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# Experimental study of heat transfer coefficient in heat exchanger reactors with square millimetric zigzag channels

Hanbin Shi, Nathalie Di Miceli Raimondi, Michel Cabassud, Christophe Gourdon

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## ► To cite this version:

Hanbin Shi, Nathalie Di Miceli Raimondi, Michel Cabassud, Christophe Gourdon. Experimental study of heat transfer coefficient in heat exchanger reactors with square millimetric zigzag channels. Chemical Engineering and Processing: Process Intensification, 2022, 182, pp.109194. 10.1016/j.cep.2022.109194. hal-03977130

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Submitted on 7 Feb 2023

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# Experimental study of heat transfer coefficient in heat exchanger reactors with square millimetric zigzag channels

Hanbin Shiª, Nathalie Di Miceli Raimondiª,*, Michel Cabassudª, Christophe Gourdonª

ª Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France

* E-mail address: nathalie.raimondi@iut-tlse3.fr

## Abstract

Heat transfer performances of compact heat exchanger reactors were experimentally characterized and measurement uncertainties were evaluated using the derivation method. Prototypes presenting two kinds of process channel configurations (zigzag and straight channels) with square cross-section were investigated, from 2 mm to 4 mm depth. Water or glycerol-water mixture were used as the process fluid, cooled by water as the utility fluid. Reynolds number ranged between 15 and 7300 and Prandtl number between 3.1 and 115. From the measurements of the fluid temperatures and flow rates, local heat transfer coefficients were calculated. In zigzag channels, it ranged from 760 to 15000 W.m⁻².K⁻¹, with an average uncertainty of 15.7%. With water as the process fluid, the heat transfer coefficients in zigzag channel were 2 to 5 times higher than in straight channel. The experimental data were then used to establish a correlation for the Nusselt number in the zigzag channels as a function of the channel geometry, the fluid properties and flow rates expressed in terms of dimensionless numbers. The average relative difference between experimental and calculated values was 9.1%. Finally, the consistency of the correlation was confirmed by comparison with data obtained in a previous work based on CFD.

**Keywords:** Compact heat exchanger reactor; Millimetric zigzag channels; Heat transfer coefficient; Nusselt number; Experimental uncertainty;

## 1. Introduction

Recently, numerous technologies of compact heat exchanger (HEX) reactors have been developed to respond to the needs of safer, cleaner, more effective and less energy-consuming processes. Basically, they integrate the miniaturization of the units and the multi-functionalization of the apparatuses, which exhibit many advantages such as good heat transfer performances, better temperature control, reactive volume confinement, etc. Indeed, this type of reactor has a large heat exchange area to fluid volume ratio (A/V) due to size reduction. This leads to a decrease in the total amount of fluid mass and heat inside the reactor which is favorable for safety concerns, particularly for highly exothermic reactions. Therefore, they are promising devices for process intensification, notably for the transposition from batch processes to continuous processes in fine chemicals units [1]. Indeed, stirred tank reactors are still very used in this domain because of their flexibility in multipurpose plants while they have a poor thermal performance. In a large tank, the volumetric heat exchange capacity (UA/V where U is the overall heat transfer coefficient) is around 1 kW.K⁻¹.m⁻³ [2], and can reasonably reached several tens of kW.K⁻¹.m⁻³ in tanks of a few liters. In comparison, compact heat exchanger reactors can offer several thousand kW.K⁻¹.m⁻³ [2]. For example, Théron et al. characterized experimentally the thermal-hydraulic performance of a compact heat exchanger reactor in stainless steel with wavy channels of 2 mm of hydraulic diameter, and obtained a volumetric heat exchange capacity between 5000 and 8000 kW.K⁻¹.m⁻³ [3]. Lavric and Whoehl present volumetric heat exchange capacity around 1000 kW.K⁻¹.m⁻³ in Corning® Advanced-Flow™ reactors made out of glass [4]. Mielke et al. obtained a
volumetric heat exchange capacity of roughly 10000 $kW.K^{-1}.m^{-3}$ in A6-sized FlowPlate® reactor manufactured by Ehrfeld Mikrotechnik in Hastelloy®. This device is composed of rectangular curved channels with an hydraulic diameter of 0.9 mm [5]. Guo et al. studied experimentally and numerically flow distribution and heat exchange performance in a multi-functional heat exchanger reactor comprising arborescent distributors and collector and straight channels with hydraulic diameter of 1 to 2 mm [6]. They found volumetric heat exchange capacities around 200 $kW.K^{-1}.m^{-3}$, considering the volume of the whole device: this value would be much higher by considering only the volume of fluid. Tochon et al. designed an innovative low-cost compact heat exchanger reactor based on several straight tubes of 8 mm of internal diameter with twisted tape inserts measured a volumetric heat exchange capacity of 580 $kW.K^{-1}.m^{-3}$ (also expressed per volume of device) [7].

In order to get sufficient residence time for the reactions, compact heat exchanger reactors are generally operated with a low fluid velocity, leading to laminar flow regime instead of turbulent flow regime, the latter favoring mass and heat transfer. Indeed, turbulence generates mixing which allows the reduction of the boundary layers at the walls and interfaces and promotes their renewal. Therefore, to enhance transfer phenomena in compact devices, different mixing enhancement techniques are used. Active methods are based on the use of an external source of energy like in pulsed reactors [8,9]. Passive mixing is the result of the interaction between the pumped fluid and the geometric structure of the effective volume of the device or fixed obstacles. This method is mainly used in compact heat exchanger reactors. For liquid flow, zigzag channels are often used as they allow performance enhancements, due to the generation of secondary flows in the curves, so-called Dean vortices, with moderate pressure drop increases [10–15]. Table 1 shows that the thermohydraulic performance of millimetric zigzag channels are comparable to other compact heat exchanger reactors with roughly the same hydraulic diameter: Corning® Advanced-Flow Reactor™ G1 (the process channel is a succession of millimetric heart-shaped cells) and Chart ShimTec® Reactor (three parallel millimetric channels composed of straight lengths and turns). In a previous work [16], flow and heat transfer in 2 mm width zigzag channels with square cross-section were studied numerically on the basis of CFD simulations. The impacts of fluid velocity and channel geometries, in terms of straight section length between two bends, on heat transfer performance were investigated for Reynolds number ranging from 112 to 560. Laminar model was used and failed to simulate the flow for Reynolds number higher than 560 or when the straight length between bends was very short, leading to significant flow instabilities. Unsteady model adapted to transitional flow between laminar and turbulent flow would be adapted but would require high CPU and time resources.

