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# Maximum heat transfer rate density in two-dimensional minichannels and microchannels

Michel Favre-Marinet, Stéphane Le Person, A. Bejan

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Michel Favre-Marinet, Stéphane Le Person, A. Bejan. Maximum heat transfer rate density in two-dimensional minichannels and microchannels. Microscale Thermophysical Engineering, 2004, 8 (3), pp.225-237. 10.1080/10893950490477419. hal-00204638

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# MAXIMUM HEAT TRANSFER RATE DENSITY IN TWO-DIMENSIONAL MINICHANELS AND MICROCHANNELS

## M. Favre-Marinet and S. Le Person

Laboratoire des Ecoulements Géophysiques et Industriels CNRS-UJF-INPG, Grenoble Cedex, France

## A. Bejan

Duke University, Department of Mechanical Engineering and Materials Science, Durham, North Carolina, USA

The objective of the present article is to compare previous experimental data of Gao et al. [20] to the predictions of Bejan and Sciubba's analysis [7] on the optimal spacing for maximum heat transfer from a package of parallel plates. Experimental investigations of the flow and the associated heat transfer were conducted in two-dimensional microchannels in order to test possible size effects on the laws of hydrodynamics and heat transfer and to infer optimal conditions of use from the measurements. The test section was designed to modify easily the channel height $e$ between 1 mm and 0.1 mm. Measurements of the overall friction factor and local Nusselt numbers show that the classical laws of hydrodynamics and heat transfer are verified for $e > 0.4$ mm. For lower values of $e$, a significant decrease of the Nusselt number is observed, whereas the Poiseuille number continues to have the conventional value of laminar developed flow. The transition to turbulence is not affected by the channel size. The experimental data were processed by using the dimensionless parameters of Bejan and Sciubba's analysis [7]. For fixed pressure drop across the channel, a maximum of heat transfer rate density is found for a particular value of $e$. The corresponding dimensionless optimal spacing and heat transfer rate density are in very good agreement with the predictions of Bejan and Sciubba. This article reports the first time that the optimal spacing between parallel plates is determined experimentally.

**Keywords** microfluidics, heat transfer, microchannels, optimization

## INTRODUCTION

The development of microscale systems has grown rapidly during the last decades, making possible many applications. In the field of heat transfer, use of microchannels

This research was supported by the CNRS and Region Rhône-Alpes. The authors gratefully acknowledge Institut National Polytechnique de Grenoble for supporting the visit of A. Bejan in Grenoble. They also thank P. Gao for performing the experiments.

This paper was presented at the First International Conference on Microchannels and Minichannels on April 24–25, 2003, at Rochester, New York.

Address correspondence to Michel Favre-Marinet, Laboratoire des Ecoulements Géophysiques et Industriels, CNRS-UJF-INPG, 1025 rue de la Piscine, Grenoble Cedex, BP 53 X, 38041, France. E-mail: Michel.Favre-Marinet@hmg.inpg.fr
## NOMENCLATURE

$b$ channel width (spanwise direction), m

$D_h$ hydraulic diameter, m

$e$ channel height, m

$f$ Fanning friction factor, dimensionless

$h$ heat transfer coefficient, $W \cdot m^{-2} \cdot K^{-1}$

$k$ thermal conductivity, $W \cdot m^{-1} \cdot K^{-1}$

$L$ channel length, m

$L^+ = \frac{L}{D_h \frac{1}{Re}}$ dimensionless channel length

$L_h$ heated channel length used in optimization analysis, m
$l_h$ heating resistance length, m

$Nu$ Nusselt number, dimensionless

$P$ electric power, W

$p$ pressure, $N \cdot m^{-2}$

$Po$ Poiseuille number, dimensionless

$Pr$ Prandtl number, dimensionless

$q''$ heat flux, $W \cdot m^{-2}$

$\bar{q}''$ dimensionless heat flux,

$Q$ total heat transfer rate, W

$Q_v$ volumetric flow rate, $m^3/s$

$Re = \frac{\rho V D_h}{\mu}$ Reynolds number, dimensionless

$T$ temperature, °C

$V$ channel bulk velocity, $m \cdot s^{-1}$

$x$ abscissa along the channel, m

$x^+ = \frac{x}{D_h \frac{1}{Re}}$ hydrodynamic entrance parameter, dimensionless

