HEAT TRANSFER ENHANCEMENT OF BACKWARD-FACING STEP FLOW BY USING NANO-ENCAPSULATED PHASE CHANGE MATERIAL SLURRY

Numerical Heat Transfer, Part A, 67: 381 400, 2015 Copyright # Taylor & Francis Group, LLC ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/10407782.2014.937229 1. INTRODUCTION HEAT TRANSFER ENHANCEMENT OF BACKWARD-FACING STEP FLOW BY USING NANO-ENCAPSULATED PHASE CHANGE MATERIAL SLURRY Hanwen Lu, Hamid Reza Seyf, Yuwen Zhang, and H. B. Ma Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, Missouri, USA Laminar forced convection of nano-encapsulated phase change material (NEPCM) slurry over a 2D horizontal backward-facing step is numerically investigated using a finite volume method based on a collocated grid. The slurry consists of water as base fluid and n-octadecane NEPCM particles with an average diameter of 100 nm. Uniform heat flux boundary condition is imposed to the downstream wall while the step and upstream walls are subjected to adiabatic boundary condition. The effects of Reynolds number ranging from 20 to 80, volume fractions of nanoparticles ranging from 0% to 30%, as well as heat flux ranging from 500 to 2,500 W/m 2 are studied. In order to understand the physics of flow and heat transfer of slurry over the backward-facing step, the streamlines and isotherms of the flow were studied. An enhancement in heat transfer coefficient up to 67% using slurry as working fluid compared with pure water can be observed. However, because of the higher viscosity of mixture compared with pure water, the slurry can cause a higher pressure drop in the system. Furthermore, as wall heat flux and Reynolds number increase, the heat transfer coefficient of the bottom wall increases until a critical heat flux is reached and heat transfer performance becomes independent of heat flux. Flow separation and subsequent reattachment due to sudden expansion in flow passages such as backward-facing steps have received considerable attention from researchers due to their importance in many applications such as electronic thermal management, energy=power systems, and chemical reaction system control. In some cases, separation of flow has a harmful effect such as uneven heat loading in thermal equipment, while in other cases separation of flow leads to enhanced heat and mass transfer or mixing. Therefore, it is important to understand the mechanics of heat and mass transfer in separating and reattaching flows. Over the past years, a considerable amount of effort has been made to study various aspects of fluid flow and heat transfer in separating flows [1]. For instance, Armaly et al. [2] Received 18 December 2013; accepted 3 May 2014. Address correspondence to Yuwen Zhang, Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA. E-mail: zhangyu@missouri.edu Color versions of one or more of the figures in the article can be found online at www.tandfonline. com/unht. 381

382 H. LU ET AL. NOMENCLATURE C volumetric concentration c p specific heat (J=kg K) D h hydraulic diameter of inlet channel (m) d p diameter (m) e shear rate H height of inlet channel (m) k thermal conductivity (W=m K) k b static thermal conductivity (W=m K) Pe p particle Péclet number q heat flux on the bottom wall (W=m 2 ) Re Reynolds number (2quH=m) T temperature (K) T in inlet temperature (K) u X component of velocity vector (m=s) U in inlet velocity (m=s) v Y component of velocity vector (m=s) ~v velocity vector (m=s) a thermal diffusivity (m 2 =s) u mass concentration m dynamic viscosity (Pa s) q density (kg=m 3 ) Subscripts eff effective liquidus liquid state of phase change materials (PCMs) in nanoparticles p particle solidus solid state of PCM in nanoparticles w water experimentally and numerically investigated the relationship between reattachment length and Reynolds number (Re) and presented the results for laminar, transitional, and turbulent air flow. They observed that for Re 1,200, wake length increased when Reynolds number was increased; however, for the case when 200 Re 5,500, this showed the opposite trend. Kondoh et al. [3] numerically studied fluid flow and heat transfer in a backward-facing step and reported the local Nusselt number and its peak value, location, and its thermal front. Their results showed that peak Nusselt number is not necessarily located near the reattachment location. Tylli et al. [4] used particle image velocimetry measurements along with three-dimensional numerical simulations to study three-dimensional effects introduced by the presence of sidewalls. Their results showed that in instances where Reynolds numbers was less than 400, the sidewall did not affect the structure of the two-dimensional flow at the channel midplane. They also studied the effects of sidewalls on the primary recirculation zone and upper secondary recirculation zone, showing the discrepancies between two-dimensional numerical simulation and experimental results. Abu-Mulaweh [5], using a cold wire anemometer two-component and LDV, measured heat transfer and flow over a vertical forward-facing step for turbulent mixed convection. Their results showed that with increasing step height, the transverse velocity fluctuation and temperature fluctuations downstream of the forward-facing step increased. Terhaar et al. [6] experimentally studied laminar unsteady heat transfer over a backward-facing step for pulsating flow at Reynolds number 300 and observed enhancement of Nusselt number up to a certain Strouhal number followed by degradation when the pulsation frequency increased. Kumar and Dhiman [7] numerically investigated the effect of insertion of an adiabatic circular cylinder on heat transfer enhancement in laminar forced convection flow over a backward-facing step. They considered different cross-stream positions of the circular cylinder for 1 Re 200 and reported 155% heat transfer enhancement compared with the no-cylinder case. One innovative method of enhancing heat transfer capability of working fluids is to add phase change materials (PCMs) to base fluids. In this method, the PCM is

