MHD Generator Research Articles

Performance evaluation of a liquid-sodium thermoacoustic engine with magnetohydrodynamic electricity generation based upon the Swift model.

Liquid sodium is an attractive working fluid for thermoacoustic conversion. Herein, a numerical study on a standing-wave thermoacoustic electricity generation system with liquid sodium as the working fluid is presented, based upon the Swift model. The characteristics of the thermoacoustic conversion and the output performance of the system have been investigated. The results show that the sodium engine can reach a power density much higher than the classical gas engine. Due to the strong acoustic coupling between components, the electricity output is significantly affected by the input heating power, the magnetic flux density, and the load ratio. In a typical case, the thermal-to-electric efficiency and the relative Carnot efficiency can reach 4.6% and 7.8%, respectively, with a temperature difference of 563 K and an input heat of 5 kW. More importantly, the output electricity density reaches 150 kW/m3, higher than some commercially available technologies. These results demonstrate the potential of such technology for small-scale electricity generation. Its extremely simple structure without any mechanical moving part endows the system with high reliability and long lifetime, if risks of corrosion and exposure to air and water can be avoided.

Experimental and numerical study of a liquid metal magnetohydrodynamic generator for thermoacoustic power generation

Combining a thermoacoustic cycle engine with a liquid metal magnetohydrodynamic (LMMHD) generator will result in a thermal power generation system with no mechanical moving parts and high reliability. This disruptive technology has drawn much attention in space nuclear power generation, especially in recent years. It requires an LMMHD generator to work at a higher frequency than conventional LMMHD generators targeted for ocean wave energy conversion. However, the operating characteristics and loss mechanisms of LMMHD generators at high operating frequencies remain poorly understood, and experimental characterization of such a generator is lacking. In this work, a three-dimensional transient numerical analysis of a high-frequency LMMHD generator is performed based on multi-physics field simulation software COMSOL, to understand the operating characteristics of the generator, and the effects of inlet velocity, load resistance, and operating frequency on the generator’s performance. Furthermore, an LMMHD generator prototype was designed, constructed, and tested under different inlet velocities, load resistances, and frequencies by using a linear compressor for the first time. When the operating frequency and inlet velocity are 15 Hz and 4.3 m/s, the output voltage and current of the generator prototype reached 113 mV and 1720 A, with an output power of 68 W at a corresponding acoustic-to-electric efficiency of 24 %. A discrepancy between the numerical predictions and the experimental results was found, which gave insight into where further improvements can be made. This work reveals the operating characteristics and losses mechanism of LMMHD generators operating at higher frequencies and contributes to the development of high-efficiency generators for thermoacoustic power generation.

Entropy generation on MHD flow of Williamson hybrid nanofluid over a permeable curved stretching/shrinking sheet with various radiations

The research aims to scrutinize the significance of MHD, radiation, Joule heating, porous medium, and heat generation impacts on the convective boundary flow. It contains blood as the base fluid, silver (Ag), and aluminum oxide (Al2O3 ) as the nanoparticles over a curved stretching and shrinking sheet. To investigate the rheology of blood, we used a Williamson fluid model with comparisons for both curved stretching/shrinking sheets for velocity and a comparison between linear and nonlinear thermal radiation for temperature, entropy production, and Bejan number. This model is used to study physiological systems, surgical procedures, and nano-pharmacological delivery systems. The governing nonlinear coupled PDEs are transformed into nonlinear associated ODEs via the similarity transformation in the Maple software solver using the Homotopy Perturbation Method. The Homotopy Perturbation Method (HPM) results are compared with the Numerical Method (NM) results, where HPM gives reliable results. The results and discussion provide graphical results with several parameters for velocity, temperature, entropy production, Bejan number, Nusselt number, and skin friction. As the magnetic field parameter (M) increases, the velocity profile decreases for curved stretching sheet. We observe the opposite behavior of the velocity profile for a curved shrinking sheet. As the volume fractions increase, the temperature profile increases for linear and non-linear thermal radiation. The temperature profile rises due to the thermal conductivity of nanoparticles, which is amplified by estimations of the nanoparticle volume fractions.

