Implementation of TOPOPT in a Microfluidic Heat Transfer Model

Transcription

1 Microfluidics Theory and Simulation, c33443 Implementation of TOPOPT in a Microfluidic Heat Transfer Model Kristian Krüger Grøndahl, (s011426) and Thomas Glasdam Jensen, (s011427) Supervisor: Henrik Bruus MIC Department of Micro and Nanotechnology Technical University of Denmark 22 June 2005

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3 Abstract The material distribution method in topology optimization can be applied in order to solve numerous physical problems numerically. we have used a modified version of a recently developed high-level programming implementation of topology optimization in order to model and simulate heat transfer in cross-flow in two dimensions. Inspired by the article Constructal multi-scale cylinders in cross-flow by T. Bello-Ochende and A.Bejan [1], where the finite elements method (FEM) is used to calculate the effect of cooling cylinders in a liquid flow. We have performed a vast amount of simulations using FEMLAB and MAT- LAB in order to determine the optimum placing of heat dissipating material in the cooling liquid. Our work has shown that it is possible to implement the model and simulate it in MATLAB with FEMLAB but fast processors and a huge (minimum 1 GB) RAM facilities are needed to carry out the iterations in a reasonable time. We have also stated that the optimum shape of the material to cool is far from cylindrical but has large surface areas which is expected in order to get the biggest contact area where the heat transport takes place. iii

4 iv ABSTRACT

5 Contents 1 Introduction The system Inspiration Distributing material in channel Theoretical Aspects Fluid Dynamics The continuum hypothesis The Navier-Stokes equation HeatEquation The γ-field Dimensionless equation Implementation of TOPOPT Setting up Femlab Adapting the code Heat dissipation The Objective function Global effects of local changes Permeable material Adjusting the material parameters The conservative heat equation Usolve Results Conservative Vs. Non-conservative Making Q independent of the γ-area Lowering Darcy s number Varying C p Varying k Varying ρ Varying the initial conditions Further work 29 v

6 vi CONTENTS 6 Conclusion 31

7 Chapter 1 Introduction In development of Lab On a Chip (LOC) systems cooling can be of high importance. Detecting bacteria on chickens requires amplification of DNA which is done with PCR (Polymerase Chain Reaction) on chip, basically copying of DNA. Cooling is needed in order to control the fluid temperature to a high precision. The denaturation of the DNA (splitting the double stranded DNA) involve one temperature and the enzymatic reactions taking care of copying DNA involves another. This has lead to the idea of optimizing the cooling process, basically the heat transfer between a cooling liquid and a heat dissipating material (e.g.iron or cobber). Looking into the problem it is clear that the shape and thermodynamically properties plays a superior role. In this report it is discovered how the transfer of heat into a cooling liquid can be optimized by choosing a best-shape of the heaters. 1.1 The system The idea is to setup a closed channel with inlet and outlet and stalactites-like tubes going from top to bottom. Inducing a flow of a cooling liquid will transfer the heat from the stalactites, out of the system by convection. The questions are how should the shape of the stalactites be and what should be their individual placement Inspiration A group of scientists have worked with heat transfer in cross-flow as well [1]. In 2 dimensions with the channel seen from above, they have placed material in center of the fluid stream with a circular shape, illustrating a tube. The characteristic of the heat transfer from tube to fluid was numerically solved and the optimal distance from tube to channel wall was found. Further on smaller tubes were placed in between the channel wall and the initial tube in order to find an optimal number of tubes for transferring the heat to the liquid, taking into account that the channel must not be blocked by tubes. Again the optimal placement was found by numerical calculations. In the article of Bejan A. the shape of the tubes were cylindrical. This is exactly the point where it is believed that further optimization can be done. 1

8 2 CHAPTER 1. INTRODUCTION Figure 1.1: Picture of dripstones hanging from ceiling to floor in a cave. This is how the channel is supposed to look a like. When a flow of liquid is induced a cross flow is established and a temperature difference between solid material and fluid makes heat transfer possible

