0.6
-0.2 0.4
-0.4 -0.6
0.2
-0.8 -1.0 0.0
0.2
0.4
0.6
T
0.8
1.0
*
Fig. 6. Normalized mean temperature distributions in different x planes at y/db = 0.1.
0.0 6.0
6.5
7.0
7.5
8.0
t /s Fig. 9. Normalized temperature variations in the tee and on the top and bottom walls.
4464
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
(2) In the present cases, buoyancy greatly influences the mixing flow resulting in back flow up into the branch duct so the thermal striping phenomena is quite obvious. The porous media in the mixing zone reduces the temperature fluctuations in the tee junction. (3) The PSD directly shows the relationship between the power spectrum density and frequency of the temperature fluctuations as one of the key parameters for thermal fatigue analysis. The PSD shows that the porous media can effectually reduce the power density of the temperature fluctuations within the frequency range of 2–3 Hz.
Power Spectrum Density [degC2/Hz]
10000 1000 100 10 1 0.1 0.01 1E-3
Acknowledgements x/db=1,y/db=0.1,z/db=-0.1 (without Porous)
1E-4
x/db=2,y/db=0.3,z/db=-0.5 (with Porous)
1E-5 0.1
1
10
f /Hz Fig. 10. Power spectrum densities of the squared instantaneous temperatures in the tee.
This work was supported by the project of National Natural Science Foundation of China (No. 50906002) and the project of Beijing Novel Program (No. 2008B16). Appendix A. SGS model The strain rate tensor is
mixing zone is much greater due to the flow resistance in the porous media than without the porous media. The maximum normalized instantaneous temperature fluctuations in the tee and on the top and bottom walls were shown in Fig. 9. The maximum normalized instantaneous temperature fluctuations in the tee without the porous media vary periodically with time from 0.01 to 1, which means that the hot and cold fluids alternate in this location. However, with the porous media, the temperature fluctuations in the tee are much smaller than without the porous media. The fluctuations on the top and bottom walls are similar. Thus, the porous media in the tee junction in the mixing substantially reduces the temperature fluctuations at the walls. However, the normalized instantaneous temperature fluctuations cannot describe the relationship between the power spectrum densities (PSD) and frequency of the temperature fluctuations. The PSD is one of the most important parameters for thermal fatigue analysis, since it directly shows the variation of the measured effect with frequency. The PSDs of the squares of the instantaneous temperatures of two points calculated using FFT are plotted in Fig. 10. Without the porous media temperature fluctuations with frequencies from 2 to 3 Hz have higher PSD than with the porous media. The temperature fluctuations on the order of several Hz are very dangerous for structures since they have the greatest effect on the thermal fatigue [12]. 4. Conclusions The temperature fluctuations in a tee junction with and without porous media were simulated using the LES turbulent flow model with the SL sub-grid scale model using FLUENT. The models predicted the temperature and velocity fields, the normalized mean temperatures and the temperature fluctuations. The predicted normalized mean temperatures and temperature fluctuations with or without porous media are compared with previous experimental data. The temperature fluctuations and the power spectrum densities at the locations having the strongest temperature fluctuations in the tee junction were then analyzed to evaluate the thermal fatigue potential. The numerical results showed that: (1) The LES model can capture the instantaneous turbulent fluctuations for the flow without the porous media in the mixing tee as shown by the good agreement between the numerical results with the corresponding experimental data [8].
Sij ¼
i @ u j 1 @u þ 2 @xj @xi
ðA-1Þ
The sub-grid scale Reynolds stress and scalar flux are
sij ¼ qui uj qui uj
ðA-2Þ
j qj ¼ huj hu
ðA-3Þ
The SGS models most often used are the eddy-viscosity models having the following form:
sij
skk dij 3
¼ 2lt Sij
ðA-3Þ
Which relates the sub-grid scale stresses, sij, to the eddy-viscosity, lt, and the resolved-scale strain rate tensor, Sij . The SGS models are distinguished from one another by the their eddy-viscosity models, lt [9,11,40]. The Smagorinsky SGS model was firstly proposed by Smagorinsky [45] and further developed by Lilly [46]. In the Smagorinsky–Lilly model, the eddy-viscosity is modeled by
lt ¼ qL2s jSj2
ðA-4Þ
where Ls is the mixing length for the sub-grid scales and qffiffiffiffiffiffiffiffiffiffiffiffi jSj ¼ 2Sij Sji . Ls is computed using:
Ls ¼ minðkd; C s V 1=3 Þ
ðA-5Þ
where k is the Von Karman constant of 0.42, d is the distance to the closest wall, Cs is the Smagorinsky constant, and V is the volume of the computational cell. Lilly derived a value of 0.17 for Cs for homogeneous isotropic turbulence in the inertial sub range. However, this value was found to cause excessive damping of large-scale fluctuations in the presence of a mean shear and in transitional flows such as near solid boundaries. The default value of Cs adopted by the FLUENT code is 0.1, which was found to yield the best results for a wide range of flow conditions. The sub-grid scale scalar flux is modeled analogously with a mixing length gradient model
qj ¼
lsgs @ h Prsgs @xj
ðA-6Þ
Here the turbulent Prandtl number, Prsgs, a constant value of 0.6 is used.The sub-grid scale heat flux in the energy equation is
qi ¼
CsV @T jSj PrT @xi
where PrT is the turbulent Prandtl number.
