Fast thermal simulation of a heated crude oil pipeline with a BFC-Based POD reduced-order model

Applied Thermal Engineering 88 (2015) 217e229

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Fast thermal simulation of a heated crude oil pipeline with a BFC-Based POD reduced-order model Dongxu Han, Bo Yu*, Yi Wang, Yu Zhao, Guojun Yu National Engineering Laboratory for Pipeline Safety, Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum, Beijing 102249, People's Republic of China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 16 June 2014 Received in revised form 30 September 2014 Accepted 7 October 2014 Available online 15 October 2014

A proper orthogonal decomposition (POD) Galerkin reduced-order model (ROM) for unsteady-state heat conduction problems is proposed based on body-fitted coordinate (BFC), and the discrete format of spatial derivative of basis function which could enhance the calculation precision is employed in the present paper. Applying the POD reduced-order model into the fast thermal calculation of oil pipeline, such as the batch transportation of cold and hot oil as well as pipeline shutdown, the computing speed of POD reduced-order model is about 259 times and 177 times faster than that of traditional finite volume method (FVM) respectively. And from the calculation results, the reduced-order model is proved to enjoy good precision. Thus the reduced-order model holds great engineering application value and would be a very helpful tool to solve the time-consuming problem involved in the optimal design and risk assessment of oil pipeline. © 2014 Elsevier Ltd. All rights reserved.

Keywords: POD reduced-order model Galerkin projection Body-fitted coordinate Unsteady Oil pipeline

1. Introduction Due to the poor fluidity of crude oil with high pour point or high viscosity, this kind of crude oil has to be heated during pipeline transportation to ensure safe operation [1]. In order to achieve this goal, the thermal characteristics involved in the heating transportation, especially the unsteady-state process, have to be studied and revealed. In recent years, a large number of numerical simulation have been carried out to study the unsteady-state thermal characteristics of buried hot oil pipeline and the corresponding results significantly contributed to the research in this field [2e5]. However, direct numerical simulation is not economic for its huge time consumption. And in the optimal design and operation as well as risk assessment of oil pipeline [6e8], the calculation for different operation conditions containing various combinations of operating parameters involves very large calculation quantity and the calculation time is likewise unacceptable. On this occasion, the low efficiency of numerical calculation could not meet the requirement of engineering practice, thus it is quite necessary and important to propose an efficient calculation method.

* Corresponding author. Tel.: þ86 10 89733849. E-mail address: [email protected] (B. Yu). http://dx.doi.org/10.1016/j.applthermaleng.2014.10.017 1359-4311/© 2014 Elsevier Ltd. All rights reserved.

POD reduced-order model has already got extensive attention among researchers as an efficient calculation method and has been widely applied to various engineering fields [9e12]. In the past two years, the applications of POD reduced-order model for thermal calculation of hot oil pipeline have been reported in the literature. For instance, the reference [13] introduced the POD reduced-order model into the thermal calculation of hot oil pipeline for the first time and studied the batch transportation of cold and hot crude oil as well as pipeline commissioning by employing the unstructured grid-based reduced-order model. Ref. [14] used the BFC-based POD reduced-order model to establish the relationship between pipeline cross-sections and achieved the rapid calculation of temperature field with different pipeline diameter and buried depth. Despite the desirable conclusions, Refs. [13 and 14] didn't take the wax layer, steel wall and corrosion protection layer of pipeline into consideration in the thermal calculation or introduce their influence into the heat transfer process of oil pipeline thermal calculation. However, in reality the heat conductivity coefficients of these structures differ greatly from each other and that would have great impact on the calculation accuracy of POD reduced-order model. Due to the employment of unstructured grid, Ref. [13] could only implement rapid calculation of hot oil pipeline with fixed buried depth and pipeline diameter. In other words, the implementation of POD reduced-order model on oil pipeline must be exactly the same with the selected samples in buried depth and

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D

Nomenclature T

r C

l d H L H0 hf

t th td win A V w P q



temperature ( C) density (Kg/m3)  specific heat capacity (J/(Kg C))  thermal conductivity (W/(m C)) thicknesses (m) heat influence region on the Y direction (m) heat influence region on the X direction (m) pipeline buried depth (m)  heat transfer coefficient (W/(m2 C)) time (s) time (h) time (d)  inlet oil temperature of the pipeline ( C) 2 flow area of the pipeline (m ) flow rate of the oil (m/s) hot oil temperature in the pipeline ( C) pressure (Pa) the heat flux between the oil and the inner wall of the three layer structure (W/m2)

diameter, while other conditions could be different. This drawback greatly limited the application range of the reduced-order model. Ref. [14] only calculated the steady-state temperature field of one specific cross-section by reduced-order model and couldn't reflect the practical situations. In addition, the difference between steadystate model and unsteady-state model not only is the unsteady term, but involves the proper treatment of spatial derivative of Jacobi factor [15]. According to the current researches mentioned above, the present paper established the unsteady-state heat conduction reduced-order model based on body-fitted coordinate and employed new scheme which could better satisfy the energy conservation during the spatial derivative discretization. The reducedorder model could also be applied to the fast calculation of the unsteady-state thermal characteristics of hot oil pipeline with different buried depth and diameter. Because in relevant studies, two typical operation situations, batch pipelining of cold and hot crude oil and shutdown of hot oil pipeline, obtained great attentions, the present paper represents the efficient calculation method to calculate these two operation conditions in the following text. The layout of this paper is as follows. The physical problem and mathematical model of batch transportation of cold and hot crude oil and pipeline shutdown are presented beforehand. Then the direct numerical simulation method is introduced followed by the establishment, solution and involved important issues of BFCbased reduced-order model. Finally, the accuracy and efficiency of reduced-order model are assessed through employing the reducedorder model to calculate the two typical operation conditions (batch pipelining of cold and hot crude oil and pipeline shutdown).

