There is a large number of different volute types and applications: centrifugal pumps, axial pumps, centrifugal compressors, axial-flow compressors, radial-inflow turbines, radial fans, and multi-stage blowers, to name a few. Within each group, there is a narrower division on volute types and every application has its own unique features as well as specific properties that can be shared among the group members. The purpose of this post is not to have a detailed discussion of every possible scenario, but rather to show a robust and proven method of volute geometry construction working as a part of aerodynamic design and analysis in a system such as AxSTREAM®.

Let’s classify this broad range of applications into categories and briefly explore the underlying physical phenomena, main affecting factors, and geometry construction main steps.

Volute underlying physics can be broadly divided into two groups:

- volutes that are part of diffusing systems such as in compressors, pumps, or fans; and
- radial-inflow turbines where volutes direct flow into the next flow path component axisymmetrically

From here, let’s move on to volutes commonly found in compressor applications. The main purpose of the volute is to collect axisymmetric flow to further deliver it to the discharge flange as well as recover static pressure. Velocities at the entrance of the scroll are relatively small as the gas was previously diffused in the diffuser. However, the later statement can not be generalized for all applications and turbomachines. For example, to reduce size and due to some assembly limitation, a volute can go right after small vaneless gap and impeller discharge – very common for centrifugal pumps and centrifugal fans.

Even though the losses can be evaluated for a volute, the author rarely considers performance of a standalone volute – a better indicator is the performance of the diffusing system, or even better, stage performance overall. A typical diffusing system consists of a diffuser, volute and discharge diffuser.

Manufacturing considerations and geometric limitations are important, nonetheless. Examples of manufacturing limitations for cast volutes include minimum wall thickness and minimum radii of fillets. Geometric limitations are typically set on the overall compressor/turbomachine size to fit into the enclosed space or save on material. The cross-section and volute type are predetermined by these important considerations as well as aerodynamic performance. Depending on a cross-section, the volutes can be:

- – Circular cross-section;
- – Trapezoidal (with round ages);
- – Rectangular (commonly fabricated);
- – Arbitrary shape cross-sectional shape; or
- – Symmetric vs Asymmetric.

Based on geometrical arrangement to the diffuser, volutes are classified as:

- – External;
- – Internal; or
- – Semi-external.

Based on geometric construction diameter, volutes can be:

- – Constant mean diameter;
- – Constant inner diameter; or
- – Constant outer diameters.

Another aspect of volute design is area scheduling as shown in the Figure 3 (Bottom Right) below. The Volute throat is determined by area at the full collection plane. Full collection plane is at 0/360 degree location as shown below. The Volute throat is typically determined during the preliminary stage based on diffuser size and flow kinematics. Additionally, diffuser cone length and area ratio is determined so there are no large separation zones at the discharge. The right bottom corner of Figure 3 shows area distribution as a function of polar angle for constant external diameter volute. By default, AxSTREAM® schedules the areas right after preliminary design according to fundamental aerodynamic principles such as conservation laws. However, the user always has the ability to modify the area distribution as required.

Typically, there is a straight diffuser cone following immediately after full collection plane which is part of the diffusing system. If the discharge cone orientation is different from tangential direction, one can design a transition to properly fit into the discharge flange as shown in the Figure 4.

As noted earlier, volute design consists of many aspects such as aerodynamic performance, manufacturing details, geometric constraint. Some of the main steps during volute geometry construction are summarized below:

- – Preliminary stage sizing
- – Iterations on aerodynamic design, structural, manufacturability, etc.
- – Determination of volute type (cross-section, internal vs external, etc.)
- – Area scheduling
- – Diffuser cone geometry
- – Orientation of discharge flange details
- – Construction of base surfaces
- – Adding features such as tongue and fillets (ref. to figure below)
- – Construction of 3D gas path
- – Construction of 3D part solid model and detailed drawing based on 3D gas path (in CAD)
- – Final manufacturing evaluation (by vendor)
- – Final stage aerodynamic evaluation in design and off-design

In AxSTREAM**®**, one can iterate between geometric construction and stage performance evaluation within one platform. **Highlighted in the list above are activities that can be accomplished in AxSTREAM****® platform. Yes, AxSTREAM® constructs volutes with tongues! **Below, the AxSTREAM® output of different volute shapes is provided. The model is parameterized, so the designer works with only few parameters and can generate complex shapes in the matter of seconds. This is the reason why our clients select AxSTREAM® – for its accurate aerodynamic design and prediction algorithms as well as robust geometric construction modules.