ðA-7Þ
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
4465
Appendix B. Numerical procedure
References
The three-dimensional tee junction model shown in Fig. 1 is arranged so that the origin of the coordinate system is at the center of the tee. The geometry was meshed with GAMBIT, a pre-processor for the FLUENT software. The LES simulation procedures both with and without porous media are summarized as follows:
[1] S. Chapuliot, C. Gourdin, T. Payen, J.P. Magnaud, A. Monavon, Hydro-thermalmechanical analysis of thermal fatigue in a mixing tee, Nucl. Eng. Des. 235 (2005) 575–596. [2] L.J. Young, Lifetime evaluation of cracked shaft sleeve of reactor coolant pump under thermal striping, Int. J. Solids Struct. 38 (46–47) (2001) 8345–8358. [3] J.I. Lee, L.W. Hu, P. Saha, M.S. Kazimi, Numerical analysis of thermal striping induced high cycle thermal fatigue in a mixing tee, Nucl. Eng. Des. 239 (5) (2009) 833–839. [4] T. Muramatsu, H. Ninikata, Development of thermohydraaulics computer programs for thermal striping phenomena, Nucl. Technol. 113 (1996) 54–72. [5] F. Claude, Thermal fatigue in mixing tees: a step simplified procedure, in: 11th International Conference on Nuclear Engineering, Tokyo, 2003. [6] J.H. Kim, R.M. Riodt, A.F. Deardorff, Thermal stratification and reactor piping integrity, Nucl. Eng. Des. 139 (1993) 83–95. [7] K.J. Metzner, U. Wilke, European THERFAT project-thermal fatigue evaluation of piping system ‘‘Tee”-connections, Nucl. Eng. Des. 235 (2005) 473–484. [8] N. Fukushima, K. Fukagata, N. Kasagi, H. Noguchi, K. Tanimoto, Numerical and experimental study on turbulent thermal mixing in a t-junction flow, in: The 6th ASME-JSME Thermal Engineering Joint Conference, Hawaii, 2003. [9] P. Majander, T. Siikonen, Large-eddy simulation of a round jet in a cross-flow, Int. J. Heat Fluid Flow 27 (2006) 402–415. [10] A. Tokuhiro, N. Kimura, An experimental investigation on thermal striping mixing phenomena of a vertical non-buoyant jet with two adjacent buoyant jets as measured by ultrasound Doppler velocimetry, Nucl. Eng. Des. 188 (1999) 49–73. [11] L.W. Hu, M.S. Kazimi, LES benchmark study of high cycle temperature fluctuations caused by thermal striping in a mixing tee, Int. J. Heat Fluid Flow 27 (2006) 54–64. [12] M. Wakamatsu, H. Nei, K. Hashiguchi, Attenuation of temperature fluctuations in thermal striping, J. Nucl. Sci. Technol. 32 (8) (1995) 752–762. [13] N. Kimura, H. Miyakoshi, H. Kamide, Experimental investigation on transfer characteristics of temperature fluctuation from liquid sodium to wall in parallel triple-jet, Int. J. Heat Mass Transfer 50 (2007) 2024–2036. [14] M. Kaviany, Laminar flow through a porous channel bounded by isothermal parallel plates, Int. J. Heat Mass Transfer 28 (1985) 851–858. [15] P.C. Huang, K. Vafai, Analysis of forced convection enhancement in a channel using porous blocks, J. Thermophys. Heat Transfer 8 (1994) 563–573. [16] J.M. Zhang, W.H. Sutton, F.C. Lai, Enhancement of heat transfer using porous convection-to-radiation converter for laminar flow in a circular duct, Int. J. Heat Mass Transfer 40 (1997) 39–48. [17] T. Rizk, C. Kleinstreuer, Forced convective cooling of a linear array of blocks in open and porous matrix channels, Heat Transfer Eng. 12 (1991) 4–47. [18] Y. Ould-Amer, S. Chikh, K. Bouhadef, G. Lauriat, Forced convection cooling enhancement by use of porous materials, Int. J. Heat Fluid Flow 19 (1998) 251– 258. [19] P.C. Huang, C.F. Yang, J.J. Hwang, M.T. Chiu, Enhancement of forced-convection cooling of multiple heated blocks in a channel using porous covers, Int. J. Heat Mass Transfer 48 (3-4) (2005) 647–664. [20] B. Zhang, Y. Zhao, A numerical method for simulation of forced convection in a composite porous/fluid system, Int. J. Heat Fluid Flow 21 (4) (2000) 432–441. [21] H.C. Chan, W.C. Huang, J.M. Leu, C.J. Lai, Macroscopic modeling of turbulent flow over a porous medium, Int. J. Heat Fluid Flow 28 (5) (2007) 1157–1166. [22] J. Braga Edimilson, J.S. de Lemos Marcelo, Simulation of turbulent natural convection in a porous cylindrical annulus using a macroscopic two-equation model, Int. J. Heat Mass Transfer 49 (23–24) (2006) 4340–4351. [23] G. Alvarez, P.E. Bournet, D. Flick, Two-dimensional simulation of turbulent flow and transfer through stacked spheres, Int. J. Heat Mass Transfer 46 (13) (2003) 2459–2469. [24] P.X. Jiang, X.C. Lu, Numerical simulation and theoretical analysis of thermal boundary characteristics of convection heat transfer in porous media, Int. J. Heat Fluid Flow 28 (5) (2007) 1144–1156. [25] P.X. Jiang, X.C. Lu, Numerical simulation of fluid flow and convection heat transfer in sintered porous plate channels, Int. J. Heat Mass Transfer 49 (9–10) (2006) 1685–1695. [26] S. Whitaker, Simultaneous heat, mass, and momentum transfer in porous media: a theory of drying, Adv. Heat Transfer 13 (1977) 119–203. [27] J.Y. Jang, J.L. Chen, Forced convection in a parallel plate channel partially filled with a high porosity medium, Int. Commun. Heat Mass Transfer 19 (1992) 263–273. [28] N. Rudraiah, Forced convection in a parallel plate channel partially filled with a porous material, ASME J. Heat Transfer 107 (1985) 331–332. [29] Y.T. Yang, M.L. Hwang, Numerical simulation of turbulent fluid flow and heat transfer characteristics in heat exchangers fitted with porous media, Int. J. Heat Mass Transfer 52 (13-14) (2009) 2956–2965. [30] Z.F. Huang, A. Nakayama, K. Yang, C. Yang, W. Liu, Enhancing heat transfer in the core flow by using porous medium insert in a tube, Int. J. Heat Mass Transfer 53 (5–6) (2010) 1164–1174. [31] M.B. Saito, J.S. de Lemos Marcelo, A macroscopic two-energy equation model for turbulent flow and heat transfer in highly porous media, Int. J. Heat Mass Transfer 53 (11–12) (2010) 2424–2433. [32] A. Amiri, K. Vafai, Analysis of dispersion effects and non-thermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media, Int. J. Heat Mass Transfer 30 (1994) 939–954.
(1) Fully-developed velocity profiles in both the main duct and the branch duct calculated using the steady Reynolds stress turbulent model (RSM) were used as the inlet velocity conditions. (2) A larger model with a coarse mesh for rapid convergence was used to select an appropriate downstream main duct length, Ldn, which was initially set at Ldn/dm = 15. The purpose of this simulation was to determine the minimum downstream main duct length needed to prevent reversed flow at the outlet. The RSM simulations gave a minimum required length of about Ldn = 10dm. (3) A smaller model with an upstream main duct length of Lup,m = 3dm, an upstream branch duct length of Lup,b = 3db and a downstream main duct length of Ldn = 10dm was generated with fine meshes concentrated in the mixing zone and the near-wall region with a first layer thickness of 0.1 mm and a growth ratio of 1.2. The maximum cell thickness in the model was 3 mm. Simulations were performed with 800,000 cells, 1,000,000 cells, 1,200,000 cells, and 1,540,000 cells. Results for the last three cases had the same accuracy with the last one chosen as the grid independent model. (4) The LES model was used with the inlet velocity boundary conditions for both the main duct and the branch duct obtained from fully-developed velocity profiles calculated by a steady RSM simulation. The outlet flow boundary condition was used for the main duct outlet with adiabatic thermal boundary conditions for all walls. The spatial discretization scheme for solving the momentum and energy equations was the central-differencing scheme with the second-order upwind scheme used for the convection term in the momentum equation. The LES was initiated using the previously converged RSM solution as the initial condition. A constant time step size of 5 ms was used for the simulations. The total simulation time for the mixing process was about 13 s. The flow became statistically quasi stable after about 3 s as the mixed hot and cold fluids discharged from the main duct outlet. The temperatures were sampled from 6 to 8 s with a sampling frequency of 100 HZ. Since the temperature fluctuations in the near-wall region are of primary concern for the thermal fatigue, the computational elements in the near-wall region need to be fine enough to resolve the small-scale turbulent motions in this region. The dimensionless parameter, y+, is used to define boundary layer regions and the turbulent flow velocity profiles in the near-wall region. The viscous sub layer, which is dominated by the viscous stresses, is defined as y+ < 5. The buffer layer which is the transition between the viscosity dominated and the turbulence-dominated regions is defined as 5 < y+ < 30. For y+ > 50 through y = 0.05d, the viscous contribution to the wall shear stress diminishes with distance from the wall [39]. Depending on the velocity and temperature, the first element thickness of 0.1 mm in the present FLUENT model corresponded to y+ < 12. The subsequent grid spacings used a growth factor of 1.2 so that the boundary layer was divided into 10 layers. The near-wall region was sufficiently resolved so that the near-wall model could be used by FLUENT in the LES simulations.
