4.1 Formulation of the Problem

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Mathematical models of gas and steam turbines stages, discussed above, allow to put the task of their geometry and gas-dynamic parameters optimization. This optimization problem is solved by the direct problem of stage calculation. The reason for this are the following considerations:

  • – it is most naturally in optimizing to vary the geometry of the blades;
  • – in the streamlines form refinement it is convenient to use well-established methods for the solution of the direct problem in the general axisymmetric formulation;
  • – only a direct problem statement allows to optimize the stage, taking into account the off-design operation;
  • – for the stages to be optimized, assumed to be given:
  • – the distribution of the flow at the stage entrance;
  • – the form of the meridian contours;
  • – the number of revolutions of the rotor;
  • – mass flow of the working fluid;
  • – averaged integral heat drop.

In general, you want to determine the distribution along the certain axial sections of angles α1 and β2 to ensure maximum peripheral efficiency of the stage:

4.1

Here the inlet geometric angle of the rotor we assume equal to the angle of the inlet flow. Selection of the optimal angle β1g can be achieved solving an optimal profiling problem.

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Study of a Supercritical CO2 Power Cycle Application in a Cogeneration Power Plant

This is an excerpt from a technical paper, presented at the ASME Power & Energy Conference in Pittsburg, Pennsylvania USA and  written by Oleksii Rudenko, Leonid Moroz, and  Maksym Burlaka.  Follow the link at the end of the post to read the full study! 

Introduction

Supercritical CO2 operating in a closed-loop recompression Brayton cycle has the potential of equivalent or higher cycle efficiency versus supercritical or superheated steam cycles at similar temperatures [2]. The current applications of the supercritical CO2 Brayton cycle are intended for the electricity production only and the questions which are related to the building of CHP plants based on Supercritical CO2 technology were not considered yet.

CHP is the concurrent production of electricity or mechanical power and useful thermal energy (heating and/or cooling) from a single source of energy. CHP is a type of distributed generation, which, unlike central station generation, is located at located at or near the point of consumption. Instead of purchasing electricity from a local utility and then burning fuel in a furnace or boiler to produce thermal energy, consumers use CHP to improve efficiency and reduce greenhouse gas (GHG) emissions. For optimal efficiency, CHP systems typically are designed and sized to meet the users’ thermal base load demand. CHP is not a single technology but a suite of technologies that can use a variety of fuels to generate electricity or power at the point of use, allowing the heat that would normally be lost in the power generation process to be recovered to provide needed heating and/or cooling. This allows for much greater improvement in overall fuel efficiency, therefore resulting in lower costs and CO2 emissions. CHP’s potential for energy saving is vast.

It should be noted that CHP may not be widely recognized outside industrial, commercial, institutional, and utility circles, but it has quietly been providing highly efficient electricity and process heat to some of the most vital industries, largest employers, urban centers, and campuses. While the traditional method of separately producing useful heat and power has a typical combined efficiency of 45 %, CHP systems can operate at efficiency levels as high as 80 % (Figure 1) [1].

Figure 1 - CHP Process Flow Diagram
Figure 1. CHP Process Flow Diagram.

Taking into consideration the high efficiency of fuel energy utilization of CHP plants and the high potential of the supercritical CO2 technology, the latter should be also considered as the base of future CHP plants. The comparison with traditional Steam based CHP plants also should be performed.

The study of CHP plant concepts were performed with the use of the heat balance calculation tool AxCYCLE™ [3].

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3.2 Preliminary Design of the Multistage Axial Flow Turbine Method Description

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In the early stages of the flow path (FP) design of the turbine, when determined the diameter, the blade heights, heat drops and other main characteristics of the stages, required to study alternatives with a view to the design solution, in the best sense of a quality criterion.

Most effectively, this problem is solved within the created turbine flow path CAD systems, because manage: to achieve a rational division of the designer, defining the strategy and computer, quickly and accurately perform complex calculations and presents the results in human readable numeric or graphical form; to take into account many different factors influencing the efficiency,
reliability, manufacturability, cost and other indicators of the quality of the design being created; organize dialogue or fully automatic determination of optimal parameters, etc [29].

Most methods of the multi-stage turbine parameters optimization is designed to select the number of gas-dynamic and geometric parameters on the basis of the known prototype, the characteristics of which are taken as the initial approximation.

When using complex mathematical models, a large number of variables and constraints, the solution of such problems requires considerable computer time and for the purposes of CAD that require quick response of the system is often unacceptable.

