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.
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; Χgc,Υgc 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; lin, lout – 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; b – 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 .
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  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):
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.
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.
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
2.2.1 Axisymmetric Flow in the Axial Turbine Stage
Assume that in the flow path of the turbine:
The flow is steady relatively to the impeller, rotating at a constant angular velocity ω about the z-axis or stationary guide vanes.
The fluid is compressible, non-viscous and not thermally conductive, and the effect of viscous forces is taken into account in the form of heat recovery in the energy and the process equations, i.e., friction losses are accounted energetically.
If the working fluid is real (wet steam) it is considered the equilibrium process of expansion.
the flow is axisymmetric, i.e., its parameters are independent of the circumferential coordinate.
Under these assumptions the system of equations describing the steady axisymmetric compressible flow motion, includes:
1. The equation of motion in the relative coordinate system in the Crocco form
2. Continuity equation
3. The equation of the process or system of equations describing the process
4. The equations of state
5. The equation of the flow surface
where n ⃗’ – normal to the S2 surface (Fig. 2.1).
6. The equation of blade force orthogonality to the flow surface
Projections of the vortex in the relative motion rot W ⃗’ = ∇ * W ⃗’ to be determined by the formulas:
Taking into account (2.12), projection of the equation of motion (2.6) on the axes of cylindrical coordinate system can be written as follows: Read More
This is an excerpt from a technical paper, presented at the ASME Turbo Expo 2018 Conference in Oslo, Norway and written by Leonid Moroz, Leonid Romanenko, Roman Kochurov, and Evgen Kashtanov. Follow the link at the end of the post to read the full study!
High-performance rotating machines usually operate at a high rotational speed and produce significant static and dynamic loads that act on the bearings. Fluid film journal bearings play a significant role in machine overall reliability and rotor-bearing system vibration and performance characteristics. The increase of bearings complexity along with their applications severity make it challenging for the engineers to develop a reliable design. Bearing modeling should be based on accurate physical effects simulation. To ensure bearing reliable operation, the design should be performed based not only on simulation results for the hydrodynamic bearing itself but also, taking into the account rotor dynamics results for the particular rotor-bearing system, because bearing characteristics significantly influence the rotor vibration response.
Numbers of scientists and engineers have been involved in a journal bearing optimal design generation. A brief review of works dedicated to various aspects of bearing optimization is presented in . Based on the review it can be concluded, that the performance of isolated hydrodynamic bearing can be optimized by proper selection of the length, clearance, and lubricant viscosity. Another conclusion is that the genetic algorithms and particle swarm optimization can be successfully applied to optimize the bearing design. Journal bearings optimizations based on genetic algorithms are also considered in [2-5]. The studies show the effectiveness of the genetic algorithms. At the same time, the disadvantages of the approach are high complexity and a greater number of function evaluations in comparison with numerical methods, which require significantly higher computational efforts and time for the optimization. A numerical evolutionary strategy and an experimental optimization on a lab test rig were applied to get the optimal design of a tilting pad journal bearing for an integrally geared compressor in . The final result of numerical and experimental optimizations was tested in the field and showed that the bearing pad temperature could be significantly decreased. Optimal journal bearing design selection procedure for a large turbocharger is described in . In this study power loss, rotor dynamics instability, manufacturing, and economic restrictions are analyzed. To optimize the oil film thickness by satisfying the condition of maximizing the pressure in a three lobe bearing, the multi-objective genetic algorithm was used in . In the reviewed studies the optimization has been performed for ‘isolated’ bearing and influence on rotor dynamics response was not considered.
