Solution of the Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges

Karamanis, Georgios; Hodes, Marc; Kirk, Toby; Papageorgiou, Demetrios T.

doi:10.1115/1.4036281

We consider convective heat transfer for laminar flow of liquid between parallel plates that are textured with isothermal ridges oriented parallel to the flow. Three different flow configurations are analyzed: one plate textured and the other one smooth; both plates textured and the ridges aligned; and both plates textured, but the ridges staggered by half a pitch. The liquid is assumed to be in the Cassie state on the textured surface(s), to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. Heat is exchanged with the liquid either through the ridges of one plate with the other plate adiabatic, or through the ridges of both plates. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). Axial conduction is neglected and the inlet temperature profile is arbitrary. We solve for the three-dimensional developing temperature profile assuming a hydrodynamically developed flow, i.e., we consider the Graetz–Nusselt problem. Using the method of separation of variables, the thermal problem is essentially reduced to a two-dimensional eigenvalue problem in the transverse coordinates, which is solved numerically. Expressions for the local Nusselt number and those averaged over the period of the ridges in the developing and fully developed regions are provided. Nusselt numbers averaged over the period and length of the domain are also provided. Our approach enables the aforementioned quantities to be computed in a small fraction of the time required by a general computational fluid dynamics (CFD) solver.

Introduction

Background.

Superhydrophobic surfaces, i.e., those with hydrophobic micro- and/or nanoscale protrusions, are of interest in the context of liquid flow through microchannels, especially in direct liquid cooling applications as a means to reduce flow and thus caloric resistance [1]. When criteria are met [1,2], the solid–liquid interfaces are confined to the tips of the structures, forming a composite interface along with the liquid–gas interfaces (menisci), as per Fig. 1, and the liquid is said to be in the unwetted or Cassie state [3,4]. Then, the solid–liquid interfaces are subjected to the no-slip [5,6] boundary condition, whereas the menisci are subjected to a low-shear boundary condition. Thus, a lubrication effect is provided which reduces caloric resistance. However, the reduction in the solid–liquid interface area reduces the Nusselt number (Nu) and thus increases the convective component of thermal resistance. A net reduction of the total, i.e., caloric plus convective, thermal resistance can be achieved with proper sizing of the structures [1] and it requires knowledge of Nusselt numbers as a function of the geometry of the channel and the structures. The surfaces can be textured with a variety of periodic structures such as pillars, transverse ridges, or parallel ridges [2].¹ The latter configuration for the ridges is examined here and it is the most favorable from a heat transfer perspective [1,7].

Fig. 1

View large Download slide

Liquid in the Cassie state and the composite interface

The hydrodynamic effects of structured surfaces with parallel ridges have been studied for flat and curved menisci [8–12]. However, there is a relatively limited body of work on heat transfer effects. Enright et al. [7] derived an expression for the Nusselt number for fully developed flow through a microchannel with isoflux structured surfaces as a function of the (apparent) hydrodynamic and thermal slip lengths. Moreover, Enright et al. [7] developed analytical expressions for slip lengths for structured surfaces with parallel or transverse ridges or pillar arrays assuming flat and adiabatic menisci. Ng and Wang [13] derived semi-analytical expressions for the thermal slip length for isothermal parallel ridges while accounting for conduction through the gas phase. Lam et al. [14] derived expressions for the thermal slip length for isoflux and isothermal parallel ridges accounting for small meniscus curvature. Hodes et al. [15] captured the effects of evaporation and condensation along menisci on the thermal slip length for isoflux ridges. Lam et al. [16] developed expressions for the Nusselt number for Couette flow as a function of the slip lengths for various boundary conditions. Also, Lam et al. [16] discussed when Nu results from the molecular slip literature can be used to capture the effects of apparent slip. Maynes et al. [17] numerically investigated the thermal transport in microchannels with isothermal transverse ridges and flat menisci taking into account the heat transfer through the gas in the cavities. Maynes et al. [18] and Maynes and Crockett [19] developed expressions for the Nusselt number and the thermal slip length for microchannels with isoflux transverse and parallel ridges, respectively, assuming flat menisci and using the Navier slip approximation for the velocity profile. Kirk et al. [20] also developed expressions for the Nusselt number for isoflux parallel ridges using the fully resolved velocity field in the thermal energy equation. Furthermore, Kirk et al. [20] accounted for small meniscus curvature using a boundary perturbation method.

The present work develops semi-analytical expressions for the Nusselt number for the case of isothermal parallel ridges for hydrodynamically developed and thermally developing flow with negligible axial conduction, i.e., for the Graetz–Nusselt problem [22–24].² It is emphasized that we do not assume diffusive heat transfer near the composite interface.

Assumptions.

We consider three different configurations for the parallel ridges: (1) one plate textured and the other one smooth, as per Fig. 2; (2) both plates textured and the ridges aligned in the transverse direction (see Fig. 14); and (3) both plates textured and the ridges staggered in the transverse direction by half a pitch (see Fig. 18). The solution approach is similar in all three configurations. Therefore, it suffices to present here the detailed analyses for the first one and the relevant parts of the analysis for the other two configurations in the Appendix.

Fig. 2

View large Download slide

Schematic of the domain when one plate is textured and the other one is smooth

The domain (D) for the first configuration is depicted in Fig. 2, where $| x | \leq d$ and 0 ≤ y ≤ H and where 2d is the pitch of the ridges and H is the distance between the parallel plates. The hydraulic diameter of the domain (D_h) is 2H. The width of the meniscus is 2a. The curvature of the meniscus is neglected [14,20] and the triple contact lines coincide with the corners of the ridges at $x = | a |$ and y = 0. The cavities may be filled with inert gas and/or vapor. Along the composite interface (y = 0), a no-shear boundary condition is applied for $| x | < a$ and a no-slip one is imposed for $a < | x | < d$ ⁠. A no-slip boundary condition is also imposed on the smooth upper plate. Symmetry boundary conditions apply at the $x = | d |$ boundaries. The flow is pressure driven, steady, laminar, hydrodynamically developed, and thermally developing with constant thermophysical properties and negligible axial conduction and viscous dissipation. The ridges on the lower plate are isothermal, whereas the upper plate and the meniscus are considered adiabatic. The temperature profile in the liquid starts developing at z = 0 from an arbitrary (unless otherwise stated) two-dimensional distribution T_in (x, y). Effects due to Marangoni stresses [25,26], evaporation and condensation [15], and gas diffusion in the liquid phase are neglected. The independent dimensionless variables are the solid fraction of the ridge $(ϕ = (d - a) / d)$ and the aspect ratio of the domain (H/d).

Analysis

Hydrodynamic Problem.

The relevant form of the streamwise momentum equation is

\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} = \frac{1}{μ} \frac{d p}{d z}

(1)

where w is the streamwise velocity, dp/dz is the prescribed pressure gradient, and μ is the dynamic viscosity. Denoting nondimensional variables with tildes, Eq. and the boundary conditions imposed on it are rendered dimensionless by defining

\tilde{x} = \frac{x}{a}

(2)

\tilde{y} = \frac{y}{a}

(3)

\tilde{H} = \frac{H}{a}

(4)

\tilde{w} = \frac{2 μ w}{a H (- d p / d z)}

(5)

Then, Eq. becomes

\frac{\partial^{2} \tilde{w}}{\partial {\tilde{x}}^{2}} + \frac{\partial^{2} \tilde{w}}{\partial {\tilde{y}}^{2}} = - \frac{2}{\tilde{H}}

(6)

subjected to

\frac{\partial \tilde{w}}{\partial \tilde{y}} = 0 \begin{matrix} for | \tilde{x} | < 1, \tilde{y} = 0 \end{matrix}

(7)

\tilde{w} = 0 \begin{matrix} for 1 < | \tilde{x} | < \tilde{d}, \tilde{y} = 0 \end{matrix}

(8)

\tilde{w} = 0 \begin{matrix} for | \tilde{x} | < \tilde{d}, \tilde{y} = \tilde{H} \end{matrix}

(9)

\frac{\partial \tilde{w}}{\partial \tilde{x}} = 0 \begin{matrix} for | \tilde{x} | = \tilde{d}, 0 < \tilde{y} < \tilde{H} \end{matrix}

(10)

where $\tilde{d} = d / a$ is the dimensionless (half) pitch of the ridges.

