**1. Introduction**

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].

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 particu‐ lar, 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 eigenval‐ ues [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.

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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:


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 conduc‐ tivity 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.

### **2. Bulk flow model**

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

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

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

**•** 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

**•** 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

and longitudinal conduction in the (thin) corrugated wall. The multi-scale asymptotic analysis proceeds in two steps:

in the laminar flow regime.

4 Heat Transfer Studies and Applications

wall thermal resistance.

operating costs.

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<sup>i</sup>* , heat capacity, *ci* , and thermal conductivity, *ki* ).

**Figure 1.** Unitary cell of the idealized counterflow parallel-plate heat exchanger considered in the bulk flow model, showing the coordinate system, the velocity profiles, the inlet conditions, and the physical properties of fluids 1 and 2. A heat conducting wall, of finite thickness, *δ*w and thermal conductivity, *k*w, separates both fluids. The domain under study is shaded gray.

The Reynolds numbers, Re*<sup>i</sup> =* 2*ai Vi* /*νi* , based on the channel half-width, *ai* , average flow velocity, *Vi* , and kinematic viscosity, *ν<sup>i</sup>* , of fluid *i*, are assumed to be sufficiently small for the flows to remain laminar and steady [23]. The Prandtl numbers, Pr*<sup>i</sup> = ν<sup>i</sup>* /*αi* , based on the thermal diffusivity, *α<sup>i</sup> = ki* /(*ρ<sup>i</sup> ci* ), 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, *ui* (*Yi* ) = ±(3/2)*Vi* [1– (*Yi* /*ai* ) 2 ], 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*<sup>i</sup>* = Re*<sup>i</sup>* Pr*<sup>i</sup>* = 2*ai Vi* / *α<sup>i</sup>* , 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*/(Pe1*a*1) and *yi* = *Yi* /*ai* , and the normalized temperature *θ<sup>i</sup>* = (*Ti* –*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

$$\begin{aligned} \frac{3}{4}(1-\mathcal{y}\_1^2)\frac{\partial\theta\_1}{\partial\xi} &= \frac{\hat{\sigma}^2\theta\_1}{\partial\boldsymbol{\hat{y}}\_1^2} \\ -m\frac{3}{4}(1-\mathcal{y}\_2^2)\frac{\partial\theta\_2}{\partial\xi} &= \frac{\hat{\sigma}^2\theta\_2}{\partial\boldsymbol{\hat{y}}\_2^2} \end{aligned} \quad \text{in} \quad 0 \le \boldsymbol{\xi} < \boldsymbol{\xi}\_L, \quad 0 \le \mathbf{y}\_i \le 1. \tag{1}$$

These equations are to be integrated with adiabatic boundary conditions at the channel symmetry planes

$$\frac{\partial \mathcal{O}\_1}{\partial y\_1} = \frac{\partial \mathcal{O}\_2}{\partial y\_2} = 0 \quad \text{at} \quad 0 < \xi < \xi\_L, \quad y\_i = 0,\tag{2}$$

the condition of continuity of wall-normal heat fluxes at the heat conducting wall

$$\begin{aligned} \partial\_z - \partial\_1 &= \frac{1}{\kappa\_w} \frac{\partial \partial\_1}{\partial \mathbf{y}\_1} = \frac{\nu(\boldsymbol{\xi})}{\kappa\_w} \\ \frac{\partial \partial\_1}{\partial \mathbf{y}\_1} &= -k \frac{\partial \partial\_2}{\partial \mathbf{y}\_2} \equiv \nu(\boldsymbol{\xi}) \end{aligned} \quad \text{at} \quad 0 < \boldsymbol{\xi} < \boldsymbol{\xi}\_L, \quad \mathbf{y}\_i = \mathbf{l}, \tag{3}$$

and the inlet conditions

$$\begin{aligned} \partial\_1 = \partial\_{1, \text{in}} = 0 \quad \text{at} \quad \underline{\xi} = 0, \quad 0 \le \underline{\nu}\_1 < 1, \\ \partial\_2 = \partial\_{2, \text{in}} = 1 \quad \text{at} \quad \underline{\xi} = \underline{\xi}\_L, \quad 0 \le \underline{\nu}\_2 < 1. \end{aligned} \tag{4}$$

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 approxi‐ mation provided the dimensionless wall thickness *δ*w/*a*1 remains sufficiently small [19].

The solution to the problem stated above provides the temperature field, *θ<sup>i</sup>* (*ξ*, *yi* ), for given values of the parameters

Wall Conduction Effects in Laminar Counterflow Parallel-Plate Heat Exchangers with Small-Scale Wall Corrugations http://dx.doi.org/10.5772/60550 7

$$m = \frac{a\_2}{a\_1} \frac{\text{Pe}\_2}{\text{Pe}\_1}, \quad k = \frac{a\_1}{a\_2} \frac{k\_2}{k\_1}, \quad \kappa\_w = \frac{a\_1}{\mathcal{S}\_w} \frac{k\_w}{k\_1}, \quad \text{and} \quad \mathcal{L}\_L = \frac{L}{a\_1} \frac{1}{\text{Pe}\_1}. \tag{5}$$

It also provides the local heat-transfer rate from fluid 2 to fluid 1, *ν*(*ξ*), the interfacial wall-fluid temperature distributions, *θ*w*<sup>i</sup>* (*ξ*) ≡ *θ<sup>i</sup>* (*ξ*, 1), the local wall temperature jump, *θ*w2(*ξ*) – *θ*w1(*ξ*) = *ν*(*ξ*)/*κ*w, and the outlet temperature profiles, *θ*1,out(*y*1) = *θ*1(*ξL*, *y*1) and *θ*2,out(*y*1) = *θ*2(*0*, *y*2). Another interesting result is the spatial distribution of the bulk (or mixing-cup) temperatures

$$
\partial\_{mi}(\xi^{\varepsilon}) = \int\_0^1 \frac{3}{2} (1 - \chi\_i^2) \partial\_i(\xi^{\varepsilon}, \chi\_i) d\mathbf{y}\_i \tag{6}
$$

which represent the uniform temperature that would eventually be attained by fluid *i* if it was allowed to evolve adiabatically downstream of a certain section *ξ*. This includes, in particular, the values of the outlet bulk temperatures of both streams, *θ*m1(*ξL*) ≡ *θ*m1,out and *θ*m2(0) ≡ *θ*m2,out = 1 – *θ*m1,out/(*mk*), which are closely related to the heat exchanger effectiveness [12].

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 dimen‐ sionless 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* = (2*a*2*V*2*ρ*2*c*2)/(2*a*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].

Following standard practice, the solution of the problem can be expanded as an infinite series of eigenfunctions of the form [19]

$$\partial\_i(\xi, \mathbf{y}\_i) = A + \sum\_{n=-\alpha}^{\ast \alpha} \mathbf{C}\_n e^{-\lambda\_n \xi} \mathbf{g}\_{n,i}(\mathbf{y}\_i) \tag{7}$$

where the eigenfunctions are given by

The Reynolds numbers, Re*<sup>i</sup> =* 2*ai*

6 Heat Transfer Studies and Applications

, and kinematic viscosity, *ν<sup>i</sup>*

/(*ρ<sup>i</sup> ci*

numbers of the flow in the channels, Pe*<sup>i</sup>*

transverse coordinates *ξ* = *X*/(Pe1*a*1) and *yi*

diffusivity, *α<sup>i</sup> = ki*

symmetry planes

and the inlet conditions

values of the parameters

*Vi*

(*Yi* /*ai* ) 2 *Vi* /*νi*

remain laminar and steady [23]. The Prandtl numbers, Pr*<sup>i</sup> = ν<sup>i</sup>*

to be a fully developed Poiseuille flow independent of the temperature field, *ui*

assume that the inlet temperature of fluid *i* is uniform and equal to T*i*,in.

2 1 1

q

x

q

x

2 2 2

1

*y*

¶ - = ¶

<sup>3</sup> (1 ) 4

<sup>3</sup> (1 )

q

2 1

q

q q

¶ -- = ¶

4

*m y*

2

1 2

1 2 1 2

¶ ¶ =- º ¶ ¶

qq

q q

*k y y*

 q

k

¶ -= = ¶

 q  = Re*<sup>i</sup>* Pr*<sup>i</sup>* = 2*ai Vi* / *α<sup>i</sup>*

 = *Yi* /*ai*

approximation, to a balance between axial convection and transverse conduction

1

2

*y*

 q

*y*

¶ ¶

1

q nx

*y*

w1 w

n x

1 ()

2

in

2 2

2

¶

¶ ¶ ¶  q

, based on the channel half-width, *ai*

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

Then, axial heat conduction can be neglected in both fluids in first approximation, and we can

To write the problem in non-dimensional form, we introduce the dimensionless axial and

(*T*2,in –*T*1,in). Then, under the assumptions stated above, the energy equation reduces, in first

These equations are to be integrated with adiabatic boundary conditions at the channel

x x

1 2 0 at 0 , 0,

= = << = ¶ ¶ *L i <sup>y</sup>*

the condition of continuity of wall-normal heat fluxes at the heat conducting wall

( )

1 1,in 1 2 2,in 2

The solution to the problem stated above provides the temperature field, *θ<sup>i</sup>*

 x

= = = £< = = = £< *<sup>L</sup>*

 xx

0 at 0, 0 1, 1 at , 0 1.

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 approxi‐ mation provided the dimensionless wall thickness *δ*w/*a*1 remains sufficiently small [19].

 k x x

<

], where the + (–) sign holds for fluid 1 (2). As a result, we shall assume that the Peclet

, of fluid *i*, are assumed to be sufficiently small for the flows to

), of fluid *i*, are also assumed to be moderately large compared to unity,

0< , 0 < y < 1.

*L i*

*y y* (2)

at 0 , 1,

*y*

<< =

*L i*

*y*

*<sup>y</sup>* (4)

(*ξ*, *yi*

), for given

x x /*αi*

, and the normalized temperature *θ<sup>i</sup>*

, average flow velocity,

, based on the thermal

(*Yi*

, are both large compared to unity.

) = ±(3/2)*Vi*

= (*Ti* –*T*1,in)/

(1)

(3)

[1–

$$\mathbf{g}\_{\boldsymbol{n},i}(\boldsymbol{\wp}\_{i}) = \left(\mathbf{I} \mp \frac{\mathbf{g}\_{\boldsymbol{n},1}'(\mathbf{I})}{2\kappa\_{\boldsymbol{w}}}\right) f\_{\boldsymbol{n},i}(\boldsymbol{\wp}\_{i}), \quad \mathbf{g}\_{\boldsymbol{n},1}'(\mathbf{I}) = \frac{f\_{\boldsymbol{n},1}'(\mathbf{I})}{1 + \frac{1}{2\kappa\_{\boldsymbol{w}}} f\_{\boldsymbol{n},1}'(\mathbf{I})} \tag{8}$$

with the – (+) sign corresponding to fluid 1 (2), and the eigenvalues are determined by the eigencondition

$$\left| f\_{n,1}'(\mathbf{l}; \boldsymbol{\lambda}\_n) + k f\_{n,2}'(\mathbf{l}; m, \boldsymbol{\lambda}\_n) \right| \left| 1 + \frac{f\_{n,1}'(\mathbf{l}; \boldsymbol{\lambda}\_n)}{\boldsymbol{\kappa}\_w} \right| = 0. \tag{9}$$