- Different modes of aircrafts:
- Ground idle mode
- Take off
- Maximum continuous mode
- Cruising mode

- Different ambient conditions
- Grid demands (for power generation engines and gas pumping (compressor) stations)

Due to the off-design/part load operating conditions, the parameters of the engines might change significantly, which influences not only the engine efficiency, but also the reliable work of the turbine (high temperature at turbine inlet) and compressor (surge zone) at joint operational points. This is why accurate predictions of the gas generator parameters are crucial at every off-design mode.

To define the joint operational point, the compressor and turbine maps which are created for specified ambient conditions can be used. For example, pressure equal 101.3kPa, temperature – 288.15K. Maps method is widely used, relatively simple and allows you to find the needed engine parameters in the shortest time. However, when cooling is present, engine operation at low power modes (ground idle) impede the accurate determination of joint operational conditions based on maps. The significant drawback to the maps based approach is that it does not give the full picture of the physical processes in turbomachine flow paths which is critical for off-design calculations.

Utilization of the digital twin concept allows significant increase of the off-design performance calculation accuracy. Use of the digital equivalent of object was introduced in 2003 [2]. Despite this, less 1% of machines that are in use today are modeled with digital twin technology [3]. Utilization of digital twin leads to a significant decrease in time and cost for developing and optimization of an object.Good joint operational conditions accuracy of turbomachines is achieved using 2D simulation in combination with well-proven loss models which are based on experimental data. The loss models of Craig&Cox [4], Kacker-Okapuu for turbine and Aungier [5] for compressor, are often default models in the industry. The designed compressor and turbine flow paths are presented below.

At the ground idle mode a compressor is at risk of surging. To avoid this, the guide vanes of the first stages of compressor are performed turning. The flow paths, above, have enough freedom degree to allow their geometry changing and recalculation to find renewed performance.To close the cycle of the gas turbine unit digital twin a thermodynamics model of combustor, intake and exhaust duct can be used. Practice has shown with such simulation models utilization does not lead to significant error in performance determination.To find the compressor and turbine joint operational point the digital twin calculation model should be based on an iterative process, which start with some initial educated guesses of unknown quantities, but are required for next step parameters.

The presence of a cooling system in a gas turbine unit does not lead to significant changes of the digital twin scheme. A cooling system can be simulated via detailed (1D) [8] and simplified (0D) methods depending from required task. For gas turbine units that do not have the strict constraint on weight (power propulsion) the optimization of cooling air mass flow rate at off-design modes also can be performed. Wherein the possibility of hot gases entering into cooled blade can be traced.

In this way, the utilization of digital twin technology leads to significant decreases in object development and improvement time. Utilizing the digital twin for off-design modes allows engineers to determine if the compressor is beyond the safe operation margin to surge and perform automatic correction of guide vanes stagger angles to satisfy the requirements for safe work.

The digital twin based method is flexible, fast and highly accurate with the ability to connection different tools/models with different complexity degree for object simulation. Utilization of the digital twin opens new capabilities in the aircraft and power propulsion industries.

To learn more, contact us at info@Softinway.com.

- https://www.google.com.ua/search?q=aircraft+compressor+map&sa=X&rlz=1C1EJFA_enUA767UA767&tbm=isch&tbo=u&source=univ&ved=2ahUKEwi-i9mSy7zcAhXKESwKHQ3XB_oQsAR6BAgDEAE&biw=1745&bih=885#imgrc=yxnyHJR695tb5M:
- http://innovate.fit.edu/plm/documents/doc_mgr/912/1411.0_Digital_Twin_White_Paper_Dr_Grieves.pdf
- Walker, M.J.: Hype Cycle for Emerging Technologies, Gartner, 2017. Report ID G00314560.
- R. M. Craig; H. J. A. Cox, 1970 “Performance Estimation of Axial Flow turbines,” Proc. Instn. Mech. Engrs. 1970-71, ol.185 32/71.
- Ronald H. Aungier, 2003. “Axial –Flow compressors: a strategy for aerodynamic design and analysis”, The American Society of Mechanical Engineers, New York, 2003, 363p.
- Moroz, Yu. Govoruschenko, P. Pagur, A uniform approach to conceptual design of axial Turbine / compressor flow path, The Future of Gas Turbine Technology. 3rd International Conference, October 2006, Brussels, Belgium.
- SoftInWay Inc, 2017, AxSTREAM ION user documentation.
- SoftInWay Inc, 2016, AxSTREAM NET user documentation.

