Thermal Entropy Research Articles

Analysis of thermal transport and entropy generation characteristics for electroosmotic flow through a hydrophobic microchannel considering viscoelectric effect

We investigate the thermo-fluidic and entropy generation characteristics for electroosmotic flow through a hydrophobic microchannel with the consideration of viscoelectric effect. A closed form expression for the velocity is obtained from the analytical solution of the momentum and continuity equations together with the Poisson-Boltzmann equation and thereafter the temperature field is computed numerically. The flow velocity, flow rate, average Nusselt number, and average total entropy generation are computed by varying the slip coefficient, viscoelectric coefficient (f), Brinkman number (Br), and thermal Peclet number(Pe). Results reveal that the viscoelectric effect decreases the flow velocity and percentage decrement in flow rate due to the viscoelectric effect is larger for the slip case and reaches up to 40.09%. The value of the average Nusselt number decreases with f at lower value of Br, and the effect is opposite at higher values of Br. Although the heat transfer enhancement is more with the interfacial slip, the augmentation decreases with f and increases with Pe. The value of average total entropy generation decreases with the increase in f, and decrement is substantial at higher values of Br.

Numerical study of entropy generation and forced convection heat transfer of a nanofluid in a channel with different fin cross-sections

PurposeThis study aims to numerically investigate the thermal-hydrodynamic performance of silicon oxide/water nanofluid laminar flow in the heat sink miniature channel with different fin cross-sections. The effect of the fin cross-section including semi-circular, rectangular and quadrant in two directions of flat and curved, and channel substrate materials of steel, aluminum, copper and titanium were examined. Finally, the analysis of thermal and frictional entropy generation in different channels is performed.Design/methodology/approachAccording to the numerical results, the highest heat transfer coefficients belong to the rectangular, quadrant 2, quadrant 1 and semi-circular fins compared to the channel without fin is 38.65%, 29.94%, 27.45% and 17.1%, respectively. Also, the highest performance evaluation criteria belong to the rectangular and quadrant 2 fins, which have 1.35 and 1.29, respectively. Based on the thermal conductivity of the substrate material, the best material is copper. According to the results of entropy analysis, the reduction of thermal irreversibility of the channel with rectangular, quadrant 1, quadrant 2 and semi-circular compared to non-finned channel is equal to 72%, 57%, 63% and 48%, respectively.FindingsThe rectangular and quadrant 2 fins are the best fins and the copper substrate material is the best material to reduce the entropy generation.Originality/valueThe silicon oxide/water nanofluid flow in the heat sink miniature channel with various fin shapes and the curvature angle against the fluid flow was simulated to increase the heat transfer performance. The whole test section is simulated in three-dimensional. Different channel materials have been investigated to find the best channel substrate material.

First and Second Law Thermodynamic Analyses of Hybrid Nanofluid with Different Particle Shapes in a Microplate Heat Exchanger

The improvement in the quantitative and qualitative heat transfer performances of working fluids is trending research in the present time for heat transfer applications. In the present work, the first and second law analyses of a microplate heat exchanger with single-particle and hybrid nanofluids are conducted. The microplate heat exchanger with single-particle and hybrid nanofluids is analyzed using the computational fluid dynamics approach with symmetrical heat transfer and fluid flow analyses. The single-particle Al2O3 nanofluid and the hybrid Al2O3/Cu nanofluid are investigated for different nanoparticles shapes of sphere (Sp), oblate spheroid (OS), prolate spheroid (PS), blade (BL), platelet (PL), cylinder (CY) and brick (BR). The first law characteristics of NTU, effectiveness and performance index and the second characteristics of thermal, friction and total entropy generation rates and Bejan number are compared for Al2O3 and Al2O3/Cu nanofluids with considered different-shaped nanoparticles. The OS- and PL-shaped nanoparticles show superior and worse first and second law characteristics, respectively, for Al2O3 and Al2O3/Cu nanofluids. The hybrid nanofluid presents better first and second law characteristics compared to single-particle nanofluid for all nanoparticle shapes. The Al2O3/Cu nanofluid with OS-shaped nanoparticles depicts maximum values of performance index and Bejan number as 4.07 and 0.913, respectively. The first and second law characteristics of the best combination of the Al2O3/Cu nanofluid with OS-shaped nanoparticles are investigated for various volume fractions, different temperature and mass flow rate conditions of hot and cold fluids. The first and second law characteristics are optimum at higher hot fluid temperature, lower cold fluid temperature, lower hot and cold fluid mass flow rates. In addition, the first and second law characteristics have improved with increase in volume fraction.