x, h a, b, g J

f ak

the diameter of the inner wall of the three layer structure (m) coordinates in a body-fitted coordinate system parameters related to grid status Jacobi factor basis function spectrum coefficient of the kth basis function

Subscripts 1, 0, 1, 2, 3, 4 regions of liquid oil, gelatinous oil, wax layer, steel wall, corrosion protection layer and soil respectively c constant e, w, s, n variable with location in east,west, south, and north faces of control volume E, W, S, N variable with location in east,west, south, and north neighbor points of the control volume P variable with location in control volume x, h partial derivatives of the variable

illustrate the corresponding models for batch transportation of cold and hot crude oil and pipeline shutdown. 2.1. Physical and mathematical model of pipeline cross-section Fig. 1(a) shows the overview of physical domain in the pipeline cross-section. In order to simplify the calculation and analysis, the following assumptions are made in the present paper.

2. Physical problem and mathematical model Hot oil in the pipeline continuously transfers its heat to the inner wall of wax layer by means of heat convection and then through the wax layer, steel wall and corrosion protection layer to the soil surrounding the oil pipeline (see the three layer in Fig. 1(b)). In the above process, the heat is also exchanged between the soil and ambient atmosphere. Before studying the thermal characteristics of typical operation conditions, we firstly abstract the physical model from the engineering problems and then on this basis,

Fig. 1. Sketch map of pipeline cross-section.

D. Han et al. / Applied Thermal Engineering 88 (2015) 217e229

(1) The soil anisotropy outside the pipelines is simplified as isotropy. (2) The axial temperature drop of surrounding soil is small enough to be neglected, thus the heat transfer in the soil area can be assumed to be two-dimensional. (3) The thickness of the wax deposition remains unchanged along the pipeline. (4) According to the literature and engineering experience, the thermal influence region of the hot crude oil pipeline is within 10 m, where L ¼ 10 m and H ¼ 10 m as shown in Fig. 1(a) [16]. On the basis of above assumptions, the mathematical models describing the thermal system of the pipeline cross-section can be obtained. Heat transfer equations for wax layer, pipeline wall and corrosion protection layer are written as follows:

rk Ck



 vT v vT v vT ¼ lk þ lk vt vx vx vy vy

k ¼ 1; 2; 3; 4

(1)

where the subscripts 1, 2, 3, 4 denotes wax layer, steel wall, corrosion protection layer and surrounding soil respectively and the thicknesses of wax layer, steel wall and corrosion protection layer are d1,d2,d3 as shown in Fig. 1(b). Boundary conditions: since the calculation domain is symmetrical, two symmetry boundary conditions are given as follows:

lk

vT ¼ 0 at vx

oil and the cross-section calculation domain and they can be found in Sections 2.2 and 2.3 treated in different way. 2.2. Physical and mathematical model of batch transportation of cold and hot oil The batch transportation of cold and hot crude oil is one of the advanced oil transportation technologies recently emerging in the world. Employing this technology, different crude oil with different physical property would be heated to different temperature and transported in order to reduce the heating energy consumption and enhance the economic performance of the project. Fig. 2 shows the inlet oil temperature of the pipeline win in different operation period. Before studying the thermal characteristics of batch transportation of cold and hot crude oil, two more assumptions should be added as follows. (1) The oil temperature on a fixed pipeline cross-section is assumed to be uniform, thus the oil temperature is only the function of time and axial position. (2) The oil exchange at the batch interface of cold and hot crude oil is neglected. On the basis of above assumptions, the governing equations of oil flow in the pipeline can be obtained. Mass conservation equation:

v v ðrAÞ þ ðrVAÞ ¼ 0 vt vz

x ¼ 0;  ðH0  R1 þ d1 Þ  y  0

219

(3)

Energy conservation equation:

vT lk ¼ 0 at vx

x ¼ 0;  H  y  ðH0 þ R1  d1 Þ:

The top of the cross-section is the ground surface which exchanges heat with the atmosphere by convection as follows:

  vT l4 ¼ hfa Ta  Tground ; at vy

y ¼ 0:

vT ¼ 0; at vx

T ¼ Tc ; at

y ¼ H:

In the boundary condition expressions above, hfa and Tground are the heat transfer coefficient between soil and atmosphere and temperature of the ground surface. Tc is the temperature of constant temperature layer which can be obtained by practical measurement. Ta stands for the atmosphere temperature and can be calculated by Eq. (2) which is.

