Yong-xing Hong,Wen Chen,Ji Lin,*,Jian Gong,Hong-da Cheng

aState Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering,Hohai University,Nanjing 210098,China

bCollege of Mechanics and Materials,Hohai University,Nanjing 211100,China

cSchool of Engineering,University of Mississippi,M ississippi38677,USA

Thermal field inwater pipe cooling concrete hydrostructures simulated w ith singular boundary method

Yong-xing Honga,b,Wen Chena,b,Ji Lina,b,*,Jian Gonga,b,Hong-da Chengc

aState Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering,Hohai University,Nanjing 210098,China

bCollege of Mechanics and Materials,Hohai University,Nanjing 211100,China

cSchool of Engineering,University of Mississippi,M ississippi38677,USA

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 boundarymethod,a sem i-analyticalmeshless technique,wasemployed to analyze the temperature distribution.A numericalalgorithm solved the transient temperature field w ith consideration of the effectsof cooling pipe specification,isolation of heatof hydration,and ambient temperature.Numerical results are verified through comparison w ith those of the finite elementmethod,demonstrating that the proposed approach is accurate in the simulation of the thermal field in concrete structuresw ith a water cooling pipe.

?2017 Hohai University.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license(http:// creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:Thermal field；Singular boundarymethod；Semi-analyticalmethod；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,1980).A water pipe cooling system is considered an efficient technology for cooling the interior temperature in mass concrete and consequently m itigating thermal stress cracks and resultantstructuralweakness(Qiang et al.,2015).As stated in Hauser et al.(2000),the flow ing water along the pipe not only absorbs the heat from the concrete,butalso upgrades the thermal storage capacity and decreaseshydration heat(Kim etal.,2001),furnishing a strategy for better thermal removal.

Since the fi rst successful application of the cooling pipe system in the Hoover Dam(Kwak etal.,2014),the prediction of the temperature history and distribution inmassive concrete structures w ith 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,2013),the thermal gradient(Sato et al.,2005), the structure restraint,and the material thermal stability (Zhang et al.,2004),have been taken into account.The equivalentequation of heat conduction in a concrete structure w ith a cooling pipe,proposed by Zhu(1991)w ithout consideration of the thermal gradient,describes the average temperature:

whereTis the temperature of concrete,T0is the initial temperature of concrete close to thewater pipe,Twis the cooling water temperature,φis a given function associated w ith theheat flux from concrete towater,ais the thermaldiffusivity of concrete,andθis theadiabatic temperature raiseof concreteat a certain time instancet.Three boundary conditions for measuring the thermal gradient on the boundary(Zhu,1999) are as follows:

whereT（t）is a prescribed temperature process,ηrepresents the coefficientofheatconvection,n is theunitoutward normal vector on the boundary of the computational domain,andβis a rational number.This system of equations can also be well utilized in the formulation of casting processes ofmass concrete containing double-layerstaggered heterogeneouscooling pipes,as reported in Yang et al.(2012).

Asmentioned above,the equivalent equation of heat conduction in the concrete-pipe system hasbeen studied using the finite elementmethod(FEM)(Chen etal.,2011).In the FEM, the structures are discretized into small elements,and a final system ofequations ismadeup ofapproximations 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(Sasakiet al.,2014).In order to tackle this bottleneck,one alternativemethod is to neglect the practical sectional shape and size of cooling pipes for simulations(Liu et al.,2015).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 new ly developed meshlessmethod proposed by Chen(2009)for the simulation of boundary value problems(Chen and Gu,2012).This method falls into the category of the boundary-type method w ith integration-free attributes,which can also be considered one type of ameliorative algorithm of the method of fundamental solutions(MFS)(ˇSarler,2009).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., 2016；Wei et al.,2015).Therefore,the SBM is a truly semianalytical boundary-typemeshlessmethod.

TheSBM hasbeenw idelyused todealw ithmanyengineering problems,such aswavepropagation problems(Lin etal.,2014), steady-state heat conduction problems(Wei et al.,2016), and time-dependent problems(Wang and Chen,2016).Moreover,the SBM,combined with the dual reciprocitymethod and inverse interpolation,is effective in solving non-homogeneous problems(Chen et al.,2014).For the problems considered in this study,only the information on the surface of the structure and pipewas required.Due to the use of the fundamental solution,the SBM is a sem i-analytical technique and has great potential for thesimulation of time-dependentproblems.

This paper is organized as follows:in Section 2,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 exam ined to show the effectivenessof the presentedmethod；and,finally, some conclusions and remarks are provided in Section 4.

2.Num erical form ulation 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 schemewas 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 domainΩof interest.

