Investigation into the heat transfer models for the hot crude oil transportation in a long‐buried pipeline

Based on reasonable simplifications of a two‐dimensional (2D) energy equation, a basic heat transfer equation in 1D form is obtained for the hot crude oil transportation in a long‐buried pipeline. To enclose the heat transfer equation, the overall heat transfer coefficient and the numerical simulation of temperature fields of pipe wall and soil are introduced, and they together with the basic heat transfer equation constitute 1D and cross‐dimensional heat transfer models of crude oil transportation, respectively. Some numerical procedures, including grid generation, partial differential equation discretization, and algebraic equation solution, are combined to solve the two heat transfer models. Based on the designed cases with different pipeline parameters, the relative deviation between the numerical results from the two heat transfer models does not exceed 1%, and the two heat transfer models agree well. Furthermore, the nonuniform natural soil temperature field is designed, and its influence on the oil temperature and the deviation of the 1D heat transfer model is investigated. Under the condition of the nonuniform natural soil temperature field, the atmospheric temperature as the ambient temperature in the 1D heat transfer model causes an apparent deviation, whereas the soil temperature at the buried depth of the pipeline in the natural soil temperature field as the ambient temperature does not. This study can provide a reference for the reasonable selection and use of the heat transfer models of the buried hot oil pipeline.

ratio, the pipeline is commonly selected to be buried under the ground surface. 2 Some kinds of crude oil, such as heavy oil and waxy crude oil, exhibit high viscosity or pour point. To reduce the large friction loss or avoid the gelling of crude oil in pipeline transportation, crude oil with poor fluidity is transported by heating in general. Due to the temperature difference between the heated crude oil and the surrounding environment, the heat of crude oil is gradually lost to the surrounding environment during the flow from upstream to downstream of the pipeline, and the temperature of crude oil gradually drops. Predicting oil temperature distribution along the pipeline is the basis of pipeline and transportation scheme designs. Accurate prediction results can effectively improve the rationality of pipeline and transportation scheme designs.
To achieve the accurate heat transfer prediction of buried pipeline transportation, many studies on the heat transfer of the buried pipeline have been conducted in the past. Initially, the natural soil temperature was assumed to be constant and equal to the ground surface temperature, and the boundary conditions at the pipeline surface and ground surface belonged to the isothermal (Dirichlet) boundary condition. 3 Based on the above assumptions, an analytical solution of heat loss was obtained by theoretical derivation. Subsequently, more complex boundary conditions were considered, including constant flux (Neumann) and convective (Robin) boundary conditions at the pipeline surface, [4][5][6] and a convective boundary condition at the ground surface, [7][8][9][10][11] and the corresponding overall heat transfer coefficient correlations were proposed. In addition, considering double-pipeline conditions, the overall heat transfer coefficient correlation with two pipelines laid in one ditch was deduced. 12 On the other hand, Sukhov first proposed a temperature drop formula along the pipeline and suggested that the overall heat transfer coefficient was obtained by reverse calculation from the field data. 7 However, this suggestion did not apply to the designed pipeline, which was not yet operational. To overcome this shortcoming, Leapienzon suggested that the overall heat transfer coefficient was calculated by the combination of the thermal conductivity of the cylinder wall and the equivalent heat transfer coefficient at the outer wall of the pipeline. 7 In addition, the effect of friction heat on oil temperature was considered in his studies. Nowadays, Sukhov's and Leapienzon's temperature drop formulas are adopted widely in engineering.
