Operation 337
Calculation of temperature fields for main turbine design elements
The stationary and unsteady temperature fields of the main turbine design elements should be known for subsequent calculations of the thermal stresses and for assessing the stress state of the elements with regard to the action of other factors, such as centrifugal forces for rotating elements, inner steam pressure for casings, and so on. These temperature fields can be obtained by means of digital solution of the direct heat conduction problem for the considered “critical” (potentially most thermally-stressed) elements—that is, by calculating the metal temperatures, relying on the known values of the heating steam temperatures and heat transfer conditions. Calculation optimization of the turbine transients can be accomplished by means of a combined solution of the direct and reverse heat conduction problems, where the reverse problem is solved by referring to the heating steam temperature. The problem is to find the value or variation of this temperature under conditions of limits set for the temperature field—for example, keeping a specific temperature difference at the upper allowable level (permanent or variable).
To solve the combined (direct and reverse) heat conduction problem for the critical turbine elements, the mathematical models of these elements should reflect their genuine geometrical forms and boundary heat conditions varying over time and lengthwise of the heated surfaces. For this purpose, it is not necessary to model the design elements upon the whole in all their three-dimensional complexity. It is usually sufficient to confine the model to only a fragment of the entire element in the vicinity of the sections where the leading indices (the temperature differences across the element thickness) occur. The influence of the heat fluxes on the side surfaces of the fragment on the temperature distribution across the thickness of the element should not exceed approximately 10%.158
For example, if the critical design element is the casing flange, it is advisable to consider the two-dimensional unsteady-state thermal conduction problem for the horizontal section at the flange midheight, ignoring the heat flux in the circular direction, between the casing’s flange and wall. For long casing chambers (with a chamber length-to-flange width ratio greater than 5–6), it is possible to model the flange area at a single stud bolt pitch, as shown in Figure 4–46b.
www.EngineeringBooksPdf.com
338 Wet-Steam Turbines for Nuclear Power Plants
On the contrary, if the maximum temperature difference across the flange width occurs in a relatively short or bent chamber with significantly different boundary conditions than in the adjacent chambers, the calculation model should encompass the flange at a length of about three or four bolt pitches on either side of the considered cross section (Fig. 4–45). In any event, it is absolutely necessary to take into account the holes in the flanges, with air gaps between the flanges and stud bolts, as well as outward heat flux on the external flange surface through the thermal insulation. This heat flux can be assessed using the equivalent heat transfer coefficient on the external surface. Although its value depends on the actual thermal insulation quality, it cannot be formally calculated by relying on the insulation material properties listed in reference books. Comparison of calculation results achieved with the use of different calculation meshes (orthogonal, triangular, and mixed) proves that it is more important to model with due accuracy the position and area of the stud bolt holes than their true form (the holes can be represented, for example, by orthogonal or hexagonal wells). Such an approach is justified by comparison of the calculated data with results of direct temperature measurements (Figs. 4–44 through 4–46).These models cannot be accurate if the flange is additionally heated by steam leaking through the split surface of the flange—for example, if the flange loses its tightness. This is the reason for the discrepancy between the calculated and measured temperature differences at the end of the start-up process shown in Figure 4–44.
If the critical design element is a rotor, whether it is a forged or welded rotor of the HP, IP, or LP cylinder, to calculate the optimized start-up diagrams, it is sufficient to consider only a fragment of its longitudinal section in the vicinity of the considered cross section with the highest radial temperature differences. For forged rotors, as applied to turbine transients, especially start-ups, more often than not, it is allowable to use one-dimensional models, in which only the radial temperature distribution in the most thermally stressed cross-section is considered.The radial temperature differences in the most thermally stressed sections as calculated using oneand two-dimensional mathematical models do not significantly differ.The explanation is that, in the most intensely heated zones, the radial heat fluxes from the steam to the rotor surface and from the heated surface into the metal thickness are much greater than the axial heat fluxes into the adjacent lessheated zones. This is especially true for the steam admission zones
www.EngineeringBooksPdf.com
Operation 339
of turbine cylinders with central steam admission (as for integrated, double-flow, and loop-flow cylinders).A typical temperature field for a solid double-flow LP rotor of a Siemens high-speed 1,000-MW turbine (see Fig. 3–7) is shown in Figure 4–60. Moving away from the zone of maximum heating steam temperatures to the rotor ends, the lines of equal metal temperatures (isotherms) change from being almost horizontal to rather vertical; that is, the dominant heat fluxes become axial rather than radial. Likewise, for HP rotors in their most thermally stressed sections, the intense heat transfer conditions from wet steam make the radial heat fluxes prevail over the axial ones.
