REPRESENTATION OF 3D PHENOMENA OF
LAMINAR FLOW AND HEAT TRANSFER IN A
HELICALLY COILED TUBE USING A 2D
GEOMETRY

Nilo Henrique Meira Fortes, Artemio Plana-Fattori, Jorge Audrey Wilhelms Gut

► To cite this version:

Nilo Henrique Meira Fortes, Artemio Plana-Fattori, Jorge Audrey Wilhelms Gut. REPRESENTATION OF 3D PHENOMENA OF LAMINAR FLOW AND HEAT TRANSFER IN A HELICALLY COILED TUBE USING A 2D GEOMETRY. International Journal of Thermal Sciences, 2024, 200, pp.108943. 10.1016/j.ijthermalsci.2024.108943. hal-04429594

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REPRESENTATION OF 3D PHENOMENA OF LAMINAR FLOW
AND HEAT TRANSFER IN A HELICALLY COILED TUBE USING A
2D GEOMETRY

Nilo Henrique Meira Fortes¹, Artemio Plana-Fattori², Jorge Andrey Wilhelms Gut¹,³,*

¹ Universidade de São Paulo, Escola Politécnica, Dept. of Chemical Engineering, 05508-000, São Paulo, Brazil

² Université Paris-Saclay, INRAE, AgroParisTech, UMR SayFood, 91120, Palaiseau, France

³ Universidade de São Paulo, FoRC – Food Research Center, 05508-080, São Paulo, Brazil

11

12 * Corresponding author. Tel.: +55-11-30912253.

13 E-mail address: jorgewgut@usp.br (J. A. W. Gut).

14

15 Abstract

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25
The coiled tube geometry has different industrial applications whereby heating and
cooling operations are required. In this work, a 2D semi-empirical model based on a
straight tube is proposed to represent the 3D phenomena of flow and heat transfer in a
coiled tube. The model considers a modified velocity profile based on residence time
distribution (RTD) experiments and a heat transfer enhancement factor based on heating
and cooling experiments. For testing and validating the model, experiments were
performed using a Newtonian fluid (glycerin/water mixture) and a non-Newtonian fluid
(carboxymethylcellulose CMC solution) at flow rates from 0.5 to 2.0 L/min in a coiled
tube with 9 turns. The y-laminar RTD model was successfully adjusted to experimental
E-curves and its parameter y was correlated with flow rate. Heating and cooling
26 experiments with the coil provided outlet temperatures at different conditions of flow rate
27 and temperature. The model was adjusted to match the outlet temperatures, providing the
28 heat transfer enhancement factor F, the second empirical parameter of the model. This
29 factor was linearly correlated with Reynolds number in a log-log plot and showed a
30 threshold for negligible enhancement (F = 1). The model was numerically solved the
31 finite difference method after a mesh dependency study. Good outlet temperature
32 predictions were obtained with computational time of about 1 min, which is much smaller
33 than usual computational times for a 3D coil geometry, and model provides velocity and
34 temperature fields useful for further simulation of non-isothermal laminar flow reactors,
35 such as bacterial inactivation in thermal processing of liquid foods.
36

36

37 **Keywords**

38 Heat transfer enhancement; Heat exchanger; Laminar flow; Residence time distribution;

39 Coiled tube; Dean vortices

40

41 NOMENCLATURE

$A(t)$ exit absorbance of tracer (-)

$A_0$ initial absorbance of tracer (-)

$A_e$ external surface area of the tube (m²)

$A_i$ internal surface area of the tube (m²)

$C(t)$ exit concentration of tracer (kg.m⁻³)

$C_0$ initial concentration of tracer (kg.m⁻³)

$C_p$ Specific heat capacity at constant pressure (J.kg⁻¹.K⁻¹)

$d_c$ coil diameter (m)

$d_e$ tube external diameter (m)
$d_{eq}$ equivalent diameter of the duct (m)

$d_i$ tube internal diameter (m)

$D$ Mass diffusivity ($m^2 \cdot s^{-1}$)

$De$ Dean number (-)

$E(t)$ exit age distribution function ($s^{-1}$)

$E_{exp}(t)$ exit age distribution function of experimental data ($s^{-1}$)

$E_{model}(t)$ exit age distribution function of the model ($s^{-1}$)

$E_{\theta}(\theta)$ dimensionless exit age distribution function (-)

$f$ Theoretical friction factor for laminar flow in tube with smooth walls (-)

$f_{eff}$ Effective friction factor (-)

$F$ heat transfer enhancement factor (-)

$h$ convective heat transfer coefficient of the internal fluid ($W \cdot m^{-2} \cdot K^{-1}$)

$h_e$ convective heat transfer coefficient of the external fluid ($W \cdot m^{-2} \cdot K^{-1}$)

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

$k_{eff}$ effective thermal conductivity ($W \cdot m^{-1} \cdot K^{-1}$)

$k_s$ thermal conductivity of the solid ($W \cdot m^{-1} \cdot K^{-1}$)

$K$ consistency coefficient ($Pa \cdot s^n$)

$l$ characteristic length of the geometry (m)

$L$ calculated linear length of the tube (m)

$m$ parameter of the *m*-laminar model (-)

$n$ number of turns of the coil or flow behavior index (-)

$Nu$ Nusselt number (-)

$p$ coil pitch (m)

$Pe$ Peclet number (-)
$Pr$ Prandtl number (-)

$q$ heat transfer rate (W)

$\dot{q}_v$ volumetric thermal energy generation (W.m⁻³)

$Q$ volumetric flow rate (m³.s⁻¹)

$r$ radial dimension (m)

$r_i$ tube internal radius (m)

$r_e$ tube external radius (m)

$r^*$ dimensionless radius (-)

$R^2$ coefficient of determination (-)

$Re$ Reynolds number (-)

$Re_{crit}$ critical Reynolds number (-)

$Re_g$ generalized Reynolds number (-)

$t_m$ mean residence time (s)

$T_b$ bulk temperature (K)

$T_i$ initial temperature (K)

$T_{in}$ inlet temperature (K)

$T_m$ average temperature (K)

$T_{out}$ outlet temperature (K)

$T_e$ external temperature (K)

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

$v_b$ bulk velocity (m.s⁻¹)

$v_{in}$ inlet velocity (m.s⁻¹)

$v_{max}$ maximum velocity (m.s⁻¹)

$v_z$ axial component of velocity (m.s⁻¹)
$\vec{v}$ velocity vector (m.s⁻¹)

$v^*$ dimensionless velocity (-)

$V$ internal volume of the tube (m³)

$V_A$ active volume of the tube (m³)

$V_S$ volume of the solid (m³)

$y$ y-laminar model parameter (-)

$z$ axial dimension (m)

42

## 43 Greek letters

$\alpha$ thermal diffusivity (m².s⁻¹) or parameter of the sinusoidal model (-)

$\alpha_{eff}$ effective thermal diffusivity (m².s⁻¹)

$\beta$ parameter of the exponential model (-)

$\delta$ curvature ratio (-)

$\varepsilon$ Friction factor enhancement factor (-)

$\theta_0$ dimensionless breakthrough time (-)

$\mu$ dynamic viscosity (Pa.s)

$\mu_g$ generalized viscosity (Pa.s)

$\nu$ kinematic viscosity (m².s⁻¹)

$\xi$ geometrical parameter of the duct (-)

$\varphi$ geometrical parameter of the duct (-)

$\rho$ density (kg.m⁻³)

$\tau$ theoretical residence time (s)

44
1. INTRODUCTION

The coiled or helical tube geometry is widely used in processes that involve continuous flow chemical reactions and/or heat transfer because of its compact structure, enhanced mixing and passive heat transfer enhancement under laminar flow [1,2]. The centrifugal forces experienced by the fluid due to the coil curvature generate secondary flow structures called Dean cells, a pair of counter-rotating symmetrical vortices at the cross-section of the tube that moves fluid outward from the curvature center of the coil. This circulation of fluid promotes local mixing and increases heat transfer rate and chemical reaction yield [3].

The effects of the tube curvature on the flow can be characterized by the Dean number (De), a dimensionless group that relates the Reynolds number (Re) with the square root of the curvature ratio ($\delta = d_i/d_c$) according to Eq. (1) for tubes with small torsion (ratio between coil pitch and coil radius) [4]. The Reynolds number is the ratio of inertial forces to viscous forces, as defined in Eq. (2):

$$De = Re\sqrt{d_i/d_c} \qquad (1)$$

$$Re = \frac{\rho d_i v_b}{\mu} \qquad (2)$$

where $d_i$ is the tube internal diameter, $d_c$ is the coil or curve diameter, $v_b$ is the bulk velocity, $\rho$ is the average fluid density and $\mu$ is the average fluid viscosity [5,6].

