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Variable Refrigerant Flow Systems

Variable Refrigerant Flow Systems
Variable Refrigerant Flow Systems

HVAC Variable Refrigerant Flow Systems

Course No: M03-014

Credit: 3 PDH

A. Bhatia

Continuing Education and Development, Inc.

9 Greyridge Farm Court

Stony Point, NY 10980

P: (877) 322-5800

F: (877) 322-4774

info@https://www.wendangku.net/doc/1a14240827.html,

Variable Refrigerant Flow (VRF) Systems

Variable refrigerant flow (VRF) is an air-condition system configuration where there is one outdoor condensing unit and multiple indoor units. The term variable refrigerant flow refers to the ability of the system to control the amount of refrigerant flowing to the multiple evaporators (indoor units), enabling the use of many evaporators of differing capacities and configurations connected to a single condensing unit. The arrangement provides an individualized comfort control, and simultaneous heating and cooling in different zones.

Currently widely applied in large buildings especially in Japan and Europe, these systems are just starting to be introduced in the U.S. The VRF technology/system was developed and designed by Daikin Industries, Japan who named and protected the term variable refrigerant volume (VRV) system so other manufacturers use the term VRF "variable refrigerant flow". In essence both are same.

With a higher efficiency and increased controllability, the VRF system can help achieve a sustainable design. Unfortunately, the design of VRF systems is more complicated and requires additional work compared to designing a conventional direct expansion (DX) system.

This course provides an overview of VRF system technology.

OVERVIEW OF VRF SYSTEMS

The primary function of all air-conditioning systems is to provide thermal comfort for building occupants. There are a wide range of air conditioning systems available, starting from the basic window-fitted units to the small split systems, to the medium scale package units, to the large chilled water systems, and currently to the variable refrigerant flow (VRF) systems.

The term VRF refers to the ability of the system to control the amount of refrigerant flowing to each of the evaporators, enabling the use of many evaporators of differing capacities and configurations, individualized comfort control, simultaneous heating and cooling in different zones, and heat recovery from one zone to another. VRF systems operate on the direct expansion (DX) principle meaning that heat is transferred to or from the space directly by circulating refrigerant to evaporators located near or within the conditioned space. Refrigerant flow control is the key to many advantages as well as the major technical challenge of VRF systems.

Note the term VRF systems should not be confused with the centralized VAV (variable air volume) systems, which work by varying the air flow to the conditioned space based on variation in room loads.

Split Air-conditioning Systems

Split type air conditioning systems are one-to- one systems consisting of one evaporator (fan coil) unit connected to an external condensing unit. Both the indoor and outdoor units are connected through copper tubing and electrical cabling.

The indoor part (evaporator) pulls heat out from the surrounding air while the outdoor condensing unit transfers the heat into the environment.

Split Air-conditioning System

Advantages of using Split Air-conditioners

?Low initial cost, less noise and ease of installation;

?Good alternative to ducted systems;

?Each system is totally independent and has its own control.

Disadvantages

?There is limitation on the distance between the indoor and outdoor unit i.e.

refrigerant piping can’t exceed the limits stipulated by the manufacturer (usually 100 to 150 ft) otherwise the performance will suffer;

?Maintenance (cleaning/change of filters) is within the occupied space;

?Limited air throw which can lead to possible hot/cold spots;

?Impact on building aesthetics of large building because too many outdoor units will spoil the appearance of the building.

Multi-Split Systems

A multi-type air conditioning system operates on the same principles as a split type air- conditioning system however in this case there are ‘multiple’ evaporator units connected to one external condensing unit. These simple systems were designed mainly for small to medium commercial applications where the installation of ductwork was either too

expensive, or aesthetically unacceptable. The small-bore refrigerant piping, which connects the indoor and outdoor units requires much lower space and is easier to install than the metal ducting. Each indoor unit has its own set of refrigerant pipe work connecting it to the outdoor unit.

Advantages of Multi-splits

?The fact that one large condenser can be connected to multiple evaporators within the building reduces and/or eliminates the need for ductwork installation completely.

?Multi-splits are suitable for single thermal zone (defined below) applications with very similar heat gains/losses.

Drawbacks

?Inability to provide individual control;

?Multi-split systems turn OFF or ON completely in response to a single thermostat/control station which operates the whole system. These systems are therefore not suitable for areas/rooms with variable heat gain/loss characteristics. (Thermal zone: A thermal zone is referred to a space or group of spaces within a building with similar heating and cooling requirements. Each thermal zone must be ‘separately controlled’ if conditions conducive to comfort are to be provided by an HVAC system.)

Any area that requires different temperature, humidity and filtration needs shall be categorized as an independent zone and shall be controlled by dedicated control or HVAC system. Few examples below illustrate and clarify the zone concept: ? A conference room designed for 50 people occupancy shall experience lower temperatures when it is half or quarterly occupied. Thus, the design should include a provision for a dedicated temperature controller for this zone;

? A smoking lounge of airport has different filtration, ventilation (air changes) and pressure requirement compared to other areas; therefore, it is considered a separate zone;

? A hotel lobby area is different from the guest rooms or the restaurant area because of occupancy variations;

?In a commercial building, the space containing data processing equipment such as servers, photocopiers, fax machines and printers are exposed to much larger heat load than the other areas; and hence, it is a different thermal zone;

? A hospital testing laboratory, isolation rooms and operation theatre demand different indoor conditions/pressure relationships than the rest of areas; and thus ,shall be treated as a separate zones;

? A control room or processing facility in industrial set up may require a high degree of cleanliness/positive pressure to prevent ingress of dust/hazardous elements, and therefore, it may be treated as separate zone.

Variable Refrigerant Flow or VRF Systems

VRF systems are similar to the multi-split systems which connect one outdoor section to several evaporators. However, multi-split systems turn OFF or ON completely in response to one master controller, whereas VRF systems continually adjust the flow of refrigerant to each indoor evaporator. The control is achieved by continually varying the flow of refrigerant through a pulse modulating valve (PMV) whose opening is determined by the microprocessor receiving information from the thermistor sensors in each indoor unit. The indoor units are linked by a control wire to the outdoor unit which responds to the demand from the indoor units by varying its compressor speed to match the total cooling and/or heating requirements.

VRF systems promise a more energy-efficient strategy (estimates range from 11% to 17% less energy compared to conventional units) at a somewhat higher cost.

