مجله مهندسی مکانیک
سال پنجاه و یکم شماره 2 (پیاپی 95، تابستان 1400)

This paper presents a novel scheme for robust controller design for teleoperation system; there are parametric uncertainties in nonlinear model and varying time-delay in the communication channel, and robots are in contact with non-passive human operator and environment. First, the structure of nonlinear control law is designed for nominal system utilizing the Passivity Based Control (PBC) scheme. Then, using the Lyapunov-Krasovskii (LK) theorem, sufficient conditions are derived for robust asymptotic stability of system in terms of matrix inequalities to tune controller parameters. Also, by definition of an appropriate performance criterion, controller parameters are obtained to minimize the position tracking errors besides guaranteed robust stability. The contributions of the proposed method are twofold. First, the developed controller is fixed-structure which can be implemented easily in practice. Second, the asymptotic stability of system is assured without any requirement for existence of passive part in the dynamical model of non-passive interaction forces. Simulation results of 3-dof teleoperation system are compared to a rival method to demonstrate the advantages of the suggested method.

Satellites weighing 1-10kg are known as Nano-satellite. With the advancement of technology and the smaller parts of the electronics, satellites become smaller and lighter every day. Due to the advancement of technology, the use of CanSats/Cubsats have become more prominent. The optimal and low cost of launching these satellites is air-launch-to-orbit method. The conceptual design and simulation of the trajectory of an air-launch-to-orbit rocket vary based on the type of aircraft, the technology available to make the rocket engine, Aircraft’s velocity, Starting Altitude, Initial (the aircraft’s) Flight Path Angle and Latitude and Longitude, with changes in each of these parameters as a new design. In this research, the feasibility study of air launching vehicle to deliver a 10kg Nano-satellite into a parking orbit using the F-4 aircraft has been performed natively. For this purpose, the conceptual design of the air launch-to-orbit rocket has been achieved after five design stages. The final design is based on the downscale of the single operating sample, weighing 2270 kg, length 6.6 m and diameter 0.65 cm. Simulation of the ascent trajectory has been done using a 3DOF code and validated by Pegasus trajectory modeling.

Flow in a centrifugal electrical submersible pump is simulated. The purpose of this study is to find the effect of changing the axial distance between impeller and scroll on pump’s operation. Flow in the impeller and scroll is simulated with the Fluent software. Finite volume method with SST k-ω turbulance model is used for numerical solution. In order to evaluate the simulation results, values of head, power and efficiency are calculated at the design conditions and are compared to the experimental results. Simulations are then performed with different impeller to scroll axial distances. The results show that by increasing the distance between impeller and scroll, each of the three properties, head, power and efficiency will decrease. Increasing the distance between impeller and scroll from 0.5 to 5.0 mm, head, power and efficiency will drop by 33%, 18% and 20% respectively. Also, effect of changing the axial distance between impeller and scroll on vortices created in the fluid passing through the impeller and the axial force imposed on the impeller are calculated.

In this paper the governing equation for incompressible flow with heat transfer are solved numerically using artificial compressibility method. The finite volume method was used to solve the governing equation with Jameson’s method on triangular unstructured grid. In order to stabilize the solution a blend of the second and fourth differences of conserved variable used in order to prevent the oscillations. An implicit method with second order accuracy was used to time integration. In order to solve the obtained equation at each time step an artificial time step and the forth order explicit runge-kutta has been used. In order to ensure the accuracy of obtained results independent of the grid size the obtained results has been compared with the standard test cases in the literature and indicated a good agreement. Comparison of results indicated that the method used in the numerical solution of incompressible flow has a desirable accuracy and convergence rate.

This paper investigates the free longitudinal vibrations of functionally graded nanorods with constant cross-section on the basis of nonlocal strain gradient theory. Functionally graded nanostructures can be defined as nonhomogeneous composites which are obtained by combining two different materials with different elasticity moduli and mass densities. Due to the special properties of the functionally graded materials (FGM), the changes of physical properties (such as the elasticity modulus and mass density along the axis of functionally graded nanorods) are expressed by an exponential relation in terms of the location and constant ratios of the material properties. In this paper, numerical results related to these changes are shown. Galerkin solution technique is utilized to obtain an approximate solution to the free longitudinal vibration of FGM nanorods with clamped–clamped boundary conditions. A parametric study is carried out to show the influences of nonlocal parameters on the fundamental natural frequencies such as location-dependent nonlocal parameter and material length scale parameter as well as the physical parameters such as the elasticity modulus and mass density of different material composition.

