**An Important Consideration in Designing a Heat Exchanger operating close to Freeze Point of a fluid – Phenomenon of “Freeze Fouling”**

This type of fouling can occur at a surface when a medium is cooled to within the vicinity of its freeze point.

It is a good engineering practice to design a heat exchanger by avoiding this phenomenon as much as possible. To achieve this, it is important to maintain an elevated level of turbulence within the tube. This avoids ice crystal formation on the surface of the tubes. Also, it is important to maintain uniform velocity profile across the bundle in each pass (in the case of multi pass heat exchangers).

Ice formation is a complex transient phenomenon; however, to explain it in a simplest form we assume a one-dimensional heat transfer across the tube wall from Ammonia on the shell side to say sea water inside the tube. Assuming a very small-time frame and performing an energy balance on an infinitesimal control volume across heat flow path, the steady state equation can be written as follows:

*h _{i} (T_{i} – T_{f}) = (T_{f} – T_{a}) / (x/k_{ice} + t/k_{tube} + 1/ h_{ammonia}) (1)*

where:

h_{i} sea water heat transfer coefficient.

T_{i} sea water bulk temperature.

T_{f} sea water-ice temperature.

T_{a} ammonia temperature.

x ice thickness.

k_{ice} ice thermal conductivity.

t tube wall thickness.

k_{tube} tube thermal conductivity.

h_{ammonia} ammonia heat transfer coefficient.

Rearranging equation (1) for ice thickness results as follows:

*x = {k _{ice} (T_{f} – T_{a}) / h_{i} (T_{i} – T_{f})} – k_{ice} (t/k_{tube} + 1/ h_{ammonia}) (2)*

Equation (2) can be further simplified by eliminating the last two terms because h_{ammonia} is comparatively high, while the tube wall thickness resistance is minimal due to thinner wall. Hence equation (2) can be further simplified to:

*x = k _{ice}/h_{i} {(1 – T_{c}/T_{f}) / (T_{i}/T_{f} – 1)} (3)*

In Equation (3), the ratio {(1 – T_{c}/T_{f}) / (T_{i}/T_{f} – 1)} is a function of temperatures and almost cancels out resulting in:

*x = k _{ice}/h_{i} (4)*

This shows that ice thickness is a function of thermal conductivity of ice and the heat transfer coefficient of sea water. Thermal conductivity of ice is a known fixed quantity. So, the only factor that affects the ice formation and growth is the heat transfer coefficient of sea water. The larger this coefficient the lower will be the ice formation.

From the heat transfer fundamentals, we know that the sea water coefficient is a direct function of a dimensionless number called Reynolds Number (Re), which in turn has direct relationship to two other dimensionless numbers called Nusselt Number (Nu) and Prandtl Number (Pr), each defined as follows:

*Re = ρ D V / µ *

*Nu = h _{i} D / k_{sw}*

*Pr = µ c _{psw} / k_{sw}*

where:

ρ density of fluid

D inside diameter of tube

V velocity inside the tube

µ kinematic viscosity

k_{sw} sea water thermal conductivity

c_{psw} specific heat of sea water

The relationship between these three non-dimensional numbers is as follows:

*Nu ~ Re ^{m} Pr^{n}*

With ‘m’ and ‘n’ empirical constants are generally equal to 0.8 and 0.3, respectively.

Since Prandtl Number is a function of transport properties it is constant for a fixed design condition. Hence, we are left with the direct one-to-one relationship between Nusselt and Reynolds. The larger the “Re” the larger will be “Nu” and as a result the larger will be “h_{i}”, the sea water coefficient and therefore the lower will be ice formation according to equation 4 above.

In conclusion, Re is a direct function of sea water density, sea water viscosity, tube inside diameter and velocity. The density and viscosity are fluid properties and will not change for design condition. The two parameters left are the velocity and tube inside diameter. Increasing the velocity or the diameter or both will increase Re. Let us take a case where velocity is constant for a ¾” and 1” tubes. The Re for 1” tube will be 37% (1/0.75) more than ¾” tube. This points out to the fact that that a heat exchanger with 1” tubes will be less prone to freeze up than a heat exchanger with ¾” tubes.