2.1.Time discretization

To beginw ith,the time interval［0,t］is divided equally intoMsub-intervals.Then,the time step isdt=t/M,andtn=ndt, wheren=0,1,…,M.The implicit Euler scheme is used to discretize Eq.(1)as follows:

whereTnis the temperature of concrete at timetn,φn+1isφat timetn+1,θn+1isθat timetn+1,and x=（x,y）for a twodimensional problem.Using the notation 1/（adt）=μ2,we come to the follow ing system of Helmholtz equations,which can be solved by the proposed SBM:

2.2.Singular boundarymethod

At fi rst,using the dual reciprocity method(Chen and Gu, 2012),the solution to the non-homogeneous Helmholtz equation(Eq.(6))can be approximated by the summation of（x）and（x）,as follows:

Using the dual reciprocity method,the non-homogeneous termgn+1（xi）in Eq.(8)at the node xiin the domainΩof interest can be expanded by the chosen radial basis function:

wherecεis the Euler constant,andcε≈0˙5772；andK0is the zero-ordermodified Bessel function of the second kind.

Oncewe obtain a particular solution,the homogenous solution（xm）can be approximated by a linear combination of fundamental solutionsGmjas follows:

whereGmj=K0μrmj/（2π）for two-dimensional problems；Nbis the number of boundary nodes for approximation；γj（j=1,2,…,Nb）is the unknown coefficient to be determ ined；andUm（m=1,2,…,Nb）is the origin intensity factor, which isproposed to avoid the singularitiesof the fundamental solutionswhen thenode xjapproaches the node xm.Theorigin intensity factors used in this study are obtained using the inverse interpolation technique(Chen et al.,2014):

whereG0mj=］is the fundamental solution of the two-dimensional Laplace operator,Tm=xm+ym+1 is a two-dimensional sample solution that satisfies the Laplace equations,andTI=xI+yI+1.It isnoted that the node xIfor approximating the origin intensity factor in the SBM is chosen random ly,and we chose xI=（1˙21,2˙72）in this study.A lso, for the heat flux,we have

where njdenotes the unit outward normal vector of node xj, andδjisa parameter thatdenotes the average arc length of two nodes linked to node xj.More details about the SBM can be found in Chen and Gu(2012).

By forcing Eqs.(20)and(22)to satisfy the boundary conditions,γj（j=1,2,…,Nb）can be obtained.After we obtain,the solution to the problem can beobtained by the combination ofandas stated in Eq.(7).

3.Numerical results and discussion

We neglected the influence of cooling water in the fi rst two examples described in this section,to illustrate the validity ofthe proposed method.A simplified cross-section of mass concrete w ith a cooling pipe was studied using the proposed method considering boundary conditions(Eqs.(2)and(3)). The accuracy of numerical results was validated w ith the follow ing average relative errorerel:

wherentrepresents the number of test nodes,andTnumkandTanakdenote the numerical approximation and analytical solution of test node xk,respectively.

3.1.Example 1

In the fi rst example,a classical two-dimensional heat conduction problem w ithout water pipes was examined to validate the described method.This exam ination was carried out on a square domain,Ω=｛（x,y）|0＜x,y＜3m｝.In the SBM,80 boundary nodes were used to model the concrete system,and 361 internalnodeswere chosen forapproximation of the particular solutions.Allof these collocation nodeswere distributed evenly and the distance between every two adjacentnodeswas0.15m.The governing equation of the general heat conduction problem is as follows:

w ith the following initial and Dirichlet boundary conditions:

whereT（x,y,t）is the thermal function to bedetermined,andΓ is the simple closed curve bounding the domainΩof interest. In this case,we considereda=0.8m2/s.Itwas assumed that θ=0 and the boundary conditions were given directly.The compared analytical solution taken from Bruch and Zyvoloski (1974)was as follows:

Numerical results shown in Fig.1 through Fig.3 were obtained using dt=0.002 s.Fig.1 presents the comparison between numerical solutionsand analyticalsolutions for three random ly chosen test nodes,node 1(2.4,1.5),node 2(2.4, 2.4),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 w ith the analytical solutions, indicating the accuracy and stability of the proposed method. Fig.2 shows the relationship between the average relative errorereland the elapsed timet,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 w ith the increasing number of boundary nodes.When the number of boundary nodes is greater than 100,the average relative error remains near 10-2. We should note here thatbetter accuracy can be obtained for a larger number of internal nodes.A ll these results demonstrate that the proposed SBM is feasible for solving heat conduction problemswithoutwater cooling pipes.

3.2.Example 2

Fig.1.Comparison of numerical and exact solutions for three test nodes and corresponding relative errors.

Fig.2.Average relative error versus elapsed time.

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 isΩ=｛（x,y,z）|0＜x,y,z＜1m｝,where the diameter of concretemodelDis 1m,and the diameter of the tiny pipedis 0.004 m.We used a total number of 1572 boundary nodes for the singular boundary method,w ith 30 nodes on thewater pipe boundary,and 3375 internalnodes 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 w ritten as

Under the follow ing initial and Dirichlet boundary conditions:

whereT（x,y,z,t）isan unknown thermal function.In this case, it was assumed thata=0.16 m2/s,and the adiabatic temperature raise of the target concrete wasθ=100× sin（πx）sin（πy）sin（πz）t.The compared analytical solution is

Fig.3.Average relative error versus number of boundary nodes.

