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| m101321 | The Redlich-Kwong equation of state is given by
where R = the universal gas constant [= 0.518 kJ/(kg K)], T = absolute temperature (K), p = absolute pressure (kPa), and v = the volume of a kg of gas (m3/kg). The parameters a and b are calculated by
where pc = critical pressure (kPa) and Tc = critical temperature (K). As a chemical engineer, you are asked to determine the amount of methane fuel (pc = 4600 kPa and Tc = 191 K) that can be held in a 3-m3 tank at a temperature of -40°C with a pressure of 65,000 kPa. Use a root-locating method of your choice to calculate y and then determine the mass of methane contained in the tank. |
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| m101348 | The velocity is equal to the rate of change of distance x (m),
(a) Substitute Eq. (1.10) and develop an analytical solution for distance as a function of time. Assume that x(0) = 0.
(b) Use Euler s method to numerically integrate Eqs. (P1.18) and (1.9) in order to determine both the velocity and distance fallen as a function of time for the first 10 s of free-fall using the same parameters as in Example 1.2.
(c) Develop a plot of your numerical results together with the analytical solution. |
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| m101362 | Three linked bungee jumpers are depicted in Fig. P25.26. If the bungee cords are idealized as linear springs (i.e., governed by Hooke s law), the following differential equations based on force balances can be developed
Where mi = the mass of jumper i (kg), kj = the spring constant for cord j (N/m), xi = the displacement of jumper i measured downward
From its equilibrium position (m), and g = gravitational acceleration (9.81 m/s2). Solve these equations for the positions and velocities of the three jumpers given the initial conditions that all positions and velocities are zero at t = 0. Use the following parameters for your calculations: m1 = 60 kg, m2 = 70 kg, m3 = 80 kg, k1 = k3 = 50, and k2 = 100 (N/m). |
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| m101363 | Three masses are suspended vertically by a series of identical springs where mass 1 is at the top and mass 3 is at the bottom. If g = 9.81 m/s2, m1 = 2 kg, m2 = 3 kg, m3 = 2.5 kg, and the k s = 10 kg/s2, solve for the displacements x. |
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| m101379 | Two masses are attached to a wall by linear springs. Force balances based on Newton s second law can be written as
Where k = the spring constants, m = mass, L = the length of the unstretched spring, and w = the width of the mass. Compute the positions of the masses as a function of time using the following parameter values: k1 = k2 = 5, m1 = m2 = 2, w1 = w2 = 5, and L1 = L2 = 2. Set the initial conditions as x1 = L1 and x2 = L1 + w1 + L2 + 6. Perform the simulation from t = 0 to 20. Construct timeseries plots of both the displacements and the velocities. In addition, produce a phase-plane plot of x1 versus x2. |
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| m101380 | Under a number of simplifying assumptions, the steadystate height of the water table in a one-dimensional, unconfined groundwater aquifer (Fig. P28.30) can be modeled with the following second-order ODE,
Where x = distance (m), K = hydraulic conductivity (m/d), h = height of the water table (m), h = the average height of the water table (m), and N = infiltration rate (m/d). Solve for the height of the water table for x = 0 to 1000 m where h(0) = 10 m and h(1000) = 5 m. Use the following parameters for the calculation: K = 1 m/d and N = 0.0001 m/d. Set the average height of the water table as the average of the boundary conditions. Obtain your solution with
(a) The shooting method and
(b) The finite difference method. |
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| m101391 | Use (a) Euler s and (b) the fourth-order RK method to solve
Over the range t = 0 to 0.4 using a step size of 0.1 with y(0) = 2 and z(0) = 4. |
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| m101397 | Use a software package to solve for the temperature distribution of the L-shaped plate in Fig. P29.18. Display your results as a contour plot with flux arrows.
Fig.P29.18. |
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| m101403 | Use Archimedes principle to develop a steady-state force balance for a spherical ball of ice floating in seawater (Fig. P1.25). The force balance should be expressed as a third-order polynomial (cubic) in terms of height of the cap above the water line (h), the seawater s density (pf), the ball s density (ps), and the ball s radius (r). |
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| m101404 | Use bisection to determine the drag coefficient needed so that an 82-kg parachutist has a velocity of 36 m/s after 4 s of free fall. Note: The acceleration of gravity is 9.81 m/s2. Start with initial guesses of xl = 3 and xu = 5 and iterate until the approximate relative error falls below 2%. Also perform an error check by substituting your final answer into the original equation. |
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| m101407 | Use Excel to model the temperature distribution of the slab shown in Fig. P31.9. The slab is 0.02 m thick and has a thermal conductivity of 3 W/(m ∙ °C).
Fig .P31.9. |
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| m101408 | Use Gauss elimination to solve:
8x1 + 2x2 - 2x3 = 22
10x1 + 2x2 + 4x3 = 4
12x1 + 2x2 + 2x3 = 6
Employ partial pivoting and check your answers by substituting them into the original equations. |
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| m101409 | Use Gauss-Jordan elimination to solve:
2x1 + x2 - x3 = 1
5x1 + 2x2 + 2x3 = 24
3x1 + x2 + x3 = 5
Do not employ pivoting. Check your answers by substituting them into the original equations. |
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| m101411 | Use MATLAB or Mathcad to determine the roots for the equations in Prob. 7.5. |
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| m101412 | Use MATLAB or Mathcad to develop a contour plot with flux arrows for the Excel solution from Prob. 31.9.
Prob.31.9. |
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| m101413 | Use MATLAB or Mathcad to develop a contour plot with flux arrows for the Excel solution from Prob. 31.7. |
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| m101414 | Use MATLAB or Mathcad to integrate
dx/dt = -θx + θy
dy/dt = rx - y - xz
dz/dt = -bz + xy
Where s = 10, b = 2.666667, and r = 28. Employ initial conditions of x = y = z = 5 and integrate from t = 0 to 20. |
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| m101415 | Use Newton s interpolating polynomial to determine y at x = 8 to the best possible accuracy. Compute the finite divided differences as in Fig. 18.5 and order your points to attain optimal accuracy and convergence.
Figure 18.5
Graphical depiction of the recursive nature of finite divided differences. |
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| m101416 | Use Newton s interpolating polynomial to determine y at x = 3.5 to the best possible accuracy. Compute the finite divided differences as in Fig. 18.5 and order your points to attain optimal accuracy and convergence.
Figure 18.5
Graphical depiction of the recursive nature of finite divided differences. |
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| m101417 | Use numerical integration to compute how much mass leaves a reactor based on the following measurements. |
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