The flow pattern inside a coiled tube is complex due to the secondary flow, especially in laminar flow with small Reynolds numbers [7]. The geometrical characteristics of the coil, particularly the curvature ratio and the pitch, influence the degree of fluid mixing. Also, Dean instabilities may arise above a threshold Dean number, which creates additional pairs of counter-rotating vortices at the outer wall of tube. This critical value is unique for each coil, since it depends on the coil curvature ratio ($\delta$) and cross-section aspect ratio (ratio between height and width) [8].
The flow pattern inside tubes can be studied using the concept of residence time distribution (RTD), which is defined as the distribution of time that fluids spend inside a vessel. Usually, RTD is determined experimentally by injecting a tracer at the inlet and measuring its concentration at the outlet over time, C(t). The exit age distribution function E(t), or E-curve, is defined in Eq. (3a), while the mean residence time is defined in Eq. (3b) and the dimensionless E-curve in Eq. (3c), where θ = t/t_m is the dimensionless time. The E-curve is important to characterize flow pattern and dispersion inside vessels [9].

$$E(t) = \frac{C(t)}{\int_{0}^{\infty} C(t) \, dt} \qquad (3a)$$

$$t_m = \int_{0}^{\infty} t E(t) \, dt \qquad (3b)$$

$$E_{\theta}(\theta) = t_m E(t) \qquad (3c)$$

The RTD in tubes with laminar flow is related to the dispersion caused by the mechanisms of convection (axial velocity profile) and molecular diffusion (in axial and radial directions), constituting the Taylor-Aris dispersion. The laminar velocity profile promotes a large spread in the E-curve, while the radial diffusion promotes mixing between layers, thus reducing this spread. The secondary flow that arises from the flow in curves and helical pipes also contributes to narrow the dispersion in the E-curve and to delay the first appearance of the tracer, because it promotes convective mixing in the cross section of the tube [9,10,11].

Florit et al. [11] studied the axial dispersion in coiled reactors with computational fluid dynamics (CFD) using the finite volume method in 3D simulations. Different geometrical parameters were considered, such as coil pitch, coil radius and tube length, while a single value of tube diameter was chosen. Transient CFD simulations were used
to evaluate the residence time distribution (RTD) in the coil, which was represented by the axial dispersion model (ADM). RTD experiments were performed to determine the dispersion number (parameter of the ADM) using a blue dye as tracer (pulse injection) and water as fluid. The CFD results were in good agreement with experimental data and a reduction was observed in the axial dispersion in the coiled tubes when compared to flow in straight tubes.

Sharma et al. [12] studied mixing in coiled tubes and coiled flow inverters in different flow regimes. RTD experiments were conducted by the colorimetric method, using water as solvent and a red dye as tracer. The results were evaluated in terms of the Peclet number, a dimensionless group that represents the ratio between advection and mass diffusion (Eq. 4a) or thermal diffusion (Eq. 4b):

$$Pe = \frac{l v_b}{D} \qquad (4a)$$

$$Pe = \frac{l v_b}{\alpha} \qquad (4b)$$

where D is the mass diffusivity or diffusion coefficient, α is the thermal diffusivity and l is the characteristic length of the geometry (diameter for circular tubes). Plots of Peclet number versus Reynolds number indicated that: in laminar flow, the Dean vortices have a strong influence in mixing; in the transition zone, the effect of the Dean vortices is balanced by the increase in the axial dispersion induced by the velocity profile; in the turbulent regime, the cross-sectional mixing dominates.

Different types of heat exchangers can be used for heating and cooling, depending on the application. Plate heat exchangers are widely used in the food industry for low viscosity fluids because of the compactness and ease of cleaning; however, pressure drop is a limiting factor. Shell and tube heat exchangers are more versatile in terms of pressure and temperature conditions [13,14]. For highly viscous fluids and small flow rates, coiled
or helical heat exchangers are a common choice due to compactness, enhanced heat transfer and lower pressure drop [15]. The coiled geometry can be used to construct different types of heat exchangers, such as coiled double-pipe heat exchangers, shell and helically coiled tube heat exchangers and helical coil heat exchangers. Kushwaha et al. [16] numerically studied heat transfer and fluid flow in a coiled double-pipe heat exchanger with a Newtonian and a non-Newtonian fluid; Reddy et al. [17] used CFD to study the improvement in heat transfer in a coiled double-pipe heat exchanger with different geometries for the inner tube; Bozzoli et al. [18] studied the local Nusselt number and showed that at the outside surface of the coil, it can be five times larger than that at the inside surface because of the secondary flow; and Wang et al. [19] studied the optimal design of a double-pipe heat exchanger based on the outward helically corrugated tube.

Examples of application of the shell and helically coiled tube heat exchangers can be found in recent literature. Maghrabie et al. [20] performed an experimental study with a shell and helically coiled tube heat exchanger considering different inclination angles; Raut et al. [21] numerically studied the effect of geometrical and operational parameters of a shell-and-tube heat exchanger in a process of paraffin melting; and Wang et al. [22] conducted an experimental and numerical study on the heat transfer and flow based on multi-objective optimization in a shell and helically coiled tube heat exchanger.

Helical coil heat exchangers have different applications. Lei and Bao [23] experimentally studied the laminar heat transfer at supercritical pressure in a helical coiled tube; Oliveira et al. [24] developed a model to calculate the pressure loss of Newtonian and non-Newtonian fluids in coiled tubes with experimental validation; and Xu et al. [25] developed a model based on the simulation code RELAP5 to study the thermo-hydraulic characteristics of helically coiled tubes.
A variety of tube configurations and accessories have been studied as additional methods to improve heat transfer in coiled tubes, e.g. flow inverter/reverser [26,27], helical tube with fins [28], tubes with corrugated wall [29,30], and twisted tapes/wired coils inside the tube [31,32].

Many of the studies on flow, heat transfer and dispersion in coiled tubes are conducted using CFD software because of the complex flow pattern generated by the secondary flow. Numerical simulations in such cases require an accurate modeling using refined 3D meshes with a large number of elements to adequately represent the mixing phenomenon [33]. Moreover, the use of accessories in tubes and/or different tube configurations further increases the model complexity and the computational cost (memory and time).

Simplified models that describe flow and heat transfer in coiled tubes are an alternative to complex 3D models, because simpler models require less computational time, which allows them to be implemented in optimization problems or in predictive control in the industry. An approach to study mixing in laminar flow in 3D and 2D models was proposed by Galaktionov et al. [34,35]. The method is known as mapping approach and consists of numerical simulations for the accurate tracking of fluid volumes in small intervals of time or space. The flow domain is subdivided into small subdomains (creating a mesh) and mixture is described in these subdomains with local averaged values, usually concentrations. Then, the flow is established as a sequence of discrete steps using the subdomains and then the concentration values are stored in a mapping matrix. This method requires knowledge of the velocity field and complex geometries may increase computational costs.

A semi-empirical approach to study flow and heat transfer in tubes was realized by Morais and Gut [36]. A 2D model was proposed for laminar flow in a straight tube
with enhanced radial heat transfer. The objective was to simulate heat transfer in systems with enhanced mixing, such as coiled or corrugated tubes. A heat transfer enhancement factor was defined in terms of a modified Peclet number. The results indicated that the effective radial thermal diffusivity in a coiled tube (experimental) was up to six times higher than the fluid property; however, the laminar flow velocity profile used in their model did not adequately represent the residence time distribution (RTD) in the coiled tube, which would compromise reaction yield calculations for a non-isothermal LFR (Laminar Flow Reactor).

Dantas and Gut [37] applied this model to simulate the continuous-flow thermal processing of a liquid food with associations of double-pipe heat exchangers for heating and cooling. Enhanced radial heat and mass diffusions were considered for straight tube laminar flow of the food product because of the high relative roughness of the tube. Simulation results provided the time-temperature history of the product along the process with the corresponding rates of inactivation of the target microorganism and of loss of quality due to high temperature. It was possible to see how enhanced heat transfer or enhanced diffusion in the radial direction can largely impact the process lethality or yield; however, there was no experimental validation of the model.

Therefore, the objective of this study was to develop and to experimentally validate a 2D semi-empirical model to represent laminar flow RTD and heat transfer in a coiled tube. The original 3D problem was reduced to a 2D axisymmetric problem in a straight tube geometry with the same linear length, but with enhanced heat transfer and adjusted velocity profile. A velocity profile deriving from a RTD model obtained from experiments was introduced to better represent the flow and dispersion in the coiled tube, which is an important improvement over previous studies.
## 2. MATHEMATICAL MODELING

The flow and heat transfer in a coiled tube is inherently a 3D phenomenon. In order to build a 2D model, a straight tube with length $L$ and internal diameter $d_i$ was considered (Fig. 1); however, the shape of the velocity profile is going to be changed to match the real E-curve of the coiled tube and a heat transfer enhancement factor in the radial direction will be introduced to match the real heat transfer rate of the coiled tube. As further explained, since these two modifications rely on experimental data for parameter estimation, this 2D model is semi-empirical.

