Kantrowitz Factor

Kantowitz Factor

Kantowitz factor, (often simply called the non-isothermal correction factor), is applied as a correction to the nucleation rate from classical nucleation theory. In most cases (the example here is taken from the wet steam model chapter of the fluent theory guide) it appears as follows:

Equation 1 –typical nucleation rate given by Classical Nucleation Theory, with Kantowitz factor 1/1+θ up front.

The non-isothermal factor is found from the enthalpy and the ratio of specific heat capacities:

Equation 2 – non isothermal correction factor.

According to Bakhtar et al (2005) this is a simplification that can be made at low pressure, when the critical radius is much smaller than the mean free path of the gas. Really it should look like:

Equation 3 – nonitsothermal correction factor is a simplification of this.

In the presence of a large volumes of inert gas, isothermal assumptions are justified because collisions with the inert gas, (which transfer heat) are much more common than those with condensing species.

The derviation of the Kantrowitz factor (and Feder’s correction factor) assume that every cluster is at the same elevated temperature. Wyslouzil and Sienfield (1992) criticise this, on the basis that the temperature of small (subcritical) clusters is more sensitive to the addition of more molecules, and the growth of these small clusters affects the final result.

Feder also came up with their own correction factor of the form shown below, that is used in a similar way to the Kantrowtiz factor.

Equation 3 – feder's factor - similar to 1/1+θ .

It results is around 0.01 or a hundredth of the isothermal (Wu, 1973), though it’s worth mentioning that most of these flows are very fast – discussions in the literature centre exclusively on nozzles and steam turbines. b and q are defined in terms of thermodynamic state of the supersaturated vapour. q is basically mostly latent heat. The b is calculated from β, gas kinetic collision frequency (Dawson et al. 1969).

Pandey 

Equation 3 – feder's factor - nuts and bolts.

(2014, p. 15) discusses the steady-state assumption in connection to the Kantrowitz factor. τ = 10-7 to 10-6 s is the timescale for ‘steady state nucleation rate’. Cooling rate ranges from 0.2-0.8 K/µs. The characteristic time in nozzle flows is typically τ = 10-3 s. Kantrowitz factor uses a steady-state assumption, so it is justified on this basis, for use even for unsteady state assumptions.

So to conclude on Kantrowitz factor, corrects for non-isothermal effects which make the nucleation rate about 50 to 100 times slower in rapidly expanding steam flows, such as steam turbines and nozzles. It assumes a steady-state conditions but this is valid as the time to reach steady state is so small. However, it’s the problem of non-isothermal particles itself is less of a big deal when the bulk of the gas is an inert component (such as air) – in these cases the inert gas collisions conduct heat much faster than the particle can grow. Also, my brief lit review hasn’t shown a Kantrowitz factor being applied to something that isn’t a turbine or a nozzle. Feder’s factor is used an alternative, but is basically for similar purposes.

Bakhtar, F. et al. 2004. Classical nucleation theory and its application to condensing steam flow calculations. Proceedings of the IMechE, 219 C. p. 1315-1333.

Pander, A (2014). Numerical modelling of non-equilibrium condensing steam flows. Delft University.

Wu, B. 1973. Possible condensation in rocket exhaust plumes 2. Yale University.

Wyslouzil, BE. Sienfield, JH. 1992. Nonisotermal homogeneous nucleation, 97 (4). pp. 2661-2670.