Besides the analytical solution of the heat transfer of the buried pipeline, some numerical simulations of the heat transfer were conducted. Based on the one-dimensional (1D) mass, momentum, and energy conservation equations of the oil stream, Wang derived the 1D heat transfer equation of the oil stream in the pipeline, and the influence of fluid expansion effect on oil temperature was considered in this equation. 13 Combining the heat transfer equation of the oil stream with the heat conduction equations of the wax deposition layer, steel pipe wall, anticorrosive coating and soil, the heat transfer process of the buried crude oil pipeline with batch transportation was numerically simulated, and the numerical model was verified by field test data of an actual pipeline. Yu applied a numerical scheme combining unstructured-finitevolume and finite difference methods to solve the heat transfer model. Numerical simulations in a wide range of operating conditions were conducted, and a good agreement between numerical simulations and field measurement indicates that the proposed numerical scheme was a suitable method to simulate the heat transfer process of the buried hot crude oil pipeline. 14 Subsequently, a numerical scheme combining the body-fitted-finitevolume method and bi-conjugate gradients stabilized method (Bi-CGSTAB) with the MILU preconditioner was presented to simulate the temperature field of the pipeline cross-section including wax deposition layer, steel pipe wall, anticorrosive coating, thermal insulation layer and soil. 15 And based on the temperature field of the pipeline cross-section, the heat loss and the oil temperature along the pipeline were further obtained. The obtained temperature field and oil temperature were verified by the commercial software FLUENT and the field data of an actual pipeline in our previous study, respectively. 15 Based on the change trend of atmospheric temperature, three atmospheric temperature models were established and their expressions were presented according to different approximations in our previous study. 2 It was found that the simplified model neglecting the atmospheric temperature changes from the alternation of day and night hardly induced the predictive deviation of oil temperature at the inlet of general buried pipelines, and the simplified model neglecting the atmospheric temperature changes within a month induced the prediction deviation within 0.5°C for 80 km buried pipelines.
The above investigations have provided significant insights into the heat transfer of the buried pipeline. However, some defects are also exposed. The natural soil temperature field is generally assumed to be uniform, and the nonuniformity of the natural soil temperature field is rarely considered in the analytical solution of the heat transfer of the buried pipeline. Some studies 7 point out that the lack of consideration of the natural soil temperature field may be able to induce a large deviation. Although the numerical simulation is an effective solution for this problem, its implementation is complex, and its solution efficiency is relatively low. Thus, it is challenging for ordinary engineers to apply numerical simulation. However, combined with numerical simulation, it is significant to investigate the deviation of the analytical solution and give suggestions on the use of the analytical solution, which can provide an essential scientific reference for the use of the analytical solution. Therefore, this study is organized as follows. The second section presents the heat transfer models from the analytical solution and numerical simulation; the third section introduces solution methods of heat transfer models; the fourth section verifies the rationality of heat transfer models by comparison with a uniform natural soil temperature field. The fifth section investigates the influence of the nonuniform natural soil temperature field on the oil temperature and the deviation induced by the analytical solution, and gives suggestions on the use of the analytical solution.

