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A numerical study on thermal mixing in narrow channels inserted rectangular bodies
International Communications in Heat and Mass Transfer 44 (2013) 69–76
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
A numerical study on thermal mixing in narrow channels inserted rectangular bodies☆ Besir Kok a, Mujdat Firat b, Hakan F. Oztop c, Yasin Varol d,⁎ a
Technical Vocational School, Firat University, 23119 Elazig, Turkey Department of Mechanical Education, Firat University, 23119 Elazig, Turkey Department of Mechanical Engineering, Technology Faculty, Firat University, 23119 Elazig, Turkey d Department of Automotive Engineering, Technology Faculty, Firat University, 23119 Elazig, Turkey b c
a r t i c l e
i n f o
Available online 9 April 2013 Keywords: Thermal mixing Parallel jets Heat transfer CFD
a b s t r a c t A numerical study has been performed on thermal mixing phenomena in a narrow channel inserted with a body by twin-jets at different temperatures. The study is carried out for three jet Reynolds numbers (Re = 200, 400 and 600). Water is used as working fluid and the channel has an exit to supply continuity of mass. Governing equations are solved by using finite volume method. Adiabatic rectangular shaped objects at three different aspect ratios and locations are inserted into the channel to control thermal mixing as a passive technique. A thermal mixing index is defined and calculated from obtained values of temperatures for different parameters. Thus, results are presented with isotherms, streamlines and mixing index graphics for studied parameters. It is found that mixing performance increases in the first half of the channel (x/L = 0–0.5) with increasing of Re number. Also, it is decreased in the second half (x/L = 0.5–1) of the channel with higher values of Re number. The best mixing performance is formed for Case 3 at Re = 200 near the exit. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Mixing is one commonly used application in chemical engineering, mechanical engineering and environmental science. It can be classified into two separate groups: flow mixing and thermal mixing. In flow mixing phenomena, the same fluid/fluids or different fluids are mixed by using mechanical devices such as a propeller or by jet mixing [1]. In this context, jet mixed tanks are very popular because they consume low energy and require low investment and there is no complex mechanical structure and noise problem. This idea also can be used for the thermal mixing process. Control of thermal mixing and design of efficient systems are very important in engineering. Turki [2] used a square cylinder to make a control mechanics for flow mixing numerically. Computations are carried out for different Reynolds numbers. In the study, the effect of splitter plate length and its location, and the drag and lift coefficients are analyzed. It is found that the presence of the splitter plate significantly reduces the lift and drag fluctuations, and also changing of the location of the splitter plate affects the flow. Patil and Tiwari [3] made a numerical work to control laminar flow in a channel behind two inclined square cylinders. The results show that for a given Reynolds number, there is a range of values of the size ratio and cylinder spacing. Farjallah et al. [4] numerically studied heat transfer and vortex shedding behind a square cylinder in a laminar ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y. Varol). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.03.019
channel flow under a magnetic field. In the study they found that, through forced convection, both drag and lift coefficients are clearly damped by a magnetic field leading to the stabilization of the square cylinder. They also found that the application of a magnetic field to the vortex shedding flow can lead to a transition to symmetric flow. Chandran et al. [5] carried out a numerical analysis to simulate the thermal striping phenomena in a 1/5 scale water model of a prototype fast breeder reactor (PFBR) primary circuit. Two non-isothermal water jets in the different temperatures are impinged on a lattice plate which was placed above the jets. Reynolds stress turbulence model is used to evaluate the temperature fluctuation near the plate. In the study it is found that the cold jet dominated and the hot jet dominated flows result in very high and low temperature fluctuations respectively. The three-dimensional flow and mixing characteristics of multiple and multi-set three dimensional confined turbulent round opposing jets in a novel in-liner mixer are examined numerically using the standard k − ε turbulence model by Wang and Mujumdar [6]. They indicated that multiple opposing jets achieve better mixing than single opposing jets in the study. Wang et al. [7] numerically studied the laminar flow in an in-line mixer based on an opposing jet impingement. They found that the unequal inlet momenta of opposing jets obtained using both equal and unequal slot widths and the addition of baffles in the exit of the channel yield better mixing over shorter distances after impact. Beuf et al. [8] studied the influence of the geometry of the cell on mixing efficiency using three different geometries as circle, square and rectangle. They indicated that the flows in the Hele-Shaw cells are generally laminar and these flows can be in a first approximation considered as quasi-two dimensional. They also showed that the rectangular geometry leads to a
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Nomenclature AR Djet e g Gr H Hp k L Lp MI N P PE Re Ri St t T Tavg Tc Th u, v, w ΔT
Aspect ratio Diameter of jet (m) Internal energy per unit mass (J/kg) Acceleration of gravity (m/s 2) Grashof number Height of the channel (m) Height of the passive element (m) Thermal conductivity (W/m K) Length of the channel (m) Length of the passive element (m) Mixing index Number of jet Pressure (Pa) Passive element Reynolds number Richardson number Standard deviation of fluid temperature (K) Time (s) Temperature (K) Average temperature (K) Cold jet temperature (K) Hot jet temperature (K) Velocity magnitudes in direction x, y, z (m/s) Temperature difference between hot and cold water (K)
Greek symbols ν Kinematic viscosity μ Dynamic viscosity ρ Fluid density (kg/m 3) β Volumetric thermal expansion coefficient
better mixing, but also that the aspect ratio of the rectangle plays unexpectedly no important role on mixing. Walker et al. [9] made both numerical and experimental studies to carry out mixing of coolant streams of different temperatures in pipe junctions. In this way thermal fatigue may be prevented in the pipe wall. They presented a distribution of time averaged mixing scalar for different velocity ratios. Wang et al. [10] studied the jet mixing problem inside a slot, experimentally. They tested the jet array effect on cooling performance. They also tested the effect of the orientation angle and H/D ratio. It is found that the acceptable uniform flow is observed for shallow flows as H/D = 1. Chang et al. [11] made a numerical analysis to investigate the thermal mixing efficiency in a Y-shaped channel. They solved two dimensional incompressible, steady state equations using the Lattice Boltzmann method. They inserted different types of passive elements to improve thermal mixing efficiency. It is demonstrated that the enhanced mixing efficiency is a result of an increased intersection angle between the velocity vector and the temperature gradient within the channel. Thermal mixing is also a very important application for T-junctions as given by Naik-Nimbalkar et al. [12]. They made an experimental work on a T-junction with water and three dimensional steady state CFD simulations were carried out to predict velocity and temperature fields. The predicted values were in good agreement with the experimental measurements. Hu and Kazimi [13] made a numerical simulation on thermal striping for a three-dimensional, unsteady turbulent model. They used two types of mixing tee configurations and they modeled it by using the commercial CFD code FLUENT. A similar work has been done by Kamide et al. [14]. They performed a numerical work using the finite difference method. They indicated that the mixing