The model uses a cylindrical coordinate system with temperature varying with $z$ and $r$, axial velocity varying with $r$, a fluid with inlet temperature $T_{in}$, inlet volumetric flow rate $Q_{in}$ and a heat transfer rate $q(z)$ on the tube wall.

The length of the straight tube is equal to the linear length of the coiled tube (center line), which can be calculated with Eq. (5) [38]:

$$L = n\sqrt{p^2 + \pi(d_c + d_e)^2} \qquad (5)$$

where $n$ is the number of turns of the coil, $p$ is the coil pitch, $d_c$ is the coil diameter and $d_e$ is the tube external diameter.

The heat transfer was evaluated using the energy conservation equation (Eq. 6) [5,39]:

$$\frac{\partial(T\rho C_p)}{\partial t} + \vec{v} \cdot \nabla(\rho C_p T) = \nabla(k\nabla T) + \dot{q}_v \qquad (6)$$

where $T$ is the temperature, $\rho$ is the fluid density, $C_p$ is the specific heat capacity, $\vec{v}$ is the velocity vector, $k$ is the fluid thermal conductivity and $\dot{q}_v$ is the volumetric thermal energy generation.

The following assumptions were considered: 1) steady state, 2) negligible heat generation (viscous dissipation), 3) laminar, incompressible and fully developed flow, 4)
velocity only in the axial direction ($v_z$), 5) negligible axial diffusion of heat and mass, 6)
homogeneous local convective heat transfer coefficient along the tube circumference, 7)
2D axisymmetric problem around the z axis with $T = T(z,r)$ and $v = v(r)$ and 8)
constant thermophysical properties evaluated at an average temperature. Consequently,
Eq. (7) showed to be analogous to the Graetz problem for heat transfer in tubular laminar
flow:

$$v_z \frac{\partial T}{\partial z} = \alpha \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right) \qquad (7)$$

where $\alpha = k/(\rho C_p)$ is the thermal diffusivity and $v_z(r)$ is the axial velocity profile.

Three boundary conditions are required to solve Eq. (7): specification of the tube
inlet temperature, symmetry condition at the center of the tube and homogeneous
convective heat transfer on the surface of the tube; the latter being a modification of the
Graetz problem that considers constant wall temperature. The boundary conditions are
summarized in Eq. (8) as:

$$\left\{
\begin{array}{l}
\text{For } z = 0 \rightarrow T(z,r) = T_{in} \\
\text{For } r = r_i \rightarrow -kA_i \frac{\partial T}{\partial r} = h_e A_e [T(z,r) \pm T_e] \\
\text{For } r = 0 \rightarrow \frac{\partial T}{\partial r} = 0
\end{array}
\right.
\qquad (8)$$

where $r_i$ is internal radius of the tube, $A_i$ is the internal surface area of the tube, $A_e$ is the external surface area of the tube, $h_e$ is the average convective heat transfer coefficient of the external fluid over the tube wall and $T_e$ is the temperature of the external fluid, which was considered uniform. In Eq. (8), the radial temperature gradient in the tube wall was neglected because of the low thermal resistance of the metal; nevertheless, this condition can be modified with the use of an overall heat transfer coefficient that considers the thermal resistances of the wall and of the external fluid.
## 2.1 Heat transfer enhancement

In a coiled tube, secondary flow due to the curves increases heat transfer within the fluid. In order to increase the heat transfer rate in the 2D straight tube model, an effective thermal diffusivity ($\alpha_{eff}$) in the radial direction was used in Eq. (7), which is based on an effective thermal conductivity ($k_{eff}$), since $\alpha_{eff} = k_{eff}/(\rho c_p)$. This empirical parameter can be determined by heat transfer experiments, which are discussed in section 3 using the approach from Morais and Gut [36].

A heat transfer enhancement factor ($F$) was defined as in Eq. (9) in terms of the ratio of thermal diffusivities or thermal conductivities, comparing the fluid property ($\alpha$ or $k$) with the effective values ($\alpha_{eff}$ or $k_{eff}$).

$$F = \frac{\alpha_{eff}}{\alpha} = \frac{k_{eff}}{k} \qquad (9)$$

For high coil pitches or high coil diameters, the $F$ value is expected to tend to unity, since the geometry approximates a straight tube and the effect of secondary flow (Dean vortices) is reduced.

The enhancement in heat transfer from the Dean vortices comes with a pressure drop penalty. For pressure drop evaluation using the 2D model, it could be calculated for laminar flow in a straight tube using Hagen-Poiseuille's equation (taking into account any height difference between tube inlet and outlet), but using an enhancement factor $\epsilon$ to increase the straight friction factor:

$$\epsilon = \frac{f_{eff}}{f} \qquad (10)$$

where $f_{eff}$ is the effective friction factor obtained from experiments with the coiled tube, $f$ is the theoretical flow friction for laminar in straight tube with smooth walls. This approach was successfully used by Rainieri et al. [40] to evaluate friction factor increase in coiled tubes with and without wall corrugation.
247 **2.2 Modified velocity profiles for laminar flow**

248
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271
The laminar velocity profile in a straight tube is not adequate to represent the flow
in a coiled tube, since the secondary flow modifies the RTD by narrowing the dispersion
in the E-curve [11]. In order for the 2D straight tube model to have the same RTD of the
coiled tube it represents, modified laminar velocity profiles are proposed herein. They
consist of modifications on the shape of the parabolic Newtonian velocity profile v* =
1 - r*^2, which comes from the application of continuity and motion equations to describe
steady-state fully developed isothermal laminar flow of a Newtonian fluid in a straight
tube [41], where r* = r/r_i is the dimensionless radius and v* = v_z/v_max is the
dimensionless velocity.
Four alternative velocity profiles and their corresponding RTD models were
selected based on the work from Pegoraro et al. [41]. Fig. 2 presents the shape of the
velocity profiles, as affected by the model parameters. Since flow is unidimensional and
incompressible, these profiles satisfy the continuity equation for mass conservation. The
laminar velocity profile of the straight tube was modified so that its theoretical RTD
matches the RTD obtained from experiments with the coiled tube. The proposed velocity
profiles and their corresponding RTD models are summarized in Table 1, where θ₀ =
v_b/v_max is the dimensionless breakthrough time or first appearance time, θ = t/t_m is
the dimensionless time, t_m is the mean residence time and E_θ(θ) is the dimensionless
exit age distribution function. The RTD function were derived directly from the velocity
profiles without considering radial or axial diffusivity. In coiled tubes, the breakthrough
time has shown to be higher than in straight tubes, e.g., the first appearance of the tracer
is delayed due to the presence of secondary flow [42].
The y-laminar RTD model is based on the time-smoothed velocity profile
characteristic of turbulent flow in tubes, which can be obtained with y = 1/7. Parameter

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271
Four alternative velocity profiles and their corresponding RTD models were
selected based on the work from Pegoraro et al. [41]. Fig. 2 presents the shape of the
velocity profiles, as affected by the model parameters. Since flow is unidimensional and
incompressible, these profiles satisfy the continuity equation for mass conservation. The
laminar velocity profile of the straight tube was modified so that its theoretical RTD
matches the RTD obtained from experiments with the coiled tube. The proposed velocity
profiles and their corresponding RTD models are summarized in Table 1, where θ₀ =
v<sub>b</sub>/v<sub>max</sub> is the dimensionless breakthrough time or first appearance time, θ = t/t<sub>m</sub> is
the dimensionless time, t<sub>m</sub> is the mean residence time and E<sub>θ</sub>(θ) is the dimensionless
exit age distribution function. The RTD function were derived directly from the velocity
profiles without considering radial or axial diffusivity. In coiled tubes, the breakthrough
time has shown to be higher than in straight tubes, e.g., the first appearance of the tracer
is delayed due to the presence of secondary flow [42].

