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Abstract

A numerical analysis is carried out for unsteady evaporation of a Newtonian fluid Couette flow confined within a cylindrical annulus with relative rotation. The inner wall is humid and adiabatic, whereas the outer one is isothermal and impermeable. The present hydrodynamic and heat-transfer fields are obtained numerically using the finite volumes method. Main attention was paid to the effect of geometrical and thermodynamic parameters on the entropy creation and air evaporation. The results show that the volumetric entropy generation due to thermal conduction and mass diffusion is distinctly developed compared to entropy generated by fluid friction.

Keywords

Couette flow entropy generation evaporation heat transfer mass diffusion

Introduction

The cylindrical Couette flow has an important place in the history of fluid mechanics. It belongs to the family of rotating systems whose utility is capital in many application fields such as chemical mixers, industrial furnaces, nuclear reactors, and heat exchangers. Several studies have thoroughly dealt with the analysis of the Couette flow. In order to evaluate the viscosity of fluids within a cylindrical annulus, Couette¹ measured the value of the torque to which the inner cylinder is suspended by a torsion wire when the outer cylinder rotates. He showed that the viscosity remains constant as the rotation speed does not exceed a critical value. Mallock² observed similar characteristics of rotating the inner cylinder. Rayleigh³ established the criterion of the circular stability between two cylinders in differential rotation known under the name of “Rayleigh criterion.” Few years later, Taylor⁴ showed theoretically and experimentally that the circular Couette flow can undergo the instability caused by the stratification of the kinetic moment if the rotational speed of the cylinder exceeds a certain critical value. Grifoll et al.⁵ investigated the mass transfer in a cylindrical Couette flow in order to characterize the mass transfer at high Schmidt’s and Reynolds’ numbers at smooth and rough surfaces. They showed that for smooth surfaces, Sherwood’s number depends on the cube root of Schmidt’s number. Sharipov and Kremer⁶ presented a numerical study of the momentum and heat transfer of a rarefied gas confined between two rotating cylinders having different temperatures. Their results showed that heat flow is determined by the angular velocities, whereas the temperature distribution is not affected by the angular velocity difference and is determined only by the temperature of each wall. Kim et al.⁷ developed a study on the convective stability of the Couette flow caused by a linearly accelerated circular inner cylinder. They showed that the velocity of the inner cylinder, which exceeds a critical value, causes the migration of the laminar flow to a turbulent flow.

However, considerable attention is given to the study of the evaporation processes. It is of practical interest in many engineering applications, such as air conditioning, drying, and desalting. Several studies have been published dealing with evaporation analysis in different geometries. Mezaache and Daguenet⁸ developed a numerical study on the evaporation of a thin liquid film flowing over an inclined plate in a forced humid-air flow. They concluded that for adiabatic wall, isothermal or traversed by a constant heat flux density, the enthalpy diffusion term is always negligible. Debbissi et al.⁹ studied the evaporation of water in a heated symmetrically vertical channel by a uniform heat flow density. They assumed that the plates with higher emissivity increase the amount of evaporated water. Ben Nejma and Slimi¹⁰ examined the effect of gas radiation and laminar forced convection on evaporation within a vertical channel formed by two parallel vertical plates.

They showed that the thermal radiation is not the dominant heat transfer mode; it acts as a major factor in the evaporation process. Lately, Sakly et al.¹¹ developed a numerical study to estimate heat and mass transfer for a steady and not fully developed flow in a duct made up of two coaxial cylinders. They accorded a special attention to gas radiation effect on the evaporation process.

Entropy generation has become one of the primary objectives in designing a thermal system. It has been used as a reference for evaluating the significance of irreversibility related to friction, heat, and mass transfer. In fact, many researchers analyzed the second law of thermodynamics in thermal engineering systems in order to minimize entropy production for higher energy efficiency. Bejan¹² introduced the concept of irreversibility distribution ratio and entropy generation number, presenting spatial distribution profiles of entropy production. Bejan¹³ analyzed the fundamental mechanisms responsible for the generation of entropy in heat and fluid flow and on the design trade-off of balancing the heat transfer irreversibility against the fluid flow irreversibility. San et al.^14,15 studied the irreversibility generated by heat and mass transfer during coupled conduction–convection transfer in a channel. They showed that entropy creation is related to the gradients of velocity, temperature, concentration, and physical properties of the fluid.