The effect of inlet/outlet number and arrangement on hydrothermal behavior and entropy generation of the laminar water flow in a pin-fin heat sink

A numerical analysis was performed to evaluate the hydrothermal performance of a pin-fin mini-channel heatsink (MCHS) with six different inlet/outlet flow arrangements from the first and second laws of thermodynamics point of views. The results are presented in terms of several important parameters such as convective heat transfer coefficient (h), CPU mean temperature (Tm, CPU), pumping power (Ẇp), frictional and thermal entropy generation rates (Ṡfr and Ṡth) as well as Performance Evaluation Criterion (PEC). The results demonstrated that configurations with two inlets and outlets have the lowest Ẇp, Ṡfr, h, and highest Tm, CPU among the studied cases. The vorticity intensification is high at the inlet/outlet headers and over the pin-fin bodies for the case with one inlet from the left and two outlets located at opposite sides, namely configuration d. The highest values of Ṡfr and h, the lowest value of Tm, CPU, and a moderate value of Ẇp are obtained for configuration d as compared to the other studied cases. In consequence, the maximum PECs (1.61–2.20) is obtained for case d at different Re numbers (500–2000). The PEC of configuration d is nearly 19.77% higher than those of configurations with two inlets and two outlets. Also, the PEC of configuration d is nearly 66.95% higher than those obtained for two other configurations with one inlet and two outlets.

Irreversibilities in natural convection inside a right-angled trapezoidal cavity with sinusoidal wall temperature

Analysis on natural convective heat transfer in different engineering systems allows optimization of the technical apparatus. For this purpose, numerical simulation of the fluid flow and heat transport within the system is combined with study of entropy generation. The latter is very important considering the Gouy–Stodola theorem of thermodynamics. The present research deals with the mathematical modeling of thermal convection and entropy generation in a right-angled trapezoidal cavity under the influence of sinusoidal vertical wall temperature distribution. Control Oberbeck–Galerkin finite element technique has solved Boussinesq equations formulated using the non-dimensional primitive variables. Analyses of flow structures, thermal and entropy generation patterns for different values of the Rayleigh number, and parameters of non-uniform wall temperature were performed. It was found that a rise in the sinusoidal wall temperature amplitude increases the average Nusselt and Bejan numbers and average entropy generation. Moreover, growth in the non-uniform wall temperature wave number decreases the energy transport strength and Bejan number.

Significant inverse magnetocaloric effect induced by quantum criticality

The criticality-enhanced magnetocaloric effect (MCE) near a field-induced quantum critical point (QCP) in the spin systems constitutes a very promising and highly tunable alternative to conventional adiabatic demagnetization refrigeration. Strong fluctuations in the low-$T$ quantum critical regime can give rise to a large thermal entropy change and thus significant cooling effect when approaching the QCP. In this work, through efficient and accurate many-body calculations, we show there exists a significant inverse MCE (iMCE) in the spin-1 quantum chain materials (${\mathrm{CH}}_{3}{)}_{4}\mathrm{NNi}{({\mathrm{NO}}_{2})}_{3}$ (TMNIN) and ${\mathrm{NiCl}}_{2}\text{\ensuremath{-}}4\mathrm{SC}{({\mathrm{NH}}_{2})}_{2}$ (DTN), where DTN has substantial low-$T$ refrigeration capacity while requiring only moderate magnetic fields. The iMCE characteristics, including the adiabatic temperature change $\mathrm{\ensuremath{\Delta}}{T}_{\mathrm{ad}}$, isothermal entropy change $\mathrm{\ensuremath{\Delta}}S$, differential Gr\"uneisen parameter, and the entropy change rate, are obtained with quantum many-body calculations at finite temperature. The cooling performance, i.e., the efficiency factor and hold time, of the two compounds is also discussed. Based on the many-body calculations on realistic models for the spin-chain materials, we conclude that the compound DTN constitutes a very promising and highly efficient quantum magnetic coolant with pronounced iMCE properties. We advocate that such quantum magnets can be used in cryofree refrigeration for space applications and quantum computing environments.

