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Research Papers: Forced Convection

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

[+] Author and Article Information
Georgios Karamanis

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155
e-mail: Georgios.Karamanis@tufts.edu

Marc Hodes

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155
e-mail: Marc.Hodes@tufts.edu

Toby Kirk

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: toby.kirk12@imperial.ac.uk

Demetrios T. Papageorgiou

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: d.papageorgiou@imperial.ac.uk

1In general, the cavities beneath the menisci are filled with inert gas and vapor on account of the vapor pressure of the liquid phase, and for brevity, we refer to this mixture as “gas.”

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 5, 2017; final manuscript received January 18, 2018; published online March 9, 2018. Assoc. Editor: George S. Dulikravich.

J. Heat Transfer 140(6), 061703 (Mar 09, 2018) (15 pages) Paper No: HT-17-1395; doi: 10.1115/1.4039085 History: Received July 05, 2017; Revised January 18, 2018

We consider convective heat transfer for laminar flow of liquid between parallel plates. The configurations analyzed are both plates textured with symmetrically aligned isothermal ridges oriented parallel to the flow, and one plate textured as such and the other one smooth and adiabatic. 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. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). We solve for the developing three-dimensional temperature profile resulting from a step change of the ridge temperature in the streamwise direction assuming a hydrodynamically developed flow. Axial conduction is accounted for, i.e., we consider the extended Graetz–Nusselt problem; therefore, the domain is of infinite length. The effects of viscous dissipation and (uniform) volumetric heat generation are also captured. Using the method of separation of variables, the homogeneous part of the thermal problem is reduced to a nonlinear eigenvalue problem in the transverse coordinates which is solved numerically. Expressions derived for the local and the fully developed Nusselt number along the ridge and that averaged over the composite interface in terms of the eigenvalues, eigenfunctions, Brinkman number, and dimensionless volumetric heat generation rate. Estimates are provided for the streamwise location where viscous dissipation effects become important.

A sessile droplet on a structured surface characterized by small periodic length scales compared to the capillary length may be stable in the Cassie state [1,2] where solid–liquid contact is exclusively at the tips of the structures. A liquid flowing through a microchannel with structured surfaces may be as well and the necessary criteria are provided by Lam et al. [3]. Then, a mixed boundary condition of no-slip [4,5] and low-shear applies along the solid–liquid and the liquid–gasff1 interfaces (menisci), respectively. The low-shear boundary condition provides a lubrication effect and thus reduces both the hydrodynamic resistance and the caloric part of the thermal resistance. However, the reduction in the solid–liquid interfacial area reduces the available area for heat transfer and thus increases the convective part of the thermal resistance. A net reduction of the total, i.e., caloric plus convective, thermal resistance can be achieved with proper sizing of the structures [3] and it requires the knowledge of the Nusselt number (Nu). Such Nusselt numbers are especially relevant to direct liquid cooling applications [3] as per Fig. 1 that depicts a structured microchannel etched into the upper portion of a microprocessor die.

The channel surfaces can be textured with a variety of periodic structures such as pillars, transverse ridges, or parallel ridges [6]. The latter configuration is examined here since it is more favorable from a heat transfer perspective [3,7]. The hydrodynamic effects of structured surfaces with parallel ridges in parallel plate channels have been studied for flat and curved menisci [815]. In terms of the 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. Their analysis applies to both symmetrically and asymmetrically heated channels with large plate spacing to structure pitch ratio. Enright et al. [7] too developed analytical expressions for apparent slip lengths for structured surfaces with parallel or transverse ridges or pillar arrays assuming flat and adiabatic menisci. Ng and Wang [16] derived semi-analytical expressions for the apparent thermal slip length for isothermal parallel ridges while accounting for conduction through the gas phase. Lam et al. [17] derived expressions for the apparent thermal slip length for isoflux and isothermal parallel ridges accounting for small meniscus curvature. Hodes et al. [18] captured the effects of evaporation and condensation along menisci on the apparent thermal slip length for isoflux ridges. Lam et al. [19] developed expressions for the Nusselt number for thermally developing Couette flow as a function of apparent slip lengths for various boundary conditions. Also, Lam et al. [19] discuss when Nu results accounting for molecular slip can be used to capture the effects of apparent slip. Maynes and Crockett [20] developed expressions for the Nusselt number and the thermal slip length for microchannels with isoflux parallel ridges assuming flat menisci and using the Navier slip approximation for the velocity profile. Kirk et al. [21] also developed expressions for the Nusselt number for this configuration, but without invoking the Navier slip approximation. Kirk et al. [21] also accounted for small meniscus curvature using a boundary perturbation method. Karamanis et al. [22] developed 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 [2325].