(2)

where m1 ¼ 5 with unit  C, m2 ¼ p/12 with unit (h1) and m1sin(m2th) denotes the temperature changes within one day and atd þ b (the value of a and b can be found in the following text) stands for the temperature variation along the change of date. Besides the above boundary condition, matching conditions are needed around y ¼ H (see Fig. 1(a)) which are the interfaces of the

(5)

Among the Eq. (4) and Eq. (5) w denotes the hot oil temperature, q in the Eq. (5) is the key variable which bind the pipeline cross-section thermal calculation with the temperature calculation

x¼L

Ta ¼ m1 sinðm2 th Þ þ atd þ b

(4)

Matching condition:

 vT  q ¼ l1 ! ¼ hf 1 ðw  Tw Þ vn w

As it is stated above the thermal affected region of the pipeline is limited, and we believe it has the feature of the natural soil land temperature field in the right (when the right half of the physical domain shown in Fig. 1(a) is concerned) and the bottom boundary conditions as follows:

l4


vw vw 4q þv ¼ CP vt vz rD

Fig. 2. Pipeline inlet oil temperature.

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of the oil in the pipe and it can be calculate by Eq. (5). Tw is temperature of interface between the wax layer and oil flow and its values are calculated by solving Eq. (1), hf1 is the forced convection heat transfer coefficient of oil flow and wax layer which is the function of oil temperature and has to be determined by experiment data [1]. Since we set the r and A as constant for clarity, the flow rate is constant. Thus through solving Eq. (1) and Eq. (4), the temperature field of pipeline cross-section and the oil temperature distribution along the pipeline can be obtained. 2.3. Physical and mathematical model of oil pipeline shutdown After the shutdown of pipeline, the temperature of crude oil decreases gradually over time and the natural convection inside the pipeline is emerging due to the temperature difference. When the crude oil cools to the temperature below the wax appearance temperature, the wax crystals begins to precipitate on the pipeline wall and latent heat of crystallization is released meanwhile. As a result, the crude oil starts to gelatinize and the gelatinization is observed to extend from the near wall area to the center area of pipeline for the oil around the pipeline wall suffering the lowest temperature. As time goes by, the oil temperature decreases continuously and the gelatinous area extends with the reducing of liquid phase area until all the crude oil in the pipeline is gelatinous. At that time, the heat is transferred from the gelatinous oil to the environment by means of heat conduction. It is worth noting that in the process of gelatinization, the gelation interface is not absolutely concentric with the center of pipeline. In order to solve this problem briefly, several assumptions need to be made beforehand.

 vT  l1 ! ¼ hf 0 ðw  Tw Þ vn w

(8)

Besides the boundary conditions in Section 2.1, additional boundary condition is given as follows which is also a symmetry boundary condition:

lk

vT ¼ 0 at x ¼ 0;  ðH0 þ R1  d1 Þ  y  ðH0  R1 þ d1 Þ vx

(9)

The calculations of temperature drop after hot oil pipeline shutdown mainly involve the solution of Eq. (1) and Eq. (7). Eq. (1) is solved to obtain the temperature field of pipeline cross-section and Eq. (7) aims to get average oil temperature and temperature distribution inside the pipeline. 3. Direct numerical simulation On the basis of mathematical models for batch transportation of cold and hot crude oil and temperature drop after pipeline shutdown, governing equations need to be solved using numerical method to obtain the thermal characteristics of practical problems. Though the present paper aims to propose the BFC-based POD reduced-order model to calculate the unsteady-state operation conditions of hot oil pipeline efficiently, the basis functions of POD method need to be obtained from the calculation results of direct numerical simulation(DNS)and there will be a comparison between these two different methods in the following text. Thus it is necessary to briefly introduce the DNS application for the two typical operation conditions. 3.1. Computational domain based on body-fitted coordinate

(1) The heat transfer in the axial direction of pipeline is neglected. (2) In the process of oil gelatinization, the gelation interface is regarded to be concentric with the center of pipeline. (3)Employing the equivalent heat conductivity coefficient method to treat the heat convection as heat conduction, that's to say, at the interface of gelatinous oil the form of heat exchange is heat convection between the liquid part and gelatinous part of crude oil, and the heat convection in the liquid part is simplified into heat conduction with an “equivalent heat conductivity coefficient”. The heat convection coefficient is measured by experiment while the heat conductivity coefficient can be obtained as follows.

l1 ¼

hf 0 ðw  Tw Þ   vT  ! vn

Considering the symmetry of the heat influence area, take half of the area as the computational domain and establish the geometric model on it. Then transform the original domain into the body-fitted coordinate plane, and we can get the new computational domain as shown in Fig. 3. It is worth pointing out that different practical region with various buried depth and pipe

(6)

w

! where Tw and vT=v n jw denote the temperature and temperature gradient at the interface of gelatinous oil, w is the average temperature of the liquid oil area and hf0 is the heat convection coefficient between the liquid oil and gelatinous oil. In addition to the governing equations given in Section 2.1, one more equation needs to be given as follows.

rk Ck



 vT v vT v vT ¼ lk þ lk þS vt vx vx vy vy

k ¼ 1; 0

(7)

where 1 and 0 denote the liquid oil area and gelatinous oil area inside the pipeline respectively; S stands for the heat release rate of wax precipitation which can be neglected due to its small value. The matching condition between the liquid oil and gelatinous oil is given as follows.

Fig. 3. Computational domain on body-fitted coordinate.

D. Han et al. / Applied Thermal Engineering 88 (2015) 217e229

221

Fig. 4. Computational grids for batch transportation of cold and hot crude oil.

diameter could all be transformed to the same domain in Fig. 3 and that builds up the desirable foundation for the subsequent employment of reduced-order model. Similar to the transformation of physical domain, the governing equations and boundary conditions should also be transformed into the body-fitted coordinate plane. Eq. (1) and Eq. (7) can be rewritten in the form of Eq. (10) or Eq. (11) as follows.

