The actual measurement is composed of truth value and noise. So when we grasp a physics
phenomenon by the actual measurement, it can't be used directly. The data is revised by
interpolation or average. These method change the data, so it is difFerent from the actual
measurernent. To make good use of the all data with the noise, in this study use Extended Kalman
Filter. Extended Kalman Filter use the leastsquares method which is used frequently in solving
noise problems. In general, Kalman Filter is a linear model, but Extended Kalman Filter
is a nonlinear model. The method presented in this paper employs the shallow water equation
as the basic equation, the method can be adaptable for the analysis of thermal conduction.
The inside of triangle element is interpolated, and as for the weighting function, the dis
creetness in space can be made by applying the Galerkin method. Also, the discreetness can
be made by the explicit Euler method.
The basic equation of Extended Kalman Filter is nolinear equations written as follows ;
i) Both system noise and observation noise are assumed to be white Gaussian noise process :
ALGORITHM
Using initial condition in Bayes rule, we can lead the Algorithm :
in this Algorithm, depend on observation data. In this study,
parameter is fix value so Q is set to 0. Therefore the value of decided is only R, so the parameter
depend on R.
NUMERICAL EXAMPLE
In this study, the computational grid (473 nodes, 840 elements) of finite element mesh is shown
in Figure 1 , the incident wave of amplitude is estimated. The flow condition is a sine wave
coming from the left side. The sine wave is as follows ;
In this case, each results which are given at observation points, are added to noise, and these
data make use of observation data (Figure 2). According to the observation data, the amplitude
is estimated. In this case, R is 0.0075, lumping parameter is 0.90 . The initial condition, in all
nodes velocity and water elevation, set to zero.
In the observation points No.l No.2 No.3(Figure 1), the results of estimate amplitude are
shown in Figure 3. In this case, the good agreement value are found by using various R.
Next, observation point is only set to No.2, and the incident wave of amplitude changes into
1.5m and 2.0m. In this case, R is used same value which is used to lead the best agreement in
case incident wave of amplitude is l.Om.
In this case, we estimate amplitude, which is fixed value. And, sensitive matrix is
CONCLUSION
We find that agreement value on incident condition is given by changing R, by this method.
From Figure 3, each observation points are given by each R. Therefore, it is said that in case of
observation points, R is set to each case. And father and father from incident points is bigger
and bigger the slope of graph at l.Om of Y scale. In case same observation point, if changed
dimension of size, R is same value.
This paper made use of shallow water equations as a basic equation, and parameter estimated
incident wave of amplitude as a parameter estimation. The feature of this method is
pointed out in the fact that Finite Element Method is incorporated into Extended Kalman
Filter. It was estimated by this model case that the propriety of this method is right and
reasonable.
REFERENCES
 M.Kawahara, Finite Element Method, Gihoudo publishing, 1983.
 T.Katayama, Applied Kalman Filter, Asakura publishing,1983.
Predictive Control Applied to Flood Control Problem
Kiyofumi SAKUMA and Mutsuto KAWAHARA
Dept. of Civil Engineering Chuo University
11327 Kasuga Bunkyouku, Tokyo,Japan
1. INTRODUCTION
It is very important to control flood flow especially like Japan where it often happens, so various approaches
have tried to the problem. Then this study presents a flood control problem operating dam gate to find optimal
discharge objectively. As the method, predictive control theory is applied, which is the method that is able to
determine the optimal one at any time, if possible to know some future data. And the purpose of this study is
set to find optimal value which lower water elevation.
Its characteristics is online control which can determine the discharge getting predicted state values, and
possible to recover ones on computing, if they have errors.
The behavior of water surface is governed by linear shallow water equation. It is discretized by finite element
method and the phenomonon is analyzed.
2. BASIC EQUATION, BOUNDARY CONDITION
The wave propagation of reservoir is governed by two dimensional shallow water equation and represented
as follows; equation as follows ;
where q is unit flux, h is water elevation, g is gravitational acceleration and h is water depth. Boundary conditions
are given as follows ;
where S_{1} and S_{2} are entrance and exit boundaries , and S_{3} is treated as land boundary condition.
3. FINITE ELEMENT METHOD
The analyzed domain is divided into triangle finite elements and weighted residual equations are given to
them. And they are interpolated by Galerkin Method, finite element equation can be obtained as follows ;
where {X} is state vector, [M], [H] are coefficient matrices which are represented as follows ;
where F is interpolation function. Twosteps explicit method is applied to finite element equation (6), and
represented as follows ;
where is lumped coefficient matrix, using lumping parameter e , mixed coefficient matrix is written as
follows ;
4. PREDICTIVE CONTROL METHOD
4.1. State Equation
To argue control theory, Eqn.(6) is changed into state equation like (12) ;
where [A] [B] [C] are coefficient matrices,and [x],[u],[f] is state, control , force vectors respectively.
4.2. Performance Function
Performance function J_{n} is determined as to reduce water elevation in the reservoir as well as possible,and
it is represented as the quadratic form.
where [R] [Q] are diagonal terms of weighting matrices.
4.3. Stationary Condition
Control vector {u}_{n} which minimizes performance function J_{n} is represented as follows ;
4.4. State Prediction
Future state equation is represented as Eqn.(15) by descetizing Eqn.(12) with explicit Euler method.
If Eqn.(15) is substitued for each state vector obtained by Eqn.(14) , control vector {u} is represented as follows ;
where each matrice is represented as follows ;
5. NUMERICAL EXAMPLE
Numerical test is experienced at rectanglar mesh and Fig.l is the finite element mesh. Total number of
nodes and elements are 33 and 40 respectively. Time increment D is 0.6 (sec), predicted state value is 60.0
(sec.). Fig.2 shows the time history of unit inflow flux at A point, and Fig.3 is the one of unit discharge flux at C
point, where solidline is control discharge and brokenline is noncontrol one. Fig.4Fig.6 show water elevation
at each point, and solidline is controlled, brokenline is noncontrolled ones.
6. RESULT
7. CONCLUSION
If it is possible to know certain state value constantly, water elevation is enough lowered compared with
noncontrolled case.