9 1.1. THE SYSTEM Distributing material in channel A special type of topology optimization, abbreviated the material distribution method (MDM), has been utilized in diverse areas such as structural mechanics, acoustics, optics and fluidics for optimizing performance. Basically the method is used for optimizing a Figure 1.2: Schematic of the setup used in the simulations. Due to horizontal symmetry only half of the model is evaluated, easing the calculation process. A pressure difference between inlet and outlet, p, drives the cooling liquid through the main design region. Heat dissipating material is modeled by both spatially varying the permeability in the design region and varying the material properties towards metallic values. Furthermore the objective function to be minimized is activated in this central region. given objective function, Φ, by mapping (i.e. distributing material) a designated design region. Recently the method has been applied to a full Navier-Stokes flow through the implementation in a high-level programming language based on FEMLAB and MATLAB [2]. The development of new and innovative fluidic and micro fluidic devices will benefit from this progress. In the present report we have applied the MDM in order to study the model of a 2D heat exchanger for use in LOC. The basic setup is simple. Fluid with a fixed inlet temperature is pumped through the design region by a pressure difference p. By iteratively varying the material distribution in the design region, the placement of heat dissipating matter is varied in order to minimize the average material temperature. Albeit the setup is simple, refer to Fig. 1.2, the task of finding the optimal material distribution is a complex Navier- Stokes conduction convection problem linked to the minimization of the temperature, which can only be solved numerically. The main objective has been to implement the TOPOPT-algorithm into a heat transfer problem. Explained in Chapter 3.

10 4 CHAPTER 1. INTRODUCTION

11 Chapter 2 Theoretical Aspects 2.1 Fluid Dynamics In this section the theory concerning the motions of fluids will be discussed. First with a brief description of the continuum hypothesis, then the different parts of the Navier-Stokes equation are explained The continuum hypothesis This work is based on the continuum hypothesis, in which a fluid is not treated as individual molecules. Instead large numbers of molecules are blocked together to form fluid particles. These fluid particles are usually cubic and have a side length, λ of about 10 nm, such a fluid particle contains about molecules. The reason why the fluid is not treated as individual molecules is the fact that these have very large fluctuation. Repeated measurements would therefore yield different result and hence not be reproducible. Statistics then gives a lower boundary to the number of molecules in a fluid particle, the upper boundary is needed to keep the fluid particle small enough to avoid large variations in external forces. When using the continuum hypothesis it is important to work on length scales much larger than λ 10 nm otherwise theory breaks down. However, this is not a big concern as most Lab-on-a-Chip systems are made with length scales in the order of several µm The Navier-Stokes equation The Navier-Stokes equation can be derived from Newton s second law m a = F. First the acceleration, a is substituted with time-derivative of the velocity, d t v. Then the force is expressed as the sum of several different external forses, F = j F j. Now Newton s second equation look as follows m d t v = F j. (2.1) j However, since this equation is going to be used on the fluid particles and not on the individual molecules it is necessary to divide with the volume of such a fluid particle. This 5

12 6 CHAPTER 2. THEORETICAL ASPECTS leaves the density, ρ instead of the mass, m, the force densities f j instead of the forces F j And the material time-derivative, D t instead of d t. The equation now takes the following form; ρ D t v = f j. (2.2) j Since Newton s second law follows the Eulerian description, where the Navier-Stokes equation follows the Lagrangian description, it is necessary to use the material time-derivative, D t = t + (v ). Now the Navier-Stokes equation can be expressed as; ( ) ρ t v + (v )v = f j, (2.3) j All that now remains is to specify the external forces. Normally four forces must be taken into account, these forces are caused by; gravity, pressure-gradient, viscosity and electricity. All forces must be used as force densities. The gravitational force density has the form f grav = ρg. The force density caused by the pressure gradient is f pres = p. When dealing with incompressible fluids, as in this entire work, the viscous force density is of the form f visc = η 2 v. The fourth and final force density is that of electricity, this force density has the form f el = ρ el E where ρ el is the charge density. If all of these force densities are substituted into Eq. (2.3) the Navier-Stokes equation looks like this: ( ) ρ t v + (v )v = p + η 2 v + ρ g + ρ el E, (2.4) It is often possible to neglect the gravitational force, as it is balanced out by an opposite normal force.

