Effect of heat transfer coefficient of steam turbine rotor on thermal stress field under off-design condition

Abstract

The precise calculation of temperature and thermal stress field of steam turbine rotor under off-design conditions is of paramount significance for safe and economic operation, in which an accurate calculation of heat transfer (HT) coefficient plays a decisive role. HT coefficient changes dramatically along with working conditions. First, a finite element analysis of rotor model, applied with ordinary rotor materials, has been conducted to investigate the temperature and thermal stress difference along with the change of HT coefficient from 20 W/(m²·°C) to 20000 W/(m²·°C). Next, the differentiation between existing empirical formulas has been analyzed from the aspect of physical significance of non-dimension parameters. Finally, a verifying case of the cold startup of a 1000MW unit has been proceeded. The result shows that the accuracy of coefficient calculation when steam parameters are low has a greater influence on that of rotor temperature and thermal stress, which means a precise empirical HT coefficient formula, like the Sarkar formula is strongly recommended. When steam parameters are high and HT coefficient is larger than 10⁴ W/(m²·°C), there will be barely any influence on the calculation of thermal stress. This research plays a constructive role in the calculation and analysis of thermal stress.

Keywords

steam turbine / rotor / thermal stress / heat transfer coefficient / empirical formula

Introduction

It is of great importance to calculate the temperature and thermal stress distribution under off-design conditions [ 1]. The steam of high temperature and pressure touches turbine rotor directly. The boundary condition of heat exchange between rotor and steam can be regarded as the third thermal boundary condition with known steam temperature and HT coefficient, which is related to the physical parameters of steam [ 2]. This coefficient changes with working conditions. To calculate thermal stress more accurately, it is important to calculate HT coefficient accurately [ 3].

Lots of experiments and theoretical attempts related to HT coefficient calculation have been made by researchers, with a few empirical formulas concluded. In Ref. [ 4], two forms of empirical formulas of cylindrical curved surface heat transfer coefficient were introduced by D. Sarkar, including the natural convection influenced by the Grashof number and the Prandtl number, and the forced convection influenced by the Reynolds number and the Prandtl number. Adinarayana et al. [ 5] built a model of three stages steam turbine and calculated the HT coefficient of key points. They concluded that the HT coefficient was larger in the early phase of cold-start because of steam condensation. Reference [ 6] introduced a set of empirical formulas differentiated by different areas of rotor: rim, rotor surface, shaft seals, etc. Reference [ 7] calculated the HT coefficient of a certain case using two different empirical formulas: Westinghouse Formula, South-east University (China) & Harbin Turbine Company Formula. The results showed that the results of HT coefficient obtained by different formulas were different, resulting in inaccurate thermal stress. Reference [ 8] did a deep research in Alstom Formula, divided by pressure level, which was a very convenient solution to calculating HT coefficient. Because little research was done on theoretical ways of calculating HT coefficient, and different companies adopt different formulas, the results are absolutely different. The conclusion in Ref. [ 9] showed that different formulas might result in two values with totally different orders of magnitude.

In this paper, different cases with the change of HT coefficient from 20 W/(m²•°C) to 20000 W/(m²•°C) were studied by using the finite element analysis (FEA) method. The difference of temperature and thermal stress field of all cases was investigated, aiming at analyzing the influence of various empirical formulas on the result of temperature and thermal stress distribution.

Finite element analysis

FEA modeling

A single stage rotor model is established according to Ref. [ 3], which is two dimensional, axial symmetric and uniformly isotropy in ANSYS. Considering higher accuracy of thermos-structural coupled model [ 10], the object is meshed by quadrilateral coupled grid PLANE 13. There are 1064 elements and 1195 nodes in total, as shown in Fig. 1.

The material for the rotor is 30 CrMoV, which is a very heat-resistant alloy steel. Table 1 lists changes of physical parameters of the material along with temperature T (°C), including density D (kg/m³), special heat C (J/(kg·K)), thermal conductivity λ (W/(m·K)), coefficient of thermal expansion α (1/(10⁻⁶K)), the Young’s modulus E(GPa) and the Poisson’s ratio γ

Boundary conditions

The common boundary conditions are as follows: The initial temperature of the rotor is set as 30°C; the temperature of steam around the rotor changes at a rate of 2°C/min from 30°C to 590°C within 280 min, and then declines to 120°C at the same rate within 120 min; the central symmetric axis is set as heat insulation, and the same condition is set to both sides of the rotor, where the axial heat transfer is ignored.