Fig. 4–60. Calculated temperature (°C) fields (1) and total stress (MPa) fields
(2) for the most stressed start-up instant and admissible sizes of possible initial macrodefects (mm) (3) for the solid double-flow LP rotor of a Siemens high-speed 1,000-MW turbine
Source :W. Engelke, J. S. Joyce, D. Lambrecht, et al.159
www.EngineeringBooksPdf.com
340 Wet-Steam Turbines for Nuclear Power Plants
These features of the temperature fields and character of their isotherms are also typical for the most-stressed sections of welded rotors with their more complicated geometrical shape.Temperature fields for the first stages’ disks of the welded HP, IP, and LP rotors of wet-steam turbines are shown in Figures 4–8b, 4–10a, and 4–12b. Under stationary operating conditions, the rotor temperature fields become essentially two-dimensional, and often, especially for forged rotors of single-flow cylinders and welded rotors, the axial heat fluxes become more influential than the radial ones (Fig. 4–10b). The same can be said for a combined LP rotor with shrunk-on wheel disks (Fig. 4–61).
These characteristics of temperature fields at different operating conditions are typical for all types of rotors.
Fig. 4–61. Calculated temperature field for a combined LP rotor (right half)
Source : H. Oeynhausen, G. Roettger, J. Ewald, et al. 160
www.EngineeringBooksPdf.com
Operation 341
While calculating metal temperature fields in welded rotors, it is important to set the proper boundary heat transfer conditions on the surfaces of internal cavities. The medium (air or residual gas left from welding) that is locked within these cavities, being heated at the cavity periphery, is forced out by heavier colder gas and moves toward the rotor axes, where it transfers its heat to the metal of the colder central part of the rotor, thus reducing the radial temperature differences in the rotor.This thermal convection process takes place in a field of artificial gravity under action of centrifugal forces. For high-speed turbine rotors with rotation speed of 3,000 rpm and an outer diameter of the internal cavity of approximately 1 m ( 40 in), which is quite typical for such turbines, the centrifugal acceleration at the periphery is more than 6×103 g. Some preliminary calculations showed that heat transfer with thermal convection in a field of centrifugal forces can significantly influence the dynamics of heating the rotor, and with regard to this effect, the temperature difference along the rotor radius could noticeably decrease, but this phenomenon needs further study.
The solution of the problem of free convection in rotor cavities is seriously hindered by several circumstances: 1) the temperature distribution at the cavity boundaries varies and is itself determined by the heat transfer conditions in the cavity, which makes it necessary to solve a conjugate non-stationary two-dimensional problem of free convection in the cavity coupled with thermal conduction in the adjacent rotor body; 2) the high acceleration of centrifugal forces in the rotor cavities corresponds to the Rayleigh number (Ra) values beyond the range for which experimental and theoretical investigations have ever been conducted, and 3) the acceleration of centrifugal forces changes significantly over the cavity height.
The considered problem was subjected to thorough analysis using specially developed mathematical models of a hollow thick-walled cylinder heated from the outside.161 In the cylinder’s internal cavity, free convection occurs, characterized by the Rayleigh number varying from 0 to 10 10. Some results of this investigation are presented in Figures 4–62 and 4–63. The calculations showed that if the Rayleigh number characterizing free convection within the rotor cavity does not exceed 1010, the influence of the effect can be neglected, and the heat fluxes on the cavity surfaces can be taken as zero. At the same time, estimates showed that if the free convection within the rotor
www.EngineeringBooksPdf.com
342 Wet-Steam Turbines for Nuclear Power Plants
cavity is artificially intensified (for example, by means of filling the cavity with a special gas or compressed air), it can result in significant decreases of the radial temperature differences and thermal stresses in the rotor.
