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Investigation of wall mass transfer characteristics downstream of an orifice
Nuclear Engineering and Design 242 (2012) 353–360
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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Investigation of wall mass transfer characteristics downstream of an orifice M. El-Gammal b , W.H. Ahmed a , C.Y. Ching b,∗ a b
Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, Saudi Arabia Department of Mechanical Engineering, McMaster University, Canada
a r t i c l e
i n f o
Article history: Received 14 April 2011 Received in revised form 22 September 2011 Accepted 4 October 2011
a b s t r a c t Numerical simulations were performed to determine the effect of Reynolds number and orifice to pipe diameter ratio (do /d) on the wall mass transfer rate downstream of an orifice. The simulations were performed for do /d of 0.475 for Reynolds number up to 70,000. The effect of do /d was determined by performing simulations at a Reynolds number of 70,000 for do /d of 0.375, 0.475 and 0.575. The momentum and mass transport equations were solved using the Low Reynolds Number (LRN) K- turbulence model. The Sherwood number (Sh) profile downstream of the orifice was in relatively good agreement with existing experimental results. The Sh increases sharply downstream of the orifice, reaching a maximum within 1–2 diameters downstream of the orifice, before relaxing back to the fully developed pipe flow value. The Sh number well downstream of the orifice was in good agreement with results for fully developed pipe flow estimated from the correlation of Berger and Hau (1977). The peak Sh numbers from the simulations were higher than that predicted from Tagg et al. (1979) and Coney (1980). © 2011 Elsevier B.V. All rights reserved.
1. Introduction Flow Accelerated Corrosion (FAC) is a serious safety and reliability problem facing aging power generation plants, especially nuclear power plants (NPP). This phenomenon results in the thinning and weakening of the pipe wall and can lead to premature component failure if its effect is underestimated or it is not detected during scheduled maintenance. FAC is essentially a three step process: (a) a series of electrochemical reactions at the metal-oxide interface, (b) chemical erosion that dissolves the oxide layer of the carbon steel pipe walls and (c) mass transfer from the wall to the flow that is accelerated by the hydrodynamics of the fluid flow within the piping component. Thus, the control of FAC in practical applications requires the identification of regions with changes in flow chemistry or regions of high local mass transfer, the latter of which is of interest here. Increases in the local mass transfer rate from the wall can occur at a number of locations in a flow where the flow pattern changes rapidly, including elbows, tee junctions, and sudden expansions. All of these can result in local mass transfer rates that are substantially higher than in the corresponding fully developed pipe flow. Orifices are frequently used in piping systems of power plants for measuring and restricting the flow rate. Pipe ruptures downstream of orifices due to FAC have been reported at several fossil and nuclear power stations. For example, there were personnel
∗ Corresponding author. Tel.: +1 905 525 9140x24998. E-mail address: [email protected] (C.Y. Ching). 0029-5493/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2011.10.009
casualties due to a hot water line explosion in the Mihama Nuclear Power plant in 2004 that was attributed to FAC downstream of an orifice (ClassNK, 2008). A similar serious pipeline explosion took place downstream of a feed water orifice at Kakrapar Atomic Power Station Unit-2 after only 10 years of service (Kain et al., 2008). The flow dynamics downstream of an orifice is complex, with flow separation at the sharp edge lip of the orifice, leading to the development of vortices that can shed downstream. This results in elevated levels of surface skin friction and turbulence production close to the wall which can enhance the wall mass transfer. Furthermore, flow impingement within the flow reattachment zone can lead to high levels of static pressure along the pipe wall. There have been several studies on FAC downstream of an orifice (ClassNK, 2008; Kimitoshi and Ryo, 2006; Hwang et al., 2009; Nagaya et al., 2010). Several factors, such as the turbulence levels, wall shear stress and pressure have been attributed as responsible for the increased mass transfer. However, the flow mechanisms responsible for FAC downstream of orifices are still not well understood. Poulson (1999) observed that the wall wear profile downstream of an orifice was well correlated to the mass transfer rate and turbulence levels, and suggested that the surface shear stress was not primarily responsible for FAC. However, Chang et al. (2009) and Nagaya et al. (2010), based on plant inspections and experiments, concluded that the wall mass transfer was well correlated to the wall shear stress. On the other hand, Okada et al. (2008) and Kimitoshi and Ryo (2006) found, based on numerical and experimental studies, that the FAC rate was well correlated to the level of flow turbulence. More recently, Hwang et al. (2009) compared the local wall thinning of carbon steel piping downstream of an orifice
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in typical NPP stations with different turbulence flow parameters from numerical simulations, and determined that the local wear rates were correlated to the vertical flow velocity component in the flow reattachment region. Several empirical correlations have been developed to predict the wall mass transfer in straight pipes (Berger and Hau, 1977) and downstream of orifices (Coney, 1980). Coney (1980) developed a correlation for the distribution of Sherwood number (Sh) downstream of an orifice, while Tagg et al. (1979) and Rizk et al. (1996) developed correlations for the maximum Sh downstream of an orifice. The maximum values from the correlation of Rizk et al. (1996) are 3.2 times higher than that from Tagg et al. (1979). Thus, there is a wide discrepancy in the prediction of the wall mass transfer rates and the causal effects for FAC are still not well understood. The objectives of this study are to investigate the effect of Reynolds number and orifice to pipe diameter ratio (do /d) on the flow hydrodynamics and wall mass transfer distribution downstream of an orifice under single-phase flow conditions. Numerical simulations were performed using the Low Reynolds Number (LRN) K- turbulence model for do /d = 0.375, 0.475 and 0.575 and Reynolds number up to 70,000. The results are compared with experimental results and mass transfer correlations from existing literature. 2. Numerical analysis Numerical simulations were performed to estimate the wall mass transfer rates downstream of an orifice for do /d = 0.5 at Re = 20,000, 40,000 and 70,000 and at Re = 70,000 for do /d = 0.375, 0.457 and 0.575 using the Fluent CFD code. Since the flow is axisymmetric, a three dimensional computational domain with an azimuthal section of 20◦ along the direction was considered to reduce the computational time. The computational domain is extended 15 diameters upstream and 40 diameters downstream of the orifice. The governing Reynolds averaged conservation equations for continuity, momentum and species are ∂ui =0 ∂xi
∂ui ∂xi
=0
∂P ∂ ∂u uj i = − + ∂xj ∂xi ∂xj uj
∂c ∂ = ∂xj ∂xj
(1)
1 ∂ui − ui uj Re ∂xj
1 ∂c − c uj Re Sc ∂xj
(2)
(3)
Here ui and ui are mean and fluctuating velocity components (i = 1, 2, 3), Re (=Uav d/) is Reynolds number, Uav is the average inlet velocity, d is the pipe diameter, is the kinematic viscosity, P is the pressure, c is the concentration of the dissolvable wall chemical species and Sc (= /D) is the Schmidt number. The simulations were performed for a Sc = 1450, which is representative for FAC in carbon steel piping in power generation plants (Pietralik and Schefski, 2011). The turbulent transport term c uj in Eq. (3) is the dominant term and is modeled based on the assumption that the turbulent transport of species c uj and turbulent shear stress ui uj are correlated (Nesic et al., 1993). Therefore, c uj is estimated from the turbulent Schmidt number (Sct ) defined as (t /Dt ), where t is the turbulent viscosity and Dt is the turbulent diffusivity defined from c uj = −Dt (∂c/∂y). The Sct is typically determined empirically and assumed constant, with a range between 0.4 and 1.7, and is found to be independent of the molecular fluid properties. Sydberger and Lotz (1982) found that a value of 1.7 produced results that were in good agreement with experimental mass
