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1、Full Terms&Conditions of access and use can be found athttp:/ Heat Transfer,Part B:FundamentalsAn International Journal of Computation and MethodologyISSN:1040-7790(Print)1521-0626(Online)Journal homepage:http:/ Boltzmann simulation of heat transfer withphase change in saturated soil during freezing

2、processZhiliang Wang,Libin Xin,Zemin Xu&Linfang ShenTo cite this article:Zhiliang Wang,Libin Xin,Zemin Xu&Linfang Shen(2017)Lattice Boltzmannsimulation of heat transfer with phase change in saturated soil during freezing process,NumericalHeat Transfer,Part B:Fundamentals,72:5,361-376,DOI:10.1080/104

3、07790.2017.1400311To link to this article:https:/doi.org/10.1080/10407790.2017.1400311Published online:27 Nov 2017.Submit your article to this journal Article views:23View related articles View Crossmark dataNUMERICAL HEAT TRANSFER,PART B 2017,VOL.72,NO.5,361376 https:/doi.org/10.1080/10407790.2017.

4、1400311 Lattice Boltzmann simulation of heat transfer with phase change in saturated soil during freezing process Zhiliang Wang,Libin Xin,Zemin Xu,and Linfang Shen Faculty of Civil Engineering and Mechanics,Kunming University of Science and Technology,Kunming,China ABSTRACT A lattice Boltzmann model

5、 is presented for simulating heat transfer with phase change in saturated soil.The model includes a quartet structure generation set for creating soil structure,double distribution functions for simulating temperature field evolution of soil particles and water,respectively,and an enthalpy-based met

6、hod for tracing phase interface.The model is validated by two cases with analytical solutions.Then,we investigate the influence of porosity on freezing process in saturated sandy loam soil.The results demonstrate that porosity is the predominant factor when the location is far from the cold source;o

7、therwise,thermal gradient is more important.ARTICLE HISTORY Received 21 August 2017 Accepted 26 October 2017 1.Introduction Heat transfer with phase change in saturated soil is of practical importance to many engineering problems,such as natural freezing of soil in cold regions,artificial freezing o

8、f ground during subway construction,soil freezing around ground heat exchangers,etc 16.If engineers or researchers cannot clearly grasp the mechanism of temperature field evolution during soil freezing,it will ser-iously affect the safety of project and may cause heavy lives and economic losses.One

9、of the most serious accidents related to soil freezing happened in Shanghai Metro Line 4 on July 1,2003.Due to the collapse of freezing wall of cross passage,mixture of water and sand from Huangpu River poured into the twin tunnels,the adjacent buildings were adversely destroyed and the completed tu

10、nnels were damaged as a result,which produced economic losses nearly 20 million dollars 7.Therefore,a better understanding of heat transfer mechanism in soil is essentially important,and special attention should be paid to the problem during freezing process.Numerous experimental and numerical studi

11、es have been done and many classic models have been developed on the issue of soil freezing process.Most works are based on the macroscopic continuum method,which regard the multiphase soil material as homogeneously dispersed system ignoring the interaction among different components 3,810.In fact,s

12、oil is a kind of granular porous media,and lots of studies indicate that heat transfer in porous media is strongly affected by its pore structure 1113.But traditional continuum theories can hardly describe the complex pore boundaries.And few experiments can get the accurate results of heat transfer

13、at microscopic scale referring to the interaction between fluid and solid.It is therefore desirable to look for alternative method no longer based on continuum assumptions,but able instead to capture the mesoscopic nature of heat transfer with phase change in soil during freezing process.The lattice

14、 Boltzmann method(LBM),as a mesoscopic numerical method,is based on the kinetic equations of particle distribution functions.In recent years,the LBM has been successfully applied to simulate the pore-scale heat transfer in porous media,owing to its numerical stability,none defined CONTACT Linfang Sh

15、en Faculty of Civil Engineering and Mechanics,Kunming University of Science and Technology,Kunming 650500,China.Color versions of one or more of the figures in the article can be found online at Taylor&Francis inherently parallelizability,simple implementation,and ability to handle complex geometry

16、and boundary conditions.To solve the temperature evolution equation coupling with the nonlinear latent heat source term,Jiaung 14 first developed a lattice Boltzmann model for simulating phase change governed by the heat conduction equation incorporated with enthalpy formation.Subsequently,Chatterje

17、e 15,Semma 16,Huber 17,Semma 18,Eshraghi 19,et al published a series of papers on simulating solidliquid phase change problem using LBM with enthalpy approach to treat latent heat effect on the temperature field.The above studies are mainly aimed at heat transfer problem of pure substance.For porous