Last month we discussed a few basic aspects of wind as a source of clean energy. We showed what wind was, how it forms and where it goes. Then after going on a tangent about the history of turbines, we showed where on the Earth we could recover the highest amount of wind energy and how this potential changes with altitude. Today’s post offer the pros and cons of wind energy while touching upon several topics discussed in the previous post before diving into the optimal where and when.

With an established worldwide potential of more than 400 TW (20 times more than what the entire human population needs) and a clean, renewable source wind is definitely attractive to the current and future generations. In terms of harvesting it, over 99% percent of wind farms in the USA are located in rural areas with 71% of them in low-income counties. Indeed, the more land is available (and the fewer buildings), the higher the possibility and interest to transform this kinetic energy into mechanical work and then most likely electricity.

Where one would see sporadic turbines on the side of the highway, these stand-alone equipment have begun to turn into actual modules (farms) that can work as an overall unit instead of individual ones. This strategy of creating a network of turbines follows the philosophy of “the Whole is Greater than the Sum of its Parts”. What this translates into is that by having 20 (arbitrary number) wind turbines working together to determine the best orientation, pitch, etc. of their blades in such a way that it least negatively impacts the downstream units we can produce more energy than if each of them were live-optimized individually (some interesting A.I. work is going into this). This means that the overall system is more efficient at converting energy and therefore it is more cost effective to provide bulk power to the electrical grid. This is similar to the concept in the post on solar energy comparing PV panels and CSP. Read the full post here.

In terms of power production per wind turbine, the utility-scale ones range from about 100 kW to several MW for the land-based units (Offshore wind turbines are typically larger and produce more power – getting ahead of myself here but check out the figure below for wind potential in Western Europe that clearly showcases coast vs. non-coast data). On the low-power end of the spectrum, we find some below 100 kW for some non-utility applications like powering homes, telecommunications dishes, water pumping, etc. Solar power (PV) is generally regarded as the first choice for homeowners looking to become energy producers themselves, but wind turbines make an excellent alternative in some situations. It would take a wind turbine of about 10 kilowatts and $40,000 to $70,000 to become a net electricity producer. Investments like this typically break even after 10 to 20 years.

One of the elements of wind formation we covered in the last post here was a different in pressure (and therefore temperature). This simplification works rather well at the macro-scale, but as we zoom in closer to the surface we can see that wind flow speeds and patterns vary quite significantly based on more than just the general location of Earth. On top of the altitude we already discussed, factors like vegetation, presence of high-rise buildings or bodies of water come into play.

If you have ever left a beach towel by the water without putting rocks (or shoes) at the corners, then you already know the winds can be pretty strong at such land-water interfaces. Looking back at the figure from the previous section or even the maps shown last month it is easy to see that coastal regions are generally the most attractive when it comes to harnessing wind power. Despite this, 41 U.S. States have utility-scale wind farms today and the USA is the number two country (behind China) with the highest installed capacity for wind power generation which contributes to about 6% of the total U.S. electricity generation (> 80GW); both countries have very vast lands and rather long coast lines. Other remarkable countries with high rates of installed wind power include Denmark, Portugal and Ireland, which each get more than 20% of their nation’s electricity from wind energy. All three of these are peninsulas. Therefore, having vast land and/or access to water definitely helps with accessing this renewable energy source. One attractive aspect of wind turbines is that one can make use of the space between them for crops or grazing livestock.

Here is a quick comparison of different places regarding wind statistics, taken from https://globalwindatlas.info/:

Now that we have seen locations of most interest for wind turbines let’s look at the reasons why it is an interesting source of energy as well as arguments made against it.

**Cost-effective.**Land-based utility-scale wind is one of the lowest-priced energy sources available today, costing between two and six cents per kilowatt-hour, depending on the wind resource and the particular project’s financing. Due to its relative stability for given locations and seasons, wind energy mitigates the price uncertainty that fuel costs add to traditional sources of energy.**Clean Fuel Source.**Wind energy does not pollute the air like power plants that rely on combustion of fossil fuels. Also, wind turbines do not produce atmospheric emissions that cause acid rain, smog, or greenhouse gases. The manufacturing of the equipment is of course as clean as the plant that produced it. Note that the release of greenhouse gases during manufacturing and installation of wind turbines are usually recouped within 9 months of clean operation.**Domestic.**It is abundant and inexhaustible, meaning renewable.**Compatible with Existing Land Use.**Wind turbines can be built on existing farms or ranches.This helps boost the economy of rural areas while allowing landowners to get a monthly rent check that would make their financial wellbeing less dependent upon yearly harvests.**Extremely Low Operational Costs.**Once the infrastructure is paid for (and beside maintenance) the power generated is almost free as the operational costs are low.

