Heat Mass Transfer (2008) 44:445–461 DOI 10.1007/s00231-007-0256-5
ORIGINAL
Unsteady fluid flow and heat transfer over a bank of flat tubes N. Benarji Æ C. Balaji Æ S. P. Venkateshan
Received: 31 July 2006 / Accepted: 7 March 2007 / Published online: 19 April 2007 Springer-Verlag 2007
Abstract Transient numerical simulations of fluid flow and heat transfer over a bank of flat tubes have been carried for both in-line and staggered configurations for the following boundary conditions: (a) isothermal and (b) isoflux. The effect of Reynolds number, Prandtl number, length ratio, and the height ratio, on the Nusselt number, and the dimensionless pressure drop are elucidated. Correlations are proposed for both pressure drop and Nusselt number and optimum configurations have been determined. List of symbols A total area of all tubes, m2 At are a of a single tube, m2 C constant in the temperature boundary condition (Eq. 7) Cp specific heat, J/kg K Da transverse diameter of flat tube, m Db longitudinal diameter of flat tube, m DH hydraulic diameter, 2H, m Pout DP dimensionless pressure drop, PinqU 2 b H height of HEM, m h heat transfer coefficient for single HEM, W/m2 K k thermal conductivity, W/m K L length of HEM, m . m mass flow rate of the fluid across the tube bank, kg/s Nu Nusselt number, h Dk H P pressure, Pa Pin module inlet pressure, Pa Pout module outlet pressure, Pa N. Benarji C. Balaji (&) S. P. Venkateshan Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036 Tamil Nadu, India e-mail:
[email protected]
DP Q q¢¢ Re ST SL T Tb Tb,m Tin TW DT t U, V Ub Uin Umax X, Y
pressure drop across the tube bank, Pa total heat transferred across the tube bank, W constant heat flux on tube surface, W/m2 Reynolds number, Umaxt DH transverse pitch for tube bank, m longitudinal pitch for tube bank, m temperature, K bulk temperature at any cross section, K average of the module inlet and module outlet temperatures, K inlet temperature of the fluid, K constant temperature of tube surface, K temperature difference, K time, s horizontal and vertical velocities, m/s RH bulk velocity at any cross section, m/s, H1 U dy 0 inlet velocity of the fluid, m/s maximum velocity at the minimum cross section, m/s horizontal and vertical co-ordinates, m
Greek symbols l dynamic viscosity, Ns/m2 q density, kg/m3 m kinematic viscosity, m2/s
Subscripts i module number Abbreviations HEM Heat exchanger module IN Inlet MI Module inlet
123
446
MO OUT
Heat Mass Transfer (2008) 44:445–461
Module outlet Outlet
1 Introduction The importance of fluid flow and heat transfer characteristics over tube banks in the design of heat exchangers is well known. Extensive experimental [1] and numerical studies [2–10] on circular tube banks have been done previously. Based on previous studies reported in the literature, one can infer that the tube shape and arrangement exert considerable influence on the heat transfer [11–15]. Flat tubes, though, have not been investigated to the same extent, notwithstanding the fact that they play an important role in many technical applications, such as modern heat exchangers and automotive radiators. Flat tube designs have recently been introduced for use in automotive air conditioning evaporators and condensers. Recent developments in automotive brazed aluminium manufacturing technology have made the cost of the flat tube heat exchanger construction more favourable [16]. Flat tube heat exchangers are expected to have lower air-side pressure drop and better air-side heat transfer coefficients compared to circular tube heat exchangers. The pressure drop is expected to be lower than that for circular tubes, because of a smaller wake area. For the same reason, vibration and noise are expected to be less in flat tube heat exchangers compared to circular tube heat exchangers. Literature on the flow over bank of flat tubes is limited, except for the recent studies of Bahaidarah et al. [17]. They numerically investigated steady, two dimensional, laminar, incompressible flow over a bank of flat tubes for both staggered as well as inline configurations and determined an optimum configuration from the point of view of heat transfer. Also, recently some researchers have been carrying out transient numerical simulations on the flow across bluff bodies, to determine the transient behavior of tube banks. Tatsutani et al. [18] presented results for incompressible, two-dimensional, unsteady flow across a pair of square cylinders, placed in tandem in a channel, by dye visualization and also by direct numerical simulation (DNS). They found that there exists an optimum cylinder separation distance, which has effect on both the fluid flow and heat transfer, over the range of Reynolds numbers 200 < Re < 1,600. Johnson et al. [19] presented results of a detailed numerical investigation of unsteady incompressible flow problems involving periodic array of staggered cylinders. They considered three types of problems: (a) uniperiodic flow with two cylinders, (b) biperiodic flow
123
with two cylinders and (c) uniperiodic flow with ten cylinders. The analyses were carried out for Re = 100. Beale and Spalding [20] carried out numerical calculations for transient flow in in-line square and rotated square tube banks, with a pitch to diameter ratio of 2.1 in the Reynolds number range of 30–3,000. Flow visualization studies of the results of the calculations were presented, together with quantitative details of pressure drop, lift, drag, and heat transfer. Rosales et al. [21] investigated, numerically, unsteady laminar flow and heat transfer from a square cylinder located in a channel, with fully developed inlet velocity profiles. The time averaged Nusselt number for each face and the time averaged cylinder Nusselt number were determined at Re = 500. Jue et al. [22] investigated heat transfer and fluid flow across three heated cylinders arranged in an isosceles right angled triangle arrangement between two parallel plates. The finite element method was used to solve the continuity, momentum and energy equations. The variations in the drag coefficient and the time averaged Nusselt number around the surface of the three cylinders and the surface averaged value of the time averaged Nusselt number for each cylinder were investigated. The analysis was carried out for: 100 £ Re £ 300 and 0.5 £ gap/diameter £ 1.25. Srinivas et al. [23] carried out unsteady numerical computations for momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel for the range of Reynolds numbers 10 < Re < 500, using the commercially available FLUENT code. Schneider and Farge [24] presented a high resolution numerical simulation of incompressible, unsteady, two dimensional flows in tube bundles for both in-line and staggered arrangements, using the ‘volume penalization’ method. Transient simulations were performed on circular, square, and double-cruciform shaped tubes at Re = 200. Horvat et al. [25] performed a detailed transient numerical simulation of fluid and heat transfer for a number of heat exchanger segments with cylindrical ellipsoidal and wing shaped tubes in staggered arrangement. DNS was done to solve the governing equations. Drag coefficient and Stanton number correlations were constructed as polynomial functions of the Reynolds number and hydraulic diameter. Horvat and Mavko [26] performed transient numerical simulations of heat transfer for heat exchanger segments with cylindrical and ellipsoidal tubes for a staggered arrangement. The time averaged values of Nusselt number were further used to construct relations for the drag coefficient and Stanton number as polynomial functions of the Reynolds number and the hydraulic diameter. From the review of literature presented above, it is seen that transient numerical simulations have been performed on all bluff bodies (cylindrical, square, ellipsoidal, cruciform, wing shaped tubes) except on flat tubes. Since flat
Heat Mass Transfer (2008) 44:445–461
447
tubes have lower pressure drop characteristics and have potential applications in modern heat exchangers and automotive radiators, it is important to know their transient fluid flow and heat transfer behavior for a better design of such heat exchangers. The objective of the present study is to numerically study two dimensional, laminar, incompressible unsteady fluid flow and heat transfer over a bank of flat tubes with constant physical properties for different kinds of thermal boundary conditions.
