Thermal field in water pipe cooling concrete hydrostructures simulated with singular boundary method

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1 Water Science and Engineering 017, 10(): 107e114 HOSTED BY Available online at Water Science and Engineering journal homepage: Thermal field in water pipe cooling concrete hydrostructures simulated with singular boundary method Yong-xing Hong a,b, Wen Chen a,b, Ji Lin a,b, *, Jian Gong a,b, Hong-da Cheng c a State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 1009, China b College of Mechanics and Materials, Hohai University, Nanjing 11100, China c School of Engineering, University of Mississippi, Mississippi 3677, USA Received 14 November 016; accepted 13 February 017 Available online 3 June 017 Abstract The embedded water pipe system is often used as a standard cooling technique during the construction of large-scale mass concrete hydrostructures. The prediction of the temperature distribution considering the cooling effects of embedded pipes plays an essential role in the design of the structure and its cooling system. In this study, the singular boundary method, a semi-analytical meshless technique, was employed to analyze the temperature distribution. A numerical algorithm solved the transient temperature field with consideration of the effects of cooling pipe specification, isolation of heat of hydration, and ambient temperature. Numerical results are verified through comparison with those of the finite element method, demonstrating that the proposed approach is accurate in the simulation of the thermal field in concrete structures with a water cooling pipe. 017 Hohai University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Thermal field; Singular boundary method; Semi-analytical method; Water cooling pipe; Concrete hydrostructure 1. Introduction Concrete hydrostructures such as dams, foundations, and pumping stations suffer from crack problems due to the hydration heat, especially at the early stages of concrete solidification (Kogan, 190). A water pipe cooling system is considered an efficient technology for cooling the interior temperature in mass concrete and consequently mitigating thermal stress cracks and resultant structural weakness (Qiang et al., 015). As stated in Hauser et al. (000), the flowing water along the pipe not only absorbs the heat from the concrete, but also upgrades the thermal storage capacity and decreases hydration heat (Kim et al., 001), furnishing a strategy for better thermal removal. This work was supported by the National Natural Science Foundation of China (Grants No and ) and the 111 Project (Grant No. B11). * Corresponding author. address: linji61103@16.com (Ji Lin). Peer review under responsibility of Hohai University. Since the first successful application of the cooling pipe system in the Hoover Dam (Kwak et al., 014), the prediction of the temperature history and distribution in massive concrete structures with a pipe cooling system has attracted much attention from engineering and science communities. Several major factors affecting cracks, such as the temperature (Ding and Chen, 013), the thermal gradient (Sato et al., 005), the structure restraint, and the material thermal stability (Zhang et al., 004), have been taken into account. The equivalent equation of heat conduction in a concrete structure with a cooling pipe, proposed by Zhu (1991) without consideration of the thermal gradient, describes the average temperature: vt ¼ av T þðt 0 T w Þ vf vt þ vq vt where T is the temperature of concrete, T 0 is the initial temperature of concrete close to the water pipe, T w is the cooling water temperature, f is a given function associated with the ð1þ / 017 Hohai University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( creativecommons.org/licenses/by-nc-nd/4.0/).