For this three-dimensional problem,Ψij,φij,Gmj,andG0mjare as follows(Chen et al.,2014):

Numerical results in Figs.4 and 5were obtained using time step dt=0.1 s.Fig.4 illustrates the comparisons between numerical solutions and analytical solutions for three random test lines,y=z=0.5m(line 1),y=z=0.2m(line 2),andy=z=0.7m(line3).Itcan be seen that thepredictionsof the proposed method agree w ith the analytical solutions.Fig.5 presents the relative errors of the three test lines,show ing 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 10-2,which is acceptable in practical engineering.The results demonstrate the validity of the proposed method for three-dimensional heat conduction problems w ithout consideration of the cooling influence of water flow.

3.3.Example 3

A fter the numerical verificationsmade in the two examples mentioned above,we applied our algorithm in the simulation of a thermal field in a concrete structure w ith a water pipe cooling system.The simulation was conducted in a square domain with a cross-section,Ω=｛（x,y）|0＜x,y＜3m｝.Considering Eq.(1)the equivalent heat conduction equation, and based on Zhu(1991),we have

Fig.4.Comparison of numerical approximations and analytical solutions for three test lines at t=1 s.

Fig.5.Relative errors for three test lines at t=1 s.

wherek1=2˙08-1˙174ξ+0˙256ξ2,w ithξ=λL/（cwqwρw）, whereλis the heat flux coefficient,Lis the length of concrete,cwis the specific heat capacity of water,qwis the flux of cooling water,andρwis the water density；s=0˙971+0˙1485ξ-0˙0445ξ2；andzis a coefficient,and can be represented asz=a′t/D2,w itha′=aln［100/ln（D/d）］. In order to simplify the problem,it is assumed thatD=3m,d=0.025m,a=0.1m2/d,k1≈2,ands≈1.Then,we have φ≈exp（-0˙024t）,andθ=38［1-exp（-0˙27t）］.Substituting the coefficients and functions above into Eq.(1),we have

In this study the number of internal nodeswas considered to be 360；the number of boundary nodeswas considered to be 312,w ith 300 nodeson the boundary of the square(Γ1)and 12 nodes on the boundary of the tiny pipe(Γ2)；and all of the

Fig.6.Average relative error versus elapsed time.

collocation nodes were distributed evenly at each physical position.The initial condition,

was considered along w ith the follow ing boundary conditions:

whereTadenotes the temperature of the air surrounding the concrete.By consideringTa=Tw=15°C,and dt=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 fort=0.1 d,1 d,2 d,4 d,12 d,and 23 d.From this figure,it can be observed that the results obtained from the proposed SBM are almost the same as the results obtained from the FEM.A ll numerical results obtained from the SBM and FEM verify that the temperature near the cooling pipe decreases sw iftly,while temperature in the m iddle of the pipe and physical boundary increases fi rst due to the rising of adiabatic temperature,and then decreases slow ly. Fig.7 also displays the distribution of thermal fields inside the interest domain w ith respect to the elapsed timet.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 consistentw ith 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 w ith a cooling pipe.

A fter the accuracy of the SBM was verified,three special test pointswere taken into account to determ inemore details of the thermal field:point1(1.54,1.54)near the cooling pipe, point 2(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.(25)and the follow ing boundary condition:

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 temperaturewith elapsed timet,at the three test points,between concrete structures w ith a cooling pipe and without cooling pipes,is shown in Fig.8.It can be seen from Fig.8 that the temperature atpoint1 in the structure including a cooling pipe sw iftly decreases as time elapses.In contrast,the temperature at point 1 in the structurew ithout cooling pipes declines very slow ly,and even rises to around 50°C at thebeginning,mainlydue 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 atpoints 2 and 3 reveal that the temperature of the system w ith 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.A ll the obtained results demonstrate the effectiveness of the cooling pipe system in control of the thermal rise.

Fig.7.Comparisons between approximations of thermal field obtained from SBM and FEM for different times.

4.Conclusions

This paper presents a sem i-analytical singular boundary method formulation for the simulation of the thermal field in a concrete structurew ith awater pipe cooling system.The real problem wassimplified into a two-dimensional cross-sectional domain considering general Dirichlet boundary conditions. Three examples were exam ined to show the validity of the proposedmethod.The fi rst two exampleswere used to verify the accuracy of the presented algorithm.The third example wasan 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 w ith the heat conduction problem in a concrete structure w ith 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 problemsw ith complex distributed water cooling pipes.

Fig.8.Comparisonsof variationsof numerical temperaturew ith time at test points between structures w ithout cooling pipes and w ith a cooling pipe.

Thismethodology presents not only the temperature inside concrete structures,butalso the thermal gradient surrounding the cooling pipes,which is an essential factor affecting thermal cracks.More real-world problemsare under intense study and w ill be exam ined in a subsequent report.

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Received 14 November 2016；accepted 13 February 2017 Available online 23 June 2017

Thiswork was supported by the National Natural Science Foundation of China(Grants No.11572111 and 11372097)and the 111 Project(Grant No. B12122).

*Corresponding author.

E-mail address:linji861103@126.com(Ji Lin).

Peer review under responsibility of Hohai University.

http://dx.doi.org/10.1016/j.wse.2017.06.004

1674-2370/?2017 Hohai University.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license(http:// creativecommons.org/licenses/by-nc-nd/4.0/).