Fig. 4–62. Results of mathematical modeling of heating for a hollow thickwalled cylinder with and without free convection within the internal cavity and traced free convection stream lines within the cavity and isotherms for the rotor body at the end of heating (numbering of temperature curves is based on notations on sketch; 1: Ra = 1010 , 2: Ra = 0)
Source :V.A. Brailovskaya,V. R. Kogan,A. S. Leyzerovich, and V.A. Polezhaev162
www.EngineeringBooksPdf.com
Operation 343
Fig. 4–63. Influence of free convection within the internal cavity of a welded rotor on its heating for a process modeling cold start-up (1: results of calculations using special mathematical model (see Fig. 4–62); 2: results of calculations using approximate engineering methodology)
Source :V.A. Brailovskaya,V. R. Kogan,A. S. Leyzerovich, and V.A. Polezhaev163
Calculated optimization of start-up diagrams
Optimization of the turbine transients is regarded as a problem of minimizing their duration with unconditional observance of limitations caused by reliability requirements. To solve this problem, it is necessary to coin the term leading indications of the turbine’s temperature and thermal stress state. Generally, the leading indications of the turbine state are those that limit (actually or potentially) the rate of transients in such a way that variation of these indications is determined only by the main external controlling actions, or in other words, varying the main operating parameters: the steam flow amount through the turbine (its load) and the main and reheat steam temperatures.164 For modern, large, superheated-steam turbines, the leading indications are generally the temperature differences in the most-stressed (critical) turbine elements, which more often than not are the HP and IP rotors and HP stop-valve steam-chests. Depending on the turbine design, output, and steam conditions and the unit’s start-up system features, certain other indications can also play such a role (for example, RRE or temperature differences across the flange width of the high-temperature cylinders).
www.EngineeringBooksPdf.com
344 Wet-Steam Turbines for Nuclear Power Plants
Eventually, the problem of optimizing is formulated as searching for the diagrams of changing the turbine’s operating parameters in such a way that the turbine’s leading indications are maintained at their top admissible levels without exceeding them over the entire course of a transient. In this case, the duration of the transient is supposed to be minimal. However, if there are more than one leading indication and more than one controlling actions, this definition of the problem is not absolutely correct, and it is necessary to use supplementary conditions to obtain a single-valued solution.
During the first stage, it is expedient to consider the problem of the optimal heating for individual turbine elements, taking the temperature differences across their thickness as the leading indications and the change of the heating steam temperatures for these elements as controlling actions.This problem can be solved using solutions for unsteady-state heat conduction with the special boundary heat conditions on the heated surface, that is, with the surface metal temperature changing in such a way that the aforesaid temperature difference across the thickness of the element is maintained at the upper admissible level, which can be generally set as a function of the current metal temperature.Then, the known heat transfer conditions make it possible to move on from the surface metal temperature to the heating steam temperature. This can be done by means of the combined digital solution for the direct and reverse problems of unsteady heat conduction for the considered elements. In so doing, the reverse problem is solved in reference to the heating steam temperature.The solution of the direct problem gives the change in the temperature fields for the considered elements. This in turn makes it possible to solve the reverse problem with regard to the heating steam temperature. Both the direct and reverse problems can be solved for several potentially critical elements simultaneously.
