2. Numerical model

The two-dimensional corrugated channel with a width (b) of 10 mm is described schematically in Figure 1. The water as heat transfer fluid enters the computational domain at a temperature of 27°C and intensity of turbulent of 5%. Also, 5% of turbulent intensity is considered at the exit. The end effects and viscous dissipation terms are ignored. The constant heat flux of 600 W/cm<sup>2</sup> is applied on the channel wall. The consideration of an axisymmetric situation reduces the size of the numerical domain for saving computational time.

The flow-thermal behavior is modeled by the governing conservation equations (continuity, momentum, and energy) in a RANS technique as

$$\frac{\partial u\_i}{\partial \mathbf{x}\_i} = \mathbf{0} \tag{1}$$

in which ρ, μ, u<sup>0</sup>

Figure 1.

, and ρu<sup>0</sup> i u0

Schematic representation of the computational domain.

DOI: http://dx.doi.org/10.5772/intechopen.84736

τij � �

∂ ∂xi

ð Þ¼ ρ εui

performance evaluation criterion (PEC).

The average Nusselt number is presented as

Nu <sup>¼</sup> <sup>q</sup>″<sup>d</sup> kt ðx 0

½ �¼ uið Þ <sup>ρ</sup><sup>E</sup> <sup>þ</sup> <sup>P</sup> <sup>∂</sup>

eff ¼ μeff

ð Þ¼ ρk ui

∂ ∂xj

and μ<sup>t</sup> is the eddy viscosity which is modeled as

The transport equations in k-e model are presented as [17]

∂ ∂xj

� �

<sup>μ</sup> <sup>þ</sup> <sup>μ</sup><sup>t</sup> σε � � ∂ε

∂xj

Thermal-Hydrodynamic Characteristics of Turbulent Flow in Corrugated Channels

∂T ∂xj

where Prt is the turbulent Prandtl number and (τij)eff is the deviatoric stress

<sup>μ</sup> <sup>þ</sup> <sup>μ</sup><sup>t</sup> σk � � ∂k

∂xj

<sup>μ</sup><sup>t</sup> <sup>¼</sup> <sup>ρ</sup> <sup>C</sup><sup>μ</sup> <sup>k</sup><sup>2</sup>

The model constants Cμ, C1<sup>ε</sup>, C2ε, σk, and σε are 0.09, 1.44, 1.92, 1.0, and 1.3,

No-slip condition and constant wall heat flux are assumed as boundary conditions. The thermal-hydrodynamic performance of the corrugated channels is assessed by dimensionless parameters which are the Nusselt number, friction factor, and

> 1 Twð Þ� x Tbð Þ x

� �

∂ui ∂xj þ ∂uj ∂xi � �

kt þ

turbulent shear stress, respectively.

∂ ∂xi

tensor which is evaluated as

∂ ∂xi

respectively.

163

<sup>j</sup> are density, viscosity, fluctuated velocity, and

Cpμ<sup>t</sup> Pr<sup>t</sup> � �

> � 2 3 μeff ∂ui ∂xj

∂xj

<sup>þ</sup> <sup>C</sup>1<sup>ε</sup> ð Þ <sup>ε</sup>=<sup>k</sup> Gk � <sup>C</sup>2<sup>ε</sup> ρ ε<sup>2</sup>

<sup>þ</sup> ui <sup>τ</sup>ij � � eff � � (3)

δij (4)

=k � � (6)

þ Gk � ρε (5)

dx (8)

<sup>ε</sup> (7)

$$\frac{\partial}{\partial \mathbf{x}\_{j}} \left( \rho \,\boldsymbol{u}\_{i} \boldsymbol{u}\_{j} \right) = -\frac{\partial P}{\partial \mathbf{x}\_{i}} + \frac{\partial}{\partial \mathbf{x}\_{j}} \left[ \mu \left( \frac{\partial \boldsymbol{u}\_{i}}{\partial \mathbf{x}\_{j}} + \frac{\partial \boldsymbol{u}\_{j}}{\partial \mathbf{x}\_{i}} - \frac{2}{3} \delta\_{\overline{\eta}} \frac{\partial \boldsymbol{u}\_{j}}{\partial \mathbf{x}\_{j}} \right) \right] + \frac{\partial}{\partial \mathbf{x}\_{j}} \left( -\rho \,\overline{\boldsymbol{u}\_{i}^{\prime} \boldsymbol{u}\_{j}^{\prime}} \right) \tag{2}$$

Thermal-Hydrodynamic Characteristics of Turbulent Flow in Corrugated Channels DOI: http://dx.doi.org/10.5772/intechopen.84736

Figure 1. Schematic representation of the computational domain.

