International Journal of Mechanical and Production Engineering Research and Development (IJMPERD ) ISSN 2249-6890 Vol.2, Issue 3 Sep 2012 36-55 © TJPRC Pvt. Ltd.,

NUMERICAL INVESTIGATION OF THE EFFECT OF FIN WAVINESS, SPACING RATIO, AND FLOW ATTACK ANGLE ON LAMINAR FLUID FLOW AND HEAT TRANSFER IN WAVY PLATE-FIN CHANNELS MORTEZA PIRADL Mechanical Engineering Department, Payame Noor University, 19395-4697 Tehran, Iran

ABSTRACT Single-phase, periodically developed, constant property, laminar forced convection in two dimensional and sinusoidal corrugated ducts, which are maintained at uniform wall temperature, are considered. The governing differential equations for continuity, momentum, and energy transfer are solved computationally using finite-volume techniques, where the pressure term is handled by the SIMPLE algorithm. The computational grid is non orthogonal and non-uniform, and it is generated algebraically. All the dependent variables are stored in a non-staggered manner. Numerical solutions are obtained for different corrugation aspect ratios (γ=2A/L), plate spacing ratio (ε=S/2A), flow rates (Re) and different flow attack angles (φ). In corrugated ducts, the flow pattern changes drastically with Reynolds number and the flow gets separated at a critical Re. This is because the pressure distribution ceases to be linear and local variations of pressure cause flow to separate. The size of the separation region is seen to be a function of Re, γ and ε and it increases with increasing Re and γ. With increasing ε, however, it first increases and then starts to decrease after a critical ε is reached. This behavior is also seen in the friction factor and Nusselt number results, which increase to peak values corresponding to the critical ε value, and then begin to decrease. Both friction factor and Nusselt number results are

presented for different γ and ε in the two-dimensional case, for a wide range

of flow conditions (100≤Re≤1000), and for different flow attack angles (φ).

KEYWORDS: Flow Rate, Friction Factor, Nusselt Number, Spacing Ratio, Wavy Flow Channel INTRODUCTION Studies on enhanced heat transfer have been reported for more than 100 years now. In recent years, due to the increasing demand by industries for heat exchangers that are more efficient, compact and less expensive, heat transfer enhancement has gained serious momentum. Generally, the study of enhanced heat transfer is focused on two areas: (1) increasing the heat transfer surface area (such as extended surfaces or fins), (2) increasing the heat transfer coefficient by modifying the flow patterns near the heat transfer surface. Extended or finned surfaces are widely used in compact heat exchangers to enhance heat transfer and

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Numerical Investigation of the Effect of fin Waviness, Spacing Ratio, and Flow Attack Angle on Laminar Fluid flow and Heat Transfer in Wavy Plate-Fin Channels

reduce their size [1–3]. Often geometrically modified fins are incorporated, which, besides increasing the surface area density of the exchanger, also improve the convection heat transfer coefficient. Some examples of such enhanced surface compact cores include offset-strip fins, louvered fins, perorated fins, and corrugated or wavy fins [1–6]. Of these, wavy fins are particularly attractive for their simplicity of manufacture, potential for enhanced thermal–hydraulic performance, and ease of usage in both plate-fin and tube-fin type exchangers. The wavy plate-fin flow channel considered in the analysis is schematically shown in Figure 1. As seen in the figure 1(b-c), the flow channel, which is formed by placing wavy fins side-by-side and bonded to a set of flat plates, can be described by the fin height H, fin spacing S, amplitude A and wavelength or pitch of waviness L. The dimensionless representations of these variables are given by the channel spacing ratio (ε=S/2A), flow cross-section aspect ratio (δ=S/H), and corrugation or waviness aspect ratio (γ=2A/L). In such a flow channel, when the fin height H is much larger than the fin spacing S ( for example H >10S ), a two-dimensional flow can be considered as a good engineering approximation. In this study, only the twodimensional flow and heat transfer problem has been modeled and simulated.

(a) (b) (c) Figure 1: Wavy-plate fins: (a) typical sinusoidally corrugated plate fins, (b) geometrical description, (c) two- dimensional representation of the inter-fin flow channel.

The wavy flow channel essentially exhibits geometric periodic characteristics, and in a computational model one can readily considered that the fluid flow attains a periodic fully developed regime. The periodic fully developed velocity field repeats itself from one-periodic-length flow cycle to another cycle. The dimensionless temperature field also shows a similar periodic behavior. Based on these characteristic, and also considering the time saving in the simulation, only one-periodic-length flow channel needs to be considered in the both 2D and 3D modeling and simulation. Figure 2 shows the typical solution domain with the mesh distribution for a 2D wavy channel.

