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.
To create FMM it is advisable to use the mathematical apparatus of the design of experiment theory to significantly reduce the number of computing experiments with OMM, i.e. obtain sufficient information with a minimum dimension of the vector of observations Y’ . We use two types of FMM – (1.2) and (1.12). To get them methods of the design of experiment theory applied (three-level Boxing-Benkin plans and saturated Rehtshafner’s plans) and cubic spline interpolation.
In a particular implementation of a formal macro modelling methodology need to perform the following steps:
1) the choice of the IMM of the flow path element;
2) the appointment of its performance criteria;
3) choice of OMM parameters, whose influence on performance criteria of the flow path element is necessary to study in detail and the formation on their basis vector of varied parameters Q ⃗;
4) macro modelling area appointment (ranges of components of the vector q ⃗ );
5) DOE matrix formation;
6) active numerical experiments conducting and evaluation of the components of observations vector
7) the processing of the experimental results and the determination of the FMM coefficients.
Steps 1–4 are not amenable to formalization and their implementation should take into account the specific features of macro modelling objects and existing experience of designing elements of axial turbines flow path.
]]>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; I_{e} and I_{n} minimum and maximum moments of inertia;I_{u} – 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; Χ_{gc},Υ_{gc} the coordinates of the center of gravity;β_{i} – stagger angle;l_{ss} – the distance from the outermost points of the edges and suction side to the axis Ε; l_{in}, l_{out} – the distance from the outermost points of the edges to the axis Ν; W_{e}, W_{ss}, W_{in}, W_{out}, – moments of profile resistance.
The listed DGCP values most essentially dependent on the following independent parameters (IGCP) β_{1g} – geometric entry angle; β_{2eff} – effective exit angle; b – chord; t/b – relative pitch; r_{1}, r_{2} – edges radii; ω_{1}, ω_{2} – 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 r_{1}, r_{2}, ω_{2} taking them equal r_{1} =0.03b; r_{2}=0.01b ; ω_{2}=0.014K_{ω}ω_{1}/(0.2 +ω_{1}) , K_{ω} = 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):
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 β_{1g }and t/b highly correlated.
It is advisable to use in place of these factors their counterparts – the flow rotation angle in the cascade θ and the parameter Δt=t/b-T, where T=1.08-0.004θ linear regression equation that specifies the statistical relationship between the relative pitch and angle of rotation of the flow, the resulting data for typical turbine cascade.
Thus, informal macromodelling as IGCP were taken:θ, β_{2e }, ω_{1}, Δt relatively in ranges 20…120, 10…30, 20…30, –0.2…0.2. In normalized form in the range of –1…1 the factors are calculated as follows:
During macromodelling were designed 25 turbine cascades with b =1 and with IGCP values, corresponding to the points in the of numerical experiment plan, were calculated DGCP values and the dependencies on the form (1.2) built for them. Calculation of flow diagrams and loss factors confirms the high aerodynamic quality of the 25 profile cascades.
In Tables 2.3, 2.4 the FMM coefficients and variance of the cascades DGCP FMM are given. In the tables FMM coefficients increased by 10^{4}. Similar relationships were also obtained for a special class of nozzle profiles with an elongated front part [26].
On the choice of cascade’s optimal gas dynamic parameters significantly affect the strength limitations, which, in turn, is largely dependent on the flow path design.
For example, the calculation of splitted diaphragms strength based on using a simplified scheme, according to which the diaphragm is considered as a semi-circle rod (band with a constant cross-section), loaded with unilateral uniform pressure and supported on the curved outer contour[27]. This approach allows us to evaluate the maximum stress in the diaphragm and is sufficient to
assess the strength of the diaphragm at the stage of conceptual and technical design.
Calculation of the blades strength is carried out using the beam theory that restrict computer time to evaluate the tensile and bending stress, for example, using statistical data on profiles, as shown in Section 2.3.1.
To ensure the vibration reliability of blading, rotor blades requires detuning from resonance, i.e., the natural frequencies of the blades should not coincide with the frequency of the disturbing forces that are multiples of the frequency of rotation. The required for detuning dynamic (depending on rotation speed) the first natural frequency of the blade is defined by a simplified formula.
]]>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.
The secondary flow system was modeled to determine the fluid mass flow rate that provides sufficient cooling for reliable operation of the bearings. The hydraulic network analysis tool was used for the secondary flow system calculation (Block 16). The calculation scheme of the secondary flow system is presented below
Heat amount produced by bearings friction (Block 15) is determined using turbopump axial load (Block 13) and bearings reaction in the radial direction. The radial reaction of bearing was determined using rotor dynamics simulation tool taking into account radial forces induced by circumferentially non-uniform flow admission on the turbine (Block 14). The rotor geometry was transferred from the turbopump parameterized CAD model (Block 11). The rotor model for bearing reaction determination is shown in the picture below
Preliminary FE stress analysis for the turbomachinery components (presented in Block 5 and Block 6) was also included in the algorithm. The stress analysis results for the pump is presented below. The results of this stress analysis were used for refinement of turbomachinery components geometry.
Overall seven different configurations of the turbopump were taken into account during the execution of the algorithm. The developed approach allows switching from a manual approach an automatic one, performing preliminary design steps described above for each configuration and recording crucial performance parameters for the selection of the best configuration. The configurations are presented in the picture below.
The preliminary design of the thrust nozzle for the engine was part of the algorithm as well. It can be seen in the picture below.
The configuration #1 was determined as the best option for the micro-launcher with 50 kg payload propelled by gas-generator cycle liquid rocket engine. The automatic search for optimum turbopump design took about 3.5 hours. The completion of the iterative process of turbopump preliminary design, including both pumps design, turbine design, turbopump preliminary layout development, secondary flows simulation, bearings simulation, rotor dynamics, and stress analysis would take 3 weeks at a minimum of experienced engineer labor time for a single configuration. Seven configuration would result in 21 weeks (840 hours) of labor time. Thus, the labor time for the preliminary design of the liquid rocket engine was reduced by 240 times utilizing the developed approach. This time reduction not only decreases labor time but also decrease the associated project cost and allows the engine to be supplied in a shorter period.
Off-The-Shelf Software Tools Utilized in this Study
The AxSTREAM® Platform was used in the design system, including:
Assume that in the flow path of the turbine:
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 S_{2} 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:
The components of the relative velocity based on the designated flow angles (Fig. 2.1) can be written as
From the relation [ n ⃗’ , F ⃗’ , ] = 0 will have:
We express the ratio of the normals projections through the flow angles (Fig. 2.2):
Than we can write
Transforming the radial equilibrium equation
transformed into
The last expression, in turn, by shifting from the coordinates s, n to coordinates s, r is represented as:
since
The projection of the equation of motion in the circumferential direction
Let us now consider the projection of the equation of motion (2.6) in the circumferential direction (2.14). Using (2.16), and the relationship between the coordinates z, r and s, r in the meridian plane
equation (2.14) becomes:
These equations enable us to determine the projection of the blade force F_{u} in the circumferential direction. The radial component F_{r} is expressed through the circumferential according to (2.17).