$x^* = \frac{x}{D_h \frac{1}{RePr}}$ thermal entrance parameter, dimensionless

### Greek Symbols

$\alpha$ thermal diffusivity, $m^2 \cdot s^{-1}$

$\mu$ dynamic viscosity, $kg \cdot m^{-1} \cdot s^{-1}$

$\Pi_L$ pressure drop number, dimensionless

$\rho$ density, $m^3 \cdot s^{-1}$

$\tau$ wall shear stress, $N \cdot m^{-2}$

### Subscripts

av average

c critical

f fluid

film film

in inlet

out outlet

w wall

seems very promising for cooling electronic equipment. Enhancement of heat transfer coefficient is obviously expected with reduction of length scales, and this has been observed many times, starting with the experiments of Tuckerman and Pease [1]. However, at the same time, pressure losses are rising very rapidly when the passage of the flow is strongly reduced. It is therefore important to consider the optimization of the design of microchannels for proper application to heat transfer problems. Since the work of Tuckerman and Pease [1], several investigations have been undertaken for optimizing heat transfer in micro heat sinks with the classical fin analysis method [2, 3]. Kim and Kim [4] used a different approach and the microchannel heat sink was modeled as a porous medium in their analysis. Very recently, a three-dimensional analysis procedure for the thermal performance of a microchannel heat sink has been developed and applied to optimize the heat sink design [5]. In this study, Ryu et al. assumed a fully developed flow, but took into account the development of thermal boundary layers. They obtained the optimum values of geometric parameters for particular working conditions (pumping power, overall dimensions), but did not bring out general rules for optimization design. The analysis was also applied to a new type of microchannel heat sink based on the principles of a manifold heat sink [6]. All these studies deal with plate-and-fin extended surfaces. They consider a heat sink base where the thermal boundary conditions are given and convective cooling by a flow in microchannels separated by fins.

In sum, most studies adopted the classical fin analysis method and used the assumption of fully developed flow and uniform heat exchange coefficient in the microchannels. On the other hand, Bejan and Sciubba [7] showed that the optimal spacing for maximum heat transfer from a package of parallel plates results from a competition between high
heat exchange coefficient of the boundary layer type for large plate spacing and large
exchange surface in the opposite situation of small plate spacing. Bejan and Sciubba
assumed that the pressure drop is fixed and showed that the optimal spacing is such that
the channel length is equal to the entrance length. This analysis raises some questions
about previous optimization studies, which do not take into account the development of
boundary layers in the channels. To our knowledge, the present article is the first attempt
to determine experimentally the optimal spacing between parallel plates and to test Bejan
and Sciubba's predictions.

A second issue of this article concerns the scaling laws pertaining to the hydrodynamics and heat transfer in microchannels, which are not clearly established, although many studies have been devoted to this subject in recent years. Published experimental results question the applicability of classical laws (friction factor, laminar-turbulent transition, heat exchange coefficient) when the characteristic dimensions of the channel are of the order of several hundred to one µm. A recent review article [8] shows the dispersion of results from last decade experiments.

The following discussion is restricted to rectangular microchannels. Microchannels of this type may be obtained by micromachining in a metal substrate. Peng et al. [9] studied such microchannels of various sizes (hydraulic diameter $D_h$: 133–367 µm) micromachined in a stainless steel substrate. In the laminar regime, they generally found increase of the friction factor *f*. On the contrary, Xu et al. [10] found that characteristics of flow in microchannels ($D_h$: 56–260 µm) agree with conventional laws of hydrodynamics. Pfund et al. [11] investigated microchannels made in a sandwich structure with plates in organic material and observed friction factors greater than the classical values.

Microchannels may also be obtained by etching in silicon wafers. In several publications, the cross-section is trapezoidal because a wet anisotropic technique was used for etching the microchannel. However, the cross-section shape is not far from rectangular when the channel height is small compared to the mean width, and comparison can be made with studies on rectangular microchannels. Flockhart and Dhariwal [12] found no scale effects on hydraulics in such trapezoidal channels (27 or 63 µm in height, 1 mm in width, Reynolds number <240). On the contrary, Qu et al. [13] observed a significant increase of *f* in the same type of channels (height 28 to 111 µm, width 380 µm for the smallest height) in all the range of Re investigated (until 1600).