In this paper, an efficient scheme is presented based on fractional order sliding mode control (FOSMC) method for tip position control of a single link flexible robot manipulator. The proposed control strategy is robust against the system parameters variations such as payload and viscous friction variations in the presence of the unknown Coulomb friction disturbances. The main objective of the proposed control scheme is to reduce the deflection due to the flexibility of the link and the precise tip positioning control of the single-link flexible manipulator. To achieve this aim, sliding mode control method is performed in two stages. In the first step, the difference between the motor angle and the tip angle of the flexible link is reduced by applying the proposed fractional order sliding mode controller. Then, in the second step, another sliding mode controller is added to obtain precise control of tip-link position. Numerical simulation results show the feasibility and effectiveness of the proposed control method.

The aim of the present work is to determine the transverse deflection of a plate on the granular material under the impact of a ball. Through the analytical solution, the granular bed is equivalent to the elastic bed. The granular materials behave elastically by with a damping reaction under the impact loading. There are suggestions by literatures for introducing these elastic and damping effects which are used in the present work. The impact force is an intensive short duration force which is considered as a function of time and impact position. The equations of motion are derived by Hamilton's principle and the stress and strain relations are written by the first order shear theory and the vibration response of the plate is derived. Moreover, an experimental apparatus is designed to measure the plate deflection. Experimental parameters include the plate type and thickness, ball speed, impact position and granular thickness. The experimental data are used to validate the theoretical evaluations. The comparison of the evaluated and measured data shows that the theoretical model is acceptably gives the plate deflection.

In this paper, the thermal characteristics of solar gas heaters were analyzed to find the effect of using radiating gases rather than air on the efficiency of converting radiative heat flux into gas enthalpy. The set of governing equations including the gas energy equation with radiative term was solved by the finite difference method (FDM). The radiative term in this equation was calculated by numerical solution of the radtaive transfer equation (RTE) using the discrete ordinate method (DOM). Based on the numerical results, it was found that using the radiating gas in the solar gas heater can lead to a considerable efficiency increase. In the studied test case, about 7% increase in the gas outlet temperature was seen when the optical thickness increased from zero (non participating medium) to 2. Comparison between the present numerical results with experimental data published in literature shows good consistency with the error less than 3%.

Performance of journal bearings as one of the common parts of industrial machines, especially rotary systems, are affected by any change in the nature of the lubricant as well as changes in geometric structure. The production of Nano fluids and textured surfaces to improve fluid and structural interaction in various mechanical systems, including bearing supports, has been provided by the recent technology advancements. In this regard, the effects of the depth of the cylindrical texture and the addition of 〖TiO〗_2 Nanoparticles on the steady state performance of hydrodynamic journal bearings are investigated in present study. Analysis is done by rewriting the governing Reynolds equation due to changes in the film thickness of the lubricant affected by the surface texture and the relative viscosity of the oil according to volume percentage of added nanoparticles. Analysis is done using the finite element method. Results show that the maximum pressure, load carrying capacity and friction increase by enhancing the volume percentage of the nanoparticles, and vice versa these characteristics are reduced by increasing the depth of surface textures created on the bearing shell and the corresponding imprisoned Nano lubricant film thickness.

In the issue of vibration suppression, the implementation of passive control systems based on eddy-current dampers and its magnetic features play the main role of lowering both the sensitivity and the power consumption. Regarding application in space systems, the eddy-current damper role is in opening of a satellite antenna. For simulation purpose of the eddy-current damper, the differential and integral forms of Maxwell equations in COMSOL software, as well as the finite element energy method in a MATLAB are used. In the electro-magnetic section by COMSOL, the simulation of electrical field and magnetic potential during the disk angle change is considered while the PDE of electro-magnetic field and energy are numerically solved through MATLAB. The application of COMSOL software based on finite-element method and Maxwell differential equations yield a discontinuity of magnetic materials effectiveness for damper. Resultant material groups were determined to be the Ferromagnetic and diamagnetic material which their pertaining substances are nickel and the copper. The results of simulation showed that applying electrical isolation leads to decrease of the damping ratio.