The projections of the friction force on the coordinate axes
The expression for the friction force
can be transformed by using the expression (2.16) and the binding ratio between the cylindrical coordinates z, r and the coordinates s, r:
whence we get the projection of the friction force on the coordinate axes:
The continuity equation is advisable to use in a form
For a free channel from the projection of the equation of motion in the absolute coordinate system to the circumferential direction, obtain, that the circulation c_{u}r = const along the meridian streamline and
Then for a free channel (χ = 1) from (2.31) we have:
The considered above in the general formulation, the problem of calculation of axisymmetric flows of a compressible fluid in the flow path of the axial turbine can be simplified and reduced to the calculation in gaps [9]. The flow in the axial gap is seen at the main proposals set out above. Within axial gap in the space free of the blades (χ = 1); because of its small length in the axial direction the entropy S locally do not changes along the meridian streamlines
it is possible to force components F_{r}= f_{r} = 0;stream keeps the direction of motion, telling him by blades (i.e., the angle of the flow β is et).
In these assumptions the radial equilibrium equation will differ from (2.24) in the absence of the right side of F_{r}= f_{r}:
Consequently, the system of equations, describing the steam flow in the axial turbine stage gaps are as follows:
The numerical realization of the stage thermal calculation problem
Mathematical models of axial turbine stages, discussed above, allow their calculation by setting some additional (closing) relations, for example, the
distribution of the angles β and α (direct problem), the quantities c_{u}r, pc_{z} et al. (inverse problems).
To solve the direct problem of stage calculation in gaps the following information is required:
There are varieties of the direct problem with a given flow rate G and with a specified back pressure P_{2} Solution of the problem with a fixed flow easier because the integration of the equations (2.39), (2.40) is made for a known ψ^{*}=G/(2π) value and mathematically formulated as a two-point boundary value problem for a system of two ordinary first order differential equations of the form:
The solution of (2.43) is made in two stages: first, the first equation is solved and the distribution of the flow in fixed axial gap is found, then, knowing the parameters entering the impeller can solve the second equation. That is, the problem is reduced to determining the roots of the equation with one unknown for each of the two equations (2.43). For the calculation of the subsonic solutions of (2.43) can be successfully used the methods of nonlinear programming. The system (2.43) is solved by sequential minimization of residuals
using one of the described one-dimensional extremum search methods.
Solution of the problem with a given back pressure P_{2} (flow rate unknown) is more complicated. To determine the unknown mass flow G to the system of equations (2.43) is necessary to add one more thing – a limit on the heat drop.
In this formulation of the problem it seems appropriate to set the mass flow averaged pressure according to the formula
In view of (2.45) to calculate the level with a given back pressure is needed to solve a system of three equations with three unknowns:
Numerically, the problem is solved to minimize the sum of squared residuals
A maked up mathematical model describing the flow in the axial gaps of turbomachine (equation (2.39), (2.40)), allows the calculation of supersonic flow (including the transition through the speed of sound), which M_{s} <1, i.e.,in the case of the meridional component of velocity less than the velocity of sound. Specified the conditions satisfy all existing stages of powerful steam turbines.
Calculation of supersonic stages must be performed with a given back pressure, because otherwise does not provide a unique solution of the equation of the form (2.41). At the same time, the system of transcendental equations (2.46) in the variables c_{1h},w_{2h,}ψ^{*}.in contrast to (2.43) has a unique root.
Another feature of the supersonic stages calculation is the need to consider the flow deflection in an oblique cut at Mach numbers higher than unity. For this purpose it is possible to use a method of determining the flow deflection angle in an oblique cut comprising in equating flow rate into the throat section and behind the blade [10, 11].
In this case, to calculate the residuals of equations (2.44), (2.46) it is necessary to integrate the system of ordinary differential equations of the form (2.41), namely (2.39), (2.40). These equations are due to the complexity of the form of the right sides in the general case can be integrated numerically. When integrating (2.39) (2.40) should be borne in mind that at each step of pressure shall be determined by solving the equations of the form (2.38), which greatly complicates the task.
Finally, we note that because of the existence of the right sides of (2.38), (2.40), a member
, the system, generally speaking, can not be considered as written in the form of Cauchy, as these non-linear supplements are some of the functions w_{2} or c_{1}, r and their derivatives. When integrating these terms are determined by successive approximations.
The important point is the choice of numerical methods for integrating systems of the form (2.41). Extensive experience in solving such problems suggests the possibility of partitioning the integration interval to a small number of steps (5–10). As a result of numerical experiments comparing different methods, preference was given to the modified Euler’s method [12], which has
the second order of accuracy for the integration step.
The leakage calculation is necessary to conduct together with a stage spatial calculation, the results of which are determined the parameters along a height in the calculation sections, including the meridian boundaries of the flow part.
The stage capacity depends on the value of clearance (or leakages), in connection with which calculation of the main stream flow is made by mass flow amplification at fixed the initial parameters and counter-pressure on the mean radius, or with counter-pressure elaboration at fixed initial parameters and mass flow.
The need for multiple steps in optimization problems requires a less labor-capacious, but well reflecting the true picture of the flow, methods of axisymmetric stage calculation. Its main point is to calculate the stage parameters in the axial gaps supplemented by the algorithm of stream lines slope and curvature refinement in the design sections.
When calculating the stage taking into account leakage, the continuity equation is convenient to take the form:
where μ– mass transfer coefficient, which allows to take into account changes in the amount of fluid passing through the crowns, and at the same time to solve a system of ordinary differential equations
As shown, the calculation of spatial flow in the stage with the known in some approximation the shape of the stream lines is reduced to the solution in the sections
(Fig. 2.3) of a system of ordinary differential equations (2.39) and (2.40), where as independent variable a stream function ψ is taken. Thus, the equations describing the flow in the axial gap, presented in the form of:
The solution of the boundary problems (2.49), (2.50) for a given mass flow rate is reduced to finding the roots of the two independent transcendental equations (2.43) with respect to the hub velocities c_{1h},w_{2h}.
For a given backpressure to the number of defined values the mass flow ψ^{*} is added and the problem reduces to solving a system of three equations. As a third equation the stage heat drop constraint is added (2.45) that can be symbolically written as
Systems of equations are solved using the methods of nonlinear programming.
