Effective Thermal Conductivity Of Nanofluids Research Articles

Cascaded lattice Boltzmann simulation of Newtonian and non-Newtonian mixture nanofluids with variable thermophysical properties in a cavity with vertical heat radiator

The central moments-based cascaded lattice Boltzmann method (CLBM) for then Newtonian and non-Newtonian Buongiorno’s model mixture nanofluids (CuO, ZnO, Al2O3-water) has been implemented and applied in a square chamber with a vertical heat radiator using accelerated graphics processing unit (GPU) computing through compute unified device architecture (CUDA) C/C++ platform. Due to the higher numerical stability, the CLBM is a superior numerical tool to the raw moments-based MRT-LBM (multiple-relaxation-time lattice Boltzmann method). Three different models for the viscosity and thermal conductivity of the nanofluids: (i) the Binkmann model for the constant viscosity and the Maxwell model for the constant thermal conductivity (ii) Binkmann and Maxwell model with temperature dependent Brownian motion and (iii) Corcione model with non-Newtonian fluid where the temperature and strain rate determine the nanofluid effective thermal conductivity and viscosity, have been used. The enclosure’s upper and bottom walls are thermally adiabatic, but the left and right walls are uniformly cold. A vertical heater is immersed in the middle position of the cavity. The benchmark results for non-Newtonian, Newtonian, and nanofluids for the various computational domains are used to validate the current code adequately. The Bingham number (Bn), the Rayleigh number (Ra), and The volume fraction of the nanoparticles (ϕ) are the three key parameters that are varied in this investigation to demonstrate the effects of natural convection on the isotherms, streamlines, isolines of nanoparticle volume fractionation, yielded and unyeilded zone, and average Nusselt number (Nu¯). The Brownian motion effects of the nanoparticles augmented the average rate of heat transfer and the use of the Bingham nanofluids reduced the heat transfer enhancement. For the CuO-water nanofluid, the augmentation of the rate of heat transfer is 15.42% from ϕ=0 to 4% while Ra=106 and the corresponding heat transfer enhancement for the ZnO-water nanofluid is 11.11%. For the Bingham fluid, the rate of heat transfer increases 7% from Ra=105 to Ra=106 while ϕ=2% and Bn=0.3. The findings can be applied to optimize automotive radiator systems, which are crucial for maintaining engine temperature.

Peristaltic transport of nanofluid with temperature dependent thermal conductivity: A numerical study

Peristaltic flow of nanofluid with temperature-dependent thermal conductivity is investigated. Variation of effective thermal conductivity of nanofluids with temperature is addressed. Mathematical modeling of momentum and energy equation is considered in a symmetric channel. Viscous dissipation, Ohmic heating, and mixed convection are accounted. Applied magnetic field and convective boundary conditions are imposed. Dimensionless expressions are simplified under long wavelength approximation. The relevant problems are solved by numerical technique. Velocity, temperature, pressure, and heat transfer characteristics are studied graphically. Heat transfer rate rapidly increases by increasing the concentration of nanomaterial. Therefore, nanomaterial is used for the cooling process in engineering and biomedical devices.

Studying Alumina–Water Nanofluid Two-Phase Heat Transfer in a Novel E-Shaped Porous Cavity via Introducing New Thermal Conductivity Correlation

Investigating natural convection heat transfer of nanofluids in various geometries has garnered significant attention due to its potential applications across several disciplines. This study presents a numerical simulation of the natural convection heat transfer and entropy generation process in an E-shaped porous cavity filled with nanofluids, implementing Buongiorno’s simulation model. Analyzing the behavior of individual nanoparticles, or even the entire nanofluid system at the molecular level, can be extremely computationally intensive. Symmetry is a fundamental concept in science that can help reduce this computational burden considerably. In this study, nanofluids are frequently conceived of as a combination of water and Al2O3 nanoparticles at a concentration of up to 4% by volume. A unique correlation was proposed to model the effective thermal conductivity of nanofluids. The average Nusselt number, entropy production, and Rayleigh number have been illustrated to exhibit a decreasing trend when the volume concentration of nanoparticles inside the porous cavity rises; the 4% vol. water–alumina NFs yield 17.35% less average Nu number compared to the base water.