13 2.2. HEATEQUATION HeatEquation The total time dependent heat equation including conduction and convection reads δ ts ρc p T t + ( k T ) = Q ρc pu T (2.5) For the steady state problem the equation is time independent and the first term disappears The properties of the elements are ( k T ) = Q ρc p u T (2.6) C p k Q u ρ heat capacity thermal conductivity tensor heat source velocity field vector the density The equation Eq. (2.6) written above is in its non-conservative form. When the nonconservative mode is chosen it is assumed that the continuity equation for the velocity field, u = 0, is fulfilled. From lectures in Theoretical microfluidics we know that this is true for incompressible fluids. The non-conservative mode also ensures that no unphysical source term arises from a flow field where the incompressibility is not fulfilled. The conservative form of the time independent heat equation allows us to alter the physical parameters; density, heat capacity, and conductivity since they are placed inside the integral as seen ( k T +ρc p ut ) = Q (2.7) The practical consequences are further explained in Section

14 8 CHAPTER 2. THEORETICAL ASPECTS 2.3 The γ-field The γ-field describes the ratio between the amount of cross-flowing liquid and heat dissipating material in any area. If γ = 1 no material is present in this area of the channel and the liquid can flow freely. If γ = 0 then this part of the channel is blocked by the material. To simulate that the material is obstructing the flow, the Darcy s force has been implemented. The Darcy s force looks like this Where F x = α v (2.8) α 1 γ, γ = (2.9) Da and Da is the Darcy s number and has a value about This results in a strong force working in the opposite direction than that of v.

15 2.4. DIMENSIONLESS EQUATION Dimensionless equation In the article Constructal multi-scale cylinders in cross-flow of T. Bello-Ochende, Bejan [1] a dimensionless heat transfer density rate is derived, in order to be able to optimize the spacing between the cylinders placed in the fluid stream (see page 1376). To set up the scenario for the optimization it has therefore been necessary to rearrange the conservation equation for mass, momentum and, energy, into dimensionless equations. To get a feeling for the method we will rearrange the momentum equation into a dimensionless momentum equation. The equation for conservation of momentum reads u v x + v v y = 1 P ρ y + ν 2 v (2.10) The dimensionless variables needed for the rearrangement are ( x, ỹ) = (x, y) D 0, (ũ, ṽ) = (u, v) P D 0 /µ, P P = P (2.11) We also need the Prandtl number defined P r = ν α (2.12) and the dimensionless pressure drop number, named the Bejan number Be = P D 0 αµ (2.13) Starting with the right-hand side of the momentum conservation equation, ρ is from its SI-units seen to be equal ν /µ. It gives u v x + v v y = ν P µ y + ν 2 v (2.14) substituting the variables leads to u v x + v v y = ν µ ( P P ) ( (ỹ D 0 ) + ν 2 ṽ P D ) 0 µ Introducing the Prandtl number and the dimensionless Bejan number gives u v x + v v ( Be α 2 y = P r ( P P ) P D0 2 (ỹ D 0 ) + αµ ) 2 ṽ P D 0 u v x + v v ( Be α 2 y = P r ( P P ) Be ) α2 P D ṽ (ỹ D 0 ) D 0 Putting the Bejan number and D 0 outside the parenthesis u v x + v v y r Be ( α 2 = P P ) D 0 D0 2 ỹ + α2 2 ṽ (2.15) (2.16) (2.17) (2.18)