| HEAT TRANSFER MODELS
The heat transfer schematic of the crude oil transportation process in a buried hot pipeline is shown in Figure 1. In the transportation process, the heated crude oil in the pipeline transfers heat by a flowing means. This flow is actually a 3D flow. During the flow process, the heated crude oil transfers partial heat to the pipe wall, and the heat of crude oil is continuously lost along the pipeline. Due to the reduction of the temperature difference between crude oil and pipe wall, the heat loss gradually drops along the pipeline. On the other hand, the heat in the pipe wall is further transferred to the soil and influences the soil temperature field. Thereafter, the heat in the soil is released into the atmosphere.

| Derivation of a basic heat transfer equation
Long-buried hot oil pipelines are generally hundreds and even thousands of kilometers. The computation cost of a fully 3D equation solution is huge, and the numerical F I G U R E 1 Heat transfer schematic of the crude oil transportation process in a buried hot pipeline. results from a fully 3D simulation cannot be obtained so far. To significantly reduce the computation cost, 3D equations need to be simplified to lower-dimensional equations. The small computation cost is conducive to engineering application.
Neglecting the azimuthal variation of parameters in the pipeline, the 2D energy equation of the oil stream can be written as follows. (1) where r and z are the radial and axial coordinates in the cylindrical coordinate system, respectively, m; u r and u z are the velocity components in the radial and axial directions, respectively, m s −1 ; ρ is the density of crude oil, kg m −3 ; U is the internal energy of unit mass crude oil, J kg −1 ; V is the composite velocity, defined as V u u = + r z 2 2 , m s −1 ; q r and q z are the components of the heat flux density in the radial and axial directions, respectively, Wm −2 ; σ rr , τ rz , τ zr , and σ zz are the four components of the stress, Pa.
Substituting the relationship U H p ρ = − / into Equation (1), Equation (1) can be transformed. Then, the continuity and momentum equations are substituted into the transformed equation, and the following expression can be obtained.
where H is the enthalpy of unit mass crude oil, where c p is the specific heat capacity of crude oil at constant pressure, J kg −1°C−1 ; β is the expansion coefficient of crude oil, defined by β = −(1/ρ)(∂ρ/∂T),°C −1 ; T is the temperature,°C.
As the T and p change little in the radial direction of the pipeline for long pipelines, radial gradients of temperature and pressure in Equation (3) can be neglected. And considering the relationship τ τ = rz zr , Equation (3) can be simplified as follows: For long pipelines, the gradients of velocity and heat flow density in the axial direction of the pipeline are far less than those in the radial direction. Thus Equation (4) can be further simplified as follows: By integrating Equation (5) on the cross-section of the pipeline and using the relationships ∂p/∂z = 2τ w /R and τ rz = r/Rτ w , Equation (6) can be obtained as follows: where u z is the average velocity on the cross-section of the pipeline, which is defined by r G R ρ d = /(π ) 2 , ms −1 ; R is the radius of the pipeline, m; τ w is the shear stress at the inner wall of the pipeline, Pa.
For the flow of Newtonian fluids in the pipeline, the τ w can be calculated by the expression τ ρf u = − /8 w z 2 , and thus Equation (6) can be rewritten as the following equation: where f is the Darcy friction coefficient, and f = 64/Re (Re is the Reynolds number, defined as Re ρdu μ = / z ) when the flow is laminar (Re ≤ 2000), whereas f = 0.3164/Re 0.25 when the flow is turbulent (Re > 2000). 16 Equation (7) is a 1D form. The first and second terms on its left-hand side are convection and expansion heat terms, and the first and second terms on its right-hand side are exchange and friction heat terms. Calculating the heat flux density q r | r=R at the inner wall of the pipeline is critical to the enclosure of 1D heat transfer Equation (7) and the accuracy of oil temperature calculation. There are two ideas in calculating q r | r=R . The first is to consider the overall heat transfer coefficient regardless of the specific heat transfer process in the pipe wall and soil; the second is to consider the specific heat transfer process in the pipe wall and soil, and the numerical simulation of temperature fields of pipe wall and soil is applied. These two ideas are introduced as follows.

| Overall heat transfer coefficient
The overall heat transfer coefficient is assumed to be a combination of the convective heat transfer coefficient at the inner wall of the pipeline, the thermal conductivities of the pipe wall, and the equivalent heat transfer coefficient at the outer wall of the pipeline, and its expression is shown in the following equation 7,16 : ; d, D, and D w are the inner diameter of the pipeline, the actual outer diameter which corresponds to the steel pipe wall of the pipeline, and the enlarged outer diameter which corresponds to the anticorrosive coating of the pipeline, m; λ 1 and λ 2 are the thermal conductivities of the steel pipe wall and anticorrosive coating of the pipeline, respectively, W m −1°C−1 ; α o is the convective heat transfer coefficient at the inner wall of the pipeline, and α o = 0.17Re 0.33 Pr 0.43 Gr 0.1 (Pr/Pr w ) 0.25 λ/d (Pr is the Prandt number, defined as Pr = c p μ/λ, and Gr is the Grashov number, defined as Gr = (T-T w )βgρ 2 d 3 /μ 2 ) when Re ≤ 2000 whereas α o = 0.021Re 0.8 Pr 0.44 Gr 0.1 (Pr/Pr w ) 0.25 λ/d when Re ≥ 10,000, and the linear interpolation is adopted when 2000<Re < 10,000, 7 W·m −2°C−1 ; α s is the equivalent heat transfer coefficient at the outer wall of the pipeline, and it can be calculated by using Equation (9) which is deduced on the basis of the assumption of uniform soil temperature field under the natural condition, 7 W m −2°C−1 .
where λ s is the thermal conductivity of the soil, W m −1°C −1 ; h b is the buried depth of the pipeline, m; Bi is the 16 (α a represents the convective heat transfer coefficient at the ground surface and v a represents the wind speed at the ground surface). According to the above equations, the overall heat transfer coefficient K can be obtained. Then, according to the relationship  q K T T = ( − ) r r R = a m b i e n c e (T ambience is the ambient temperature), the heat flux density  q r r R = can be obtained.