In the above expressions, *fn,i*(*yi* ) represent the eigenfunctions of the thermally thin wall limit *κ<sup>w</sup>* → ∞ (or 1/*κ<sup>w</sup>* → 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 *yi* = 1 to give the derivative of the eigenfunctions at the heat conducting wall

$$f\_{n,i}'(1) = -\frac{1}{2} + 2\left[\kappa\_{n,i} + \frac{2\sqrt{\pi}\left(\frac{3}{4} + \kappa\_{n,i}\right)M\_{\kappa\_{n,i}+1,\frac{1}{4}}(4\kappa\_{n,i}) - \Gamma\left(\frac{1}{4} - \kappa\_{n,i}\right)W\_{\kappa\_{n,i}+1,\frac{1}{4}}(4\kappa\_{n,i})}{2\sqrt{\pi}M\_{\kappa\_{n,i}+\frac{1}{4}}(4\kappa\_{n,i}) + \Gamma\left(\frac{1}{4} - \kappa\_{n,i}\right)W\_{\kappa\_{n,i}+\frac{1}{4}}(4\kappa\_{n,i})}\right],\tag{10}$$

where *κn*,1 = 3*λ<sup>n</sup>* / 8, *κn*,2 =**i** 3*mλ<sup>n</sup>* / 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 *λn* 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 *Cn* involved in the series expansion (7), which must be truncated to a finite number of terms in order to get an approx‐ imate 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*<sup>i</sup>* at the inlet section of fluid *i*. This leads to analytic expressions for the outlet bulk temperatures of the form

$$\begin{aligned} \Theta\_{m1,\text{out}} & \equiv \Delta\_1 + \left( 1 - \Delta\_1 - \Delta\_2 \right) \left[ \frac{1 - e^{-\lambda\_0 \tau\_L}}{1 - (mk)^{-1} e^{-\lambda\_0 \tau\_L}} \right] + mk\Delta\_2\\ \Theta\_{m2,\text{out}} & \equiv \Delta\_1 + \left( 1 - \Delta\_1 - \Delta\_2 \right) \left[ \frac{1 - (mk)^{-1}}{1 - (mk)^{-1} e^{-\lambda\_0 \tau\_L}} \right] + (mk)^{-1} \Delta\_1 \end{aligned} \tag{11}$$

where Δ*<sup>i</sup>* 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

$$\Delta\_1 = \left[\frac{1 - (mk)^{-1}}{1 - (mk)^{-1}e^{-\lambda\_0 \varepsilon\_L}}\right] \frac{\Delta\_1^{\{0\}}}{2}, \quad \Delta\_2 = (mk)^{-1} \left[1 - \frac{1 - e^{-\lambda\_0 \varepsilon\_L}}{1 - (mk)^{-1}e^{-\lambda\_0 \varepsilon\_L}}\right] \frac{\Delta\_1^{\{0\}}}{2} \tag{12}$$

with

$$\Delta\_{\rm i}^{(0)} = \frac{2(\mathbf{l} + mk)\mathbf{g}\_{0,1}'^{2}(\mathbf{l}) + mk \left[\boldsymbol{\Omega}\_{0}^{-} - \frac{\mathbf{g}\_{0,1}'(\mathbf{l})\boldsymbol{\Omega}\_{0}^{+}}{2\kappa\_{\rm w}}\right] \boldsymbol{\lambda}\_{0}^{2}}{8\mathbf{g}\_{0,1}'^{2}(\mathbf{l}) + (\mathbf{l} + mk) \left[\boldsymbol{\Omega}\_{0}^{-} - \frac{\mathbf{g}\_{0,1}'(\mathbf{l})\boldsymbol{\Omega}\_{0}^{+}}{2\kappa\_{\rm w}}\right] \boldsymbol{\lambda}\_{0}^{2}}.\tag{13}$$

For given values of *m*, *k*, *κ*w, and *ξL*, 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

$$
\Omega \Omega\_0^{\pm} = \frac{\partial}{\partial \mathcal{A}\_0} \Big[ \mathbf{g}\_{0,1}'(\mathbf{l}; \mathcal{A}\_0) \pm k \mathbf{g}\_{0,2}'(\mathbf{l}; m, \mathcal{A}\_0) \Big], \tag{14}
$$

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 andbulktemperaturedistributions,localNusseltnumbers,andoverallheat-transfercoefficient.

, ,

*n i n i*

1 , , 1, , , 4 4

 k

è ø

, 1, , 1, 1, 1, 4 4

*n i n i n i n i*

+ +

)

*i*

at the inlet section of fluid *i*. This leads to

(11)

(13)

1


0

*L*

l x

l x

l

l

+

*n*

 k

 k

> k

 k

 k

, ,

*n i n i*

*n i n i*

4

*M W*

<sup>1</sup> <sup>2</sup> (4 ) (4 ) <sup>1</sup> 4 4

ê ú ÷ ç÷ èø èø - + æ ö +G - ç ÷

where *κn*,1 = 3*λ<sup>n</sup>* / 8, *κn*,2 =**i** 3*mλ<sup>n</sup>* / 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 *λn* for given

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 *Cn* involved in the series expansion (7), which must be truncated to a finite number of terms in order to get an approx‐ imate 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

> 0 0

*L L*

l x


é ù -D -D <sup>ê</sup> + D - <sup>ú</sup> ë û

*mk e*

0 0 1 [0] [0] 1 1 1

*L L*

2 2 0,1 0 0,1 0 0


2 2 0,1 0 0,1 0 0

For given values of *m*, *k*, *κ*w, and *ξL*, the above expressions provide closed-form approximate expressions for the outlet bulk temperatures of both streams in terms of the computed values

2 0 (1; ) (1; , ) ,


(1) 2(1 ) (1) <sup>2</sup> . (1) 8 (1) (1 ) <sup>2</sup>

+ + W- ¢ ê ú

1( ) <sup>1</sup> , ( )1 1( ) 2 1( ) 2

<sup>û</sup> ê ú

*mk <sup>e</sup> mk*


é ù ê ú ë û

*mk e*

0

*L*

*mk e mk e* (12)

w

 l

<sup>±</sup> ¶ W = é ù ¢ ¢ <sup>±</sup> ë û ¶ *g kg m* (14)

k

ë û

é ù ¢ W

k

+

*mk mk*

l x


represents the inlet bulk temperature offset of fluid *i* associated with the neglected

1

l x

*<sup>e</sup> mk*

k

k

pk

2 , <sup>2</sup> <sup>1</sup> 2 (4 ) (4

 k

é ù æö æö ç + -G -

ë û

*M W f* (10)

,

*n i*

+

terms through apparent temperature offsets Δ*<sup>i</sup>*

q

*m*

q

where Δ*<sup>i</sup>*

with

of *λ*0, *g* ′

0,1(1), and

*m*

analytic expressions for the outlet bulk temperatures of the form

( )

<sup>1</sup> <sup>1</sup>

1,out 1 1 2 1 2


2,out 1 1 2 1 1

1( )

1( ) <sup>1</sup> ( ) 1( )

( )

higher-order eigenfunctions at both ends of the heat exchanger, given by

l x

é ù

ë û

1

1 2 1 1

[0] w

ë û D = é ù ¢ <sup>W</sup> ¢ + + W- ê ú

*<sup>g</sup> g mk*

0 0,1 0 0,

l

0

l

*<sup>g</sup> mk g mk*


k

3

pk

,

*i*

¢

*n*

(1)=

8 Heat Transfer Studies and Applications

values of *m*, *k*, and *κ*w.

Figure 2 shows the variation of the outlet bulk temperature of fluid 1, *θ*m1,out, with the dimen‐ sionless wall thermal resistance, 1/*κ*w, for increasing values of the heat exchanger length, *ξL* = [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* =(*m*˙ <sup>2</sup>*c*2) / (*m*˙ <sup>1</sup>*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, *ξL*, 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<sup>L</sup>* ≥ 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.

Figure 2. Variation of the outlet bulk temperature of fluid 1 with the wall thermal resistance 1/*κ*w for several heat exchanger lengths *ξ<sup>L</sup>* corresponding to *m* = 1, *k* = 2 (left) and *m* = 2, *k* = 1(right), as obtained from the numerical integration of Eqs. (1) – (4) (solid lines) and from the approximate two-term solution (symbols). The aim of the following sections is to extend the present analysis to the study of laminar counterflow heat exchangers **Figure 2.** Variation of the outlet bulk temperature of fluid 1 with the wall thermal resistance 1/*κ*w for several heat ex‐ changer lengths *ξL* corresponding to *m* = 1, *k* = 2 (left) and *m* = 2, *k* = 1(right), as obtained from the numerical integration of Eqs. (1) – (4) (solid lines) and from the approximate two-term solution (symbols).

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\* ξL\** 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. **3. Wall corrugation model** 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

the figure.

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, Δw = *δ*w /*λ* ൏൏ 1, emerges naturally as a small

shear stress far from the wall, *τi*, the corrugation wavelength, *λ*, and the fluid density, *ρi*, and viscosity, *μi*, of fluid *i*, are

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

parameter. In addition, the Reynolds numbers of the shear flows at both sides of the wall, Re*τi* = *ρiλ*2*τi* /*μ<sup>i</sup>*

also assumed to be sufficiently small for the flows to remain laminar and steady.

, *k\** , \* <sup>w</sup> , and

2, based on the

analysis will provide the apparent values of the dimensionless parameters *m\** , *k\** , *κ*<sup>w</sup> \* , and *ξ<sup>L</sup> \** 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.

#### **3. Wall corrugation model**

**10** 15 Note that due, Note that, due

**11** 6 parameter, the parameter; the

**11** 24 (or finite-volume) or finite-volume

**12** 5 square domain square computational domain

**11** 27 convection convective

**10** 16 in fluid 1 of fluid 1

**11** 22 *y*=0 *y* = 0

**12** 1 can be was

**12** 11 in this section above

**12** 12 numerical study analysis

**12** 23-24 Note the growth of the low-velocity

**12** 25

recirculation regions inside the wall

**12** 24-25 surface obtained from the three simulations shown in the figure are

**12** 13 restricted to restricted here to

**12** 16 troughs that troughs, which

corrugations for increasing Reynolds numbers.

**8** Fig. 3

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, Δw = *δ*<sup>w</sup> /*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<sup>i</sup> λ*2 *τi* /*μi* 2 , based on the shear stress far from the wall, *τ<sup>i</sup>* , the corrugation wavelength, *λ*, and the fluid density, *ρ<sup>i</sup>* , and viscosity, *μi* , 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 dimension‐ less 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.

**9** 25 −∞ < *y < y*<sup>w</sup> −∞ < y < <sup>w</sup>*y* **10** 1 *y*w < *y <* ∞ <sup>w</sup>*y* < y < ∞ **Figure 3.** Schematic representation of the wall geometry defined by the functions *x*w(*α*) and *y*w(*α*) given by Eq. (15). The up‐ per plot illustrates the apparent wall position for Stokes flow (i.e., Re*<sup>i</sup>* =0) parallel (*ywi* <sup>∥</sup> , dashed lines) and transverse (*ywi* ⊥, solid lines) to the corrugations for the three corrugation amplitudes *A*/2π = [0.5, 1, 1.5]/2π considered in the figure.

**10** 3 *x=*0 *x =* 0 For illustrative purposes, we define the corrugation shape using the parametric representation

[Could you please remove this sentence from

surface, obtained from the three simulations

fits the font size in the figure caption?]

the figure caption?]

shown in the figure, are

w2 w1 *y y* [ ] 0.1032, 0.1084, 0.1167 [Could you please reduce the font size so that it

2

Wall Conduction Effects in Laminar Counterflow Parallel-Plate Heat Exchangers with Small-Scale Wall Corrugations http://dx.doi.org/10.5772/60550 11

analysis will provide the apparent values of the dimensionless parameters *m\**

**3. Wall corrugation model**

10 Heat Transfer Studies and Applications

density, *ρ<sup>i</sup>*

**9** 25 −∞ < *y < y*<sup>w</sup> −∞ < y < <sup>w</sup>*y*

**10** 1 *y*w < *y <* ∞ <sup>w</sup>*y* < y < ∞

**10** 15 Note that due, Note that, due

**11** 6 parameter, the parameter; the

**11** 24 (or finite-volume) or finite-volume

**12** 5 square domain square computational domain

**11** 27 convection convective

**10** 16 in fluid 1 of fluid 1

**11** 22 *y*=0 *y* = 0

**12** 1 can be was

**12** 11 in this section above

**12** 12 numerical study analysis

**12** 23-24 Note the growth of the low-velocity

**12** 25

recirculation regions inside the wall

**12** 24-25 surface obtained from the three simulations shown in the figure are

**12** 13 restricted to restricted here to

**12** 16 troughs that troughs, which

corrugations for increasing Reynolds numbers.

**10** 3 *x=*0 *x =* 0

**8** Fig. 3

based on the shear stress far from the wall, *τ<sup>i</sup>*

y

per plot illustrates the apparent wall position for Stokes flow (i.e., Re*<sup>i</sup>*

, and viscosity, *μi*

to remain laminar and steady.

plane shown in the figure.