4466
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
[33] S. Whitaker, Improved constraints for the principle of local thermal equilibrium, Ind. Eng. Chem. Res. 30 (1991) 983–997. [34] M. Quintard, S. Whitaker, Local thermal equilibrium for transient heat conduction: theory and comparison with numerical experiments, Int. J. Heat Mass Transfer 38 (1995) 2779–2796. [35] F. Kuwahara, M. Shirota, A. Nakayama, A numerical study of interfacial convective heat transfer coefficient in two-energy equation model for convection in porous media, Int. J. Heat Mass Transfer 44 (2001) 1153– 1159. [36] B.S. Marcelo, J.S. de Lemos Marcelo, Interfacial heat transfer coefficient for non-equilibrium convective transport in porous media, Int. Commun. Heat Mass Transfer 32 (5) (2005) 667–677. [37] B.S. Marcelo, J.S. de Lemos Marcelo, Laminar heat transfer in a porous channel simulated with a two-energy equation model, Int. Commun. Heat Mass Transfer 36 (2009) 1002–1007. [38] N. Wakao, S. Kaguei, T. Funazkri, Effect of fluid dispersion coefficients on particle-to-fluid heat transfer coefficients in packed bed, Chem. Eng. Sci. 34 (1979) 325–336. [39] S. Pope, Turbulence Flows, Cambridge University Press, 2000.
[40] L. Temmerman, M.A. Leschziner, C.P. Mellen, J. Fröhlich, Investigation of wallfunction approximations and subgrid-scale models in large eddy simulation of separated flow in a channel with streamwise periodic constrictions, Int. J. Heat Fluid Flow 24 (2003) 157–180. [41] B. Wegner, Y. Huai, A. Sadiki, Comparative study of turbulent mixing in jet in cross-flow configurations using LES, Int. J. Heat Fluid Flow 25 (2004) 767–775. [42] Y.H. Dong, X.Y. Lu, Large eddy simulation of a thermally stratified turbulent channel flow with temperature oscillation on the wall, Int. J. Heat Mass Transfer 47 (10–11) (2004) 2109–2122. [43] S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog. 48 (2) (1952) 89–94. [44] K. Kimura, Thermal striping in mixing tees with hot and cold water (Type A: Characteristics of flow visualization and temperature fluctuations in collision type mixing tees with same pipe diameter), in: NTHAS3: Third Korea-Japan symposium on Nuclear Thermal Hydraulics and Safety, Korea, 2002. [45] J. Smagorinsky, General circulation experiments with the primitive equations. I. The basic experiment, Mon. Weather Rev. 91 (1963) 99–164. [46] D.K. Lilly, On the application of the eddy viscosity concept in the inertial subrange of turbulence, NCAR Manuscript 123 (1966).
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Large-eddy simulations (LES) of temperature fluctuations in a mixing tee with/without a porous medium T. Lu a, P.X. Jiang b,*, Z.J. Guo a, Y.W. Zhang b, H. Li b a b
School of Mechanical and Electrical Engineering, Beijing University of Chemical Technology, Beijing 100029, China Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 30 April 2009 Received in revised form 21 June 2010 Accepted 22 June 2010
Keywords: Large-eddy simulations Temperature fluctuations Mixing Porous media
a b s t r a c t Temperature fluctuations in a mixing tee were simulated with and without a porous media in FLUENT using the LES turbulent flow model with the sub-grid scale (SGS) Smagorinsky–Lilly (SL) model with buoyancy. The normalized mean and fluctuating temperatures are used to describe the time-averaged temperatures and the time-averaged temperature fluctuation intensities. For the tee junction without the porous media, the predicted normalized mean temperature and temperature fluctuations compare well with previous experimental data. Comparison of the numerical results with the porous media with both experimental and numerical data without porous media shows that the porous media significantly reduces the temperature fluctuations. Moreover, analysis of the temperature fluctuations and the power spectrum densities (PSD) at the locations having the strongest temperature fluctuations in the tee junction shows that the porous media significantly reduces the thermal fatigue effects and can be useful in various structures such as tee junctions, elbows, piping systems. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Thermal fatigue is still not fully understood. One of the main obstacles to a full understanding resides in the multi-domain nature of the loading and associated damage, involving three complementary scientific disciplines: thermal-hydraulics, thermo-mechanics and materials science [1]. Prediction of thermal fatigue in mixing tees is a challenging subject that is needed for life cycle management of nuclear power plant piping systems. Thermal striping is one phenomenon that has been identified as a cause of thermal fatigue failures [2,3]. Thermal striping characterizes the phenomenon where hot and cold fluid mixing results in temperature fluctuations of the mixing fluids near a wall. The temperature fluctuations of the mixing fluids may then cause cyclical thermal stresses and subsequent fatigue cracking of the pipe wall. Research on thermal striping was initially carried out for liquidmetal-cooled fast breeder reactors in the 1980’s because of the high thermal conductivity of the liquid metal coolant [4]. Areas susceptible to thermal striping include components in the core outlet region, such as the core upper plenum, flow guide tube, and control rod upper guide tubes. Outside the core region, components where hot and cold fluids come in contact or mix, such as tee junctions, elbows, and leakage from valves may also be affected. The focus of thermal striping studies shifted to light water reactors * Corresponding author. E-mail address: [email protected] (P.X. Jiang). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.07.001
after several incidents of piping failure at some nuclear power plants [5,6]. Following the discovery of a leak due to a longitudinal crack at the outer edge of a bend in the mixing zone of the residual heat removal system of the Civaux nuclear power plant unit in 1998, metallurgical studies showed that the origin of this degradation was thermal fatigue cracking [1,7]. This incident was caused by thermal fatigue associated with temperature fluctuations in the fluid at the mixing tee. The piping system components that are most susceptible to thermal striping fatigue cracking are mixing tees in the residual heat removal systems in both boiling water reactors and pressurized-water reactors. Therefore, the European commission funded the international project ‘‘Thermal Fatigue Evaluation of Piping System Tee-connections” with the main objective being to advance the accuracy and reliability of thermal fatigue load determinations in engineering tools and to formulate research oriented approaches to outline a science based practical methodology for managing thermal fatigue risks [7]. To reach these objectives some of technological and scientific aims are focused on assessments of the fatigue significance of turbulent thermal stratification and mixing effects in the identified piping system tee-connections and determination of the effects of variable mass flow rates in the ‘‘run” and ‘‘branch” pipes by experimental and numerical simulations, e.g. computational fluid dynamics (CFD). The main goal of thermal striping evaluations is to identify the temperature fluctuation magnitudes and frequencies. This can be achieved by mock-up experiments or three-dimensional, unsteady turbulent modeling that resolves both large and small-scale
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