Numerical study of nanoparticles aggregation on radiative 3D flow of maxwell fluid over a permeable stretching surface with thermal radiation and heat source/sink

The flow of fluid past a stretching sheet under heat and mass transfer analysis is significant because it has numerous applications in engineering and technology, including metal spinning, polymer extrusion, the manufacture of glass fibres, and metal casting etc. The current article investigates the effects of thermal radiation and heat source/sink on 3D nanofluid flow over an elongated surface embedded in a porous medium with nanoparticles aggregation. Further, the significance of variable magnetic field is one of the striking features of the present study. We have considered copper (metallic) and titania (metallic oxide) as nanoparticles and water as base fluid. The effective thermal conductivity of nanofluid is predicted by using a modified Maxwell model. Applying similarity transformations, the governing PDEs are transformed into nonlinear ODEs and solved numerically by MATLAB software using bvp4c and the shooting method. Graphs and tables are plotted to depict the significance of the effective parameters on velocity and temperature profiles as well as skin friction coefficient and Nusselt number. The existence of radiative parameter effects is more advantageous for improving heat transmission. The temperature distribution is enhanced due to presence of nanoparticles aggregation. Thus, the impact of nanoparticles aggregation is supportive theoretical tool in future bioengineering and industrial applications.

Heat Conduction and Microconvection in Nanofluids: Comparison between Theoretical Models and Experimental Results

A nanofluid is a suspension consisting of a uniform distribution of nanoparticles in a base fluid, generally a liquid. Nanofluid can be used as a working fluid in heat exchangers to dissipate heat in the automotive, solar, aviation, aerospace industries. There are numerous physical phenomena that affect heat conduction in nanofluids: clusters, the formation of adsorbate nanolayers, scattering of phonons at the solid–liquid interface, Brownian motion of the base fluid and thermophoresis in the nanofluids. The predominance of one physical phenomenon over another depends on various parameters, such as temperature, size and volume fraction of the nanoparticles. Therefore, it is very difficult to develop a theoretical model for estimating the effective thermal conductivity of nanofluids that considers all these phenomena and is accurate for each value of the influencing parameters. The aim of this study is to promote a way to find the conditions (temperature, volume fraction) under which certain phenomena prevail over others in order to obtain a quantitative tool for the selection of the theoretical model to be used. For this purpose, two sets (SET-I, SET-II) of experimental data were analyzed; one was obtained from the literature, and the other was obtained through experimental tests. Different theoretical models, each considering some physical phenomena and neglecting others, were used to explain the experimental results. The results of the paper show that clusters, the formation of the adsorbate nanolayer and the scattering of phonons at the solid–liquid interface are the main phenomena to be considered when φ = 1 ÷ 3%. Instead, at a temperature of 50 °C and in the volume fraction range (0.04–0.22%), microconvection prevails over other phenomena.

Effective Thermal Conductivity of Nanofluids Containing Silicon Dioxide or Zirconium Dioxide Nanoparticles Dispersed in a Mixture of Water and Glycerol

The present study investigates the effective thermal conductivity of nanofluids containing crystalline or amorphous silicon dioxide (SiO2), or zirconium dioxide (ZrO2) nanoparticles dispersed in a mixture of water and glycerol with a mass ratio of 60:40. Such fluids are relevant as potential cutting fluids in tribology and feature a broad distribution of irregularly shaped non-spherical particles of dimensions on the order of (100 to 200) nm that were produced by comminution of larger particles or particle aggregates. A new steady-state guarded parallel-plate instrument was applied for the absolute measurement of the effective thermal conductivity of the nanofluids with an expanded uncertainty (coverage factor k = 2) of 3% for temperatures from (293 to 353) K and particle volume fractions up to 0.1. For a constant volume fraction of 0.03 for the three particle types, the measured thermal-conductivity ratios, i.e. the effective thermal conductivity of the nanofluids relative to the thermal conductivity of the base fluid, are less than 1.05 and not affected by temperature. In the case of the nanofluids with crystalline SiO2, with increasing particle volume fraction from 0.03 to 0.10 the thermal-conductivity ratios increase up to values of about 1.18 for all temperatures. A comparison of the measurement results with the Hamilton-Crosser model and an analytical resistance model for the effective thermal conductivity of nanofluids shows that the former one allows for better predictions for the present nanofluids with a relatively large viscosity. In this context, it could be shown that detailed knowledge about the sphericity and thermal conductivity of the dispersed nanoparticles is required for the modeling approaches.

