Wall Conduction Effects in Laminar Counterflow Parallel-Plate Heat Exchangers with Small-Scale Wall Corrugations

Parallel-plate heat exchangers are widely used in chemical, pharmaceutical, food process‐ ing, and many other industrial applications. More recently, they have also found applica‐ tion in a variety of emerging branches of thermal engineering. They are currently used in miniaturized reaction systems involving heterogeneously catalyzed gas-phase reactions [1], in thermoelectric generators that convert low-grade thermal energy into electrical power [2, 3], and in thermoacoustic engines [4]. In addition, they are a key component of many cryogenic systems [5,6,7].


Introduction
Parallel-plate heat exchangers are widely used in chemical, pharmaceutical, food processing, and many other industrial applications. More recently, they have also found application in a variety of emerging branches of thermal engineering. They are currently used in miniaturized reaction systems involving heterogeneously catalyzed gas-phase reactions [1], in thermoelectric generators that convert low-grade thermal energy into electrical power [2,3], and in thermoacoustic engines [4]. In addition, they are a key component of many cryogenic systems [5,6,7].
Progress in the analysis of parallel-plate heat exchangers has been significant in the last decades due to their simple geometry and well established flow conditions [8]. In particular, the analysis of the steady-state laminar heat transfer between different streams coupled through compatibility conditions at the boundaries constitutes the so-called conjugated Graetz problem. Under certain simplifying assumptions, i.e., constant property fluids and fully developed laminar flow, the problem becomes linear and is amenable to an elegant solution based on eigenfunction expansions, which in counterflow systems involves sets of positive and negative eigenfunctions associated with sets of positive and negative eigenvalues [9,10,11,12].
It is well known that the use of corrugated plates is a suitable method for increasing the thermal performance of heat exchangers and provide higher compactness [13]. In many practical applications the flow is turbulent, and the corrugated surface has a significant effect on the enhancement of heat transfer rates. Because of the presence of highly unsteady recirculation zones, the heat transfer coefficients obtained with corrugated walls are higher than those with flat plates. However, the pressure drop is also significantly higher.
In some other cases the Reynolds number remains small due to the small dimensions of the system, hence laminar flow conditions prevail. In this case the corrugation troughs exhibit lowvelocity swirling cells [14] that are both weak and steady. As a result, the convective heat enhancement process is poor and the effect on heat exchanger performance is much smaller. For certain corrugation shapes and/or flow conditions no enhancement of the heat transfer rate may even occur [15]. Thus, even if the heat transfer surface between the wall and the fluid is increased, the global heat transfer effectiveness may not overcome that of a flat wall channel of comparable size [16]. This difficulty has motivated a considerable amount of work on methods to increase the rates of heat and mass transfer in compact heat exchangers operated in the laminar flow regime.
Durmuş et al. [17] demonstrated that under laminar flow conditions the heat exchanged by a plate heat exchanger with a corrugated sinusoidal surface is higher than for other geometries (flat plate and asterisk plate), although corrugated plate heat exchangers exhibit also higher pressure drops. Specifically, the pressure drop in corrugated channels is higher than in flat channels of the same external size. Therefore, a trade-off exists between the heat gained and the pressure drop, because increasing the heat exchanger efficiency leads to smaller system dimensions, which reduces production costs, whereas larger pressure drops result in increased operating costs.
The aim of this work is to extend recent analyses of laminar counterflow parallel-plate heat exchangers [12,18,19] to quantify the effect of small scale wall corrugations. In particular, the amplitude and wavelength of the corrugations will be assumed to be comparable to each other, but small compared to the channel half-widths. Although such separation of scales hardly occurs in practical devices, it enables the problem to be solved by a multi-scale asymptotic approach similar to that proposed by Woollard et al. [20] for mass transfer in channels with small-scale wall corrugations. In this case the problem exhibits two main asymptotic regions: a bulk flow region where the walls appear flat at leading order, and a wall region where there is a full interaction between convection and conduction in the fluids, coupled by transverse and longitudinal conduction in the (thin) corrugated wall.
The multi-scale asymptotic analysis proceeds in two steps: • On the large (i.e., bulk flow) scale, a theoretical study of laminar counterflow parallel-plate heat exchangers [19] provides the temperature field as an infinite series of eigenfunctions associated with sets of positive and negative eigenvalues. The analysis, which ignores the effects of axial conduction both in the fluids and in the plates, is used to construct an approximate two-term solution that incorporates the effect of higher order eigenfunctions through apparent temperature offsets introduced at the inlet/outlet sections [12]. The approximate solution yields an accurate description of the temperature field away from the thermal entrance regions, thereby enabling simplified expressions for the wall and fluid temperatures, local Nusselt numbers, and overall heat-transfer coefficient in terms of the wall thermal resistance.