A method to optimize the external magnetic field to suppress the end current in liquid metal magnetohydrodynamic generators

Liquid metal magnetohydrodynamic (LMMHD) generators can generate electricity from the movement of liquid metal in an external magnetic field, without any mechanical moving parts. They can be applied in scenarios that require a highly reliable generator, including harvesting ocean wave energy and space nuclear power generation. However, in practice, the presence of end current in LMMHD generators limits the generator efficiency to a relatively low level, which restricts their wide application. After analyzing the generating mechanism of the end current, we report herein a method to optimize the external magnetic field that cancels the electrostatic field with the motional electromotive force and suppresses the end current in LMMHD generators. We use two-dimensional numerical simulations with the proposed method to demonstrate its effectiveness, which we validate with three-dimensional numerical simulations. Both the two- and three-dimensional numerical results show that this method should suppress the end current in the LMMHD generator, thus cutting the internal joule heating nearly in half and improving the generator efficiency by 9.5%. This work provides in-depth insights into both the generating mechanism and the suppression of the end current and contributes to the development of high-efficiency LMMHD generators.

Development of capacitive‐coupled hall‐type MHD generator

AbstractWe have demonstrated a capacitively coupled Hall‐type MHD generator using ECR plasma. To clarify the characteristics of the fabricated MHD generator, we measured the power generation characteristics as a function of magnetic field strength using a DC Hall‐type MHD power generation experiment. The results showed that the output power decreased due to magnetic pressure at the higher magnetic field. However, the output power corresponded to the theoretical value at the lower magnetic field. An AC Hall‐type MHD power generation experiment was conducted using an AC magnetic field. As a result, full‐wave rectification voltage was observed as per theory. Finally, capacitively coupled Hall‐type MHD power generation experiments were conducted, and full‐wave rectified waveforms were observed as in AC Hall‐type MHD power generation. These waveforms were similar to the output waveforms predicted from theory. These results show that the capacitively coupled Hall‐type MHD generator is feasible.

Numerical Study of Plasma Behavior and Power Generation Characteristics in a Disk MHD Generator With Xenon-Seeded Neon Gas

Plasma behavior and power generation characteristics in a disk-shaped magnetohydrodynamic (MHD) power generator with xenon (Xe)-seeded neon (Ne) working gas have been examined by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$</tex-math> </inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta $</tex-math> </inline-formula> 2-D numerical simulation under the assumptions of isentropic gas flow and a non-MHD interaction. The seed fraction (mole% of Xe in the mixture) is varied from 0.0% (pure Ne) to 100.0% (pure Xe) to assess the effect that Xe seeding has on plasma behavior and power generation characteristics. Adding a small amount of Xe to Ne improves the power performance due to the resulting enhancement of electrical conductivity, whereas adding an excessive amount (over 5.0%) deteriorates the performance. At an appropriate load resistance, uniform plasma structure occurs for low seed fractions. However, the plasma structure becomes nonuniform for excessively high seed fractions (over 5.0%). This result suggests that adding an excessive amount of Xe can reduce the uniformity of the plasma structure in the generator. On the other hand, at low load resistance ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$&lt;$</tex-math> </inline-formula> the appropriate load resistance) and high load resistance ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$&gt;$</tex-math> </inline-formula> the appropriate load resistance), a nonuniform plasma structure occurs for any seed fraction and results in the deterioration of generator performance. The electron temperature ranges for maintaining uniform plasma structure become narrower as the seed fraction increases which can be attributed to the low ionization potential of Xe and its large atomic weight.

Analysis of unsteady thermo-solutal MoS[formula omitted]-EO Brinkman electro-conductive reactive nanofluid transport in a hybrid rotating Hall MHD generator

MHD rotating generators offer a plausible renewable energy mechanism. New designs are emerging in which nanotechnology is contributing. Such systems are increasingly deploying more complex functional fluid materials such as base fluids containing magnetic nanoparticles which constitute electromagnetic nanofluids and can be tuned to enhance efficiencies. Motivated by these developments, a mathematical model is presented for the combined effects of Hall current, heat source, chemical reaction and radiative flux on the unsteady rotating thermo-solutal magnetohydrodynamic transport of a Molybdenum disulphide (MoS 2)-EO oil electroconductive Brinkman nanofluid to study the boundary layer characteristics in the vicinity of the side wall of an MHD generator system. The governing dimensional conservation equations are scaled using appropriate transformations into a system of dimensionless coupled partial differential equations. Under appropriate initial and boundary conditions, solutions are derived using the Laplace Transform Method (LTM) and complex variables. The physical impacts of the magnetic, nanoscale, thermal and species control parameters on primary and secondary velocity, temperature and concentration are visualized graphically. The judicious doping of the base fluid with MoS 2 nanoparticles is shown to achieve superior thermal performance for MHD rotating energy generators.