In short,it is said that predictive control theory is applicable for the problem which represents natural
phenomena as basic equations so that it is useful for Hear Transfer Problem as real time control.
PROPER ORTHOGONAL DECOMPOSITION ANALYSIS OF TRANSITIONAL CONVECTIVE FLOW SYSTEMS IN COMPLEX GEOMETRIES
R. A. Sahan, A. Liakopoulos and H. Gunes
Department of Mechanical Engineering and Mechanics, Lehigh University,
200 W. Packer Ave. Packard Lab. No: 19, Bethlehem, PA 180153085 USA
ABSTRACT
The use of computational fluid dynamics (CFD) codes in understanding the flow behavior in complex transitional and/or turbulent thermofluids systems has been extensively increased in recent years. Research on lowdimensional modeling (LDM) of nonlinear dynamical systems has been also gaining considerable attention. Certain fluid systems exhibit chaotic behavior through the nonlinear interaction of small number of degrees of freedom, enabling us to represent transition process in phase space by lowdimensional dynamical behavior. This remarkable outcome furnishes the basis for the development of LDMs of transitional thermofluid systems. These developed models replace the system of governing partial differential equations (PDEs) with a relatively small set of ordinary differential equations (ODEs) that captures the underlying dynamical behavior of the thermofluids system. Direct numerical simulations (DNSs) of transitional and turbulent convective flows in complex geometries, such as grooved and vertical channels, require tremendous computational time and power. Thus, accurate and valid loworder approximations to the full model may enable us to employ the techniques of dynamical system theory in investigating bifurcation, stability and control characteristics of complex transitional thermofluid systems.
Methods, such as FourierGalerkin, ChebyshevGalerkin, trigonometric functions and orthogonal polynomials are widely used to transform a system of PDEs into a set of ODEs. However, these methods use general basis functions in the expansion leading to large system of ODEs. It is important that LDMs are constructed by expanding the unknown functions in terms of basis functions that are constructed for each system separately and reflects the dynamical behavior of the system in the vicinity of some flow controlling parameters. Proper orthogonal decomposition (POD, the method of empirical eigenfunctions, KarhunenLoeve expansion) first introduced in the fluid mechanics community by Lumley [1], is a candidate for obtaining a set of optimal basis functions. Berkooz et al. [2] give a rigorous review of POD in the analysis of turbulent flows. The method of snapshots has been introduced by Sirovich [3] as an efficient way of extracting the empirical eigenfunctions of large data sets. Using snapshots POD, Deane et al. [4] and Sahan et al. [56] constructed lowdimensional dynamical models for transitional isothermal and nonisothermal flow in a periodically grooved channel. Gunes et al. [78] analyzed buoyancydriven convective flow system in a vertical channel with discrete heaters. Liakopoulos et al. [9] studied free convective flow in a differentially heated cavity.
This paper focuses on DNS and loworder dynamical representation of transitional flow and heat transfer in a periodically grooved channel relevant to forced convective air (Prandtl number, Pr=0.71) cooling of electronic systems. The governing PDEs (conservation of mass, momentum and energy) of the thermofluid system are solved by a spectral element method. Spontaneously oscillatory solutions are calculated for a supercritical value of Reynolds number, Re=750. POD is used to extract empirical eigenfunctions, identify and define the spatiotemporal (coherent) structures of the thermofluid system. POD enables us to represent the velocity and temperature fields in an optimal way. The 6 largest normalized eigenvalues of the most energetic eigenmodes and their respective contribution to the flow and temperature fluctuation energy are given in Table 1.
Table 1
Eigenvalues of the 6 Most Energetic Velocity and Temperature Modes and Their
Respective Contributions to the Total Flow and Temperature Fluctutation Energy

Velocity Modes 
Temperature Modes 
Modes 
Normalized Eigenvalue 
Cumulative Energy, % 
Normalized Eigenvalue 
Cumulative Energy, % 
1 
0.5116 
51.16 
0.4916 
49.16 
2 
0.4562 
96.78 
0.4694 
96.10 
3 
0.0123 
98.01 
0.0137 
97.47 
4 
0.0113 
99.14 
0.0131 
98.77 
5 
0.0030 
99.43 
0.0044 
99.22 
6 
0.0029 
99.73 
0.0042 
99.65 
As observed in Table 1, eigenvalues come in pairs of comparable magnitude and the major portion of the fluctuation energy is captured by the first two modes of velocity and temperature. The first four modes for both velocity and temperature fields contain almost all flow and temperature fluctuation energy. It is found that first two most energetic eigenfunctions for each field variable contain the large scale features of the thermofluid system while higher modes, with lesser energy level, capture the small scale features. These organized spatiotemporal structures occur in pairs with a phaseshift of approximately a quarterwavelength in the streamwise direction. The same observations are true for all velocity and temperature eigenfunctions. Eigenfunctions and coherent structures of the thermofluid system are identified as standing and travelling waves respectively.
A lowdimensional dynamical system of nonlinear ordinary differential equations is obtained by using the empirical eigenfunctions as basis functions in a truncated series expansion and applying Galerkin projection (GP). The system of nonlinear ODEs is the representative form of the LDMs of the thermofluid system under investigation. The number of modes retained in the series expansions for both modes determines the size of the reduced models. It is found that expansion coefficients computed based on lowdimensional dynamical model's predictions and by direct projection of the full model simulation data are found to be in very good agreement. Four modes for each field variable is the smallest set capable of predicting stable, selfsustained oscillations with correct amplitude and frequency. Retaining fewer modes in the truncated series expansion either does not produce stable selfsustained oscillations in time or fails to predict the correct amplitude of the oscillations. Keeping more modes than necessary in the expansion may reduce the accuracy and restrict the validity of the LDMs due to the noise introduced by the lowenergy level higher modes. At Reynolds numbers close to the "design" conditions, LDMs successfully estimate the dynamical attributes of the thermofluid system. At higher Reynolds numbers, the reduced models predict quasiperiodic route to chaos.