behavior in the tee was characterized by the relatively large vortex structures defined by the diameters and the velocities in the pipes. Using jets to enhance thermal mixing performance is a very useful way and many authors have worked on this subject such as Shi et al. [15], Lou et al. [16], Chua et al. [17] and Devahastin and Mujumdar [18]. These authors mostly studied the jet mixing phenomena using numerical techniques. They observed that geometric parameters are very important and thermal mixing is mostly a function of the temperature of jets. Sometimes, different shaped passive elements are used to control flow field, heat transfer and thermal mixing. Shan and Zhang [19] made a numerical calculation to investigate the different mixer configurations for an exhaust system of a turbo-fan engine. They observed that the lobed forced mixer can increase the mixing efficiency by 65%, and decrease the thrust coefficient by 3% only. Oztop [20] worked on the laminar mixed convection heat transfer for mixing in a partially open cavity. The main objective of this work is to understand the phenomena of flow and thermal mixing in a narrow channel with square objects inserted. Twin jets are used at different temperatures to supply water into the channel. This is a good application for some sanitary systems and mixture tanks. Based on the abovementioned literature and the authors' knowledge, there is no numerical work on thermal mixing in a narrow channel with a passive element inserted. 2. Problem description A schematic view of the parallel slot jet configuration in a channel is shown in Fig. 1(a). In this study, the effects of a passive element (PE) on thermal mixing in a rectangular cross-section channel have been investigated, numerically. Water was used as the working fluid. Two parallel slot jets at different temperatures were set in the channel in Re = 200, 400 and 600. Cold water jet (Tc) and hot water jet (Th) temperatures were defined as 293 K and 313 K, respectively. Calculations were made for three different aspect ratios such as AR = 0.5, 1 and 1.5. The aspect ratio was calculated from the proportions of height (Hp) and length (Lp) as shown in the figure. While the length of the PE is constant and equal to L/8, the height of the PE is chosen for three different values (Hp = H/4, H/2 and 3H/4). The passive elements were located in the channel for three different cases (Case 1 = L/4, Case 2 = 3 L/8 and Case 3 = L/2) as shown in the figure. There is a relationship between the sizes of the channel as L = 4H. The channel is divided into eight equal sections along the X coordinate. Thus there are seven columns considered and called as stations inside the channel. Using these stations, the thermal mixing index (MI) is calculated. 2.1. Grid independency Fig. 1(b) shows the distribution of a typical grid that is used in this study. Different grid elements, such as square and triangular shapes were tried. As its geometry is not very complicated, a square-element grid structure has a more appropriate result. The results were obtained from different grid densities such as 2500, 4500, 6500 and 8000. These results were compared with each other as given in Fig. 2 and the 6500 grid distribution was chosen for this study. 3. Numerical method In-channel flow was simulated using the commercial code FLUENT. This code uses finite volume method in order to solve Navier–Stokes and energy equations and it is widely used in heat transfer and fluid flow studies. The finite volume method can accommodate any type of grid. The CFD code is based on the pressure-correction and uses the SIMPLE algorithm of Patankar [21]. The first order upwind difference scheme (UDS) is used to discretize the momentum and energy equations.
B. Kok et al. / International Communications in Heat and Mass Transfer 44 (2013) 69–76
71
(a)
(b)
Fig. 1. (a) Schematical configuration of the channel, (b) grid distribution.
accepted as a forced convection. The mass conservation equation can be written as follows:
4. Governing equations Calculation of the temperature and flow field in a channel requires the obtainment of the solution of the governing equations. Incompressible, unsteady and laminar flows can be described by differential equations of continuity, momentum and energy. The radiation mode of heat transfer is neglected according to the other modes of heat transfer. Buoyancy forces are also neglected and the heat transfer regime is
304.5 2500 grid 4500 grid 6500 grid 8000 grid
304.0
ð1Þ
Momentum equations can be written in two directions as 2 ∂ðρuÞ ∂ ρu ∂ðρuvÞ ∂p ∂ ∂v ∂u ∂u þ ¼ þ λ þ V þ 2μ þ ∂t ∂y ∂x ∂x ∂x ∂y ∂x ∂x ∂ ∂v ∂u μ þ þ ∂y ∂x ∂y 2 ∂ðρvÞ ∂ ρv ∂ðρuvÞ ∂p ∂ ∂v ∂u þ ¼− þ μ þ þ ∂t ∂x ∂y ∂x ∂x ∂y ∂y ∂ ∂v λ∇V þ 2μ : þ ∂y ∂x
303.5
T
∂ρ þ ∇⋅ðρV Þ ¼ 0: ∂t
ð2Þ
ð3Þ
303.0
The energy equation can be written as ∂ðρeÞ ∂ ∂T ∂ ∂T þ ∇ðρeV Þ ¼ k þ k : ∂t ∂x ∂x ∂y ∂y
302.5
302.0
301.5 0.0
0.2
0.4
0.6
0.8
1.0
h/H Fig. 2. Effect of grid densities on the temperature at 7 L/8 location along the Y coordinate for Re = 400.
ð4Þ
Reynolds number is chosen based on the inlet jet's velocity and the Prandtl number is fixed at 5.43 for water. Other dimensionless numbers can be written as follows, Re ¼
u:Djet ν
2
; Gr ¼
g:β:d3 :ΔT Gr ; Ri ¼ 2 : ν2 Re
ð5Þ
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B. Kok et al. / International Communications in Heat and Mass Transfer 44 (2013) 69–76
Richardson number is calculated by Eq. (5) and, Ri b 1 is obtained. This result shows that the effect of natural convection can be neglected in this study.
(a) 29 5
29
8
30
2
303
4.1. Boundary conditions The inlet boundary conditions were obtained from the calculated instantaneous mass flow rate. This can be done due to the acceptation of the incompressibility. Also, no-slip boundary conditions were applied for all velocities at the walls. Boundary conditions for the considered physical model (Fig. 1(a)) are given as
31
1
30
30 4
8
(b) 296
29 8
302
On all solid boundaries; u = v = 0
303
∂T On the all horizontal boundaries; ¼0 ∂y ∂T ¼0 On the all vertical boundaries; ∂x ∂u ¼0 At the exit; P = 0, ∂x Temperatures, Tc = 293 K, Th = 313 K Inlet Reynolds numbers are; Re = 200, 400, and 600.
311
304
306
(c) 299
300
302 3 03
4.2. Definition of mixing index
309
The definition of the mixing index is the main function for this study. It is defined by Eq. (6) as used by many others earlier [6]. MI ¼
St 100: ΔT
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 n ∑i¼1 T h −T avg =ðn−1Þ
305
4
Fig. 3. Isotherms along the channel at different Reynolds numbers without any inserted PE, a) Re = 200, b) Re = 400, c) Re = 600.
ð6Þ
In this equation, St shows the standard deviation between hot and cold water temperatures. Thus, the standard deviation is given by
St ¼
30
ð7Þ
where ΔT is the temperature difference between hot and cold water temperatures and Tavg is the average temperature. Physically, this mixing index is a measure of how much the bulk temperature at any specific cross-section represents the set of temperatures that it comes from. An MI = 0 implies complete mixing which corresponds to a perfectly flat temperature profile [7].
the channel. It means that flow is developed at the middle part of the channel. The increase in Reynolds number affects the flow distribution and the flow developed again after mid-axis in the x-direction. The flow from the top inlet is suppressed to the bottom and it impinges on the bottom. Again, a circulation cell is formed near the exit but it is not shown in the figure. Fig. 5 displays the isotherms at different Reynolds numbers for AR = 1 and Case 1. As given in Fig. 5(a) an adiabatic body is less effective on isotherms for whole channels because of the low Reynolds number. It was noticed that the cold flow and hot flow through the inlets to the channel were at 293 and 313 K, respectively.
