Gaseous hydrogen at a constant pressure of 0.658 MPa 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.
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
CH = 30.8√(p_(H_2 ) ) exp^((12,300J/mol)/RT) (6.34)
Furthermore, the diffusion coefficient for the diffusion of H in Ni depends on temperature as
D_H (m^2/s) = (4.76×〖10〗^(-7) )exp(-(39,560J/mol)/RT) (6.35)
To calculate the minimum wall thickness, we need to determine the diffusion flux and then use it in conjunction with the given equations to solve for the wall thickness.
The diffusion flux (J) is given by Fick's first law of diffusion, which states that the flux is proportional to the concentration gradient. In this case, the concentration gradient is the difference in hydrogen concentration between the inside and outside of the tube.
J = -D_H * dCH/dx
Where:
- J is the diffusion flux (mol/m^2/s)
- D_H is the diffusion coefficient (m^2/s)
- dCH/dx is the concentration gradient (moles hydrogen per cubic meter of Ni per meter)
To determine the concentration gradient, we need to find the difference in hydrogen concentrations between the inside and outside of the tube. Given that the pressure of hydrogen outside the tube is 0.0127 MPa, we can use equation 6.34 to calculate the hydrogen concentration outside (C_H_outside) as follows:
C_H_outside = 30.8 * sqrt(P_H2_outside) * exp((12,300 J/mol) / (R * T))
Where:
- C_H_outside is the concentration of hydrogen outside the tube (moles hydrogen per cubic meter of Ni)
- P_H2_outside is the pressure of hydrogen outside the tube (MPa)
- R is the gas constant (8.314 J/(mol*K))
- T is the temperature in Kelvin
Similarly, we can calculate the hydrogen concentration inside the tube (C_H_inside) using equation 6.34 with the given pressure and temperature conditions.
C_H_inside = 30.8 * sqrt(P_H2_inside) * exp((12,300 J/mol) / (R * T))
Where:
- C_H_inside is the concentration of hydrogen inside the tube (moles hydrogen per cubic meter of Ni)
- P_H2_inside is the pressure of hydrogen inside the tube (MPa)
Now, we have the concentration gradient:
dCH/dx = (C_H_inside - C_H_outside) / thickness
Where thickness is the wall thickness of the tube (meters).
Substituting the values, we can now calculate the diffusion flux J:
J = -D_H * (C_H_inside - C_H_outside) / thickness
Given that J should be no greater than 1.25 × 10^(-7) mol/m^2/s, we can rearrange the equation to solve for the minimum wall thickness (thickness_min):
thickness_min = -D_H * (C_H_inside - C_H_outside) / (J_max)
Now, we substitute the values for the equations:
- Use equation 6.34 to calculate C_H_outside using P_H2_outside = 0.0127 MPa and T = 350 °C.
- Use equation 6.34 to calculate C_H_inside using P_H2_inside = 0.658 MPa and T = 350 °C.
- Use equation 6.35 to calculate D_H using the given temperature T = 350 °C.
- Substitute the values into the equation for thickness_min.
Keep in mind that all temperatures must be converted to Kelvin (K) before performing any calculations.
I hope this helps you solve the problem!