J

      vðrk ck TÞ v lk  v lk  ¼ aTx  bTh þ gTh  bTx þ JS vt vx J vh J

Eq. (10) and Eq. (11) are mathematically equivalent but the reduced-order models derived from these two different equations hold different grid adaptability which would be discussed in the following text. The directional derivatives in boundary conditions can be transformed to the computational domain as follows.

In x direction :

vT aTx  bTh vT gTh  bTx pffiffiffi ; In h direction : ¼ ¼ pffiffiffi : vnh J g vnx J a

k ¼ 1; 0; 1; 2; 3; 4 (10)       vðrk ck TÞ 1 v lk  1 v lk  ¼ aTx  bTh þ gTh  bTx þ S vt J vx J J vh J k ¼ 1; 0; 1; 2; 3; 4 (11) where a ¼ x2h þ y2h ; b ¼ xx xh þ yx yh ; g ¼ x2x þ y2x ; J ¼ xx yh  xh yx

Fig. 5. Grids of the pipeline.

3.2. Calculation of batch transportation of cold and hot crude oil by FVM In the process of direct numerical simulation, Eq. (1) is solved by FVM on body-fitted grids (as shown in Fig. 4). The total number of computational grids is 3761 and the heat conductivities of different physical interfaces are obtained by harmonic mean interpolations [15]. The temperature distributions are obtained by solving the model utilizing Gauss-Seidel iterative method and the iteration is thought to reach convergence when the maximum error of corresponding control points between two adjacent iterative layers is smaller than 109. Fig. 5 shows the grids used in the axial direction of the pipeline. The method of thermal characteristic line is employed to describe the unsteady flow and obtain the oil temperature distribution along the pipeline. The heat transfer between hot oil pipeline and surrounding region is coupled by means of balancing the release and adsorption of heat along the interface in an iterative procedure.

Fig. 6. Computational grids for oil temperature drop after pipeline shutdown.

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3.3. Calculation of oil temperature drop after pipeline shutdown by FVM In the calculation of oil temperature drop after pipeline shutdown, the cross-sections in different position along the pipeline are independent from each other due to oil flow velocity and heat conduction in the axial direction can be neglected. For the sake of simplicity, the present paper only focuses on one specific crosssection along the pipeline and employs BFC-based FVM to calculate the temperature change after pipeline shutdown. The total number of computational grids is 4111 (Fig. 6) and the heat conductivities of different physical interfaces are obtained by harmonic mean interpolations. The iterative method and convergence criteria are the same as those in Section 3.2. 4. POD reduced-order models based on BFC When it comes to the temperature field calculations for batch transportation of cold and hot crude oil and temperature drop after pipeline shutdown, the solution of Eq. (1) and Eq. (7) occupy more than 99.5% of the total computation time. To improve this issue, the present paper proposes the BFC-based POD reducedorder model to speed up solving the equations. The reason utilizing BFC-based POD reduced-order model lays on its advantage that after one sampling operation, the BFC-based POD reducedorder model could predict thermal processes of different pipelines with various combinations of buried depth and pipe diameter very quickly. The establishment and solution of unsteadystate BFC-based POD reduced-order model will be discussed in detail as follows. It is worth noting that the spatial derivative of Jacobi factor on computational domain should be avoided in the process of establishing POD reduced-order model, especially when the heat conductivity coefficients between adjacent control volumes differ significantly from each other. On this occasion, the spatial derivative of basis function should be discretized using special scheme. According to the definition of POD, the temperature field can be written as the combination of a set of basis functions and the corresponding coefficients as Eq. (12) shows where N denotes the total number of basis functions. N   X T x; h z ak fk ðx; hÞ

(12)

k¼1

(13)

k¼1

As mentioned above, though Eq. (10) and Eq. (11) are mathematically equivalent, the reduced-order model which is derived from Eq. (11) is only applicable to uniform grids. On the contrary, the reduced-order model from Eq. (10) can be applied to different body-fitted grids even when the sizes of adjacent grids vary greatly (Ref. [15]). Thus the present paper only introduces the process of establishing reduced-order model from Eq. (10). Based on Eq. (10), the residual R can be defined as follows.

        v rcp T v l v l  R¼J aTx  bTh  gTh  bTx  JS vx J vh J vt

ðR; fi Þ ¼ 0

i ¼ 1; :::; M

(14)

(15)

The symbol ( , ) stands for the Hilbert Inner Product. Synthesize Eq. (13), Eq. (14) and Eq. (15), we can get:

0

1   PM B v rcp k¼1 ak fk C B C   C J ; f 0¼B i C  JS; fi B vt @ A 2

0

B 6 v B v 6al 6 B Bvx 6 J @ 4

 PM

k¼1 ak fk

 

vx

bl J

v

3  a f k¼1 k k 7 7 7 7 vh 5

 PM

2

3 1  PM  PM   a f v a f 6 v C k¼1 k k k¼1 k k 7 7 C v 6gl bl 7; fi C þ 6  7 C vh 6 J vh vx 4J 5 A

(16)

