﻿ 锅筒疲劳寿命计算中的径向温差计算方法
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 哈尔滨工程大学学报  2020, Vol. 41 Issue (7): 1005-1009, 1042  DOI: 10.11990/jheu.201905010 0

### 引用本文

ZHENG Xinwei. Method for calculating radial temperature difference in the fatigue life of steam drum[J]. Journal of Harbin Engineering University, 2020, 41(7): 1005-1009, 1042. DOI: 10.11990/jheu.201905010.

### 文章历史

Method for calculating radial temperature difference in the fatigue life of steam drum
ZHENG Xinwei
College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: In this study, we sought to verify the accuracy of the radial-temperature-difference calculation for the steam drum, which is used as the Chinese national standard method for the fixed power boiler, and the feasibility of its application to the marine power boiler. To do so, we designed a calculation method for determining the radial temperature difference of the steam drum and applied it to the above two types of boilers. We derived formulas for both an analytic solution and a method of extremely approaching analytic solution for the radial temperature difference of the steam drum, along with an expected value formula for calculating the temperature damping coefficient of the steam drum. We also investigated the method of extremely approaching analytic solution, and a number of engineering examples were calculated and analyzed. The results reveal that the Chinese national standard method for the fixed power boiler cannot be used to calculate the radial temperature difference of the steam drum in the marine power boiler, whereas the method of extremely approaching analytic solution can be used to calculate the radial temperature difference of the steam drum for both the fixed power boiler and the marine power boiler. Whether or not the Chinese national standard method for the fixed power boiler can be converted into the method of extremely approaching analytic solution depends on the temperature damping coefficient of the steam drum. The proposed calculation method is consistent with current engineering practice, and therefore has theoretical and applied values.
Keywords: boiler    steam drum    fatigue life    radial temperature difference    Chinese national standard method of fixed power boiler    analytic solution    method of extremely approaching analytic solution

1 固定式动力锅炉国家标准法

 $\Delta {t_1} = {t_{\rm{o}}} - {t_{\rm{i}}} = - \frac{{{C_{\rm{t}}}{\delta ^2}v}}{{{a_{\rm{t}}}}}[1 - {\rm{exp}}( - \chi t/\tau )]$ (1)

Ct 为与按名义厚度确定的锅筒外径与内径的比值β有关的结构系数，可按式计算：

 ${C_{\rm{t}}} = \frac{{2{\beta ^2}{\rm{ln}}\beta - {\beta ^2} + 1}}{{4{{(\beta - 1)}^2}}}$ (2)

 $\chi = \sqrt {\frac{{\beta - 1}}{{{\beta _1}}}}$ (3)
 $\begin{array}{*{20}{l}} {{\beta _1} = \frac{{{\beta ^5} - 1}}{5} - 4{\beta ^2}\left( {\frac{{{\beta ^3}{\rm{ln}}\beta }}{3} - \frac{{{\beta ^3} - 1}}{9}} \right) + }\\ {{\kern 1pt} {\kern 1pt} 4{\beta ^4}[\beta {{({\rm{ln}}\beta - 1)}^2} + \beta - 2] + \beta - 1 + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2\left\{ {2{\beta ^2}[\beta ({\rm{ln}}\beta - 1) + 1] - \frac{{{\beta ^3} - 1}}{3}} \right\}} \end{array}$ (4)

 $\tau = \frac{{D_{\rm{i}}^2}}{{16{a_{\rm{t}}}}}$ (5)

 $\Delta {t_1} = - vt$ (6)

2 非常逼近解析解计算方法的建立

 $\begin{array}{*{20}{c}} {\Delta {t_2} = \frac{v}{{4{a_{\rm{t}}}}}\left( {R_2^2 - R_1^2 - 2R_2^2{\rm{ln}}\frac{{{R_2}}}{{{R_1}}}} \right) + \frac{v}{{4{a_{\rm{t}}}}}(R_1^2 + }\\ {\left. {2R_2^2{\rm{ln}}\frac{{{R_2}}}{{{R_1}}} - R_2^2} \right){\rm{exp}}( - {a_{\rm{t}}}n_i^2t) = \frac{{vR_1^2}}{{4{a_{\rm{t}}}}}({\beta ^2} - }\\ {1 - 2{\beta ^2}{\rm{ln}}\beta )[1 - {\rm{exp}}( - {a_{\rm{t}}}n_i^2t)]} \end{array}$ (7)

 ${{\rm{J}}_0}({n_i}{R_1}){{\rm{Y}}_1}({n_i}{R_2}) - {{\rm{J}}_1}({n_i}{R_2}){{\rm{Y}}_0}({n_i}{R_1}) = 0$ (8)

 ${ - \frac{{{C_{\rm{t}}}{\delta ^2}v}}{{{a_{\rm{t}}}}} = \frac{{vR_1^2}}{{4{a_{\rm{t}}}}}({\beta ^2} - 1 - 2{\beta ^2}{\rm{ln}}\beta )}$ (9)

 ${1 - {\rm{exp}}( - \chi t/\tau ) = 1 - {\rm{exp}}( - {a_{\rm{t}}}n_i^2t)}$ (10)