An approximate method of meridian stream lines form amplification using their coordinates in the three sections, is to construct an interpolation cubic spline at a given slopes at the flow path boundaries. In order to accelerate the convergence the stream line curvature is specified with lower relaxation. Previous calculations showed that the interpolation process converges with sufficient accuracy in 3…5 iterations.
Mass flow rates through crowns carried out in parallel with the streamlines construction. The algorithm allows to solve the direct problem of the spatial stage calculation in the gaps in various statements, with given or variable in the process of calculating the streamlines, velocity and flow coefficients of crowns, at various ways of flow angles distribution along the height, for a perfect gas or steam.
The algorithm was tested by comparing the calculation results with the exact solutions, as well as with the experimental data obtained for a large number of stages of the experimental air turbines in the turbine department of NTU “KhPI” [13–14, 15]. The results of calculations and experiments illustrated in Fig. 2.4–2.12. It should be stated a good calculations agreement with the experimental result for the various stages of the different elongation, meridian shape contours, twist laws and the reaction degree at the mean radius.
The greatest difficulty to calculate present stages with the steep opening of the flow path (Fig. 2.12), and the cylindrical stages with inversely twisted guide vanes (Fig. 2.4, 2.6–2.8).
The calculation of stages with inverse twist using the proposed method allows to obtain a valid gradient of reaction degree and circumferential velocity component of the stage, while the calculation provided in assumption of cylindrical flow gives results that differ significantly from the experimental data (Fig. 2.6–2.8). The technique allows to take into account also the effect of the
law of the impeller’s twist on the distribution of parameters in the gap between guide vane and rotor. This is evidenced by the comparison stages 41 and 42 (Fig. 2.7, 2.8) with the same nozzle unit, the first of which has a cylindrical impeller, and the second – twisted by constant circulation law.
Formulation of the problem
The off-design analysis problem is to determine the gas-dynamic characteristics derived from the design calculation such as the size of the flow path (FP) and the parameters that determine the long-term (steady) operation of the turbine. The need to analyze FP off-design modes arises when assessing aero- and thermodynamic, power, strength parameters of the turbine in extreme operating conditions, the choice of method for control and calculation of steam distribution, for turbines designed to operate at changing the regime parameters (speed, unregulated steam extraction and so on).
The specifics of these problems requires a gas-dynamic calculations in a direct statement, which is more labor intensive than the calculations commonly used in the design stage. In connection with this methods designed for use with a computer optimization procedures must meet several requirements:
To calculate high, medium and, to a lesser extent, the low-pressure parts of powerful steam turbines, justified the use of one-dimensional gas dynamics calculations using the simplified radial equilibrium equations in a axial clearance, the leaks balance at the root of the diaphragm design stages and the calculation method of the FP moisture separation. Accounting for the loss of
kinetic energy and efficiency assessment should be carried out by successive approximations based on the current results of the gas-dynamic calculation and empirical relationships, and reliability of the results – achieved by comparison with experimental results and the introduction of necessary adjustments.
Should be regarded as a satisfactory the accuracy of coincidence of calculated and experimental values of the relative losses in the range of 5…7% for FP made with straight or twisted by constant circulation law blading in the absence of the sharp curvature of the meridian contours. When the actual loss levels of 10…30% error in determining the efficiency, thus lies in the range of 0.5…2% [15].
Method of calculation
One-dimensional steady-state equilibrium adiabatic motion of water vapor in the flow path in a coordinate system rotating with angular velocity ω, sought a system of equations:
The solution to this system of equations for an isolated axial turbine stage in a direct statement requires:
The two main statements involve mass flow G_{0} determination at certain stagnated pressure P^{*}_{0} at stage inlet, or the P^{*}_{0} definition at known flow rate. It is also possible the solution of the problem with given at the same time G_{0} and P^{*}_{0} changing angles α_{1e} or β_{2e} in particular, makes it possible to simulate the nozzle assembly with rotary blades. In all cases, subject to the definition of the flow speed c_{1} and w_{2}.
For definiteness we shall consider the problem with fixed P^{*}_{0} and mass flow determination. We transform the equation of continuity (2.52) for the nozzle in view of (2.51), (2.53), (2.54):
Under α_{1} and β_{2} at subsonic flow understood the cascade’s effective angles, and at supersonic – flow angles in the oblique cut-off by the Ber formula.
The third equation is:
Under certain velocity factors φ and ψ to determine the unknown c_{1} , G, w_{1, } there are three equations (2.55)–(2.57), which in general terms be written as follows:
The system (2.58) is solved numerically by minimizing the sum of squared residuals g_{1 }^{ 2} +g_{2}^{ 2} + h^{ 2} using the conjugate gradient method.
Calculation of multistage flow path does not differ systematically from the stage calculation. An equation of (2.58) is written for each of the stages, which leads to a system of the form
where j – stage index; n – number of stages in the FP.
The numerical solution is carried out by minimizing the function
by 2n + 1 unknowns c_{1j},w_{2j}, (j = 1, …,n) + h^{2}
Sections may have different mass flows because of the leaks, district heating or regenerative steam extraction, moisture separation and so on. In the equations (2.59) in this case instead of a mass flow rate G in the relevant sections should take the current value
where ΔG_{k}– given or confirmed in iterations the mass flow change in the transition from (k – 1) section to the k-th (k=1…2n).
The unknown is considered the G_{0} mass flow at the FP entrance.
After the solution of (2.58) or (2.59) all the parameters of the flow calculated, loss factors and the actual mass flows in sections adjusted. The required number of iterations is usually equal to 3…4.
Kinetic energy loss determination
Losses associated with the leakage of the working fluid are considered separately. The remaining components are divided into losses in cascade and auxiliary, which are allocable to the stage heat drop.
Methods of assessing the losses in cascades based on research [8, 16] with a corresponding adjustment of empirical dependencies using test data about profiles used in the turbine building [17, 18].
Corrections for the Reynolds number, angle of attack, the thickness of the trailing edge, at supersonic flow are taken over without change [19]. The amendment to the angle of attack in the profiles provided with an extension of the leading edge, is estimated according to experimental studies on the standard nozzle profiles and the impact of the extension on the profile loss – NPO CKTI the procedure [20].
The basic component of the profile X_{pb} obtained by a corresponding adjustment to the loss level of graphic dependence [16]. Basic secondary loss is determined by the corrected chart [16], an amendment to the ratio of the chord to the height of the blade N_{b/l } – according to [16], and the coefficient N_{s} taking into account the length of the visor hanging over the trailing edge of the blade – based on experimental data on nozzle standard profiles test data.
