**3. Single droplet drying**

The study of mechanisms that describe single droplet drying is a challenging issue since it involves many disciplines: heat and mass transfer, fluid mechanics and chemical kinetics. The main objective of this section is to attempt to further understand the mechanisms in‐ volved in the drying of a single droplet, and more specifically to formulate a mathematical model. The model should predict temperature profiles for both the inner core and the outer surface of the droplet under simulated conditions that might be encountered in spray drying equipment. Experimental work will also involve the measuring the moisture contents against time to provide more information for the droplet drying process.

The experimental apparatus was comprised of a horizontal wind tunnel 2.2 m long. The wind tunnel supplied a forced drying air into the working section where the droplet was suspended from a glass nozzle. The apparatus and the flow system are shown in Figure 12. A gate valve at the inlet to the wind tunnel controlled airflow rate. Air was heated to the desired temperature, using a 3 KW electric heating element controlled by a rotary voltage regulator.

Through the wind tunnel, a controlled flow of hot, dry air, with an average velocity of 1 m/s, was passed across the droplet suspended from the glass nozzle. A forced dry air was ob‐ tained by using a centrifugal fan and a molecular sieve air dryer containing silica gel and calcium silicate.

**Figure 12.** Experimental apparatus: 1-fan 2-molecular sieve 3-voltage regulator 4-air heater 5-Digital balance 6-Tem‐ perature recorder 7-Glass nozzle 8-Observation port.

The experiment was initiated by switching on the centrifugal fan and then the electric air heater. The voltage regulator was adjusted to provide the desired air temperature. A ther‐ mocouple was fixed in the wind tunnel near to the glass nozzle to measure the air tempera‐ ture with an accuracy of ± 1o C. The air temperature was monitored until it reached a steady state. This state requires between 1 to 2 hours to be achieved. When the apparatus achieved a constant air temperature, the drying process was initiated.

As a part of this study, a droplet suspension device was specially designed to measure the weight and temperature of the droplet. The droplet suspension device is illustrated in Fig‐ ure 13. It consisted of a glass nozzle with the dimensions of 180 mm in length and 9 mm outside diameter. The upper section of the glass nozzle was fixed by a small electric rotator device able to provide a rotational speed range of 1- 30 rpm. In order to reduce the contact area between the free end of the glass nozzle and the droplet surface, the lower section of the nozzle was shaped as a small cone with a free end diameter of 4 mm. The droplet re‐ ceives a relatively small amount of heat transferred by conduction through the glass nozzle, which has a thermal conductivity of 0.480 W/m K. The heat transferred by conduction was taken into account in the proposed model represented by Equations 4&5. The glass nozzle was rotated at constant low speed of 5 rpm. This rotation speed had no effect on the shape or stability of the droplet. The upper section of the suspension device was fixed by a metallic clip installed beneath the analytical balance. The lower section of the suspension device, i.e. the glass nozzle, was inserted through a hole in the wind tunnel.

**Figure 13.** Droplet suspension device; glass nozzle and the connected thermocouples.

The experimental apparatus was comprised of a horizontal wind tunnel 2.2 m long. The wind tunnel supplied a forced drying air into the working section where the droplet was suspended from a glass nozzle. The apparatus and the flow system are shown in Figure 12. A gate valve at the inlet to the wind tunnel controlled airflow rate. Air was heated to the desired temperature, using a 3 KW electric heating element controlled by a rotary voltage

Through the wind tunnel, a controlled flow of hot, dry air, with an average velocity of 1 m/s, was passed across the droplet suspended from the glass nozzle. A forced dry air was ob‐ tained by using a centrifugal fan and a molecular sieve air dryer containing silica gel and

**Figure 12.** Experimental apparatus: 1-fan 2-molecular sieve 3-voltage regulator 4-air heater 5-Digital balance 6-Tem‐

The experiment was initiated by switching on the centrifugal fan and then the electric air heater. The voltage regulator was adjusted to provide the desired air temperature. A ther‐ mocouple was fixed in the wind tunnel near to the glass nozzle to measure the air tempera‐ ture with an accuracy of ± 1o C. The air temperature was monitored until it reached a steady state. This state requires between 1 to 2 hours to be achieved. When the apparatus achieved