18、 media,simulation at both pore scale and representative elementary volume(REV)scale can be conducted with LBM.Guo 20 extended the LBM to model the convection heat transfer in porous media at the REV scale.Gao 21 proposed an enthalpy-based LBM model to investigate natural convection with solidliquid

19、phase change in porous media at the REV scale.Liu 22 developed a double MRT-LB model for simu-lating transient solidliquid phase change problems in porous media at the REV scale.Wang 23 presented a LBM model to simulate fluidsolid conjugate heat transfer by implementing an inter-face boundary.Wang 2

20、4 studied the effective thermal conductivity of two-phase porous media for a certain porosity based on the pore scale.Song 6 studied the heat and mass transfer phenomenon with phase transformation in soil during freezing process.Almost all the works relevant are either at the REV scale neglecting th

21、e influence of pore structure or at the pore scale without considering the phase change problem for simplifying the calculation.The aim of this work is to study the heat transfer with phase change in saturated soil during freez-ing process with LBM.For this purpose,we use the quartet structure gener

22、ation set(QSGS)to create the stochastic pore structure of soil,and then based on the LBM with two-dimensional four-speed(D2Q4)model,the double distribution functions are applied to simulate temperature field evolution,one is for soil particles,and the other is for water.To reflect the phase change p

23、rocess during water freezing into ice,the enthalpy approach is used to trace the solidliquid interface by updating the liquid-phase fraction.The model is subsequently tested to simulate solidliquid phase change of pure substance and fluidsolid conjugate heat transfer of dual-component materials with

24、 two basic structures:series mode and parallel mode.At last,we discussed the temperature field evolution of the saturated sandy loam soil with different porosities during freezing process.Nomenclature c lattice speed Cp specific heat ei discrete velocity in the direction i in the lattice f liquid-ph

25、ase fraction g(r,t)temperature distribution function H total enthalpy k thermal conductivity L characteristic length La latent heat of phase change pc initial distribution probability of solid phase pi growth probability in the direction i r lattice site Sr heat source term Ste Stefan number T tempe

26、rature t time Tf phase change temperature T0 temperature of cold source Ti initial temperature x axis coordinate Greek symbols thermal diffusivity H the amount of heat released due to phase change dx lattice space step dt lattice time step porosity density dimensionless relaxation time weight factor

27、 Subscripts eff effective variable f fluid i direction i in the lattice l liquid phase ll water in the liquid phase ls water in the solid phase s solid phase Superscripts eq equilibrium function nm growth of nth phase on the mth phase*dimensionless variable 362Z.WANG ET AL.2.Reconstruction of soil m

28、icrostructure In nature,soil is a heterogeneous assemblage of materials,forming a porous structure.In general,it is difficult to accurately describe the microstructure of soil due to its complexity and randomicity.To simulate the heat transfer problem with phase change in saturated soil at the pore

29、scale,it is necessary to acquire the microstructure and the distribution information of different phases 23.2.1.Description of soil microstructure The saturated soil consists of two parts:the pore(or water)and the solid particles.The distribution of particles and pore is disordered,so the binary ran

30、dom function Z(x)is usually used to model the soil microstructure,Z(x)is defined as follows:Zx 1if x is pore0othewise1It is obvious that the mean of Z(x)is the porosity of soil.2.2.Soil microstructure generation Several methods have been put forward to reconstruct the microstructure of porous media

31、in the past decades:multiple-point geostatistics 25,particle packing model 26,network generation model 27,fractal model 28,the QSGS method 2931,etc.Among all of these,the QSGS method,proposed by Wang in 2007 29,has been widely used in the reconstruction of porous media due to its clear physical mean

32、ing,capability of considering the solid particles random distribution,and statistical characteristics.In this paper,the QSGS method is applied for soil microstructure generation according to its porosity.Assuming the solid particles are the growth phase,and the whole lattice nodes of construction ar

33、ea are all pores at the beginning.The generation process follows the steps below:i.Assign pc as the initial distribution probability to generate the solid-phase core randomly,and promise that pc is less than the volume fraction of solid phase(i.e.,pc1-).ii.Take solid-phase core as growing center and

34、 apply the growth probability pi in different directions to make solid-phase core grow on the neighboring lattice nodes.The subscript i is the direction of growth,i 1,2,3 8,which is shown in Figure 1.iii.Repeat the growing process of(ii)until volume fraction of pores reaches the given porosity.For t

