基于积分变换方法的含热源海底单层管瞬态导热研究

朱晓青; 何杨烨; 苏健

doi:10.21656/1000-0887.460114

基于积分变换方法的含热源海底单层管瞬态导热研究

doi: 10.21656/1000-0887.460114

朱晓青^1,,
何杨烨^1, ,,
苏健²

1.
中国石油大学(北京) 安全与海洋工程学院，北京 102200
2.
巴西里约热内卢联邦大学核工程系，巴西里约热内卢 24941-927

基金项目:

中国石油大学(北京)基金资助项目 2462023QNXZ007

详细信息

作者简介:
朱晓青(1997—)，女，博士生(E-mail: xiaoqing_zhucup@163.com)

通讯作者:
何杨烨(1991—)，女，副教授，博士，博士生导师(通信作者. E-mail: yangyehe@cup.edu.cn)

中图分类号: O241.82
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- 被引次数: 0
出版历程
- 收稿日期: 2025-06-04
- 修回日期: 2025-07-30
- 刊出日期: 2026-08-01

Transient Heat Conduction Analysis on Subsea Single-Layer Pipes Under Internal Heat Sources With the Integral Transform Method

Zhu Xiaoqing^1
,,
He Yangye^{1
, ,},
Su Jian²

1.
College of Safety and Ocean Engineering, China University of Petroleum (Beijing), Beijing 102200, P.R. China
2.
Department of Nuclear Engineering, Federal University of Rio de Janeiro, Rio de Janeiro 24941-927, Brazil

摘要

摘要: 海洋输送高温油气的管道受管内流体对流换热及内部热源的影响，其管体温度升高，热量也会从管壁面传递到外部低温海水中，在管道径向及轴向形成温度梯度.本研究采用积分变换方法，对非齐次边界条件下且含热源的轴对称单层管进行二维瞬态热传导分析.首先，将二维瞬态温度表示为含热源的轴对称一维稳态、滤波齐次二维稳态及滤波均质二维瞬态温度的组合，并对其进行无量纲处理.其次，依据分离变量方法确定径向、轴向的积分变换对，对热传导控制方程及边界条件、初始条件进行积分变换处理，分离瞬态温度的时间和空间依赖性，得到关于时间的一阶线性常微分方程组.最后，通过积分变换对的逆变换确定二维瞬态温度分布的理论解，并将二维瞬态温度分布与稳态温度分布结果对比验证.在此基础上，研究不同内外Biot数(Bi)组合、不同热源强度G对管道温度分布的影响.结果表明：内外Biot数组合相同且热源强度大的，管道时空维度上的温度梯度及温度值均较大；热源强度相同但内外Biot数组合不同时，时空维度上的温度梯度和整体温度分布存在差异；在径向方向和时间维度上，不同内外Biot数组合的温度分布曲线存在交点，其温度梯度和温度值相对大小在该点发生变化，但在轴向方向上没有出现该现象.
- 瞬态热传导 /
- 轴对称单层管 /
- 积分变换 /
- 非齐次边界条件
Abstract: The pipeline for marine transportation of high-temperature oil and gas is affected by the convective heat transfer of the fluid inside the pipe and the internal heat source. With the rise of the pipe body temperature, the heat will also transfer from the pipe wall surface to the external low-temperature seawater, to form a temperature gradient in the pipeline radial and axial direction. The 2D transient heat transfer in an axisymmetric single-layer pipe with non-homogeneous boundary conditions and an internal heat source was analyzed with the integral transform technique. First, the 2D transient temperature was expressed as a combination of axisymmetric 1D steady-state temperature with a heat source, a filtered 2D steady-state temperature, and a filtered homogeneous 2D transient temperature, and was non-dimensionalized. Next, based on the method of separating variables to determine the integral transform pairs in radial and axial directions, the heat conduction control equations and boundary conditions plus initial conditions were subjected to the integral transform process to separate the time and space dependence of transient temperatures, and the 1st-order linear ordinary differential equations with respect to time were obtained. Finally, the theoretical solution of the 2D transient temperature distribution was obtained through the inverse transformation of the integral transform pair, and the 2D transient temperature distribution was verified through comparison of the results with the steady-state temperature distribution. On this basis, the effects of different combinations of inner and outer Biot numbers and different heat source strengths G on the temperature distributions of pipes were investigated. The results show that, the temperature gradients and temperature values in the spatial and temporal dimensions of the pipe are larger for the same combination of internal and external Biot numbers and for the high intensity of the heat source. There are differences in the temperature gradient and overall temperature distribution in the spatial and temporal dimensions when the intensity of the heat source is the same but the combinations of the internal and external Biot numbers are different. In the radial direction and the temporal dimension, the temperature distribution curves of different combinations of inner and outer Biot numbers have an intersection point where the temperature gradient and the relative magnitude of the temperature values change, but this phenomenon does not occur in the axial direction.
- transient heat conduction /
- axisymmetric single-layer pipe /
- integral transform /
- non-homogeneous boundary condition