| Numerical simulation of temperature fields of pipe wall and soil
The axial heat flux in a long-buried hot oil pipeline is much less than the radial heat flux in the pipe wall and soil. Thus, the pipe wall and soil can be simplified into a series of slices, as shown in Figure 2.
For a series of slices, the heat transfer process in the pipe wall and soil can be mathematically described by conduction equations, as shown in the following equations: where m = 1, 2, representing the steel pipe wall and anticorrosive coating.
where x and y are the horizontal and vertical coordinates in the Cartesian coordinate system, respectively, m. To write the above heat conduction equations into a unified form, the body-fitted coordinates 15,17 are adopted, and the unified form is shown in Equation (12). In addition, the irregular physical slice and its boundary conditions are transformed into the regular slice and its corresponding boundary conditions, as shown in Figure 3.
where l = 1, 2, 3, and represent the steel pipe wall, anticorrosive coating and soil; ξ and η are coordinates at two different directions in the body-fitted coordinate system; α, β, γ, and J are some intermediate parameters due to the coordinate transformation from the Cartesian coordinates to the body-fitted coordinates, which are defined by α and J x y x y = − ξ η η ξ , m 2 .
F I G U R E 2 Schematic of a series of slices for the pipe wall and soil.
F I G U R E 3 Irregular physical slice, regular transformed slice and their boundary conditions. YUAN ET AL.

| 2173
Based on the governing equation as shown in Equation (12) and the boundary conditions as shown in Figure 3, the temperature fields of the pipe wall and soil can be obtained by the numerical simulation. Then, according to the relationship (T w is the temperature of the inner wall of the pipeline), the heat flux density  q r r R = can be obtained.
As the combination of Equations (7) and (8) is a 1D calculation, the model with the overall heat transfer coefficient is called the 1D heat transfer model in this study. On the other hand, Equation (7) is a 1D equation, whereas Equation (12) is a 2D equation, and thus the model with the numerical simulation of the pipe wall and soil temperature field is called a cross-dimensional heat transfer model.

| NUMERICAL METHODS
To obtain the temperature of hot oil along the pipeline, some numerical methods need to be adopted, mainly including the grid generation method, partial differential equation discretization method and solution method, which will be introduced in the following.
Based on a fixed spatial step, the flow zone inside the pipeline is discretized into a series of 1D grid points, as shown in Figure 4. In addition, based on the Laplace method and the algebraic auxiliary method, 15 the slices of pipe wall and soil are discretized into a series of 2D grid points, as shown in Figure 5. The Laplace method is used to generate inner grid points on the physical slice by solving Laplace equations in which the coordinates on the physical slice are related to those on the transformed slice. Before the Laplace method is used, the coordinates of some internal grid points on the physical slice need to be specified through the proportional function, which is an algebraic auxiliary method. The use of the algebraic auxiliary method can make the grid distribution generated by the Laplace method more reasonable. 15 After the above grids are generated, the governing equations with partial differential form are discretized on grids. Equation (7) is transformed into the discrete algebraic equation by the finite difference method (FDM), 18 as shown in Equation (13). For the convenience of the linearized solution, the values of some parameters related to the temperature are expressed approximately by the values obtained at the upstream grid point.  Besides the discretization of Equation (7), Equation (12) also needs to be discretized for the crossdimensional heat transfer model. Based on the grid points in body-fitted coordinates, a series of control volumes are generated, as shown in Figure 6. Then, Equation (12) where a a = +     According to Equation (8), the overall −heat transfer coefficient is obtained easily for the 1D heat transfer model. Then, the oil temperature on each gird point is obtained one by one from the upstream to the downstream according to Equation (13). On the other hand, the numerical simulation of the temperature fields of pipe wall and soil is required for the cross-dimensional model. Equation (14) is solved by using the Bi-CGSTAB algorithm 19 with MILU preconditioner, 20 and then the heat flux density from the heated crude oil into the inner wall of the pipeline is further obtained. Finally, the oil temperature on each gird point is obtained according to Equation (13).

