Background
In real life as a process engineer we face situations where we need to calculate the velocity distribution and film thickness of falling film outside of a circular tube. For example, falling film evaporator, falling film gas absorption, where a viscous fluid flows upwards through a small circular tube and after that it overflow outside of tube as shown in below figure.
For the steady-state flow, the momentum balance equation can be written as
We consider a shell of thickness of ∆r and length L in falling liquid film at the radius r. In this case end effects are neglected and fluid is in-compressible.
The various contributions to momentum balance in the z-direction are as below
Putting all above values in momentum balance equation we will get as below
In above equation 3rd and 4th term will be cancelled out, because fluid is incompressible and velocity is z-direction is constant. So, we can rewrite the above equation as
Divide the above equation by 2πL∆r and rearrange we will get
Take the limit as ∆r → 0; this gives
The expression on the left side is the definition of the first order derivative. Hence, we can write this as below
Integration of above equation will give as follows
To estimate the integration constant, we apply the following boundary condition at gas-fluid interface. At r = aR, the sheer stress τrz = 0, hence we can solve for C1
Substituting value of C1, we can write the equation for momentum flux distribution
The Newton’s law of viscosity is given by
Replacing the value of τrz in momentum flux distribution equation we get
Integration of above equation will give
To estimate the value of integration coefficient of C2 we use boundary condition at r = R, vz = 0,
Replacing value of C2 in above equation we will get velocity distribution equation for the system
The volume rate flow in the thin film can be given by Q = Flow are of film* Velocity
References
TRANSPORT PHENOMENA, R. Byron Bird, Warren E. Stewart and Edwin N. Lightfoot, Chapter 2, Problem 2.G2.