When assessing the energy losses in the rotor blades, can be taken into account the effect of the periodic incident flow unsteadiness caused by the presence of traces of the previous nozzle cascade, as amended N_{y}. The degree of non-uniformity of the incoming flow is taken over [8].
Additional energy losses are the disc friction and ventilation, extortion, humidity, the presence of the wire bonding and friction in the open and closed axial clearance in accordance with the guidelines [21].
Leak sand leakage losses calculation
It is estimated that losses caused by leakage of the working fluid into the gaps of the flow path, associated with a decrease in the mass flow rate through the crowns, aerodynamic and thermodynamic mixing with the main flow losses, as well as the deviation of the kinematic parameters in the gaps comparing to the design.
To determine the thermodynamic parameters near the flow path margins, needed to calculate the leaks mass flows, a simplified equation of radial equilibrium
is involved. In the gap between vanes considered that c_{u}r = const, P_{1} = const , and behind the stage c2_{u} = const, P_{2} = const.
Leaks in the root area of multistage flow paths are the solution of the mass flow balance equations through diaphragm, root seals and discharge holes taking into account given dependences of the gaps flow factors and friction coefficients of the regime and geometrical parameters, changes in pressure and flow swirling in the disk chambers along the radius at the presence of the working fluid flow etc.
Evaluation of leakages based on a calculation of the anterior chamber only, first, does not allow correct balance the mass flows along the FP, and secondly, may lead to considerable errors as the leakage values and axial forces, particularly at the off-design operation.
The algorithm is developed for the calculation of leakages in multistage FP, in which can be built leaks circuit within the cylinder based on the majority of the factors, influencing them [14]. Calculation of mixing the main flow with leaks through tip and root gaps is based on the balance equations for flow, enthalpy, and entropy. Raising the equations of motion for the evaluation of aerodynamic mixing losses allows, under certain assumptions, take into account the impact on the mixing loss of the blowing working fluid angle.
The third group of losses factors, caused by leaks, mainly, through a change of velocity coefficient of cascades after gaps, where mixing occurs, due to variations of inlet flow angles.
To solve this problem, we used a combined one-dimensional and axisymmetric approach.
A mathematical model of a coaxial flow of the working fluid in the flow part of a multi-stage axial turbine
This model belongs to the class of quasi- two-dimensional models, and is a logical continuation of the one-dimensional model of the FP shown in subsection (2.2.3). All equations, methods and techniques of assessment of energy dissipation in the elements of FP used in the one-dimensional model, have been fully utilized in the development of quasi- two-dimensional model of the coaxial FP.
A distinctive feature of the coaxial model is the fact that the system of equations (2.59) are determined not to cross-sections corresponding to the mean radius of the multistage FP crowns, and for each current streams along its midline.
The system of equations (2.59) in a coaxial FP model in a general form as follows:
where m – is equal to increased by two the number given sections (streamlines) along the radius of the blades; j – number of cross-section along the blade height (the first cross section is located at the root level).
Accordingly, the dimension of the system of equations in a mathematical model of a coaxial flow in the FP is equal to (n + 1)m.
The marked increase in the number of sections required for a significant approaching of the root and near-the-tip stream lines to the root level and the peripheral area, respectively. With the same purpose the cross sectional area of the extreme stream lines assigned minimum values (1% of area of the corresponding vane). For the first iteration the remaining cross-sectional areas between the stream lines are equal and are determined as follows:
S_{(k,j)}=0.98S_{(k)}(M-2),
Where S_{(k) } – cross-section area of k-th vane.
After determining the S_{(k,j)} are determined the radii of mean lines of all flow streams, angular velocities and the values of all the geometric characteristics of the cascades at those radii. In subsequent iterations, the average radius of the stream lines, and all the characteristics of cascades and the working flow determined in accordance with the obtained distribution of the mass flow the radius of corresponding vanes. This ensures the equality of the working fluid (including the extractions and leakages) along the respective stream lines.
Considering that the system of equations (2.62) is based on the onedimensional flow theory for each stream line, where there is no equation of radial equilibrium, it becomes apparent that the above-described method of stream tubes sizing, is most accurate by using this model, it will be possible to evaluate the characteristics of the axial turbines, which vane’s twist corresponds
to the S_{u}r = const law, or close to it. For practical tasks coaxial mathematical model is most suitable when assessing the characteristics of the high pressure cylinder (HPC) flow path.
Despite the fact that the flow of working fluid along each stream tube in consideration of coaxial mathematical model of the FP is modeled in accordance with the one-dimensional theory, when calculating the flow kinematics the slope angles of each stream line are taken into account (curvature of the streamlines is not considered) and identifies all components of the flow velocity in axial gaps. To determine the angles of the middle line of the stream tubes cubic spline interpolation is used. A well-known feature of these splines is the coincidence of the first and second derivatives of the neighboring areas in the nodes of the spline coupling. It allows us to describe the midline of a stream line using dependence, which provides its most smooth shape.
Because in the outer iteration loop of the multistage axial turbine FP coaxial mathematical model (as well as in the one-dimensional mathematical model of the FP), the quantities of moisture separation, tip leakage and near-the-hub leakages and the working fluid extractions to the heating system and feed-water heating refer to the entire stage, and not to each stream tube, the question of adequate distribution of the marked mass flow changes between the stream tubes arise.
In this case, there are two variants of distribution of leaks and the working fluid extractions between the stream tubes:
Additionally, there are also two versions of the distribution of secondary loss of height of the blade:
Integral indicators of each stage in the coaxial model are determined by the relationships below:
A mathematical model of an axisymmetric flow of the real working fluid in a multi-stage axial turbine FP
Despite the fact that the coaxial mathematical model of the flow of the working fluid in the FP, as described in the previous section, has a fairly narrow range of independent use, yet it has a sufficiently high potential. If the formation of the transverse dimensions of the stream tubes to carry the light of the decision of the radial equilibrium equation (sections 2.2.1, 2.2.2), this model can be successfully used in the calculation of axisymmetric flow in a multistage axial turbine FP with virtually any kind of its crowns twists.
The use of coaxial FP model to evaluate the distribution of the static pressure behind the rotor blades to determine disposable heat drops of each stage that you need to solve the axisymmetric problem “with a given back pressure”. It is known that only in such a setting is possible to find the correct solution to supersonic stages. Marked problems for each crown of multi-stage flow path solved by the means of stream line curvature method. Thus, in view of (2.46) and the system of equations (2.62), the scheme for solving the problem “with a specified back pressure” for a multi-stage axial flow turbine parts will be as follows:
For clarity, the above-described sequence of solving axisymmetric problem “with a given back pressure” for multi-stage FP is shown in Fig. 2.13.