This paper is aimed to produce a Kevlar/PP weft knitted composite lamina with high absorbed energy under high velocity impact loading and propose a finite element model for studying the initiation and evolution of interlaminar damage with focusing on the experimental extraction of the required properties for the simulation. In the fabrication procedure, the fabrics produced on a modern flat-knitting machine were being processed via hot-pressure molding technique until the thermoplastic component formed the composite matrix phase. Based on a progressive damage model and using Hashin criterion in composite layers, damage initiation was predicted while the damage evolution was predicted by reducing the value of stiffness based on fracture toughness energy that is available in ABAQUS. Parameters required for modeling such as fracture toughness energy, were experimentally measured through some tests such as compact tension. The proposed model was validated by making comparison between the experimental and numerical results of the impactor velocity before and after the test of 120 J impact loading and 11 percent difference between the experimental and numerical impact results was observed

The purpose of the present research is to perform the energy analysis of a Combined Heat and Power (CHP) plant using a Solid Oxide Fuel Cell (SOFC) system fed by biogas from a Waste Water Treatment Plant (WWTP) and a solar system. CHP is a reliable and cost-effective option for WWTPs where anaerobic digesters are equipped. Biogas produced in WWTPs can be utilized to fuel a CHP system to produce electricity and useful thermal energy using a variety of prime movers. The thermal energy produced by the CHP system can be used to meet a part of the anaerobic digestion process heat demand in the WWTP digester. In the present study, a steady-state model is developed for the fuel cell simulation and the integrated plant system. Tabriz WWTP is studied in this work. Three SOFC modules produce, in cogeneration mode, 180 kW electricity and 125 kW thermal energy. The results show that supplying a part of the digester heat demand through solar energy can save natural gas combustion in the plant.

In this study the strongly coupled method (two-way fluid structural interaction (FSI)) is presented to simulate the membrane wings used in ultralight aircraft. In the present study, numerical modeling is performed by commercial software ANSYS using two separate solvers one for fluid and one for structure. Unlike the one-way method, the two-way coupled method, membrane wing deformation is considered at each time step, which increases the accuracy of the simulation. The membrane wings are made of fabric covering on the naca 2418 plan. fabric is considered to be orthotropic and non-porous. Influence of angle of attack, Young's modulus and thickness of fabric on the aerodynamic coefficients and membrane wing deformation were investigated. All simulations were performed at Re=100000. The results show that increasing the Young’s modulus from 180.25 to 270 MPa reduces the maximum membrane displacement in X and Y direction by 81% and 78%, respectively. Increasing the fabric thickness from 0.4 to 0.6 mm also reduced the maximum displacement of the membrane wings in the X and Y directions by 85 and 80%, respectively.

Lower unit cost and acceptable efficiency are two important factors in choosing a domestic solar collector. Due to its simplicity and integrated structure, the passive photovoltaic-thermal (PVT) solar system offers a promising part of the co-generation of water heating and electricity in varied climates. In this paper to the aim of looking at solar costs and improving energy conversion efficiency, a new passive PVT solar system has been proposed. This system includes a semi-transparent a-Si solar module, which in addition to generating electricity acts as a solar absorber. The semi-transparent a–Si solar cell is selected because of its useful features that help reduce module cost such as the relatively small amount of silicon and energy used in their manufactures. In the present work, the four different cases of passive PVT systems have been simulated and compared. The results show that the case number one includes glass cover and semi-transparent a-Si module has simpler design and better performance than other designs. The results show that the case number one, which includes glass cover and semi-transparent a-Si module, has higher thermal efficiency than case number 3 and 4 and higher power output than the other 3 cases. For case number one thermal efficiency and electrical efficiency is achieved as 76-84% and 8.94% -9.87%, respectively.

Air intake system of a 160 MW gas turbine cycle has been simulated using computational fluid dynamic. Fluent software is used to solve the continuity and momentum conservation equations.  The temperature effect in anti-icing system has been neglected.  According to the large dimension of the Intake body, the appropriate kind of mesh has been selected in every section and element sizes are tuned with parameters gradient. The first goal of this study is to suggest a way to reduce the pressure loss by accurate calculation of the flow parameters in all sections of the air intake system. Vortex as an effective source in pressure loss assessment has been studied locally in various part of the system. Results show that some vortex has been created in the corner of the filter house. Elimination of these vortexes can reduce the pressure loss and optimize the energy recovering from fuel. One of the important parameter that has high effect on the compressor efficiency is the pressure distortion of the inlet flow. In current study four case studies for transition piece has been suggested and the best one has been selected.