• On the small (i.e., wall corrugation) scale, the shear-induced flow of two Newtonian fluids that exchange heat through a thin corrugated wall is studied numerically, retaining both transverse and longitudinal conduction effects in the heat conducting wall. The analysis shows that far from the wall the effect of the corrugations is equivalent to that of a thickened virtual plane surface with partial slip along the corrugations [21,22]. This virtual plane wall exhibits an apparent thermal resistance that incorporates i) the intrinsic thermal resistance of the corrugated wall, ii) an additional resistance due to the apparent wall thickening, and iii) a negative contribution associated with the heat-transfer enhancement induced by the corrugations (due to the enlarged heat exchange surface area and, when the wall conductivity is larger than those of the fluids, also to the longitudinal heat conduction along the wall). Apparent wall thicknesses and thermal resistances will be presented here for the reference case of sinusoidal corrugations.
Evaluating the approximate solution of the bulk flow model presented in Section 2 with the apparent thermal resistance derived from the wall corrugation model of Section 3, a physically sound multi-scale description of laminar counterflow parallel-plate heat exchangers with small-scale wall corrugations is proposed in Section 4. Illustrative results are presented in Section 5, and some conclusions are drawn in Section 6.

Bulk flow model
In this section we analyze the heat exchange process between two constant-property fluids flowing through a multilayered counterflow parallel-plate heat exchanger composed by a relatively large number of channels. A detailed account of this problem can be found elsewhere [19], but we shall summarize here its main aspects for completeness of the presentation. The flow channels are separated by flat plates of finite thickness, δ w , and thermal conductivity, k w . The heat conducting plates allow the exchange of heat through a section of length L. Figure  1 shows a sketch of the theoretical configuration under study, showing the coordinate system, the velocity profiles, the boundary conditions, and the physical properties of fluids i = 1, 2 (namely, fluid density, ρ i , heat capacity, c i , and thermal conductivity, k i ).  The Reynolds numbers, Re i = 2a i V i /ν i , based on the channel half-width, a i , average flow velocity, V i , and kinematic viscosity, ν i , of fluid i, are assumed to be sufficiently small for the flows to remain laminar and steady [23]. The Prandtl numbers, Pr i = ν i /α i , based on the thermal diffusivity, α i = k i /(ρ i c i ), of fluid i, are also assumed to be moderately large compared to unity, a good approximation for most non-metallic liquids in applications. In this case, the thermal entry length is large compared to the hydrodynamic entry length, and the flow can be assumed to be a fully developed Poiseuille flow independent of the temperature field, , where the + (-) sign holds for fluid 1 (2). As a result, we shall assume that the Peclet numbers of the flow in the channels, Pe i = Re i Pr i = 2a i V i / α i , are both large compared to unity. Then, axial heat conduction can be neglected in both fluids in first approximation, and we can assume that the inlet temperature of fluid i is uniform and equal to T i,in .
To write the problem in non-dimensional form, we introduce the dimensionless axial and transverse coordinates ξ = X/(Pe 1 a 1 ) and y i = Y i /a i , and the normalized temperature θ i = (T i -T 1,in )/ (T 2,in -T 1,in ). Then, under the assumptions stated above, the energy equation reduces, in first approximation, to a balance between axial convection and transverse conduction These equations are to be integrated with adiabatic boundary conditions at the channel symmetry planes the condition of continuity of wall-normal heat fluxes at the heat conducting wall and the inlet conditions It should be noted that in writing the compatibility condition at the heat conducting wall (3) we have ignored the effect of axial conduction in the wall, which constitutes a good approximation provided the dimensionless wall thickness δ w /a 1 remains sufficiently small [19].
It is interesting to note that the inverse of κ w , which appears explicitly in Eq. (3) represents the dimensionless wall thermal resistance, 1/κ w = (δ w /a 1 )/(k w /k 1 ), defined as the ratio of the dimensionless wall thickness, δ w /a 1 , to the dimensionless heat conductivity, k w /k 1 ; the dimensionless wall temperature jump being proportional to 1/κ w . Note also that the product of m and k represents the ratio of heat capacity flow rates, mk = (2a 2 V 2 ρ 2 c 2 )/(2a 1 V 1 ρ 1 c 1 ), which will hereafter be assumed, without loss of generality, to be greater than one, so that fluid 1 has the lowest heat capacity flow rate [12].
where the eigenfunctions are given by In the above expressions, f n,i (y i ) represent the eigenfunctions of the thermally thin wall limit κ w → ∞ (or 1/κ w → 0) which can be expressed in terms of Whittaker functions as shown by Vera & Liñán [12]. These written can be derived analytically and evaluated at y i = 1 to give the derivative of the eigenfunctions at the heat conducting wall  where κ n,1 = 3λ n / 8, κ n,2 = i 3mλ n / 8, Γ(z) is the complex Gamma function and i is the imaginary unit. Using this expression in Eq. (9) it is possible to compute the eigenvalues λ n for given values of m, k, and κ w .