For higher reliability and longer life of rotating mechanical equipment, the vibration of the rotor-bearing system and of the entire drivetrain should be as low as possible. A good practice
for safe rotor design typically involves the avoidance of any resonance situation at operating speeds with some margins. One common method of designing low vibration equipment is to have a separation margin between the critical natural frequencies and operating speed, as required by API standard . The bearing design and parameters significantly influence rotor-bearing system critical speeds. Thus, to guarantee low rotor vibrations, the critical speeds separation margins should be ensured at rotor-bearing system design/optimization stage
Conjugated optimization for the entire rotor-bearing system is a challenging task due to various conflicting design requirements, which should be fulfilled. In  parameters of
rotor-bearing systems are optimized simultaneously. The design objective was the minimization of power loss in bearings with constraints on system stability, unbalance sensitivities, and
bearing temperatures. Two heuristic optimization algorithms, genetic and particle-swarm optimizations were employed in the automatic design process.
There are several objective functions that are considered by researchers to optimize bearing geometry, such as:
– Optimum load carrying capacity ;
– Minimum oil film thickness and bearing clearance optimization [1, 6, 8];
– Power losses minimization [6, 7];
– Rotor dynamics restrictions;
– Manufacturing, reliability and economics restrictions 
The most common design variables which are considered in reviewed works are clearance, bearing length, diameter, oil viscosity, and oil supply pressure.
Finding the minimum power loss or optimal load carrying capacity together with the entire rotor-bearing system dynamics restrictions, require to employ optimization techniques, because accounting the effects from all considered parameters significantly enlarge the analysis process. Several numerical methods, such as FDM and FEM are usually employed to solve this complex problem and calculation process can sometimes be time-consuming and takes a large amount of computing capacity. To leverage this optimization tasks, efficient algorithms are needed.
In the current study, the optimization approach, which is based on DOE and best sequences method (BSM) [11, 12] and allows to generate journal bearings with improved characteristics was developed and applied to 13.5 MW induction motor application. The approach is based on coupled analysis of bearing and entire rotor-bearing system dynamics to satisfy API standard requirements.
Problem Formulation and Analysis Methods Description
The goal of the work is to increase reliability and efficiency for the 13.5 MW induction motor prototype (Fig. 1) by oil hydrodynamic journal bearings optimization.
The motor operating parameters and rotor characteristics are presented below:
– Rated speed rpm: 1750
– Minimum operating speed rpm: 1750
– Maximum operating speed rpm: 1750
– Mass of the rotor kg: 6509
– Length of the rotor mm: 3500
Initially, for the motor application, plain cylindrical journal bearings were chosen to support the rotor. The scheme of the DE (drive end) and NDE (non-drive end) baseline bearings designs
is presented in Fig. 2. For baseline designs, bearing loads were 35 kN for DE and 28 kN for NDE bearing.
The methodology for the bearing characteristics simulation is based on the mass-conserving mathematical model, proposed by Elrod & Adams , which is by now well-established as the
accurate tool for simulation in hydrodynamic lubrication including cavitation.
This is an excerpt from a technical paper, presented at the ASME ORC 2015 Conference in Brussels, Belgium and written by Oleksii Rudenko, Leonid Moroz, Maksym Burlaka, and Clement Joly. Follow the link at the end of the post to read the full study!
Internal combustion piston engines are among the largest consumers of liquid and gaseous fossil fuels all over the world. Despite the introduction of new technologies and constant improving of engines performances they still are relatively wasteful. Indeed, the efficiency of modern engines rarely exceeds 40-45% (Seher et al. (2012), Guopeng et al. (2013)) and the remainder of the fuel energy usually dissipates into the environment in the form of waste heat. The heat balance diagram of typical engine is given in Figure 1. As is evident from Figure 1, besides the mechanical work energy the heat balance includes a heat of exhaust gas, a heat of charge air, a Jacket Water (JW) heat, a heat of lubricating oil and a radiation heat. The energy from all the heat sources except the last one (radiation), due to its ultra-low waste heat recovery potential, can be used as heat sources for WHRS (Paanu et al. (2012)) and are considered here.
Waste heat utilization is a very current task because it allows to reduce the harmful influence of ICPE operation on the environment as well as to obtain additional energy and to reduce the load on the engine’s cooling system. Different WHRS can produce heat energy, mechanical energy or electricity and combinations of the converted energy forms exist as well. In general, the type of WHRS to be used is determined by the engine type, fuel cost, available energy customers and other factors. In the presented paper, only WHRS for mechanical power and electricity production were considered because these kinds of energy are preferable for this type of applications and they can be easily converted into other forms of energy.
