26-th ECMI Modelling Week Final Report Dresden, Germany

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1 26-th ECMI Modelling Week Final Report Dresden, Germany

2 Group 3 Modeling of a storage water heater Stefan Eberhard Dresden University of Technology, Germany Marina Ferreira University of Coimbra, Portugal Nicolai S. Johnsen Technical University of Denmark, Kongens Lyngby, Denmark Laura S. Mendoza University of Strasbourg, France Igor Zarvansky National Technical University of Ukraine, Kiev, Ukraine Instructor: Joachim Krenciszek Kaiserslautern University of Technology, Germany 2

3 Abstract Nowadays, hot water is needed on a daily basis. In many households the hot water is provided by a storage water heater. As the water is heated at a relatively slow rate, the question arises how long it takes to heat fresh water up to a desired temperature when the supply of hot water is running low in the tank. The present work documents our attempts to find a model for the heating process as well as an optimal shape for the heating element, such that the time required to heat up the water is minimized.

2 Modeling of a storage water heater 3.1 Introduction Hot water has long become indispensable for domestic use, such as cooking, cleaning or personal hygiene.

4 2 Modeling of a storage water heater 3.1 Introduction Hot water has long become indispensable for domestic use, such as cooking, cleaning or personal hygiene. Storage water heaters are appliances providing a sufficiently constant supply of hot water. They consist of a water tank which is used for storing the hot water. It is equipped with a heating element that often is a closed water circuit, e.g. a pipe. That is due to the high heat capacity of water and the fact that it is also non-toxic and low in cost. However, the water is heated at a relatively slow rate, necessitating an optimization of the shape of the heating element, such that time required for the heating process is minimized. Because of various assumptions that we have to make, we optimize the heating element in terms of the radius of the pipes versus the number of loops in the coil. We investigate a case in which the starting temperature of the water in the tank is 15 C while the desired average temperature at the end of the heating process is 45 C. The water inside the heating element is assumed to be 80 C constant. We initially consider the general heat equation describing the diffusion of the temperature inside the tank. We solve this equation by discretizing the domain in order to implement a finite element method in Matlab. We modify that model by then mimicking the advection, which is the bulk motion of the fluid, in accordance with our expectations. Finally, we draw conclusions on our results and propose possible ways to improve our model. 3.2 The model Description of the industrial problem we are considering The water tank in question is of cylindric shape. The heating element is a regularly-shaped coil revolving around a straight tube in its center (cf. Figure 3.1). Figure 3.1: dimensions of the water tank

5 26th ECMI modelling week 3 The diameter of the heating pipes is the same for both the coil and its straight center piece. We further specify our problem as follows: volume of the tank: 200l surface area of the heating element : 1m 2 temperature of the heating pipes : 80 C starting temperature of the water : 15 C desired average temperature of the water : 45 C Assumptions For reasons of simplicity we assume that the tank is perfectly insulated which means that the heat can neither leak from the tank nor be absorbed from the outside. Likewise, we assume that the temperature of the heating element is not affected by the surrounding water in the tank. As we also neglect the fact that the heating pipes would realistically need time to heat up, we can state that the temperature of the heating element is 80 C constant. Since computing three-dimensional models is very costly we decided to limit our model to two dimensions. We study three different 2D cross sections. In order to keep the results as comparable as possible, we assume that the heating system - that is, the coil and the central straight pipe - goes all the way through the tank. This can be modeled in any of the considered cross sections. Mathematical setup To begin with, we only consider the diffusion of heat which means that we study the heat distribution while neglecting the bulk motion of the fluid, from now on referred to as advection. This is described by the heat equation where : ρ = 1000 kg m 3 W κ = 0.58 m K J c = 1000 kg K u t = κ ρc u density thermal conductivity specific heat capacity One might expect a source term in the heat equation. We neglected that as we chose to define the heat sources as boundaries. The sources are therefore

Additionally, we set the initial condition for all non-boundary points of the domain Ω R 2.