The y-laminar RTD model is based on the time-smoothed velocity profile
characteristic of turbulent flow in tubes, which can be obtained with y = 1/7. Parameter
y varies between 0 and 1, and the piston flow condition is obtained when it approaches 0
272
[41]. The *m*-laminar RTD model is based on the laminar flow of a power-law fluid in a
273
tube. The power-law velocity profile equation was generalized by introducing the
274
parameter *m* > 1 as the exponent of *r**. As the value of *m* increases, the velocity profile
275
gets flatter (piston flow), while the classic Newtonian profile is obtained with *m* = 2 [41].
276

A sinusoidal velocity profile was also suggested to represent non-ideal laminar
277
flow in tubes. A cosine function was scaled to fit the (0,0)–(1,1) square of the $v^* \times r^*$ plot
278
and the exponent $\alpha$ was introduced as a parameter. As the value of $\alpha$ decreases, the
279
velocity profile gets flatter (piston flow) and a very close match of classic Newtonian
280
profile is obtained with $\alpha = 0.430$. An exponential velocity profile was additionally
281
proposed to represent non-ideal laminar flow in tubes. The exponential function was
282
scaled to fit the (0,0)-(1,1) square of the $v^* \times r^*$ plot and the exponent $\beta$ was introduced
283
as a parameter. As the value of $\beta$ decreases, the velocity profile gets flatter (piston flow).
284

RTD experiments with the coiled tube provide the E-curve, as further described
285
in Section 3.1. The RTD models in Table 1 will then be adjusted in order to match this
286
experimental E-curve. The velocity profile $v(r)$ corresponding the best match is going to
287
be substituted in eq. (6) as $v_z$, thus changing the RTD of the 2D straight tube to mimic
288
the RTD from the coiled tube. The experimental RTD is generally obtained under
289
isothermal conditions at room temperature. In the proposed 2D straight tube model, it was
290
assumed that the velocity profiles adjusted from the experimental RTD data are not
291
significantly affected by temperature changes (effect of temperature on viscosity and
292
density were not considered).
293

## 3. MATERIAL AND METHODS

In order to test and to validate the proposed model, the stainless-steel helically
295
coiled tube shown in Fig. 3 (Armfield, Hampshire, UK) was used in experiments. The
296
297 tube has the following dimensions: internal diameter $d_i$ = 9.3 mm, external diameter
298 $d_e$ = 12.7 mm, diameter of the coil $d_c$ = 107 mm, pitch $p$ = 12.7 mm and nine turns ($n$ =
299 9).
300

The calculated linear length of the helically coiled tube included the helix length
301 calculated with Eq. (4), the length of the straight tube section at the inlet (measured with
302 a tape measure) and the length of the curved tube at the outlet calculated with Eq. (4),
303 assuming $n = 1/4$. Thus, the total linear length was $L = 2.85$ m and the calculated
304 internal volume was $V = 194$ mL.
305

Two model fluids with different rheological behaviors were chosen: 1) a
306 glycerin/water mixture with a glycerin mass fraction of 80 %, which is a Newtonian fluid,
307 and 2) a carboxymethylcellulose (CMC) aqueous solution with mass fraction 1 %, which
308 is a non-Newtonian pseudoplastic fluid (power law).
309

The thermophysical properties of pure water and pure glycerin were obtained from
310 Yaws [43] and the properties for the glycerin/water mixture were weighted based on the
311 volume fraction for $k$ and $\rho$ or on the mass fraction for $C_p$. The viscosity ($\mu$) of the mixture
312 was obtained from Cheng [44].
313

Properties of CMC solution were obtained from Carezzato et al. [45]. The concept
314 of generalized viscosity ($\mu_g$) was used for calculating the generalized Reynolds number
315 ($Re_g$) and the generalized Prandtl numbers ($Pr_g$). The generalized viscosity for power-
316 law flow in a duct is defined in Eq. (11) as:
317

$$ \mu_g = K \cdot \xi^{n-1} \cdot \left( \frac{v_m}{d_{eq}} \right)^{n-1} \left( \frac{\varphi \cdot n + 1}{(\varphi + 1) \cdot n} \right)^n \quad (11) $$

where $K$ is the consistency coefficient, $n$ is the flow behavior index $d_{eq}$ is the equivalent
diameter of the duct and $\xi$ and $\varphi$ are the geometrical parameters of the duct. For a circular
318
319 tube, $d_{eq} = 2r_i$, $\xi = 8$ and $\varphi = 3$ [46]. All thermophysical properties were calculated at
320 the average temperature between measurements at the inlet and outlet of the tube.

321 Critical Reynolds number was evaluated for all the experiments to determine the
322 flow regime. The transition from laminar flow to turbulent flow is delayed in coiled tubes
323 because the Dean vortices reduces the axial dispersion and stabilizes the flow suppressing
324 turbulent fluctuations [40]. The correlation used was proposed by El-Genk and Schriener
325 [47] for coiled tube (Eq. 12), which is a function of the curvature ratio $\delta = d_i/d_c$ and
326 valid for $0.001 < \delta < 0.124$. Values under the $Re_{crit}$ indicate laminar flow.

$$Re_{crit} = 2300 (1 + 51640 \delta^{1.575})^{0.2} \qquad (12)$$

327 First, RTD experiments at room temperature were necessary to adjust the velocity
328 profile from RTD models. Next, heat exchange experiments were performed to obtain the
329 outlet temperatures for different conditions. The product pump and the heat exchangers
330 of the pasteurization unit Microwave Lab25-UHT/HTST EHVH (MicroThermics,
331 Raleigh, USA) were used to provide the feed for the coiled tube at the desired flow rate
332 and temperature.

333 **3.1 Residence time distribution experiments**

334 The RTD experiments were carried out by colorimetric method using methylene
335 blue as tracer. The glycerin/water mixture and the CMC solution were pumped at room
336 temperature (between 20 and 25 °C) and at four flow rates (0.5, 1.0, 1.5 and 2.0 L/min).
337 Three repetitions were made for each flow rate.

338 The tracer used in the experiments was methylene blue dissolved in the carrier
339 fluid (glycerin/water mixture or CMC solution) at the concentration of 160 mg/L, which
340 was introduced in the system by the pulse technique. A syringe with 1 mL capacity was
341 used to perforate a silicone tube connected at the inlet of the coiled tube of Fig. 3 and 0.3
342 mL of tracer was quickly injected. Samples were collected at the outlet of the tube every
second and each sample was homogenized before spectrophotometric reading to measure the absorbance of the fluid (A). A spectrophotometer (model 700 Plus, FEMTO, São Paulo, Brazil) was used setting the wavelength to 665 nm with a quartz cuvette of 10-mm optical length.

The concentration of the tracer dissolved in the carrier fluid was tested to verify the validity of the Beer-Lambert law. Tests with both fluids showed that absorbance values were smaller than one ($A < 1$), thus, the Beer-Lambert law was valid in the RTD experiments (concentration of the tracer is proportional to the absorbance) [48]. Hence, the concentrations in the computation of $E(t)$ were replaced by the absorbances according to Eq. (13), where $A_0$ is the background absorbance and $A(t)$ is the absorbance of the sample. The integrals were numerically evaluated with the trapezoidal method.

$$E(t) = \frac{C(t) - C_0}{\int_0^\infty (C(t) - C_0) dt} = \frac{A(t) - A_0}{\int_0^\infty (A(t) - A_0) dt} \quad (13)$$

The four RTD models presented in section 2.2 (y-laminar, m-laminar, exponential and sinusoidal) were adjusted to the $E(t)$ values calculated from the experimental data. The parameter of each model and the corresponding mean residence time were iteratively adjusted by non-linear regression on $E(t)$, in terms of the minimization of the sum of squared errors (MSSE) between the experimental values ($E_{exp}(t)$) and the adjusted model ($E_{model}(t)$), as in Eq. (14).

$$MSSE = \min \left[ \sum_{i=1}^{n} \left( E_{exp,i}(t) - E_{model,i}(t) \right)^2 \right] \quad (14)$$

Starting with an initial guess for the model parameter ($\theta_0$, $y$, $m$, $\alpha$ or $\beta$) and $t_m$, the SSE was minimized using the generalized reduced gradient method (GRG) in the Solver tool of Excel 2016 (Microsoft, Redmond, USA). The obtained RTD parameter was correlated with the volumetric flow rate and the correlation was later used for model simulation (Section 3.4).
## 3.2 Cooling and heating experiments

Cooling and heating experiments were conducted using an ultrathermostatic bath with agitation (model MA184/30/-15A150 - Marconi, São Paulo, Brazil). The coiled tube was immersed in hot or cold water, whilst the glycerin/water mixture or CMC solution flowed inside the tube at flow rates of 0.5, 1.0, 1.5 and 2.0 L/min. An additional mechanical agitator (model 715 - Fisatom, Brazil) was used to intensify mixing in the bath.

Thermocouples with exposed junctions (IOPE, Brazil) were placed at the inlet ($T_{in}$) and outlet ($T_{out}$) of the tube using union tees (John Guest, UK), and two other thermocouples of the same type were placed in different regions of the water bath to measure the external temperature ($T_e$). The experimental setup is shown in Fig. 4.

The thermocouples were connected to a cDAQ-9172 data acquisition system and a computer running LabView data-logger code (National Instruments, Texas, EUA). Once steady-state operation was verified, the temperatures were recorded every 1 s for 1 min. The experimental conditions are detailed in Table 2.

## 3.3 Lumped capacitance experiments

The coiled tube in the water bath was treated as a heat exchanger with convection on the external surface, which means that the temperature on the surface of the tube was not equal to the bulk temperature of the water bath ($T_w \neq T_e$). The wall boundary condition for heat transfer on the wall (Eq. 8) requires the convective heat transfer coefficient of the surrounding fluid over the external tube wall ($h_e$). The lumped capacitance method was used to determine $h_e$ experimentally, as an average value for the whole surface of the coil.