For vehicle engines the WHRS based on Organic Rankine Cycle (ORC) are the most commercially developed (Paanu et al. (2012)). Because of strict restrictions on weight and dimensions, the
mentioned systems typically operate on the base of a simple or recuperated ORC and utilize only high temperature waste heat from the exhaust gases and the exhaust gas recirculation. They usually produce mechanical power or electricity. More complex cycles and a larger number of heat sources are used for waste heat recovery from powerful internal combustion engines where additional weight and dimensions are not crucial factors. Waste heat from stationary, marine and another more powerful ICPE can be recovered using a typical steam bottoming cycle. Steam WHRS allow utilizing almost all a high temperature waste heat and partially utilizing a low temperature heat. The high efficiency steam WHRS are presented in (MAN Diesel & Turbo (2012), Petrov (2006)), they provide up to 14.5% of power boost for the engine.
Addition of the internal heat recuperation to a WHR cycle:
Appropriate working fluid selection;
Increment of initial parameters of bottoming cycle up to supercritical values;
Maximize waste heat utilization due to the usage of low temperature heat sources;
Bottoming cycle complexification or usage of several bottoming cycles with different fluids
This paper focuses on the development of new WHRS as an alternative to high efficiency steam bottoming cycles by accounting for the latest progress in the field of waste heat recovery. The
application range of the proposed system extends to powerful and super powerful ICPEs.
The goal of the presented work is the development of a new, high efficiency WHRS for powerful and super powerful ICPEs based on ORC principles. To solve the assigned task, a thorough study of the currently existing works was performed and the best ideas were combined. The principles of the maximum waste heat utilization, maximum possible initial cycle parameters, recuperation usage and single working fluid were assumed as a basis for the new WHRS design.
The equation of state can be written in different forms depending on the independent variables taken. Numerical algorithms should allow to calculate and optimize the axial turbine stages, both with an ideal and a real working fluid. It uses a single method of calculating the parameters of the state of the working fluid, in which as the independent variables are taken enthalpy i and pressure P:
For a perfect gas equation of state with P and i variables are very simple:
For the water steam approximation formula proposed in  is used, which established a procedure to calculate parameters of superheated and wet fluid. It is easy to verify that the knowledge of the value of the velocity coefficient Read More
To solve demanded by practice of axial turbines design multi-criteria problems, multi-parameter and multi-mode optimization of the multistage flow path further development and improvement of appropriate numerical methods and approaches required.
It should be noted some features of numerical solution of problems related to the optimization of design objects based on their modes of operation, multi-modal objective functions, as well as issues related to the multi-objective optimization problems.
Some aspects of the above problems solutions are given below.
1.5.1 Solution of the Multi-Criteria Optimization Problems
Set out in section 1.4 are the basic optimization techniques. However, depending on the formulation of the optimization problem, as well as the selected design object there are some features of numerical implementation of these methods and their applications.
It is known that the actual design object is usually characterized by a number of quality indicators and improvement in one of them leads to a deterioration in values of other quality criteria (Pareto principle). In such cases it is necessary to consider the optimization problem from many criteria.
The authors offer a well-established practice in solving multi-objective optimization problems – “convolution” of partial objective function weighted by u i depending on the importance of a particular quality criteria in a comprehensive quality criteria based on the following:
1.4.1 General Information About the Extremal Problems
To solve problems with the single criterion of optimality rigorous mathematical methods are developed.
Direct methods of the calculus of variations – one of the branches of the theory of extreme problems for functional – reduce the problem of finding the functional extremum to the optimization of functions.
There are analytical and numerical methods for finding optimal solutions. As a rule, the real problems are solved numerically, and only in some cases it is possible to obtain an analytical solution.