Transition to turbulence is also controversial. Peng et al. [9] observed much earliest transition ($Re_c$ ~ 200–700) in their smallest microchannels than in conventional channels. This trend was not observed in other studies [10, 14]. A smaller reduction in transition Reynolds number was found by Pfund et al. [11]. Some possible microscale effects were suggested by Mala et al. [15], Guo and Li [16], and Koo and Kleinstreuer [17].

Results on single-phase heat transfer are also contradictory. In the laminar regime, Peng et al. [9], Qu et al. [18] obtained data for the Nusselt number well below the values predicted by classical theory. The same trend was observed by Hegab et al. [14] in microchannels milled into an aluminium plate. On the contrary, Qu and Mudawar [19] conducted both experimental and numerical investigations and found good agreement of the measured temperature distributions with the numerical predictions based on the conventional mass, momentum, and energy equations.

All these studies refer to microchannels of moderate aspect ratio (width/height ~ 3–4), and the results may be affected by the shape of cross-section, in addition to size effects. Recent investigations were conducted by Gao et al. [20, 21] in very high aspect
ratio (25–250) rectangular microchannels in order to approach a two-dimensional situation and to eliminate the cross-section shape from the possible factors likely to affect the flow and heat transfer. They measured the friction factor and the heat transfer coefficient in mini/microchannels of various heights (0.1–1 mm). Their experimental conditions and results are summarized in the next sections of the article.

The objective of the present article is to compare the experimental data from Gao et al. [20] to the predictions of Bejan and Sciubba's analysis [7] and to describe the optimal conditions for use in the design of micro heat exchangers.

# EXPERIMENTAL FACILITY AND PROCEDURE

## Experimental Setup

Flows of demineralized water in channels of large-span rectangular cross-section and adjustable height were investigated. Details on the facility may be found in Gao et al. [20]. In the experimental setup (Figure 1), the active channel walls are two plane bronze blocks, which are separated by a foil (thickness $e$) with a hollowed-out central part of width $b$ (= 25 mm). The thickness of this foil fixes the channel height $e$, which can be varied in the range 0.1–1 mm by steps of 0.1 mm. The other dimensions of the channel are $b$ and the length $L$ (= 82 mm). The main advantage of this arrangement is that the channel walls are represented by the same surfaces and that only the channel height $e$ is varied during all the experiments. The two blocks are rounded off in the upstream part to form a convergent channel entrance. They were hand-polished (measured roughness <0.1 µm). Two sumps are machined in the working section at the channel inlet/outlet. Heating was provided by four electric cartridges (maximum total power of $4 \times 250$ W; typical value of power used in the experiments: 200–500 W), which were inserted inside the two blocks and surrounded by an insulating material. The active length of the resistances $l_h$ (= 62 mm) was smaller than the channel length.

Representative flow conditions are presented in Table 1. The origin of abscissa $x$ along the channel is the beginning of the straight section.

## Instrumentation

Flow rates were measured with two high-precision turbine flowmeters. The pressure drop $\Delta p$ was determined by piezo resistive strain gauge transmitters (full scale: $10^6$ N·m$^{-2}$) at the channel inlet/outlet, or by a differential gauge (full scale: $10^4$ N·m$^{-2}$). Thermocouples of type K were placed in the two sumps to determine the fluid inlet/outlet temperatures, denoted respectively $T_{in}$ and $T_{out}$. Wall temperatures were measured by five thermocouples (denoted TC$_1$ to TC$_5$) of type T (diameter = 0.5 mm) placed in 1.5 cm increments along the channel (position of the first thermocouple $x = 0.6$ cm). The corresponding sections are denoted “1” to “5.” The thermocouples were located in the bronze blocks 1 mm away from the metal-fluid interface.