It is desirable to have a method of design that combines simplicity, reliability and speed of obtaining results with an accuracy of the mathematical model, a large number of factors taken into account and optimized, the depth of finding the optimal variant. This inevitably certain assumptions, the most important of which are: the synthesis parameters of “good”, competitive structure without attracting accurate calculation models; in-depth analysis and refinement of the parameters are not taken into account at the first stage; optimization of the basic parameters by repeatedly performing the steps of the synthesis and analysis.

Design of the FP in such a formulation will be called preliminary (PD). PD does not claim to such a detailed optimization of parameters, as in the above-mentioned methods of optimal design. Its goal – to offer a workable, effective enough design, the characteristics of which, if necessary, can be selected as the initial approximation for more accurate calculations.

Major challenges in creating a PD method are:

  • – a rational approach to the problem of the preliminary design, the selection of the quality criteria and the constraints system;
  • – development of a method for the multi-stage flow path basic parameters selection;
  • – formation of requirements for a mathematical models complex describing different aspects of turbines and their efficient numerical implementation;
  • – selection of the appropriate algorithm for finding the optimal solution;
  • – a flexible software creation for a dialog based solution of the design problems in various statements and visual representation of the results.

(more…)

3.1 Analytical Solutions

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An important objective in the design of a multi-stage axial turbine is to determine the optimal number of stages in the module and the distribution of heat drop between stages.

Typically, a given quantity is the module’s heat drop, and should vary the number of stages and the rotational speed (diameter). It should be understood that the circumferential velocity reduction, and hence the diameters of the stages, reduces the disc friction losses, increase height of the blades (and therefore reduce the proportion of end losses), decrease the flow path leakage. At the
same time it leads to an increase in the optimal number of stages, which causes an increase in losses due to discs friction and an additional amount of the turbine rotor elongation. Immediately aggravated questions of reliability and durability (the critical number of revolutions), materials consumption, increase cost of turbine production and power plant construction.

A special place in the problem of the number of stages optimization is the correct assessment of the flow path shape influence, keeping its meridional disclosure in assessing losses in stages. As you know, the issue is most relevant for the powerful steam turbines LPC. It is therefore advisable for the problem of determining the optimal number of stages to be able to fix the form of the flow path for the LPC and at the same time to determine its optimal shape in the HPC and IPC.

It should also be noted that the choice of the degree of reaction at the stages mean radius (the amount of heat drop also associated with it) must be carried out with a view to ensuring a positive value thereof at the root. Formulated in this section methods and algorithms:

  • – May serve as a basis for further improvement of the mathematical model and complexity of the problem with the accumulation of experience, methods and computer programs used in the algorithm to optimize the flow of the axial turbine;
  • – Allow the analysis of the influence of various factors on the optimal characteristics of the module, which gives reason for their widespread use in teaching purposes, the calculations for the understanding of the processes taking place in stages, to evaluate the impact of the various losses components on a stage operation;
  • – Allow to perform heat drop distribution between stages and to determine the optimal number of stages in a module within the modernization of the turbine, i.e. at fixed rotational speeds (diameters) and a given flow path shape or at the specified law or the axial velocity component change along the cylinder under consideration.

A possible variant of the form setting of n stages group of the flow path can be carried out by taking the known axial and circumferential velocity components in all cross-sections, which the numbering will be carried out as shown in Fig. 3.1.

The sections numbering in the turbine flow part section,

The axial velocity components we refer to the axial velocity at the entrance to the stages group: Read More

Role of AxSTREAM® in Radial Turbine Design

Radial turbines are quite popular for turbochargers and micro-gas turbines. They can also be found in compact power sources like in auxiliary power units of aircrafts. In short, they are suitable in power generation applications where expansion ratios are high and mass flow rates are relatively small. In a radial turbine, the flow enters radially and exits either axially or radially depending on whether it is an inflow or outflow type radial turbine. The most commonly used type of radial turbine is a radial-inflow turbine, in which the working fluid flows from a larger radius to a smaller radius. A centripetal turbine is very similar in appearance to the centrifugal compressor, but the flow direction is reverse. Figure 1 shows the radial-inflow turbine on the left and radial-outflow turbine on the right.