REVIEW OF EXPERIMENTAL AND NUMERICAL METHODS TO DETERMINE THERMAL CONDUCTIVITY OF NANOFLUIDS

Study and research around heat conduction mechanisms of nanofluids and suspensions are active for a decade. Several models of effective thermal conductivity of nanofluids have been published. This review summarizes recent research for thermal conductivity of a solid/liquid suspension and unfolds the effects of interfacial layers, shells, and aggregate structure on this property. Basically, experimental, and numerical, both, ways to determine thermal conductivity are summarized here in respective sections along with an introductory section named preparation of nanofluids. Key parameters affecting the preparation and stability of nanofluids are reviewed here and further scope of research is assessed. Numerical or theoretical thermal conductivity models are validated against experimental ones by some of the authors, which are reviewed and summarized here. The effect of different parameters pertaining to both nanoparticles and base fluids are reviewed and summarized here for different combination of metal/oxides as particles and water/oils as base fluids. Other properties such as viscosity, convective heat transfer, boiling heat transfer of nanofluids and suspensions are equally important and will be reviewed in subsequent reports.

Three-dimensional numerical analysis of flow and heat transfer of bi-directional stretched nanofluid film exposed to an exponential heat generation using modified Buongiorno model

The heat transfer characteristics of copper/water nanofluid flow over a bi-directional stretched film are theoretically studied. The used mathematical model accounts for nanofluid effective dynamic viscosity and thermal conductivity. The model of the current study utilizes the modified Buongiorno model to scrutinize the effect of haphazard motion, nanoparticles' thermo-migration, and effective nanofluid properties. 3D flow is driven by having the nanofluid film elongation in two directions. The thermal analysis of the problem considers the nonlinear internal heat source and Newton heating conditions. In modeling the problem, the Prandtl boundary layer approximations are employed. Moreover, the nonlinear problem set of governing equations for investigating the transport of water conveying copper nanoparticles was non-dimensionalized before being treated numerically. The current parametric study investigates the impact of governing parameters on nanoparticles velocities, temperature, and concentration distributions. The presence of copper nanoparticles leads to a higher nanofluid temperature upon heating. The temperature enhances with the nanoparticles Brownian movement and thermo-migration aspects. Furthermore, involving a heat source phenomenon augments the magnitude of the heat transfer rate. Moreover, the velocity ratio factor exhibits decreasing behavior for x-component velocity and increasing behavior for y-component velocity. In conclusion, the study results proved that for larger values of Nb and Nt the temperature is higher. In addition, it is clear from the investigations that the Lewis number and Brownian motion factor decline the nanoparticle concentration field.

Comparison of Different Empirical Models for Analysing the Thermal Conductivities of Various Materials for Use in Nanofluid Preparation

For over the past ten years, various empirical models have been studied for analysing the thermal conductivity of different materials. In this analysis, different empirical correlations and models of thermal conductivity have been compared. The thermal conductivities of four types of oxide materials (SiO2, TiO2, CuO, Al2O3) and MWCNTs with volume fractions from 0.5 to 5% in a temperature range of 273–373 K and various nanoparticle shapes were compared. The results illustrated that the thermal conductivity of nanofluids based upon various nanoparticles increased with increasing volume fraction. In comparison, the effective thermal conductivity of nanofluids based on MWCNTs was enhanced much more than that of other types of nanofluids. Furthermore, Maxwell’s model was considered the basis for predicting the effective thermal conductivity of nanofluids. According to the basic models, a new correlation was proposed for predicting the effective thermal conductivity as a function of temperature and nanoparticle volume concentration. The nanoparticle shape has a great impact on the thermal conductivity of nanofluids. Regarding the precise heat transfer enhancement, the analysed nanofluids are suggested to be heat carriers in thermal systems, particularly solar thermal collectors. Typically, the use of nanofluids in solar thermal collectors improves the thermal efficiency of collectors in terms of the precise heat transference of nanofluids.