transfer results in straight pipes. Wang (1997), however, used a value of 0.9 for flow in straight pipes. In the present study the default value of Sct = 0.7 is used. In order to resolve the mass transfer of the wall species, the concentration has to be resolved all the way up to the wall surface. This requires modeling the turbulent flow in the near wall region. The conservation equations were solved using the Low Reynolds Number (LRN) K- turbulence model (Wang and Mujumdar, 2005; Hrenya et al., 1995; Kechiche et al., 2004). This turbulence model incorporates damping functions to consider the viscous effects in the near-wall region. Thus the transport equations of the turbulence kinetic energy and turbulence dissipation are integrated in the solution through the viscous sublayer, and has been used for wall-bounded flows. Kechiche et al. (2004) used various LRN K- models to study the heat transfer across a wall with a turbulent jet. The models were found to be very effective in resolving the thermal characteristics close to the wall. Wang (1997) utilized the Abe–Kondoh–Nagano (AKN) version of LRN K- model (Abe et al., 1994) to predict the concentration gradient of the wall species in straight pipe sections. There was very good agreement between the numerical and experimental results. Therefore, the AKN version of LRN K- is selected for the present study. Park and Sung (1995) simulated the separated and reattached flow downstream of a backward facing step using AKN version of LRN K- model and found a very good agreement between their and DNS results. The convection terms in the equations are approximated by a second order bounded upwind scheme. The convergence criterion applied to the residuals of the main flow parameters was less than 10−8 . A fully developed turbulent flow velocity profile is selected as the entrance condition for the inlet pipe. Symmetry boundary conditions are applied along the sides of the flow domain. The gage pressure at the exit was set to zero. The concentration of wall species in the entrance flow is assumed zero, and assumed fully saturated along the walls downstream of the orifice. The Sherwood number (Sh) is defined as Sh = hm d/D, where hm is the local mass transfer coefficient. The local mass transfer coefficient is estimated as hm = (−D(∂c/∂n)w )/(cw − cb ), where n is a normal vector to the pipe wall. The value for D was taken as 6.45 × 10−10 m2 /s, and cw and cb , the species concentration along the wall and bulk flow was set to 3.46 and 0 kg/m3 , respectively. These values were chosen to match the experimental conditions of Ching et al. (2011), who used gypsum test sections to measure the wall mass transfer rate downstream of orifices. The computational mesh was significantly refined in areas where high flow and wall species concentration gradients are expected. Close to the surface, the refined boundary layer cells are applied to capture the very thin diffusive boundary layer of the dissolved wall species as shown in Fig. 1. The maximum y+ value for the first cell close to the wall did not exceed 0.125. This is important so that the wall mass transfer rate can be accurately estimated from the near-wall concentration profile (Wang, 1997). The solution domain is divided into 800,000 control volumes. Grid independence studies were performed by increasing the number of control volumes up to a factor 2, especially in the near-wall region. The results showed that the Sh number is independent of grid resolution to within ±6% while the flow quantities are independent of grid resolution to within ±2%.
3. Results and discussion 3.1. Reynolds number effect The simulations were performed for Re of 20,000, 40,000 and 70,000 and do /d = 0.475. The flow characteristics for the three values of Re is qualitatively similar, therefore, only the representative flow
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0
Re=20 ,00 0 Re=40 ,00 0 Re=70 ,00 0
Cp
-2
-4
-6
-8 Fig. 1. Segment of the computational domain for do /d = 0.375 along a mid-plane of the pipe-orifice arrangement.
-10
characteristics for Re = 40,000 are presented here. The mean velocity vectors normalized by the averaged inlet velocity (Uo ) across the flow domain are presented in Fig. 2a. The flow accelerates as it approaches the orifice due to the reduction in the flow cross sectional area. It then separates at the sharp edge of the orifice, forming large vortices behind the orifice. The flow in the central region of the orifice accelerates further downstream of the orifice due to the reduction in the flow cross sectional area due to the separating vortices. The flow reattaches at a distance x/d ≈ 2 downstream of the orifice. The mean horizontal velocity (Ux ) profiles for the three values of Re at different axial locations are shown in Fig. 2b. Here, x is measured from the orifice and r is measured from the centreline of the pipe. As Re increases, the maximum velocity along the centreline increases within the circulation zone. The relative increase in the centreline velocity at x/d = 1.25 is about 7% as Re increases from
0
2
4
6
8
10
x/d Fig. 3. Surface pressure distribution downstream of the orifice for do /d = 0.475.
20,000 to 70,000. The velocity profiles relax and are almost similar at x/d = 7.5 with the flow relaxing back to the fully developed turbulent flow at x/d ≈ 30. The surface pressure distribution along the pipe walls expressed as a pressure coefficient Cp defined as Cp = (P − Po )/(1/2Uo2 ), where P is the static pressure and Po a reference pressure taken at the outflow, for the three values of Re is illustrated in Fig. 3. The variation of Cp is similar for the three Re, with it slightly decreasing with an increase in the Re. There is a significant decrease in the pressure immediately downstream of the orifice due to the flow separation at the lip of the orifice. The pressure reaches a minimum at x/d = 0.5, and then increases and reaches a maximum value in the flow reattachment region at x/d = 2.1. The Cp thereafter decreases with the axial distance. The wall shear stress distribution downstream of the orifice, represented as a skin friction coefficient Cf defined as Cf = w /(1/2Uo2 ) where w is the wall shear stress, for the three values of Re is presented in Fig. 4. There is a steep increase in the skin 0.07 Re=20,00 0 Re=40,00 0 Re=70,00 0
0.06 0.05
Cf
0.04 0.03 0.02 0.01 0 Fig. 2. (a) Normalized mean velocity vectors along a mid-plane of the pipe-orifice arrangement for Re = 40,000 and (b) profiles of normalized horizontal velocity component at Re = 20,000, 40,000 and 70,000 for do /d = 0.475.
0
2
4
6
8
x/d Fig. 4. Wall skin friction coefficient downstream of the orifice for do /d = 0.475.
10
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1 Re=20 ,00 0 Re=40 ,00 0 Re=70 ,00 0
0.9 0.8
Tke U o2
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8
10
x d Fig. 6. Distribution of the turbulent kinetic energy downstream of the orifice at r/d = 0.45 for do /d = 0.475.
Fig. 5. (a) Contours of normalized turbulent kinetic energy at half-plane of the orifice-pipe arrangement for Re = 40,000 and (b) radial profiles of the normalized turbulence kinetic energy for do /d = 0.475.
friction immediately downstream of the orifice due to the reversed flow generated from the separating vortices, and it reaches a peak at x/d ≈ 1.1. The surface shear stress then steeply decreases and reaches a minimum value within the flow reattachment region at x/d ≈ 2.1. The peak value of Cf decreases by 23% as the Reynolds number increases from 20,000 to 70,000. Downstream of the flow reattachment region, the surface shear stress increases due to the
Fig. 7. (a) Contours of normalized wall species concentration field at half-plane of the orifice-pipe arrangement for Re = 40,000 and (b) radial profiles of the normalized wall species concentration for do /d = 0.475.
