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This investigation contributes to a better understanding of condensation heat transfer in horizontal non-circular microchannels. For this purpose, the conservation equations of mass, momentum and energy have been numerically solved in both phases (liquid and vapor), and all the more , so the film thickness analytical expression has been established. Numerical results relative to variations of the meniscus curvature radius, the condensate film thickness, the condensation length and heat transfer coefficients, are analyzed in terms of the influencing physical and geometrical quantities. The effect of the microchannel shapes on the average Nusselt number is highlighted by studying condensation of steam insquare, rectangular and equilateral triangular microchannels with the same hydraulic diameter of 250 μm.

Understanding the heat transfer behavior of condensation flow in microchannels is important for a broad variety of engineering applications. Although there have been a number of investigations on boiling flow in microchannels, there are relatively little experimental data and theoretical analyses relative to condensation processes available in the literature, especially, for condensation inside a noncircular microchannels. In the chapter 6 of the reference [

Most of the physical and mathematical models that focused on annular condensation heat transfer in circular channel were developed in the previous works. Begg et al. [

On the other hand, various theoretical models have been proposed to predict the local heat transfer related to the condensation annular flow in non-circular channel where the surface tension plays a predominant effect on the condensate flow, more specifically, in the channel corners. Indeed, Zhao and Liao [

The main purpose of the present work is to determine the heat transfer coefficient during the steam condensation inside horizontal various non-circular microchannels (rectangle, square, or equilateral triangle). Indeed, the classical mathematical model of the annular condensation flow is retained in both phases (liquid and vapor). An appropriate numerical method is used to solve the differential equations system obtained from different conservation equations (mass, momentum and energy). To compute the heat transfer coefficient, a new and simple geometrical method is used to express the condensate film thickness.

The physical model investigated in this paper is illustrated on the

nel having wall temperature lower than vapor saturation temperature. The film thickness on the heat exchange surface varies along the axial direction with the vapor quality.

When total condensation occurs, the end of the condensation zone has a hemispherical meniscus. Its length L is one of the unknown parameters in the physical model. The condensate film flows along the axial direction under effects of the pressure, surface tension, and shear stress.

The mathematical formulation of the problem is based on the following principal assumptions:

• For liquid and vapor phases, thermophysical properties are assumed to be constant.

• In the microchannel, the flow is supposed to be steady-state, laminar, one-dimensional and axis-symmetrical.

• The free surface of the condensate film is smooth.

• Gravity forces are negligible compared to the effects of surface tension.

• The heat transfer from the cooling fluid to the condensate flow is assumed to be one-dimensional.

• In the condensate, the temperature profile is supposed linear.

• The saturation temperature of the vapor is assumed to remain constant along the microchannel.

The modeling approach developed here describes the liquid and vapor phases separately. The governing equations are used in the Cartesian coordinates as shown in the

The average parameters over a cross-section are used in liquid and vapor phases of the condensation flow respecting continuity conditions at the liquid-vapor interface. The equation of the mass conservation can be written for each local cross section as follows:

where represents the volumic rate of phase change. By convention, its sign is negative for condensation and positive for evaporation. So, we can write:

, and are respectively the microchannel cross section, the density and the axial velocity. The subscript refers to the considered phase (= v or L).

For a fixed position z along the microchannel, the condensate film thickness in noncircular microchannels is much thicker into the microchannel corners than elsewhere, especially; at the internal considered circumference because of the surface tension effect. This is the reason that the axial flow in the film region between the corners is neglected.

From the forces balance illustrated on the

• In the liquid phase:

• In the vapor phase:

, and are respectively the microchannel cross-section, the pressure and the axial velocities in the phase . Furthermore, we indicate that is the liquid–vapor interface surface along , is the wet heat exchange surface along , and is the shear stress at liquid–vapor interface, is the shear stress at the microchannel heat exchange surface.

The local energy equation in the liquid phase can written as:

The total energy equation defined in the length of the total condensation zone as:

where q is the heat flux density, is the local liquid mass flow rate is the microchannel perimeter, and z is the microchannel abscissa.