The HT coefficients in different cases are presented in Table 2. In Cases 1 to 4, the HT coefficient increases geometrically from 20 to 20000, while those in Cases 5 to 10 increases arithmetically from 500 to 1750 with an increment of 250.

Results and analysis

The temperature-time and thermal stress-time curves of the key point of Cases 1 to 4 are shown in Fig. 2, with the steam temperature drawn for comparison.

From Fig. 2 (a), it can be seen that the temperature-time curve increases with HT coefficient, which means the heat transfer between the steam and the rotor enhances as expected. The curve of Case 2 almost doubles that of Case 1, with HT coefficient increasing from 20 to 200. However, the curve of Case 3 is only a little bit bigger than that of Case 2, with HT coefficient changing from 200 to 2000. The curve of Case 4 almost coincides with that of the steam. A speculation can be made that when HT coefficient exceeds 20000, the temperature-time curve no longer increases dramatically.

Figure 2 (b) indicates that all the thermal stress-time curves have a similar tendency as the temperature-time curves. With the temperature of steam increasing, the temperature difference between the surface and internal increases. That is the cause for the rapid rise in thermal stress at the beginning. When steam temperature decreases, the temperature difference between the surface and inside the rotor also reduces and the thermal stress of the key point decreases. After around 20000s, the temperature of the surface of the rotor becomes lower than that of the inside of the rotor. Therefore, the thermal stress rises slightly again. Table 3 lists the maximal temperature and thermal stress of the key point in Cases 1 to 4. Besides, the increment of temperature and thermal stress in each case are also listed in Table 1.

Considering that the HT coefficient in lots of practical engineering situations is between 200 and 2000, and the increases of temperature and thermal stress from Case 2 to Case 3 are larger than that from Case 3 to Case 4, as narrated in 2.1, some new cases are set to make comparisons clearer. The temperature-time curves and thermal stress-time curves of all new cases are plotted in Fig. 3.

The curves in Fig. 3 have a similar trend as those in Fig. 2 as expected. The maximal temperature and thermal stress of Cases 5 to 10 are tabulated in Table 4.

From Tables 3 and 4, it can be seen that when HT coefficient is 10 times larger (rising from 2000 to 20000), the increments of temperature and thermal stress (which are 5.22% and 10.70%, respectively) are almost the same as those when the coefficient rises from 500 to 750 (which are 4.86% and 11.31%, respectively). Besides, if the actual HT coefficient of a practical engineering case were 2000, and a little error were made when calculating this coefficient with a certain equation and got 1750, the difference between the actual thermal stress and the calculated stress (which are 355 MPa and 350 MPa, respectively) would be only 1.43%. It can be speculated that when the real HT coefficient of an engineering problem is a lot larger than 20000, and calculated a little bit imprecisely, there will be barely any influence on calculating thermal stress.

In short, with HT coefficient increasing, the inaccuracy in calculating this coefficient has less effect on the calculation of temperature and thermal stress field. When the HT coefficient is less than thousands, it is important to calculate it precisely, where a proper empirical formula should be chosen to make minimal error in calculating thermal stress. When the HT coefficient is greater than 10 thousands, little error should be made in calculating thermal stress.

Analysis of empirical formula of HT coefficient

Types of empirical formula

Because of the various factors affecting the value of HT coefficient, there is no exact mathematical equation but only some empirical formulas to be used. Different turbine manufacturing companies adopt different empirical formulas. At present, there are some frequently-used formulas such as the Sarkar formula used in Ref. [ 4], the Soviet Union formula used in Ref. [ 6], the South-east University & Harbin Turbine Company formula (SU&HTC) and Westinghouse formula used in Ref. [ 7], and the Alstom formula used in Ref. [ 8]. The results of all empirical formulas have a positive correlation with the running load of the steam turbine. However, there are great differences of even several orders of magnitudes between these formulas. That is also the reason for the study of the HT coefficient between 20 and 20000 in this paper.