Figure 5 shows typical HEM configurations considered in this study. To save space only six configurations have been shown out of nineteen that have been considered. The parameters considered for transient numerical simulation are 25 £ Re £ 400, Pr (0.7 and 7.0). Reynolds number is defined as Re ¼ Umaxm DH where DH = 2H. To ensure fully developed conditions at the inlet and the outlet, the computational domain is taken to be sufficiently long at both the places. The thermophysical properties are taken at the inlet temperature of the fluid.
2 Mathematical formulation
2.1 Governing equations
Both in-line and staggered configurations are considered for the analysis. A total of ten longitudinal rows including inlet and outlet sections have been considered. Both air (Pr = 0.7) and water (Pr = 7.0) are considered as the working media. The configurations considered in this study are shown in Figs. 1 and 2. Shaded parts define a heat exchanger module (HEM). The flat tubes are characterised by the dimensions shown there on. The tube bank considered for analysis along with the boundary conditions for different configurations is shown in Figs. 3 and 4. These figures show only five HEMs for convenience, though a typical configuration contains ten HEMS. Table 1 lists the various configurations considered in the present study.
For unsteady, two-dimensional, incompressible, laminar flow of a fluid, with constant property fluid, the governing equations are: continuity equation: @U @V þ ¼0 @X @Y
ð1Þ
X-momentum equation:
@U @U @U q þU þV @t @X @Y
2 @P @ U @2U þl þ ¼ @X @X 2 @Y 2 ð2Þ
Y-momentum equation: 2 @V @V @V @P @ V @2V þU þV þl q þ ¼ @t @X @Y @Y @X 2 @Y 2 ð3Þ Energy equation: q Cp
@T @T @T þU þV @t @X @Y
¼k
@2T @2T þ @X 2 @Y
ð4Þ
2.2 Boundary conditions Fig. 1 In-line configuration of flat tube bank
2.2.1 Inlet condition The fluid is assumed to enter with a uniform horizontal velocity, Uin, and temperature of Tin. U ¼ Uin;
T ¼ Tin;
V¼0
ð5Þ
2.2.2 Symmetry condition
Fig. 2 Staggered configuration of flat tube bank
For the top and bottom surfaces of the computational domain excluding the tube surfaces, symmetry boundary condition is used. The mathematical form of this condition is
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Heat Mass Transfer (2008) 44:445–461
Fig. 3 Typical in-line geometry considered for simulation with boundary conditions
Fig. 4 Typical staggered geometry considered for simulation with boundary conditions
Isothermal or isoflux
U = V = 0, T = TW or q = C "
Symmetry,
∂U ∂T = =V = 0 ∂Y ∂Y
Y, V Velocity Inlet,
Interior
U = U in , T = Tin ,V = 0
Outflow,
∂U ∂V ∂T = = =0 ∂X ∂X ∂X
X, U
@U ¼ 0; @Y
V ¼ 0;
@T ¼0 @Y
ð6Þ
2.2.4 Outlet condition Zero diffusion flux implemented for all variables at the outlet boundary.
2.2.3 Wall condition At the tube surface no slip condition is applied and as far as energy equation is concerned the tube is at ‘isothermal or isoflux’ condition. U ¼ V ¼ 0;
T ¼ TW;
or
Table 1 Geometric parameter for the various in-line and staggered configurations considered in this study
123
q00 ¼ C
ð7Þ
@U @V @T ¼ ¼ ¼0 @X @X @X
ð8Þ
The governing equations of fluid flow and heat transfer (Eqs. 1–4) along with boundary conditions (5–8) are solved simultaneously by using a finite volume based solver.
Configuration
H/Da
L/Da
Configuration
H/Da
L/Da
In-line, H*2-L*4
2
4
Stagg, H*2-L*5
2
5
In-line, H*2-L*5
2
5
Stagg, H*2-L*6
2
6
In-line, H*2-L*6
2
6
Stagg, H*2-L*7
2
7
In-line, H*3-L*4
3
4
Stagg, H*3-L*5
3
5
In-line, H*3-L*5
3
5
Stagg, H*3-L*6
3
6
In-line, H*3-L*6
3
6
Stagg, H*3-L*7
3
7
In-line, H*4-L*4
4
4
Stagg, H*4-L*5
4
5
In-line, H*4-L*5
4
5
Stagg, H*4-L*6
4
6
In-line, H*4-L*6
4
6
Stagg, H*4-L*7
4
7
Heat Mass Transfer (2008) 44:445–461
449
Fig. 5 Typical configurations considered in the study a inline H2-L4, b inline-H3-L5, c inlineH4-L6, d stagg-H2-L5, e staggH3-L6, f stagg-H4-L7
3 Numerical methodology
3.1 Grid independence test
The governing continuity, fluid flow and energy equations are solved using FLUENT 6.2 [27], which employs a finite volume method. In the present study, structured quadrilateral mapped meshes were created with fine meshes near the flat tube and coarse meshes away from the flat tube, with the help of a successive ratio scheme using the gridding software GAMBIT 2.2.30 [28]. Unsteady segregated solver was used with second order upwinding scheme for the convective terms in the momentum equation. For pressure–velocity coupling, pressure implicit with splitting of operators (PISO) scheme was used. A convergence criterion of 1 · 10–6 was applied to the residuals of the continuity and the momentum equations and 1 · 10–9 to the residuals of the energy equation.