The modern VRF technology uses an inverter-driven scroll compressor and permits as many as 48 or more indoor units to operate from one outdoor unit (varies from manufacturer to manufacturer). The inverter scroll compressors are capable of changing

the speed to follow the variations in the total cooling/heating load as determined by the suction gas pressure measured on the condensing unit. The capacity control range can be as low as 6% to 100%.

Refrigerant piping runs of more than 200 ft are possible, and outdoor units are available in sizes up to 240,000 Btuh.

A schematic VRF arrangement is indicated below:

Figure (Source: Fujitsu)

VRF systems are engineered systems and use complex refrigerant and oil control circuitry. The refrigerant pipe-work uses a number of separation tubes and/or headers (refer schematic figure above).

A separation tube has 2 branches whereas a header has more than 2 branches. Either of the separation tube or header, or both, can be used for branches. However, the separation tube is NEVER provided after the header because of balancing issues.

Compared to multi-split systems, VRF systems minimize the refrigerant path and use less copper tubing. Minimizing the refrigerant path allows for maximizing the efficiency of refrigerant work.

Types of VRF

VRV/VRF systems can be used for cooling only, heat pumping or heat recovery. On heat pump models there are two basic types of VRF system: heat pump systems and energy-recovery.

VRF heat pump systems

VRF heat pump systems permit heating or cooling in all of the indoor units but NOT simultaneous heating and cooling. When the indoor units are in the cooling mode, they act as evaporators; when they are in the heating mode, they act as condensers. These are also known as two-pipe systems.

Cooling Type VRF System

Figure (Source: Fujitsu)

VRF heat pump systems are effectively applied in open plan areas, retail stores, cellular offices and any other area that require cooling or heating during the same operational periods.

Heat Recovery VRF system (VRF-HR)

Variable refrigerant flow systems with heat recovery (VRF-HR) capability can operate simultaneously in heating and/or cooling mode, enabling heat to be used rather than rejected as it would be in traditional heat pump systems. VRF-HR systems are equipped with enhanced features like inverter drives, pulse modulating electronic expansion valves and distributed controls that allow system to operate in net heating or net cooling mode, as demanded by the space.

Each manufacturer has its own proprietary design (2-pipe or 3-pipe system), but most uses a three-pipe system (liquid line, a hot gas line and a suction line) and special valving arrangements. Each indoor unit is branched off from the 3 pipes using solenoid valves. An indoor unit requiring cooling will open its liquid line and suction line valves and act as an evaporator. An indoor unit requiring heating will open its hot gas and liquid line valves and will act as a condenser.

Typically, extra heat exchangers in distribution boxes are used to transfer some reject heat from the superheated refrigerant exiting the zone being cooled to the refrigerant that is going to the zone to be heated. This balancing act has the potential to produce significant energy savings.

Heat Recovery Type VRF System - Figure (Source: Fujitsu)

VRF-HR mixed mode operation leads to energy savings as both ends of the thermodynamic cycle are delivering useful heat exchange. If a system has a cooling COP (Coefficient of Performance) of 3, and a heating COP of 4, then heat recovery operation could yield a COP as high as 7. It should be noted that this perfect balance of heating and cooling demand is unlikely to occur for many hours each year, but whenever mixed mode is used energy is saved. Units are now available to deliver the heat removed from space cooling into hot water for space heating, domestic hot water or leisure applications, so that mixed mode is utilized for more of the year.

VRF-HR systems work best when there is a need for some of the spaces to be cooled and some of them to be heated during the same period. This often occurs in the winter in medium-sized to large sized buildings with a substantial core or in the areas on the north and south sides of a building.

The Coefficient of Performance (COP) is the performance rating used primarily in heat pumps. It is defined as the ratio of heat output to the amount of energy input of a heat pump. It compares the heat produced by the heat pump to the heat obtained from resistance heat. COPs vary with the outside temperature; as the temperature falls the COP falls also since the heat pump is less efficient at lower temperatures. ARI standards compare equipment at two temperatures, 47°F and 17°F, to give you an idea of the COP in both mild and colder temperatures.

Refrigerant Modulation in a VRF System

VRV/VRF technology is based on the simple vapor compression cycle (same as conventional split air conditioning systems) but gives you the ability to continuously control and adjust the flow of refrigerant to different internal units, depending on the heating and cooling needs of each area of the building. The refrigerant flow to each evaporator is adjusted precisely through a pulse wave electronic expansion valve in conjunction with an inverter and multiple compressors of varying capacity, in response to changes in the cooling or heating requirement within the air conditioned space.

We will discuss this further but before that let’s refresh the basic refrigeration cycle.

The fundamental of an air conditioning system is the use of a refrigerant to absorb heat from the indoor environment and transfer it to the external environment. In the cooling mode, indoor units are supplied with liquid refrigerant. The amount of refrigerant flowing through the unit is controlled via an expansion valve located inside the unit. When the refrigerant enters the coil, it undergoes a phase change (evaporation) that extracts heat from the space, thereby cooling the room. The heat extracted from the space is exhausted to the ambient air.

Refrigeration systems can operate on reverse cycle mode with an inclusion of special 4-way reversing valve, enabling the absorption of heat from the external environment and using this heat to raise the internal temperature. When in the heating mode, indoor units are supplied with a hot gas refrigerant. Again, the amount of hot gas flowing through the unit is controlled via the same electronic expansion valve. As with the liquid refrigerant, the hot gas undergoes a phase change (condensation), which releases heat energy into the space. These are called heat pump systems. Heat pumps provide both heating and cooling from the same unit and due to added heat of compression, the efficiency of a heat pump in the heating mode is higher compared to the cooling cycle.

Expansion valve is the component that controls the rate at which liquid refrigerant can flow into an evaporator coil. The conventional refrigeration cycle uses “thermostatic expansion valve (TXV)” that uses mechanical spring for control. It has its drawbacks.

?TXV operation is totally independent of compressor operation;

?TXV is susceptible to hunting, i.e. overfeeding and starvation of refrigerant flow to the evaporator.

As the evaporator load increases, available refrigerant will boil off more rapidly. If it is completely evaporated prior to exiting the evaporator, the vapor will continue to absorb heat (superheat). Although superheating ensures total evaporation of the liquid refrigerant before it goes into the compressor, the density of vapor which quits the evaporator and enters the compressor is reduced leading to reduced refrigeration capacity. The inadequate or high super heat in a system is a concern.