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Figure 2: Typical non-uniform and non-orthogonal grid in the computational domain A large body of literature related to wavy flow channels or corrugated flow channels is available. The fluid flow and heat transfer behavior have been studied experimentally and numerically. Nishimura et al. [7– 9] have experimentally and numerically studied laminar flow mass transfer enhancement in two-dimensional wavy-plate channels by varying the pitch, amplitude, and the channel spacing in the range of 0.215≤ε≤1.33 and γ=0.25. Asako and Faghri [10] developed a finite volume methodology to predict fully developed heat transfer coefficients, friction factor, and streamline flow fields for flow in a channel with triangular corrugation. The numerical solutions were performed for laminar flow in the Reynolds number range between 100 and 1500. Metwally [11–12] has numerically studied the fluid flow and heat transfer behaviors in two-dimensional wavy flow channel with spacing ratio equal to one, and corrugation aspect ratios in the range of 0.25≤γ≤1.0. Newtonian, power-law type non-Newtonian fluid, and Herschel-Bulkley fluid flows were investigated. Motamed et al. [13] have numerically investigated the flow characteristics and heat transfer for periodic fully-developed steady laminar flow in various wavy sinusoidal channels with phase angle θ= 0˚ degree. The channel geometries were varied for wave lengths of 1.5, 2.0, 3.0, and 4.0, and the amplitude-to-wave length ratios of 0.25, 0.375, 0.50, and 0.75. The influence of the sinusoidal wavy-surface plate-fin geometry (γ and ε) on laminar airflow and convective heat transfer in the inter-fin channels is computationally investigated in this study. Rush [14] experimentally investigated local heat transfer and flow behavior for laminar and transitional flows in sinusoidal wavy passages. Flow visualization methods were used to characterize the flow field and detect the onset of macroscopic mixing. The entire channel exhibited unsteady, macroscopic mixing at Re ~ 1600 and the onset of this mixing is linked directly to the significant increases in local heat transfer. Focke [15] conducted flow visualization experimental studies in channels with wavy walls formed in a plate heat exchanger. The channel plates have corrugation angles of 0˚, 45˚, 80˚, and 90˚ relative to the channel axis. In their study, the complex flow patterns in flow rate of Re = 10 to 1000 were reported. Bereiziat [16] experimentally studied local flow structure for non-Newtonian fluids in periodically corrugated wall channels. In the present study a two-dimensional flow geometry (H>>S, or α→0) is considered, and the role of inter-fin spacing and severity of wall waviness in promoting the wall-curvature-induced periodically developed steady swirl flows is highlighted. With plate surfaces maintained at a uniform temperature (UWT condition) for the heat transfer problem, a wide range of air (Pr = 0:7) flow rates in the low Reynolds number regime (100≤Re≤1000) are considered. velocity vectors are mapped, and the parametric variations in the

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Numerical Investigation of the Effect of fin Waviness, Spacing Ratio, and Flow Attack Angle on Laminar Fluid flow and Heat Transfer in Wavy Plate-Fin Channels

average isothermal Fanning friction factor f and Nusselt number (Nu) results with γ, ε, and Re are outlined along with an assessment of the enhanced thermal–hydraulic performance.

MATHEMATICAL FORMULATION FLOW GOVERNING EQUATIONS The governing equations to be considered are continuity, momentum and energy equation. Laminar flow and convective heat transfer prevail. Steady state and constant thermo-physical proprieties are assumed and natural convection is excluded. The governing equations in Cartesian coordinates can be expressed as:

Continuity: ∂u ∂v + =0 ∂x ∂y (1) Momentum: ∂u ∂u ∂u 1 ∂p ∂2u ∂2u ∂2u +v +w = − +υ + + ρ ∂x ∂x2 ∂y2 ∂z2 ∂x ∂y ∂z (2a) u

u

∂2v ∂2v ∂2v ∂v ∂v ∂v 1 ∂p +v +w =− +υ + + ρ ∂y ∂x2 ∂y2 ∂z2 ∂x ∂y ∂z

(2b) Energy: u

∂T ∂T ∂T k ∂ 2T ∂ 2T ∂ 2T +v +w = + + ∂x ∂y ∂z ρ c ∂x2 ∂y 2 ∂z 2 p

(3)

PERIODIC FULLY DEVELOPED FLOW In periodic flow regime, the velocity profile is periodically repeating over some periodic length L, with an accompanying periodically constant pressure drop along the periodic length. The periodic thermally fully developed regime also exists for boundary conditions such as constant wall temperature and constant wall heat flux. In periodic thermal fully developed regime, the dimensionless temperature profiles (based on the constant wall temperature or constant wall heat flux) behavior in a periodic manner. The wavy channel geometry repeat itself in a periodic length L, the wavy channel.

