Two Dimensional Numerical Simulation of Temperature Field During the Construction of the Gravity Dams with

Cooling Pipe

Saeed-Reza Sabbagh-Yazdi Assistant Professor Kamiar Davallou

M.Sc. Hydraulic Structures Civil Engineering Department

KNT University of Technology, 19697 Tehran IranEmail: SYazdi@kntu.ac.ir

AbstractBecause of the cement hydration in mass concrete the temperature grows up. Weak heat conduction property of the concrete usually gives rise to thermal stresses which may cause cracking. Using cooling pipe may help overcoming the problem. In order to analysis temperature field and the effects of cooling pipe during the concrete hardening and gradual construction of the mass concrete structures, a temperature simulation program is developed using finite volume formulation which is derived by application of the Galerkin method on the domain discretized into triangular elements. The solution algorithm considers gradual moving the top boundary of the solution domain. The accuracy of the numerical model is assessed by comparison of the results with available analytical solutions and experimental measurements of two-dimensional square domain. The developed computer model is utilized to model the transient temperature field due to effects of cement hydration, cooling pipes, heat conduction, laminar construction, boundary conditions and ambient weather in various stages of construction of a typical gravity dam.

1 IntroductionThe concrete purring of neighboring concrete blocks of the gravity dams usually proceeds with short time delay. Hence, the heat exchange in gravity dams principally takes place in the transverse direction of the dam axis and two-dimensional models are generally adopted for temperature studies [1]Although application of the techniques for pre-cooling of the materials in the concrete mixture and the use of additives or low heat generation cements are effective methods for reducing maximum temperature of the mass concrete, technical difficulties, unwanted side effects and time delay encourage the use of post cooling water pipe coils which provides flexible control over temperature variations. However, the use of the post cooling system in mass concrete may not properly control the temperature variation tensile and consequently danger of cracking may appear. This fact makes temperature profile simulation an important part of the

design and construction process of mass concrete structures with embedded cooling pipes. In this paper, the results of a numerical solver developed for heat generation and conduction in layered concrete bodies [5] and completed for considering two dimensional post cooling effects [8], are presented. The quality of the results for cement heat generation and heat conduction in concrete are compared with the available experimental and analytical results reported in the literature [4,6]. The computational effects of absorbed heat by numerical modeling of the cooling pipes are compared with the numerical solutions of previous researchers [8]. In order to present the ability of the developed model to investigate the effects of using post-cooling system in gravity dams, the developed model is applied for prediction of the transient temperature field in a typical gravity dam section during the construction.

2 Governing EquationsAssuming isotropic thermal properties for the solid materials, the familiar two-dimensional equation defining heat generation and transfer is of the form,

TQT && =+∇καα 2

(1) Here, T )( co and Q& )( 3hmJK are temperature and the rate of heat generation per unit volume, respectively. If thermal diffusion is defined as Cρκα /= , where the parameters areρ )( 3mKg density, C )( cKgJK o specific heat, κ )( cmW o heat conduction coefficient, respectively.The natural boundary condition for the equation on concrete external surface is taken as,

( ) 0ˆ. =+∇ qnTκ(2)Where, q is the rate of heat exchange per unit volume of concrete surface with surrounding ambient and the unit vector n represents the boundary surface normalTo compute the source of the heat generation rate in the concrete body, Q& , the considerable influence of the temperature on hydration rate of cementation materials should be properly considered. For this propose, due to effects of ambient temperature on the rate of heat generation of cements, it is necessary to take the temperature history of various points in concrete body into account. Various functions have been proposed for considering this effect and are referred to as maturity function [2,3,4]. The function proposed by Rastrup [2], which is a well-known example is defined as,

( )[ ][ ])(1.01 2)(exp)()()(

)()( rTTae

ae

e

e

ee tbtEbatd

tdtd

tdQtQ −−−− −=

=&

(3)Where, E, a and b are constants obtained by regression of experimental heat evolution data. Note that using this expiration the rate of heat generation Q& can be computed for every single point of the concrete body, by considering as a function of equivalent time et . The equivalent time ∫= tdTHte )( is determined using the relative

rate of reaction concept by using a reference temperature rT and taking the temperature history into account. Note worthy that, for the points in the fields of

other materials rather than the concrete (i.e. rock foundations), the rate of heat generation rate &Q can be considered with zero value. For absorbing heat by cooling pipe or coil, the equation that derived by Cai Giano and Zhu Boofang can be used [7].