Causality in heliophysics: Magnetic fields as a bridge between the Sun’s interior and the Earth’s space environment

Our host star, the Sun, is a middle-aged main sequence G type star whose activity varies. These variations are primarily governed by solar magnetic fields which are produced in the Sun’s interior via a magnetohydrodynamic dynamo mechanism. Solar activity manifests across different timescales, spanning transient phenomena such as flares, energetic particle events and coronal mass ejections to short to long-term modulation of solar irradiance, plasma winds, open flux and cosmic ray flux in the heliosphere. Collectively, these phenomena define space weather and space climate, which impact the state of the near-Earth space environment, the Earth’s magnetosphere, atmosphere and our space-reliant technologies. Understanding physical processes that are at the heart of solar variability and which causally connect the Sun–Earth system is therefore of immense importance to humanity. Such understanding leads to predictions of the impact of solar activity on our planet and provides a window to explore the plasma universe and other star–planet systems, including assessing the habitability of (exo)planets. In this review, based on our research on the solar–terrestrial system and extant scientific literature, we illuminate processes related to the genesis of solar magnetic fields in the Sun’s interior, their emergence and evolution, their manifestation as solar eruptive events, and their eventual impact on the geospace environment mediated via solar winds and storms. We focus on few phenomena that establish causal connections and demonstrate how our current understanding can lead to development of predictive capabilities encompassing the domain of heliophysics.

Contribution of the Hall term in small-scale magnetohydrodynamic dynamos

The Hall effect plays an interesting role in the context of astrophysical dynamos. Using direct numerical simulation, the dynamo action in Hall magnetohydrodynamic turbulence has been systematically studied. Unlike previous studies, the total Hall contribution has been decomposed, separating the contribution of magnetic and current fields in feeding the large-scale magnetic field. Using scale-specific flux relations, it is shown that the small-scale magnetic and current fields contribute to the large-scale magnetic field growth. Unlike small-scale magnetic fields and large-scale current fields, the small-scale current fields appear to be the leading contributors to small-scale dynamo action.

Development of Capacitive-coupled Hall-type MHD Generator

We have demonstrated a capacitively coupled Hall-type MHD generator using ECR plasma. To clarify the characteristics of the fabricated MHD generator, we measured the power generation characteristics as a function of magnetic field strength using a DC Hall-type MHD power generation experiment. The results showed that the output power decreased due to magnetic pressure at the higher magnetic field. However, the output power corresponded to the theoretical value at the lower magnetic field. An AC Hall-type MHD power generation experiment was conducted using an AC magnetic field. As a result, full-wave rectification voltage was observed as per theory. Finally, capacitively coupled Hall-type MHD power generation experiments were conducted, and full-wave rectified waveforms were observed as in AC Hall-type MHD power generation. These waveforms were similar to the output waveforms predicted from theory. These results show that the capacitively coupled Hall-type MHD generator is feasible.

Analysis of a novel concentrated solar power and magnetohydrodynamic liquid metal units integrated system with hydrogen production

The engineers are very interested in concentrated solar power (CSP) due to its renewable energy source nature. However, for this technology to grow, it is crucial to integrate efficient, cost-effective subsystems. On the other hand, since liquid metal magnetohydrodynamic (LMMHD) power generation systems can operate at high temperatures of 600 °C–3000 °C, they are ideal for use as a subsystem of a CSP-based plant to improve efficiency. The use of waste heat recovery units is another method of increasing efficiency and preventing exergy losses. Taking these points into consideration, the proposed trigeneration system includes an LMMHD, a CSP, and humidification-dehumidification and proton exchange membrane units to produce power, freshwater, as well as hydrogen, respectively. Performance evaluation of the presented system includes thermodynamic and thermoeconomic considerations. The results show that the presented system produces 11.87 kW of power, 6.1 m3/h of hydrogen, and 860.2 L/h of freshwater with an energy utilization factor of 45.81%, a total exergy efficiency of 4.63%, and a unit cost of 19.57 $/kWh. The receiver is the most destructive component of the system, with 256.9 kW of exergy destruction. Further, the parametric study indicates that it is possible to maximize the energy efficiency of the system by changing the concentration ratio of the receiver.