6 4 Modeling of a storage water heater inner boundaries on which a Dirichlet boundary condition is imposed. On the outer boundary we impose a Neumann boundary condition which relates to the aforementioned perfect insulation of the tank which enables us to disregard potential flux. Additionally, we set the initial condition for all non-boundary points of the domain Ω R 2. outer boundary Γ 1 : Neumann boundary condition u n = 0, x Γ 1 inner boundary Γ 2 : Dirichlet boundary condition u(x, t) = 80, t 0, x Γ 2 initial condition : u(x, 0) = 15, x Ω The different cross sections The diffusion - and optionally the advection - of heat is considered in three different sections of the tank in order to obtain different viewpoints, hopefully yielding information on the approximate behaviour of the actual threedimensional tank. These three sections are (a) the circular cross section, (b) the horizontal rectangular cross section and (c) the vertical rectangular cross section of the tank. See figures 3.2a, 3.2b and 3.2c, respectively. (a) circular (b) horizontal rectangular (c) vertical rectangular Figure 3.2: the three sections to be considered In the circular cross section, the coil is represented by nine circular inner boundaries Γ 2 with radius r. (cf. Figure 3.3.) The central circle corresponds to the straight tube in the coil. Although the coil only intersects twice with the cross section plane, the additional heat from the nearby parts of the coil should be taken into account. This fact is represented by the 8 surrounding circles which all impact the heat distribution equally. This only provides a crude estimate and more realistic setups may be derived by accounting for

7 26th ECMI modelling week 5 a smooth descent of the magnitude of heat from the two circles at which the coil in fact intersects to the remaining heat sources. Due to lack of time, however, we stuck with the initial setup which was also more easily implemented in MatLab. In the rectangular cross sections, the heating element is represented by an inner rectangular boundary and circular boundaries on each side, corresponding to the straight central tube and the intersections of the coil, respectively. The approach is of a simplified nature yet again as we did not address the lack of a bond between the heat sources. The diffusion in the cross sections 3.2a and 3.2b is modeled in MatLab and solved by applying a finite element method. In addition to the diffusion of heat, it was suggested by the instructor to modify the heat flow in order to imitate the behaviour of advection. That is, to create an artificial flow of heat allowing hot water to circulate to the top of the tank and cold water to the bottom of the tank. We realized this by implementing a MatLab function which rotates the water on predefined elliptic paths, cf. Figure 3.3b. In the vertical rectangular cross section (cf. Figure 3.2c) a completely different approach was used. We modeled the problem in Python by solving a conservation equation with a dedicated advection term using a finite volume method. 3.3 Numerical analysis and implementation FEM - a description of the general procedure In the cross sections 3.2a and 3.2b, we solve the heat equation using the Finite Element Method. Before explaining the procedure, we will state some important concepts, namely the space L 2 and the Hilbert space H0 1. Definition 1 Let Ω R n be measurable. We define: L 2 (Ω) = {f : Ω R f is measurable and f(x) 2 dx < } and H 1 0 = {u L 2 (Ω) xi u L 2 (Ω), i = 1,..., n}, where xi u denotes the i-th weak derivative of u. From the heat equation u t = κ ρc u we obtain the weak formulation by multiplying by the test functions v (Ω) and by integrating by parts H 1 0 Ω u t v = κ ρc Ω Ω u v, v H 1 0 (Ω). (3.1)

8 6 Modeling of a storage water heater As we can see, the second order derivative disappeared and we only have the gradient, which makes the problem much easier to compute. The solutions of the initial equation are also solutions of the new equation, so the task now is to find u(t) H0 1 (Ω), for all t (0, T ) such that it satisfies (3.1). The next step is to find an approximation of the solution using a Galerkin method. To do that we first discretize Ω using triangulation and then define a finite dimensional subspace V h H0 1(Ω) to find an approximation u h(t) V h, for all t (0, T ), so that with Ω u h t v h = κ ρc Ω u h v h, v h V h, (3.2) u h (x, 0) = 15. The space V h with dimension N h and basis {ϕ j } j=1,...,nh where is defined by V h = {v h C(Ω) v T h Π 1, T T h, v Γ2 h = 80, v h n Γ1 = 0}. Π 1 is the set of the polynomial functions of degree 0 or 1, T h is the set of the triangles of the triangulation, T is a triangle in T h. Each function v h V h is uniquely determined by its values at the points {P j } j=1,...,nh of T h. These functions can therefore have the form N h v h (x, t) = α j (t)ϕ j (x), x Ω, t (0, T ) j=1 with α j = v h (P j ), j = 1,..., N h. Substituting that into (3.2) leads to N h j=1 α j t (t) N h ϕ j (x)ϕ i (x)dx = κ Ω ρc }{{} j=1 M ij Rewritten in matricial form we obtain α j (t) M α t (t) = κ ρc A α(t) ϕ j (x) ϕ i (x)dx, i = 1,..., N h. Ω } {{ } A ij which is a linear system of ODEs. We solved this system using a MatLab function that implements the Runge-Kutta method. For a more detailed description [3] and [5] are recommended.