∫ ∂∂∆

=∆0Bwww

w dsrT

qcLT

ρλ

(4)Where, wT∆ is temperature of water in pipe at L∆ ,λ is thermal conductivity, wc is specific heat of water, wρ is density of water, wq is rate of water flow in pipes, 0B is

boundary of cooling pipe (the outer edge of the pipe) and dsrT

B∫ ∂∂

0

is radial gradient of

the temperature field of the concrete at the outer edge of the pipe. Equation (5) can be derived, if the equation (4) discretizes.

∑=

∆⋅⋅∆∆∆

=∆n

iwwww D

rT

qcLT

1

θρλ

(5)Figure 1 shows the algorithm for cooling the block by coil.

Figure 1: Layout of coil on cooling pipe

It’s assumed that the pipes embedded in the concrete are connected outside concrete so that L,, 11 wCwBwBnwAn TTTT == . For using equation (5), two sections (at two ends of block) have been used.

3 Numerical SolutionHere, an efficient numerical technique is introduced which enables fast and accurate solution of the temperature field in two-dimensional domains with complex and moving boundaries by the use of triangular meshes [5]. For this propose, first the governing equation for heat generation and transfer can be written in the following two-dimensional form,

SFT d =∇+vv

& .(6)Where T (temperature) is the unknown parameter and nS is the heat source, and temperature flux in i direction is defined as

i

Ti x

TF ∂∂α= ( )2,1=i .

The governing equation is multiplied by a piece wise linear test function on triangular element meshes and then it is integrated over all triangles surrounded every computational nodes (vertices of the triangular elements). By application of Guass divergence theorem and using the property of the test function, which satisfies homogeneous boundary condition on the dependent variable, the boundary integral terms can be omitted. After some manipulations, the resulted equation can be written as,

∫ ∫∫Ω ΩΩ

Ω+Ω=Ω dSdx

FdtT

nni

di

n φ∂

φ∂φ∂∂ )()( ( )i =1 2,

(7)After some manipulations, the procedure of deriving algebraic equation will end up with an efficient explicit numerical algorithm by using of following formula within each sub-domain Ω consisting the triangles associated with every node n [5],

∆

Ω−∆+= ∑

=

∆+ni

N

k

di

nn

tn

ttn lFStTT )(

23

1( )i =1 2,

(8)Where, il∆ is the i direction component of normal vector of mth edge of every triangular element. The area of sub-domain, Ω can be computed by summation of the area of the triangles Λ associated with node n, using

kk iiii xdx ][)( 3∑∫ ≈Λ=ΛΛ

lδ . The resulting discrete equation can be solved

explicitly with rather computational efforts.Note that, using the linear interpolation function for the temperature, the algorithm takes advantage from the fact that the first derivatives of T

iF are constant inside each triangular element. By application of the Gauss divergence theorem, the temperature flux in i direction, Fi

d , at each triangular element can be calculated using following algebraic formula,

( )∑=

∆Λ

=3

1

1m

midi lTF

(9)Where, T is the average temperature of its boundary edges. The resulted numerical model, which is similar to Non-Overlapping Scheme of the Cell-Vertex Finite Volume Method [5], can explicitly be solved for every node n (the center of the sub-domain Ω which is formed by gathering triangles sharing node n).Using meshes with various sub-domain sizes of sub-domains, to maintain the stability of the explicit time stepping the minimum time step of the domain of interest should be as,

min)(n

nktαΩ

=∆

(10)Remember, the heat source for each node n in concrete body is defined by

( ) nnenn tQS κα &= .If α and κ at node n are considered independent of time and temperature and maturity of the concrete, then for determination of the heat source in every time step, only the value of heat generation rate ( )etQ& at nodes located in concrete body must be updated for the nodes using equation (3). It should be noted that the value of

( )etQ& in each node is a function of et for that node. Therefore, et should be updated at each time step using following formula,

∑∆+=

=

− ∆=tNtt

tt

TTe tt r

0

0

)2( )(1.0

(11)

Two types of boundary conditions are usually applied in this numerical modeling. The essential and natural boundary conditions are used for temperature at boundary nodes and temperature diffusive flux (gradients) at boundary elements, respectively [5,6]. For those boundary nodes where nodal temperatures are to be imposed (essential boundary conditions), there is no need to compute the terms with spatial derivative of the equation (5). For the boundaries where the natural boundary conditions are to be applied, the temperature gradients d

iF can be imposed at boundary elements using equation (2) in the following form,

=

== ii

i

di K

qndTd

dxdTF γαγαα

(12)Here,

si

i δδγ l= is used to project the normal gradient to the i direction. This factor

can be computed using sδ and ilδ the length of the desired boundary edge and its projection perpendicular to the i direction, respectively.