turbulent motions. This requires CFD simulations using either direct numerical simulations (DNS) [8] or large-eddy simulations (LES) [9]. Thermal striping experiments have been performed using flow visualizations, temperature measurements, and velocity measurements [8,10]. In experiments where fluid and pipe wall temperatures were measured, both the frequencies and the magnitudes of the temperature fluctuations were obtained at selected locations in the pipes. There are few studies using analytical approaches or computer modeling of thermal striping due to the extensive computational capacity required for unsteady, three-dimensional turbulent flow simulations. Thermal striping was modeled previously using LES and DNS [4,8,11], however, these studies were performed for only a small volume around the mixing zone due to the computational constraints. The DNS method requires extreme grid resolution and computational expense that is not viable for realistic geometries and flow conditions. However, there have been many recent develops in LES methods for turbulent flows. The present work simulates mixing flows with relatively high Reynolds numbers and low Richardson numbers both in the main duct and the branch duct using the FLUENT LES model with the sub-grid scale (SGS) Smagorinsky–Lilly (SL) model. The normalized mean temperatures and the normalized temperature fluctuations are used to describe time-averaged temperatures and time-averaged temperature fluctuation intensities. The numerical results for the normalized mean temperature and the normalized temperature fluctuation are compared with experimental data published in the literature [8]. Wakamatsu et al. [12] reported that the temperature oscillations causing thermal striping have relatively high frequencies on the order of several Hz, but there is little information about the power spectrum densities (PSD) of temperature fluctuations in the literature [13]. The PSD of temperature fluctuations, which statistically represent how power densities change with the temperature fluctuation frequency, is one of the key parameters for characterizing thermal fatigue. Therefore, the present work presents the PSD of the normalized temperatures versus frequency using Fast Fourier Transforms (FFT) of the time series results to analyze the energy density. In many fields such as for piping systems of nuclear power plants and in the cooling of turbine blades, engineers try to design the pipe or blades so that they do not experience large temperature fluctuations resulting in thermal fatigue. The designers then seek to reduce the temperature fluctuations to improve the structural integrity. There have been many theoretical and numerical investigations [14–25] of fluid flow and heat transfer in channels fully or partially packed with porous media in recent years due to the interest in a wide range of engineering applications such as heat exchangers, heat pipes, electronic cooling, thermal insulation, geothermal energy systems, enhanced oil recovery, and drying. The porous material improves the convective heat transfer due to the high surface area to volume ratio in the system and the enhanced flow mixing caused by the tortuous path through the porous matrix which improves the thermal dispersion. Studies of the flow and heat transfer in porous media have resulted in great progress in recent decades. Most mathematical models are based on the volume-averaging theory popularized by Whitaker [26]. The Darcy, Darcy-Brinkman or Darcy–Brinkman– Forchheimer models are often applied to describe the fluid momentum conservation in a porous media [27–31]. There are two basic energy conservation models used for porous media analyses. One is the local thermal equilibrium model which uses one energy balance equation for both the fluid and the solid assuming thermal equilibrium between the solid matrix and the fluid [32–34], while the other is the local non-thermal equilibrium model which uses separate energy balance equations for the solid and fluid phases with different temperatures [35–37]. The local thermal equilibrium model is no valid when the particles or pores are not very small,
4459
when the thermal properties differ widely, when convective transport is not important, or when there is significant heat generation. The local non-thermal equilibrium model using two temperature equations should then be used to model the heat transfer between the solid phase and the fluid phase. However, the local non-thermal equilibrium model requires additional information on the interfacial heat transfer coefficient between the solid phase and the fluid phase [35,38], which limits application of the local non-thermal equilibrium model. Various investigations have studied the heat transfer enhancement in tubes or ducts with porous media. Numerical simulations by Yang and Hwang [29] investigated the turbulent heat transfer enhancement in a pipe filled with a porous media using the k–e turbulence model to calculate the fluid flow and heat transfer characteristics in a pipe filled with porous media. The flow resistance and heat transfer characteristics of the air flow from laminar to fully turbulent ranges of Reynolds numbers in a tube inserted with porous media were investigated experimentally and numerically by Huang et al. [30]. The porous media is also expected to reduce the temperature fluctuations in a tee by improving the flow mixing in the tee. In the present work, large-eddy simulations were used to investigate for temperature fluctuations during mixing of hot and cold fluids in a mixing tee partially packed with porous media in the mixing zone of the main duct. Comparisons of the numerical results for the porous media with numerical and experimental results without porous media show how the porous media reduces the temperature fluctuations. 2. Governing equations 2.1. Fluid flow region without porous media Fig. 1 shows a schematic of the mixing of hot and cold fluids in a tee junction as numerically simulated on the FLUENT platform. The tee junction is constructed of square ducts [8]. The main duct is 100 mm by 100 mm while the branch duct is 50 mm by 50 mm. Hot water at a temperature of 343.48 K and velocity of 0.15 m/s enters the main duct in the horizontal direction while cold water at a temperature of 296.78 K and velocity of 0.3 m/s enters the branch duct in the vertical direction. The filtered LES equations for isothermal incompressible flows with a passive scalar h transport are [9,39–41]:
i @ q @ qu ¼0 þ @t @xi i @ qu i u j @ qu @p @ þ ¼ q0 bðT T 0 Þg þ 2lSij sij @t @xj @xi @xj j @ h @ hu @ @h þ ¼ a qj @t @xj @xj @xj
ð1Þ ð2Þ ð3Þ
i are the velocity components, p is the pressure, q is the where u density, b is the thermal expansion coefficient, and q0 is the reference density at the reference temperature, T0. The molecular dynamic viscosity is denoted by l. a is diffusivity. The fluid mechanics are combined with the energy transport in the energy equation [42]
@T @ðui TÞ 1 @ 2 T @qi þ ¼ @t @xi RePr @x2i @xi
ð4Þ
where Re represents the Reynolds number and Pr is the molecular Prandtl number. qi represents sub-grid scale heat flux, which is modeled by the SGS model. As shown in Fig. 1, no-slip and adiabatic conditions were imposed at the walls as the velocity and thermal boundary conditions. The inlet velocity boundary conditions for both the main duct and the
4460
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
Fig. 1. Schematic of hot and cold fluids mixing in a tee junction.