The present work extends the analysis in Ref. [22] to the case of flow with finite axial conduction, i.e., to the extended Graetz–Nusselt problem [2628]. Viscous dissipation [29,30] and (uniform) volumetric heat generation [31] are also captured. The menisci are assumed to be flat [17] and adiabatic. The configurations for the isothermal ridges are either both plates textured, as per Fig. 2, or one plate textured and the other one smooth and adiabatic, as per Fig. 3. The solution approach is similar in both configurations. It therefore suffices to present the detailed analysis for the first configuration. The second configuration is considered in Appendix A.

The cross-sectional view of one period of the domain (D) considered is depicted in Fig. 2. It extends from minus to plus infinity in the streamwise direction z, and |x|d and 0yH, where 2d is the pitch of the ridges and H is the distance between the ridge tips on opposing plates. The hydraulic diameter of the domain (Dh) is 2H. The width of the meniscus is 2a. The triple contact lines coincide with the corners of the ridges at |x|=a at both y = 0 and y = H. Along the composite interfaces at y = 0 and y = H, a no-shear boundary condition applies for |x|<a and a no-slip one is imposed for a<|x|<d [4,5]. Symmetry boundary conditions apply along the boundaries at x=|d|. The temperature of the ridges is Tr and Tr+ for z0 and z > 0, respectively. The flow is pressure driven, steady, laminar, hydrodynamically developed, and thermally developing with constant thermophysical properties. The temperature profile becomes uniform throughout the cross section as z and z+, where it is Tr and Tr+, respectively. Effects due to Marangoni stresses [32,33], evaporation and condensation [18], and gas diffusion in the liquid phase are neglected. The independent dimensionless geometric variables are the solid fraction of the ridge, ϕ=(da)/d, and the aspect ratio of the domain, H/d. Finally, the analysis utilizes the symmetry of the domain with respect to the yz and zx planes through x = 0 and y=H/2, respectively, and therefore, we further restrict to 0xd and 0yH/2.

Hydrodynamic Problem.

Assuming hydrodynamically developed laminar flow with constant thermophysical properties, the streamwise-momentum equation takes the formDisplay Formula

(1)2w=1μdpdz

where w is the streamwise velocity, μ is the dynamic viscosity, and dp/dz is the prescribed (constant) pressure gradient. Denoting nondimensional variables with tildes, Eq. (1) is rendered dimensionless by definingDisplay Formula

(2)x̃=xDh
Display Formula
(3)ỹ=yDh
Display Formula
(4)w̃=μDh2(dpdz)1w

It becomesDisplay Formula

(5)2w̃=1
subjected toDisplay Formula
(6)w̃ỹ=0for0<x̃<ã,ỹ=0
Display Formula
(7)w̃=0forã<x̃<d̃,ỹ=0
Display Formula
(8)w̃ỹ=0for0<x̃<d̃,ỹ=1/4
Display Formula
(9)w̃x̃=0forx̃=0,x̃=d̃,0<ỹ<1/4

where d̃=d/Dh and ã=a/Dh are the dimensionless (half) pitch of the ridges and width of the meniscus, respectively. This hydrodynamic problem has been studied analytically [15], semi-analytically [11,14], and numerically [10] in the past. Here, we solve it numerically (see Appendix B) to facilitate the solution of the thermal energy equation.

To proceed with the formulation of the Nusselt number, we first compute the Poiseuille number fRe whereDisplay Formula

(10)Re=ρw¯Dhμ
Display Formula
(11)f=2Dhρw¯2(dpdz)
Display Formula
(12)w¯=2dH0H/20dwdxdy
are the Reynolds number based on hydraulic diameter, the friction factor and the mean velocity of the liquid, respectively, and ρ is the density. Combining Eqs. (2)(4) and (10)(12), it follows thatDisplay Formula
(13)fRe=2w̃¯

whereDisplay Formula

(14)w̃¯=4d̃01/40d̃w̃dx̃dỹ
is the dimensionless mean velocity of the flow. Henceforth, w̃(x̃,ỹ) and thus fRe are considered to be known given that the hydrodynamic problem can be solved independently from the thermal one.

Thermal Problem.

Capturing axial conduction, viscous dissipation, and volumetric heat generation, the relevant form of the thermal energy equation isDisplay Formula

(15)ρcpwTz=k(2Tx2+2Ty2+2Tz2)+μ[(wx)2+(wy)2]+q˙

where T, k, and cp are the temperature, thermal conductivity, and specific heat at constant pressure of the liquid, respectively, and q˙ is the (constant) volumetric heat generation rate within the liquid.