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.

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, Δw = *δ*<sup>w</sup> /*λ* << 1, emerges naturally as a small parameter. In

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 dimension‐ less 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*

> y⊥ <sup>w</sup><sup>2</sup> y|| w2 y|| <sup>w</sup><sup>1</sup> y<sup>⊥</sup> w1

<sup>123</sup> <sup>x</sup> <sup>z</sup> <sup>0</sup>

**Figure 3.** Schematic representation of the wall geometry defined by the functions *x*w(*α*) and *y*w(*α*) given by Eq. (15). The up‐

For illustrative purposes, we define the corrugation shape using the parametric representation

solid lines) to the corrugations for the three corrugation amplitudes *A*/2π = [0.5, 1, 1.5]/2π considered in the figure.

[Could you please remove this sentence from

surface, obtained from the three simulations

fits the font size in the figure caption?]

the figure caption?]

shown in the figure, are

w2 w1 *y y* [ ] 0.1032, 0.1084, 0.1167 [Could you please reduce the font size so that it

A/2π = (0.5, 1, 1.5)/2π

addition, the Reynolds numbers of the shear flows at both sides of the wall, Re*τi* = *ρ<sup>i</sup>*

, *k\** , *κ*<sup>w</sup> \* , and *ξ<sup>L</sup> \**

> *λ*2 *τi* /*μi* 2 ,

, the corrugation wavelength, *λ*, and the fluid

−2 3

<sup>∥</sup> , dashed lines) and transverse (*ywi*

⊥,

=0) parallel (*ywi*

−1 3

1 3

2 3

γ

2

, of fluid *i*, are also assumed to be sufficiently small for the flows

$$\mathbf{x} = \mathbf{x}\_w(\alpha) = \frac{\alpha + \chi \sin(\alpha)}{2\pi}, \quad \mathbf{y} = \mathbf{y}\_w(\alpha) = \frac{A}{2\pi} \cos(\alpha), \quad \alpha \in \mathbb{R}, \tag{15}$$

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<sup>w</sup> << 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*<sup>w</sup> *=* (*A*/2π)cos(*2πx*) obtained for γ = 0.

The velocity of fluid *i* is measured with the characteristic shear velocity *λτ<sup>i</sup>* /*μi* to give the dimensionless velocity field *u* = (*u*, *v, w*). The variations in pressure with respect to its unper‐ turbed value far from the wall are scaled in fluid *i* with the corresponding shear stress, *τ<sup>i</sup>* , 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 temper‐ ature 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

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial \mathbf{v}}{\partial \mathbf{y}} = \mathbf{0}$$

$$\text{Re}\_{rl} \left( u \frac{\partial u}{\partial \mathbf{x}} + \nu \frac{\partial u}{\partial \mathbf{y}} \right) = -\frac{\partial p}{\partial \mathbf{x}} + \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial \mathbf{y}^2}$$

$$\text{Re}\_{rl} \left( u \frac{\partial \mathbf{v}}{\partial \mathbf{x}} + \nu \frac{\partial \mathbf{v}}{\partial \mathbf{y}} \right) = -\frac{\partial p}{\partial \mathbf{y}} + \frac{\partial^2 \mathbf{v}}{\partial \mathbf{x}^2} + \frac{\partial^2 \mathbf{v}}{\partial \mathbf{y}^2} \tag{16}$$

$$\text{Re}\_{rl} \left( u \frac{\partial \mathbf{w}}{\partial \mathbf{x}} + \nu \frac{\partial \mathbf{w}}{\partial \mathbf{y}} \right) = \frac{\partial^2 w}{\partial \mathbf{x}^2} + \frac{\partial^2 w}{\partial \mathbf{y}^2}$$

$$\text{Re}\_{rl} \mathbf{Pr}\_l \left( u \frac{\partial T}{\partial \mathbf{x}} + \nu \frac{\partial T}{\partial \mathbf{y}} \right) = \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial \mathbf{y}^2}$$

where viscous dissipation has been neglected in the energy balance and Pr*<sup>i</sup> = ν<sup>i</sup>* /*α<sup>i</sup> = μi* /(*ki* /*ci* ) 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*<sup>w</sup> - ) and above (*y*<sup>w</sup> + < *y <* ∞) the wall. Hereafter we shall use *y*w to denote the wall position, *y*<sup>w</sup> ± 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

$$
\mathcal{Y} = \mathcal{Y}\_{\text{w}} : \quad \mathfrak{u} = \mathcal{v} = \mathcal{w} = 0,\tag{17}
$$

and a uniform shear flow far from the wall

$$\mathbf{y} \gg \mathbf{l} \, : \quad \frac{\partial u}{\partial \mathbf{y}} = \sin \theta\_2, \quad \mathbf{v} = \mathbf{0}, \quad \frac{\partial w}{\partial \mathbf{y}} = \cos \theta\_2, \quad p = \mathbf{0}, \tag{18}$$

$$\mathbf{u} - \mathbf{y} \gg \mathbf{l} \, ; \quad \frac{\partial \mathbf{u}}{\partial \mathbf{y}} = -\sin \theta\_{\mathbf{l}} , \quad \mathbf{v} = \mathbf{0} , \quad \frac{\partial \mathbf{w}}{\partial \mathbf{y}} = -\cos \theta\_{\mathbf{l}} , \quad \mathbf{p} = \mathbf{0} . \tag{19}$$

In the far field boundary conditions *θ<sup>i</sup>* denotes the angle of misalignment between the wall corrugations and the uniform shear imposed on fluid *i* far from the wall. So that *θ<sup>i</sup>* =0, *π*, and *θi* = ±*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

$$\mathbf{y} \gg \mathbf{l} : \quad \frac{\partial T}{\partial \mathbf{y}} = \mathbf{K}^{-1}, \tag{20}$$

where *K = k2/k1* denotes the fluid-to-fluid heat conductivity ratio, whereas in fluid 1 we impose a linear temperature profile of the form

$$-\mathbf{y} \gg \mathbf{l} \colon \quad T = \mathbf{y}.\tag{21}$$

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, Δw << 1, the jump conditions for the temperature and the wall-normal heat fluxes at the heat conducting wall can be written as

$$\mathcal{Y} = \mathcal{Y}\_{\text{w}} : \begin{cases} \mathcal{T}\_{\text{w}}^{+} - \mathcal{T}\_{\text{w}}^{-} = \frac{\Delta\_{\text{w}}}{K\_{\text{w}}} \left[ \frac{\nu\_{\text{w}}^{+} + \nu\_{\text{w}}^{-}}{2} + \mathcal{O}\left(\Lambda\_{\text{w}}^{2}\right) \right] \\\\ \nu\_{\text{w}}^{+} - \nu\_{\text{w}}^{-} = -\Delta\_{\text{w}}K\_{\text{w}} \left[ \frac{1}{2} \left( \frac{\mathbf{d}^{2}T\_{\text{w}}^{+}}{\text{ds}^{2}} + \frac{\mathbf{d}^{2}T\_{\text{w}}^{-}}{\text{ds}^{2}} \right) + \mathcal{O}\left(\Lambda\_{\text{w}}^{2}\right) \right] \end{cases} \tag{22}$$

where *T*<sup>w</sup> <sup>±</sup> <sup>=</sup>*<sup>T</sup>* (*x*w(*α*), *<sup>y</sup>*<sup>w</sup> ± (*α*)) denotes the temperature at the upper (+) and lower (‒) wall surfaces, and

Wall Conduction Effects in Laminar Counterflow Parallel-Plate Heat Exchangers with Small-Scale Wall Corrugations http://dx.doi.org/10.5772/60550 13

$$\nu\_{\text{w}}^{+} = K \frac{\partial T\left(\mathbf{x}\_{\text{w}}(\boldsymbol{\alpha}), \mathbf{y}\_{\text{w}}^{+}(\boldsymbol{\alpha})\right)}{\partial \boldsymbol{n}}, \quad \nu\_{\text{w}}^{-} = \frac{\partial T\left(\mathbf{x}\_{\text{w}}(\boldsymbol{\alpha}), \mathbf{y}\_{\text{w}}^{-}(\boldsymbol{\alpha})\right)}{\partial \boldsymbol{n}} \tag{23}$$

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

$$\mathbf{ds} = \sqrt{\mathbf{x}\_{\mathbf{w}}'^2(\alpha) + \mathbf{y}\_{\mathbf{w}}'^2(\alpha)} \,\mathrm{d}a.\tag{24}$$

In addition to the dimensionless wall thickness, Δw, the jump conditions (22) involve the wallto-fluid heat conductivity ratio, *K*<sup>w</sup> *= k*w*/k*1, as an additional dimensionless parameter; the wall temperature difference being proportional to Δw/*K*w, whereas the jump in wall-normal heat fluxes is proportional to Δw*K*w. As a result, in the formal limit Δ<sup>w</sup> → 0 both Δw and *K*<sup>w</sup> disappear from the formulation and the jump conditions (22) reduce to the continuity of temperatures, *T*<sup>w</sup> <sup>+</sup> <sup>=</sup>*T*<sup>w</sup> − , and wall-normal heat fluxes, *ν*<sup>w</sup> <sup>+</sup> <sup>=</sup>*ν*<sup>w</sup> − .

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*<sup>i</sup>* , the fluid-to-fluid and wall-to-fluid heat conductivity ratios, *K = k*2*/k*1 and *K*<sup>w</sup> *= k*w*/k*1, the dimensionless wall thickness, Δ<sup>w</sup> *= δ*w*/λ* << 1, the angles of misalignment between the imposed shear and the direction of the corrugations, *θi* , 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*τ*1, Re*τ*2, Pr1, Pr2, *K*, *K*w, Δw, *θ1*, and *θ2*.

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*w*<sup>i</sup>* <sup>∥</sup> ) and transverse (*y*w*<sup>i</sup>* <sup>⊥</sup>) to the corrugations

$$\begin{split} \mathcal{Y}\_{\mathbf{w}\_{2}}^{\parallel} &= \lim\_{\mathbf{y} \to \boldsymbol{\nu}} \left( \mathbf{y} - \frac{\boldsymbol{\nu}}{\cos \theta\_{2}} \right), \quad \mathcal{Y}\_{\mathbf{w}\_{2}}^{\perp} = \lim\_{\mathbf{y} \to \boldsymbol{\nu}} \left( \mathbf{y} - \frac{\boldsymbol{\nu}}{\sin \theta\_{2}} \right), \\ \mathcal{Y}\_{\mathbf{w}\_{1}}^{\parallel} &= \lim\_{\mathbf{y} \to \boldsymbol{\nu}} \left( \mathbf{y} + \frac{\boldsymbol{\nu}}{\cos \theta\_{1}} \right), \quad \mathcal{Y}\_{\mathbf{w}\_{1}}^{\perp} = \lim\_{\mathbf{y} \to \boldsymbol{\nu}} \left( \mathbf{y} + \frac{\boldsymbol{\nu}}{\sin \theta\_{1}} \right), \end{split} \tag{25}$$

as well as the apparent temperature jump at *y* = 0

(*y*<sup>w</sup> +

*θi*

at the wall

lower (–) and upper (+) wall surfaces.

12 Heat Transfer Studies and Applications

and a uniform shear flow far from the wall

In the far field boundary conditions *θ<sup>i</sup>*

a linear temperature profile of the form

current flow arrangements.

< *y <* ∞) the wall. Hereafter we shall use *y*w to denote the wall position, *y*<sup>w</sup>

As boundary conditions we impose periodicity at *x =*0 and 1, the no-slip boundary condition

2 2 1: sin , 0, cos , 0

1 1 1: sin , 0, cos , 0.