This method consists of a transient heat transfer problem, in which a solid experiences a sudden change in its thermal environment and the change in the temperature
of the solid is registered in function of time. The main assumptions of the lumped capacitance method were: 1) uniform external temperature ($T_e$); 2) uniform convective heat transfer coefficient over the solid ($h_e$) and 3) the temperature of the solid was spatially uniform at any instant in time during the experiment, i.e., the temperature gradients within the solid were negligible [39].

The validity of the latter assumption can be verified by the Biot number ($Bi$), a dimensionless group defined in Eq. (15) as the ratio between the thermal resistances inside the solid (conduction) and at the interface with the fluid (convection):

$$Bi = \frac{h l}{k_s} \qquad (15)$$

where $k_s$ is the thermal conductivity of the solid, $l$ is its characteristic length, calculated as the ratio between the solid volume ($V_s = \pi(r_e^2 - r_i^2)L$) and its external superficial area ($A_e = 2\pi r_e L$). The assumption of uniform temperature in the solid is reasonable for small Biot numbers ($Bi < 0.1$) [39].

Considering a solid losing heat by convection to a fluid media in a situation of small Biot number, the overall energy balance on the solid yields Eq. (16).

$$\rho V C_p \frac{dT}{dt} = -h_e A_e (T - T_e) \qquad (16)$$

Separating variables and integrating for an initial state $t = 0$ with the solid at the initial temperature $T_i$ to an instant $t$ with temperature $T(t)$, Eq. (17) is obtained. With experimental data of temperature as a function of time, the heat transfer coefficient of the surrounding fluid can be obtained by curve fitting.

$$\frac{T(t) - T_e}{T_i - T_e} = \exp \left( -\frac{h_e A_e}{\rho V C_p} t \right) \qquad (17)$$

The empty coiled tube in Fig. 3, at a given initial temperature, was submerged in the ultrathermostatic water bath used in heating and cooling experiments, with the same
410 agitation conditions. For the heating experiment, the settings were $T_i = 10$ °C and $T_e = 80$ °C; while for the cooling experiment: $T_i = 80$ °C and $T_e = 10$ °C.

412 A thermocouple with exposed junction (IOPE, Brazil) was fixed in the middle part of the coil with thermal paste for better contact ($k = 11$ W.m⁻¹.K⁻¹). Temperature was recorded every second until the temperature of the tube reached the value of the external temperature ($T_e$). Next, parameter $h_e$ in Eq. (17) was iteratively adjusted by non-linear regression, using the minimization of the sum of squared errors (MSSE) between $T(t)_{calc}$ and $T(t)_{exp}$. The value of $T_i$ was as well adjusted (pseudo initial temperature) for a better curve fitting.

## 3.4 Model simulation

420 The 2D model proposed in Section 2, including the velocity profile adjusted from the RTD experiments (Table 1, Section 3.1), was solved by the finite difference method with the differential-algebraic solver DASOLV built in software gPROMS 6.0.2 (Process Systems Enterprise, UK). The differential and algebraic equations for the model were inputted using the Model Builder interface. For domain discretization, the geometry domain was divided into rectangular elements to compose the mesh. An aspect ratio of 5.0 was considered for each element, i.e., the length of each rectangle was five times its height, because of the large length of the tube.

428 A mesh independency study was conducted to obtain a mesh independent solution, which is a way to reduce discretization errors [49]. The mixing cup temperature or bulk temperature ($T_b$) at the tube outlet was the variable selected for this study. The bulk temperature can be calculated in function of the axial position according to Eq. (18) [5].

$$T_b(z) = \frac{2}{v_b r_i^2} \int_0^{r_i} T(z, r) v_z(r) r dr \qquad (18)$$

432 The details of the meshes tested are shown in Table 3. The mesh independency study was realized for both fluids (glycerin/water mixture or CMC solution), at flow rate
0.5 L/min and $F = 1$. The temperatures were based on the heating condition H1 and cooling condition C4 because of the larger temperature differences. The best mesh was obtained when the absolute error of temperature between simulations was lower than 0.1 °C.

After establishing the mesh, the simulations of the heating and cooling experiments were performed. Equations (7) and (18) were discretized using the first order backward finite difference for the axial components and with the second order centered finite difference for the radial components.

Since the outlet bulk temperature is unknown before simulation, an initial guess was needed for calculating the average thermo-physical properties ($\rho$, $C_p$ and $k$). The initial guess used for the outlet temperature was the experimental measurement. An iterative procedure was used to converge the fluid outlet bulk temperature: 1) the 2D model was solved for $T(z,r)$, 2) the outlet value of $T_b$ was used to calculate the new average temperature, and 3) the values of the thermophysical properties were updated. Steps 1 to 3 were repeated until convergence for a precision of 0.1 °C on the outlet bulk temperature.

To test the model, simulations were first carried out using given values of $F$ to determine the corresponding temperature profile $T(z,r)$ and the outlet bulk temperature $T_b(z=L)$. Next, simulations were run specifying the experimental outlet temperatures of the fluid obtained from the 32 experimental conditions in Table 2, while $\alpha_{eff}$ was set as a free variable in the model. Then, the corresponding $F$ value was calculated using Eq. (9) and correlated with the Reynolds number. The diagram in Fig. 5 shows the information flows between the experimental and simulation steps for adjusting the parameters of the semi-empirical model and using the model to simulate heat and mass transfer.
To verify results, the correlations for F were substituted in the 2D model and it was simulated to predict the outlet temperatures for the 32 heating and cooling experiments, which were compared with the measurements. Additionally, the Nusselt correlations from Janssen and Hoogendoorn [50] for laminar convective heat transfer in helical coiled tubes were used to predict the outlet temperatures for all heating and cooling experiments. Since different flow rates were considered, the range of Dean number vary between the upper and lower limits of Eqs. (19a) to (19c):

$$Nu = 1.7 (De^2 Pr)^{1/6} \quad \text{for } De < 20 \quad (19a)$$

$$Nu = 0.9 (Re^2 Pr)^{1/6} \quad \text{for } 20 < De < 1 \times 10^2 \quad (19b)$$

$$Nu = 0.7 Re^{0.43} Pr^{1/6} \left( \frac{d_i}{d_c} \right)^{0.07} \quad \text{for } 1 \times 10^2 < De < 8.3 \times 10^2 \quad (19c)$$

were $Nu = h d_i / k$ is the peripherally averaged asymptotic Nusselt number for the fully developed thermal region. The outlet temperature of the coiled tube then was calculated from:

$$\rho Q C_p |T_{out} - T_{in}| = U A_e \frac{(T_{out} - T_e) - (T_{in} - T_e)}{\ln \frac{(T_{out} - T_e)}{(T_{in} - T_e)}} \quad (20)$$

in which the left-side term is the heat transfer rate based on the temperature change of the stream and the right-side term is the heat load based on the log-mean temperature difference and on the overall heat transfer coefficient U [39]:

$$\frac{1}{U A_e} = \frac{1}{h A_i} + \frac{\ln\left(\frac{d_e}{d_i}\right)}{2\pi L k_s} + \frac{1}{h_e A_e} \quad (21)$$

## 4. RESULTS AND DISCUSSION

The RTD experimental results are presented first, since the velocity profile in the model is defined according to the RTD model with the best fit, followed by the results
from the heating and cooling experiments and lumped capacitance experiments. Next, the simulation results of the 2D model are presented and discussed.

## 4.1 Results from the RTD experiments

The average values of the sum of squared errors (SSE) of the RTD model adjustment for the four flow rates (0.5, 1.0, 1.5 and 2.0 L/min) are shown in Table 4 for the glycerin/water mixture and the CMC solution. The y-laminar model was considered as the best fit because of the lowest value of SSE. An example of the y-laminar model fitting to experimental RTD data is shown in Fig. 6A for the CMC solution and in Fig. 6B for the glycerin/water mixture, at the flow rate of 0.5 L/min. Consequently, the velocity profile of the y-laminar model (Eq. 22) was included in the 2D model.

$$v_z(r) = v_{max} \left(1 - \frac{r}{r_i}\right)^y \qquad (22)$$

A linear correlation was adjusted between the RTD parameter y and the flow rate, as shown in Fig. 7A for the CMC solution, and in Fig. 7B for the glycerin/water mixture. Values of y were smaller than one and linearly increased with the Reynolds number for the Reynolds range herein. A similar behavior can be seen for the two fluids. The correlations in Fig. 7 were used to obtain the values of y for Eq. (22), which varied from 0.12 to 0.31 for the CMC solution and from 0.11 to 0.39 for the glycerin/water mixture.

The critical Reynolds number for the laminar flow calculated with Eq. (12) was 9337. For all RTD experiments, the highest Reynolds number value was 112 for the glycerin/water mixture and 61 for the CMC solution (generalized Reynolds number). Since these values were below the critical Reynolds, the flow regime was assumed to be laminar.