Next, we introduce the dimensionless streamwise coordinate z̃ and temperature T̃, as perDisplay Formula

(16)z̃=zPeDh
Display Formula
(17)T̃=TTrTr+Tr
respectively, whereDisplay Formula
(18)Pe=RePr
is the Péclet number, i.e., the scale of the ratio of advective to axially diffusive heat transfer andDisplay Formula
(19)Pr=cpμk
is the Prandtl number of the liquid. Combining Eqs. (2)(4), (13), and (15)(19), the dimensionless form of the (inhomogeneous) thermal energy equation isDisplay Formula
(20)fRe2w̃T̃z̃=2T̃x̃2+2T̃ỹ2+1Pe22T̃z̃2+Br(fRe)24[(w̃x̃)2+(w̃ỹ)2]+q˙̃

whereDisplay Formula

(21)Br=μw¯2k(Tr+Tr)
Display Formula
(22)q˙̃=q˙Dh2k(Tr+Tr)
are the Brinkman number and the dimensionless heat generation rate, respectively. It is subjected to the following boundary conditions:Display Formula
(23)T̃ỹ=0for0<x̃<ã,ỹ=0
Display Formula
(24)T̃=0forã<x̃<d̃,ỹ=0,z̃0
Display Formula
(25)T̃=1forã<x̃<d̃,ỹ=0,z̃>0
Display Formula
(26)T̃ỹ=0for0<x̃<d̃,ỹ=1/4
Display Formula
(27)T̃x̃=0forx̃=0,x̃=d̃,0<ỹ<1/4
Display Formula
(28)T̃=0for0<x̃<d̃,0<ỹ<1/4,z̃
Display Formula
(29)T̃=1for0<x̃<d̃,0<ỹ<1/4,z̃+

The dimensionless temperature field is decomposed asDisplay Formula

(30)T̃=T̃h+T̃p

where T̃h(x̃,ỹ,z̃,H/d,ϕ,Pe) and T̃p(x̃,ỹ,H/d,ϕ,Br,q˙̃) are the homogeneous and particular solutions, respectively. Thus, T̃h satisfies the homogeneous form of Eq. (20), with viscous dissipation and heat generation absent, and T̃p satisfies Eq. (20) but with homogeneous boundary conditions.

Homogeneous Solution.

Here, we consider the homogeneous form of Eq. (20), i.e., Br and q˙̃ are set to zero. We seek solutions of the form

T̃h(x̃,ỹ,z̃)={ψ(x̃,ỹ)g(z̃)forz̃01ψ+(x̃,ỹ)g+(z̃)forz̃>0
which separate the dependence of T̃ on the streamwise coordinate z̃ from that on the transverse coordinates x̃ and ỹ. It follows thatDisplay Formula
(32)g±(z̃)=exp(λ±z̃)

and ψ±(x̃,ỹ) satisfiesDisplay Formula

(33)2ψ±=λ±(fRe2w̃+λ±Pe2)ψ±
subject to the boundary conditionsDisplay Formula
(34)ψ±ỹ=0for0<x̃<ã,ỹ=0
Display Formula
(35)ψ±=0forã<x̃<d̃,ỹ=0
Display Formula
(36)ψ±ỹ=0for0<x̃<d̃,ỹ=1/4
Display Formula
(37)ψ±x̃=0forx̃=0,x̃=d̃,0<ỹ<1/4

Therefore, (ψ+,λ+) and (ψ,λ) are solutions of the same nonlinear eigenvalue problem, given by Eqs. (33)(37). The eigenvalues are real. We assume that they are discrete and there are infinitely many and let λi and ψi denote the ith eigenvalue and eigenfunction, respectively, ordered such that <<λ2<λ1<0<λ1<λ2<+. Then, the eigensolutions for z̃>0 and z̃0 correspond to those with λi>0 and λi<0, respectively, so that there is exponential decay in the upstream (z̃) and downstream (z̃+) directions. The set of ψi and λi is determined numerically (see Appendix B) and henceforth assumed to be known.

We proceed by expressing the general homogeneous solution T̃h(x̃,ỹ,z̃) as a linear combination of the appropriate eigenfunctions in each region, i.e.,Display Formula

(38)T̃h(x̃,ỹ,z̃)={i=1ciψi(x̃,ỹ)exp(λiz̃)forz̃01i=1+ciψi(x̃,ỹ)exp(λiz̃)forz̃>0

The expansion coefficients ci follow from the requirement that both temperature and heat flux are continuous at z̃=0, for 0<x̃<d̃ and 0<ỹ<1/4, i.e.,Display Formula

(39)limz̃0T̃h=limz̃0+T̃h
Display Formula
(40)limz̃0T̃hz̃=limz̃0+T̃hz̃

Substituting Eq. (38) in Eqs. (39) and (40), the latter become, respectively,Display Formula

(41)i=,i0+ciψi=1
Display Formula
(42)i=,i0+ciλiψi=0

Given that the eigenvalue problem is nonlinear, we lack a natural orthogonality condition for the eigenfunctions ψi. Therefore, we modify the analysis by Deavours [27] to derive an orthogonality condition, to enable us to determine the expansion coefficients from Eqs. (41) and (42). First, we multiply both sides of Eq. (33) for the ith eigenvalue by λjψj to giveDisplay Formula