For the unsteady term. M X dðak Þ f ;f Jrcp dt k i

! þ

k¼1

Z rcp

¼

M X k¼1

U

!   M v rcp X ak fk ; fi J vt k¼1

dðak Þ f $f JdU þ dt k i

Z U

  M v rcp X J ak fk $fi dU vt k¼1

Z  X Z X M v rcp M dðak Þ fk $Qi dU þ ak fk $Qi dU ¼ dt vt U

k¼1

k¼1

U

Rearrange it, we can have:

Jrcp

M X dðak Þ fk ; fi dt k¼1

Due to the first M basis functions of POD possess most of the energy, thus the first M basis functions are usually used to achieve the precise construction of original physical field as follows. M   X T x; h z ak fk ðx; hÞ

It should be note that the residual R is generated by the approximation of Eq. (13). Because the employment of basis functions fin reconstructed equations could achieve the maximum approximation based on least square between the predicted physical field and the actual one, the residual is projected onto the space spanned by the first M basis functions and the value of the projection is set as 0, then we have:

where R Htik ¼ fk $Qi dU;

! ¼

M X

Htik

k¼1

(17)

k¼1

Qi ¼ fi rcp J;

U

M dðak Þ X þ Hnik ak dt

Hnik ¼

R U

vðrcp Þ=vtfk $Qi dU

Table 1 Samples of batch transportation of cold and hot crude oil. No.

Oil temperature ( C)

Heat conductivity of pipeline (W/(m2  C))

Atmosphere temperature ( C)

Heat transfer coefficient of atmosphere (W/(m2  C))

Geometry

1 2 3 4 5 6 7 8 9

20 40 60 20 40 60 20 40 60

25 55 90 25 55 90 25 55 90

5 0 15 15 0 5 15 0 15

10 25 25 10 10 25 25 25 25

Geo1 (see Table 2) Geo2 (see Table 2) Geo3 (see Table 2)

D. Han et al. / Applied Thermal Engineering 88 (2015) 217e229

223

For the convective term.

2

0

B 6 B v 6al B 6 Bvx 6 J @ 4

¼

M P

 PM  v k¼1 ak fk vx

bl J

v



3

2

ak fk 7 6 6 7 7 þ v 6gl 7 vh 6 J vh 4 5

k¼1

 PM  v k¼1 ak fk vh

U

Using Green's theorem, we can have:






I




H al vfk bl vfk gl vfk bl vfk  ,4i dx þ  ,fi dh J J J vh J vx vx vh k¼1 
 Z
M P al vfk bl vfk v4i gl vfk bl vfk vfi  , þ  , dU  ak J vx J vh J vh J vx vx vh k¼1 ¼

M P

ak

U

(18) Z ðJS; fi Þ ¼ 

JSfi dU

(19)

U

Substitute Eq. (17), Eq. (18) and Eq. (19) into Eq. (16), we can get: M P

Htik

k¼1

dðak Þ  dt

Z JSfi dU U

 
I
H al vT bl vT gl vT bl vT  $4i dx þ  $fi dh  J vx J vh J vh J vx 
 Z
M P al vfk bl vfk v4i gl vfk bl vfk vfi  $ þ  $ dU ak þ J vx J vh J vh J vx vx vh k¼1 U

(20) According to the definition of heat flux at interface of computational domain, the evolution equation can be written as follows.

0¼

M X da

k

k¼1

I

þ

bl J

v



3

1

ak fk 7 C 7 C 7; fi C 7 C vx 5 A

k¼1

dt

$Htik þ

M X k¼1

  f þ BE =BP1 fE fP þ BW =BP1 fW  fx P ¼ P 1 þ BE =BP1 1 þ BW =BP1

(22)

  f þ BN =BP2 fN fP þ BS =BP2 fS  fh P ¼ P 1 þ BN =BP2 1 þ BS =BP2

(23)

pffiffiffi pffiffiffi pffiffiffi where BW ¼ la=J ajW ; BE ¼ la=J ajE ; BN ¼ lg=J gN ; BS ¼ lg= pffiffiffi pffiffiffi pffiffiffi J g S ; BP1 ¼ la=J ajP ; BP2 ¼ lg=J g P 5. Applications and discussions

For the source term.

0¼



 PM

    v al vfk bl vfk v gl vfk bl vfk  $fi þ  $fi dU vx J vx J vh vh J vh J vx

Z ak

k¼1



 PM

ak $Hnik þ

M X k¼1

pffiffiffi x pffiffiffi aq fi dx þ gqh fi dh þ Fi

ak $Hnik (21)

pffiffiffi pffiffiffi where Hnik ¼ ðvfi =vx; l avfk =vnx Þ þ ðvfi =vh; l avfk =vnh Þ; Fi ¼ ðS; Qi Þ When solving the reduced-order model, the detailed treatments of initial conditions and boundary conditions can be found in Ref. [13]. It is worth noting that because the spatial derivative of basis functions stands for the heat flux, the central difference scheme is no longer applicable when the heat conductivity coefficients of different parts vary largely (to be specific, the heat conductivities of wax layer, steel wall, insulation layer and surrounding soil are 0.15W/m2  C, 50W/m2  C, 0.15 W/m2  C and 0.85 W/m2  C respectively). To improve this point, the present paper applies the discrete scheme employing the concept of harmonic mean [15]. For point P.

