Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 20

CFD simulation Related Abstracts

20 Effect of Highly Pressurized Dispersion Arc Nozzle on Breakup of Oil Leakage in Offshore

Authors: N. M. M. Ammar, S. M. Mustaqim, N. M. Nadzir

Abstract:

The most important problem occurs on oil spills in sea water is to reduce the oil spills size. This study deals with the development of high pressurized nozzle using dispersion method for oil leakage in offshore. 3D numerical simulation results were obtained using ANSYS Fluent 13.0 code and correlate with the experimental data for validation. This paper studies the contribution of the process on flow speed and pressure of the flow from two different geometrical designs of nozzles and to generate a spray pattern suitable for dispersant application. Factor of size distribution of droplets generated by the nozzle is calculated using pressures ranging from 2 to 6 bars. Results obtain from both analyses shows a significant spray pattern and flow distribution as well as distance. Results also show a significant contribution on the effect of oil leakage in terms of the diameter of the oil spills break up.

Keywords: droplets, Oil Spills, arc nozzle, CFD simulation

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19 Effect of Fuel Injection Discharge Curve and Injection Pressure on Upgrading Power and Combustion Parameters in HD Diesel Engine with CFD Simulation

Authors: Saeed Chamehsara, Seyed Mostafa Mirsalim, Mehdi Tajdari

Abstract:

In this study, the effect of fuel injection discharge curve and injection pressure simultaneously for upgrading power of heavy duty diesel engine by simulation of combustion process in AVL-Fire software are discussed. Hence, the fuel injection discharge curve was changed from semi-triangular to rectangular which is usual in common rail fuel injection system. Injection pressure with respect to amount of injected fuel and nozzle hole diameter are changed. Injection pressure is calculated by an experimental equation which is for heavy duty diesel engines with common rail fuel injection system. Upgrading power for 1000 and 2000 bar injection pressure are discussed. For 1000 bar injection pressure with 188 mg injected fuel and 3 mm nozzle hole diameter in compare with first state which is semi-triangular discharge curve with 139 mg injected fuel and 3 mm nozzle hole diameter, upgrading power is about 19% whereas the special change has not been observed in cylinder pressure. On the other hand, both the NOX emission and the Soot emission decreased about 30% and 6% respectively. Compared with first state, for 2000 bar injection pressure that injected fuel and nozzle diameter are 196 mg and 2.6 mm respectively, upgrading power is about 22% whereas cylinder pressure has been fixed and NOX emission and the Soot emissions are decreased 36% and 20%, respectively.

Keywords: CFD simulation, HD diesel engine, upgrading power, injection pressure, fuel injection discharge curve, combustion process

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18 Improvement of Heat Pipe Thermal Performance in H-VAC Systems Using CFD Modeling

Authors: H. Shokouhmand, A. Ghanami

Abstract:

Heat pipe is simple heat transfer device which combines the conduction and phase change phenomena to control the heat transfer without any need for external power source. At hot surface of heat pipe, the liquid phase absorbs heat and changes to vapor phase. The vapor phase flows to condenser region and with the loss of heat changes to liquid phase. Due to gravitational force the liquid phase flows to evaporator section. In HVAC systems the working fluid is chosen based on the operating temperature. The heat pipe has significant capability to reduce the humidity in HVAC systems. Each HVAC system which uses heater, humidifier or dryer is a suitable nominate for the utilization of heat pipes. Generally heat pipes have three main sections: condenser, adiabatic region and evaporator.Performance investigation and optimization of heat pipes operation in order to increase their efficiency is crucial. In present article, a parametric study is performed to improve the heat pipe performance. Therefore, the heat capacity of heat pipe with respect to geometrical and confining parameters is investigated. For the better observation of heat pipe operation in HVAC systems, a CFD simulation in Eulerian- Eulerian multiphase approach is also performed. The results show that heat pipe heat transfer capacity is higher for water as working fluid with the operating temperature of 340 K. It is also showed that the vertical orientation of heat pipe enhances it’s heat transfer capacity used in the abstract.