(43)λjψj2ψi=λjψjλi(fRe2w̃+λiPe2)ψi

Interchanging i and j in Eq. (43) and subtracting the result from Eq. (43), it follows thatDisplay Formula

(44)λjψj2ψiλiψi2ψj=λiλjPe2(λjλi)ψiψj

Using the identityDisplay Formula

(45)ψj2ψi=·(ψjψi)ψj·ψi

Equation (44) can be rewritten asDisplay Formula

(46)λj·(ψjψi)λi·(ψiψj)=(λjλi)(λiλjPe2ψiψj+ψi·ψj)

Integrating Eq. (46) over the domain yieldsDisplay Formula

(47)λj01/40d̃·(ψjψi)dx̃dỹλi01/40d̃·(ψiψj)dx̃dỹ=(λjλi)01/40d̃(λiλjPe2ψiψj+ψi·ψj)dx̃dỹ

However, employing the Divergence Theorem and utilizing Eqs. (34)(37), we can show thatDisplay Formula

(48)01/40d̃·(ψiψj)dx̃dỹ=0

Hence, from Eq. (48) and given that λjλi, Eq. (47) yieldsDisplay Formula

(49)01/40d̃(λiλjPe2ψiψj+ψi·ψj)dx̃dỹ=0,ij

or, in vector notationDisplay Formula

(50)01/40d̃[λiψiψi/x̃ψi/ỹ]T[1Pe200010001][λjψjψj/x̃ψj/ỹ]dx̃dỹ=0

Thus, the required orthogonality condition is that the vectors [λiψi,ψi/x̃,ψi/ỹ]T and [λjψj,ψj/x̃,ψj/ỹ]T are orthogonal with respect to the matrixDisplay Formula

(51)B=[1Pe200010001]

With this orthogonality relation, we can now proceed to compute the expansion coefficients ci. We multiply each vector [λjψj,ψj/x̃,ψj/ỹ]T by the corresponding expansion coefficient cj and sum the resulting expressions over all indices j. Then, it follows from Eq. (42) thatDisplay Formula

(52)j=,j0+cj[λjψjψj/x̃ψj/ỹ]=j=,j0+cj[0ψj/x̃ψj/ỹ]

Next, taking the dot product of both sides of Eq. (52) with the vector B[λiψi,ψi/x̃,ψi/ỹ]T and integrating the resulting expression over the domain, it follows thatDisplay Formula

(53)01/40d̃j=,j0+cj(λjλiPe2ψjψi+ψj·ψi)dx̃dỹ=01/40d̃j=,j0+cj(ψj·ψi)dx̃dỹ
Switching the order of integration and summation on the left-hand side of Eq. (53) and employing Eq. (49) yieldsDisplay Formula
(54)ci01/40d̃(λi2Pe2ψi2+ψi·ψi)dx̃dỹ=01/40d̃j=,j0+cj(ψj·ψi)dx̃dỹ

Then, using Eq. (45) (which is valid for i = j as well as ij), Eq. (54) becomesDisplay Formula

(55)ci01/40d̃[λi2Pe2ψi2+·(ψiψi)ψi2ψi]dx̃dỹ=01/40d̃j=,j0+cj[·(ψjψi)ψj2ψi]dx̃dỹ

Switching the order of integration and summation on the right-hand side of Eq. (55) and employing Eqs. (33) and (48) yieldsDisplay Formula

(56)ci01/40d̃(2λiPe2+fRe2w̃)ψi2dx̃dỹ=01/40d̃(λiPe2+fRe2w̃)ψi(j=,j0+cjψj)dx̃dỹ

Then, using condition Eq. (41), it follows thatDisplay Formula

(57)ci01/40d̃(2λiPe2+fRe2w̃)ψi2dx̃dỹ=01/40d̃(λiPe2+fRe2w̃)ψidx̃dỹ

Thus, rearranging Eq. (57) gives the expansion coefficients as perDisplay Formula

(58)ci=01/40d̃[w̃+2λi/(fRePe2)]ψidx̃dỹ01/40d̃[w̃+4λi/(fRePe2)]ψi2dx̃dỹ

Finally, our attention shifts to compute the integral over the ridge of ψi/ỹ|ỹ=0 that is used later in the formulation of the Nusselt number. Integrating Eq. (33) over the domain yieldsDisplay Formula

(59)01/40d̃2ψidx̃dỹ=λi01/40d̃(fRe2w̃+λiPe2)ψidx̃dỹ

Applying the Divergence Theorem on the left-hand side and utilizing Eqs. (34)(37), Eq. (59) becomesDisplay Formula

(60)ãd̃ψiỹ|ỹ=0dx̃=λi01/40d̃(fRe2w̃+λiPe2)ψidx̃dỹ

Particular Solution.