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14000 Re=20,00 0 Re=40,00 0 Re=70,00 0 Experimental, Re=40 ,00 0 Coney (19 80), Re=40 ,00 0
12000 10000
Sh
8000
6000 Fig. 9. Profiles of normalized horizontal velocity component at Re = 70,000 for different do /d.
4000
2000
0
0
2
4
6
8
10
x/d Fig. 8. Distribution of the Sherwood number downstream of the orifice for do /d = 0.475.
development of the boundary layer and reaches a second local peak at x/d ≈ 3. This local peak in Cf increases as Re increases. The shear stress then decreases as the boundary layer thickness increases with axial distance. The normalized turbulent kinetic energy (Tke) contours within the flow domain for Re = 40,000 and do /d = 0.475 are shown in Fig. 5a. The turbulent kinetic energy increases significantly within the flow separation zone (x/d = 1–2 and r/d = 0.2–0.3). This is attributed to the high velocity gradients within this region. The normalized Tke profiles downstream of the orifice are shown in Fig. 5b. The profiles are qualitatively similar for all three Re, and collapse to a single line at x/d = 7.5. The axial distribution of Tke downstream of the orifice at the radial location r/d = 0.45 is plotted in Fig. 6. There is a steep increase in the Tke immediately downstream of the orifice and it reaches a maximum at x/d = 1.3, and then decreases as the flow develops downstream. The contours of the normalized concentration field of the dissolved wall species within the flow domain for Re = 40,000 and do /d = 0.475 is shown in Fig. 7a. In the region of the flow reversal, the wall species concentration decreases and reaches a minimum at x/d = 1–2, resulting in a reduction in the diffusive boundary layer thickness. The concentration of the wall species increases again from x/d = 2 and is almost homogenous within the pipe as the flow evolves further downstream. The normalized concentration profiles are shown in Fig. 7b for the three values of Re. Due to the relatively small thickness of the diffusive boundary layer compared to the velocity boundary layer, the profiles are shown from r/d = 0.49925 up to the wall. The diffusive boundary layer thickness from x/d = 1–2 is within the order of 1/900 of the thickness of the velocity boundary layer. The diffusive boundary layer grows thereafter as the flow develops downstream of the orifice. As Re number increases, the diffusive boundary layer thickness decreases, and the near wall concentration gradient increases.
The distribution of the Sherwood number along the pipe surface downstream of the orifice for the three Re values is plotted in Fig. 8. The experimental results from Ching et al. (2011) for the mass transfer downstream of an orifice for Re = 40,000 is also plotted in this figure. The experiments in Ching et al. (2011) were performed with test sections cast from gypsum (Sc = 1450) using water as the working fluid. The experimental methodology is described in detail by El-Gammal et al. (2009). There is, in general, good agreement between the numerical and experimental axial profiles of Sh. The Sh increases steeply downstream of the orifice and reaches a maximum within the flow recirculation region, and then decreases as the flow evolves downstream. The peak value from the simulations is approximately 10% higher than the experimental results. The axial location of the peak is slightly under-predicted by the simulations when compared to the experimental value. The magnitude of the peak value of Sh is dependent on Re; it increases from 5560 to 13460 as Re is increased from 20,000 to 70,000. The axial location of the peak Sh moves downstream from x/d = 1.24 to x/d = 1.38 as Re increases from 20,000 to 70,000. The Sh profiles are well correlated to the Tke profiles, and there is a good correspondence between the axial locations for the peak values of Sh and Tke. After reaching a peak, the Sh relaxes back to the fully developed pipe flow value. The
Table 1 Sherwood number in fully developed pipe flow. Re 20,000 40,000 70,000
Sh (Berger–Hau)
Sh (LRN K-)
% Difference
911 1654 2676
863 1680 2858
5.5 1.5 6.3
Fig. 10. (a) Radial profiles of the normalized turbulence kinetic energy at Re = 70,000 and (b) distribution downstream of the orifice at r/d = 0.45.
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Table 2 Peak Sherwood number downstream of the orifice. Re
LRN K- 5562 10,233 13,465
20,000 40,000 70,000
Tagg et al. (1979)
% Difference
Coney (1980)
% Difference
Experimental
% Difference
3839 6108 8886
31 40 34
2850 4474 6737
49 56 50
9102
11
Note: % Difference denotes discrepancy from LRN K-.
0 do/d=0.375 do/d=0.475 do/d=0.575
-2 -4
Cp
-6 -8 -10 -12
Fig. 13. Radial profiles of the normalized wall species concentration at Re = 70,000.
-14 -16 -18
0
2
4
6
8
10
x/d Fig. 11. Surface pressure distribution downstream of the orifice for Re = 70,000.
Sh values at x/d = 35 where the flow has relaxed back to the fully developed pipe flow are compared to the values obtained for a fully developed pipe flow from Berger and Hau (1977) in Table 1. There is very good agreement between the numerical results using the LRN K- turbulence model and from the Berger and Hau (1977) correlation for all three Re. The Sh profile using the correlation of Coney (1980) at Re = 40,000 is also shown in Fig. 8. There is, however, a large discrepancy between the correlation and the numerical and experimental results, with the correlation values lower by about 60%
The magnitude of the peak Sherwood number (Shp ) downstream of the orifice obtained from the numerical simulations, Tagg et al. (1979) and Coney (1980) correlations along with experimental value at Re = 40,000 are presented in Table 2. The Shp predicted from LRN K- is generally higher than those obtained from Tagg et al. (1979). The Shp were higher by 30–40% for the range of Re number studied here. The Shp values obtained from Coney (1980) were even lower than those from Tagg et al. (1979), with the values being 50–56% lower than that computed from the LRN K- simulations. The experimental Shp for Re = 40,000 was, however, only lower by 11% indicating relatively good agreement with the numerical analysis. 3.2. Orifice to pipe diameter ratio The effect of the orifice to pipe diameter ratio was investigated by performing the simulations for do /d = 0.375, 0.475 and 0.575 at Re = 70,000. The normalized horizontal velocity component Ux profiles along the axial direction are shown in Fig. 9. As expected, the
0.1
20000 do/d=0.375 do/d=0.475 do/d=0.575
0.09 0.08
do/d=0.375 do/d=0.475 do/d=0.575
16000
12000
0.06 Sh
Cf
0.07
0.05
8000
0.04 0.03
4000
0.02 0.01 0
0
0
2
4
x/d
6
8
10
Fig. 12. Wall skin friction coefficient downstream of the orifice for Re = 70,000.
0
2
4
6
8
10
x/d Fig. 14. Distribution of the Sherwood number downstream of the orifice for Re = 70,000.