| MUTUAL VERIFICATION OF TWO HEAT TRANSFER MODELS
Some heat transfer models of pipeline transportation have been verified by the field data from the actual pipelines in our previous studies. [13][14][15] However, due to the complexity of the actual environments and the difficulty in accurate measurement of instruments, it is very difficult to verify the heat transfer models very accurately. Therefore, mutual verification of two heat transfer models is conducted in this section, and more accurate comparisons can be obtained.

| Mutual verification based on a basic case
The transportation process of a long-buried hot oil pipeline is taken as the investigated object. The transported crude oil is heavy oil, and its thermal conductivity and specific heat capacity are 0.14 W m −1°C −1 and 2200 J kg −1°C−1 , respectively. The dynamic viscosity and density of crude oil meet the curves shown in Figure 7. In addition, some feature parameters of the long-buried hot oil pipeline are shown in Figure 8. There is no thermal insulation layer on the surface of the pipeline. In Figure 8, the atmospheric temperature refers to the average atmospheric temperature of a month (the atmospheric temperature change within a month is neglected 2,21 ), and the soil temperature at the thermostatic layer of soil is equal to the average atmospheric temperature of a year. 22 The thermal-influenced region of the hot oil pipeline is within 10 m, which implies that the soil temperature beyond 10 m from the pipeline is not influenced by the pipeline. This has been demonstrated by both experimental data and numerical analysis for the buried pipelines with buried depths of 1.2-1.8 m. 13,23 As the value of atmospheric temperature at the ground surface is the same as the value of soil temperature at the thermostatic layer of soil, it can be inferred that the natural soil temperature field is uniform and equal to the zero field under the steady state condition. However, influenced by the hot oil pipeline, the original uniform natural soil temperature field is broken down, and the actual (or comprehensive) soil temperature field generally exhibits a nonuniform distribution, as shown in Figure 9.
For the cross-dimensional heat transfer model, it is important to select the appropriate grid number on the slice. A large number of grids generally cause low computational efficiency, while a small number of grids may cause large errors in the numerical solution. Thus, grid independence verification is carried out. According  to heat transfer models and numerical methods introduced in Sections 2 and 3, the numerical results based on different grid numbers from the cross-dimensional model can be obtained, as shown in Table 1. It can be found that relative deviations of the numerical results corresponding to N ξ × N η = 30 × 30 and N ξ × N η = 60 × 60 are small. Considering the fact that the small enough deviation is more favorable for the error analysis of model comparison in the following text, the grid number of N ξ × N η = 60 × 60 is selected.
The numerical results from the cross-dimensional model are compared with those from the 1D model, as shown in Table 2. It can be seen from Table 2 that the obtained outlet temperatures of the pipeline from the 1D and cross-dimensional heat transfer models match well under the condition of the uniform natural soil temperature field, and the relative deviation of the temperature drops is 0.7%. Although the 1D model and cross-dimensional model are two different models, almost the same results can be obtained. This comparison verifies the rationality of the two heat transfer models to some extent.