2 10 Yong-xing Hong et al. / Water Science and Engineering 017, 10(): 107e114 heat flux from concrete to water, a is the thermal diffusivity of concrete, and q is the adiabatic temperature raise of concrete at a certain time instance t. Three boundary conditions for measuring the thermal gradient on the boundary (Zhu, 1999) are as follows: T ¼ TðtÞ h vn ¼ bðt w TÞ ð3þ vn ¼ 0 ð4þ where TðtÞ is a prescribed temperature process, h represents the coefficient of heat convection, n is the unit outward normal vector on the boundary of the computational domain, and b is a rational number. This system of equations can also be well utilized in the formulation of casting processes of mass concrete containing double-layer staggered heterogeneous cooling pipes, as reported in Yang et al. (01). As mentioned above, the equivalent equation of heat conduction in the concrete-pipe system has been studied using the finite element method (FEM) (Chen et al., 011). In the FEM, the structures are discretized into small elements, and a final system of equations is made up of approximations in each subelement, which can be arduous, time consuming, and computationally expensive, especially due to the fact that a considerable number of elements are needed to model the small pipes of a cooling system (Sasaki et al., 014). In order to tackle this bottleneck, one alternative method is to neglect the practical sectional shape and size of cooling pipes for simulations (Liu et al., 015). Although it can be used to approximate the equivalent temperature of the target structure efficiently, this method still lacks the capacity to model the thermal field surrounding the inner water pipes. Instead of the FEM, the singular boundary method (SBM) was introduced in this study to solve the heat conduction problem with a water pipe. The SBM is a newly developed meshless method proposed by Chen (009) for the simulation of boundary value problems (Chen and Gu, 01). This method falls into the category of the boundary-type method with integration-free attributes, which can also be considered one type of ameliorative algorithm of the method of fundamental solutions (MFS) (Sarler, 009). The SBM uses the fundamental solutions of the governing equations as the basis functions. To avoid the singularity of the fundamental solutions, the origin intensity factors are introduced (Li et al., 016; Wei et al., 015). Therefore, the SBM is a truly semianalytical boundary-type meshless method. The SBM has been widely used to deal with many engineering problems, such as wave propagation problems (Lin et al., 014), steady-state heat conduction problems (Wei et al., 016), and time-dependent problems (Wang and Chen, 016). Moreover, the SBM, combined with the dual reciprocity method and inverse interpolation, is effective in solving non-homogeneous problems (Chen et al., 014). For the problems considered in this study, only the information on the surface of the structure ðþ and pipe was required. Due to the use of the fundamental solution, the SBM is a semi-analytical technique and has great potential for the simulation of time-dependent problems. This paper is organized as follows: in Section, the mathematical formulation of the SBM for the thermal field in a concrete structure with a water pipe cooling system is introduced; in Section 3, three benchmark examples are examined to show the effectiveness of the presented method; and, finally, some conclusions and remarks are provided in Section 4.. Numerical formulation of pipe water cooling system In order to apply the SBM, the time-dependent problems can be transformed into a system of steady-state problems using the time difference method. In this study, the implicit Euler scheme was employed to discretize the time derivatives in Eq. (1) and transform the considered problems into a system of Helmholtz equations. The SBM could then be used to carry out the spatial discretization in the domain U of interest..1. Time discretization To begin with, the time interval ½0; tš is divided equally into M sub-intervals. Then, the