 = ±*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 counter-

We also impose the overall heat flux transferred by conduction from fluid 2 to fluid 1 by

1 1: , ¶ - <sup>=</sup> ¶

where *K = k2/k1* denotes the fluid-to-fluid heat conductivity ratio, whereas in fluid 1 we impose

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

In the thin wall limit considered here, Δw << 1, the jump conditions for the temperature and

ww w 2

2

+ -

<sup>D</sup> <sup>D</sup>

w w ww 2 2 w 1d d 2 d

*<sup>T</sup> <sup>K</sup> s s*

ë û

O

w w w w <sup>w</sup> 2 2

<sup>ì</sup> é ù <sup>+</sup> <sup>ï</sup> -= + ê ú

n n

( )

+ -

é ù æ ö D +

d

ê ú ç ÷ + D ë û è ø

*T*

w w 2

(*α*)) denotes the temperature at the upper (+) and lower (‒) wall surfaces,

*<sup>T</sup> Ky*

? *u w y vp*

 ¶ ¶ - =- = =- = ¶ ¶ ? *u w y vp*

corrugations and the uniform shear imposed on fluid *i* far from the wall. So that *θ<sup>i</sup>*

, ¶ ¶ = == = ¶ ¶

q

q

specifying a uniform temperature gradient in fluid 2 far from the wall

at the mean plane (*y* = 0) of the corrugated wall equal to zero.

+ -

*T T*

+ -

<sup>ï</sup> - =- <sup>ï</sup>

:

<sup>ï</sup> <sup>=</sup> <sup>í</sup>

±

*y y*

<sup>±</sup> <sup>=</sup>*<sup>T</sup>* (*x*w(*α*), *<sup>y</sup>*<sup>w</sup>

where *T*<sup>w</sup>

and

n n

î

the wall-normal heat fluxes at the heat conducting wall can be written as

*K*

?

±

<sup>w</sup> *yy uvw* = == = : 0, (17)

*y y* (18)

*y y* (19)

denotes the angle of misalignment between the wall

*<sup>y</sup>* (20)


( )

O

 q

> q

representing the

=0, *π*, and

(22)

$$
\Delta T = \lim\_{\mathbf{y} \to \mathbf{x}} \left( T - K^{-1} \mathbf{y} \right). \tag{26}
$$

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 twodimensional, a vorticity-streamfunction formulation was adopted for the numerical integra‐ tion, 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*w*<sup>i</sup>* <sup>∥</sup> , *y*wi ⊥, and Δ*T*.

Although the formulation presented above is of great generality, the large number of param‐ eters 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 presents illustrative results for Pr*<sup>i</sup>* = 5, *K* = 2, *K*<sup>w</sup> = 0.1, Δw = 0.1, *A* = 1, and different values of Re*τi* = [32, 128, 512]. As can be seen, the flow exhibits the typical low-velocity recirculation regions within the corrugation troughs, which grow spatially with the Reynolds number. The presence of this quasi-stagnant fluid in the vicinity of the wall modifies the overall wall thermal resistance, thereby changing the effectiveness of the heat exchange process in a way that is to be determined below.

**Figure 4** Streamlines (white contours) obtained for *A* = 1, *γ* = 0, *θ*2 = *θ*1 = π/2, and Re*τi* = [32, 128, 512], and temperature field (black contours and colored background; see colorbars for the color scale) corresponding to Pr*i* = 5, *K* = 2, *K*w = 0.1, and Δw = 0.1. Note the growth of the low-velocity recirculation regions inside the wall corrugations for increasing Reynolds numbers. The parameters defining the **Figure 4.** Streamlines (white contours) obtained for *A* = 1, *γ* = 0, *θ*2 = –*θ*1 = π/2, and Re*τi* = [32, 128, 512], and temperature field (black contours and colored background; see colorbars for the color scale) corresponding to Pr*<sup>i</sup>* = 5, *K* = 2, *K*w = 0.1, and Δw = 0.1. The parameters defining the thickened virtual plane surface, obtained from the three simulations shown in the figure, are *y*w2 <sup>⊥</sup> <sup>=</sup> <sup>−</sup> *<sup>y</sup>*w1 <sup>⊥</sup> = 0.1032, 0.1084, 0.1167 and Δ*T* = 0.8092, 0.8081, 0.7688].

thickened virtual plane surface obtained from the three simulations shown in the figure are w2 w1 *y y* [ ] 0.1032, 0.1084, 0.1167 and *T* [0.8092, 0.8081, 0.7688] . Figure 5 top shows the effect of the Reynolds number on the virtual no-slip plane wall location for flow transverse to the corrugations, *ywi*<sup>⊥</sup>,given by Eq. (25). As can be seen, the growth of

the recirculation regions observed in 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 *ywi*<sup>⊥</sup> 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 Δ*T* at *y* = 0, defined by Eq. (26). As can be seen, for low wall conductivities Δ*T* takes positive values due to the correspondingly large wall thermal resistance. However, for high wall conductivities Δ*T* 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*<sup>w</sup> >> 1, leading to the negative values of Δ*T* reported in the last plot of Figure 5. It is also interesting to note that although Δ*T* 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.

dimensional, a vorticity-streamfunction formulation was adopted for the numerical integra‐ tion, 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

Although the formulation presented above is of great generality, the large number of param‐ eters 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 presents illustrative results for Pr*<sup>i</sup>* = 5, *K* = 2, *K*<sup>w</sup> = 0.1, Δw = 0.1, *A* = 1, and different values of Re*τi* = [32, 128, 512]. As can be seen, the flow exhibits the typical low-velocity recirculation regions within the corrugation troughs, which grow spatially with the Reynolds number. The presence of this quasi-stagnant fluid in the vicinity of the wall modifies the overall wall thermal resistance, thereby changing the effectiveness of the heat

**Figure 4** Streamlines (white contours) obtained for *A* = 1, *γ* = 0, *θ*2 = *θ*1 = π/2, and Re*τi* = [32, 128, 512], and temperature field (black contours and colored background; see colorbars for the color scale)

**Figure 4.** Streamlines (white contours) obtained for *A* = 1, *γ* = 0, *θ*2 = –*θ*1 = π/2, and Re*τi* = [32, 128, 512], and temperature

and Δw = 0.1. The parameters defining the thickened virtual plane surface, obtained from the three simulations shown

Figure 5 top shows the effect of the Reynolds number on the virtual no-slip plane wall location for flow transverse to the corrugations, *ywi*<sup>⊥</sup>,given by Eq. (25). As can be seen, the growth of

−0.5 0 0.5 1

x

0 0.2 0.4 0.6 0.8 1

Re = 128

regions inside the wall corrugations for increasing Reynolds numbers. The parameters defining the thickened virtual plane surface obtained from the three simulations shown in the figure are

<sup>⊥</sup> = 0.1032, 0.1084, 0.1167 and Δ*T* = 0.8092, 0.8081, 0.7688].

w2 w1 *y y* [ ] 0.1032, 0.1084, 0.1167 and *T* [0.8092, 0.8081, 0.7688] .

field (black contours and colored background; see colorbars for the color scale) corresponding to Pr*<sup>i</sup>*

= 5, *K* = 2, *K*w = 0.1, and Δw = 0.1. Note the growth of the low-velocity recirculation

y

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x

−0.5 0 0.5 1

= 5, *K* = 2, *K*w = 0.1,

0 0.2 0.4 0.6 0.8 1

Re = 512

of *y*w*<sup>i</sup>* <sup>∥</sup> , *y*wi

⊥, and Δ*T*.

14 Heat Transfer Studies and Applications

corresponding to Pr*<sup>i</sup>*

in the figure, are *y*w2

x

−0.5 0 0.5 1

<sup>⊥</sup> <sup>=</sup> <sup>−</sup> *<sup>y</sup>*w1

0 0.2 0.4 0.6 0.8 1

Re = 32

y

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

exchange process in a way that is to be determined below.

y

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

**Figure 5** Top: Variation with the Reynolds number of the apparent plane wall location for flow transverse to the corrugations obtained numerically for *A* = 1, *γ* = 0. Bottom: Variation with the Reynolds number of **Figure 5.** Top: Variation with the Reynolds number of the apparent plane wall location for flow transverse to the corru‐ gations obtained numerically for *A* = 1, *γ* = 0. Bottom: Variation with the Reynolds number of the normalized apparent

low (left) and high (right) conductive walls. The open symbols indicate values obtained from the

= 5, *K*, *Κ*w and Δw representative of

the normalized apparent wall temperature jump Δ*T* for values of Pr*<sup>i</sup>*

simulations of Figure 4.

wall temperature jump Δ*T* for values of Pr*<sup>i</sup>* = 5, *K*, *Κ*w and Δw representative of low (left) and high (right) conductive walls. The open symbols indicate values obtained from the simulations of Figure 4.

#### **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

$$
\boldsymbol{\Delta}\_{\rm w}^{\rm \!\!\!} = \boldsymbol{\mathcal{y}}\_{\rm w}^{\rm \!\!\!} + \boldsymbol{\Delta}\_{\rm w} - \boldsymbol{\mathcal{y}}\_{\rm w}^{\rm \!\!\!\!} > \boldsymbol{\Delta}\_{\rm w}, \tag{27}
$$

with an apparent wall temperature jump

$$
\Delta T^\prime = K^{-1} \mathbf{y}\_{\text{w2}}^\perp + \Delta T - \mathbf{y}\_{\text{w1}}^\perp > \Delta T. \tag{28}
$$

This virtual wall separates two fluid channels of apparent half-widths

$$\left| a\_i^\dagger = a\_i - \lambda \left| \mathbf{y}\_{\mathbf{w},i}^\perp \right| = a\_i \left( 1 - \frac{\lambda}{a\_i} \middle| \mathbf{y}\_{\mathbf{w},i}^\perp \right| \right) < a\_i,\tag{29}$$

smaller than the average channel half-widths, *ai* , 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*<sup>i</sup>* = *λ*/*ai* , 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

$$k^\* = \frac{\mathbf{k}\_2 a\_1^\*}{\mathbf{k}\_1 a\_2^\*} = \frac{\mathbf{k}\_2 a\_1}{\mathbf{k}\_1 a\_2} \left(\frac{a\_1^\* \;/\; a\_1}{a\_2^\* \;/\; a\_2}\right) = k \frac{\left(1 - \frac{\mathcal{\lambda}}{a\_1} \left|\mathbf{y}\_{\mathbf{w},1}^\perp\right|\right)}{\left(1 - \frac{\mathcal{\lambda}}{a\_2} \left|\mathbf{y}\_{\mathbf{w},2}^\perp\right|\right)}.\tag{30}$$

The apparent value of *κ*<sup>w</sup> \* =(*k*<sup>w</sup> / *<sup>δ</sup>w*)\* (*a*1 \* / *k*1) can be determined from the dimensional apparent wall temperature jump (*q<sup>λ</sup>* / *<sup>k</sup>*1)Δ*<sup>T</sup>* \* , where Δ*<sup>T</sup>* \* is given by Eq. (28)in terms of the computed

Wall Conduction Effects in Laminar Counterflow Parallel-Plate Heat Exchangers with Small-Scale Wall Corrugations http://dx.doi.org/10.5772/60550 17

wall temperature jump Δ*T* for values of Pr*<sup>i</sup>*

16 Heat Transfer Studies and Applications

**4. Combined multi-scale model**

with an apparent wall temperature jump

smaller than the average channel half-widths, *ai*

Section 2.

parameter

The apparent value of *κ*<sup>w</sup>

walls. The open symbols indicate values obtained from the simulations of Figure 4.

corrugated wall behaves as a virtual plane wall of apparent thickness

\* 1

This virtual wall separates two fluid channels of apparent half-widths

l

\* \* \* 21 21 1 1 \* \* 12 12 2 2

> (*a*1 \*

\* =(*k*<sup>w</sup> / *<sup>δ</sup>w*)\*

plane wall concept will be denoted by a superscript asterisk (\*

Note that the dimensionless corrugation wavelength, Λ*<sup>i</sup>*

\*

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

> w,i w,i \* 1 ,

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

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

Using Eq. (29) to evaluate the apparent channel half-widths, we obtain the apparent value of

kk / . kk /

wall temperature jump (*q<sup>λ</sup>* / *<sup>k</sup>*1)Δ*<sup>T</sup>* \* , where Δ*<sup>T</sup>* \* is given by Eq. (28)in terms of the computed

ç ÷ - æ ö è ø = = ç ÷ <sup>=</sup> æ ö è ø ç ÷ -

*<sup>y</sup> a aaa <sup>a</sup> k k a aaa <sup>y</sup> <sup>a</sup>*

 ^ ^ æ ö =- = - < ç ÷

l

è ø *<sup>i</sup>*

= 5, *K*, *Κ*w and Δw representative of low (left) and high (right) conductive

w w 2 w w1 , ^ ^ D + D - > D= *<sup>w</sup> y y* (27)

w 2 w1 . - ^ ^ *T Ky* = +D D - *yT* D> *T* (28)

*i i <sup>i</sup> <sup>i</sup> aa y a y a <sup>a</sup>* (29)

).