This hydrodynamic problem has been solved analytically [8] and semi-analytically [9,11]. However, no analytical solutions have been found for the problems in the Appendix. Therefore, the velocity field is numerically determined here for all cases, which also facilitates the numerical solution of the thermal energy equation in Sec. 2.4. The numerical results were validated against the analytical solution [8] using the computed Poiseuille number (fRe), where

Re = \frac{ρ \bar{w} D_{h}}{μ}

(11)

f = - \frac{d p}{d z} \frac{2 D_{h}}{ρ {\bar{w}}^{2}}

(12)

\bar{w} = \frac{1}{2 d H} \int_{0}^{H} \int_{- d}^{d} w d x d y

(13)

are the Reynolds number based on the hydraulic diameter, the friction factor and the mean velocity of the flow, respectively, and ρ is the liquid density. From Eqs. and –, it follows that the Poiseuille number is given by

f Re = 16 \frac{\tilde{H}}{\bar{\tilde{w}}}

(14)

where

\bar{\tilde{w}} = \frac{1}{2 \tilde{d} \tilde{H}} \int_{0}^{\tilde{H}} \int_{- \tilde{d}}^{\tilde{d}} \tilde{w} d \tilde{x} d \tilde{y}

(15)

is the dimensionless mean velocity of the flow.

Thermal Problem.

The relevant form of the dimensional thermal energy equation is

w \frac{\partial T}{\partial z} = α (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}})

(16)

where T and α are the temperature and the thermal diffusivity of the liquid, respectively. Defining the dimensionless temperature

\tilde{T}

and the dimensionless streamwise coordinate

\tilde{z}

as

\tilde{T} = \frac{T - T_{sl}}{T_{ref} - T_{sl}}

(17)

\tilde{z} = \frac{2 α μ z}{a^{3} H (- d p / d z)}

(18)

where T_sl is the constant temperature of the ridge and T_ref is a reference temperature for the problem, Eq. becomes

\tilde{w} \frac{\partial \tilde{T}}{\partial \tilde{z}} = \frac{\partial^{2} \tilde{T}}{\partial {\tilde{x}}^{2}} + \frac{\partial^{2} \tilde{T}}{\partial {\tilde{y}}^{2}}

(19)

It is subject to the following boundary conditions:

\frac{\partial \tilde{T}}{\partial \tilde{y}} = 0 \begin{matrix} for | \tilde{x} | < 1, \tilde{y} = 0 \end{matrix}

(20)

\tilde{T} = 0 \begin{matrix} for 1 < | \tilde{x} | < \tilde{d}, \tilde{y} = 0 \end{matrix}

(21)

\frac{\partial \tilde{T}}{\partial \tilde{y}} = 0 \begin{matrix} for | \tilde{x} | < \tilde{d}, \tilde{y} = \tilde{H} \end{matrix}

(22)

\frac{\partial \tilde{T}}{\partial \tilde{x}} = 0 \begin{matrix} for | \tilde{x} | = \tilde{d}, 0 < \tilde{y} < \tilde{H} \end{matrix}

(23)

\tilde{T} = {\tilde{T}}_{in} \begin{matrix} at \tilde{z} = 0 \end{matrix}

(24)

where ${\tilde{T}}_{in} (\tilde{x}, \tilde{y})$ is the prescribed dimensionless temperature profile at the inlet of the domain $(\tilde{z} = 0)$ ⁠.

Seeking separable solutions of the form

\tilde{T} = ψ (\tilde{x}, \tilde{y}) g (\tilde{z})

⁠, which separate the streamwise coordinate

\tilde{z}

from the transverse coordinates

\tilde{x}

and

\tilde{y}

⁠, it can be shown that

g (\tilde{z}) = exp (- λ \tilde{z})

and

ψ (\tilde{x}, \tilde{y})

satisfies

\nabla^{2} ψ = - λ \tilde{w} ψ

(25)

with λ real and positive.³ Note that

ψ (\tilde{x}, \tilde{y})

cannot be separated further into a product of a function of

\tilde{x}

and one of

\tilde{y}

since the velocity field,

\tilde{w} = \tilde{w} (\tilde{x}, \tilde{y})

⁠, is not separable in such a way. Equation satisfies the boundary conditions

\frac{\partial ψ}{\partial \tilde{y}} = 0 \begin{matrix} for | \tilde{x} | < 1, \tilde{y} = 0 \end{matrix}

(26)

ψ = 0 \begin{matrix} for 1 < | \tilde{x} | < \tilde{d}, \tilde{y} = 0 \end{matrix}

(27)

\frac{\partial ψ}{\partial \tilde{y}} = 0 \begin{matrix} for | \tilde{x} | < \tilde{d}, \tilde{y} = \tilde{H} \end{matrix}

(28)

\frac{\partial ψ}{\partial \tilde{x}} = 0 \begin{matrix} for | \tilde{x} | = \tilde{d}, 0 < \tilde{y} < \tilde{H} \end{matrix}

(29)

and so constitutes a two-dimensional Sturm–Liouville eigenvalue problem for λ and the corresponding eigenfunction ψ, with weight function

\tilde{w} (\tilde{x}, \tilde{y})

⁠. Assuming that the eigenvalues are discrete and there are infinitely many, let λ_i and ψ_i denote the ith eigenvalue and eigenfunction, respectively, ordered such that

0 < λ_{1} < λ_{2} < \dots < λ_{i} < \dots \to \infty

⁠. The eigenfunctions are orthogonal with respect to the inner product defined by

〈 F_{1}, F_{2} 〉 = \int_{0}^{\tilde{H}} \int_{- \tilde{d}}^{\tilde{d}} \tilde{w} F_{1} F_{2} d \tilde{x} d \tilde{y}

(30)

that is

〈 ψ_{i}, ψ_{j} 〉 = 0 for i \neq j

(31)

Moreover, the eigenfunctions are unique up to a multiplication by a constant. Thus, for the rest of the present analysis, we normalize ψ_i such that

〈 ψ_{i}, ψ_{i} 〉 = 1

(32)

The eigenvalue problem is solved numerically. The calculation of ψ_i and λ_i is detailed in Sec. 2.4, and for the rest of the present analysis, they are assumed to be known.

We proceed by expressing the solution

\tilde{T} (\tilde{x}, \tilde{y}, \tilde{z})

as a linear combination of the eigenfunctions

\tilde{T} (\tilde{x}, \tilde{y,} \tilde{z}) = \sum_{i = 1}^{\infty} c_{i} ψ_{i} (\tilde{x}, \tilde{y}) exp (- λ_{i} \tilde{z})

(33)

using the expansion coefficients c_i. The c_i are determined by taking the inner product of Eq. with ψ_i at the inlet

\tilde{z} = 0

⁠, where

\tilde{T} (\tilde{x}, \tilde{y}, 0) = {\tilde{T}}_{in} (\tilde{x}, \tilde{y})

⁠, giving

c_{i} = 〈 ψ_{i}, {\tilde{T}}_{in} 〉

(34)

Substituting Eq. into Eq. , the dimensionless temperature profile takes the form

\tilde{T} (\tilde{x}, \tilde{y}, \tilde{z}) = \sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 ψ_{i} (\tilde{x}, \tilde{y}) exp (- λ_{i} \tilde{z})

(35)

Nusselt Number Expressions.