We choose the particular solution to be the solution constant in z̃ of Eq. (20) (the inhomogeneous equation) with homogeneous boundary conditions. Viscous dissipation and volumetric heat generation are considered separately and the solutions are superimposed. Therefore, we express the particular solution asDisplay Formula

(61)T̃p=Br(fRe)24T̃p,Br+q˙̃T̃p,q˙̃
The quantity T̃p,Br satisfiesDisplay Formula
(62)2T̃p,Br=|w̃|2
with boundary conditionsDisplay Formula
(63)T̃p,iỹ=0for0<x̃<ã,ỹ=0
Display Formula
(64)T̃p,i=0forã<x̃<d̃,ỹ=0
Display Formula
(65)T̃p,iỹ=0for0<x̃<d̃,ỹ=1/4
Display Formula
(66)T̃p,ix̃=0forx̃=0,x̃=d̃,0<ỹ<1/4

where i=Br. Recall that once the velocity field is computed from Eqs. (5)(9), the right-hand side of Eq. (62) is known. An important result for the formulation of the Nusselt number follows by combining Eqs. (5), (45), and (62) to show thatDisplay Formula

(67)2T̃p,Br=[·(w̃w̃)+w̃]

Then, integrating both sides of Eq. (67) over the domain, applying the Divergence Theorem utilizing the boundary conditions in Eqs. (6)(9) and (63)(66), and using Eq. (13), it follows thatDisplay Formula

(68)ãd̃T̃p,Brỹ|ỹ=0dx̃=d̃2fRe

It follows, from Eqs. (20) and (61), that T̃p,q˙̃ satisfiesDisplay Formula

(69)2T̃p,q˙̃=1
with boundary conditions given by Eqs. (63)(66) where i=q˙̃. Comparing Eq. (69) with Eq. (5), it follows that the problem for T̃p,q˙̃ is identical to the hydrodynamic problem. However, this is not the case for the second configuration considered in Appendix A, where one plate is smooth. Integrating both sides of Eq. (69) over the domain and applying the Divergence Theorem utilizing the boundary conditions in Eqs. (63)(66) yieldsDisplay Formula
(70)ãd̃T̃p,q˙̃ỹ|ỹ=0dx̃=d̃4

We note that T̃p,Br and T̃p,q˙̃ are only functions of the transverse coordinates, the aspect ratio, and the solid fraction of the domain. They are determined numerically (see Appendix C), and for the rest of the analysis, they are assumed to be known.

Local Nusselt Number.

The local Nusselt number is defined asDisplay Formula

(71)Nul±=hl±Dhk

where hl± is the local heat transfer coefficient for z̃>0 and z̃0, respectively. An energy balance at a point along the ridges yieldsDisplay Formula

(72)kTy|y=0=hl(TrTb)forz0
Display Formula
(73)kT+y|y=0=hl+(Tr+Tb+)forz>0
where Tb± is the bulk temperature of the liquid defined asDisplay Formula
(74)Tb±=2dHw¯0H/20dwT±dxdy

Substituting Eqs. (72)(74) into Eq. (71) yieldsDisplay Formula

(75)Nul=1T̃bT̃ỹ|ỹ=0forz̃0
Display Formula
(76)Nul+=1(T̃b+1)T̃+ỹ|ỹ=0forz̃>0

whereDisplay Formula

(77)T̃b=2fRed̃i=1ciexp(λiz̃)01/40d̃w̃ψidx̃dỹ+T̃p,b
Display Formula
(78)T̃b+=12fRed̃i=1+ciexp(λiz̃)01/40d̃w̃ψidx̃dỹ+T̃p,b
are the dimensionless bulk temperatures of the liquid for z̃0 and z̃>0, respectively, andDisplay Formula
(79)T̃p,b=2fRed̃01/40d̃w̃[Br(fRe)24T̃p,Br+q˙̃T̃p,q˙̃]dx̃dỹ
is the contribution to these quantities from the particular solution.

Thus, from Eqs. (30), (38), (61), and (75)(79), it follows that the local Nusselt number is given byDisplay Formula

(80)Nul±=d̃(Fl,1±Fl,2)2fRe(F3±F4)

where2Display Formula

(81)Fl,1±=i=±1±ciexp(λiz̃)ψiỹ|ỹ=0
Display Formula
(82)Fl,2=Br(fRe)24T̃p,Brỹ|ỹ=0+q˙̃T̃p,q˙̃ỹ|ỹ=0
Display Formula
(83)F3±=i=±1±ciexp(λiz̃)01/40d̃w̃ψidx̃dỹ
Display Formula
(84)F4=01/40d̃w̃[Br(fRe)24T̃p,Br+q˙̃T̃p,q˙̃]dx̃dỹ

We note that Fl,1± and Fl,2 are functions of x̃, but that F3± and F4 are not; therefore, only the former have subscript l.