On the basis of the theory of non-equilibrium thermodynamics, Carrington and Sun¹⁶ derived expressions for the local rate of entropy generation in a fluid mixture subject to heat and mass transfer. They compared their expressions by those previously obtained by Bejan¹² and San et al.^14,15 Carrington and Sun¹⁷ used the control volume method to establish the rate of entropy generation due to heat and mass transfer in a fluid stream, accompanied by fluid friction. Yilbas¹⁸ carried out a numerical study to determine the heat transfer characteristics and resulting entropy generation across annuli with outer cylinder rotating. He showed that the point of minimum entropy generation in the fluid moves away from the outer cylinder wall as the Brinkman number increases. Tasnim and Mahmud¹⁹ investigated entropy production in a vertical cylindrical annulus. They determined irreversibility characteristic analyses through the evaluation of entropy generation in the case of laminar mixed convection flow. Mahmud and Fraser^20,21 performed the second law analysis of the rotating concentric cylinder problem in the presence of heat transfer and fluid motion. They obtained general expressions for the velocity and temperature distributions, entropy generation number, and Bejan’s number. Haddad et al.²² analyzed entropy creation due to laminar forced convection in the entrance region of concentric annulus. They found that raising the Eckert number causes an increase in entropy production. Ko and Ting^23,24 performed various investigations to calculate entropy creation and irreversibility profiles for different geometrical configurations, thermal boundary conditions, and flow situations. Based on the minimal entropy generation principal, they discussed the relevant design parameters to induce the best exergy utilization with the minimal entropy generation and least irreversibility. Ben Nejma et al.²⁵ established a numerical study of entropy generation due to the volumetric radiation of a non-gray gas confined between two parallel plates. They paid special attention to entropy production and its dependence on geometric and thermodynamic parameters. Mazgar et al.²⁶ extended this last work to develop entropy production through combined non-gray gas radiation and natural convection in vertical pipe. Ben Nejma et al.^27,28 performed numerical analysis to establish the profiles of entropy creation components through non-gray gas radiation inside a cylindrical and a spherical enclosure, respectively. They noted that entropy production due to gas radiation is more developed in heating configuration, whereas radiative entropy produced at wall is more developed in case of cooling. Mazgar et al.²⁹ evaluated entropy generation due to combined gas radiation and mixed convection through a vertical cylindrical annulus. They concluded that the volumetric radiative entropy production is distinctly developed compared to wall radiative entropy production and to entropy generation due to friction and conduction. Elazhary and Soliman³⁰ conducted a theoretical study of the rate of entropy production for steady, laminar, and fully developed liquid flow in a micro-channel formed by two parallel plates. They found that entropy generation due to heat transfer reaches a maximum, while entropy production due to viscous dissipation reaches a minimum within a specific range of zeta-potential. Shojaeian and Kosar³¹ analytically examined convective heat transfer and entropy production in Newtonian and non-Newtonian fluid flows between parallel plates with velocity slip boundary condition. They deduced that the global entropy generation rate increases with increasing both power-law index and the Brinkman number, whereas a reduction in the global entropy generation rate is observed with the existence of slip condition and an increase in slip coefficient. Recently, Jarray et al.³² analyzed entropy production due to non-gray gas radiation through a concentric cylindrical annulus. They showed that radiative entropy generation is greatly affected by gas and wall temperatures.

The main objective of this work is to perform evaporation process and entropy generation due to heat and mass transfer within a cylindrical Couette flow.