The local Nusselt number is defined as

{Nu}_{l} = \frac{h_{l} D_{h}}{k}

(36)

where h_l is the local heat transfer coefficient and k is the thermal conductivity of the liquid. An energy balance at a point along the ridges yields

- k {\frac{\partial T}{\partial y} |}_{y = 0} = h_{l} (T_{sl} - T_{b})

(37)

where T_b is the bulk temperature of the liquid defined as

T_{b} = \frac{1}{2 d H \bar{w}} \int_{0}^{H} \int_{- d}^{d} w T d x d y

(38)

Combining Eqs. –, the local Nusselt number can be written in terms of dimensionless quantities as

{Nu}_{l} = 2 \frac{\tilde{H}}{{\tilde{T}}_{b}} {\frac{\partial \tilde{T}}{\partial \tilde{y}} |}_{\tilde{y} = 0}

(39)

where

{\tilde{T}}_{b}

is the dimensionless bulk temperature of the liquid defined as

{\tilde{T}}_{b} = \frac{1}{2 \tilde{d} \tilde{H} \bar{\tilde{w}}} 〈 \tilde{T}, 1 〉

(40)

Next, combining Eqs. , , , and yields

{Nu}_{l} = 64 \frac{{\tilde{H}}^{3} \tilde{d}}{f Re} \frac{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 {\frac{\partial ψ_{i}}{\partial \tilde{y}} |}_{\tilde{y} = 0} exp (- λ_{i} \tilde{z})}{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 〈 ψ_{i}, 1 〉 exp (- λ_{i} \tilde{z})}

(41)

The local Nusselt number for the limiting case of fully developed flow (Nu_l,fd) follows from the evaluation of Eq. as

\tilde{z} \to \infty

⁠. Given that the λ_i are real and 0 < λ₁ < λ₂ < … < λ_i < … → ∞, upon dividing both the numerator and the denominator of Eq. by

e^{- λ_{1} \tilde{z}}

and letting

\tilde{z} \to \infty

⁠, only the first term of each sum remains. It follows that

{Nu}_{l, fd} = 64 \frac{{\tilde{H}}^{3} \tilde{d}}{f Re} \frac{{\frac{\partial ψ_{1}}{\partial \tilde{y}} |}_{\tilde{y} = 0}}{〈 ψ_{1}, 1 〉}

(42)

Nu_l,fd is a function only of the first eigenfunction and it is independent of the inlet temperature profile. However, in the thermally developing region, Nu_l is a function of ${\tilde{T}}_{in}$ ⁠.

The Nusselt number averaged over the composite interface is

Nu = \frac{1}{2 d} \int_{- d}^{d} {Nu}_{l} d x

(43)

Substituting Eq. into Eq. and utilizing the symmetry of the eigenvalue problem with respect to the y-axis and the boundary condition given by Eq. yields

Nu = 64 \frac{{\tilde{H}}^{3}}{f Re} \frac{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 \int_{1}^{\tilde{d}} {\frac{\partial ψ_{i}}{\partial \tilde{y}} |}_{\tilde{y} = 0} d \tilde{x} exp (- λ_{i} \tilde{z})}{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 〈 ψ_{i}, 1 〉 exp (- λ_{i} \tilde{z})}

(44)

Next, we express the integral in the numerator of Eq. as a function of the inner product

〈 ψ_{i}, 1 〉

⁠. This is because it is more accurate to numerically evaluate

〈 ψ_{i}, 1 〉

than

\int_{1}^{\tilde{d}} {\partial ψ_{i} / \partial \tilde{y} |}_{\tilde{y} = 0} d \tilde{x}

as the latter requires numerical differentiation in order to evaluate the derivative at the boundary. The steps are as follows: First, we rearrange Eq. and integrate it over the cross section of the domain to obtain

\int_{0}^{\tilde{H}} \int_{- \tilde{d}}^{\tilde{d}} - \frac{1}{λ_{i}} \nabla^{2} ψ_{i} d \tilde{x} d \tilde{y} = 〈 ψ_{i}, 1 〉

(45)

Then, applying the divergence theorem to the left-hand side of Eq. , we find that

- \frac{1}{λ_{i}} \oint_{\partial D} \nabla ψ_{i} \cdot \hat{n} d \tilde{S} = 〈 ψ_{i}, 1 〉

(46)

where

\hat{n}

is the outward pointing unit normal vector on the boundary ∂D and

\tilde{S}

is a dimensionless coordinate along ∂D. Then, imposing boundary conditions –, we obtain

\int_{1}^{\tilde{d}} {\frac{\partial ψ_{i}}{\partial \tilde{y}} |}_{\tilde{y} = 0} d \tilde{x} = \frac{λ_{i}}{2} 〈 ψ_{i}, 1 〉

(47)

Inserting this result into Eq. yields

Nu = 32 \frac{{\tilde{H}}^{3}}{f Re} \frac{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 〈 ψ_{i}, 1 〉 λ_{i} exp (- λ_{i} \tilde{z})}{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 〈 ψ_{i}, 1 〉 exp (- λ_{i} \tilde{z})}

(48)

The Nusselt number averaged over the composite interface for the limiting case of fully developed flow (Nu_fd) follows in the same manner as Eq. and it is given by

{Nu}_{fd} = 32 \frac{{\tilde{H}}^{3} λ_{1}}{f Re}

(49)

Nu_fd is a function only of the first eigenvalue λ₁ and it is independent of ${\tilde{T}}_{in}$ ⁠.

The Nusselt number averaged over the composite interface and the streamwise length of the domain is defined as

\bar{Nu} = \frac{1}{\tilde{z}} \int_{0}^{\tilde{z}} Nu d \tilde{z}

(50)

Substituting Eq. into Eq. , it follows that

\bar{Nu} = 32 \frac{{\tilde{H}}^{3}}{f Re} \frac{1}{\tilde{z}} \ln (\frac{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 〈 ψ_{i}, 1 〉}{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 〈 ψ_{i}, 1 〉 exp (- λ_{i} \tilde{z})})

(51)

In the case of a uniform inlet temperature (UIT) at T_ref, we have

{\tilde{T}}_{in} = 1

⁠. Then, the foregoing expression reduces to

{\bar{Nu}}_{UIT} = 32 \frac{{\tilde{H}}^{3}}{f Re} \frac{1}{\tilde{z}} \ln (\frac{1}{{\tilde{T}}_{b}})

(52)

It is emphasized that Eqs. (41), (48), (51), and (52) hold for all streamwise locations $\tilde{z}$ ⁠; however, to achieve a given accuracy, more terms are required in the evaluation of each sum as $\tilde{z}$ is decreased. Moreover, expressions for the Nusselt number averaged only over the width of the ridge rather than the composite interface follow by dividing Eqs. (48), (49), (51), and (52) by the solid fraction.

Solution of the Eigenvalue Problem.

The two-dimensional eigenvalue problem defined by Eqs. (25)–(29) was numerically solved for multiple values of the aspect ratio and the solid fraction of the domain using a finite element method. The solution process is iterative and it was coded in MATLAB^® employing the partial differential equation (PDE) toolbox [27]. The algorithm exploits the symmetry of the hydrodynamic and the eigenvalue problems with respect to the y-axis in order to increase computational efficiency; therefore, the boundary conditions given by Eqs. (10) and (29) were both modified to apply at the $\tilde{x} = 0$ and $\tilde{x} = \tilde{d}$ boundaries.

The steps of the algorithm are as follows: First, the half domain is discretized with an initial number of finite elements. Next, Eq. (6) is solved subject to the new form of the boundary conditions given by Eqs. (7)–(9) to determine the two-dimensional velocity profile $\tilde{w} (\tilde{x}, \tilde{y})$ required in Eq. (25). Then, Eq. (25) subject to the new form of the boundary conditions (26)–(29) is solved to determine all the eigenvalues in the interval 0 ≤ λ_i ≤ UB along with their corresponding eigenfunctions ψ_i [27]. The upper bound (UB) was varied depending on the number of the eigenvalues sought (MATLAB^® requires prescription of the aforementioned interval for the sought λ_i because it solves the discretized eigenvalue problem by applying the Arnoldi algorithm to a shifted and inverted version of the original pencil [27]). Next, mesh refinement is implemented and the algorithm proceeds from step two until the change in the computed value of Nu_fd is less than 0.01%—typically this required 3.5 × 10⁵ elements that were adaptively placed in regions of sharp gradients. Finally, the computed eigenfunctions are normalized to satisfy Eq. (32).