branch duct are specified as the inlet flow boundary conditions with the out flow boundary condition used at the main duct outlet. The operating pressure was ambient pressure, 101,325 Pa. There, three governing equations for the flow and thermal boundary conditions were solved for the simulations. Eq. (1) is the conservation of mass. Eq. (2) is the Navier–Stokes equations which represent the momentum conservation. Eq. (4) represents the energy conservation. The LES approach was then used to simulate the fluid mixing mechanics. The details of SGS model are given in Appendix A. 2.2. Fluid flow region with porous media The center of the mixing zone is assumed to be the coordinate system origin. The copper porous media is then packed in part of the main duct from x/db = 2.5 to x/db = 2.5. With the assumption of constant porosity, /, the continuity equation is given by
/
p;i @ q @ qu ¼0 þ @t @xi
ð5Þ
Assuming local thermal equilibrium, which means that the solid and fluid temperatures are the same at every location, the governing energy equation can be written as: o @ðq cp;f u p;i TÞ @ n @ f ½/qf cp;f þ ð1 /Þqs cp;s T þ ¼ @t @xi @xi
keff ¼ /kf þ ð1 /Þks
gbðT cold T hot Þdb u2m
p;i @ qu p;i u p;j @ qu @p @ ¼ q0 bðT T 0 Þg þ ð2lSij sij Þ þ @xi @xj @t @xj
3. Numerical results
The first part of the last term on the right side (Darcy’s term) accounts for the microscopic viscous drag while the second part accounts for the form drag due to inertial effects (direction changes) inside the pores and to turbulent dissipation. For the particular case of stacked spheres, Ergun [43] suggested the following expressions for the constants C1 and C2
C1 ¼ C2 ¼
150ð1 /Þ
2
/3 dp 1:75ð1 /Þ /3 dp
ð7Þ
@qi @xi
ð9Þ
ð10Þ
ð11Þ
The water properties such as the dynamic viscosity, the specific heat at constant pressure, and the thermal diffusivity were assumed to be constants in the simulations.
The numerical procedure is described in Appendix B. The normalized mean temperatures and normalized temperature fluctuations are used to describe the time-averaged temperatures and time-averaged temperature fluctuation intensities [11,44]. The normalized temperature at a given location is defined as
T i ¼
T i T cold T hot T cold
ð12Þ
where T is the instantaneous temperature at a given location, Tcold is the cold fluid inlet temperature, and Thot is the hot fluid inlet temperature. The time-averaged normalized mean temperature at a given location is
ð8Þ
where dp is the equivalent diameter of the packed spheres. Here, we assumed dp = 28 mm and a porosity of about 30%.
The Smagorinsky–Lilly SGS model is used to simulate the turbulence in the porous media. The buoyancy force due to the temperature difference is incorporated by adding a body force term to the Navier–Stokes equations based on the Boussinesq approximation since the Reynolds numbers of 36, 300 in the main duct and 16, 300 in the branch duct are both relatively high in the turbulent flow range. The effects of buoyancy are related to the Richardson number defined as
Ri ¼
ð6Þ
!
where the effective thermal conductivity in the porous region, keff, is simplify computed as the volume average of the fluid conductivity, kf, and the solid conductivity, ks
p;i ¼ /u in;i , has been where the Dupuit–Forchheimer relationship, u in;i is the intrinsic (liquid) average of the local velocity used and u vector. For a simple homogeneous porous media, the momentum equations for the porous media are modeled by the addition of a momentum source term to the standard fluid flow model in Eq. (2). The momentum equation is then
p;i ju p;i Þ p;i þ C 2 qju ðC 1 lu
keff
@T @xi
T ¼
N 1 X T N n¼1 i
where N is the total number of sampling points.
ð13Þ
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
4461
Fig. 2. Combined temperature distribution and velocity vectors in the y = 0 mm plane at 6 s. (a) Without porous media; (b) with porous media.
The normalized temperature fluctuation is defined as the rootmean square (RMS) of the instantaneous temperature at a given location
T RMS
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 2 u1 X ¼t T i T N i¼1
ð14Þ
The temperature and velocity fields both with and without the porous media were calculated using the LES simulations with the SL SGS model. The combined temperature and velocity distributions on the y = 0 mm plane are shown in Figs. 2 and 3 for both cases at two different times. The temperature and velocity distributions on the plane of y = 0 mm at 6 s are quite different from those at 10 s
Fig. 3. Combined temperature distribution and velocity vectors in the y = 0 mm plane at 10 s (a) without porous media; (b) with porous media.
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
a
1.0 0.8 0.6 0.4 0.2
z/db
due to the unsteady turbulent flow. The results show that the buoyancy influences the mixing of the hot and cold fluids in the tee junction, with the hot fluid in the main duct rising upwards into the branch duct and mixing with the cold fluid in the branch duct. Moreover, the cause of the thermal striping phenomena can be seen with the fluid temperatures varying with time in the downstream section of the main duct after the hot and cold fluids mix. The temperature distributions correspond to the flow field shown by the velocity vectors in Figs. 2 and 3. Many vortices occur in both ducts downstream of the mixing zone and even upstream of the mixing tee, which improve the temperature distributions and affect the influence of the buoyancy on the mixing in the tee junction. However, the back flow up into the branch duct with the porous media is more serious than without the porous media due to the higher flow resistance in the main duct packed with the porous media. The mixing region from x/db = 0 to x/ db = 2.5 with the porous media is much narrower than without the porous media. Because the porous media restricts the flow, and results in more uniform temperatures and velocities with the porous media than without the porous media. The instantaneous temperatures and velocities at the intersections between the plane of y/db = 0.3 and the planes of x/db = 1 and 2 along the z direction were abstracted from the temperature and velocity fields in the tee junction to compare with experimental data without the porous media [8]. The normalized mean temperatures, T , and the normalized temperature fluctuations, T RMS , were obtained using Eqs. (13) and (14). The predicted normalized mean temperatures and normalized temperature fluctuations along these three lines are compared with the experimental data in Figs. 4 and 5. Firstly, the numerical results without the porous media in Fig. 4 for the normalized mean temperatures at different intersections agree well with the experimental data, which verifies the validity of the LES model for predicting the mixing of hot and cold fluids in a tee junction. For x/db = 1 (Fig. 4(a)), the fluid near the top wall (1 < z/db < 0.6) has a relatively high normalized mean temperature, which means that hot fluid is pushed up along the top wall as shown in Figs. 2 and 3 due to the downward flow of the cold fluid from the branch duct and the effect of buoyancy. The fluid in the middle height (0.6 < z/db < 0.2) has relatively low normalized mean temperatures, which means that the hot and cold fluids are well mixed. However, the normalized mean temperature of the fluid in the range of 0.6 < z/db < 1.0 is equal to 1, which indicates that the hot fluid from the main duct does not mix with the cold fluid from the branch duct in this region. The normalized mean temperature distribution at the intersection for x/db = 1 has a large range, which suggests that thermal striping will occur downstream of the mixing zone in the main duct. With further mixing of the hot and cold fluids in the main duct, the normalized mean temperature distributions along the lines of intersection for x/db = 2 (Fig. 4(b)) become smoother and the range of the normalized mean temperatures is narrower. The variations of the normalized mean temperatures with the porous media along the lines of intersection for x/db = 1 and 2 are much smaller than those without the porous media because the porous media restricts the turbulence and enhances the heat transfer. Although the normalized mean temperatures can describe how the fluid temperature varies at a location with time, they cannot explain the fluid temperature fluctuations. The distributions of the normalized temperature fluctuations along the same lines of intersection are shown in Fig. 5. The numerical results for the normalized temperature fluctuations without the porous media agree reasonably well with the experimental data. The normalized temperature fluctuations are most intensive for x/db = 1 (Fig. 5(a)) than in the other locations for both cases which implies that the hot and cold fluids are mixing strongly here with the strongest velocity fluctuations. The largest normalized temperature fluctuations at each location near the middle height are reduced as the flow
0.0 -0.2 -0.4
y/db=0.3 x/db=1 Exp.[8] Without Porous With Porous
-0.6 -0.8 -1.0 0.0
b
0.2
0.4
T
*
T
*
0.6
0.8
1.0
0.6
0.8
1.0
1.0 0.8 0.6 0.4 0.2
z/db
4462
0.0 -0.2 -0.4
y/db=0.3 x/db=2
-0.6
Exp.[8] Without Porous With Porous
-0.8 -1.0 0.0
0.2
0.4
Fig. 4. Normalized mean temperature distributions in different x planes at y/db = 0.3 (a) x/db = 1; (b) x/db = 2.