**Steadiness.**The main disadvantage of wind power is that the wind does not blow consistently or steadily. Indeed, wind is a fluctuating (intermittent) source of energy and is therefore not suited to meet the base load energy demand unless some form of energy storage is utilized (e.g. batteries, pumped hydro).**Wildlife Impact**. Not only birds, but bats have experienced fatalities (~10,000-440,000 per year in the USA). The same argument can be made against vehicles and stationary buildings (up to 976M) though.**Localized Impact.**Due to a change of the wind pattern, wind turbines may cause a localized impact on night-time temperatures and weather.**Visual Pollution.**This seemed to be the first complaint when wind turbines started to sprout, but it appears people are getting used to them.**Noise.**This is not a problem with offshore wind turbines at all. New designs for onshore installations show significant improvements compared to older models and generate less noise.**Competition.**Conventional power generation sources are currently cheaper in most places, so wind farming may not be cost-competitive even though prices have decreased significantly over the last decade.**Location.**Wind turbines need land or water. The electricity generated is usually used in urban areas, so transportation of the energy is required.**Land Investments.**Despite the fact that inter-turbines space can be used they may not be the most profitable land investment for the locals, especially in places where land is scarce like in Japan for example.**Initial Capital Investment.**The manufacturing, delivery (ever seen these oversized trucks carrying a single propeller on the road) and installation of wind turbines requires heavy upfront investments.**Seasonal Potential**. Winds are seasonal. The figure below showing regional wind trends across the USA throughout a selected year.

Each of these unsteady factors make it difficult for companies to provide reliable estimates on wind power generation which represent a higher risk factors for some lenders/buyers.

In conclusions, we have seen where and when it was best to consider wind power and, we looked at the pros and cons of harnessing this renewable resource. Overall, wind speeds seem higher during spring and fall unlike the potential for solar energy, which was higher in the summer months. Based on the US national average, using solar and wind together seems to be the best option for clean energy generation throughout the year (see figure below).

]]>- In the oil and gas or chemical industries, converting crude oil to products requires a complex process. Pumps play an important role in transferring these liquids, providing the required pressure and flow rate for chemical reactions. Sometimes, pumps are used to adjust temperature in certain parts of the system.
- In agriculture, centrifugal pumps are used in the majority of irrigation machinery. Agriculture pumps make up half of the total amount of centrifugal pumps being used today.
- In mining and metallurgy industries, centrifugal pumps are the most widely used equipment, for draining, and cooling of water supplies, etc.
- For power generation, the nuclear power plants need large amounts of primary, and secondary system pumps, while the thermal power plants also need boiler feed pumps, condensate pumps, loop pumps and as well as ash pumps.
- In military applications, the adjusting of airplane wings and rudders, turning of turret on ships and tanks, the up and down of submarines, all rely on pumps for hydraulic fluids.
- In shipbuilding, there are more than 100 different types of pumps in one typical ocean ship.
- Other applications include municipal water supplies and drainage; water supplies of locomotives; lubricating and cooling of machining equipment; bleach and dye transfer of textile industry; and milk and beverage pumping and sugar refining in the food industry.

Centrifugal pumps can be classified based on the number of impellers in the pump:

**A single-stage pump**, with only one impeller, is commonly used for high flow and low to moderate total dynamic head, as in Figure 1.

**A multi-stage pump** has two or more impellers working in a series to achieve higher total dynamic head.

All of these different applications and scenarios ask for different pumps, which means pump suppliers need to preform many designs for their clients. The majority of these pumps can be designed quickly with AxSTREAM Hydro, an integrated turbomachinery design software. Whether it is a new design starting from scratch, or the design analysis and optimization of an existing pump, AxSTREAM^{® }can handle the complete flow path design, map generation (Figure 2.), blade profiling, CFD and stress analysis, rotor dynamics and bearing design.

References:

https://www.powerzone.com/centrifugal-pumps-working-applications-types-power-zone

]]>As noted, the FMM is an approximation of the original model, which means it can be obtained by statistical processing of the results of numerical experiment using OMM. The complexity of solving the equations of the original model forces minimize the number of sampling points, which is practically achieved by using methods of the theory of experiment design. Get the response function in the form (1.2) can, in particular, on the basis of three-level Box and Benken plans [1]. Special selection of sampling points on the boundary of the approximation:

and in its center possible in accordance with the least squares method to obtain the values of the coefficients according to (1.2), without resorting to the numerical solution of the normal equations. The number of sampling points is in the range from 13 at N = 3 to 385 at N = 16.