To express the curvature radius derivative, we need to use the Laplace–Young equation:

Combining equations (1)-(8), we get:

To establish the dimensionless equations, the following variables are used:

Then the dimensionless form of the derivative curvature radius, the velocity gradients and the pressure gradients in both phases, are given by:

are respectively the boiling number, the capillary number, the vapor Reynolds number and the vapor mass flux.

Definitions of other parameters appearing in the above relations (11-15) are such that:

1) The liquid friction factor for laminar is given by:

C is the Poiseuille number given in [

For turbulent flow, the liquid friction coefficient is determined from the Blasius equation [

In this expression, the liquid Reynolds number is calculated assuming the liquid single phase:

2) The interfacial frictional coefficient taking into account the effect of the condensation process on the interfacial shear stress is defined by [

where f_{v}_{,0} is the friction factor for single phase vapor flow defined for laminar flow as :

For turbulent flow, is determined from:

The factor is defined as the ratio of the local condensation mass flow rate to the vapor mass flow rate rebounding from the liquid-vapor interface [

Equations (11) to (15) are solved using the following dimensionless boundary conditions:

1) At the microchannel inlet (z^{*} = 0):

• the flow mass flux G is imposed;

• the temperature and pressure at the saturated state are given;

• the non-dimensional vapor velocity is

• the non-dimensional liquid velocity is

;

• the non-dimensional curvature radius is

2) At the position z^{*} = L^{*} corresponding to the end of the condensation zone:

• the non-dimensional outlet liquid pressure is:

;

• for the rectangular cross section, the dimensionless curvature radius is [

where ;

• for the equilateral triangle cross section, the dimensionless curvature radius a is [

The above boundary conditions combined with nondimensional equations (11)-(15) constitute the mathematical model of two-phase flow in capillary regime with a vapor-liquid phase change. Solution of this mathematical model is not trivial since one of the limit positions remains to be found, the value of L being one of the unknowns of the problem. So, the iteration process is used to solve the mathematical model. To start the calculations, the saturated vapor at the inlet of the tube is assumed having the known mass flux, temperature, pressure, and vapor quality. For the next axial step , the following steps are executed:

1) To start calculation, an arbitrary total condensation length is assumed;

2) Calculation of the curvature radius from equation (11);

3) Calculation of the local liquid velocity from equation (12);

4) Calculation of the local vapor velocity from equation (13);

5) Calculation of the liquid pressure from equation (14);

6) Calculation of the local vapor pressure from equation (15);

7) Calculation of the total condensation length from equation (26).

Steps 2) to 7) are repeated until the value of the condensation length obtained at the iteration number “it” is approximately equal to the one determined at iteration number “it-1”.

Knowing the total condensation length, calculations are then made for the next values of z locations. For each z location, steps 2) to 7) are made and calculation is stopped when z value is approximately equal to the total condensation length L.

Assuming that the temperature profile is linear in the condensate film, the local heat transfer is expressed by:

and hence the average heat transfer coefficient in the microchannel condensation length L is performed by the relationship:

Two last quantities are used to determine the average and the local Nusselt numbers respectively:

is the average condensate film thickness for each z location along the microchannel. Using the geometrical considerations from the

The parameter dR/dz given by the equation (9) is expressed in terms of the limit curvature radius .

At z = L location, dR/dz is to be infinite which causes a calculation problem. To avoid this complication, we set where is an infinitesimal parameter. The limit values of the curvature radius and its first derivative are defined as [

To estimate the numerical value of the condensation length a dichotomy method was performed using equation (26), between the inlet of the microchannels which corresponds to and the end of the condensation zone where

The validation of the results obtained in the present work for condensation of water in a square microchannel, are compared to the predictions of various correlations available in the literature. These correlations are proposed for condensation heat transfer in microchannels and macrochannels. Among these predictive correlations, those of condensation in microchannels are defined by Wang et al. [

vapor mass fluxes ranging from 70 to 220 kg/m²s. It is found that the best predictions of the present average Nusselt number are obtained by the correlations of Dobson et al. [

Numerical results are given in the Figures 5(a) and (b) and the Figures 6(a) and (b) for steam condensation in square section microchannel with hydraulic diameter of 110 and 250 µm. Computations were conducted for vapor mass flux G = 90 kg/m²s, contact angle q = 15˚, heat flux density q = 100 kW/m². For the conditions used in the present computation, the boiling number, the capillary number, the inlet steam temperature and pressure are maintained constant.