Consider some features of the numerical solution of axisymmetric problem for multi-stage FP. First, in dealing with this problem it is necessary to determine the parameters of the working fluid along the streamlines for multistage FP with variable from crown to crown mass flow of the working fluid. The marked change often occurs in the steam turbines FP, where the extraction of the working fluid is carried out between the stages, for example, for the feed water heating or a heat supply needs.
As the result of boundary problems solutions (2.49) and (2.50), the mean radii of stream tubes for all the crowns of multistage FP are transmitted to the coaxial model, where for the new stream tube’s cross-sectional areas, angular velocities, and all the geometric characteristics of the nozzle and working blades, the FP calculation is carried out and the new static pressure distribution after
working stages crowns is determined.
The FP calculation results using the algorithm corresponding to the coaxial model again transferred to the block of boundary problems solutions (2.49) and (2.50). Described iterative process continues as long as the results of the calculation for both FP calculation algorithms differ less than a prescribed accuracy. Thus, the FP coaxial model and boundary value problems (2.49) and (2.50) complement each other in solving the axisymmetric problem, eliminating the “alignment” on the results of the one-dimensional calculation and more adequately assess the value of disposable heat drop of FP stages.
It should be noted that the numerical implementation of the axisymmetric mathematical model of the working fluid flow in the FP in the form of alternate use of coaxial mathematical model and boundary problems, can with a high degree of adequacy and accuracy to model the processes in the FP with stages with relatively long blades and having a twisted crowns substantially different
from the law c_{u}r = const. As an example, in Fig. 2.14 are shown the shape of the flow lines resulting from the calculation of LPC FP of powerful steam turbine using the above axisymmetric mathematical model.
For the design of high efficiency axial flow turbines flow path it is important to have accurate, reliable and fast method for calculation of cascade flow and friction loss on the profile surfaces.
In the calculation of subsonic flows of an ideal liquid in the cascades long used an approach based on the reduction of partial differential equations to Fredholm integral equation of the 1-st or 2-nd kind [8, 22]. Available numerical implementation of solutions to these equations are facing a number of problems that do not allow a sufficient degree of reliability or accuracy of the calculated
arbitrary configuration cascades.
For example, for a long time, we used the method of calculation [22] reduces to the solution of the integral equation of the second kind with respect to the speed potential. It is possible to solve a number of important practical problems of cascade optimization, but had important shortcomings: the complexity of the integral equation kernel normalization, which led to difficulties in calculating thin and strongly curved profiles, as well as the need for numerical differentiation calculated potentials, which brings an additional error in the profile velocity distribution.
Later, we developed a method for cascades potential flow numerical calculating with an approximate view of the ideal gas compressibility based on the solution of the Fredholm equation of the 2-nd kind with respect to speed on the rigid surface, and a program for the PC is designed to work interactively.
Friction loss on the profile is carried out by calculating the compressible laminar, transitional and turbulent boundary layers using one-parameter Loitsiansky method [23]. To improve the accuracy of the results obtained on the basis of the recommendations given in the literature, calculated buckling points and end of the transition from laminar to turbulent boundary layer depending on the pressure gradient, the degree of free-stream turbulence and of profile surface roughness.
The developed algorithms for an ideal fluid flow calculation in the cascade and the boundary layer on the surface of the profile give a good qualitative and quantitative agreement between the calculated and experimental data for different types of cascades at different inlet angles, relative pitch, Mach and Reynolds numbers, characterized by high speed and are therefore suitable for
use in problems of optimizing the axial turbomachinery blades shape.
Aerodynamic optimization of turbine cascades is directed search a large number (hundreds to thousands) variants for their geometry, which increases with the number of variable parameters. The most reliable source of objective data on the flow of gas in a turbine cascade – physical experiment – obviously can not provide a sufficiently deep extreme.
Therefore, currently in the works for aerodynamic optimization it is the most popular approach in which to obtain data on the nature and parameters of the flow of the working fluid in the tested inter-blade channels numerically solve the Navier-Stokes equations, or their modifications [24].
Navier-Stokes equations written in conservative form is as follows:
Since the analytical solution of this system of equations associated with insurmountable mathematical difficulties, such a direction as computational fluid dynamics (CFD) arose, which deals with the numerical solution of the Navier-Stokes equations. The numerical solution of the equations of fluid dynamics involves replacing the differential equations of discrete analogs. The main criteria for the quality of the sampling scheme are: stability, convergence, lack of nonphysical oscillations. Computational fluid dynamics is a separate discipline, distinct from theoretical and experimental fluid dynamics and complement them. It has its own methods, its own sphere of applications, and its own difficulties.
Given the speed of modern computers, the most appropriate approach is based on a system of Reynolds-averaged Navier-Stokes (RANS) equations. It involves some additional turbulence modeling using some complimentary to the system (2.69) equations, which are called turbulence model.
The reliability obtained by the CFD results requires a separate analysis. As an example, compare the results of experimental studies of stages with D/l = 3.6 (Fig. 2.15–2.17) with the calculations in one-dimensional, axisym-metric (for gaps) and 3D CFD statements [19].
Flow parameters distribution along nozzle vane and blade height:
a – reaction; b – axial velocity component after nozzle vane;
c – tangential velocity component after nozzle vane; d – nozzle vane velocity coefficient;
e – blade exit flow angle in relative motion; f – axial velocity component after blade
Flow parameters distribution along nozzle vane and blade height:
a – reaction; b – axial velocity component after nozzle vane;
c – tangential velocity component after nozzle vane; d – nozzle vane velocity coefficient;
e – blade exit flow angle in relative motion;
f – axial velocity component after blade
It was shown that proper unidimensional and axisymmetric models combined with proven empiric methods of loss calculation provide the accuracy of the turbine flow path computation sufficient for optimization procedures in a bulk of practice valuable cases. Comparative analysis of the experiment and simulation results indicates an untimely nature of the assertion that 3D CFD analysis is already capable to substitute physical experiments.
]]>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 [1]. 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 [6]. 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 [7]. 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 [8]. 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 [9]. 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 [10] 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 [5];
– Minimum oil film thickness and bearing clearance optimization [1, 6, 8];
– Power losses minimization [6, 7];
– Rotor dynamics restrictions;
– Manufacturing, reliability and economics restrictions [7]
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.
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 [13], which is by now well-established as the
accurate tool for simulation in hydrodynamic lubrication including cavitation.