moves downward, especially with the porous media. The normalized temperature fluctuations decrease due to mixing of the hot and cold fluids. The normalized temperature fluctuations with the porous media are much less than those without the porous media, which again shows that the porous media strongly reduces the temperature fluctuations to improve the thermal conditions in the tee junction. The distributions of the normalized mean temperatures and the normalized temperature fluctuations for the planes of y/db = 0.1 are shown in Fig. 6. The y/db = 0.1 plane is closer to the middle of the main duct than the plane of y/db = 0.3. For the case the normalized mean temperature distributions at y/db = 0.1 (Fig. 6) are quite different from those at y/db = 0.3 (Fig. 4). The normalized mean temperatures with the porous media are not always lower than those without the porous media. As shown in Fig. 6, the normalized mean temperatures with the porous media for x/db = 1 and x/db = 2 are higher in the middle than without the porous media. The corresponding normalized temperature fluctuations for y/db = 0.1 are shown in Fig. 7. The temperature fluctuations with the porous media are less in all three planes than without the porous media which shows that the porous media in the tee junction effectively reduces the temperature fluctuations for hot and cold mixing in a tee junction. Fig. 8 shows the area averaged pressure drops in the x direction for both cases. The pressure drop with the porous media in the
4463
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
a
1.0
1.0 y/db=0.3 x/db=1
0.8
0.8
Exp.[8] Without Porous With Porous
0.6 0.4
0.6 0.4 0.2
z/db
z/db
0.2 0.0
0.0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1.0 0.0
0.1
0.2
y/db=0.1 without Porous with Porous x/db=1 x/db=1
-1.0 0.0
0.3
0.1
0.2
x/db=2
x/db=4
x/db=4
0.4
0.5
* TRMS
* RMS
T
b
0.3
x/db=2
Fig. 7. Normalized temperature fluctuations in different x planes at y/db = 0.1.
1.0 y/db=0.3 x/db=2
0.8
Exp.[8] Without Porous With Porous
0.6 0.4
0
z/db
0.2 -500
0.0 -0.2 Δ p/Pa
-1000
-0.4 -0.6 -0.8
-1500 -2000
without Porous with Porous
-1.0 0.0
0.1
0.2
0.3
-2500
* TRMS
-3000
Fig. 5. Normalized temperature fluctuations in different x planes at y/db = 0.3 (a) x/db = 1; (b) x/db = 2.
-4
-3
-2
-1
0
1
2
3
4
x/db Fig. 8. Area averaged pressure drops in x direction.
1.0 0.8 0.6
without Porous x/db=2,y/db=0.9,z/db=1
on top wall
y/db=0.1
on bottom wall
without Porous with Porous x/db=1 x/db=1
x/db=2
x/db=2
x/db=4
x/db=4
in tee
with Porous x/db=2,y/db=0.7,z/db=1
x/db=2,y/db=0.3,z/db=-1
x/db=4,y/db=0.7,z/db=-1
x/db=1,y/db=0.1,z/db=-0.1
x/db=2,y/db=0.3,z/db=-0.5
1.0
0.8
0.4
0.0
T*
z/db
0.2 0.6
-0.2 0.4
-0.4 -0.6
0.2
-0.8 -1.0 0.0
0.2
0.4
0.6
T
0.8
1.0
*
Fig. 6. Normalized mean temperature distributions in different x planes at y/db = 0.1.
0.0 6.0
6.5
7.0
7.5
8.0
t /s Fig. 9. Normalized temperature variations in the tee and on the top and bottom walls.
4464
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
(2) In the present cases, buoyancy greatly influences the mixing flow resulting in back flow up into the branch duct so the thermal striping phenomena is quite obvious. The porous media in the mixing zone reduces the temperature fluctuations in the tee junction. (3) The PSD directly shows the relationship between the power spectrum density and frequency of the temperature fluctuations as one of the key parameters for thermal fatigue analysis. The PSD shows that the porous media can effectually reduce the power density of the temperature fluctuations within the frequency range of 2–3 Hz.
Power Spectrum Density [degC2/Hz]
10000 1000 100 10 1 0.1 0.01 1E-3
Acknowledgements x/db=1,y/db=0.1,z/db=-0.1 (without Porous)
1E-4
x/db=2,y/db=0.3,z/db=-0.5 (with Porous)
1E-5 0.1
1
10
f /Hz Fig. 10. Power spectrum densities of the squared instantaneous temperatures in the tee.
This work was supported by the project of National Natural Science Foundation of China (No. 50906002) and the project of Beijing Novel Program (No. 2008B16). Appendix A. SGS model The strain rate tensor is
mixing zone is much greater due to the flow resistance in the porous media than without the porous media. The maximum normalized instantaneous temperature fluctuations in the tee and on the top and bottom walls were shown in Fig. 9. The maximum normalized instantaneous temperature fluctuations in the tee without the porous media vary periodically with time from 0.01 to 1, which means that the hot and cold fluids alternate in this location. However, with the porous media, the temperature fluctuations in the tee are much smaller than without the porous media. The fluctuations on the top and bottom walls are similar. Thus, the porous media in the tee junction in the mixing substantially reduces the temperature fluctuations at the walls. However, the normalized instantaneous temperature fluctuations cannot describe the relationship between the power spectrum densities (PSD) and frequency of the temperature fluctuations. The PSD is one of the most important parameters for thermal fatigue analysis, since it directly shows the variation of the measured effect with frequency. The PSDs of the squares of the instantaneous temperatures of two points calculated using FFT are plotted in Fig. 10. Without the porous media temperature fluctuations with frequencies from 2 to 3 Hz have higher PSD than with the porous media. The temperature fluctuations on the order of several Hz are very dangerous for structures since they have the greatest effect on the thermal fatigue [12]. 4. Conclusions The temperature fluctuations in a tee junction with and without porous media were simulated using the LES turbulent flow model with the SL sub-grid scale model using FLUENT. The models predicted the temperature and velocity fields, the normalized mean temperatures and the temperature fluctuations. The predicted normalized mean temperatures and temperature fluctuations with or without porous media are compared with previous experimental data. The temperature fluctuations and the power spectrum densities at the locations having the strongest temperature fluctuations in the tee junction were then analyzed to evaluate the thermal fatigue potential. The numerical results showed that: (1) The LES model can capture the instantaneous turbulent fluctuations for the flow without the porous media in the mixing tee as shown by the good agreement between the numerical results with the corresponding experimental data [8].