HEAT TRANSFER ENHANCEMENT OF BACKWARD-FACING STEP FLOW 383 either nano- or micro-encapsulated and suspended in a heat transfer fluid to form a phase change suspension. The phase change process of PCM particles in the base fluid drastically increases the thermal energy storage capability of slurry and thus enhances the ability of working fluid to absorb high heat fluxes [9 13, 16 19]. Due to the capability of high convective heat transfer and thermal storage performance of slurry, utilizing nano-encapsulated phase change material (NEPCM) slurry as heat transfer fluid is gaining interest. In general, NEPCM particles are composed of a core of paraffin wax phase change material coated with a cross-linked polymer wall. The wall material is usually 14 20% of the total capsule mass and is sufficiently flexible to accommodate volume changes that accompany solid liquid phase change [8]. The latent heat of the NEPCM particles can be used to store energy and enhance heat transfer as the particles experience phase change. Therefore, by utilizing high latent heat and effective thermal conductivity of the mixture due to micro-convection induced by PCM nanoparticles, better heat transfer performance with minimal temperature variation can be achieved. Many experimental and numerical studies have shown that PCM slurry improved the thermal performance of the carrier fluid and consequently enhanced heat transfer [8 18]. Sabbah et al. [8] used a three-dimensional numerical model to investigate the performance of water-based slurry in microchannel heat sinks. Their results showed that heat transfer enhancement depended on the melting temperature range of the PCM, and the channel inlet and outlet temperatures. Goel et al. [9] conducted an experimental study using a suspension of n-eicosane microcapsules in water under laminar, hydrodynamically fully developed flow condition in a circular tube with a constant heat flux boundary condition. Their results showed that water with PCM suspension can reduce the increase in wall temperature by up to 50% as compared with a single-phase fluid with the same nondimensional parameters. Goel et al. [10] experimentally examined laminar forced convection heat transfer in a circular tube with constant heat flux boundary condition using PCM slurry as working fluid. They reported significant reduction in wall temperature when PCM slurry, instead of water, was used. Rao et al. [11] experimentally studied the heat transfer characteristic of micro-encapsulated phase change material (MEPCM) in microchannels and reported high cooling performance even at lower flow rates. Inaba et al. [12] studied the turbulent and laminar heat transfer characteristics of MEPCM slurry with 20% mass fraction and different particle sizes flowing in a circular tube with constant wall heat flux; they reported 2 2.8-fold higher heat transfer coefficient with the MEPCM suspension compared with pure water. They also observed a lower friction factor for slurry in the turbulent flow region. Wang et al. [13] conducted an experimental study to investigate heat transfer enhancement of MEPCM in a horizontal tube and observed higher heat transfer coefficient of slurry flow compared with water for all experiments in the laminar regime. Kuravi et al. [14] numerically analyzed the thermal performance of NEPCM slurry in manifold microchannel heat sinks and found that the system showed better performance with MEPCM slurry than one with pure fluid even in thermally developing flows. Sabbah et al. [15] numerically investigated the effect of MEPCM on natural convection heat transfer inside a rectangular cavity. The results showed significant enhancement in heat transfer (up to 80%) when using MEPCM in the system. Wu et al. [16] studied the heat transfer enhancement of a microchannel heat exchanger using two types of