<table><caption>Table 1. Thermohydraulic performance of different heat exchanger reactors for Reynolds number $Re = 500$ with water.</caption><thead><tr><th>Technology</th><th>Material</th><th>d<sub>h</sub><br>(mm)</th><th>UA/V<br>(kW.m<sup>-3</sup>.K<sup>-1</sup>)</th><th>Fanning friction<br>factor f</th><th>Ref.</th></tr></thead><tbody><tr><td>Zigzag channels</td><td>Aluminum/PMMA</td><td>2</td><td>1500-2500</td><td>0.04-0.12</td><td>[17]</td></tr><tr><td>Corning® Advanced-<br>Flow Reactor™ G1</td><td>Glass</td><td>2</td><td>900</td><td>0.27</td><td>[18,19]</td></tr><tr><td>Chart ShimTec® Reactor</td><td>Stainless steel</td><td>2.2</td><td>3000</td><td>0.10</td><td>[19,20]</td></tr></tbody></table>

In order to study the performances of the zigzag channels at higher Reynolds numbers, an experimental method has been applied in the present work, covering laminar to turbulent flow regime. Six plate-type heat exchanger reactor prototypes with millimetric zigzag channels have been constructed. Each prototype is composed of a process plate (with a millimetric channel engraved) and a utility plate (with a larger channel engraved). The overall heat transfer coefficient and the heat transfer coefficient of
process fluid are characterized. Two kinds of process channel configurations are investigated which are zigzag channel and straight channel (as reference) with square cross-sections. For that purpose, a test bench has been developed to allow the circulation of a hot fluid and a cold fluid. Heat exchange between both fluids occurs and a global heat transfer coefficient is calculated and discussed in function of the operating conditions and the millimetric channel geometry. A special care is brought to the results reliability and the measurement uncertainties are considered. The mathematical method used to evaluate the uncertainties induced on the calculated data is described in details. The results are then analyzed and a correlation of Nusselt number for process fluid in zigzag channels is then proposed in terms of hydraulic diameter, curvature radius, fluid properties and Reynolds number. Finally, the experimental results are compared to those obtained by CFD simulations in a previous work [18].

## 2. Materials

### 2.1 Experimental prototypes

The prototypes consist of two plates: a process plate and a utility plate separated by a heat transfer wall. The wall is made of a 1 mm (±0.1 mm) thickness anodized aluminum plate with a high thermal conductivity ($\lambda_w$ = 237 W.m⁻¹.K⁻¹), and therefore a very low thermal resistance ($e_w/\lambda_w$ = 4.2 × 10⁻⁶ m².K.W⁻¹). The process plate and utility plate are made of PolyMethylMethAcrylate (PMMA) of 10 mm thickness. Its thermal conductivity is very low which can significantly limit heat losses ($\lambda$ = 0.19 W.m⁻¹.K⁻¹, $e/\lambda$ = 0.05 m².K W⁻¹). The process and utility plates are engraved with a numerically controlled milling machine at the Laboratoire de Génie Chimique (LGC, France). The two PMMA plates are pasted to the aluminum wall using double-sided adhesives cut to fit the channel designs (Fig. 1).

<figure><img src="image_3.png" /><figcaption>Fig. 1. Prototype composed of a PMMA process plate with a zigzag channel engraved (a) and a PMMA utility plate with a rectangular straight channel engraved (c) separated by an aluminum plate (middle in (b)). Top view of 65 cm total length and 3 cm width (d).</figcaption></figure>

In this study, two kinds of channel configuration are investigated for the process plate: straight channel and zigzag channel with three different hydraulic diameters ($d_h$ = 2 mm, 3 mm, 4 mm), equal to the channel widths since the cross-section is square. The total straight length is nearly 0.6 m. The zigzag channels consist of periodic straight sections and bends (Fig. 2), where the ratio of the curvature radius of bends (at the channel center) to the hydraulic diameter $R_c/d_h$ is kept constant, giving a curvature radius $R_c$ of 1.5 mm, 2.25 mm and 3 mm respectively. The straight section length $L_s$ is fixed at 7 mm. The
angle $\theta$ between two straight sections is 90°. This geometry pattern has been previously studied by CDF simulation [16] and was used to manufacture compact heat exchanger reactors designed by the LGC and the Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA, France). Such devices made of silicon carbide or stainless steel have been successfully used to carry out highly exothermic reactions [21-23].

<figure><img src="image_4.png" /><figcaption>Fig. 2. Schematic of the zigzag channel: $d_h = 2 - 4$ mm, $L_s = 7$ mm, $R_c = 1.5 - 3$ mm, $\theta = 90^\circ$</figcaption></figure>

In the utility plate, a straight channel with a rectangular cross-section of 12 mm width and 6 mm depth is engraved, giving a hydraulic diameter of 8 mm. Its geometry is unique for all the prototypes, whatever the process channel design is. Its width and length are slightly higher than the total width and length of all the zigzag channels in the process plate (the largest process channel is 10.7 mm, obtained for the zigzag channel of 4 mm width). The utility and process channels are face to face and centered in the PMMA plates. This way, the utility channel entirely covers the zigzag channel or straight channel in the process plate.

<table><caption>Table 2. Geometry parameters of the zigzag process channel and the straight utility channel ($L$ is the total developed length of the channel)</caption><thead><tr><th></th><th>w (mm)</th><th>d (mm)</th><th>d<sub>h</sub> (mm)</th><th>R<sub>c</sub> (mm)</th><th>θ (°)</th><th>L<sub>s</sub> (mm)</th><th>L (mm)</th></tr></thead><tbody><tr><td rowspan="3">Zigzag process channel</td><td>2</td><td>2</td><td>2</td><td>1.5</td><td>90</td><td>7</td><td>778</td></tr><tr><td>3</td><td>3</td><td>3</td><td>2.25</td><td>90</td><td>7</td><td>759</td></tr><tr><td>4</td><td>4</td><td>4</td><td>3</td><td>90</td><td>7</td><td>747</td></tr><tr><td rowspan="3">Straight process channel</td><td>2</td><td>2</td><td>2</td><td>-</td><td>-</td><td>-</td><td>588</td></tr><tr><td>3</td><td>3</td><td>3</td><td>-</td><td>-</td><td>-</td><td>588</td></tr><tr><td>4</td><td>4</td><td>4</td><td>-</td><td>-</td><td>-</td><td>588</td></tr><tr><td>Utility channel</td><td>12</td><td>6</td><td>8</td><td>-</td><td>-</td><td>-</td><td>588</td></tr></tbody></table>

## 2.2 Experimental setup

The experimental setup is shown in Fig. 3. The hot fluid at about 50 °C (60 °C for small flow rates) is introduced in the process channel of the prototype. It is obtained thanks to an external heat exchanger where hot water produced by a thermostated bath (Julabo, model Corio CD, filled with tap water) is used as the heating fluid. In the prototype, cold water temperature is circulating in the utility channel. The two fluids circulate co-currently. The prototype is surrounded by an insulation foam in order to limit the thermal losses. The inlet and outlet temperatures of both fluids are measured by thermocouples (Ecotherm Mesures, model 96MIJC with type T thermocouple CL.1, tolerance ±0.5 °C). The thermocouple input signals are connected to acquisition modules (National Instruments, NI 9213) inserted in an USB acquisition rack (National Instruments, cDAQ-9178). The process fluid flow is provided by a gear pump (Verder, model SLH-712A, 0 - 100 L.h⁻¹) and the utility fluid flow by a
centrifugal pump (Iwaki, model MDT-20L, 1200 L.h⁻¹ max). The mass flow rate of the process fluid is measured by a Coriolis flowmeter (Micro Motion, model Elite CMF010M, 100 L.h⁻¹ max; with transmitter Micro Motion 1700). The mass flow rate of utility channel is measured by a rotameter (Georg Fischer, model SK 730, 100-1000 L.h⁻¹). Temperature and process flow rate measurements are recorded by a data acquisition system based on LabVIEW software (National Instruments, USA) which can also control the gear pump. To limit the temperature increase of the cold water, due to closed-loop recirculation, a tank of 500 L is used. The increase in temperature is roughly 1°C.h⁻¹. The overall heat transfer coefficient is calculated as the experiment progressed, with the data acquired (temperatures and flow rates are measured every 10 s). Steady state is assumed when this coefficient is stable. One experiment lasts at least one hour.