The developed lowdimensional models may be used in the description of dynamical behavior of spatiotemporal structures, in the bifurcation and stability analysis of transitional thermofluids systems in complex configurations, in the study of "offdesign" system dynamics and in exploring the active and passive intelligent flow control ideas with the use of artificial neural networks (ANNs).
Key Words: transitional flow; forced convection in grooved channels; proper orthogonal decomposition; KarhunenLoeve expansion; empirical eigenfunctions; spatiotemporal (coherent) structures; lowdimensional (order) models; quasiperiodic route to chaos.
REFERENCES
 J.L. Lumley, 'The structure of inhomogeneous turbulent flow', In Atmospheric Turbulence and Radio Wave Propagation, (ed. A.M. Yaglom and V.I. Tatarski), Nauko, Moscow, 160178 (1967).
 G. Berkooz, P. Holmes and J.L. Lumley, 'The proper orthogonal decomposition in the analysis of turbulent flows', Annual Review of Fluid Mechanics, 25, 539575 (1993).
 L. Sirovich, 'Turbulence and dynamics of coherent structures: IIII.', Quarterly of Applied Mathematics, 45, 561590 (1987).
 A.E. Deane, I.G. Kevrekidis, G.E. Karniadakis, S.A. Orszag, 'Lowdimensional models for complex geometry flows: application to grooved channels and circular cylinders', Physics of Fluids A, 3(10), 23372354 (1991).
 R.A. Sahan, H. Gunes and A. Liakopoulos, 'Lowdimensional models for coupled momentum and energy transport problems', In Cooling and Thermal Design of Electronic Systems, (ed. C. Amon), ASME HTD319/EEP15 , 115 (1995).
 R.A. Sahan, A. Liakopoulos and H. Gunes, 'Reduced dynamical models of nonisothermal transitional grooved channel flow', Physics of Fluids A, 9(3), 551565 (1997).
 H. Gunes, R.A. Sahan and A. Liakopoulos, 'Lowdimensional representation of buoyancy driven flow in a vertical channel with discrete heaters', In Enhancing Natural Convection Cooling of Electronic Systems and Components, (eds. A. Ortega and S.P. Mulay), ASME HTD303, 125137 (1995).
 H. Gunes, A. Liakopoulos and R. A. Sahan, 'Lowdimensional description of oscillatory thermal convection: the small Prandtl number limit', Theoretical and Computational Fluid Dynamics, inpress, (1997).
 Liakopoulos, P.A. Blythe and H. Gunes, 'A reduced dynamical model of convective flows in tall laterally heated cavities', Proceedings of the Royal Society of London A, 453, 663672 (1997).
INTEGRAL TRANSFORM SOLUTION OF THE TURBULENT BOUNDARY LAYER EQUATIONS IN STREAMFUNCTIONONLY FORMULATION
L.C.G. Pimentel ^{(*)}, RM. Cotta ^{(*)}, and S. Kakaç
Department of Mechanical Engineering
University of Miami
P.O. Boz 248294  Coral Gables
Florida  331240624  USA
^{(*)}Permanent address:  Laboratory of Transmission & Technology of Heat
Mechanical Engineering Dept.  Universidade Federal do Rio de Janeiro
EE/COPPE/UFRJ
Cidade Universitária  C.P. 68503  Rio de Janeiro, RJ  21945970  BRASIL
ABSTRACT. Developing turbulent flow between parallel plates is analyzed, by making use of the
generalized integral transform technique to obtain a hybrid numericalanalytical solution of the
governing boundary layer equations for incompressible steady flow. The streamfunctiononly
formulation is preferred over the primitive variables one, in light of the improved convergence rates
achievable, for the related eigenfunction expansions, with the former choice. Also, a wellestablished
algebraic turbulence model is adopted for the required flow diffusivities, and numerical examples are
employed to illustrate the computational performance of the proposed approach.
THE UNSTEADY ONEDIMENSIONAL ADVECTION MODELLING USING CCMC SCHEME
HuanLin Luo
Dept. of General Courses
Kaohsiung Polytechnic Institute, Taiwan, R.O.C.
INTRODUCTION
In recent years, many advances have been made in the study of the advective modeling methods with higher accuracy and computational efficiency for the advection equation. These methods are sometimes named as 'shockcapturing' schemes, or as 'TVD(totalvariation diminution)' which are referring to the oscillationsuppression . To achieve a satisfactory numerical modeling of convection is always a dilemma to the researchers in the field of computational fluid dynamics. On the one hand, the secondorder methods or other higherorder methods give highorder accuracy, but at the same time, nonphysical oscillatory behavior in regions where steep gradients exist may be generated. Central difference scheme(CDS) is one of the example. This scheme may lead to unrealistic oscillatory behavior in an implicit solution or to nonconvergent solutions in an explicit computation in regions where convection strongly dominates . On the other hand, using the classical firstorder upwind difference scheme(UDS), the oscillations will be replaced by the unacceptable global artificial diffusion. In 1979, proposed the quadratic upstream interpolation for convective kinematics scheme (QUICK) for quasisteady flow situations. The methods have the desirable simultaneous properties of high accuracy, inherent numerical convective stability, algorithmic simplicity and a physically appealing foundation. Bram Van proposed the MUSCL approach, which stands for monotone upstreamcentered schemes for conservation laws. In this method, in order to generate higher spatial approximations, the state variables at the interfaces are obtained from an extrapolation between the neighboring cell averages.
In this paper, a new technique, Characteristic Convective with Monotonic Conservation (CCMC) Scheme, is proposed. The procedures of the method in solving the pure one dimensional convective equation are presented. Finally, a set of benchmark test problems, which includes three critical test profiles and has been tested in Leonard's , is used to exam the new scheme.
THE MATHEMATICAL FORMULATION
Consider the unsteady onedimensional pure convection equation of a scalar, ,
(1)
Where u is velocity. We consider a 1D domain divided uniformly in the xdir at points for with . Eqn. (1) can be integrated in time over a time step and in space over a control volume at station from to . This gives
where is the Courant number. The subscript notation refers to the west and east side of the control volume faces. The superscript designates the time level. How to determine a better convective interpolation of the control volume faces is a key in solving advective equation. In the following section, CCMC provides another technique in finding find face values.