The computations were validated in four ways. First, computed Poiseuille numbers were compared against those that follow from an analytical solution for the velocity profile by Philip [8] at a solid fraction $ϕ = 0.25$ and various values of H/d as per Fig. 3; agreement was within 0.006%. Second, fRe and Nu_fd were computed in the limit of $ϕ \to 1$ ⁠, i.e., for fully developed flow between two smooth parallel plates where one is isothermal and the other is adiabatic and fRe_s = 96 and Nu_fd,_s = 4.86 [28]. Agreement was within 0.03% of fRe and 0.009% of Nu_fd. Thirdly, the boundary condition at $\tilde{y} = \tilde{H}$ was changed to an isothermal one and Nu_UIT and ${\bar{Nu}}_{UIT}$ were computed at various streamwise locations in the limit $ϕ \to 1$ ⁠. The results were compared with those provided by Shah [29]. We find that if only ten terms are used in the series given by Eqs. (48) and (52), the difference between our results and those provided in Ref. [29] was found to be less than 1.4% and 0.3%, respectively, even down to thermal entrance lengths $z^{*} = z / (D_{h} Pe) = \tilde{z} / (4 {\tilde{H}}^{2} \bar{\tilde{w}}) = 1.5 \times 10^{- 4}$ ⁠. It is noted that we choose to present the results for the Nusselt number in the thermal entrance region as function of z* instead of $\tilde{z}$ to enable direct comparison of the results with those for nonstructured channels. However, $\tilde{z}$ is a more appropriate quantity for the case at hand given that $\bar{\tilde{w}}$ is a function of the solid fraction and the aspect ratio in the case of channels with textured surfaces.

Fig. 3

View large Download slide

fRe versus $ϕ$ for selected values of H/d when one plate is textured and the other one is smooth

Finally, semi-analytical values of Nu_fd were compared with those obtained using FLUENT [30], which is a general three-dimensional CFD solver. This was done for the present case and for those in the Appendix when only one plate has isothermal ridges. Conditions of hydrodynamically and thermally developed flow were imposed by using translational periodic boundary conditions between the inlet and outlet—for details, see Refs. [26], [30], and [31]. The governing equations were discretized using a second-order upwind scheme and were solved using the pressure-based coupled algorithm provided by FLUENT. The aspect ratio, solid fraction, Reynolds number, and Péclet number were taken to be $H / d = 4, ϕ = 0.3$ ⁠, Re = 2342.89, and Pe = 100.87, respectively. This value of the Péclet number was chosen to enable comparisons with the present analysis which assumes Pe ≫ 1 given that FLUENT accounts for axial conduction. This study considers steady flows only and so the solutions obtained are laminar even at Reynolds numbers as high as 2342.89. Adaptive mesh refinement was employed, with the final computational mesh containing as many as 9 × 10⁵ hexahedral elements. The computed Nu_fd for the three geometries mentioned above are 4.124, 3.836, and 3.836 correct to three decimals, and the discrepancy with the predicted values from the analysis are 0.12%, 0.19%, and 0.19%. (The aligned and staggered values are almost identical since H/d is large enough and makes the alignment unimportant—see the Appendix for more details.)

It is important to note that the present analysis produces results for Nu_fd in less than 3 min on a desktop computer, whereas FLUENT requires several hours to converge. Furthermore, it provides the means to evaluate the Nusselt number averaged over the composite interface and, additionally, the streamwise length of the domain at any $\tilde{z}$ ⁠, quantities which are prohibitively expensive to compute using a general CFD code.

Results

In this section, we present the results for the case at hand and some representative ones for the cases in the Appendix for comparison. The additional results are presented in the Appendix.

Figure 4 plots the fully developed Nusselt number averaged over the composite interface, Nu_fd, versus the solid fraction $ϕ$ for aspect ratios of H/d = 1, 1.5, 2, 4, 6, 10, and 100, when the lower plate is textured with isothermal ridges and the upper one is smooth and adiabatic. The dashed curve corresponds to smooth plates with Nusselt number Nu_fd,_s = 4.86. The results obey the expected asymptotic behavior as $ϕ \to 1$ ⁠, with ${Nu}_{fd} \to {Nu}_{fd, s}$ ⁠, irrespective of the aspect ratio. Additionally, as $ϕ \to 0, {Nu}_{fd}$ tends to zero because the available area for heat transfer vanishes. Moreover, for a given $ϕ$ (excluding the aforementioned limits) as H/d → 0 and $H / d \to \infty, {Nu}_{fd}$ tends to zero and to Nu_fd,_s, respectively. This is because for H/d → 0 heat is mainly advected by the part of the flow above the shear-free meniscus as opposed to the relatively stagnant liquid above the ridges degrading the heat transfer. In the other limit, as H/d → ∞ the difference between the temperature of the ridge and the mean temperature of the composite interface becomes significantly smaller than the difference between the temperature of the ridge and the bulk temperature of the flow.

Fig. 4

View large Download slide

Nu_fd versus $ϕ$ for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Figure 5 plots the fully developed local Nusselt number, Nu_l,fd, versus the normalized coordinate along the ridge $(\tilde{x} - 1) / (\tilde{d} - 1)$ for H/d = 10 and $ϕ = 0.01$ ⁠, 0.1 and 0.99. The results show that Nu_l,fd increases with decreasing $ϕ$ ⁠, indicating a local enhancement of heat transfer. The same trend has been observed in previous studies [18] and it is due to the fact that as $ϕ \to 0$ ⁠, the velocity of the liquid close to the ridge increases. Figure 6 plots the fully developed Nusselt number averaged over the width of the ridge, Nu_fd,ridge, versus the solid fraction. In summary, the overall effect of the decrease in the available heat transfer area and the local enhancement of heat transfer for $ϕ < 1$ is an increase in the convective portion of the total thermal resistance that is completely captured in Fig. 4.

Fig. 5

Nul,fd versus the normalized coordinate (x̃−1)/(d̃−1) along the ridge for H/d = 10 and selected values of ϕ when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

View large Download slide

Nu_l,fd versus the normalized coordinate $(\tilde{x} - 1) / (\tilde{d} - 1)$ along the ridge for H/d = 10 and selected values of $ϕ$ when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Fig. 6

View large Download slide

Nu_fd,ridge versus $ϕ$ for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

For the case of uniform inlet temperature (UIT), Figs. 7 and 8 plot Nu_UIT and ${\bar{Nu}}_{UIT}$ versus z* for $ϕ = 0.01$ and 0.1, respectively. The results were computed using the first 29 eigenvalues.⁴ The results exhibit the correct asymptotic behavior as z* → 0 and z* → ∞; in the former case, both Nu_UIT and ${\bar{Nu}}_{UIT}$ increase monotonically with decreasing z*, and in the latter case, they tend to Nu_fd. The first ten eigenvalues and the corresponding expansion coefficients that were computed for H/d = 4 at $ϕ = 0.01$ and 0.1 are provided in Table 1.