<figure><img src="image_5.png" /><figcaption>Fig. 3. Experimental setup for the determination of heat transfer coefficients</figcaption></figure>

## 2.3 Fluids

Two process fluids characterized by highly different Prandtl number are used: water and a glycerol-water mixture (74% of glycerol in mass fraction). Tap water and ReagentPlus® glycerol (Sigma-Aldrich, ≥99.0%) are used. The physicochemical properties of both fluids at 20 °C are presented in Table 3. They present a Newtonian rheological behavior.

<table><caption>Table 3. Physicochemical properties of water and glycerol-water mixture (74 wt% of glycerol) at 20 °C</caption><thead><tr><td>Process fluids</td><td>ρ (kg m<sup>-3</sup>)</td><td>C<sub>P</sub> (J kg<sup>-1</sup> K<sup>-1</sup>)</td><td>μ (mPa s)</td><td>λ (W m<sup>-1</sup> K<sup>-1</sup>)</td><td>Pr</td></tr></thead><tbody><tr><td>Water</td><td>998</td><td>4184</td><td>1.0</td><td>0.6</td><td>7</td></tr><tr><td>Glycerol-water mixture</td><td>1192</td><td>2840</td><td>32.6</td><td>0.34</td><td>272</td></tr></tbody></table>
The physicochemical properties (density ρ, specific heat capacity C<sub>p</sub>, viscosity μ and thermal conductivity λ) of water at different temperatures are estimated with the correlations given by Patek et al. [24]. The physicochemical properties of the water-glycerol mixture as a function of temperature are obtained by different methods. Its density has been measured by a pycnometer (Anton Paar, DMA 38) and the viscosity by a Kinexus ultra+ rheometer (Malvern Instruments). They have been correlated between 20°C and 60 °C as follows, with T expressed in K:

$$ \rho[\text{kg m}^{-3}] = 831.7 + 2.905 T - 5.714 \times 10^{-3} T^2 \quad (1) $$

$$ \mu[\text{Pa} \cdot \text{s}] = 13.59161 - 1.23407 \times 10^{-1} T + 3.74365 \times 10^{-4} T^2 - 3.79246 \times 10^{-7} T^3 \quad (2) $$

The heat capacity of the water-glycerol mixture is calculated by adding the heat capacity of each component multiplied by its mass fraction. The heat capacity of glycerol at different temperatures is estimated by the correlation given by Righetti et al. [25]. The expression suggested by Bates [26] for the thermal conductivity of glycerol-water mixture at 75% in weight of glycerol is used, with T expressed in K:

$$ \lambda[W \cdot m^{-1} \cdot K^{-1}] = 0.3344 + 1.254 \times 10^{-4}(T - 273.15) \quad (3) $$

Prandtl number is commonly used in correlation for heat transfer coefficient calculation to consider the impact of the fluid properties. It is expressed as follows:

$$ Pr = \frac{C_p \mu}{\lambda} \quad (4) $$

## 2.4 Operating conditions

In the process channel, the range of flow rate varies with the channel cross-section and reaches a maximum of 60 L.h<sup>-1</sup>. In the utility plate, three flow rates of 300, 400 and 500 L.h<sup>-1</sup> have been studied. The flow rates in process and utility channels as well as the corresponding velocity and Reynolds number are presented in Table 4 for the six prototypes. Reynolds number of the process fluid ranges from 20 to 7 300, which allows to study heat transfer from laminar to turbulent flow. The utility fluid flows in the turbulent regime (Re<sub>u</sub> > 10 000) to avoid heat transfer limitation by the utility side. Reynolds number is expressed as a function of the hydraulic diameter of the channel and the mean velocity, the density and the viscosity of the fluid:

$$ Re = \frac{ud_H \rho}{\mu} \quad (5) $$

<table><caption>Table 4. Flow rates in process channel and utility channel for three prototypes with corresponding velocity and Reynolds number</caption><thead><tr><th rowspan="2"></th><th colspan="3">Process channel (straight or zigzag)</th><th>Utility channel</th></tr><tr><th>d<sub>h</sub> = 2 mm</th><th>d<sub>h</sub> = 3 mm</th><th>d<sub>h</sub> = 4 mm</th><th>d<sub>h</sub> = 8 mm</th></tr></thead><tbody><tr><th rowspan="3">Water</th><td>Q<sub>p</sub> = 1.5 – 15 L.h<sup>-1</sup></td><td>Q<sub>p</sub> = 2.8 – 34 L.h<sup>-1</sup></td><td>Q<sub>p</sub> = 5 – 60 L.h<sup>-1</sup></td><td>Q<sub>u</sub> = 300 – 500 L.h<sup>-1</sup></td></tr><tr><td>u<sub>p</sub> = 0.1 – 1 m.s<sup>-1</sup></td><td>u<sub>p</sub> = 0.09 – 1 m.s<sup>-1</sup></td><td>u<sub>p</sub> = 0.09 – 1 m.s<sup>-1</sup></td><td>u<sub>u</sub> = 1.1 – 1.9 m.s<sup>-1</sup></td></tr><tr><td>Re<sub>p</sub> = 280 – 4000</td><td>Re<sub>p</sub> = 430 – 6200</td><td>Re<sub>p</sub> = 640 – 7300</td><td>Re<sub>u</sub> = 10000 – 20000</td></tr><tr><th rowspan="3">Glycerol-water mixture</th><td>Q<sub>p</sub> = 1.2 – 11 L.h<sup>-1</sup></td><td>Q<sub>p</sub> = 2.4 – 28 L.h<sup>-1</sup></td><td>Q<sub>p</sub> = 2.4 – 31 L.h<sup>-1</sup></td><td>Q<sub>u</sub> = 500 L.h<sup>-1</sup></td></tr><tr><td>u<sub>p</sub> = 0.08 – 0.7 m.s<sup>-1</sup></td><td>u<sub>p</sub> = 0.07 – 0.9 m.s<sup>-1</sup></td><td>u<sub>p</sub> = 0.04 – 0.4 m.s<sup>-1</sup></td><td>u<sub>u</sub> = 1.9 m.s<sup>-1</sup></td></tr><tr><td>Re<sub>p</sub> = 20 – 350</td><td>Re<sub>p</sub> = 25 – 420</td><td>Re<sub>p</sub> = 15 – 220</td><td>Re<sub>u</sub> = 20000</td></tr></tbody></table>
## 3. Methods

