This article was downloaded by: [Eastern Michigan University] On: 21 July 2013, At: 01:06 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/unht20 Numerical Investigation of Evaporation in the Developing Region of Laminar Falling Film Flow Under Constant Wall Heat Flux Conditions a a a R. S. Maurya , S. V. Diwakar , T. Sundararajan & Sarit K. Das a a Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, India Published online: 21 Jul 2010. To cite this article: R. S. Maurya , S. V. Diwakar , T. Sundararajan & Sarit K. Das (2010) Numerical Investigation of Evaporation in the Developing Region of Laminar Falling Film Flow Under Constant Wall Heat Flux Conditions, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology, 58:1, 41-64, DOI: 10.1080/10407782.2010.490174 To link to this article: http://dx.doi.org/10.1080/10407782.2010.490174 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. 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Terms & Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Numerical Heat Transfer, Part A, 58: 41–64, 2010 Copyright # Taylor & Francis Group, LLC ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/10407782.2010.490174 NUMERICAL INVESTIGATION OF EVAPORATION IN THE DEVELOPING REGION OF LAMINAR FALLING FILM FLOW UNDER CONSTANT WALL HEAT FLUX CONDITIONS Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 R. S. Maurya, S. V. Diwakar, T. Sundararajan, and Sarit K. Das Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, India A ﬁnite-volume-based incompressible ﬂow algorithm on Cartesian grid is presented for the simulation of evaporation phenomena in a falling liquid ﬁlm under low wall superheat conditions. The model employs the PLIC–VOF method to capture the free surface evolution, and the continuum surface force (CSF) approximation to emulate the effects of interfacial tension. The phase change process is represented through a source term in the interfacial cells, which is evaluated from the normal temperature gradients on either side of the interface. In order to evaluate these discontinuous temperature gradients across the interface accurately, a simple and efﬁcient ghost ﬂuid method has been implemented, which properly takes into account the dynamic evolution of the interface. The overall numerical model, including the phase change algorithm, has been validated against standard benchmark analytical results. Finally, the model is used to simulate the evaporating ﬂow of a 2-D laminar, developing ﬁlm falling over an inclined plane surface, subjected to constant wall heat ﬂux. The results thus obtained, clearly illustrate the signiﬁcance of inertial effects in the developing region of the falling ﬁlm, which are generally neglected in the available analytical models. It is also observed, that the evaporation of ﬂuid commences only after the growing thermal boundary layer reaches the interface, and the length of the nonevaporating section reduces with the increase in wall heat ﬂux value. 1. INTRODUCTION Liquid ﬁlm emerging from a distributor and ﬂowing over a surface under the action of gravity is commonly encountered in systems such as distillation columns, evaporators, concentrators, food processors, and water desalinators. Generally, liquid ﬁlms undergo a development phase where the ﬂow pattern changes continuously before a steady developed behavior is established. This developing region plays a signiﬁcant role in the optimal design of many chemical process equipments and, hence, a proper understanding of its hydrodynamic and thermal behavior has a great impact on the system development. Received 27 October 2009; accepted 8 April 2010. Address correspondence to Sarit K. Das, Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India. E-mail: [email protected] 41 42 R. S. MAURYA ET AL. NOMENCLATURE Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 Cp F(x, y, t) Fv g hy k L Lp _ m ^ n n p q Re Sc Si t T u v v V x, y speciﬁc heat color function volume force acceleration due to gravity local heat transfer coefﬁcient thermal conductivity latent heat of vaporization length of phase interfacial mass transfer rate per unit area unit normal face normal pressure heat ﬂux ﬁlm Reynolds number per unit width control volume surface control volume interface time temperature x-component of velocity vector velocity vector y-component of velocity vector volume coordinate directions Le h b d / m q r s Subscripts 1 0 av e i l lv sat v w nondimensional distance along plate from the slit angle of inclination of plane from horizontal ratio of ﬁlm thickness to developed ﬁlm thickness (nondimensional ﬁlm thickness) ﬁlm thickness ﬂuid property dynamic viscosity mass density surface tension stress developed region value value at the inlet (slit) average evaporation interface liquid ghost node of liquid region in the vapor region saturation condition vapor wall Generally, the liquid to vapor phase transformation in a falling ﬁlm can occur via two modes. The ﬁrst mode prevails in situations where a high degree of wall superheat is available in the system that causes an onset of nucleation at the wall surface. Subsequently, vapor bubbles formed at the nucleation sites grow continuously until the buoyancy and inertial forces make them depart and migrate to the liquid– vapor interface. The other mode of phase change involves a pure evaporation process at the liquid–vapor interface, where no vapor bubble is formed at the heating wall. Of these processes, nucleate