16 10 CHAPTER 2. THEORETICAL ASPECTS Extracting D 2 0 and α2 u v x + v v y P r Be ( α2 P ) = D0 3 ỹ + D2 0 2 ṽ (2.19) Keeping in mind that 2 ṽ is equal to ṽ reads u v x + v v y x, ṽ y, we get D 2 0 with Eq. (2.11) the right side P r Be ( α2 P ) = D0 3 ỹ + 2 ṽ Substituting the variables with dimensionless variables on the left hand side ) ) ( P D0 ) ṽ( P D0 ( µ P D0 ) ṽ( P D0 µ P r Be ( α2 P ) ũ + ṽ = µ x D 0 µ ỹd 0 D0 3 ỹ + 2 ṽ (2.20) (2.21) Collecting the tilde variables P 2 D0 2 1 ( µ 2 ũ ṽ D 0 x + ṽ ṽ ) ỹ () Collecting terms outside the on the left side = P r Be ( α2 P ) D0 3 ỹ + 2 ṽ P 2 D 0 D 3 ( 0 µ 2 P r Be α 2 ũ ṽ x + ṽ ṽ ) ( P ) = ỹ ỹ + 2 ṽ (2.22) (2.23) and the total dimen- With the definition in Eq. (2.13) the prefactor can be written as Be P r sionless equation for conservation of momentum is Be ( ũ ṽ P r x + ṽ ṽ ) ỹ ( P ) = ỹ + 2 ṽ (2.24) This agrees with the dimensionless equation for conservation of momentum seen in the article of T. Bello-Ochende and A. Bejan

17 Chapter 3 Implementation of TOPOPT Our main objective have been to modify the source code of the high-level programming implementation of topology optimization in order to simulate heat transfer in a well defined design region. 3.1 Setting up Femlab The MDM is used to model the heat dissipating material density. The problem has a complex multi physical nature composed of a Poiseuille flow in which both convection and conduction of the heat is present. Furthermore the temperature is evaluated in the main design region, in which the heat dissipating material density is iteratively altered by the algorithm in order to achieve the minimal integral value of the total temperature. In FEMLAB we have modeled the heat transfer by choosing appropriate subdomain and boundary settings. For the incompressible Navier-Stokes part, which is activated in all regions, we have defined the pressure at the inlet and outlet and imposed the no slip boundary conditions on all outer walls, the slip/symmetry condition was chosen for the all symmetry boundaries. In the case of the convection and conduction of the heat we have defined T 0 = 0 at the inlet and furthermore imposed a convective flux boundary condition at the outlet. The heat dissipation is solely activated in the main design region and as a function of γ in order to simulate that the material are not allowed to move in or out of the system. In order to include the design variable γ, describing the spatial permeability (i.e. the material density), we finally define a PDE, General Form with linear lagrange elements in the design region. By modifying the source code as describe in notes handed out by Fridolin Okkels and adapting the functionality (as described below) to heat transfer, rather than metabolism, the program was able to calculate heat transfer. In Fig. 3.1 we have included an example of the graphs produced by running the simulation. By running numerous simulations we can thus investigate the impact of both geometrical and physical parameters on the outlet concentration. 11

18 12 CHAPTER 3. IMPLEMENTATION OF TOPOPT Figure 3.1: Example of the output from running a simulation. The output displays the material distribution, the flow velocity, the temperature. The mesh size is in the design region and on the boundaries between the inlet/design region and design region/outlet. 3.2 Adapting the code Heat dissipation The first problem that was encountered during the implementation was how to simulate heat dissipation. It seemed that Femlab offered two different possibilities, either one should set the surface temperature, T of the material to a fixed value and then let Femlab work out how this would affect the surrounding liquid, or one could set the amount of energy dissipated,q from the material into the liquid. The problem with fixing the surface temperature, is that the system no longer works as a cooling devise, instead it works as a water heater. In the case of fixing Q the same amount of energy is released into the liquid no matter how fast it is moving past the material or how hot it already is. It was decided to use the fixed Q value, but since Q is a energy flux it is dependent on the area of the material it is dissipating from. This means that as the MDM algorithm places more or less materiel in the design region the amount of dissipated energy will increase or decrease, respectively. Unfortunately this leads to the obvious conclusion that if no material is placed at all, then no energy will be dissipated and hence the overall temperature will be the lowest possible. thereby, tricking the algorithm to believe that the ultimate cooling structure has been achieved. This was fixed by adding two modifications; the first was to set a minimum value of material volume, the second was to divide the Q by the area of the materia, thereby fixing the amount of energy dissipated no matter the amount of