?Too little: liquid refrigerant entering a compressor washes out the oil causing premature failure.

?Too much: valuable evaporator space is wasted and possibly causing compressor overheating problems.

The shortcomings of TXV are offset by the modern electronic expansion valve.

Electronic Expansion Valve (EEV)

With an electronic expansion valve (EEV), you can tell the system what superheat you want and it will set it up. The primary characteristic of EEV is its ability to rotate a prescribed small angle (step) in response to each control pulse applied to its windings. EEV consists of a synchronous electronic motor that can divide a full rotation into a large number of steps, 500 steps/rev. With such a wide range, an EEV valve can go from full open to totally closed and closes down when system is satisfied.

EEV in a VRF system functions to maintain the pressure differential and also distribute the precise amount of refrigerant to each indoor unit. It allows for the fine control of the refrigerant to the evaporators and can reduce or stop the flow of refrigerant to the individual evaporator unit while meeting the targeted superheat.

Design Considerations for VRF Systems

Deciding what HVAC system best suits your application will depend on several variables such as building characteristics; cooling and heating load requirements; peak

occurrence; simultaneous heating and cooling requirements; fresh air needs; accessibility requirements; minimum and maximum outdoor temperatures; sustainability; and acoustic characteristics.

Building Characteristics

VRF systems are typically distributed systems – the outdoor unit is kept at a far off location like the top of the building or remotely at grade level and all the evaporator units are installed at various locations inside the building. Typically the refrigerant pipe-work (liquid and suction lines) is very long, running in several hundred of feet in length for large multi-story buildings. Obviously, the long pipe lengths will introduce pressure losses in the suction line and, unless the correct diameter of pipe is selected, the indoor units will be starved of refrigerant resulting in insufficient cooling to the end user. So it is very important to make sure that the pipe sizing is done properly, both for the main header pipe as well as the feeder pipes that feed each indoor unit.

The maximum allowable length varies among different manufacturers; however the general guidelines are as follows:

?The maximum allowable vertical distance between an outdoor unit and its farthest indoor unit is 164 ft;

?The maximum permissible vertical distance between two individual indoor units is

49 feet,

?The maximum overall refrigerant piping lengths between outdoor and the farthest indoor unit is up to 541 ft.

Note: The longer the lengths of refrigerant pipes, the more expensive the initial and operating costs.

Figure Source: ASHRAE

As stated, the refrigerant piping criteria varies from manufacturer to manufacture; for

example for one of the Japanese manufacturer (Fujitsu), the system design limits are:

Source: Fujitsu

?L1: Maximum height difference between outdoor unit and indoor unit = 50m

?L2: Maximum height difference between indoor unit and indoor unit = 15m

?L3: Maximum piping length from outdoor unit to first separation tube = 70m

?[L3+L4+L5+L6]: Maximum piping length from outdoor unit to last indoor unit = 100m

?L6 & L7: Maximum piping length from header to indoor unit = 40m

?Total piping length = 200m (Liquid pipe length)

Building Load Profile

When selecting a VRF system for a new or retrofit application, the following assessment tasks should be carried out:

?Determine the functional and operational requirements by assessing the cooling load and load profiles including location, hours of operation, number/type of

occupants, equipment being used, etc.

?Determine the required system configuration in terms of the number of indoor units and the outdoor condensing unit capacity by taking into account the total

capacity and operational requirements, reliability and maintenance considerations

Building a load profile helps determine the outdoor condensing unit compressor capacity. For instance, if there are many hours at low load, it is advantageous to install multiple compressors with at least one with inverter (speed adjustment) feature. The figure below shows a typical load profile for an office building.

The combined cooling capacity of the indoor sections can match, exceed, or be lower than the capacity of the outdoor section connected to them. But as a normal practice: ?The indoor units are typically sized and selected based on the greater of the heating or cooling loads in the zone it serves, i.e. maximum peak load expected in any time of the year.

?The outdoor condensing unit is selected based on the load profile of the facility which is the peak load of all the zones combined at any one given time. The important thing here is that it is unlikely that all zones will peak at a given time so an element of diversity is considered for economic sizing. Adding up the peak load for each indoor unit and using that total number to size the outdoor unit will result in an unnecessarily oversized condensing unit. Although an oversized condensing unit with multiple compressors is capable of operating at lower capacity, too much over sizing sometimes reduces or ceases the modulation function of the expansion valve. As a rule of thumb, an engineer can specify an outdoor unit with a capacity anywhere between 70% and 130% of the combined capacities of the indoor units.

Example:

Given a commercial office building with 3 zones. Zone #1 has a peak load of 3 tons, Zone #2 has a peak load of 6 tons and Zone #3 has a peak load of 7 tons.

Combined zone load = 3 + 6 + 7 = 16 tons

Building Peak load = 12 tons

Nearest available sizes for outdoor units are 12.5 tons and 15 tons

Select an outdoor condensing unit of 12.5 tons

Sustainability

One attractive feature of the VRF system is its higher efficiency compared to conventional units. Cooling power in a VRF system is regulated by means of adjusting the rotation speed of the compressor which can generate an energy saving around 30%.

A VRF system permits easy future expansion when the conditions demand. Oversizing however, should be avoided unless a future expansion is planned.

Other sustainability factors include:

?Use of non-ozone depleting environment-friendly refrigerants such as R-410a.

?Opting for heat pump instead of electrical resistance heating in areas demanding both cooling and heating. Heat pumps offer higher energy efficiency.

Simultaneous Heating and Cooling

Some manufacturers offer a VRF system with heat recovery feature which is capable of providing simultaneous heating and cooling. The cost of a VRF-HR is higher than that of a normal VRF heat pump unit and therefore its application should be carefully evaluated. More economical design can sometimes be achieved by combining zones with similar heating or cooling requirements together. For example, the areas that may require simultaneous heating and cooling are the perimetric and interior zones. Perimetric areas with lot of glazing and exposure especially towards west and south will have high load variations. A VRF heat pump type system is capable of providing simultaneous heating and cooling exceeding 6 tons cooling requirement.