(a)
5. Results and discussion A numerical analysis is made to investigate the effects of the parameters of the Reynolds number, aspect ratio and location of the inserted passive element (PE) on thermal mixing performance in a rectangular cross-section channel. Two parallel slot jets are determined at different temperatures to supply fluid into the channel. The inserted body is chosen as adiabatic and a channel whose length is long enough is chosen. Fig. 3 presents the isotherms along the channel at different Reynolds numbers without any inserted PE. As indicated earlier the cold fluid flows through the inlets to the channel from the top port and the hot one through the inlets to the channel from the bottom. The exit port is located at the middle of the opposite wall. The temperature shows the symmetric distribution according to the middle y axis at lower values of Reynolds number (Re = 200) as seen from Fig. 3(a). This symmetrical shape is destroyed with an increase in the Reynolds number and a plume like temperature distribution is observed from the top port. Isotherms are cumulated around the exit of the hot port for both Re = 400 and Re = 600. Fig. 4 illustrates the streamline to see the effects of Reynolds number on fluid flow with the same parameters of the previous figure. Flow is symmetric according to the mid-axis at low Reynolds number. Two circulating cells are formed around the inlet jets. Streamlines are parallel to each other at the middle part of
(b)
(c)
Fig. 4. Streamlines along the channel at different Reynolds numbers without any inserted PE, a) Re = 200, b) Re = 400, c) Re = 600.
B. Kok et al. / International Communications in Heat and Mass Transfer 44 (2013) 69–76
35
(a) 295
29 8
301
308
305
30
25
MI (%)
3 04
(b)
20 15 10
298
3 01
29 5
302
5
303 310
Re=200 Re=400 Re=600
30
2
303 3 11
73
304 306
0 0.0
305
0.2
0.4
0.6
0.8
1.0
x/L
(c) 298
Fig. 7. Variation of mixing index along the channel (L) without any PE inserted.
298
310
301
302 303
304
304
306 Fig. 5. Isotherms along the channel at different Reynolds numbers for AR = 1 and Case 2, a) Re = 200, b) Re = 400, c) Re = 600.
Each jet forms their own circulation at the exit which turns at different directions. A stagnation point occurs in front of the body due to a combination of cold and hot jets. Temperature distributions are almost parallel to the horizontal walls at the middle-section of the channel. The interaction between hot and cold jets increases with the increase of Reynolds number as seen from Fig. 5(b) and (c). The isotherms especially inclined down the wall between the jet inlets and PE. Isotherms have a linear shape along the channel and the temperature difference between the hot and cold water became lower after PE in all Re numbers.
(a)
This means that PE has a positive effect on thermal mixing efficiency. As indicated earlier, the symmetricity on flow is destroyed with an increase in Reynolds number. A minimum circulation cell is formed behind the PE at low Reynolds number (Re = 200) as given in Fig. 6(a). With an increasing Reynolds number a huge cell is formed at the top side of the hot jet and the flow goes from the top of the body. This cell is formed because of the impinged and deviated flow to the body. The hot jet hits the cold jet and it almost moves to the bottom side of the object. The center of the cell of the biggest notation moves to the right with increasing Reynolds number as seen from Fig. 6(c). And a circulation cell is formed behind the body. The dimensions of the circulated flow on the top and bottom sides of the cold jet become almost the same for Re = 400 and 600. The clustered streamlines under the body indicate the increase of flow rate at that port. Fig. 7 presents the mixing index (MI) along the channel with no PE inserted which was calculated from Eq. (6). MI starts from 100% at x/L = 0 and decreases along the channel (L) when the thermal mixing efficiency of the cold and hot fluid gets better. The values of MI decrease at the middle part of the channel (x/L = 0.5) with increasing jet Re number. After here higher values of mixing index are formed for higher values of Reynolds number. When compared with the other values of Re numbers a more uniform MI decrease was observed at Re = 200 and after x/L = 0.3 the MI showed nearly the same decrease in shape for Re = 400 and 600. A variation of MI along the channel (L) for different Re numbers for the aspect ratio (AR) = 1 and Case 2 is given in Fig. 8. MI shows a sharp decrease for all Re numbers between jets and PE (x/L = 0 and 0.4) and MI decreases with increasing Re number in 35
(b)
Re=200 Re=400 Re=600
30
MI (%)
25 20 15
(c) 10 5 0 0.0
0.2
0.4
0.6
0.8
1.0
x/L Fig. 6. Streamlines along the channel at different Reynolds number for AR = 1 and Case 2, a) Re = 200, b) Re = 400, c) Re = 600.
Fig. 8. Variation of mixing index along the channel (L) for AR = 1 and Case 2.
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B. Kok et al. / International Communications in Heat and Mass Transfer 44 (2013) 69–76
(a)
35
300 296
30
302
302
303 3 07
25
3 04
MI (%)
309
3 04
(b)
3 07
15
302 5
303 309
20
10
3 01
296
Empty channel AR=0.5 AR=1 AR=1.5
30
4
304 0 0.0
0.2
0.4
0.6
0.8
1.0
x/L
(c) 296
302
4
307
Fig. 11. Variation of mixing index along the channel (L) for different aspect ratios for Case 1 and Re = 200.
30
309
30 0
302 303 304
Fig. 9. Isotherms along the channel for different aspect ratios for Case 3, Re = 400, a) AR = 0.5, b) AR = 1, c) AR = 1.5.
this region. MI becomes almost constant for Re = 400 and 600 between x/L = 0.4 and 1. But decreasing is going on along the channel and the lowest MI is formed at the exit at Re = 200. It means that thermal mixing efficiency becomes worse for higher values of Reynolds number due to lower velocity of the flow at the middle of the channel after the block. Figs. 9 and 10 show the isotherms and streamline for different ARs for Case 3 and Re = 400, respectively. As seen from the figures the blockage effect of the inserted body increases with increasing AR. The effect of AR on temperature distribution becomes insignificant at the left and right sides of the block. Temperature values become constant
(a)
on the top and bottom sides of the block. Jet flows incline toward the top of the vertical wall of the channel and a big circulation cell occurs on the left-bottom region of the channel and it is stuck there with increasing AR. Circulation cells are formed behind the PE with increasing AR. Effects of the different aspect ratios on thermal mixing efficiency in the channel are presented in Fig. 11 for Case 1 and Re = 200. As seen from the figure MI is lower and more uniform in no PE inserted channel while all values of MI of different ARs present similar a trend and are higher. A sharp decline is observed until x/L = 0.2 and it is almost kept constant until the PE for all AR values. This means that the insertion of PE makes a negative effect on thermal mixing performance in the channel in the PE region. After the flow passes the passive element a uniform decrease is formed for all AR values. As seen from the figure, MI values decrease in the left side of the PE and increase in the right side of the PE with the enhancement of AR values. This means that an increase in the aspect ratio of the inserted body is positive only in the right half of the channel. Fig. 12 gives a variation of the mixing index along the channel (L) for different ARs for Case 3 and Re = 400. A regular decrease is observed in the MI curve in the no PE inserted case and all other AR values present a similar decrease trend along the channel. Different AR values have only a slight effect on thermal mixing effectiveness for present parameters. Thermal mixing efficiency is better in the PE inserted condition until x/L = 0.7, but in the exit region better thermal mixing is observed in the no PE inserted case. An interesting 35
(b)
Empty channel AR=0.5 AR=1 AR=1.5
30
MI (%)
25
(c)
20 15 10 5 0 0.0
0.2
0.4
0.6
0.8
1.0
x/L Fig. 10. Streamlines along the channel for different aspect ratios for Case 3 and Re = 400, a) AR = 0.5, b) AR = 1, c) AR = 1.5.
Fig. 12. Variation of mixing index along the channel (L) for different aspect ratios for Case 3 and Re = 400.