A grid independence test was carried out to assess the sensitivity of the result to the grid sizes distribution. The following case of in-line configuration has been considered for grid independence test, Db/Da = 2, H/Da = 4, L/Da = 4, Pr = 0.7, Re = 400. Grid sizes of 20 · 25, 51 · 51 and 101 · 101 per module were considered by including the inlet and outlet sections. A typical grid structure for a single HEM is shown in Figs. 6 and 7, for both inline as well as staggered configurations. For the in-line configuration, half of the geometry and for staggered configuration, full geometry need to be considered. The results of the grid independence test are shown in Table 2. From the table, it can be observed that the grid size of 51 · 51 acceptable values. The grid sizes
Fig. 6 Typical grid patterns for an in-line arrangement: inline-H4L4-Grids per module
Fig. 7 Typical grid pattern for a staggered arrangement: stagg-H4-L5 Grids per module
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450
Heat Mass Transfer (2008) 44:445–461
Table 2 Grid independence test for the various modules considered in this study HEM
20 · 25 Nu
51 · 51 Nu
% Diff
1
26.535
26.726
2
18.121
3 4
hi DH j
Nui ¼
101 · 101 Nu
% Diff
where hi is given by
0.715
26.800
0.2738
18.356
1.280
18.442
0.4678
hi ¼
16.551
16.657
0.632
16.666
0.056
14.481
14.460
0.149
14.518
0.4005
5
13.574
13.552
0.168
13.543
0.0622
6
14.043
13.086
7.319
13.142
0.4305
7
12.878
13.111
1.776
12.934
1.3717
8
12.159
12.705
4.296
12.836
1.0213
9
12.878
12.753
0.975
12.809
0.4366
10
13.982
11.033
26.731
10.587
4.2079
% Diff = [(Grid I)–(Grid II)] · 100/(Grid II)
Qi Tb;m Tw At
ð9Þ
ð10Þ
and Qi ¼ m: CP ½Tb ðMIÞ Tb ðMOÞ
ð11Þ
where MI and MO refer to module inlet and module outlet respectively. Calculation of overall Nusselt number: Nu ¼
h DH j
ð12Þ
where h is given by mentioned in the table are for a single module. From the above table, one can say the solution is fully grid independent. To minimize the computational time and resources, a grid size of 51 · 51 has been considered for further analysis. For the complete configuration, the total number of grid points will be 51 · 51 · 10 = 26,010, excluding the inlet and outlet sections. 3.2 Validation In ref [17], steady state numerical simulations have been performed for flat tube banks. In order to validate with the same reference, an in line H*4-L*4 configuration was selected and steady state simulations have been performed at Re = 400. With the selected grid size of 51 · 51 per module, the results of the present study are validated with the results of [17]. The calculated Nusselt number and the dimensionless pressure drop per module for interior modules are shown in Table 3. The agreement between the two is generally good with a maximum deviation of 15%, in so far as Nusselt number is concerned. As regards the pressure drop, the agreement is even better. Energy balance in all cases is accurate to within 0.01% of total heat input.
h¼
ð13Þ
and Q ¼ m: CP ½Tb ðOUTÞ Tb ðINÞ
ð14Þ
For isoflux boundary condition: Calculation of module average Nusselt number: hi DH k
Nui ¼
ð15Þ
where hi is given by hi ¼
q00 Tw Tb;m
ð16Þ
Calculation of overall Nusselt number: Nu ¼
h DH k
ð17Þ
where h is given by
4 Results and discussion
h¼
A total of 360 cases have been considered for the analysis, and due to paucity of space only selected cases have been taken up for discussion. Before proceeding to the discussion of the results, the calculation of module average Nusselt number and overall Nusselt number is for two boundary conditions given below. For isothermal boundary condition: Calculation of module average Nusselt number:
and
123
Q A ðTw Tin Þ
Q00 ðDT Þ
DT ¼ Tb ðINÞ Tb ðOUTÞ
ð18Þ
ð19Þ
The analysis has been carried out for each case with a time step size of 0.001 s for a total time steps of 20,000, i.e. up to 20 s of operation. This is representative of a typical blow down time in a cooler of a hypersonic wind tunnel.
Heat Mass Transfer (2008) 44:445–461 Table 3 Validation with the results of Bahaidarah et al.
451
HEM
Nu—present work
Nu—Bahaidarah et al.