Keywords: CFD simulation, heat pipe, HVAC system, grooved heat pipe

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17 Heat Pipe Thermal Performance Improvement in H-VAC Systems Using CFD Modeling

Authors: H. Shokouhmand, A. Ghanami

Abstract:

Heat pipe is a simple heat transfer device which combines the conduction and phase change phenomena to control the heat transfer without any need for external power source. At hot surface of the heat pipe, the liquid phase absorbs heat and changes to vapor phase. The vapor phase flows to condenser region and with the loss of heat changes to liquid phase. Due to gravitational force, the liquid phase flows to evaporator section. In HVAC systems, the working fluid is chosen based on the operating temperature. The heat pipe has significant capability to reduce the humidity in HVAC systems. Each HVAC system which uses heater, humidifier or dryer is a suitable nominate for the utilization of heat pipes. Generally, heat pipes have three main sections: condenser, adiabatic region, and evaporator.Performance investigation and optimization of heat pipes operation in order to increase their efficiency is crucial. In the present article, a parametric study is performed to improve the heat pipe performance. Therefore, the heat capacity of the heat pipe with respect to geometrical and confining parameters is investigated. For the better observation of heat pipe operation in HVAC systems, a CFD simulation in Eulerian- Eulerian multiphase approach is also performed. The results show that heat pipe heat transfer capacity is higher for water as working fluid with the operating temperature of 340 K. It is also showed that the vertical orientation of heat pipe enhances its heat transfer capacity.

Keywords: CFD simulation, heat pipe, HVAC system, grooved heat pipe

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16 CFD Simulation of Forced Convection Nanofluid Heat Transfer in the Automotive Radiator

Authors: Sina Movafagh, Younes Bakhshan

Abstract:

Heat transfer of coolant flow through the automobile radiators is of great importance for the optimization of fuel consumption. In this study, the heat transfer performance of the automobile radiator is evaluated numerically. Different concentrations of nanofluids have been investigated by the addition of Al2O3 nano-particles into the water. Also, the effect of the inlet temperature of nanofluid on the performance of radiator is studied. Results show that with an increase of inlet temperature the outlet temperature and pressure drop along the radiator increase. Also, it has been observed that increase of nono-particle concentration will result in an increase in heat transfer rate within the radiator.

Keywords: Heat Transfer, Nanofluid, CFD simulation, car radiator

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15 Heat Transfer Enhancement via Using Al2O3/Water Nanofluid in Car Radiator

Authors: S. Movafagh, Y. Bakhshan

Abstract:

In this study, effect of adding Al2O3 nanoparticle to base fluid (water) in car radiator is investigated numerically. Radiators are compact heat exchangers optimized and evaluated by considering different working conditions. The cooling system of a car plays an important role in vehicle's performance, consists of two main parts, known as radiator and fan. Improving thermal efficiency of engine leads to increase the engine's performance, decline the fuel consumption and decrease the pollution emissions. In this study, the effects of fluid inlet flow rate and nanoparticle volume fraction on heat transfer and pressure drop of acar radiator are studied.

Keywords: Nanofluid, forced convection, CFD simulation, radiator

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14 Designing an Exhaust Gas Energy Recovery Module Following Measurements Performed under Real Operating Conditions

Authors: Piotr Lijewski, Jerzy Merkisz, Pawel Fuc, Andrzej Ziolkowski, Pawel Czarkowski

Abstract:

The paper presents preliminary results of the development of an automotive exhaust gas energy recovery module. The aim of the performed analyses was to select the geometry of the heat exchanger that would ensure the highest possible transfer of heat at minimum heat flow losses. The starting point for the analyses was a straight portion of a pipe, from which the exhaust system of the tested vehicle was made. The design of the heat exchanger had a cylindrical cross-section, was 300 mm long and was fitted with a diffuser and a confusor. The model works were performed for the mentioned geometry utilizing the finite volume method based on the Ansys CFX v12.1 and v14 software. This method consisted in dividing of the system into small control volumes for which the exhaust gas velocity and pressure calculations were performed using the Navier-Stockes equations. The heat exchange in the system was modeled based on the enthalpy balance. The temperature growth resulting from the acting viscosity was not taken into account. The heat transfer on the fluid/solid boundary in the wall layer with the turbulent flow was done based on an arbitrarily adopted dimensionless temperature. The boundary conditions adopted in the analyses included the convective condition of heat transfer on the outer surface of the heat exchanger and the mass flow and temperature of the exhaust gas at the inlet. The mass flow and temperature of the exhaust gas were assumed based on the measurements performed in actual traffic using portable PEMS analyzers. The research object was a passenger vehicle fitted with a 1.9 dm3 85 kW diesel engine. The tests were performed in city traffic conditions.