M. El-Gammal et al. / Nuclear Engineering and Design 242 (2012) 353–360
centreline velocity increases as do /d decreases. At x/d = 1, the centreline velocity decreases by 50% as do /d increases from 0.375 to 0.575. The magnitude of the velocity in the flow reversal region, from r/d = 0.37 up to r/d = 0.5, also increases as do /d decreased, indicating an increased strength of the circulation vortices as do /d decreases. The velocity profiles have nearly relaxed back to the fully developed flow by x/d = 7.5 for all three do /d. The normalized turbulent kinetic energy (Tke) profiles for the three do /d at Re = 70,000 is shown in Fig. 10a. There is a significant effect of do /d on the turbulent kinetic energy distribution up to x/d ≈ 7.5, with an increase in the Tke as do /d is decreased. There is a local maximum in the Tke around r/d ≈ 0.2 for x/d up to about 1.5. The variation of the turbulent kinetic energy with downstream distance at the radial location r/d = 0.45 is plotted in Fig. 10b. For all do /d, the Tke increases immediately downstream of the orifice, reaches a maximum at x/d in the range 1.29–1.34, before decreasing with streamwise distance. The maximum in the Tke at this r/d location decreases by 60% as do /d increases from 0.375 to 0.475 and by 50% as do /d increases from 0.475 to 0.575. Along r/d = 0.45, the axial location of the maximum in Tke varies with do /d. As do /d increases from 0.375 to 0.475 the location moves downstream from x/d = 1.29 to 1.34, and it then moves upstream from 1.34 to 1.2 as do /d is further increased from 0.475 to 0.575. The distribution of the surface pressure, presented as Cp , downstream of the orifice for the three do /d is illustrated in Fig. 11. There is a large pressure drop immediately downstream of the orifice, the magnitude of which is significantly dependent on do /d. The minimum in Cp decreases from about −6 to −16.5 as do /d decreases from 0.575 to 0.375, indicating the increased strength of the circulating vortices as do /d is decreased. The pressure recovers by x/d ≈ 2.0 at the point of the flow reattachment. The distribution of the wall shear stress along the pipe wall is presented in Fig. 12. The peak value of Cf within the recirculation region decreases by 45% as do /d increases from 0.375 to 0.475, and a further 45% as do /d increases from 0.475 to 0.575. The location of the peak Cf remains at x/d ≈ 1.1 for do /d of 0.375 and 0.475. It, however, moves upstream to x/d = 0.85 for do /d = 0.575. The location of the minimum Cf for do /d = 0.375 and 0.475 is at x/d = 2.1 and moves upstream to x/d = 1.89 for do /d = 0.575, indicating the location of the flow reattachment decreases as do /d increases. The normalized concentration profiles shown in Fig. 13 are qualitatively similar for all three do /d. The diffusive boundary layer thickness increases and the concentration gradient at the wall decreases as do /d increases. The distribution of Sh downstream of the orifice for the three values of do /d is illustrated in Fig. 14. The peak value of Sh increases as do /d is decreased. The peak value of Sh increases by 7% and 40% as do /d decreases from 0.575 to 0.475 and from 0.475 to 0.375, respectively. There is a slight shift in the axial location where the peak value of Sh occurs, and this location is well correlated with the location of the peak Tke along r/d = 0.45. The location moves upstream from x/d = 1.25 to 1.4 as do /d increases from 0.375 to 0.475, and then downstream from x/d = 1.4 to 1.24 as do /d increases from 0.475 to 0.575.
4. Conclusions Numerical simulations were performed to determine the wall mass transfer rate downstream of an orifice. The Reynolds averaged mass, momentum and concentration conservation equations were solved using the Low Reynolds Number (LRN) K- turbulence model. The simulations were performed for Re = 20,000, 40,000 and 70,000 with orifice to pipe diameter ratio of do /d = 0.475, and at Re = 70,000 for do /d of 0.375, 0.475 and 0.575. The simulation results for the distribution of Sh downstream of the orifice were in good agreement with existing experimental results at Re = 40,000. The
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simulation results for the Sh in the fully developed region well downstream of the orifice were in good agreement with values from the correlation of Berger and Hau (1977) for fully developed pipe flow. The Sh steeply increases downstream of the orifice and reaches a maximum within the flow circulation region at x/d in the range 1–2, and then decreases as the flow evolves downstream. The peak Sh increases by 60% as Re increases from Re = 20,000 to 70,000 for do /d = 0.475. The peak values of Sh obtained from the correlations of Tagg et al. (1979) and Coney (1980) were relatively low compared to the numerical values. The magnitude of Sh is significantly dependent on do /d, with the peak value decreasing as do /d is increased. At Re = 70,000, the peak Sh decreased by 60% and 50% as do /d increased from 0.375 to 0.475 and from 0.475 to 0.575, respectively. The Sh profile downstream of the orifice and the location of the peak value in Sh was well correlated with the turbulence kinetic energy.
Acknowledgements The support of the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Deanship of Scientific Research at King Fahd University of Petroleum & Minerals (under project IN090038) are gratefully acknowledged.
References Abe, K., Kondoh, T., Nagano, Y., 1994. A new turbulence model for predicting fluid flow and heat transfer in separating and reattaching flows-I. Flow field calculations. International Journal of Heat and Mass Transfer 37, 139–151. Berger, F.P., Hau, K.F., 1977. Mass transfer in turbulent pipe flow measured by the electrochemical method. International Journal of Heat and Mass Transfer 20, 1185–1194. Chang, Y.S., Kim, S.H., Chang, H.S., Lee, S.M., Choi, J.B., Kim, Y.J., Choi, Y.H., 2009. Fluid effects on structural integrity of pipes with an orifice and elbows with a wall-thinned part. Journal of Loss Prevention in the Process Industries 22, 854–859. Ching, C.Y., Ewing, D., Sun, L., 2011. Flow accelerated corrosion in piping components: a collaborative project between McMaster University and AECL, CANDU Owners Group report COG-10-4056. ClassNK, 2008. Guidelines on pipe wall thinning, taking into account flow accelerated corrosion of water and wet steam pipelines in ships, Nippon Kaiji Kyokai, September 2008. Coney, M., 1980. CREL Report RD/L/N197/80, November 1980. El-Gammal, M., Mazhar, H., Cotton, J.S., Shefski, C., Pietralik, J., Ching, C.Y., 2009. The hydrodynamic effects of single-phase flow on flow accelerated corrosion in a 90-degree elbow. Nuclear Engineering and Design 240, 1589– 1598. Hrenya, C.M., Bolio, E.J., Chakrabarti, D., Sinclair, L., 1995. Comparison of Low Reynolds Number K-turbulence models in predicting fully developed pipe flow. Chemical Engineering Science 50 (12), 1923–1941. Hwang, K.M., Jin, T.E., Kim, K.H., 2009. Identification of the relationship between local velocity components and local wall thinning inside carbon steel piping. Journal of Nuclear Science and Technology 46 (5), 469–478. Kain, V., Roychowdhury, S., Mathew, T., Bhandakkar, A., 2008. Flow accelerated corrosion and its control measures for the secondary circuit pipelines in Indian nuclear power plants. Journal of Nuclear Materials 383, 86–91. Kechiche, J., Mhiri, H., Le Palec, G., Bournot, 2004. Application of Low Reynolds Number K-epsilon turbulence models to the study of turbulent wall jets. International Journal of Thermal Sciences 43, 201–211. Kimitoshi, Y., Ryo, M., 2006. Investigation of the flow in orifice downstream related to pipe wall thinning. Nihon Kikai Gakkai Nenji Taikai Koen Ronbunshu 3, 9–10. Nagaya, Y., Nakamura, A., Murase, M., Utanohara, Y., 2010. Evaluation of the flow accelerated corrosion downstream of an orifice (measurement of corrosion rate). In: FAC2010 Conference, Lyon, France. Nesic, S., Adamopoulos, G., Postlethwaite, J., Bergstrom, D.J., 1993. Modelling of turbulent flow and mass transfer with wall function and low-Reynolds number closures. Canadian Journal of Chemical Engineering 71, 28–34. Okada, H., Uchida, S., Naitoh, M., Uehara, Y., Koshizuka, S., 2008. Evaluation method on flow accelerated corrosion of components in cooling systems of nuclear power plants by coupling analysis of corrosion and flow dynamics. In: International Conference on the Properties of Water and Steam ICPWS XV, Berlin. Park, T.S., Sung, H.J., 1995. A nonlinear low-Reynolds-number K eplsion model for turbulent separated and reattaching flows – I. Flow field computations. International Journal Heat and Mass Transfer 38, 2657–2666.