Using VRF heat pump units for heating and cooling can increase building energy efficiency. The designer must evaluate the heat output for the units at the outdoor design temperature. Supplemental heating with electric resistors shall be considered only when the heating capacity of the VRF units is below the heating capacity required by the application. Even though supplemental heating is considered, the sequence of operation and commissioning must specify and prevent premature activation of supplemental heating.

First Costs

The installed cost of a VRF system is highly variable, project dependent, and difficult to pin down. Studies indicate that the total installed cost of a VRF system is estimated to be 5% to 20% higher than air or water cooled chilled water system, water source heat pump, or rooftop DX system providing equivalent capacity. This is mainly due to long refrigerant piping and multiple indoor evaporator exchanges with associated controls. Building owners often have no incentive to accept higher first costs, even if the claimed payback period is short, as the energy savings claims are highly unpredictable.

VRF ADVANTAGES

Comfort

?The main advantage of a variable refrigerant flow system is its ability to respond individually to fluctuations in space load conditions. The user can set the ambient

有限元网格划分心得

有限元网格划分的基本原则 划分网格是建立有限元模型的一个重要环节,它要求考虑的问题较多,需要的工作量较大,所划分的网格形式对计算精度和计算规模将产生直接影响。为建立正确、合理的有限元模型,这里介绍划分网格时应考虑的一些基本原则。 1网格数量 网格数量的多少将影响计算结果的精度和计算规模的大小。一般来讲,网格数量增加,计算精度会有所提高,但同时计算规模也会增加,所以在确定网格数量时应权衡两个因数综合考虑。 图1中的曲线1表示结构中的位移随网格数量收敛的一般曲线,曲线2代表计算时间随网格数量的变化。可以看出,网格较少时增加网格数量可以使计算精度明显提高,而计算时间不会有大的增加。当网格数量增加到一定程度后,再继续增加网格时精度提高甚微,而计算时间却有大幅度增加。所以应注意增加网格的经济性。实际应用时可以比较两种网格划分的计算结果,如果两次计算结果相差较大,可以继续增加网格,相反则停止计算。 图1位移精度和计算时间随网格数量的变化 在决定网格数量时应考虑分析数据的类型。在静力分析时,如果仅仅是计算结构的变形,网格数量可以少一些。如果需要计算应力,则在精度要求相同的情况下应取相对较多的网格。同样在响应计算中,计算应力响应所取的网格数应比计算位移响应多。在计算结构固有动力特性时,若仅仅是计算少数低阶模态,可以选择较少的网格,如果计算的模态阶次较高,则应选择较多的网格。在热分析中,结构内部的温度梯度不大,不需要大量的内部单元,这时可划分较少的网格。 2网格疏密 网格疏密是指在结构不同部位采用大小不同的网格,这是为了适应计算数据的分布特点。在计算数据变化梯度较大的部位(如应力集中处),为了较好地反映数据变化规律,需要采用比较密集的网格。而在计算数据变化梯度较小的部位,为减小模型规模,则应划分相对稀疏的网格。这样,整个结构便表现出疏密不同的网格划分形式。 图2是中心带圆孔方板的四分之一模型,其网格反映了疏密不同的划分原则。小圆孔附近存在应力集中,采用了比较密的网格。板的四周应力梯度较小,网格分得较稀。其中图b中网格疏密相差更大,它比图a中的网格少48个,但计算出的孔缘最大应力相差1%,而计算时间却减小了36%。由此可见,采用疏密不同的网格划分,既可以保持相当的计算精度,又可使网格数量减小。因此,网格数量应增加到结构的关键部位,在次要部位增加网格是不必要的,也是不经济的。

_基于ANSYS的有限元法网格划分浅析

文章编号:1003-0794(2005)01-0038-02 基于ANSYS的有限元法网格划分浅析 杨小兰,刘极峰,陈 旋 (南京工程学院,南京210013) 摘要:为提高有限元数值的计算精度和对复杂结构力学分析的准确性,针对不同分析类型采用了不同的网格划分方法,结合实例阐述了ANSYS有限元网格划分的方法和技巧,指出了采用ANSYS有限元软件在网格划分时应注意的技术问题。 关键词:ANSYS;有限元;网格;计算精度 中图号:O241 82;TP391 7文献标识码:A 1 引言 ANSYS有限元分析程序是著名的C AE供应商美国ANSYS公司的产品,主要用于结构、热、流体和电磁四大物理场独立或耦合分析的CAE应用,功能强大,应用广泛,是一个便于学习和使用的优秀有限元分析程序。在ANSYS得到广泛应用的同时,许多技术人员对ANSYS程序的了解和认识还不够系统全面,在工作和研究中存在许多隐患和障碍,尤为突出的是有限元网格划分技术。本文结合工程实例,就如何合理地进行网格划分作一浅析。 2 网格划分对有限元法求解的影响 有限元法的基本思想是把复杂的形体拆分为若干个形状简单的单元,利用单元节点变量对单元内部变量进行插值来实现对总体结构的分析,将连续体进行离散化即称网格划分,离散而成的有限元集合将替代原来的弹性连续体,所有的计算分析都将在这个模型上进行。因此,网格划分将关系到有限元分析的规模、速度和精度以及计算的成败。实验表明:随着网格数量的增加,计算精确度逐渐提高,计算时间增加不多;但当网格数量增加到一定程度后,再继续增加网格数量,计算精确度提高甚微,而计算时间却大大增加。在进行网格划分时,应注意网格划分的有效性和合理性。 3 网格划分的有效性和合理性 (1)根据分析数据的类型选择合理的网格划分数量 在决定网格数量时应考虑分析数据的类型。在静力分析时,如果仅仅是计算结构的变形,网格数量可以少一些。如果需要计算应力,则在精度要求相同的情况下取相对较多的网格。同样在响应计算中,计算应力响应所取的网格数应比计算位移响应多。在计算结构固有动力特性时,若仅仅是计算少数低阶模态,可以选择较少的网格。如果计算的模态阶次较高,则应选择较多的网格。在热分析中,结构内部的温度梯度不大,不需要大量的内部单元,可划分较少的网格。 (2)根据分析数据的分布特点选择合理的网格疏密度 在决定网格疏密度时应考虑计算数据的分布特点,在计算固有特性时,因为固有频率和振型主要取决于结构质量分布和刚度分布,采用均匀网格可使结构刚度矩阵和质量矩阵的元素不致相差很大,可减小数值计算误差。同样,在结构温度场计算中也趋于采用均匀的网格形式。在计算数据变化梯度较大的部位时,为了更好地反映数据变化规律,需要采用比较密集的网格,而在计算数据变化梯度较小的部位,为了减小模型规模,则应划分相对稀疏的网格,这样整个结构就表现出疏密不同的网格划分形式。 以齿轮轮齿的有限元分析模型为例,由于分析的目的是求出齿轮啮合传动过程中齿根部分的弯曲应力,因此,分析计算时并不需要对整个齿轮进行计算,可根据圣文男原理将整个区域缩小到直接参与啮合的轮齿。虽然实际上参与啮合的齿数总大于1,但考虑到真正起作用的是单齿,通常只取一个轮齿作为分析对象,这样作可以大大节省计算机内存。考虑到轮齿应力在齿根过渡圆角和靠近齿面处变化较大,网格可划分得密一些。在进行疏密不同网格划分操作时可采用ANSYS提供的网格细化工具调整网格的疏密,也可采用分块建模法设置网格疏密度。 图1所示即为采用分块建模法进行网格划分。图1(a)为内燃机中重要运动零件连杆的有限元应力分析图,由于连杆结构对称于其摆动的中间平面,其厚度方向的尺寸远小于长度方向的尺寸,且载荷沿厚度方向近似均匀分布,故可按平面应力分析处 38 煤 矿 机 械 2005年第1期