| Mutual verification based on different pipeline parameters
Different pipeline parameters generally correspond to different outlet temperatures and temperature drops. To make the verification more convincing, more cases based on different pipeline parameters are designed, as shown in Table 3. Except for the length, diameter and buried depth of the pipeline, the values of other parameters in these cases are the same as those in the case mentioned in Section 4.1. These cases are designed under the condition of the uniform natural soil temperature field. In addition, cases 2, 5, and 8 are actually the same case, and they are also the same as the case mentioned in Section 4.1. In these cases, cases 1-3 exhibit different lengths of the pipeline; cases 4-6 exhibit different diameters of the pipeline; cases 7-9 exhibit different buried depths of the pipeline.
Based on the above-designed cases, the numerical results from the 1D and cross-dimensional heat transfer models are obtained, as shown in Table 4. Although these cases exhibit different lengths, diameters and buried depths of the pipeline, the outlet temperatures obtained by the 1D model are very close to those by the cross-dimensional model under the condition of the uniform natural soil temperature field. For cases 1-8, the relative deviations of temperature drops are less than 1%, F I G U R E 7 Viscosity-temperature and density-temperature curves of crude oil.
F I G U R E 8 Feature parameters of a long-buried hot oil pipeline.
F I G U R E 9 An actual soil temperature field. which is deemed to be a very small value. However, the relative deviation is 3.0% for case 9. Although this deviation is also a small value, it is obviously different from the deviations in other cases. To clarify this difference, the analysis is done in the following.
The buried depth of the pipeline in case 9 is 2.5 m, which is greater than that in other cases. Greater burial depth means greater thermal resistance from the pipeline to the ground surface. The heat from the pipeline is more easily transferred to the soil region on both sides of the pipeline and below the pipeline, and the thermalinfluenced region of the pipeline becomes wider and deeper, as shown in Figure 10. For this reason, using the computational domain within the range of 10 m for the soil may bring some deviation. To confirm such an analysis, a wider computational domain is adopted. Under the condition of the computational domain within the ranges of 15 m and 20 m, the numerical results are obtained for case 9, as shown in Table 5. It can be found that the relative deviations are less than 1% and are obviously smaller than the previous deviation. In addition, the thermal-influenced region of the pipeline obviously becomes wider and deeper, as shown in Figure 11. Therefore, the compared results well confirm the above analysis. On the other hand, besides the relative deviation from case 9, all relative deviations of temperature drops are less than 1%. Therefore, these comparisons further verify the rationality of the two models under the condition of the uniform natural soil temperature field.
The above analysis is helpful in finding the source of deviation. Considering that the relative deviation of 3.0% is still small, the computational domain within the range of 10 m can still be selected for buried pipelines with buried depths of less than 2.5 m.

| RESULTS AND DISCUSSION
Influenced by the change of atmospheric temperature, the soil temperature field under the natural condition is generally not uniform. However, Equation (9) is obtained on the basis of the assumption of a uniform soil temperature field under natural conditions. To clarify the influence of nonuniform natural soil temperature field on the oil temperature and the deviation induced by Equation (9), the nonuniform natural soil temperature field is designed first, and then its influence is investigated.

| Design of nonuniform natural soil temperature field
After the atmospheric temperature at the ground surface and the soil temperature at the thermostatic layer of soil are set, the nonuniform soil temperature field can be obtained according to the heat transfer theory. Figure 12 shows two kinds of nonuniform natural soil temperature fields. The soil temperature gradient shown in Figure 12A is a positive value, whereas the soil temperature gradient shown in Figure 12B is a negative value. The atmospheric temperature in both Figure 12A,B is set to 0°C. The soil temperature at the thermostatic layer of soil in Figure 12A,B is set to −30°C and 30°C, respectively. The soil temperature at any position T A B L E 4 Numerical results from two models for cases 1-9.  in the computational domain can be obtained by Equation (15). If the α a and λ s are set to be the same as those in the case mentioned in Section 4.1, the soil temperature gradient can be further obtained. According to the parameters specified above, it can be obtained that the soil temperature gradient in Figure 12A equals 2.98°C m −1 , and the soil temperature gradient in Figure 12A equals −2.98°C m −1 .