Analysis of the efficiency of MHD cycle supported by nanosecond pulsed discharge pre-ionization

A numerical analysis of plasma generation by a nanosecond pulsed discharge in a strong magnetic field was performed. A 2D model of the plasma formation and decay in magnetohydrodynamic (MHD) channel has been elaborated. A comparison with experimental results demonstrates good agreement between calculated and measured plasma distribution in the channel. Preliminary analysis of the efficiency of the MHD-generator based on the nanosecond dielectric barrier discharge flow pre-ionization shows that the ratio of the power required to maintain ionization in the flow to the power extracted from the flow in MHD process is ∼3% for an ideal Faraday MHD generator with segmented electrodes with BY ∼ 5 T and supersonic flow speed uZ ∼ 1600 m s−1. Such a high energy efficiency of the generator allows us to consider this concept of the generator as promising for energy generation and flow control.

Simulation of Prandtl Nanofluid in the Mixed Convective Flow of Activation Energy with Gyrotactic Microorganisms: Numerical Outlook Features of Micro-Machines

The physiological systems and biological applications that have arisen during the past 15 years depend heavily on the microscale and nanoscale fluxes. Microchannels have been utilized to develop new diagnostic assays, examine cell adhesion and molecular transport, and replicate the fluid flow microenvironment of the circulatory system. The various uses of MHD boundary flow in engineering and technology are extensive, ranging from MHD power generators and the polymer industry to MHD flow meters and pumps and the spinning of filaments. In this investigation, the (Magnetohydrodynamic) MHD flow of Prandtl nanofluid is investigated along with mixed convection, energy activation, microorganism, and chemical reaction. The flow model is considered through partial differential equations in dimensionless form which is then integrated numerically via considering the Bvp4c technique. The outcome is numerous emerging physical parameters over velocity profile, temperature, mass concentration, and microorganism with the separate pertinent quantities such as the Prandtl fluid parameter, elastic fluid parameter, magnetic field, mixed convection parameter, activation energy, chemical reaction, Brownian motion, thermophoretic force, Prandtl number, and Schmidt number. The friction factor, rate of heat transfer and Sherwood number, and density of microbes are revealed numerically and graphically. The outcomes indicate that the Prandtl fluid parameter and elastic fluid parameter tend to enhance the velocity profile. It is also noted that the Prandtl fluid parameter depreciates the thermal rate with the addition of the concentration profile while the opposite trend is recorded for activation energy. Obtained numerical outcomes are correspondingly compared with the current statistics in limiting cases and a close match is obtained.

Investigation of induced magnetic field on MHD radiative flow across an exponentially stretching sheet

In the analysis of magneto-hydrodynamics (MHD) flow, the induced magnetic field (i.m.f) is of critical concern owing to its diverse applications in technological and scientific phenomena such as MRI, glass manufacturing, geophysics, and MHD generators, magneto-hydrodynamic pumps, magneto-hydrodynamic generators, blood flow, etc. The configuration of fluid flow is impacted by the i.m.f, also it is essential to more precisely regulate the rate of flow nature. With this motivation, the present article is focused on the exploration of boundary layer flow behaviour and heat transfer characteristics in a viscous and conducting fluid moving over a stretching surface susceptible to heat radiation and i.m.f effects. Transformation variables are employed in order to translate the PDEs into non-linear ODEs. The BVP4C strategy is implemented to compute computational outcomes of a non-linear system. Consequences of the existing physical model are shown in tabular and graphical form. Some of the key findings are as the induced magnetic field intensifies for augmented values of magnetic parameter and vice-versa for the reciprocal magnetic field parameter. Temperature is accelerated by radiation parameter and reciprocal magnetic field. Moreover, a noteworthy verification between published data and the present outcomes in special cases. It perceived that our outcomes are in perfect harmony.