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TREATMENT OF PRESSURE TERM The pressure term in periodic flows is assumed to consist of a periodic part and a constant pressure gradient part. The pressure gradient part is responsible for driving the flow and the periodic part is responsible for the local variations in the flow field. Thus, the pressure term is expressed as follows: p ( x, y ) = − β x + p* ( x, y )

(4) where β is the constant global pressure gradient and is defined as:

β=

p( x, y ) − p( x + L, y ) L

(5) P* is the local pressure that exhibits periodicity, and their respective dimensionless forms are B = (β S ρu 2 ) , m

p = ( p* ρ 2 ) m

(6)

.TEMPERATURE FIELD For uniform wall temperature boundary conditions, The following dimensionless temperature variable is introduced. T ( x, y ) − T w θ ( x, y ) = T ( x) − T b w (7) Where Tb(x) is the local bulk temperature, and it is expressed as: ∫∫ u (T ( x, y ) − Tw ) dy T ( x) − T = b w ∫∫ u dy (8) And θ(x,y) explicitly satisfies the periodic boundary condition.

COORDINATE TRANSFORMATION AND NON DIMENSIONALIZATION For arbitrary-shaped geometries, generalized body- fitted coordinates are employed, denoted by (ξ,η), (Figure 3), where ξ=ξ(X,Y), η=η(X,Y), that (ξ,η) are defined as:

ξ = ( x S ), η = ( y S ) − ( A S )sin(2π Sξ L) (9)

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Numerical Investigation of the Effect of fin Waviness, Spacing Ratio, and Flow Attack Angle on Laminar Fluid flow and Heat Transfer in Wavy Plate-Fin Channels

Figure 3: Illustration of unit vector n and tangential vector t in the two-dimensional physical domain In the computational domain (ξ,η), the geometry of wavy plate-fin flow channels appears as a rectangular for

2-D problem.

The governing equations in the computational domain for the wavy plate-fin channel are expressed as the following: ∂U ∂V + =0 ∂ξ ∂η (10)

In a compact form, the momentum and energy equation can be written as follows: ∂2φ ∂φ ∂φ ∂2φ +S +V =Γ + (1 + ω 2 ) ϕ 2 2 φ ∂ξ ∂η ξ η ∂ ∂ (11) U

Where U φ = V , θ

1 Re S Γ = 1 Re , S φ 1 (Re Pr) S

(12) And ReS is defined as: Su ρ Re = m S µ (13) And ф represents a scalar variable, Γф represents a diffusion coefficient, and Sф represents the source term. Source terms Sф are defined as follows: 1 ∂ ∂U ∂ ∂U S =− ω +ω U Re ∂ξ ∂η ∂η ∂ξ S

(14a)

∂P ∂P +ω + B − ∂ ξ ∂ η

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1 ∂ S =− V Re ∂ξ S −ω B − U 2

∂V ω ∂η

∂ ∂V ∂P ∂P 1 − (1 + ω 2 ) + +ω + ω η ξ ξ η ∂ ∂ ∂ ∂ Re S

∂ω ∂U ∂U 2 −ω ∂η ∂ξ ∂ξ

∂ 2U +U ∂ξ 2

∂ω ∂ξ

(14b)

S =−

θ

∂ Re Pr ∂ξ S 1

∂θ ∂ ∂θ ω +ω ∂η ∂ξ ∂η

d (T − T ) dξ 2 ∂θ ∂θ θ b w + −ω + − Uθ . ∂η (Tb − Tw ) Re S Pr Re S Pr ∂ξ

2 d d (Tb − Tw ) d ξ d (Tb − Tw ) d ξ . + T −T dξ T −T b w b w

(14c) In the computational domain (ξ,η), the dimensionless velocity components (U,V) are given by: U = (u um ) , (15)