Cases
where T a is the atmospheric temperature at the ground surface,°C; T c is the soil temperature at the thermostatic layer of soil,°C.
5.2 | Influence of nonuniform natural soil temperature field

| Nonuniform natural soil temperature fields with the same atmospheric temperature
Based on the design method of the nonuniform natural soil temperature field, more tested cases are designed, as shown in Table 6. As the ambient temperature is the atmospheric temperature at the ground surface for the buried hot oil pipeline, the atmospheric temperature is set to the same for the designed cases. In these designed cases, nonuniform natural soil temperature fields are different, and their gradients vary. Except for the parameters in Table 6, the values of other parameters in these cases are the same as those in the case mentioned in Section 4.1. Based on the above-designed cases, the numerical results from the 1D and cross-dimensional heat transfer models are obtained, as shown in Table 7. It can be seen from Table 7 that numerical results from the 1D model are the same for different tested cases. These identical results are attributed to the dependence of the 1D model on the selected ambient temperature (e.g., the atmospheric temperature at the ground surface) rather than the nonuniform natural soil temperature fields. It can be found that the outlet temperature obtained from the cross-dimensional model gradually rises with the decrease of the soil temperature gradient. In addition, the relative deviation between the temperature drops obtained by 1D and cross-dimensional models rises with the increase of the absolute value of the soil temperature T A B L E 6 Designed cases based on different nonuniform natural soil temperature fields with the same atmospheric temperature. gradient. As the cross-dimensional model can be well applicable to the numerical simulation with nonuniform natural soil temperature fields, its corresponding numerical results are assumed to be more reliable. Therefore, the comparative results indicate that the deviation of numerical results from the 1D model can be displayed obviously under the condition of nonuniform natural soil temperature fields, and it rises with the increase of nonuniformity.

Cases
The soil temperature field at the outlet of the pipeline under the condition of different nonuniform natural soil temperature fields with the same atmospheric temperature is shown in Figure 13. Comparing Figure 13 and Figure 10A, it can be found that the soil temperature fields near the hot oil pipeline exhibit an obvious difference. Comparing Figure 13 and Figure 12, it can be seen that the isotherm of soil in Figure 13A is lower, and the isotherm of soil in Figure 13B is upper. The above phenomena can be explained as the interaction between a hot oil pipeline and a natural soil temperature field to form a comprehensive (or an actual) soil temperature field. 5.2.2 | Nonuniform natural soil temperature fields with the same soil temperature at the buried depth of the pipeline In the above section, the atmospheric temperature at the ground surface is used as the ambient temperature for the buried hot oil pipeline. However, the pipeline is in direct contact with the soil. The soil temperature at the buried depth of the pipeline in the natural soil temperature field may directly influence the outlet temperature of the pipeline. To clarify the influence of ambient temperature selection on the outlet oil temperature of the pipeline, five cases based on different nonuniform natural soil temperature fields with the same soil temperature at the buried depth of the pipeline are designed, as shown in Table 8. In these designed cases, the soil temperature at the buried depth of the pipeline in natural soil temperature fields is set to the same. The atmospheric temperature at the ground surface and the gradients of natural soil temperature fields vary. Except for the parameters in Table 8, the values of other parameters in these cases are the same as those in the case mentioned in Section 4.1.
For cases 15-19, the outlet temperature and temperature drop from the 1D and cross-dimensional heat transfer models are shown in Table 9. It can be seen from Table 9 that the outlet temperature and temperature drop from the 1D model are in good agreement with those from the cross-dimensional model under the condition of different nonuniform natural soil temperature fields. All relative deviations of the temperature drops do not exceed 1.0%, which is a pretty small value. The comparative results indicate that the numerical results obtained by the 1D model rarely exhibit deviation under the condition of different nonuniform natural soil temperature fields. Therefore, it is suggested that the soil F I G U R E 13 Soil temperature fields at the outlet of the pipeline under the condition of different nonuniform natural soil temperature fields with the same atmospheric temperature. (A) Natural soil temperature field with a positive gradient and (B) natural soil temperature field with a negative gradient. temperature at the buried depth of the pipeline in the natural soil temperature field is used as the ambient temperature for the 1D heat transfer computation of the buried hot oil pipeline. Under this suggestion, the accuracy of the 1D model is almost independent of nonuniform natural soil temperature fields.
The soil temperature field at the outlet of the pipeline under the condition of different nonuniform natural soil temperature fields with the same soil temperature at the buried depth of the pipeline is shown in Figure 14. It can be found from Figure 14 that the actual soil temperature field exhibits obvious different features. This phenomenon indicates that the actual soil temperature fields near the hot oil pipeline are obviously influenced by the nonuniform natural soil temperature fields, although the outlet temperature and temperature drop of the pipeline are rarely influenced.
According to the above comparative results, it can be known that the measurement of the natural soil temperature at the buried depth of the pipeline is very important for the 1D heat transfer computation. If this temperature is measured, the 1D heat transfer computation is recommended due to its easy implementation process and high solution efficiency; otherwise, it is not recommended due to its large deviation. Under the condition that the natural soil temperature at the buried depth of the pipeline is not measured, the atmospheric temperature can generally be easily obtained according to some historical meteorological data, and crossdimensional heat transfer computation can be used. If you still want to use the 1D computation, it is suggested to first calculate the natural soil temperature at the buried depth of the pipeline according to the atmospheric temperature.