As pointed out by Nunge and Gill [9], the exact solution to the problem stated in Eqs. (1) -(4) involves an infinite set of linear equations for the coefficients A and C n involved in the series expansion (7), which must be truncated to a finite number of terms in order to get an approximate numerical solution. An alternative approach, initially proposed by Vera and Liñán [12] in the thermally thin wall limit and recently extended by Quintero et al. [19] to the case of nonzero wall thermal resistance, proceeds by constructing an approximate two-term solution that retains only the first two eigenvalues (0 and λ 0 ) but still incorporates the effect of higher order terms through apparent temperature offsets Δ i at the inlet section of fluid i. This leads to analytic expressions for the outlet bulk temperatures of the form  (11) where Δ i represents the inlet bulk temperature offset of fluid i associated with the neglected higher-order eigenfunctions at both ends of the heat exchanger, given by with 0,1 0 For given values of m, k, κ w , and ξ L , the above expressions provide closed-form approximate expressions for the outlet bulk temperatures of both streams in terms of the computed values of λ 0 , g ′ 0,1 (1), and 0 0,1 0 0, 0 2 0 Heat Transfer Studies and Applications which are to be obtained numerically from Eqs. (8) - (10). As shown by Quintero et al. [19], the approximate two-term solution provides accurate representations for the temperature field away from the thermal entrance regions, thereby enabling simplified expressions for the wall and bulk temperature distributions, local Nusselt numbers, and overall heat-transfer coefficient. Figure 2 shows the variation of the outlet bulk temperature of fluid 1, θ m1,out , with the dimensionless wall thermal resistance, 1/κ w , for increasing values of the heat exchanger length, ξ L = [0.125, 0.25, 0.5, 1, 2]. As previously discussed, although m and k take different values in the left and right plots, in both cases mk = (ṁ 2 c 2 ) / (ṁ 1 c 1 ) > 1, so that θ m1,out coincides by definition with the heat exchanger effectiveness.
The model results indicate that the heat exchanger effectiveness increases monotonously with the dimensionless heat exchanger length, ξ L , and decreases with the wall thermal resistance, 1/κ w . The results presented in Figure 2 were obtained from the numerical integration of Eqs.
(1) -(4) (solid lines) and from the approximate two-term solution presented above (symbols). As can be seen, the agreement is excellent even for the shortest heat exchangers under study, with errors below 1% for ξ L ≥ 0.25 [19]. This demonstrates that the approximate two-term solution constitutes a fast evaluation tool that could be readily used for the optimization of heat exchanger performance. The aim of the following sections is to extend the present analysis to the study of laminar counterflow heat exch with small-scale wall corrugations. It will be shown that when the corrugation wavelength and amplitude ar compared to the channel half-widths, the effect of the corrugated wall on the bulk flow is similar to that of a th virtual plane surface with an apparent thermal resistance that has to be obtained numerically. We shall begin in S investigating the local conjugate heat-transfer problem that provides this apparent thermal resistance for a fix design, characterized by its shape, thickness, and thermal conductivity, and specified values of the Reynolds and numbers of both streams. The analysis will provide the apparent values of the dimensionless parameters m * , k * ,  ξL * that should be used in the bulk flow model to incorporate the effect of small-scale wall corrugations and evalu corresponding changes in heat exchanger performance.

Wall corrugation model
In this section, we analyze the incompressible two-dimensional shear-induced flow of two constant property flui 2, that exchange heat through a thin corrugated wall. The wall thickness, δw, is assumed to be small compared corrugation wavelength, λ, so that the dimensionless wall thickness, Δw = δw /λ 1, emerges naturally as parameter. In addition, the Reynolds numbers of the shear flows at both sides of the wall, Reτi = ρiλ 2 τi /μi 2 , based shear stress far from the wall, τi, the corrugation wavelength, λ, and the fluid density, ρi, and viscosity, μi, of flu also assumed to be sufficiently small for the flows to remain laminar and steady.