THE PROPOSED METHOD
For purely convective problems, as the case here, the solution of from the Eqn.(1) will read
If the scalar versus position x is plotted at different time periods, the curve of will move along the characteristic direction with a convective speed u at different time steps. The west face of the ith node of control volume is the face of interest and is denoted as . The variation of the versus time t can be plotted into a curve. This curve is called the convective characteristic curve. The convective characteristic curves for slope and curvature of can be construct in the similar way. These two curves provide a tool to construct the convective interpolation,, without phase angle shift. The details will be discussed in the following section. However, the overshoot and undershoot can be reduced by the consideration of the local behavior of the profile. This could be solved by the help of the normalized variable diagram and the 'limiter'.
BENCHMARK TEST PROBLEMS AND RESULTS
There are three bench mark test problems examined in Leonard's . They are unit step function, sinesquared wave and semiellipse function. These are selected on the basis of simplicity and ease of reproducibility, and are intended to represent basic characteristics of behavior that might be encountered in practice. Finally, the results obtained from the proposed method, characteristic convective with monotonic conservation, are compared with those results presented in Leonard's , CDS, UDS, and MUSCL. Overall performance is slightly better than the MUSCL at the smaller Courant number. This technique gives sharp results for the step simulation at all Courant numbers. It also generates a degree of artificial staircasing for sinesquared and semiellipse simulation at all Courant numbers.
REFERENCES
 Roe (1985). Some Contributions to the Modelling of Discontinuous Flows. In E. Engquist, S. Odher, and R.J.C. Sommerville, editors, Proc. 1983 AMSSIAM Summer Seminar on Large Scale Computing in Fluid Mechanics, Lectures in Applied Mathematics., pages 163193. Philadelphia, 1983. SIAN.
 Roache. Computational Fluid Dynamics. Hermosa Publishers, Albuquerque, NM, 1972.
 Leonard (1979). A Stable and Accurate Convective Modelling Procedures Based on Quadratic Upstream Interpolation. Comp. Math. Appl. Mech. Engng., 19:5998, 1979.
 Van Leer (1977) Towards the Ultimate Conservative Difference Scheme. V. A Second Sequel to Godunov's Method. J. Comput. Phys., 32:101136, 1979.
 Leonard (1991). The ULTIMATE Conservative Difference Scheme Applied to Unsteady OneDimensional Advection,. Computer Methods in Applied Mechanics and Engineering, 88:1774, 1991.
ADOMIAN'S METHOD APPLIED TO HAMMERSTEIN INTEGRAL EQUATIONS ARISING FROM CHEMICAL REACTOR THEORY
N.M. Madbouly,
G.F. Roach,
D.F. McGhee
Department of Mathematics ,
University of Strathclyde
Glasgow, G1 1XH.
ABSTRACT
We consider the mathematical model for an adiabatic tubular chemical reactor which processes an irreversible exothermic chemical reaction. For steady state solutions, the model can be reduced to the ordinary differential equation
with boundary conditions
where
(see [1]). The unknown represents the steady state temperature of the reaction, and the parameters , and represent the Peclet Number, the Damkohler Number and the dimensionless adiabatic temperature rise respectively. This problem has been studied by numerous authors (e.g. [1], [2], [3]) who have demonstrated numerically the existence of solutions (sometimes multiple solutions), for particular parameter ranges.
In order to develop general results concerning the solution of (1)(2), we convert the problem into an Hammerstein integral equation
where is defined by
and
Existence and uniqueness results are achieved for specific ranges of the parameter .
Our main concern is to consider the application of a novel numerical procedure known as Adomian's Method to approximate this unique solution. It has been claimed that this method is particularly well suited for the solution of Hammerstein integral equations ([4]). An advantage of this method is that it produces an analytic approximation to the solution, i.e. a function defined on [0,1] , rather than approximate numerical values at a discrete set of points.
Adomian's method is used for solving solving operator equations of the form
where is a nonlinear mapping from a Banach space into itself and is known. It is assumed that and can be expanded as infinite series
Here, , are the socalled Adomian's polynomials, (although, as we demonstrate, they are not always polynomials). depends on . By setting
*we formally solve (4), and have an iterative procedure allows the successive calculation of , .
This method is applied to the Hammerstein integral equation (3) in two cases: (i) and (ii) b = 0 . In the first case the method appear to converge rapidly, (although no convincing proof of convergence is known to us) and in the latter case the method gives the known unique trivial solution.
The solutions achieved in case (i) using Adomian's method is compared with numerical solutions achieved by application of the Contraction Mapping Principle and by using a simple shooting method applied to the original boundary value problem (1)(2). For all calculations, the Computer Algebra Software, Maple, is used. Good agreement between the results of Adomian's method and the other methods is demonstrated. Since the Contraction Mapping Principle gives an iteration procedure which is known to converge to the unique solution, this strongly suggests that Adomian's method converges to the unique solution in this case.
References
 Poore, A.B. A Tubular Chemical Reactor Model. A Collection of Nonlinear Model Problems Contributed to the Proceeding of the AMSSIAM. p2831, 1989.
 Heinemann, R. Poore, A. Multiplicity, Stability, and Oscillatory Dynamics of the Tubular Reactor. Chemical Engineering Science. 36, 14111419. 1981.
 Heinemann, R. Poore, A. The Effect of Activation Energy on Tubular Reactor Multiplicity. Chemical Engineering Science. 37, 128131. 1982.
 Some, B. Some Recent Numerical Methods for Solving Nonlinear Hammerstein Integral Equations. Mathematical Computer Modelling. 18, 5562. 1993.
COMPUTATION OF TRANSIENT BUOYANCY DRIVEN FLOWS USING A FINITE ELEMENT PROJECTION TECHNIQUE
R.L. Sani* and D. Veyret**
*Dept. of Chemical Engineering, University of Colorado,
Boulder, CO 803090424 USA
** I.U.S.T.I., Université de Provence
5, rue Enrico Fermi 13453 Marseille cedex 13 France
A Galerkin finite element projection method for the numerical modeling of transient
buoyancy driven flows is presented. Time accurate solutions are obtained by using a
semiconsistent mass projection algorithm which is shown to have better phase speed accuracy
than similar finite difference or lumped mass finite element algorithms. Its accuracy is
discussed and validated by comparison of numerical experiments with an exact solution.