384 H. LU ET AL. slurry with bare and silica-encapsulated indium nano-pcms in poly-a-olefin (PAO). They reported that both NEPCMs provided nearly the same heat transfer performance and was better than pure PAO. Chen et al. [17] experimentally studied the effect of a mixture of water as carrier fluid and bromohexadecane as PCM inside a circular tube under constant heat flux. They reported that PCM slurry reduced wall temperature by 30% and enhanced heat transfer by 40% compared with pure water. Their results also showed 67% reduction in pressure drop for the PCM slurry compared with pure water. Recently, Seyf et al. [18] numerically studied the effect of NEPCM slurry on thermal and hydrodynamics performance of a tangential microchannel heat sink. They used PAO as base fluid and octadecane as nanoparticles and observed higher heat transfer, better temperature uniformity, and lower thermal resistance for NEPCM slurry coolant compared with pure PAO. Using high-performance working fluids such as nanofluids [19] and NEPCM slurry is among the methods used to alter flow structure and temperature distribution and consequently enhance heat transfer of the backward-facing step. To the best of the authors knowledge, there is no research on NEPCM slurry flow and heat transfer on the backward-facing step in the literature, and this has motivated the present study. This study deals with two-dimensional laminar force convection of NEPCM slurry over a backward-facing step and investigates the effect of wall heat flux, Reynolds number, and volume fraction of nanoparticles on flow and heat transfer behaviors of slurry in the system. The results of interest, such as local and average heat transfer coefficient on the bottom wall, pressure drop in the system, as well as temperature contours and streamlines, are reported to illustrate the effects of NEPCM slurry on these parameters. We used water as the carrier fluid and n-octadecane as the PCM particles, with volume concentrations ranging from 0% to 30%. The thermophysical properties of carrier fluid are assumed to be temperature dependent while those for NEPCM are assumed to be constants. 2. MATHEMATICAL MODEL The physical flow system for the backward-facing step used in this study is presented in Figure 1. The expansion ratio and step height of the backward-facing step are 2 and H, respectively; the inlet temperature and velocity are uniform. The downstream distance between the edge of the step and the exit plane of the computational domain is 20H while the upstream wall length is 6H. The step wall and the wall upstream were considered to be adiabatic. Diffusion fluxes of all quantities at the duct exit were set at zero. The top wall is adiabatic and constant heat flux is prescribed at the bottom wall. Nano-encapsulated PCM enters the system and its temperature increases as it moves over the hot surface and reaches the melting temperature of the PCM. There is no mass transfer between the carrier fluid and capsules because the melted PCM will not mix with the carrier fluid and it remains in the capsules. The carrier fluid shows a lower temperature change during phase change inside the particles. It is assumed that: 1. Coolant flow is incompressible and laminar, and the radiation is negligible. 2. Particle concentration is less than 0.3; therefore, the fluid is Newtonian [20].

