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About 3307 results. 1294 free access solutions
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| m100361 | (a) Apply the Newton-Raphson method to the function f(x) = tanh(x2 - 9) to evaluate its known real root at x = 3. Use an initial guess of x0 = 3.2 and take a minimum of four iterations.
(b) Did the method exhibit convergence onto its real root? Sketch the plot with the results for each iteration shown. |
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| m100363 | A biofilm with a thickness Lf (cm) grows on the surface of a solid. After traversing a diffusion layer of thickness
L (cm), a chemical compound A diffuses into the biofilm where it is subject to an irreversible first-order reaction that converts it to a product B.
Steady-state mass balances can be used to derive the following ordinary differential equations for compound A:
Where D = the diffusion coefficient in the diffusion layer = 0.8 cm2/d, Df = the diffusion coefficient in the biofilm = 0.64 cm2/d, and k = the first-order rate for the conversion of A to B = 0.1/d. The following boundary conditions hold:
Where ca0 = the concentration of A in the bulk liquid = 100 mol/L. Use the finite-difference method to compute the steady-state distribution of A from x = 0 to L + Lf, where L = 0.008 cm and Lf = 0.004 cm. Employ centered finite differences with ∆x = 0.001 cm. |
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| m100390 | A group of 35 students attend a class in a room that measures 11 m by 8 m by 3 m. Each student takes up about 0.075 m3 and gives out about 80 W of heat (1 W = 1 J/s). Calculate the air temperature rise during the first 20 minutes of the class if the room is completely sealed and insulated. Assume the heat capacity, Cy, for air is 0.718 kJ/(kg K). Assume air is an ideal gas at 20∘C and101.325 kPa. Note that the heat absorbed by the air Q is related to the mass of the air m, the heat capacity, and the change in temperature by the following relationship:
The mass of air can be obtained from the ideal gas law:
where P is the gas pressure, V is the volume of the gas, Mwt is the molecular weight of the gas (for air, 28.97 kg/kmol), and R is the ideal gas constant [8.314 kPa m3/(kmol K)]. |
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| m100396 | A linearized groundwater model was used to simulate the height of the water table for an unconfined aquifer. A more realistic result can be obtained by using the following nonlinear ODE:
Where x = distance (m), K = hydraulic conductivity (m/d), h = height of the water table (m), and N = infiltration rate (m/d). Solve for the height of the water table for the same case as in Prob. 28.30. That is solve from x = 0 to 1000 m with h(0) = 10 m, h(1000) 5 = m, K = 1 m/d, and N = 0.0001 m/d. Obtain your solution with
(a) The shooting method and
(b) The finite-difference method. |
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| m100417 | A series of first-order, liquid-phase reactions create a desirable product (B) and an undesirable byproduct (C)
If the reactions take place in an axially-dispersed plug-flow reactor (Fig. P28.14), steady-state mass balances can be used to develop the following second-order ODEs,
Use the finite-difference approach to solve for the concentration of each reactant as a function of distance given: D = 0.1 m2/min, U = 1 m/min, k1 = 3/min, k2 = 1/min, L = 0.5 m, ca,in = 10 mol/L. Employ centered finite-difference approximations with ∆x = 0.05 m to obtain your solutions and assume Danckwerts boundary conditions, as described in Prob. 28.14. Also, compute the sum of the reactants as a function of distance. Do your results make sense? |
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| m100442 | A version of the Poisson equation that occurs in mechanics is the following model for the vertical deflection of a bar with a distributed load P(x):
where Ac = cross-sectional area, E = Young s modulus, u = deflection, and x = distance measured along the bar s length. If the bar is rigidly fixed (u = 0) at both ends, use the finite-element method to model its deflections for Ac = 0.1 m2, E = 200 × 109 N/m2, L = 10 m, and P(x) = 1000 N/m. Employ a value of Δx = 2 m. |
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| m100448 | According to Archimedes principle, the buoyancy force is equal to the weight of fl uid displaced by the submerged portion of an object. For the sphere depicted in Fig. P5.19, use bisection to determine the height h of the portion that is above water. Employ the following values for your computation: r = 1 m, ps = density of sphere 5 200 kg/m3, and pw = density of water = 1000 kg/m3. Note that the volume of the above-water portion of the sphere can be computed with |
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| m100453 | Although we did not mention it in Sec. 8.2, Eq. (8.10) is actually an expression of electro neutrality; that is, that positive and negative charges must balance. This can be seen more clearly by expressing it as
[H+] = [HCO-3 ] + 2[CO2-3 ] + [OH-]
In other words, the positive charges must equal the negative charges. Thus, when you compute the pH of a natural water body such as a lake, you must also account for other ions that may be present. For the case where these ions originate from nonreactive salts, the net negative minus positive charges due to these ions are lumped together in a quantity called alkalinity, and the equation is reformulated as
Alk + [H+] = [HCO-3] + 2[CO2-3 ] + [OH-] (P8.28)
where Alk 5 alkalinity (eq/L). For example, the alkalinity of Lake Superior is approximately 0.4 3 1023 eq/L. Perform the same calculations as in Sec. 8.2 to compute the pH of Lake Superior in 2008. Assume that just like the raindrops, the lake is in equilibrium with atmospheric CO2, but account for the alkalinity as in Eq. (P8.28). |