To better understand the behavior of the peripheral local heat transfer coefficient, Figures 5(a) and (b) show predicted square channel condensate film thickness (24) plotted as a function of the peripheral coordinate W at four different axial locations. It can be seen from these figures that for the same z location on the microchannel, the condensate film thickness is higher for 250 µm (

To indicate the influence of the channel cross-section shape, we start by the presentation of the results concerning curvature radius. In fact,

The annular condensation length is one of the most important parameter which influences the thermal performance of the microchannel studied here.

In the heat transfer exchange point of view, the influence of the microchannel shape on the average Nusselt number is highlighted by studying condensation of steam in a square, equilateral triangular, and rectangular microchannels with the same hydraulic diameter of 250 µm. The sides of the equilateral triangular and square microchannels are 433 µm and 250 µm respectively. For rectangular microchannels having the same hydraulic diameter, the aspect ratio is about 2, 3 and 4. Three different rectangular cross sections are investigated: 375 × 187.5 µm², 500 × 166.6 µm² and 625 × 156.25 µm².

Under the same conditions,

for microchannel with high aspect ratio (equal to 4) and low confinement (a = 156.25 µm). It is interesting to note that for the same hydraulic diameter, the microchannels perimeter and cross section increase with the aspect ratio leading to a reduction of the condensate film thickness. This is due to the predominant effect of the surface tension in the microchannel that increases with microchannel perimeter and thins the condensate film along the sides of the microchannel. For the same Reynolds number, average Nusselt number for equilateral triangular microchannel is between Nusselt number for rectangular microchannel with aspect ratio of 2 and that with aspect ratio of 3. Recall that the perimeter of the triangular microchannel is about 1299 µm and those of rectangular microchannels with aspect ratios of 2 and 3 are about 1125 µm and 1333 µm respectively.

The numerical model characterizing local heat and mass transfer for condensation in microchannels has been developed by including the effects of wall and liquid vapor interface shear stresses, surface tension, pressure forces, contact angle, etc. Numerical results are compared for steam condensation in square, rectangular, and equilateral triangular microchannels with the same hydraulic diameter. They are compared with the correlations available in the literature and it is shown that correlations for annular condensation in macro-channels predict heat transfer with significant deviations from those obtained for microchannels. The best predictions are obtained with the correlations of Dobson et al. [

By using the established condensate film thickness expression [

Finally, to show the major influence of cross-section shape on the condensation heat transfer, comparison of the average Nusselt number is conducted for different microchannel shapes with the same hydraulic diameter. It can be concluded that condensation average heat transfer increases with aspect ratio for rectangular microchannels. The lowest average Nusselt numbers are obtained for the square microchannel because its perimeter and cross section are lower than those of triangular and rectangular microchannels.

List of Latin symbols A: Area (m²)

a: Width of rectangular microchannels (m)

b: Depth of rectangular microchannels (m)

c = 1/4, 1/3: Inverse relationship of number of corners

D: Hydraulic diameter (m)

f: Friction factor hfg: Latent heat (J∙kg^{−1})

l: Annular condensation length (m)

l: Length of the end part in condensation (m)

M: Mass flux (kg∙s^{−1})

P: Pressure (Pa)

Q: Heat flux density (W∙m^{−2})

R: Curvature radius (m)

: Perimeter (m)

U: Velocity (m∙s^{−1})

Greek symbols

Β: Half of right angle (˚)

ε = min(a,b) (m)

μ: Viscosity (kg m^{−1}s^{−1})

θ: Contact angle (˚)

ρ: Density (kg∙m^{−3})

σ: Surface tension coefficient (N∙m^{−}^{1})

τ: Shear stress (N∙m^{−2})

Ω: Peripheral angle (˚)

Subscripts

L: Liquid

Lw: Liquid-wall interface

V: Vapor

Vl: Liquid-vapor interface

O: Inlet