The mathematical model used for the system is the coupled linear momentum and energy
equations employing the Boussinesq approximation.
plus appropriate boundary and initial conditions.
The basis of the finite element based algorithm is the use of a projection method to
approximate the solution of the Boussinesq equations. One rewrites the equations in the
form:
plus appropriate boundary and initial conditions where
Then one can interpret the solution of the system in the following manner: given u, the
vector F(u,T) is known and can be projected on to the subspace of solenoidal vectors
and the subspace of curlfree vectors , i.e.,
where P is a projection operator. This viewpoint is equivalent to restricting the solution of
the equations to the space of solenoidal velocity vectors. The scheme utilized in the finite
element algorithm is an approximation wherein one utilizes an approximation to , for
example, and computes the intermediate velocity field from
then projects onto the space of solenoidal vectors via
and then accepts v as the solution at the current time. While this description contains the
essential features of the scheme, the actual scheme must address such issues as the
appropriate boundary conditions for and the appropriate form of .
A Galerkin finite element spatial discretization coupled with a semiimplicit temporal
discretization is used to discretize the equations. The noteworthy features are that only a
sequence of uncoupled, symmetric, positive definite algebraic systems must be solved for
the temperature field and each velocity component and a discrete Poisson equation for the
projection and update. Thus, this forms the basis of an efficient scheme which is amenable
to the implementation of iterative solution techniques. Another noteworthy feature is that
the consistent mass matrices have been in the discretized linear momentum and energy
equations. This leads to an increased phase accuracy which is essential for accurate
modeling of convectively dominated flows and concomitant thermal transport.
Some numerical experiments validating the temporal accuracy of the scheme as well as the
numerical modeling of some complex 2D and 3D transient buoyancy driven flows will be
presented.
ON THE INTEGRAL TRANSFORM SOLUTION OF THE HEAT DIFFUSION EQUATION IN A SLAB WITH RADIANT FLUX AT THE OUTER SURFACE
A. G. Oliveira Filho. * and C. M. Hackenberg. **
* Military Institute of Engineering (IME), Chem. Eng. Dept.,
Praça Gen. Tiburcio 80, 22290270Rio de JaneiroRJ, Brazil.
** PEQ/COPPE/UFRJ, Chem. Eng. Dept.,
P.O. Box 68502, 21945970Rio de JaneiroRJ, Brazil.
ABSTRACT
In this work, some known solutions of the transient heat transfer equation in a slab with
thermal radiation are reviewed. One notices that the traditional way to deal with this kind
of problem is to obtain a lumped parameter transport equation from thermal energy flux
balance at the body's surface and to include global radiation as one of its terms.
Boundary conditions are either constant or homogeneous and the transient temperature
profile is thereafter obtained through integration.
Although many applications are found in the literature that utilize these classical
solutions, they will not provide accurate resuits in high temperature processes, such as in
elevated pressure combustion chambers. Lumped analysis is not possible due to Biot
Number limitations when diffusive transient behaviour plays an important role inside the
solid.
Here, the diffusion equation in a solid slab is analitically solved with radiant flux at the
outer boundary by the Integral Transform Technique and also by a functional method
that uses the BassetDuhamel Theorem for arbitrary boundary value problems. Thus,
treating the surface temperature as an arbitrary function of time, the problem may be
solved through computational codes. It's important to point out that the Integral
Transform Technique generates a SturmLiouville problem whose eigenvalues and
eigenfunctions are calculated by computation.
In a first step, the outer surface temperature function is solved utilizing a linearized form
of the boundary condition. The nonlinear boundary condition is also analyzed and the
solution is then worked out through a first order approximation of the resulting
functional integral equation at the outer surface. It is shown that the solutions converge
to the existing lumped parameter solutions for highly thermal conductive materials, as
expected. Hence, the procedure developed here is general enough to include all ranges of
Biot Numbers.
STEADY TEMPERATURE FIELDS IN 2D PARQUETTYPE MEDIA
Anvar R. Kacimov* and Yurii V. Obnosov *
*Institute of Mathematics and Mechanics,
Kazan University, Kazan, 420008, Tatarstan, Russia
Media composed of doubleperiodic phases with different (arbitrary) conductivity are ñonsidered. Within each phase temperature is a harmonic function and along interfaces two conjugation conditions hold (continuity of temperature and normal flux component) as well as periodicity conditions along the boundaries of elementary cells. For specific examples of composites (chequerboard, regular triangles, rectangular inclusions) explicit analytic solutions are derived in terms of the thermogradients. For other cases standard FDM procedures in terms of temperature are used and compared with analytic formulae. Bulk values of the field (total dissipation, effective conductivity) are calculated and compared with the approximate results of Rayleigh, Wiener, Odelevskii, Dul'nev and Zarichnjak, Bakhvalov and Panasenko. For the chequerboard composite the hodograph of effective conductivity is shown to coincide exactly with an ellipse while effective resistivity is a curve inversely symmetric with this ellipse about a circle. For the case of quadrangular inclusions, and for the specific case of a field oriented along one of the sides, the value of effective conductivity is shown to vary from to 1 /.
As limiting cases of parquets, the temperature fields are studied above the cooling surfaces. For the case of ideally conducting walls periodic fingers of optimal dissipation potential are derived and shown to coincide with the classical contours of PolubarinovaKochina. In the class of semielliptic protrusions with an arbitrary conductivity ratio of the wall and environment, two nontrivial local extrema of total flux exist alongside two global ones. Rectangular fingers are shown to exhibit intuitively obvious behaviour at the large scale but less trivial field distribution exist near these tips. Another limiting pattern derived from the chequerboard composite is a fault in the standard layering. This defect is shown to provide more intensive conduction in one area of the composite and less intensive in others. For isolated inclusions in homogeneous matrixes (which model caverns or macropores) the nontrivial interaction between two neighboring cavities is studied. For the case of heat dissipation from a wall underlying flow of a viscous fluid the Couettetype regime is considered, and optimal riblets are established. Shape optimization is performed for inclusions placed in 2D flows with the corresponding total heat flux depending on the Peclet number.