Keywords: Waste Heat Recovery, heat exchanger, CFD simulation, PEMS

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13 Numerical Analysis of the Turbulent Flow around DTMB 4119 Marine Propeller

Authors: K. Boumediene, S. E. Belhenniche

Abstract:

This article presents a numerical analysis of a turbulent flow past DTMB 4119 marine propeller by the means of RANS approach; the propeller designed at David Taylor Model Basin in USA. The purpose of this study is to predict the hydrodynamic performance of the marine propeller, it aims also to compare the results obtained with the experiment carried out in open water tests; a periodical computational domain was created to reduce the unstructured mesh size generated. The standard kw turbulence model for the simulation is selected; the results were in a good agreement. Therefore, the errors were estimated respectively to 1.3% and 5.9% for KT and KQ.

Keywords: CFD simulation, RANS, propeller flow, hydrodynamic performance

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12 Experimental and CFD Simulation of the Jet Pump for Air Bubbles Formation

Authors: L. Grinis, N. Lubashevsky, Y. Ostrovski

Abstract:

A jet pump is a type of pump that accelerates the flow of a secondary fluid (driven fluid) by introducing a motive fluid with high velocity into a converging-diverging nozzle. Jet pumps are also known as adductors or ejectors depending on the motivator phase. The ejector's motivator is of a gaseous nature, usually steam or air, while the educator's motivator is a liquid, usually water. Jet pumps are devices that use air bubbles and are widely used in wastewater treatment processes. In this work, we will discuss about the characteristics of the jet pump and the computational simulation of this device. To find the optimal angle and depth for the air pipe, so as to achieve the maximal air volumetric flow rate, an experimental apparatus was constructed to ascertain the best geometrical configuration for this new type of jet pump. By using 3D printing technology, a series of jet pumps was printed and tested whilst aspiring to maximize air flow rate dependent on angle and depth of the air pipe insertion. The experimental results show a major difference of up to 300% in performance between the different pumps (ratio of air flow rate to supplied power) where the optimal geometric model has an insertion angle of 600 and air pipe insertion depth ending at the center of the mixing chamber. The differences between the pumps were further explained by using CFD for better understanding the reasons that affect the airflow rate. The validity of the computational simulation and the corresponding assumptions have been proved experimentally. The present research showed high degree of congruence with the results of the laboratory tests. This study demonstrates the potential of using of the jet pump in many practical applications.

Keywords: Applications, CFD simulation, jet pump, air bubbles

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11 Thermal Analysis and Computational Fluid Dynamics Simulation of Large-Scale Cryopump

Authors: Yong Wang, Wei Sun, Yue Shuai Zhao, Rong Ping Shao, Guo Hua Ren, Li Chen Sun

Abstract:

A large-scale cryopump (DN1250) used in large vacuum leak detecting system was designed and its performance experimentally investigated by Beijing Institute of Spacecraft Environment Engineering. The cryopump was cooled by four closed cycle helium refrigerators (two dual stage refrigerators and two single stage refrigerators). Detailed numerical analysis of the heat transfer in the first stage array and the second stage array were performed by using computational fluid dynamic method (CFD). Several design parameters were considered to find the effect on the temperature distribution and the cooldown time. The variation of thermal conductivity and heat capacity with temperature was taken into account. The thermal analysis method based on numerical techniques was introduced in this study, the heat transfer in the first stage array and the second stage cryopanel was carefully analyzed to determine important considerations in the thermal design of the cryopump. A performance test system according to the RNEUROP standards was built to test main performance of the cryopump. The experimental results showed that the structure of first stage array which was optimized by the method could meet the requirement of the cryopump well. The temperature of the cryopanel was down to 10K within 300 min, and the result of the experiment was accordant with theoretical analysis' conclusion. The test also showed that the pumping speed for N2 of the pump was up to 57,000 L/s, and the crossover was over than 300,000 Pa•L.