HEAT TRANSFER ENHANCEMENT OF BACKWARD-FACING STEP FLOW 385 Figure 1. Schematic of flow over backward-facing step and computational domain. 3. The micro-convection caused by particle wall, particle fluid, and particle particle interactions is lumped together, and its effect is accounted by an effective thermal conductivity. 4. Particle distribution is homogeneous and densities, viscosity, and slurry heat capacity are functions of temperature, and thermal conductivity is a function of space and temperature [14]. 5. Encapsulated particles are spherical and their melting range is between T 1 and T 2 [14, 18]. 6. The particle melts instantaneously and there is no temperature gradient within [14]. 7. The difference in particle and fluid velocities is negligible (i.e., particles follow the fluid with no lag) [14]. 8. The effect of shell material and the particle depletion layer is negligible. The particle depletion layer is of the order of the particle radius if the size ratio of channel to particle is large [21, 22]. The two-dimensional incompressible Navier Stokes and energy equations in the Cartesian coordinate system with temperature-dependent thermophysical properties can be written as follows: rv ¼ 0 rðq eff VVÞ ¼ rp þrðm eff rvþ ð2þ r q eff Vc p;eff T ¼r ð keff rtþ ð3þ ð1þ where p, T, and V are pressure, temperature, and velocity vector, respectively. q eff, k eff, m eff, and c p, eff are density, thermal conductivity, dynamic viscosity, and heat capacitance of the NEPCM slurry, respectively. The Reynolds number is defined as: Re ¼ q effu in D h m eff ð4þ

386 H. LU ET AL. where U in and D h are inlet velocity and hydraulic diameter, respectively, of the inlet channel which is twice channel height. The slurry consists of water and n-octadecane phase change particles with diameter, melting point, density, specific heat, thermal conductivity, and latent heat of 100 nm, 296.15 K, 815 kg=m 3, 2,000 J=kg K, 0.18 W=m K, and 24, 400 J=K, respectively [20]. The density of the bulk fluid is: q eff ¼ Cq p þ ð1 CÞq w ð5þ The dynamic viscosity of slurry is calculated using the following equation [23, 24]: m eff ¼ m w 1 C 1:16C 2 2:5 ð6þ where C is the volume concentration of NEPCM and m w is the water viscosity, which is a function of temperature. The static thermal conductivity of the suspension or thermal conductivity of slurry at rest (no shear rate) can be expressed as [25]: 2 þ kp k w þ 2 C k p k w 1 k b ¼ k w ð7þ 2 þ k p k w C k p k w 1 Due to particle particle, particle liquid, and particle wall interactions, the effective thermal conductivity of slurry under motion increases. These interactions play a major role in enhancement of the thermal conductivity of the mixture. The effective thermal conductivity of slurry flow is specified by the following correlation [14]: k eff ¼ k w 1 þ BCPe m p B ¼ 3; m ¼ 1:5; Pe p < 0:67 B ¼ 1:8; m ¼ 0:18; 0:67 < Pe p < 250 B ¼ 3; m ¼ 1 11 ; Pe p > 250 The particle Péclet number is defined as: Pe p ¼ ed2 p a w where a w is the thermal diffusivity of water. The shear rate is a function of all the spatial coordinates and corresponding velocity components. The magnitude of the shear rate can be calculated using the following equation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " 1 e ¼ 2 2 qu qy þ qv 2 þ 2 qu 2 þ 2 qv # u 2 t ð10þ qx qx qy ð8þ ð9þ