## 4.2 Results from the cooling and heating experiments

The results of the heating and cooling experiments are shown in Table 5 for the CMC solution and in Table 6 for the glycerin/water mixture, in flow rates of 0.5, 1.0, 1.5
and 2.0 L/min. In total, 32 experiments without repetitions were carried out for each fluid, combining different conditions of flow rate, inlet temperature and external temperature (Table 2). Further in this work, the experimental outlet temperatures were specified as a boundary condition in the 2D model, setting $\alpha_{eff}$ as a free variable to be calculated in the simulations.

The Reynolds number for the experimental conditions was calculated using thermophysical properties calculated for the average temperature of the fluid $T_m = (T_{in} + T_{out})/2$. The range of Reynolds obtained for each experimental condition is shown in Table 7 for the glycerin/water mixture and in Table 8 for the CMC solution, along with the ranges of Dean number and viscosity ($K$ and $n$ for the CMC solution).

The highest Reynolds number was 892 for the glycerin/water mixture (Table 7) and 102 for the CMC solution (Table 8). The critical Reynolds number calculated by Eq. (12) was not exceeded ($Re_{crit} = 9337$); thus, the flow was assumed laminar in all experiments.

The Dean number varied from 20 to 263 for the glycerin/water mixture (Table 7) and from 3 to 30 for the CMC solution (Table 8). According to Dravid et al. [7], the secondary flow is significant for $De > 100$. This value was exceeded at the higher flow rates (1.5 L/min and 2.0 L/min) for the glycerin/water mixture; thus, the secondary flow effects may have been stronger in these cases.

## 4.3 Results from the lumped capacitance experiments

Results of the lumped capacitance method are shown in Fig. 8A for the heating experiment and in Fig. 8B for the cooling experiment, where $\theta = (T(t) - T_e)/(T_i - T_e)$ is the dimensionless temperature. A good fitting was obtained with a clear log-linear dependence between dimensionless temperature and time.
For the heating experiment, the adjusted value of the convective coefficient was $h_e = 962 \text{ W} \cdot \text{m}^{-2} \cdot \text{K}^{-1}$ with $Bi = 0.11$; for the cooling experiment, the adjusted value was $h_e = 753 \text{ W} \cdot \text{m}^{-2} \cdot \text{K}^{-1}$ with $Bi = 0.08$. The requirement of $Bi < 0.1$ of the lumped capacitance method was acceptable given the limiting value obtained. The values of $h_e$ were used in the 2D model in the wall boundary condition of heat transfer.

## 4.4 Simulations and model validation

Results of the model mesh refinement study are shown in Table 9 for the CMC solution, $Q = 0.5 \text{ L/min}$ and $F = 1$. The absolute errors for outlet temperature were lower than $0.1 \ ^{\circ}\text{C}$ for meshes M9 and M10, which were considered acceptable for this study. Mesh M10 with 1000 axial points and 200 radial points was then selected to represent the computational domain because of the small relative error of temperature and the small computational time (ca. 1 min). Similar results were obtained with the glycerin/water mixture.

The 2D model was simulated for the 32 experimental conditions in Table 2, using the velocity profile of the y-laminar model and the correlations in Fig. 7 to calculate the value of $y$. First, different values of $F$ (1.0 and 3.0) were tested to verify its influence on the bulk temperature along the tube length, as shown in Fig. 9A for heating condition H1 and in Fig. 9B for cooling condition C2. When $F = 1$, the effective thermal diffusivity is equal to the fluid property ($\alpha_{eff} = \alpha$); when setting $F = 3$, the heat transfer rate was increased up to 93 % for the glycerin/water mixture and up to 99% for the CMC solution in comparison to the base case.

Another analysis was conducted to verify the influence of the enhancement factor on the radial temperature profile at the tube outlet, for the experimental conditions H1 (Fig. 10A) and C2 (Fig. 10B). Flatter temperature profiles (small difference between
center and wall temperatures) can be observed as $F$ increases, because of the higher heat transfer rate.

Simulations of the 2D model with the specification of the experimental outlet temperatures provided the values of $\alpha_{eff}$. The corresponding $F$ values were calculated using Eq. (8), which varied from 0.84 to 2.62 for the CMC solution and from 1.25 to 3.18 for the glycerin/water mixture.

The $F$ values were correlated with the Reynolds number in log scale, as shown in Fig. 11. It was possible to group data from both fluids in a single plot; however, the combination of heating and cooling experiments in Fig. 11C increased the scattering of points and was not considered appropriate. The dispersion of points seen in Fig. 11 may be related to the assumption that the velocity profile derived from the experimental RTD was valid at all temperatures in the experiments, since viscosity has an important temperature dependence and affects flow. Trends show that the value of $F$ tends to unity with decreasing Reynolds number, with a limiting Reynolds at $F = 1$. This was expected, since the influence of secondary flow becomes negligible for decreasing Reynolds and Dean numbers. The linear log-log correlation in Fig. 11A is valid for $Re \ge 6.8$ and the correlation of Fig. 11B is valid for $Re \ge 16.1$ ($F = 1$ for lower values). As Dean numbers, these thresholds are $De \ge 2.0$ (heating) and $De \ge 4.7$ (cooling), which are lower than the limit of 20, under which the heat transfer enhancement is negligible according to Dravid et al. [7].

The adjusted $F$ correlations in Fig. 11A and 11B were substituted in the model, and the experimental conditions were simulated to calculate the outlet bulk temperature for all the heating and cooling experiments in Table 2 to evaluate the model prediction error for outlet temperature. Additionally, the Janssen and Hoogendoorn [50] correlations in Eqs. (19a) to (19c) were used to predict the outlet temperatures. The range of Dean
numbers presented in Tables 7 and 8 confirmed that Eqs. (19a) to (19c) were used for the glycerin/water mixture, while for the CMC solution only Eqs. (19a) and (19b) were used.

The values obtained were compared with the measured outlet temperatures, as shown in the parity charts in Fig. 12 for the CMC solution (Fig. 12A) and for the glycerin/water mixture (Fig. 12B). A good agreement between experimental and simulated values was obtained for the CMC solution with the $R^2 = 0.939$ and all deviations under 5 °C. Predictions from the Nusselt correlations had $R^2 = 0.647$ with 11 conditions with deviations over 5 °C. For the glycerin/water mixture, the errors were larger, yielding $R^2 = 0.791$, but with most of the deviations under 5 °C. Predictions from Nusselt correlations were mostly over 5 °C, as can be seen in Fig. 12A.

## 5. CONCLUSIONS

The proposed 2D semi-empirical model allowed the determination of the velocity and temperature fields inside the straight tube in a way that its RTD mimics the E-curve of the coiled tube, and the predicted outlet temperature matches the experimental value using a heat transfer enhancement factor. This model depends on inputting two parameters: y and F, which were obtained from experiments using two model fluids with different rheological behavior. Velocity profile parameter y was correlated with the flow rate (Reynolds was not used since temperature effects were neglected for the RTD) and provided a flatter profile in comparison with the theoretical laminar profile for a straight tube. Heat transfer enhancement factor F linearly correlated with the Reynolds number in a log-log plot, combining data from the two model fluids, but with a small difference in trend between the heating and cooling experiments. These plots provided a threshold value for negligible heat transfer enhancement in the coiled tube.
Simulation time for this 2D model was very small in comparison with the values
596
expected for typical 3D CFD simulations, and it can provide a reliable outlet temperature
597
for the coiled heat exchanger (Fig. 12). The advantage of this approach over using Nusselt
598
correlations to predict the outlet temperature is that the 2D model provides velocity and
599
temperature fields in the tube that can be used for the simulation of non-isothermal
600
reactions. The downside of the model is the previous requirement of experiments to
601
provide correlations for y and F. In the case of design changes, a new set of parameters
602
will be necessary. Some ideas for further investigation of this 2D modeling approach
603
would be: 1) this model could use virtual experiments from a rigorous 3D model in CFD
604
to adjust not only the outlet temperature, but the bulk temperature distribution along the
605
tube as well; 2) to study the dependence of RTD parameter y with temperature, thus
606
correlating it with the Reynolds number; 3) to use this approach to model a corrugated
607
coiled tube so y and F would take into account two passive heat transfer enhancement
608
techniques; 4) to introduce the differential species mass balance equation in the model to
609
evaluate a non-isothermal LFR in a coiled tube to predict not only temperature
610
distribution, but reaction yield as well, as in the bacterial inactivation in thermal
611
processing of liquid foods.
612

613

614 DECLARATION OF COMPETING INTEREST

The authors declare that they have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this paper.
615
616

617

618 DATA AVAILABILITY

Data will be made available on request.
619

620
621 ACKNOWLEDGEMENTS

622 This research was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo
623 – FAPESP (grants 2006/01628-0, 2013/07914-8), Conselho Nacional de
624 Desenvolvimento Científico e Tecnológico – CNPq (grants 140573/2019-4,
625 316388/2021-1) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior –
626 CAPES (Finance Code 001).