Keywords: Thermal analysis, temperature distribution, CFD simulation, cryopump

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10 A Computational Study of the Effect of Intake Design on Volumetric Efficiency for Best Performance in Motorsport

Authors: Shian Gao, Dominic Wentworth-Linton

Abstract:

This project was aimed at investigating the effect of velocity stacks on the intakes of internal combustion engines for motorsport applications. The intake systems in motorsport are predominantly fuel injection with a plate mounted for the stacks. Using Computational Fluid Dynamics software, the relationship between the stack length and power and torque delivery across the engine’s rev range was investigated and the results were used to choose the best option for its intended motorsport discipline. The test results are expected to vary with engine geometry and its natural manufacturer characteristics. The test was also relevant in bridging between computational data and real simulation as the results show flow, pressure and velocity readings but the behaviour of the engine is inferred from the nature of each test. The results of the data analysis were tested in a real-life simulation on a dynamometer to prove the theory of stack length on power and torque delivery, which helps determine the most suitable stack for the Vauxhall engine for rallying in the Caribbean.

Keywords: CFD simulation, internal combustion engine, intake system, Dynamometer test

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9 A Computational Fluid Dynamics Study of Turbulence Flow and Parameterization of an Aerofoil

Authors: Shian Gao, Mohamed Z. M. Duwahir

Abstract:

The main objective of this project was to introduce and test a new scheme for parameterization of subsonic aerofoil, using a function called Shape Function. Python programming was used to create a user interactive environment for geometry generation of aerofoil using NACA and Shape Function methodologies. Two aerofoils, NACA 0012 and NACA 1412, were generated using this function. Testing the accuracy of the Shape Function scheme was done by Linear Square Fitting using Python and CFD modelling the aerofoil in Fluent. NACA 0012 (symmetrical aerofoil) was better approximated using Shape Function than NACA 1412 (cambered aerofoil). The second part of the project involved comparing two turbulent models, k-ε and Spalart-Allmaras (SA), in Fluent by modelling the aerofoils NACA 0012 and NACA 1412 in conditions of Reynolds number of 3 × 106. It was shown that SA modelling is better for aerodynamic purpose. The experimental coefficient of lift (Cl) and coefficient of drag (Cd) were compared with empirical wind tunnel data for a range of angle of attack (AOA). As a further step, this project involved drawing and meshing 3D wings in Gambit. The 3D wing flow was solved and compared with 2D aerofoil section experimental results and wind tunnel data.

Keywords: CFD simulation, aerofoil, shape function, turbulent modelling

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8 Characterization of Plunging Water Jets in Crossflows: Experimental and Numerical Studies

Authors: Mina Esmi Jahromi, Mehdi Khiadani

Abstract:

Plunging water jets discharging into turbulent crossflows are capable of providing efficient air water interfacial area, which is desirable for the process of mass transfer. Although several studies have been dedicated to the air entrainment by water jets impinging into stagnant water, very few studies have focused on the water jets in crossflows. This study investigates development of the two-phase flow as a result of the jet impingements into crossflows by means of image processing technique and CFD simulations. Investigations are also conducted on the oxygen transfer and a correlation is established between the aeration properties and the oxygenation capacity of water jets in crossflows. This study helps the optimal design and the effective operation of the industrial and the environmental equipment incorporating water jets in crossflows.

Keywords: Image Processing, CFD simulation, two-phase flow, air entrainment, jet in crossflow, oxygen transfer

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7 Internal and External Overpressure Calculation for Vented Gas Explosion by Using a Combined Computational Fluid Dynamics Approach

Authors: Hong Hao, Jingde Li

Abstract:

Recent oil and gas accidents have reminded us the severe consequences of gas explosion on structure damage and financial loss. In order to protect the structures and personnel, engineers and researchers have been working on numerous different explosion mitigation methods. Amongst, venting is the most economical approach to mitigate gas explosion overpressure. In this paper, venting is used as the overpressure alleviation method. A theoretical method and a numerical technique are presented to predict the internal and external pressure from vented gas explosion in a large enclosure. Under idealized conditions, a number of experiments are used to calibrate the accuracy of the theoretically calculated data. A good agreement between the theoretical results and experimental data is seen. However, for realistic scenarios, the theoretical method over-estimates internal pressures and is incapable of predicting external pressures. Therefore, a CFD simulation procedure is proposed in this study to estimate both the internal and external overpressure from a large-scale vented explosion. Satisfactory agreement between CFD simulation results and experimental data is achieved.