As mentioned above, the computing cost of unsteady-state thermal calculation of buried hot oil pipeline is usually quite high. To overcome this shortcoming, the present paper employs POD reduced-order model to achieve the efficient calculation. Typically, the implementation of POD reduced-order model consists of three parts: sampling, calculation of basis functions and rapid prediction. In this section, the procedures of these three parts will be discussed in detail and the accuracy and speed advantage of reduced-order model will also be assessed through test cases. As the accuracy of FVM has been testified by the field data [2], accuracy of POD ROM is testified by comparing with the results of FVM. 5.1. Batchtransportation of cold and hot crude oil In the design and optimization of operation scheme of batch transportation of cold and hot crude oil, the thermal characteristics of pipeline under different buried depth, pipe diameter, pipeline inlet oil temperature, throughput, alternate frequency of cold and hot oil as well as environment temperature need to be understood clearly first. To achieve this goal, lots of operation conditions should be calculated but direct numerical simulation is huge timeconsuming, thus POD reduced-order model is used to solve this problem. The first and essential step of POD reduced-order model is sampling which aims to obtain the temperature field information of different pipeline cross-section (values of involved parameters are shown in Table 1 and Table 2). Setting the time step as 1200 s, different samples are calculated from the initial state to the steady state. At the beginning when the temperature field changes dramatically, it needs to increase the density of sampling. When the variables change slowly, sampling density should be reduced to some extent. What should be noted is that the calculation of the basis function can be time consuming. However, the basis function calculation is a one-off effort. Once the basis function is obtained, it can be employed to the fast thermal calculation of the oil pipeline with different lengths, diameters and buried depths as well as in different conditions. This is also true for the basis calculation of temperature drop after pipeline shutdown in Section 5.2.

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Table 2 Geometric parameters of samples in Table 1. Buried Pipe Thickness of Thickness of Thickness depth (m) diameter (m) steel wall (m) wax layer (m) of corrosion protection layer (m) Geo1 1.5 Geo2 1 Geo3 1.3

0.8 0.5 0.6

0.007 0.012 0.09

0.022

0.004

The second step is calculating the basis functions. After obtaining the sample matrix, ‘snapshot’ method is used to calculate the basis functions and from the accumulative energy distribution of POD bases, it can be concluded that the first eight basis functions possess 99.96% of the total energy. Due to the influence caused by geometric parameters change, some basis functions which have low energy contribution contain very important local information of the physical field. A problem caused by this is what's the proper number of basis to be used and is there a principle to determine it? So far it is not clear but a good direction to study on. A simple way is to reconstruct the base data with different numbers of basis function and evaluate the deviations, and then obtain the proper number of basis function to be used. For this particular problem, the first twenty-six basis functions are employed in the practical calculation of POD reduced-order model in order to gain the high precision temperature field of the pipeline cross-section. Then the basis functions are used to predict the temperature field of batch transportation of cold and hot crude oil. In the specific operation condition, there are two pipelines, Pipe1 and Pipe2, laying parallel to each other with large enough separation distance and both of the two pipelines have reached the thermal steady state. The buried depth, diameter and thickness of steel wall of Pipe1 and Pipe2 differ greatly from each other and other operating parameters are listed in Table 3. Due to the change of engineering practice, one of the two pipelines needs to employ the batch transportation of cold and hot oil. The calculations are carried out for two operation schemes (as shown in Table 4). In Scheme 1,Pipe1 employs the batch transportation of two kinds of oil whose temperature is 30  C and 60  C respectively with the alternate time of 10 h and flow velocity of 1 m/s. When the atmosphere temperature is 20  C, the Pipe1 changes its operation scheme. In Scheme 2, Pipe2 employs the batch transportation of two kinds of oil whose temperature is 20  C and 50  C respectively with the alternate time of 20 h and flow velocity of 2 m/s. When the atmosphere temperature is 20  C, the Pipe2 changes its operation scheme. Utilizing FVM and POD reduced-order model to calculate the above two operation schemes, the time step is set to be 1200 s. The

Fig. 7. Oil temperature distribution along the pipeline.

computational grid, iterative method and convergence criteria are all the same as mentioned in previous sections. In Fig. 7, the curves of oil temperature distribution along the pipeline at some typical periods after pipeline commissioning are given. In Scheme 1, it needs about 30 h to completely replace the

Table 3 Parameters of two parallel pipelines. Name

Distance (km)

Buried depth (m)

Pipe diameter (m)

Thickness of steel wall (m)

Pipeline inlet oil temperature ( C)

Atmosphere temperature ( C)