Mathematical formulation

The geometry of the problem under consideration is shown in Figure 1. We admit a non-stationary air flow between two coaxial cylinders. The working fluid is considered Newtonian and thermo-dependent gas. The cylinders are considered infinite in length where the inner wall is assumed wet, adiabatic, and rotates at an angular velocity Ω₁, while the outer one is supposed to be isothermal, impermeable, and rotates at an angular velocity Ω₂. A relative rotational motion is generated through the inner and the outer cylinders, inducing the fluid flow which is assumed to be axisymmetric and bidirectional. The Dufour and Soret effects, as well as the pressure–volume work, are neglected. Moreover, only radial pressure gradients are considered. On the basis of these assumptions, mass, momentum, species concentration, and energy conservation equations are written as follows

\frac{\partial ρ}{\partial t} + \frac{1}{r} \cdot \frac{\partial (ρ \cdot r \cdot u_{r})}{\partial r} = 0

(1)

\begin{array}{l} ρ \frac{\partial u_{r}}{\partial t} + ρ \cdot u_{r} \cdot \frac{\partial u_{r}}{\partial r} = - \frac{\partial P}{\partial r} \\ + \frac{1}{r} \frac{\partial}{\partial r} (r \cdot μ \cdot \frac{\partial u_{r}}{\partial r}) - μ \frac{u_{r}}{r^{2}} + ρ \cdot \frac{u_{θ}^{2}}{r} \end{array}

(2)

\begin{array}{l} ρ \frac{\partial u_{θ}}{\partial t} + ρ \cdot u_{r} \cdot \frac{\partial u_{θ}}{\partial r} = \frac{1}{r} \frac{\partial}{\partial r} (r \cdot μ \cdot \frac{\partial u_{θ}}{\partial r}) \\ - μ \frac{u_{θ}}{r^{2}} - ρ \cdot \frac{u_{θ} \cdot u_{r}}{r} \end{array}

(3)

ρ \frac{\partial c}{\partial t} + ρ \cdot u_{r} \cdot \frac{\partial c}{\partial r} = \frac{1}{r} \frac{\partial}{\partial r} (r \cdot ρ \cdot D \cdot \frac{\partial c}{\partial r})

(4)

\begin{array}{l} ρ C_{p} \frac{\partial T}{\partial t} + ρ C_{p} u_{r} \frac{\partial T}{\partial r} = \frac{1}{r} \frac{\partial}{\partial r} (r k \frac{\partial T}{\partial r}) \\ + \frac{1}{r} \frac{\partial}{\partial r} (r (C_{p v} - C_{p a}) ρ D T \frac{\partial c}{\partial r}) \end{array}

(5)

Figure 1.

Physical description of the system.

Subjected to the following initial conditions

{\begin{matrix} u_{r} (r, t = 0) = 0 \\ u_{θ} (r, t = 0) = 0 \\ c (r, t = 0) = c_{0} \\ P (r, t = 0) = P_{0} \\ T (r, t = 0) = T_{0} \end{matrix}

(6)

The boundary conditions associated with the problem are as follows.

At the inner cylinder, the radial velocity of gas is deduced by considering that the air–water interface is semi-permeable as given by Ben Nejma and Slimi¹⁰

u_{r} (r = R_{1}, t) = \frac{- D}{1 - c (r = R_{1}, t)} {\frac{\partial c}{\partial r} |}_{r = R_{1}}

(7)

The tangential velocity at the inner cylinder is given as follows

u_{θ} (r = R_{1}, t) = R_{1} Ω_{1}

(8)

Considering the air–water interface to be at a thermodynamic equilibrium and the air–vapor mixture as an ideal gas, the mass fraction of vapor can be expressed in accordance with Ben Nejma and Slimi¹⁰ as

c (r = R_{1}, t) = \frac{M_{v} / M_{a}}{p / p_{vs} + M_{v} / M_{a} - 1}

(9)

The saturated vapor pressure is given by Vachon³³

\begin{array}{l} \log_{10} (p_{v s}) = 28.59051 - 8.2 \times \log_{10} (T) \\ + 2.4804 \times 10^{- 3} T - \frac{3142.32}{T} \end{array}