Keywords: CFD simulation, internal pressure, external pressure, Ansys fluent, vented gas explosion, FLACS

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6 Implementation of a Low-Cost Instrumentation for an Open Cycle Wind Tunnel to Evaluate Pressure Coefficient

Authors: Cristian P. Topa, Esteban A. Valencia, Victor H. Hidalgo, Marco A. Martinez

Abstract:

Wind tunnel experiments for aerodynamic profiles display numerous advantages, such as: clean steady laminar flow, controlled environmental conditions, streamlines visualization, and real data acquisition. However, the experiment instrumentation usually is expensive, and hence, each test implies a incremented in design cost. The aim of this work is to select and implement a low-cost static pressure data acquisition system for a NACA 2412 airfoil in an open cycle wind tunnel. This work compares wind tunnel experiment with Computational Fluid Dynamics (CFD) simulation and parametric analysis. The experiment was evaluated at Reynolds of 1.65 e5, with increasing angles from -5° to 15°. The comparison between the approaches show good enough accuracy, between the experiment and CFD, additional parametric analysis results differ widely from the other methods, which complies with the lack of accuracy of the lateral approach due its simplicity.

Keywords: Wind Tunnel, CFD simulation, experimental testing, low cost instrumentation

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5 Multi-Stage Optimization of Local Environmental Quality by Comprehensive Computer Simulated Person as Sensor for Air Conditioning Control

Authors: Sung-Jun Yoo, Kazuhide Ito

Abstract:

In this study, a comprehensive computer simulated person (CSP) that integrates computational human model (virtual manikin) and respiratory tract model (virtual airway), was applied for estimation of indoor environmental quality. Moreover, an inclusive prediction method was established by integrating computational fluid dynamics (CFD) analysis with advanced CSP which is combined with physiologically-based pharmacokinetic (PBPK) model, unsteady thermoregulation model for analysis targeting micro-climate around human body and respiratory area with high accuracy. This comprehensive method can estimate not only the contaminant inhalation but also constant interaction in the contaminant transfer between indoor spaces, i.e., a target area for indoor air quality (IAQ) assessment, and respiratory zone for health risk assessment. This study focused on the usage of the CSP as an air/thermal quality sensor in indoors, which means the application of comprehensive model for assessment of IAQ and thermal environmental quality. Demonstrative analysis was performed in order to examine the applicability of the comprehensive model to the heating, ventilation, air conditioning (HVAC) control scheme. CSP was located at the center of the simple model room which has dimension of 3m×3m×3m. Formaldehyde which is generated from floor material was assumed as a target contaminant, and flow field, sensible/latent heat and contaminant transfer analysis in indoor space were conducted by using CFD simulation coupled with CSP. In this analysis, thermal comfort was evaluated by thermoregulatory analysis, and respiratory exposure risks represented by adsorption flux/concentration at airway wall surface were estimated by PBPK-CFD hybrid analysis. These Analysis results concerning IAQ and thermal comfort will be fed back to the HVAC control and could be used to find a suitable ventilation rate and energy requirement for air conditioning system.

Keywords: Indoor Environmental Quality, CFD simulation, computer simulated person, HVAC control

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4 A Study on Mesh Size Dependency on Bed Expansion Zone in a Three-Phase Fluidized Bed Reactor

Authors: Liliana Patricia Olivo Arias

Abstract:

The present study focused on the hydrodynamic study in a three-phase fluidized bed reactor and the influence of important aspects, such as volume fractions (Hold up), velocity magnitude of gas, liquid and solid phases (hydrogen, gasoil, and gamma alumina), interactions of phases, through of drag models with the k-epsilon turbulence model. For this purpose was employed a Euler-Euler model and also considers the system is constituted of three phases, gaseous, liquid and solid, characterized by its physical and thermal properties, the transport processes that are developed within the transient regime. The proposed model of the three-phase fluidized bed reactor was solved numerically using the ANSYS-Fluent software with different mesh refinements on bed expansion zone in order to observe the influence of the hydrodynamic parameters and convergence criteria. With this model and the numerical simulations obtained for its resolution, it was possible to predict the results of the volume fractions (Hold ups) and the velocity magnitude for an unsteady system from the initial and boundaries conditions were established.