time step is dt ¼ t=m, and t n ¼ ndt, where n ¼ 0; 1; /; M. The implicit Euler scheme is used to discretize Eq. (1) as follows: V 1 adt T nþ1 ðxþ¼ 1 adt T nðxþ 1 a ðt 0 T w Þ vf nþ1 þ vq nþ1 vt vt where T n is the temperature of concrete at time t n, f nþ1 is f at time t nþ1, q nþ1 is q at time t nþ1, and x ¼ðx; yþ for a twodimensional problem. Using the notation 1=ðadtÞ ¼m,we come to the following system of Helmholtz equations, which can be solved by the proposed SBM: V m T nþ1 ðxþ¼g nþ1 ðxþ ð6þ where g nþ1 ðxþ ¼m T n ðxþðt 0 T w Þvf nþ1 =ðavtþvq nþ1 =ðavtþ... Singular boundary method At first, using the dual reciprocity method (Chen and Gu, 01), the solution to the non-homogeneous Helmholtz equation (Eq. (6)) can be approximated by the summation of T p nþ1 ðxþ and Th nþ1ðxþ, as follows: T nþ1 ðxþ¼t p nþ1 ðxþþt h nþ1 ðxþ where T p nþ1 ðxþ represents the particular solution and Th nþ1 ðxþ represents the homogeneous solution. The particular solution T p nþ1ðxþ satisfies the non-homogeneous equation (Eq. ()), but does not necessarily satisfy the boundary conditions, and Tnþ1 h ðxþ satisfies the homogeneous equation as follows: ð5þ ð7þ

3 Yong-xing Hong et al. / Water Science and Engineering 017, 10(): 107e V m T p nþ1 ðxþ¼g nþ1ðxþ V m T h nþ1 ðxþ¼0 ð9þ Using the dual reciprocity method, the non-homogeneous term g nþ1 ðx i Þ in Eq. () at the node x i in the domain U of interest can be expanded by the chosen radial basis function: g nþ1 ðx i Þ¼ XN a j 4 ij x i U ð10þ where 4 ij ¼ 4ð xi x j Þ, with 4 being the chosen radial basis function; a j ðj ¼ 1; ; /; NÞ is the unknown coefficient to be determined; and N denotes the number of collocation nodes in the domain of interest. By collocating Eq. (10) at all collocation nodes, we come to the following equations: g nþ1 ¼ 4a ðþ ð11þ where g nþ1 ¼½g nþ1 ðx 1 Þ; g nþ1 ðx Þ; /; g nþ1 ðx N ÞŠ T, 4¼½4 ij Š NN, and a¼½a 1 ; a ; /; a N Š T. The unknown coefficients can be easily determined from Eq. (11): a ¼ 4 1 g nþ1 ð1þ In the end, the particular solution in Eq. () can be obtained: Tnþ1ðx p i Þ¼ XN a j J ij x i U ð13þ where J ij is determined by 4 ij, and the following relationship always holds: V m J ij ¼ 4 ij ð14þ Eq. (13) can be written as T p nþ1 ¼ Ja ð15þ where T p nþ1 ¼½Tp nþ1 ðx 1Þ; T p nþ1 ðx Þ; /; T p nþ1 ðx NÞŠ T, and J ¼½J ij Š NN. Through substitution of a into Eq. (15), T p nþ1 can take the following form: T p nþ1 ¼ J4 1 g nþ1 The heat flux can be approximated by p nþ1 >< J ij ¼ >: ð16þ vn ¼ vj vn 41 g nþ1 ð17þ Using the notation r ij ¼ xi x j, J ij and 4 ij were defined as follows in this study: 4 m 4lnr ij r ij ln r ij 4K 0mr ij r 4 m 4 m m 4 ij > 0 ð1þ 4 m 4 þ 4c ε m 4 þ 4 m 4 ln m r ij ¼ 0 4 ij ¼ r ij ln r ij ð19þ where c ε is the Euler constant, and c ε z0:577; and K 0 is the zero-order modified Bessel function of the second kind. Once we obtain a particular solution T p nþ1, the homogenous solution Tnþ1 h ðx mþ can be approximated by a linear combination of fundamental solutions G mj as follows: T h nþ1 ðx mþ¼ >< >: X N b X N b jsm g j G mj x m sx j g j G mj þ g m U m x m ¼ x j ð0þ where G mj ¼ K 0 mr mj =ðpþ for two-dimensional problems; N b is the number of boundary nodes for approximation; g j ðj ¼ 1; ; /; N b Þ is the unknown coefficient to be determined; and U m ðm ¼ 1; ; /; N b Þ is the origin intensity factor, which is proposed to avoid the singularities of the fundamental solutions when the node x j approaches the node x m. The origin intensity factors used in this study are obtained using the inverse interpolation technique (Chen et al., 014):! T m XN a j G 0mj T I XN a j G 0Ij U m ¼ jsm a m 1 p ln m c ε p ð1þ where G 0mj ¼ln ½r mj =ðpþš is the fundamental solution of the two-dimensional Laplace operator, T m ¼ x m þ y m þ 1isa two-dimensional sample solution that satisfies the Laplace equations, and T I ¼ x I þ y I þ 1. It is noted that the node x I for approximating the origin intensity factor in the SBM is chosen randomly, and we chose x I ¼ð1:1; :7Þ in this study. Also, for the heat flux, we have nþ1 h ðx mþ ¼ vn m Q m ¼ XN jsm >< >: X N b X N b d j vg 0mj vn j jsm g j vg mj vn m x m sx j g j vg mj vn m þ g m Q m x m ¼ x j ðþ ð3þ where n j denotes the unit outward normal vector of node x j, and d j is a parameter that denotes the average arc length of two nodes linked to node x j. More details about the SBM can be found in Chen and Gu (01). By forcing Eqs. (0) and () to satisfy the boundary conditions, g j ðj ¼ 1; ; /; N b Þ can be obtained. After we obtain Tnþ1 h, the solution to the problem can be obtained by the combination of Tnþ1 h and Tp nþ1 as stated in Eq. (7). 