EBK NUMERICAL METHODS FOR ENGINEERS
7th Edition
ISBN: 8220100254147
Author: Chapra
Publisher: MCG
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Textbook Question
Chapter 6, Problem 31P
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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2. Consider a polymeric membrane within a 6 cm diameter stirred ultrafiltration cell. The membrane is
30 μm thick. The membrane has pores equivalent in size to a spherical molecule with a molecular weight
of 100,000, a porosity of 80%, and a tortuosity of 2.5. On the feed side of the membrane, we have a
solution containing a protein at a concentration of 8 g L-1 with these properties: a = 3 nm and DAB = 6.0 ×
10-7 cm² s¹. The solution viscosity is 1 cP. The hydrodynamic pressure on the protein side of the
membrane is 20 pounds per square inch (psi) higher than on the filtrate side of the membrane. Assume
that the hydrodynamic pressure difference is much larger than the osmotic pressure difference
(advection >> diffusion). Determine the convective flow rate of the solution across the membrane.
1. Calculate the filtration flow rate (cm³ s¹) of a pure fluid across a 100 cm² membrane. Assume the
viscosity (µ) of the fluid is 1.8 cP. The porosity of the membrane is 40% and the thickness of the
membrane is 500 μm. The pores run straight through the membrane and these pores have a radius of
0.225 μm. The pressure drop applied across the membrane is 75 psi. (Note: 1 cP = 0.001 N s m²² = 0.001
Pa s.)
3. Tong and Anderson (1996) obtained for BSA the following data in a polyacrylamide gel for the
partition coefficient (K) as a function of the gel volume fraction (4). The BSA they used had a molecular
weight of 67,000, a molecular radius of 3.6 nm, and a diffusivity of 6 × 10-7 cm2 s-1. Compare the
Ogston equation
K=exp
+
to their data and obtain an estimate for the radius of the cylindrical fibers (af) that comprise the gel.
Hint: You will need to plot Ink as a function of gel volume fraction as part of your analysis. Please include
your MATLAB, or other, code with your solution.
Gel Volume Fraction (4)
KBSA
0.00
1.0
0.025
0.35
0.05
0.09
0.06
0.05
0.075
0.017
0.085
0.02
0.105
0.03
Chapter 6 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
Ch. 6 - 6.1 Use simple fixed-point iteration to locate the...Ch. 6 - 6.2 Determine the highest real root of...Ch. 6 - Use (a) fixed-point iteration and (b) the...Ch. 6 - Determine the real roots of f(x)=1+5.5x4x2+0.5x3:...Ch. 6 - 6.5 Employ the Newton-Raphson method to determine...Ch. 6 - Determine the lowest real root of...Ch. 6 - 6.7 Locate the first positive root of
Where x...Ch. 6 - 6.8 Determine the real root of, with the modified...Ch. 6 - 6.9 Determine the highest real root of:...Ch. 6 - 6.10 Determine the lowest positive root...
Ch. 6 - 6.11 Use the Newton-Raphson method to find the...Ch. 6 - 6.12 Given
Use a root location technique to...Ch. 6 - You must determine the root of the following...Ch. 6 - Use (a) the Newton-Raphson method and (b) the...Ch. 6 - 6.15 The “divide and average” method, an old-time...Ch. 6 - (a) Apply the Newton-Raphson method to the...Ch. 6 - 6.17 The polynomial has a real root between 15...Ch. 6 - Use the secant method on the circle function...Ch. 6 - You are designing a spherical tank (Fig. P6.19) to...Ch. 6 - 6.20 The Manning equation can be written for a...Ch. 6 - 6.21 The function has a double root at. Use (a)...Ch. 6 - 6.22 Determine the roots of the following...Ch. 6 - 6.23 Determine the roots of the simultaneous...Ch. 6 - Repeat Prob. 6.23 except determine the positive...Ch. 6 - A mass balance for a pollutant in a well-mixed...Ch. 6 - Fir Prob. 6.25, the root can be located with...Ch. 6 - 6.27 Develop a user-friendly program for the...Ch. 6 - Develop a user-friendly program for the secant...Ch. 6 - 6.29 Develop a user-friendly program for the...Ch. 6 - 6.30 Develop a user-friendly program for Brent’s...Ch. 6 - 6.31 Develop a user-friendly program for the...Ch. 6 - 6.32 Use the program you developed in Prob. 6.31...
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