4.1 Why is the Equilibrium Diameter Important?

Transcription

1 Chapter 4 Equilibrium Calculations 4.1 Why is the Equilibrium Diameter Important? The model described in Chapter 2 provides information on the thermodynamic state of the system and the droplet size distribution. It does not provide any information on the state of the droplet itself. Therefore, a few questions arise: Is the droplet in equilibrium with its surroundings? If the answer is no, what is the state of the droplet regarding the state of a droplet in equilibrium at the ambient conditions? Is the droplet activated? These questions can be answered by determining the equilibrium diameter,,eq, for the thermodynamic conditions occuring inside the tube using the Köhler theory. By comparing,eq to the actual tube droplet diameter,, the questions above can be answered. 4.2 How is the Equilibrium Diameter Determined? The starting point in the calculation of,eq is the single particle growth law, Equation (2.10). When there is no net p,i =0, and the droplet is in equilibrium with surroundings. The equation that needs to be solved therefore = 2 ß (S r S p (,eq )) R v T (1 ο v M Mv ) D v e sat,w f Mass + L2 w/v Sp(dp,eq ) R v T 2 k mix f Heat =0 (4.1) 33

2 34 Equilibrium Calculations which corresponds to S r S p (,eq )=0 or S r = S p (,eq ) The equilibrium saturation ratio at the droplet surface, S p, is determined using Equation (2.12): S p =exp 4 Mw ff s/a ν' s " V m s M w =M s ß RT% w % sol 6 d3 p m s (4.2) In the model, the actual particle diameter,, is calculated as a function of the droplet mass and density, which in turn depend on the mass of liquid water in the droplet, m w, since the mass of solute in the droplet, m s, is constant. Therefore, S r S p (,eq ) = 0 should more accurately read S r S p (m w,eq )=0 (4.3) The root to (4.3) is determined with the help of the bisection method which has been realised as a DEFINE_ON_DEMAND macro in the UDF (Fluent, 2001). Such a macro is executed on demand only, i. e. FLUENT does not call subfunctions of this kind during the calculation. Using the bisectional scheme, the Köhler equilibrium diameters are calculated for the saturation and temperature values occuring along the tube axis. The starting interval for the scheme is chosen in such a manner that the theoretical existence of at least one root in the interval is ensured. The lower boundary is set to 1: kg which corresponds to the dry particle diameter. The upper boundary is set to 6: kg which corresponds to a droplet diameter of about 100 μm. This value is generally regarded as the threshold between cloud and rain droplets. The starting interval is split by the droplet diameter,, calculated by the model. This procedure has been chosen because the equilibrium diameter is assumed to be located in the vicinity of the actual droplet diameter. Afterwards, the interval is split by the geometric average of its boundaries. In this manner, nested intervals are formed.

3 4.2. Determination of the Equilibrium Diameter 35 S w Figure 4.1: Illustration of nomenclature used in the bisection module. Figure 4.1 illustrates the nomenclature used below. For any given iteration, the lower boundary of the interval investigated is denoted by m i;l, the upper boundary by m i;l, and the split by m i;s S p Figure 4.2: Arbitrary Köhler curve (brown graph) to illustrate the different cases occuring in the bisection module. Three different cases have to be considered. They are depicted by the colourearallels to the abscissa in Figure 4.2 which shows an arbitrary Köhler curve (the brown graph). The easiest case is for S r < 1 (purple line) because the parallel intersects the arbitrary Köhler curve only once there is only one root. An additional simplification arises in the

4 36 Equilibrium Calculations a) S w b) S w Figure 4.3: Illustration of the two possibilities for 1 <S r <S crit. part of the flow tube where the particle / droplet is in equilibrium with its surroundings in any case, that is in the first centimetres downstream of the tube inlet. Then, no iterative procedure to determine,eq is necessary. The condition for which the iteration is forgone is Sr Sp S p For S r > 1, two cases have to be differentiated. On the one hand, the droplet is no longer in equilibrium with its surroundings once the critical saturation is surpassed by the actual saturation in the tube. Then, the parallel no longer intersects the curve (blue line in Figure 4.2). If this is the case, the scheme should not start an iteration at all, a characteristic exploited to set,eq to zero in the part of the tube where such conditions occur. On the other hand, there may be two roots (olive line in Figure 4.2). Then, the position of the first split is crucial, as is illustrated in Figure 4.3. Either no root is determined since