19 3.2. ADAPTING THE CODE 13 material The Objective function The objective function is the function the MDM algorithm tries to optimize by distributing material. In the code this function is called Phi. In our case of a cooling system, the obvious choice would be to maximize the amount of energy carried out of the system by the liquid. However, this is not possible since it was chosen to use fixed Q instead of fixed T. When specifying Q and working with steady state systems, it is given that the same energy which comes into the system must come out. Hence the amount of energy carried away by the liquid is independent of material distribution. To avoid this problem the objective function was chosen to depend on the temperature in the design region. This also caused some troubles since it would be better to place the material as late in the design region as possible. This was not the desired solution, so instead of looking at the temperature in the entire design region, only the temperature of the material should be evaluated. This was done by changing the objective function to: Where γ = 1 is pure liquid and γ = 0 is solid material Global effects of local changes Φ = T (1 γ) (3.1) During the simulations it was discovered that a important feature had been over looked. Namely the fact that when dividing the amount of dissipated heat, Q with the area of the γ-field, then local changes causes Global effects. In other words, when the γ-field is changed in one place, the temperature is changed everywhere. This might make it very difficult for MDM-algorithm to optimize the structure. However, in the relative simple cases studied here the MDM-algorithm seems to handle the job just fine. If this code is further expanded, one should keep an eye on this effect Permeable material Many of the initial simulations were made with a Darcy s number of Da = This made the γ-field settle at some value between 0 and 1. Thereby meaning that the optimal structure would be a permeable heat dissipating material. This problem could be avoided in two ways; one idea was simply to decrease the darcy s number to Da = making the permeability of the material much less. Another idea would simply be to refine the mesh size, but due to the lack of computational powers and limited time, the mesh was not further decreased. With these changes it becomes much easier for the MDM-algorithm to decide on whether γ should be one or zero Adjusting the material parameters In order to make the simulation as physical as possible it is necessary to adjust the material parameters as γ is varied between zero and one. When γ = 1 the material is the liquid

20 14 CHAPTER 3. IMPLEMENTATION OF TOPOPT which in this case is water. All material parameters such as the density ρ, the specific heat capacity C p and the thermal conductivity k should then take values corresponding to those of water. On the other hand when γ = 0 the parameters should match those of the material (copper in our case). Hence all material parameters should be functions of γ. Below is shown how this has been implemented in the case of the thermal conductivity, the procedure is exactly the same for the other parameters. k water = 0.6 k copper = 385 k material = k vand γ+k copper (1 γ), γ = (3.2) All these parameters was never implemented at the same time, due to several crashes in the algorithm The conservative heat equation To prevent the algorithm from crashing due to changes in the material parameters, it was necessary to change the non-conservative heat equation to a conservative one, see Section 2.2. Unfortunately the conservative equation give much rougher result than the nonconservative, this means that in order to obtain the same resolution as earlier the mesh must be further refined. This was however not possible due to the lack of computational powers and limited time. The switch from the default non-conservative equation to the conservative is made by the follwing line. fem.appl{- x -}.prop.equform = cons ; where - x - is the application number corresponding to convection and conduction. With this line implemented it is possible change the material parameters as a function of γ Usolve FEMLAB has a built in scaling function resulting in division by a high number ( ) when running our main-file. This forces the pc to calculate with numbers close to its uncertainty value, making it crash. To avoid the problem we have disabled the scaling by the following script, when calling the femlin function fem.sol=femnlin(fem, init,fem.sol, solcomp,{ u, v, p },... outcomp,{ u, v, p, gamma },... nonlin, on, uscale, none );