Considering the fact that there is resemblance between the Soviet Union formula and the SU&HI formula, the Soviet Union formula will not be used for comparison. Besides, after verification, it is found out that the Alstom formula is not suitable for the steam with high temperature and pressure of large-scale thermal plant. So, it will also be excluded. Each empirical formula has several equations of its own, varying according to the place where the steam and rotor contacts: rim, rotor surface, shaft seals, etc. In this paper, all rotor surface empirical formulas are listed for comparison.

1)SU&HTC formula

(1) \alpha =Nu\left({\displaystyle \frac{\lambda }{r}}\right),

where Nu=0.1Re^0.68, Re=ur/υ; λ is the thermal conductivity of steam, W/(m·°C); r is the radius of the rotor, m; u is the linear velocity of the rotor surface, m/s; and υ is the kinematic viscosity, m²/s.

2)Sarkar formula

(2) \alpha =\mathrm{0.026}\times {\displaystyle \frac{R{e}^{\mathrm{0.805}}P{r}^{\mathrm{1}/\mathrm{3}}\lambda }{d}},

where Pr=C_pμ/λ, d is the hydraulic radius of the rotor, m; μ is the dynamic viscosity, Pa·s; and C_p is the special heat at constant pressure, J/(kg·°C). The other parameters are the same as those described above.

3)Westinghouse formula

(3) \alpha =\mathrm{0.1139}\times \left({\displaystyle \frac{\lambda }{r}}\right){\left({\displaystyle \frac{\omega r^{\mathrm{2}}}{\upsilon }}\right)}^{\mathrm{0.7}}{\left({\displaystyle \frac{\mu C_p}{\lambda }}\right)}^{\mathrm{0.35}},

where ω is angular velocity, rad/s. The other parameters are the same as those described above.

Analysis of formulas

Because

(4) u=\omega r,

the hydraulic radius of the rotor equals

(5) d=\mathrm{Area}/\mathrm{Perimeter}=\pi r^{\mathrm{2}}/\mathrm{2}\pi r=r/\mathrm{2}.

Substituting Eq. (4) into Eq. (1), the SU&HTC formula becomes

(6) \alpha =\mathrm{0.1}\left({\displaystyle \frac{\lambda }{r}}\right){\left({\displaystyle \frac{\omega r^{\mathrm{2}}}{\upsilon }}\right)}^{\mathrm{0.68}}.

Substituting Eqs. (4) and (5) into Eq. (2), the Sarkar formula becomes

(7) \alpha =\mathrm{0.052}\left({\displaystyle \frac{\lambda }{r}}\right){\left({\displaystyle \frac{\omega r^{\mathrm{2}}}{\upsilon }}\right)}^{\mathrm{0.85}}{\left({\displaystyle \frac{\mu C_p}{\lambda }}\right)}^{\mathrm{1}/\mathrm{3}}.

Comparing Eqs. (3), (6) and (7), it is observed that all empirical formulas have (λ/r) and (ωr²/υ). Apart from Eq. (6), the other formulas have (μC_p/λ). Besides the different constant term, the only distinction among them is the exponent of latter terms.

From the aspect of physical significance, the dimension of the HT coefficient α is identical to (λ/r), which can be considered as a basic component of any empirical formula. (ωr²/υ) is a Reynolds number, which represents the intensity of steam flow and is also a paramount factor of heat transfer. Therefore, it is a necessary part of all empirical equations. (μC_p/λ) is a Prandtl number, which represents the inner thermal and physical state of the steam at certain temperature and pressure. So, it should also be considered in all formulas.

After comparing all empirical equations listed above, it is discovered that the SU&HTC formula ignores the Prandtl number. Compared to the Sarkar formula, the Westinghouse formula has a smaller exponent of Reynolds number and a similar exponent of Prandtl number. The difference between them will be studied through a cold start-up case study of a certain 1000 W unit.