Experimental investigation and artificial intelligent estimation of thermal conductivity of nanofluids with different nanoparticles shapes

Determining and modelling the effective thermal conductivity of nanofluids is always a concern of investigators. The effect of nanoparticles' shape on the effective thermal conductivity of nanofluids has not been accurately correlated with other influencing factors such as temperature and nanoparticle concentration. The main purpose of this study is to reveal the effects of influencing factors on the effective thermal conductivity of nanofluids by experimental investigation and artificial intelligence (AI). We prepared the TiO2/water nanofluids with four shapes of TiO2 nanoparticles (spherical, ellipsoidal, clubbed, and sheet) and measured the effective thermal conductivities of the samples with nanoparticles concentrations from 0.5 to 4.0 vol% as a function of temperature from 20 to 60 °C. Besides, 389 experimental datasets collected from literature and 80 datasets from our experiment were used to determine the optimum structure of AI-based models. Temperature, concentrations, shape factor, and thermal conductivity of nanoparticles were selected as independent variables, and the relative thermal conductivity of nanofluids was selected as the dependent variable. Six AI-based models were examined, including four artificial neural networks of multi-layer perceptron, cascade feedforward, radial basis function, and generalized regression neural network, adaptive neuro-fuzzy inference systems, and least-squares support vector machines, and the estimating performances were compared. The experimental results showed that increasing temperature and nanoparticles concentrations led to a remarkable improvement in the relative thermal conductivity of nanofluids, attributed to the intensified Brownian motion of nanoparticles with the high temperature and the effective collision of nanoparticles with the high concentrations. Besides, nanoparticles with a large aspect ratio provided the fast pathway with seldom crossing an interparticle boundary or junction point for effective heat transfer in the nanofluids, resulting in higher relative thermal conductivity of nanofluids. Statistical analyses confirmed that the cascade feed-forward neural network with ten hidden neurons and the Levenberg–Marquardt training algorithms was the optimized AI-based model for estimating the relative thermal conductivity of nanofluids. This model estimated allover experimental data by MSE = 0.0039, RMSE = 0.0622, AARD% = 2.66%, and R2 = 0.9908.

Effects of surface modification and surfactants on stability and thermophysical properties of TiO2/water nanofluids

TiO2/water nanofluids are prepared by adding different surfactants and utilizing surface modification technique. The dispersion stability of TiO2 in water is evaluated, and better stability is found in Tetramethylammonium hydroxide (TMAH) added nanofluids and 3-aminpropyltriethoxysilane (APTS) treated nanofluids. Then the effects of nanoparticle concentration and temperature on thermophysical properties of the above two nanofluids are investigated. The viscosity of nanofluids increases with the increase of nanoparticle concentration, and decreases with temperature. The viscosity of APTS treated TiO2/water nanofluids is obvious higher than that of TMAH added TiO2/water nanofluids, because the grafting of APTS enhances the adhesion of nanoparticles to water. The thermal conductivity and effective thermal conductivity of nanofluids increases with the increase of temperature and nanoparticle concentration. And the changing trends and enhancement ratios of thermal conductivity for two nanofluids are similar. Finally, by applying properties enhancement ratio (PER), the thermal performance of TiO2/water nanofluids is evaluated.

A reconstruction of Hamilton-Crosser model for effective thermal conductivity of nanofluids based on particle clustering and nanolayer formation

Although many models have been proposed to estimate the effective thermal conductivity (ETC) of nanofluids, the thermal conduction mechanisms need to be further addressed to improve the prediction accuracy of ETC model. In this paper, by fully considering the effects of particle clustering, Brownian motion, Kapitza resistance and nanolayer of particle, Hamilton-Crosser model is reconstructed to establish an improved model for the ETC of nanofluids. To develop this model, the particle clustering is characterized by the particle size distribution analysis, and the thermal conductivity distribution in the nanolayer is represented as a specific function of the distance from the nanoparticle. The influences of temperature, viscosity, particle size and other factors on the ETC of nanofluids are also included in this model. The results show that the accuracy of this model can be improved as compared to those considering only several of these factors, and the maximum error is 3% against the available experimental data. With this model, the inconsistency phenomena in ETC data of nanofluids can be explained in the view of the agglomeration and Brownian motion with the system conditions.