### 3.1 Heat transfer coefficient calculation

As the temperature difference between the channel inlet and outlet for process fluid is much higher than for utility fluid due to their respective flow rates, the calculation of heat fluxes (heat loss and heat exchanged) are performed with the process fluid data. From the temperature difference $\Delta T_p$, the heat flux exchanged between the two fluids $q$ can be calculated by Eq. 6. The thermal losses $q_l$ were previously estimated by introducing only the process fluid in the prototype (Eq. (6) with $q = 0$), with no flow in the utility channel.

$$q = \dot{m}_p C_{p,p} \Delta T_p - q_l \qquad (6)$$

$\dot{m}_p$ is the mass flow rate and $C_{p,p}$ is the specific heat of the process fluid. The overall heat transfer coefficient $U$ of the prototype can be calculated by:

$$U = \frac{q}{A \Delta T_{ml}} \qquad (7)$$

$A$ is the overall heat exchange area, $\Delta T_{ml}$ is the logarithmic mean temperature difference calculated by:

$$\Delta T_{ml} = \frac{(T_{p,i} - T_{u,i}) - (T_{p,o} - T_{u,o})}{\ln\left(\frac{T_{p,i} - T_{u,i}}{T_{p,o} - T_{u,o}}\right)} \qquad (8)$$

For the prototype, the overall conductance can be expressed as:

$$\frac{1}{UA} = \frac{1}{h_p A_p} + \frac{e_w}{\lambda_w A} + \frac{1}{h_u A_u} \qquad (9)$$

$e_w$ is the aluminum wall thickness and $\lambda_w$ its thermal conductivity, $h_p$ and $h_u$ are the convection heat transfer coefficients of the process fluid and utility fluid, $A_p$ and $A_u$ are the heat exchange areas in the process plate and the utility plate. They are calculated by:

$$A_p = w_p L_p \qquad (10)$$

$$A_u = w_u L_u \qquad (11)$$

$w_p$ and $w_u$ are the process and utility channel widths, respectively, $L_p$ and $L_u$ are their lengths (developed length for zigzag channels), given in Table 2. In this work, $A$ is considered equal to $A_p$.

The heat transfer coefficient $h_u$ of the utility fluid can be deduced from common correlations of Nusselt number ($Nu = hd_p/\lambda$) in straight channels. The ranges in terms of Reynolds number and Prandtl number investigated in the present study are as follows: $10000 \le Re_u \le 20000$ and $4 \le Pr_u \le 7$. It corresponds to turbulent flow. It has been chosen to use the simplified correlation proposed by Gnielinski for $Pr$ ranging from 1.5 to 500 and $Re$ from 2300 to $10^6$ [27,28]:

$$Nu_u = \frac{h_u d_{h,u}}{\lambda_{p,u}} = \alpha_{cor} 0.012 (Re_u^{0.87} - 280) Pr_u^{0.4} \left(1 + \left(\frac{d_{h,u}}{L_u}\right)^{2/3}\right) \left(\frac{Pr_u}{Pr_{w,u}}\right)^{0.11} \qquad (12)$$

$\alpha_{cor}$ is a factor introduced to consider the uncertainty of $Nu_u$ related to the use of this correlation. It is equal to one, with an uncertainty discussed in the next paragraph. The maximum temperature that the wall on the utility side can reach is process fluid temperature. Less than 5% of difference in average is obtained on $Nu_u$ when $Pr_{w,u}$ is calculated at the mean process fluid temperature ($Pr_{w,u}@T_p$) or at the mean utility fluid temperature ($Pr_{w,u}@T_u = Pr_u$). Thus, $(\frac{Pr_u}{Pr_w})^{0.11}$ can be reasonably assumed to be equal to 1.
Once $U$ and $h_u$ are determined, then the heat transfer coefficient of the process fluid $h_p$ can be deduced from Eq. (9). The Nusselt number $Nu_p$ can be furtherly obtained by:

$$Nu_p = \frac{h_p d_h}{\lambda_p} \qquad (13)$$

The physicochemical properties of the fluids used to calculate the dimensionless numbers are estimated at the average temperature of the fluids from their inlet to their outlet of the prototype.

## 3.2 Uncertainty estimation

In this work, the uncertainties of the overall heat transfer coefficient $U$, the heat transfer coefficient $h_p$, Nusselt number $Nu_p$ and Reynolds number $Re_p$ of the process fluid determined in the experiments are estimated in order to evaluate the experimental results accuracy. The derivation method is used as proposed in previous works [29]. For a quantity $X$ that is calculated as a function of input factors $X_i$ ($X = f(X_1, X_2, ...)$), the total uncertainty $\Delta X$ is approximately:

$$(\Delta X)^2 = \sum_{i} \left( \frac{\partial f}{\partial X_i} \right)^2 (\Delta X_i)^2 \qquad (14)$$

$X_i$ could be either primary experimental inputs or tabulated data. It is provided that the uncertainty in each input, $\Delta X_i$, is small and that each input is uncorrelated with all others. The uncertainty calculation method for the different parameters of interest is detailed in Appendix A.1. The uncertainty values of the raw data used to calculate the uncertainty of the quantities of interest (heat transfer coefficients, Reynolds numbers and Nusselt numbers) are provided in Table 5. Experimental uncertainties are estimated thanks to a prior calibration work for all devices (maximum deviation between the measured data and the reference). The uncertainty values considered and presented in Table 5 are the maximum value between this estimation and manufacturer technical data. Based on the uncertainties of the temperatures and of the correlations used, density, heat capacity and thermal conductivity of process and utility fluids are considered with an uncertainty of $\pm 1\%$ while the uncertainty of viscosity is considered equal to $\pm 2\%$ as it is more sensitive to temperature. An uncertainty due to the use of simplified Gnielinski correlation, Eq. (12), for the calculation of the utility side heat transfer coefficient is considered. Other correlations could have been used such as the original Gnielinski correlation (for $2300 \le Re \le 10^6$; $0.5 \le Pr \le 200$) or Dittus-Boelter correlation (for $Re \ge 10000$; $0.6 \le Pr \le 160$; $L/d_h > 10$). The first one provides values of $Nu_u$ 10% higher; The second one provides values very close to those considered (less than 5% of relative difference). Taking into account the accuracy of these different correlations and the simplification made on $Pr$ ratio $((Pr_u/Pr_{u,w})^{0.11} = 1)$, a 20% uncertainty on the factor $\alpha_{cor}$ introduced in Eq. (12) is considered. The uncertainty values calculated for each experiment are reported on the figures that present the experimental results.