V = (v um − ωU )

where um is the mean axial velocity, and

ω = (2π A L)cos(2π Sξ L) (16) 2.4. Boundary conditions 1. No slip condition on the walls 2. Constant wall temperature 3. periodicity conditions The latter essentially requires that

φ (U ,V ,θ ) = φ (U ,V ,θ ) ξ =0 ξ =1 (17) Where

φ = U ,V ,θ (18) 2.5. friction factor The Fanning friction factor is often used to calculate the pressure drop; the Fanning friction factor f is obtained from its usual definition as: f =−

(19)

BD 1 dp Dh h = 2 dx ρ u 2 2S m

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NUSSELT NUMBER The overall Nusselt number is computed from the temperature field by applying the energy balance over the one-period flow domain, and the log-mean temperature difference (LMTD) as: Nu =

& (T mc −T ) L d (T − T ) d ξ p b, o b, i b w = A A (Re Pr) − ∫ dx c h kA ( LMTD ) ( T −T ) h b w 0

(

)

(20) where Ah is the heat transfer area, LMTD is the log mean temperature and Ac is the cross-section area of the flow duct. T ( x) = b

∫ u T ( x, y )dy ∫ u dy

(21) LMTD =

(T − T ) − (T − T ) w b, o w b, i ln (T − T ) (T − T ) w b, i w b, o

(22)

DISCRETIZATION AND DIFFERENCING SCHEMES To obtain numerical solutions, the governing differential equations were discretized on a structured, non-orthogonal grid using the finite-volume method. Both the diffusion and convection terms are treated by the power-law differencing scheme, and the source terms by central differencing. The SIMPLE algorithm was applied to evaluate the coupling between the pressure and velocity.

VALIDATION OF NUMERICAL RESULTS The numerical results were first validated with analytical solutions for a parallel-plate flow channel, which is obtained by setting the amplitude A=0 in the simulation. The variation of Fanning friction factor and Nusselt number with Re, for a parallel-plate flow channel, for both Numerical and analytical (Metwally & Manglik, 2000) results, are presented in Figure 4(a), and (b), respectively. results are also listed in table 1a.

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(a) (b) Figure 4: Comparison of numerical and analytical Results for a parallel-plate flow channel: (a) f.Re-Re , (b) Nu-Re. Also for the case of γ=0.25, and ε=1.0, friction factor and Nusselt number are compared in Numerical and experimental [11-12], [17] results, Figure 5 and table 1b. represent the results of this comparison.

(a) (b) Figure 5: Comparison of numerical and experimental Results for a wavy flow channel with ε =1.0, and γ=0.25: (a) f-Re (b) Nu-Re.

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Table 1: Comparison of numerical and experimental Results for : (a) parallel-plate flow channel, (b) wavy flow channel with ε =1.0, and γ=0.25. (a) Re f.Re(Ana) f.Re(Num) Nu(Ana) Nu(Num) 100

24

24.02834

7.54

5.9474179

200

24

24.09244

7.54

6.3340524

400

24

24.22755

7.54

6.8162444

600

24

24.35358

7.54

7.0742280

800

24

24.48402

7.54

7.2134776

1000

24

24.62656

7.54

7.3039890

Re

f(exp)

(b) f(Num)

Nu(exp)

Nu(Num)

100

0.34487

0.34122

8.280670

6.62816870

200

0.17819

0.17662

8.817590

7.30639140

400

0.11394

0.10663

10.18411

9.43803476

600

0.10056

0.08829

11.71926

11.5568515

800

0.09414

0.07864

13.31012

13.8222700

1000

0.08982

0.07136

14.91055

16.0559861

Investigation the Figure 4, Figure 5, and table 1, show that the numerical model have the adequate accuracy for modeling other geometries of the duct.

RESULTS AND DISCUSSIONS The computational results for periodically developed flows in the parallel wavy-plate channels were obtained; Different cases were run to study the parametric effects of aspect ratio (γ=2A/L), spacing ratio (ε=S/2A), Reynolds Number (Re) on the flow behavior and heat transfer characteristics. The numerical programs, which include the fluid flow solver and heat transfer solver, were developed in C++ to model and simulate the convective behaviors in the 2-D wavy plate flow channels. For the heat transfer problem, uniform wall temperature condition is considered. Results for velocity distributions, friction factor, and Nusselt number with different flow rates are presented, and the effects of channel geometry are discussed in detail.