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| m100467 | Apply the conservation of volume (see Prob. 1.9) to simulate the level of liquid in a conical storage tank (Fig. P1.11). The liquid flows in at a sinusoidal rate of Qin = 3 sin2(t) and flows out according to
Q out = 3(y - yout)1.5y > yout
Q out = 0 y ≤ yout
where flow has units of m3/d and y 5 the elevation of the water surface above the bottom of the tank (m). Use Euler s method to solve for the depth y from t 5 0 to 10 d with a step size of 0.5 d. The parameter values are rtop = 2.5 m, ytop = 4 m, and yout = 1 m. Assume that the level is |initially below the outlet pipe with y(0) 5 0.8 m. |
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| m100468 | As a member of Engineers Without Borders, you are working in a community that has contaminated drinking water. At t = 0, you add a disinfectant to a cistern that is contaminated with bacteria. You make the following measurements at several times thereafter:
If the water is safe to drink when the concentration falls below 5 #/100 mL, estimate the time at which the concentration will fall below this limit. |
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| m100469 | As depicted in Fig. P1.15, an RLC circuit consists of three elements: a resistor (R), and inductor (L) and a capacitor (C). The flow of current across each element induces a voltage drop. Kirchhoff s second voltage law states that the algebraic sum of these voltage drops around a closed circuit is zero,
where i = current, R = resistance, L = inductance, t = time, q = charge, and C = capacitance. In addition, the current is related to charge as in
(a) If the initial values are i(0) = 0 and q(0) = 1 C, use Euler s method to solve this pair of differential equations from t = 0 to 0.1 s using a step size of Δt = 0.01 s. Employ the following parameters for your calculation: R = 200 Ω, L = 5 H, and C 5 10-4 F.
(b) Develop a plot of i and q versus t. |
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| m100470 | As depicted in Fig. P1.22, a spherical particle settling through a quiescent fluid is subject to three forces: the downward force of gravity (FG), and the upward forces of buoyancy (FB) and drag (FD). Both the gravity and buoyancy forces can be computed with Newton s second law with the latter equal to the weight of the displaced fluid. For laminar flow, the drag force can be computed with Stokes s law,
FD = 3 π µ d ν
where µ = the dynamic viscosity of the fluid (N s/m2), d = the particle diameter (m), and ν = the particle s settling velocity (m/s). Note that the mass of the particle can be expressed as the product of the particle s volume and density ps (kg/m3) and the mass of the displaced fluid can be computed as the product of the particle s volume and the fluid s density p (kg/m3). The volume of a sphere is πd3/6. In addition, laminar flow corresponds to the case where the dimensionless Reynolds number, Re, is less than 1, where Re = p d ν / µ.
(a) Use a force balance for the particle to develop the differential equation for dy/dt as a function of d, p, ps, and µ.
(b) At steady-state, use this equation to solve for the particle s terminal velocity.
(c) Employ the result of (b) to compute the particle s terminal velocity in m/s for a spherical silt particle settling in water: d = 10 mm, p = 1 g/cm3, ps = 2.65 g/cm3, and µ = 0.014 g/(cm∙s).
(d) Check whether flow is laminar.
(e) Use Euler s method to compute the velocity from t = 0 to 2-15 s with Δt = 2-18 |
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| m100471 | As depicted in Fig. P1.24, the downward deflection y (m) of a cantilever beam with a uniform load w (kg/m) can be computed as
Where x = distance (m), E = the modulus of elasticity = 2 ×1011 Pa, I= moment of inertia = 3.25 × 10-4 m4, w = 10,000 N/m, and L = length = 4 m. This equation can be differentiated to yield the slope of the downward deflection as a function of x:
If y = 0 at x = 0, use this equation with Euler s method (Δx = 0.125 m) to compute the deflection from x = 0 to L. Develop a plot of your results along with the analytical solution computed with the first equation. |
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| m100472 | As depicted in Fig. P2.27, a water tank consists of a cylinder topped by the frustum of a cone. Develop a well structured function in the high-level language or macro language of your choice to compute the volume given the water level h (m) above the tank s bottom. Design the function so that it returns a value of zero for negative h s and the value of the maximum filled volume for h s greater than the tank s maximum depth. Given the following parameters, H1 = 10 m, r1 = 4 m, H2 = 5 m, and r2 = 6.5 m, test your function by using it to compute the volumes and generate a graph of the volume as a function of level from h = - 1 to 16 m. |
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| m100473 | As depicted in Fig. P5.15, the velocity of water, y (m/s), discharged from a cylindrical tank through a long pipe can be computed as
where g = 9.81 m/s2, H = initial head (m), L = pipe length (m), and t = elapsed time (s). Determine the head needed to achieve v = 5 m/s in 2.5 s for a 4-m-long pipe (a) graphically, (b) by bisection, and (c) with false position. Employ initial guesses of xl = 0 and xu = 2 m with a stopping criterion of Es = 1%. Check you results. |
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| m100474 | As described in Prob. 1.22, in addition to the downward force of gravity (weight) and drag, an object falling through a fluid is also subject to a buoyancy force that is proportional to the displaced volume. For example, for a sphere with diameter d(m), the sphere svolume is V = π d3/6 and its projected area is A = π d2/4. The buoyancy force can then be computed as Fb = -pVg. We neglected buoyancy in our derivation of Eq. (1.9) because it is relatively small for an object like a parachutist moving through air. However, for a more dense fluid like water, it becomes more prominent.