(10)

The energy balance at the inner wall is expressed by the following equation

- k {\frac{\partial T}{\partial r} |}_{r = R_{1}} + q_{r} + ρ L_{v} u_{r} (r = R_{1}, t) = 0

(11)

where q_r is the radiative heat flux density depending on temperature and optical characteristics of walls³⁴

q_{r} = \frac{ε_{1} ε_{2} σ (T_{w 1}^{4} - T_{w 2}^{4})}{ε_{2} (1 - ε_{1}) + ε_{1} ε_{2} + \frac{R_{1}}{R_{2}} ε_{1} (1 - ε_{2})}

(12)

At the outer cylinder, the wall is considered isothermal, dry, and impermeable, verifying the following conditions

{\begin{matrix} u_{r} (r = R_{2}, t) = 0 \\ u_{θ} (r = R_{2}, t) = R_{2} Ω_{2} \\ \frac{\partial c}{\partial r} (r = R_{2}, t) = 0 \\ T (r = R_{2}, t) = T_{w} \end{matrix}

(13)

The physical properties of the working fluid are given by Ben Nejma and Slimi.^10,35

The discretization of the governing equations with the relevant boundary conditions is carried out with the classical finite volume method. Moreover, the partial differential equations are resolved using an implicit scheme. A uniform 6000-node grid in accordance with time for an interval equal to 0.01 s and a uniform 100-node grid in accordance with the radial direction have been used. This grid has been chosen as a trade-off between accuracy and computational time.

The evaporating flow rate is expressed in equation (14) as shown by Ben Nejma and Slimi¹⁰

m (0 - t) = \int_{0}^{t} ρ u_{r} (r = R_{1}, t) dt

(14)

The local and average conductive Nusselt numbers, indicating energy characterization of heat between the outlet wall and the moving fluid, are given in equations (15) and (16), respectively

Nu (t) = {| \frac{2 (R_{2} - R_{1})}{T_{w} - T_{0}} \frac{\partial T}{\partial r} |}_{r = R_{2}}

(15)

Nu (0 - t) = \frac{1}{t} \int_{0}^{t} Nu (t) dt

(16)

At a given location, the local entropy generation is expressed as follows

{S ‴}_{gen} = {S ‴}_{c} + {S ‴}_{f} + {S ‴}_{m}

(17)

${S ‴}_{c}$ represents the local entropy generation due to thermal conduction and is given in the following equation (Bejan¹³)

{S ‴}_{c} = k \frac{{[\nabla T]}^{2}}{T^{2}} = \frac{k}{T^{2}} {(\frac{\partial T}{\partial r})}^{2}

(18)

${S ‴}_{f}$ is the local entropy generation due to fluid friction and its expression is given by Haddad et al.²² as follows

{S ‴}_{f} = \frac{μ}{T} ϕ

(19)

where ϕ is the viscous dissipation, expressed as follows

ϕ \approx [{(r \frac{\partial}{\partial r} (\frac{u_{θ}}{r}))}^{2}]

(20)

It is worth noting that irreversibility due to viscous effect is neglected compared to entropy production due to heat transfer.

${S ‴}_{m}$ is the local entropy production due to mass diffusion. In binary mixture (A, B), it is given by Carrington and Sun¹⁶ as follows

{S ‴}_{m} = \frac{CRD}{x_{A} x_{B}} {(\nabla x_{A})}^{2}

(21)

According to equation (18), the global entropy production due to thermal conduction can be written as

S'_{c} (0 \to t) = 2 π \int_{0}^{t} \int_{R_{1}}^{R_{2}} \frac{k}{T^{2}} {[\frac{\partial T}{\partial r}]}^{2} rdrdt

(22)

According to equation (21), the total entropy production due to mass diffusion is given as follows

S'_{m} (0 \to t) \approx 2 π \int_{0}^{t} \int_{R_{1}}^{R_{2}} \frac{CRD}{x_{v} x_{a}} {(\nabla x_{v})}^{2} rdrdt

(23)