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:
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 [7] 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
allows to determine the value of losses at the expansion
and obtain an expression that relates the enthalpies i_{T} and i at the end of the isentropic and the actual process of expansion, as well as stagnated enthalpy in relative motion
The last expression in combination with isentropic process equation from point 1 with parameters P_{1},i_{1} and the value of the relative velocity
allows to come, deleting from (2.3), for example i_{T}, to the following process equation with unknowns P, i :
With the help of the equation (2.5) can be solved a number of problems related to the thermal calculations of stages, which statement depends on which parameter of the unknown is a given. If we assume a known specific enthalpy i at the end of expansion, we obtain the equation (2.5) relative to the pressure P. This problem arises, for example, based on a predetermined degree of reaction or determining the counter pressure by the theoretical enthalpy drop per stage.
Solution of equations of the form (2.5) with one unknown is carried out by means of minimizing the residual square using one-dimensional search of extreme.
]]>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.
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:
where B^{*} _{i} the components of the vector criterion (partial indicators of quality of the object); x ⃗’_{d},x ⃗’_{p} vectors of design parameters and operational parameters, respectively, which together define a design decision.
In fact, (1.37) is the magnitude of the partial criteria of quality, taking into account their weights u _{ i }.
Thus, in the n-dimensional normalized criterial space each variant of definitely best design object is characterized by a corresponding so-called Pareto point, whose distance to the center of coordinate proportional to the value of the module:
of vector quality criterion.
The experience of steam turbines cylinder optimization with the flow extraction for the purposes of regeneration and heating shows that there is needed to consider at least two criteria of quality – the efficiency of the flow of the cylinder and the power, generated by them.
In some cases it is necessary to check the objective function on multi-modality.
In the developed subsystem of multi-criterial and multi-level multi-parameter optimization of design objects to find the optimal solution the search is always performed in two stages whether uni-modal or multi-modal objective function.
Thus, the first (preliminary) stage is used to determine suspicious extremum points, to find which method is used ideas swarm (Bees Algorithm), the first work of which were published in 2005 [5, 6]. The method is an iterative heuristic multi-agent random search procedure, which simulates the behavior of bees when looking for nectar.
The criterion for the selection of points and their respective sub-areas, in which will be specified by the relevant decision of optimization problems, is the Euclidean distance:
in the space of optimized parameters between the compared points from the set LP_{t} sequence.
If the Euclidean distance R_{ab} between two points of LP_{t} sequence (x ⃗’_{a},x ⃗’_{b}), less than some fixed value R _{ set} then point with the large value of the objective function is selected.
Criteria evaluation for quality and functional limitations at the preliminary stage is performed by using FMM (of the form (1.2) or (1.12)). After processing all of the set of LP_{t} sequence points by a “swarm” algorithm suspicious extremum point are defined.
These points are then used as initial approximations of the final (refining) stage of the optimal solution finding. When refining the optimal solutions around the extremum suspicious spot, in a recursive optimize algorithm it is provided the transition from the evaluation criteria of quality and functional limitations by using FMM to their evaluation by appropriate OMM. It uses a method of coordinate descent or conjugate gradient method, for example, Fletcher-Reeves. Thus found several points of local optima are sorted by the value of the objective function, and the best solution given the status of optimal.
The above (1.37) convolution vector type of the objective function allows to take into account the specific feature of the problem of optimal design of facilities intended for use as a constant, and the variable modes. In the case of optimization taking into account the variability of operating loads, function (1.37), on the one hand, carries information about the overall effectiveness of the design in all modes of operation, and on the other hand, it emphasizes the Pareto signs of the competitive effect of ‘individual’ quality criteria for each of the operating modes on the final result.
Below is a description of the developed method, which provides the solution of problems of optimum design of turbomachinery, operated at a predetermined range of modes.
This method is based on the integration of formal macro-models of the objective functions.
When included in the examination of the alleged operation modes, created FMM criteria of quality and functional limitations are functions of the design and operational parameters. Ranges of change of regime parameters are selected in accordance with the proposed schedule changes and they do not change in the course of iterations to refine the optimal solutions.
Such FMM usage at the step of finding the optimal solutions necessitates multiple evaluation of quality criteria and functionality limitations for each sampling point (corresponding to a combination of structural parameters), the number of calculations of each FMM considered equivalent to the number of operating modes. Obviously, the increased number of calculations requires additional computing resources in the search for the best design.
The decision of the problem marked can be achieved by eliminating the regime parameters of the vector of varied FMM parameters (1.2). To eliminate the regime parameters it is necessary to carry out the FMM integration. In this case, the new FMM coefficients of integral quality criterion obtained from the following relationship:
Where N_{c},N_{m}– numbers of structural and operational parameters, respectively; t – time.
The new FMM of form (1.38) contains integrals of regime parameters, which can be calculated from the charts of regime parameters (q_{j} (t)) and converted to the form:
FMM form (1.39) is more convenient to use in the optimization algorithms for quality criteria and functional constraints evaluation, as presented macromodel depends only on the design parameters that do not change their values when changing the operating mode of the FP. Thus, the account of the expected schedule change duty operation is performed due to the fact, that the operating parameters are integrally included in the new coefficients FMM (1.40).
]]>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 X^{T} = Hx>0(<0) for every set of real numbers x_{1 ,} x_{2, ….} x_{n,} not all of which are zero.
The necessary conditions for that x̂ – the point of local extremum of n variables function f(x), x ƐE^{ n } are as follows:
If the Hessian is positive (negative) defined for all xƐE^{ n}, it is a sufficient condition of unimodality of the function. To test matrix A definiteness, Sylvester criterion is applied, according to which the necessary and sufficient condition for positive certainty are the inequalities:
This case involves determining the extremum in an infinite change range of variables x_{1 ,} x_{2, ….} x_{n. } If optimized function imposed additional conditions (restrictions), talk about the problem of conditional extremum. In general, you want to find extremum f(x), xƐE^{ n} under the constraints
To solve the problem (1.14) only with restrictions in the form of equations a method of Lagrange multipliers is used, which is based on the conduct of the Lagrange’s function:
where λ_{j} – undetermined Lagrange multipliers. We write the necessary conditions for optimality in the problem of conditional extremum with equality constraints:
It is a system of n + m equations from which can be determined x _{ i} , i=1,…,n, λ_{j}, j=1,…,m. A rigorous proof of the Lagrange conditions set out in the specific manuals. Explain the meaning of the method as follows. On the one hand, for all of x which satisfy the constraints h_{j} (x)=0, j=1,…,m, obviously L=(x, λ) = f(x). On the other hand, the extreme point of the Lagrange function also satisfies these conditions (the second equation (1.14), and therefore, finding an extremum L(x, λ), we simultaneously obtain a conditional f (x) extremum. To address the issue of the presence of a stationary point to be a local extremum in the problem of conditional extremum, let us expand Lagrange function in a Taylor series with a subject to the satisfaction of relations h_{j}(x)=0.
Optimization with constraints in the form of inequalities
Classical methods of finding the conditional and unconditional extrema of functions discussed above, in some cases, allow to solve problems with inequality constraints.