Fig. 7

NuUIT and Nu¯UIT versus z* for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic for ϕ=0.01

View large Download slide

Nu_UIT and ${\bar{Nu}}_{UIT}$ versus z* for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic for $ϕ = 0.01$

Fig. 8

NuUIT and Nu¯UIT versus z* for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic for ϕ=0.1

View large Download slide

Nu_UIT and ${\bar{Nu}}_{UIT}$ versus z* for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic for $ϕ = 0.1$

Table 1

First ten eigenvalues and corresponding expansion coefficients for H/d = 4 and $ϕ = 0.01$ or 0.1 when one plate is textured with isothermal ridges, the other one is smooth and adiabatic and ${\tilde{T}}_{in} = 1$

H/d = 4
$ϕ = 0.01$		$ϕ = 0.1$
λ_i	c_i	λ_i	c_i
4.375 × 10⁻²	3.441	6.578 × 10⁻²	3.428
6.078 × 10⁻¹	4.248 × 10⁻¹	6.923 × 10⁻¹	6.707 × 10⁻¹
1.976	1.309 × 10⁻¹	2.039	2.504 × 10⁻¹
4.174	4.645 × 10⁻²	4.130	9.351 × 10⁻²
5.478	5.006 × 10⁻²	5.453	5.008 × 10⁻²
6.832	3.990 × 10⁻²	6.791	2.538 × 10⁻²
7.400	4.909 × 10⁻²	7.367	7.305 × 10⁻²
9.063	5.237 × 10⁻²	9.289	8.425 × 10⁻²
1.115 × 10¹	7.095 × 10⁻³	1.075 × 10¹	7.194 × 10⁻³
1.197 × 10¹	4.493 × 10⁻²	1.230 × 10¹	8.377 × 10⁻²

H/d = 4
$ϕ = 0.01$		$ϕ = 0.1$
λ_i	c_i	λ_i	c_i
4.375 × 10⁻²	3.441	6.578 × 10⁻²	3.428
6.078 × 10⁻¹	4.248 × 10⁻¹	6.923 × 10⁻¹	6.707 × 10⁻¹
1.976	1.309 × 10⁻¹	2.039	2.504 × 10⁻¹
4.174	4.645 × 10⁻²	4.130	9.351 × 10⁻²
5.478	5.006 × 10⁻²	5.453	5.008 × 10⁻²
6.832	3.990 × 10⁻²	6.791	2.538 × 10⁻²
7.400	4.909 × 10⁻²	7.367	7.305 × 10⁻²
9.063	5.237 × 10⁻²	9.289	8.425 × 10⁻²
1.115 × 10¹	7.095 × 10⁻³	1.075 × 10¹	7.194 × 10⁻³
1.197 × 10¹	4.493 × 10⁻²	1.230 × 10¹	8.377 × 10⁻²

Figures 9 and 10 compare the computed values of fRe and Nu_fd, respectively, for the case when one plate is textured and the other one is smooth (solid curves), to the case in the Appendix when both plates are textured and the ridges are aligned in the transverse direction (dashed curves). In both cases, one plate has isothermal ridges and the other one is adiabatic. Although fRe is significantly reduced if both plates are textured, especially as $ϕ \to 0$ ⁠, Nu_fd changes by only a small fraction due to texturing. More importantly, as per Fig. 10, Nu_fd decreases if both plates are textured and heat is exchanged through the domain only through the isothermal ridges of one plate. This can be explained by comparing Figs. 11 and 12 that present the contour plots of the scaled dimensionless streamwise velocity $\tilde{w} / \bar{\tilde{w}}$ for the cases at hand for H/d = 4 and $ϕ = 0.3$ ⁠. Indeed, when both plates are textured and the ridges are aligned, as per Fig. 12, the flow exhibits higher velocities closer to the center of the domain, but lower velocities closer to the ridge. Thus, the convective thermal transport is degraded. When one plate is smooth, however (see Fig. 11), the velocity in the vicinity of the ridge is higher and so enhances heat transfer. This can be quantified by considering the ratio (R) of the average velocity of the flow in an area close to the ridge, i.e., $0 \leq \tilde{x} \leq \tilde{d}$ and $0 \leq \tilde{y} \leq \tilde{H} / 2$ ⁠, over the mean velocity of the flow

Fig. 9

View large Download slide

fRe versus $ϕ$ for selected H/d when one plate is textured with isothermal ridges and the other one is adiabatic and either smooth (t–s) or textured with aligned ridges (al)

Fig. 10

View large Download slide

Nu_fd versus $ϕ$ for selected H/d when one plate is textured with isothermal ridges and the other one is adiabatic and either smooth (t–s) or textured with aligned ridges (al)

Fig. 11

Contour plot of w̃/w̃¯ when one plate is textured with isothermal ridges and the other one is adiabatic and smooth (t–s) for H/d = 4 and ϕ=0.3

View large Download slide

Contour plot of $\tilde{w} / \bar{\tilde{w}}$ when one plate is textured with isothermal ridges and the other one is adiabatic and smooth (t–s) for H/d = 4 and $ϕ = 0.3$

Fig. 12

Contour plot of w̃/w̃¯ when one plate is textured with isothermal and the other one with adiabatic ridges and the ridges are aligned (al) for H/d = 4 and ϕ=0.3

View large Download slide

Contour plot of $\tilde{w} / \bar{\tilde{w}}$ when one plate is textured with isothermal and the other one with adiabatic ridges and the ridges are aligned (al) for H/d = 4 and $ϕ = 0.3$

R = \frac{2 \int_{0}^{\tilde{H} / 2} \int_{0}^{\tilde{d}} \tilde{w} d \tilde{x} d \tilde{y}}{\tilde{d} \tilde{H} \bar{\tilde{w}}}

(53)

When both plates are textured and the ridges are aligned, R is equal to 1 due to symmetry, but, when one plate is smooth, R becomes 1.127 for the prescribed values of H/d and $ϕ$ ⁠, which indicates higher velocities close to the ridge. The same observations can be made for the case when both plates are textured, but the ridges are staggered in the transverse direction. The corresponding plots for fRe and Nu_fd and the contour plot of the scaled dimensionless streamwise velocity are presented in the Appendix.

Finally, Fig. 13 compares the computed values of Nu_fd when both plates are textured with isothermal aligned ridges against those calculated for the same configuration but for isoflux ridges (Nu_fd,if) by Kirk et al. [20]. The results show that depending on the aspect ratio H/d, there is a range for $ϕ$ where the fully developed Nusselt number averaged over the composite interface for isothermal ridges is slightly higher than for isoflux ridges despite the fact that the fully developed Nusselt number for smooth isothermal plates is smaller than that for smooth isoflux plates.

Fig. 13

View large Download slide

Nu_fd and Nu_fd,if versus $ϕ$ for selected H/d when both plates are textured with aligned isothermal and isoflux ridges, respectively

Fig. 14

View large Download slide

Schematic of the domain when both plates are textured and the ridges are aligned

Conclusions

We developed semi-analytical expressions for the Nusselt number for the case of hydrodynamically developed and thermally developing flow between parallel plates that are textured with ridges oriented parallel to the flow. The ridges of one plate are isothermal and the other plate can be smooth and adiabatic, or textured with adiabatic or isothermal ridges. When both plates are textured, the ridges can be aligned or staggered by half a pitch in the transverse direction. The menisci between the ridges were considered to be flat and adiabatic. The solid–liquid interface and the menisci were subjected to no-slip and no-shear boundary conditions, respectively. Using separation of variables, we expressed the three-dimensional temperature field as an infinite sum of the product of an exponentially decaying function of the streamwise coordinate and a second eigenfunction depending on the transverse coordinates. The latter eigenfunctions satisfy a two-dimensional Sturm–Liouville problem from which the eigenvalues and eigenfunctions follow numerically.

The derived expressions for the local Nusselt number, the Nusselt number averaged over the composite interface, and the Nusselt number averaged over the composite interface and the streamwise length of the domain indicate that the Nusselt number is a function of the dimensionless streamwise coordinate, the aspect ratio of the domain, the solid fraction, and the inlet temperature profile. Expressions were also derived for the fully developed local Nusselt number and for the fully developed Nusselt number averaged over the composite interface in terms of the first eigenfunction and of the first eigenvalue, respectively.

The results indicate that the Nusselt number averaged over the composite interface decreases as the aspect ratio and/or the solid fraction decrease. Moreover, it was observed that when one plate is adiabatic, the configuration where the adiabatic plate is smooth provides a higher Nusselt number than when it is textured. Finally, using the present analysis, the fully developed local Nusselt number and the fully developed Nusselt number averaged over the composite interface can be computed in a small fraction of the time that is required by a general CFD solver. More importantly, the analysis provides semi-analytical expressions to evaluate the Nusselt number averaged over the composite interface and, additionally, the streamwise length of the domain at any location, quantities which are prohibitively expensive to compute using a general CFD code.