4 Domain Discretization The solution comprises of the concrete dam body and the foundation. Considering the regular geometry of dam cross section and in order to facilitate the movement of the upper boundary of concrete as layered sequences of construction progresses, the use of structured mesh was considered. Since the age of concrete varies from layer to layer in the body of gravity dam, the use of regular mesh spacing allows moving the top boundary by gradual increasing the number of layers with constant thickness. The structured mesh is therefore quite appropriate because of offering control on the layer thickness and sequence of construction.

5 Verification of the Numerical SolutionFirstly, in order to evaluate the accuracy of the source term representing the rate of the heat generation, a set of experimental measurements on a concrete cube with 60 (cm) dimensions cast by 450 kilogram per cubic meter is used [4]. The concrete block was insolated all over the faces. The properties of concrete is considered as 0038.0=cα

( hm 2 ) and )/(9 hcmKJ o=κ . The average experimental data were determined using measurements at three points of concrete block (one point at the center and two points near the faces). A two dimensional rectangular triangular mesh with 6 (cm) spacing is utilized for numerical simulations. The results of the computer model using various placing temperature are compared with the experimental data for the permanently isolated concrete block present reasonable agreements, in figure 2. It can be seen that the result of the formula with the 13.5oC concrete placing temperature provides acceptable agreements with experimental measurements.

0 2 4 6 8 10T im e (day)

10

15

20

25

30

35

40

45

50

55

60

65

70T('c)

E xpe rim enta lT 0 = 1 3 .5 'cT 0 = 1 6 'cT 0 = 1 9 'c

0 0 .25 0 .5 0 .75 1X (m )

0 .05

0 .1

0 .15

0 .2

0 .25

T('c)

Ana lytica lC om puta tiona l

Figure 2: Comparison of the computed results (with various T0) with the experimental data [8]

Figure 3: Comparison of computed and analytical results on the middle axis of the square [8]

Secondly, the accuracy of the solution of spatial derivative terms is investigated by comparison of the results of the numerical solver with the analytical solution of the following two-dimensional boundary value problem with a constant source term as,

12 =∇ T (13)in the spatial field of ( ) 1,0 21 <<=Ω xx . The boundary conditions are considered 0=T at

11 =x , 12 =x and 0=∂∂

nT at 01 =x , 02 =x . The analytical solution is given by [6],

( ) ( ) ( ) ( )[ ] ( )[ ]( ) ( )

−−−−−

+−= ∑∞

=13

123

2221 2/12cosh12

2/12cosh2/12cos132121,

n

n

nnxnxnxxxT

πππ

π (14)

The problem is numerically solved on 2020× grid for triangular meshing. In figure 3, the results of numerical computations are compared with the analytical solution [8].For verification of cooling system the numerical software compares with Boofang and Gianbo solutions. They used their numerical software to achieve the heat distribution for a concrete block. The layers place every 7days and the height and width of layers are 1.5 (m). The ambient temperature is 10 degree of Celsius. Cement content of concrete block is 210-kilo gram per cubic meter and length of block is 200 (m). Water temperature at inlet is 0 degree of Celsius. As shown in figure 5, boundary conditions in walls are thermal insulation and top of block has connective boundary condition. Figure 6 shows the mean temperature along horizontal line in cross section that achieved by numerical software and Boofang and Ginbo software (note that in numerical solution, foundation with 5 (m) height has been used.Comparison between the results of the results of present and previous works in figure 4presents good agreements.

a) Problem illustration b) results reported by Boofang and Ginbo c) results of present workFigure 4: Mean temperature along cross section of multi layer block with cooling pipes [8]