Pipe1 Pipe2

108 108

1.3 0.9

0.711 0.457

0.010 0.006

50 30

(Eq. (2)) b ¼ 20, a ¼ 0.4 (Eq. (2)) b ¼ 20, a ¼ 0.4

Table 4 Schemes of batch transportation of cold and hot crude oil. Pipeline name

Oil temperature

Flow velocity

Alternate frequency

Atmosphere temperature

Scheme 1

Pipe1

(Fig. 4) TH ¼ 60  C,TL ¼ 30  C

1 m/s

(Fig. 4) h ¼ 10,l ¼ 10

Scheme 2

Pipe2

(Fig. 4) TH ¼ 50  C,TL ¼ 20  C

2 m/s

(Fig. 5) h ¼ 20,l ¼ 20

(Eq. (2)) b ¼ 20, a ¼ 0.4 (Eq. (2)) b ¼ 20, a ¼ 0.4

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original oil in pipeline and at that time, the two alternation of different oil are finished to have the temperature distribution curve shown in Fig. 7(a). In Scheme 2, the time cost of oil replacement is only 15 h and there will be the temperature distribution curve as shown in Fig. 7(b). It is easy to conclude that the calculation results of POD reduced-order model reflect the thermal characteristics of both Scheme 1 and Scheme 2 clearly and meet well with the results obtained by FVM. Generally speaking, we mainly focus on the pipeline outlet oil temperature of pipeline. Fig. 8 gives the changing curves of pipeline outlet oil temperature of Pipe1 and Pipe2 in a period of 30 days. From the figure, we can see that the pipeline outlet oil temperatures of Scheme 1 and Scheme 2 present a periodic variation pattern and the period is 10 h and 20 h respectively which are in accord with the changing pattern of pipeline outlet oil temperature. In Scheme 1, the pipeline outlet oil temperature shows a tendency of decreasing first then increasing while, in Scheme 2, the pipeline outlet oil temperature maintains the increasing trend all the time. This is because in Scheme 1 the initial temperature field is the steady-state temperature field when the pipeline inlet oil temperature is 50  C. In addition, the cold oil temperature is 20  C and the hot oil temperature is 60  C under the batch transportation condition. At the beginning, the temperature difference between

the environment and cold oil is larger than that between the environment and hot oil which would give rise to the heat release from the hot oil to surrounding soil which is smaller compared with that from cold oil. Along with the drop of atmosphere temperature, the pipeline outlet oil temperature in different period decreases continuously. Followed by the rising atmosphere temperature which increases from initial 20  C to the final 32  C, the ambient temperature around oil pipeline also increases at the same time resulting in the increasing of pipeline outlet oil temperature. In Scheme 2, the initial temperature field is the steady-state temperature field when the pipeline inlet oil temperature is 50  C with the same cold oil temperature and hot oil temperature as in Scheme 1. In the same period, the surrounding environment absorbs more heat from the hot oil pipeline than its heat release to the cold oil pipeline and that results in the rising temperature of environment. Though the pipeline outlet oil temperature is also increasing meanwhile, the increasing rate is decreasing thus the drop of atmosphere temperature is not the main influential factor during the calculation period of 30 days. As seen from Fig. 8, the calculation results of FVM and POD clearly reflect the thermal characteristics of batch transportation discussed in the above paragraph and in accord well with each

Fig. 8. Pipeline outlet oil temperature change over time.

Fig. 9. Calculation error of reduced-order model.

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Fig. 10. Temperature fields of pipeline outlet cross-section in Scheme 1.

Fig. 11. Temperature fields of pipeline outlet cross-section in Scheme 2.

other. To further illustrate the calculation accuracy of POD method, the absolute calculation errors of reduced-order model are presented in Fig. 9(a) and Fig. 9(b). It is found that the biggest calculation error is no greater than 0.85  C and the average errors are 0.35  C and 0.25  C respectively which are acceptable in the petroleum industry. In order to illustrate the calculation accuracy of POD method more visually, Fig. 10 and Fig. 11 show the temperature fields of pipeline outlet cross-section in Scheme 1 and Scheme 2 after running for 5 days and 30 days and they are shown in physical domain (Cartesian coordinate). It can be concluded that for different pipeline with various buried depth and diameter, the POD method could solve Eq. (10) successfully and predict the temperature field accurately. With the addition of solving Eq. (4), the temperature distribution along the entire pipeline could also be calculated efficiently. In Scheme 1 and Scheme 2, the pipelines are under different buried depth, diameter, pipeline inlet oil temperature, throughput, alternate frequency of cold and hot oil as well as environment temperature. POD reduced-order model is proved to be capable of obtaining the thermal characteristics of pipeline in batch transportation of cold and hot oil with the various operation conditions accurately.

Table 5 Calculating speed comparison between FVM and POD.

Scheme 1 Scheme 2

Table 6 Parameters of samples. Atmosphere Buried Oil Thickness Thickness of Pipe wax layer (m) diameter depth temperature of steel wall temperature ( C) ( C) (m) (m) (m) 1 0.055 2 3 4

0.529 0.529 0.72 0.72

1.2 1.5 1.2 1.5

60 35 60 35

0.006 0.011 0.006 0.011

30 10 15.0 0.0

In order to demonstrate the speed advantage of POD reducedorder model over FVM, the present paper shows the calculating speed of these two different methods in the form of CPU time (as shown in Table 5). All the tests were performed on a 3.50-GHz computer. From the Table 5, it can be concluded that computation time of FVM and POD in calculating Scheme 1 are 3900 s and 14.8 s respectively and in calculating Scheme 2 the respective time consumption are 2220 s and 8.7 s where the calculating speed of POD is 260 times faster than that of FVM. It needs to be noted that the less time consumption of Scheme 2 is caused by the high flow velocity in Scheme 2, that is to say, the number of pipeline cross-section needs to be calculated is smaller than that of Scheme 1. Furthermore, if there are more than 200 practical operation conditions to Table 7 Parameters of cases.

FVM (s)

POD(s)

Ratio

3900 2220

14.8 8.7

263 255

Case1 Case2

Buried depth (m)

Pipe diameter (m)

Oil temperature ( C)

1.5 1

0.8 0.426

43 35

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Fig. 12. Comparison of the calculation results between the POD ROM and FVM in Case1.