<table><caption>Table 5. Uncertainties of the raw data and correlation</caption><thead><tr><th>Parameters</th><th>Range</th><th>Uncertainty</th></tr></thead><tbody><tr><td><strong>Geometry</strong></td><td></td><td></td></tr><tr><td>Channel width <i>w</i></td><td>2-12 mm</td><td>0.1 mm</td></tr><tr><td>Channel length <i>L</i></td><td>588-778 mm</td><td>1 mm</td></tr><tr><td>Hydraulic diameter <i>d</i><sub>H</sub></td><td>2-8 mm</td><td>0.1 mm</td></tr><tr><td>Wall thickness <i>e</i><sub>w</sub></td><td>1 mm</td><td>0.1 mm</td></tr><tr><td><strong>Measurement (sensor)</strong></td><td></td><td></td></tr></tbody></table>
<table><tbody><tr><td>Temperature T (thermocouples)</td><td>20-60 °C</td><td>0.5 °C</td></tr><tr><td>Process flowrate ṁ<sub>p</sub> (Coriolis flowmeter)</td><td>1.2-60.7 kg.h<sup>-1</sup></td><td>0.2 kg.h<sup>-1</sup></td></tr><tr><td>Utility flowrate ṁ<sub>u</sub> (rotameter)</td><td>300-500 kg.h<sup>-1</sup></td><td>3 kg.h<sup>-1</sup></td></tr><tr><td colspan="3"><strong>Physicochemical properties</strong></td></tr><tr><td>Heat capacity C<sub>p</sub></td><td>Water: 4179-4183 J.kg<sup>-1</sup>.K<sup>-1</sup><br>Glycerol-water: 2899-2960 J.kg<sup>-1</sup>.K<sup>-1</sup></td><td>1%</td></tr><tr><td>Thermal conductivity λ</td><td>Water: 0.602-0.652 W.m<sup>-1</sup>.K<sup>-1</sup><br>Glycerol-water: 0.414-0.420 W.m<sup>-1</sup>.K<sup>-1</sup><br>Aluminum: 237 W.m<sup>-1</sup>.K<sup>-1</sup></td><td>1%</td></tr><tr><td>Density ρ</td><td>Water: 985-998 kg.m<sup>-3</sup><br>Glycerol-water: 1170-1183</td><td>1%</td></tr><tr><td>Viscosity μ</td><td>Water: 0.50-0.96 cP<br>Glycerol-water: 7.0-14 cP</td><td>2%</td></tr><tr><td colspan="3"><strong>Correlation</strong></td></tr><tr><td>α<sub>cor</sub></td><td>1</td><td>20%</td></tr></tbody></table>

# 4. Results

## 4.1 Heat transfer performances of zigzag channels vs straight channels

### 4.1.1 Overall heat transfer coefficients

The overall heat transfer coefficient *U* as a function of process fluid flow rate (water) at different utility fluid flow rates is presented in Fig. 4. As expected, in Fig. 4(a), *U* increases with increasing both flow rates. The curves tend to asymptotic values as that the thermal resistance of the process side becomes negligible for high process flow rates compared to that of the utility side. The same trends can be seen for the straight channels, as shown in Fig. 4(b). However, the increase to the asymptotic value is less obvious for the range of flow rate studied since the thermal resistance of process side is higher in the straight channel than in the zigzag channel under the same flow condition. Indeed, by comparing both figures, it is observed that *U* in the prototypes with zigzag channels (between 2400 and 10000 W.m<sup>-2</sup>.K<sup>-1</sup>) is more than 2 times larger than with straight channels (between 750 and 5600 W.m<sup>-2</sup>.K<sup>-1</sup>) for the same flow rates. It indicates that the presence of bends in the zigzag channel enhances the heat transfer compared to straight channels, particularly at low flow rates, where the overall heat transfer coefficient can even be 4 times higher in zigzag channels than in straight channels. This result was expected as the zigzag channel geometry generates secondary flows tangential to the main flow direction that intensify tangential mixing and reduce the thermal boundary layer even at low Reynolds number [12–16]. Regarding the influence of the channel cross-section, *U* increases when the cross-section decreases in both zigzag and straight channels, due to the process fluid velocity increases at constant flow rate.

In the zigzag channels' dataset, the maximal uncertainty of *U* is 22.3%, 10.6% in average; In the straight channels' datasets, the maximal uncertainty of *U* is 37.7%, 14.6% in average. From Fig. 4, it can be observed that the experimental error increases with the process fluid flow rate. That can be explained by the fact that when the flow rate increases, the difference between inlet and outlet temperatures decreases. As the heat flux is proportional to the temperature difference, the measurement uncertainty is highest at small temperature difference as can be deduced from Eq. (17). The uncertainties are globally higher in the straight channels' dataset as the heat fluxes are lower, hence lower temperature differences.
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

Δd<sub>hp</sub> = 2 mm ; Q<sub>u</sub> = 300 L·h<sup>-1</sup>

◊d<sub>hp</sub> = 2 mm ; Q<sub>u</sub> = 400 L·h<sup>-1</sup>

○d<sub>hp</sub> = 2 mm ; Q<sub>u</sub> = 500 L·h<sup>-1</sup>

Δd<sub>hp</sub> = 3 mm ; Q<sub>u</sub> = 300 L·h<sup>-1</sup>

◊d<sub>hp</sub> = 3 mm ; Q<sub>u</sub> = 400 L·h<sup>-1</sup>

○d<sub>hp</sub> = 3 mm ; Q<sub>u</sub> = 500 L·h<sup>-1</sup>

▲d<sub>hp</sub> = 4 mm ; Q<sub>u</sub> = 300 L·h<sup>-1</sup>

◆d<sub>hp</sub> = 4 mm ; Q<sub>u</sub> = 400 L·h<sup>-1</sup>

●d<sub>hp</sub> = 4 mm ; Q<sub>u</sub> = 500 L·h<sup>-1</sup>

Fig. 4. Overall heat transfer coefficient as a function of the process fluid flow rate (water) in (a) the zigzag channels and (b) the straight channels. The legend gives the hydraulic diameter of the process channel ($d_{h,p} = 2, 3$ and $4$ mm) and the utility fluid flow rate ($Q_u = 300, 400$ and $500$ L·h$^{-1}$ in (a); $Q_u = 300$ and $500$ L·h$^{-1}$ in (b)).