REFERENCES
 Kacimov, A.R. Optimization of the protrusion shape for a Couette type flow, Optimal Control Applications and Methods, Vol. 15, pp 93203, 1994.
 Kacimov, A.R., and Obnosov ,Yu.V., Explicit, rigorous solutions to 2D heat transfer: Two component media and optimization of cooling fins, International J. Heat and Mass Transfer, Vol. 40, pp 11911196, 1997.
 Kasimov, A.R., and Obnosov, Yu.V. Explicit solutions to problems of shape and regime optimization in steady heat transfer, Proceedings of I Russian Natl. Conference in Heat Transfer, Moscow, Vol.8, pp 9196, 1994.
 Obnosov ,Yu.V., Solution of a Markushevitch problem in the class of doublyperiodic functions with orthogonal periods. Sov.Phys.Dokl., Vol. 36, pp 7374, 1991.
 Obnosov , Yu.V., Solution of a problem of Rlinear conjugation for a triangular regular checkerboard field, Sov.Phys.Dokl., Vol. 37, pp 4648, 1992.
 Obnosov ,Yu.V., Exact solution of a problem of Rlinear conjugation for a rectangular checkerboard field. Proc. Royal Soc. London A, Vol.452, pp 24232442, 1996.
VERIFICATION OF TEMPERATURE PROFILES IN POROUS MEDIA WITH EXPERIMENTAL DATA
S. S. Helvacı, S. Yapar
Abstract is not available.
THREEDIMENSIONAL MODEL OF HEAT TRANSFER IN FOAMED CHARS
Igor S. Reshetnikov and Nikolay A. Khalturinskij
Polymer Burning Laboratory,
Institute of Synthetic Polymeric Materials, 70 Profsoyuznaya str., Moscow, 117393, Russia
Introduction
Among the fire retardant systems^{1} there exist a peculiar class  socalled "intumescent systems", which posses the property to produce on their surface under conditions of external heat flow protective coke layer  foamed char (FC)^{2}. This layer can be tens times thicker than initial coating and protect polymer from the external influence. Practically all authors, who investigate behaviour of intumescent systems, agree that heatprotective function of FC is one of the main factors which explain great efficiency of intumescent coatings. Moreover, it has been shown^{3} that efficiency of the intumescent fire retardant system correlate with the thermoprotection properties of foamed char.
There exist a number of works devoted to numerical simulation on intumescent system burning^{47} and investigation of thermal conductivity of foamed char layer^{89}. However practically in all existing works foamed char cap was considered as a uniform medium. But it is known that radiation can play a significant role and affect on the heat transfer processes in porous media^{9}. So we need in mathematical model of heat transfer in foamed chars which can accurately takes into consideration as all mechanisms of heat transfer (conduction, radiation etc.) as well as influence of external factors  heat losses through substrate, external heat flow and so on. The aim of present work is to present threedimensional model of heat transfer into porous media at high temperatures and conduct some analysis of contribution of each heat transfer mechanisms into total heat flux.
Experimental
Thermoprotection properties (TPP) of FC the was investigated by the following method. The sample (char plate 37 cm thickness) was placed on the asbestos table. Up surface was subjected by the continuous heat flow from CO_{2}laser (10.6 nm wavelength, 5 W/cm^{2} power of the beam). The responce of the thermocouple (Pt  Pt+10%Rh, 50 mkm wire diameter), located on the back side of the sample, vs. time was registered.
Theoretical model
FC was considered as a porous medium. Total volume was separated on the cube "element cells". The edge length of each cell was 10^{2} mm. It was Each cell can be either empty or filled by char material. The initial distribution of filled cells was made in accordance with integral foamed char characteristics  their effective density, average pores diameters, etc. Pores with fixed diameter d_{0} were distributed in the model char cap in random order. Vertical cut of the model char cap presented on the fig. 1.
Heat transfer was computed in the follows way. If two filled cells are separated by some (one or more) empty then heat flow between them will be
,
where l _{aer}  heat conductivity of the air, n  number of empty cells, T_{i}  temperatures of the cells, e  blackness coefficient of the char, s  StefanBoltzman constant.
In the case of contacted filled cells, their heat exchange will be
,
where l _{c}  heat conductivity of the char material.
For the cells, located on the up surface external heat flow Q will give their addition to the heat flow to the cell. For these cells heat flux from the up side will be the next:
Note, that in this expression external heat flow supposed to be recalculated on the cell facet area, i.e. d_{0}^{2}.
The lowest layer, which contact with substrate supposed totally filled. In the expression of heat flow to this layer we need to introduce the member, which will describe heat losses through substrate. In present model this member was written in the follows manner:
,
where g is the coefficient of heat losses, T_{0}  environmental temperature.
For the each time step cells temperatures was computed. Than in accordance with above equations we can compute total heat flux to the each cell and from these data we can compute new cells temperatures. Initial distribution of temperatures was supposed stationary and for each cell equal to T_{0}. Time interval in calculations was 10^{2} s.
In such model the average temperature on the lowest layer will represent the response of thermocouple, located on the back side of the sample. In further discussion as TPP we'll use the maximal temperature of this layer. The char in the model is considered as indestructible  it provide existence of this temperature.
Results and discussion
In order to compare experimental and theoretical results TPP of real intumescent system on the base of carbamideformaldehyde resin with intumescent addition (PER+APP) have been investigated. This systems is notable for the uniform foamed char, which it produce during burning. Comparison of the experimental and theoretical data indicates that supposed model provide good agreement with experiment. All calculations were made on the base of real char characteristics.