HEAT TRANSFER ENHANCEMENT OF BACKWARD-FACING STEP FLOW 387 where e is the shear rate and u and v are the velocity components. The effective thermal conductivity correlation above shows that thermal conductivity largely depends on (i) shear rate, which can be increased by decreasing the conduction dimension or increasing slurry flow rate, and (ii) particle size due to interactions (drag force, lift force, and virtual mass) between the liquid and the particles. The particle s phase change is modeled by varying the specific heat of particles across the liquidus and solidus temperatures. For temperatures higher or lower than melting range, the value of the specific heat of particles is equal to C p,pcm and for temperatures in the melting range, the specific heat of particles is calculated using a sine profile. Therefore, the slurry specific heat capacity of bulk fluid can be calculated using the following equations: For T p < T solidus : For T solidus < T p < T liquidus : C p;eff ¼ ð1 / ÞC p;w þ / C p;pcm þ p 2 For T p > T liquidus : where C p;eff ¼ ð1 / ÞC p;w þ / C p;p ð11þ L C p;pcm T Mr ð sin p T T 1Þ T Mr ð12þ C p;eff ¼ ð1 / ÞC p;w þ / C p;p ð13þ C q p / ¼ ð14þ q water þ C q p q w where C p,water, q water, u, q p, and T Mr are the specific heat and density of water, mass concentration or loading fraction of slurry, density of NEPCM, and melting range of PCM particles (T Mr ¼ T 2 T 1 ), respectively. According to Alisetti and Roy [27], the difference between using various profiles for calculating the specific heat of PCM is less than 4%. Therefore, in this study we used the sine profile to represent NEPCM particle-specific heat. 3. NUMERICAL SOLUTION AND VALIDATION In the present study, a validated code was employed as the numerical solver [27 30]. The two-dimensional governing equations are discretized using the finite volume method with the SIMPLE algorithm [31] to deal with the linkage between velocity and pressure. A collocated grid was used to discretize the computational domain and hence the pressure, temperature, velocity components, and all properties are stored on the main grid, which is at the center of the control volume. The convection diffusion terms in the momentum and energy equations are discretized by a QUICK scheme. The resulting block systems of equations for all dependent variables are solved using a point implicit linear equation in conjunction with an algebraic multigrid (AMG) method. The numerical solution is considered to be converged

388 H. LU ET AL. Table 1. Comparison to data in the literature of reattachment length divided by step X R =S Relative error (%) Present numerical results 5.03 Lin et al. [34] 4.91 2.64 El-Refaee et al. [35] 4.77 5.66 Dyne et al. [36] 4.89 3.06 Acharya et al. [37] 4.98 1.20 Cochran et al. [38] 5.31 5.08 at an iteration in which the summation of absolute values of relative errors for velocity components and temperature as well as continuity equation reduces by four to five orders of magnitude. The overall energy and mass balance are also checked as a second criterion for convergence. The computational domain is discretized into small rectangular elements. A fine, nonuniform quadrilateral grid is used in the regions near the step and point of reattachment as well as in the region of boundary layers where the gradients of velocity and temperature are steeper, while a coarser grid is used downstream of the channel. Furthermore, due to the relatively high velocity and temperature gradients on the surface-limiting height of the channel, a very fine grid is proposed to accurately capture flow and heat transfer near this region. The calculations were carried out for a grid with a total number of 86,259 cells. Finer grid sizes (126,987 cells) were also used in the calculations, but the difference in heat transfer coefficient and shear stress over the bottom wall for the present grid size is less than 0.8% and 0.3%, respectively. Therefore, the grid with 86,259 cells is fine enough to capture the flow and heat transfer behavior over the backward-facing step. The grid independency study was conducted for the most critical case (i.e., highest Reynolds number). The validation of results consists of two parts: validation of codes for fluid flow and heat transfer in the backward-facing step and validation of the slurry model. Regarding the first part, Table 1 shows the comparison to results in the literature Figure 2. Comparison of results among the current model, the numerical model from [14], and the experimental data from [10].

