21. chapter 21 - thermal analyses _a4

8/11/2019 21. Chapter 21 - Thermal Analyses _a4

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Chapter 21 Thermal Analyses_____ __________________________________________

204

CHAPTER 2

THERMAL ANALYSES

The temperature variation induces supplementary stresses into civil

engineering structures with constrained displacements. The temperature

variation effect becomes more important for massive or long structures,

sometimes exceeding the payload effect. It is the case of hydraulic

structures, as concrete arch dams, subjected to season-depending ambient

temperatures. The concrete hydration is also a heat generation process

affecting the thermal field of massive structures after pouring.

The thermal analyses are usually applied to calculate the thermal field

distribution, as a preliminary step for structural analyses. In case of assigned

material properties that change with the temperature level, a thermal

analysis is also needed. The amount of heat lost, the thermal gradients or the

thermal flux assessments are less important in structural design, except the

case when the thermal insulation is the analysis object.

In order to perform a thermal analysis, the domain is mashed into 2D or 3Dthermal finite elements, which have a similar shape as 2D or 3D solids. In

case of thermal analysis, the primary unknowns of the problem (the nodal

degrees of freedom) are the nodal temperatures, which are scalar values

(with no oriented components). The other thermal quantities are derived

from the nodal temperatures.

The heat transfer is taken into account by means of conduction, convection

and radiation. The conduction properties of materials are defined by the

conductivity matrix Kq. Depending on its definition, the materials may

exhibit isotropic or anisotropic properties. The convection is usually

specified as a surface load, but dedicated convection elements are also

available. The radiation heat transfer is solved by using radiation finiteelements, surface effect elements or by generating supplementary radiation

matrices.


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______________________Basics of the Finite Element Method Applied in Civil Engineering

205

21.1 STEADY-STATE THERMAL ANALYSES

The steady-state thermal analysis is used to determine the temperature field,

heat flow rates and heat fluxes due to thermal loads that do not vary in time.

The thermal loads are applied as boundary conditions, such as specified

temperatures, convection, heat fluxes or heat flow rates. If the material

properties (thermal conductivities) are constant, the analysis is linear. If the

material properties are temperature-dependent, the analysis is nonlinear. If

the radiation effect is taken into account, the analysis also becomes

nonlinear.

The first step in performing a thermal analysis is to build the model. If thethermal analysis is a precursory step of a stress analysis, it is recommended

to use the same mesh, only by changing the element type from structural to

thermal. Thus, the nodal points being the same, the calculated nodal

temperatures may be used directly in the stress analysis.

To apply the boundary conditions as prescribed nodal temperatures, all

nodes with similar temperature values should be selected at once. The

selection criteria are specified in chapter 26. If the prescribed temperatures

are different from node to node, they must be assigned manually.

The heat flow rate boundary conditions are concentrated nodal loads, used

in line-elements models where convection and heat fluxes can not bespecified. The convection boundary conditions are applied on exterior

element faces, as the amount of lost or gained heat from the surrounding

medium. They are available only for solid thermal elements. Similar

convection boundary conditions should be applied after a suitable element

face selection process (see chapter 26). The heat flux boundary condition is

used when the amount of heat transfer across a surface is known. It is also

interpreted as a surface load, applied on previously selected element faces.

The self generated heat due to internal chemical reactions can be

represented as a body load of selected thermal elements. The heat generation

rate has units of heat flow rate per unit volume.


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21.2 TRANSIENT THERMAL ANALYSES

The transient thermal analysis is used to determine the temperature field,

heat flow rates and heat fluxes due to time-varying thermal loads. The

calculated temperatures are used in structural analyses for thermal effects

evaluation. Generally, the transient analysis follows the same steps as a

steady-state analysis. The difference is due to the fact that thermal loads are

functions of time.

Unlike the steady-state analysis, the material properties that should be

assigned are the conductivity, the density and the specific heat. If they are

temperature-dependent, the analysis becomes nonlinear.

A transient analysis has always a multi-step solution phase. The loads

versus time functions should be divided into suitable load steps. For each

load step, the corresponding load and time values are assigned.

Before applying the transient loads the initial conditions should be set up.

These conditions may be the result of a steady-state analysis, or a uniform

assigned temperature in all nodal points of the model. The subsequent load

steps are usually written into a batch file, indicating the time interval, the

corresponding load value and the stepping option (ramped or stepped load).

21.3 FINITE ELEMENT EQUATIONS

The conduction and convection relationships are based on the general

energy conservation principle

qqLLv =+

+

TTT

t

Tc (21.1)

where the materials density;

c the materials specific heat;

T the temperature;

t time


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=

z

y

x

L - the derivative operator;

=

z

y

x

v

v

v

v - the velocity vector of heat transport;

=

z

y

x

q

q

q

q - the heat flux vector;

q - the heat generation rate per unit volume.

The heat flux vector is related to the thermal gradients by the Fouriers law

TLKq q= (21.2)

with

=

zz

yy

xx

k

k

k

00

00

00

qK - the conductivity matrix

By replacing equation (21.2) into (21.1) and expanding, yields

=

+

+

+

z

Tv

y

Tv

x

Tv

t

Tc zyx

+

+

+=

z

Tk

zy

Tk

yx

Tk

x

zzyyxxq (21.3)

The boundary conditions presumed to cover the element are:


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a. assigned temperatures acting on node-defined surfaces, T= T*;b. assigned heat flow crossing element surface

= qTq (21.4)

where - the cosine orientation vector of the element face;

q* - the specified heat flow.

c. assigned convection acting over elements surface

( )smfT

TTh =q (21.5)

where hf the film coefficient;

Tm the temperature of the surrounding fluid;

Ts the temperature of the element surface.