In this research, the compact power substation with 800KVA capacity that is produced by Irantransfo Substations Development Company, has been modeled and its thermal class is determined. Modeling of this compact power substation that include a three-phase cast resin dry transformer placed in the enclosure chamber carried out in three-dimension. Then the desired parameters such as fluid flow, pressure drop and temperature distribution in significant parts are investigated. In order to verify the accuracy of the results, four sensors were placed in suitable locates to compare the results of the analysis with the measured values of these sensors. The results of the modeling in comparison with the experimental results show a difference of up to 5.8K. By applying data according to IEC62271-202 standard and appropriate boundary conditions, the maximum temperature difference which occurs in low voltage bobbin is 12K. So the thermal class will be 15K according standard definition. Although it was 14% lower than the laboratory results, the thermal class of this substation also determined in the Laboratory of Electrical Industries of Iran 15K and indicate the good results of this study.

One of the important aspect of internal combustion engines (ICEs) heat transfer is their warm up period. The importance of engine warm up period, especially in terms of emissions production, has always been of interest to researchers. In this study, the concept of using electric water pump in cooling circuit as a solution for reducing this period is investigated. In the present study, the thermodynamic performance of an ICE is first simulated in the GT-SUITE software and then coupled to the simulated cooling circuit of engine in the same software. After comparing the experimental and numerical results of the engine and validation, the effect of using an electric water pump on the thermal behavior of different engine components during engine warm-up period is observed. The results show that in the early stages of engine operation, it’s possible to manage the coolant flow rate in order to provide faster heating of the engine components which in turn leads to decreasing of the warm up period.

In the present paper, a comparison was made among four carbon dioxide (CO2) horizontal direct-expansion ground source heat pumps by using the thermo-economic analysis. For four study cycles, the unit cost of heat (LCOH, $/kWhth) was determined for two cases with constant a) Space Heating (SH) and b) Domestic Hot Water (DHW) loads with the peak of 8kW. Upon the obtained results, the (EVC+HE) and (EC+IHE) were found to have the lowest unit cost of heat and the highest COP, respectively. Also, it was shown that for the SH/DHW loads, 19% /11% increase in the COP results in 11% /7.5% increase in the LCOH for the electricity cost of 0.025$/kWh. Also, for the mentioned percentages increase in the COP and considering the electricity cost of 0.25$/kWh, LCOH is increased by about 6% for the SH and DHW cases, respectively. It was also concluded that the configuration with the highest COP is not that with the lowest unit cost of heat.

In this paper, an integrated combined cycle power plant structure and thermal water desalination system have been developed. The integrated design has the capacity of producing 188.6 MW power, 34.77 kg/s fresh water and 94.57 kg/s liquid carbon dioxide. In order to supply the input heat to combined cycle power plant and thermal water desalination, heat recovery operations are carried out on the out flow of the oxy-fuel power plant and solar collector cycle. Also, in order to provide cooling of the air separation unit and natural gas used in the integrated structure, the operation of converting liquefied natural gas (LNG) to natural gas is used. In the integrated structure, the overall thermal energy efficiency is 75.83% and total exergy efficiency is 93.67%, respectively. The total thermal energy efficiency is 13.62% and the total exergy efficiency is 34.55% higher than the integrated structure developed compared to other similar structures in this paper. The period of return is 2.282 years and prime cost of freshwater product is 0.2269 US$/m3.

Nowadays, as the wind turbines become widespread, it is important to study the turbine behavior in real conditions which may be oscillatory or unsteady. Identifying the characteristics of airfoils used in HAWT blades in different operating conditions can predict the turbine’s aerodynamic performance with acceptable accuracy. In this research, NACA 4412 airfoil has been studied and its performance characteristics such as lift coefficients drag coefficient, surface pressure distributions in subsonic flow field at different angles of attack and Reynolds numbers have been investigated. For this purpose, RANS & URANS governing equations are employed for analysis of static & oscillating conditions respectively using two different turbulence models (Spallart- Almaras and "K-ω" ) have been used. Numerical solution results show acceptable agreement with experimental results. Furthermore, the aerodynamic characteristics with oscillating flow field and the effect of Reynolds number and reduced domain variation has been investigated. The results show that as the value of the reduced domain parameter increases, the lift coefficient increases, but drag coefficient, especially at low angles of attack, does not change significantly.