## Data Processing

The data were interpreted by using the Reynolds number $Re$ and the Nusselt number $Nu$, which were defined based on the hydraulic diameter $D_h$. $Re$ was based on the
<figure><img src="image_3.png" /><figcaption>Figure 1. Sketch of the test section.</figcaption></figure>

<table><caption>Table 1. Typical flow conditions</caption><thead><tr><th>e</th><th>b<br>(mm)</th><th>L<br>(mm)</th><th>U<br>(m/s)</th><th>Q<sub>v</sub><br>(1/mn)</th><th>Re</th></tr></thead><tbody><tr><td>100 µm</td><td>25</td><td>82</td><td>0.1–10</td><td>0.015–1.5</td><td>20–2400</td></tr><tr><td>200 µm</td><td>25</td><td>82</td><td>0.16–15</td><td>0.05–4.5</td><td>57–6000</td></tr><tr><td>1 mm</td><td>25</td><td>82</td><td>0.02–4</td><td>0.03–6</td><td>40–7400</td></tr></tbody></table>

bulk velocity in the channel. The water properties, such as density, dynamic viscosity, and Prandtl number, were determined at the average of the inlet/outlet temperatures $T_{av}$ (= [1/2][$T_{in}$ + $T_{out}$]). The thermal conductivity used in Nu was determined at the local film temperature $T_{film}$ (average of wall and fluid bulk temperatures; $T_{film}$ = [1/2][$T_{w}$ + $T_{f}$]). The friction factor $f$ was determined from the inlet/outlet pressure measurements. The
inlet pressure was corrected for the inertia term accounting for flow acceleration in the converging channel entrance. The classical hypothesis of constant pressure in the outlet sump was made to evaluate the pressure at the channel outlet. In other words, the head losses were neglected in the converging inlet channel and were concentrated at the channel outlet. The pressure drop across the microchannels was therefore estimated by:

$$ \Delta p = p_{\text{in}} - p_{\text{out}} - (1/2) \rho V^2 \quad (1) $$

where $p_{\text{in}}$ and $p_{\text{out}}$ are measured at inlet/outlet sumps, respectively (Figure 1). The friction factor was given by:

$$ f = \frac{\tau}{(1/2)\rho V^2} = \frac{\Delta p}{2\rho V^2} D_h L \quad (2) $$

The local wall heat flux $q''$ was deduced from the electric power $P$ assumed uniformly distributed along the resistances. The local fluid bulk temperature was deduced from a heat balance between the heat rate exchanged with the walls and the rate of enthalpy convected by the stream. Close agreement was found for the rate of enthalpy convected by the stream across the channel and the electric power dissipated in the blocks, except for very small flow rates. In this last case, incorrect measurements of $T_{\text{out}}$ are suspected. A one-dimensional model was developed to account for conduction in the metallic slab, which is located between the resistances and the channel. From these computations it was found that the measurements in sections 2 and 3 are not affected by conductive effects and may be used to determine the heat transfer coefficient in the channel (for more details, see Gao et al. [20]). It was also verified that the local fluid bulk temperature $T_f$ may be assumed to vary linearly along the channel over the length $l_h$. The heat exchange surface was assumed to correspond to the active length of the resistances. The local Nusselt number was then computed from the measurements by using the following relations

$$ q'' = \frac{P}{2bl_h} \quad (3) $$

$$ T_f = T_{\text{in}} + (T_{\text{out}} - T_{\text{in}}) \frac{x}{l_h} \quad (4) $$

$$ h = \frac{q''}{T_w - T_f} \quad (5) $$

$$ \text{Nu}_x = \frac{h D_h}{k} \quad (6) $$

When $Q$ was smaller than the electric power $P$, $Q$ was used instead of $P$ in Eq. (3). Uncertainties were analyzed and estimated. The results are given in Table 2.

## RESULTS

The next two sections summarize the results found by Gao et al. [20].
<table><caption>**Table 2.** Uncertainty analysis for 100 < Re < 8000</caption><thead><tr><th></th><th>e (mm)</th><th>1 (%)</th><th>0.5 (%)</th><th>0.1 (%)</th></tr></thead><tbody><tr><td rowspan="2">dRe/Re</td><td>Mini</td><td>0.4</td><td>0.4</td><td>0.4</td></tr><tr><td>Maxi</td><td>0.8</td><td>0.7</td><td>0.7</td></tr><tr><td rowspan="2">dNu/Nu</td><td>Mini</td><td>0.6</td><td>0.7</td><td>0.4</td></tr><tr><td>Maxi</td><td>7.8</td><td>4.9</td><td>1.2</td></tr></tbody></table>