CATIA有限元高级划分网格教程

CATIA有限元高级网格划分教程 盛选禹李明志 1.1进入高级网格划分工作台 (1)打开例题中的文件Sample01.CATPart。 (2)点击主菜单中的【开始】→【分析与模拟】→【Advanced Meshing Tools】(高级网格划分工具),就进入【Advanced Meshing Tools】(高级网格划分工具)工作台,如图1-1所示。进入工作台后,生成一个新的分析文件,并且显示一个【New Analysis Case】(新分析算题)对话框,如图1-2所示。 图1-1【开始】→【分析与模拟】→【Advanced Meshing Tools】(高级网格划分工具)(3)在【New Analysis Case】(新分析算题)对话框内选择【Static Analysis】(静力分析)选项。如果以后打开该对话框的时候均希望是计算静力分析,可以把对话框内的【Keep as default starting analysis case】(在开始时保持为默认选项)勾选。这样,下次进入本工作台时,将自动选择静力分析。 (4)点击【新分析算题】对话框内的【确定】按钮,关闭对话框。 1.2定义曲面网格划分参数 本节说明如何定义一个曲面零件的网格类型和全局参数。 (1)点击【Meshing Method】(网格划分方法)工具栏内的【高级曲面划分】按钮

,如图1-3所示。需要在【Meshing Method】(网格划分方法)工具栏内点击中间按钮的下拉箭头才能够显示出【高级曲 面划分】按钮。 图1-2【New Analysis Case】(新分析算题)对话框图1-3【高级曲面划分】按钮

有限元网格划分

有限元网格划分 摘要:总结近十年有限元网格划分技术发展状况。首先,研究和分析有限元网格划分的基本原则;其次,对当前典型网格划分方法进行科学地分类,结合实例,系统地分析各种网格划分方法的机理、特点及其适用范围,如映射法、基于栅格法、节点连元法、拓扑分解法、几何分解法和扫描法等;再次,阐述当前网格划分的研究热点,综述六面体网格和曲面网格划分技术;最后,展望有限元网格划分的发展趋势。 关键词:有限元网格划分;映射法;节点连元法;拓扑分解法;几何分解法;扫描法;六面体网格 1 引言 有限元网格划分是进行有限元数值模拟分析至关重要的一步,它直接影响着后续数值计算分析结果的精确性。网格划分涉及单元的形状及其拓扑类型、单元类型、网格生成器的选择、网格的密度、单元的编号以及几何体素。在有限元数值求解中,单元的等效节点力、刚度矩阵、质量矩阵等均用数值积分生成,连续体单元以及壳、板、梁单元的面内均采用高斯(Gauss)积分,而壳、板、梁单元的厚度方向采用辛普生(Simpson)积分。 2 有限元网格划分的基本原则 有限元方法的基本思想是将结构离散化,即对连续体进行离散化,利用简化几何单元来近似逼近连续体,然后根据变形协调条件综合求解。所以有限元网格的划分一方面要考虑对各物体几何形状的准确描述,另一方面也要考虑变形梯度的准确描述。为正确、合理地建立有限元模型,这里介绍划分网格时应考虑的一些基本原则。 2.1 网格数量 网格数量直接影响计算精度和计算时耗,网格数量增加会提高计

算精度,但同时计算时耗也会增加。当网格数量较少时增加网格,计算精度可明显提高,但计算时耗不会有明显增加;当网格数量增加到一定程度后,再继续增加网格时精度提高就很小,而计算时耗却大幅度增加。所以在确定网格数量时应权衡这两个因素综合考虑。 2.2 网格密度 为了适应应力等计算数据的分布特点,在结构不同部位需要采用大小不同的网格。在孔的附近有集中应力,因此网格需要加密;周边应力梯度相对较小,网格划分较稀。由此反映了疏密不同的网格划分原则:在计算数据变化梯度较大的部位,为了较好地反映数据变化规律,需要采用比较密集的网格;而在计算数据变化梯度较小的部位,为减小模型规模,网格则应相对稀疏。 2.3 单元阶次 单元阶次与有限元的计算精度有着密切的关联,单元一般具有线性、二次和三次等形式,其中二次和三次形式的单元称为高阶单元。高阶单元的曲线或曲面边界能够更好地逼近结构的曲线和曲面边界,且高次插值函数可更高精度地逼近复杂场函数,所以增加单元阶次可提高计算精度。但增加单元阶次的同时网格的节点数也会随之增加,在网格数量相同的情况下由高阶单元组成的模型规模相对较大,因此在使用时应权衡考虑计算精度和时耗。 2.4 单元形状 网格单元形状的好坏对计算精度有着很大的影响,单元形状太差的网格甚至会中止计算。单元形状评价一般有以下几个指标: (1)单元的边长比、面积比或体积比以正三角形、正四面体、正六面体为参考基准。 (2)扭曲度:单元面内的扭转和面外的翘曲程度。 (3)节点编号:节点编号对于求解过程中总刚矩阵的带宽和波前因数有较大的影响,从而影响计算时耗和存储容量的大小 2.5 单元协调性 单元协调是指单元上的力和力矩能够通过节点传递给相邻单元。为保证单元协调,必须满足的条件是: (1)一个单元的节点必须同时也是相邻点,而不应是内点或边界