= *λ*/*ai*

2

è ø

l

l

æ ö

1

1

w,1 1

^

w,2

/ *k*1) can be determined from the dimensional apparent

^

, as a result of the apparent wall displacement

, appearing explicitly in Eq. (29)

(30)

**Figure 6.** A corrugated heat conducting wall of finite thickness, *δ*<sup>w</sup> and thermal conductivity, *k*<sup>w</sup> separates the channels of fluids 1 and 2 with average half-widths *a*1 and *a*2, respectively. The wall corrugation amplitude and wavelength are both assumed to be small compared to *ai* .

values of *y*w*<sup>i</sup>* <sup>⊥</sup> and Δ*<sup>T</sup>* . Multiplying the apparent wall temperature jump by (*k*<sup>w</sup> / *<sup>δ</sup>w*)\*=*κ*<sup>w</sup> \* (*k*<sup>1</sup> / *a*<sup>1</sup> \* ) we should recover the overall heat flux transferred by conduction from fluid 2 to fluid 1, i.e., *<sup>q</sup>* =(*q<sup>λ</sup>* / *<sup>k</sup>*1)Δ*<sup>T</sup>* \* *κ*w \* (*k*<sup>1</sup> / *a*<sup>1</sup> \* ), whence solving for *κ*<sup>w</sup> \* we get

$$\boldsymbol{\kappa}\_{\boldsymbol{w}}^{\*} = \frac{\boldsymbol{a}\_{1}^{\*}}{\lambda \Delta T} = \frac{\boldsymbol{a}\_{1}}{\lambda \Delta T} \left( 1 - \frac{\lambda}{\boldsymbol{a}\_{1}} \left| \boldsymbol{\mathcal{y}}\_{\text{w},1}^{\perp} \right| \right) = \underbrace{\left( \frac{\boldsymbol{a}\_{1}}{\mathcal{S}\_{\text{w}}} \frac{\boldsymbol{k}\_{\text{w}}}{\boldsymbol{k}\_{1}} \right)}\_{\boldsymbol{\kappa}\_{\boldsymbol{w}}} \underbrace{\left| \left( \frac{\boldsymbol{\mathcal{S}}\_{\text{w}}}{\mathcal{A}} \frac{\boldsymbol{k}\_{1}}{\boldsymbol{k}\_{w}} \right) \frac{1}{\Delta T} \left( 1 - \frac{\lambda}{\boldsymbol{a}\_{1}} \left| \boldsymbol{\mathcal{y}}\_{\text{w},1}^{\perp} \right| \right) \right|}\_{\boldsymbol{\kappa} \Delta T} = \frac{\boldsymbol{\kappa}\_{\boldsymbol{w}}}{\Delta T} \left( 1 - \frac{\lambda}{\boldsymbol{a}\_{1}} \left| \boldsymbol{\mathcal{y}}\_{\text{w},1}^{\perp} \right| \right). \tag{31}$$

In the above expression we have expressed *κ*<sup>w</sup> \* in terms of the normalized apparent wall temperature jump

$$\overline{\Delta T} = \frac{\Delta T^\circ}{\Delta\_\text{w} \,/\, K\_\text{w}} \tag{32}$$

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 <sup>⊥</sup> also tends to zero, Eq. (31) reduces to *κ*w \* <sup>=</sup>*κ*<sup>w</sup> <sup>≡</sup>(*k*<sup>w</sup> / *<sup>δ</sup>*w)(*a*<sup>1</sup> / *<sup>k</sup>*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, Δw << 1, and Λ << 1 verify the following relation

$$\boldsymbol{\kappa}\_{\rm w} = \frac{\boldsymbol{a}\_{\rm l}}{\boldsymbol{\delta}\_{\rm w}} \frac{\boldsymbol{k}\_{\rm w}}{\boldsymbol{k}\_{\rm l}} = \frac{\boldsymbol{a}\_{\rm l}}{\boldsymbol{\lambda}} \frac{\boldsymbol{\lambda}}{\boldsymbol{\delta}\_{\rm w}} \frac{\boldsymbol{k}\_{\rm w}}{\boldsymbol{k}\_{\rm l}} = \frac{\boldsymbol{K}\_{\rm w}}{\boldsymbol{\Lambda} \boldsymbol{\Delta}\_{\rm w}} \tag{33}$$

which provides *κ*w explicitly in terms of *K*w, Δw, and Λ.

Figure 7 shows the influence of the Reynolds number, Re*τi*, on the normalized apparent wall temperature jump, Δ¯ *T* , for the same values of *K*, *Κ*w and Δ<sup>w</sup> considered in Figure 5. As can be seen, for low-conductive walls Δ¯ *T* 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 Δ¯ *T* 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, Δ¯ *T* exhibits a non-monotonous behavior with Re*τi* that ends in an abrupt reduction for Reynolds numbers Re*τi* above 300 or so.

**Figure 7** Variation with the Reynolds number of the normalized apparent wall temperature jump *T* for values of Pr*i* = 5, *K*, *Κ*w and Δw representative of low (left) and high (right) conductive walls. The open **Figure 7.** Variation with the Reynolds number of the normalized apparent wall temperature jump <sup>Δ</sup>¯ *T* for values of Pr*<sup>i</sup>* = 5, *K*, *Κ*w and Δ<sup>w</sup> representative of low (left) and high (right) conductive walls. The open symbols correspond to results obtained from the simulations of Figure 4.

symbols indicate values obtained from the simulations of Figure 4. For the plane Poiseuille flow considered in this work, the wall shear stress is given by *τi* = 3*μ<sup>i</sup> Vi* /*ai* , which upon substitution in the expression for Re*τi* yields the relation

$$\text{Re}\_{\boldsymbol{\pi}i} = \frac{\rho\_i \boldsymbol{\lambda}^2 \boldsymbol{\pi}\_i}{\mu\_i} = \frac{3}{2} \left(\frac{\boldsymbol{\lambda}}{\boldsymbol{a}\_i}\right)^2 \frac{2\rho\_i V\_i \boldsymbol{a}\_i}{\mu\_i} = \frac{3}{2} \boldsymbol{\Lambda}\_i^2 \text{Re}\_i,\tag{34}$$

stating that the ratio between the Reynolds numbers Re*τi* and Re*<sup>i</sup>* equals 3/2 times the square of the dimensionless wall corrugation wavelength, Λ*<sup>i</sup>* = *λ*/*a<sup>i</sup>* << 1. Since in plane Poiseuille flow finite-amplitude disturbances are known to drive transition to turbulence at Reynolds numbers as low as Re*<sup>i</sup>*,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*<sup>i</sup>* considered below (Λ*<sup>i</sup>* = 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 Δ¯ *T* remains almost constant, with convective effects accounting only for small variations of Δ¯ *T* of order 1% for *Κ*<sup>w</sup> = 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*<sup>w</sup> \* using Eq. (31) with the value of Δ¯ *T* corresponding to Re*τi* = 0.

which provides *κ*w explicitly in terms of *K*w, Δw, and Λ.

in an abrupt reduction for Reynolds numbers Re*τi* above 300 or so.

symbols indicate values obtained from the simulations of Figure 4.

rlt

m

<sup>100</sup> <sup>101</sup> <sup>10</sup><sup>2</sup> <sup>10</sup><sup>3</sup> 0.91

obtained from the simulations of Figure 4.

Reτ i

t

of the dimensionless wall corrugation wavelength, Λ*<sup>i</sup>*

K = 2, K<sup>w</sup> = 0.1, Δ<sup>w</sup> = 0.1

temperature jump, Δ¯

18 Heat Transfer Studies and Applications

0.92 0.93 0.94 0.95 0.96 0.97 0.98

ΔT

3*μ<sup>i</sup> Vi* /*ai*

seen, for low-conductive walls Δ¯

temperature jump. In both cases, Δ¯

Figure 7 shows the influence of the Reynolds number, Re*τi*, on the normalized apparent wall

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 Δ¯

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

**Figure 7.** Variation with the Reynolds number of the normalized apparent wall temperature jump <sup>Δ</sup>¯

, which upon substitution in the expression for Re*τi* yields the relation

<sup>2</sup> <sup>2</sup> 32 3 <sup>2</sup> Re Re , 2 2

æ ö = = ç ÷ = L è ø *i i iii i i i i ii*

 l

**Figure 7** Variation with the Reynolds number of the normalized apparent wall temperature jump *T* for values of Pr*i* = 5, *K*, *Κ*w and Δw representative of low (left) and high (right) conductive walls. The open

= 5, *K*, *Κ*w and Δ<sup>w</sup> representative of low (left) and high (right) conductive walls. The open symbols correspond to results

For the plane Poiseuille flow considered in this work, the wall shear stress is given by *τi* =

 r

stating that the ratio between the Reynolds numbers Re*τi* and Re*<sup>i</sup>* equals 3/2 times the square

finite-amplitude disturbances are known to drive transition to turbulence at Reynolds numbers as low as Re*<sup>i</sup>*,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*<sup>i</sup>* considered below (Λ*<sup>i</sup>* = 0.25 << 1) the maximum allowable value of Re*τi* turns out to

 m

*V a*

= *λ*/*a<sup>i</sup>*

ΔT

*T* , for the same values of *K*, *Κ*w and Δ<sup>w</sup> considered in Figure 5. As can be

*T* remains slightly below unity regardless of the Reynolds

*T* exhibits a non-monotonous behavior with Re*τi* that ends

<sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup> <sup>100</sup>

*<sup>a</sup>* (34)

<< 1. Since in plane Poiseuille flow

Reτ i

*T* for values of Pr*<sup>i</sup>*

K = 2, K<sup>w</sup> = 100, Δ<sup>w</sup> = 0.05

*T*

To quantify this effect, let us consider the ratio between the overall heat transfer rate, *m*˙ *c*1(*T*m1,out−*T*1,in), and the mechanical power required to drive the flow, characterized here by the pumping power, Δ*P*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

$$R\_u = \left[\frac{\frac{\dot{m}\_1 c\_1 \left(T\_{\text{m1,out}} - T\_{\text{t,in}}\right)}{\Delta P\_1 Q\_1}}{\left[\frac{\dot{m}\_1 c\_1 \left(T\_{\text{m1,out}} - T\_{\text{t,in}}\right)}{\Delta P\_1 Q\_1}\right]\_{\text{flat}}}\right]\_{\text{flat}}\tag{35}$$

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 = (Tm1,out–T1,in)/(T2,in–T1,in), when fluid 1 has the smallest heat capacity flow rate. For the plane Poiseuille flow considered in this work the pressure drop, Δ*P<sup>i</sup>* , the volume flow rate, *Q<sup>i</sup>* , and the mass flow rate, *m*˙ , are given respectively by

$$
\Delta P\_i = \frac{3\mu\_i V\_i}{a\_i^2} L, \quad \underline{Q}\_i = 2a\_i V\_i, \quad \dot{m}\_i = \rho\_i \underline{Q}\_i. \tag{36}
$$

Using these expressions in Eq. (35), the overall augmentation heat-transfer number can be expressed as

$$R\_a = \left(\frac{a\_1^\*}{a\_1}\right)^2 \frac{V\_1}{V\_1^\*} \frac{\varepsilon^\*}{\varepsilon}. \tag{37}$$

