Session 12d5 Formulation And Development Of Mathematical Models

Session 12d5 Formulation And Development Of Mathematical Models For Engineering Problems: An Active-Inductive Approach Marcius F. Giorgetti School of Engineering at São Carlos-SHS University of São Paulo 13560-970 São Carlos, SP, Brazil. Abstract - The paper describes the advances in methodology, materials, and status for an engineering undergraduate course designed to fill the gap between the basic freshman year (two years in Brazil, for five-year engineering programs) and the following Engineering Science courses. A preliminary report on this effort has been presented at FIE-97 under the title “Formulation and Development of Mathematical Models for Engineering Problems: an Experience on the Integration of Theory and the Laboratory”. Up to 1997 the course had been offered as an elective in two Brazilian universities. Three different engineering schools have now incorporated the course into their regular curriculum. The project persists on its investment on the poor integration between the so called “theoretical courses” and “laboratory courses”, by addressing the two issues consecutively. Moreover, it opts, whenever possible, for a methodology that could be described as both inductive and active. Experiments are planned to be easily performed in class with intensive student participation without, necessarily, a prior knowledge of a theoretical approach to the phenomenon under analysis. Emphasis is given to the fundamental sequence of observation of facts or phenomena, election of parameters to quantify them, measurement and data acquisition, and finally, modeling (formulation) and simulation. The methodology of modeling or formulation is established with a strong appeal to the laws of conservation. Emphasis is given to the splitting of the process into two steps, the use of a fundamental law, followed by the use of a subsidiary law or a constitutive relation. The problems are unsteady and discrete (lumped) in the first part of the course; then, continuous steady problems are analyzed, and finally continuous unsteady systems are tackled. The students are encouraged to develop and use their own numerical methods, from the beginning, for the solution of the equations of formulation. Only the part relative to the unsteady discrete problems was presented in the previous paper. The second and third parts of the program are explored in the present paper.

Introduction In the first part of the course the problems analyzed are unsteady and discrete (lumped); the ones presented in full in

the paper given at FIE-97 were: a) the emptying of the water in a reservoir (mass balance); and, b) the cooling of a cylindrical solid exposed to an air current (internal-energy balance). Several other examples were cited, such as, the change of the speed of a solid sliding against friction over a horizontal plane (balance of linear momentum), and the discharge of a condenser in a RC electric circuit (balance of electric charge). Other cases have been added, such as the washout of a tracer, quickly introduced into a constantvolume well-mixed reactor, by a steady solvent flow rate throughout the reactor (mass balance). In all of these cases the system response is characterized by the decreasing of a single parameter with time, with a common characteristic: fast initial time rate of variation, followed by a gradual slowing down and a tendency towards stabilization. For all of the examples discussed above, this is what one observes, respectively, for the behavior of different properties: water level, temperature, speed, electric charge, and concentration, all associated with some sort of potential difference. To each of these examples there correspond more complex problems, involving a continuum, in such a way that the observable property may be a function of time and position. Steady one-dimensional problems in continuous media are analyzed in the second part of the course. Numerical methods (finite differences), introduced in the first phase, are further developed and compared with methods based on physical discretization (lumped parameter analysis). As the two formulations produce systems of linear algebraic equations, some concepts of Linear Algebra and Matrix Algebra, as well as numerical methods for the solution of systems of linear equations are reviewed. In the third part of the course, unsteady non-uniform problems in continuous media are analyzed as a synthesis of the problems in the first two phases, making use of the same tools developed for them.

Permanent Non-Homogeneous Systems. In the previous examples a single parameter, for example, the depth in the water tank h or the temperature T of a solid, varied as a function of time. The systems were (supposed to be) homogeneous with respect to these parameters, that is, in the same instant all the points of the

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Session 12d5 water surface were at the same level and all the points of the solid ( an aluminum cylinder) were at the same temperature. The next category of problems to be discussed is that of the non-homogeneous systems, but at the steady state. Each of the previous examples can be used to generate the new, more complex, cases. The discussion is centered on the analysis of the problem of heat conduction in a fin (figure 1), exchanging heat by convection with the surrounding air, an extension of the energy balance problem of the first part. Fins are protuberances at the surface of a solid, used to increase the heat transfer rate between the solid and the fluid in which it is immersed.

