# THEORY OF DEAERATION

## Principle of physical deaeration

The equilibrium between gas dissolved in water and gas in steam is given by Raoult´s law. This law states that the ratio between the partial gas pressure in the steam and the product of the coefficient of activity and the concentration of the gas in the water at a given temperature is constant, if the gas in the steam fulfillls the condition P.V/T = constant, then:

Pg = h · a · Cg         (1)

Where:
Pg = the partial gas pressure in the steam
h = distribution coefficient
a = the coefficient of activity
Cg = the concentration of the gas in the water

At low, partial pressures, oxygen and carbon dioxide in steam behave according to the relationship P.V/T = constant.

The coefficient of activity is a measure for the deviation from the ideal behavior of the gas dissolved in water or, in other words, a measure for the interaction between the dissolved gas and the water. For oxygen the coefficient of activity is 1.0, if the concentration is smaller than 50 ppm. That of carbon dioxide greatly depends on the dissolved quantity of carbon dioxide and the chemical composition of the water.

The ratio constant h is determined by the kind of gas and the temperature. Table 1 shows the values of h for oxygen and carbon dioxide.

 T (˚C) 02 C02 0 14.3 x 10-3 0.298 x 10-3 10 18.4 x 10-3 0.427 x 10-3 20 22.6 x 10-3 0.581 x 10-3 30 26.8 x 10-3 0.767 x 10-3 40 30.2 x 10-3 0.963 x 10-3 50 33.4 x 10-3 1.170 x 10-3 60 35.8 x 10-3 1.420 x 10-3 70 38.2 x 10-3 80 39.7 x 10-3 90 40.6 x 10-3 100 40.7 x 10-3

Table 1. Distribution coefficient h of oxygen and carbondioxide for water at different temperatures in bara/ppm.

The principle on which the physical deaeration process is based is the tendency towards restoring the equilibrium as defined by the above-mentioned law after that equilibrium has previously been disturbed. The disturbances of equilibrium that bring about deaeration of the water are the decrease in the partial gas pressure in the steam and the increase of the temperature of the water. At water temperatures below boiling point the decrease in the partial gas pressure is brought about by removing gas from the steam. By increasing the temperature the solubility of the gas in the steam decreases at a constant partial pressure. If the water temperature is raised to boiling point at the prevailing pressure in the deaerator, the total pressure is equal to the water-vapor pressure, which implies that the partial gas pressure is zero. The water temperature rises as a result of condensation of the steam.

Decrease of the partial gas pressure and/or increase of the temperature are not the only factors of importance in the deaeration process, since the transport velocity of the gas also plays a role. This velocity is determined by:

1. the diffusion of the gas in the water
2. the flow of the water and of the steam
3. the ratio between the area of the contact surface water-steam and the volume of water.

### Diffusion of the gas in the water

When water is deaerated, the concentration of the gas molecules in the water is lower at the contact surface than elsewhere in the water. In the steam, this is exactly the other way a round. Owing to the differences in concentration and the thermal movement of the gas molecules, transportation of gas molecules takes place towards the contact surface in the water and away from the contact surface in the steam. At the contact surface Raoult´s law invariably applies. It will be clear that the deaeration rate is determined by the phase in which the transportation of the gas molecules is slowest. As the ratio between the diffusion coefficient, (a measure for the thermal movement), of the gas in water and that of the gas in steam is approximately 10-4, the deaeration rate is fully determined by the transportation of the gas molecules in the water.

Fig. 2 Schematic display of gas concentration in the steam/water contact area with convection and diffusion.

### Flow of water and steam

Flow of water and steam raises the deaeration rate. In the interfacial layer on either side of the water-steam contact surface flow in a direction perpendiular to the contact surface can not occur. The gas transport there is still exclusively only possible due to diffusion. In places further away from the contact surface the flow equalizes the concentration of the gas, see figure 2. Hence, the gas transport in the two media is controlled by a narrow diffusion zone along the contact surface. Owing to the great difference between the diffusion coefficient of the gas in water and that of the gas in steam the deaeration rate will be determined by the diffusion zone on the water side of the contact surface. The gas transport in this diffusion zone is maximum at the lowest possible partial pressure in the steam. This condition is fulfillled when the gas is rapidly diskharged from the deaerator by flow in the steam.