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

$$N\_a = \frac{\dot{S}\_{\text{corr}}}{\dot{S}\_{\text{flat}}} = \frac{\left(\dot{S}/\dot{m}\_\text{l}c\_1\right)\_{\text{corr}}}{\left(\dot{S}/\dot{m}\_\text{l}c\_1\right)\_{\text{flat}}} \frac{\left(\dot{m}\_\text{l}c\_1\right)\_{\text{corr}}}{\left(\dot{m}\_\text{l}c\_1\right)\_{\text{flat}}} = \frac{\left(\dot{S}/\dot{m}\_\text{l}c\_1\right)\_{\text{corr}}}{\left(\dot{S}/\dot{m}\_\text{l}c\_1\right)\_{\text{flat}}} \left(\frac{Q\_1^\*}{Q\_1}\right) = \frac{\left(\dot{S}/\dot{m}\_\text{l}c\_1\right)\_{\text{corr}}}{\left(\dot{S}/\dot{m}\_\text{l}c\_1\right)\_{\text{flat}}} \left(\frac{a\_1^\*V\_1^\*}{a\_1V\_1}\right) \tag{38}$$

defined as the ratio of the entropy generation rates with corrugated and flat plates. In the above expression *S*˙ / *<sup>m</sup>*˙ <sup>1</sup>*c*1 denotes the dimensionless entropy generation rate, which is approximated by

$$\begin{split} \frac{\dot{S}}{\dot{m}\_{l}c\_{1}} \stackrel{\scriptstyle \dot{S}}{=} \frac{\dot{S}\_{1} + \dot{S}\_{2}}{\dot{m}\_{l}c\_{1}} \stackrel{\scriptstyle \dot{m}}{=} \ln \frac{T\_{\text{m1,out}}}{T\_{1,\text{in}}} + \frac{\widetilde{\dot{m}\_{l}c\_{2}}}{\dot{m}\_{l}c\_{1}} \ln \frac{T\_{\text{m2,out}}}{T\_{2,\text{in}}} = \ln \left[ \frac{T\_{\text{m1,out}}}{T\_{1,\text{in}}} \left( \frac{T\_{\text{m2,out}}}{T\_{2,\text{in}}} \right)^{\text{vol}} \right] \\ &= \ln \left[ \left( 1 + \mathcal{E} \frac{\Delta T}{T\_{1,\text{in}}} \right) \left( \frac{1}{1 + \Delta T/T\_{1,\text{in}}} \right)^{\text{vol}} \left( 1 + \left( 1 - \frac{\mathcal{E}}{mk} \right) \frac{\Delta T}{T\_{1,\text{in}}} \right)^{\text{vol}} \right], \end{split} \tag{39}$$

where entropy changes associated with viscous dissipation (i.e., pressure drop irreversibility) and heat conduction in the wall have both been neglected. These changes should be retained in a detailed second-law analysis of heat exchanger performance, which is not to be pursued in this work, as they may be critical for the thermodynamic optimization of the heat transfer equipment [29]. It should be noted that the evaluation of the dimensionless entropy generation rate requires the introduction of the dimensionless temperature difference Δ*T*/*T*1,in = (*T*2,in–*T*1,in)/ *T*1,in or, alternatively, the temperature ratio *T*2,in/*T*1,in = Δ*T*/*T*1,in + 1, as an additional parameter of the problem.

Combining Eqs. (38) and (39), we obtain the final expression for the augmentation entropy generation number

$$N\_{a} = \frac{\ln\left[\left(1+\varepsilon^{\*}\frac{\Delta T}{T\_{1,\text{in}}}\right)\left(\frac{1}{1+\Delta T/T\_{1,\text{in}}}\right)^{n^{\*}\lambda^{\*}}\left(1+\left(1-\frac{\varepsilon^{\*}}{m^{\*}\lambda^{\*}}\right)\frac{\Delta T}{T\_{1,\text{in}}}\right)^{n^{\*}\lambda^{\*}}\right]}{\ln\left[\left(1+\varepsilon\frac{\Delta T}{T\_{1,\text{in}}}\right)\left(\frac{1}{1+\Delta T/T\_{1,\text{in}}}\right)^{\text{int}}\left(1+\left(1-\frac{\varepsilon}{mk}\right)\frac{\Delta T}{T\_{1,\text{in}}}\right)^{\text{int}}\right]}\right] \tag{40}$$

The remaining two apparent parameters of the bulk flow model, i.e., *m\** and *ξ<sup>L</sup> \** , involve by definition the Peclet numbers of the fluid streams. To evaluate their apparent values we must therefore specify the apparent Peclet numbers, Pe*<sup>i</sup>* \* , 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 *Qi* = 2*ai Vi* of fluid *i* is the same with flat and corrugated walls, the average flow velocity *Vi* \* must grow to compensate for the reduction of the channel half-width

Wall Conduction Effects in Laminar Counterflow Parallel-Plate Heat Exchangers with Small-Scale Wall Corrugations http://dx.doi.org/10.5772/60550 21

$$V\_i^\* \mathbf{a}\_i^\* = V\_i \mathbf{a}\_i \quad \Longrightarrow \quad V\_i^\* = V\_i \frac{\mathbf{a}\_i}{a\_i} = V\_i \left(1 - \frac{\mathcal{L}}{a\_i} \left| \mathbf{y}\_{\mathbf{w},i}^\perp \right| \right)^{-1}. \tag{41}$$

In this case, the apparent value of Pe*<sup>i</sup>* , defined with the apparent velocities and channel halfwidths, remains unchanged

$$\text{Pe}\_{i}^{\*} = \frac{\text{2}a\_{i}^{\*}V\_{i}^{\*}}{\alpha\_{i}} = \frac{\text{2}a\_{i}V\_{i}}{\alpha\_{i}} = \text{Pe}\_{i},\tag{42}$$

while the apparent values of *m* and *ξL* are given by

defined as the ratio of the entropy generation rates with corrugated and flat plates. In the above expression *S*˙ / *<sup>m</sup>*˙ <sup>1</sup>*c*1 denotes the dimensionless entropy generation rate, which is

*mk mk*

where entropy changes associated with viscous dissipation (i.e., pressure drop irreversibility) and heat conduction in the wall have both been neglected. These changes should be retained in a detailed second-law analysis of heat exchanger performance, which is not to be pursued in this work, as they may be critical for the thermodynamic optimization of the heat transfer equipment [29]. It should be noted that the evaluation of the dimensionless entropy generation rate requires the introduction of the dimensionless temperature difference Δ*T*/*T*1,in = (*T*2,in–*T*1,in)/ *T*1,in or, alternatively, the temperature ratio *T*2,in/*T*1,in = Δ*T*/*T*1,in + 1, as an additional parameter

Combining Eqs. (38) and (39), we obtain the final expression for the augmentation entropy

1,in 1,in 1,in

*T T T TT mk T*

e

, in the heat exchanger with corrugated

of fluid *i* is the same with flat and corrugated

\* must grow to compensate for the reduction of the channel

.

 and *ξ<sup>L</sup> \** (39)

(40)

, involve by

*mk mk*

<sup>1</sup> ln 1 1 1 , <sup>1</sup>

é ù æ öæ ö æ D D <sup>ö</sup> æ ö = + ê ú ç ÷ç ÷ ç + - <sup>÷</sup> ç ÷ + D <sup>÷</sup> ê ú è ø è øè ø è <sup>ø</sup> ë û

\* \* \* \* \*

*m k m k*

e

e

1,in 1,in 1,in \* \*

*a V T T*

\*

1,in 1,in 1,in

definition the Peclet numbers of the fluid streams. To evaluate their apparent values we must

walls. To this end, we shall consider three different flow constraints, namely: i) constant flow

 = 2*ai Vi*

*T TT mk T*

æ öæ ö æ D D æ ö <sup>ö</sup> ê ú ç ÷ç ÷ ç <sup>+</sup> + - ç ÷ <sup>÷</sup> ê ú + D <sup>÷</sup> è øè ø è è ø <sup>ø</sup> æ ö ë û <sup>=</sup> ç ÷ é ù æ öæ ö æ D D <sup>ö</sup> è ø æ ö ê ú ç ÷ç ÷ ç <sup>+</sup> + - ç ÷ <sup>÷</sup> + D <sup>÷</sup> ê ú è ø è øè ø è <sup>ø</sup> ë û

*T T*

é ù

*T TT mk T a V <sup>N</sup>*

\* \*

ç ÷ ê ú è ø ë û

}

*S SS T T TT m c mc mc T mc T T T*

11 11 1,in 1 1 2,in 1,in 2,in

ln ln ln

1 2 m1,out 2 2 m2,out m1,out m2,out

e

é ù <sup>+</sup> æ ö @@ + = ê ú ç ÷

approximated by

20 Heat Transfer Studies and Applications

of the problem.

generation number

\*

e

therefore specify the apparent Peclet numbers, Pe*<sup>i</sup>*

**i. Constant flow rate**

half-width

To ensure that the volume flow rate *Qi*

walls, the average flow velocity *Vi*

e

<sup>1</sup> ln 1 1 1 1

<sup>1</sup> ln 1 1 1 1

*a mk mk*

The remaining two apparent parameters of the bulk flow model, i.e., *m\**

rate, ii) constant pumping power, and iii) constant pressure drop.

& && & && &

$$m^\* = \frac{\operatorname{Pe}\_2^\* a\_2^\*}{\operatorname{Pe}\_1^\* a\_1^\*} = \frac{\operatorname{Pe}\_2 a\_2}{\operatorname{Pe}\_1 a\_1} \left( \frac{a\_2^\* \mid a\_2}{a\_1^\* \mid a\_1} \right) = m \frac{\left( 1 - \frac{\lambda}{a\_2} \left| \mathbf{y}\_{w,2}^\perp \right| \right)}{\left( 1 - \frac{\lambda}{a\_1} \left| \mathbf{y}\_{w,1}^\perp \right| \right)} \implies m^\* k^\* = mk,\tag{43}$$

$$\boldsymbol{\xi}\_{\boldsymbol{L}}^{\*} = \frac{\boldsymbol{L}}{\boldsymbol{a}\_{\boldsymbol{l}}^{\*} \mathbf{P} \mathbf{e}\_{\boldsymbol{l}}^{\*}} = \frac{\boldsymbol{L}}{\boldsymbol{a}\_{\boldsymbol{l}} \mathbf{P} \mathbf{e}\_{\boldsymbol{l}}} \left( \frac{\boldsymbol{a}\_{\boldsymbol{l}}}{\boldsymbol{a}\_{\boldsymbol{l}}^{\*}} \right) = \boldsymbol{\xi}\_{\boldsymbol{L}}^{\*} \left( \boldsymbol{l} - \frac{\boldsymbol{\mathcal{L}}}{\boldsymbol{a}\_{\boldsymbol{l}}} \left| \boldsymbol{\mathcal{y}}\_{\boldsymbol{w},\boldsymbol{l}}^{\perp} \right| \right)^{-1}. \tag{44}$$

Using (29) and (41) in (37), the overall augmentation heat-transfer parameter takes the form

$$R\_a = \left(1 - \frac{\mathcal{L}}{a\_1} \left| \boldsymbol{\nu}\_{\rm w, 1}^\perp \right| \right)^3 \frac{\boldsymbol{\mathcal{E}}}{\boldsymbol{\mathcal{E}}}. \tag{45}$$

#### **ii. Constant pumping power**

The pumping power is equal to the product of the volume flow rate, 2*ai Vi* , by the pressure drop, 3*μi LVi* /*ai* 2 . To ensure that the pumping power remains the same with flat and corrugated walls, the average flow velocity *Vi* \* of fluid *i* must satisfy the condition

$$\frac{V\_i^{\*2}}{a\_i^{\*}} = \frac{V\_i^{\*2}}{a\_i} \quad \Longrightarrow \quad V\_i^{\*} = V\_i \left(\frac{a\_i^{\*}}{a\_i}\right)^{\frac{1}{2}} = V\_i \left(1 - \frac{\mathcal{L}}{a\_i} \left| \boldsymbol{\nu}\_{\text{w},i}^{\perp} \right| \right)^{\frac{1}{2}}.\tag{46}$$