Therefore, the total entropy generation is expressed as follows

S'_{t} (0 \to t) = S'_{c} (0 \to t) + S'_{m} (0 \to t) + S'_{f} (0 \to t)

(24)

In order to validate our numerical code, we have drawn the transitory evolution of pressure and tangential velocity of an isothermal flow (Figure 2), using the analytical solutions of pressure and tangential velocity obtained from the resolution of the equations of momentum in the radial and tangential directions, respectively.

\begin{array}{l} P = P (r = R_{1}) + ρ (\frac{α^{2}}{8} r^{2} + α β L n (r) - \frac{β^{2}}{2 r^{2}}) \\ - ρ (\frac{α^{2}}{8} R_{1}^{2} + α β L n (R_{1}) - \frac{β^{2}}{2 R_{1}^{2}}) \end{array}

(25)

u_{θ} = \frac{α}{2} r - \frac{β}{r}

(26)

where α and β are given as follows

{\begin{matrix} α = 2 \frac{R_{2}^{2} \cdot Ω_{2} - R_{1}^{2} . Ω_{1}}{R_{2}^{2} - R_{1}^{2}} \\ β = - (Ω_{2} - Ω_{1}) \frac{R_{2}^{2} \cdot R_{1}^{2}}{R_{2}^{2} - R_{1}^{2}} \end{matrix}

(27)

Figure 2.

Variation of pressure and tangential velocity with P₀ = 1 atm, T₀ = T_w = 300 K, R₁ = 0.1 m, R₂ = 0.2 m, Ω₁ = 1 rad s⁻¹, Ω₂ = 0 rad s⁻¹, c₀ = 0.

The profiles of pressure and tangential velocity given in Figure 2 are compared to those plotted for a stationary flow. It can be noted that the results are very close where our values converge with those calculated for a stationary flow.

A review of the asymptotic trends of the pressure profiles (Figure 2(a)) shows that the rotational movement of the inner cylinder initiates the movement of the fluid particles in vicinities of this wall and tends to drag the adjacent fluid layers, engendering the apparition of fictitious forces and causing the development of a radial pressure gradient. Far from the rotating wall, the viscosity effect is not yet felt and practically no particle movement is signaled (Figure 2(b)), inducing uniform pressure profiles. As you can see, there is an increase over time in the asymptotic limit of the pressure. In fact, augmenting well the time of the fluid within the annulus provides more opportunities to shear stresses to induce the rotation of the fluid particles and therefore to increase the fluid pressure.

Results and discussions

A selected set of graphical results are presented in Figures 3 –9 to provide an easy understanding of the effect of some thermodynamic and geometric parameters on entropy generation and the evaporation process.

Figure 3.

Local volumetric entropy generation components with P₀ = 1 atm, T₀ = 300 K, T_w = 400 K, R₁ = 0.1 m, R₂ = 0.2 m, Ω₁ = 1 rad s⁻¹, Ω₂ = 0 rad s⁻¹, c₀ = 0, ε₁ = 0, ε₂ = 1.

Figure 4.

Effect of the outer radius with P₀ = 1 atm, T₀ = 300 K, T_w = 400 K, R₁ = 0.1 m, Ω₁ = 1 rad s⁻¹, Ω₂ = 0 rad s⁻¹, c₀ = 0, ε₁ = 0, ε₂ = 1.

Figure 5.

Effect of rotation velocity of the inner cylinder with P₀ = 1 atm, T₀ = 300 K, T_w = 400 K, R₁ = 0.1 m, c₀ = 0, Ω₂ = 0 rad s⁻¹, ε₁ = 0, ε₂ = 1.

Figure 6.

Effect of the initial temperature with P₀ = 1 atm, T_w = 500 K, R₁ = 0.1 m, R₂ = 0.2 m, Ω₁ = 1 rad s⁻¹, Ω₂ = 0 rad s⁻¹, c₀ = 0, ε₁ = 0, ε₂ = 1.

Figure 7.