627

628 ORCID

629 Nilo H. M. Fortes - https://orcid.org/0000-0003-1763-8329

630 Artemio Plana-Fattori - https://orcid.org/0000-0001-8326-7007

631 Jorge A. W. Gut - https://orcid.org/0000-0001-6447-8490

632

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785
786 **FIGURE CAPTIONS**

787 **Figure 1** – Representation of the geometry in the 2D axisymmetric model

788 **Figure 2** – Dimensionless velocity profiles derived from laminar tube flow in function of the model parameter

790 **Figure 3** – Geometrical characteristics of the coiled tube used in the experiments

791 **Figure 4** – Experimental setup of the cooling/heating experiments

792 **Figure 5** –Experimental and simulation steps for adjusting the parameters of the semi- empirical model

794 **Figure 6** – Fitting of the y-laminar model to the experimental RTD data for the CMC solution (A) and for the glycerin/water mixture (B) at the flow rate of 0.5 L/min

796 **Figure 7** – Parameter y in function of the volumetric flow rate for the CMC solution (A) and for the glycerin/water mixture (B)

798 **Figure 8** – Results of the lumped capacitance experiments for the heating (A) and cooling (B) conditions

800 **Figure 9** – Simulation results of the bulk temperature ($T_b$) in function of the axial position (z) at the flow rate of 0.5 L/min for $F = 1$ and for $F = 3$ in the heating condition H1 (A) and in the cooling condition C2 (B). Non-dashed lines represent the CMC solution and dashed lines represent the glycerin/water mixture

804 **Figure 10** – Simulation results of the temperature profile at the tube outlet ($T_{z=L}$) in function of the radial position (r) at the flow rate of 0.5 L/min for $F = 1$ and for $F = 3$ in the heating condition H1 (A) and in the cooling condition C2 (B). Non-dashed lines represent the CMC solution and dashed lines represent the glycerin/water mixture

808 **Figure 11** – Logarithm of the heat transfer enhancement factor ($F$) in function of the logarithm of the Reynolds number for the heating experiments (A), cooling experiments (B) and assembled results (C). Triangles represent the CMC solution for heating (filled
811 triangles) and cooling cases (empty triangles), while circles represent the glycerin/water
812 mixture for the heating (filled circles) and cooling cases (empty circles).

813 **Figure 12** – Parity charts for predicting the outlet temperature of the CMC solution (A)
814 and the glycerin/water mixture (B) with the 2D model and using Nusselt correlations from
815 literature. Dashed lines indicate deviations of ±5 %.

816
817 TABLE CAPTIONS

818 **Table 1** – Details of the adjusted RTD models: velocity profiles, $E_{\theta}(\theta)$ functions,
819 parameters and breakthrough times. Adapted from [33]

820 **Table 2** – Conditions of the heating and cooling experiments for the glycerin/water
821 mixture and for the CMC solution: inlet temperatures, external temperatures and flow
822 rates

823 **Table 3** – Mesh independence study of the 2D axisymmetric model

824 **Table 4** – Sum of squared errors (SSE) between the adjusted RTD models and the
825 experimental data for the glycerin/water mixture and for the CMC solution

826 **Table 5** – Outlet temperatures obtained in the heating and cooling experiments for the
827 CMC solution

828 **Table 6** – Outlet temperatures obtained in the heating and cooling experiments for the
829 glycerin/water mixture

830 **Table 7** – Viscosity, Reynolds number and Dean number ranges of the heating and
831 cooling experiments with the glycerin/water mixture for the flow rates of 0,5 to 2,0 L/min

832 **Table 8** – Viscosity, Reynolds number and Dean number ranges of the heating and
833 cooling experiments with the CMC solution for the flow rates of 0,5 to 2,0 L/min

834 **Table 9** – Results of the mesh independence study for the CMC solution: outlet bulk
835 temperature, absolute errors of temperature and computational time

836
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863

<table><caption>Table 1 – Details of the adjusted RTD models: velocity profiles, $E_\theta(\theta)$ functions, parameters and breakthrough times.</caption><thead><tr><th>RTD Model</th><th>Velocity profile</th><th>RTD function</th><th>Parameter</th><th>Breakthrough time</th></tr></thead><tbody><tr><td>m-laminar</td><td>v* = 1 - r*<sup>m</sup></td><td>E<sub>θ</sub>(θ) = <sup>1</sup>/<sub>θ<sup>3</sup></sub> <sup>2θ<sub>0</sub></sup>/<sub>m</sub> (1 - <sup>θ<sub>0</sub></sup>/<sub>θ</sub>)<sup>2-m/m</sup></td><td>m</td><td>θ<sub>0</sub> = <sup>m</sup>/<sub>m + 2</sub></td></tr><tr><td>y-laminar</td><td>v* = (1 - r*)<sup>y</sup></td><td>E<sub>θ</sub>(θ) = <sup>1</sup>/<sub>θ<sup>2</sup></sub> <sup>2</sup>/<sub>y</sub> (<sup>θ<sub>0</sub></sup>/<sub>θ</sub>)<sup>1/y</sup> [1 - (<sup>θ<sub>0</sub></sup>/<sub>θ</sub>)<sup>1/y</sup>]</td><td>y</td><td>θ<sub>0</sub> = <sup>2</sup>/<sub>y<sup>2</sup> + 3y + 2</sub></td></tr><tr><td>Sinusoidal</td><td>v* = (<sup>1 + cos(πr*)</sup>/<sub>2</sub>)<sup>α</sup></td><td>E<sub>θ</sub>(θ) = <sup>1</sup>/<sub>θ<sup>2</sup></sub> <sup>2</sup>/<sub>απ<sup>2</sup></sub> tan(Φ/2)</td><td>α</td><td>θ<sub>0</sub> ≈ <sup>α<sub>0</sub><sup>p</sup></sup>/<sub>α<sup>p</sup> + α<sub>0</sub><sup>p</sup></sub><br>α<sub>0</sub> = 0.43072<br>p = 1.0183</td></tr><tr><td>Exponential</td><td>v* = (<sup>e - e<sup>r*</sup></sup>/<sub>e - 1</sub>)<sup>β</sup></td><td>E<sub>θ</sub>(θ) = <sup>1</sup>/<sub>θ<sup>2</sup></sub> <sup>2</sup>/<sub>β</sub> (<sup>α</sup>/<sub>(e - Ω)</sub>) ln(e - Ω)</td><td>β</td><td>θ<sub>0</sub> ≈ <sup>a</sup>/<sub>β<sup>2</sup> + bβ + a</sub><br>a = 5.4204<br>b = 6.5342</td></tr><tr><td></td><td></td><td>with Ω = (<sup>α</sup>/<sub>θ</sub>)<sup>1/β</sup></td><td></td><td></td></tr></tbody></table>

864
<table><caption>Table 2 – Conditions of the heating and cooling experiments for the glycerin / water mixture and for the CMC solution: inlet temperatures, external temperatures and flow rates</caption><thead><tr><th>Experiment</th><th>T<sub>in</sub> (°C)</th><th>T<sub>e</sub> (°C)</th><th>Q (L/min)</th></tr></thead><tbody><tr><td>H1</td><td>20</td><td>80</td><td>0.5 / 1.0 / 1.5 / 2.0</td></tr><tr><td>H2</td><td>30</td><td>80</td><td>0.5 / 1.0 / 1.5 / 2.0</td></tr><tr><td>H3</td><td>40</td><td>80</td><td>0.5 / 1.0 / 1.5 / 2.0</td></tr><tr><td>H4</td><td>50</td><td>80</td><td>0.5 / 1.0 / 1.5 / 2.0</td></tr><tr><td>C1</td><td>60</td><td>10</td><td>0.5 / 1.0 / 1.5 / 2.0</td></tr><tr><td>C2</td><td>70</td><td>10</td><td>0.5 / 1.0 / 1.5 / 2.0</td></tr><tr><td>C3</td><td>80</td><td>10</td><td>0.5 / 1.0 / 1.5 / 2.0</td></tr><tr><td>C4</td><td>90</td><td>10</td><td>0.5 / 1.0 / 1.5 / 2.0</td></tr></tbody></table>
<table><caption>Table 3 – Mesh independence study of the 2D axisymmetric model</caption><thead><tr><td>Mesh</td><td>N° of axial points</td><td>N° of radial points</td><td>N° of elements</td></tr></thead><tbody><tr><td>M1</td><td>100</td><td>20</td><td>2 000</td></tr><tr><td>M2</td><td>200</td><td>40</td><td>8 000</td></tr><tr><td>M3</td><td>300</td><td>60</td><td>18 000</td></tr><tr><td>M4</td><td>400</td><td>80</td><td>32 000</td></tr><tr><td>M5</td><td>500</td><td>100</td><td>50 000</td></tr><tr><td>M6</td><td>600</td><td>120</td><td>72 000</td></tr><tr><td>M7</td><td>700</td><td>140</td><td>98 000</td></tr><tr><td>M8</td><td>800</td><td>160</td><td>128 000</td></tr><tr><td>M9</td><td>900</td><td>180</td><td>162 000</td></tr><tr><td>M10</td><td>1 000</td><td>200</td><td>200 000</td></tr></tbody></table>

