Fully developed laminar flow in a tube of radius \(R\) and length \(L\) has the parabolic velocity profile
\[ v(r) = 2\bar v\left[1-\left(\tfrac{r}{R}\right)^{2}\right], \qquad \tau = \frac{L}{\bar v} \]
For steady, isothermal, constant-density flow with radial diffusion and no axial diffusion, the mole balance for A with rate \(-r_A = kC_A^{\,n}\) is
\[ v(r)\frac{\partial C_A}{\partial z} = \frac{D}{r}\frac{\partial}{\partial r}\!\left(r\frac{\partial C_A}{\partial r}\right) - kC_A^{\,n} \]
In dimensionless form, with \(c = C_A/C_{A0}\), \(\rho = r/R\) and \(\zeta = z/L\):
\[ 2(1-\rho^{2})\frac{\partial c}{\partial \zeta} = \Phi\,\frac{1}{\rho}\frac{\partial}{\partial\rho}\!\left(\rho\frac{\partial c}{\partial\rho}\right) - \mathrm{Da}\,c^{\,n}, \qquad \mathrm{Da}=k\tau C_{A0}^{\,n-1},\quad \Phi=\frac{D\tau}{R^{2}} \]
The equation is solved numerically (finite volumes in \(\rho\), implicit marching in \(\zeta\)). The outlet conversion is the flow-weighted (mixing-cup) average:
\[ X = 1-\frac{\int_0^1 c\,(1-\rho^2)\,\rho\,d\rho}{\int_0^1 (1-\rho^2)\,\rho\,d\rho} \]
No radial diffusion (Φ = 0). Each streamline behaves as a batch reactor whose reaction time is its residence time, \(t = \tau/[2(1-\rho^2)] \ge \tau/2\). The exit-age distribution is
\[ E(t)=\frac{\tau^{2}}{2t^{3}}\quad (t\ge \tau/2), \qquad X = 1-\int_{\tau/2}^{\infty} c_{\text{batch}}(t)\,E(t)\,dt \]
Without radial diffusion, no fluid leaves before \(\tau/2\), and the tail of slow wall fluid extends far past \(\tau\). Conversion is \(1-\int C_{\text{batch}}(\theta)\,E(\theta)\,d\theta\), where \(\theta = t/\tau\).
Radial diffusion (Φ > 0). There is no closed form for \(E(\theta)\) once molecules can cross streamlines, so the curve shown is built by tracking several thousand tracer particles: each advects along its streamline at \(v(r)/\bar v\) and takes a random radial step every \(d\theta\) with variance \(2\Phi\,d\theta\), reflecting at the wall, until it exits at \(\zeta=1\). As Φ increases, fluid that would have been trapped near the wall mixes toward the centerline, so the distribution narrows and shifts toward the plug-flow spike at \(\theta=1\).
Reference reactors. Plug flow: \(c_{\text{PFR}}\) is the batch solution evaluated at \(t=\tau\). CSTR: \(1-c = \mathrm{Da}\,c^{\,n}\). Very large Φ makes the concentration uniform across the tube, so the laminar reactor approaches the PFR result.
Assumptions: constant density and temperature, fully developed laminar flow (Re below about 2100), no axial dispersion, no reaction volume change.
Nomenclature.
- \(C_A\)
- concentration of A
- \(C_{A0}\)
- inlet concentration of A
- \(c\)
- dimensionless concentration, \(C_A/C_{A0}\)
- \(c_{\text{batch}}\)
- dimensionless batch-reactor concentration at time \(t\)
- \(D\)
- radial diffusion coefficient
- \(\mathrm{Da}\)
- Damköhler number, \(k\tau C_{A0}^{\,n-1}\)
- \(E(\theta)\)
- exit-age (residence-time) distribution
- \(k\)
- reaction rate constant
- \(L\)
- tube length
- \(n\)
- reaction order
- \(r\)
- radial position
- \(R\)
- tube radius
- \(t\)
- time
- \(v(r)\)
- local axial velocity at radius \(r\)
- \(\bar v\)
- average (mean) velocity
- \(X\)
- conversion
- \(z\)
- axial position
- \(\zeta\)
- dimensionless axial position, \(z/L\)
- \(\theta\)
- dimensionless time, \(t/\tau\)
- \(\rho\)
- dimensionless radial position, \(r/R\)
- \(\tau\)
- mean residence time, \(L/\bar v\)
- \(\Phi\)
- dimensionless radial diffusion, \(D\tau/R^{2}\)