Similarly, relations (1.2) can also be obtained by using the three-level saturated plans by Rehtshafner [2]. In this case, the dimension of the observation vector will vary from 16 at N = 4 to 232 at N = 20. The feature of these plans is that it is the most economical plans that require a minimum number of calculations to generate a vector of observations, i.e. the number of calculations (experiments) equal to the number of the coefficients according to (1.2).

When creating subsystems FMM quality criteria, should be noted, that at lower levels increases the degree of detailed description of the design objects, which leads to an increase in the dimension of Q ⃗ _{k} vectors. If the dimension exceeds the permissible (N = 20), or for any reason is limited, for example, due to the complexity of OMM, it can be reduced by replacing a number of components of the control parameters vector defined by the laws of their change, by numbers of the same type subsystems (objects) at the considered design level. For example, in the formal macromodelling of the multi-stage turbine flow path efficiency, may be appropriate to change the degree of reaction, disposable heat drop and so forth linearly from stage to stage. To ensure information

consistency between FMMs of adjacent levels, in a number of components of Q ⃗ _{k+1 } should be required to include parameters that uniquely determine the position of the subsystems in the settings space of a higher k-level. It should be noted that in addressing the increasingly complex, multi-parameter, multi-mode and multi-criteria problems of optimal design increases the likelihood of multimodal objective functions. Using the dependency of the form (1.2) for the approximation of the objective functions and functional limitations in this case can lead to a decrease in the accuracy and adequacy of the obtained with its help optimal solutions for the projected objects or subsystems.

The analysis of the structure of formula (1.2) shows, that its second term is a superposition of the parabola from each independent parameter that mainly determines the failure of functions of the form:

taking into account the more complex nature of real dependencies, having, for example, bends and local extremes. We will use a second member according to (1.2) to reflect the independent effect of the parameters on the approximated function, and replace it with a more perfect form of addiction. It is obvious that in the general case, the shape and structure of dependency, reflecting the influence of each parameter, is unique. Given that a priori a kind of dependency is not known, to solve this problem and ensure that the principle of universality, the second term of the form

should be replaced with the superposition of interpolation cubic splines. As known, the interpolation cubic splines allow with a high degree of accuracy and adequacy to describe features of varying complexity, including multi-extremal. Thus, taking into account this replacement, the formal macromodel of the form (1.2) will be as follows:

where a_{ij}, b_{ij}, c_{ij}, d_{ij} cubic spline coefficients of current (j-th) interpolation section of the i-th independent variable. For each independent normalized variable q_{i} there are several areas in the interpolation range between –1 and +;change in q_{ij } – the distance between the current value q_{i} and coordinate of the initial node of j-th section of the spline, which q_{i} coordinate value is between the initial coordinates of (j-th) and final (j + 1-th) of its nodes.

Of course, for the coefficients a_{ij}, b_{ij}, c_{ij}, d_{ij} of dependance (1.12) determination additional computational experiment is needed. This experiment carried out at the points of a normed space of independent variables q_{i} The length of the interpolation areas and their nodes coordinates are the same for all the independent variables. The number of sections is given. The minimum required number of sections is four. In this case, an additional calculation of the objective functions at four points (1; 0.5; 0.5; 1) by each variable q_{i} is needed.

To ensure the principle of an independent effect of each variable, other variables in the calculation are assigned by the value 0 (q_{j} = 0 ), which corresponds to the center of the accepted range of their changes. It should also be noted that in the case of Rehtshafner’s design plans to create more accurate FMM of form (1.12) the number of computations by OMM is reduced for each independent parameter of FMM by two and equal, accordingly, two, since the other two points coincide with the points of the Rehtshafner’s plan and their corresponding calculations for OMM performed at the stage of creating a FMM of form (1.2). For clarity, in Fig. 1.3 shows a comparison of the accuracy and the adequacy of the approximation of test functions of the form:

by formal macromodel of the form (1.2) and the form (1.12).

]]>

Such a 720°-periodic function can be created in AxSTREAM RotorDynamics, which provides a transient approach to determine the response torque in the shaft after a respective torque excitation. In this example, a rotor speed of 3000 rpm is considered. With this information, the total time for two crankshaft-revolutions (720°) reads:

By multiplying the tangential force in Figure 1 with the length of the connecting rod of 0.1 m, the stepwise function can be defined.