T = Ta + (Tb − Ta )

coshβ(L − x) coshβL

(2)

where cosh is the function hyperbolic cosine and β2=h’p/kA. Temperatures of 43.5, 32.6, and 29.5 oC were measured, respectively, with thermocouples embedded in the fin at the positions x/L=1/4, 1/2, and 3/4. When the model given by equation 2 is adjusted to these data, a value of β=28.9 results, or, assuming the value k=16W/moC, found in handbooks for nickel-chrome steel, the value h'=26.7 W/m2.oC follows. In figure 2 the four experimental points are shown, as well as the curve predicted by equation 2 with β=28.9 . 70 60

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Figure 1. Heat conduction in fin with heat convection to the air.

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Figure 1 illustrates an experiment done in class involving the transfer of heat from a hot wall to the otherwise undisturbed ambient air, by conduction, through a stainless steel rod of length L=12.0cm, and by convection from the rod to the air. The curve shown in the figure represents the temperature profile T(x) along the fin. The normal section of the rod is square, with side a=8.0mm. The wall temperature was set equal to Tb=69oC, and the air temperature Ta=25oC. The differential formulation for this problem is a classic in the heat transfer literature and will not be developed here. The final result is the following differential equation,

d 2T h' p (T − Ta ) = 0 − dx2 kA

(1)

where k is the thermal conductivity of the fin, A is the area and p the perimeter of its normal section. The parameter h’ is the film coefficient, associated with the heat convection. One boundary condition is T=Tb at x=0. There are different possibilities for the second boundary condition; in this case, given the characteristics of the problem, the condition dT/dx=0 at x=L is adopted. The classical analytical solution to this formulation is:

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Figure 2. Steady state temperature profile along fin axis. The main function of this example, however, is the introduction of discrete methods for formulation and solution of problems of this kind, in particular aiming at the development of numerical schemes. Thus, in this point of the course a revision is made of a few numerical methods for the solution of systems of linear algebraic equations, including the use of matrix inversion and multiplication. For the present example, two different ways are used to produce approximate formulations based on discrete values of the temperature along the length of the fin. The first is mathematical discretization using the method of finite differences in equation 1. The result, a finite difference equation, is employed at determined number n of points (nodes) along the axis of the fin. Thus, n algebraic equations are obtained, involving n unknowns, the values of the temperature at the n nodes. The second way, the main objective of this section of the course, is the introduction of physical discretization (lumped parameters or finite volumes), and the reduction of the continuous problem onto a group of discrete problems similar to the one studied in the first part of the course. The fin is imagined as a set of independent elements, each one of them with its own homogeneous temperature. In this example these temperatures are constant (steady state), but the method will evolve to be able to handle each element as


Session 12d5 having a time dependent temperature as in the case of the aluminum cylinder in the first part of the course. Here, from the thermal energy balance for each element, an algebraic equation is produced. Overall, a set of n equations involving the temperatures of the n elements (attributed to the element’s center) is generated, a final result formally equivalent to the preceding one. Substituting finite differences for the derivatives in equation 1 one obtains:

Tn + 1 − 2Tn + Tn − 1 − β2 ∆x2 Tn = β2 ∆x 2Ta

(3)

An arrangement for the application of equation 3 to 9 nodes is shown in figure 3.

the positions x/L=1/4, 1/2, and 3/4, for which the experimental values of the temperature are available. As the state is steady, that is, the temperatures do not vary as functions of time, the models are very simple. For each element, an equation describing the conservation of energy is written, stating that during a time interval ∆t, the heat that flows into the element, by conduction, through its left face is equal to the sum of the heat leaving by conduction through the right face plus the heat leaving by convection through the sides into the air. As a particular law for the conduction of heat between neighbor elements one can use the equation for the heat transfer across a wall of surface area A and thickness equal to the distance from the center of an element to the center of the next. For the convection between the elements and the air, Newton’s law of cooling, already validated in the first part of the course, is used. This method, similarly to the first one, produces a system of nine algebraic equations with nine unknowns, the temperatures of the nine elements.

Non-Homogeneous Non-Steady Systems. Figure 3. Discretization with 9 nodes. Finite element method Equation 3 is easily applied to nodes 2 through 8. Temperature T1 is equal to Tb, thus known. The boundary condition at x=L is met by considering an auxiliary node, number 10, to the right of node 9, and making its temperature equal to T8. A system of nine equations and nine unknowns is thus obtained, and can be solved using the methods previously revised. Physical discretization is a bit more elaborate, but one does not need to previously know a differential formulation, a great advantage in the more complex problems of the third part of the course. Only the pertinent physical laws, general and particular, have to be known, as in the first part of the course, to obtain a formulation for the problem. Figure 4 shows a possible physical discretization for the present problem.