The mathematical formulation of the problem can be written in non-dimensional form using the corrugation wav λ as appropriate length scale. The resulting Cartesian dimensionless coordinates (x, y, z) are shown in Figure 3. T The aim of the following sections is to extend the present analysis to the study of laminar counterflow heat exchangers with small-scale wall corrugations. It will be shown that when the corrugation wavelength and amplitude are small compared to the channel half-widths, the effect of the corrugated wall on the bulk flow is similar to that of a thickened virtual plane surface with an apparent thermal resistance that has to be obtained numerically. We shall begin in Section 3 investigating the local conjugate heat-transfer problem that provides this apparent thermal resistance for a fixed wall design, characterized by its shape, thickness, and thermal conductivity, and specified values of the Reynolds and Prandtl numbers of both streams. The analysis will provide the apparent values of the dimensionless parameters m * , k * , κ w * , and ξ L * that should be used in the bulk flow model to incorporate the effect of small-scale wall corrugations and evaluate the corresponding changes in heat exchanger performance.

Wall corrugation model
In this section, we analyze the incompressible two-dimensional shear-induced flow of two constant property fluids i = 1, 2, that exchange heat through a thin corrugated wall. The wall thickness, δ w , is assumed to be small compared to the corrugation wavelength, λ, so that the dimensionless wall thickness, Δ w = δ w /λ << 1, emerges naturally as a small parameter. In addition, the Reynolds numbers of the shear flows at both sides of the wall, Re τi = ρ i λ 2 τ i /μ i 2 , based on the shear stress far from the wall, τ i , the corrugation wavelength, λ, and the fluid density, ρ i , and viscosity, μ i , of fluid i, are also assumed to be sufficiently small for the flows to remain laminar and steady.
The mathematical formulation of the problem can be written in non-dimensional form using the corrugation wavelength λ as appropriate length scale. The resulting Cartesian dimensionless coordinates (x, y, z) are shown in Figure 3. The wall geometry is assumed to be a periodic function of the x-coordinate, while the y-coordinate is defined normal to the mean plane of the corrugated wall pointing from fluid 1 to fluid 2. The z-coordinate is perpendicular to the x-y plane shown in the figure.  For illustrative purposes, we define the corrugation shape using the parametric representation a g a a p a a a p adopted from [24]. Here, A ≥ 0 measures the amplitude of the corrugations, A/2π being an effective amplitude-to-wavelength ratio, and −1 ≤ γ ≤ 1 represents a suitable scallop parameter.
As previously discussed, we further assume that the dimensionless wall thickness is small, Δ w << 1, as often occurs in the applications. The surface thus defined is periodic and symmetric with respect to x, with the x-axis located midway between the highest and lowest points of the corrugations, and with values of y w (α) ranging between ±A/2π. The parameter γ has a strong influence on the corrugation shape. Thus, for γ = 0 the surface adopts a wavy (i.e., sinusoidal) shape devoid of any irregularities, whereas for 0 < |γ| < 1 it develops a scalloped geometry that becomes more and more pronounced for increasing values of |γ|, with downward (upward) cusps appearing in the surface as γ → 1 (−1). Note that the amplitude parameter A also represents the maximum slope of the sinusoidal surface y w = (A/2π)cos(2πx) obtained for γ = 0.
The velocity of fluid i is measured with the characteristic shear velocity λτ i /μ i to give the dimensionless velocity field u = (u, v, w). The variations in pressure with respect to its unperturbed value far from the wall are scaled in fluid i with the corresponding shear stress, τ i , to give the dimensionless pressure, p. Finally, the characteristic temperature qλ/k 1 , based on the corrugation wavelength, λ, the heat conductivity of fluid 1, k 1 , and the overall heat flux, q, transferred from fluid 2 to fluid 1, is used to define the dimensionless temperature T. Note that the thermal coupling through the heat conducting wall enforces the use of a common temperature scale for both fluids.
Assuming that the pressure, velocity, and temperature fields are translation invariant along the z-axis, the continuity, momentum, and energy conservation equations take the form where viscous dissipation has been neglected in the energy balance and Pr i = ν i /α i = μ i /(k i /c i ) denotes again the Prandtl number of fluid i. The above equations must be integrated in the periodic domain 0 < x < 1 for fluids 1 and 2, located, respectively, below (−∞ < y < y w -) and above (y w + < y < ∞) the wall. Hereafter we shall use y w to denote the wall position, y w ± representing the lower (-) and upper (+) wall surfaces.
As boundary conditions we impose periodicity at x =0 and 1, the no-slip boundary condition at the wall and a uniform shear flow far from the wall 2 2 sin , 0, cos , In the far field boundary conditions θ i denotes the angle of misalignment between the wall corrugations and the uniform shear imposed on fluid i far from the wall. So that θ i =0, π, and θ i = ±π/2 correspond, respectively, to flows parallel (u = v = p = 0) and transverse (w = 0) to the corrugations, with θ 2 = θ 1 and θ 2 = θ 1 + π corresponding, respectively, to co-current and countercurrent flow arrangements.