A single piston engine is considered in this example, for which reason only one firing cycle is present during the two revolutions. The rotor model consists of a massless shaft with applied torsional stiffness and mass inertia elements, representing the piston as well as inertia effects of the remaining shaft sections, (see Figure 2). Please refer to Blog 1: Torsional Analysis of a Four-Stroke Engine for further information about modeling the piston’s geometry.

The transient response in each shaft section can be calculated by using the system’s natural frequencies and shapes. A superposition of a finite number of torsional modes is used in this example to evaluate the vibration behavior of the crankshaft. To describe the complex interrelations, 1000 time steps between 0 and 0.04s were chosen.

The torque function in Figure 3 corresponds to a particular Finite Element-node of the rotor. The influence of the firing process can clearly be seen after approximately 0.02s, which leads to higher torque amplitudes. However, due to the system’s damping (modal damping of ξ = 0.02 for each mode respectively) the torque tends to zero. Notable angular deflections can also be seen after 0.02s.

Effects of varying speeds of the crankshaft also need to be investigated to find optimal speed ranges or identify possible collision of torsional natural frequencies and operating speeds. Therefore, the speed range was varied from 3000-9000rpm, which is a common speed range of internal combustion engines. As it can be seen in Figure 4, the maximum shaft torque occurs at 5500rpm, whereas the minimal value is stated at 7000rpm. Either the operating speed or the geometry and components can be changed/added (e.g. torsional vibration dampers) in order to achieve satisfactory results.

]]>In the __first post in this series__, we discussed clean energy as a whole. After describing what it is and what it is not, we pointed out some of the energy sources we would analyze in subsequent articles.

The __second post in this series__ took us on an extraterrestrial journey for two reasons: we looked at solar energy and we also went on a tangent about the rovers operating on planet Mars. I got so many “Likes” on these little droids that I figured I would keep going with them (that or I found a cool article that I’ll be sharing here) for this current post on one of the fastest-growing energy sources in the world: Wind Energy. What’s the link between Mars equipment and wind? See this recent discovery – https://www.space.com/41023-mars-wind-power-landers-experiment.html

Side note: ever wondered what would happen if the sun just blinked out? Check it out here – https://what-if.xkcd.com/49/

The wind we are looking at in today’s post is somewhere in between bovine flatulence and hurricanes in terms of intensity. Wind as we know it is created by air (or any fluid) moving from a zone of high pressure to one of low pressure. This high-to-low concentration migration might sound tricky, but it is easy to understand if you think of cars on a highway. It is more likely that cars stuck in a slow lane on the highway would move on to a lane with less traffic rather than the other way around.

Pressure varies with things like irregularities on the Earth surface, AKA altitude (“in case loss of cabin pressure occurs, oxygen masks will drop […]”), but also with temperature. This means that two people at the same altitude but in areas of different temperatures would experience different pressures. For example, think of standing at the North Pole vs. standing on a Caribbean beach vs. standing on a paddleboard in the Great Lakes. This example of standing at different places demonstrates the uneven heating of the Earth from the sun due to its shape (not flat), its rotation and its tilt, as we introduced in the __previous post__. But which location is under the most pressure? Colder temperature equals higher pressure. Let me explain with another analogy, (even though this example has nothing to do with pressure, it will help the information stick). When people get stressed, we say they are under pressure. We can imagine somebody above the Arctic Circle is more stressed (cold, where to find food, shelter, etc.) than somebody enjoying a Mai Tai on the beach at an all-inclusive resort in Aruba. So here is your mnemonics; colder equals higher pressure.

Now that we have seen what wind was and the theory behind how it forms, we can start thinking about how to utilize this energy. Today we will talk about the aerodynamic aspect of wind turbines while in a future post we will be focusing on the assessment of such technology as wind power; pros, cons, where, what, etc.

We talked about wind being a moving fluid and we know we want to extract energy from it. This is done by slowing it down and converting the difference of velocity into mechanical work.

This is far from being a new technological concept. Indeed, in the first century CE the first wind wheel was invented by Hero of Alexandria in Greece to power a musical organ; he did also invent the concept of the coin-operated vending machine, but due to a lack of sodas and gelatinous candies at the time it provided a specific amount of holy water for each coin inserted instead.

Windmills are another more recent example of pre-wind turbines which were putting the wind to use. The concept was to grind grains therefore utilizing the kinetic energy available and converting it to mechanical energy through the panes/sails attached to the shaft and rotating a moving mill stone on top of a stationary stone. This type of wind-powered equipment traces its origin to the twelfth century in Europe with possible Middle-Eastern influences.