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Pietralik, J.M., Schefski, C.S., 2011. Flow and mass transfer in bends under flow accelerated corrosion wall thinning conditions. Journal of Engineering for Gas Turbines and Power 133, 012902 (7 pages). Poulson, B., 1999. Complexities in predicting wear corrosion. Wear 235, 497–504. Rizk, T.Y., Thompson, G.E., Dawson, J.L., 1996. Mass transfer enhancement associated with sudden flow expansion. Corrosion Science 38 (10), 1801–1814. Sydberger, T., Lotz, U., 1982. Relation between mass transfer and corrosion in a turbulent pipe flow. Journal of the Electrochemical Society 129 (2), 276–283.
Tagg, D.J., Patrick, M.A., Wragg, A.A., 1979. Heat and mass transfer downstream of abrupt nozzle expansions in turbulent flow. Chemical Engineering Research and Design 57, 176–181. Wang, Y., 1997. Flow dependent corrosion in turbulent pipe flow. Ph.D. Thesis, University of Saskatchewan, Sakatoon, Canada. Wang, S.J., Mujumdar, A.S., 2005. A comparative study of five Low Reynolds Number k-epsilon models for impingement heat transfer. Applied Thermal Engineering 25, 31–44.
Contents lists available at SciVerse ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Investigation of wall mass transfer characteristics downstream of an orifice M. El-Gammal b , W.H. Ahmed a , C.Y. Ching b,∗ a b
Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, Saudi Arabia Department of Mechanical Engineering, McMaster University, Canada
a r t i c l e
i n f o
Article history: Received 14 April 2011 Received in revised form 22 September 2011 Accepted 4 October 2011
a b s t r a c t Numerical simulations were performed to determine the effect of Reynolds number and orifice to pipe diameter ratio (do /d) on the wall mass transfer rate downstream of an orifice. The simulations were performed for do /d of 0.475 for Reynolds number up to 70,000. The effect of do /d was determined by performing simulations at a Reynolds number of 70,000 for do /d of 0.375, 0.475 and 0.575. The momentum and mass transport equations were solved using the Low Reynolds Number (LRN) K- turbulence model. The Sherwood number (Sh) profile downstream of the orifice was in relatively good agreement with existing experimental results. The Sh increases sharply downstream of the orifice, reaching a maximum within 1–2 diameters downstream of the orifice, before relaxing back to the fully developed pipe flow value. The Sh number well downstream of the orifice was in good agreement with results for fully developed pipe flow estimated from the correlation of Berger and Hau (1977). The peak Sh numbers from the simulations were higher than that predicted from Tagg et al. (1979) and Coney (1980). © 2011 Elsevier B.V. All rights reserved.
1. Introduction Flow Accelerated Corrosion (FAC) is a serious safety and reliability problem facing aging power generation plants, especially nuclear power plants (NPP). This phenomenon results in the thinning and weakening of the pipe wall and can lead to premature component failure if its effect is underestimated or it is not detected during scheduled maintenance. FAC is essentially a three step process: (a) a series of electrochemical reactions at the metal-oxide interface, (b) chemical erosion that dissolves the oxide layer of the carbon steel pipe walls and (c) mass transfer from the wall to the flow that is accelerated by the hydrodynamics of the fluid flow within the piping component. Thus, the control of FAC in practical applications requires the identification of regions with changes in flow chemistry or regions of high local mass transfer, the latter of which is of interest here. Increases in the local mass transfer rate from the wall can occur at a number of locations in a flow where the flow pattern changes rapidly, including elbows, tee junctions, and sudden expansions. All of these can result in local mass transfer rates that are substantially higher than in the corresponding fully developed pipe flow. Orifices are frequently used in piping systems of power plants for measuring and restricting the flow rate. Pipe ruptures downstream of orifices due to FAC have been reported at several fossil and nuclear power stations. For example, there were personnel
∗ Corresponding author. Tel.: +1 905 525 9140x24998. E-mail address: [email protected] (C.Y. Ching). 0029-5493/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2011.10.009
casualties due to a hot water line explosion in the Mihama Nuclear Power plant in 2004 that was attributed to FAC downstream of an orifice (ClassNK, 2008). A similar serious pipeline explosion took place downstream of a feed water orifice at Kakrapar Atomic Power Station Unit-2 after only 10 years of service (Kain et al., 2008). The flow dynamics downstream of an orifice is complex, with flow separation at the sharp edge lip of the orifice, leading to the development of vortices that can shed downstream. This results in elevated levels of surface skin friction and turbulence production close to the wall which can enhance the wall mass transfer. Furthermore, flow impingement within the flow reattachment zone can lead to high levels of static pressure along the pipe wall. There have been several studies on FAC downstream of an orifice (ClassNK, 2008; Kimitoshi and Ryo, 2006; Hwang et al., 2009; Nagaya et al., 2010). Several factors, such as the turbulence levels, wall shear stress and pressure have been attributed as responsible for the increased mass transfer. However, the flow mechanisms responsible for FAC downstream of orifices are still not well understood. Poulson (1999) observed that the wall wear profile downstream of an orifice was well correlated to the mass transfer rate and turbulence levels, and suggested that the surface shear stress was not primarily responsible for FAC. However, Chang et al. (2009) and Nagaya et al. (2010), based on plant inspections and experiments, concluded that the wall mass transfer was well correlated to the wall shear stress. On the other hand, Okada et al. (2008) and Kimitoshi and Ryo (2006) found, based on numerical and experimental studies, that the FAC rate was well correlated to the level of flow turbulence. More recently, Hwang et al. (2009) compared the local wall thinning of carbon steel piping downstream of an orifice
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in typical NPP stations with different turbulence flow parameters from numerical simulations, and determined that the local wear rates were correlated to the vertical flow velocity component in the flow reattachment region. Several empirical correlations have been developed to predict the wall mass transfer in straight pipes (Berger and Hau, 1977) and downstream of orifices (Coney, 1980). Coney (1980) developed a correlation for the distribution of Sherwood number (Sh) downstream of an orifice, while Tagg et al. (1979) and Rizk et al. (1996) developed correlations for the maximum Sh downstream of an orifice. The maximum values from the correlation of Rizk et al. (1996) are 3.2 times higher than that from Tagg et al. (1979). Thus, there is a wide discrepancy in the prediction of the wall mass transfer rates and the causal effects for FAC are still not well understood. The objectives of this study are to investigate the effect of Reynolds number and orifice to pipe diameter ratio (do /d) on the flow hydrodynamics and wall mass transfer distribution downstream of an orifice under single-phase flow conditions. Numerical simulations were performed using the Low Reynolds Number (LRN) K- turbulence model for do /d = 0.375, 0.475 and 0.575 and Reynolds number up to 70,000. The results are compared with experimental results and mass transfer correlations from existing literature. 2. Numerical analysis Numerical simulations were performed to estimate the wall mass transfer rates downstream of an orifice for do /d = 0.5 at Re = 20,000, 40,000 and 70,000 and at Re = 70,000 for do /d = 0.375, 0.457 and 0.575 using the Fluent CFD code. Since the flow is axisymmetric, a three dimensional computational domain with an azimuthal section of 20◦ along the direction was considered to reduce the computational time. The computational domain is extended 15 diameters upstream and 40 diameters downstream of the orifice. The governing Reynolds averaged conservation equations for continuity, momentum and species are ∂ui =0 ∂xi