Functions optimization using differentiation
Finding the extremum of the function of one or more variables possible by means of differential calculus methods. It’s said that the X̂ point gives to function f (x) local maximum, if there is a number Ɛ>0 at which from the inequality | x-x̂| < Ɛ the inequality f (x) ≤ f (x̂) comes after.
The function is called one-extremal (unimodal) if it has a single extremum and multi-extremal (multimodal), if it has more than one extremum. The point at which the function has a maximum or minimum value of all local extrema, called a point of the global extremum.
A necessary condition for an extremum of a differentiable function of one variable gives the famous Fermat’s theorem: let f (x) – function of one variable, differentiable at the point x̂. If x̂ – local extreme point, then f’ (x̂) = 0.
The points at which this relationship is satisfied, called stationary. The stationary points are not necessarily the point of extreme. Sufficient conditions for the maximum and minimum functions of one variable – respectively f” (x̂) <0, f” (x̂) > 0.
Before proceeding to the necessary and sufficient conditions for extrema of functions of several variables, we introduce some definitions. The gradient of function f (x) is a vector
The real symmetric matrix H is called positive (negative) defined if XT = Hx>0(<0) for every set of real numbers x1 , x2, …. xn, not all of which are zero. Read More
Pumps are important for many common systems which deal with water, such as heating circulating flows, consumer or industrial water supply, fountains, and fire protection systems.
Pumps are classified into two major categories: Rotodynamic pumps and positive displacement pumps (piston pumps). Rotodynamic pumps can be further classified as axial pumps, centrifugal (radial) pumps, or mixed pumps.
Centrifugal pumps are the devices which impart energy to the fluid (liquid) by means of rotating impeller vanes, and the fluid exits radially from the pump impeller. Such pumps are simple, efficient, reliable, relatively inexpensive, and easily meet the pumping system requirements for filtration. This is a great pump choice for moving liquids from one place to another using pressure.
Centrifugal Pump Design
A centrifugal pump is a very common component in turbomachines, but as with any component, it still needs continual improvement in the design methodology, from conceptual level to the final product development including testing at different levels. The challenge is to design a pump with improved efficiency while minimizing the possibility of cavitation.
Need of Numerical Simulation
Years ago, engineers performed prototype testing at each level of design to check the performance (which was very costly and time consuming). Now with advancements in the computation technology and resources, it is comparatively easier to design high efficiency pumps within a short duration of time. These simulations can be done with a computer, so, the number of physical prototypes required is greatly reduced. The main advantage of numerical simulation is that it allows engineers to virtually test the CAD model early in the design process, and provides flexibility for engineers to iterate the design until getting the required performance.
Computational Fluid Dynamics for Centrifugal Pumps
Computational fluid dynamics (CFD) replaces the huge number of testing requirement. This not only shortens the design cycle time but also significantly reduces development cost.
In a CFD model, the region of interest, a pump impeller flow-path for example, is subdivided into a large number of cells which form the grid or mesh. The PDEs (partial differential equations) can be rewritten as algebraic equations that relate the velocity, pressure, temperature, etc. in a cell to those in all of the cell’s immediate neighbors. The resulting set of equations can then be solved iteratively, yielding a complete description of the flow throughout the domain.
To accomplish CFD simulations, there are several software programs available, but user must select a very well validated software that can provide and easy user interface, automatic mesh generation and flexibility to modify the geometry to perform optimization without needing to move to some other software platform.
In the current trend, automatic mesh generation tools like AxCFD™ are employed in the AxSTREAM® software platforms which reduces the turbomachines meshing complications and generate good quality mesh in considerably short timeframe which can capture the accurate flow features needed. Figure 2 shows the discretized impeller and pressure contour after CFD analysis.
AxCFD™, in AxSTREAM® platform, provides user an opportunity to perform CFD analysis by applying standard methods of full three-dimensional CFD, axisymmetric CFD (meridional), and blade-to-blade analysis. User can even perform optimization of the blade profiles and other geometrical parameters within the AxSTREAM® platform and perform CFD simulation without altering any CFD settings.