In the third part of the course, several spatially onedimensional problems, involving boundary conditions or internal terms variable with time, are worked out. Transient, harmonic, and general perturbations are included. As a last example in this paper, the fin of the preceding item is analyzed under a transient perturbation in the wall temperature. The initial experimental condition is a constant and uniform temperature at the wall and the fin, equal to the air temperature Ta=25oC. At time t=0 the wall temperature is brusquely risen to Tb=69oC and maintained there. Figure 5 shows the temperature evolution at the points of the fin coincident with nodes 3 and 5, as picked up by two thermocouples. The final measurements were T3=43.4oC and T5=33.5oC, very close to the corresponding values observed in the experiment run is steady state. 45 40 35 30 25

Figure 4. Discretization with 9 volumes. Lumped-parameter analysis. In this case the subdivision is not uniform; it has been conceived this way to force a positional coincidence for the centers of most elements and the corresponding nodes, in particular of the nodes 3, 5, and 7, which are coincident with

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Figure 5. Transient heating of fin. Experimental values at x/L=1/4 ( ) and x/L=1/2 ( ∆ ).


Session 12d5 Computational simulations can be developed, as in the former case, from the mathematical discretization of a differential equation, or through a physical discretization of the problem, followed by a lumped parameter analysis. The mathematical discretization is not a good option, since the basic differential model, a non-ordinary differential equation, is too advanced for the target students of the course. Physical discretization, on the other hand, with the use of the method of finite volumes, offers a very good pedagogical opportunity for the synthesis of the first and second parts of the course. For each fin element, the heat balance is now a little more elaborate than in the steady state case, and more similar to what has been done in the analysis of the cooling cylinder. Consider a time interval ∆t; the difference between the heat input through the left face and the heat output through the right face and the sides of the element equals the change ∆E in the energy content within the element, that is a function of its temperature. As before, with ∆t→ 0, for each element a differential equation of the following type results:

dTi = a i Tb + b i Ta + c i Ti − 1 + d i Ti + e i Ti + 1 dt

(4)

The coefficients ai, ..., ei are the same for elements 3 through 7, but differ for the other elements, as determined by the boundary conditions and by the sizes of elements 1 and 9. Substituting ∆Ti /∆t for dTi /dt, with ∆Ti = Ti,n+1 -Ti,n follows the relation

Ti, n + 1 = Ti,n + (a i Tb + b i Ta + c i Ti − 1,n + ... ... + d i Ti, n + e i Ti + 1,n )∆t

(5)

that permits the explicit computation of the temperature of the element i at the future time n+1, when its temperature and the temperature of the other elements are known at time n; or the relation

Ti, n + 1 = Ti,n + (a i Tb + b i Ta + c i Ti − 1,n + 1 + ... ... + d i Ti, n + 1 + e i Ti + 1, n + 1 )∆t

(6)

that does not produce an explicit value for Ti,n+1. The first scheme is simpler but has stability limitations. The second scheme is stable, but demands the solution of a system of 9 algebraic equations and nine unknowns at each time interval ∆t. This task is facilitated when the system is solved by matrix inversion and multiplication. The present example is solved this way using for computation an Excel spreadsheet, a simple but powerful tool. Figures 6 and 7 show the longitudinal temperature profiles T(x,t) at two different instants, t=200sec and t=800sec, and the temperature vs. time sequences for elements 1, 3, and 5 from the beginning to the two instants considered. The results match very adequately the data shown in figure 5.

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Figure 6. Instant temperature profile and time series at and up to t=200 sec.


Session 12d5

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Figure 7. Instant temperature profile and time series at and up to t=800 sec.

Bibliography Adams, Alan J. and Rogers, David F., Computer Aided Heat Transfer Analysis, McGraw-Hill, 1973.

Chapra, Steven C. and Canale, Raymond P., Numerical Methods for Engineers, McGraw-Hill, 1985.

Doebelin, Ernest O., Dynamic Analysis and Feedback Control, MacGraw-Hill, 1962.

Giorgetti M.F., Formulation and Development of Mathematical Models for Engineering Problems: an Experience on the Integration of Theory and the Laboratory, Proceedings, FIE-97, 1997. Simon, William, Mathematical Techniques for Biology and Medicine, Dover Publications, Inc., 1977. Russell T.W.E. and Denn M.M., Introduction to Chemical Engineering Analysis, John Wiley, 1972.