3. Numerical results and discussion We neglected the influence of cooling water in the first two examples described in this section, to illustrate the validity of

4 110 Yong-xing Hong et al. / Water Science and Engineering 017, 10(): 107e114 the proposed method. A simplified cross-section of mass concrete with a cooling pipe was studied using the proposed method considering boundary conditions (Eqs. () and (3)). The accuracy of numerical results was validated with the following average relative error e rel : e rel ¼ " 1 n t X nt k¼1 # 1= Tnumk T anak T anak ð4þ where n t represents the number of test nodes, and T numk and T anak denote the numerical approximation and analytical solution of test node x k, respectively Example 1 In the first example, a classical two-dimensional heat conduction problem without water pipes was examined to validate the described method. This examination was carried out on a square domain, U ¼ fðx; yþj0 < x; y < 3mg. In the SBM, 0 boundary nodes were used to model the concrete system, and 361 internal nodes were chosen for approximation of the particular solutions. All of these collocation nodes were distributed evenly and the distance between every two adjacent nodes was 0.15 m. The governing equation of the general heat conduction problem is as follows: vt ¼ av T þ vq ð5þ vt with the following initial and Dirichlet boundary conditions: indicating the accuracy and stability of the proposed method. Fig. shows the relationship between the average relative error e rel and the elapsed time t, which indicates that the proposed method can continue to provide reasonable results. Fig. 3 presents the relationship between the average relative error and the number of boundary nodes. The average relative error decreases quickly with the increasing number of boundary nodes. When the number of boundary nodes is greater than 100, the average relative error remains near 10. We should note here that better accuracy can be obtained for a larger number of internal nodes. All these results demonstrate that the proposed SBM is feasible for solving heat conduction problems without water cooling pipes. 3.. Example In the second example, a heat conduction problem in a cubic concrete structure with a tiny pipe in the center of the domain was examined to validate the described method. The interest domain is U ¼ fðx; y; zþj0 < x; y; z < 1mg, where the diameter of concrete model D is 1 m, and the diameter of the tiny pipe d is m. We used a total number of 157 Tðx; y; 0Þ¼30 C ðx; yþu ð6þ Tðx; y; tþ¼0 C ðx; yþg ð7þ where Tðx; y; tþ is the thermal function to be determined, and G is the simple closed curve bounding the domain U of interest. In this case, we considered a ¼ 0. m /s. It was assumed that q ¼ 0 and the boundary conditions were given directly. The compared analytical solution taken from Bruch and Zyvoloski (1974) was as follows: Tðx;y;tÞ¼ X X mpx npy B mn sin sin 3 3 m¼1 n¼1 1:5p ðm þ n Þt exp where 3 ðþ B mn ¼ 4 30 ½ð1Þm 1Š½ð1Þ n 1Š ð9þ mnp Numerical results shown in Fig. 1 through Fig. 3 were obtained using dt ¼ 0.00 s. Fig. 1 presents the comparison between numerical solutions and analytical solutions for three randomly chosen test nodes, node 1 (.4, 1.5), node (.4,.4), and node 3 (1.5, 1.5), and their corresponding relative errors. From this figure, we can see that the results obtained from the SBM are consistent with the analytical solutions, Fig. 1. Comparison of numerical and exact solutions for three test nodes and corresponding relative errors.