| CONCLUSIONS
The equivalent heat transfer coefficient at the outer wall of the pipeline can be obtained under the assumption of uniform natural soil temperature fields, which can be used to calculate the overall heat transfer coefficient of the pipeline. Combining a basic heat transfer equation with the overall heat transfer coefficient, a 1D heat T A B L E 8 Designed cases based on different nonuniform natural soil temperature fields with the same soil temperature at the buried depth of the pipeline. transfer model of crude oil transportation in a longburied hot pipeline is presented. On the other hand, considering the numerical simulation of temperature fields of the pipe wall and soil, the heat flux density from the oil stream to the inner wall of the pipeline can be obtained. Combining a basic heat transfer equation with the numerical simulation of temperature fields of pipe wall and soil, a cross-dimensional heat transfer model is presented. Many cases are designed in this study to compare these two heat transfer models, and the main results of the comparison are summarized as follows:

Cases
1. Under the condition of the uniform natural soil temperature fields, the temperature drops obtained by the 1D and cross-dimensional heat transfer models match well, and the deviation between them is less than 1%. The small deviation mutually verifies the rationality of the two models. 2. Under the condition of the nonuniform natural soil temperature fields with the same atmospheric temperature, taking the temperature drops obtained by the cross-dimensional heat transfer model as the benchmark solution, the temperature drops obtained by the 1D heat transfer model exhibit an obvious deviation, and the deviation rises with the increase of the nonuniformity of natural soil temperature fields. 3. Under the condition of the nonuniform natural soil temperature fields with the same soil temperature at the buried depth of the pipeline, although the actual soil temperature field exhibits obviously different features, the temperature drops obtained by the 1D heat transfer model hardly exhibit the deviation and are rarely influenced by nonuniform natural soil temperature fields. It is suggested that the soil temperature at the buried depth of the pipeline in the natural soil temperature field is used as the ambient temperature for the 1D heat transfer computation of the buried hot oil pipeline. 4. If the natural soil temperature at the buried depth of the pipeline is measured, the 1D heat transfer computation of the buried hot oil pipeline is recommended. Otherwise, the natural soil temperature at the buried depth of the pipeline needs to be calculated first according to the atmospheric temperature. However, cross-dimensional heat transfer computation can be used directly.

ACKNOWLEDGMENTS
The study is supported by the fund of the Beijing Municipal Education Commission (No. 22019821001) and Award Cultivation Foundation from Beijing Institute of Petrochemical Technology (No. BIPTACF-002).

CONFLICT OF INTEREST STATEMENT
The authors declare no conflict of interest.