## Friction Factor

For low values of the Reynolds number the flow is laminar, but entrance effects are present and affect the total pressure drop across the channel. The wall boundary layers merge at a distance $x^+ \approx 0.04$ [22]. Because the length of the tested channels is fixed, the entrance length depends on the channel height and on the flow velocity. It is therefore helpful to plot the pressure drop as a function of the dimensionless channel length $L^+$. The Poiseuille number

$$ Po = fRe \quad (7) $$

deduced from the measurements is drawn in Figure 2 for the lowest values of $e$. The graph also displays the following formula proposed by Shah [23]:

$$ Po = \frac{3.44}{\sqrt{L^+}} + \frac{24 + 0.674/4L^+ - 3.44/\sqrt{L^+}}{1 + 2.9 \times 10^{-5}/L^+} \quad (8) $$

Figure 2 shows that the friction law for the developed laminar flow regime ($f$ Re = 24) is confirmed very well by the present experiment, regardless of the channel height. Equation (8) slightly overestimates the pressure drop in the range $0.1 < L^+ < 1$. A clear departure from Eq. (8) is observed when $L^+$ is lower than a critical value $L_c^+$,

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Effect of channel size on the nondimensional pressure drop.</figcaption></figure>
which depends on the channel height. $L_c^+$ obviously corresponds to a critical value of the Reynolds number $Re_c$. It was found that $Re_c$ does not depend significantly on $e$ and is approximately equal to 3500, close to the accepted value of 4000 for transition in plane wall channels. This result was also confirmed by visualizations obtained by dye injection at the channel inlet. Solid lines in Figure 2 correspond to a hybrid formula using Eq. (8) in the first half of the channel and the Blasius equation

$$f = 0.079Re^{-1/4} \qquad (9)$$

in the second part of the channel. The good agreement with the experimental data also confirms the existence of the turbulent regime for $Re > Re_c$.

## Heat Transfer Coefficient

For a given flow rate, four values of the local Nusselt number $Nu_x$ were obtained from the thermocouples $TC_1$ to $TC_4$ by using Eqs. (3) to (6). As for pressure losses, the local Nusselt number $Nu_x$ was plotted as a function of $x^* (= x/[D_h Re Pr])$ in order to identify entrance effects and transition to turbulence in the heat transfer results. Figure 3 shows the results for $e = 0.5$ mm and $e = 0.1$ mm. The data of Shah and London [24] for simultaneous developing flow and heat transfer were interpolated for $Pr = 4$ and displayed in the same figure.

The general trend observed for all the channels tested is in good agreement with the data of Shah and London [24] for $e \ge 0.5$ mm. There is a significant decrease in $Nu_x$ when the channel height is decreased below 0.5 mm. A satisfying overlap of the results deduced from the four thermocouples was observed, which confirms that $x^*$ is the relevant parameter for the heat transfer results. However, for each channel height, two systematic deviations from the general trend were observed. A deviation toward low values of $Nu_x$ and, more generally, an important scatter of the results occur when $x^*$ is increased for a given thermocouple; that is, when the flow rate is decreased. In fact, the observed rapid decrease of $Nu_x$ is not linked to $x^*$, as can be seen in Figure 3, but to

<figure><img src="image_5.png" /><figcaption>Figure 3. Local Nusselt number.</figcaption></figure>
the small flow rates used in these conditions since it occurs for the same flow rate at the four measurement sections. This reduction of $Nu_x$ corresponds to the low values of the total enthalpy rise $Q$ measured for the smallest flow rates, as mentioned before. It is likely that $Q$ is incorrectly determined in these conditions and that the corresponding results for $Nu_x$ are not reliable. Consequently, they were discarded in the optimization analysis. On the other hand, the values of $Nu_x$ given by $TC_4$ are significantly above the other results for small values of $x^* (<0.002$ for $e = 0.5$ mm in Figure 3) for all the channels. This range of $x^*$ corresponds to $TC_4$ at the highest flow rates and highest $Re$ used in the experiments. The Reynolds number found for the change of slope in the $Nu_x$-curve is close to the value deduced from pressure losses measurements. Consequently, this change of regime for heat transfer may be attributed to the transition to turbulence in the microchannel.