 ${C_{\rm{t}}} = \frac{{2{\beta ^2}{\rm{ln}}\beta - {\beta ^2} + 1}}{{4{{(\beta - 1)}^2}}}$ (11)

 $\chi = {\left( {\frac{{{n_i}{D_{\rm{i}}}}}{4}} \right)^2} = {\left( {\frac{{{n_i}{R_1}}}{2}} \right)^2}$ (12)

 $\Delta {t_2} = - \frac{{{C_{\rm{t}}}{\delta ^2}v}}{{{a_{\rm{t}}}}}[1 - {\rm{exp}}( - {a_{\rm{t}}}n_i^2t)]$ (13)

 ${{{\rm{J}}_0}({n_1}{R_1}){{\rm{Y}}_1}({n_1}{R_2}) - {{\rm{J}}_1}({n_1}{R_2}){{\rm{Y}}_0}({n_1}{R_1}) = 0}$ (14)
 ${\chi = {{\left( {\frac{{{n_1}{R_1}}}{2}} \right)}^2}}$ (15)
 ${\Delta {t_2} = - \frac{{{C_{\rm{t}}}{\delta ^2}v}}{{{a_{\rm{t}}}}}[1 - {\rm{exp}}( - {a_{\rm{t}}}n_1^2t)]}$ (16)

n1R1=x，利用β=R2/R1，所以n1R2=βx。这样，式(14)可以写成：

 ${{\rm{J}}_0}(x){{\rm{Y}}_1}(\beta x) - {{\rm{J}}_1}(\beta x){{\rm{Y}}_0}(x) = 0$ (17)

 ${{{\rm{J}}_0}(x) = {x^{ - 0.5}}{f_0}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\theta _0}}$ (18)
 ${{{\rm{Y}}_0}(x) = {x^{ - 0.5}}{f_0}{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\theta _0}}$ (19)
 $\begin{array}{*{20}{l}} {\;\:{f_0} = 0.797{\kern 1pt} {\kern 1pt} {\kern 1pt} 884{\kern 1pt} {\kern 1pt} {\kern 1pt} 56 - 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 000{\kern 1pt} {\kern 1pt} {\kern 1pt} 77(3/x) - }\\ {0.005{\kern 1pt} {\kern 1pt} {\kern 1pt} 527{\kern 1pt} {\kern 1pt} {\kern 1pt} 40{{(3/x)}^2} - 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 095{\kern 1pt} {\kern 1pt} {\kern 1pt} 12{{(3/x)}^3} + }\\ {0.001{\kern 1pt} {\kern 1pt} {\kern 1pt} 372{\kern 1pt} {\kern 1pt} {\kern 1pt} 37{{(3/x)}^4} - 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 728{\kern 1pt} {\kern 1pt} {\kern 1pt} 05{{(3/x)}^5} + }\\ {0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 144{\kern 1pt} {\kern 1pt} {\kern 1pt} 76{{(3/x)}^6} + \varepsilon (|\varepsilon | < 1.6 \times {{10}^{ - 8}})} \end{array}$ (20)
 $\begin{array}{*{20}{l}} {{\theta _0} = x - 0.785{\kern 1pt} {\kern 1pt} {\kern 1pt} 398{\kern 1pt} {\kern 1pt} {\kern 1pt} 16 - 0.041{\kern 1pt} {\kern 1pt} {\kern 1pt} 663{\kern 1pt} {\kern 1pt} {\kern 1pt} 97(3/x) - }\\ {0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 039{\kern 1pt} {\kern 1pt} {\kern 1pt} 54{{(3/x)}^2} + 0.002{\kern 1pt} {\kern 1pt} {\kern 1pt} 625{\kern 1pt} {\kern 1pt} {\kern 1pt} 73{{(3/x)}^3} - }\\ {0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 541{\kern 1pt} {\kern 1pt} {\kern 1pt} 25{{(3/x)}^4} - 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 293{\kern 1pt} {\kern 1pt} {\kern 1pt} 33{{(3/x)}^5} + }\\ {0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 135{\kern 1pt} {\kern 1pt} {\kern 1pt} 58{{(3/x)}^6} + \varepsilon (|\varepsilon | < 7 \times {{10}^{ - 8}})} \end{array}$ (21)
 ${{{\rm{J}}_1}(x) = {x^{ - 0.5}}{f_1}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\theta _1}}$ (22)
 ${{{\rm{Y}}_1}(x) = {x^{ - 0.5}}{f_1}{\rm{sin}}{\kern 1pt} {\kern 1pt} {\theta _1}}$ (23)
 $\begin{array}{*{20}{c}} {{f_1} = 0.797{\kern 1pt} {\kern 1pt} {\kern 1pt} 884{\kern 1pt} {\kern 1pt} 56 + 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 001{\kern 1pt} {\kern 1pt} {\kern 1pt} 56(3/x) + }\\ {0.016{\kern 1pt} {\kern 1pt} {\kern 1pt} 596{\kern 1pt} {\kern 1pt} {\kern 1pt} 67{{(3/x)}^2} + 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 171{\kern 1pt} {\kern 1pt} {\kern 1pt} 05{{(3/x)}^3} - }\\ {0.002{\kern 1pt} {\kern 1pt} {\kern 1pt} 495{\kern 1pt} {\kern 1pt} {\kern 1pt} 11{{(3/x)}^4} + 0.001{\kern 1pt} {\kern 1pt} {\kern 1pt} 136{\kern 1pt} {\kern 1pt} {\kern 1pt} 53{{(3/x)}^5} - }\\ {0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 200{\kern 1pt} {\kern 1pt} {\kern 1pt} 33{{(3/x)}^6} + \varepsilon (|\varepsilon | < 4 \times {{10}^{ - 8}})} \end{array}$ (24)
 $\begin{array}{*{20}{c}} {{\theta _1} = x - 2.356{\kern 1pt} {\kern 1pt} {\kern 1pt} 194{\kern 1pt} {\kern 1pt} 49 + 0.124{\kern 1pt} {\kern 1pt} 996{\kern 1pt} {\kern 1pt} {\kern 1pt} 12(3/x) + }\\ {0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 056{\kern 1pt} {\kern 1pt} {\kern 1pt} 50{{(3/x)}^2} - 0.006{\kern 1pt} {\kern 1pt} {\kern 1pt} 378{\kern 1pt} {\kern 1pt} {\kern 1pt} 79{{(3/x)}^3} + }\\ {0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 743{\kern 1pt} {\kern 1pt} {\kern 1pt} 48{{(3/x)}^4} + 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 798{\kern 1pt} {\kern 1pt} 24{{(3/x)}^5} - }\\ {0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 291{\kern 1pt} {\kern 1pt} {\kern 1pt} 66{{(3/x)}^6} + \varepsilon (|\varepsilon | < 9 \times {{10}^{ - 8}})} \end{array}$ (25)