21 Chapter 4 Results In this chapter the process is described and the results are discussed. The first attempt to run the MatLab-code modeling our heat exchanger failed. Every parameter of the material had to be disabled or smeared out giving the heat dissipating material the same physical properties as the cooling fluid (water), in order to ease the calculations and making it possible for the PC to find a solution. Restarting with the simplest setup, and adding one parameter at a time gave us the ability to learn how Femlab responded to different codes. Parameters: for both the fluid and the material distributed the parameters are; Da = k = 0.6 C p = 4186 ρ = Conservative Vs. Non-conservative The first test was conducted to check that two simulations, one using the conservative heat equation and one using the non-conservative, would yield the same result. In all three simulations the exact same parameters were used and the simulations lasted for 100 iterations. The first test was made with a grid size of and using the conservative equation. The result is shown in Fig The second simulation was made with the non-conservative heat equation, The result is shown in Fig When comparing Fig. 4.1 and Fig. 4.2 it can be seen that the overall structure in the γ-field is almost the same. However, when comparing the temperature figures it is seen that the non-conservative seems to be of much higher resolution. This was later found out to an effect of using the conservative equation. To compensate for this a third simulation was conducted with a smaller grid. The result is shown in Fig When comparing Fig. 4.3 and Fig. 4.2 it seems that the smaller grid made the two solution more alike, but since neither of the simulations did converge within the 100 iterations it could not be concluded that the conservative equation could be implemented for later use. 15

22 16 CHAPTER 4. RESULTS Figure 4.1: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ = 1000, size grid = and the conservative heat equation. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature. Figure 4.2: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ = 1000, size grid = and the nonconservative heat equation. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature.

23 4.2. MAKING Q INDEPENDENT OF THE γ-area 17 Figure 4.3: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ = 1000, size grid = and the conservative heat equation. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature. 4.2 Making Q independent of the γ-area As described in Section it was believed that dividing Q with the area of the γ-field could confuse the algorithm, it was therefore decided to make a pair of simulations without this factor. The results are shown in Fig. 4.4 and Fig Strangely enough the removal of the area dependence, did not help at all. If the two results are compared it can be seen that they are much more different than those of Section 4.1. The area dependence was added again.

24 18 CHAPTER 4. RESULTS Figure 4.4: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ = 1000, size grid = and the conservative heat equation. Q is independent the γ-area. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature. Figure 4.5: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ = 1000, size grid = and the nonconservative heat equation. Q is independent the γ-area. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature.

25 4.3. LOWERING DARCY S NUMBER Lowering Darcy s number In order to make the solutions converge faster further simulations were made with Da = In the hope that this would yield a much nicer outcome. The results are shown in Fig. 4.6 and Fig. 4.7 As expected the results were much better, if the results are Figure 4.6: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ = 1000, size grid = and the conservative heat equation. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature. compared, it can be seen that they now are very alike. Hence, it was concluded that the conservative heat equation could be used in further simulations.

26 20 CHAPTER 4. RESULTS Figure 4.7: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ = 1000, size grid = and the nonconservative heat equation. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature. 4.4 Varying C p Once it had been verified that the conservative heat equation could be used, it was possible to change the material parameters. The first test was made with a slightly varying heat capacity C p. The difference between the value for the liquid and for the material was only about 5%. This test was conducted both with Q dependent and independent of the γ-area. the results are shown in Fig and Fig In both cases the algorithm seems to work fine.

27 4.4. VARYING C P 21 Figure 4.8: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p 4186, ρ = 1000, size grid = and the conservative heat equation. Q is independent the γ-area. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature. Figure 4.9: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p 4186, ρ = 1000, size grid = and the conservative heat equation. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature.

28 22 CHAPTER 4. RESULTS 4.5 Varying k A pair of simulations were also conducted with the thermal conductivity being varied about 5% between the liquid and the material. Again the simulation was tested both with and without Q depending on the γ-area. The results are shown in Fig.?? and Fig.??. Figure 4.10: This figure shows the result after 100 iteration using the following parameters: Da = , k 0.6, C p = 4186, ρ = 1000, size grid = and the conservative heat equation. Q is independent the γ-area. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature. These results also show that the algorithm actually works better when Q is divided with the area of γ. Fig.?? also shows that the algorithm can work when k is changing.

29 4.6. VARYING ρ 23 Figure 4.11: This figure shows the result after 100 iteration using the following parameters: Da = , k 0.6, C p = 4186, ρ = 1000, size grid = and the conservative heat equation. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature. 4.6 Varying ρ For the last of the three parameters two test was also made, again both with and without Q depending on the area of γ. ρ is varied about 5%. The results are shown in Fig.?? and Fig.??. Again the results show that the algorithm works better when Q is dependent of the area of γ. Fig.?? also shows that the algorithm can work when ρ is changing.