1
26.726
27.925
4.292
0.748
0.727
2.914
2
18.356
16.141
13.722
0.230
0.233
1.391
3
16.657
14.499
14.881
0.202
0.202
0.122
4
14.460
13.670
5.776
0.181
0.18
0.803
5
13.552
13.270
2.122
0.171
0.167
2.187
6
13.086
13.074
0.089
0.165
0.159
3.768
7
13.111
12.978
1.024
0.162
0.154
5.284
8
12.705
12.932
1.757
0.161
0.151
6.930
9
12.753
12.910
1.214
0.161
0.149
7.719
10
11.033
12.720
13.264
0.140
0.137
2.433
Figure 8 shows the variation of overall Nusselt number with time, for two cases in the in-line configuration, with isothermal boundary condition and air as the medium. Though a total time of 20 s is considered for the analysis, these two cases reach the steady state condition with in 0.5 s. For the remaining cases, the time to reach varies and may be as high as 5 s. 4.1 The isothermal tube The effect of a constant temperature boundary condition on the unsteady fluid flow and heat transfer is analysed here. This corresponds to a phase change taking place on the inside of the tube, which is not uncommon. For some cases, the variation of dimensionless pressure drop and Nusselt number are shown here to illustrate the fluid flow and heat transfer characteristics, with air and water as the fluids for both configurations. In Fig. 9a, b, the variation of dimensionless pressure drop with HEM number for in-line con110 100 90
In-H2-L6-Re-400
Nu
80
In-H2-L6-Re-200
70 60 50 40 30 0
0.5
1
1.5
2
2.5
3
Time, s Fig. 8 Variation of overall Nusselt number with time for isothermal boundary condition
% Difference
Pressure drop—present work
Pressure drop—Bahaidarah et al.
% Difference
figurations with air as the fluid is shown. Figure 9 shows the variation of dimensionless pressure drop with module length, whereas Fig. 9b shows the variation with module height. From these plots, it is clear that from the third module onwards fully developed velocity conditions exist. Figure 9c, d shows the variation of Nusselt number with HEM number for in-line configurations for air as the medium. Again, from these figures, it can be inferred that the Nusselt number is constant from the third module and remains constant up to module number 9 and varies slightly for the 10th module. The reason for this is the end effect for the last module. From the above figures, it can be concluded that at a particular Reynolds number the dimensionless pressure drop increases as the length of the module increases, as expected, and decreases as the height of module increases. The Nusselt number decreases as the length of the module increases, where as it increases as the height of the module increases. Figure 10a, b shows the variation of dimensionless pressure drop with HEM number for in-line configuration with water as the medium. From these figures, it can be concluded that fully developed condition exists from 4th and 3rd modules, respectively. Figure 10c, d shows the variation of Nusselt number with HEM number for water with in-line configuration. The figure clearly shows that the Nusselt number varies continuously for all configurations without attaining a constant value. The Nusselt number value is highest for module 1 and decreases slowly. The Nusselt number is constant from the 6th module for one configuration and this is shown in Fig. 10d. For the remaining cases, the Nusselt number decreases continuously from the 1st to the 10th modules. A comparison of Fig. 10c, d with Fig. 9c, d show a much higher Nusselt number for water, as opposed to air for reasons well known. From Fig. 10, it can be concluded that with water as the fluid there exists a fully developed velocity condition.
123
452 3.5
3.0
(a)
In-H2-L6-Re-400 In-H2-L5-Re-400 In-H2-L4-Re-400
3.0
(b)
In-H3-L6-Re-400 In-H4-L6-Re-400
2.0
2.5
Dp
1.5
2.0
1.0 1.5
0.5 0.0
1.0 1
2
3
4
5
6
7
8
9
10
1
27 In-H2-L4-Re-400
In-H2-L6-Re-400
Nu
Nu
5
6
7
8
9
10
In-H2-L6-Re-400 In-H3-L6-Re-400 In-H4-L6-Re-400
23
21
21
19
19
17
17
15
15 1
2
3
4
5
6
7
8
9
10
1
Heat Exchanger Module (HEM) Number
2
3
4
5
6
7
8
9
10
Heat Exchanger Module (HEM) Number
Fig. 11, it can be observed that the dimensionless pressure drop decreases as both the length and the height of the module increase. The Nusselt number decreases as the length of the module increases and it increases as the height of the module increases. Figure 12a, b shows the variation of dimensionless pressure drop with HEM number for water with staggered configuration. Figure 12c, d shows the variation of Nusselt number with HEM number for water with staggered configuration. These results are similar to those obtained for in-line configuration.
It exhibits dimensionless pressure drop characteristics similar to that of air, whereas the Nusselt number increases as the length of the module increases, and decreases as the height of module increasing, different from the trends seen for air. Figure 11a, b shows the variation of dimensionless pressure drop with HEM number for staggered configuration with air as the medium. The variation of Nusselt number with HEM number for air with staggered configuration is shown in Fig. 11c, d. Trends are similar to that those observed with the in-line configuration for air. From 1.4
3.0
(a)
1.2
In-H3-L6-Re-400
1.0
(b)
2.5
In-H3-L5-Re-400 In-H3-L4-Re-400
In-H2-L5-Re-400 In-H3-L5-Re-400 In-H4-L5-Re-400
2.0
0.8
DP
DP
4
(d)
25
In-H2-L5-Re-400
23
0.6
1.5 1.0
0.4
0.5
0.2 0.0
0.0 1
2
3
4
5
6
7
8
10
9
1
Heat Exchanger Module (HEM) Number
2
3
4
5
6
7
8
9
10
Heat Exchanger Module (HEM) Number 60
55
(c)
50
In-H3-L6-Re-400
45
(d)
55
In-H3-L5-Re-400
In-H2-L5-Re-400
In-H3-L5-Re-400
50
In-H3-L4-Re-400
In-H4-L5-Re-400
45
40
Nu
Nu
3
27
(c)
25
35
40 35
30
30
25
25
20
20 1
2
3
4
5
6
7
8
9
Heat Exchanger Module (HEM) Number
123
2
Heat Exchanger Module (HEM) Number
Heat Exchanger Module (HEM)Number