Combining equations (21.2) with (21.4) and (21.5)

= qTT LK q (21.6)

)(q smfT

TThT =LK (21.7)

Integrating equation (21.3) over the volume of the element and combining

with (21.6) and (21.7), yields:

=

+

+

VTTT

t

TTc

Ve

TT dqLKLLv

+ += Vee e smf VTTTThTq dd)(d q (21.8)

If radiation is also taken into account, the heat transfer rate between two

surfaces iandj, is

)( jiiijii TTAFQ = (21.9)

with Qi heat transfer rate from surface i;

Stefan-Boltzman constant


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i effective emissivityFij view factor from itoj

Ai area of surface i

Ti, Tj absolute temperatures at surfaces iandj

The temperature Tinside the finite element, in a point located at coordinates

(x,y,z) and time tis expressed using the shape functions:

e

TT TN= (21.10)

where N= N(x,y,z) element shape functions vector;

Te= Te(t) the nodal temperature vector

Thus, the time derivatives of equation (21.10) are:

eT

t

TT

=

= TN (21.11)

and

ee

TT BTTLNL == (21.12)

The variational equation (21.8) yields

=++

eVe q

T

eVe

T

eVe

T

VVcVc TBKBTBNvTNN ddd

+ += Vee e e eT

ffm VhhTq dddd* NqTNNNN (21.13)

with eT and e

T constant nodal values that may be removed from the

integrals. Equation (21.13) can be rewritten as

eeeee QTKTC =+

q (21.14)

where Ce element specific heat matrix;

Kqe total element conductivity matrix;

Qe total element heat flow vector.

For steady-state thermal analyses, the equation (21.14) turns into


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eee QTK =q (21.15)

After assembling the contribution of all elements in the mesh, by merging

the elemental conductivity matrices into the global conductivity matrix Kq,

the global equation system is ready to receive the boundary conditions.

They are prescribed values of temperature and/or heat flow as it was shown

before. The solution of the algebraic system represents the nodal

temperature vector, as primary unknowns of the problem.

21.4 EXAMPLE OF USING THE THERMAL ANALYSIS

Finite element models are used for back-analyses of existing structures, in

order to explain some abnormal behavior attested by site measurements and

recordings. It is the case of massive hydraulic structures subjected to

simultaneous water pressure and temperature field effects, both variable in

time. Using the periodical records of reservoir level, as well as the

temperature records of air and water, the boundary condition of the problem

can be determined. The results of the numerical simulation (usually

displacements) are compared with the recordings of the monitoring system.

A numerical beck-analysis was performed for Acena arch-gravity dam in

Spain, on both 3D and 2D finite element models*. During the operation, the

records at the pendulums installed in the dam body pointed out a significantand unexpected increase of displacements toward downstream. Some

aspects of the 2D analysis are presented below.

The finite element mesh used is illustrated in figure 21.1. It was developed

on the dam radial direction, crossing the pendulum position. For the

structural analysis, linear isoparametric structural solid elements were used,

with included incompatible modes. For the steady-state thermal analysis, the

element type was changed into thermal solids. Only elements of the dam

body were used.

*Popvici, A.; Sarghiu, R. Numerical simulation of an arch gravity dambehavior during operation. The 9thICOLD Benchmark Workshop on Numerical

Analysis of Dams, 2006


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In order to determine the interior temperature distribution corresponding tovarious dates and operation conditions, the boundary conditions were

prescribed. The monthly average air temperature was applied on the dam

faces in contact with the atmosphere. The water temperature at the surface

of the reservoir generally follows the air temperature with some delay,

increasing with depth. At 60 m, it reaches a constant value of 4.5C, which

is also maintained below. For the dates taken into account (with water levels

of 30 and 58 m), the water temperature was considered constant and equal to

8C. They were applied on the upstream face of the dam, from the bottom to

the actual water level. The same temperature was applied on the dam-

foundation contact line.

Two sets of results were chosen, corresponding to the following conditions:

- October 1999, water level 30 m, average air temperature 3.7C,

water temperature 8C;

- February 2001, water level 58 m, average air temperature 11C,

water temperature 8C.

Fig. 21.1 Mesh of the 2D model of Acena arch-gravity dam

The results of the steady-state thermal analysis are shown in figure 21.2.

The inside temperature distribution is represented for the two different dateswith their corresponding boundary temperatures.


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Fig. 21.2 Temperature distribution over the dam cross section (C):a. October 1999; b. February 2001.

Fig. 21.3 Radial displacements over the dam cross section (m):

a. October 1999; b. February 2001.

a. b.

a. b.


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The nodal temperature values calculated during the thermal analysis wereused as input data for the structural analysis. The date of joints grouting was

chosen as reference temperature (10C). Out of the calculated results, the

contours of the radial displacements due to upstream water pressure and the

temperature field at dates mentioned before are shown in figure 21.3.

Even if the displacement scale is not similar for both drawings, it is obvious

that the deformed shape has a different pattern. The main reason for

achieving smaller horizontal displacements for a higher level of water

pressure is the body temperature distribution. The temperature field, with

lower values on the downstream face of the dam, induces strains in opposite

direction as the structural load.

21. chapter 21 - thermal analyses _a4

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