30 24 CHAPTER 4. RESULTS Figure 4.12: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ 1000, size grid = and the conservative heat equation. Q is independent the γ-area. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature. Figure 4.13: This figure shows the result after 100 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ 1000, size grid = and the conservative heat equation. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature.

31 4.7. VARYING THE INITIAL CONDITIONS Varying the initial conditions In order to see how results depend on the initial conditions of γ three simulations were made. All parameters was independent of γ. All the simulation were given the time to converge. The first simulation is shown in Fig This simulation was started with γ as a sine function of x. The second simulation is shown in Fig This simulation was started with γ as a sine function of y. The third simulation is shown in Fig This simulation was started with γ as a constant. Figure 4.14: This figure shows the result after 986 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ = 1000, size grid = and the conservative heat equation. The initial conditions for the γ-field is a sin(x)-wave. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature. If these three results are compared it is easy to see that the final result is dependent on the initial conditions for γ. Hence, one cannot be certain that the optimal structure is achieved when using the MDM-algorithm.

32 26 CHAPTER 4. RESULTS Figure 4.15: This figure shows the result after 145 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ = 1000, size grid = and the conservative heat equation. The initial conditions for the γ-field is a sin(y)-wave. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature.

33 4.7. VARYING THE INITIAL CONDITIONS 27 Figure 4.16: This figure shows the result after 900 iteration using the following parameters: Da = , k = 0.6, C p = 4186, ρ = 1000, size grid = and the conservative heat equation. The initial conditions for the γ-field is a constant γ 0. The top window shows the γ-field, in the middle the flow velocity is shown and in the bottom the temperature.

34 28 CHAPTER 4. RESULTS

35 Chapter 5 Further work During this work we did not have sufficient time to do all the worked we would have liked. We only managed to simulate a small change in each of the three parameters C p, kandρ, but not on the same time. Had there been more time, it would be of great benefit to add one parameter at a time, and watching how the algorithm handles it. Finally it should hopefully be possible to change all the material parameters, thereby giving the simulations a real physical relevance. It would also be interesting to see how the structures would change if the simulations were made with a much smaller mesh, this task could not be accomplished as the computational time increases greatly as the mesh is decreased. It would also be interesting to see how efficient self-designed-structures would be, and how the MDM-algorithm would try to improve those. 29

36 30 CHAPTER 5. FURTHER WORK

37 Chapter 6 Conclusion We have successfully implemented the physics of heat transfer in cross-flow in FEMLAB with MATLAB. We have learned the importance of starting from scratch adding more and more parameters making the solutions physical. We have gathered knowledge about the way FEMLAB iterates and how to take this into account when implementing the physics of the model. Setting γ 0 to the value zero resulted in a zero denominator which were avoided by adding a sine function to describe the initial value of gamma in the design field. Unfortunately the initial guess seemed to influence the solution. This suggests that the FEMLAB solution is not singular. The built in scaling function described in Chapter 3 needed to be enabled in order to hinder the pc to crash this was done by adding usolve in the script calling the femlin function. To prevent the algorithm from crashing due to changes in the material parameters, we changed the heat equation to the conservative version, allowing a less refined resolution. Acknowledgement We thank Fridolin Okkels, Laurits Olesen, Niels Asger Mortensen, and Henrik Bruus for interesting discussions on the functionality of the developed high-level program used for the implementation of the algorithm and the parameters making our model physical realistic for simulating the 2D Heat transfer model. 31

38 32 CHAPTER 6. CONCLUSION

39 Bibliography [1] Constructal multi-scale cylinders in cross-flow, T. Bello-Ochende and A. Bejan, International Journal of Heat and Mass Transfer. 48, (2005). [2] A high-level programming implementation of topology optimization applied to steadystate Navier-Stokes flow, Laurits Højgaard, Fridolin Okkels and Henrik Bruus, Phys. Rev. E. (11 Oct. 2004). 33