1. The stress induced in a column subjected to an eccentric load is given by the following equation;

P/A = (S_yc/[1+(ex/k²)sec((l/2k)(√(P/AE)))])

where P is the axial load (lb_f), A is the cross-sectional area of the column, (in²), S_yc is the yield stress of the material in compression, e is the eccentricity of the load, E is Young's modulus, c is the distance of the outermost fiber from the neutral axis of the column, and k is the radius of gyration of the cross section of the column. Find the value of P/A for the following data; S_cy = 40,000psi, (ec/k²) = 0.2, (l/k) = 50 and E = 30x10⁶ psi. Use the bisection method and any other method to confirm you answer. How would you find consistent units on the various terms in this equation?

2. The velocity (u) of a non-Newtonian fluid flowing in a circular tube can be expressed as

u/u_mean = (3n+1/n+1) [1- (r/R)^(n+1)/n]

where u_mean is the mean velocity of the fluid, r is the radial distance from the tube center, R is the radius of the tube, and n is a constant whose value depends on the fluid (for example n = 1 for a Newtonian fluid, n = 3 for a dilatant fluid, and n = 1/3 for a pseudo-plastic fluid). Determine the value of n for a fluid for which u/u_mean = 0.8 and r/R = 0.8 and the fluid type using the secant method.  Verify you answer using any other method(s).

3. A fin, with a uniform circular section has a root temperature of 140^oC and ambient temperature of 40^oC. It has a thermal conductivity k = 70 Watts/cm-K and a heat transfer coefficient of h = 5 Watts/cm²-K. When the convective losses from the ends are considered, the nodal temperatures T₁, T₂ and T₃ are governed by the following equation

Determine the values of the nodal temperatures using a program you write that performs the Gauss-Jordan technique described in the text, and any other method of choice.

4. Three equally sized chemical reactors are connected by large pipes such that pressure drops within the pipes can be neglected.  The mass transfer through each pipe is equal to the product of the flow Q and the reactor concentration c of the reactor from which the flow originates.  The inlet concentration to reactor 1 is 400 mg/sec and reactor 3, 200 mg/sec.  Flow rates from reactor 1 to reactor 3, or Q_1,3 = 40 m³/sec, Q_1,2= m³/sec, Q_2,3= 60 m³/sec, Q_2,1= 40 m³/sec, finally Q_3,3= 120 m³/sec.  Develop a set of equations to determine all flow rates and compositions for each pipe within the network using any method of choice.  Finally determine the mass transfer thru each pipe. You must use the units in the problem.

5. An isomerizer is a catalytic reactor that simply tends to rearrange isomers. The number of moles entering an isomerizer is equal to the number of moles leaving. A process, as shown in the following figure, has been designed to produce a p-xylene-rich product from an aromatic feed charge. All compositions on the flow sheet are in mole percent. The components are indicated as follows:

A = ethyl benzene,                          B = o-xylene

C = m-xylene                                     D = p-xylene

Eighty percent of the ethyl benzene entering the distillation tower is removed in the top stream from the tower. The ratio of the moles of fresh feed to the process as a whole, to the moles of product from the crystallizer is 1.63. Find:

(a) The reflux ratio (ratio of moles of stream from the bottom of the distillation tower per mole of feed to the tower).

(b) The composition (in mole percent) of the product from the crystallizer.

(c) The moles leaving the isomerizer per mole of feed.