Use separation of variables to solve the 2D transient heat equation on a rectangular domain with homogeneous temperature boundary conditions. ди Ot C 2u მ12მ2 (1) u(0, y, t) = u(a, y, t) = 0 (2) 13 ۵ u(x, 0,t) = u(x, b,t) = 0 u(x, y, 0) = f(x, y)
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- ZVA @ e SIM a A moodle1.du.edu.om Solve the second order homogeneous differential equation y " -6y'-7y = 0 with intial values y(0) = 0; y'(0) = 1 %3D Maximum file size: 200MB, maximum number of files: 1 Files IIYou have a 1-D steady-state conduction problem, constant thermal properties, with energy generation q_dot. The material is 4.0 cm thick and has a constant thermal conductivity of k = 65.0 (W/m-k). The temperature distribution within the object is: T(x) = a + bx^2 a = 100 Celcius b = -1000 Celcius/m^2 Starting with the Heat Diffusion Equation and using the data given above, determine the following: • Determine the heat generation rate q_dot within the wall. • Determine the heat flux q" at x=0 and at x=L.Use FEM to calculate the temperature at nodes 2 and 3, and the heat at nodes 1 and 4. Assume A is one unit area.
- Eng MathAn unmanned submarine is equipped with temperature and depth measuring instruments and hasradio equipment that can transmit the output readings of these instruments back to the surface.The submarine is initially floating on the surface of the sea with the instrument output readingsin steady state. The depth measuring instrument is approximately zero order and the temperaturetransducer first order with a time constant of 50 seconds. The water temperature on the seasurface, T0, is 20C and the temperature Tx at a depth of x metres is given by the relation:Tx = T0- 0.01x(a) If the submarine starts diving at time zero, and thereafter goes down at a velocity of 0.5metres/second, draw a table showing the temperature and depth measurements reported atintervals of 100 seconds over the first 500 seconds of travel. Show also in the table the error ineach temperature reading(b) What temperature does the submarine report at a depth of 1000 metres?Thermocouples are devices used to measure temperature of a given sample or the surroundingmedium. These devices feature a “bead”, which has a spherical shape, and produces voltage upon changein temperature. For an engineering application at a pharmaceutical company, thermocouple devices arebeing tested for their responsiveness, i.e. how fast it can detect temperature changes in the environment(surrounding air). The engineers at this company have specified design constraints for an idealthermocouple: it must detect temperature changes no later than 1.5 minute, and the reportedtemperature value must be reasonably correct: at most %3 difference between measured and actualtemperature values is allowed.Four thermocouples from different vendors are being tested. Relevant properties of these devices arelisted below: The experiment involves placing the thermocouple from air at 20 °C to air at 150 °C, and monitoring thetime it takes for the measured values to reach 150 °C, which is the…
- 1 - Explain Van-Neumann stability by finite difference schemes for this equations a- Laplace equation. J²u dyz (x, y) = 0 J²u dx2(x, y) + b- poison equation. J²u J²u { (x, y) + əy² (x, y) = f (x, y) əx²Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) f(x, y) = 9 sin(x) sin(y), −? < x < ?, −? < y < ? local maximum value(s) local minimum value(s) saddle point(s) (x, y) =1. The general form of linear second-order differential equation can be written in the form: و بار / كلية الهندسة Q4)/ grap dy q(x)y = r(x) d'y +p(x) dx dy b. dx - F(x)y = F(x) x2 dy dx - xy = C. d. r2 d?y dx2 -f(x)y = F(x) 2431)(5-1) 3 (3-21)2 a. (221 -91i) / 169 b. (21 + 52i)/ 13 c. (-90+220i)/169 d. (-7+17i)/ 13 2. Simplify: الحدار المك المراغة 3. If the roots of second order differential equation is complex conjugate, then the gene contain: a. sinusoidal functions and exponentials b. constant and two exponentials c. two constants and two exponentials d. two constants and one exponential 5 4. The order and degree of the differential: 3(3 - + 4y = sinx* are: d²y a. First-order, First-degree- b. First-order, second-degree Second -order, First -degree d. Second -order, second-degree dx2 lo - 2i tisi. 8- 12i 5. The particular solution of (D² + 4)y = cos 2x is equal to: a. sin 2x b. cos 2x 13+159 C. 4 cos 2x d. 4 sin 2x 5-12 lo Best wishes الامتحانية د. مازن ياسین عبود رئيس القسم بن فاضل…
- (3) For the given boundary value problem, the exact solution is given as = 3x - 7y. (a) Based on the exact solution, find the values on all sides, (b) discretize the domain into 16 elements and 15 evenly spaced nodes. Run poisson.m and check if the finite element approximation and exact solution matches, (c) plot the D values from step (b) using topo.m. y Side 3 Side 1 8.0 (4) The temperature distribution in a flat slab needs to be studied under the conditions shown i the table. The ? in table indicates insulated boundary and Q is the distributed heat source. I all cases assume the upper and lower boundaries are insulated. Assume that the units of length energy, and temperature for the values shown are consistent with a unit value for the coefficier of thermal conductivity. Boundary Temperatures 6 Case A C D. D. 00 LEGION Side 4 z episA bar of length 1 m is insulated along its sides. It is initially at a uniform temperature of 15°C and a t=0 the ends are plunged into ice and maintained at a temperature of 0°C. 1. State all Initial and boundary conditions 2. Determine an expression for the temperature distribution at a point P a distance x from one end at any subsequent time t seconds after t-0, 3. write all the Eigen values and Eigen function possible 1 ди %3D dx? c2 atThe term dT/dx Where T is temperature and x is length, in Fourier law conduction, is called Temperature GradientRequired to answer. Single choice. OPTIONS: 1.May be 2.May not be 3.True 4.False