Sij ¼
i @ u j 1 @u þ 2 @xj @xi
ðA-1Þ
The sub-grid scale Reynolds stress and scalar flux are
sij ¼ qui uj qui uj
ðA-2Þ
j qj ¼ huj hu
ðA-3Þ
The SGS models most often used are the eddy-viscosity models having the following form:
sij
skk dij 3
¼ 2lt Sij
ðA-3Þ
Which relates the sub-grid scale stresses, sij, to the eddy-viscosity, lt, and the resolved-scale strain rate tensor, Sij . The SGS models are distinguished from one another by the their eddy-viscosity models, lt [9,11,40]. The Smagorinsky SGS model was firstly proposed by Smagorinsky [45] and further developed by Lilly [46]. In the Smagorinsky–Lilly model, the eddy-viscosity is modeled by
lt ¼ qL2s jSj2
ðA-4Þ
where Ls is the mixing length for the sub-grid scales and qffiffiffiffiffiffiffiffiffiffiffiffi jSj ¼ 2Sij Sji . Ls is computed using:
Ls ¼ minðkd; C s V 1=3 Þ
ðA-5Þ
where k is the Von Karman constant of 0.42, d is the distance to the closest wall, Cs is the Smagorinsky constant, and V is the volume of the computational cell. Lilly derived a value of 0.17 for Cs for homogeneous isotropic turbulence in the inertial sub range. However, this value was found to cause excessive damping of large-scale fluctuations in the presence of a mean shear and in transitional flows such as near solid boundaries. The default value of Cs adopted by the FLUENT code is 0.1, which was found to yield the best results for a wide range of flow conditions. The sub-grid scale scalar flux is modeled analogously with a mixing length gradient model
qj ¼
lsgs @ h Prsgs @xj
ðA-6Þ
Here the turbulent Prandtl number, Prsgs, a constant value of 0.6 is used.The sub-grid scale heat flux in the energy equation is
qi ¼
CsV @T jSj PrT @xi
where PrT is the turbulent Prandtl number.
ðA-7Þ
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
4465
Appendix B. Numerical procedure
References
The three-dimensional tee junction model shown in Fig. 1 is arranged so that the origin of the coordinate system is at the center of the tee. The geometry was meshed with GAMBIT, a pre-processor for the FLUENT software. The LES simulation procedures both with and without porous media are summarized as follows:
[1] S. Chapuliot, C. Gourdin, T. Payen, J.P. Magnaud, A. Monavon, Hydro-thermalmechanical analysis of thermal fatigue in a mixing tee, Nucl. Eng. Des. 235 (2005) 575–596. [2] L.J. Young, Lifetime evaluation of cracked shaft sleeve of reactor coolant pump under thermal striping, Int. J. Solids Struct. 38 (46–47) (2001) 8345–8358. [3] J.I. Lee, L.W. Hu, P. Saha, M.S. Kazimi, Numerical analysis of thermal striping induced high cycle thermal fatigue in a mixing tee, Nucl. Eng. Des. 239 (5) (2009) 833–839. [4] T. Muramatsu, H. Ninikata, Development of thermohydraaulics computer programs for thermal striping phenomena, Nucl. Technol. 113 (1996) 54–72. [5] F. Claude, Thermal fatigue in mixing tees: a step simplified procedure, in: 11th International Conference on Nuclear Engineering, Tokyo, 2003. [6] J.H. Kim, R.M. Riodt, A.F. Deardorff, Thermal stratification and reactor piping integrity, Nucl. Eng. Des. 139 (1993) 83–95. [7] K.J. Metzner, U. Wilke, European THERFAT project-thermal fatigue evaluation of piping system ‘‘Tee”-connections, Nucl. Eng. Des. 235 (2005) 473–484. [8] N. Fukushima, K. Fukagata, N. Kasagi, H. Noguchi, K. Tanimoto, Numerical and experimental study on turbulent thermal mixing in a t-junction flow, in: The 6th ASME-JSME Thermal Engineering Joint Conference, Hawaii, 2003. [9] P. Majander, T. Siikonen, Large-eddy simulation of a round jet in a cross-flow, Int. J. Heat Fluid Flow 27 (2006) 402–415. [10] A. Tokuhiro, N. Kimura, An experimental investigation on thermal striping mixing phenomena of a vertical non-buoyant jet with two adjacent buoyant jets as measured by ultrasound Doppler velocimetry, Nucl. Eng. Des. 188 (1999) 49–73. [11] L.W. Hu, M.S. Kazimi, LES benchmark study of high cycle temperature fluctuations caused by thermal striping in a mixing tee, Int. J. Heat Fluid Flow 27 (2006) 54–64. [12] M. Wakamatsu, H. Nei, K. Hashiguchi, Attenuation of temperature fluctuations in thermal striping, J. Nucl. Sci. Technol. 32 (8) (1995) 752–762. [13] N. Kimura, H. Miyakoshi, H. Kamide, Experimental investigation on transfer characteristics of temperature fluctuation from liquid sodium to wall in parallel triple-jet, Int. J. Heat Mass Transfer 50 (2007) 2024–2036. [14] M. Kaviany, Laminar flow through a porous channel bounded by isothermal parallel plates, Int. J. Heat Mass Transfer 28 (1985) 851–858. [15] P.C. Huang, K. Vafai, Analysis of forced convection enhancement in a channel using porous blocks, J. Thermophys. Heat Transfer 8 (1994) 563–573. [16] J.M. Zhang, W.H. Sutton, F.C. Lai, Enhancement of heat transfer using porous convection-to-radiation converter for laminar flow in a circular duct, Int. J. Heat Mass Transfer 40 (1997) 39–48. [17] T. Rizk, C. Kleinstreuer, Forced convective cooling of a linear array of blocks in open and porous matrix channels, Heat Transfer Eng. 12 (1991) 4–47. [18] Y. Ould-Amer, S. Chikh, K. Bouhadef, G. Lauriat, Forced convection cooling enhancement by use of porous materials, Int. J. Heat Fluid Flow 19 (1998) 251– 258. [19] P.C. Huang, C.F. Yang, J.J. Hwang, M.T. Chiu, Enhancement of forced-convection cooling of multiple heated blocks in a channel using porous covers, Int. J. Heat Mass Transfer 48 (3-4) (2005) 647–664. [20] B. Zhang, Y. Zhao, A numerical method for simulation of forced convection in a composite porous/fluid system, Int. J. Heat Fluid Flow 21 (4) (2000) 432–441. [21] H.C. Chan, W.C. Huang, J.M. Leu, C.J. Lai, Macroscopic modeling of turbulent flow over a porous medium, Int. J. Heat Fluid Flow 28 (5) (2007) 1157–1166. [22] J. Braga Edimilson, J.S. de Lemos Marcelo, Simulation of turbulent natural convection in a porous cylindrical annulus using a macroscopic two-equation model, Int. J. Heat Mass Transfer 49 (23–24) (2006) 4340–4351. [23] G. Alvarez, P.E. Bournet, D. Flick, Two-dimensional simulation of turbulent flow and transfer through stacked spheres, Int. J. Heat Mass Transfer 46 (13) (2003) 2459–2469. [24] P.X. Jiang, X.C. Lu, Numerical simulation and theoretical analysis of thermal boundary characteristics of convection heat transfer in porous media, Int. J. Heat Fluid Flow 28 (5) (2007) 1144–1156. [25] P.X. Jiang, X.C. Lu, Numerical simulation of fluid flow and convection heat transfer in sintered porous plate channels, Int. J. Heat Mass Transfer 49 (9–10) (2006) 1685–1695. [26] S. Whitaker, Simultaneous heat, mass, and momentum transfer in porous media: a theory of drying, Adv. Heat Transfer 13 (1977) 119–203. [27] J.Y. Jang, J.L. Chen, Forced convection in a parallel plate channel partially filled with a high porosity medium, Int. Commun. Heat Mass Transfer 19 (1992) 263–273. [28] N. Rudraiah, Forced convection in a parallel plate channel partially filled with a porous material, ASME J. Heat Transfer 107 (1985) 331–332. [29] Y.T. Yang, M.L. Hwang, Numerical simulation of turbulent fluid flow and heat transfer characteristics in heat exchangers fitted with porous media, Int. J. Heat Mass Transfer 52 (13-14) (2009) 2956–2965. [30] Z.F. Huang, A. Nakayama, K. Yang, C. Yang, W. Liu, Enhancing heat transfer in the core flow by using porous medium insert in a tube, Int. J. Heat Mass Transfer 53 (5–6) (2010) 1164–1174. [31] M.B. Saito, J.S. de Lemos Marcelo, A macroscopic two-energy equation model for turbulent flow and heat transfer in highly porous media, Int. J. Heat Mass Transfer 53 (11–12) (2010) 2424–2433. [32] A. Amiri, K. Vafai, Analysis of dispersion effects and non-thermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media, Int. J. Heat Mass Transfer 30 (1994) 939–954.