870
Table 4 – Sum of squared errors (SSE) between the adjusted RTD models and the experimental data for the glycerin/water mixture and for the CMC solution

<table><thead><tr><th rowspan="2">Model</th><th colspan="2">SSE</th></tr><tr><th>Glycerin/water</th><th>CMC</th></tr></thead><tbody><tr><td>m-laminar</td><td>3.2×10<sup>-2</sup></td><td>2.9×10<sup>-2</sup></td></tr><tr><td>y-laminar</td><td>1.0×10<sup>-2</sup></td><td>1.3×10<sup>-2</sup></td></tr><tr><td>Sinusoidal</td><td>3.3×10<sup>-2</sup></td><td>2.4×10<sup>-2</sup></td></tr><tr><td>Exponential</td><td>1.1×10<sup>-2</sup></td><td>1.4×10<sup>-2</sup></td></tr></tbody></table>

873
<table><caption>Table 5 – Outlet temperatures obtained in the heating and cooling experiments for the CMC solution</caption><thead><tr><th rowspan="2">Experiment</th><th><i>T</i><sub>out</sub> (°C)</th><th><i>T</i><sub>out</sub> (°C)</th><th><i>T</i><sub>out</sub> (°C)</th><th><i>T</i><sub>out</sub> (°C)</th></tr><tr><th>0.5 L/min</th><th>1.0 L/min</th><th>1.5 L/min</th><th>2.0 L/min</th></tr></thead><tbody><tr><td>H1</td><td>56.4</td><td>49.5</td><td>45.6</td><td>43.7</td></tr><tr><td>H2</td><td>63.8</td><td>59.5</td><td>54.5</td><td>52.2</td></tr><tr><td>H3</td><td>64.3</td><td>61.2</td><td>56.2</td><td>55.5</td></tr><tr><td>H4</td><td>67.1</td><td>63.1</td><td>61.6</td><td>59.7</td></tr><tr><td>C1</td><td>34.2</td><td>41.2</td><td>41.8</td><td>43.5</td></tr><tr><td>C2</td><td>36.8</td><td>48.4</td><td>51.6</td><td>55.8</td></tr><tr><td>C3</td><td>39.5</td><td>53.3</td><td>55.1</td><td>58.2</td></tr><tr><td>C4</td><td>42.7</td><td>54.4</td><td>59.5</td><td>62.5</td></tr></tbody></table>

876
<table><caption>Table 6 – Outlet temperatures obtained in the heating and cooling experiments for the glycerin/water mixture</caption><thead><tr><th rowspan="2">Experiment</th><th><i>T</i><sub>out</sub> (°C)</th><th><i>T</i><sub>out</sub> (°C)</th><th><i>T</i><sub>out</sub> (°C)</th><th><i>T</i><sub>out</sub> (°C)</th></tr><tr><th>0.5 L/min</th><th>1.0 L/min</th><th>1.5 L/min</th><th>2.0 L/min</th></tr></thead><tbody><tr><td>H1</td><td>61.6</td><td>53.3</td><td>45.9</td><td>42.5</td></tr><tr><td>H2</td><td>67.4</td><td>58.4</td><td>53.5</td><td>50.4</td></tr><tr><td>H3</td><td>68.1</td><td>61.2</td><td>57.9</td><td>55.8</td></tr><tr><td>H4</td><td>71.9</td><td>67.0</td><td>63.9</td><td>62.9</td></tr><tr><td>C1</td><td>31.6</td><td>38.0</td><td>41.1</td><td>43.1</td></tr><tr><td>C2</td><td>32.4</td><td>41.3</td><td>43.1</td><td>48.3</td></tr><tr><td>C3</td><td>33.5</td><td>47.0</td><td>49.9</td><td>52.3</td></tr><tr><td>C4</td><td>35.0</td><td>47.4</td><td>53.7</td><td>58.8</td></tr></tbody></table>

879
<table><caption>Table 7 – Viscosity, Reynolds number and Dean number ranges of the heating and cooling experiments with the glycerin/water mixture for the flow rates of 0.5 to 2.0 L/min</caption><thead><tr><th>Experiment</th><th>μ (mPa·s)</th><th>Re (-)</th><th>De (-)</th></tr></thead><tbody><tr><td>H1</td><td>20.4 – 32.3</td><td>67 – 169</td><td>20 – 50</td></tr><tr><td>H2</td><td>14.5 – 21.0</td><td>93 – 259</td><td>28 – 76</td></tr><tr><td>H3</td><td>11.8 – 15.0</td><td>115 – 361</td><td>34 – 106</td></tr><tr><td>H4</td><td>9.19 – 10.7</td><td>146 – 502</td><td>43 – 148</td></tr><tr><td>C1</td><td>12.9 – 16.3</td><td>83 – 420</td><td>25 – 124</td></tr><tr><td>C2</td><td>9.73 – 13.1</td><td>104 – 553</td><td>31 – 163</td></tr><tr><td>C3</td><td>7.69 – 10.6</td><td>127 – 697</td><td>37 – 205</td></tr><tr><td>C4</td><td>5.98 – 8.67</td><td>155 – 892</td><td>46 – 263</td></tr></tbody></table>

882
<table><caption>Table 8 – Viscosity, Reynolds number and Dean number ranges of the heating and cooling experiments with the CMC solution for the flow rates of 0.5 to 2.0 L/min</caption><thead><tr><th>Experiment</th><th><i>K</i> (Pa·s<sup>n</sup>)</th><th><i>n</i> (-)</th><th><i>Re</i><sub>g</sub> (-)</th><th><i>De</i> (-)</th></tr></thead><tbody><tr><td>H1</td><td>1.63 – 1.95</td><td>0.421 – 0.436</td><td>9 – 70</td><td>3 – 21</td></tr><tr><td>H2</td><td>1.27 – 1.49</td><td>0.444 – 0.460</td><td>10 – 79</td><td>3 – 23</td></tr><tr><td>H3</td><td>1.11 – 1.24</td><td>0.462 – 0.474</td><td>11 – 85</td><td>3 – 25</td></tr><tr><td>H4</td><td>0.94 – 1.03</td><td>0.481 – 0.492</td><td>12 – 91</td><td>3 – 27</td></tr><tr><td>C1</td><td>1.12 – 1.27</td><td>0.460 – 0.472</td><td>10 – 88</td><td>3 – 26</td></tr><tr><td>C2</td><td>0.84 – 1.07</td><td>0.478 – 0.506</td><td>11 – 96</td><td>3 – 28</td></tr><tr><td>C3</td><td>0.71 – 0.91</td><td>0.496 – 0.525</td><td>12 – 100</td><td>3 – 29</td></tr><tr><td>C4</td><td>0.61 – 0.78</td><td>0.515 – 0.546</td><td>12 – 102</td><td>4 – 30</td></tr></tbody></table>

885
<table><caption>Table 9 – Results of the mesh independence study for the CMC solution: outlet bulk temperature, absolute errors of temperature and computational time</caption><thead><tr><th rowspan="2">Mesh</th><th colspan="3">CMC heating</th><th colspan="3">CMC cooling</th></tr><tr><th>T<sub>b,out</sub> (K)</th><th>Abs. Error (K)</th><th>time (s)</th><th>T<sub>b,out</sub> (K)</th><th>Abs. Error (K)</th><th>time (s)</th></tr></thead><tbody><tr><td>M1</td><td>322.64</td><td>-</td><td>0.58</td><td>312.75</td><td>-</td><td>0.75</td></tr><tr><td>M2</td><td>326.28</td><td>3.64</td><td>2.30</td><td>315.70</td><td>2.95</td><td>2.33</td></tr><tr><td>M3</td><td>327.41</td><td>1.12</td><td>5.25</td><td>316.60</td><td>0.91</td><td>4.88</td></tr><tr><td>M4</td><td>327.94</td><td>0.54</td><td>8.92</td><td>317.04</td><td>0.43</td><td>8.92</td></tr><tr><td>M5</td><td>328.25</td><td>0.31</td><td>14.36</td><td>317.29</td><td>0.25</td><td>14.19</td></tr><tr><td>M6</td><td>328.45</td><td>0.20</td><td>21.83</td><td>317.45</td><td>0.16</td><td>20.39</td></tr><tr><td>M7</td><td>328.59</td><td>0.14</td><td>29.22</td><td>317.56</td><td>0.11</td><td>30.00</td></tr><tr><td>M8</td><td>328.70</td><td>0.10</td><td>40.11</td><td>317.64</td><td>0.08</td><td>39.42</td></tr><tr><td>M9</td><td>328.78</td><td>0.08</td><td>52.23</td><td>317.71</td><td>0.06</td><td>51.42</td></tr><tr><td>M10</td><td>328.84</td><td>0.06</td><td>64.27</td><td>317.76</td><td>0.05</td><td>67.50</td></tr></tbody></table>

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