The significant decrease observed in the Nusselt number with the size of the channel is in agreement with previously published results [18]. It was also observed in experimental investigations on heat transfer with a microtube [25]. Figure 3 shows that it can reach a factor of about 30% for $e = 0.1$ mm (Nu ≈ 3 instead of 8.23 for the fully developed laminar regime). A numerical modeling of the fluid flow and heat transfer in this experiment is now in progress [26].

## Optimal Spacing of the Plates in a Micro Heat Exchanger

The optimal spacing for maximum heat transfer from a package of parallel plates was investigated both theoretically and numerically by Bejan and Sciubba [7] (see also Bejan [27]). In this study, the authors considered an arrangement of parallel plates cooled by forced convection and found a maximum heat transfer rate for fixed pressure drop and fixed inlet and wall temperatures. They also considered the case of uniform heat flux surfaces. It should be stressed that this arrangement is different from a heat sink with parallel fins, which is used in most optimization studies [1–5]. In this last case, the total heat flux is released at the heat sink surface and the thermal resistance of the structure depends on the fin efficiency, whereas the wall temperature or the heat transfer rate density are assumed uniform along the plates in Bejan and Sciubba’s study. These authors showed that the relevant parameters are the dimensionless heat transfer rate density

$$ \tilde{q}'' = \frac{2q''L_h^2}{e \cdot k(T_{\text{max}} - T_{\text{min}})} \quad (10) $$

and the pressure drop number (which earlier Bhattacharjee and Grosshandler [28] had termed the Bejan number; see also Petresen [29]):

$$ \Pi_L = \frac{\Delta p_{L_h} L_h^2}{\mu \alpha} \quad (11) $$

Here $\Delta p_{L_h}$ is the pressure drop over a channel of length $L_h$. $T_{\text{max}}$ is the wall temperature for $x = L_h$ and $T_{\text{min}}$ is the fluid inlet temperature. The theoretical analysis suggests that the optimal spacing for maximum heat transfer is given by

$$ e/L_h = a_1 \Pi_L^{-1/4} \quad (12) $$
and that the maximum power density satisfies

$$ \tilde{q}'' = a_2 \Pi_L^{1/2} \qquad (13) $$

The constants $a_1$ and $a_2$ depend slightly on Pr and on the thermal boundary conditions. Bejan and Sciubba [7] found: $a_1 = 3.077$, $a_2 = 0.522$ for Pr = 6 and isothermal surfaces, and $a_1 = 3.287$, $a_2 = 0.424$ for Pr = 10 and uniform heat flux surfaces.

The present experimental data were analyzed by using this scale analysis. The actual conditions of heating were close to the case of a constant flux channel. However, the maximum temperature did not occur at the channel exit, owing to the reduced heating resistance length, as mentioned before. Section 3 of the thermocouple TC₃ ($x = L_h = 3.6$ cm) was chosen to analyze the data because it is not affected by wall conduction effects regardless of the flow conditions and may be assumed to be in a region of constant heat flux for all the channels [20]. The pressure drop between inlet and section 3 was deduced from the measured total pressure drop $\Delta p$ across the channel (Eq. (1)) by using the following relation:

$$ \Delta p_{\text{inlet-"3"}} = \Delta p - \Delta p_{\text{"3"-outlet}} \qquad (14) $$

In most cases, section 3 was located in the fully developed region and it was possible to estimate the pressure drop $\Delta p_{\text{"3"-outlet}}$ by the classical laws of friction for fully developed flow in the laminar or the turbulent regime, depending on the value of Re ($f Re = 24$ for Re < Re_c and Eq. (9) for Re > Re_c).

When the resulting $\Delta p_{\text{inlet-"3"}}$ was less than 100 N·m⁻², the data were discarded because the resulting pressure difference corresponds to the accuracy of the differential pressure transmitter. This occurred especially for the largest channels because the pressure drop is small in this case. All the data were kept for $e = 0.1$, 0.2 mm, but 25% were eliminated for $0.3 \text{ mm} \le e \le 0.7$ mm and 50% for $e = 1$ mm.