有限元网格划分和收敛性

一、基本有限元网格概念 1.单元概述?几何体划分网格之前需要确定单元类型.单元类型的选择应该根据分析类型、形状特征、计算数据特点、精度要求和计算的硬件条件等因素综合考虑。为适应特殊的分析对象和边界条件,一些问题需要采用多种单元进行组合建模。? 2.单元分类选择单元首先需要明确单元的类型,在结构有限元分析中主要有以下一些单元类型:平面应力单元、平面应变单元、轴对称实体单元、空间实体单元、板单元、壳单元、轴对称壳单元、杆单元、梁单元、弹簧单元、间隙单元、质量单元、摩擦单元、刚体单元和约束单元等。根据不同的分类方法,上述单元可以分成以下不同的形式。?3。按照维度进行单元分类 根据单元的维数特征,单元可以分为一维单元、二维单元和三维单元。?一维单元的网格为一条直线或者曲线。直线表示由两个节点确定的线性单元。曲线代表由两个以上的节点确定的高次单元,或者由具有确定形状的线性单元。杆单元、梁单元和轴对称壳单元属于一维单元,如图1~图3所示。 ?二维单元的网 格是一个平面或者曲面,它没有厚度方向的尺寸.这类单元包括平面单元、轴对称实体单元、板单元、壳单元和复合材料壳单元等,如图4所示。二维单元的形状通常具有三角形和四边形两种,在使用自动网格剖分时,这类单元要求的几何形状是表面模型或者实体模型的边界面。采用薄壳单元通常具有相当好的计算效率。

??三维单元的网格具有空间三个方向的尺寸,其形状具有四面体、五面体和六面体,这类单元包括空间实体单元和厚壳单元,如图5所示.在自动网格划分时,它要求的是几何模型是实体模型(厚壳单元是曲面也可以)。 ? 4.按照插值函数进行单元分类 根据单元插值函数多项式的最高阶数多少,单元可以分为线性单元、二次单元、三次单元和更高次的单元。 线性单元具有线性形式的插值函数,其网格通常只具有角节点而无边节点,网格边界为直线或者平面.这类单元的优点是节点数量少,在精度要求不高或者结果数据梯度不太大的情况下,采用线性单元可以得到较小的模型规模.但是由于单元位移函数是线性的,单元内的位移呈线性变化,而应力是常数,因此会造成单元间的应力不连续,单元边界上存在着应力突变,如图6所示。

比较PageRank算法和HITS算法的优缺点

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ANSYS有限元网格划分的基本要点

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有限元网格划分

本文讨论了有限元网格的重要概念,包括单元的分类、有限元误差的分类与影响因素;并讨论分析结果的收敛性控制方法,并由实例说明了网格质量及收敛性对取得准确分析结果的重要性。同时讨论了一些重要网格控制的建议及其他网格设定的说明。 一、基本有限元网格概念 1.单元概述 几何体划分网格之前需要确定单元类型。单元类型的选择应该根据分析类型、形状特征、计算数据特点、精度要求和计算的硬件条件等因素综合考虑。为适应特殊的分析对象和边界条件,一些问题需要采用多种单元进行组合建模。 2.单元分类 选择单元首先需要明确单元的类型,在结构有限元分析中主要有以下一些单元类型:平面应力单元、平面应变单元、轴对称实体单元、空间实体单元、板单元、壳单元、轴对称壳单元、杆单元、梁单元、弹簧单元、间隙单元、质量单元、摩擦单元、刚体单元和约束单元等。根据不同的分类方法,上述单元可以分成以下不同的形式。 3.按照维度进行单元分类 根据单元的维数特征,单元可以分为一维单元、二维单元和三维单元。 一维单元的网格为一条直线或者曲线。直线表示由两个节点确定的线性单元。曲线代表由两个以上的节点确定的高次单元,或者由具有确定形状的线性单元。杆单元、梁单元和轴对称壳单元属于一维单元,如图1~图3所示。

二维单元的网格是一个平面或者曲面,它没有厚度方向的尺寸。这类单元包括平面单元、轴对称实体单元、板单元、壳单元和复合材料壳单元等,如图4所示。二维单元的形状通常具有三角形和四边形两种,在使用自动网格剖分时,这类单元要求的几何形状是表面模型或者实体模型的边界面。采用薄壳单元通常具有相当好的计算效率。 三维单元的网格具有空间三个方向的尺寸,其形状具有四面体、五面体和六面体,这类单元包括空间实体单元和厚壳单元,如图5所示。在自动网格划分时,它要求的是几何模型是实体模型(厚壳单元是曲面也可以)。 4.按照插值函数进行单元分类 根据单元插值函数多项式的最高阶数多少,单元可以分为线性单元、二次单元、三次单元和更高次的单元。 线性单元具有线性形式的插值函数,其网格通常只具有角节点而无边节点,网格边界为直线或者平面。这类单元的优点是节点数量少,在精度要求不高或

有限元网格划分和收敛性

一、基本有限元网格概念 1.单元概述 几何体划分网格之前需要确定单元类型。单元类型的选择应该根据分析类型、形状特征、计算数据特点、精度要求和计算的硬件条件等因素综合考虑。为适应特殊的分析对象和边界条件,一些问题需要采用多种单元进行组合建模。 2.单元分类 选择单元首先需要明确单元的类型,在结构有限元分析中主要有以下一些单元类型:平面应力单元、平面应变单元、轴对称实体单元、空间实体单元、板单元、壳单元、轴对称壳单元、杆单元、梁单元、弹簧单元、间隙单元、质量单元、摩擦单元、刚体单元和约束单元等。根据不同的分类方法,上述单元可以分成以下不同的形式。 3.按照维度进行单元分类 根据单元的维数特征,单元可以分为一维单元、二维单元和三维单元。 一维单元的网格为一条直线或者曲线。直线表示由两个节点确定的线性单元。曲线代表由两个以上的节点确定的高次单元,或者由具有确定形状的线性单元。杆单元、梁单元和轴对称壳单元属于一维单元,如图1~图3所示。 二维单元的网格是一个平面或者曲面,它没有厚度方向的尺寸。这类单元包括平面单元、轴对称实体单元、板单元、壳单元和复合材料壳单元等,如图4所示。二维单元的形状通常具有三角形和四边形两种,在使用自动网格剖分时,这类单元要求的几何形状是表面模型或者实体模型的边界面。采用薄壳单元通常具有相当好的计算效率。

