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| m100526 | Beyond fluids, Archimedes principle has proven useful in geology when applied to solids on the earth s crust. Figure P1.26 depicts one such case where a lighter conical granite mountain "floats on" a denser basalt layer at the earth s surface. Note that the part of the cone below the surface is formally referred to as a frustum. Develop a steady-state force balance for this case in terms of the following parameters: basalt s density (pb), granite s density (pg), the cone s bottom radius (r), and the height above (h1) andbelow (h2) the earth s surface. |
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| m100527 | Beyond the Colebrook equation, other relationships, such as the Fanning friction factor f, are available to estimate friction in pipes. The Fanning friction factor is dependent on a number of parameters related to the size of the pipe and the fluid, which can all be represented by another dimensionless quantity, the Reynolds number Re. A formula that predicts f given Re is the von Karman equation,
Typical values for the Reynolds number for turbulent flow are 10,000 to 500,000 and for the Fanning friction factor are 0.001 to 0.01. Develop a function that uses bisection to solve for f given a user-supplied value of Re between 2500 and 1,000,000. Design the function so that it ensures that the absolute error in the result is Ea,d < 0.000005. |
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| m100553 | Compute the temperature distribution for the L-shaped plate in Figure. |
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| m100554 | Compute work as described in Sec. 24.4, but use the following equations for F(x) and θ(x):
F(x) = 1.6x - 0.045x2
u(x) = -0.00055x3 + 0.0123x2 + 0.13x
The force is in newtons and the angle is in radians. Perform the integration from x = 0 to 30 m. |
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| m100644 | Design the optimal cylindrical tank with dished ends (Fig. P16.3). The container is to hold 0.5 m3 and has walls of negligible thickness. Note that the area and volume of each of the dished ends can be computed with Figure P16.3
(a) Design the tank so that surface area is minimized. Interpret the result.
(b) Repeat part (a), but add the constraint L ( 2h? |
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| m100645 | Determine the Frobenius and the row-sum norms for the systems in Probs. 10.3 and 10.4. Scale the matrices by making the maximum element in each row equal to one? |
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| m100646 | Determining the velocity of particles settling through fluids is of great importance of many areas of engineering and science. Such calculations depend on the flow regime as represented by the dimensionless Reynolds number,
Re =pdv/μ (P8.48.1)
where ρ = the fluid s density (kg/m3), d = the particle diameter (m), y = the particle s settling velocity (m/s), and μ = the fluid s dynamic viscosity (N s/m2). Under laminar conditions (Re, 0.1), the settling velocity of a spherical particle can be computed with the following formula based on Stokes law,
where g = the gravitational constant (5 9.81 m/s2), and ps = the particle s density (kg/m3). For turbulent conditions (i.e., higher Reynolds numbers), an alternative approach can be used based on the following formula:
where CD = the drag coefficient, which depends on the Reynolds number as in
(a) Combine Eqs. (P8.48.2), (P8.48.3), and (P8.48.4) to express the determination of y as a roots of equations problem. That is, express the combined formula in the format f(y) = 0.
(b) Use the modified secant method with δ = 10-3 and εs = 0.05% to determine v for a spherical iron particle settling in water, where d = 200 μm, ρ = 1 g/cm3, ρs = 7.874 g/cm3, and μ = 0.014 g/(cm∙s). Employ Eq. (P8.48.2) to generate your initial guess.
(c) Based on the result of (b), compute the Reynolds number and the drag coefficient, and use the latter to confirm that the flow regime is not laminar.
(d) Develop a fixed-point iteration solution for |
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| m100647 | Develop a user-friendly computer program for adaptive quadrature based on Fig. 22.5. Test it by solving Prob. 22.10.
Fig 22.5. |
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| m100648 | Develop a user-friendly program for Brent s root location method based on Fig. 6.12. Test it by solving Prob. 6.6. |
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| m100649 | Develop a user-friendly program for the Newton-Raphson method based on Fig. 6.4 and Sec. 6.2.3. Test it by duplicating the computation from Example 6.3. |
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| m100650 | Develop a user-friendly program for the two-equation Newton-Raphson method based on Sec. 6.6.2. Test it by solving Example 6.12. |
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| m100651 | Develop a user-friendly program to model the steady-state distribution of temperature in a rod with a constant heat source using the finite-element method. Set up the program so that unequally spaced nodes may be used. |
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| m100652 | Develop a well-structured function to determine the elapsed days in a year. The function should be passed three values: mo = the month (1-12), da = the day (1-31) and leap = (0 for non-leap year and 1 for leap year). Test it for January 1, 1999; February 29, 2000; March 1, 2001; June 21, 2002; and December 31, 2004. |
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| m100653 | Develop a well-structured function to determine the elapsed days in a year. The first line of the function should be set up as function nd = days(mo, da, year) where mo = the month (1-12), da = the day (1-31) and year = the year. Test it for January 1, 1999; February 29, 2000; March 1, 2001; June 21, 2002; and December 31, 2004. 2.21 Manning s equation can be used to compute the velocity of water in a rectangular open channel, |
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| m100654 | Develop an M-file function for the Newton-Raphson method based on Fig. 6.4 and Sec. 6.2.3. Along with the initial guess, pass the function and its derivative as arguments. Test it by duplicating the computation from Example 6.3. |
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| m100655 | Develop an M-file that is expressly designed to locate a maximum with the golden-section search algorithm. In other words, set it up so that it directly finds the maximum rather than finding the minimum of -f(x). Test your program with the same problem as Example 13.1. The function should have the following features:
• Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations.
• Return both the optimal x and f(x)? |
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| m100656 | Develop an M-file to implement parabolic interpolation to locate a minimum. Test your program with the same problem as Example 13.2. The function should have the following features:
• Base it on two initial guesses, and have the program generate the third initial value at the midpoint of the interval.
• Check whether the guesses bracket a maximum. If not, the function should not implement the algorithm, but should return an error message.
• Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations.
• Return both the optimal x and f(x).
• Use a bracketing approach (as in Example 13.2) to replace old values with new values? |
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| m100657 | Develop an M-file to locate a minimum with the golden section search. Rather than using the standard stopping criteria (as in Fig. 13.5), determine the number of iterations needed to attain a desired tolerance? |
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| m100658 | Develop well-structured function procedures to determine
(a) The factorial;
(b) The minimum value in a vector; and
(c) The average of the values in a vector. |
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| m100666 | Employ two- through six-point Gauss-Legendre formulas to solve
Interpret your results in light of Eq. (22.32) |
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