Although windmills do include a turbine, the terminology was only changed once the mechanical energy was transformed into electric energy making it easier to store, transport and includes a very wide range of applications. This last component of the energy conversion is done by a generator that sits at the top of our current wind turbines and is linked to the blades through a gearbox which is used to bring the turbine rotation speed of about 18 rpm to ~1800 rpm where it can be distributed. This means that each turbine tower is technically independent and self-sustainable to deliver electricity where needed.

The first wind turbines, as we recognize them today, were created at the end of the 19^{th} century and in 1896 one of them was used for the first time to provide power to a Danish village.

Today’s wind turbines are actually very sophisticated machines that include more than 8,000 parts. We typically see the modern wind turbines as the horizontal-axis type, but vertical-axis types do exist with limited popularity.

Several important technological changes have allowed us to make even better use of this free energy than by using windmills. Examples include:

**2 or 3 aerodynamically designed, propeller-like blades that increase the lift of the blades**(similar concept as for airplane wings) due to the uneven pressure on either side of a blade (pressure and suction sides) resulting from its shape and curvature. Wind turbine blades these days include several 3D features that make them well optimized for their working conditions; twist, tapering, sweep, curvature of the camberline, etc.**Turbine rotation**along the tower axis which allows orienting the face of the turbine in the direction of the wind to extend its range of applicability as winds are not always coming from the same direction (Where I live, the wind is coming from: due South 17% of the time; 14% from the Southwest; 10% from the Southeast; and the 59% left is scattered around wind frequency rose which would make for significant losses/non-use of the wind turbine if it were fixed in one given direction).

**Variable pitch rotor blades**also allow extending the range of use and the performances of wind turbines by adjusting the blade angle of attack to provide better lift and therefore convert more energy.**Taller towers**give access to stronger winds and provide the opportunity to use longer blades and therefore sweep a bigger area while also being harder to rotate due to the added mass. The images below show the wind power density potential in the Northern hemisphere at altitudes of 100m vs 200m, respectively from top to bottom, using the same scale.

Despite the recent improvements in blade design, reaching new heights for the towers, using longer blades, etc. one must understand that converting 100% of the wind energy is unrealistic. Indeed, a theoretical limit to how much energy could be transformed was derived by Albert Betz in 1919 and this yielded the following efficiency based on the laws of conservation of mass and energy: 59.3%. With the current designs reaching between 70 and 80% of this theoretical, maximum value more efficient systems are possible for the future with a strong emphasis on wind turbine farms working smartly to control each turbine as part of a larger group instead of individually.

]]>Block-hierarchical representation of the design process, implemented with the creation of complex technical devices, leads to a problem of such complexity that can be effectively resolved by means of modern computing, and the results of the decision – understood and analyzed by experts. Typically, the design hierarchy of tasks is formed along functional lines for turbine can have the form shown in Fig. 1.1.

The uniformity of mathematical models of the subsystems of the same level and local optimality criteria make it possible to organize the process of multi-level design, providing maximum global quality criterion of the whole system, in our case – the turbine. This process is based on the idea of so-called multilevel optimization approximation scheme that involves aggregation of mathematical models of the subsystems in the hierarchy when moving upward and disaggregation based on optimization results when moving downwards.

The problem of optimization the subsystem parameters described by OMM has the form (1.5). It can be solved by the methods of nonlinear programming and optimal control, depending on the form of the equations and the optimality criterion of the OMM.

Consider the solution order for the problems hierarchy of the system parameters optimization. Input parameters of k-level subsystem form of the set of internal and external parameters of the higher (k–1)-level subsystem. Feedback is carried out at the expense of the influence of the output parameters B ⃗ “_{k–1 }of the subsystem of k-level which with respect to the (k–1)-th subsystem is external. Complete vector of (k–1)-level external parameters, thus consists of a vector of external parameters B ⃗ ‘_{k–1} coming from the higher-level and lower-level subsystems of vectors B ⃗ “_{k–1} ( Fig. 1.2).

Moving from the bottom up, we solve the problem of the form (1.6) at each k-level for all possible values of the vector of external parameters coming from a higher level. In this phase k-level variables are excluded from the internal parameters of the (k–1)-level model by effect of equations describing the k-level subsystem, and control – as a result of optimization. Thus, at each level above information is transmitted not about all, but only about the best projects of lower-level subsystems:

At the top, the 1-st level, from the problem (1.5) output parameters found, and predetermine external parameters of the level 2 subsystems, which makes it possible to restore the optimum parameters of the 2-nd level, solving the same problem (1.5). This disaggregation process extends to the lowest level, with the result that the optimal parameters are determined by all the subsystems that make up the complex technical systems.