Experimental investigation of heat transfer of Al2O3 nanofluid in a microchannel heat sink

Microchannel heat sink (MCHS) with nanofluids has been widely utilized to improve heat transfer performance in micro-scale mechanical systems. Although various studies have demonstrated the heat transfer phenomena of Al2O3 nanofluid in MCHS, the flow and heat transfer characteristics have not been fully elucidated due to a insufficiency of experimental results. To investigate these heat transfer phenomena, measurements of the temperature and velocity fields in the MCHS are important because the flow characteristics influence the heat transfer performance. Therefore, in the present study, velocity and temperature fields of Al2O3 nanofluid and deionized(DI) water in a MCHS were systematically analyzed by employing the particle image velocimetry(PIV) and laser induced fluorescence(LIF) measurement techniques. The heat transfer characteristics were explained using flow behaviors and temperature distribution in the MCHS. Moreover, the entropy generation rate in the MCHS was also evaluated to demonstrate heat transfer performance. When the nanofluid was used as a working fluid in the MCHS, the generation rate of thermal entropy was 6.3% smaller than that of water. In conclusion, we experimentally demonstrated the heat transfer of nanofluid by using the velocity and temperature measurement technique. The results of this study will advance the study of heat transfer using Al2O3 nanofluids.

Entropy generation in MHD free convection of nanoliquid within a square open chamber with a solid body

PurposeThe study aims to investigate magnetohydrodynamics thermal convection energy transference and entropy production in an open chamber saturated with ferrofluid having an isothermal solid block.Design/methodology/approachAnalysis of thermal convection phenomenon was performed for an open chamber saturated with a nanofluid having an isothermal solid unit placed inside the cavity with various aspect ratios. The left border temperature is kept at Tc. An external cooled nanofluid of fixed temperature Tc penetrates into the domain from the right open border. The nanofluid circulation is Newtonian, incompressible, and laminar. The uniform magnetic field of strength B at the tilted angle of γ is applied. The finite volume technique is used to work out the non-linear equations of liquid motion and energy transport. For Rayleigh number (Ra=1e+7), numerical simulations were executed for varying the solid volume fractions of the nanofluid (ϕ = 0.01–0.04), the aspect ratios of a solid body (As = 0.25–4), the Hartmann number (Ha = 0–100), the magnetic influence inclination angle (γ = 0–π/2) and the non-dimensional temperature drop (Ω = 0.001–0.1) on the liquid motion, heat transference and entropy production.FindingsNumerical outcomes are demonstrated by using isolines of temperature and stream function, profiles of mean Nusselt number and entropy generations. The results indicate that the entropy generation rate and mean Nu can be decreased with an increase in Ha. The inner solid block of As = 0.25 reflects the maximum heat transfer rate in comparison with other considered blocks. The addition of nano-sized particles results in a growth of energy transport and mean entropy generations.Originality/valueAn efficient computational technique has been developed to solve natural convection problem for an open chamber. The originality of this research is to scrutinize the convective transport and entropy production in an open domain with inner body. The outcomes would benefit scientists and engineers to become familiar with the investigation of convective energy transference and entropy generation in open chambers with inner bodies, and the way to predict the energy transference strength in the advanced engineering systems.

Darcy Forchheimer bioconvection flow of Casson nanofluid due to a rotating and stretching disk together with thermal radiation and entropy generation

Present work is pursued to examine the 3D bioconvection Casson nanofluid flow between two rotating and stretchable disks. The consequences of Darcy Forchheimer flow along with thermal radiation, magnetic field and heat source/sink are analyzed. Effects of thermophoresis and Brownian motion are also considered. Flow related partial differential equations are converted to the ordinary differential equations by using similarity transformations. Shooting technique is applied to solve the prevailing equations with the support of MATLAB Packages. Radial and tangential components of velocity for different values of Casson parameter, magnetic field parameter and porosity parameter are depicted through graphs. Temperature profile on varying the values of Prandtl number is also discussed. Moreover, thermal profile for increasing radiation parameter, thermophoresis parameter and Brownian movement parameter is analyzed. Nanoparticle's concentration effects on micro-organisms with growing Lewis number and Peclet number are studied graphically.