In order to study heat transfer in foamed chars we should answer some questions. First, we need to know the influence of FC structure on their TPP. Secondly, the role of substrate heat conductivity should be examined. And the last  we have to know the contribution of each heat transfer mechanism (radiation, conduction, etc.) to the total heat flux. When we'll be able to answer all these questions we'll can to conclude that we know all features of heat transfer in FC. Some aspects of these questions already have been discussed in detail^{9}.
First of all let us study the influence of char structure on their TPP (maximum temperature of the last layer). From the analysis it can be seen the decreasing of pore diameters lead to decreasing of back side temperature. It should be noted here that contents of air in FC was supposed more than 70%, this point has been previously examined^{10}. Behaviour of described dependence shows us one important aspect of heat transfer  radiation play a significant role. Indeed, if we separate two heated plate by third (with intermediate temperature), it will lead to decreasing of radiation flow, but the conductive flow remain practically unchanged. Increasing of FC cap height lead to decreasing of back side temperature. In first rough approximation we can speak about liner dependence of back side temperature vs. char cap height and average pores diameter.
Some words about influence of substrate. In order to take it into account we introduced in our model coefficient of heat losses through substrate g . In fact this term represent thermal conductivity of the substrate. From the examination of temperature response for equivalent char caps on different substrates we can conclude that in first approximation value TT_{0} is reverse proportional to the g . On the base of this expression we can estimate efficiency of the intumescent coating on any substrate in the case when we know it for any particular substrate.
References
 Aseeva, R.M. and Zaikov, G.E., Combustion of polymeric materials, Nauka, Moscow, 1981
 Camino, G., Costa, L., Martinasso, G., Intumescent Fireretardant Systems, Polymer Degrad. and Stab., Vol. 23, pp. 359376, 1989
 Reshetnikov, I., Antonov, A., Rudakova, T., et al., Some aspects of intumescent fire retardant systems, Polymer Degrad. and Stab., Vol. 54, pp. 137141, 1996
 Gibov, K.M., Zhubanov, B.A., Dovlichihin, T.H., et al., About fire retardation mechanism of intumescent paints, Proceedings of Int. conf. "Nehorlavast Polymernych Materiolov", Bratislava, 1976, pp. 6971
 Buckmaster, J., Anderson, C., Nachman, A., A Model for Intumescent Paints, Int. J. Eng. Sci., Vol. 24, No. 3, pp. 263276, 1986
 Zverev, V.G., Isakov, G.N., Nesmelov, V.V., et al., Heat Transfer Mechanism and Fire Insulation Properties of Some Intumescent Materials, J. of Polym. Mat., Vol. 20, No. 12, pp. 91101, 1993
 Butler, K., Baum, H., Kashiwagi, T., ThreeDimensional Modeling of Intumescent Behavior in Fires, Proceedings of 5^{th} Int. Symp. on Fire Saf. Sci., Melbourne, 1997
 Anderson, C.E., Ketchum, D.E., Mountain, W.P., Thermal Conductivity of Intumescent Chars, J. of Fire Sci., Vol. 6, pp. 390410, 1988
 Reshetnikov, I.S., Khalturinskij, N.A., About modelling of intumescent systems burning, Russian Journal Chem. Phys., Vol. 16, No. 2, pp. 102107, 1997
 Gnedin, Ye., Kozlova, N. et al. Structure of foam cokes formed in pyrolysis and burning of polymers containing foaming fireretardant systems, Vysokomolek. Soed., Vol. 33(A), pp. 15681575, 1991
 Kanary, K., Heat conductivity of high polymers, Denki sikinse hose daigaku, Vol. 176, 1973
SOIL DRYING PROCESS IN PERIODIC AND NONPERIODIC GEOMETRIES: APPLICATION TO BURIED POWER CABLES
Ligia D.F. Marczak and Simone Sebben
Chemical Engineering Department  UFRGS
Rua Luis Englert s/n, Porto Alegre, RS, 90.040040, Brazil
The work reported in this paper is concerned with the moisture migration process that causes drying of the soil involving buried electrical power cables. Numerical simulations have been performed for periodic and nonperiodic geometries, and for two types of soils: one natural and other artificial. The results showed that, for initial times, the two geometries give very similar temperature and moisture content profiles in the vicinity of the cable. It was also observed that for the natural soil used the drying process is more intense than for the artificial soil.
INTRODUCTION
The study of simultaneous heat and mass transfer processes in soils is of great interest in different engineering and environmental applications. The work reported in this paper is mainly concerned with the heat and mass transfer process that occurs in soils that involve underground electrical power cables. The electrical current passing through these cables generates heat which has to be dissipated by the surrounding soil in order to keep the cable temperature at an acceptable level. The heat generated and dissipated by the cable induces a moisture migration in the soil surrounding it. This, in turn, increases the thermal resistance of the soil which starts to behave as a thermal insulator. As a consequence, the temperature at the surface of the cable can reach values which are higher than the maximum operating level, causing damage, and even rupture, of the cable. Considering the high costs involved in buried power cable systems, it is of interest to understand and to predict the behavior of the moisture front migration in the surrounding soil. In this work, two types of soils, one natural and other artificial (backfill), and two distinct geometries, one periodic and other nonperiodic, were investigated. The main objective was to compare the behavior of the temperature and moisture content profiles for the two geometries studied.
PROBLEM STATEMENT
The governing equations of the problem are the energy and moisture content transport equations, which for unsaturated porous media can be written as^{1}:
(1)
(2)
where T is temperature and q is the volumetric moisture content (ratio between the liquid volume and the total volume of the medium); t is time and x and y are the spatial coordinates; C is the volumetric heat capacity; k_{*} is the soil apparent thermal conductivity; r is the density of the liquid; h_{lv} is the latent heat of vaporization; DTv and Dq v are the difusivities of the vapor associated with the temperature and moisture content gradients, respectively; DTl and Dq l are the difusivities of the liquid associated with the temperature and moisture content gradients, respectively; and KH is the hydraulic conductivity of the porous media. These governing equations are based on the Philip and De Vries model^{2 }. The physical properties and diffusivities were considered to vary with both temperature and moisture content of the porous material. In this work, the expressions used for the computation of these properties were obtained experimentally by Hartley et al.^{3}
For simplicity, only cables with square cross section were considered. The geometries of interest are schematically represented in Figs. 1(a) and (b). For the periodic situation (Fig. 1 (a)), the calculation domain was restricted to a single cable, with cyclic conditions applied along lines AB and CD. Fig. 1 (b) represents the nonperiodic situation with three cables buried. For this case, due to symmetry along line EF, only half of the geometry was included in the calculation domain. The surface of the cable is impermeable to mass flux and is maintained at a constant temperature T_{c}. Other boundary conditions can be seen in Figs. 1 (a) and (b). Initially the soil is at a uniform temperature T_{i} and uniform moisture content q _{i}. Two initial values of q _{i}, representing intermediate and practically saturated soils, were analyzed. Results are presented in terms of moisture content isocurves, and nondimensional temperature and moisture content profiles for different stages in time and for the two geometries considered.
