International Journal of Chemical Engineering

Research Article

A Solution of the Convective-Diffusion Equation for Solute Mass Transfer inside a Capillary Membrane Bioreactor

Table 1

The boundary conditions of the MBR.
B.C. 𝑅 , 𝑍 𝐶 , 𝑈 , 𝑉 , 𝑃 𝐿 RangeEquation
B.C.1 𝑍 = 0 𝐶 = 1 0 ≤ R ≤ 1(6a)
B.C. 2 𝑅 = 0 𝜕 𝐶 𝜕 𝑅 = 0 0 ≤ Z ≤ 1(6b)
B.C. 3 𝑅 = 1 𝐶 = 𝐶 ( 𝑍 ) 0 ≤ Z ≤ 1(6c)
B.C. 4 𝑅 = 1 − 𝜕 𝐶 𝜕 𝑅 + 𝑃 𝑒 𝑣 𝑉 𝐶 = S h [ 𝐶 ( 𝑍 , 1 ) − 𝐶 ( 𝑍 , 𝑅 2 ) ] 0 ≤ Z ≤ 1(6d)
B.C. 5 𝜑 𝑍 ≥ ( 1 6 𝛽 𝜅 R e F r − 𝑎 𝜑 ) C(Z) = constant0 ≤ R ≤ 1(6e)
B.C. 6 𝑅 = 0 C = finite0 ≤ Z ≤ 1(6f)
B.C. 7 𝑅 = 1 𝑈 = 0 0 ≤ Z ≤ 1(6g)
B.C. 8 𝑅 = 0 𝜕 𝑈 𝜕 𝑅 = 0 0 ≤ Z ≤ 1(6h)
B.C. 9 𝑅 = 0 𝑉 = 0 0 ≤ Z ≤ 1(6i)
B.C.10 𝑅 = 1 𝑉 = 𝑉 𝑀 0 ≤ Z ≤ 1(6j)
B.C.11 𝑍 = 0 𝑃 = 𝑃 0 ; 𝑃 = 𝜑 𝑎 0 ≤ R ≤ 1(6k)