三维单元的网格具有空间三个方向的尺寸,其形状具有四面体、五面体和六面体,这类单元包括空间实体单元和厚壳单元,如图5所示。在自动网格划分时,它要求的是几何模型是实体模型(厚壳单元是曲面也可以)。 4.按照插值函数进行单元分类 根据单元插值函数多项式的最高阶数多少,单元可以分为线性单元、二次单元、三次单元和更高次的单元。 线性单元具有线性形式的插值函数,其网格通常只具有角节点而无边节点,网格边界为直线或者平面。这类单元的优点是节点数量少,在精度要求不高或者结果数据梯度不太大的情况下,采用线性单元可以得到较小的模型规模。但是由于单元位移函数是线性的,单元内的位移呈线性变化,而应力是常数,因此会造成单元间的应力不连续,单元边界上存在着应力突变,如图6所示。

pagerank算法实验报告

PageRank算法实验报告 一、算法介绍 PageRank是Google专有的算法,用于衡量特定网页相对于搜索引擎索引中的其他网页而言的重要程度。它由Larry Page 和Sergey Brin在20世纪90年代后期发明。PageRank实现了将链接价值概念作为排名因素。 PageRank的核心思想有2点: 1.如果一个网页被很多其他网页链接到的话说明这个网页比较重要,也就是pagerank值会相对较高; 2.如果一个pagerank值很高的网页链接到一个其他的网页,那么被链接到的网页的pagerank值会相应地因此而提高。 若页面表示有向图的顶点,有向边表示链接,w(i,j)=1表示页面i存在指向页面j的超链接,否则w(i,j)=0。如果页面A存在指向其他页面的超链接,就将A 的PageRank的份额平均地分给其所指向的所有页面,一次类推。虽然PageRank 会一直传递,但总的来说PageRank的计算是收敛的。 实际应用中可以采用幂法来计算PageRank,假如总共有m个页面,计算如公式所示: r=A*x 其中A=d*P+(1-d)*(e*e'/m) r表示当前迭代后的PageRank,它是一个m行的列向量,x是所有页面的PageRank初始值。 P由有向图的邻接矩阵变化而来,P'为邻接矩阵的每个元素除以每行元素之和得到。 e是m行的元素都为1的列向量。 二、算法代码实现

三、心得体会 在完成算法的过程中,我有以下几点体会: 1、在动手实现的过程中,先将算法的思想和思路理解清楚,对于后续动手实现 有很大帮助。 2、在实现之前,对于每步要做什么要有概念,然后对于不会实现的部分代码先 查找相应的用法,在进行整体编写。 3、在实现算法后,在寻找数据验证算法的过程中比较困难。作为初学者,对于 数据量大的数据的处理存在难度,但数据量的数据很难寻找,所以难以进行实例分析。

有限元分析的基本步骤

一个典型的ANSYS分析过程可分为以下6个步骤: 1定义参数 2创建几何模型 3划分网格 4加载数据 5求解 6结果分析 1定义参数 1.1指定工程名和分析标题 启动ANSYS软件,选择File→Change Jobname命令 选择File→Change Title菜单命令 1.2定义单位 (2) 设置计算类型 ANSYS Main Menu: Preference→Material Props →Material Models →Structural →OK (3) 定义分析类型 ANSYS Main Menu: Preprocessor →Loads →Analysis Type →New Analysis→STATIC →OK 1.3定义单元类型 选择Main Menu→Preprocessor→Element Type→Add/Edit/Delete命令 单击[Options]按钮,在[Element behavior]下拉列表中选择[Plane strs w/thk]选项,单击确定 1.4定义单元常数 在ANSYS程序主界面中选择Main Menu→Preprocessor→Real Constants→Add/Edit/Delete命令 单击[Add]按钮,进行下一个[Choose Element Type]对话框 1.5定义材料参数 在ANSYS程序主界面,选择Main Menu→Preprocessor→Material Props→Material Models命令 (1)选择对话框右侧Structural→Linear→Elastic→Isotropic命令,并单击[Isotropic]选项,接着弹出如下所示[Linear Isotropic Properties for Material Number 1]对话框。 在[EX]文本框中输入弹性模量“200000”,在[PRXY]文本框中输入泊松比“0.3”,单击OK 2创建几何模型 在ANSYS程序主界面,选择Main Menu→Preprocessor→Modeling→Creat→Areas→Rectangle →By 2Corners命令 选择Main Menu→Preprocessor→Modeling→Creat→Areas→Circle→Solid Circle命令 3网格划分(之前一定要进行材料的定义和分配) 选择Main Menu→Preprocessor→Modeling→Operate→Booleans→Subtract→Arears Circle命令 选择Main Menu→Preprocessor→Meshing→Mesh→Areas→Free命令,弹出实体选择对话框,单击[Pick All]按钮,得到如下所示网格 4加载数据 (1)选择Main Menu→Preprocessor→Loads→Define Loads→Apply→Structural→Displacement→On Lines命令, 出现如下所示对话框,选择约束[ALL DOF]选项,并设置[Displacement value]为0,单击OK。