Effect of the outer wall temperature with P₀ = 1 atm, T₀ = 300 K, R₁ = 0.1 m, R₂ = 0.2 m, Ω₁ = 1 rad s⁻¹, Ω₂ = 0 rad s⁻¹, c₀ = 0, ε₁ = 0, ε₂ = 1.

Figure 8.

Effect of the outer wall emissivity with P₀ = 1 atm, T₀ = 300 K, T_w = 400 K, R₁ = 0.1 m, R₂ = 0.2 m, Ω₁ = 1 rad s⁻¹, Ω₂ = 0 rad s⁻¹, c₀ = 0, ε₁ = 0.1.

Figure 9.

Effect of the mass fraction with P₀ = 1atm, T₀ = 300 K, T_w = 400 K, R₁ = 0.1 m, R₂ = 0.2 m, Ω₁ = 1 rad s⁻¹, Ω₂ = 0 rad s⁻¹, ε₁ = 0, ε₂ = 1.

Profiles of the local volumetric entropy generation

Figure 3(a) presents the local variation of volumetric entropy generation due to thermal conduction, mass diffusion, and viscosity effect. In the vicinity of the dry surface, we can see that the existence of an extremum tends to reduce and migrate to the central zone of the flow as we advance in time. This extremum results in high temperature gradient, generating high entropy creation. On the contrary, temperature variations are not important in the vicinity of the humid wall, inducing a low entropy production. Entropy generation profiles resulting from mass transfer and illustrated in Figure 3(b) have two behaviors. In the first phase, when the effect of conduction due to the external cylinder is not felt yet and the concentration limit value is stagnant, the mass entropy creation shows profiles having a maximum which is attenuated for sections close to the inner wall when we move forward in the time. In the second step, these profiles seem to have a decreasing appearance with recovering slopes and without entropy generation in the vicinity of the outer wall. In fact, mass entropy generation, conditioned by the existence of concentration gradients, is then absent near the outer surface.

By examining Figure 3(c), we can note a decreasing profile of viscous entropy creation according to the radial position. In fact, the wide variations in velocity cause the appearance of high viscous stresses and therefore greater energy dissipation. Comparing the different sources of entropy generation, we can remark that the component due to viscous effect is negligible compared to the conductive and the mass entropy production. This is always the case when temperature differences are important and the velocities are relatively weak. That is why we will focus in our analyses only on entropy creation due to heat and mass transfers.

Effect of the outer radius

The effect of the outer radius on the evaporated mass flow rate, mass, and entropy creation due to thermal conduction is shown in Figure 4. Numerical results show that increasing the outer radius has two behaviors on the evaporated mass flow rate. The first phase is defined by the fact that heat flux propagating from the outer wall is not felt yet by the wet surface. The profile of the evaporated mass flow rate seems independent of the outer radius and displays an asymptotic trend corresponding to medium saturation. In the second phase, the inner wall is heated, causing progressively an increase in its temperature and a modification in the limit saturation of the fluid. In addition, we can remark a change in the behavior of the evaporated mass flow rate and mass entropy creation. This change is observed after a certain time is needed for the heat flux to propagate across the medium. This time is all the more important as the outer radius increases.

Effect of the rotation velocity of the inner cylinder

A close look at Figure 5 clearly shows that the rotation velocity of the inner cylinder has no effect on the evaporated mass flow rate and entropy generation fields. In fact, increasing the rotation velocity of the inner wall considerably raises the centrifugal inertial forces, causing the thickening of the fluid in the vicinity of the external cylinder. In addition, using low rotation velocities to ensure flow stability conditions generates negligible pressure variation compared to the statistic pressure and the fluid physical properties remain insensible to the intensity of rotation. That is why all profiles of Figure 5 are independent of the rotation velocity of the inner cylinder.