Each series of experiments was conducted at constant channel height, for obvious practical reasons, and constant heat flux. As a consequence, the pressure drop and the maximum temperature were variable during the experiments, but the data may then be processed at constant pressure drop, as in the theoretical analysis. In fact, following the results of the theoretical analysis (Eqs. (12)–(13)), the present data are plotted by using $\tilde{q}''/\Pi_L^{1/2}$ as a function of $e/L_h\Pi_L^{1/4}$ in Figure 4. In spite of the lack of measurements for very low values of $e/L_h\Pi_L^{1/4}$, the present results clearly show an optimal spacing for $e/L_h\Pi_L^{1/4} \approx 3$ for the narrowest microchannels investigated ($e = 0.1–0.4$ mm). The maximum values of $\tilde{q}''/\Pi_L^{1/2}$ are slightly less than predicted by the theory, but the agreement with Bejan and Sciubba’s prediction is good. The curve corresponding to $e = 0.1$ mm is shifted toward lower values of $\tilde{q}''/\Pi_L^{1/2}$. This is due to the strong reduction in the Nusselt number, which was observed for these conditions (Figure 3). Very few points were obtained on the left branch of the curves of Figure 4, especially for $e > 0.5$ mm, because they correspond to a very low pressure drop across the channel and consequently to the low range of the differential pressure transmitter.

The theoretical analysis shows that the position of the $\tilde{q}''$ maximum corresponds to $x^* \approx 0.04$. Bejan and Sciubba [7] remarked that the optimal spacing is therefore such that the channel length is equal to the thermal entrance length. The present experiments
<figure><img src="image_6.png" /><figcaption>Figure 4. Optimal spacing for maximum heat transfer.</figcaption></figure>

provide strong support to the theoretical prediction for $e = 0.1$ mm ($x^* = 0.046$) and for $e = 0.2$ mm ($x^* = 0.041$). The agreement deteriorates at $e = 0.4$ mm ($x^* = 0.018$), but the position of the maximum is not well defined in this case (Figure 4).

## CONCLUSION

Two-dimensional microchannels were investigated in the present study. The design of the test section enabled variations of the channel height by steps of 0.1 mm, from 1 mm, which corresponds to minichannels to the smallest height of 0.1 mm, where size effects were suspected to affect the flow dynamics and heat transfer. Flow and heat transfer measurements were first interpreted by using a dimensionless distance to the channel inlet (respectively, $L^+$ or $x^* = x/[D_h \text{ Re Pr}$]). This presentation clearly identifies entrance effects and transition to turbulence. The results show that the classical laws of friction in two-dimensional ducts are well verified both in the laminar and the turbulent regimes. In fact, no scale effects were found in the present experiments for the flow hydrodynamics. Heat transfer laws are also well verified for the minichannels of the present study in the laminar regime ($e \ge 0.5$ mm). However, for $e < 0.5$ mm, the plots of $Nu_x$ ($x^*$) show a departure from the classical heat transfer law, which grows when the channel height is decreased. The reduction of Nu is very strong for the narrowest microchannels. For $e = 0.1$ mm, Nu is roughly 60% smaller than the conventional value for large-scale channels.

The objectives of the study were the optimization of two-dimensional channels and to test the theoretical and numerical results of Bejan and Sciubba [7]. For this purpose, the experimental data were interpreted by using the dimensionless parameters first introduced by these authors. Although the range of the dimensionless spacing $e/L_h \Pi_L^{1/4}$ is rather limited in the present study, especially for small spacings, an optimal spacing is clearly identified when the dimensionless heat flux $\tilde{q}''/\Pi_L^{1/2}$ and spacing $e/L_h \Pi_L^{1/4}$ are
used to interpret the data. The present experiments support well the predictions of Bejan and Sciubba [7], except the smaller values of maximum heat transfer rate density, which may be attributed to microscale effects. This is the first time that the optimal spacing between parallel plates was determined experimentally.

Further experiments will be conducted with other surface finishes in order to identify the phenomena governing this flow. In particular, it is planned to carry out investigations of microchannel flows with rough walls. Further insight in the flow phenomena should be provided by visualizations, which will be soon performed in a test section with transparent walls. Measurements with microsensors are also planned in the near future.

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