B. Kok et al. / International Communications in Heat and Mass Transfer 44 (2013) 69–76
(a)
35
298
295
3 04
305
25
MI (%)
310
(b)
Empty channel Case 1 Case 2 Case 3
30
302.5 303
20 15 10
297
300
298 310
302 303
5
304
0
306
0,0
0,2
0,4
310
0,6
0,8
1,0
x/L
(c) 296
75
299
Fig. 15. Variation of mixing index along the channel (L) for different locations of the passive element for Re = 200 and AR = 1.
3 02
302 303
304
3 06
Fig. 13. Isotherms along the channel for different locations of the passive element for Re = 600, AR = 0.5, a) Case 1, b) Case 2, c) Case 3.
observation is that, thermal mixing efficiency is far better in higher Re numbers when compared with a prior figure (Fig. 11) in the passive element inserted channel. Also maybe it is caused by a higher distance between jets and the passive element. But for both cases better thermal mixing efficiency is plotted in the no PE inserted in the exit port of the channel. Figs. 13 and 14 show the isotherms and streamlines for different cases at AR = 0.5 and Re = 600, respectively. As indicated above thermal mixing parameter is a function of inlet temperatures. Low and high temperature fluids started to mix above the part of the high temperature
(a)
inlet. The isotherms are clustered at that point for all cases. After the body mixing performance is decreased. The flow temperature becomes almost constant above the body due to low velocity of the fluid at that region. This can be seen also from the streamlines. The temperature of the mixing fluid can be controlled via the body and the changing location of the body. The location of the body affects the flow field and the location of the main cell changes with the changing of the location of the PE. It affects the thermal mixing far from the inlet. The variation of mixing index along the channel (L) for different locations of the PE is plotted under operation conditions of Re = 200 and AR = 1 in Fig. 15. A parabolic decrease is seen in MI along the channel and the best thermal mixing performance is achieved at the no PE inserted case. For the compared effects of different locations of the PE on thermal mixing efficiency, Case 3 gives the best thermal mixing performance, Case 2 followed and Case 1 gives the least thermal mixing performance. As it is resulted from almost all prior figures, here the same condition is determined that in the exit part of the channel better thermal mixing performance is in the no PE inserted condition. Fig. 16 shows a variation of the mixing index along the channel for the different locations of the passive element for Re = 600 and AR = 0.5. In contrast to the prior figures, MI shows a wavy decrease along the channel in Fig. 16 because of high Reynolds number (Re = 600) for the no PE inserted case. All other MI curves exhibit a similar trend along the channel (L) except Case 1. A sharp decline is seen at Case 1 between PE and jets (x/L = 0–0.4). This resulted from
35
(b)
Empty channel Case 1 Case 2 Case 3
30
MI (%)
25
(c)
20 15 10 5 0 0.0
0.2
0.4
0.6
0.8
1.0
x/L Fig. 14. Streamlines along the channel for different locations of the passive element for Re = 600, AR = 0.5, a) Case 1, b) Case 2, c) Case 3.
Fig. 16. Variation of mixing index along the channel (L) for different locations of the passive element for Re = 600 and AR = 0.5.
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B. Kok et al. / International Communications in Heat and Mass Transfer 44 (2013) 69–76
35
the PE with increasing aspect ratio values. Higher AR values have a positive effect on thermal mixing efficiency on the second half of the channel. • The temperature of the mixing fluid can be controlled via the changing of location of the PE. Also, the location of PE affects the flow field and the location of the main cell. The effects of the location of PE on thermal mixing performance were compared. • Case 3 gives the best mixing performance and Case 1 gives the worst. A uniform decrease of the MI curve is observed along the channel and the lowest value of MI is formed near the exit of almost all the parameters in the no PE inserted case.
Empty channel Case 1 Case 2 Case 3
30
MI (%)
25 20 15 10 5
References
0 0.0
0.2
0.4
0.6
0.8
1.0
x/L Fig. 17. Variation of mixing index along the channel (L) for different locations of the passive element for Re = 400 and AR = 1.5.
the fact that the small PE-jet distance formed a strong vortex in the flow under high Reynolds (Re = 600) number operation conditions. MI is almost constant until the exit of the channel for this case. These results show that thermal mixing performance is the best and more stable for Case 1. Fig. 17 presents a variation of mixing index along the channel (L) for the different locations of passive element for Re = 400 and AR = 1.5. As it is plotted in the figure there is a huge difference between MI values of Case 1 and other. It is far higher than the other and fluctuating along the channel. The lowest MI value is formed at Case 3. The same with most of the former conditions, also in this figure the best thermal mixing efficiency is observed at the no PE inserted case in the exit of the channel and MI shows a uniform decrease along the channel. Based on the findings of the last three figures, it is concluded that to locate PE in Case 2 in the channel, gives optimum thermal mixing performance. 6. Conclusions In the present study, the influence of the jet Reynolds number, aspect ratio and different locations of the passive element on flow and thermal mixing performance in a rectangular cross-section channel was investigated numerically. The main findings from the studied works can be listed as follows: • The interaction between hot and cold jets increases with the increasing of Reynolds number and isotherms decline to the bottom wall of the channel. • Flow is symmetric according to the mid-axis at low Reynolds (Re = 200) number and this symmetricity is destroyed with an increase in Re number. Reynolds number has an important effect on thermal mixing performance in the channel. • It is found that mixing performance increases in the first half of the channel (x/L = 0–0.5) with an increase in Re number. Also, it is decreased in the second half (x/L = 0.5–1) of the channel with higher values of Re number. The best mixing performance is formed at Re = 200 near the exit. • The blockage effect of PE increases and circulation cells occurred behind
[1] K.L. Wasewar, J.V. Sarathi, CFD modeling and simulation of jet mixed tanks, Engineering Applications of Computational Fluid Mechanics 2 (2008) 155–171. [2] S. Turki, Numerical simulation of passive control on vortex shedding behind square cylinder using splitter plate, Engineering Applications of Computational Fluid Mechanics 2 (2008) 514–524. [3] P.P. Patil, S. Tiwari, Numerical investigation of laminar unsteady wakes behind two inclined square cylinders confined in a channel, Engineering Applications of Computational Fluid Mechanics 3 (2009) 369–385. [4] H. Farjallah, H. Abbasi, S. Turki, Vortex shedding of electrically conducting fluid flow behind square cylinder under magnetic field, Engineering Applications of Computational Fluid Mechanics 5 (2011) 349–356. [5] R.K. Chandran, I. Banerjee, G. Padmakumar, K.S. Reddy, Numerical analysis of thermal striping phenomena using a two jet water model, Engineering Applications of Computational Fluid Mechanics 4 (2010) 209–221. [6] S.J. Wang, A.S. Mujumdar, Flow and mixing characteristics of multiple and multi-set opposing jets, Chemical Engineering and Processing 46 (2007) 703–712. [7] S.J. Wang, S. Devahastin, A.S. Mujumdar, A numerical investigation of some approaches to improve mixing in laminar confined impinging streams, Applied Thermal Engineering 25 (2005) 253–269. [8] A. Beuf, J.N. Gence, P. Carriére, F. Raynal, Chaotic mixing efficiency in different geometries of Hele-Shaw cells, International Journal of Heat and Mass Transfer 53 (2010) 684–693. [9] C. Walker, M. Simiano, R. Zboray, H.