HEE and HSC for flavors: perturbative structure in open string geometries

Introduction of electric field in the D-brane worldvolume induces a horizon in the open string geometry perceived by the brane fluctuations. We study the holographic entanglement entropy (HEE) and subregion complexity (HSC) in these asymptotically AdS geometries in three, four and five dimensions aiming to capture these quantities in the flavor sector introduced by the D-branes. Both the strip and spherical subregions have been considered. We show that the Bekenstein-Hawking entropy associated with the open string horizon, which earlier failed to reproduce the thermal entropy in the boundary, now precisely matches with the entanglement entropy at high temperatures. We check the validity of embedding function theorem while computing the HEE and attempt to reproduce the first law of entanglement thermodynamics, at least at leading order. On the basis of obtained results, we also reflect upon consequences of applying Ryu-Takayanagi proposal on these non-Einstein geometries.

Nanoparticles shape effect on the efficiency of microheat sinks with tightly packed pin-fins

In this paper, the effects of the nanoparticles shape on thermal efficiency, entropy generation for the application of a /boehmite alumina nanofluid in a microheat sink (MHS) are studied. The MHS is composed of a number of microchannels, the walls of which consist of circular pin-fins attached together. A constant heat flux enters the HS from the bottom and heat it up. On the other hand, the nanofluid is responsible for cooling the HS. To study the thermal efficiency and entropy, a simulation using the control volume method has been performed in Fluent software. The effect of the NP size has been reported in this research. The simulation results showed that the average HS temperature strongly depends on the inlet fluid velocity, and an increase in the nanofluid velocity decreases it. Moreover, the addition of NPs reduces the average temperature of the HS, but its impact is smaller than that of the fluid velocity. Adding 5% of the cylindrical nanoparticles reduces the average temperature of the heatsink by 0.72 K. In addition, increasing the velocity and volume fraction of NPs decreases the thermal resistance () and increases the pumping power. The addition of 5% platelet-shaped nanoparticles increases the pumping power required by up to 80%. Using cylindrical NPs leads to the best temperature uniformity and least thermal resistance in the HS. Adding NPs decreases entropy generation. Application of blade NPs in high volume fractions, as well as brick NPs in average volume fractions, leads the highest Figure of Merit (FOM).

Oriented magneto-conjugate heat transfer and entropy generation in an inclined domain having wavy partition

Energy transmission in an efficient mode has become a crucial challenge in both industrial and biomedical systems because of the worldwide energy crisis. An initiative has been taken by the present research for designing energy-efficient industrial and biomedical systems using different fluids. This numerical study focuses on investigating magneto-hydrodynamic conjugate natural convection and entropy generation of a hybrid nanofluid (Ag-MgO-water) in a differentially heated square domain including heat-generating sinusoidal solid partition. The partition is mid-positioned with a finite thickness and divides the computational regime into two flow domains. The FEM has been applied for solving the governing PDEs. The effects of magnetic field intensity and orientation, cavity inclination, heat generation from the partition and volume fraction of hybrid nanoparticles on temperature distribution, flow field and local entropy generation have been investigated and displayed by isotherms, streamlines and isentropic lines, respectively. The contours have been plotted at the highest Rayleigh number for better visualization of thermo-fluid interaction. Considering a wide variation of free convection, thermal performance is determined through the computation of the average Nusselt number along the hot wall and system energy loss has been evaluated through the calculation of average total entropy generation as well as average Bejan number in both fluid zones. The obtained numerical results show that thermal performance and entropy generation are significantly influenced by the magnetic field intensity and cavity inclination. Thermo-fluid energy transfer is reduced by 11.62% for the highest magnetic field strength whereas the increase of cavity tilting is responsible for a 56.72% decrease in thermal performance. A new regression equation has been derived from the obtained results. The numerical study carried out in this research has superior results compared to the existing methodology and/or experimental/ numerical research.

