Fluid-dynamic study of the behavior of the air inside a textile Stenter

In the textile industry, drying is one of the most important processes. This process requires large investments and high energy consumption, which generates high costs for companies in this sector. In this work, a modeling of the behavior of the air was carried out in a textile Stenter, under real operating conditions through the development of fluid-dynamic simulations. For the computational modeling of the problem, a 3D geometry was constructed based on measurements taken from an injector of a textile Stenter. The standard k-ε turbulence model was used in the turbulent flow solution. The equations of the model were solved numerically using the finite element method. The standard k-ϵ turbulence model proved to be a model capable of reproducing the behavior of the air in the injectors of the textile Stenter.


Introduction
In the textile industry, drying is one of the most important processes. From a physical point of view, the drying of the fabric can be defined as a balance between the transfer of heat from the air stream to the surface of the fabric and the transfer of moisture from the surface of the fabric to the air stream. This process requires large investments and high energy consumption, which generates significant costs for companies in this sector. In industrialized countries, around 7% to 15% of the energy generated for industrial purposes is generally used for drying with relatively low thermal efficiency [1].
Therefore, it is of great interest to find ways to reduce energy demand without damaging the final characteristics of the fabric. Textile Stenters are thermal equipment used to dry fabrics in a continuous process. In the drying chambers, hot air flows over the fabric and dries it by convection with the exchange of heat and mass. The exhaust system removes the vapors resulting from the drying of the fabric, reducing the humidity of the air in the system and, consequently, increasing the drying efficiency in the process [2].
In the literature there are several studies on turbulent air jets, in which they carry out numerical studies, comparing different types of turbulence models with experimental values, this in order to study the heat and mass transfer applied in industrial processes, including drying. For example, Shi Y, et al, [3] studied the used of the standard turbulence model and the Reynolds stress model (RSM), model to calculate the heat transfer for a single semi-confined turbulent jet, which impinges on a flat plate, using different airflow parameters and different geometric parameters. For low heat flux values (1000 W/m 2 -6000 W/m 2 ), the physical properties of air were independent of 2 temperature and heat flux; however, for high heat flux (50000 W/m 2 ), they observed that the physical properties of air were dependent on temperature.
Colucci and Viskanta [4] studied the effect of the geometry of a hyperbolic nozzle on local heat transfer coefficients, for this, they used a thermochromic liquid crystal technique to visualize and record isotherms on a uniformly heated impact surface. The authors concluded that the heat transfer coefficients at the site of incidence of the confined jets are more sensitive to the Reynolds number and the space between the nozzle and the plate, than the respective coefficients for the unconfined jets. Other authors who used the liquid crystal technique were Lee J. and Lee S [5]. They studied the effects of the nozzle outlet configuration on improvements in heat transfer, experimenting with three different types of nozzles.
Wang and Mujumdar [6] used 5 low Reynolds number models to study a two-dimensional turbulent jet, most of the models showed good agreement with the experimental data, both in the stagnant regions and in the jet walls. Gulati P, et al [7]. Studied the effects of the nozzle shape, the space between the plate and the jet, and Reynolds number, on the local heat transfer distribution of an air jet on a smooth and flat surface, for this they used three nozzles with different geometries: circular, square and rectangular, each with an equivalent diameter of 20 mm.
In this work the objective was to study the behavior of the air inside the drying chambers of a textile Stenter under real operating conditions, by obtaining speed fields, turbulent kinetic energy profiles and dissipation rate of turbulent air in the injectors, with in order to relate them to the phenomena of heat and mass transfer.

Materials and methods
This section presents the mathematical methodologies used to carry out the study this work.

Mathematical formulation
The simulations were carried out in COMSOL Multiphysics® version 4.4, a phenomenological simulation and modeling software. COMSOL Multiphysics® internally compiles the set of equations that represent the model in its entirety, generating a system of linear equations. The resolution of the model is obtained by the finite element method (FEM), which is numerical procedure used to solve systems of partial differential equations (PDE), and consists of dividing the domain of the solution into regions of simple form or by elements. Then, a rough solution for electronic data processing can be developed for each of these elements. The total solution is generated by joining the individual solutions and, at the same time, continuity at the boundaries between the elements is ensured [8]. For this study, the standard k-ε turbulence model was used, which is the most used for computational simulations, since it presents good convergence and requires relatively few computational resources.

k-ԑ turbulence model
Turbulence is a phenomenon in which fluid particles mix in a non-linear way, which is described through various models of turbulence; the turbulence model (k-ε), used in this work is governed by two parameters: k related to the kinetic energy of the turbulence (defined as the variation of the fluctuations in speed) and ε the dissipation of the turbulence eddy (the rate at which speed fluctuations dissipate) [9]. Turbulent stresses are modeled through turbulent viscosity that include the parameters k and ε. For the steady state, the values of k and ε come directly from the differential equations for turbulence kinetic energy transport and the turbulence dissipation rate (see Equation (1) and Equation (2), respectively [10]).
where u is the velocity, ρ the density and µ the viscosity of the fluid; C $% , C $& and C ! are constants equal to 1.44, 1.92 and 0.09 respectively. σ # = 1.0 is the turbulent Prandlt number for k and σ $ is theturbulent Prandlt number for ε. The Equation (3) calculate P # is the turbulent kinetic energy production due to buoyancy and viscous forces [10]).