9 26th ECMI modelling week 7 A short description of the programming tools As mentioned before, the diffusion in the cross sections 3.2a and 3.2b is modeled in MatLab, where an FEM procedure is applied to solve the 2D heat equation. By triangulating the mesh and using a weak formulation of the PDE a system of ODEs is obtained. The system is solved by the built-in MatLab routine ode45, which is based on an explicit one-step Runge-Kutta (4,5) method. (Source: MatLab help, ode23, ode45, ode113, ode15s, ode23s, ode23t, ode23tb ). The triangulation of the domain is obtained by the distmesh mesh generator, which applies a Delaunay triangulation method (cf. [4]). The important property of the Delaunay triangulation is that it maximizes the minimum angle among all faces, yielding a rather regular mesh. For reasons of simplicity a uniform mesh was chosen with a initial edge lengths of 0.010m and 0.015m for the circular cross section and rectangular cross sections, respectively. The initial lengths are used to create the triangles, before the algorithm iterates to obtain a Delaunay triangulation. The MatLab implementation allows for the possibility of generating a short film, which displays the heat distribution at any specified time step. In order to make our model more adaptable and functional on arbitrary meshes, the average temperature at a given time t, u(t) is determined by the weighted sum of the mean temperature of each face (triangle), that is u(t) = 1 3 n i=1 A i (v i1 (t) + v i2 (t) + v i3 (t)) n i=1 A i where A i is the face area of the i-th face with vertex heat values v ij, j = 1, 2, 3 and n is the number of faces. Due to the Delaunay triangulation all triangles are shaped quite regularly, which makes it reasonable to use the mean of the vertex values as the face value. For information on the Python implementation refer to [1] and [2]. Motivation and description of the artificial convection function As previously stated it is of interest to modify the heat distribution to obtain an approximation of the advection in the circular cross section (cf. Figure 3.2a). The water is assumed to follow a pattern, where hot water moves upwards from the center of the tank section and cold water moves down on each side of the cross section. The flow is depicted in Figure 3.3a. We modeled this pattern by circulation on elliptic paths, meaning we implemented a MatLab function that rotates the temperature values of vertices along the ellipses. The function takes the shape, center point and rotation of an ellipsis along with the mesh data as input. It then detects the vertices closest to

8 Modeling of a storage water heater 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 (a) sketch of heat flow 0.25 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 (b) implementation to mimic heat flow by rotation of heat along ellipses Figure 3.

10 8 Modeling of a storage water heater (a) sketch of heat flow (b) implementation to mimic heat flow by rotation of heat along ellipses Figure 3.3: implementing the advection the specified ellipse and rotates the respective temperature values without modifying the position of the vertices. We did use a rotation by 180 degrees, although, naturally, any value could be set. In total, twelve elliptic rotations of heat were applied in order to obtain a subjectively satisfying imitation of the advection, cf. Figure 3.3b. Since the rotation of heat is performed independently of the FEM procedure, a merging of the two procedures was required. For every 100 seconds (with 200 time steps in each interval) the advection procedure is called to rotate the heat as depicted in Figure 3.3b, corresponding to an advection pulse. Optimization Using the notations in Figure 3.4 we can fix H coil = H tank = 1m while we remember that the surface area of the heating element is 1m 2. Within the Figure 3.4: Model notations

11 26th ECMI modelling week 9 system of assumptions we work under, there are three parameters we could potentially modify : N, the number of loops, W coil, the width of the coil and r, the radius of the heating pipes. As W coil is dependent on the two other parameters, we will only study two cases. First of all, let us state a few reminders of the relationships between the different parameters: A T otal = L T otal 2 π r = 1 L T otal = H coil + L coil Where L coil the length of the coil and L T otal the length of the entire heating system. We notice that the formula for the area of the coil is the formula for the area of a cylinder. If we want to study the influence of the radius (r) with a fixed number of loops (N) over the width of the coil W coil we can use the following formula : (Lcoil ) 2 ( ) 2 Hcoil W coil = 1 π N N L coil can be calculated in function of r as we can see in the formula of the area of the coil. That is : L coil = 1 2 π r H coil If we wanted to study the influence of the number of loops of the coil (N) with a fixed radius of the pipe (r) over the width of the coil (W coil ) we could naturally use the same formula. 3.4 Results As far as the artificial convection function is concerned, we can see that the heat distribution is no longer symmetrical as before (cf. Figure 3.5a), but does instead lean towards the top of the tank (cf. Figure 3.5b), which is what we expected. The time required to reach the 45 C termination condition was reduced from 197 minutes to 130 minutes, accordingly (cf. Figure 3.6a). Both of these times were obtained with the optimal radius r = m (cf. Figure 3.6b). We can therefore conclude that the advection plays a major role. We would, however, have surmised that the difference would even be larger. That is a strong indication that the artificial convection function is not really an adequate substitute for more precise mathematical descriptions, although it might have seemed so at first glance. When we optimize the coil parameters for the horizontal rectangular cross section, we find that for N = 7 loops the optimal radius is r = The heating process takes 139 minutes in this case (cf. figures 3.7a and 3.8). We have found, however, that