(a) Derive a differential equation in the same fashion as Eq. (1.9), but include the buoyancy force and represent the drag force as described in Prob. 1.21.
(b) Rewrite the differential equation from (a) for the special case of a sphere.
(c) Use the equation developed in (b) to compute the terminal velocity (i.e., for the steady-state case). Use the following parameter values for a sphere falling through water: sphere diameter =1 cm, sphere density = 2700 kg/m3, water density = 1000 kg/m3, and Cd = 0.47.
(d) Use Euler s method with a step size of Δt = 0.03125 s to numerically solve for the velocity from t = 0 to 0.25 s with an initial velocity of zero. |
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| m100475 | As described in Sec. PT3.1.2, linear algebraic equations can arise in the solution of differential equations. For example, the following differential equation results from a steady-state mass balance for a chemical in a one-dimensional canal,
Where c = concentration, t=time, x=distance, D=diffusion coefficient, U=fluid velocity, and k=a first-order decay rate. Convert this differential equation to an equivalent system of simultaneous algebraic equations. Given D=2, U=1, k=0.2, c(0)=80 and c(10)=20, solve these equations from x=0 to 10 with (x=2, and develop a plot of concentration versus distance? |
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| m100478 | As noted in Prob. 1.3, drag is more accurately represented as depending on the square of velocity. A more fundamental representation of the drag force, which assumes turbulent conditions (i.e., a high Reynolds number), can be formulated as
where Fd = the drag force (N), r = fluid density (kg/m3), A = the frontal area of the object on a plane perpendicular to the direction of motion (m2), ν = velocity (m/s), and Cd = a dimensionless drag coefficient.
(a) Write the pair of differential equations for velocity and position (see Prob. 1.18) to describe the vertical motion of a sphere with diameter d (m) and a density of ps (kg/km3). The differential equation for velocity should be written as a function of the sphere s diameter.
(b) Use Euler s method with a step size of Δt = 2 s to compute the position and velocity of a sphere over the first 14 s. Employ the following parameters in your calculation: d = 120 cm, p = 1.3 kg/m3, ps = 2700 kg/m3, and Cd = 0.47. Assume that the sphere has the initial conditions: x(0) = 100 m and ν(0) = -40 m/s.
(c) Develop a plot of your results (i.e., y and ν versus t) and use it to graphically estimate when the sphere would hit the ground.
(d) Compute the value for the bulk second-order drag coefficient Cd (kg/m). Note that, as described in Prob. 1.3, the bulk second order drag coefficient is the term in the final differential equation for velocity that multiplies the term ν | ν |. |
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| m100482 | As was done in Sec. 24.4, determine the work performed if a constant force of 1 N applied at an angle u results in the following displacements. Use the MATLAB function cumtrapz to determine the cumulative work and plot the result versus u. |
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| m100498 | Bacteria growing in a batch reactor utilize a soluble food source (substrate) as depicted in Fig. P28.16. The uptake of the substrate is represented by a logistic model with Michaelis-Menten limitation. Death of the bacteria produces detritus which is subsequently converted to the substrate by hydrolysis. In addition, the bacteria also excrete some substrate directly. Death, hydrolysis, and excretion are all simulated as first-order reactions. Mass balances can be written as
Where X, C, and S = the concentrations (mg/L) of bacteria, detritus, and substrate, respectively; mmax = maximum growth rate (/d), K = the logistic carrying capacity (mg/L); Ks = the Michaelis-Menten half-saturation constant (mg/L), kd = death rate (/d); ke = excretion rate (/d); and kh = hydrolysis rate (/d). Simulate the concentrations from t = 0 to 100 d, given the initial conditions X(0) = 1 mg/L, S(0) = 100 mg/L, and C(0) = 0 mg/L. Employ the following parameters in your calculation: mmax = 10/d, K = 10 mg/L, Ks = 10 mg/L, kd = 0.1/d, ke = 0.1/d, and kh = 0.1/d. |
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