35、hree phases of porous media,the abovementioned steps can be applied to generate each phase ignoring the interaction between these phases.Otherwise,the probability pinm should be introduced to characterize the growth probability of the nth phase on the mth phase in the ith direction.There-fore,the QS

36、GS method can control the microstructure of porous media by setting the four parameters(pc,pi,pinm,and).For the two-dimensional saturated soil in this paper,the microstructure is Figure 1.Solid-phase core growth direction.The central black square is the growing core of solid.The numbers 18 represent

37、 the 8 growth directions of solid core.NUMERICAL HEAT TRANSFER,PART B363 produced by setting the parameters(pc,pi,and),without considering the interaction between both phases.Figure 2 is an example of reconstructed soil with the QSGS method.3.Heat transfer model of saturated soil during freezing pro

38、cess Based on the assumptions that soil is saturated,and composed of two parts:soil particles(solid phase)and water(liquid phase),the energy equations for heat transfer with phase change during freezing process in the multiphase system can be written,respectively,as:Energy equation of solid phase qC

39、p?sqTqt ksr2T2Energy equation of liquid phase with phase change qCp?fqTqt kfr2T qfqDHqt3where subscript f represents the fluid,and s the solid;T is the temperature.,Cp,k are density,specific heat,and thermal conductivity,respectively;t is the time;H is the amount of heat released due to phase change

40、.To simplify the numerical calculations,the energy equation with phase change for liquid phase is generally expressed in terms of total enthalpy,thus the governing Equation,Eq.(3),can be written as qfqHqt kfr2T4the total enthalpy H has two parts,namely,the sensible enthalpy and latent heat enthalpy,

41、so it can be written as H CpT flLa5where fl is the liquid phase fraction,and is zero for the solid region,1 for the liquid region.La is the latent heat of phase change.Figure 2.The microstructure of reconstructed soil with QSGS method.White for pore and black for solid.Note:QSGS,quartet structure ge

42、neration set.364Z.WANG ET AL.Substituting Eq.(5)into Eq.(4)yields qTqt afr2T Sr6where af is the thermal diffusivity,af kf=qCpf;Sr is the heat source term Sr?qfl=qtLa=Cpf When the phase change temperature is Tf,the total enthalpy of solid region Hs equals to CpTf,and the liquid region Hl is CpTf La.S

43、o the relationship between H and T can be established as T HCpH Hl8:7Once the total enthalpy H is determined,the liquid phase fraction fl can be expressed as flH?HsHl?Hs8Thus,Eq.(2)can be used to simulate heat transfer of the solid phase in the soil,and Eq.(6)Eq.(8)could describe heat transfer probl

44、em with phase change for the liquid phase.To have a clear understanding of physical problem and to simplify the number of parameters,the dimensionless variables are introduced as xxLtastL2TT?T0Ti?T0Ste CpTf?T0La8:9where x*,t*,T*are the dimensionless coordinate,time,and temperature,respectively.Ste i

45、s the Stefan number.L is the characteristic length.T0 is the temperature of cold source.Ti is the initial temperature of soil.4.Lattice Boltzmann model The LBM is a mesoscopic approach based on the evolution of statistical particle distribution of a lattice gas whose density represents the physical

46、quantities to be modeled,such as temperature.Its main idea 3233 is to establish the relationship between microscale and macroscale by not consider-ing each particle behavior alone but behavior of a collection of particles as a unit.This method is effective to study the macroscopic relation and the m

47、icroscopic mechanism of energy transport process in porous media,and it has been extensively adopted for investigating the transport problems at the pore scale.4.1.Lattice Boltzmann equation The D2Q4 LB model(Figure 3)for single relaxation time(or LBGK),which was proposed by Qian 34,is used to solve

48、 the heat transfer problem in the saturated soil,which is governed by Eq.(2)and Eq.(3).In the present study,we choose the double distribution functions to model the temperature field evolution in the saturated soil at the pore scale:one is for soil particles(solid phase),the other is for water(liqui

49、d phase).4.1.1.Evolution equation for heat transfer of solid phase For the solid phase without phase change,the temperature evolution equation is generally written as gsir eidt;t dt?gsir;t?gsir;t?geqsir;tssi 0;1;2;310NUMERICAL HEAT TRANSFER,PART B365 where is the dimensionless relaxation time,its va

50、lue should insure to be within(0.5,2)23,24;gi(r,t)is the temperature distribution function of the ith direction at the lattice site r and time t;geqir;t represents the equilibrium distribution function;ei is the discrete velocity in the lattice ei c 1;0;?1;0;0;1;0;?111where c is the lattice speed,an