Page 223 - Numerical methods for chemical engineering

Problems 209

cnstant
diaeter L = 50 cm
3
D = 100 cm
3
z 5 = 150 cm
z = 125 cm, meas. D(z ) = 85 cm
variae 4 4
diaeter L = 100 cm z = 100 cm, meas. D(z ) = 55 cm
2
D(z) 3 3
z = 75 cm, meas. D(z ) = 40 cm
2
2
cnstant z 1 = 50 cm
diaeter L 1 = 50 cm
D = 35 cm
1
z = 0 cm
0
cindrica ie L = 20 cm
p
dia D = 1 cm.
p
ie tet t atseric ressre
Figure 4.21 Tank of variable cross-sectional area.
4.A.2. Compute the value of the following definite integral using both dblquad and Monte
Carlo integration,
√
2 x
' '
2
2
I D = [(x − 1) + y ]dydx (4.248)
1 0
4.A.3. Consider the 1-D motion of a point mass, connected to the origin by a harmonic
spring, that experiences a frictional drag force and a time-dependent external force. The
equation of motion is

2
d x dx
m =−Kx − ζ + F ext (t) (4.249)
dt 2 dt
√
Let the mass be 1 kg, and let the harmonic frequency ω c = K/m be 2π rad/s. Let the
external force be F ext (t) = (10 −2 N)sin (ωt), where ω is set at 0.01ω c , 0.1ω c , 0.5ω c , 0.9ω c
0.99ω c , and 0.999ω c . Plot x(t) for each of these cases when ζ = 0 and describe what
happens as ω → ω c . Repeat for non zero ζ, and describe how the behavior changes as ζ
increases.
4.B.1. Consider the axisymmetric tank shown in Figure 4.21. We want to estimate the time
3
that it takes for the tank to drain if filled with water (density 1000 kg/m , viscosity 0.001
Pa s). Let h(t) be the height of water in the tank, h(0) = 2m.v p (t). The mean velocity in
the drain pipe and the volumetric flow rate out of the tank is
π 2 π dh
υ out (t) = D v p = [D(h)] 2 (4.250)
p
4 4 dt
To obtain v p , we use the engineering Bernoulli equation to relate the velocity and pressure