We also impose the overall heat flux transferred by conduction from fluid 2 to fluid 1 by specifying a uniform temperature gradient in fluid 2 far from the wall 1 1: , where K = k 2 /k 1 denotes the fluid-to-fluid heat conductivity ratio, whereas in fluid 1 we impose a linear temperature profile of the form 1: .
Note that, due to the invariance of the problem under arbitrary shifts in temperature, we can always add a constant to the temperature distribution of fluid 1 so as to set its apparent value at the mean plane (y = 0) of the corrugated wall equal to zero.
In the thin wall limit considered here, Δ w << 1, the jump conditions for the temperature and the wall-normal heat fluxes at the heat conducting wall can be written as where T w ± = T (x w (α), y w ± (α)) denotes the temperature at the upper (+) and lower (-) wall surfaces, a a a a n + - are the local wall-normal heat fluxes transferred by conduction from fluid 2 to the wall, and from the wall to fluid 1, respectively. In the above expressions, n represents the local wallnormal coordinate pointing from fluid 1 to fluid 2, and s is the arc-length coordinate measured along the wall, defined by In addition to the dimensionless wall thickness, Δ w , the jump conditions (22) involve the wallto-fluid heat conductivity ratio, K w = k w /k 1 , as an additional dimensionless parameter; the wall temperature difference being proportional to Δ w /K w , whereas the jump in wall-normal heat fluxes is proportional to Δ w K w . As a result, in the formal limit Δ w → 0 both Δ w and K w disappear from the formulation and the jump conditions (22) reduce to the continuity of temperatures, T w + = T w − , and wall-normal heat fluxes, ν w Inspection of the mathematical problem stated above shows that the steady conjugate heat transfer between two shear-induced flows separated by a corrugated wall can be characterized by the Reynolds and Prandtl numbers of the two fluid streams, Re τi and Pr i , the fluid-to-fluid and wall-to-fluid heat conductivity ratios, K = k 2 /k 1 and K w = k w /k 1 , the dimensionless wall thickness, Δ w = δ w /λ << 1, the angles of misalignment between the imposed shear and the direction of the corrugations, θ i , and the dimensionless wall geometry, determined here by the parameters A and γ. Thus, for a fixed wall geometry the solution of the problem depends on the nine dimensionless parameters: Re τ1 , Re τ2 , Pr 1 , Pr 2 , K, K w , Δ w , θ 1 , and θ 2 .
In addition to the velocity, pressure, and temperature fields, the solution of the problem determines the virtual no-slip plane wall location for the flows parallel (y wi ∥ ) and transverse (y wi ⊥ ) to the corrugations  (25) as well as the apparent temperature jump at y = 0 The conjugate heat transfer problem stated in Eqs. (16) - (22) can be solved numerically using standard finite-difference or finite-volume methods. A closer inspection reveals that the transverse flow problem (u, v, p) is uncoupled from the longitudinal flow (w) and thermal (T) problems, so that it can be integrated first; the resulting velocity field being required to evaluate the convective terms in the equations for w and T. Since the transverse flow field is two-dimensional, a vorticity-streamfunction formulation was adopted for the numerical integration, thereby avoiding the determination of the pressure field, which is not required here. Fourth-order finite differences were used for the spatial discretization, resulting in a five-point explicit scheme. An algebraic non-orthogonal coordinate transformation was previously employed to map the complex fluid domain onto a square computational domain with uniform grid spacing, thereby enabling a simple treatment of the wall curvilinear boundaries. The far field boundary conditions (18) − (21) were imposed sufficiently far from the wall to ensure the independence of the solution on the size of the numerical domain. The solution was then marched in pseudo-time until a converged (i.e., steady) solution was obtained. A grid independence study was carried out to ensure at least 4 significant digits in the reported values of y wi ∥ , y wi ⊥ , and ΔT.
Although the formulation presented above is of great generality, the large number of parameters involved makes unfeasible a thorough parametric study. As a result, the analysis will be restricted here to the case of two-dimensional counter-current flows, θ 2 = -θ 1 = π/2, transverse to sinusoidal corrugations, γ = 0.     Figure 4 results in an apparent thickening of the virtual plane wall for increasing values of Re τi . On the other hand, for decreasing Re τi the value of y wi ⊥ approaches the asymptotic (i.e., Stokes-flow) limit provided by the semi-analytical solution of Luchini et al. [25], recalculated here as an additional check of the accuracy of the numerical method. Figure 5 (bottom) shows the influence of the Reynolds number Re τi on the apparent temperature jump ΔT at y = 0, defined by Eq. (26). As can be seen, for low wall conductivities ΔT takes positive values due to the correspondingly large wall thermal resistance. However, for high wall conductivities ΔT becomes negative due to the effect of longitudinal heat conduction along the wall, which provides an efficient way of transferring heat from the corrugation crests to the troughs, making the wall to behave as an effective thermal shortcut. This mechanism, which is absent for K w << 1, becomes dominant for K w >> 1, leading to the negative values of ΔT reported in the last plot of Figure 5. It is also interesting to note that although ΔT tends to decrease for increasing values of Re τi due to the enhancement of heat transfer by convection, in poorly conductive walls this trend is partially reversed for Re τi in the range 100 -200, when the downstream stagnation points of the swirling cells that develop at each side of the wall come closer to each other; see, e.g, Figure 4.