3 实例计算分析 3.1 对于固定式动力锅炉 3.1.1 基本参数

3.1.2 计算结果

3.1.3 计算结果分析

3.2 对于船舶动力锅炉 3.2.1 锅筒材料物理性能和基本参数

1) 文献[10]给出的锅筒材料BHW35的热扩散率at随温度变化的数据：20 ℃时为648 mm2/min，100 ℃时为618 mm2/min，200 ℃时为576 mm2/min，300 ℃时为534 mm2/min；

2) 锅筒内直径Di为1 300 mm，名义厚度δ为75 mm。这样，外径与内径比值β为1.115 4，按式(2)或式(11)得到的结构系数Ct为0.519，按式(3)和式(4)得到的温度阻尼系数χ为49.383，按式(17)~(25)得到的参数n1为20.74 m-1

3.2.2 船舶动力锅炉典型工况

1) 锅炉冷态启动工况。

2) 锅炉热备用启动和停炉工况。

① 锅炉启动时间为50 s[12]，锅筒工作压力由3.5 MPa升为6.47 MPa，则锅筒内工质饱和温度由244.22 ℃升为281.61℃[11]。这样，锅筒内工质平均变化速率为44.868 ℃/min；

② 锅炉停炉时间为30 s[12]，锅筒工作压力由6.47 MPa降为3.5 MPa，则锅筒内工质饱和温度由281.61 ℃降为244.22℃[11]。这样，锅筒内工质平均变化速率为-74.78 ℃/min。

3.2.3 计算结果和分析

1) 对于锅炉冷态启动工况。

 Download: 图 1 冷态启动时的t-Δt1或t-Δt2 Fig. 1 The relation of t-Δt1 or t -Δt2 during the start-up of cold state
 Download: 图 2 冷态启动时的t-ε Fig. 2 The relation of t-ε during the start-up of cold state

2) 对于锅炉热备用启动和停炉工况。

 Download: 图 3 热备用启动时的t-Δt1或t-Δt2 Fig. 3 The relation of t-Δt1 or t-Δt2during the start-up of stand-by heat
 Download: 图 5 热备用停炉时的t-Δt1或t-Δt2 Fig. 5 The relation of t-Δt1 or t -Δt2 during the furnace shut down of stand-by heat
 Download: 图 4 热备用启动时的t-ε Fig. 4 The relation of t-ε during the start-up of stand-by heat
 Download: 图 6 热备用停炉时的t-ε Fig. 6 The relation of t-ε during the furnace shut down of stand-by heat

3.2.4 国家标准法对其他计算工况的适用性与分析

4 结论

1) 当船舶动力锅炉与固定式动力锅炉具有相同的锅筒导热方程和定解条件情况下，提出的基于非常逼近解析解的锅筒径向温差计算公式和计算方法对固定式动力锅炉和船舶动力锅炉锅筒径向温差计算结果都具有足够高的计算精度；

2) 现行固定式动力锅炉国家标准法中的锅筒径向温差计算的2个公式是近似公式，该方法是一种近似计算方法，导致无法对船舶动力锅炉锅筒在任何可能工况下的径向温差进行合理计算；

3) 得到了基于非常逼近解析解的锅筒径向温差计算中的锅筒温度阻尼系数计算公式，明确指出了其在应用上的价值。

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