boiling has a higher heat transfer coefﬁcient as compared to the pure evaporation process. Despite this fact, in several special industrial applications the evaporative mode of phase change is the preferred process than the former. Fruit and vegetable juice evaporators are examples of such applications where the low degree of wall superheat is preferred. This is mainly due to the sensitivity of juice to high temperature, which tends to degrade its taste and nutrients. So, fruit-juice processing devices generally operate at a very low range of temperature difference (3–10 C) depending on the heat transfer surface and the nature of the liquid. Hence, quantifying the operational characteristics of such devices requires a clear understanding of the behavior of falling ﬁlm evaporation process, which is the focal point here. In the past, major work on modeling of hydrodynamics and the other transport processes in evaporating falling ﬁlm has been carried out through analytical approaches, which mainly correspond to the fully developed region. A detailed review Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 EVAPORATION OF LAMINAR FALLING FILM FLOW 43 of the available mathematical methods concerning the phase change process can be found in the paper by Bankoff [1]. A fully developed evaporating liquid ﬁlm ﬂowing along an inclined wall has been addressed by Murty and Sastri [2]. Many other works [3–6] have also focused on the developed evaporating ﬁlm region subjected to various thermal and ﬂow conditions. However, analyses pertaining to the fully developed region alone do not provide a complete representation of falling ﬁlm behavior, as in actual systems the development region covers a signiﬁcant portion. The characteristics of developing ﬁlm region are governed by nonlinear interactions between the ﬂow, thermal, and phase change processes, which cannot be treated analytically and hence, the focus needs to be shifted towards numerical simulations. Several approaches have been introduced in the last two decades to address the numerical challenges in phase change problems arising due to an evolving interface with discontinuous variation of properties across it. The ﬁrst attempt in this direction was by Welch [7], who simulated the two-dimensional ﬁlm boiling process using moving triangular grids, and this was further extended by Son and Dhir [8, 9]. Another landmark mathematical model for simulating the liquid–vapor phase transition was presented by Juric and Tryggvason [10], to which further improvements in handling the interfacial conditions were suggested by Son [11]. A volume of ﬂuid (VOF) method-based adaptation of Juric and Traggvason’s [10] phase change model was developed by Welch and Wilson [12], in which a mass source term was introduced to emulate the phase change process. The capability of these approaches has been demonstrated in droplet vaporization situations by Lacas et al. [13]. Recent developments in the modeling techniques, include the energy of ﬂuid (EOF) method by Anghaie et al. [14] and the enthalpy-based formulation by Shin and Juric [15]. In the current work, we choose an approach similar to that of Welch and Wilson [12], where the VOF method has been used to track the dynamics of interfacial ﬂuid structures. In view of the small mass transfer rates observed in the evaporation process, the interface tracking process has to be very accurate since any numerically-induced mass imbalance would hamper the effective characterization of phase change process. Hence, in this regard, the VOF method with its perfect volume conservation characteristics has a deﬁnite edge over other multiphase tracking and capturing techniques. However, even upon using the VOF method, special care is required for the implementation of interfacial condition (say, the prescription of saturation temperature) and the accurate estimation of mass source pertaining to the evaporation process. Particularly, with respect to the estimation of temperature gradients on either side of the interface, the approach should be consistent with the drastic change in ﬂuid topology. Such sophistication is possible using the ghost node technique suggested by Fedkiw et al. [16]. Morgan [17] and Gibou et al. [18] have used the ghost node technique coupled with level-set to implement the interfacial condition and to capture the interfacial jump conditions accurately. The numerical formulation for imposing accurate jumps in this method was further improved and simpliﬁed by Tanguy et al. [19]. In the current work, a numerical model has been developed that couples the VOF method and the ghost ﬂuid technique to estimate the mass source arising out of the evaporation process at the interface accurately. A simple and efﬁcient way to extrapolate information of ghost nodes on either side of the interface has been 44 R. S. MAURYA ET AL. suggested. This coupled technique has been used to simulate the transient evaporating characteristics in the developing region of a laminar liquid ﬁlm. The simulations highlight the effects of wall heat ﬂux on the temperature and velocity distributions across the ﬁlm, and the variation of wall temperature, ﬁlm thickness, and evaporation rate. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 2. MATHEMATICAL MODEL Consider a liquid ﬁlm emerging from a 2-D slit and ﬂowing over an inclined plane wall under the action of gravity (Figure 1). The ﬂow is assumed to be in the laminar regime and a constant heat ﬂux condition is imposed at the wall. As a result, both the thermal and hydrodynamic boundary layers develop simultaneously until they achieve asymptotic developed proﬁles. The behavior of the ﬁlm in the developing region is fundamentally governed by gravitational, viscous, and surface tension forces whose interactions determine the local features of development. In the case of an evaporating liquid ﬁlm, these forces render the hydrodynamics complex and nonlinear, while the phase change introduces complexities in the associated heat and mass transfer. The following assumptions are invoked to simplify the complex interactions related to these processes. The ﬂuids are assumed to be incompressible and Newtonian with constant thermophysical properties. . Buoyancy effects are neglected in both phases. . Figure 1. Schematic diagram of evaporating falling liquid ﬁlm. EVAPORATION OF LAMINAR FALLING FILM FLOW 45 The interface is assumed to be at the saturation temperature corresponding to the prevailing pressure. Hence, a constant surface tension value is employed for the whole interface. . The effects of kinetic energy, viscous dissipation, and surface tension work are neglected in the energy equation. . The gaseous medium is assumed to consist only of the vapor of liquid under consideration. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 . Subjected to these assumptions, the equations governing the laminar, incompressible ﬂows of an evaporating liquid ﬁlm and the ambient vapor medium can be written for each control volume (computational cell) as Mass balance ZZ Z ðqv nÞdS þ ½qðv vi Þ n dS ¼ 0 ð1Þ Sc Si Here, [/] indicates a jump in / across the interface. Momentum balance ZZZ ZZ ZZZ q qv dV þ qðv nÞv dS ¼ ðqg rpÞ n dV qt V Sc V ZZ þ ðs nÞdS Sc ZZZ þ F v n dV ð2Þ V Energy balance q qt ZZZ V qCp T dV þ ZZ qCp Tðv nÞdS ¼ Sc ZZ q n dS ð3Þ Sc The second term of Eq. (1) corresponds to the interfacial mass source due to the evaporation process, and it assumes signiﬁcance only for interface cells where the local thermal conditions favor the phase change process. The last term in Eq. (2) represents the effects of surface tension, which is approximated as a volumetric force in the vicinity of the interface, as discussed later. The computational domain used in the current analysis of evaporating falling ﬁlm ﬂow is shown in Figure 1. The boundaries of the domain are represented by broken lines, and various conditions applied at these surfaces are listed below. Boundary—A A parabolic velocity proﬁle is imposed at the inlet as follows. " # x x 2 Inlet parabolic velocity; u ¼ 6uav ð4Þ d0 d0 Boundary—B qu qv qT ¼ 0; ¼ 0; ¼0 qy qy qy and p¼0 ð5Þ 46 R. S. MAURYA ET AL. Boundary—C qu qv qT ¼ 0; ¼ 0; ¼ 0 and qx qx qx p¼0 ð6Þ qu qv qT ¼ 0; ¼ 0; ¼0 qy qy qy p¼0 ð7Þ Boundary—D and Boundary—E Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 u ¼ 0; v ¼ 0; and q ¼ constant ð8Þ At the interface T ¼ Tsat (saturation temperature corresponding to the prevalent system pressure) 3. SOLUTION ALGORITHM In the current work, the uniﬁed governing equations (Eqs. (1)–(3)) corresponding to incompressible ﬂow of the falling ﬁlm and the vapor medium are discretized using the ﬁnite-volume method. These discretized equations are marched explicitly in time and proper pressure-velocity coupling is obtained by the use of SIMPLE algorithm. The evolution of the liquid free surface is tracked with the help of a density-based VOF method, and the associated interfacial tension is modeled as a localized volumetric force based on the continuum surface model (CSF) approximation of Brackbill et al. [20]. Finally, modeling of the phase change process has been carried out using the approach of Welch and Wilson [12], which employs a mass source term to emulate the evaporation process at the interface. The details of these models are discussed in the following subsections. 3.1. Density-Based VOF Approach to Model the Evaporation Process A detailed review of the general VOF method can be found from the works of Rider and Kothe [21] and Scardovelli and Zaleski [22]. In this method, the ﬂuid structures in the computational domain are identiﬁed with the help of a scalar phase indicator called the liquid volume fraction, F. The basic relationship between the F–value and the phase contained in the cell is given as Cell with liquid phase F ¼ 1 Mixed cell (including interface) Cell with gas phase F ¼ 0 0<F <1 ð9Þ Here, the evolution of free surface in the domain is achieved by means of two steps: namely, interface reconstruction and ﬂuid advection. In the reconstruction process, the 2-D interface segments are assumed to be straight lines (piecewise linear interface calculation (PLIC)) in each cell whose approximate orientation and EVAPORATION OF LAMINAR FALLING FILM FLOW 47 position are obtained through Youngs’ multidimensional stencil [23]. The ﬂuid advection process then geometrically estimates the liquid ﬂux through the cell faces and effectively tracks the dynamics of liquid structures by employing the volume conservation equation given below. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 DF qF ¼ þ ðv rÞF ¼ 0 Dt qt ð10Þ The essential features of the current VOF implementation and the procedure involved in the evaluation of the surface tension force are essentially similar to those used in the work concerning the hydrodynamics of falling liquid ﬁlm by Maurya et al. [6]. However, slight modiﬁcations in the conservation process have been incorporated here, since in situations involving change of phase, a procedure incorporating the direct mass balance is more advisable than the volume conservation process (Eq. (10)). Hence, a density-based approach has been employed to model the evolution of evaporating interface, where the ﬂuxes obtained from the advection process are used to evaluate the density directly through the mass conservation equation given below. q qt Z qdV þ Z V qv n dS ¼ 0 ð11Þ Sc With the new density ﬁeld thus obtained, the volume fraction in each of the cells is updated through the relation F¼ q qv ql qv ð12Þ 3.2. Numerical Estimation of Phase Change Process In a typical multiphase problem involving phase change process, additional jump conditions pertaining to interfacial normal velocity and heat ﬂux are encountered along with the general discontinuity of properties. Hence, the numerical formulation concerning the conservation of mass, momentum, and energy should be adequately sophisticated to handle these jump conditions accurately. The discontinuity in the velocity component normal to the interface is given by, _e vv n vl n ¼ m 1 1 qv ql ð13Þ _ e Þ can be expressed in terms of the difference in normal where the evaporation rate ðm heat ﬂux on either side of the interface, as _e¼ m ql qv L ð14Þ 48 R. S. MAURYA ET AL. With the knowledge of the interface orientation and the segment length obtained from the PLIC interface reconstruction procedure, the velocity jump evaluated through Eq. (13) can be used to estimate the mass source (second term on the left-hand side of Eq. (1)) in an interfacial cell as ðql qv Þ ^n 1 1 ^ dS ¼ q Mass source ¼ ½qðv vi Þ n interface area L qv q l ð15Þ Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 In the above equation, the jump in normal heat ﬂux can be written in terms of the temperature gradients as qT qT ðql qv Þ ^ n ¼ kv ðrTv ^ nÞ kl ðrT1 ^nÞ ¼ kv kl qn v qn l ð16Þ From Eqs. (15) and (16), it is evident that the estimation of jump in heat ﬂux requires accurate computation of normal temperature gradients on either side of the interface. Unfortunately, this information cannot be directly extracted from the uniﬁed energy equation (Eq. (3)), which considers the two phases as a single continuous medium with spatially varying thermophysical properties. Also, due to the discontinuity of temperature gradient at the interface, the information used to estimate the normal gradients should be conﬁned to the respective phases and this involves preserving the identity of mesh points with respect to the phases under consideration. Hence, the temperature gradients evaluated in the individual phases can be extrapolated to the interface in order to estimate the mass source term in Eq. (15). However, such an approach is not trouble-free, particularly when the ﬂuid structures are subjected to drastic topological changes with respect to time. Figure 2 shows a typical interface whose movement changes the association of certain nodes from the liquid phase to vapor phase during the time step Dt, in a ﬁxed Cartesian mesh. As a result, the liquid temperature information that was associated with these nodes becomes irrelevant, which in turn gives rise to an ambiguous situation when these nodes are used to estimate the normal temperature gradient in the vapor phase region. However, such an awkward situation can be successfully handled using the ghost ﬂuid method (GFM) proposed by Fedkiw et al. [16]. The GFM implicitly treats the interfacial discontinuity by introducing ghost ﬂuid nodes in a thin band of Cartesian mesh points across the interface where the information pertaining to one ﬂuid is extrapolated to the ghost ﬂuid point on the other side of the interface. This can be explained with the help of Figure 2a, where the vapor and liquid ghost nodes derive information by extrapolation of values from the actual vapor and liquid region at a particular instant. In the case of drastic changes in ﬂuid distributions, the nodes are self-contained with the information corresponding to the newly associated phase and this paves way for a proper estimation of temperature gradients in the individual phases without the inﬂuence of the complementary phase. The extrapolation procedure for the nodal values and the estimation of temperature gradients are explained in the following subsection. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 EVAPORATION OF LAMINAR FALLING FILM FLOW 49 Figure 2. (a) Ghost node and moving interface, and (b) ghost node construction and estimation of normal temperature gradient. 3.3. Estimation of Ghost Node Values and Normal Temperature Gradient The current implementation of the ghost ﬂuid method uses linear extrapolation to estimate the values of temperature at the ghost nodes, which lie within the range of max(2dx, 2dy) distance normal to the interface, where dx and dy are the cell dimensions in the x- and y-directions, respectively. At each time step, the extrapolation process starts after the interface reconstruction procedure of VOF method, which identiﬁes the orientation and location of interface in each of the mixed cells in the domain. The estimation of the ghost node value is then based on a nine cell stencil, as shown in Figure 2b, where one reference node is chosen from each of the phases. The selected reference nodes may be two of the extreme diagonal nodes in the nine cell stencil, which are designated in Figure 2b, as fsw, fse, fne, or fnw corresponding to the farthest southwest, southeast, northeast, and northwest nodes, respectively. For the illustrative ﬂuid conﬁguration shown in Figure 2b, node fsw and fne can be chosen as reference points for the calculation of the liquid side ðdT=dnÞjl and vapor side ðdT=dnÞjv temperature gradients. Since the interface is assumed to be at saturated condition, the liquid side normal temperature gradient can be written as dT Tfsw Tsat ¼ dn l dn ð17Þ where, dn is the normal distance from node fsw to the interface. Using the normal temperature gradient thus obtained, the temperature value at the ghost liquid nodes 50 R. S. MAURYA ET AL. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 lying at a normal distance of dnl from the interface in the vapor region can be evaluated as dT Tlv ¼ Tfsw þ ðdn þ dnl Þ ð18Þ dn l A similar approach can be applied on the vapor side to estimate the vapor phase ghost node temperatures and the normal gradients which can be used to effectively handle the ambiguous situations where grid points undergo a sudden change in their phase association. Also, with the estimated temperature gradients ðdT=dnÞjl and ðdT=dnÞjv , the source term in the mass conservation equation, Eq. (3), can be directly calculated without any extra computational effort. Despite the use of linear extrapolation procedure, the current technique yields second order accuracy as evident from the work of Gibou et al. [18]. Also, it is more convenient compared to the probe method suggested by Udaykumar et al. [24]. 3.4. Interfacial Velocity Conditions Akin to the issues faced during the evaluation of temperature gradients, estimation of interfacial velocity also suffers due to the change in the association of grid points with individual phases. In order to avoid this difﬁculty, the methodology introduced by Nguyen et al. [25] has been used in the current work where a band of ghost node velocities is deﬁned on both sides of the interface. At every grid location corresponding to the vapor phase, a liquid ghost velocity is obtained using the evaporation rate, as follows. ! 1 1 1 1 _e _e ulv ¼ uv m ð19Þ ny nx vlv ¼ vv m qg ql qv ql Here, nx and ny are the direction cosines of the interface normal. Similarly, the vapor phase ghost velocities can also be calculated at appropriate points. The complete algorithm to model the ﬂow of an evaporating liquid ﬁlm can now be summarized as follows. 1. Based on the initial ﬂuid conﬁguration for a time step, the approximate orientation and position of the PLIC interfaces are obtained using Youngs’ multidimensional stencil. 2. Using the known velocity ﬁeld vn, the liquid ﬂux through each cell face is estimated. This, in turn, is used to track the dynamics of ﬂuid structures by estimating the density ﬁeld at the new time level as q nþ1 dt ¼q þ V n Z qðvn nÞ dS ð20Þ Sc 3. The liquid volume fraction distribution (F) in the domain for the new time level is calculated using Eq. (12) and, once again, the interfaces are reconstructed in all of mixed cells. EVAPORATION OF LAMINAR FALLING FILM FLOW 51 4. The various thermophysical and transport properties are evaluated in all of the cells using the volume fraction F as weighting function. For transport properties such as viscosity and thermal conductivity, the relation is 1 F ð1 F Þ ¼ þ u ul uv ð21Þ and for the other thermophysical properties, the relation is given as Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 u ¼ ul F þ uv ð1 F Þ ð22Þ 5. With the known interface positions in the mixed cells, liquid and vapor nodes are properly identiﬁed in the domain and their vapor and liquid ghost node counterparts are created in the vicinity of the interface. 6. The temperature ﬁeld in the domain is updated using the energy conservation equation of the form T nþ1 k 2 ¼ T þ dt v rT þ r T qCp n ð23Þ Here, necessary care is taken to select either the original node or the ghost node value depending upon the change in phase association of nodes over the time step Dt. 7. The temperature gradients on either side of the interface are accurately evaluated in all mixed cells and the resultant mass source term is calculated using Eq. (12). 8. Finally, the iterative solution procedure of the SIMPLE algorithm is used to update the pressure and velocity ﬁelds. Here, using the body force obtained from the CSF approximation and a guessed value of pressure, the momentum equations are marched in time to arrive at the guessed velocity ﬁeld. The resultant mass imbalance corresponding to the guessed velocity ﬁeld is used to obtain the correct pressure value iteratively by solving the pressure correction equation. However, with respect to the phase change process, the mass source term evaluated in Eq. (15) is subtracted from the overall mass imbalance, and a revised residue is used at each iteration. 4. VALIDATION OF THE EVAPORATIVE FLOW MODEL In order to access the efﬁcacy of the basic evaporation model, two standard benchmark problems are considered here for which analytical solutions are readily available from the work of Jamet and Duquennoy [26]. The ﬁrst problem deals with the phase change process in a saturated liquid layer caused because of external heat supplied at the bottom boundary of the domain. The second test problem involves a superheated liquid layer and a saturated vapor layer with no other source of heat. Such a conﬁguration forms a meta-stable system where the phase change process is driven by the superheat available with the liquid. The results obtained for these test cases using the current model are discussed in the remainder of this section. 