### 4.1.2 Process side heat transfer coefficients

The local heat transfer coefficient of the process side $h_p$ in zigzag channel as a function of Reynolds number $Re_p$ at different utility flow rates and hydraulic diameter is shown in Fig. 5. $h_p$ is calculated using Eq. (9) to Eq. (12), with a maximal uncertainty estimated at 46.3%, 17.2% in average. As expected, for a same channel, $h_p$ is independent of the utility flow rate and increases with increasing Reynolds number. $h_p$ also increases with the decrease of the cross-section size at constant Reynolds number, showing the positive effect of size reduction on the heat transfer coefficients. In straight channel, for established laminar flow and turbulent flow, the theory gives that Nusselt number (ratio of total heat flux to conductive heat flux) is constant at constant Reynolds number and Prandtl. Therefore, $h$ increases when the channel diameter decreases. This is due to both lower distances that makes thermal diffusion more efficient and higher fluid velocity that increases advective heat transfer. These effects appear also in zigzag channel and allow to counterbalance the fact that the secondary vortices are less intense when the diameter decreases regarding Dean number.
<figure><img src="image_8.png" /><figcaption>Fig. 5. Local heat transfer coefficient of the zigzag channel as a function of Reynolds number with water as the process fluid. The legend gives the hydraulic diameter of the process channel ($d_{h,p} = 2, 3$ and 4 mm) and the utility fluid flow rate ($Q_u = 300, 400$ and $500$ L.h$^{-1}$).</figcaption></figure>

To further show the heat transfer enhancement in zigzag channels compared to straight channels, the ratio $h_{p,zigzag}/h_{p,straight}$ is calculated and presented in Fig. 6 as a function of the $Re_p$ averaged between the two experiments that are compared (zigzag channel and straight channel of same $d_{h,p}$ and $Q_u$, and roughly the same $\dot{m}_p$). The horizontal error bar in Fig. 6 represents the difference in $Re_p$ in both experiments (maximal value of 16.3% for low flow rates, 5.4% in average). $h_{p,zigzag}/h_{p,straight}$ varies between 2 and around 5 for the studied Reynolds number range. $h_p$ in zigzag channel is much larger than in straight channel, due to the bends which generate secondary flow and decrease the thermal boundary layer thickness for a global heat transfer enhancement. It also appears that this ratio seems to follow a bell-shaped trend, with a maximal value around $Re_p = 1000$. In laminar flow regime, $h_{p,zigzag}/h_{p,straight}$ first increases with Reynolds number as the intensity and the numbers of vortices in the fluid induced by the bends increase [30,31]. Then, this ratio decreases: the effect of the bends is still positive but the enhancement ratio diminishes. From $Re_p = 2000$, the flow regime in straight channel is transitional and tends toward a turbulent flow at $Re_p > 10000$. It is expected that the enhancement factor tends to 1 in turbulent regime as the effect of the bends on the fluid hydrodynamics will be compensated by the turbulence. The influence of bends in zigzag channel with square cross-sections on the flow and heat transfer has been previously studied based on Computational Fluid Dynamics (CFD) simulations in square zigzag channel of 2 mm depth for $Re_p < 600$ and $Pr_p = 6.13$ [22]. For the same geometry than the experimental mock-up ($d_{h,p} = 2$ mm, $L_s = 7$ mm, $R_c = 1.5$ mm), it was found that $h_{p,zigzag}/h_{p,straight}$ increases from 2 to 5.5 when $Re_p$ increases from 112 to 448. It is consistent with the values obtained experimentally and confirms the bell-shaped trend observed.
<figure><img src="image_9.png" /><figcaption>Fig. 6. $h_{p,zigzag}/h_{p,straight}$ as a function of Reynolds number at two utility flow rates. The legend gives the hydraulic diameter of the process channel ($d_{h,p} = 2, 3$ and 4 mm) and the utility fluid flow rate ($Q_u = 300$ and $500$ L.h$^{-1}$). The error bars represent the difference in $Re_p$ in the experiments with the zigzag and the straight channels that are compared.</figcaption></figure>

## 4.2 Influence of fluid properties on heat transfer coefficients in zigzag channels

In the previous experiments, only the influence of the process flow rate was affecting Reynolds number. To extend the generalization of the results, a second process fluid (glycerol-water mixture, 74 wt% of glycerol) is also considered. The local heat transfer coefficient $h_p$ for both fluids as a function of the process fluid flow rate in Fig. 7(a) and Reynolds number in Fig. 7(b) is shown. These curves correspond to the data obtained in the zigzag channels of different hydraulic diameter for a constant utility flow rate $Q_u = 500$ L.h$^{-1}$. Concerning the dataset with glycerol-water mixture, maximal uncertainty of $h_p$ is 18.3%, 11.2% in average; maximal uncertainty of $Re_p$ is 14.8%, 5.5% in average. $h_p$ obtained for water as the process fluid is much higher than for glycerol-water mixture at the same flow rate, around 3 to 4 times higher. This is due to the fact that glycerol-water mixture is much more viscous than water, resulting in a smaller Reynolds number for the same flow rate which is unfavorable to heat transfer. In terms of channel size influence, $h_p$ increases when the channel cross-section decreases for both fluids.
<figure><img src="image_10.png" /><figcaption>Fig. 7. Local heat transfer coefficient for two process fluids in the zigzag channels as a function of (a) process flow rate and (b) Reynolds number, $Q_u = 500 \text{ L.h}^{-1}$. The legend gives the hydraulic diameter of the process channel, and the process fluid used.</figcaption></figure>

## 4.3 Dimensionless correlation for heat transfer coefficient in zigzag channels

The effects of different parameters on the heat transfer coefficient in millimetric zigzag channels has been studied in the present paper: the channel hydraulic diameter, the process flow rate and the fluid properties. In order to lump all these effects into a simple expression, a correlation is proposed based on the following dimensionless numbers: Nusselt number $Nu_p$ to characterize the heat transfer, Reynolds number $Re_p$ to characterize the flow hydrodynamics, Prandtl number $Pr_p$ to characterize the physicochemical properties of the fluid, and the ratio of the hydraulic diameter to the straight section length between two bends ($d_{h,p}/L_s$) as a dimensionless length. This ratio allows to consider the effect of the hydraulic diameter on the heat transfer coefficients. In curved channels, the ratio of the hydraulic diameter to the curvature radius ($d_{h,p}/R_c$) is often used through the use of Dean number ($De = Re(d_{h,p}/R_c)^{0.5}$). However, this ratio is constant in the different zigzag channels used in the present study. Therefore, its use fails to explain the impact of the hydraulic diameter observed on the Nusselt number. The ratio ($d_{h,p}/L_s$) has been proposed in a previous numerical study of heat transfer by CFD to explain the influence of $L_s$ on flow hydrodynamics and heat transfer in a square zigzag channel of 2 mm depth [16]. ($d_{h,p}/L_s$) is the inverse of the dimensionless length typically used for flow establishment length calculation. In the zigzag channels, it characterizes the length between two bends left to the flow to
reestablish. When ($d_{h,p}/L_s$) increases, the flow disturbance due to the successive bends is more intense. This is confirmed by the pressure drop increase with this ratio [16].