(1) Fully-developed velocity profiles in both the main duct and the branch duct calculated using the steady Reynolds stress turbulent model (RSM) were used as the inlet velocity conditions. (2) A larger model with a coarse mesh for rapid convergence was used to select an appropriate downstream main duct length, Ldn, which was initially set at Ldn/dm = 15. The purpose of this simulation was to determine the minimum downstream main duct length needed to prevent reversed flow at the outlet. The RSM simulations gave a minimum required length of about Ldn = 10dm. (3) A smaller model with an upstream main duct length of Lup,m = 3dm, an upstream branch duct length of Lup,b = 3db and a downstream main duct length of Ldn = 10dm was generated with fine meshes concentrated in the mixing zone and the near-wall region with a first layer thickness of 0.1 mm and a growth ratio of 1.2. The maximum cell thickness in the model was 3 mm. Simulations were performed with 800,000 cells, 1,000,000 cells, 1,200,000 cells, and 1,540,000 cells. Results for the last three cases had the same accuracy with the last one chosen as the grid independent model. (4) The LES model was used with the inlet velocity boundary conditions for both the main duct and the branch duct obtained from fully-developed velocity profiles calculated by a steady RSM simulation. The outlet flow boundary condition was used for the main duct outlet with adiabatic thermal boundary conditions for all walls. The spatial discretization scheme for solving the momentum and energy equations was the central-differencing scheme with the second-order upwind scheme used for the convection term in the momentum equation. The LES was initiated using the previously converged RSM solution as the initial condition. A constant time step size of 5 ms was used for the simulations. The total simulation time for the mixing process was about 13 s. The flow became statistically quasi stable after about 3 s as the mixed hot and cold fluids discharged from the main duct outlet. The temperatures were sampled from 6 to 8 s with a sampling frequency of 100 HZ. Since the temperature fluctuations in the near-wall region are of primary concern for the thermal fatigue, the computational elements in the near-wall region need to be fine enough to resolve the small-scale turbulent motions in this region. The dimensionless parameter, y+, is used to define boundary layer regions and the turbulent flow velocity profiles in the near-wall region. The viscous sub layer, which is dominated by the viscous stresses, is defined as y+ < 5. The buffer layer which is the transition between the viscosity dominated and the turbulence-dominated regions is defined as 5 < y+ < 30. For y+ > 50 through y = 0.05d, the viscous contribution to the wall shear stress diminishes with distance from the wall [39]. Depending on the velocity and temperature, the first element thickness of 0.1 mm in the present FLUENT model corresponded to y+ < 12. The subsequent grid spacings used a growth factor of 1.2 so that the boundary layer was divided into 10 layers. The near-wall region was sufficiently resolved so that the near-wall model could be used by FLUENT in the LES simulations.
4466
T. Lu et al. / International Journal of Heat and Mass Transfer 53 (2010) 4458–4466
[33] S. Whitaker, Improved constraints for the principle of local thermal equilibrium, Ind. Eng. Chem. Res. 30 (1991) 983–997. [34] M. Quintard, S. Whitaker, Local thermal equilibrium for transient heat conduction: theory and comparison with numerical experiments, Int. J. Heat Mass Transfer 38 (1995) 2779–2796. [35] F. Kuwahara, M. Shirota, A. Nakayama, A numerical study of interfacial convective heat transfer coefficient in two-energy equation model for convection in porous media, Int. J. Heat Mass Transfer 44 (2001) 1153– 1159. [36] B.S. Marcelo, J.S. de Lemos Marcelo, Interfacial heat transfer coefficient for non-equilibrium convective transport in porous media, Int. Commun. Heat Mass Transfer 32 (5) (2005) 667–677. [37] B.S. Marcelo, J.S. de Lemos Marcelo, Laminar heat transfer in a porous channel simulated with a two-energy equation model, Int. Commun. Heat Mass Transfer 36 (2009) 1002–1007. [38] N. Wakao, S. Kaguei, T. Funazkri, Effect of fluid dispersion coefficients on particle-to-fluid heat transfer coefficients in packed bed, Chem. Eng. Sci. 34 (1979) 325–336. [39] S. Pope, Turbulence Flows, Cambridge University Press, 2000.
[40] L. Temmerman, M.A. Leschziner, C.P. Mellen, J. Fröhlich, Investigation of wallfunction approximations and subgrid-scale models in large eddy simulation of separated flow in a channel with streamwise periodic constrictions, Int. J. Heat Fluid Flow 24 (2003) 157–180. [41] B. Wegner, Y. Huai, A. Sadiki, Comparative study of turbulent mixing in jet in cross-flow configurations using LES, Int. J. Heat Fluid Flow 25 (2004) 767–775. [42] Y.H. Dong, X.Y. Lu, Large eddy simulation of a thermally stratified turbulent channel flow with temperature oscillation on the wall, Int. J. Heat Mass Transfer 47 (10–11) (2004) 2109–2122. [43] S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog. 48 (2) (1952) 89–94. [44] K. Kimura, Thermal striping in mixing tees with hot and cold water (Type A: Characteristics of flow visualization and temperature fluctuations in collision type mixing tees with same pipe diameter), in: NTHAS3: Third Korea-Japan symposium on Nuclear Thermal Hydraulics and Safety, Korea, 2002. [45] J. Smagorinsky, General circulation experiments with the primitive equations. I. The basic experiment, Mon. Weather Rev. 91 (1963) 99–164. [46] D.K. Lilly, On the application of the eddy viscosity concept in the inertial subrange of turbulence, NCAR Manuscript 123 (1966).