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EFFECT OF REYNOLDS NUMBER (Re) Figure 6. shows velocity vectors for different flow Re , in a wavy-plate channel of fixed aspect ratio (γ=0.5), and plate separation (ε=1.0).

(a)

(b)

(c) Figure 6: Velocity vector plots for different flow rates in a wavy-plate channel with γ=0.5, and ε=1.0: (a) Re = 100, (b) Re = 400, and (c) Re = 800

With increasing flow rate or Re, it can be seen that the size of the separated region increases with Re. Also, the recirculation tends to grow laterally as well and occupy a larger portion of the flow cross section. The corresponding velocity vectors for the different Re are depicted in Figure 6. It can be seen from this figure that the strength of the recirculation region increases with increasing Re. Also, the velocity gradient at the wall, upstream of point of separation, increases sharply thereby increasing the local shear stress. The variation of Fanning friction factor and Nusselt number with Re, for the case of γ=0.5, ε=1.0, are presented in Figure 7. it can be seen from this figure that with increasing Re, the friction factor decreases but the Nusselt number increases.

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(a) (b) Figure 7: Variation of (a) Fanning friction factor and (b) Nusselt number with Re, for the case of γ=0.5, and ε=1.0

EFFECT OF ASPECT RATIO (γ) Figure 8 shows velocity vectors for different γ = 0.25, 0.375, and 0.5, in a wavy-plate channel of fixed flow rate (Re=400) and plate separation (ε=1.0). With increasing γ, It can be seen that the flow separation and the size of this separation-reattachment and recirculation region, increase.

(a)

(b)

(c) Figure 8: Velocity vector plots for different wall waviness aspect ratio with Re=400, and ε=1.0: (a) γ=0.25, (b) γ=0.375, and (c) γ=0.5.

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The variation of Nusselt number with γ , for the case of, ε=1.0, Re=400 are presented in Figure 9. with increasing γ, it can be seen from this figure that the Nusselt number increases.

Figure 9: Variation of Nusselt number with γ, for the case of ε=1.0, and Re=400

EFFECT OF SPACING RATIO (ε) The velocity vectors for different plate spacing ratios (ε) but with fixed Re and wall waviness aspect ratio of 0.5, are presented in Figure 10.

(a)

(b)

(c)

(d) Figure 10: Velocity vector plots for different plate separation with Re =400, and γ=0.5: (a) ε=0.5, (b) ε=1.0, (c) ε=1.5, and (d) ε=2.0.

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The increasing channel spacing is seen to decrease the penetration of viscosity into the flow thereby promoting early separation. As flow separates near one wall, there is a resulting local acceleration at the same location on the

opposite wall because of continuity. This localized flow acceleration produces regions of

high wall shear stress, which lend to higher friction factors in the flow channels.

Figure 11(a), and (b) graph the variation of f , and Nu with ε, for γ=0.25, 0.375, and 0.5, and flow Re=400. As the spacing ratio increases, both the friction factor and Nusselt number is seen to increase to a peak value and then begin to decrease. The decrease in friction factor and Nusselt

number beyond a

specific value of ε is because the size of the separated region remains constant after this critical value, and any further increase in plate spacing does not affect the magnitude of the wall shear stress.

(a) (b) Figure 11: Variation of: (a) f, and (b) Nu with ε, for γ=0.25, 0.375, and 0.5, Re = 400. Variation of Nu/f (also called ‘Area goodness factor) with ε is plotted in Figure 12. The higher end of ε is not taken into account because in practice ε>3.0 is not viable, so the actual range of ε, is between ε=0.5, and ε=1.0. the peak value of Nu/f occurs between ε ~ 1.0 and ε ~ 2.0 depending on the value of γ.

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Figure 12: Variation of Nu/f with ε, for for γ=0.25, 0.375, and 0.5, Re = 400.

EFFECT OF FLOW ATTACK ANGLE (φ) In the previous sections, we investigated the effects of Reynolds number (Re), aspect ratio (γ), and spacing ratio (ε), on Nusselt number and friction factor, for φ=0˚. in this section, we are going to investigate the effect of different flow attack angles (φ). Negative and positive angles of attack are shown in Figure 13.