Case study

Modeling and boundary conditions

Taking the HP rotor of a 1000 MW steam turbine as an example, the thermal stress of the rotor under startup condition is calculated. The HP rotor has a total length of 4.885 m, with 14 pressure stages. The radius of the rotor surface is 0.44 m. The 2D axisymmetric model is depicted in Fig. 4. The coupled grid PLANE 13 is used to mesh the model. Some girds in the impeller root are refined as displayed in Fig. 5. The meshed model contains 6364 elements and 6793 nodes.

Figure 6 demonstrates the cold startup curves, including steam temperature, steam pressure, rotational speed, and running load. The initial temperature of the rotor is considered as the initial temperature of steam. The temperature and pressure of every stage is calculated by the off-design thermal calculation method.

There are three cases with the same temperature condition but different HT coefficients calculated by three empirical formulas, as listed in Table 5. Figure 7 exhibits the results of HT coefficients of the first stage.

Figure 7 describes the temporal change of HT coefficient. The values of the SU&HTC formula are almost twice that of the Sarkar formula, four times that of the Westinghouse formula. Besides, it can be obviously seen that at the initial phase of startup, the HT coefficient is very small. Then, it increases with the load, the steam temperature and pressure.

FEA and results

The numerical computation shows that the temperature and thermal stress fields in all cases are similar. Case a is a typical example. Figures 8 and 9 exhibit the temperature and thermal stress distribution at the end of cold startup, respectively.

As can be seen from Figs. 8 and 9, the maximal temperature and maximal thermal stress are all located at the root of the first stage rotor wheel. The node with the maximal thermal stress is chosen to be the key point. The temperature-time curves of Cases a, b and c are shown in Fig. 10, in which the main steam temperature is also given for comparisons.

From Fig. 10, it is obviously observed that temperature-time curves of all cases have the same trend with that of the main steam. Besides, there is not much difference in the temperatures between the key points in all cases, because there is no difference in orders of magnitude among all HT coefficients calculated by different formulas.

The thermal stress-time curves of Cases a, b and c are shown in Fig. 11.

It can obviously be seen from Fig. 11 that the thermal stress of the key points keeps rising along with the proceeding of startup until the unit runs stably, and then decreases. In the initial phase of startup, the thermal stress of the key points is proportional to the HT coefficient. After around 18000 s (300 min), the relative positions of three curves switched upside down. The reasons for this is that, first, from around 250 min, all the HT coefficients of different formulas begins to rise rapidly; next, the values of the Westinghouse HT coefficient are the smallest (no more than several hundreds); and finally, as has been discussed in Section 2, a smaller HT coefficient will have more influence on the calculation of the thermal stress when this coefficient changes. Therefore, the thermal stress calculated in the latter stage of startup in Case c is bigger than those in Cases a and b. The mean errors of the three cases are shown in Fig. 12.

As can be seen from Fig. 12 that at the beginning of startup, the mean error of each case is relatively bigger than that at the end of startup. That is because the running load is low at the start and the HT coefficients are relatively low, leading to a bigger error when the coefficients keep rising. When the load keeps increasing, as is expected, the error of thermal stress will get lower, which accords with Fig. 12. Generally speaking, the accuracy of thermal stress is more sensitive to the HT coefficient when the running load of the unit is low. From Fig. 12, it is seen that the mean error of Case b is the least in every startup stage, so the Sarkar formula is recommended to calculate the HT coefficient when solving practical engineering problems.

Conclusions

In conclusion, with the increase of the HT, the inaccuracy in calculating this coefficient will have less effect on calculating the temperature and thermal stress field. When the HT coefficient is less than thousands, it is important to calculate it precisely, where a proper empirical formula should be chosen to avoid big errors on calculating thermal stress. When the HT coefficient is greater than 10 thousands or even larger, little effect will be made on calculating the thermal stress.

When the unit runs with low load and the parameters of the steam are small, the accuracy of HT coefficient will have a significant effect on the calculation of thermal stress. An appropriate empirical formula with a higher accuracy should be chosen.

There are various kinds of empirical formulas for HT coefficient. Meticulous comparisons should be made to analyze the influence of the Reynolds number and Prandtl number. The Sarkar formula has been suggested to calculate the thermal stress in solving similar engineering problems.

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