5 4.3. Results of the Equilibrium Calculations 37 both (S r S p (m i;l )) (S r S p (m i;s )) and (S r S p (m i;s )) (S r S p (m i;u )) are positive (Figure 4.3a), or two roots are determined (Figure 4.3b). By varying the position of the split it is ensured that all roots in the interval are found. If there are two roots, they will be referred to as the left and the right equilibrium diameter below. This distinction is attributed to the fact that the Köhler curve can be considered to consist of a left and a right branch. The two branches are separated by the critical droplet diameter,,crit.for diameters smaller than,crit, i. e. to the left of,crit in a graph, the droplets are not yet activated. Droplets on the right branch of the Köhler curve, i. e. with diameters to the right of,crit in a graph, continue to grow as long as the ambient saturation is larger or equal to their equilibrium saturation. The terminating condition for the bisection is defined as m i 1;s m i;s m i;s (4.4) Other numerical schemes have been considered for the calculation of the equilibrium diameter, in particular the secant, false position and Newton-Raphson methods (Press et al., 1992). However, they have been abandoned in favour of the bisection method due to problems encountered in the numerical calculation of w Which Results Are Obtained? Since the current diploma thesis is meant as a proof of principle, the results of the bisectional scheme are only illustrated for an example case. The tube wall material is aluminium. The droplet is assumed to be an ideal solution. Two different inlet saturations, 0:94 and 0:87, are considered.

6 38 Equilibrium Calculations Figure 4.4 depicts the saturation ratio as a function of tube length (a) as well as the actual and equilibrium diameters at the tube axis (b and c) for the higher inlet saturation (Si = 0:94). Figure 4.4c is an enlarged view of Figure 4.4b to better recognise the differences between dp and dp,eq,left. A Köhler curve has been superimposed in Figure 4.4a. The curve has been determined for an NaCl particle with a dry diameter of 100 nm and a temperature of 10 ffi C using Equation (4.2). The routine to calculate the curve has been realised as a DEFINE_ON_DE- MAND macro in the UDF. It accesses the parameters that characterise the particles entering the flow tube so that it is not necessary to define them again in the routine. These parameters are the mass, molar mass, and density of the dry particles, which act as condensation nuclei. The temperature has been chosen somewhat arbitrarily since the graph is only intended to give an impression of how the Köhler curves of the particles in the tube would look like. The critical saturation is reached when Sr surpasses the Köhler curve which occurs at about 20 cm downstream of the tube inlet (see Figure 4.4a). From this point onwards, the droplets experience rapid condensational growth which is clearly discernible in the steep rise of the black curve in Figure 4.4c. Past 40 cm downstream of the inlet the saturation starts to decrease slowing the droplet growth. In the last third of the tube, from x ß 1 m to x =1:5 m, the droplet size decreases. The red curve in Figures 4.4b and 4.4c depicts the left equilibrium diameter. At the beginning of the tube, the two curves are identical, i. e. Sr Sp S p At about 15 cm, the curves start to diverge, dp is smaller than dp,eq,left. This divergence shows that the response of the system to the changing thermodynamic conditions is delayed compared to the theoretically possible response. The discontinuities in dp,eq,left are due to the fact that the equilibrium diameter is set to zero when the saturation inside the tube exceeds the critical saturation given by the Köhler curve. Hence, there is no need to especially determine the Köhler curve in order to decide whether the droplets are activated or not they are if there exists a discontinuity in dp,eq,left. The model has also been run for an inlet saturation of 0:86. In this case, there is no gap in the graph of the equilibrium diameter

7 4.3. Results of the Equilibrium Calculations 39 a) S S r Köhler curve b) ,eq,left,eq,right c) ,eq,left Figure 4.4: Equilibrium considerations for an inlet saturation of 0:94. a) Saturation profile at the tube axis (S versus x), superimposed is the Köhler curve for an NaCl particle with a dry diameter of 100 nm at 10 ffi C(S versus dp). At S =1a line has been drawn to help differentiate between saturated and supersaturated conditions. b) and c) Actual and equilibrium diameters at the tube axis. Here dp is the actual droplet diameter in the tube, dp,eq,left refers to the equilibrium diameter on the left branch of the Köhler curve, and dp,eq,right refers to the equilibrium diameter on the right branch of the Köhler curve.)

