Task DetailsThroughout this assignment, all variables are non-dimensionalised. The heat transfer system can beapproximated by the following partial differential equation (PDE) in time and space:𝜕Θ𝜕𝑡 + 𝑐 𝜕Θ𝜕𝑥 − 𝑑 𝜕 ! Θ𝜕𝑥 ! = 0, 𝑥 ∈ [0, 1], 𝑡 ∈ [0, 3], (1)where Θ is a non-dimensional temperature, 𝑡 is time, 𝑥 is a spatial coordinate with inlet at 𝑥 = 0 andoutlet at 𝑥 = 1, 𝑐 is a convection speed, and 𝑑 is a positive thermal diffusivity. Boundary conditionsare specified below, and they are different for tasks 1 and 2. The following initial condition applies:Θ(𝑥, 𝑡 = 0) = 𝜏 cos(2𝜋 𝑘 𝑥) , (2)where 𝑘 is an integer wave number. The PDE defined by equations (1) and (2), should be solved by aMATLAB script and/or functions using a suitable Runge-Kutta time-integration scheme and finite-difference spatial discretization of 𝑥 (do not exceed 4th order in time and space).1. (a) Take 𝑑 = 0, 𝜏 = 1 and apply periodic boundary conditions in 𝑥 as well as convectionspeeds c ∈ [0.1,0.3]. Run a total of 𝑁 simulations for 𝑘 = 2, each one with a different c, andrepeat the same for 𝑘 = 3, then store the temperatures Θ(𝑡 = 3, 𝑥 = 1), respectively.Choose appropriate spatial and temporal step sizes, and hence CFL number, and keep thischoice for all simulations, respectively, and briefly explain this choice in the written report,considering numerical stability and accuracy.

The homogeneous form of the heat equation requires the boundary conditions to be identically zero.Group of answer choicesTrueFalse

Exercice - 1 Discrétisation de l'équation de la chaleur 1DConsidérons le problème monodimensionnel de la conduction de la chaleur dans une barre de 1mde longueur. Le champ de température T (x, t) vérie l'équation de la chaleur :∂T∂t = α ∂2T∂x2où α est la diusivité thermique.Á cette EDP s'ajoute deux conditions aux limites aux extrémités de la barre T (0, t) = Tg etT (1, t) = Td ainsi qu'une condition initiale T (x, 0) = T0.L'intervalle [0, 1] est discrétisé en N n÷uds de coordonnées xi régulièrement espacés. Notons∆x le pas d'espace. Le temps est discrétisé en intervalles de pas constant ∆t avec M noeud. NotonsT ni la température au n÷ud i à l'instant n.1. Calculer les pas ∆x et ∆t. Préciser la natures des conditions aux limites.2. Écrire la formulation discrétisée centrée.3. Écrire la forme matricielle du problème4. Écrire un script MATLAB qui implémente la méthode explicite, et qui trace les prols àdivers instants.5. Comparer la solution théorique Tth et numérique Tnum avec les deux schéma par rapportau temps.Exercice - 2 Considérons le problème bidimensionnel instationnaire de la conduction de lachaleur dans un domaine rectangulaire [−L1, L1] × [1, L2] × [0, tmax]. La fonction U (x, y, t) vériel'équation de Laplace :∂U∂t = α ∂2U∂x2 + α ∂2U∂y2U (−L1, y, t) = Ug , U (L1, y, t) = UdU (x, 1, t) = Ub, U (x, L2, t) = UhU (x, y, 0) = u0(x, y) = U0Le domaine de calcul est discrétisé en N × P n÷uds et M n÷uds par rapport au temps. Onsupposera que les pas d'espace dans chaque direction ∆x, ∆y et du temps ∆t sont constants. Lafonction discrète au n÷ud (xni , ynj ) sera notée U nij = U (xni , ynj ).1. Calculer les pas ∆x, ∆y et ∆t. Préciser la natures des conditions aux limites.2. Écrire la formulation discrétisée.3. Écrire la forme matricielle.4. Écrire un script MATLAB qui implémente la méthode explicite, et qui trace les prols àdivers instants.1

Consider a brick wall (Fig. P10.1) of thicknessL=30 cm,k=0.7 W/m ∘ C. The inner surface is at28 ∘ Cand the outer surface is exposed to cold air at−15 ∘ C. The heat-transfer coefficient associated with the outside surface ish=40 W/m 2 ⋅ ∘ C. Determine the steadystate temperature distribution within the wall and also the heat flux through the wall. Use a two-element model, and obtain the solution by hand calculations. Assume onedimensional flow. Then prepare input data and run program HEAT1D.

Steel ball bearings (k= 50 W/mK, =1.3 10-5 m2/s) having a diameter of 40 mm are heated to atemperature of 650oC and then quenched in a tank of oil at 55oC. If the convective heat transfercoefficient between the ball bearings and oil is 300 W/m2K, determine (a) the duration of time thebearings must remain in oil to reach a temperature of 200 oC and (b) the total amount of heat removedfrom each bearing during this time.

4: A wall 0.12 m thick having thermal diffusivity of 15 10-6 m2/s is initially at temperature of 85oC.Suddenly one face temperature lowered to a temperature of 20 oC while other face is perfectly insulated.Using explicit finite difference method, write descretization equation for space and time steps of 30 mmand 300 s respectively.