Thermal transport dynamics in active heat transfer fluids (AHTF)

We present results of molecular dynamics calculations of the effective thermal conductivity of nanofluids containing self-propelled nanoparticles. The translational and rotational dynamics observed in the simulations follow the behavior expected from the standard theoretical analysis of Brownian and self-propelled nanoparticles. The superposition of self-propulsion and rotational Brownian motion causes the behavior of the self-propelled nanoparticles to resemble Brownian diffusion with an effective diffusivity that is larger than the standard Brownian value by a factor of several thousand. As a result of the enhanced diffusion (and the convective mixing resulting from the motion), we observe a discriminable increase of the effective thermal conductivity of the solution containing self-propelled nanoparticles. While the increases we observe are in the range of several percent, they are significant considering that, without propulsion, the nanofluid thermal conductivity is essentially not affected by the Brownian motion and can be understood within the effective medium theory of thermal conduction. Our results constitute a proof of concept that self-propelled particles have the potential to enhance thermal conductivity of the liquid in which they are immersed, an idea that could ultimately be implemented in a broad variety of cooling applications.

ANFIS modelling of effective thermal conductivity of ethylene glycol and water nanofluids for low temperature heat transfer application

Nanofluids have been proven to be promising coolants for heat transfer application due to their superior thermal properties. In the present work, experimental studies were carried out to investigate the thermal conductivity of ethylene glycol (EG) and water mixture (35:65 v/v in ratio) based CuO, Al2O3 and TiO2 nanofluids. The study has considered nanofluid containing nanoparticles in the concentration range of 0.2 to 2 wt%. The effect of nanoparticle concentration on thermal conductivity was investigated in terms of effective thermal conductivity (ratio of thermal conductivity of nanofluid to that of base fluid) at low temperatures (5 to 25 °C). Results indicated significant improvement in thermal conductivity at lower temperature of nanofluids. Enhancement of 6.34%, 4.87% and 3.59% in thermal conductivity was obtained for 2 wt% of concentration at 5 °C for CuO, Al2O3 and TiO2 nanoparticles respectively compared to EG:Water. Correlations for effective thermal conductivity were developed by regression considering two input (temperature, concentration) and three input (temperature, concentration and nanoparticles thermal conductivity) parameters using experimental results. An intelligent model, adaptive neuro-fuzzy inference system (ANFIS) approach was used to model the effective thermal conductivity of nanofluids. ANFIS model performed better than the correlations developed.

Thermal Conductivity Calculations for Nanoparticles Embedded in a Base Fluid

The Prasher analytical model was used for calculating the thermal conductivity of the embedded nanoparticles of Al2O3, CuO, ZnO, and SiO2 in conventional fluids, such as water and ethylene glycol. The values that were obtained were used in the nanofluid theoretical models for comparison with experimental data, where good agreement was obtained. Liang and Li’s theoretical model was also used to calculate the thermal conductivity of these nanoparticles, where the results agreed with those obtained using the Prasher model. The effect of the liquid nanolayer thickness around the nanoparticles that was used to enhance the effective thermal conductivity of nanofluids was explained. The role of the nanoparticles’ surface specularity parameter, which was size-dependent, was clarified. This theoretical trend provides a simple method for estimating the thermal conductivity of nanoparticles and nanofluids.

Numerical simulation of nanoparticles size/aspect ratio effect on thermal conductivity of nanofluids using lattice Boltzmann method

Recent developments in nanofluids have led to a renewed interest in the geometrical effects of nanoparticles on the effective thermal conductivity (ETC) of nanofluids. Although the experimental data for the effect of nanoparticles shapes (the aspect ratio) on ETC are quite consistent, there are still controversial results regarding the effect of the nanoparticles size. Theoretical approaches have been proposed to shed light on the experimental observations in both macroscopic and microscopic scales. However, these approaches cannot be generalized due to the collective interrelated behaviours of nanoparticles. In this paper, a mesoscale numerical simulation, Lattice Boltzmann method (LBM), is implemented to study the effect of nanoparticle geometry on the nanofluids thermal conductivity. The results demonstrate that the nanofluids thermal conductivity and the nanoparticles volume fraction can be linearly correlated. In essence, increasing the nanoparticles aspect ratio can improve ETC of nanofluids. However, the nanoparticle size is not statistically significant (P-value>0.05) to be considered as an influencing parameter (as observed in the controversial results of the experiments) when the interfacial phenomena are not taken into account. The surface to volume ratio of the particle is the parameter that should be studied as all the solid/liquid interfacial interactions depends on this ratio.