Fig. 14. Comparison of the calculation results between the POD ROM and FVM in Case2.

be calculated, employing FVM will cost about 7 days which is not acceptable in engineering practice, while the needed computation time of POD is only 40 min which will bring great convenience to practical design and optimization without any doubt.

5.2. Temperature drop after pipeline shutdown In the flow assurance evaluation of hot oil pipeline, the restart evaluation after pipeline shutdown holds great significance.

Fig. 13. Temperature fields of winter shutdown in Case1.

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Fig. 15. Temperature fields of summer shutdown in Case2.

Considering the uncertainty of involved parameters, hundreds of thousands of operation conditions need to be calculated to study the temperature drop characteristic. On this occasion, it is obvious that FVM doesn't meet the calculation requirement thus the present section will introduce how to use the POD reduced-order model to achieve this goal. Sampling is first carried out and the relevant parameters are listed in Table 6. The wax precipitation point is 30  C. Utilize BFCbased FVM to calculate the cross-section temperature fields of the four operation conditions listed in Table 6 and set the time step as 1200 s. The calculation period is 200 h which is the maximum shutdown time in the engineering practice. Put the temperature field of every cross-section into the sample matrix and then use ‘snapshot’ method to obtain the basis functions. With the basis functions, two cases are designed to verify the calculation efficiency of POD as presented in Table 7. In these two cases, the temperature drop curves of crude oil after pipeline shutdown in the seasons of summer (Eq. (2), a ¼ 0.4,b ¼ 20), spring (Eq. (2), a ¼ 0.4,b ¼ 0) and winter (Eq. (2), a ¼ 0.4,b ¼ 20) are calculated by FVM and POD reduced-order model respectively. As stated above, though the first six basis functions occupy most of the total energy, the POD reduced-order model still uses the first thirty basis functions to gain the accurate prediction of temperature field. Fig. 12(a) gives the oil temperature drop curves in summer, spring and winter. It is found that the calculation results of POD reduced-order model are in accord with the results obtained by FVM very well. The oil temperature drops fastest in winter while slowest in summer and after 193 h, 71 h and 43 h in summer, spring and winter respectively, the gelatinization extends to all the crude oil in pipeline. It can be concluded from these results that the atmosphere temperature has great influence on the oil temperature dropping rate thus the higher the atmosphere temperature is, the slower oil temperature drops and the longer the secure shutdown time is. The calculation error curves of POD reduced-order model are shown in Fig. 12(b). The biggest calculation error is less than 0.84  C and the average error is no greater than 0.55  C which could completely satisfy the requirements of engineering practice. In order to further illustrate the calculation accuracy of POD reducedorder model, the temperature fields of pipeline cross-section after shutdown 50 h and 200 h in winter are presented in Fig. 13 (Cartesian coordinate) from which we could see the great accordance between the calculation results of POD and FVM. Fig. 14(a) presents the oil temperature drop curves of Case2 in different seasons. After shutdown 19 h, 7 h and 5 h, the oil in pipeline are all gelatinous. The calculation error curves of POD reduced-order model are shown in Fig. 14(b). The biggest calculation error is less than 0.68  C and the average error is less than 0.20  C. The temperature fields of pipeline cross-section in summer

are shown in Fig. 15(Cartesian coordinate). From the Table 7, it can be concluded that under different operation conditions, POD reduced-order model could calculate the temperature fields in different seasons accurately and efficiently. From Table 8, it can be seen that average computation time of FVM and POD in calculating the temperature field of pipeline crosssection after shutdown 200 h are 438 s and 2.5 s respectively, that is to say, the calculation speed of POD reduced-order model is about 170 times faster than that of FVM. In practical flow assurance evaluation, if there are about 10,000 operation conditions to be calculated, employing FVM will cost about 50.7 days which is not applicable in engineering practice, while the needed computation time of POD is only 7 h which has high value of engineering application. To sum up, in calculating the batch transportation of cold and hot oil and temperature drop after hot oil pipeline shutdown, the POD reduced-order model presents desirable accuracy and efficiency, and has great application potential in future engineering practice. 6. Conclusions (1) The present paper proposes the BFC-based unsteady-state heat conduction POD reduced-order model and in the process of calculating basis functions, the newly scheme which reflects the physical meaning better is used to discretize the spatial derivative of basis functions instead using central difference scheme. With one operation of sampling, the unsteady-state thermal characteristics of different pipelines can be predicted all at once. (2) Through applying the newly proposed method to the fast calculation of hot oil pipeline, the research results showed that the POD reduced-order model enjoys satisfying accuracy and can meet the requirements of engineering practice. And in the calculations of two typical operation conditions of crude oil pipeline, the calculation speed of POD reducedorder model is 259 times and 177 times faster than that of FVM respectively. Table 8 Calculating speed comparison between FVM and POD reduced-order model.

Test1

Test2

Season

FVM(s)

POD(s)

Ratio

Summer Spring Winter Summer Spring Winter

428.4 412.4 452.8 443.5 428.1 465.2

2.5 2.4 2.7 2.5 2.3 2.8

171 171 168 177 186 186

D. Han et al. / Applied Thermal Engineering 88 (2015) 217e229

Acknowledgements The study is supported by National Science Foundation of China (No.51325603, No. 51134006 and No. 51176204). The study is also supported by Science Foundation of China University of Petroleum, Beijing (No. 2462013YXBS009).

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