The Nusselt number averaged over the composite interface isDisplay Formula

(85)Nu±=1d0dNul±dx

Combining Eqs. (80)(85) and utilizing Eqs. (60), (68), and (70) yieldsDisplay Formula

(86)Nu±=F1±F24(F3±F4)

whereDisplay Formula

(87)F1±=i=±1±λiciexp(λiz̃)01/40d̃(w̃+2λifRePe2)ψidx̃dỹ
Display Formula
(88)F2=d̃4(Br+2q˙̃fRe)

Fully Developed Nusselt Number.

In this section, our attention shifts to the asymptotic values that the Nusselt number attains in the streamwise direction as a function of the Péclet and Brinkman numbers and the dimensionless volumetric heat generation rate. Two regions can be identified where the Nusselt number does not depend on z̃. First, where aside from geometrical effects, those of Pe are dominant and, second, when those of Br and q˙̃ are dominant. First, notice that Fl,1±,F1± and F3± decay exponentially with increasing |z̃|. Comparing the two leading terms of Fl±, it follows that when |z̃||z̃Pe±|, whereDisplay Formula

(89)z̃Pe±=ln|λ±2|ln|λ±1|(λ±2λ±1)

Fl,1±,F1±, and F3± can be approximated with their leading term. Next, comparing the leading terms of F1± and F3± with F2 and F4, respectively, it follows that when |z̃|min(|z̃Br±|,|z̃q˙̃±|) whereDisplay Formula

(90)z̃Br±=1λ±1ln(1Br)
Display Formula
(91)z̃q˙̃±=1λ±1ln(1q˙̃)
respectively, F1±F2 and F3±F4.

Regions Where Pe Effects are Dominant.

From Eqs. (80)(84) and (86)(88), it follows that when |z̃Pe±||z̃|min(|z̃Br±|,|z̃q˙̃±|), the fully developed local Nusselt number (Nul,fd,Pe±) and the fully developed Nusselt number averaged over the composite interface (Nufd,Pe±) are given byDisplay Formula

(92)Nul,fd,Pe±=d̃ψ±1ỹ|ỹ=02fRe01/40d̃w̃ψ±1dx̃dỹ

andDisplay Formula

(93)Nufd,Pe±=λ±14(1+2λ±1fRePe201/40d̃ψ±1dx̃dỹ01/40d̃w̃ψ±1dx̃dỹ)
respectively.

Regions Where Br and q˙̃ Effects are Dominant.

When |z̃|max(|z̃Br±|,|z̃q˙̃±|) such that the effects of Br and/or q˙̃ are dominant, the corresponding fully developed local Nusselt number (Nul,fd,Br,q˙̃±) and fully developed Nusselt number averaged over the composite interface (Nufd,Br,q˙̃±) are found to beDisplay Formula

(94)Nul,fd,Br,q˙̃±=d̃Fl,22fReF4

andDisplay Formula

(95)Nufd,Br,q˙̃±=F24F4

respectively.

Equations (94) and (95) indicate that when |z̃|max(|z̃Br±|,|z̃q˙̃±|), the fully developed Nusselt number is independent of the Péclet number. Moreover, when q˙̃Br, Eq. (95) becomesDisplay Formula

(96)Nufd,q˙̃±=d̃8fRe01/40d̃w̃T̃p,q˙̃dx̃dỹ

and if q˙̃=0, it becomesDisplay Formula

(97)Nufd,Br±=d̃4(fRe)201/40d̃w̃T̃p,Brdx̃dỹ

Equations (96) and (97) state that when volumetric heat generation is either much larger than viscous dissipation or absent, and |z̃||z̃Br±|, the fully developed Nusselt number is only function of the geometry of the domain.

This section contains three subsections. The first two consider separately the effects of axial conduction, and of viscous dissipation and volumetric heat generation, respectively, on the fully developed (local and averaged over the composite interface) Nusselt number. The third one considers the combined effects of axial conduction and viscous dissipation on the developing Nusselt number averaged over the composite interface. The results are for the first configuration of the ridges and those for the second configuration are presented in Appendix A. When a variable (Pe,Br,q˙̃) appears as a subscript of Nu, it signifies that the corresponding physical effects are dominant in that scenario or at that streamwise location. Two subscript variables signify that both are equally important. Note, however, that a subscript variable may not appear in the corresponding Nu expression, see for example Eqs. (96) and (97).

Effects of Axial Conduction on Fully Developed Nusselt Number
Effects of ϕ and H/d for Pe=1.