The current upper admissible value of the heating steam temperature for each element can be calculated as follows:
[tst] = tm +[∆t ] × (1+ λ × ∂∂xt |x=0 ) = tm + [∆t ] × (1+ BΦi ) |
(4.4) |
where ∆t = ts – tm, and ts and tm are the metal temperatures of the considered element on the heated surface and at a characteristic
www.EngineeringBooksPdf.com
Operation 345
point at distance H from the heated surface. This point can be fixed (for example, on the external insulated surface or at the rotor axis) or
sliding (if it refers to the average integral temperature). In this case,
~
the Biot number, Bi , is also calculated for H . In this equation, Φ is the dimensionless temperature gradient on the heated surface. Generally speaking, all of the values in Equation 4.4 are variable quantities. For nonstationary processes, 
is also variable. However, if the process is close to optimal and heat removal from the insulated surface is much smaller than the heat flux from steam to metal, it is expedient to take 
as a constant.This assumption assures a stable solution even in the case of outside disturbances of the process.
The value of the dimensionless temperature gradient, Φ , for simple-shaped elements under conditions of quasi-stationary heating can be found analytically. For a hollow cylinder with a radius ratio typical for forged rotors, the value of Φ , in reference to the admissible effective temperature difference, [∆t ], is approximately 1.17. For more complex cases, this value can be calculated digitally. For the HP flanges of wet-steam turbines, this value in reference to the entire temperature difference, ∆t, across the flange width is approximately 1.2.
Even after finding the optimal change of the heating steam temperature for every critical design element, the problem of optimizing turbine transients cannot be considered solved. First, it is necessary to calculate the changes of the turbine operating parameters—the controlling actions influencing the heating steam temperatures for different elements. In addition, two or more leading indications of the turbine state can be influenced by different controlling actions to different extents.As this takes place, the change of some indications that are influenced by the same external actions can feature in their dynamics, and the change of one indication can simultaneously depend on different actions. Finally, it is necessary to take into account the existence of technological limitations in changing turbine operating parameters. These limitations can concern their absolute values, the rate of variations, and functional interrelations.
Actual transients under manual control run with almost inevitable occasional deviations from the optimized schedules.These deviations are caused by external disturbances and operator errors. Thus, the schedule must be optimized with regard to these potential deviations.
www.EngineeringBooksPdf.com
346 Wet-Steam Turbines for Nuclear Power Plants
The simplest way is to decrease the upper admissible value of the leading indications by the value of the expected deviations of the heating steam temperature. However, in fact, in order to optimize the process, it is desirable to take into account the influence of the current Biot number for the considered turbine elements on dynamics of heating them, the frequency spectrum for the expected disturbances, as well as other factors.
The optimal diagrams of the transients are usually calculated using special computer programs, and their algorithms must involve all of the previously mentioned circumstances. The greater the number of interrelated controlling actions, the more complex the set problem and its solution. In the case of start-ups for superheated-steam turbines of fossil fuel units under variable steam conditions, at least four independent main controlling actions occur.They are (1) the changes in the steam flow amount through the turbine, (2) the main steam temperatures, (3) the reheat steam temperatures, and (4) the position of the turbine’s control valves, even without considering the control of bypass valves.165 By contrast, wet-steam turbines of nuclear power units are started-up under constant main steam conditions and have only two main external controlling actions: the changes in the steam flow amount through the turbine and the steam temperature after the MSR. Because of invariable steam parameters in the start-up process of wet-steam turbines, the heating of the HP sections practically depends solely on the steam flow amount through the turbine, that is, the turbine load, N. The steam temperature downstream from the MSR, tMSR, can be governed during start-ups by means of changing the heating main-steam flow to the MSR.Thus, the changes in both of the main operating parameters (N and tMSR) can be optimized separately in accordance with variations of their “own” leading indications. The loading diagram is formed according to the temperature state of the HP cylinder, and under this condition, the diagram for raising tMSR is dictated by the temperature state of the cylinder located downstream from the MSR with regard to the change of its boundary heat conditions due to loading. For wet steam in the HP cylinder, the surface metal temperature for the critical elements can be considered practically equal to the heating-steam temperature, that is, in Equation 4.4, Bi , and Φ/Bi 0.
The transition from optimizing the heating steam temperature diagrams, θ( ),to the loading diagram, N( ),is carried out through the static
www.EngineeringBooksPdf.com