∂ui ∂xi
=0
∂P ∂ ∂u uj i = − + ∂xj ∂xi ∂xj uj
∂c ∂ = ∂xj ∂xj
(1)
1 ∂ui − ui uj Re ∂xj
1 ∂c − c uj Re Sc ∂xj
(2)
(3)
Here ui and ui are mean and fluctuating velocity components (i = 1, 2, 3), Re (=Uav d/) is Reynolds number, Uav is the average inlet velocity, d is the pipe diameter, is the kinematic viscosity, P is the pressure, c is the concentration of the dissolvable wall chemical species and Sc (= /D) is the Schmidt number. The simulations were performed for a Sc = 1450, which is representative for FAC in carbon steel piping in power generation plants (Pietralik and Schefski, 2011). The turbulent transport term c uj in Eq. (3) is the dominant term and is modeled based on the assumption that the turbulent transport of species c uj and turbulent shear stress ui uj are correlated (Nesic et al., 1993). Therefore, c uj is estimated from the turbulent Schmidt number (Sct ) defined as (t /Dt ), where t is the turbulent viscosity and Dt is the turbulent diffusivity defined from c uj = −Dt (∂c/∂y). The Sct is typically determined empirically and assumed constant, with a range between 0.4 and 1.7, and is found to be independent of the molecular fluid properties. Sydberger and Lotz (1982) found that a value of 1.7 produced results that were in good agreement with experimental mass
transfer results in straight pipes. Wang (1997), however, used a value of 0.9 for flow in straight pipes. In the present study the default value of Sct = 0.7 is used. In order to resolve the mass transfer of the wall species, the concentration has to be resolved all the way up to the wall surface. This requires modeling the turbulent flow in the near wall region. The conservation equations were solved using the Low Reynolds Number (LRN) K- turbulence model (Wang and Mujumdar, 2005; Hrenya et al., 1995; Kechiche et al., 2004). This turbulence model incorporates damping functions to consider the viscous effects in the near-wall region. Thus the transport equations of the turbulence kinetic energy and turbulence dissipation are integrated in the solution through the viscous sublayer, and has been used for wall-bounded flows. Kechiche et al. (2004) used various LRN K- models to study the heat transfer across a wall with a turbulent jet. The models were found to be very effective in resolving the thermal characteristics close to the wall. Wang (1997) utilized the Abe–Kondoh–Nagano (AKN) version of LRN K- model (Abe et al., 1994) to predict the concentration gradient of the wall species in straight pipe sections. There was very good agreement between the numerical and experimental results. Therefore, the AKN version of LRN K- is selected for the present study. Park and Sung (1995) simulated the separated and reattached flow downstream of a backward facing step using AKN version of LRN K- model and found a very good agreement between their and DNS results. The convection terms in the equations are approximated by a second order bounded upwind scheme. The convergence criterion applied to the residuals of the main flow parameters was less than 10−8 . A fully developed turbulent flow velocity profile is selected as the entrance condition for the inlet pipe. Symmetry boundary conditions are applied along the sides of the flow domain. The gage pressure at the exit was set to zero. The concentration of wall species in the entrance flow is assumed zero, and assumed fully saturated along the walls downstream of the orifice. The Sherwood number (Sh) is defined as Sh = hm d/D, where hm is the local mass transfer coefficient. The local mass transfer coefficient is estimated as hm = (−D(∂c/∂n)w )/(cw − cb ), where n is a normal vector to the pipe wall. The value for D was taken as 6.45 × 10−10 m2 /s, and cw and cb , the species concentration along the wall and bulk flow was set to 3.46 and 0 kg/m3 , respectively. These values were chosen to match the experimental conditions of Ching et al. (2011), who used gypsum test sections to measure the wall mass transfer rate downstream of orifices. The computational mesh was significantly refined in areas where high flow and wall species concentration gradients are expected. Close to the surface, the refined boundary layer cells are applied to capture the very thin diffusive boundary layer of the dissolved wall species as shown in Fig. 1. The maximum y+ value for the first cell close to the wall did not exceed 0.125. This is important so that the wall mass transfer rate can be accurately estimated from the near-wall concentration profile (Wang, 1997). The solution domain is divided into 800,000 control volumes. Grid independence studies were performed by increasing the number of control volumes up to a factor 2, especially in the near-wall region. The results showed that the Sh number is independent of grid resolution to within ±6% while the flow quantities are independent of grid resolution to within ±2%.
3. Results and discussion 3.1. Reynolds number effect The simulations were performed for Re of 20,000, 40,000 and 70,000 and do /d = 0.475. The flow characteristics for the three values of Re is qualitatively similar, therefore, only the representative flow
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355
0
Re=20 ,00 0 Re=40 ,00 0 Re=70 ,00 0
Cp
-2
-4
-6
-8 Fig. 1. Segment of the computational domain for do /d = 0.375 along a mid-plane of the pipe-orifice arrangement.
-10
characteristics for Re = 40,000 are presented here. The mean velocity vectors normalized by the averaged inlet velocity (Uo ) across the flow domain are presented in Fig. 2a. The flow accelerates as it approaches the orifice due to the reduction in the flow cross sectional area. It then separates at the sharp edge of the orifice, forming large vortices behind the orifice. The flow in the central region of the orifice accelerates further downstream of the orifice due to the reduction in the flow cross sectional area due to the separating vortices. The flow reattaches at a distance x/d ≈ 2 downstream of the orifice. The mean horizontal velocity (Ux ) profiles for the three values of Re at different axial locations are shown in Fig. 2b. Here, x is measured from the orifice and r is measured from the centreline of the pipe. As Re increases, the maximum velocity along the centreline increases within the circulation zone. The relative increase in the centreline velocity at x/d = 1.25 is about 7% as Re increases from
0
2
4
6
8
10
x/d Fig. 3. Surface pressure distribution downstream of the orifice for do /d = 0.475.