8 40 Equilibrium Calculations (not shown) which leads to the conclusion that the droplets have not been activated. At x ß 1 m, Sr falls below 1 which means that the saturation inside the tube is smaller than the equilibrium saturation above the droplets. Hence, droplet growth is inhibited. Instead, the droplet size decreases due to evaporation from the droplet surface. Theoretically, the system again supports an equilibrium which is evidenced by the fact that the bisection module computes the left equilibrium diameter again. The droplets, though, are slow to respond to the changes in their ambient conditions and do not reach their equilibrium state before they exit the flow tube. The comparison of the Köhler curve to the actual tube saturation (Figure 4.4a) shows that only in the vicinity of x ß 0:2 m and x ß 1 m the tube saturation is in a range where two roots to (4.3) exist. In the other parts of the tube, Sr is either larger than the critical saturation so that the system is no longer in equilibrium, or it is smaller than 1 so that only one root to (4.3) exists. At x ß 0:2 m, the right equilibrium diameters can be discarded since the critical saturation has not yet been reached. Towards the end of the tube, one should not expect the curve of the actual particle diameter to coincide with either equilibrium diameter first because dp does not follow the Köhler curve and second because of the delayed response of the system. That dp and dp,eq,right do coincide for x =1:00973 mis literally a coincidence (see Figure 4.4b, dp,eq,right is depicted by blue squares). It indicates that the droplets pass through one of their equilibrium states, i. e. the one on the right branch of the Köhler curve, on the way to their other equilibrium state, i. e. one on the left branch of the curve. The first equilibrium cannot be maintained because the ambient saturation is too low. At the grioints where dp,eq,right is calculated near x = 1 m,it increases with increasing x since Sr decreases (Figure 4.4b; dp,eq,right equals 4:18712 μm at x = 1:00973 m and 15:8274 μmatx = 1:02494 m). The lower inlet saturation (Si =0:87) has been chosen to verify the bisection module. For this saturation ratio, dp and dp,eq should coincide not only at the beginning, but also at the end of the tube. Or, if they do not coincide at the end of the tube, at least they should not differ too much. Figure 4.5c clearly reflects this behaviour.

9 4.3. Results of the Equilibrium Calculations 41 a) S S r Köhler curve b) ,eq,left,eq,right c) ,eq,left Figure 4.5: Equilibrium considerations for an inlet saturation of 0:87. a) Saturation profile at the tube axis (S versus x), superimposed is the Köhler curve for an NaCl particle with a dry diameter of 100 nm at 10 ffi C(S versus dp). At S =1a line has been drawn to help differentiate between saturated and supersaturated conditions. b) and c) Actual and equilibrium diameters at the tube axis. Here dp is the actual droplet diameter in the tube, dp,eq,left refers to the equilibrium diameter on the left branch of the Köhler curve, and dp,eq,right refers to the equilibrium diameter on the right branch of the Köhler curve.)

10 42 Equilibrium Calculations The comparison of the Köhler and saturation curves (Figure 4.5a) again shows that dp,eq,right can be determined only for a few points in the tube (near x ß 0:35 m and x ß 0:7 m). The right equilibrium diameter can be discarded at the first location due to the same considerations as given in the case of the higher inlet saturation. At the second location, the increase in dp,eq,right with increasing x is slightly better discernible than in Figure 4.4b. In Figure 4.6, the saturation ratio in the tube, Sr, is plotted versus the actual droplet diameter, dp. Furthermore, the Köhler curve at 10 ffi C is depicted. The droplet diameters where the saturation values are larger than the Köhler curve values belong to growing droplets. When the saturation values are smaller, the droplets shrink. At the intersection of the curves, i. e. at dp = 1:2 μ m for Si = 0:87 and at dp = 4:3 μ m for Si = 0:94, the tube saturation falls below the saturation necessary to maintain equilibrium. Therefore, the droplet size starts to decrease. In the figure, the different behaviour of the system for the two inlet saturations is discernible. For both Si = 0:87 and Si = 0:94, the droplets first are in equilibrium. Next, they experience growth, as indicated by the orange and cyan curves being larger than the S Köhler curve S i = 0.87 S i = Figure 4.6: Saturation versus droplet diameter at the tube axis.

11 4.3. Results of the Equilibrium Calculations 43 brown Köhler curve. At the intersections given in the previous paragraph, the droplets pass through their equilibrium state on the right branch of the Köhler curve, but they continue to shrink since S r is smaller than the equilibrium saturation ratio. The cyan curve approaches the Köhler curve again indicating that the droplets reach equilibrium once more for the lower inlet saturation. Considering the higher inlet saturation, a similar behaviour could be expected if the tube were longer. In this Chapter, it has been shown that the equilibrium diameters both below and above the critical diameter defined by the Köhler equation can be determined with the bisection module that has been developed as part of this diploma thesis. Three cases can be differentiated. In all cases, the droplets are in equilibrium near the inlet. In the first case, the saturation in the tube is lower than the critical saturation throughout the domain. In the second case, the critical saturation is exceeded and the droplets approach equilibrium again near the tube outlet. The third case differs from the second case in that the droplets are only initially in equilibrium, even though the system again theoretically supports an equilibrium towards the end of the tube. Another general characteristic of all cases is the fact that the the actual droplet diameter starts to deviate from the equilibrium diameter, the droplet is smaller than it would be at the equilibrium conditions determined by the ambient thermodynamic conditions inside the tube. This delay indicates that the system is slow to respond to the changes in the thermodynamic conditions occuring as the vapourcarrier gas mixture is cooled on its way through the tube. The most important factor in determining which of the cases described above is appropriate is the inlet saturation. With the help of the equilibrium diameter calculated by the bisection module one can also discern whether the droplets in the tube are activated or not. If they are, a discontinuity in the equilibrium diameter occurs which indicates that the critical saturation given by the Köhler theory has been exceeded, i. e. that the droplet has been activated.