Combined multi-scale model
The virtual plane wall concept introduced in Section 3 can be used in numerical simulations of corrugated-plate, or -tube, heat exchangers to account for the effect of small-scale wall corrugations in a computationally effective way. According to the results presented above, the corrugated wall behaves as a virtual plane wall of apparent thickness with an apparent wall temperature jump This virtual wall separates two fluid channels of apparent half-widths smaller than the average channel half-widths, a i , as a result of the apparent wall displacement introduced by the corrugations, which effectively reduces the hydraulic diameter of the channels. From here on, all apparent quantities obtained from the application of the virtual plane wall concept will be denoted by a superscript asterisk ( * ).
Note that the dimensionless corrugation wavelength, Λ i = λ/a i , appearing explicitly in Eq. (29) must be small compared to unity for the present multi-scale asymptotic analysis to hold. Otherwise the flow perturbations induced by the corrugations would significantly alter the flow in the channels, which will no longer resemble a planar Poiseuille flow, as assumed in Section 2.
Using Eq. (29) to evaluate the apparent channel half-widths, we obtain the apparent value of parameter a a a k k a a a a y a (30) The apparent value of κ w * = (k w / δ w ) * (a 1 * / k 1 ) can be determined from the dimensional apparent wall temperature jump (qλ / k 1 )ΔT * , where ΔT * is given by Eq. (28) In the above expression we have expressed κ w * in terms of the normalized apparent wall a quantity defined so that it tends to unity as the amplitude of the corrugations tends to zero (A → 0). In this limiting case, when y w,1 ⊥ also tends to zero, Eq. (31) reduces to κ w * = κ w ≡ (k w / δ w )(a 1 / k 1 ), and the original expression for κ w given in Eq. (5) for a flat wall of finite thickness, δ w , and thermal conductivity, k w , is recovered. Note also that, by definition, the values of κ w , K w , Δ w << 1, and Λ << 1 verify the following relation  which provides κ w explicitly in terms of K w , Δ w , and Λ. Figure 7 shows the influence of the Reynolds number, Re τi , on the normalized apparent wall temperature jump, ΔT , for the same values of K, Κ w and Δ w considered in Figure 5. As can be seen, for low-conductive walls ΔT remains slightly below unity regardless of the Reynolds number. In this case, the relatively high conductivity of the recirculating fluids combined with the higher surface area of the corrugated wall results in a smaller apparent wall temperature jump as compared to the originally flat wall. On the other hand, for high-conductive walls ΔT takes values well above unity, indicating that in this case the presence of low-conductive recirculating fluids within the corrugation troughs significantly increases the apparent wall temperature jump. In both cases, ΔT exhibits a non-monotonous behavior with Re τi that ends in an abrupt reduction for Reynolds numbers Re τi above 300 or so.  For the plane Poiseuille flow considered in this work, the wall shear stress is given by τ i = 3μ i V i /a i , which upon substitution in the expression for Re τi yields the relation stating that the ratio between the Reynolds numbers Re τi and Re i equals 3/2 times the square of the dimensionless wall corrugation wavelength, Λ i = λ/a i << 1. Since in plane Poiseuille flow finite-amplitude disturbances are known to drive transition to turbulence at Reynolds numbers as low as Re i,crit ≅ 1330 [23], the above expression severely limits the maximum value of Re τi compatible with steady laminar flow in the channels. Thus, for the moderately large value of Λ i considered below (Λ i = 0.25 << 1) the maximum allowable value of Re τi turns out to be 125. From the results shown in Figure 7, it is clear that below this critical Reynolds number the normalized wall temperature jump ΔT remains almost constant, with convective effects accounting only for small variations of ΔT of order 1% for Κ w = 0.1 and 7% for Κ w = 100. Thus, to get a first idea of the effect of the corrugations on the performance of the heat exchanger we may simply evaluate κ w * using Eq. (31) with the value of ΔT corresponding to Re τi = 0.