Let the task of finding the maximum of a function of one variable f(x) on the interval a≤x≤b. Using the necessary optimality conditions, we find the roots of f'(x) = 0 which lie in the interval [a, b]; We check the sufficient conditions for maximum f” (x̂) <0 and choose the points corresponding to the maximum. Also, we compute the function values at the borders of a segment, where it can take higher values than the interval (Fig. 1.4). We turn now to the case of several variables and consider the optimization problem: find a maximum f(x), xƐE^{ n }, subject to the constraints:
In the first stage of the solution by the method of Lagrange multipliers, we find all stationary points lying in the positive octant of n-dimensional space and isolate the maximum points on the basis of sufficient conditions for an extremum. Then we explore the positive octant boundary, in turn equating to zero in all sorts of combinations of 1, 2,…,n-m+1 variables, and each time solving the optimization problem with equality constraints. As a result of the computing process, the complexity of which is obvious, the largest of all the extrema should be selected.
A more general problem, find the maximum
For the problem can be written necessary optimality conditions (generalized Lagrange multiplier rule), however, it is rarely used because of the complexity of solving the resulting system of equations.
Subject of nonlinear programming
Nonlinear programming – branch of applied mathematics dealing with finding the extremum of function of many variables in the presence of nonlinear constraints in the form of equalities and inequalities, i.e. solution of the problem (1.14), discussed in the previous section [3].
Classical methods of optimization are part of it, along with disciplines such as linear, quadratic, separable programming. However, of the greatest practical interest to us are the numerical or direct methods of nonlinear programming, especially intensively developed in recent years.
None of the proposed algorithms is absolutely the best, so the choice of a numerical method is dictated by the content of a specific problem, which must be solved. Computational methods are classified according to some peculiarity of the problem (no restrictions, with equality constraints, inequalities and so on), the nature of methods of solutions (e.g., with or without the use of derivatives), the type of computers, programming language, and so on.
Search of one variable function extremum
A number of methods of finding an extremum of function of many variables use as a part the procedure for the one-dimensional optimization. In the case then function of one variable is multi-extremal, the only correct method of finding the global extremum is a direct enumeration of a number of values with some step in its change.
Obviously, the function can vary sharply, the smaller should be chosen the grid. After a rough determination of the neighborhood of extremum, begin to search its exact value. For this purpose, one-dimensional algorithms for searching the extremes of unimodal functions in a given interval are used.
One of the most effective methods is the so-called golden section. Recall that if a segment divided into two parts, so that the ratio of the lengths at a greater relative length equal to the length of most of all segment, obtain the so-called golden ratio (is approximately 0.38: 0.62). Golden section method just based on the multiple division of uncertainty interval, i.e. the interval in which the
extremum enclosed in an appropriate ratio.
To reduce the range of uncertainty in the 100 times 11 calculations is required, in the 10000 times – 21 calculations. For comparison, the bisection method (dichotomy) leads to a corresponding narrowing of the range of 14 and 28 function evaluations.
The advantage of the golden section is that it works equally well for smooth and non-smooth functions. It was found that, in the case of smooth functions by a polynomial approximation possible to quickly determine the number of extreme at the same accuracy as that by the golden section.
If the optimized function is defined and unimodal on the entire real axis, there is no need to worry about selecting the initial uncertainty interval. For example, in the method of Davis, Sven and Campy (abbreviated as DSC), from a certain point, it becomes increasing steps until extremum is passed, and then made quadratic interpolation on the basis of information about the functions in the past three points is determined extremum of the interpolation polynomial.
The Powell’s algorithm of quadratic polynomial interpolation is carried out in three arbitrary points, approximate extremum is found, dropped one of the four points and the procedure is repeated until convergence. The most effective is a combination of the described algorithms, or the so-called method of DSC-Powell. In accordance with this first algorithm DSC sought interval in which the extremum, three points are selected and carried there through parabola. Approximate value at an extremum is calculated as in the method of Powell:
If the value of the function at the point x̂ of optimum values f(x_{1}), f(x_{2}), f(x_{3}) differ by less than a predetermined accuracy, complete calculations,
otherwise discard the worst of the points x_{1}, x_{2}, x_{3},x̂ and carry out a new parabola. For functions that are sufficiently close to quadratic efficiency DSC-Powell is very high: as a rule, the decision to an accuracy 10^{-5}…10^{-6} is achieved 6–8 calculations of the objective function.
Methods for unconstrained optimization
Consider the problem of finding the maximum of a function of several variables without restrictions. Find maximum f(x), xƐE^{ n} One of the most famous is the gradient methods to solve this problem. They are based on the fact that the promotion of the objective function to the extreme in the space E^{ n} made by the rule:
Vector s_{k} sets another search direction and λ_{j} – the length of a step in this direction. Obviously,λ_{j} should be chosen so as to move as close as possible to the extreme. Various methods of selecting the direction of the search are used. The simplest of these is that the movement of the point x_{k} is made in the direction of the greatest magnification of f(x_{k}), i.e. in the direction of a gradient function at a given point.
According to this method, called the method of steepest descent,
The theory shows, and the practice of calculation confirms that the steepest descent method is not very effective in cases where the level curves of the objective function is strongly stretched, i.e. there are deep ravines while searching a minimum or ranges when searching maximum. The steepest descent direction is almost orthogonal to the best direction of the search, as a consequence, the optimal step reduced all the time, and the algorithm “get stuck” without reaching the extreme. The way out of this situation can be a scaling of variables, at which the level lines would get kind of close to the circle.
In order to reduce the amount of computations of the objective function, associated with a numerical definition of partial derivatives, sometimes used method of coordinate descent, which is also called a relaxation or Gauss-Seidel method. Let e_{i} – axis x_{i} unit vector, and x={x_{1}…,x_{n} – the starting point of the search. One international of coordinate descent is to take steps: x_{k-1}=x_{k} +λ_{k}e_{k}, k = 1,…,n.
Step as in the method of steepest descent is determined by the condition max f(x_{k} +λ_{k}s_{k} The Gauss-Seidel method suffered from the same flaw as the steepest descent method, – a bad convergence in the presence of ravines.
One way to overcome the computational difficulties associated with the gully structure of the objective function involves the use of information not only on its first derivative, but also higher order, contained in the second partial derivatives. An arbitrary function can be represented by its quadratic expansion in a Taylor series in the neighborhood of point x:
Equations (1.26) or (1.27) are applied iteratively until the end calculation process criterion is reached, called Newton’s method. Difficulties of using Newton algorithm associated, firstly, with Hessian matrix inversion, and secondly, with the computation of the second partial derivatives, which restricts its practical use.