-M. Prasser, Investigations on mixing phenomena in single-phase flow in a T-junction geometry, Nuclear Engineering and Design 239 (2009) 116–126. [10] T. Wang, S. Chintalapati, R.S. Bunker, C.P. Lee, Jet mixing in a slot, Experimental Thermal and Fluid Science 22 (2000) 1–17. [11] C.C. Chang, Y.T. Yang, T.H. Yen, C.K. Chen, Numerical investigation into thermal mixing efficiency in Y-shaped channel using Lattice Boltzmann method and field synergy principle, International Journal of Thermal Sciences 48 (2009) 2092–2099. [12] V.S. Naik-Nimbalkar, A.W. Patwardhan, I. Banerjee, G. Padmakumar, G. Vaidyanathan, Thermal mixing in T-junctions, Chemical Engineering Science 65 (2010) 5901–5911. [13] L.W. Hu, M.S. Kazimi, LES benchmark study of high cycle temperature fluctuations caused by thermal striping in a mixing tee, International Journal of Heat and Fluid Flow 27 (2006) 54–64. [14] H. Kamide, M. Igarashi, S. Kawashima, N. Kimura, K. Hayashi, Study on mixing behavior in a tee piping and numerical analyses for evaluation of thermal striping, Nuclear Engineering and Design 239 (2009) 58–67. [15] Y. Shi, M.B. Ray, A.S. Mujumdar, Effects of Prandtl number on impinging jet heat transfer under a semi-confined turbulent slot jet, International Communications in Heat and Mass Transfer 29 (2002) 929–938. [16] Z.Q. Lou, A.S. Mujumdar, C. Yap, Effect of geometric parameters on confined impinging jet heat transfer, Applied Thermal Engineering 25 (2005) 2687–2697. [17] L.P. Chua, S.C.M. Yu, X.K. Wang, Flow visualization and measurements of a square jet with mixing tabs, Experimental Thermal and Fluid Science 27 (2003) 731–744. [18] S. Devahastin, A.S. Mujumdar, A numerical study of flow and mixing characteristics of laminar confined impinging streams, Chemical Engineering Journal 85 (2002) 215–223. [19] Y. Shan, J.Z. Zhang, Numerical investigation of flow mixture enhancement and infrared radiation shield by lobed forced mixer, Applied Thermal Engineering (2009) 3687–3695. [20] H.F. Oztop, Laminar mixed convection heat transfer for mixing in partially open cavity, Proceedings of the 2nd Int. Exergy, Energy and Environment symposium, 2005. [21] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980.
Contents lists available at SciVerse ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
A numerical study on thermal mixing in narrow channels inserted rectangular bodies☆ Besir Kok a, Mujdat Firat b, Hakan F. Oztop c, Yasin Varol d,⁎ a
Technical Vocational School, Firat University, 23119 Elazig, Turkey Department of Mechanical Education, Firat University, 23119 Elazig, Turkey Department of Mechanical Engineering, Technology Faculty, Firat University, 23119 Elazig, Turkey d Department of Automotive Engineering, Technology Faculty, Firat University, 23119 Elazig, Turkey b c
a r t i c l e
i n f o
Available online 9 April 2013 Keywords: Thermal mixing Parallel jets Heat transfer CFD
a b s t r a c t A numerical study has been performed on thermal mixing phenomena in a narrow channel inserted with a body by twin-jets at different temperatures. The study is carried out for three jet Reynolds numbers (Re = 200, 400 and 600). Water is used as working fluid and the channel has an exit to supply continuity of mass. Governing equations are solved by using finite volume method. Adiabatic rectangular shaped objects at three different aspect ratios and locations are inserted into the channel to control thermal mixing as a passive technique. A thermal mixing index is defined and calculated from obtained values of temperatures for different parameters. Thus, results are presented with isotherms, streamlines and mixing index graphics for studied parameters. It is found that mixing performance increases in the first half of the channel (x/L = 0–0.5) with increasing of Re number. Also, it is decreased in the second half (x/L = 0.5–1) of the channel with higher values of Re number. The best mixing performance is formed for Case 3 at Re = 200 near the exit. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Mixing is one commonly used application in chemical engineering, mechanical engineering and environmental science. It can be classified into two separate groups: flow mixing and thermal mixing. In flow mixing phenomena, the same fluid/fluids or different fluids are mixed by using mechanical devices such as a propeller or by jet mixing [1]. In this context, jet mixed tanks are very popular because they consume low energy and require low investment and there is no complex mechanical structure and noise problem. This idea also can be used for the thermal mixing process. Control of thermal mixing and design of efficient systems are very important in engineering. Turki [2] used a square cylinder to make a control mechanics for flow mixing numerically. Computations are carried out for different Reynolds numbers. In the study, the effect of splitter plate length and its location, and the drag and lift coefficients are analyzed. It is found that the presence of the splitter plate significantly reduces the lift and drag fluctuations, and also changing of the location of the splitter plate affects the flow. Patil and Tiwari [3] made a numerical work to control laminar flow in a channel behind two inclined square cylinders. The results show that for a given Reynolds number, there is a range of values of the size ratio and cylinder spacing. Farjallah et al. [4] numerically studied heat transfer and vortex shedding behind a square cylinder in a laminar ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y. Varol). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.03.019
channel flow under a magnetic field. In the study they found that, through forced convection, both drag and lift coefficients are clearly damped by a magnetic field leading to the stabilization of the square cylinder. They also found that the application of a magnetic field to the vortex shedding flow can lead to a transition to symmetric flow. Chandran et al. [5] carried out a numerical analysis to simulate the thermal striping phenomena in a 1/5 scale water model of a prototype fast breeder reactor (PFBR) primary circuit. Two non-isothermal water jets in the different temperatures are impinged on a lattice plate which was placed above the jets. Reynolds stress turbulence model is used to evaluate the temperature fluctuation near the plate. In the study it is found that the cold jet dominated and the hot jet dominated flows result in very high and low temperature fluctuations respectively. The three-dimensional flow and mixing characteristics of multiple and multi-set three dimensional confined turbulent round opposing jets in a novel in-liner mixer are examined numerically using the standard k − ε turbulence model by Wang and Mujumdar [6]. They indicated that multiple opposing jets achieve better mixing than single opposing jets in the study. Wang et al. [7] numerically studied the laminar flow in an in-line mixer based on an opposing jet impingement. They found that the unequal inlet momenta of opposing jets obtained using both equal and unequal slot widths and the addition of baffles in the exit of the channel yield better mixing over shorter distances after impact. Beuf et al. [8] studied the influence of the geometry of the cell on mixing efficiency using three different geometries as circle, square and rectangle. They indicated that the flows in the Hele-Shaw cells are generally laminar and these flows can be in a first approximation considered as quasi-two dimensional. They also showed that the rectangular geometry leads to a
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Nomenclature AR Djet e g Gr H Hp k L Lp MI N P PE Re Ri St t T Tavg Tc Th u, v, w ΔT
Aspect ratio Diameter of jet (m) Internal energy per unit mass (J/kg) Acceleration of gravity (m/s 2) Grashof number Height of the channel (m) Height of the passive element (m) Thermal conductivity (W/m K) Length of the channel (m) Length of the passive element (m) Mixing index Number of jet Pressure (Pa) Passive element Reynolds number Richardson number Standard deviation of fluid temperature (K) Time (s) Temperature (K) Average temperature (K) Cold jet temperature (K) Hot jet temperature (K) Velocity magnitudes in direction x, y, z (m/s) Temperature difference between hot and cold water (K)