In this case, the apparent values of Pe*<sup>i</sup>* , *m,* and *ξL* are given by

$$\mathbf{Pe}\_{i}^{\*} = \frac{\mathbf{2}a\_{i}^{\*}V\_{i}^{\*}}{\alpha\_{i}} = \frac{\mathbf{2}a\_{i}V\_{i}}{\alpha\_{i}} \left(\frac{a\_{i}^{\*}}{a\_{i}}\frac{V\_{i}^{\*}}{V\_{i}}\right) = \mathbf{Pe}\_{i} \left(1 - \frac{\mathcal{A}}{a\_{i}} \left|\mathbf{y}\_{\mathbf{w},i}^{\perp}\right|\right)^{\frac{3}{2}},\tag{47}$$

$$m^\* = m \left[ \frac{1 - \frac{\mathcal{L}}{a\_2} \left| \boldsymbol{\mathcal{y}}\_{\boldsymbol{w}, \boldsymbol{z}}^\perp \right|^\frac{1}{2}}{1 - \frac{\mathcal{L}}{a\_1} \left| \boldsymbol{\mathcal{y}}\_{\boldsymbol{w}, \boldsymbol{l}}^\perp \right|} \right]^\frac{1}{2} \implies \quad m^\* \boldsymbol{k}^\* = mk \left[ \frac{1 - \frac{\mathcal{L}}{a\_2} \left| \boldsymbol{\mathcal{y}}\_{\boldsymbol{w}, \boldsymbol{z}}^\perp \right|}{1 - \frac{\mathcal{L}}{a\_1} \left| \boldsymbol{\mathcal{y}}\_{\boldsymbol{w}, \boldsymbol{l}}^\perp \right|} \right]^\frac{1}{2},\tag{48}$$

$$
\xi\_L^\* = \xi\_L^\* \left( 1 - \frac{\mathcal{A}}{a\_1} \left| \boldsymbol{\nu}\_{\mathbf{w}, 1}^\perp \right| \right)^{-\frac{\mathcal{S}}{2}},
\tag{49}
$$

and, using (29) and (46) in (37), the overall augmentation heat-transfer number takes the form

$$R\_a = \left(1 - \frac{\lambda}{a\_1} \left| \boldsymbol{\nu}\_{\rm w, 1}^\perp \right| \right)^{\frac{3}{2}} \frac{\boldsymbol{\mathcal{E}}}{\boldsymbol{\mathcal{E}}}. \tag{50}$$

#### **iii. Constant pressure drop**

To ensure that pressure drop Δ*Pi* =3*μi LVi* /*ai* 2 of fluid *i* remains the same with flat and corrugated walls, the apparent average flow velocity *Vi \** of fluid *i* must satisfy the condition

$$\frac{V\_i^\*}{a\_i^{\*\prime 2}} = \frac{V\_i}{a\_i^2} \quad \Longrightarrow \quad V\_i^\* = V\_i \left(\frac{a\_i^\*}{a\_i}\right)^2 = V\_i \left(1 - \frac{\mathcal{L}}{a\_i} \left|\mathbf{y}\_{w,i}^\perp\right|\right)^2. \tag{51}$$

In this case, the apparent values of *Pei* , *m,* and *ξL* are given by

$$Pe\_i^\* = Pe\_i \left( 1 - \frac{\mathcal{L}}{a\_i} \left| \boldsymbol{\nu}\_{\text{w},i}^\perp \right| \right)^3,\tag{52}$$

$$m^\* = m \left[ \frac{1 - \frac{\mathcal{L}}{a\_2} \left| \boldsymbol{\mathcal{V}}\_{\boldsymbol{w}, \boldsymbol{z}}^\perp \right|}{1 - \frac{\mathcal{L}}{a\_1} \left| \boldsymbol{\mathcal{V}}\_{\boldsymbol{w}, \boldsymbol{l}}^\perp \right|} \right]^4 \implies \quad m^\* k^\* = mk \left[ \frac{1 - \frac{\mathcal{L}}{a\_2} \left| \boldsymbol{\mathcal{V}}\_{\boldsymbol{w}, \boldsymbol{z}}^\perp \right|}{1 - \frac{\mathcal{L}}{a\_1} \left| \boldsymbol{\mathcal{V}}\_{\boldsymbol{w}, \boldsymbol{l}}^\perp \right|} \right]^3,\tag{53}$$

Wall Conduction Effects in Laminar Counterflow Parallel-Plate Heat Exchangers with Small-Scale Wall Corrugations http://dx.doi.org/10.5772/60550 23

$$
\xi\_L^\* = \xi\_L \left( 1 - \frac{\mathcal{A}}{a\_1} \left| \mathbf{y}\_{\mathbf{w},l}^\perp \right| \right)^{\prec d}, \tag{54}
$$

and, using (29) and (51) in (37), the overall augmentation heat-transfer number takes the form

$$R\_a = \frac{\mathcal{E}}{\mathcal{E}}.\tag{55}$$

The expressions derived above provide the apparent values *m*\* , *k*\* , *κ*<sup>w</sup> \* , and *ξ<sup>L</sup>* \* in terms of the parameters of the bulk flow model *m*, *k*, *κ*w, and *ξ*<sup>L</sup> corresponding to a flat heat exchanger, and of the numerically computed values of *y*w,*<sup>i</sup>* <sup>⊥</sup> and Δ¯ *T* . These apparent values can then be used in the expressions of the bulk flow model provided in Section 2 to determine the outlet bulk temperature of fluid 1, which provides, in particular, the heat exchanger effectiveness *ε\** = *θ*m1,out(*m*\* , *k*\* , *κ*<sup>w</sup> \* , *ξ<sup>L</sup>* \* ) needed to evaluate *Ra* and *Na*. Analytic expressions for *Na* can be easily derived from Eq. (40) by substituting the values of *a*<sup>1</sup> \* *V*1 \* / *a*1*V* =Pe1 \* / Pe1 and *m*\* *k*\* reported in each case, but they have been omitted here for brevity.

#### **5. Results and discussion**

3

,

(47)

(48)

(53)

w,

*i*

l

 l

 l

è ø *L L <sup>y</sup> <sup>a</sup>* (49)

*R y <sup>a</sup> <sup>a</sup>* (50)

of fluid *i* must satisfy the condition

*Pe Pe y i i <sup>a</sup>* (52)

 l

 l ,

l^ æö æ ö

*i i V V <sup>a</sup> VV V y aa a a* (51)

èø è ø

3

4 3

, *m,* and *ξL* are given by

w,

w,2 w,2 2 2

^ ^

w,1 w,1

^ ^

of fluid *i* remains the same with flat and corrugated

è øè ø

5 3 2 2

5


*i ii*

*<sup>y</sup> aV a*

w,2 w,2 2 2

^ ^

w,1 w,1 \*

w,1

and, using (29) and (46) in (37), the overall augmentation heat-transfer number takes the form

w,1 1

*\**

2 2 \* \* \*2 w,

= Þ = =- ç÷ ç ÷

\* 1 , l ^ æ ö = - ç ÷ è ø *<sup>i</sup> i*

1 1

*y y a a*

éù éù - - êú êú

1 1

*y y a a m m m k mk*

1 1


*i i ii*

<sup>2</sup> 1 .

^ æ ö = - ç ÷ è ø

3 \*

2 1 . l

 e

e

^ ^

1 1

*y y a a*

éù éù - - êú êú

\* \* \* \* <sup>2</sup> \*

^ æ öæ ö = = ç ÷ç ÷ = -

2 2 Pe Pe 1 ,

1 1

*y y a a m m m k mk*

1 1


> <sup>2</sup> \* 1 1 , l

æ ö = - ç ÷

^

a

l

l

*i*

22 Heat Transfer Studies and Applications

**iii. Constant pressure drop**

To ensure that pressure drop Δ*Pi*

walls, the apparent average flow velocity *Vi*

*i*

In this case, the apparent values of *Pei*

 a

\* \*

x x

=3*μi LVi* /*ai* 2

*i i i i*

\* \* \*

= Þ=

l

l

\*

= Þ=

*i ii*

*i i ii i i*

*aV aV a V*

In this work we shall restrict our attention to the case of flow transverse (|*θ<sup>i</sup>* | = *π*/2) to sinusoidal corrugations (*γ* = 0, *y*w,2 <sup>⊥</sup> <sup>=</sup> <sup>−</sup> *<sup>y</sup>*w,1 <sup>⊥</sup> ) with equal channel half-widths (*a*1 = *a*2). The analysis of other flow configurations is left for future work. Due to the geometrical symmetries of the problem, the apparent channel half-width can be expressed as

$$a\_i^\star = a\_i - \mathcal{A} \left| \mathbf{y}\_{w,i}^\perp \right| = a\_i \left( 1 - \Lambda \left| \mathbf{y}\_{w,i}^\perp \right| \right) \tag{56}$$

in terms of the single small parameter Λ=*λ* / *a*<sup>1</sup> =*λ* / *a*2≪1 representing the dimensionless corrugation wavelength. In this case, the expressions for *m*\* , *k*\* , *κ*<sup>w</sup> \* , and *ξ<sup>L</sup>* \* , as well as those for the heat-transfer and entropy augmentation numbers *Ra* and *Na*, reduce to those given in Table 1.

Table 2 shows the variation of *κ*<sup>w</sup> \* with the dimensionless corrugation amplitude, *A*/2π, the dimensionless wall thickness, Δw, and the wall-to-fluid and fluid-to-fluid heat conductivity ratios, *Kw* and *K*. The results are presented in terms of the ratio *κ*<sup>w</sup> \* / *<sup>κ</sup>*<sup>w</sup> =(1−Λ<sup>|</sup> *<sup>y</sup>*w,1 <sup>⊥</sup> |)/ <sup>Δ</sup>¯ *T* , with *κ*w \* expressed as percentage of its reference value *κ*<sup>w</sup> =(*k*<sup>w</sup> / *δ*w)(*a*<sup>1</sup> / *k*1) for a flat wall of finite thickness, *δ*w, and thermal conductivity, *k*w. Different wall designs and conductivity ratios are


**Table 1.** Expressions determining the apparent values of *m*\* , *k*\* , *κ*<sup>w</sup> \* , and *ξ<sup>L</sup>* \* to be used in the bulk flow model to evaluate the effect of the wall corrugations in the case of flow transverse (|*θ<sup>i</sup>* | = *π*/2) to sinusoidal corrugations (*γ* = 0, *y*w,2 <sup>⊥</sup> <sup>=</sup> <sup>−</sup> *<sup>y</sup>*w,1 <sup>⊥</sup> ) with equal channel half-widths (*a*1 = *a*2). Also given are the expressions for the heat-transfer and entropy augmentation numbers *Ra* and *Na*.

considered, ranging from low to moderate amplitudes and from low- to high-conductive materials. For these and all subsequent calculations, the dimensionless corrugation wave‐ length has been set equal to Λ=*λ* / *ai* =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*<sup>w</sup> \* (i.e., to increase the wall thermal resistance 1/ *κ*<sup>w</sup> \* ). This effect is accentuated for highly conductive walls (Δ<sup>w</sup> << 1, *K*<sup>w</sup> >> 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, *Kw* << 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<sup>L</sup>* \* . The results are presented in terms of the ratio *ξ<sup>L</sup>* \* / *<sup>ξ</sup><sup>L</sup>* and also as percentage increment of the flat wall value *ξ<sup>L</sup>* for the three cases under study. As can be seen, the presence of the corrugations always increases the apparent dimensionless heat-exchanger length *ξ<sup>L</sup>* \* <sup>=</sup> *<sup>L</sup>* / *<sup>a</sup>*<sup>1</sup> \* *Pe*<sup>1</sup> \* due to the simultaneous reduction of *a*<sup>1</sup> \* and *Pe*<sup>1</sup> \* (except in the case of constant flow rate), with larger increments corresponding to larger corrugation amplitudes under the constraint of constant pressure drop.

