Question 9. The following non-homogeneous heat equation Ut = 3uxx + π-x models the distribution of heat in a metal rod. Solve the equation if the initial temperature u(x, 0) = x, and the the boundary conditions are: u(0, t) = u(π, t) = = 0.
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- Question # 3: Design a heat equation model to determine the solution. The physical data needed to model for copper includes: density p= 5.82g/cm', thermal conductivity K = 0.095cal / cm sec C, %3D specific heat O =0.097cal / g C. The initial temperature is 70 sin(T x/100) C and the ends %3D are kept at 0 C of a laterally insulated copper bar of 120 cm long. Determine the time its goint to take for the maximum temperature in the bar to drop to 100 C ?differential equation A metal bar is placed in a room at constant temperature of 0°C. If after 25 mins the temperature of bar drops from 120°C to 55°C, find how many minutes the metal bar will continue to cool from 55°C to 50°C.Question 12. The temperature u(x, t) of a narrow metal rod of length L = 1 with a heat source is modelled by the following non-homogeneous heat equation: Ut = Urr +1– x. Solve the equation if the initial temperature u(x, 0) = 4+ x, and the boundary conditions are: u(0, t) = 3, u(1, t) = 5, t> 0. Hint: Transform the non-homogeneous boundary conditions into homogeneous ones for w — и — Сіх — C2.
- A wall of a house is made from two layers of bricks enclosing a layer of insulation. A radiator is positioned to cover the whole internal surface, and used intermittently when the internal temperature is low. The external surface is exposed to the outside air. Which of the following assumptions could be used to identify the relevant reduced form of the conduction equation to find the temperature in the wall. a. Conduction is mainly in two directions. b. Conduction is mainly in one direction. c. The wall properties are homogeneous. d. Steady conditions exist. e. Unsteady conditions exist. f. There is an internal volumetric heat generation in the wall.The thermal current density of a solid varies as: JT(x) = 1000e W m² W = If the solid starts at x = 0, the thermal conductivity is 2, and the temperature at x = 0 is To = 1000 K, find the equation for the temperature profile as a function of x.Solve for the temperature distribution of a long, thin circular solid tube with a length of 10 cm (shown in Figure2), using the Finite Difference Method: Parabolic Equation (Explicit Method). The following values of the circular solid tube is as provided: k’= 0.49 cal/ (s · cm · oC), Δt = 0.1 s and Δx = 2 cm. When t = 0, the temperature of the circular solid tube is zero and the boundary conditions are fixed for all times at temperature T, at x = 0 cm is 100oC where as the temperature T, at x =10 is 50oC. From this information it is given that the circular solid rod is made of material with heat capacity, C = 0.2174 cal/(g· oC) and density (ρ) = 2.7 g/cm3. Use Explicit Method to solve this problem. Task: A Visual Basic of the FDM solution for (a) has been initially developed based on the required calculation of task Q2(a). The source code is as provided below (may need some minor tweak): Option Explicit Sub Explicit() Dim i As Integer, j As Integer, np As Integer, ns As Integer…
- Question: USE CRANCK NICKLISON METHOD TO Find the steady state temperature distribution at all points of a heated steel plate .The plate has 30 cm width and 30 cm in length . The edges of the sheet are kept at constant temperature: 100 , 80 , 60 ,40 degrees. Assume the sheet is divided into 3x3 grids.The temperature distribution for the plane wall is given in Fig. and here T1 and T2 are the temperatures on both sides of the 2 walls. The thermal conductivity of the wall is constant and its thickness is L. T=T1 Subtract the expression for heat generation per unit volume based on x, which represents the distance from the wall for which the equation is satisfied. At x=0, take the heat generation rate as q0. T -T, - = C, +C,x² +C¸x' T, -T, | %3D |8m long rod is at an initial temperature of 80c the left side of the rod is at temperature equal to 39c and the right side is at temperature equal to 68c thermal diffusivity is equal to 10^-4 and grid space is equal to 2m find the temperature distribution at 30_60_90 seconds
- 2. A rectangular block has thickness B in the x-direction. The side at x = 0 is held at temperature T, while the side at x = B is held at T2. The other four sides are well insulated. Heat is generated in the block at a uniform rate per unit volume of [. (a) Use the conduction equation to derive an expression for the steady-state temperature profile, T(x). Assume constant thermal conductivity. (b) Use the result of part (a) to calculate the maximum temperature in the block for the following values of the parameters: T₁-120 °C k-0.2 W/(m K) B-1.0 m T₂-0 F-100 W/m³10. From Newton's law of cooling, the boiled egg submersed in water in the diagram below will cool according to dI = -0. 1(T – a), T(0) = 100°C . Solve this differential equation and express T in terms of a. Do not use the integrating factor method. Present your solution with T as the subject. Use your solution to solve the following: If after 1 = 10 time units this egg cools to T = 2ª°C , what was the value of a (a is the temperature of the water in the glass which is assumed to have remained constant)?Q3: Consider evaluation of different temperatures of solar photovoltaic/thermal system (PVT) as shown in Figure 1(a). The following set of differential equations represent energy balance equations to be solve using matrices and eigenvalues dTglass = -0.75Tglass + 0.75TPVT (1) dt - 1.18Tglass – 22TpyT + 237wax (2) dt dTwax 12Tglass + 18TpyT – 19 Twax (3) dt Where, Tptass, TPVT, and Twax, are temperatures illustrated in Figure 1(b). At time t-0 the initial conditions are Tglass = 35 , Tpyr = 33, and Twax = 31 °C. Cold sappty In frem water Tank Glass PVT Enpann Nane-PCMPVT Collector Wax Tubes Sterg Tank Mat Nanofluid Heat Exchanger Tepe Contalner Tuek et Pump for drain