20,000 to 70,000. The velocity profiles relax and are almost similar at x/d = 7.5 with the flow relaxing back to the fully developed turbulent flow at x/d ≈ 30. The surface pressure distribution along the pipe walls expressed as a pressure coefficient Cp defined as Cp = (P − Po )/(1/2Uo2 ), where P is the static pressure and Po a reference pressure taken at the outflow, for the three values of Re is illustrated in Fig. 3. The variation of Cp is similar for the three Re, with it slightly decreasing with an increase in the Re. There is a significant decrease in the pressure immediately downstream of the orifice due to the flow separation at the lip of the orifice. The pressure reaches a minimum at x/d = 0.5, and then increases and reaches a maximum value in the flow reattachment region at x/d = 2.1. The Cp thereafter decreases with the axial distance. The wall shear stress distribution downstream of the orifice, represented as a skin friction coefficient Cf defined as Cf = w /(1/2Uo2 ) where w is the wall shear stress, for the three values of Re is presented in Fig. 4. There is a steep increase in the skin 0.07 Re=20,00 0 Re=40,00 0 Re=70,00 0
0.06 0.05
Cf
0.04 0.03 0.02 0.01 0 Fig. 2. (a) Normalized mean velocity vectors along a mid-plane of the pipe-orifice arrangement for Re = 40,000 and (b) profiles of normalized horizontal velocity component at Re = 20,000, 40,000 and 70,000 for do /d = 0.475.
0
2
4
6
8
x/d Fig. 4. Wall skin friction coefficient downstream of the orifice for do /d = 0.475.
10
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1 Re=20 ,00 0 Re=40 ,00 0 Re=70 ,00 0
0.9 0.8
Tke U o2
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8
10
x d Fig. 6. Distribution of the turbulent kinetic energy downstream of the orifice at r/d = 0.45 for do /d = 0.475.
Fig. 5. (a) Contours of normalized turbulent kinetic energy at half-plane of the orifice-pipe arrangement for Re = 40,000 and (b) radial profiles of the normalized turbulence kinetic energy for do /d = 0.475.
friction immediately downstream of the orifice due to the reversed flow generated from the separating vortices, and it reaches a peak at x/d ≈ 1.1. The surface shear stress then steeply decreases and reaches a minimum value within the flow reattachment region at x/d ≈ 2.1. The peak value of Cf decreases by 23% as the Reynolds number increases from 20,000 to 70,000. Downstream of the flow reattachment region, the surface shear stress increases due to the
Fig. 7. (a) Contours of normalized wall species concentration field at half-plane of the orifice-pipe arrangement for Re = 40,000 and (b) radial profiles of the normalized wall species concentration for do /d = 0.475.
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357
14000 Re=20,00 0 Re=40,00 0 Re=70,00 0 Experimental, Re=40 ,00 0 Coney (19 80), Re=40 ,00 0
12000 10000
Sh
8000
6000 Fig. 9. Profiles of normalized horizontal velocity component at Re = 70,000 for different do /d.
4000
2000
0
0
2
4
6
8
10
x/d Fig. 8. Distribution of the Sherwood number downstream of the orifice for do /d = 0.475.
development of the boundary layer and reaches a second local peak at x/d ≈ 3. This local peak in Cf increases as Re increases. The shear stress then decreases as the boundary layer thickness increases with axial distance. The normalized turbulent kinetic energy (Tke) contours within the flow domain for Re = 40,000 and do /d = 0.475 are shown in Fig. 5a. The turbulent kinetic energy increases significantly within the flow separation zone (x/d = 1–2 and r/d = 0.2–0.3). This is attributed to the high velocity gradients within this region. The normalized Tke profiles downstream of the orifice are shown in Fig. 5b. The profiles are qualitatively similar for all three Re, and collapse to a single line at x/d = 7.5. The axial distribution of Tke downstream of the orifice at the radial location r/d = 0.45 is plotted in Fig. 6. There is a steep increase in the Tke immediately downstream of the orifice and it reaches a maximum at x/d = 1.3, and then decreases as the flow develops downstream. The contours of the normalized concentration field of the dissolved wall species within the flow domain for Re = 40,000 and do /d = 0.475 is shown in Fig. 7a. In the region of the flow reversal, the wall species concentration decreases and reaches a minimum at x/d = 1–2, resulting in a reduction in the diffusive boundary layer thickness. The concentration of the wall species increases again from x/d = 2 and is almost homogenous within the pipe as the flow evolves further downstream. The normalized concentration profiles are shown in Fig. 7b for the three values of Re. Due to the relatively small thickness of the diffusive boundary layer compared to the velocity boundary layer, the profiles are shown from r/d = 0.49925 up to the wall. The diffusive boundary layer thickness from x/d = 1–2 is within the order of 1/900 of the thickness of the velocity boundary layer. The diffusive boundary layer grows thereafter as the flow develops downstream of the orifice. As Re number increases, the diffusive boundary layer thickness decreases, and the near wall concentration gradient increases.
The distribution of the Sherwood number along the pipe surface downstream of the orifice for the three Re values is plotted in Fig. 8. The experimental results from Ching et al. (2011) for the mass transfer downstream of an orifice for Re = 40,000 is also plotted in this figure. The experiments in Ching et al. (2011) were performed with test sections cast from gypsum (Sc = 1450) using water as the working fluid. The experimental methodology is described in detail by El-Gammal et al. (2009). There is, in general, good agreement between the numerical and experimental axial profiles of Sh. The Sh increases steeply downstream of the orifice and reaches a maximum within the flow recirculation region, and then decreases as the flow evolves downstream. The peak value from the simulations is approximately 10% higher than the experimental results. The axial location of the peak is slightly under-predicted by the simulations when compared to the experimental value. The magnitude of the peak value of Sh is dependent on Re; it increases from 5560 to 13460 as Re is increased from 20,000 to 70,000. The axial location of the peak Sh moves downstream from x/d = 1.24 to x/d = 1.38 as Re increases from 20,000 to 70,000. The Sh profiles are well correlated to the Tke profiles, and there is a good correspondence between the axial locations for the peak values of Sh and Tke. After reaching a peak, the Sh relaxes back to the fully developed pipe flow value. The
Table 1 Sherwood number in fully developed pipe flow. Re 20,000 40,000 70,000
Sh (Berger–Hau)
Sh (LRN K-)
% Difference
911 1654 2676
863 1680 2858
5.5 1.5 6.3
Fig. 10. (a) Radial profiles of the normalized turbulence kinetic energy at Re = 70,000 and (b) distribution downstream of the orifice at r/d = 0.45.
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Table 2 Peak Sherwood number downstream of the orifice. Re
LRN K- 5562 10,233 13,465
20,000 40,000 70,000
Tagg et al. (1979)
% Difference
Coney (1980)
% Difference
Experimental
% Difference
3839 6108 8886
31 40 34
2850 4474 6737
49 56 50
9102
11
Note: % Difference denotes discrepancy from LRN K-.