Figures 4 and 5 plot the fully developed Nusselt number averaged over the composite interface versus the solid fraction ϕ for aspect ratios of H/d=1,2,4,6,10, and 100, and Pe=1. They apply when min(z̃Br,z̃q˙̃)z̃z̃Pe and z̃Pe+z̃min(z̃Br+,z̃q˙̃+), i.e., they provide Nufd,Pe and Nufd,Pe+, respectively. Recall that in this part (when it exists) of the fully developed region, the effects of Br and q˙̃ on the fully developed Nusselt number are negligible. The dashed lines correspond to smooth plates with Nusselt numbers Nufd,Pe,s=8.26 and Nufd,Pe+,s=8.01 [28], respectively. This difference between Nufd,Pe and Nufd,Pe+ was also observed by Agrawal [26] for the case of smooth parallel plates. Physically, this is expected since advection prevents symmetry arguments to be used pertaining to the upstream and downstream portions of the domain. Moreover, the difference between the computed Nufd,Pe+,s=8.01 and the corresponding value of 7.54 when Pe is a manifestation of the effects of axial conduction which provides an additional path to heat transfer as discussed in Sec. 3.1.2.

The results obey the same trends with respect to H/d and ϕ as observed in Ref. [22]. In the limit as ϕ1, Nufd,Pe±Nufd,Pe±,s, irrespective of the aspect ratio, as they should. Additionally, as ϕ0, both Nufd,Pe and Nufd,Pe+ tend to zero because the available area for heat transfer vanishes. Furthermore, and excluding the aforementioned limits, for fixed ϕ as H/d0 and H/d, both Nufd,Pe and Nufd,Pe+ tend to zero and to their corresponding counterparts for smooth plates, respectively. This is because as H/d0 and H/d, the difference between the temperature of the ridge and the mean temperature of the composite interface becomes significant and negligible, respectively, compared to the difference between the temperature of the ridge and the bulk temperature of the liquid.

Figure 6 plots the fully developed local Nusselt number (Nul,fd,Pe+) versus the normalized coordinate along the ridge (x̃ã)/(d̃ã) for Pe=1,H/d=10, and ϕ=0.01,0.1 and 0.99. The maximum and minimum values of Nul,fd,Pe+ in each case are observed at the triple contact line (x̃=ã) and at the center of the ridge (x̃=d̃), respectively. Moreover, Nul,fd,Pe+ increases with decreasing ϕ indicating a local enhancement of heat transfer due to the higher velocities of the liquid close to the ridge as ϕ0. Both trends are consistent with the previous studies [22,34]. 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 Figs. 4 and 5.

Effects of ϕ and Pe for H/d=1 and 10.

Figures 7 and 8 plot the computed Nufd,Pe and Nufd,Pe+, respectively, versus the solid fraction for Pe=0.01,1, and 10 for H/d=1. The latter also includes the Pe limit [22] for comparison. Figures 9 and 10 apply when H/d=10. The results show that as Pe0, Nufd,Pe,s approaches Nufd,Pe+,s and they become approximately equal to 8.12 [28]. This is expected as in this limit the primary mode of heat transfer is conduction and thus the problem becomes antisymmetric with respect to z̃=0, where T̃=0.5.

Comparing Figs. 7 and 9 with Figs. 8 and 10, respectively, shows that as the Péclet number increases, Nufd,Pe and Nufd,Pe+ respond differently. Nufd,Pe tends to infinity as Pe increases because the temperature field for z̃0 becomes essentially isothermal and thus T̃0 and T̃/ỹ|ỹ=00. Note that despite the fact that Nufd,Pe tends to infinity in this case, there is no heat transfer from the ridge to the liquid given that T̃/ỹ|ỹ=00. This behavior is consistent with the trends observed by Agrawal [26] for the case of smooth isothermal plates. Contrary, Nufd,Pe+ decreases as Pe increases because the axial conduction enhancement to heat transfer is reduced, and in the limit Pe, Nufd,Pe+,s tends to finite values. These trends are reversed, however, when only one plate is textured with isothermal ridges and the other one is smooth and adiabatic. The slower velocity field in this case causes T̃/ỹ|ỹ=0 to tend to zero faster than T̃ does [22] and therefore Nufd,Pe0, as Pe as per the corresponding results in Appendix A. Also, the adiabatic boundary condition along the smooth plate leads to convection dominated heat transfer and thus Nufd,Pe+ increases as Pe increases with Nufd,Pe+ tending to finite values as Pe. For both ridge configurations, the change of Nufd,Pe and Nufd,Pe+ for an increase of the Péclet number is small for Pe<1 as in this region the heat transfer is predominantly diffusive, but the change becomes large when Pe>1 and advection becomes important.

Finally, comparing Figs. 7 and 8 with Figs. 9 and 10, respectively, it follows that the effects of Péclet number become important as the solid fraction increases, and for Nufd,Pe+, the range of values of ϕ where change is observed increases with H/d. Moreover, the effects are more pronounced on Nufd,Pe which, as explained earlier, has a stronger dependence on Pe than Nufd,Pe+.

Effects of Viscous Dissipation and Volumetric Heat Generation on Fully Developed Nusselt Number.