To quantify this effect, let us consider the ratio between the overall heat transfer rate, ṁ c 1 (T m1,out − T 1,in ), and the mechanical power required to drive the flow, characterized here by the pumping power, ΔP 1 Q 1 , delivered to fluid 1. Normalizing this ratio with that of a flat-wall heat exchanger, we introduce the overall augmentation heat-transfer number where T m1,out denotes the outlet bulk temperature of fluid 1. As previously discussed, this temperature is closely related to the heat exchanger effectiveness, ε = (T m1,out -T 1,in )/(T 2,in -T 1,in ), when fluid 1 has the smallest heat capacity flow rate. For the plane Poiseuille flow considered in this work the pressure drop, ΔP i , the volume flow rate, Q i , and the mass flow rate, ṁ , are given respectively by Using these expressions in Eq. (35), the overall augmentation heat-transfer number can be expressed as A thorough optimization of the heat exchange process requires also comparison of the entropy generation rates achieved with corrugated and flat walls. As a first attempt to use the entropy minimization method [26,27,28] for the thermodynamic optimization of corrugated-wall heat exchangers, we introduce the augmentation entropy generation number defined as the ratio of the entropy generation rates with corrugated and flat plates. In the above expression Ṡ / ṁ 1 c 1 denotes the dimensionless entropy generation rate, which is approximated by } m1,out m2,out m1,out m2,out 1 2 2 2 The remaining two apparent parameters of the bulk flow model, i.e., m * and ξ L * , involve by definition the Peclet numbers of the fluid streams. To evaluate their apparent values we must therefore specify the apparent Peclet numbers, Pe i * , in the heat exchanger with corrugated walls. To this end, we shall consider three different flow constraints, namely: i) constant flow rate, ii) constant pumping power, and iii) constant pressure drop.

i. Constant flow rate
To ensure that the volume flow rate Q i = 2a i V i of fluid i is the same with flat and corrugated walls, the average flow velocity V i * must grow to compensate for the reduction of the channel half-width 1 * * * * w , In this case, the apparent value of Pe i , defined with the apparent velocities and channel halfwidths, remains unchanged while the apparent values of m and ξ L are given by w ,2 2 w * * * * * * 2 2 2 2 , 2 2 * * * 1 1 1 a a a m m m k mk a a a a y a (43)

Constant pumping power
The pumping power is equal to the product of the volume flow rate, 2a i V i , by the pressure drop, 3μ i LV i /a i 2 . To ensure that the pumping power remains the same with flat and corrugated walls, the average flow velocity V i * of fluid i must satisfy the condition w , In this case, the apparent values of Pe i , m, and ξ L are given by   , and, using (29) and (46)  . lae In this case, the apparent values of Pe i , m, and ξ L are given by , and, using (29) and (51)

Results and discussion
In in terms of the single small parameter Λ = λ / a 1 = λ / a 2 ≪ 1 representing the dimensionless corrugation wavelength. In this case, the expressions for m * , k * , κ w * , and ξ L * , as well as those for the heat-transfer and entropy augmentation numbers R a and N a , reduce to those given in Table  1. Table 2 shows the variation of κ w * with the dimensionless corrugation amplitude, A/2π, the dimensionless wall thickness, Δ w , and the wall-to-fluid and fluid-to-fluid heat conductivity ratios, K w and K. The results are presented in terms of the ratio κ w considered, ranging from low to moderate amplitudes and from low-to high-conductive materials. For these and all subsequent calculations, the dimensionless corrugation wavelength has been set equal to Λ = λ / a i = 0.25 ≪ 1.
The results shown in Table 2 indicate that the effect of the corrugations is almost invariably to reduce the apparent wall conductivity κ w * (i.e., to increase the wall thermal resistance 1/ κ w * ).