CFD simulation of the impact of tip clearance on the hydrothermal performance and entropy generation of a water-cooled pin-fin heat sink

In this paper, the 3D numerical calculations of the water flow inside a pin-fin heat sink (PFHS) to determine the influence of tip clearance on the hydrothermal behavior and frictional & thermal irreversibilities of the fluid flow. The numerical simulations were conducted for five different fin heights (ξ = 1.5 mm to 2.5 mm) considering four different Reynolds numbers (Re=500, 1000, 1500, and 2000). The results demonstrated that the highest heat transfer coefficient is associated with ξ = 2.25 mm due to more uniform vorticity distributions, while the minimum central processing unit (CPU) average temperature is obtained for ξ = 2.5 mm due to the higher heat transfer surface area. The minimum thermal and frictional entropy generation rates are associated with PFHS with ξ = 1.5 mm. However, the PFHS with ξ = 2.25 mm has the highest Performance Evaluation Criterion (PEC) of nearly equal to 1.28 as compared to the other ξs. Besides, the fluid flow from the pin-fin base to the top and vice versa leads to create the vortices especially at the entrance of the PFHSs with tip clearance.

Atomistic Insights Into the Degradation of Inorganic Halide Perovskite CsPbI3: A Reactive Force Field Molecular Dynamics Study.

Halide perovskites make efficient solar cells but suffer from several stability issues. The characterization of these degradation processes is challenging because of the limited spatiotemporal resolution in experiments and the absence of efficient computational methods to study these reactive processes. Here, we present the first reactive force field for molecular dynamics simulations of the phase instability and the defect-induced degradation in CsPbI3. We find that the phase transitions are driven by the anharmonic fluctuations of the atoms in the perovskite lattice. At low temperatures, the Cs cations tend to move away from their preferential positions, resulting in worse contacts with the surrounding metal halide framework which initiates the conversion to a nonperovskite phase. Moreover, our simulations of defective structures reveal that, although both iodine vacancies and interstitials are mobile in the perovskite lattice, the vacancies have a detrimental effect on the stability, leading to the decomposition of perovskites to PbI2.

Predictions of Conjugate Heat Transfer in Turbulent Channel Flow Using Advanced Wall-Modeled Large Eddy Simulation Techniques.

In this paper, advanced wall-modeled large eddy simulation (LES) techniques are used to predict conjugate heat transfer processes in turbulent channel flow. Thereby, the thermal energy transfer process involves an interaction of conduction within a solid body and convection from the solid surface by fluid motion. The approaches comprise a two-layer RANS–LES approach (zonal LES), a hybrid RANS–LES representative, the so-called improved delayed detached eddy simulation method (IDDES) and a non-equilibrium wall function model (WFLES), respectively. The results obtained are evaluated in comparison with direct numerical simulation (DNS) data and wall-resolved LES including thermal cases of large Reynolds numbers where DNS data are not available in the literature. It turns out that zonal LES, IDDES and WFLES are able to predict heat and fluid flow statistics along with wall shear stresses and Nusselt numbers accurately and that are physically consistent. Furthermore, it is found that IDDES, WFLES and zonal LES exhibit significantly lower computational costs than wall-resolved LES. Since IDDES and especially zonal LES require considerable extra work to generate numerical grids, this study indicates in particular that WFLES offers a promising near-wall modeling strategy for LES of conjugated heat transfer problems. Finally, an entropy generation analysis using the various models showed that the viscous entropy production is zero inside the solid region, peaks at the solid–fluid interface and decreases rapidly with increasing wall distance within the fluid region. Except inside the solid region, where steep temperature gradients lead to high (thermal) entropy generation rates, a similar behavior is monitored for the entropy generation by heat transfer process.

Impacts of Uniform Magnetic Field and Internal Heated Vertical Plate on Ferrofluid Free Convection and Entropy Generation in a Square Chamber.

The heat transfer enhancement and fluid flow control in engineering systems can be achieved by addition of ferric oxide nanoparticles of small concentration under magnetic impact. To increase the technical system life cycle, the entropy generation minimization technique can be employed. The present research deals with numerical simulation of magnetohydrodynamic thermal convection and entropy production in a ferrofluid chamber under the impact of an internal vertical hot sheet. The formulated governing equations have been worked out by the in-house program based on the finite volume technique. Influence of the Hartmann number, Lorentz force tilted angle, nanoadditives concentration, dimensionless temperature difference, and non-uniform heating parameter on circulation structures, temperature patterns, and entropy production has been scrutinized. It has been revealed that a transition from the isothermal plate to the non-uniformly warmed sheet illustrates a rise of the average entropy generation rate, while the average Nusselt number can be decreased weakly. A diminution of the mean entropy production strength can be achieved by an optimal selection of the Lorentz force tilted angle.