Fig. 10 a, b Variation of dimensionless pressure drop with HEM number for water with in-line configuration and isothermal boundary condition. c, d Variation of Nusselt number with HEM number for water with in-line configuration and isothermal boundary condition
In-H2-L6-Re-400
2.5
Dp
Fig. 9 a, b Variation of dimensionless pressure drop with heat exchanger module (HEM) number for air with inline configuration and isothermal boundary condition. c, d Variation of Nusselt number with HEM number for air with in-line configuration and isothermal boundary condition
Heat Mass Transfer (2008) 44:445–461
10
1
2
3
4
5
6
7
8
9
Heat Exchanger Module (HEM) Number
10
3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0
0.9
(a)
Stagg-H2-L5-Re-200 Stagg-H2-L6-Re-200 Stagg-H2-L7-Re-200
(b)
0.8 0.6 0.5 0.4 0.3 0.2 0.1 0.0
1
2
3
4
5
6
7
8
9
10
1
Heat ExchangerModule (HEM) Number
Stagg-H2-L5-Re-200
5
6
7
8
9
10
Stagg-H4-L6-Re-400
Stagg-H3-L6-Re-400
24
Stagg-H2-L7-Re-200
Nu
Nu
4
(d)
26
Stagg-H2-L6-Re-200
16 15 14
Stagg-H2-L6-Re-400
22 20 18
13
16 14
12 1
6 7 9 10 8 Heat Exchanger Module (HEM) Number 2
3
4
Table 4 shows the dimensionless pressure drop values for interior modules of the tube bank with air (Pr = 0.7) as the fluid. From Table 4, the following conclusions can be drawn. (1) For in-line configurations the dimensionless pressure drop values decrease as the height of module increases and increases as the length of module increases. (2) The dimensionless pressure drop values for staggered configuration decrease with length and the height of modules. (3) At a height ratio of (H/Da = 2, 3), the dimensionless pressure drop values for in-line configura3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0
5
1
6 7 9 10 2 3 4 5 8 Heat Exchanger Module (HEM) Number
tion are higher than those for the staggered configuration, whereas at a height ratio of (H/Da = 4), both configurations exhibit the same values of dimensionless pressure drop. The Nusselt number values for the interior modules of tube bank with air (Pr = 0.7) as the fluid are shown in Table 5. From Table 5, for in-line configuration it can be seen that for a low height ratio of (H/Da = 2), the Nusselt number decreases as the length of module increases, and for the remaining height ratios, it increases 2.5
(a)
(b)
Stagg-H2-L5-Re-200 Stagg-H2-L6-Re-200
Stagg-H2-L5-Re-400 Stagg-H3-L5-Re-400
2.0
Stagg-H2-L7-Re-200
Stagg-H4-L5-Re-400
DP
1.5 1.0 0.5 0.0 1
31
2 3 4 5 6 7 8 9 Heat Exchanger Module (HEM) Number
10
1
(d)
Stagg-H2-L5-Re-200
29
Stagg-H2-L6-Re-200
27
Stagg-H2-L7-Re-200
21
20
4
5
6
7
8
9
Heat Exchanger Module (HEM) Number
Stagg-H4-L5-Re-200
30 25
3
Stagg-H3-L5-Re-200
35
23
2
Stagg-H2-L5-Re-200
40
25
19 1
2 3 4 5 6 7 8 9 10 Heat Exchanger Module (HEM) Number
45
(c)
Nu
DP
3
28
(c)
17
Nu
2
Heat Exchanger Module (HEM) Number
18
Fig. 12 a, b Variation of dimensionless pressure drop with HEM number for water with staggered configuration and isothermal boundary condition. c, d Variation of Nusselt number with HEM number for water with staggered configuration and isothermal boundary condition
Stagg-H2-L6-Re-400 Stagg-H3-L6-Re-400 Stagg-H4-L6-Re-400
0.7
DP
Fig. 11 a, b Variation of dimensionless pressure drop with HEM number for air with staggered configuration and isothermal boundary condition. c, d Variation of Nusselt number with HEM number for air with staggered configuration and isothermal boundary condition
453
DP
Heat Mass Transfer (2008) 44:445–461
10
15
1
2
3
4
5
6
7
8
9
10
Heat Exchanger Module (HEM) Number
123
454
Heat Mass Transfer (2008) 44:445–461
Table 4 Dimensionless pressure drop values for interior modules with air (Pr = 0.7) for isothermal boundary condition Db/Da = 2 Re
H/Da
2
In-line arrangement
Staggered arrangement
L/Da=4
L/Da=5
2nd
3rd
4th
5th
2nd
3rd
4th
5th
25
16.228
15.758
15.993
16.228
12.794
12.747
12.771
12.771
100
4.394
4.394
4.394
4.409
3.721
3.720
3.707
3.733
400
1.430
1.372
1.366
1.370
1.415
1.314
1.314
1.314
3
25
4.464
4.464
4.167
4.464
4.167
4.167
4.167
4.167
100
1.154
1.172
1.154
1.172
1.228
1.210
1.191
1.210
400
0.417
0.361
0.337
0.327
0.494
0.390
0.375
0.367
25
2.090
2.090
2.508
2.090
2.508
2.508
2.090
2.508
100
0.600
0.627
0.600
0.600
0.679
0.653
0.679
0.653
400
0.230
0.206
0.185
0.173
0.274
0.225
0.207
0.194
3rd
4th
5th
4
L/Da = 5
2
L/Da = 6
2nd
3rd
4th
5th
2nd
25
15.993
16.228
15.993
16.228
7.291
7.291
7.056
7.291
100
4.527
4.541
4.527
4.527
2.307
2.307
2.322
2.307
400
1.558
1.507
1.508
1.507
0.483
0.462
0.460
0.459
25
4.375
4.464
4.464
4.464
3.274
3.572
3.572
3.274
100
1.247
1.265
1.265
1.265
1.098
1.098
1.098
1.098
400
0.466
0.402
0.383
0.371
0.491
0.395
0.377
0.371
25
2.508
2.508
2.090
2.508
2.508
2.090
2.508
2.090
100
0.653
0.705
0.653
0.679
0.705
0.679
0.679
0.679
400
0.265
0.230
0.204
0.196
0.305
0.240
0.219
0.212
3
4
L/Da = 6
L/Da = 7
2nd
3rd
4th
5th
2nd
3rd
4th
5th
25
15.993
16.228
16.228
15.993
4.939
5.174
4.704
5.174
100
4.586
4.586
4.586
4.600
1.778
1.808
1.793
1.793
400
1.653
1.615
1.613
1.613
0.724
0.700
0.699
0.697
25
4.762
4.167
4.762
4.167
2.679
2.976
2.679
2.976
100
1.321
1.303
1.340
1.303
0.986
0.986
0.986
0.986
400
0.508
0.434
0.420
0.410
0.439
0.376
0.361
0.356
25
2.508
2.508
2.508
2.508
2.090
2.090
2.090
2.090
100 400
0.731 0.291
0.705 0.247
0.731 0.225
0.705 0.216
0.679 0.307
0.679 0.247
0.679 0.227
0.653 0.220
2
3
4
as the length increases. As the height of module increases, the Nusselt number increases, except for two Reynolds numbers (200, 400) at a length ratio of (L/ Da = 4, 5). For staggered configuration, the Nusselt number increases as module height increases, and the Nusselt number decreases as the module length increases except at a height ratio of (H/Da = 4). The in-line configuration performs better compared to the staggered configuration at a height ratio of (H/Da = 2), and for the remaining height ratios, the staggered configuration performs better.