The experimental data obtained in the present study are fitted by a least square procedure (63 points for water and 21 points for glycerol-water mixture as the process fluid) and the following correlation for Nusselt number is established:

$$Nu_p = 0.098(Re_p^{0.69} + 20)Pr_p^{0.18}\left(1 + \left(\frac{d_{h,p}}{L_s}\right)^{1.28}\right) \quad (15)$$

The expression of the correlation proposed is similar to that of the simplified Gnielinski correlation given by Eq. (12) established in straight tubes for transitional and turbulent flow. Comparison between $Nu_p$ calculated with the correlation and experimental values is shown in Fig. 8. The error bar on each point represent the uncertainty of experimental $Nu_p$. In average, the uncertainty is 16.3% with a maximal value of 46.3%. In this figure, all the data of $Nu_p$ with water (black triangles) show less than 25% of relative difference with the values calculated with the correlation. The average relative difference is 5.5% and the maximum value is 22.9%. There are several points for glycerol-water mixture off the 25% relative difference lines (grey squares), with an average of 19.8% and a maximum value of 37.1%. This may be due to the fact that the flow regime is different with this fluid ($Re_p < 420$). Moreover, the viscosity of glycerol-water mixture is more important than that of water, resulting in a $Pr_p$ that probably varies over the channel cross-section. Therefore, the ratio between the average value of $Pr_p$ and this value at the temperature of the wall may be significantly different with glycerol-water mixture. For instance, considering a wall at the temperature of the utility fluid, ($Pr_p/Pr_{p,w}$) is higher than 0.5 in the experiments with water, while it can be as low as 0.3 in the experiments with glycerol-water mixture. However, it can be conclude that the proposed correlation provides a good approximation of the experimental data in a large range of characteristic parameters ($15 \le Re_p \le 7100$; $3.1 \le Pr_p \le 115$; $0.29 \le d_{h,p}/L_s \le 0.57$), although the effect of the ratio $d_{h,p}/L_s$ should be confirmed using data obtained with different combinations of $d_{h,p}$ and $L_s$.

<figure><img src="image_11.png" /><figcaption>Fig. 8. Comparison between experimental Nusselt number ($Nu_{p,exp}$) or simulated Nusselt number ($Nu_{p,simu}$) from and that calculated by the correlation Eq. (15) ($Nu_{p,calc}$) in zigzag channels. Dashed lines represent 25% of relative difference. Error bars represent the uncertainty of $Nu_{p,exp}$.</figcaption></figure>
This is also confirmed by the comparison between the Nusselt number values obtained by CFD simulations in a previous work [16] shown in Fig. 8 (white circles). The 21 simulated points are obtained in a square zigzag channel of 2 mm depth in laminar flow regime ($112 \le Re_p \le 560$; $Pr_p=6.13$; $0.17 \le d_{h,p}/L_s \le 1$). The average relative difference with the values obtained by the correlation given by Eq. (15) is 17.5%, with a maximal value of 86.2%. All the data are within the 25% difference lines, except 3 points at very low Nusselt number obtained for $Re_p = 112$.

# 5. Conclusion

In this work, the heat transfer performance of compact heat-exchanger reactor prototypes is experimentally characterized, in terms of overall heat transfer coefficient and local heat transfer coefficient of the process side. Two kinds of process channel configuration are studied: millimetric zigzag and straight channels of 2, 3 and 4 mm depth. Two process fluids are investigated which are water and a glycerol-water mixture characterized by very different Prandtl numbers (7 and 272 respectively at 20 °C). Reynolds number ranges from 15 to 7300, covering laminar to turbulent flow regime. The obtained local heat transfer coefficient in the zigzag channels ranges from 760 to 15000 W.m⁻².K⁻¹. It increases with Reynolds number and decreases when the channel cross-section is increased. The calculation of the uncertainties due to experimental errors is detailed, using the derivation method. The average uncertainty of the Nusselt number is 16.3% over the 84 experimental points, and the maximal value is 46.4%. With water as the process fluid, it is found that the heat transfer coefficient in zigzag channel is much larger than in straight channel, with a ratio between 2 and 5 in the range of Reynolds number studied, demonstrating the heat transfer enhancement due to the presence of alternating bends in zigzag channels compared to straight channels.

A correlation of Nusselt number in zigzag channels is then proposed with the consideration of fluid properties and flow rate (through Reynolds and Prandtl number) and of channel geometries (ratio of the hydraulic diameters to the straight length between two bends $d_h/L_s$). The correlation is established for a large range of characteristic dimensionless numbers ($15 \le Re_p \le 7100$; $3.1 \le Pr_p \le 115$; $0.29 \le d_{h,p}/L_s \le 0.57$). It provides a reasonable approximation of the experimental data, with 9.1% of relative difference in average. Finally, the consistency of the correlation is confirmed by comparison with data obtained in a previous work based on CFD, where the impact of the ratio $d_{h,p}/L_s$ on Nusselt number was particularly studied. However, in both studies either $d_h$ or $L_s$ are constant. Other experiments using prototypes presenting different combinations of $d_h$ and $L_s$ must be carried out to confirm the influence of this ratio. The proposed correlation for Nusselt number would be very useful for the modeling and the scale-up of heat exchanger reactors.

# Acknowledgement

The experimental pilot was partly funded by BPIFRANCE FINANCEMENT through the project DEEPER (FUI-AAP20).

# Appendix

## A.1 Mathematical method for uncertainty estimation
To perform uncertainty estimation, some simplifications have been considered. Firstly, the thermal loss $q_i$ which always represents less than 10% of the total flux is neglected in the uncertainty calculations. Therefore, the combination of Eq. (6) to Eq. (8) gives the following approximation for the overall heat transfer coefficient:

$$U = \frac{\dot{m}_p C_{p,p}}{A} \cdot \frac{(T_{p,i}-T_{p,o})}{(T_{p,i}-T_{u,i})-(T_{p,o}-T_{u,o})} \cdot \ln \left[ \frac{(T_{p,i}-T_{u,i})}{(T_{p,o}-T_{u,o})} \right] \quad (16)$$

Then the detailed uncertainty of overall heat transfer coefficient calculated from an experiment can be evaluated by the following expression, in agreement with [32,33]:

$$\frac{\Delta U}{U} = \sqrt{\left(\frac{\Delta \dot{m}_p}{\dot{m}_p}\right)^2 + \left(\frac{\Delta C_{p,p}}{C_{p,p}}\right)^2 + \left(\frac{\Delta w_p}{w_p}\right)^2 + \left(\frac{\Delta L_p}{L_p}\right)^2 + \left(\frac{1}{U} \frac{\partial U}{\partial T_{p,i}} \Delta T_{p,i}\right)^2 + \left(\frac{1}{U} \frac{\partial U}{\partial T_{p,i}} \Delta T_{p,o}\right)^2 + \left(\frac{1}{U} \frac{\partial U}{\partial T_{u,i}} \Delta T_{u,i}\right)^2 + \left(\frac{1}{U} \frac{\partial U}{\partial T_{u,o}} \Delta T_{u,o}\right)^2} \quad (17)$$