### Preliminary deaeration

Ratio between contact surface water-steam and water volume

To ensure efficient deaeration it is necessary to make the transport route that the gas molecules have to travel by diffusion as short as possible. This can be achieved by formation of the water into very tiny droplets before the steam is passed through the water.

### A. Droplet deaeration

In the Stork deaerator the water is divided into droplets by means of a sprayer, which is regulated according to the volume of water to be sprayed. The droplet size remains constant at different throughputs. The droplets are passed through the steam compartment of the sprayer at high velocity and then impinge on the wall of this compartment, which causes the droplets to break up into even smaller droplets. The residence time of the water in the compartment is a few tenths of a second. Constant replacement of the steam in the steam compartment ensures that the partial gas pressure remains very low.

As the water droplets travel through the hotter steam, condensation occurs at the surface of the droplet, as a result of which the water is warmed up. This heating process proceeds very rapidly owing to the favorable ratio of contact surface to water volume. The temperature of the water cannot reach boiling point, since the surface tension furnishes an extra pressure.

The droplet surface acts like a tight elastic membrane, as a result of which the pressure on the concave side is higher than on the convex side. This extra pressure is 4 x O/D, where O is the surface tension of the water and D the diameter of the droplet. After the droplets have entered the steam compartment of the sprayer the partial gas pressure in the steam is very low with respect to the gas content of the water. This low, partial pressure and the reduced solubility owing to the temperature rise compel the gas to leave the water.

### B. Bubble-formation in droplets

It is argued by some authors that the gas leaves the water due to the formation of gas bubbles in the droplets, but it is very doubtful that this could be the case. In order to maintain a gas bubble in a water droplet the sum of the partial pressures associated with the dissolved gases and the water vapor pressure must at least be equal to the pressure in the droplet plus the pressure caused by the interface of the water and the gas bubble. Hence, expressed as an inequality.

Pgases + PH2O ≥ Psteam + 4 · O/D + 4 · O/d …..(2)

Pgases =    the sum of the partial pressure of the dissolved gases
PH2O =     the water vapor pressure; if the water temperature is equal to the steam temperature: PH2O = Psteam.
d =         the diameter of the gas bubble.

As the formation of a gas bubble is only possible via nucleus-bubble formation and the expansion of such nucleus-bubbles, (very tiny gas bubbles), it follows from inequality (2) and equation (1) that a lot of gas must be dissolved in a water droplet to give rise to the formation of a gas bubble, since with nucleus-bubble formation d in inequality (2) is very small and thus 4 x O/d is very large.

The difficulty of the formation of gas bubbles in droplets can be clearly illustrated with a bottle of soda-water. On opening the bottle gas bubbles are only formed on the wall. If gas-bubble formation is also seen to occur elsewhere in the soda-water, it can be taken for granted that a particle of some contaminating substance is suspended in the water. The formation of bubbles on the wall and on the particle is possible, because owing to the presence of an interface the nucleus-formation energy is much less there than elsewhere in the soda-water. As is evident from the example, no nucleus or bubble formation can taken place in a water droplet in the absence of an interface. On the strength of the foregoing it must be assumed, therefore, that the gas escapes solely by diffusion and/or flow.

### C. Theoretically unexplainable effects in droplet deaeration

Material transfer measurements performed in respect of falling drops of liquid in liquids that do not mix with the liquid of the droplets, have revealed that the transport of material from the droplets to the liquids takes place very quickly. This transport increases as the velocity of the droplets increases. There is no theoretical explanation for this fast transport. An explanation for this phenomenon on the strength of solid spheres is impossible, as the surfaces of the droplets are not rigid. The idea that a better transport of material is obtained by flow in the droplets is very attractive at first sight, but according to the literature this flow increases the rate of transport only by a factor of 2 or 3. Moreover, it is most improbable that flow occurs in such small droplets.

Another effect that might increase the rate of material transport from droplets is the agitation of the droplet’s surface by the medium flowing round it. This agitation chiefly occurs on the side of the droplet not facing the direction of flow owing to the eddies formed there by the surrounding medium.