123
Similarly, the dimensionless pressure drop values for the interior modules of tube bank with both in-line and staggered configuration, for water exhibit the same behaviour as was observed for air. Here the in-line configurations show higher dimensionless pressure drop values compared to the staggered configurations. From the Nusselt number values for the interior modules of the tube bank with water (Pr = 7.0) as the fluid, the following conclusions can be drawn. (1) For in-line configuration the Nusselt number increases as the module length increases, and it decreases as the module height increases, where as for staggered
Heat Mass Transfer (2008) 44:445–461
455
Table 5 Nusselt number values for interior modules with air (Pr = 0.7) for isothermal boundary condition Db/Da = 2 H/Da
Re
Inline arrangement
Staggered arrangement
L/Da = 4
L/Da = 5
2nd 2
3rd
4th
5th
6.435
6.436
6.384
6.832
6.758
6.759
6.732
6.667
13.302
13.298
13.324
12.912
12.916
12.916
12.916
400
20.203
20.302
20.302
20.318
16.633
16.414
16.422
16.421
25
9.291
9.297
9.291
9.292
10.167
10.167
10.160
10.142
100
14.307
14.311
14.314
14.322
14.274
14.249
14.257
14.257
400
18.068
17.479
17.368
17.339
18.549
17.943
17.902
17.894
25
12.472
12.471
12.471
12.466
10.695
10.700
10.699
10.699
100
14.472
14.432
14.432
14.432
14.789
14.757
14.767
14.768
400
18.369
17.089
16.606
16.421
19.170
17.980
17.629
17.515
4th
5th
2nd
L/Da = 6 3rd
4th
5th
4
3rd
25
6.445
6.446
6.496
6.832
5.932
5.933
5.938
5.914
13.822
13.150
13.000
13.072
11.534
11.536
11.536
11.536
400
19.285
19.289
19.286
19.286
15.778
15.441
15.418
15.412
25
9.501
9.501
9.500
9.508
9.759
9.756
9.753
9.754
100
14.215
14.212
14.207
14.220
14.163
14.152
14.162
14.163
400
18.411
17.955
17.886
17.880
18.890
18.276
18.247
18.249
25
11.714
11.712
11.716
11.714
11.819
11.826
11.825
11.823
100
14.645
14.626
14.635
14.632
15.096
15.081
15.099
15.102
400
19.015
17.915
17.551
17.428
19.990
18.770
18.504
18.426
3rd
4th
5th
2nd
3
2nd
100
L/Da = 6
2
5th
13.316
L/Da = 5
4
4th
25
4
3
3rd
100 3
2
2nd
L/Da = 7 3rd
4th
5th
2nd
25
6.447
6.446
6.395
6.832
5.565
5.565
5.568
5.545
100
13.763
13.762
13.763
13.761
10.639
10.639
10.639
10.639
400
17.584
17.675
17.598
17.568
15.297
14.959
14.926
14.920
25
9.558
9.554
9.555
9.557
9.267
9.268
9.265
9.255
100
14.287
14.291
14.288
14.285
13.904
13.959
13.969
13.971
400
18.854
18.482
18.454
18.424
18.868
18.429
18.404
18.414
25
11.328
11.328
11.327
11.322
11.561
11.568
11.568
11.562
100 400
14.909 19.626
14.897 18.640
14.894 18.354
14.897 18.280
15.240 20.565
15.277 19.422
15.296 19.194
15.300 19.138
configuration the Nusselt number increases as the module height increases, and the Nusselt number increases as the module length increases except at a height ratio of (H/ Da = 2). (2) In-line configuration performs better than staggered one except at a height ratio of (H/Da = 4). For both fluids with in-line as well as staggered configurations, as the Reynolds number increases the Nusselt number increases and the dimensionless pressure drop decreases. This can be observed from data given in Tables 4 and 5.
4.2 Results for isoflux boundary condition A similar analysis has been conducted for the ‘isoflux’ condition on the tube surface. For brevity only the conclusions from this study are given here (1) For both fluids (air and water) there exists a fully developed velocity condition, whereas for some cases in the staggered configuration, fully developed condition as far as heat transfer is concerned is not attained. (2) With air, the dimensionless pressure drop values for in-line configuration are higher
123
456
Heat Mass Transfer (2008) 44:445–461
than that of staggered configuration at a height ratio of (H/ Da = 2, 3), for the remaining cases both have equal dimensionless pressure drop values and from the heat transfer point of view staggered configuration performs better than the in-line configuration. (3) With water, the dimensionless pressure drop values for in-line configuration are higher than that of staggered configuration, and for a height ratio of (H/Da = 4), and a length ratio of (L/ Da = 5) both configurations have equal dimensionless pressure drop values and from a heat transfer point of view staggered configuration performs better than in-line configuration except at a lower height ratio of (H/Da = 4).