5 Yong-xing Hong et al. / Water Science and Engineering 017, 10(): 107e Tðx; y; z; tþ¼ 100 3p sinðpxþsinðpyþsinðpzþ 1 exp 3ap t ð33þ For this three-dimensional problem, J ij, 4 ij, G mj, and G 0mj are as follows (Chen et al., 014): >< r ij m þ exp mrij r m 4 r ij m 4 ij > 0 r ij J ij ¼ >: ð34þ r m 3 ij ¼ 0 Fig.. Average relative error versus elapsed time. boundary nodes for the singular boundary method, with 30 nodes on the water pipe boundary, and 3375 internal nodes for approximation of the particular solution. All the collocation nodes were distributed evenly at each physical position. Without consideration of the cooling influence of the water pipe, the governing equation of the general heat conduction equation can be written as vt ¼ av T þ vq vt ð30þ Under the following initial and Dirichlet boundary conditions: Tðx; y; z; 0Þ¼0 ðx; y; zþu ð31þ Tðx; y; z; tþ¼0 ðx; y; zþg ð3þ where Tðx; y; z; tþ is an unknown thermal function. In this case, it was assumed that a ¼ 0.16 m /s, and the adiabatic temperature raise of the target concrete was q ¼ 100 sinðpxþsinðpyþsinðpzþt. The compared analytical solution is 4 ij ¼ r ij ð35þ G mj ¼ cosh m xm x j 4p xm x j ð36þ G 0mj ¼ 1 p ln xm x j ð37þ Numerical results in Figs. 4 and 5 were obtained using time step dt ¼ 0.1 s. Fig. 4 illustrates the comparisons between numerical solutions and analytical solutions for three random test lines, y ¼ z ¼ 0.5 m (line 1), y ¼ z ¼ 0. m (line ), and y ¼ z ¼ 0.7 m (line 3). It can be seen that the predictions of the proposed method agree with the analytical solutions. Fig. 5 presents the relative errors of the three test lines, showing the accuracy of the proposed method. In addition, it can be seen from Fig. 6 that, as time goes by, the average relative error for this case is less than 10, which is acceptable in practical engineering. The results demonstrate the validity of the proposed method for three-dimensional heat conduction problems without consideration of the cooling influence of water flow Example 3 After the numerical verifications made in the two examples mentioned above, we applied our algorithm in the simulation of a thermal field in a concrete structure with a water pipe cooling system. The simulation was conducted in a square domain with a cross-section, U ¼ fðx; yþj0 < x; y < 3 mg. Fig. 3. Average relative error versus number of boundary nodes. Fig. 4. Comparison of numerical approximations and analytical solutions for three test lines at t ¼ 1s.

6 11 Yong-xing Hong et al. / Water Science and Engineering 017, 10(): 107e114 collocation nodes were distributed evenly at each physical position. The initial condition, T 0 ðxþ¼30 C xu G 1 G t ¼ 0 ð40þ was considered along with the following boundary conditions: >< T a þðt 0 T a Þexpð0:16tÞ xg 1 Tðx; tþ¼ >: T w þ ðt 0 T w Þ 3ðT 0 t þ 1Þ þ 1 3 ðt 0 T w Þexpð0:5tÞ xg ð41þ Fig. 5. Relative errors for three test lines at t ¼ 1s. Considering Eq. (1) the equivalent heat conduction equation, and based on Zhu (1991), we have f ¼ expðk 1 z s Þ ð3þ where k 1 ¼ :0 1:174x þ 0:56x, with x ¼ ll=ðc w q w r w Þ, where l is the heat flux coefficient, L is the length of concrete, c w is the specific heat capacity of water, q w is the flux of cooling water, and r w is the water density; s ¼ 0:971 þ 0:145x 0:0445x ; and z is a coefficient, and can be represented as z ¼ a 0 t=d, with a 0 ¼ aln½100=lnðd=dþš. In order to simplify the problem, it is assumed that D ¼ 3m, d ¼ 0.05 m, a ¼ 0.1 m /d, k 1 z, and sz1. Then, we have fzexpð0:04tþ, and q ¼ 3½1 expð0:7tþš. Substituting the coefficients and functions above into Eq. (1), we have vt ¼ 0:1V T 0:04ðT 0 T w Þexpð0:04tÞþ 10:6expð0:7tÞ ð39þ In this study the number of internal nodes was considered to be 360; the number of boundary nodes was considered to be 31, with 300 nodes on the boundary of the square (G 1 ) and 1 nodes on the boundary of the tiny pipe (G ); and all of the where T a denotes the temperature of the air surrounding the concrete. By considering T a ¼ T w ¼ 15 C, and dt ¼ 0:05 d, several numerical results were obtained. Fig. 7 presents