The computed fully developed Nusselt numbers averaged over the composite interface when |z̃|max(|z̃Br±|,|z̃q˙̃±|), Nufd,Br± and Nufd,q˙̃±, are presented in Figs. 11 and 12, respectively. The results present the same trends with respect to ϕ and H/d as those described for Nufd,Pe±, i.e., irrespective of H/d as ϕ0, both Nufd,Br± and Nufd,q˙̃± tend to zero and as ϕ1,Nufd,Br±Nufd,Br±,s=17.5 [29], and Nufd,q˙̃±Nufd,q˙̃±,s=10 [31], respectively.

Combined Effects of Axial Conduction and Viscous Dissipation on Developing Nusselt Number.

Here, we present results for the combined effects of the Péclet and Brinkman numbers on the Nusselt number averaged over the composite interface for z̃>0(Nu+). Figure 13 presents Nu+ versus the dimensionless streamwise coordinate for ϕ=0.01,H/d=10, Pe=1,q˙̃=0 and for three different values of the Brinkman number, namely Br=2.71×105,2.71×108 and as Br0. The second value of Br is relevant to flow of liquid metals through textured microchannels [3]. In this figure, we can identify the two asymptotic values of Nu+. First, as z̃ increases and becomes larger than z̃Pe+, Nu+ approaches Nufd,Pe+=4.33. Then, as z̃ continues to increase, the effects of the step change of the ridge temperature decay significantly and become of the same order as the viscous dissipation effects. Thus, in the region where z̃z̃Br+,Nu+ starts to increase until z̃z̃Br+ where Nu+Nufd,Br±=8.92. Also, as Br increases, the location of the transition moves further upstream but its limiting value remains 8.92. The same trends were reported in Ref. [30] for the case of smooth plates. Figure 14 presents Nu+ versus z̃ for the same domain geometry and Br as in Fig. 13, but for Pe=10. Comparing Figs. 13 and 14, we see that as Pe increases, the transitions of Nu+ to Nufd,Pe+ and then to Nufd,Br± occur further upstream given that the flow becomes thermally developed faster with increasing Pe.

We considered the extended Graetz–Nusselt problem, i.e., hydrodynamically developed and thermally developing flow with finite axial conduction, for the case of textured plates (or plate) with isothermal parallel ridges. We developed semi-analytical expressions for the Nusselt number in an infinite domain, before and after a jump in ridge temperature. Effects of viscous dissipation and volumetric heat generation were included. Two different configurations for the ridges were analyzed: (1) both plates textured and (2) one plate textured and the other one smooth and adiabatic. The menisci between the ridges were considered to be flat and adiabatic. The solid–liquid interfaces and the menisci were subjected to no-slip and no-shear boundary conditions, respectively. Using separation of variables, we expressed the homogeneous part of the solution 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 nonlinear eigenvalue problem from which the eigenvalues and eigenfunctions follow numerically. The particular solution accounting for viscous dissipation and volumetric heat generation is also determined numerically.

The derived expressions for the local Nusselt number and the Nusselt number averaged over the composite interface indicate that the Nusselt number is a function of the transverse (along the ridge) and streamwise coordinates, the aspect ratio of the domain, the solid fraction of the ridges, the Péclet and Brinkman numbers, and the dimensionless volumetric heat generation rate. Expressions were also derived for the fully developed local Nusselt number and for the fully developed Nusselt number averaged over the composite interface. Two asymptotic limits were identified for the fully developed Nusselt number and expressions were derived to estimate the streamwise locations where they occur. The first limit is relevant to the effects of axial conduction, and the corresponding fully developed Nusselt number is a function of the geometry and the Péclet number. The second limit is relevant to viscous dissipation and volumetric heat generation effects, and the corresponding fully developed Nusselt number is a function of the geometry, the Brinkman number, and the dimensionless volumetric heat generation rate. If volumetric heat generation is absent, the aforementioned Nusselt number is a function of the geometry only.

The results indicate that the Nusselt number averaged over the composite interface decreases as the aspect ratio and/or the solid fraction decreases. Moreover, the fully developed Nusselt number averaged over the composite interface in the region after the temperature change tends to a finite value as the Péclet number tends to infinity for both geometries studied. On the contrary, in the region before the temperature change, the fully developed Nusselt number averaged over the composite interface tends to infinity when both plates are textured with isothermal ridges, and to zero when one plate is smooth and adiabatic, as the Péclet number tends to infinity.

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 computational fluid dynamics solver. More importantly, the analysis provides semi-analytical expressions to evaluate the local Nusselt number and the Nusselt number averaged over the composite interface at any location, which are prohibitively expensive to compute using a general computational fluid dynamics code.

The work of TK was supported by an EPSRC-UK doctoral scholarship. The computations in this paper were run on the Tufts High-performance Computing Research Cluster at Tufts University.

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