Acknowledgment

The work of GK and MH was supported by the National Science Foundation under Grant No. 1402783. The work of DTP was supported in part by the Engineering and Physical Sciences Research Council (UK) Grant Nos. EP/K041134 and EP/L020564. The work of TK was supported by an EPSRC-UK doctoral scholarship.

Nomenclature

a =: half meniscus width, m
c_i =: expansion coefficients
d =: half ridge pitch, m
D =: domain
$\tilde{d}$ =: dimensionless half ridge pitch, d/a
D_h =: hydraulic diameter, 2H
dp/dz =: prescribed pressure gradient, Pa/m
f =: friction factor, $2 D_{h} (- d p / d z) / (ρ {\bar{w}}^{2})$
fRe =: Poiseuille number
H =: distance between parallel plates, m
$\tilde{H}$ =: dimensionless distance between parallel plates, H/a
h_l =: local heat transfer coefficient, W/(m² K)
k =: thermal conductivity of liquid, W/(m K)
LB =: lower bound of λ_i
$\hat{n}$ =: outward pointing unit normal vector on boundaries
Nu =: Nusselt number averaged over the composite interface
$\bar{Nu}$ =: Nusselt number averaged over the composite interface and the streamwise length of the domain
Nu_l =: local Nusselt number, h_lD_h/k
Pe =: Péclet number, $\bar{w} D_{h} / α$
R =: average velocity ratio close to the ridge, $(2 \int_{0}^{\tilde{H} / 2} \int_{0}^{\tilde{d}} \tilde{w} d \tilde{x} d \tilde{y}) / (\tilde{d} \tilde{H} \bar{\tilde{w}})$
Re =: Reynolds number, $ρ \bar{w} D_{h} / μ$
$\tilde{S}$ =: dimensionless coordinate along ∂D
T =: temperature, °C
$\tilde{T}$ =: dimensionless temperature, $(T - T_{sl}) / (T_{in} - T_{sl})$
T_b =: bulk temperature
${\tilde{T}}_{b}$ =: dimensionless bulk temperature
T_in =: inlet temperature, °C
${\tilde{T}}_{in}$ =: dimensionless inlet temperature
T_ref =: reference temperature, °C
T_sl =: ridge temperature, °C
UB =: upper bound of λ_i
w =: streamwise velocity, m/s
$\bar{w}$ =: mean velocity, m/s
$\tilde{w}$ =: dimensionless velocity, $2 μ w / [a H (- d p / d z)]$
$\bar{\tilde{w}}$ =: dimensionless mean velocity
x =: lateral coordinate, m
$\tilde{x}$ =: dimensionless lateral coordinate, x/a
y =: vertical coordinate, m
$\tilde{y}$ =: dimensionless vertical coordinate, y/a
z =: streamwise coordinate, m
$\tilde{z}$ =: dimensionless streamwise coordinate, $2 α μ z / [a^{3} H (- d p / d z)]$
z* =: dimensionless streamwise coordinate for the thermal entrance region, z/(D_h Pe)
∂D =: boundary of the dimensionless domain

Greek Symbols

α =: thermal diffusivity, m²/s
λ_i =: ith eigenvalue
μ =: dynamic viscosity, Pa·s
ρ =: density, kg/m³
$ϕ$ =: solid fraction, (d − a)/d
ψ_i =: ith eigenfunction

Subscripts

al =: textured plates with aligned ridges
fd =: fully developed flow
if =: isoflux ridges
ridge =: indicates quantity based on the width of the ridge
s =: smooth plates
t–s =: one textured and one smooth plate
UIT =: uniform inlet temperature

Appendix

Sections A.1 and A.2 provide the necessary information for the extension of the present analysis to the configurations when both plates are textured with parallel ridges and the ridges are either aligned or staggered, respectively. Each subsection covers the cases when the ridges of one plate are isothermal and those of the other one are either adiabatic or isothermal.

Both Plates Textured, Aligned Ridges

When both plates are textured and the ridges are aligned as indicated in Fig. 14, the boundary conditions for the hydrodynamic problem given by Eqs. (7) and (8) apply rather than Eq. (9) at $\tilde{y} = \tilde{H}$ ⁠. The computed Poiseuille numbers are presented in Fig. 15.

Fig. 15

View large Download slide

fRe versus $ϕ$ for selected H/d when both plates are textured and the ridges are aligned

If only the lower plate has isothermal ridges and the upper one has adiabatic ridges, the boundary conditions for the thermal problem and for the eigenvalue problem are identical to those in Sec. 2.2. The expressions for Nu_l, Nu_l,fd, Nu, Nu_fd, $\bar{Nu}$ ⁠, and ${\bar{Nu}}_{UIT}$ are identical to those given by Eqs. (41), (42), (48), (49), (51), and (52) and the reader is referred to those expressions for their detailed form. The computed Nu_fd is presented in Fig. 16.

Fig. 16

View large Download slide

Nu_fd versus $ϕ$ for selected H/d when the ridges of one plate are isothermal and the ridges of the other one are aligned and adiabatic

If the ridges of both plates are isothermal, the thermal boundary conditions given by Eqs. (20) and (21) apply rather than Eq. (22) at $\tilde{y} = \tilde{H}$ ⁠. In terms of the eigenvalue problem, the boundary conditions given by Eqs. (26) and (27) apply rather than Eq. (28) at $\tilde{y} = \tilde{H}$ ⁠.

In this case, the expressions for Nu_l and Nu_l,fd are identical to those given by Eqs. and . However, the Nusselt number averaged over the composite interfaces is

Nu = \frac{1}{2 d} \int_{- d}^{d} {{Nu}_{l} |}_{\tilde{y} = 0} d x + \frac{1}{2 d} \int_{- d}^{d} {{Nu}_{l} |}_{\tilde{y} = \tilde{H}} d x

(A1)

Moreover, following the same steps as in Sec. 2.3, it follows that

\int_{1}^{\tilde{d}} {\frac{\partial ψ_{i}}{\partial \tilde{y}} |}_{\tilde{y} = 0} d \tilde{x} = \frac{λ_{i}}{4} 〈 ψ_{i}, 1 〉

(A2)

Thus, the expressions for the averaged values of the Nusselt number take the form

Nu = 16 \frac{{\tilde{H}}^{3}}{f Re} \frac{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 〈 ψ_{i}, 1 〉 λ_{i} exp (- λ_{i} \tilde{z})}{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 〈 ψ_{i}, 1 〉 exp (- λ_{i} \tilde{z})}

(A3)

{Nu}_{fd} = 16 \frac{{\tilde{H}}^{3}}{f Re} λ_{1}

(A4)

\bar{Nu} = 16 \frac{{\tilde{H}}^{3}}{\tilde{z} f Re} \ln (\frac{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 〈 ψ_{i}, 1 〉}{\sum_{i = 1}^{\infty} 〈 ψ_{i}, {\tilde{T}}_{in} 〉 〈 ψ_{i}, 1 〉 exp (- λ_{i} \tilde{z})})

(A5)

{\bar{Nu}}_{UIT} = 16 \frac{{\tilde{H}}^{3}}{\tilde{z} f Re} \ln (\frac{1}{{\tilde{T}}_{b}})

(A6)

The computed Nu_fd is presented in Fig. 17.