By derivation of Eq. (16), the following expressions are obtained:

$$\frac{\partial U}{\partial T_{p,i}} = U \left( \frac{1}{(T_{p,i}-T_{p,o})} + \frac{-1}{(T_{p,i}-T_{u,i})-(T_{p,o}-T_{u,o})} + \frac{1}{\ln\left[\frac{(T_{p,i}-T_{u,i})}{(T_{p,o}-T_{u,o})}\right](T_{p,i}-T_{u,i})} \right) \quad (18)$$

$$\frac{\partial U}{\partial T_{p,o}} = U \left( \frac{-1}{(T_{p,i}-T_{p,o})} + \frac{1}{(T_{p,i}-T_{u,i})-(T_{p,o}-T_{u,o})} + \frac{-1}{\ln\left[\frac{(T_{p,i}-T_{u,i})}{(T_{p,o}-T_{u,o})}\right](T_{p,o}-T_{u,o})} \right) \quad (19)$$

$$\frac{\partial U}{\partial T_{u,i}} = U \left( \frac{1}{(T_{p,i}-T_{u,i})-(T_{p,o}-T_{u,o})} + \frac{-1}{\ln\left[\frac{(T_{p,i}-T_{u,i})}{(T_{p,o}-T_{u,o})}\right](T_{p,i}-T_{u,i})} \right) \quad (20)$$

$$\frac{\partial U}{\partial T_{u,o}} = U \left( \frac{-1}{(T_{p,i}-T_{u,i})-(T_{p,o}-T_{u,o})} + \frac{1}{\ln\left[\frac{(T_{p,i}-T_{u,i})}{(T_{p,o}-T_{u,o})}\right](T_{p,o}-T_{u,o})} \right) \quad (21)$$

The heat transfer coefficient ($h_p$) of process fluid can be obtained from Eq. (9):

$$h_p = \frac{\lambda_w U h_u}{\lambda_w h_u - e_w U h_u - C \lambda_w U} \quad (22)$$

where

$$C = \frac{A_p}{A_u} = \frac{L_p w_p}{L_u w_u} \quad (23)$$

Then, the detailed uncertainty of the heat transfer coefficient $h_p$ can be evaluated by:

$$\frac{\Delta h_p}{h_p} = \sqrt{\left(\Delta C\right)^2 \left(\frac{\lambda_w U}{(\lambda_w h_u - \lambda_w CU - e_w U h_u)}\right)^2 + (\Delta e_w)^2 \left(\frac{U h_u}{(\lambda_w h_u - \lambda_w CU - e_w U h_u)}\right)^2} \\ + (\Delta U)^2 \left(\frac{1}{U} + \frac{(\lambda_w C + e_w h_u)}{(\lambda_w h_u - \lambda_w CU - e_w U h_u)}\right)^2 \\ + (\Delta h_u)^2 \left(\frac{1}{h_u} - \frac{(\lambda_w - e_w U)}{(\lambda_w h_u - \lambda_w CU - e_w U h_u)}\right)^2 \\ + (\Delta \lambda_w)^2 \left(\frac{1}{\lambda_w} - \frac{(h_u - CU)}{(\lambda_w h_u - \lambda_w CU - e_w U h_u)}\right)^2} \quad (24)$$
$$ \frac{\Delta C}{C} = \sqrt{\left(\frac{\Delta L_p}{L_p}\right)^2 + \left(\frac{\Delta w_p}{w_p}\right)^2 + \left(\frac{\Delta L_u}{L_u}\right)^2 + \left(\frac{\Delta w_u}{w_u}\right)^2} \quad (25) $$

The heat transfer coefficient $h_u$ of the utility fluid can be obtained from Eq. (12). In this equation, $Re^{0.87}$ is much larger than 280 ($Re > 10^4$ in the experiment). Consequently, for evaluating the uncertainty of $h_u$, the following simplified expression of Eq. (12), is considered:

$$ h_u = \frac{Nu_u \lambda_u}{d_{h,u}} = \alpha_{cor} 0.12 \left(\frac{4}{\pi}\right)^{0.87} \dot{m}_u^{0.87} C_{P,u}^{0.4} d_{h,u}^{-1.87} \lambda_u^{0.6} \mu_u^{-0.47} \quad (26) $$

Then, the uncertainty of heat transfer coefficient $h_u$ can be calculated:

$$ \frac{\Delta h_u}{h_u} = \sqrt{\left(\frac{\Delta \alpha_{cor}}{\alpha_{cor}}\right)^2 + \left(0.87 \frac{\Delta \dot{m}_u}{\dot{m}_u}\right)^2 + \left(1.87 \frac{\Delta d_{h,u}}{d_{h,u}}\right)^2 + \left(0.4 \frac{\Delta C_{P,u}}{C_{P,u}}\right)^2 + \left(0.6 \frac{\Delta \lambda_u}{\lambda_u}\right)^2 + \left(0.47 \frac{\Delta \mu_u}{\mu_u}\right)^2} \quad (27) $$

The detailed uncertainty of Nusselt number of the process fluid ($Nu_p$) is evaluated from Eq. 13, according to the following expression:

$$ \frac{\Delta Nu_p}{Nu_p} = \sqrt{\left(\frac{\Delta h_p}{h_p}\right)^2 + \left(\frac{\Delta d_h}{d_h}\right)^2 + \left(\frac{\Delta \lambda_p}{\lambda_p}\right)^2} \quad (28) $$

Reynolds number of process fluid ($Re_p$) is calculated by:

$$ Re_p = \frac{\rho_p u_p d_h}{\mu_p} = \frac{\dot{m}_p}{d_h \mu_p} \quad (29) $$

Its uncertainty can then be evaluated by:

$$ \frac{\Delta Re_p}{Re_p} = \sqrt{\left(\frac{\Delta \dot{m}_p}{\dot{m}_p}\right)^2 + \left(\frac{\Delta d_h}{d_h}\right)^2 + \left(\frac{\Delta \mu_p}{\mu_p}\right)^2} \quad (30) $$

## Nomenclature

$A$ heat exchange area (m²)

$C_p$ specific heat (J.kg⁻¹.K⁻¹)

$d$ channel depth (m)

$d_h$ hydraulic diameter (m)

$e$ thickness of aluminum (m)

$h$ local heat transfer coefficient (W.m⁻².K⁻¹)

$L$ channel length (m)

$L_s$ straight section length (m)

$\dot{m}$ mass flow rate (kg.h⁻¹)

$Nu$ Nusselt number
Pr Prandtl number

Q volumetric flow rate (m³.h⁻¹)

q heat flux (W)

q_l thermal loss (W)

R_c radius of curvature (m)

Re Reynolds number

T temperature (K)

ΔT_{ml} logarithmic mean temperature difference (K)

u velocity (m.s⁻¹)

U overall heat transfer coefficient (W.m⁻².K⁻¹)

V volume (m³)

w channel width (m)

### Greek symbols

α_{cor} factor introduced in Eq. (12)

θ angle between two straight sections (°)

λ thermal conductivity (W.m⁻¹.K⁻¹)

μ dynamic viscosity (Pa.s)

ρ fluid density (kg.m⁻³)

### Subscripts

calc calculated with a correlation

exp from an experiment

i inlet

o outlet

p process fluid

simu from a simulation

straight in straight channel

u utility fluid

w wall

zigzag in zigzag channel
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