To implement practically the described scheme is possible using FMM subsystems. In terms of the FMM problem (1.5) is written in a form similar to (1.6):

which immediately follows

which is quite similar to (1.8), but has the advantage that it is a known polynomial of the form (1.2).

Methods based on the use of FMM is characterized in that before starting to solve the optimization problem on (k–1)-th level, it is replaced from the OMM to FMM according to the condition (1.9). Driving multilevel optimization using FMM, is very flexible, allowing you to change the setting if necessary optimization tasks at any level due to changes in the components of vectors Q ⃗ _{k }(u ⃗_{k }, B ⃗’_{k1}).

The current level of possibilities of computer technology and mathematics allow for a new approach to the organization of the block-hierarchical representation of the process of optimal design of axial turbine flow path (Fig. 1.1) and the information exchange between adjacent levels (Fig. 1.2). The essence of this approach lies in the application of the principle of recursion, provides automatic bypass facilities at all levels and solution for each object its local optimization problem in accordance with a predetermined scenario.

On the basis of this method created invariant subsystem of recursive object-oriented multi-criteria, multi-mode and multi-parameter optimization, providing solution of optimization problems, taking into account various types of parametric, structural, technological and functional limitations. Designed for its optimization techniques are universal, and the search for the optimal solution for each object is carried out in accordance with the scenarios of computing processes optimization.

Optimization scripts for all objects of all levels are formed and defined by set of components of the following vectors and lists:

- optX – address list parameters to be optimized
- lXmin, lXmax – vectors defining the allowable range of variation of parameters to be optimized
- lYcq – address list of the object settings and quality criteria
- lYw – object quality criteria weight vector
- lYfl – address list of parameters and functional limitations
- flMin, flMin – functional limitations permissible change vectors
- lYd – address list of settings – parametric constraints
- dlMin, dlMin – parametric constraints permissible change vectors
- lReg – list of regime (changing during the operation of the facility) parameters
- sRegim – list of lines with the data on the values of operating parameters and the appropriate time of the object for these values;
- lLine – address list of parameters whose values are changed in the process of optimization by linear interpolation between the same type of parameters to be optimized nearby objects
- optM – method for solving the optimization problem of the local object

Forming all the lists, enumerated above, for all level objects and calling a recursive function, which includes a set of corresponding optimization algorithms, an automatic objects bypass and solving optimization problems for each of them is carried out.

]]>Below, a common way to express a crankshaft assembly with massless shaft and mass-inertia elements is presented, whereas the reciprocating and revolving mass around the crack can be expressed as follows:

The mass-inertias caused by given masses depend partly on the crank is:

whereas I_{1} I_{2} and correspond to the reciprocating and revolving part, respectively. Due to differing inertias of the whole assembly, a mean value is introduced to express the contribution of the inertia elements in a dynamic process:

Another simplification is made to describe the torsional stiffness of the crankshaft, which consists mainly of the main shaft and crankpin. If the shaft and crankpin diameter are of approximately same diameter, the stiffness can be expressed as:

with Shear modulus, shaft diameter and section length . As a consequence, a substitutive model can be created. The AxSTREAM RotorDynamics software developed by SoftInWay Inc. provides a 1D and 2D FE-approach to calculate torsional natural frequencies of the shaft system. This incorporates the mass-inertia elements besides the torsional stiffness of the shaft sections, which can be seen in Figure 1 for a four-stroke engine.

The resulting 3D mode shapes in Figure 2 can be evaluated, whereas the red contour line corresponds to the torsion of the shaft with applied mass-inertia elements. High angular deviations from the undeformed contour indicate critical sections when excited by a torque close to the natural frequency.

Additionally, a Campbell diagram can be plotted, with natural frequencies distributed over a chosen speed range. Interactions of natural frequencies with excitation lines or frequencies, caused by e.g. misfiring or coupling with the clutch, may lead to improper operating conditions or failures and need to be investigated.

A transient torque function, to simulate excitation introduced by the clutch (see Figure 5), can be included in AxSTREAM RotorDynamics with the following formula:

whereas M_{i} (T_{i}) is the torque value at a specific time and the engine rotational speed. Besides this particular excitation case, phenomena such as short circuit or switch-on processes can be simulated.

Interested in learning more about rotordynamics? Join us on September 5th-7th for a three day training on Rotor Dynamics and Bearing Analysis. Read more and register here

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