HTML全文

图 1 含内部热源且内外表面存在对流的单层管

Figure 1. The single-layer pipe with internal heat source and subjected to convection at both surfaces

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图 2 单层管二维瞬态无量纲温度的空间和时间分布

Figure 2. Spatiotemporal distributions of 2D dimensionless transient temperatures in a single-layer pipe

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图 3 无量纲瞬态温度与稳态温度分布对比

Figure 3. Comparison of dimensionless transient temperatures and steady-state temperature distributions

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图 4 不同Biot数组合和热源强度下的时空维度温度分布

Figure 4. The temperature distributions in spatial and temporal dimensions for different combinations of Biot numbers and heat source intensities

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表 1 不同截断项阶数和τ=10⁴时，管道不同位置处的无量纲温度θ(η, ξ, τ)

Table 1. Dimensionless temperatures θ(η, ξ, τ) at spatial locations of the pipe for varying eigenvalue orders at fixed time τ=10⁴

	η	1.2	1.5	1.6	1.8	2.0
ξ=0.001	N=20	0.713 8	0.706 0	0.702 1	0.686 8	0.513 2
	N=25	0.714 2	0.706 9	0.702 5	0.688 4	0.517 0
	N=30	0.714 4	0.706 8	0.702 7	0.687 6	0.518 8
	N=35	0.714 4	0.706 6	0.702 8	0.688 0	0.519 8
	N=40	0.714 5	0.706 6	0.702 8	0.687 8	0.520 3
ξ=0.002	N=20	0.706 6	0.691 0	0.683 4	0.653 6	0.425 1
	N=25	0.706 7	0.691 2	0.683 5	0.654 0	0.426 1
	N=30	0.706 8	0.691 2	0.683 6	0.653 9	0.426 4
	N=35	0.706 8	0.691 2	0.683 6	0.653 9	0.426 5
	N=40	0.706 8	0.691 2	0.683 6	0.653 9	0.426 6
ξ=0.005	N=20	0.683 4	0.645 2	0.626 9	0.558 2	0.280 6
	N=25	0.683 4	0.645 2	0.626 9	0.558 2	0.280 6
	N=30	0.683 4	0.645 2	0.626 9	0.558 2	0.280 6
	N=35	0.683 4	0.645 2	0.626 9	0.558 2	0.280 6
	N=40	0.683 4	0.645 2	0.626 9	0.558 2	0.280 6
ξ=0.01	N=20	0.643 8	0.572 8	0.539 7	0.430 0	0.177 0
	N=25	0.643 8	0.572 8	0.539 7	0.430 0	0.177 0
	N=30	0.643 8	0.572 8	0.539 7	0.430 0	0.177 0
	N=35	0.643 8	0.572 8	0.539 7	0.430 0	0.177 0
	N=40	0.643 8	0.572 8	0.539 7	0.430 0	0.177 0
ξ=0.1	N=20	0.342 9	0.212 4	0.173 2	0.100 0	0.032 5
	N=25	0.342 9	0.212 4	0.173 2	0.100 0	0.032 5
	N=30	0.342 9	0.212 4	0.173 2	0.100 0	0.032 5
	N=35	0.342 9	0.212 4	0.173 2	0.100 0	0.032 5
	N=40	0.342 9	0.212 4	0.173 2	0.100 0	0.032 5

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表 2 不同截断项阶数和ξ=0.5时，管道不同时间无量纲径向温度θ(η, ξ, τ)

Table 2. Dimensionless radial temperatures θ(η, ξ, τ) at axial position ξ=0.5 for varying eigenvalue orders and times