Keywords: CFD simulation, three-phase fluidized bed system, mesh dependency study, hydrodynamic study

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3 Impact of Wheel-Housing on Aerodynamic Drag and Effect on Energy Consumption on an Bus

Authors: Amitabh Das, Yash Jain, Mohammad Rafiq B. Agrewale, K. C. Vora

Abstract:

Role of wheel and underbody aerodynamics of vehicle in the formation of drag forces is detrimental to the fuel (energy) consumption during the course of operation at high velocities. This paper deals with the CFD simulation of the flow around the wheels of a bus with different wheel housing geometry and pattern. Based on benchmarking a model of a bus is selected and analysis is performed. The aerodynamic drag coefficient is obtained and turbulence around wheels is observed using ANSYS Fluent CFD simulation for different combinations of wheel-housing at the front wheels, at the rear wheels and both in the front and rear wheels. The drag force is recorded and corresponding influence on energy consumption on an electric bus is evaluated mathematically. A comparison is drawn between energy consumption of bus body without wheel housing and bus body with wheel housing. The result shows a significant reduction in drag coefficient and fuel consumption.

Keywords: Energy Consumption, Drag Coefficient, CFD simulation, wheel-housing

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2 Study on Optimization of Air Infiltration at Entrance of a Commercial Complex in Zhejiang Province

Authors: Yujie Zhao, Jiantao Weng

Abstract:

In the past decade, with the rapid development of China's economy, the purchasing power and physical demand of residents have been improved, which results in the vast emergence of public buildings like large shopping malls. However, the architects usually focus on the internal functions and streamlines of these buildings, ignoring the impact of the environment on the subjective feelings of building users. Only in Zhejiang province, the infiltration of cold air in winter frequently occurs at the entrance of sizeable commercial complex buildings that have been in operation, which will affect the environmental comfort of the building lobby and internal public spaces. At present, to reduce these adverse effects, it is usually adopted to add active equipment, such as setting air curtains to block air exchange or adding heating air conditioners. From the perspective of energy consumption, the infiltration of cold air into the entrance will increase the heat consumption of indoor heating equipment, which will indirectly cause considerable economic losses during the whole winter heating stage. Therefore, it is of considerable significance to explore the suitable entrance forms for improving the environmental comfort of commercial buildings and saving energy. In this paper, a commercial complex with apparent cold air infiltration problem in Hangzhou is selected as the research object to establish a model. The environmental parameters of the building entrance, including temperature, wind speed, and infiltration air volume, are obtained by Computational Fluid Dynamics (CFD) simulation, from which the heat consumption caused by the natural air infiltration in the winter and its potential economic loss is estimated as the objective metric. This study finally obtains the optimization direction of the building entrance form of the commercial complex by comparing the simulation results of other local commercial complex projects with different entrance forms. The conclusions will guide the entrance design of the same type of commercial complex in this area.

Keywords: CFD simulation, air infiltration, commercial complex, heat consumption

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1 Data Centers Temperature Profile Simulation Optimized by Finite Elements and Discretization Methods

Authors: Jose Alberto Garcia Fernandez, Zhimin Du, Xinqiao Jin

Abstract:

Nowadays, data center industry faces strong challenges for increasing the speed and data processing capacities while at the same time is trying to keep their devices a suitable working temperature without penalizing that capacity. Consequently, the cooling systems of this kind of facilities use a large amount of energy to dissipate the heat generated inside the servers, and developing new cooling techniques or perfecting those already existing would be a great advance in this type of industry. The installation of a temperature sensors matrix distributed in the structure of each server would provide the necessary information for collecting the required data for obtaining a temperature profile instantly inside them. However, the high number of temperature probes required to obtain the temperature profiles with sufficient accuracy is very high and expensive. Therefore, other less intrusive techniques are employed where each point that characterizes the server temperature profile is obtained by solving differential equations through simulation methods, simplifying data collection techniques, but increasing the time to obtain results. In order to reduce these calculation times, complicated and slow computational fluid dynamics simulations are replaced by simpler and faster finite element methods simulations which solve the Burgers' equations by backward, forward, and central discretization techniques after simplifying the energy and enthalpy conservation differential equations. The discretization methods employed for solving the first and second order derivatives of the obtained Burgers' equation after these simplifications are the key for obtaining results with greater or lesser accuracy regardless of the characteristic truncation error.

Keywords: Data center, FEM simulation, CFD simulation, temperature profile, Burgers' equations, discretization methods

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