HEAT TRANSFER ENHANCEMENT OF BACKWARD-FACING STEP FLOW 389 between reattachment length divided by step height for an expansion ratio of 2 and Reynolds number of 100 [27 29, 32, 33] for the case of water as working fluid. As shown, there is good agreement between the results of our numerical simulations and previous studies in the literature. For the second part, since there are no experimental data available regarding NEPCM slurry flow through the backward-facing step, validation of the code was performed with respect to the experimental results presented by Goel et al. [10] as well as the numerical simulation results presented by Kurvai et al. [14]. It is important to note that the formulations adopted in the present study have been widely used in previous published works and showed good agreement with both nano- and micro-encapsulated PCM simulation and experimental data [14, 18, 26, 32, 33]. Figure 2 shows validation of the present model with Kuvrai s model and Goel s experimental data. The results show good agreement. 4. RESULTS AND DISCUSSION Numerical simulations are carried out to study the effect of Reynolds number from 20 to 80, NEPCM slurry, heat flux range of 500 2,500 W=m 2, and volume fraction of nanoparticles ranging from 0% to 30% on the hydrodynamic and heat transfer characteristics of flow over a backward-facing step. The inlet temperature is maintained at the melting point of NEPCM (i.e., 296.15 K for n-octadecane) while a constant heat flux is applied on the bottom wall, downstream of the step. It is known that flow over the backward-facing step channel is very sensitive to abrupt change in channel geometry (i.e., sudden expansion at the step). The uniform flow at the entrance of the channel becomes a fully developed parabolic upstream and then separates and forms a recirculation region. Subsequently, the velocity profile reattaches and becomes a fully developed flow at the channel exit. NEPCM slurry affects the thermal and hydrodynamic behaviors of the working fluid over the step and consequently the pressure drop and heat transfer coefficient. Figures 3 6 show temperature contour and streamlines for Re ¼ 20 and Re ¼ 80, for water and NEPCM slurry with C ¼ 0.3. For all Reynolds numbers investigated in the present study, the working fluid uniformly flows in the inlet channel until it reaches the separation point, where it separates and forms a recirculation cell. With a constant inlet volumetric flow rate, the fluid flow in the system becomes slower due to the higher viscosity of slurry compared with pure water; therefore, the shape of the recirculation zone changes with different working fluids. However, in the present study, Reynolds numbers are kept constant in order to study the effect of volume fractions and wall heat flux. Therefore, as shown in the following figures, the inlet velocities are different for different volume fractions due to different effective density and effective viscosity under the same Re, higher volume fraction slurry has higher inlet velocities and consequently higher velocity gradient and velocity magnitude in the computational domain, as shown in Figures 3 and 4, respectively. However, NEPCM particles increase the viscosity of mixture, which has a significant effect on the size of the recirculation zone. For the conditions studied in this article, the increase in velocity negates the effect of increasing viscosity and results in insignificant size change in the recirculation zone compared with pure water. But generally when Reynolds number increases, the size of the recirculation zone is greater and the flow is reattached further downstream from the step.

390 H. LU ET AL. Figure 3. Streamlines and velocity contours of water and slurry at Re ¼ 20. In the case with slurry as working fluid, it will be observed that the isotherms are more condensed both near the step and the bottom surface, which indicates better overall heat transfer at the bottom wall. Also, the higher volume fraction of nanoparticles leads to reduced thickness of the thermal boundary layer near the surface, indicating the steep temperature gradient and thus the higher total heat transfer coefficient. This is because during the phase change process, the latent heat of NEPCM particles functions as a heat sink and consequently reduces the thickness of the thermal boundary layer. It will also be seen that for all cases with increasing Reynolds number the wake length increases and the thermal boundary layer thickness on the bottom wall decreases, so heat transfer increases. Furthermore, with increasing Reynolds number, the average bulk temperature of working fluid decreases while the heat transfer rate between fluid and heated surface increases. The opposite trends can be explained as follows. Convection heat transfer occurring in the fluid is composed of two mechanisms: energy transfer due to the bulk motion of the fluid and diffusion throughout the fluid. At low Reynolds number, the fluid mean velocity is low and the fluid has more time to absorb and spread heat; therefore, diffusive heat transfer is the dominant factor, resulting in a higher working fluid bulk temperature. On the other hand, as Reynolds number increases, mean fluid velocity increases and forced convection contributes more to heat transfer, thus transferring more heat without significant increase in temperature. Figure 7 shows the heat transfer coefficient on the bottom wall, downstream of the step for Re ¼ 20 and 80 at different volume fractions of NEPCM slurry for

HEAT TRANSFER ENHANCEMENT OF BACKWARD-FACING STEP FLOW 391 Figure 4. Streamlines and velocity contours of water and slurry at Re ¼ 80. q ¼ 500 and 2,500 W=m 2. In regard to the trend of each individual curve, it will be seen that local heat transfer coefficient is low at the corner of the bottom wall and it gradually increases until reaching maximum where it coincides with the flow reattachment point and where significant compression of the thermal boundary layer Figure 5. Temperature contours of water and slurry at Re ¼ 20.