The methods of conjugate directions are without drawbacks of gradient methods and have the convergence rate close to Newton’s method. At the same time, they are the methods of the first order, as the gradient. Positive defined quadratic form of n variables is minimized conjugate gradient method for no more than n steps. The conjugate gradient method is suitable for minimization of non-quadratic functions, only when they are iterative.
There are different ways of constructing conjugate directions. In particular, Fletcher and Reeves proposed a method, called the conjugate gradients method, according to which the subsequent direction of the search is a linear combination of the direction of steepest descent and the previous direction, i.e.:
Some methods do not use the derivatives of functions, and the optimization direction in which tis determined only on the basis of successive calculations of the objective function. In cases where the determination of the objective function derivatives is difficult, search algorithms may be preferable. In the case of one-dimensional analogue of the search method is the method of golden section, and the method of using derivatives – DSC-Powell method.
Methods of optimization with constraints
In addition to the previously described method of Lagrange multipliers for finding the extremum of functions with restrictions a number of numerical methods developed. The first approach to the construction of algorithms for constrained optimization is monotonous motion to the optimum of the objective function and at the same time striving to meet the exact or approximate limits.
Methods of this type are numerous, but the complexity, lack of flexibility and a large amount of computational work limit their use in practical calculations.
More elegant, easy to implement and effective the methods based on the reduction of problems with constraints to the solution of a sequence of
unconstrained optimization – the so-called penalty function methods. There are several variations of these methods.
Let’s begin their consideration with the interior point method for problems with inequality constraints:
The organization of numerical maximum search of (1.29) must be such that the point does not leave the feasible region. This shortage is deprived the external penalty function method, which for the problem of the form (1.33) involves the construction of the associated objective function.
It can be shown that the sequence of points x̂_{k } converges to the solution of the problem (1.33), but in contrast to the interior point method to the extreme movement takes place outside the feasible set, and is taken from the name of the method of exterior penalty functions. This method is also applicable to the general problem of nonlinear programming (1.14), for which used attached objective function:
Algorithm of solution is the same as for the problem (1.34).
The solution of nonlinear programming problems with constraints using penalty function method is complicated by the fact that as the penalty function coefficient is increasing, (1.35) expressed gully structure. As previously shown, not all the methods of unconditional optimization solution can cope with such problems, and therefore the choice of the method of finding the extremum of the attached objective function is of fundamental importance.
An important role is also played the strategy of the penalty factor change, because if you choose it immediately large, constraints of the problem satisfied well, but the objective function does not improve. In contrast, if too small values of Λ _{k}, motion occurs in the direction of improvement of the objective function, but practically does not take into account the constraints that can lead to failure in the E^{ n }areas where the objective function and constraints are not defined.
For example, if in the objective function or in limitations members of the form x^{a} are present, it is unacceptable entering the zone x≤0 Get rid of the
zone uncertainty, resulting in the computer calculations for emergencies can sometimes be the introduction of a suitable change of variables. In particular, to meet conditions x>0 the replacement x=e is suitable, which already zƐE^{1}. If such a reception is impossible, it should be carefully selected constants of unconditional search methods as the length of the step in the direction of descent, change of this step in the process to find a one-dimensional vector of variables did not leave the area where the objective function and constraints of
the problem identified.
In conclusion, we consider the possibility of nonlinear optimization methods usage in order to solve systems of nonlinear equations. Suppose that in the problem (1.14) there are no restrictions in the form of inequalities, and the number of variables equal to the number of restrictions in the form of equations, i.e., in fact, the task of solving the system of m equations with m unknowns. We form the function
and find its maximum. If the system of equations h_{j}(x)=0, j=1,…m , has a solution, then, obviously, at the same time with the maximum of I^{*} is the root of the system of equations. In particular, if the functions h_{j}(x) are linear, function (1.36) is obtained quadratic and can be effectively solved by Newton’s and conjugate gradient method.
Replacement of the problem of systems of linear equations solution to extremum problems is justified in cases where the matrix of the system is ill-conditioned (e.g., in the problem of approximation by least squares) and can not be solved by conventional methods, in particular, by process of elimination.
The values h_{j}(x) in (1.35) are called residuals, and the solution of nonlinear equations is replaced by minimizing the sum of squared residuals.
In some problems, when the calculation of the value of the objective function may take minutes, hours or even days of the computer, the range of acceptable methods of optimization significantly narrowed
These problems, in particular, include aerodynamic optimization of turbine blades using CFD.
The Nelder-Mead method (Nelder A.-Mead R.), also known as the flexible polyhedron method or the simplex method is a method of unconditional optimization of functions of several variables. Without requiring computation of the gradient function, it is applicable to non-smooth, noisy functions, and is particularly effective in small (up to 6) number of variable parameters. Its essence lies in the follow-successive movement and deformation of the simplex around the point of extreme. The method is a local extremum and can “get stuck” in one of them. If you still need to find a global extremum, one can try to select other initial simplex.
A more developed approach to the exclusion of local extrema offered algorithm based on the Monte-Carlo method, as well as evolutionary algorithms.
The genetic algorithm (GA) – is a global search heuristic method, used to solve optimization problems and modeling, by random selection, combination and variation of the required parameters with the use of mechanisms that resemble biological evolution. GA usage assumes its careful adjustment on special test functions, which, however, does not guarantee the effectiveness of the algorithm and the accuracy of decisions of the function.
This algorithm is well suited to the study of noisy functions, but requires a large number of CFD – calculations and therefore more time on optimization. The last forcing researchers to use coarse meshes and not quite accurate, but easily calculated turbulence models, which will inevitably leads to loss of the numerical calculations precision.
Monte-Carlo (random search) methods allows you to find the extremes of multimodal and noisy functions; use various constraints during optimization; is particularly effective when a large number of variable parameters; requires careful adjustment for test functions; it is one of the most common methods of optimization and solution of various problems in mathematics, physics,
economics, etc. However, the method requires tens of thousands of the objective function computing and practically not applicable for direct optimization based on CFD – calculations. To improve the efficiency of random search used quasi-random sequence of numbers (LPτ [4] Sobol), Faure, Halton et al.). Increased efficiency is achieved by eliminating clustering that occurs in a random search that is by more even distribution of points in the search study area of the function extremum.
Recently, in the optimization algorithms the methods of experiment planning are widely used. Using the methods of the theory of experimental design
(Design of the Experiment – DOE), the original mathematical model can be approximated by a quadratic polynomial. One of the relevant planning schemes of the experiment described in Section 1.3. These quadratic polynomials can be used to further optimization with the use of a universal and reliable global search method using a quasi-random sequences.
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.
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.
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 (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.
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