Greek symbols ν Kinematic viscosity μ Dynamic viscosity ρ Fluid density (kg/m 3) β Volumetric thermal expansion coefficient
better mixing, but also that the aspect ratio of the rectangle plays unexpectedly no important role on mixing. Walker et al. [9] made both numerical and experimental studies to carry out mixing of coolant streams of different temperatures in pipe junctions. In this way thermal fatigue may be prevented in the pipe wall. They presented a distribution of time averaged mixing scalar for different velocity ratios. Wang et al. [10] studied the jet mixing problem inside a slot, experimentally. They tested the jet array effect on cooling performance. They also tested the effect of the orientation angle and H/D ratio. It is found that the acceptable uniform flow is observed for shallow flows as H/D = 1. Chang et al. [11] made a numerical analysis to investigate the thermal mixing efficiency in a Y-shaped channel. They solved two dimensional incompressible, steady state equations using the Lattice Boltzmann method. They inserted different types of passive elements to improve thermal mixing efficiency. It is demonstrated that the enhanced mixing efficiency is a result of an increased intersection angle between the velocity vector and the temperature gradient within the channel. Thermal mixing is also a very important application for T-junctions as given by Naik-Nimbalkar et al. [12]. They made an experimental work on a T-junction with water and three dimensional steady state CFD simulations were carried out to predict velocity and temperature fields. The predicted values were in good agreement with the experimental measurements. Hu and Kazimi [13] made a numerical simulation on thermal striping for a three-dimensional, unsteady turbulent model. They used two types of mixing tee configurations and they modeled it by using the commercial CFD code FLUENT. A similar work has been done by Kamide et al. [14]. They performed a numerical work using the finite difference method. They indicated that the mixing
behavior in the tee was characterized by the relatively large vortex structures defined by the diameters and the velocities in the pipes. Using jets to enhance thermal mixing performance is a very useful way and many authors have worked on this subject such as Shi et al. [15], Lou et al. [16], Chua et al. [17] and Devahastin and Mujumdar [18]. These authors mostly studied the jet mixing phenomena using numerical techniques. They observed that geometric parameters are very important and thermal mixing is mostly a function of the temperature of jets. Sometimes, different shaped passive elements are used to control flow field, heat transfer and thermal mixing. Shan and Zhang [19] made a numerical calculation to investigate the different mixer configurations for an exhaust system of a turbo-fan engine. They observed that the lobed forced mixer can increase the mixing efficiency by 65%, and decrease the thrust coefficient by 3% only. Oztop [20] worked on the laminar mixed convection heat transfer for mixing in a partially open cavity. The main objective of this work is to understand the phenomena of flow and thermal mixing in a narrow channel with square objects inserted. Twin jets are used at different temperatures to supply water into the channel. This is a good application for some sanitary systems and mixture tanks. Based on the abovementioned literature and the authors' knowledge, there is no numerical work on thermal mixing in a narrow channel with a passive element inserted. 2. Problem description A schematic view of the parallel slot jet configuration in a channel is shown in Fig. 1(a). In this study, the effects of a passive element (PE) on thermal mixing in a rectangular cross-section channel have been investigated, numerically. Water was used as the working fluid. Two parallel slot jets at different temperatures were set in the channel in Re = 200, 400 and 600. Cold water jet (Tc) and hot water jet (Th) temperatures were defined as 293 K and 313 K, respectively. Calculations were made for three different aspect ratios such as AR = 0.5, 1 and 1.5. The aspect ratio was calculated from the proportions of height (Hp) and length (Lp) as shown in the figure. While the length of the PE is constant and equal to L/8, the height of the PE is chosen for three different values (Hp = H/4, H/2 and 3H/4). The passive elements were located in the channel for three different cases (Case 1 = L/4, Case 2 = 3 L/8 and Case 3 = L/2) as shown in the figure. There is a relationship between the sizes of the channel as L = 4H. The channel is divided into eight equal sections along the X coordinate. Thus there are seven columns considered and called as stations inside the channel. Using these stations, the thermal mixing index (MI) is calculated. 2.1. Grid independency Fig. 1(b) shows the distribution of a typical grid that is used in this study. Different grid elements, such as square and triangular shapes were tried. As its geometry is not very complicated, a square-element grid structure has a more appropriate result. The results were obtained from different grid densities such as 2500, 4500, 6500 and 8000. These results were compared with each other as given in Fig. 2 and the 6500 grid distribution was chosen for this study. 3. Numerical method In-channel flow was simulated using the commercial code FLUENT. This code uses finite volume method in order to solve Navier–Stokes and energy equations and it is widely used in heat transfer and fluid flow studies. The finite volume method can accommodate any type of grid. The CFD code is based on the pressure-correction and uses the SIMPLE algorithm of Patankar [21]. The first order upwind difference scheme (UDS) is used to discretize the momentum and energy equations.
B. Kok et al. / International Communications in Heat and Mass Transfer 44 (2013) 69–76
71
(a)
(b)
Fig. 1. (a) Schematical configuration of the channel, (b) grid distribution.
accepted as a forced convection. The mass conservation equation can be written as follows:
4. Governing equations Calculation of the temperature and flow field in a channel requires the obtainment of the solution of the governing equations. Incompressible, unsteady and laminar flows can be described by differential equations of continuity, momentum and energy. The radiation mode of heat transfer is neglected according to the other modes of heat transfer. Buoyancy forces are also neglected and the heat transfer regime is
304.5 2500 grid 4500 grid 6500 grid 8000 grid
304.0
ð1Þ
Momentum equations can be written in two directions as 2 ∂ðρuÞ ∂ ρu ∂ðρuvÞ ∂p ∂ ∂v ∂u ∂u þ ¼ þ λ þ V þ 2μ þ ∂t ∂y ∂x ∂x ∂x ∂y ∂x ∂x ∂ ∂v ∂u μ þ þ ∂y ∂x ∂y 2 ∂ðρvÞ ∂ ρv ∂ðρuvÞ ∂p ∂ ∂v ∂u þ ¼− þ μ þ þ ∂t ∂x ∂y ∂x ∂x ∂y ∂y ∂ ∂v λ∇V þ 2μ : þ ∂y ∂x
303.5
T
∂ρ þ ∇⋅ðρV Þ ¼ 0: ∂t
ð2Þ
ð3Þ
303.0
The energy equation can be written as ∂ðρeÞ ∂ ∂T ∂ ∂T þ ∇ðρeV Þ ¼ k þ k : ∂t ∂x ∂x ∂y ∂y
302.5
302.0
301.5 0.0
0.2
0.4
0.6
0.8
1.0
h/H Fig. 2. Effect of grid densities on the temperature at 7 L/8 location along the Y coordinate for Re = 400.
ð4Þ
Reynolds number is chosen based on the inlet jet's velocity and the Prandtl number is fixed at 5.43 for water. Other dimensionless numbers can be written as follows, Re ¼
u:Djet ν
2
; Gr ¼
g:β:d3 :ΔT Gr ; Ri ¼ 2 : ν2 Re
ð5Þ
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Richardson number is calculated by Eq. (5) and, Ri b 1 is obtained. This result shows that the effect of natural convection can be neglected in this study.
(a) 29 5
29
8
30
2
303
4.1. Boundary conditions The inlet boundary conditions were obtained from the calculated instantaneous mass flow rate. This can be done due to the acceptation of the incompressibility. Also, no-slip boundary conditions were applied for all velocities at the walls. Boundary conditions for the considered physical model (Fig. 1(a)) are given as
31
1
30
30 4
8
(b) 296
29 8
302
On all solid boundaries; u = v = 0
303
∂T On the all horizontal boundaries; ¼0 ∂y ∂T ¼0 On the all vertical boundaries; ∂x ∂u ¼0 At the exit; P = 0, ∂x Temperatures, Tc = 293 K, Th = 313 K Inlet Reynolds numbers are; Re = 200, 400, and 600.