The ejector is the most important part of the ejection refrigeration cycle. Due to the complexity and different phenomena, such as suction, mixing, and shock that occur inside the ejector, it requires separate and accurate analysis. This study, for the first time, numerically was investigated the effects of secondary inlet pressure on the ejector flow. The effects of this parameter on pressure, Mach number and temperature inside the ejector were studied. The governing equations were solved by the finite volume method and compressible, two-dimensional, axisymmetric and fully turbulent flow. By comparing numerical results with experimental and analytical results, there is a reasonable agreement between them. In order to obtain the best performance of an ejector, at different secondary inlet pressure, the entrainment ratio and exergy efficiency were computed and compared to reach the best condition. The results show that there is inverse flow at low pressure less than 0.8 bar, but it becomes better by increasing the pressure up to 1.5 bar and more than that, no improvement was found. By reducing the secondary inlet pressure from the optimal pressure of 1.5 bar until the reversal flow occurs, the entrainment ratio decreases by about 105% and the exergy efficiency by about 64%.

In current research, a new optimal hybrid guidance law is developed based on classic guidance laws and fuzzy control approach. An optimized line of sight guidance law is designed for the midcourse phase. Proportional navigation method is also used for terminal phase. In order to smoothly switching between midcourse and terminal phases, an optimal fuzzy approach is developed. Two excising multi-objective heuristic optimization algorithms are evaluated for optimal fuzzy switch problem and the non-dominated sorting gravitational search algorithm is chosen due to its less computational run time. A data bank of optimal switching conditions in several flight scenarios is derived and based on these optimal points, the optimal switching plan is derived for all missions. Also, the practical zone is determined. The advantage of the proposed approach is that it has been derived in presence of vehicle full nonlinear dynamic model. Different scenarios simulation results revealed the desirable performance of the proposed approach.

In this paper, compressible flows inside converging-diverging nozzles with different shapes at off-design conditions are investigated. Shock waves and boundary layer separation are present in the flows. The area ratio (outlet to the throat) and the total combustion chamber pressure are constant while the nozzle shape and its length are variable. Compressible Reynolds averaged Navier Stokes equations were solved using the finite volume method. By comparing the results with available experimental data, the Spalaret-Almars model has been selected for simulations. Axisymmetric, unsteady and viscous flow is considered in all simulations. The characteristics method is used to design the nozzle shape and important parameters such as Mach number, wall shear stress, Thrust efficiency, wall pressure, axial pressure and boundary layer separation due to shock wave are studied in detail. By comparing the results for four shapes of the nozzles (i.e. perfect nozzles, 15° conical nozzle, 30° conical nozzle, and minimum length nozzle) it is concluded that at design condition, perfect nozzle gives higher thrust (7889 N). However, at off-design conditions (when the shock wave and boundary layer separation occurs), the conical nozzle gives higher thrust (9700 N).

Water based CuO nanofluids at low concentrations were used to improve the heat transfer performance and also decrease the water consumption in an experimental cooling tower. Nanofluids were prepared at the concentrations lower than 0.2 wt.% and stabilized completely. Then, the experiments were conducted at different water circulating flowrates and different water return temperatures. Air flowrate was considered to be constant at all the experiments. Results showed that using nanofluid improved the cooling performance and enhance the temperature difference between the water supply and return in the cooling tower in comparison with pure water. For example, using CuO nanofluid at the concentration of 0.2 wt.% and at the return temperature of 41 oC and water to air flow rate ratio (L/G) of 0.8 decreased the water outlet temperature from the cooling tower up to %21 compare with pure water, and decrease the water loss up to %16. Results showed that using nanofluid decreased the water loss due to evaporation considerably, and therefore, it is possible to design the cooling tower in smaller size on the basis of this heat transfer enhancement.

Mg-Zn alloys system due to the hardening ability have a crucial importance, therefore many researches have been done to increase creep resistance and maintain the alloy strength at elevated temperature. Some researches has been done about the effect of alloying elements, heat treatment and improving mechanical properties. In these studies, the effect of alloying elements with different weight and different heat treatment cycles on microstructure and mechanical properties have been studied. Investigation has been done mainly by optical microscopy, hardness, tensile and impression creep. The results of these study showed that alloying elements can be created high thermal stable compounds with and high creep resistance increase. On the other hand, solutionizing and aging treatment can cause grain refinement and precipitation dispersion. Because, dispersion barriers against motion of dislocations, they can increase creep resistance.