Dp_Predicted
180 160
DP = 11058.51 (Re)-0.8625
140
(H/Da)-2.8008 (L/Da)0.2614
120
R2= 0.99
100 80 60 40 20 0 0
4.3 Proposed correlations
20
40
60
80
100 120 140 160 180
Dp_Calculated
Correlations for overall Nusselt number and dimensionless pressure drop have been developed as functions of Reynolds number, length ratio (L/Da), and height ratio (H/Da) for the entire flat tube bank with ten rows for both inline and staggered configurations with isothermal boundary condition, using multiple linear regression. The general forms of the correlations are H Da
Nu ¼ a Reb
DP ¼ e Re
f
H Da
c
L Da
g
L Da
10 0.9454
Nu = 0.0059 (Re)
8
d
Nu_Predicted
Fig. 13 Parity plots showing agreement of dimensionless pressure drop (correlated) with the respective data for inline configuration
ð20Þ h ð21Þ
1.4968
(H/Da)
0.0312
(L/Da)
2
R = 0.99
6
4
2
Two overall correlations have been now developed for inline and staggered configurations for DP. The coefficients are given in Table 6. The excellent R2 values confirm the statistical goodness of the fit. Table 7 shows all the constants for the Nusselt number correlation for two different boundary conditions. Typical parity plots are shown in
0 0
4
2
8
6
10
Nu_Calculated Fig. 14 Parity plots showing agreement of Nusselt number (correlated) with the respective data for air with isothermal boundary condition for the in-line configuration
Table 6 Constants for proposed correlations for dimensionless pressure drop R2
Configuration
e
f
g
h
Inline
11058.51
–0.8625
–2.8008
0.2614
0.99
Staggered
23659.72
–0.8030
–1.8734
–1.0407
0.92
Figs. 13 and 14 with the calculated and predicted values of the dimensionless pressure drop values and the Nusselt number values. These show reasonable agreement between data and the predictions.
Table 7 Constants for correlations proposed for Nusselt number for both boundary conditions Configuration
Isothermal boundary condition a
b
Isoflux boundary condition c
d
a
b
c
d
Air-in-line
0.0059
0.9454
1.4968
0.0312
0.0046
0.9999
1.5932
0.0001
Air-staggered
0.0073
0.9156
1.4263
0.0211
0.0046
0.9998
1.5929
–0.0020
Water-in-line
0.3711
0.6056
0.4070
0.0941
0.0013
1.9494
0.7754
–0.4286
Water-staggered
0.3500
0.6322
0.5360
–0.0325
0.0110
2.0278
0.5281
–0.4103
123
Heat Mass Transfer (2008) 44:445–461 24.0
6.0E-03
(a)
23.5
(b) Pumping Power,W
22.5 1 4 7
22.0 21.5
2 5 8
3 6 9
21.0 20.5
8
9
2.0E-03
2
3
4
5
6
7
8
9
1
10
2
3
Configuration number 6.0E-04
(c)
1600
1 4 7
1500
4
5
6
10
Configuration number 2 5 8
3 6 9
Pumping Power,W
1700
1400
Q, W
7
3.0E-03
0.0E+00
1
1300 1200 1100
(d)
5.0E-04
1 4 7
2 5 8
3 6 9
7
8
9
4.0E-04 3.0E-04 2.0E-04 1.0E-04
1000 900 1
2
3
4
5
6
7
8
9
10
0.0E+00
1
2
3
4
5
6
10
Configuration number
Configuration number
9.0E-03
50 2 5 8
46 44 42 40
(a)
38
1
2
3
4
5
6
7
8
9
7.0E-03 5.0E-03 4.0E-03 3.0E-03 2.0E-03 1.0E-03
(b)
0.0E+00
10
1
2
2 5 8
Pu,Wmping Power
3 6 9
2700 2500 2300 2100 1900
(c)
1700 1
2
3
4
5
6
7
8
9
10
Configuration number
The correlations obtained with the numerical simulation were verified with experimental data available in the literature. Grimison [29] proposed correlations for heat transfer in circular tubes long ago. The flat tube under consideration in this study has been converted to an equivalent circular tube and the comparison has been made. The correlation proposed (for air with inline configuration and isothermal boundary condition) in this study has been compared with [29] for inline configurations up to
3
4
5
6
7
8
9 10
Configuration number
3300 1 4 7
3 6 9
6.0E-03
Configuration number
3100 2900
2 5 8
1 4 7
8.0E-03
3 6 9
Pumping Power,W
1 4 7
48
Q, W
3 6 9
1.0E-03
19.5
Q, W
2 5 8
4.0E-03
20.0
Fig. 16 a–d Variation of heat transfer and pumping power with respect to staggered configurations for air and water with isothermal boundary condition, respectively
1 4 7
5.0E-03
23.0
Q, W
Fig. 15 a–d Variation of heat transfer and pumping power with respect to in-line configurations for air and water with isothermal boundary condition, respectively
457
1.0-03E 9.0E-04 8.0E-04 7.0E-04 6.0E-04 5.0E-04 4.0E-04 3.0E-04 2.0E-04 1.0E-04 0.0E+00
1 4 7
2 5 8
3 6 9
(d) 1
2
3
4
5
6
7
8
9 10
Configuration number
ten rows of tubes. The Grimison correlation is given below for ready reference. NuD ¼ C1 Rem D;max ; Pr ¼ 0:7
Nr 10 2000\ReD;max \40; 000 ; ð22Þ
Using the above formula, for inline configuration at transverse pitch to the diameter ratio of 3 and longitudinal pitch to the diameter ratio of 3 the overall Nu is 29.94.