Numerical investigation of a nanofluidic heat exchanger by employing computational fluid dynamic

Shell-and-tube heat exchanger is widely applied in different industrial processes and energy systems. Utilized fluid in the heat exchanger and its thermophysical properties are among the key factors that influence the thermal performance and fluid flow in heat exchangers. Employing nanofluids, with enhanced thermal properties, can improve heat transfer rate of heat exchanger. In this regard, the current article concentrates on the numerical simulation of a shell-and-tube heat exchanger with baffle, utilizing multi-walled carbon nanotube/water nanofluid. In this regard, computational fluid dynamic is used for modeling. The applied turbulence model in this study is k- $$\varepsilon$$ which is selected based on the literature review. Heat transfer rate comparison in cases of utilizing the nanofluid in shell side of heat exchanger revealed high potential of the nanofluid in thermal performance improvement. Moreover, it was noticed that by using higher concentration of the nanofluid, more enhancement would be obtained which is owing to higher increment in the nanofluid effective thermal conductivity. Average increment in the heat transfer rate of the HE in case of using multi-walled carbon nanotube/water with 4% concentration is around 29.5% in comparison with the condition water is used as the operating fluid.

Thermal Conductivity of Nanofluids-A Comprehensive Review

The present study deals with a comprehensive review on the enhancement of effective thermal conductivity of nanofluids. The present article summarizes the recent research developments regarding the theoretical and experimental investigations about thermal conductivity of different nanofluids. The current study analyzes several factors those strongly affecting thermal conductivity of nanofluids include solid volume fraction, temperature, particle size, particle type, particle shape, different base fluids, magnetic field, pH, surfactant and ultrasonic time. In addition, different reasonably attractive models contributing augmentation of thermal conductivity of nanofluids are invoked. Finally, important heat transfer mechanisms namely Brownian motion, nanoclustering, thermophoresis, osmophoresis and interfacial nano-layer responsible for significant role in ameliorating the thermal conductivity and therefore the heat transfer characteristics of nanofluids are discussed.

ANN modelling and experimental investigation on effective thermal conductivity of ethylene glycol:water nanofluids

In this study, ethylene glycol (EG)–water (35:65 %v)-based nanofluids have been prepared to study enhancement in thermal conductivity. Nanofluids containing nanoparticles of materials CuO, Al2O3 and TiO2 of different mass concentrations from 0.2 to 2% were prepared using ultrasonication. Thermal conductivity measurement was carried using KD2 Pro thermal properties analyser in the temperature range of 30–60 °C. The study investigated the effect of concentration of nanoparticles, temperature and nanoparticle material on effective thermal conductivity of nanofluids. The results showed a significant improvement in effective thermal conductivity due to the addition of nanoparticles to the base fluid. Correlations were developed for predicting the effective thermal conductivity considering each material separately, and a generalized correlation considering the three materials. Subsequently, ANN modelling was carried out for predicting the effective thermal conductivity of nanofluids and compared with developed correlations. The modelling work carried out in this study is more generalized as literature results were considered in addition to the results from the present study. ANN modelling predicts the effective thermal conductivity better than the proposed correlations.

Effective Thermal Conductivity of Nanofluids: Measurement and Prediction

In the present study, the effective thermal conductivity of nanoparticle dispersions, so-called nanofluids, is investigated experimentally and theoretically. For probing the influence of the nanoparticles on the effective thermal conductivity of dispersions with water as liquid continuous phase, nearly spherical and monodisperse titanium dioxide (TiO2), silicon dioxide (SiO2), and polystyrene (PS) nanoparticles with strongly varying thermal conductivities were used as model systems. For the measurement of the effective thermal conductivity of the nanofluids with particle volume fractions up to 0.31, a steady-state guarded parallel-plate instrument was applied successfully at temperatures between (298 and 323) K. For the same systems, dynamic light scattering (DLS) was used to analyze the collective translational diffusion, which provided information on the dispersion stability and the distribution of the particle size as essential factors for the effective thermal conductivity. The measurement results for the effective thermal conductivity show no temperature dependency and only a moderate change as a function of particle volume fraction, which is positive or negative for particles with larger or smaller thermal conductivities than the base fluid. Based on these findings, our theoretical model for the effective thermal conductivity originally developed for nanofluids containing fully dispersed particles of large thermal conductivities was revisited and also applied for a reliable prediction in the case of particles of relatively low thermal conductivities.