12 10 Modeling of a storage water heater (a) final state - diffusion only (b) final state - artificial advection Figure 3.5: comparison: diffusion only vs. artificial advection 80 Mean temperature as a function of time 260 Time required to reach 45 degrees as a function of coil radius Mean temperature [C] Time [min] Convection Diffusion only Time [min] (a) comparison - with and without artificial advection Radius of coil [m] (b) optimizing the radius of the pipe for N = 7 Figure 3.6: circular cross section for N = 8 loops we get an even better setting with a radius of r = 0.25m. For these parameters, it takes 134 minutes for the tank to heat up to the desired average temperature of 45 C. (cf. Figure 3.7b) These findings raise the question why the results for the circular cross section and the horizontal rectangular cross section, respectively, differ so distinctly when only diffusion is taken into account. We suspected this to be down to different ratios of source boundary length versus domain area. For the circular cross section, the inner boundaries have a combined length of L c = m. The area of the domain is A c = m 2 which leads to a ratio of Lc A c = For the rectangular cross sections we get L r = m and A r = m 2 leading to a ratio of Lr A r = We therefore find our suspicions confirmed, as the sources - or, more precisely,

13 26th ECMI modelling week Time required to reach 45 degrees as a function of coil width 280 Time required to reach 45 degrees as a function of the number of loops Time [min] 200 Time [min] Radius of the coil [m] (a) optimizing the radius of the pipe for N = Number of loops in the coil (b) optimizing the number of loops for r = 0.025m Figure 3.7: horizontal rectangular cross section Figure 3.8: final state - horizontal rectangular cross section the inner boundaries - indeed have a greater impact in the rectangular cross sections. Finally, we also found an optimal setting for the Python implementation in the vertical rectangular cross section. For N = 7 loops, we found a radius r = m to be optimal. This setup leads to a heating time of 71 minutes (cf. Figure 3.9).

14 12 Modeling of a storage water heater Figure 3.9: optimizing the radius of the pipe for N = 7 in the vertical rectangular cross section

15 26th ECMI modelling week Conclusion and perspective We proposed two models for which we could also optimize the geometry of the heating element. We did, however, have to make a lot of concessions. The assumption that the heating element has to be of equal height to the tank was only made to facilitate working on the different cross sections with more or less the same setup. We do now see that there is no real comparability of the results we obtain for the different cross sections. The results support similar statements, e.g. that the time required to heat up the water drops significantly when considering a form of advection in addition to the general diffusion of heat. That was to be expected, though, and on reflection we have to accept that the actual values obtained for the different cross sections do not relate greatly. Therefore, the circular cross section could be abandoned for future work which would lead to a lessened need for assumptions on the coil. Additionally to that, the circular cross section is particularly inaccurate near the circular end pieces of the tank, as strong turbulence is to be expected in these areas. Another aspect that could be improved is the termination condition. It is not awfully realistic to aspire to an average temperature if only a small part of the water in the tank might be demanded at any given time. One could assume that the tank is filled with water at the bottom while the hot water is removed at the top which would speed up the heating process as the warm water naturally moves upwards, anyway. This would clearly necessitate a different type of termination condition, however. One should also aspire to solve more accurate mathematical equations, e.g. the incompressible Navier Stokes equations. However, there already exists a multitude of tailor-made software for both 2D and 3D simulations, respectively, as it is a rather well-researched field - thus raising the question, whether or not future work is expedient.

16 Bibliography [1] General conservation equation URL equation.html [2] et al., JEG. Fipy manual, release , URL [3] Larsson, S. Partial Differential Equations with Numerical Methods. Texts in Applied Mathematics, Vol. 45. Springer, Berlin, Corrected 2nd printing ISBN [4] Per-Olof Persson, UB, Department of Mathematics. Distmesh - a simple mesh generator in matlab URL [5] Stoffel, A. Finite Elemente und Wärmeleitung: Eine Einführung. VCH, ISBN URL 14

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