Simulation parameters
During the development of this work, a technical visit was made to a company in the textile sector that uses direct heating textile Stenters; on this visit, we were able to access the textile Stenter injectors (see Figure 1) and their measurements were also taken with a tape measure. With the dimensions measured to the injectors during the technical visit, it was possible to build a geometry closer to the real one; thus, a 3D geometry was constructed consisting of two injectors (upper and lower) and a rectangular section volume in the middle of them, which represents the fabric in motion at a translation speed of the fabric (V )*+,-. ) of 0.4 m/s entering longitudinally in the dryer (x-axis), as shown in Figure 2 and at a temperature of 30 °C. The following physical properties were considered for the cotton fabric: density of 1500 kg/m 3 , specific heat of 1339.78 J/kg K and thermal conductivity of 0.029 W/(m K) values used by [11]. In this case, the air enters the injectors at a speed (V *-, ) of 5.0 m/s and 180 °C of temperature, operating parameters used in industry [2,11]. The air injector is made up of a duct 2.82 m long, 0.31 m wide and 0.06 m high at one end and 0.19 m at the other (see Figure 2).
Each injector has 170 nozzles (distributed in 5 rows of 34, not paired laterally). The nozzles used in textile Stenters have a hemi-toroidal shell rim; the hemi-toroidal shell shape of the nozzle aims to distribute the jet of air more efficiently over the tissue surface. The nozzles have an inlet diameter   Figure 2. 3D geometry of the upper and lower injectors of a direct heating textile stenter.

Geometry and discretization mesh
The construction of the mesh is one of the most important aspects to develop computational models, since the good construction of the mesh depends to a great extent on the convergence of a simulation, as well as the convergence time; due to the complexity of the domain, since it is composed of several subdomains, a mesh of eight hexahedral layers and 1018054 elements was built, which can be seen in Figure 3.

Results and discussions
The following is the analysis of the results obtained during the dynamic fluid study of the behavior of the air inside the textile Stenter used in the textile industry.
The objective to qualify and quantify the air flow in the region of the injectors of the dryer under study, by analyzing the velocity profiles, the streamlines and the turbulence parameters obtained from the k-ε turbulence model. As mentioned in section 2.3, velocities of 5.0 m/s were adopted for the air inlet velocity in the reference injectors and 0.4 m/s for the fabric displacement velocity.
In Figure 4, which shows the flow of air developed at steady state, it is possible to discern the velocity fields along a longitudinal section plane of the injectors, including the tissue surface. When analyzing the behavior of the air in the injector, it is observed that the highest velocity values are reached at the end nozzles where values between 8 m/s and 10 m/s are observed, showing that, in these areas of the injectors, the air reaches up to twice the inlet velocity. In the nozzles of the central region, the velocity reaches values between 4 m/s and 7 m/s. It can also be observed that, together with the extreme nozzles up to the surface of the tissue, the jets promote an intensification of the air velocity, inducing the appearance of vortices that characterize the behavior of the turbulent flow.  The air recirculation areas near the inlet of the ducts that lead the air to the injectors are shown in Figure 6. Note that the velocity values in these regions are high; according to [12], this is due to the fact that recirculation restricts the passage of air freely, forming a stagnant region. Thus, due to the laws of conservation of mass and motion, the air accelerates in this region and moves towards the interior of the injector, subsequently losing speed.  In Figure 7 it can be seen that the current lines inside the injectors are quite orderly and uniform, as they leave them and meet the tissue, which is in motion, the lines begin to behave in a disorderly way and form vortices displaced in the direction of tissue movement. In Figure 8, it can be seen that the turbulent kinetic energy has the highest values at the ends of the injectors; this is because the greatest flow accelerations occur in these regions. In the central region of the injectors, the turbulent kinetic energy tends to be lower. The rate at which energy dissipates is shown in Figure 9. It is observed that the behavior is very similar to that of turbulent kinetic energy, that is, the highest values are found at the ends of the injectors and are lowest in the central region. A similar behavior was observed by Janzen J G, et al [13]. Who studied the turbulence properties of a viscous fluid through the oscillation of two grids; they observed the existence of two regions: a region closes to the grids, where turbulence-producing effects are relevant (eventually also the advection resulting from the jets that form on the grids); and a second region (further from the gratings) in which there is a decrease in turbulent kinetic energy.