0 do/d=0.375 do/d=0.475 do/d=0.575
-2 -4
Cp
-6 -8 -10 -12
Fig. 13. Radial profiles of the normalized wall species concentration at Re = 70,000.
-14 -16 -18
0
2
4
6
8
10
x/d Fig. 11. Surface pressure distribution downstream of the orifice for Re = 70,000.
Sh values at x/d = 35 where the flow has relaxed back to the fully developed pipe flow are compared to the values obtained for a fully developed pipe flow from Berger and Hau (1977) in Table 1. There is very good agreement between the numerical results using the LRN K- turbulence model and from the Berger and Hau (1977) correlation for all three Re. The Sh profile using the correlation of Coney (1980) at Re = 40,000 is also shown in Fig. 8. There is, however, a large discrepancy between the correlation and the numerical and experimental results, with the correlation values lower by about 60%
The magnitude of the peak Sherwood number (Shp ) downstream of the orifice obtained from the numerical simulations, Tagg et al. (1979) and Coney (1980) correlations along with experimental value at Re = 40,000 are presented in Table 2. The Shp predicted from LRN K- is generally higher than those obtained from Tagg et al. (1979). The Shp were higher by 30–40% for the range of Re number studied here. The Shp values obtained from Coney (1980) were even lower than those from Tagg et al. (1979), with the values being 50–56% lower than that computed from the LRN K- simulations. The experimental Shp for Re = 40,000 was, however, only lower by 11% indicating relatively good agreement with the numerical analysis. 3.2. Orifice to pipe diameter ratio The effect of the orifice to pipe diameter ratio was investigated by performing the simulations for do /d = 0.375, 0.475 and 0.575 at Re = 70,000. The normalized horizontal velocity component Ux profiles along the axial direction are shown in Fig. 9. As expected, the
0.1
20000 do/d=0.375 do/d=0.475 do/d=0.575
0.09 0.08
do/d=0.375 do/d=0.475 do/d=0.575
16000
12000
0.06 Sh
Cf
0.07
0.05
8000
0.04 0.03
4000
0.02 0.01 0
0
0
2
4
x/d
6
8
10
Fig. 12. Wall skin friction coefficient downstream of the orifice for Re = 70,000.
0
2
4
6
8
10
x/d Fig. 14. Distribution of the Sherwood number downstream of the orifice for Re = 70,000.
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centreline velocity increases as do /d decreases. At x/d = 1, the centreline velocity decreases by 50% as do /d increases from 0.375 to 0.575. The magnitude of the velocity in the flow reversal region, from r/d = 0.37 up to r/d = 0.5, also increases as do /d decreased, indicating an increased strength of the circulation vortices as do /d decreases. The velocity profiles have nearly relaxed back to the fully developed flow by x/d = 7.5 for all three do /d. The normalized turbulent kinetic energy (Tke) profiles for the three do /d at Re = 70,000 is shown in Fig. 10a. There is a significant effect of do /d on the turbulent kinetic energy distribution up to x/d ≈ 7.5, with an increase in the Tke as do /d is decreased. There is a local maximum in the Tke around r/d ≈ 0.2 for x/d up to about 1.5. The variation of the turbulent kinetic energy with downstream distance at the radial location r/d = 0.45 is plotted in Fig. 10b. For all do /d, the Tke increases immediately downstream of the orifice, reaches a maximum at x/d in the range 1.29–1.34, before decreasing with streamwise distance. The maximum in the Tke at this r/d location decreases by 60% as do /d increases from 0.375 to 0.475 and by 50% as do /d increases from 0.475 to 0.575. Along r/d = 0.45, the axial location of the maximum in Tke varies with do /d. As do /d increases from 0.375 to 0.475 the location moves downstream from x/d = 1.29 to 1.34, and it then moves upstream from 1.34 to 1.2 as do /d is further increased from 0.475 to 0.575. The distribution of the surface pressure, presented as Cp , downstream of the orifice for the three do /d is illustrated in Fig. 11. There is a large pressure drop immediately downstream of the orifice, the magnitude of which is significantly dependent on do /d. The minimum in Cp decreases from about −6 to −16.5 as do /d decreases from 0.575 to 0.375, indicating the increased strength of the circulating vortices as do /d is decreased. The pressure recovers by x/d ≈ 2.0 at the point of the flow reattachment. The distribution of the wall shear stress along the pipe wall is presented in Fig. 12. The peak value of Cf within the recirculation region decreases by 45% as do /d increases from 0.375 to 0.475, and a further 45% as do /d increases from 0.475 to 0.575. The location of the peak Cf remains at x/d ≈ 1.1 for do /d of 0.375 and 0.475. It, however, moves upstream to x/d = 0.85 for do /d = 0.575. The location of the minimum Cf for do /d = 0.375 and 0.475 is at x/d = 2.1 and moves upstream to x/d = 1.89 for do /d = 0.575, indicating the location of the flow reattachment decreases as do /d increases. The normalized concentration profiles shown in Fig. 13 are qualitatively similar for all three do /d. The diffusive boundary layer thickness increases and the concentration gradient at the wall decreases as do /d increases. The distribution of Sh downstream of the orifice for the three values of do /d is illustrated in Fig. 14. The peak value of Sh increases as do /d is decreased. The peak value of Sh increases by 7% and 40% as do /d decreases from 0.575 to 0.475 and from 0.475 to 0.375, respectively. There is a slight shift in the axial location where the peak value of Sh occurs, and this location is well correlated with the location of the peak Tke along r/d = 0.45. The location moves upstream from x/d = 1.25 to 1.4 as do /d increases from 0.375 to 0.475, and then downstream from x/d = 1.4 to 1.24 as do /d increases from 0.475 to 0.575.
4. Conclusions Numerical simulations were performed to determine the wall mass transfer rate downstream of an orifice. The Reynolds averaged mass, momentum and concentration conservation equations were solved using the Low Reynolds Number (LRN) K- turbulence model. The simulations were performed for Re = 20,000, 40,000 and 70,000 with orifice to pipe diameter ratio of do /d = 0.475, and at Re = 70,000 for do /d of 0.375, 0.475 and 0.575. The simulation results for the distribution of Sh downstream of the orifice were in good agreement with existing experimental results at Re = 40,000. The
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simulation results for the Sh in the fully developed region well downstream of the orifice were in good agreement with values from the correlation of Berger and Hau (1977) for fully developed pipe flow. The Sh steeply increases downstream of the orifice and reaches a maximum within the flow circulation region at x/d in the range 1–2, and then decreases as the flow evolves downstream. The peak Sh increases by 60% as Re increases from Re = 20,000 to 70,000 for do /d = 0.475. The peak values of Sh obtained from the correlations of Tagg et al. (1979) and Coney (1980) were relatively low compared to the numerical values. The magnitude of Sh is significantly dependent on do /d, with the peak value decreasing as do /d is increased. At Re = 70,000, the peak Sh decreased by 60% and 50% as do /d increased from 0.375 to 0.475 and from 0.475 to 0.575, respectively. The Sh profile downstream of the orifice and the location of the peak value in Sh was well correlated with the turbulence kinetic energy.
Acknowledgements The support of the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Deanship of Scientific Research at King Fahd University of Petroleum & Minerals (under project IN090038) are gratefully acknowledged.
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