123
458 23.8
23.4 23.2 23.0 22.8
1300
In-H3-L4 In-H3-L5 In-H3-L6 In-H4-L4 In-H4-L5
Q, W
23.6
Q, W
Fig. 17 a–d Pareto plots showing the variation of heat transfer and pumping power trade off for in-line configurations for air and water with isothermal and isoflux boundary conditions, respectively
Heat Mass Transfer (2008) 44:445–461
In-H4-L6
In-H3-L4
1260
In-H3-L5
1240
In-H3-L6
1220 1200 1180
22.6 22.4
1280
1160
(a)
22.2 0.0E+00
(b)
1140 5.0E-04
1.0E-03
1.2E-04 1.3E-04 1.4E-04 1.5E-04 1.6E-04
1.5E-03
Pumping Power, W
Pumping Power,W
25.692
70
Q, W
25.688 25.686 25.684
60 55 50 45 40
25.682 25.68
In-H3-L4 In-H3-L5 In-H3-L6 In-H4-L4 In-H4-L5 In-H4-L6
65
Q, W
In-H3-L4 In-H3-L5 In-H3-L6 In-H4-L4 In-H4-L5 In-H4-L6
25.69
35 30
(c)
25.678 0.0E+00
5.0E-04
1.0E-03
(d)
25 0.0E+00
1.5E-03
5.0E-05
48.0 St-H3-L5 47.8 St-H3-L6 47.6 St-H3-L7 47.4 St-H4-L5 47.2 St-H4-L6 47.0 46.8 St-H4-L7 46.6 46.4 46.2 46.0 (a) 45.8 0.0E+00 1.0E-03
St-H3-L5
2540
St-H3-L6
2520
St-H3-L7
2500 2480 2460
(b)
2.0E-03
2440 2.4E-04 2.5E-04 2.6E-04 2.7E-04 2.8E-04
3.0E-03
Pumping Power, W St-H3-L5 St-H3-L6 St-H3-L7 St-H4-L5 St-H4-L6 St-H4-L7
170 150 St-H3-L5 St-H3-L6 St-H3-L7 St-H4-L5 St-H4-L6 St-H4-L7
(c) 1.0E-03
2.0E-03
3.0E-03
Q, W
Q, W
2.0E-04
190
0.0E+00
130 110 90 70
(d)
50 0.0E+00
Pumping Power, W
From the newly proposed correlation the overall Nu is 43.72, for Re = 2,100. The agreement between the two can be termed as good, considering the fact that in this study the correlations are based on Nusselt data for Re £ 400 and for the purposes of comparison, the correlation has been extrapolated to a range in which one is not sure of its adequacy.
123
1.5E-04
2560
Pumping Power, W
51.384 51.382 51.38 51.378 51.376 51.374 51.372 51.37 51.368 51.366
1.0E-04
Pumping Power, W
Q, W
Fig. 18 a–d Pareto plots showing the heat transfer and pumping power trade off for staggered configurations for air and water with isothermal and isoflux boundary conditions, respectively
Q, W
Pumping Power, W
1.0E-04
2.0E-04
3.0E-04
Pumping Power,W
4.4 Optimum configuration To determine the optimum configuration two independent parameters: (1) pumping power (defined in Eq. 23), (2) total heat transfer has been selected. Pareto or trade off plots are drawn with the total heat transfer being the ordinate and pumping power being abscissa.
Heat Mass Transfer (2008) 44:445–461
459
Fig. 19 a, b Stream lines for the best configuration In-H4-L4 with air as the medium and isoflux boundary condition for the entire tube bank and for the sixth HEM, respectively. c, d Stream lines for the worst configuration In-H3-L6 with air as the medium and isoflux boundary condition for the entire tube bank and for the sixth HEM, respectively
Fig. 20 a, b Isotherms for the best configuration In-H4-L4 with air as the medium and isoflux boundary condition for the entire tube bank and for the sixth HEM, respectively. c, d Isotherms for the worst configuration In-H3-L6 with air as the medium and isoflux boundary condition for the entire tube bank and for the sixth HEM, respectively
Pumping power ¼
m: DP q
ð23Þ
Figures 15 and 16 show the variation of total heat transfer and pumping power, with different configurations. Total heat transfer and pumping power are taken as the
ordinate while the configuration number is taken as the abscissa. From these figures, the configurations that have the ‘maximum heat transfer and minimum pumping power’ values have been considered as important configurations for further analysis to determine the range of optimum configurations. The variation of total heat transfer with
123
460
Heat Mass Transfer (2008) 44:445–461
pumping power for both configurations with two boundary conditions has been shown in Figs. 17 and 18. One can observe form these figures that, few configurations have low pumping power and high total heat transfer values. As the principal objective is to find the optimum from the view points of both pressure drop and heat transfer, it is possible to select the optimum configurations from these Pareto plots. Based on this criterion, some configurations have been selected as the optimum from both in-line and staggered configurations. The in line configuration with a height ratio of (H/Da = 4) and a length ratio of (L/Da = 4, 5, 6) and the staggered configuration with a height ratio of (H/Da = 5) and a length ratio of (L/Da = 5, 6, 7) give the ‘maximum heat transfer and minimum pressure drop’ values. Figures 19a, b and 20a, b show the stream lines and isotherms for the best and the worst configurations. The bending of the streamlines in Fig. 19b, denoting a higher pressure drop, and the crowding of the isotherms in Fig. 20a, denoting a higher heat transfer rate are concordant with what one would intuitively expect. So, these two configurations may be selected as the optimum configurations for the cases studied in the present unsteady simulation. If one wishes, such trade off plots can be generated for other boundary conditions on the tube surfaces like convective boundary condition as well.
5 Conclusions A detailed transient numerical simulation of fluid flow and heat transfer over a bank of flat tubes has been performed. The following conclusions have been drawn from the present analysis. •
•
•
•
For most of the cases there exists a fully developed condition from the pressure drop and the heat transfer point of view. With water, for some cases, the fully developed thermal condition is not reached with the isoflux boundary condition. The in-line configuration performs better than that of the staggered configuration for most of the cases, from the point of view of heat transfer. At the same time, the dimensionless pressure drop values are higher for inline configuration than that of the staggered configuration. Correlations are proposed for the overall Nusselt number and dimensionless pressure drop for the entire tube bank with ten rows for in-line and staggered configurations with air and water as the fluids for both thermal boundary conditions. Optimum configurations have been selected from both pressure drop as well as heat transfer point of view using Pareto plots and these are: (1) in-line configura-
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tion having a height ratio of (H/Da = 4) and a length ratio of (L/Da = 4, 5, 6), (2) staggered configuration having a height ratio of (H/Da = 5) and a length ratio of (L/Da = 5, 6, 7).
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