Fig. 17

View large Download slide

Nu_fd versus $ϕ$ for selected H/d when the ridges of both plates are isothermal and aligned

Fig. 18

View large Download slide

Schematic of the domain when both plates are textured and the ridges are staggered by half a pitch

Both Plates Textured, Staggered Ridges

When both plates are textured and the ridges are staggered in the transverse direction by half a pitch as shown in Fig. 18, the relevant boundary conditions for the hydrodynamic problem become

\frac{\partial \tilde{w}}{\partial \tilde{y}} = 0 \begin{matrix} for | \tilde{x} | < 1, \tilde{y} = 0 \end{matrix}

(A7)

\tilde{w} = 0 \begin{matrix} for 1 < | \tilde{x} | < \tilde{d}, \tilde{y} = 0 \end{matrix}

(A8)

\frac{\partial \tilde{w}}{\partial \tilde{y}} = 0 \begin{matrix} for \tilde{d} - 1 < | \tilde{x} | < \tilde{d}, \tilde{y} = \tilde{H} \end{matrix}

(A9)

\tilde{w} = 0 \begin{matrix} for | \tilde{x} | < \tilde{d} - 1, \tilde{y} = \tilde{H} \end{matrix}

(A10)

\frac{\partial \tilde{w}}{\partial \tilde{x}} = 0 \begin{matrix} for | \tilde{x} | = \tilde{d}, 0 < \tilde{y} < \tilde{H} \end{matrix}

(A11)

The computed Poiseuille numbers are presented in Fig. 19. Figure 20 presents the contour plot of the scaled dimensionless streamwise velocity $\tilde{w} / \bar{\tilde{w}}$ for this case.

Fig. 19

View large Download slide

fRe versus $ϕ$ for selected H/d when both plates are textured and the ridges are staggered by half a pitch

Fig. 20

Contour plot of w̃/w̃¯ when both plates are textured and the ridges are staggered by half a pitch for H/d = 4 and ϕ=0.3

View large Download slide

Contour plot of $\tilde{w} / \bar{\tilde{w}}$ when both plates are textured and the ridges are staggered by half a pitch for H/d = 4 and $ϕ = 0.3$

If only the lower plate has isothermal ridges and the upper one has adiabatic ridges, the boundary conditions for the thermal problem and for the eigenvalue problem are identical to those in Sec. 2.2. The expressions for Nu_l, Nu_l,fd, Nu, Nu_fd, $\bar{Nu}$ ⁠, and ${\bar{Nu}}_{UIT}$ are identical to those given by Eqs. (41), (42), (48), (49), (51), and (52). The computed Nu_fd is presented in Fig. 21.

Fig. 21

View large Download slide

Nu_fd versus $ϕ$ for selected H/d when the ridges of one plate are isothermal and the ridges of the other one are staggered by half a pitch and adiabatic

When the ridges on both plates are isothermal, the boundary conditions for the thermal problem become

\frac{\partial \tilde{T}}{\partial \tilde{y}} = 0 \begin{matrix} for | \tilde{x} | < 1, \tilde{y} = 0 \end{matrix}

(A12)

\tilde{T} = 0 \begin{matrix} for 1 < | \tilde{x} | < \tilde{d}, \tilde{y} = 0 \end{matrix}

(A13)

\frac{\partial \tilde{T}}{\partial \tilde{y}} = 0 \begin{matrix} for \tilde{d} - 1 < | \tilde{x} | < \tilde{d}, \tilde{y} = \tilde{H} \end{matrix}

(A14)

\tilde{T} = 0 \begin{matrix} for | \tilde{x} | < \tilde{d} - 1, \tilde{y} = \tilde{H} \end{matrix}

(A15)

\frac{\partial \tilde{T}}{\partial \tilde{x}} = 0 \begin{matrix} for | \tilde{x} | = \tilde{d}, 0 < \tilde{y} < \tilde{H} \end{matrix}

(A16)

\tilde{T} = {\tilde{T}}_{in} \begin{matrix} for \tilde{z} = 0 \end{matrix}

(A17)

while those of the eigenvalue problem read

\frac{\partial ψ_{i}}{\partial \tilde{y}} = 0 \begin{matrix} for | \tilde{x} | < 1, \tilde{y} = 0 \end{matrix}

(A18)

ψ_{i} = 0 \begin{matrix} for 1 < | \tilde{x} | < \tilde{d}, \tilde{y} = 0 \end{matrix}

(A19)

\frac{\partial ψ_{i}}{\partial \tilde{y}} = 0 \begin{matrix} for \tilde{d} - 1 < | \tilde{x} | < \tilde{d}, \tilde{y} = \tilde{H} \end{matrix}

(A20)

ψ_{i} = 0 \begin{matrix} for | \tilde{x} | < \tilde{d} - 1, \tilde{y} = \tilde{H} \end{matrix}

(A21)

\frac{\partial ψ_{i}}{\partial \tilde{x}} = 0 \begin{matrix} for | \tilde{x} | = \tilde{d}, 0 < \tilde{y} < \tilde{H} \end{matrix}

(A22)

The expressions for Nu_l, Nu_l,fd, Nu, Nu_fd, $\bar{Nu}$ ⁠, and ${\bar{Nu}}_{UIT}$ are identical to those given by Eqs. (41), (42), and (A3)–(A6). The computed Nu_fd is presented in Fig. 22.

Fig. 22

View large Download slide

Nu_fd versus $ϕ$ for selected H/d when the ridges of both plates are isothermal and staggered by half a pitch

1

Parallel and transverse relative to the flow direction.

2

We use the term Graetz–Nusselt problem rather than Graetz problem because they refer to flow between parallel plates and through circular duct, respectively, as per the distinction made in Shah and London [21].

3

To show that λ is real and positive we multiply Eq. (25) by the complex conjugate of ψ, integrate over the domain, and use the divergence theorem.

4

If 28 eigenvalues are used instead the maximum discrepancies for the presented values of Nu_UIT and ${\bar{Nu}}_{UIT}$ are less than 0.002% and 0.0003%, respectively, and if 25 eigenvalues are used instead the maximum discrepancies are less than 0.09% and 0.02%, respectively.

References

1.

Lam

,

L. S.

,

Hodes

,

M.

, and

Enright

,

R.

,

2015

, “

Analysis of Galinstan-Based Microgap Cooling Enhancement Using Structured Surfaces

,”

ASME J. Heat Transfer

,

137

(

9

), p.

091003

.

Google Scholar

Crossref

2.

Lobaton

,

E. J.

, and

Salamon

,

T. R.

,

2007

, “

Computation of Constant Mean Curvature Surfaces: Application to the Gas–Liquid Interface of a Pressurized Fluid on a Superhydrophobic Surface

,”

J. Colloid Interface Sci.

,

314

(

1

), pp.

184

–

198

.

Google Scholar

Crossref

PubMed

3.

Quéré

,

D.

,

2005

, “

Non-Sticking Drops

,”

Rep. Prog. Phys.

,

68

(

11

), p.

2495

.

Google Scholar

Crossref

4.

Cassie

,

A. B. D.

, and

Baxter

,

S.

,

1944

, “

Wettability of Porous Surfaces

,”

Trans. Faraday Soc.

,

40

, pp.

546

–

551

.

Google Scholar

Crossref

5.

Huang

,

D. M.

,

Sendner

,

C.

,

Horinek

,

D.

,

Netz

,

R. R.

, and

Bocquet

,

L.

,

2008

, “

Water Slippage Versus Contact Angle: A Quasiuniversal Relationship

,”

Phys. Rev. Lett.

,

101

(

22

), p.

226101

.

Google Scholar

Crossref

PubMed

6.

Cottin-Bizonne

,

C.

,

Steinberger

,

A.

,

Cross

,

B.

,

Raccurt

,

O.

, and

Charlaix

,

E.

,

2008

, “

Nanohydrodynamics: The Intrinsic Flow Boundary Condition on Smooth Surfaces

,”

Langmuir

,

24

(

4

), pp.

1165

–

1172

.

Google Scholar

Crossref

PubMed

7.

Enright

,

R.

,

Hodes

,

M.

,

Salamon

,

T.

, and

Muzychka

,

Y.

,

2013

, “

Isoflux Nusselt Number and Slip Length Formulae for Superhydrophobic Microchannels

,”

ASME J. Heat Transfer

,

136

(

1

), p.

012402

.

Google Scholar

Crossref

8.