Figure13: Negative and positive flow attack angles For investigating the effects of different flow attack angles on Nusselt number and friction factor, the dimensionless form of flow attack angle can be defined as:

α=

ϕ

tan −1(4 A L)

(23) We consider the following values for α:

α = ±0.5, ±1.0 (24) The variation of Nu with α, for the case of Re=400, and ε=1.0, for two different γ (γ=0.25, and 0.5), is presented in Figure 14(a). it can be seen from this figure that the Nusselt number change due to the attack angle change, is negligible. The variation of Nu/f , with α, for the case of Re=400, and ε=1.0, for two different γ (γ=0.25, and 0.5), is presented in Figure 14(b). as seen in this figure, by changing attack angle,

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Nu/f increases significantly, due to decreased pressure drop or friction coefficient, as seen in Figure 14(c).

(a)

(b)

(c) Figure 14: Variation of : (a) Nu, (b) Nu/f , and (c) f , with α, for the case of Re=400, and ε=1.0, for two different γ. (γ=0.25, and 0.5).

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As we saw in section 5.2, and Figure 12, with increasing γ , Nu/f decreases, due to increased pressure drop. As seen in Figure15, for the case of γ = 0.25 , and Re=400, Nu/f can be improved by changing dimensionless attack angle α, to 1.0,

Figure 15: Variation of Nu/f with ε, for the case of Re=400, and γ=0.5, for α=0, and 1. The variation of Nu, and Nu/f with ε, for the wavy flow channel with γ=0.25, and α=0, are compared with the case of γ=0.5, and α=1, in Figure 16.

(a)

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(b) Figure 16: Variation of: (a) Nu, and (b) Nu/f, with ε, for wavy flow channels with γ=0.25, α=0, and γ=0.5, α=0. Comparison of Figure 12, 15, and 16, shows that by

changing the angle of attack, due to decreased

pressure drop or friction coefficient, Nu/f increases.

CONCLUSIONS Laminar air flow (Pr = 0.7) and forced convection heat transfer in uniform-wall-temperature, two dimensional wavy plate-fin channels has been computationally simulated. Constant property, and periodically developed fluid flow and heat transfer are considered. The steady-state governing equations (continuity, momentum, and energy equations) were solved using finite-volume techniques, and the SIMPLE algorithm was used to couple the pressure-velocity fields. Periodic boundary conditions were applied to the solution domain, and the effects of thermal and hydraulic entry lengths were neglected. Numerical results for a wide range of steady laminar flows (100≤Re≤1000) and duct-geometry variations (0.25≤γ≤0.5, and 0.5≤ε≤4.0), and different flow attack angles (φ), are presented. The wavy-wall curvature induces lateral vortices in the trough region, which grow in magnitude and spatial flow coverage with increasing Re and/or γ. The inter-plate separation, however, is critical for the development of this flow structure. With small separation (ε<1.0), viscous forces dominate and a streamline, fully developed duct flow type behavior prevails. With larger inter-plate gap (ε≥1.0), this effect

diminishes and the boundary layer separation

downstream of corrugation peaks gives rise to a vortex flow structure in the valley region. The re-circulation is enveloped in the near-wall axial flow separation bubble, and its spatial growth is governed by Re, γ, and ε. The consequent local fluid mixing and coreflow acceleration results in enhanced convective heat transfer, though the associated flow friction also increases. Finally, we saw in section 5.4 that the angle of attack change, will lead to increasing Nu/f ratio, due to decreased pressure drop or friction coefficient.

NOMENCLATURE A B CP

: amplitude of wall waviness : dimensionless pressure gradient : specific heat

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Eh f L Nu Pr P P* Re ReS S Sф T U, V u, v x, y

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: hydraulic diameter : Fanning friction factor : pitch of fin waviness : Nusselt number : Prandtl number : pressure : local pressure : hydraulic-diameter-based Reynolds number : plate-spacing based Reynolds number : fin spacing : source terms : temperature : dimensionless axial and lateral velocity components : axial and lateral velocity components : Cartesian coordinates

GREEK SYMBOLS β : global pressure gradient δ : flow cross-section aspect ratio ε : channel spacing ratio γ : channel corrugation ratio υ : dynamics viscosity θ : dimensionless temperature ρ : density ξ, η : dimensionless body-fitted coordinates

SUBSCRIPTS b c i m o w

: bulk or mixed-mean value : pertaining to the cross-section area : at inlet conditions : mean or average value : at outlet conditions : at wall conditions

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Bergles, A.E., 1998. Techniques to enhance heat transfer, in: W.M. Rohsenow, J.P. Hartnett, Y.I. Cho (Eds.), Handbook of Heat Transfer, third ed., McGraw-Hill, NewYork, (Chapter 11).

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