show the performance of a turbulent flow inside a two-sided corrugated channel with an in-line and staggered arrangements. He showed the important effect of corrugation on the augmentation of heat transfer and pressure loss. Eiamsa-ard and Promvonge [10] experimentally examined the thermal-hydrodynamic performance of the three types of ribbed-grooved ducts. They reported that the maximum rate of heat exchange and pressure drop exist in the ducts with a rectangular rib and a triangular groove. Elshafei et al. [11] conducted experiments to examine the thermal-hydraulic performance of corrugated channels under the influence of variations of phase shift and channel spacing. The corrugated channels exhibit a compound increase in heat transfer and pressure loss. Mohammed et al. [12] performed a computational model to investigate the effects of wavy tilt angle, channel height, and channel height on the flow-thermal fields in a corrugated channel. A threedimensional numerical model to investigate the employing baffles on the heat transfer-flow in the corrugated channels was presented by Li and Gao [13].

Boundary Layer Flows - Theory, Applications and Numerical Methods

Increasing the baffle height enhances heat transfer effectively but leads to dramatic penalty in pressure drop. Pehlivan et al. [14] experimentally investigated the rate of heat exchange for sharp corrugation peak fins of corrugated channel for three different types and sinusoidal converging–diverging channels. It is reported that the rate of heat transfer increases with the corrugated angle. The numerical results showed that the wavy channel is an efficient method to increase the heat transfer. Ravi et al. [15] numerically studied the impact of different rib configurations on the heat transfer-flow characteristics of the turbulent flow inside corrugated channels. Shubham et al. [16] numerically investigated the thermal-hydrodynamic transport characteristics of non-Newtonian fluids in corrugated channels. It was found that using of shear thinning fluids is more convenient for maximum augmentation of

The present study offers a numerical model to investigate the thermal flow attributes of turbulent flow in corrugated channels. The performance of corrugated channels are examined under the effects of corrugation arrangement (inward, outward, and inward-outward rib distribution), corrugation configuration, corrugation roughness parameters (rib pitch, rib width, and rib height), and rib shapes (rectangular, trapezoidal, and semicircular). The comparisons between the predicted thermal flow performance of corrugated channels and that of smooth ones are fulfilled

The two-dimensional corrugated channel with a width (b) of 10 mm is described schematically in Figure 1. The water as heat transfer fluid enters the computational domain at a temperature of 27°C and intensity of turbulent of 5%. Also, 5% of turbulent intensity is considered at the exit. The end effects and viscous dissipation terms are ignored. The constant heat flux of 600 W/cm<sup>2</sup> is applied on the channel wall. The consideration of an axisymmetric situation reduces the size of the numer-

The flow-thermal behavior is modeled by the governing conservation equations

¼ 0 (1)

�ρ ui 0 uj <sup>0</sup> (2)

þ ∂ ∂xj

∂ui ∂xi

thermal performance with a minimum penalty in pressure drop.

under a large range of Reynolds number (5000–60,000).

(continuity, momentum, and energy) in a RANS technique as

ical domain for saving computational time.

∂xi þ ∂ ∂xj μ ∂ui ∂xj þ ∂uj ∂xi � 2 3 δij ∂uj ∂xj

2. Numerical model

∂ ∂xj

162

ρ uiuj ¼ � <sup>∂</sup><sup>P</sup>

in which ρ, μ, u<sup>0</sup> , and ρu<sup>0</sup> i u0 <sup>j</sup> are density, viscosity, fluctuated velocity, and turbulent shear stress, respectively.

$$\frac{\partial}{\partial \mathbf{x}\_i} \left[ u\_i (\rho E + P) \right] = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \frac{\partial T}{\partial \mathbf{x}\_j} \left( kt + \frac{\mathbf{C}\_p \mu\_t}{\mathbf{Pr}\_t} \right) + u\_i \left( \tau\_{ij} \right)\_{\mathrm{eff}} \right] \tag{3}$$

where Prt is the turbulent Prandtl number and (τij)eff is the deviatoric stress tensor which is evaluated as

$$\left(\left(\tau\_{\vec{\eta}}\right)\_{\epsilon\vec{\mathcal{f}}} = \mu\_{\epsilon\vec{\mathcal{f}}'} \left(\frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i}\right) - \frac{2}{3} \mu\_{\epsilon\vec{\mathcal{f}}'} \frac{\partial u\_i}{\partial \mathbf{x}\_j} \delta\_{ij} \tag{4}$$

The transport equations in k-e model are presented as [17]

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho k u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + \mathbf{G}\_k - \rho \varepsilon \tag{5}$$

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho \,\varepsilon u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right] + \mathbf{C}\_{1\epsilon} \left( \varepsilon/k \right) \mathbf{G}\_k - \mathbf{C}\_{2\epsilon} \rho \left( \varepsilon^2/k \right) \tag{6}$$

and μ<sup>t</sup> is the eddy viscosity which is modeled as

$$
\mu\_t = \frac{\rho \text{ C}\_{\mu} \text{ k}^2}{\varepsilon} \tag{7}
$$

The model constants Cμ, C1<sup>ε</sup>, C2ε, σk, and σε are 0.09, 1.44, 1.92, 1.0, and 1.3, respectively.