the comparisons between approximations of the thermal field obtained from the SBM and the results obtained from the FEM for t ¼ 0.1d,1d,d,4d,1d,and3d.Fromthis figure, it can be observed that the results obtained from the proposed SBM are almost the same as the results obtained from the FEM. All numerical results obtained from the SBM and FEM verify that the temperature near the cooling pipe decreases swiftly, while temperature in the middle of the pipe and physical boundary increases first due to the rising of adiabatic temperature, and then decreases slowly. Fig. 7 also displays the distribution of thermal fields inside the interest domain with respect to the elapsed time t. As shown in this figure, the maximum temperature gradient appears around the cooling pipe and the SBM can provide a detailed history of the temperature and thermal gradient at the early stages of the concrete construction, which is consistent with the facts. These results demonstrate that the SBM can achieve as accurate results as the FEM, which can be considered an alternative tool to simulating the heat conduction problems in a concrete structure with a cooling pipe. After the accuracy of the SBM was verified, three special test points were taken into account to determine more details of the thermal field: point 1 (1.54, 1.54) near the cooling pipe, point (0.03, 0.03) near the boundary of the concrete, and point 3 (1.01, 1.01) between the pipe and the boundary of concrete. For comparison, the water pipe cooling system was dislodged from the structure, considering the governing equation as shown in Eq. (5) and the following boundary condition: Tðx; tþ¼t a þðt 0 T a Þexpð0:0tÞ xg 1 ð4þ Fig. 6. Average relative error versus elapsed time. with the same initial conditions, adiabatic temperature raise, and coefficients as the case containing a water pipe cooling system. The comparison of the variations of numerical temperature with elapsed time t, at the three test points, between concrete structures with a cooling pipe and without cooling pipes, is shown in Fig.. It can be seen from Fig. that the temperature at point 1 in the structure including a cooling pipe swiftly decreases as time elapses. In contrast, the temperature at point 1 in the structure without cooling pipes declines very slowly, and even rises to around 50 C at the beginning, mainly

7 Yong-xing Hong et al. / Water Science and Engineering 017, 10(): 107e Fig. 7. Comparisons between approximations of thermal field obtained from SBM and FEM for different times. due to the adiabatic temperature raise. The results at point 1 illustrate that the water cooling pipe can decrease the temperature around the pipe efficiently. Also, the results at points and 3 reveal that the temperature of the system with a cooling pipe decreases more quickly than it does for those without cooling pipes, and the highest temperature of the former is lower than that of the latter. All the obtained results demonstrate the effectiveness of the cooling pipe system in control of the thermal rise. 4. Conclusions This paper presents a semi-analytical singular boundary method formulation for the simulation of the thermal field in a concrete structure with a water pipe cooling system. The real problem was simplified into a two-dimensional cross-sectional domain considering general Dirichlet boundary conditions. Three examples were examined to show the validity of the proposed method. The first two examples were used to verify the accuracy of the presented algorithm. The third example was an experiment to approximate the thermal field against the elapsed time. Numerical results show that this methodology can be considered an alternative competitive tool for dealing with the heat conduction problem in a concrete structure with a water pipe cooling system. Due to the semi-analytical and meshless features, the present method also possesses the potential to solve three-dimensional mass concrete heat conduction problems with complex distributed water cooling pipes.

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