Effect of the initial temperature

As shown in Figure 6(a), increasing the initial temperature considerably rises the humid surface temperature and leads to a notable increase in the evaporated mass flow rate. In fact, this raises the equilibrium vapor pressure and amplifies the concentration in the vicinities of the humid surface. For the same reason, the fluid has a greater ability to absorb the vapor, resulting in a prominent mass transfer and enhances the entropy production due to mass diffusion (Figure 6(b)). In contrast, the reduction in temperature difference when increasing the fluid initial temperature results in a lower thermal conduction due to temperature gradients. The conductive exchanges are then reduced, decreasing the entropy generation due thermal conduction (Figure 6(c)).

Effect of the outer wall temperature

Figure 7 shows the influence of the dry wall temperature on the evaporated mass flow rate and entropy generation due to mass and energy exchanges. At first blush, we can note that the variations of the outer wall temperature seem to affect the evaporated mass flow rate and entropy generation due to mass diffusion, only after a critical time depending on the dimensions of the annular duct. This critical time represents the propagation time of heat flux to reach the inner wall. Beyond this critical time, using a higher dry wall temperature increases the evaporated mass flow rate and entropy generation due to mass diffusion. As can be seen in Figure 7(c), there is a significant effect of the dry wall temperature on entropy generation due to thermal conduction. This is due to the presence of high temperature gradients, involving important heat exchanges and enhancing the entropy generation due to heat conduction.

Effect of the outer wall emissivity

Figure 8 illustrates the effects of the outer wall emissivity on the evaporated mass flow rate and entropy production. It is worth noting that when the surfaces of both cylinders are not completely reflective, thermal radiation provides a direct and an instantaneous exchange between these two surfaces. This is more important when the emissivities of walls are close to unity. This causes an increase in the humid surface temperature, enhancing air humidification process. The mass transfer is then accentuated by diffusion, resulting in a rise in the evaporated mass flow rate (Figure 8(a)) and developing entropy generation due to mass diffusion (Figure 8(b)). Knowing that air is regarded as a transparent gas, it does not participate in radiation transfer. In addition, from the moment that the conductive entropy generation is caused by the presence of conductive exchanges, there is no effect of the outer wall emissivity on entropy creation due to thermal conduction (Figure 8(c)).

Effect of the initial mass fraction

The impact of the initial mass fraction of the water vapor in air on the evaporated mass flow rate, mass, and conductive entropy generation is shown in Figure 9. It can be concluded that the greater the initial mass fraction, the lower the mass exchanges in the vicinities of the humid wall. In fact, the latent heat of vaporization extracted from the fluid is less important with a significant impact on the profiles of the evaporated mass flow rate and the mass entropy creation. Moreover, the effects of the initial mass fraction on entropy generation due to heat conduction, signaled in Figure 9(c), are essentially due to the physical properties related to the local mass fraction.

Concluding remarks

A numerical investigation is performed to analyze the unsteady evaporation of a Newtonian fluid Couette flow for a thermo-dependent gas confined between two rotating coaxial cylinders, one being adiabatic and humid and the other being isothermal and dry. The main results can be summarized as follows:

The conductive entropy generation is more pronounced when increasing the outer radius.

The increase in the rotation velocity of the inner cylinder does not affect the evaporated mass flow rate and the global entropy generation due to heat conduction and mass diffusion.

The use of higher initial temperature results in a significant evaporated mass flow rate and mass entropy generation.

Varying the dry surface temperature, even if it has a strong influence on the conductive entropy creation, affects the evaporated mass flow rate and the mass entropy creation only after a critical time depending on the dimensions of the duct.

The greater the outer wall emissivity, the higher the evaporated mass flow rate and the mass entropy generation.

Entropy creation due to mass diffusion is reduced when rising the initial mass fraction.

Footnotes

Appendix 1

Academic Editor: Bo Yu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Unsteady evaporation and entropy generation in a Couette flow

Abstract

Keywords

Introduction

Mathematical formulation

Results and discussions

Profiles of the local volumetric entropy generation

Effect of the outer radius

Effect of the rotation velocity of the inner cylinder

Effect of the initial temperature

Effect of the outer wall temperature

Effect of the outer wall emissivity

Effect of the initial mass fraction

Concluding remarks

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References