COMPUTATIONAL DETAILS
The governing equations were discretized using a Finite Volume Method with the fully implicit scheme for time discretization. Computations were performed with nonuniform grids and varying time steps. Two linebyline iterative algorithms were used to solve the set of governing equations: The tridiagonal and cyclic tridiagonal matrix algorithms (TDMA and CTDMA). The cyclic algorithm is required for lines of nodes along which the periodic boundary conditions apply. At each time step, convergence of the iterative solution procedure was assumed when the sum of the absolute values of the residues of the governing equations were less than 10^{8}.
RESULTS AND DISCUSSION
Fig. 2 shows a comparison of the moisture migration front (along a vertical line passing through the center of the cable) for the case of the native soil with an intermediate initial moisture content (q _{i} = 0.15), and the periodic and nonperiodic geometries studied. Immediately close to the cable, and for all times, the moisture content always presents the smallest values due to the high temperature and the heat dissipated by the cable. For initial times (2.4 hours), it is observed an increase of the moisture content above the initial value in the region nearby the cable. This is because, at the beginning of the process, the water present in the soil close to the cable evaporates and starts to migrate. Eventually, this vapor encounters colder regions and condenses, thus increasing the moisture content of that particular region. For longer periods of time (300 hours), the increase of the moisture content above the initial value, occurs in regions further away from the cable. It is also observed in Fig. 2 that, initially, the two geometries present practically the same moisture profiles. However, as time evolves, the periodic geometry presents slightly higher levels of moisture content in the adjacent vicinity of the cables. This happens because, in reality, the periodic geometry represents a situation of several cables buried in an inline arrangement, rather than the three cables represented by the nonperiodic geometry. Thus, it would be expected that with the periodic conditions imposed, the moisture content would be somewhat higher in the cable surroundings.
CONCLUSION
A numerical investigation of the drying process that occurs in soils in the neighborhood of buried power cables considering periodic and nonperiodic geometries has been presented. The results have shown that, for initial times, the two geometries give very similar temperature and moisture content profiles. However, as time is advanced, the periodic geometry gives rise to higher temperature and moisture content of the soil in the neighborhood region of the cables. It was also observed that for the natural soil used, the drying process is more intense than for the artificial soil or backfill.
ACKNOWLEDGMENTS
The authors wish to thank the support of the Brazilian government (CNPq) and the Research Foundation of Rio Grande do Sul (FAPERGS).
REFERENCES
 Damasceno Ferreira, L.S., Heat Transfer and Moisture Content Migration in Soils Surrounding Buried Power Cables, Ph.D. Thesis, Mech. Eng. Dept., UFSC, Brazil, 1993 (in Portuguese).
 Philip, J.R., De Vries, D.A. , Water Movement in Porous Media under Temperature Gradients, Trans. American Geophysics Union, Vol. 29, pp. 222232, 957.
 Hartley, J.G., Black, W., Bush, R.A., Martin Jr, M.A., Thermal Stability of Soils Adjacent to Underground Transmission Power Cables, Technical Report, Georgia Institute of Technology, 1982.
Fig. 1  Geometries investigated (not to scale): (a) periodic, and (b) non periodic.
Fig. 2  Moisture migration front along the y coordinate.
A STUDY ON THE BOUNDARY ELEMENT SOLUTION OF HEAT CONDUCTION PROBLEMS
İ. Ekmekçi, N. Sözbir, Ü. Uysal, İ. Çallı
Department of Mechanical Engineering, Sakarya University, Sakarya, Turkiye
ABSTRACT This paper is concerned with the numerical solution of two dimensional steady state heat conduction problems with Boundary Element Method. In order to obtain an approximated solution of a boundary value problem with Boundary Element Method, influence type functions which are called fundamental solutions are used as weighing functionsand in this manner, approximate solution is fully satisfy our governing equation in the domain exactly, while this approximated solution is being forced to satisfy weighted integral on the boundary. The development of the method is generalized to include the first, second and third kind of boundary conditions as well as nonlinear radiation conditions. A variety of illustrative problems are analyzedwith this method and their solutions are compared to those obtained analytically. The solutions for the examples indicate that the Boundary Element Method is very efficient and accurate for solving most of the condition problems. This method has no inherent limitations such as the geometric complexity, kind of boundary conditions. As in most of the practical calculations of heat transfer, boundary fluxes and temperatures are only needed informations. However complete temperature distribution can be directly obtained with minimum effort.
A NEW ALGORITHM FOR THE SOLUTION OF HEAT CONDUCTION PROBLEMS WITH THE FINITE DIFFERENCE METHOD
H. İ. Saraç* , N. Sözbir**, İ. Ekmekçi**, Ü. Uysal**, İ. Çallı**
* Department of Mechanical Engineering,
Kocaeli University, Kocaeli, Turkiye
** Department of Mechanical Engineering,
Sakarya University, Sakarya, Turkiye
ABSTRACT In this paper, a new application of solving systems of linear equations obtained by applying the finitedifference method to twodimensional steady state conduction problems is proposed. This proposed new algorithm is composed of three steps: (i) transformation of the original system to an auxiliary linear system, (ii) solution of this auxiliary system by GaussSiedel iteration method, and (iii) determination of the roots of original system using the solution vector of the auxiliary linear system.