No-slip condition and constant wall heat flux are assumed as boundary conditions.

The thermal-hydrodynamic performance of the corrugated channels is assessed by dimensionless parameters which are the Nusselt number, friction factor, and performance evaluation criterion (PEC).

The average Nusselt number is presented as

$$Nu = \frac{q^\*d}{kt} \int\_0^\infty \frac{1}{T\_w(\infty) - T\_b(\infty)} d\infty \tag{8}$$

where <sup>q</sup>″ and Tw(x) and Tb(x) act as the supplied heat flux and wall and local bulk temperatures, respectively.

The friction factor is defined as

$$f = \frac{2\,\Delta P\,d}{L\,\rho\,u\_m^2} \tag{9}$$

3. Results and discussion

DOI: http://dx.doi.org/10.5772/intechopen.84736

of corrugated channels.

Figure 3.

165

3.1 The effect of rib arrangements

The flow-thermal features of turbulent flow in corrugated channels are evaluated numerically. The enhanced heat transfer and an accompanied pressure loss are assessed for corrugated channels under the influences of rib arrangement, rib configuration, rib roughness parameters, and rib shapes. The dimensionless parameters Nu, f, and PEC through a wide range of Re are presented to assess the performance

Thermal-Hydrodynamic Characteristics of Turbulent Flow in Corrugated Channels

Corrugated channels exist in three layouts depending on rib arrangements, IOCC, ICC, and OCC, as described in Figure 1a. The variations of Nu and f with the Re number of all rectangular rib arrangements of corrugated channels and smooth one are presented in Figure 3a and b, respectively. The rate of heat that is transferred in corrugated channels is higher than that of the smooth channel. The heat transfer varies insignificantly with the rib distribution at the low Re. The rib distribution experiences a pronounced influence on the Nusselt number when Re increases. The ICC shows a maximum ability to exchange the heat, while the OCC has a lower thermal performance. On the other hand, there is an additional pressure loss associated with corrugated channels compared with smooth ones as exhibited in Figure 3b. The friction factor decreases slightly with the Re. Also, the OCC has a

(a) Different rib arrangements of corrugated channels and the influence of rib configuration on Nu, f, and PEC

as described in (b), (c), and (d), respectively, for the different values of Re.

The comparison between the enhancement in thermal performance and a penalty in the pressure drop is assessed by introducing the performance evaluation criteria (PEC) of corrugated channels with different roughness dimensions. The PEC can be calculated as

$$\text{PEC} = \frac{\text{Nu} / \text{Nu}\_{\text{s}}}{\left(f / f\_{\text{s}}\right)^{1/3}} \tag{10}$$

where fs and Nus are the friction factor and the Nusselt number of smooth channel, respectively.

The performance of corrugated channels is estimated according to different values of the Reynolds number which is introduced as

$$\text{Re} = \frac{\rho \, u\_m \, d\_h}{\mu} \tag{11}$$

where μ, ρ, dh, and um are dynamic viscosity, density, hydrodynamic diameter, and mean fluid velocity.

The ANSYS Fluent CFD package-based control volume method is adopted to discretize the governing equations and simulate thermal flow behavior of corrugated channels. The SIMPLE algorithm is utilized for solving the flow field. The diffusion terms and other resulting terms are discretized by employing the firstorder upwind scheme. The residuals lower than 10�<sup>6</sup> is chosen to achieve the convergence criterion for all variables. A fine grid discretization close to the wall is adopted. Also, the meshing system of 23,964 grids is sufficient for solution accuracy. On the other hand, the numerical code that is validated through a reasonable agreement is shown (Figure 2a) between the Nusselt number of the present work and the same number which is obtained from the well-known Gnielinski correlation [18]. Furthermore, good agreement is indicated for the friction factor (Figure 2b) between the present work and the work of San and Huang [5].

#### Figure 2.

(a) Numerical Nu of the present work and that obtained from Gnielinski's correlation [17] and (b) Numerical f and that of San and Huang [5].

Thermal-Hydrodynamic Characteristics of Turbulent Flow in Corrugated Channels DOI: http://dx.doi.org/10.5772/intechopen.84736