52 R. S. MAURYA ET AL. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 4.1. One-Dimensional Liquid-Vapor Phase Change Figure 3 shows the conﬁguration of the ﬁrst test problem with the corresponding boundary conditions. Here, the stratiﬁed liquid and vapor layers are initially at rest (no ﬂow) and at saturation temperature corresponding to the system pressure (Patm). The bottom boundary of the domain is subjected to constant heat ﬂux condition. Initially, the major part of the heat supplied is utilized in sensible heating of the vapor layer during which the phase change process is sluggish or nonexistent. Once temperature gradients are established in the vapor phase, the phase change process at the interface slowly picks up and attains an asymptotic behavior. The vapor mass formed due to the phase change process gently pushes the liquid layer in the upward direction. Analytical estimates of this liquid velocity and the interfacial velocity can be obtained from the works of Jamet and Duquennoy [26]. In the current simulations, the ﬂuid properties are selected such that the Peclet number in vapor phase is constrained to be much less than one (Pe51). This leads to the dominance of conduction process in the vapor region, as considered by Jamet and Duquennoy [26]. The computational domain chosen for the current analysis is of size 1 4 units and is discretized using a grid of 20 80 cells. The initial height of the vapor column is chosen to be 3.5% of the vertical length in order to shorten the initial transients in the simulation. Various nondimensional parameters and property ratios used in the simulation are given below. qref ¼ 1:0; ¼ 4:0; Tsat ¼ 0:0; L ¼ 1; q ¼ 0:05; ql =qv ¼ 2:0; ml =mv ¼ 1:0; Cpl =Cpv ¼ 1:0 kl =kv ð24Þ The temporal variations of interfacial and liquid region velocities are plotted in Figures 4a and 4b. Evidently, the velocities exhibit an initial transient growth and an Figure 3. One-dimensional phase change problem with imposed boundary heat ﬂux. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 EVAPORATION OF LAMINAR FALLING FILM FLOW 53 Figure 4. Comparison of transient prediction with 1-D steady state results. (a) Interface velocity and (b) liquid velocity. asymptotic convergence to the steady state values obtained from the analytical results [26]. 4.2. Unsteady Phase Change of a Superheated Liquid In the second test case, stratiﬁed ﬂuid layers are also considered. The liquid column here is at superheated condition and the bottom boundary is assumed to be adiabatic. Despite the absence of an active heat source, the phase change process is triggered and sustained in this problem mainly by the superheat of the liquid column. Akin to the previous problem, both the phases are initially at rest and the vapor phase is at saturation temperature (Tsat). The liquid region in the system is initialized to a uniform superheated temperature (Tsat þ DT), which induces a sharp temperature jump at the interface. The uniform initial conditions in the individual phases give rise to zero temperature gradients on either side of Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 54 R. S. MAURYA ET AL. the interface. However, any small perturbation to this meta-stable system initiates the phase change process at the interface, and the sharp interfacial temperature jump is smoothened by the diffusion process. The dynamics of this system has been analytically and numerically studied by Welch & Wilson [12], who recommended the use of analytical solution at time t0 > 0 as the initial condition. In line with this suggestion, the initial temperature perturbation in the current simulations is spread over a thickness of three grid points near the interface and the proﬁle is initialized using the detailed analytical solutions presented by Jamet and Duquennoy [26]. Figure 5 shows the computational domain and the various boundary conditions of the simulation. The computational domain chosen for the simulation is of size 1 4 units and the domain is discretized into 20 80 cells. The various nondimensional parameters and ﬂuid property ratios used in the simulation are as follows. qref ¼ 1:0; ¼ 4:0; Tsat ¼ 0:0; L ¼ 1; DT ¼ 0:2; ql =qv ¼ 2:0; ml =mv ¼ 1:0; Cpl =Cpv ¼ 1:0 kl =kv ð25Þ The evolution of thermal boundary layer in the liquid region is shown in Figure 6a. The comparison between the predicted temperature proﬁle and the corresponding analytical result shows an excellent match. This is further corroborated by the comparison between numerical and analytical solutions in Figure 6b, which show the variations of interface location and interface velocity with time. Figure 5. One-dimensional phase change problem for initially superheated liquid. 55 Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 EVAPORATION OF LAMINAR FALLING FILM FLOW Figure 6. (a) Evolution of temperature proﬁle in liquid, and (b) Evolution of interface position. 