Wall Conduction Effects in Laminar Counterflow Parallel-Plate Heat Exchangers with Small-Scale Wall Corrugations http://dx.doi.org/10.5772/60550 25


**Table 2.** Variation of the normalized ratio *κ*<sup>w</sup> \* / *<sup>κ</sup>*<sup>w</sup> =(1−Λ<sup>|</sup> *<sup>y</sup>*w,1 <sup>⊥</sup> |)/ <sup>Δ</sup>¯ *T* (expressed as %) with the dimensionless corrugation amplitude, *A*/2π, the dimensionless wall thicknesses, Δw, and the fluid-to-fluid and wall-to-fluid conductivity ratios, *K* and *K*w, for fixed values of *γ* = 0, Pr*<sup>i</sup>* = 5, Re*<sup>i</sup>* = 0, and Λ = 0.25. The apparent wall positions corresponding to *A* = [0.25, 0.5, 1] are *y*w2<sup>⊥</sup>= – *y*w1<sup>⊥</sup>= [0.0095, 0.0339, 0.1013].

considered, ranging from low to moderate amplitudes and from low- to high-conductive materials. For these and all subsequent calculations, the dimensionless corrugation wave‐

, *k*\* , *κ*<sup>w</sup> \* , and *ξ<sup>L</sup>*

**Constant pumping power (n = 3/2)**

*m\* = m ( =*

<sup>⊥</sup> |) <sup>1</sup> Δ¯

\* <sup>=</sup>*ξ<sup>L</sup>* (1−Λ<sup>|</sup> *<sup>y</sup>*w,1

*Ra* =(1−Λ| *y*w,1

) *mk*

) *mk*

*k* \* =*k* ( = *k*2 *k*1 ≡ *K*)

\* <sup>=</sup>*κ*w(1−Λ<sup>|</sup> *<sup>y</sup>*w,1

*ξL*

)( <sup>1</sup> 1 + Δ*T* / *T*1,in

)( <sup>1</sup> 1 + Δ*T* / *T*1,in

*κ*w

ln (1 <sup>+</sup> *<sup>ε</sup>* \* <sup>Δ</sup>*<sup>T</sup> T*1,in

ln (1 <sup>+</sup> *<sup>ε</sup>* <sup>Δ</sup>*<sup>T</sup> T*1,in

evaluate the effect of the wall corrugations in the case of flow transverse (|*θ<sup>i</sup>*

*Pe2 Pe1 )*

*<sup>T</sup>* with *<sup>κ</sup>*<sup>w</sup> <sup>=</sup> *<sup>K</sup>*<sup>w</sup>

*mk* ) <sup>Δ</sup>*<sup>T</sup> T*1,in ) *mk*

*mk* ) <sup>Δ</sup>*<sup>T</sup> T*1,in )

<sup>⊥</sup> |)−(*n*+1)

<sup>⊥</sup> |)3−*<sup>n</sup> <sup>ε</sup>* \* *ε*

(1 <sup>+</sup> (1<sup>−</sup> *<sup>ε</sup>* \*

(1 <sup>+</sup> (1<sup>−</sup> *<sup>ε</sup>*

<sup>⊥</sup> ) with equal channel half-widths (*a*1 = *a*2). Also given are the expressions for the heat-transfer and entropy

ΛΔ*<sup>w</sup>*

The results shown in Table 2 indicate that the effect of the corrugations is almost invariably to

This effect is accentuated for highly conductive walls (Δ<sup>w</sup> << 1, *K*<sup>w</sup> >> 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, *Kw* << 1) the recirculating fluids have significantly larger conductivity than the wall itself and the wall thermal resistance remains virtually unaffected,

Table 3 shows the effect of the dimensionless corrugation amplitude, *A*/2π, on the apparent

and also as percentage increment of the flat wall value *ξ<sup>L</sup>* for the three cases under study. As can be seen, the presence of the corrugations always increases the apparent dimensionless

in the case of constant flow rate), with larger increments corresponding to larger corrugation

(i.e., to increase the wall thermal resistance 1/ *κ*<sup>w</sup>

*mk* (1−Λ| *y*w,1

<sup>⊥</sup> |)*<sup>n</sup>*

\* to be used in the bulk flow model to


\* . The results are presented in terms of the ratio *ξ<sup>L</sup>*

\* and *Pe*<sup>1</sup> \* (except

**Constant pressure drop (n = 3)**

\* due to the simultaneous reduction of *a*<sup>1</sup>

\* ).

\* / *<sup>ξ</sup><sup>L</sup>*

\*

showing a small reduction as the corrugation amplitude is increased.

\* <sup>=</sup> *<sup>L</sup>* / *<sup>a</sup>*<sup>1</sup> \* *Pe*<sup>1</sup>

amplitudes under the constraint of constant pressure drop.

length has been set equal to Λ=*λ* / *ai* =0.25≪1.

**Table 1.** Expressions determining the apparent values of *m*\*

reduce the apparent wall conductivity *κ*<sup>w</sup>

*Na* =

augmentation numbers *Ra* and *Na*.

*y*w,2 <sup>⊥</sup> <sup>=</sup> <sup>−</sup> *<sup>y</sup>*w,1

**Constant flow rate (n = 0)**

24 Heat Transfer Studies and Applications

dimensionless heat-exchanger length *ξ<sup>L</sup>*

heat-exchanger length *ξ<sup>L</sup>*


**Table 3.** Variation of the ratio *ξ<sup>L</sup>* \* / *<sup>ξ</sup>L* (expressed both in absolute value and in percentage change) with the dimensionless corrugation amplitude, *A*, for the three cases under consideration.

As previously discussed, use of the computed values of *κ*<sup>w</sup> \* and *ξ<sup>L</sup>* \* 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/ΛΔw), *ξL*, Λ, Re*τi* (= 1.5Λ<sup>2</sup> Re*<sup>i</sup>* ), Pr*<sup>i</sup>* , *A*, Δw, *K*w, and *K* (*= k*). This procedure allows to evaluate the augmentation heat transfer and entropy generation numbers, *R*<sup>a</sup> 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*<sup>a</sup> ≅ 1) under constant flow rate, but takes values smaller than unity for constant pumping power (*N*<sup>a</sup> ≅ 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.


Wall Conduction Effects in Laminar Counterflow Parallel-Plate Heat Exchangers with Small-Scale Wall Corrugations http://dx.doi.org/10.5772/60550 27

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

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*<sup>a</sup> ≅ 1) under constant flow rate, but takes values smaller than unity for constant pumping power (*N*<sup>a</sup> ≅ 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.

*m*& *m\** **1** 1 *k* & *k\** **2** 2

*\** **4** 4.0141

**(b) Effect of Kw Flat wall Corrugated wall - A = 1, Reτi = 0, Kw = 10**

Constant flow rate

*m*& *m\** **1** 1 *k* & *k\** **2** 2

*\** **400** 42.4926

*\** **1** 1.0260 1.0662 1.1081

*Q*1/*Q*1flat 1 1.0000 0.9623 0.9259

/*ε*flat 1 1.0090 1.0219 1.0346 *εQ*1/(*εQ*1)flat 1 1.0090 0.9833 0.9580 Δ*P* <sup>1</sup> *Q* <sup>1</sup>/(Δ*P* <sup>1</sup> *Q* 1)flat 1 1.0800 1.0000 0.9259

*Ra* 1 0.9343 (–6.57%) 0.9833 (–1.67 %) 1.0346 (+3.46%) *Na* 1 0.9961 (–0.39%) 0.9528 (–4.72%) 0.9109 (–8.91 %)

*\** **1** 1.0260 1.0662 1.1081

*Q*1/*Q*1flat 1 1.0000 0.9623 0.9259

**(a) Ref. Case Flat wall Corrugated wall - A = 1, Reτi = 0, Kw = 0.1**

Constant flow rate

/ *ε* > 1, although the overall heat transfer rate is reduced due to the significant

*Q*1/(*εQ*1)flat < 1. It is interesting to note that the

Constant pumping power

Constant pumping power

Constant pressure drop

Constant pressure drop

effectiveness *ε*\*

26 Heat Transfer Studies and Applications

reduction in volume flow rate, making *ε*\*

*ε*flat **0.7831**

*εflat* **0.8562**

*κ<sup>w</sup>* & *κ<sup>w</sup>*

*ξ<sup>L</sup>* & *ξ<sup>L</sup>*

*ε\**

*κ<sup>w</sup>* & *κ<sup>w</sup>*

*ξ<sup>L</sup>* & *ξ<sup>L</sup>*



**Table 4.** (a)(b)(c): Influence of the wall-to-fluid heat conductivity ratio, *K***w = [0.1**, 10, 100]; (d)(a)(e): influence of the dimensionless corrugation amplitude, *A* = [0.5,**1**, 1,5]; and (a)(f): influence of the Reynolds number, Reτ*<sup>i</sup>* = [0, 64], on the performance of laminar counterflow parallel-plate heat exchangers with small scale-wall corrugations. In all reported cases Pr*<sup>i</sup>* = 5, Λ = 0.25, Δw = 0.1, and *K* = 2.

#### **6. 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.

*εflat* **0.7831**

28 Heat Transfer Studies and Applications

*εflat* **0.7831**

= 5, Λ = 0.25, Δw = 0.1, and *K* = 2.

*κ<sup>w</sup>* & *κ<sup>w</sup>*

*ξ<sup>L</sup>* & *ξ<sup>L</sup>*

*ε\**

*κ<sup>w</sup>* & *κ<sup>w</sup>*

*ξ<sup>L</sup>* & *ξ<sup>L</sup>*

*ε\**

cases Pr*<sup>i</sup>*

**6. Conclusion**

Constant flow rate

*m*& *m\** **1** 1 *k* & *k\** **2** 2

*\** **4** 5.5132

*\** **1** 1.0459 1.1187 1.1966

*Q*1/*Q*1flat 1 1.0000 0.9349 0.8741

/*ε*flat 1 1.0395 1.0607 1.0810 *εQ*1/(*εQ*1)flat 1 1.0395 0.9917 0.9449 Δ*P* <sup>1</sup> *Q* <sup>1</sup>/(Δ*P* <sup>1</sup> *Q* 1)flat 1 1.1441 1.0000 0.8741

*Ra* 1 0.9086 (–9.14%) 0.9917 (–0.83 %) 1.0810 (+8.10%) *Na* 1 0.9813 (–1.87%) 0.9063 (–9.37%) 0.8363 (–16.37 %)

*\** **1** 1.0270 1.0690 1.1126

*Q*1/*Q*1flat 1 1.0000 0.9608 0.9231

/*ε*flat 1 1.0098 1.0232 1.0363 *εQ*1/(*εQ*1)flat 1 1.0098 0.9831 0.9566 Δ*P* <sup>1</sup> *Q* <sup>1</sup>/(Δ*P* <sup>1</sup> *Q* 1)flat 1 1.0833 1.0000 0.9231

*Ra* 1 0.9321 (–6.79%) 0.9831 (–1.69 %) 1.0363 (+3.63%) *Na* 1 0.9957 (–0.43%) 0.9507 (–4.93%) 0.9073 (–9.27 %)

**Table 4.** (a)(b)(c): Influence of the wall-to-fluid heat conductivity ratio, *K***w = [0.1**, 10, 100]; (d)(a)(e): influence of the

performance of laminar counterflow parallel-plate heat exchangers with small scale-wall corrugations. In all reported

Laminar counterflow parallel-plate heat exchangers with small-scale wall corrugations have been studied theoretically and numerically following a multi-scale asymptotic approach. The

dimensionless corrugation amplitude, *A* = [0.5,**1**, 1,5]; and (a)(f): influence of the Reynolds number, Reτ*<sup>i</sup>*

**(f) Effect of Reτi Flat wall Corrugated wall - A = 1, Reτi = 64, Kw = 0.1**

Constant flow rate

*m* & *m\** **1** 1 *k*& *k\** **2** 2

*\** **4** 4.0351

Constant pumping power

Constant pumping power

Constant pressure drop

Constant pressure drop

= [0, 64], on the

In the bulk flow region, an approximated semi-analytical two-term solution based on eigen‐ function expansions provides accurate predictions of the outlet fluid temperatures in terms of the bulk parameters *m*, *k*, *κ*w, and *ξL*.

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 differ‐ ence method, involves the parameters Reτ*<sup>i</sup>* , Pr*<sup>i</sup>* , *K*, *K*w, Δw, *θ<sup>i</sup>*, as well as the corrugation shape, defined here by *A* and *γ*. The analysis shows that, far from the wall, the effect of the corruga‐ tions 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 oscilla‐ tory 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*<sup>w</sup> *\** and *ξ<sup>L</sup> \** 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 corruga‐ tions 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*<sup>w</sup> *\** (or an increased wall thermal resistance, 1/ *κ*<sup>w</sup> *\** ). 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.

#### **Acknowledgements**

This work was supported by Project ENE2011-24574 of Spanish *Ministerio de Economía y Competitividad*.