However, if the transport in the droplet determines the material transport from the droplet, the high rate of transport cannot be ascribed to the agitation of the surface of the droplet.

It is thought that vibration of the droplets has a substantial effect, but it has not yet been possible to prove this with any certainty.

There is, in principle, no difference between the behavior of falling droplets in liquids and that of droplets moving in a steam space and therefore on the strength of the foregoing it must be concluded that droplet deaeration is subject to a theoretically unknown influence, of which we only observe the outcome. This effect is a higher rate of deaeration at a higher velocity of the droplets in the steam space.

### D. Effect of the droplet size

A better deaeration effect is obtained by reducing the size of the water droplets. To give an impression of the effect of the droplet size, a derivation is given below for the deaeration rate of spherical droplets in which the transport of the gas molecules takes place exclusively by diffusion.

A diffusion process for which the concentration gradient is not constant can be represented by the following differential equation:

(3)

Where:
C = the concentration of the gas molecules at the distance r from the center of a spherical droplet
t = the time in which deaeration of the droplets occurs
Dv = the diffusion coefficient; this coefficient is assumed to be independent of the concentration. For solving the differential equation the following conditions are available:
C = C1, when t = 0 and r/R ≤ 1
C = Co, when t > 0 and r/R = 1
Co = the gas concentration according to Raoult´s law with an activation coefficient of 1.0;
C1 = the gas concentration before deaeration;
R = the radius of the droplet.

The solution of the differential equation is:

(4)

represents a function with as a variable.

It is also possible to substitute for C, Cavg (= average concentration in the droplet at time t.)

(5)

Fig. 3. Influence of droplet size on the rate of deaeration with diffusion only (Dv = 10-9m2/sec).

The functions (4) and (5) are known. With the aid of data from literature the relation between (Cavg – Co / C1 – Co), R and t is represented in figure 3. According to this figure the deaeration rate of the water increases with decreasing droplet size. For a deaerator this means that the greatest deaeration effect is attained in the stage in which the smallest droplets have formed.

### E. The effect of the surface tension of the droplet

Reference is sometimes made to the effect of the surface tension of the droplet on the deaeration rate, the idea being that the surface tension retards the gas transport in the water droplets. The advocates of this theory recommend, therefore, that in order to attain a high deaeration rate the droplets should be allowed to form over and over again.

It is difficult to imagine that the escape of the gas molecules is impeded by the contact surface between water and steam. The transport of the gas molecules in the water is not affected, because the slight increase in pressure through the surface tension of the droplets does not reduce the thermal motion of the gas molecules.

The repeated reformation of water droplets has a favorable effect on the deaeration process for another reason. By collecting the droplets in a reservoir and allowing them to reform, the concentration gradient in the droplets present before collection is averaged out. In the new droplets the optimum gas concentration occurs at the surface of the droplet, thus promoting a high deaeration rate.

It is known from practice that in the case of a cascade deaerator, for example, in which the droplets must be repeatedly reformed, the water is formed into films or jets instead of into droplets. The thickness of the films and jets increases accordingly as the throughout increases. Consequently, deaeration in a cascade type deaerator is less effective than with the use of a regulated sprayer. The reformation of films and jets has a great influence on the process in the cascade type deaerator. Despite the fact that in the Stork deaerator the water droplets are reformed only once, namely when impinging on the wall of the steam compartment of the sprayer, the deaeration effect at different through-outs is very large. The droplets produced by the sprayer and originating due to impingement are of such a size that deaeration takes place very rapidly.

### Final deaeration

The residence time of the water in the steam compartment of the sprayer is too short in some cases to ensure optimum deaeration and for this reason provision is made for post deaeration, which takes place in the water reservoir. The gas that has remained behind is expelled by conducting steam through the water by means of a steam rake.

As the steam bubbles through the water, there is a tendency according to Raoult´s law for an equilibrium to be established between the gas in the steam bubbles and the dissolved gas in their immediate vicinity. Adequate contact time is required to attain this equilibrium. This is the case when process conditions are so arranged that the steam bubbles are small and the distance they have to travel is great. A steam rake is the best means to achieve this.