Research on precise and stable seed flow delivery control for plasma MHD power generation

In this paper, a technique for precise and stable delivery control of alkali metal seed flow is proposed and experimentally verified. A high-pressure syringe driven by a stepping motor delivers liquid alkali metal seeds through a pneumatic atomizing spray gun. The seed jet is uniformly mixed, heated, vaporized and ionized by the high-temperature main flow in the plasma generator to produce a plasma with high conductivity. Based on the high-temperature arc wind tunnel test platform, an experimental study of the alkali metal cesium seed flow measure and plasma ionization characteristics was carried out. The results show that the fluctuation range of the measured seed flow was 5.93-10.39% at a small flow rate of 19 mL/min. For the back pressure of 6 atm, the fluctuation range of the measured seed flow is 6.04-11.55%, which proves that the back pressure of the plasma generator has insignificant effect on the seed flow. The plasma ionization characteristics test shows that the conductivity of the argon cesium equilibrium plasma is between 26.93-31.69 S/m, indicating that the ionization characteristics of the plasma are relatively stable, which can indirectly prove that the seed system has a precise and stable flow rate during the actual power generation. Key words: MHD power generation, alkali metal seed, seed fraction, precise and stable injection. Tables 1, Figs 9, Refs 11.

Design and Numerical Simulation of Gas-liquid Two-phase Mixer for Liquid Metal Magnetohydrodynamic Power Generation System

Design and Numerical Simulation of Gas-liquid Two-phase Mixer for Liquid Metal Magnetohydrodynamic Power Generation System

НЕЗВИЧАЙНА СОНЯЧНА АКТИВНА ОБЛАСТЬ NOAA 13088/13102

Background. Under certain conditions, deep fluctuating magnetic fields lead to violations of Hale’s and Joy’s laws of observed magnetism on the surface of the Sun. These magnetic fluctuations can be excited by two qualitatively different mechanisms of a small-scale dynamo. The first mechanism is a macroscopic MHD dynamo, while the second mechanism is a classical MHD diffusion dynamo. An important difference between the two mechanisms is the percentage of observed anti-Hale sunspot groups (relative to the total number of sunspots) in solar cycle minima. In the case of the first mechanism, the percentage of anti-Hale groups does not depend on the phase of the cycle, while the specified percentage associated with the second mechanism should reach its maximum value in solar minima. To separate the minor contributions of the two named sources of magnetic fluctuations, the researchers proposed a theoretical test based on statistical analysis of observational data over long periods of time (Sokoloff, &amp; Khlystova, 2010). According to the proposed test, the percentage of anti-Hale groups of spots increases during the minima of the cycles, which indicates the favor of the diffusion dynamo. In order to confirm the dominant contribution of the diffusion dynamo to the surface magnetism, this work investigates a specific anomalous active region near the solar minimum. Methods. Macroscopic and classical MHD, which study the behavior of electromagnetic and hydrodynamic fields in turbulent plasma. Analysis of data from observations of the surface magnetism of the Sun. Results. We investigated the evolution of the NOAA 13088/13102 active region observed on August 24, 2022, shortly after the cycle 25 minimum. For the analysis, data from observations using instruments installed on board space observatories were used. A feature was revealed, which consists in the deviation of the surface magnetic configuration of this active region from Hale’s law of the magnetic polarity of groups of spots and Joy’s law of the inclination of the axes of bipolar groups to the latitudinal direction. In addition, it was established that the active region of NOAA 13088/13102 is characterized by rather high flare activity. Conclusions. We believe that the magnetic anomalies of the active region of NOAA 13088/13102 that we found were caused by the influence of magnetic fluctuations excited by the mechanism of the deep small-scale diffusion dynamo, since it is this source that gives the most noticeable contribution to the surface magnetism near the cycle minima. The detection and study of unusual anti-Hale’s AOs with increased eruptive activity, similar to NOAA 13088/13102, may find application in predicting periods of dangerous manifestations of space weather and in forecasting the dynamics of solar cycles.