	η	1.2	1.5	1.6	1.8	2.0
τ=0.1	N=20	0.135 3	0.043 9	0.028 9	0.011 8	0.003 3
	N=25	0.131 4	0.040 5	0.026 0	0.009 9	0.002 7
	N=30	0.130 4	0.039 6	0.025 2	0.009 4	0.002 5
	N=35	0.131 2	0.040 3	0.025 8	0.009 8	0.002 7
	N=40	0.131 4	0.040 4	0.025 9	0.009 9	0.002 7
τ=0.3	N=20	0.242 9	0.126 7	0.098 4	0.052 6	0.016 5
	N=25	0.242 7	0.126 5	0.098 2	0.052 5	0.016 5
	N=30	0.242 7	0.126 5	0.098 2	0.052 5	0.016 5
	N=35	0.242 7	0.126 5	0.098 2	0.052 5	0.016 5
	N=40	0.242 7	0.126 5	0.098 2	0.052 5	0.016 5
τ=0.5	N=20	0.290 6	0.167 5	0.134 0	0.075 1	0.024 1
	N=25	0.290 6	0.167 4	0.133 9	0.075 1	0.024 1
	N=30	0.290 6	0.167 4	0.133 9	0.075 1	0.024 1
	N=35	0.290 6	0.167 4	0.133 9	0.075 1	0.024 1
	N=40	0.290 6	0.167 4	0.133 9	0.075 1	0.024 1
τ=0.7	N=20	0.312 5	0.186 3	0.150 4	0.085 5	0.027 6
	N=25	0.312 5	0.186 3	0.150 4	0.085 5	0.027 6
	N=30	0.312 5	0.186 3	0.150 4	0.085 5	0.027 6
	N=35	0.312 5	0.186 3	0.150 4	0.085 5	0.027 6
	N=40	0.312 5	0.186 3	0.150 4	0.085 5	0.027 6
τ=0.9	N=20	0.322 6	0.194 9	0.157 9	0.090 3	0.029 2
	N=25	0.322 6	0.194 9	0.157 9	0.090 3	0.029 2
	N=30	0.322 6	0.194 9	0.157 9	0.090 3	0.029 2
	N=35	0.322 6	0.194 9	0.157 9	0.090 3	0.029 2
	N=40	0.322 6	0.194 9	0.157 9	0.090 3	0.029 2

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表 3 不同截断项阶数和η=1.5时，管道不同时间无量纲轴向温度θ(η, ξ, τ)

Table 3. Dimensionless axial temperatures θ(η, ξ, τ) at radial position η=1.5 for varying eigenvalue orders and times

	ξ	0.001	0.002	0.005	0.01	0.1
τ=0.1	N=20	0.697 8	0.674 7	0.604 7	0.493 1	0.054 2
	N=25	0.697 5	0.672 4	0.598 5	0.481 5	0.043 7
	N=30	0.696 7	0.671 0	0.595 0	0.475 1	0.037 7
	N=35	0.696 1	0.670 2	0.593 3	0.472 1	0.039 7
	N=40	0.696 0	0.669 9	0.592 5	0.470 8	0.040 7
τ=0.3	N=20	0.703 7	0.686 5	0.634 0	0.550 7	0.129 7
	N=25	0.704 6	0.686 6	0.633 8	0.550 2	0.129 3
	N=30	0.704 5	0.686 6	0.633 7	0.550 1	0.129 3
	N=35	0.704 3	0.686 5	0.633 7	0.550 1	0.129 3
	N=40	0.704 3	0.686 5	0.633 7	0.550 1	0.129 3
τ=0.5	N=20	0.705 2	0.689 5	0.641 5	0.565 4	0.175 0
	N=25	0.706 2	0.689 7	0.641 5	0.565 4	0.175 0
	N=30	0.706 1	0.689 7	0.641 5	0.565 4	0.175 0
	N=35	0.705 9	0.689 7	0.641 5	0.565 4	0.175 0
	N=40	0.705 9	0.689 7	0.641 5	0.565 4	0.175 0
τ=0.7	N=20	0.705 7	0.690 4	0.643 8	0.570 1	0.196 0
	N=25	0.706 7	0.690 7	0.643 8	0.570 1	0.196 0
	N=30	0.706 6	0.690 6	0.643 8	0.570 1	0.196 0
	N=35	0.706 4	0.690 6	0.643 8	0.570 1	0.196 0
	N=40	0.706 4	0.690 6	0.643 8	0.570 1	0.196 0
τ=0.9	N=20	0.705 9	0.690 7	0.644 7	0.571 7	0.205 3
	N=25	0.706 8	0.691 0	0.644 7	0.571 7	0.205 3
	N=30	0.706 7	0.691 0	0.644 7	0.571 7	0.205 3
	N=35	0.706 5	0.690 9	0.644 7	0.571 7	0.205 3
	N=40	0.706 5	0.690 9	0.644 7	0.571 7	0.205 3

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