Philip

,

J. R.

,

1972

, “

Flows Satisfying Mixed No-Slip and No-Shear Conditions

,”

Z. Angew. Math. Phys.

,

23

(

3

), pp.

353

–

372

.

Google Scholar

Crossref

9.

Sbragaglia

,

M.

, and

Prosperetti

,

A.

,

2007

, “

A Note on the Effective Slip Properties for Microchannel Flows With Ultrahydrophobic Surfaces

,”

Phys. Fluids

,

19

(

4

), p.

043603

.

Google Scholar

Crossref

10.

Rothstein

,

J. P.

,

2010

, “

Slip on Superhydrophobic Surfaces

,”

Annu. Rev. Fluid Mech.

,

42

(

1

), pp.

89

–

109

.

Google Scholar

Crossref

11.

Teo

,

C. J.

, and

Khoo

,

B. C.

,

2009

, “

Analysis of Stokes Flow in Microchannels With Superhydrophobic Surfaces Containing a Periodic Array of Micro-Grooves

,”

Microfluid. Nanofluid.

,

7

(

3

), pp.

353

–

382

.

Google Scholar

Crossref

12.

Teo

,

C. J.

, and

Khoo

,

B. C.

,

2010

, “

Flow Past Superhydrophobic Surfaces Containing Longitudinal Grooves: Effects of Interface Curvature

,”

Microfluid. Nanofluid.

,

9

(

2

), pp.

499

–

511

.

Google Scholar

Crossref

13.

Ng

,

C.-O.

, and

Wang

,

C. Y.

,

2014

, “

Temperature Jump Coefficient for Superhydrophobic Surfaces

,”

ASME J. Heat Transfer

,

136

(

6

), p.

064501

.

Google Scholar

Crossref

14.

Lam

,

L. S.

,

Hodes

,

M.

,

Karamanis

,

G.

,

Kirk

,

T.

, and

MacLachlan

,

S.

,

2016

, “

Effect of Meniscus Curvature on Apparent Thermal Slip

,”

ASME J. Heat Transfer

,

138

(

12

), p.

122004

.

Google Scholar

Crossref

15.

Hodes

,

M.

,

Lam

,

L. S.

,

Cowley

,

A.

,

Enright

,

R.

, and

MacLachlan

,

S.

,

2015

, “

Effect of Evaporation and Condensation at Menisci on Apparent Thermal Slip

,”

ASME J. Heat Transfer

,

137

(

7

), p.

071502

.

Google Scholar

Crossref

16.

Lam

,

L. S.

,

Melnick

,

C.

,

Hodes

,

M.

,

Ziskind

,

G.

, and

Enright

,

R.

,

2014

, “

Nusselt Numbers for Thermally Developing Couette Flow With Hydrodynamic and Thermal Slip

,”

ASME J. Heat Transfer

,

136

(

5

), p.

051703

.

Google Scholar

Crossref

17.

Maynes

,

D.

,

Webb

,

B. W.

, and

Davies

,

J.

,

2008

, “

Thermal Transport in a Microchannel Exhibiting Ultrahydrophobic Microribs Maintained at Constant Temperature

,”

ASME J. Heat Transfer

,

130

(

2

), p.

022402

.

Google Scholar

Crossref

18.

Maynes

,

D.

,

Webb

,

B. W.

,

Crockett

,

J.

, and

Solovjov

,

V.

,

2012

, “

Analysis of Laminar Slip-Flow Thermal Transport in Microchannels With Transverse Rib and Cavity Structured Superhydrophobic Walls at Constant Heat Flux

,”

ASME J. Heat Transfer

,

135

(

2

), p.

021701

.

Google Scholar

Crossref

19.

Maynes

,

D.

, and

Crockett

,

J.

,

2013

, “

Apparent Temperature Jump and Thermal Transport in Channels With Streamwise Rib and Cavity Featured Superhydrophobic Walls at Constant Heat Flux

,”

ASME J. Heat Transfer

,

136

(

1

), p.

011701

.

Google Scholar

Crossref

20.

Kirk

,

T. L.

,

Hodes

,

M.

, and

Papageorgiou

,

D. T.

,

2017

, “

Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges Accounting for Meniscus Curvature

,”

J. Fluid Mech.

,

811

(

1

), pp.

315

–

349

.

Google Scholar

Crossref

21.

Shah

,

R.

, and

London

,

A.

,

1978

, “

Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data

,”

Advances in Heat Transfer: Supplement

,

Academic Press

,

New York

.

22.

Nusselt

,

W.

,

1923

, “

Der wärmeaustausch am berieselungskühler

,”

VDI-Z

,

67

(

9

), pp.

206

–

216

.

23.

Brown

,

G. M.

,

1960

, “

Heat or Mass Transfer in a Fluid in Laminar Flow in a Circular or Flat Conduit

,”

AIChE J.

,

6

(

2

), pp.

179

–

183

.

Google Scholar

Crossref

24.

Mikhaĭlov

,

M. D.

, and

Öz iş ik

,

M. N.

,

1984

,

Unified Analysis and Solutions of Heat and Mass Diffusion

,

Wiley

,

New York

.

25.

Panton

,

R. L.

,

2006

,

Incompressible Flow

,

Wiley

, New York.

26.

Hodes

,

M.

,

Kirk

,

T. L.

,

Karamanis

,

G.

, and

MacLachlan

,

S.

,

2017

, “

Effect of Thermocapillary Stress on Slip Length for a Channel Textured With Parallel Ridges

,”

J. Fluid Mech.

,

814

(

3

), pp.

301

–

324

.

Google Scholar

Crossref

27.

MathWorks

,

2016

, “

Partial Differential Equation Toolbox User's Guide

,”

MathWorks Inc.

, Natick, MA.

28.

McCuen

,

P.

,

Kays

,

W.

, and

Reynolds

,

W.

,

1962

,

Heat Transfer With Laminar and Turbulent Flow Between Parallel Planes With Constant and Variable Wall Temperature and Heat Flux

,

Stanford University

,

Stanford, CA

.

29.

Shah

,

R.

,

1975

, “

Thermal Entry Length Solutions for the Circular Tube and Parallel Plates

,”

Third National Heat Mass Transfer Conference

, Indian Institute of Technology, Bombay, Mumbai, India, Dec. 11–13, Vol.

1

, pp.

11

–

75

.

30.

ANSYS

,

2013

, “

ANSYS Fluent Theory Guide

,” ANSYS Inc., Canonsburg, PA.

31.

Patankar

,

S. V.

,

Liu

,

C. H.

, and

Sparrow

,

E. M.

,

1977

, “

Fully Developed Flow and Heat Transfer in Ducts Having Streamwise-Periodic Variations of Cross-Sectional Area

,”

ASME J. Heat Transfer

,

99

(

2

), pp.

180

–

186

.

Google Scholar

Crossref

Solution of the Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges

Introduction

Background.

Assumptions.

Analysis

Hydrodynamic Problem.

Thermal Problem.

Nusselt Number Expressions.

Solution of the Eigenvalue Problem.

Results

Conclusions

Acknowledgment

Nomenclature

Greek Symbols

Subscripts

Appendix

Both Plates Textured, Aligned Ridges

Both Plates Textured, Staggered Ridges

References

Get Email Alerts

Cited By

Journals

Conference Proceedings

eBooks

Resources

Opportunities

Solution of the Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges

Introduction

Background.

Assumptions.

Analysis

Hydrodynamic Problem.

Thermal Problem.

Nusselt Number Expressions.

Solution of the Eigenvalue Problem.

Results

Conclusions

Acknowledgment

Nomenclature

Greek Symbols

Subscripts

Appendix

Both Plates Textured, Aligned Ridges

Both Plates Textured, Staggered Ridges

References

Get Email Alerts

Cited By

Related Articles

Related Proceedings Papers

Related Chapters

Journals

Conference Proceedings

eBooks

Resources

Opportunities

This Feature Is Available To Subscribers Only