Radial-inflow turbine on the left; Radial-outflow turbine on the right
Figure 1: Radial-inflow turbine on the left; Radial-outflow turbine on the right

Nowadays, the popularity of radial-outflow turbines, in which the flow moves in the opposite direction (from the center to the periphery), is growing. With recent advancement in waste heat recovery applications, there has been a renewed interest in this type of turbines. These radial-outflow turbines are most commonly used in applications based on organic Rankine cycles (ORC).

The radial-outflow turbine design was first invented by the Ljungström brothers in 1912, however it was rarely used for a number of reasons. One of which was related to the decrease of turbine-specific work due to the increase of the peripheral velocity from inlet to outlet while expanding the vapor. Another reason was the usage of steam as a working fluid. It is known from thermodynamics that the expansion of steam is characterized by high enthalpy drops, high volumetric flows and high volumetric ratios. Thus, a significant number of stages are needed to convert the enthalpy drop of the fluid into mechanical energy.

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Modern Approaches and Significance of Multiphase Flow Modeling

Introduction

Corresponding with the development of industrial technology in the middle of the nineteenth century, people  dealt with multiphase flows but the decision to describe them in a rigorous mathematical form was first made only 70 years ago. As the years progressed, development of computers and computation technologies led to the revolution in mathematical modeling of mixing and multiphase flows. There are a few periods, which could describe the development of this computation:

«Empirical Period» (1950-1975)

There were a lot of experiments which were done during this period. All models were obtained from experimental or industrial facilities which is why using them was difficult for different cases.

«Awakening Period» (1975-1985)

Because of sophisticated, expensive and not universal experiments, the researchers’ attention was directed to the physical processes in multiphase flows.

«Modeling Period» (1985-Present)

Today, the models for multi-flow calculation using the equations of continuity together with equations of energy conservation are obtained, which allow describing phase’s interaction for different flow regimes. (A.V. Babenko, L. B. Korelshtein – Hydraulic calculation two phase gas liquid course: modern approach // Calculations and modeling journal. – 2016. – TPА 2 (83) 2016. – P.38-42.)

Technology Development

Since the time of industrial development, installation designs have undergone great changes. For example, there are shell and tube evaporators for freeze systems where the heat transfer coefficient has increased 10 times over during the last 50 years. These results are a consequence of different innovation decisions. Developments led to research into mini-channels systems, which is the one of the methods to increase intensification of phase transition. Research has shown that heat exchange systems with micro and nano dimensions have a much greater effect than the macrosystems with channels dimensions ≤3-200 mm.

In order to organize fundamental research, it is very important to understand hydro, gas dynamics and heat changes in two-phase systems with the phase transition. At present, the number of researchers using advanced CFD-programs has increased. Our team is one of the lead developers of these program complexes.

Mathematical modeling of compressible multiphase fluid flows is interesting with a lot of scientific directions, and has big potential for practical use in many different engineering fields. Today it is no secret that environmental issues are some of the most commonly discussed questions in the world. People are trying to reduce the emissions of combustion products. One of the methods to decrease emissions is the organization of an environmentally acceptable process of fuel burning with reduced yields of nitrogen and sulfur. The last blog (http://blog.softinway.com/en/modern-approach-to-liquid-rocket-engine-development-for-microsatellite-launchers/) discussed numerical methods, which can calculate these tasks with minimal time and cost in CFD applications.

Waste Heat Boiler
Picture 1 – Waste heat boiler http://tesiaes.ru/?p=6291

For more effective use of energy resources and low-potential heat utilization, the choice of the Organic Rankine Cycle (ORC) is justified. Due to the fact that heat is used and converted to mechanical work, it is important to use a fluid with a boiling temperature lower than the boiling temperature of water at atmospheric pressure (with working flow-boiling temperature about 100⁰C). The usage of freons and hydrocarbons in these systems makes a solution impossible without taking into account the changes of working fluid phases. Read More

2.5 Thermal Cycles Modelling

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Imagine the process of analyzing the thermal cycle in the example of gas turbine unit (GTU) (Fig. 2.18) in the following sequence:

  1. the structure diagram presentation as a set of standard elements and connections between them;
  2. entering the input data on the elements;
  3. generation of computer code in the internal programming language based on the chosen problem statement;
  4. processing;
  5. post-processing and analysis of results.

 

Figure 2.18 Thermal schemes graphical interactive editor window.
Figure 2.18 Thermal schemes graphical interactive editor window.