This effect is accentuated for highly conductive walls (Δ w << 1, K w >> 1) with moderately large corrugation amplitudes (A ∼ 1), when the quasi-stagnant low-conductive recirculating flow established in the corrugation troughs acts as an effective barrier to heat transfer. By contrast, for poorly conductive walls (Δw = 0.1, K w << 1) the recirculating fluids have significantly larger conductivity than the wall itself and the wall thermal resistance remains virtually unaffected, showing a small reduction as the corrugation amplitude is increased. Table 3 shows the effect of the dimensionless corrugation amplitude, A/2π, on the apparent dimensionless heat-exchanger length ξ L * . The results are presented in terms of the ratio ξ L * / ξ L and also as percentage increment of the flat wall value ξ L for the three cases under study. As can be seen, the presence of the corrugations always increases the apparent dimensionless heat-exchanger length ξ L * = L / a 1 * Pe 1 * due to the simultaneous reduction of a 1 * and Pe 1 * (except in the case of constant flow rate), with larger increments corresponding to larger corrugation amplitudes under the constraint of constant pressure drop.   As previously discussed, use of the computed values of κ w * and ξ L * as input parameters for the bulk flow model yields the effectiveness ε * of the corrugated wall heat exchanger for given values of m, k, κ w (= K w /ΛΔ w ), ξ L , Λ, Re τi (= 1.5Λ 2 Re i ), Pr i , A, Δ w , K w , and K (= k). This procedure allows to evaluate the augmentation heat transfer and entropy generation numbers, R a and N a , thereby providing a methodology that could be used to optimize the heat exchanger performance from both an energy and/or an entropy (i.e., energy devaluation [30]) point of view.   Table 4 shows preliminary results corresponding to corrugated walls with different parameter sets. The results indicate that in most situations the corrugations increase the heat exchanger effectiveness ε * / ε > 1, although the overall heat transfer rate is reduced due to the significant reduction in volume flow rate, making ε * Q 1 /(εQ 1 ) flat < 1. It is interesting to note that the augmentation heat transfer parameter R a is smaller than one with constant flow rate, but grows above unity when the flow rate is reduced (e.g., constant pumping power or constant pressure drop). Note also that the improvement in R a is accompanied by a reduction in entropy generation rate N a , which is almost unaffected (N a ≅ 1) under constant flow rate, but takes values smaller than unity for constant pumping power (N a ≅ 0.95) and constant pressure drop (N a ≅ 0.9) conditions. These results confirm the ability of the multi-scale asymptotic analysis developed here to provide useful estimations of the heat exchanger performance. This method could therefore be used in future work to perform a thermodynamic optimization of the heat exchanger in terms of the different parameters involved in the model.

Conclusion
Laminar counterflow parallel-plate heat exchangers with small-scale wall corrugations have been studied theoretically and numerically following a multi-scale asymptotic approach. The analysis assumes that the corrugation wavelength and amplitude are comparable to each other but are much smaller than the channel half-width. In this case, the problem exhibits two main asymptotic regions: a bulk flow region where the walls appear flat at leading order, and where axial heat conduction can be neglected in first approximation provided the Peclet numbers are large enough, and a near-wall region where a full interaction exists between heat convection and conduction in the fluids, coupled by transverse and longitudinal conduction in the thin corrugated wall.
In the bulk flow region, an approximated semi-analytical two-term solution based on eigenfunction expansions provides accurate predictions of the outlet fluid temperatures in terms of the bulk parameters m, k, κ w , and ξ L .
The heat transfer process in the near wall region is investigated numerically by specifying the overall heat flux transverse to the wall and calculating the apparent wall temperature jump seen by the outer bulk flow. The numerical solution, based on a boundary fitted finite difference method, involves the parameters Re τi , Pr i , K, K w , Δ w , θ i, as well as the corrugation shape, defined here by A and γ. The analysis shows that, far from the wall, the effect of the corrugations is similar to that of a thickened virtual plane surface with an appropriately defined apparent thermal resistance.
Although for sufficiently large Reynolds numbers the flow in the channels and past the corrugations is known to become unstable and undergo transition to a self-sustained oscillatory and later turbulent regime [31,32] resulting in enhanced heat transfer rates, this study has focused on the role of the wall corrugations on the overall heat transfer process before this transition occurs.
The effect of the corrugations is incorporated into the bulk flow model defining apparent values of the bulk flow parameters m * , k * , κ w * and ξ L * in terms of those of the originally flat wall. Three different constraints have been considered for the determination of the apparent parameters: i) constant flow rate, ii) constant pumping power, and iii) constant pressure drop. The results, restricted to the case of sinusoidal wall geometries (γ = 0) with equal channel halfwidths (a 1 = a 2 ) show that the most relevant parameter determining the effect of the corrugations is the wall thermal conductivity K w . For highly conductive (e.g., metallic) walls, the addition of the corrugations always leads to a reduced wall conductivity κ w * (or an increased wall thermal resistance, 1/ κ w * ). On the other hand, for poorly conductive (e.g., polymeric) walls, the addition of the corrugations may lead to a slightly larger apparent wall conductivity κ w * (or an smaller wall thermal resistance) under certain flow conditions, particularly for relatively thick walls and large corrugation amplitudes.
The aim of this work has been to introduce the methodology followed by the multi-scale asymptotic analysis and to present preliminary calculations that illustrate the type of results that can be obtained with the most simple geometry and flow conditions. Corrugations of different shapes, involving for instance scalloped geometries, γ ≠ 0, with different channel half widths, a 1 ≠ a 2 , remain underexplored and warrant future work.