A 320 MW steam power plant is studied and the energy and exergy analysis for the components as well as for the whole cycle of the plant is based on real data. Studies show that due to exergy dependence on ambient temperature, overall exergy efficiency increases by 5.93% with decreasing ambient temperature. Exergy degradation due to condenser temperature also leads to a 2.8% reduction in nominal load exergy degradation and 1.4% energy efficiency reduction in condenser temperature reduction. Also, the total exergy losses of the power plant cycle are about nominal output power, which however decreases by approximately 5% at nominal loads relative to the half load. The highest exergy losses were related to boilers and about 80% of total losses and the lowest exergy losses were for pumps and about 0.3%. Increasing the steam temperature of the boiler will also result in a 1.39% increase in energy efficiency and a 1.64% increase in the overall exergy efficiency of the plant. Therefore, the most cost efficient way of operating the power plant is at high and rated load conditions.

This paper aims to develop a model for the growth of reliability with a normal distribution based on the Non-Homogeneous Poisson Process (NHPP). To this end, firstly, the framework for modeling the NHPP for the reliability growth equation with the assumption of the normal distribution for failure data is extracted and the equations for the reliability growth based on the nonhomogeneous Poisson process with normal distribution are obtained. Then, to evaluate the reliability model with the given failure data, the maximum likelihood estimation technique was used to estimate the effective parameter in reliability growth. To estimate reliability parameters, the repetitive mathematical expectation methods are used to solve the equations derived from the maximum likelihood estimation. Finally, the proposed model, by implementing the failure data of an aerospace system, shows that the present approach is highly accurate in comparison with the basic reliability growth models, and simulates the process of growth or deterioration of the system with high precision.

In order to optimize the performance of valves including conical poppet with sharp seat in hydraulic power transmission circuits, it is to necessary examine the properties of the oil flow passing through them. In this paper, influence of the vertex angle and displacement of the poppet conical and pressure of the valve inlet oil on the distribution of the pressure and velocity of the oil flow in the valve chamber and the axial force applied to the conical poppet, are evaluated. To ensure the accuracy of numerical analysis, the results of the measured force on the conical poppet are compared with the numerical results. Studies showed that increasing the angle of cone and displacement of the poppet, the maximum speed of the flow of oil through the valve is reduced. Also, increasing the angle of poppet leads to an increase in the axial force on the movable member of the conical valve. A good agreement between the experimental results with the numerical results indicates the accuracy of the numerical solution of the problem. On the other hand, the displacement of the poppet in valves with a sharp edges seat has an inverse effect on the force applied to the moving poppet member.

The importance of changing of the longitudinal and transverse layout of the side hulls of a trimaran ship on the total resistance in calm water has been of great interest. Wave making resistance can be reduced by changing the layout of side hulls. In this study, for verification of numerical solution, some changes in the side hull arrangements are vinvestigated by experimental method and the amount of resistance and its related variables have been carefully recorded. Then, a numerical model is developed in OpenFOAM code, and in a numerical tank, the details of the fluid flow around the ship are investigated. In addition to the proper accuracy of numerical results, it can be seen that transverse position of the side hulls has a significant effect on ship total resistance. However, the variation in drag force due to the changing in longitudinal position of side hulles is. The effect of the transverse distance between the side hulls and the central line of the main hull has been studied dimensionlessly on four different positions. A similar analysis has been performed for changes in the longitudinal position of the side hulls. Changing the transverse layout of the side hulls can result in a 10% reduction in the total resistance of trimaran.

The aim of this study is to use graphene oxide (GO) with a volumetric concentration of 0.2, 0.4 and 0.6 in paraffin as phase change material (PCM) to improve the efficiency of solar desalinization for concentrated water desalination programs. The results show that by adding graphene oxide to paraffin, the melting temperature decreases. The solar desalination, which uses paraffin and graphene oxide, has a 25% improvement in sub cooling factor compare to solar desalination, which only works with phase change materials. The Nusselt number obtain during melting also suggests that the free heat transfer in the melting zone of the solar still by adding graphene oxide with a suitable weight percentage in the phase change material compare with that in which paraffin alone is used as a phase change material has been increased. Also, increasing the temperature of the upper hot wall (Th) increases the Nusselt number. Finally, an empirical equation for the relationship between the average Nusselt number as a function of Rayleigh number (Ra), Stefan number (Ste), cooling factor (Sb) and Fourier number (Fo) is presented. The obtain formula shows that increasing the Nusselt number has an inverse relationship with the Fourier number.