311
304
306
(c) 299
300
302 3 03
4.2. Definition of mixing index
309
The definition of the mixing index is the main function for this study. It is defined by Eq. (6) as used by many others earlier [6]. MI ¼
St 100: ΔT
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 n ∑i¼1 T h −T avg =ðn−1Þ
305
4
Fig. 3. Isotherms along the channel at different Reynolds numbers without any inserted PE, a) Re = 200, b) Re = 400, c) Re = 600.
ð6Þ
In this equation, St shows the standard deviation between hot and cold water temperatures. Thus, the standard deviation is given by
St ¼
30
ð7Þ
where ΔT is the temperature difference between hot and cold water temperatures and Tavg is the average temperature. Physically, this mixing index is a measure of how much the bulk temperature at any specific cross-section represents the set of temperatures that it comes from. An MI = 0 implies complete mixing which corresponds to a perfectly flat temperature profile [7].
the channel. It means that flow is developed at the middle part of the channel. The increase in Reynolds number affects the flow distribution and the flow developed again after mid-axis in the x-direction. The flow from the top inlet is suppressed to the bottom and it impinges on the bottom. Again, a circulation cell is formed near the exit but it is not shown in the figure. Fig. 5 displays the isotherms at different Reynolds numbers for AR = 1 and Case 1. As given in Fig. 5(a) an adiabatic body is less effective on isotherms for whole channels because of the low Reynolds number. It was noticed that the cold flow and hot flow through the inlets to the channel were at 293 and 313 K, respectively.
(a)
5. Results and discussion A numerical analysis is made to investigate the effects of the parameters of the Reynolds number, aspect ratio and location of the inserted passive element (PE) on thermal mixing performance in a rectangular cross-section channel. Two parallel slot jets are determined at different temperatures to supply fluid into the channel. The inserted body is chosen as adiabatic and a channel whose length is long enough is chosen. Fig. 3 presents the isotherms along the channel at different Reynolds numbers without any inserted PE. As indicated earlier the cold fluid flows through the inlets to the channel from the top port and the hot one through the inlets to the channel from the bottom. The exit port is located at the middle of the opposite wall. The temperature shows the symmetric distribution according to the middle y axis at lower values of Reynolds number (Re = 200) as seen from Fig. 3(a). This symmetrical shape is destroyed with an increase in the Reynolds number and a plume like temperature distribution is observed from the top port. Isotherms are cumulated around the exit of the hot port for both Re = 400 and Re = 600. Fig. 4 illustrates the streamline to see the effects of Reynolds number on fluid flow with the same parameters of the previous figure. Flow is symmetric according to the mid-axis at low Reynolds number. Two circulating cells are formed around the inlet jets. Streamlines are parallel to each other at the middle part of
(b)
(c)
Fig. 4. Streamlines along the channel at different Reynolds numbers without any inserted PE, a) Re = 200, b) Re = 400, c) Re = 600.
B. Kok et al. / International Communications in Heat and Mass Transfer 44 (2013) 69–76
35
(a) 295
29 8
301
308
305
30
25
MI (%)
3 04
(b)
20 15 10
298
3 01
29 5
302
5
303 310
Re=200 Re=400 Re=600
30
2
303 3 11
73
304 306
0 0.0
305
0.2
0.4
0.6
0.8
1.0
x/L
(c) 298
Fig. 7. Variation of mixing index along the channel (L) without any PE inserted.
298
310
301
302 303
304
304
306 Fig. 5. Isotherms along the channel at different Reynolds numbers for AR = 1 and Case 2, a) Re = 200, b) Re = 400, c) Re = 600.
Each jet forms their own circulation at the exit which turns at different directions. A stagnation point occurs in front of the body due to a combination of cold and hot jets. Temperature distributions are almost parallel to the horizontal walls at the middle-section of the channel. The interaction between hot and cold jets increases with the increase of Reynolds number as seen from Fig. 5(b) and (c). The isotherms especially inclined down the wall between the jet inlets and PE. Isotherms have a linear shape along the channel and the temperature difference between the hot and cold water became lower after PE in all Re numbers.
(a)
This means that PE has a positive effect on thermal mixing efficiency. As indicated earlier, the symmetricity on flow is destroyed with an increase in Reynolds number. A minimum circulation cell is formed behind the PE at low Reynolds number (Re = 200) as given in Fig. 6(a). With an increasing Reynolds number a huge cell is formed at the top side of the hot jet and the flow goes from the top of the body. This cell is formed because of the impinged and deviated flow to the body. The hot jet hits the cold jet and it almost moves to the bottom side of the object. The center of the cell of the biggest notation moves to the right with increasing Reynolds number as seen from Fig. 6(c). And a circulation cell is formed behind the body. The dimensions of the circulated flow on the top and bottom sides of the cold jet become almost the same for Re = 400 and 600. The clustered streamlines under the body indicate the increase of flow rate at that port. Fig. 7 presents the mixing index (MI) along the channel with no PE inserted which was calculated from Eq. (6). MI starts from 100% at x/L = 0 and decreases along the channel (L) when the thermal mixing efficiency of the cold and hot fluid gets better. The values of MI decrease at the middle part of the channel (x/L = 0.5) with increasing jet Re number. After here higher values of mixing index are formed for higher values of Reynolds number. When compared with the other values of Re numbers a more uniform MI decrease was observed at Re = 200 and after x/L = 0.3 the MI showed nearly the same decrease in shape for Re = 400 and 600. A variation of MI along the channel (L) for different Re numbers for the aspect ratio (AR) = 1 and Case 2 is given in Fig. 8. MI shows a sharp decrease for all Re numbers between jets and PE (x/L = 0 and 0.4) and MI decreases with increasing Re number in 35
(b)
Re=200 Re=400 Re=600
30
MI (%)
25 20 15
(c) 10 5 0 0.0
0.2
0.4
0.6
0.8
1.0
x/L Fig. 6. Streamlines along the channel at different Reynolds number for AR = 1 and Case 2, a) Re = 200, b) Re = 400, c) Re = 600.
Fig. 8. Variation of mixing index along the channel (L) for AR = 1 and Case 2.
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B. Kok et al. / International Communications in Heat and Mass Transfer 44 (2013) 69–76
(a)
35
300 296
30
302
302
303 3 07
25
3 04
MI (%)
309
3 04
(b)
3 07
15
302 5
303 309
20
10
3 01
296
Empty channel AR=0.5 AR=1 AR=1.5
30
4
304 0 0.0
0.2
0.4
0.6
0.8
1.0
x/L
(c) 296
302
4
307
Fig. 11. Variation of mixing index along the channel (L) for different aspect ratios for Case 1 and Re = 200.