392 H. LU ET AL. Figure 6. Temperature contours of water and slurry at Re ¼ 80. Figure 7. Local heat transfer coefficients along the bottom wall for Re ¼ 20 and 80 and q ¼ 500 and 2,500 W=m 2 for water and slurry.

HEAT TRANSFER ENHANCEMENT OF BACKWARD-FACING STEP FLOW 393 occurs. The local heat transfer coefficient decreases along the bottom wall after flow reattachment. In cases with the same heat flux, the increase in Reynolds number resulted in significant heat transfer coefficient enhancement. Higher Reynolds numbers not only increase convective heat transfer due to higher slurry inertia, but also introduce additional absorption with lower temperature rise for two cases with the same heat flux. An additional observation is that the location showing the highest value of heat transfer coefficient shifted to the exit along with the reattachment point of the flow as Reynolds number increased. It is also concluded that for all cases, an increase in volume fraction of NEPCM results in an increase in the heat transfer coefficient on the bottom wall. As previously established, the addition of NEPCM particles substantially enhances the heat capacity of the slurry because of high latent heat of the NEPCM. It is worth noting that based on the definition of Reynolds number, the inlet velocity is adjusted for different cases of NEPCM slurries with different effective density in order to achieve constant Reynolds number throughout simulations. In addition, at different volume fractions of nanoparticles, the values of specific heat, density, and thermal conductivity are adjusted accordingly. In other words, higher volume fractions of nanoparticles resulted in higher effective specific heat and thermal conductivity. As a result, in order to maintain constant Reynolds number, both inlet velocity and velocity gradient increase as volume fraction increases; the temperature gradient in the system also varies due to its dependency on the viscosity and specific heat of slurry. When the volume fraction of particles increases, the viscosity of slurry increases, which consequently leads to a decrease in velocity gradient near the bottom wall. These opposite behaviors should be considered to better understand the heat transfer characteristics of NEPCM. Figures 8 and 9 show the average heat transfer coefficient on the bottom wall, downstream of the step for two different heat fluxes. It is clear that NEPCM particles can significantly enhance heat transfer; higher thermal performance can be seen as the volume fraction of NEPCM increases. In this study, because the inlet temperature of working fluid is maintained at the melting temperature of the PCM particles, PCM particles within the computational domain will undergo phase change as soon as fluid reaches the backward-facing step, and no additional heat is required for Figure 8. Average heat transfer coefficients for water and NEPCM slurry at q ¼ 500 W=m 2.

394 H. LU ET AL. Figure 9. Average heat transfer coefficients for water and NEPCM slurry at q ¼ 2,500 W=m 2. PCM particles to reach melting temperature. Under this initial condition, the slurry flow over the step can be divided into two distinct regions according to the temperature of the slurry. The first region is where the temperature of the carrier fluid is within the melting range of the NEPCM particles. In this region, because NEPCM is undergoing phase change, the effective heat capacity of slurry within this region is equivalent to the summation of the latent heat of the PCM and the sensible heat of the slurry. The second region is where the temperature of the carrier fluid exceeds the maximum melting temperature of the NEPCM. This region is typically very close to the heated wall and PCM is completely melted in this region. The specific heat of n-octadecane is constant and is equivalent to the liquid-specific heat of the NEPCM slurry, and hence minimum heat transfer enhancement can be observed in this region. Figure 10 shows the pressure drop in the system verses Reynolds number for two different heat fluxes. As noted, pressure drop increases as the Reynolds number and volume fraction of nanoparticles increase and an even more drastic increase in pressure drop can be observed at higher Reynolds numbers. It is worth mentioning that pressure drop is related to the change in NEPCM volume fraction. As mentioned above, a higher volume fraction of NEPCM results in higher effective viscosity which, in consequence, directly influences the overall pressure drop of the system. Although the thermophysical properties of working fluid are temperature sensitive, variation in average bulk temperature of the computational domain remains minimal; therefore, no significant difference between pressure drops for different heat fluxes was observed. Figure 11 shows the effect of heat flux on local heat transfer coefficients at Re ¼ 20 for both water and slurry. It is obvious that at higher heat fluxes, the system shows a significantly higher heat transfer coefficient. For example, with 30% NEPCM concentration and q ¼ 1; 000 W=m 2 the system shows 67% higher heat transfer coefficient at the point of reattachment of flow while at q ¼ 2; 500 W=m 2 it showed 79% enhancement; both cases were compared to the lowest heat flux studied (i.e., q ¼ 500 W=m 2 ). The difference between heat transfer coefficients at q ¼ 500 W=m 2 and at higher heat fluxes (i.e., q ¼ 1,000, 1,500, 2,000 W=m 2, etc.)

