Mechanical Engineering Answers

This problem consist of analyzing the accuracy of the finite element method in modeling the plane wall thermal problem. The plane wall is 1 m thick. The left surface of the wall (x=0) is maintained at a constant temperature of 200˚C, and the right surface (x=L=1 m) is insulated. The thermal conductivity is Kxx=25 W/(m ˚C) and there is a uniform generation of heat inside the wall of Q=400 W/m3 .

4. Use the FE interpolation function, 𝑇𝑇(𝑥𝑥) = ∑ 𝑁𝑁(𝑥𝑥)𝑑𝑑 = [𝑁𝑁]{𝑑𝑑} to plot the FE results along the length of the wall. Compare your results against the analytical solution. Nodal points should be labeled using a Marker (o, +, d, s, etc). Interpolation points should be connected with a line.

5. Develop a function that computes the second norm of the error (the difference between FE and the analytical solution throughout the length). The error should be computed at both the nodal points and interpolation points.  

Answer needs to be in matlab coding

Problem-2: This problem consist of analyzing the accuracy of the finite element method in modeling the plane wall thermal problem. The plane wall is 1 m thick. The left surface of the wall (x=0) is maintained at a constant temperature of 200˚C, and the right surface (x=L=1m) is insulated. The thermal conductivity is Kxx=25 W/(m˚C) and there is a uniform generation of heat inside the wall of Q=400 W/m3 .

1. Use the Matlab code provided to you in class to model the problem. Also, use 5 elements for Hand Solution.  

2. Compare your results to the analytical solution, given by: T(x)=QxL/K_xx(1-x/2L)+T(0)

3. Prepare a table that reports the nodal temperature results of Matlab, and the analytical solution evaluated at the nodal points.

The boom in the figure supports a load W = 1400 lb. Compute the forces in the cables AE and BD and also the components exerted by the ball and socket joint C.

Answers: AE = 512 lb; BD = 878 lb; CX = -250 lb; CY = 600 lb; CZ = 800 lb

Determine the center of gravity and area moment of inertia about centroidal axes of shaded area as shown in Figure 1

One kg of fluid enters the steady flow apparatus at a pressure of P1, velocity 16m/s and

specific volume 0.4 m3/kg. The Inlet is 30 m above the ground level. The fluid leaves the

apparatus at Pressure of P2, velocity of 275 m/s and specific volume 0.6 m3/kg. The outlet is

at the ground level. The total heat loss between the inlet and outlet is 10 KJ/kg of fluid. If 140

KJ/kg of work is done by the system, find the change in specific internal energy and indicate

whether this is a increase or decrease.

P1 = 6 Bar − {

√X5

849 } Pa ; P2 = 1 Bar + {+√99 − X

4

}Pa

x= 51

Using powers of 10, estimate the number of quantum states accessible to an outer electron on an atom, by estimating the size of an atom and knowing that typical binding energies are a few eV. (That is, if the kinetic energy is greater than a few eV then the electron is no longer on the atom.) The number of states = VrVp/h3.) 

White dwarf stars are essentially plasmas of free electrons and free protons

(hydrogen atoms that have been stripped of their electrons). Their densities

are typically 5 × 10^9 kg/m3 (i.e., 10^6 times the average density of Earth).

Their further collapse is prevented by the fact that the electrons are highly

degenerate, that is, all the low-lying states are filled and no two identical

electrons can be forced into the same state.

(a) Estimate the temperature of this system. (Assume nonrelativistic electrons and that p2

f /2m = (3/2)kT .)

(b) Now suppose that the gravity is so strong that the electrons are forced

to combine with the protons, forming neutrons (with the release of a

neutrino). If the temperature remains the same, what would be the density

of this degenerate neutron star?

Thermal energies for nucleons in large nuclei are comparable with their

binding energies of about 6 MeV.

(a) To what temperature does this correspond?

(b) Within nuclear matter, identical nucleons are separated by about 2.6 ×

10^−15 m. What is roughly the minimum temperature needed for them

not to be degenerate (i.e., for the number of accessible states to be much

larger than the number of particles.)?

Consider a system of nonrelativistic electrons in a white dwarf star at a

temperature of 10^9 K. Very roughly, what would be their density if the system

is degenerate? How does this compare with typical electron densities in

ordinary matter of about 10^30 electrons/m3?