6 Application to a Typical CaseThe ability of the model to deal with real cases is presented by application of numerical model to a typical arch-gravity dam and foundation section. The base of the dam is 58 (m) and the height of the section is 125 (m) while the dam crest width is 7 (m). Upstream and downstream slops are vertical and 1.0:2.45, respectively. Following assumptions were made for concrete placing program. The layers are constructed every 288 hours at the thickness of 3 (m). The cement content of the concrete is 217 kilogram per cubic meter. The concrete placing temperature is considered 16.75 co . The specifications of cooling system are assumed as follows. The diameter of pipe is 2.54(cm). The water flow rate is 0.9 ( hrm /3 ). The horizontal distance between two pipes is 1.5 (m). The time for cooling the layer by pipes was 2.5 month. The water flow direction has been changed every 6 hours. For cooling water of the river has been used. The water temperature variations are derived by equation (15):

)180

)22((75.95.21 π−+= tSINT (15)

Where, t is the time (by day) that t=0 at 21st of March and T is water temperature. The ambient temperature variations are derived by equation (16):

)180

)86.10((2.121.22 π−+= tSINT (16)

Where, t is the time (by day) that t=0 at 21st of March and T is ambient temperature. The forming has been placed for 12 days for each concrete layer. The conductivity of forms are )/(1 hcmKJ o and thickness of forms are 4 cm . The properties of concrete is considered as

0047.0=α ( hm2 ) and )/(28.11 hcmKJ o=κ . Rock foundation is considered to be limestone with 0038.0=α ( hm 2 ) and )/(36.9 hcmKJ o=κ .

Figure 5: Snap shots from temperature contours at 4 stages of a typical gravity dam construction [8]

Figure 6: The mean temperature history in a typical gravity dam with cooling and no cooling [8]

Figure 7: The maximum temperature history in a typical gravity dam with cooling and non-cooling [8]

The results of the computational model for five different stages of layer-based construction of a gravity dam are shown at figure 5. Figure 6 shows the mean temperature of central gravity dam profile in cooling and non-cooling situation. The history of the maximum temperature during construction of the typical gravity dam in cooling and non-cooling situation is presented at figure 7.

7 ConclusionsThe equation of heat transfer and generation due to cementation is solved on triangular element mesh. Proper implementation of natural boundary conditions was introduced. The resulted algorithm provides light explicit computation of time dependent problems. The simplicity of the algorithm makes it easy to program and extension for further developments.By application of structured mesh for the concrete dam part of the domain, an efficient modeling of the layered concrete structure is achieved. By ignoring the desired upper part of the dam, gradual movement of the top boundary of the structure during the construction period is simulated.The results of the developed model present reasonable agreements with the analytical and experimental data for heat conduction and cement heat generation, respectively. The model was applied to a typical gravity dam section with cooling pipes and the results of the temperature fields obtained showed the general pattern expected for such structures.

8 References [1] Springenschmid,R.,’ Prevention of Thermal Cracking in Concrete at Eearly Ages,’ E & FN Spon , Rilem

Report 15 . pp. 276-287, 1999

[2] Bagheri A.R., “Early Age Thermal Effects in Conventional and Micro-Silica Concrete Linings, Ph.D. Thesis, University of Newcastle, U.K. 1990.

[3 Lachemi M., Aitcin P.C., “Influence of Ambient and Fresh Concrete Temperatures on Maximum Temperature and Thermal Gradient in a High Performance Concrete”, ACI Materials Journal, PP 102-110, March-April 1997.

[4] Branco F.A., Mendes P.A., Mirambell E., “Heat of Hydration Effects in Concrete Structures”, ACI Materials Journal, PP 139-145, March April 1992.

[5] ] Sabbagh Yazdi S.R and Bagheri A.R., “Thermal Numerical Simulation of Laminar Construction of RCC Dams”, Moving Boundaries VI, WIT Press, pp 183-192, 2001.

[6] Reddy J.N. , “An Introduction to the Finite Element Method”, McGraw-Hill, Mathematics and Statistics Series, 1984.

[7] Bofang Zhu and Jianbo Cai, ‘ Finite Element Analysis of Effect of Pipe Cooling in Concrete Dams’, ASCE, Journal of Construction Engineering and Management, Vol. 115, No. 4, December, pp 487-498,1989

[8] Davallou, Kamiar, “The analysis of heat distribution in mass concrete by using post-cooling system in construction time”, M.sc. Thesis, Faculty of Civil Engineering KNT university of technology, Iran , Feb. 2004