30
309
30 0
302 303 304
Fig. 9. Isotherms along the channel for different aspect ratios for Case 3, Re = 400, a) AR = 0.5, b) AR = 1, c) AR = 1.5.
this region. MI becomes almost constant for Re = 400 and 600 between x/L = 0.4 and 1. But decreasing is going on along the channel and the lowest MI is formed at the exit at Re = 200. It means that thermal mixing efficiency becomes worse for higher values of Reynolds number due to lower velocity of the flow at the middle of the channel after the block. Figs. 9 and 10 show the isotherms and streamline for different ARs for Case 3 and Re = 400, respectively. As seen from the figures the blockage effect of the inserted body increases with increasing AR. The effect of AR on temperature distribution becomes insignificant at the left and right sides of the block. Temperature values become constant
(a)
on the top and bottom sides of the block. Jet flows incline toward the top of the vertical wall of the channel and a big circulation cell occurs on the left-bottom region of the channel and it is stuck there with increasing AR. Circulation cells are formed behind the PE with increasing AR. Effects of the different aspect ratios on thermal mixing efficiency in the channel are presented in Fig. 11 for Case 1 and Re = 200. As seen from the figure MI is lower and more uniform in no PE inserted channel while all values of MI of different ARs present similar a trend and are higher. A sharp decline is observed until x/L = 0.2 and it is almost kept constant until the PE for all AR values. This means that the insertion of PE makes a negative effect on thermal mixing performance in the channel in the PE region. After the flow passes the passive element a uniform decrease is formed for all AR values. As seen from the figure, MI values decrease in the left side of the PE and increase in the right side of the PE with the enhancement of AR values. This means that an increase in the aspect ratio of the inserted body is positive only in the right half of the channel. Fig. 12 gives a variation of the mixing index along the channel (L) for different ARs for Case 3 and Re = 400. A regular decrease is observed in the MI curve in the no PE inserted case and all other AR values present a similar decrease trend along the channel. Different AR values have only a slight effect on thermal mixing effectiveness for present parameters. Thermal mixing efficiency is better in the PE inserted condition until x/L = 0.7, but in the exit region better thermal mixing is observed in the no PE inserted case. An interesting 35
(b)
Empty channel AR=0.5 AR=1 AR=1.5
30
MI (%)
25
(c)
20 15 10 5 0 0.0
0.2
0.4
0.6
0.8
1.0
x/L Fig. 10. Streamlines along the channel for different aspect ratios for Case 3 and Re = 400, a) AR = 0.5, b) AR = 1, c) AR = 1.5.
Fig. 12. Variation of mixing index along the channel (L) for different aspect ratios for Case 3 and Re = 400.
B. Kok et al. / International Communications in Heat and Mass Transfer 44 (2013) 69–76
(a)
35
298
295
3 04
305
25
MI (%)
310
(b)
Empty channel Case 1 Case 2 Case 3
30
302.5 303
20 15 10
297
300
298 310
302 303
5
304
0
306
0,0
0,2
0,4
310
0,6
0,8
1,0
x/L
(c) 296
75
299
Fig. 15. Variation of mixing index along the channel (L) for different locations of the passive element for Re = 200 and AR = 1.
3 02
302 303
304
3 06
Fig. 13. Isotherms along the channel for different locations of the passive element for Re = 600, AR = 0.5, a) Case 1, b) Case 2, c) Case 3.
observation is that, thermal mixing efficiency is far better in higher Re numbers when compared with a prior figure (Fig. 11) in the passive element inserted channel. Also maybe it is caused by a higher distance between jets and the passive element. But for both cases better thermal mixing efficiency is plotted in the no PE inserted in the exit port of the channel. Figs. 13 and 14 show the isotherms and streamlines for different cases at AR = 0.5 and Re = 600, respectively. As indicated above thermal mixing parameter is a function of inlet temperatures. Low and high temperature fluids started to mix above the part of the high temperature
(a)
inlet. The isotherms are clustered at that point for all cases. After the body mixing performance is decreased. The flow temperature becomes almost constant above the body due to low velocity of the fluid at that region. This can be seen also from the streamlines. The temperature of the mixing fluid can be controlled via the body and the changing location of the body. The location of the body affects the flow field and the location of the main cell changes with the changing of the location of the PE. It affects the thermal mixing far from the inlet. The variation of mixing index along the channel (L) for different locations of the PE is plotted under operation conditions of Re = 200 and AR = 1 in Fig. 15. A parabolic decrease is seen in MI along the channel and the best thermal mixing performance is achieved at the no PE inserted case. For the compared effects of different locations of the PE on thermal mixing efficiency, Case 3 gives the best thermal mixing performance, Case 2 followed and Case 1 gives the least thermal mixing performance. As it is resulted from almost all prior figures, here the same condition is determined that in the exit part of the channel better thermal mixing performance is in the no PE inserted condition. Fig. 16 shows a variation of the mixing index along the channel for the different locations of the passive element for Re = 600 and AR = 0.5. In contrast to the prior figures, MI shows a wavy decrease along the channel in Fig. 16 because of high Reynolds number (Re = 600) for the no PE inserted case. All other MI curves exhibit a similar trend along the channel (L) except Case 1. A sharp decline is seen at Case 1 between PE and jets (x/L = 0–0.4). This resulted from
35
(b)
Empty channel Case 1 Case 2 Case 3
30
MI (%)
25
(c)
20 15 10 5 0 0.0
0.2
0.4
0.6
0.8
1.0
x/L Fig. 14. Streamlines along the channel for different locations of the passive element for Re = 600, AR = 0.5, a) Case 1, b) Case 2, c) Case 3.
Fig. 16. Variation of mixing index along the channel (L) for different locations of the passive element for Re = 600 and AR = 0.5.
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35
the PE with increasing aspect ratio values. Higher AR values have a positive effect on thermal mixing efficiency on the second half of the channel. • The temperature of the mixing fluid can be controlled via the changing of location of the PE. Also, the location of PE affects the flow field and the location of the main cell. The effects of the location of PE on thermal mixing performance were compared. • Case 3 gives the best mixing performance and Case 1 gives the worst. A uniform decrease of the MI curve is observed along the channel and the lowest value of MI is formed near the exit of almost all the parameters in the no PE inserted case.
Empty channel Case 1 Case 2 Case 3
30
MI (%)
25 20 15 10 5
References
0 0.0
0.2
0.4
0.6
0.8
1.0
x/L Fig. 17. Variation of mixing index along the channel (L) for different locations of the passive element for Re = 400 and AR = 1.5.
the fact that the small PE-jet distance formed a strong vortex in the flow under high Reynolds (Re = 600) number operation conditions. MI is almost constant until the exit of the channel for this case. These results show that thermal mixing performance is the best and more stable for Case 1. Fig. 17 presents a variation of mixing index along the channel (L) for the different locations of passive element for Re = 400 and AR = 1.5. As it is plotted in the figure there is a huge difference between MI values of Case 1 and other. It is far higher than the other and fluctuating along the channel. The lowest MI value is formed at Case 3. The same with most of the former conditions, also in this figure the best thermal mixing efficiency is observed at the no PE inserted case in the exit of the channel and MI shows a uniform decrease along the channel. Based on the findings of the last three figures, it is concluded that to locate PE in Case 2 in the channel, gives optimum thermal mixing performance. 6. Conclusions In the present study, the influence of the jet Reynolds number, aspect ratio and different locations of the passive element on flow and thermal mixing performance in a rectangular cross-section channel was investigated numerically. The main findings from the studied works can be listed as follows: • The interaction between hot and cold jets increases with the increasing of Reynolds number and isotherms decline to the bottom wall of the channel. • Flow is symmetric according to the mid-axis at low Reynolds (Re = 200) number and this symmetricity is destroyed with an increase in Re number. Reynolds number has an important effect on thermal mixing performance in the channel. • It is found that mixing performance increases in the first half of the channel (x/L = 0–0.5) with an increase in Re number. Also, it is decreased in the second half (x/L = 0.5–1) of the channel with higher values of Re number. The best mixing performance is formed at Re = 200 near the exit. • The blockage effect of PE increases and circulation cells occurred behind
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