5. RESULTS FOR EVAPORATING FALLING FILM The thermal hydraulic analysis of an evaporating liquid ﬁlm, falling over a surface inclined at an angle of 60 to the horizontal (Figure 1) is now carried out. _ =m (where m _ is the ﬁlm In all simulations, the Reynolds number of the ﬁlm Re ¼ 4m ﬂow rate per unit width) is maintained at a constant value of 5. The various parameters and thermophysical properties of the ﬂuids considered are listed in Tables 1 and 2, respectively. A relatively smaller value of 100 kJ=kg has been assumed for the latent heat of vaporization in order to reduce the computational requirement. Also, 56 R. S. MAURYA ET AL. Table 1. Computational parameters Parameters Details Computational domain size Grid Time step Convergence criterion 0.048 m 0.012 m 80 40 Adaptive time, based on Courant number criterion (C < 0.001) Max. residue (R 1 1012) Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 in order to prevent the onset of nucleate boiling, low wall heat ﬂuxes (5 W=m2, 10 W= m2, and 15 W=m2) are considered. The various nondimensional parameters employed for the analysis of falling ﬁlm behavior [5] are listed below. T Tsat ðqw d1 =kÞ qw Local heat transfer coefficient ¼ ðT Tw Þ ð26Þ Nondimensinal temperature ¼ Local Nusselt number ¼ ð27Þ hy d1 k ð28Þ 5.1. Grid Independence Study The sensitivity of the predicted results with respect to the grid employed has been investigated using meshes with 40 20, 60 30, 80 40, 100 50, and 160 80 cells. Relatively ﬁner mesh has been employed in the y-direction, in view of the steeper velocity gradients in that direction. Here, the fully developed ﬁlm thickness at Re ¼ 5 has been used as the parameter to check the sensitivity of the predicted results on the computational grid. Table 3 shows the fully developed ﬁlm thickness obtained for various meshes mentioned above. It can be concluded that 80 40 is an optimum choice based on order of accuracy of result and, hence, it has been used for all simulations presented in the current work. 5.2. Hydrodynamic Development of Falling Film The hydrodynamic development of a non-evaporating falling ﬁlm has been analyzed ﬁrst. The results obtained from these analyses have been compared with the various experimental and analytical results available in the literature, which further validates the capability of the current numerical model. The variation of ﬁlm Table 2. Thermophysical properties (selected) of ﬂuids Property Liquid Vapor Density (kg=m3) Dynamic viscosity (Ns=m2) Thermal conductivity (W=mk) Speciﬁc heat (J=kgK) Surface tension (N=m) 800 0.16 1.0 2,000 1.2 5 105 0.1 1,000 0.072 EVAPORATION OF LAMINAR FALLING FILM FLOW 57 Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 Table 3. Sensitivity of results to grid size Sl. no. Grid size Developed ﬁlm thickness (mm) 1 2 3 4 5 40 20 60 30 80 40 100 50 160 80 3.3158 3.1295 3.0213 3.0132 3.0093 thickness (b) with nondimensional distance from the slit has been shown in Figure 7a. The result obtained for b0 ¼ 1.80 using the current model shows a close agreement with the experimental data of Fullford [27]. Figure 7b compares the variation of developed ﬁlm thickness with Re obtained from the present work with that of Figure 7. Validations. (a) Comparison of developing ﬁlm thickness (with experimental result), and (b) developed ﬁlm thickness variation with Reynolds number. 58 R. S. MAURYA ET AL. Nusselt’s theory. The difference in the ﬁlm thickness values predicted from these two approaches increases with Re due to the increasing effect of interfacial shear, which has not been accounted in Nusselt’s theory. Detailed investigations of falling ﬁlm hydrodynamics in the absence of evaporation for various ranges of ﬂow parameters can be found in the work of Maurya et al. [6]. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 5.3. Thermal Development of Falling Film 5.3.1. Temporal evolution. Upon supply of heat to the inclined surface, there is a simultaneous development of the thermal and hydrodynamic boundary layers. The growth of the thermal boundary layer can be quantiﬁed through the normal temperature gradients at the wall. The transient evolution of wall temperature gradient is presented in Figure 8 at a location of y ¼ 0.024 m from the ﬂow inlet. Due to the assumptions of pure conduction and nonaccelerating (negligible inertia) ﬁlm, Nusselt’s theory predicts a steady value of this temperature gradient at the given Figure 8. Variation of wall temperature gradient with time. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 EVAPORATION OF LAMINAR FALLING FILM FLOW 59 wall location. However, in the case of an actual developing ﬁlm such effects are non-trivial and they manifest in the form of a transient decay of temperature gradient which asymptotically approaches the value predicted by Nusselt’s theory. In any case, the temperature gradients at the wall (and, hence, the heat transfer coefﬁcient) are underestimated by Nusselt’s theory which ignores the inertial effects. Similar features can also be observed from the transient evolution of temperature proﬁles across the falling ﬁlm shown in Figure 9a. The transient growth of the parabolic temperature proﬁles is in stark contrast to the linear temperature proﬁle considered in Nusselt’s theory. Here, the interfacial temperature gradient (at x ¼ 0.003 m) is initially zero and assumes signiﬁcance only after the thermal boundary layer transiently reaches up to the interface, thereby leading to the interfacial evaporation process. The variations of wall temperature for various supplied heat ﬂuxes are shown in Figure 9b. The ﬁgure shows that all the curves transiently tend to attain an asymptotic state. Figure 9. (a) Transient evolution of temperature proﬁle at y ¼ 0.024 m, and (b) transient evolution of wall temperature at mid plane y ¼ 0.024 m. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 60 R. S. MAURYA ET AL. 5.3.2. Spatial evolution. Figure 10 shows the spatial evolution of the nondimensional temperature proﬁle across the ﬁlm thickness at steady state. The proﬁles clearly illustrate that the thermal boundary layer within the liquid ﬁlm attains a fully developed state at a distance of Le ¼ d=(d1Re) 1. The parabolic proﬁle of temperature in the fully developed state clearly suggests the importance of the convective (inertial) effects in the liquid ﬁlm which were neglected in Nusselt’s approach. On the whole, the liquid ﬁlm can be ascribed to be thermally developing in the spatial range of 0 Le 1. The variation of nondimensional wall temperature along the ﬂow direction at steady state is depicted in Figure 11 for various wall heat ﬂux conditions. Once again, the temperature value slowly increases along the ﬂow direction and attains an asymptotic condition. This asymptotic condition represents a state of thermal energy balance wherein most of the heat supplied at the wall is transferred to interface for the phase change process. The spatial decay of the Nusselt number along the ﬂow direction is shown in Figure 12. The Nusselt number value at the fully developed condition slightly deviates from the corresponding analytical result of Yih and Lee [5]. This small deviation can be attributed to convective effects which have been neglected in the analytical work. 5.4. Film Evaporation in Developing Region The evaporative characteristics of a laminar ﬁlm in the developing region can be partially understood by examining the temperature proﬁles of Figure 10. In order to further quantify the ﬁlm evaporation process, the variation of dimensionless evaporation rate (evaporating mass ﬂux=ﬂow rate at inlet) along the plate length is depicted in Figure 13. It is obvious from this ﬁgure that the evaporation process is Figure 10. Spatial variation of temperature across ﬁlm thickness in developing region at wall heat ﬂux ¼ 5 W=m2. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 EVAPORATION OF LAMINAR FALLING FILM FLOW Figure 11. Wall temperature along plate length. Figure 12. Nusselt number along plate length. Figure 13. Dimensionless evaporation rate in the developing region of ﬁlm. 61 62 R. S. MAURYA ET AL. nonexistent in the initial regions of the ﬁlm where the thermal boundary layer has not grown to the full extent of the local ﬁlm thickness. After the temperature gradient reaches up to the interface, the evaporation process slowly picks up and attains an asymptotic developed condition. As expected, the evaporation rate ampliﬁes with an increase in the applied heat ﬂux and the inception of evaporation process also occurs at a distance closer to the inlet. For the conditions considered in the current work, the evaporative heat ﬂux (meL) is just a small fraction (6–8%) of the total heat supplied at the solid wall and the remaining part is convected away with the falling ﬁlm. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 5.5. Effect of Evaporation on Film Thickness Figure 14a shows the effect of wall heat ﬂux on the thickness of liquid ﬁlm in the developing region. As evident from the ﬁgure, the ﬁlm thickness decays exponentially to a constant value mainly due to its acceleration along the sloping surface. Figure 14. Effect of evaporation on ﬁlm thickness. EVAPORATION OF LAMINAR FALLING FILM FLOW 63 Such decay=growth behavior of the ﬁlm is mainly governed by its inlet velocity and the angle of surface inclination, and the effect of heat ﬂux is relatively minor. The magniﬁed view shown in Figure 14b illustrates that the case with higher ﬂux undergoes larger reduction in ﬁlm thickness due to a higher evaporation rate. Downloaded by [Eastern Michigan University] at 01:06 21 July 2013 6. CONCLUSION In the present work, the numerical study of an evaporating falling liquid ﬁlm ﬂow subjected to constant wall heat ﬂux has been carried out, including the effects of ﬂow development from the inlet and surface tension force at the interface. A new methodology based on the ghost ﬂuid method has been developed on a Cartesian grid to accurately evaluate the discontinuous normal temperature gradients on either side of the interface even in the case of dynamic ﬂow evolution. Using these gradients, a mass source term which emulates the phase change process in the interface is evaluated. The present numerical model has been validated with the analytical results of various benchmark test cases available in the literature. Using the current numerical model, the characteristics of an evaporating falling ﬁlm in the developing region have been clearly brought out. The thermal boundary layer within the evolving liquid ﬁlm undergoes both transient and spatial growth before a steady, developed temperature proﬁle is established. Interestingly, the inertial effects signiﬁcantly inﬂuence the growth rate of the thermal boundary layer, particularly over the dimensionless distance of 0 Le 1. Surface evaporation does not commence until the growing thermal boundary layer reaches the free surface of the ﬁlm. Once the effect of wall heating penetrates up to the interface, the evaporation process slowly picks up and attains an asymptotic steady value in the fully developed region. With an increase in the wall heat ﬂux, the steady state evaporation rate increases and the inception point of evaporation also shifts towards the ﬂow inlet. 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