This sequence of actions combines a high degree of automation of routine operations (input-output and storage of data, programming, presentation of the results of calculations, and so on) with the possibility of human intervention in the process of calculations at any stage (editing of data, changing the program code in the domestic language, writing additional custom code for non-standard calculations performing, etc.).

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2.4 Flow Path Elements Macro Modelling

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Macromodels are dependencies of the “black box” type with a reduced number of internal relations. This is most convenient to create such dependence in the form of power polynomials. Obtaining formal macromodels (FMM) as a power polynomial based on the analysis of the results of numerical experiments conducted with the help of the original mathematical models (OMM).

Therefore, the problem of formal macro modelling includes two subtasks:

1. The FMM structure determining.
2. The numerical values of the FMM parameters (polynomial coefficients) finding.

As is known, the accuracy of the polynomial and the region of its adequacy greatly depend on its structure and order. At the same time, obtaining polynomials of high degrees requires analysis of many variants of the investigated flow path elements, which leads to significant computer resources cost and complicates the process of calculating the coefficients of the polynomial.

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2.3 Geometric and Strength Model

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2.3.1 Statistical Evaluation of Geometric Characteristics of the Cascade Profiles

For accurate estimates of the size of the blades, which takes into account not only their aerodynamic properties and conditions of safe operation, it is required to calculate the set of dependent geometric characteristics of the profiles (DGCP) as a function of a number of parameters that determine the shape of the profile. When the shape of the profiles is not yet known, to assess DGCP should use statistical relations. From the literature are known attempts to solve a similar problem [25, 26] on the basis of the regression analysis.

The DGCP include: f – area; Ie and In minimum and maximum moments of inertia;Iu – moment of inertia about an axis passing through the center of gravity of the cross section parallel to the axis of rotation u; φ the angle between the central axis of the minimum moment of inertia and the axis u; Χgcgc the coordinates of the center of gravity;βi – stagger angle;lss – the distance from the outermost points of the edges and suction side to the axis Ε; linlout – the distance from the outermost points of the edges to the axis Ν; We, Wss, Win, Wout, – moments of profile resistance.

The listed DGCP values most essentially dependent on the following independent parameters (IGCP) β1g – geometric entry angle; β2eff – effective exit angle; – chord; t/b  – relative pitch; r1, r2 – edges radii; ω1, ω2 – wedges angles.

Formal macromodelling techniques usage tends to reduce the IGCP number, taking into account only meaningful and independent parameters. In this case, you can exclude from consideration the magnitude of  r1, r2, ω2 taking them equal r1 =0.03b; r2=0.01b ; ω2=0.014Kωω1/(0.2 +ω1) , Kω = 1…3 , depending on the type of profile [26].

We obtained basic statistical DGCP relationships using profiles class, designed on the basis of geometric quality criteria – a minimum of maximum curvature of high order power polynomials [15] involving the formal macromodelling technique. Approximation relations or formal macromodel (FMM) are obtained in the form of a complete quadratic polynomial of the form (1.2):

Formula for chapter 2.3

The response function y(q ⃗’) values (DGCP) corresponding to the points of a formal macromodelling method, calculated by the mathematical model of cascades profiling using geometric quality criteria.

Analysis of profiles used in turbine building reveals, that two of remaining four IGCP β1g and t/b highly correlated.

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Modern Approach to Liquid Rocket Engine Development for Microsatellite Launchers

Microsatellites have been carried to space as secondary payloads aboard larger launchers for many years. However, this secondary payload method does not offer the specificity required for modern day demands of increasingly sophisticated small satellites which have unique orbital and launch-time requirements. Furthermore, to remain competitive the launch cost must be as low as $7000/kg. The question of paramount importance today is how to design both the liquid rocket engine turbopump and the entire engine to reduce the duration and cost of development.

The system design approach applied to rocket engine design is one of the potential ways for development duration reduction. The development of the design system which reduces the duration of development along with performance optimization is described herein.

The engineering system for preliminary engine design needs to integrate a variety of tools for design/simulation of each specific component or subsystem of the turbopump including thermodynamic simulation of the engine in a single iterative process.

The process flowchart, developed by SoftInWay, Inc., integrates all design and analysis processes and is presented in the picture below.

Execution Process Flow Chart
Execution Process Flow Chart

The preliminary layout of the turbopump was automatically generated in CAD tool (Block 11). The developed sketch was utilized in the algorithm for mass/inertia parameters determination, secondary flow system dimensions generations, and for the visualization of the turbopump configuration. The layout was automatically refined at every iteration. Read More