Page curve from non-Markovianity

In this paper, we use the exactly solvable Sachdev-Ye-Kitaev model to address the issue of entropy dynamics when an interacting quantum system is coupled to a non-Markovian environment. We find that at the initial stage, the entropy always increases linearly matching the Markovian result. When the system thermalizes with the environment at a sufficiently long time, if the environment temperature is low and the coupling between system and environment is weak, then the total thermal entropy is low and the entanglement between system and environment is also weak, which yields a small system entropy in the long-time steady state. This manifestation of non-Markovian effects of the environment forces the entropy to decrease in the later stage, which yields the Page curve for the entropy dynamics. We argue that this physical scenario revealed by the exact solution of the Sachdev-Ye-Kitaev model is universally applicable for general chaotic quantum many-body systems and can be verified experimentally in near future.

Irreversibility features of a shell-and-tube heat exchanger fitted with novel trapezoidal oblique baffles: Application of a nanofluid with different particle shapes

The influences of nanoparticle shape on the Entropy Generation Rate (EGR) of a boehmite nanofluid in a shell and tube heat exchanger fitted with trapezoidal oblique baffles are investigated. The hot fluid is the nanofluid with five particle shapes (i.e., platelet, cylinder, blade, brick, and oblate spheroid), whereas the cold fluid is selected to be water. The hot fluid streams at the tube-side, while the cold fluid streams at the shell-side. By the elevation of the Reynolds number (Re), in the warm fluid, the thermal and frictional entropy productions increase, while the Bejan number reduces. The nanofluid with platelet-shaped nanoparticles results in the greatest frictional EGR and thermal EGR in the hot fluid, pursued by the suspensions with the cylinder-, blade-, brick-, and Os-shaped nanoparticles, respectively. By rising the Reynolds number, in the cold fluid, the thermal EGR reduces, and the frictional EGR increases. The nanofluid with Os-shaped particles shows the highest thermal and frictional entropy productions in the cold fluid. All in all, with considering the whole heat exchanger regarding the second law of thermodynamics, the platelet-shaped particles are the best ones to be utilized in the heat exchanger, resulting in the lowest irreversibility.

Improving energy storage and performance through implementing artificial neural network modeling to forecast heat transfer and entropy generation within a wavy‐wall microchannel under discontinuous‐boundary condition–hybrid nanofluid utilization

Numerous industries deal with heat transfer looking for a new technique to boost efficiency. Using nonuniform condition and making a wavy wall and microchannel are the conventional techniques to increase heat transfer which consequently lead to intensification in energy storage potential. In the present study, forced convection heat transfer of nanofluid through the slippage microchannel is investigated by utilizing numerical methods. Moreover, the results of the artificial neural network in determining the parameters involved in the first and second laws of thermodynamics are discussed; so it is necessary to determine the optimal number of neurons in the middle layer. In addition, the current study aims to examine the effects of Reynolds number, slip wall, and locations of thermal boundary condition on heat transfer and irreversibility. However, thermal entropy generation becomes 1.52 times worse. It is also revealed that the slippage wall can enhance the heat transfer up to 15.6%. The best thing about slippage microchannel is creating slip wall raises the thermal entropy generation and declines the viscous entropy generation.

Second law of thermodynamic analysis of 40:60% propylene glycol and water mixture based nanodiamond nanofluid under transition flow

Second law efficiency (exergy) was studied experimentally for transition flow in a tube of 40:60% (weight) p4ropylene glycol and water mixture based nanodiamond nanofluid. Prior to prepare the stable nano-diamond (ND) nanofluid, its soot was rinsed by strong chemicals. The experiments were carried out for several values of particle loading (0.2% to 1.0%) and of Reynolds number (2000 to 8000). The experimental analysis was also conducted for the heat transfer, friction factor and pumping power characteristics of the ND nanofluid. By increasing the values of particle concentration and Reynolds number, the results indicate an increase of heat transfer, Nusselt number, friction factor, pumping power and thermal performance factor, and a decrease of the thermal entropy generation. In comparison with the base fluid data, the heat transfer coefficient, Nusselt number, pressure drop, pumping power, friction factor and the thermal performance factor of the ND nanofluid augment by 36.83%, 24.16%, 15.38%, 13.18%, 19.9%, and 16.94%, respectively, and the thermal entropy generation decreases by 26.92% for 1.0 vol% and Reynolds number of 5321.16.