PageRank算法的核心思想

如何理解网页和网页之间的关系,特别是怎么从这些关系中提取网页中除文字以外的其他特性。这部分的一些核心算法曾是提高搜索引擎质量的重要推进力量。另外,我们这周要分享的算法也适用于其他能够把信息用结点与结点关系来表达的信息网络。 今天,我们先看一看用图来表达网页与网页之间的关系,并且计算网页重要性的经典算法:PageRank。 PageRank 的简要历史 时至今日,谢尔盖·布林(Sergey Brin)和拉里·佩奇(Larry Page)作为Google 这一雄厚科技帝国的创始人,已经耳熟能详。但在1995 年,他们两人还都是在斯坦福大学计算机系苦读的博士生。那个年代,互联网方兴未艾。雅虎作为信息时代的第一代巨人诞生了,布林和佩奇都希望能够创立属于自己的搜索引擎。1998 年夏天,两个人都暂时离开斯坦福大学的博士生项目,转而全职投入到Google 的研发工作中。他们把整个项目的一个总结发表在了1998 年的万维网国际会议上(WWW7,the seventh international conference on World Wide Web)(见参考文献[1])。这是PageRank 算法的第一次完整表述。 PageRank 一经提出就在学术界引起了很大反响,各类变形以及对PageRank 的各种解释和分析层出不穷。在这之后很长的一段时间里,PageRank 几乎成了网页链接分析的代名词。给你推荐一篇参考文献[2],作为进一步深入了解的阅读资料。

PageRank 的基本原理 我在这里先介绍一下PageRank 的最基本形式,这也是布林和佩奇最早发表PageRank 时的思路。 首先,我们来看一下每一个网页的周边结构。每一个网页都有一个“输出链接”(Outlink)的集合。这里,输出链接指的是从当前网页出发所指向的其他页面。比如,从页面A 有一个链接到页面B。那么B 就是A 的输出链接。根据这个定义,可以同样定义“输入链接”(Inlink),指的就是指向当前页面的其他页面。比如,页面C 指向页面A,那么C 就是A 的输入链接。 有了输入链接和输出链接的概念后,下面我们来定义一个页面的PageRank。我们假定每一个页面都有一个值,叫作PageRank,来衡量这个页面的重要程度。这个值是这么定义的,当前页面I 的PageRank 值,是I 的所有输入链接PageRank 值的加权和。 那么,权重是多少呢?对于I 的某一个输入链接J,假设其有N 个输出链接,那么这个权重就是N 分之一。也就是说,J 把自己的PageRank 的N 分之一分给I。从这个意义上来看,I 的PageRank,就是其所有输入链接把他们自身的PageRank 按照他们各自输出链接的比例分配给I。谁的输出链接多,谁分配的就少一些;反之,谁的输出链接少,谁分配的就多一些。这是一个非常形象直观的定义。

CATIA有限元高级划分网格教程

C A T I A有限元高级划 分网格教程 -CAL-FENGHAI.-(YICAI)-Company One1

CATIA有限元高级网格划分教程 盛选禹李明志 1.1进入高级网格划分工作台 (1)打开例题中的文件。 (2)点击主菜单中的【开始】→【分析与模拟】→【Advanced Meshing Tools】(高级网格划分工具),就进入【Advanced Meshing Tools】(高级网格划分工具)工作台,如图1-1所示。进入工作台后,生成一个新的分析文件,并且显示一个【New Analysis Case】(新分析算题)对话框,如图1-2所示。 图1-1【开始】→【分析与模拟】→【Advanced Meshing Tools】(高级网格划分 工具) (3)在【New Analysis Case】(新分析算题)对话框内选择【Static Analysis】(静力分析)选项。如果以后打开该对话框的时候均希望是计算静力分 析,可以把对话框内的【Keep as default starting analysis case】(在开始 时保持为默认选项)勾选。这样,下次进入本工作台时,将自动选择静 力分析。 (4)点击【新分析算题】对话框内的【确定】按钮,关闭对话框。 1.2定义曲面网格划分参数 本节说明如何定义一个曲面零件的网格类型和全局参数。 (1)点击【Meshing Method】(网格划分方法)工具栏内的【高级曲面划分】按钮,如图1-3所示。需要在【Meshing Method】(网格划分方法)工 具栏内点击中间按钮的下拉箭头才能够显示出【高级曲面划分】按钮。

有限元网格划分的基本原则

有限元网格划分的基本原则 杜平安 《机械设计与制造》 划分网格是建立有限元模型的一个重要环节,它要求考虑的问题较多,需要的工作量较大,所划分的网格形式对计算精度和计算规模将产生直接影响。为建立正确、合理的有限元模型,这里介绍划分网格时应考虑的一些基本原则。 1网格数量 网格数量的多少将影响计算结果的精度和计算规模的大小。一般来讲,网格数量增加,计算精度会有所提高,但同时计算规模也会增加,所以在确定网格数量时应权衡两个因数综合考虑。 图1中的曲线1表示结构中的位移随网格数量收敛的一般曲线,曲线2代表计算时间随网格数量的变化。可以看出,网格较少时增加网格数量可以使计算精度明显提高,而计算时间不会有大的增加。当网格数量增加到一定程度后,再继续增加网格时精度提高甚微,而计算时间却有大幅度增加。所以应注意增加网格的经济性。实际应用时可以比较两种网格划分的计算结果,如果两次计算结果相差较大,可以继续增加网格,相反则停止计算。 图1位移精度和计算时间随网格数量的变化 在决定网格数量时应考虑分析数据的类型。在静力分析时,如果仅仅是计算结构的变形,网格数量可以少一些。如果需要计算应力,则在精度要求相同的情况下应取相对较多的网格。同样在响应计算中,计算应力响应所取的网格数应比计算位移响应多。在计算结构固有动力特性时,若仅仅是计算少数低阶模态,可以选择较少的网格,如果计算的模态阶次较高,则应选择较多的网格。在热分析中,结构内部的温度梯度不大,不需要大量的内部单元,这时可划分较少的网格。 2网格疏密 网格疏密是指在结构不同部位采用大小不同的网格,这是为了适应计算数据的分布特点。在计算数据变化梯度较大的部位(如应力集中处),为了较好地反映数据变化规律,需要采用比较密集的网格。而在计算数据变化梯度较小的部位,为减小模型规模,则应划分相对稀疏的网格。这样,整个结构便表现出疏密不同的网格划分形式。 图2是中心带圆孔方板的四分之一模型,其网格反映了疏密不同的划分原则。小圆孔附近存在应力集中,采用了比较密的网格。板的四周应力梯度较小,网格分得较稀。其中图b中网格疏密相差更大,它比图a中的网格少48个,但计算出的孔缘最大应力相差1%,而计算时间却减小了36%。由此可见,采用疏密不同的网格划分,既可以保持相当的计算精度,又可使网格数量减小。因此,网格数量应增加到结构的关键部位,在次要部位增加网格是不必要的,也是不经济的。

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