(a) Gaseous hydrogen at a constant pressure of 0.658 MPa (5 atm) is to flow within the ins...

(a) Gaseous hydrogen at a constant pressure of 0.658 MPa (5 atm) is to flow within the inside of a thin-walled cylindrical tube of nickel that has a radius of 0.125 m. The temperature of the tube is to be 350°C and the pressure of hydrogen outside of the tube will be maintained at 0.0127 MPa (0.125 atm). Calculate the minimum wall thickness if the diffusion flux is to be no greater than 1.25 × 10–7 mol/m2 •s. The concentration of hydrogen in the nickel, CH (in moles hydrogen per cubic meter of Ni), is a function of hydrogen pressure, PH2 (in MPa), and absolute temperature T according to

Furthermore, the diffusion coefficient for the diffusion of H in Ni depends on temperature as

(b) For thin-walled cylindrical tubes that are pressurized, the circumferential stress is a function of the pressure difference across the wall (≤p), cylinder radius (r), and tube thickness (≤x) according to Equation 6.25—that is,

Compute the circumferential stress to which the walls of this pressurized cylinder are exposed. (Note: The symbol t is used for cylinder wall thickness in Equation 6.25 found in Design Example 6.2; in this version of Equation 6.25 (i.e., 6.25a) we denote wall thickness by Δx.)

(c) The room-temperature yield strength of Ni is 100 MPa (15,000 psi), and σy diminishes about 5 MPa for every 50°C rise in temperature. Would you expect the wall thickness computed in part (b) to be suitable for this Ni cylinder at 350°C? Why or why not?

(d) If this thickness is found to be suitable, compute the minimum thickness that could be used without any deformation of the tube walls. How much would the diffusion flux increase with this reduction in thickness? However, if the thickness determined in part (c) is found to be unsuitable, then specify a minimum thickness that you would use. In this case, how much of a decrease in diffusion flux would result?

 (6.25)

DESIGN EXAMPLE 6.2

Materials Specification for a Pressurized Cylindrical Tube

(a) Consider a thin-walled cylindrical tube having a radius of 50 mm and wall thickness 2 mm that is to be used to transport pressurized gas. If inside and outside tube pressures are 20 and 0.5 atm (2.027 and 0.057 MPa), respectively, which of the metals and alloys listed in Table 6.8 are suitable candidates? Assume a factor of safety of 4.0.

For a thin-walled cylinder, the circumferential (or “hoop”) stress (σ) depends on pressure difference (Δp), cylinder radius (ri), and tube wall thickness (t) as follows:

These parameters are noted on the schematic sketch of a cylinder presented in Figure 6.21.

(b) Determine which of the alloys that satisfy the criterion of part (a) can be used to produce a tube with the lowest cost.

(a) In order for this tube to transport the gas in a satisfactory and safe manner, we want to minimize the likelihood of plastic deformation. To accomplish this, we replace the circumferential stress in Equation 6.25 with the yield strength of the tube material divided by the factor of safety, N—that is,

And solving this expression for σy leads to

Table 6.8 Yield Strengths, Densities, and Costs per Unit Mass for Metal Alloys That Are the Subjects of Design Example 6.2

Alloy	Yield Strength, σy (MPa)	Density, ρ (g/cm3)	Unit mass cost, c ($US/kg)
Steel	325	7.8	1.75
Aluminum	125	2.7	5.00
Copper	225	8.9	7.50
Brass	275	8.5	10.00
Magnesium	175	1.8	12.00
Titanium	700	4.5	85.00

We now incorporate into this equation values of N, ri, Δp, and t given in the problem statement and solve for σy. Alloys in Table 6.8 that have yield strengths greater than this value are suitable candidates for the tubing. Therefore,

Four of the six alloys in Table 6.8 have yield strengths greater than 197 MPa and satisfy the design criterion for this tube—that is, steel, copper, brass, and titanium.

(b) To determine the tube cost for each alloy, it is fi rst necessary to compute the tube volume V, which is equal to the product of cross-sectional area A and length L—that is,

Here, ro and ri are, respectively, the tube inside and inside radii. From Figure 6.21, it may be observed that ro = ri +t, or that

Next, it is necessary to determine the mass of each alloy (in kilograms) by multiplying this value of V by the alloy’s density, r (Table 6.8) and then dividing by 1000, which is a unit-conversion factor because 1000 mm = 1 m. Finally, cost of each alloy (in $US) is computed from the product of this mass and the unit mass cost (τ) (Table 6.8). This procedure is expressed in equation form as follows:

For example, for steel,

Cost values for steel and the other three alloys, as determined in the same manner are tabulated below.

Hence, steel is by far the least expensive alloy to use for the pressurized tube.

Figure 6.21 Schematic representation of a cylindrical tube, the subject of Design Example 6.2.