Наблюдения обратного сейсмоэлектрического эффекта II рода при электрозондированиях в районе Центрально-Сахалинского разлома

The results of experiments on electrical sounding of the near-surface layer of the Earth's crust in the fault zone, which have involved a recording of seismoacoustic and seismic noise in the close zone near the source (the primary dipole source), are represented. The experiments were carried out in 2021-2022 in the southern part of the Central Sakhalin fault with the use of the generator of electric pulses developed at IMGG FEB RAS, output electric power being up to 3 kW. The aim was to reveal seismoacoustic signatures of the medium reaction to the soundings with current pulses of 5–13 A. The generator provided significantly higher current in the dipole than its typical characteristics in the case of soundings for electrical exploration by resistance methods, as well as in the case of conventional seismic and electrical exploration. At the same time, the range of current amplitudes was much smaller in comparison with the case of a deep sounding based on application of geophysical MHD generators or other extra high-power electric pulses units. Up to now, the inverse seismoelectric effect has remained practically unexplored at currents in the “intermediate” range of ~10 A and scale lengths of the order of few hundreds of meters. The presence or absence of the medium reaction to electrical soundings was distinguished by the records of molecular-electronic sensors developed by R-sensors LLC: the CME-6111 broadband seismometer and the hydrophone, installed at a distance of about 50 m from one of the poles of the electric dipole source. An increase in the average level of seismoacoustic noise during electrical soundings was revealed, which is essentially a variety of the inverse seismoelectric effect of the second kind (excitation of elastic waves during an electric current run in a two-phase medium). Previously, no similar signature of medium reaction to the current pulses was noted in the close zone adjacent to one of the dipole electrodes. The noise level increase occurs almost without delay after the start of electrical soundings, and this is in accordance with the previously obtained results on the responses of seismic acoustic emission to powerful current pulses, which were used for a deep sounding in the Northern Tien Shan.

A Computational Study of Magneto-Convective Heat Transfer Over Inclined Surfaces With Thermodiffusion

In this article, the ocean energy generator system is analysed. The need for sustainable renewable energy systems is continually growing, given the situation of the world’s energy supplies. Numerous similar systems, including photovoltaic solar collectors, biomass, and wind turbines are used for energy generation. The ocean energy generator system uses the magnetohydrodynamics transformation concept to convert kinetic energy to electrical energy. Similar to conventional generators, an ocean generator requires an applied magnetic field to generate current, making it a critical component of the system. To optimize the performance and efficiency of ocean generators, various devices utilizing superconducting magnets have been developed, including Hall current generators, rotating channels, rotating disc magnetohydrodynamics generators, and helicoid generators. However, these systems also involve complex heat, momentum, and mass transfer, which can be better understood through mathematical modeling. The similarity transformation are introduced to transform the mathematical model from partial differential equation system to ordinary differential equation system. By adopting this approach, the numerical solution is significantly simplified while still preserving numerous crucial physical aspects of the studied heat and material transport phenomena. The physical characteristics of sea waves are governed by the three variables of seawater: temperature, salinity, and pressure. Small dispersed particles also affect the generation of hydroelectric power from surface water. The behaviour of velocity, temperature, and salinity profile is observed for the variations of different parameters such as magnetic, Grashof number and heat source. The system is converted into an optimization problem and solved by a neural network procedure. The solutions are compared with reference solutions for validation. The errors, performance, testing and training data are also presented graphically. The data is typically visualized using histograms, line graphs, and other visual aids. This allows for easier comprehension and analysis of the data.

Hybrid Nanofluid Flow and Heat Transfer Past a Vertical Cylinder in the Presence of MHD and Heat Generation

The basic objective of this work is to investigate the effects of Lorentz force and internal heat generation on the unsteady flow of a hybrid nanofluid and the heat transfer caused by a moving semi-infinite vertical cylinder. The governing partial differential equations are solved numerically using a robust implicit finite difference approach with proper boundary conditions. The current work is corroborated by existing literature on the subject of special situations of the problem. The effects of the magnetic parameter, the Grashof number, and the heat generation parameter on the Nusselt number, the skin friction coefficient, as well as the velocity and temperature fields, have been investigated and graphed. The results obtained can be applied to a variety of engineering devices, such as chemical reactors, heat exchangers, and solar collectors.