A second function of the steam bubbles is to start and maintain the circulation in the water reservoir, whereby the diffusion path of the gas is considerably shortened. Compared with any other method of supplying the steam, the steam rake construction has the advantage that the circulation in the reservoir is more intensive.

### Removal of carbon dioxide water

With a pH lower than 4 the activity coefficient of the total carbon dioxide dissolved in water is set at 1.0; with higher pH values the coefficient is smaller. Owing to the interaction with the water the dissolved carbon dioxide manifests itself not only as CO2, but also in the form of H2CO3 and the ions HCO3 and CO3 2-. The equilibria formulae are:

CO2 + H2O ↔ H2CO3                 (6)
H2CO3 ↔ H+ + HCOˉ3               (7)
HCOˉ3 ↔ H+ + CO32-                 (8)

As the quantity of H2CO3, in comparison with that of the ions, is negligibly small, the equilibria (6) and (7) can be combined as follows:

CO2 + H2O ↔ H+ + HCOˉ3       (9)

With the aid of the equilibrium constants it is possible to calculate the ratios, at a given temperature, of the quantities of CO2, HCOˉ3 and CO32- with respect to the total dissolved quantity of carbon dioxide. In figure 9 these ratios at 25oC are graphically represented as a function of the acid value.

This graph was drawn up on the assumption that no other substances than CO2 were dissolved in the water. At temperatures above 25oC the curves in figure 4 shift a little to the left.

Fig. 4. Relation between molar ratio and pH.

Fig. 5. Relation between the partial pressure of carbon dioxide and oxygen in the steam and the amount of dissolved carbon dioxide and oxygen in the water.

During the deaeration process the dissolved carbon dioxide is removed from the water in the form of CO2. The ion HCOˉ3 indirectly takes part in the formation of the CO2process, because when the CO2 content drops it is converted into CO2 according to equation (9). If CO32 is present (pH > 8.3 at 25oC), it is technically absolutely impossible – owing to the absence of CO2 to drive off the carbon dioxide dissolved in the water by partially lowering the carbon-dioxide pressure in the steam. The pH of pure water containing dissolved carbon dioxide lies between 4 and 7, depending on the quantity of carbon dioxide. For a total carbon dioxide content in excess of 1 ppm at 25oC the following approximation formula applies:

pH = 5.5 – ½ log Ctot. CO2

Ctot. CO2 = the total carbon dioxide concentration in ppm. Between 0.01 and 1 ppm total carbon dioxide the pH calculated according to the approximation formula is 0.2 higher than the actual pH.

In pure water the carbon dioxide is only dissolved in the form of CO2 and HCOˉ3 ions. According to figure 4 the carbon dioxide is chiefly present as CO2 at pH = 4 and as HCOˉ3 at pH = 7. This implies that the less carbon dioxide is dissolved in the water, the greater the HCOˉ3 content is with respect to that of CO2. Figure 5, which represents the relationship between the partial carbon dioxide pressure in the steam and the total dissolved quantity of carbon dioxide at 20oC, illustrates how great the deviation from the equation Pco2-gas = h.C tot. CO2 is with small quantities of carbon dioxide. The deviation can be calculated with the aid of the activity coefficient of the total dissolved carbon dioxide, which has been made equal here to CO2/tot. CO2.

For the sake of comparison figure 5 also shows the relationship between solubility and partial gas pressure for oxygen. This illustrates how much more difficult it is to remove carbon dioxide from the water than oxygen. To be able to lower the oxygen content in water at 20 °C to 0.005 ppm a partial oxygen pressure of 1.13 x 10-4 bara is essential. At the same carbon dioxide pressure it is only possible to lower the carbon dioxide content in the water to 0.280 ppm. If by adding acid care is taken to keep the pH lower than 4, a content of 0.195 ppm can be attained. The carbon dioxide is then chiefly dissolved as CO2.

The lowering of the total carbon dioxide content to very low values cannot be achieved with a deaerator. To attain pressures in the steam are essential and these are hard or impossible to realize. Owing to the lack of data it is impossible to give a diagram as shown in figure 10 also at 100 °C. On the strength of the data mentioned in the present article, however, it can be predicted that the removal of carbon dioxide will be easier at temperatures above 20 °C.