HEAT TRANSFER ENHANCEMENT OF BACKWARD-FACING STEP FLOW 395 Figure 10. Pressure drop in the system for water and NEPCM slurry. started to increase once flow reached the step and it kept increasing until all PCM nanoparticles were melted completely when heat transfer coefficient curves converged (see Figure 10). Furthermore, it will be seen that in this case there is only a small difference between heat transfer coefficients at q ¼ 1,500 W=m 2 and q ¼ 2,000 W=m 2 as they are very close to each other. In other words, there are optimum heat fluxes and NEPCM volume fractions at different Reynolds numbers when heat transfer is maximized. This is due to the fact that higher heat fluxes increase the temperature of slurry in the system; once slurry temperature exceeds the melting

396 H. LU ET AL. Figure 11. Local heat transfer coefficient for different heat fluxes. range of PCM particles, all particles will be melted in the system. Further increase in heat flux will have a minimal effect on the specific heat capacity of the slurry since no more phase change is taking place. Therefore, after slurry temperature exceeds the maximum melting temperature of NEPCM, no enhancement in heat transfer can be observed, hence optimal heat flux is reached. The average heat transfer coefficient versus heat flux is shown in Figure 12. The average heat transfer coefficient increases as the heat flux and Reynolds number increase. This trend agrees with the results for local heat transfer coefficient in Figure 11. In regard to water, the heat transfer coefficient shows a linear relation with heat flux. For NEPCM slurry at higher Reynolds numbers, heat transfer shows a growing trend with heat flux while for lower Reynolds numbers, the heat transfer coefficient remains constant once heat flux exceeds a specific value. For lower Reynolds numbers, the flow velocity within the system is relatively low and convective heat transfer is relative weak; to transfer a certain amount of heat, more PCM nanoparticles are completely melted to compensate the lack of convective heat transfer. After phase change is complete, nanoparticles continue to absorb thermal energy that results in temperature rise. Based on the effective heat

HEAT TRANSFER ENHANCEMENT OF BACKWARD-FACING STEP FLOW 397 transfer coefficient calculation, higher slurry bulk temperature will result in a lower heat transfer coefficient for constant heat flux, and this is why for lower Reynolds numbers, the heat transfer coefficient curve plateaues at higher heat flux. 5. CONCLUSION Figure 12. Average heat transfer coefficient for different heat fluxes. The effects of NEPCM volume fraction, heat flux, and Reynolds number on flow and heat transfer are numerically studied for two-dimensional laminar NEPCM slurry force convection flow adjacent to a backward-facing step in a rectangular duct. The range of Reynolds numbers based on the inlet channel hydraulic diameter of the channel ranges from 20 to 80 while NEPCM volume fraction ranges from 0% to 30%. It can be concluded that by using NEPCM slurry as the working fluid, the overall heat transfer coefficient is significantly improved by up to 79%. The maximum value of local heat transfer coefficient at the bottom wall of the backward-facing step is also increased by up to 90%. Higher NEPCM volume fraction and Reynolds number result in heat transfer enhancement while pressure drop is also increased. Furthermore, when using NEPCM slurry as the working fluid, local heat transfer coefficient at the heated wall is substantially higher until all NEPCM particles are melted in the system; thereafter, further increase in heat flux has no significant effect on the local heat transfer coefficient.

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