As a part of this study, a droplet suspension device was specially designed to measure the weight and temperature of the droplet. The droplet suspension device is illustrated in Fig‐ ure 13. It consisted of a glass nozzle with the dimensions of 180 mm in length and 9 mm outside diameter. The upper section of the glass nozzle was fixed by a small electric rotator device able to provide a rotational speed range of 1- 30 rpm. In order to reduce the contact area between the free end of the glass nozzle and the droplet surface, the lower section of the nozzle was shaped as a small cone with a free end diameter of 4 mm. The droplet re‐ ceives a relatively small amount of heat transferred by conduction through the glass nozzle, which has a thermal conductivity of 0.480 W/m K. The heat transferred by conduction was taken into account in the proposed model represented by Equations 4&5. The glass nozzle was rotated at constant low speed of 5 rpm. This rotation speed had no effect on the shape or stability of the droplet. The upper section of the suspension device was fixed by a metallic

regulator.

calcium silicate.

perature recorder 7-Glass nozzle 8-Observation port.

180 Wind Tunnel Designs and Their Diverse Engineering Applications

a constant air temperature, the drying process was initiated.

Two thermocouple sensor types, NiCr-NiAl, were used to measure the core and surface temperatures of the droplet. One of the thermocouples was placed inside the glass nozzle and extended to the center of the droplet. The other thermocouple was fixed outside and along the glass nozzle. The outer thermocouple rotates simultaneously with the rotation of the glass nozzle. The end tip of the thermocouple was positioned in a manner to touch the outer surface of the droplet. The rotation process that made both the nozzle and the droplet rotate together assisted in avoiding any separation between them that might have been caused by the force of air drying. The core and the surface temperatures of the droplet were easily recorded by a temperature recorder at 50 sec. intervals. The drying process of the dif‐ ferent material droplets was investigated under air temperature of 75o C and 140o C.

The procedure for weighing the droplet was carried out quickly and intermittently by caus‐ ing the suspension device to be freely-suspended. The gate valve was closed to cut off the airflow to the working section and diverted to an outlet 20 mm valve in order to prevent any vibration of the glass nozzle during the weighing process. The nozzle rotation was simulta‐ neously stopped. A metallic clip was opened manually to allow the suspension device to be freely-suspended from a hook connected beneath the balance. This arrangement made the weight measurement readings more accurate. The weighing procedure was repeated during the droplet drying experiment at 100 secintervals. Thus, the weight loss of droplet was re‐ corded and moisture content was determined versus time. The required time for each weighing procedure step was about 10 sec. This time was not included in the recorded dry‐ ing time and had no noticeable effect on the results.

Three types of liquids were selected for the drying process experiments. The first type was sodium sulphate decahydrate solution (60 wt % solid). The second type was a concentrated fruit juice (60 wt % fruit juice powder of apple, peach and blueberry, MTC product). The third type was an organic paste (20% sodium chloride, 25% dispersal pigment; Goteks prod‐ uct) used for adhesive and coating applications. Table 2 shows the physical properties of the sample materials. Droplets ranging from 9 mm to 14 mm diameter were subjected to the drying process. However, the actual size of the droplets in typical spray drying applications is much smaller. The droplets with small sizes require developing a more accurate technique to measure both core and surface temperatures. Therefore, the current research assumes that the mechanisms of the drying large and small droplets are similar.


**Table 2.** Physical properties of sodium sulphate decahydrate solution, fruit juice and organic paste.

#### **3.1. Mathematical model**

In the drying process, the droplet is first heated by the hot air flow with significant evapora‐ tion from the surface. The temperature of the surface increases and approaches the wet-bulb temperature, indicating the constant drying rate period. In order to propose a mathematical model, the droplet was assumed to have a fixed size with no change during this period. Al‐ so, the droplet was assumed to have a uniform initial temperature and moisture content. Temperature distribution within the droplet can be represented as

$$\frac{\partial \text{ T}}{\partial \text{ }t} = a(\frac{\partial^2 \text{T}}{\partial \text{ }r^2} + \frac{2}{r}\frac{\partial \text{T}}{\partial r}) \tag{8}$$

Equation 8 was solved with the following boundary conditions using explicit finite differen‐ ces,

$$-k\frac{\partial T}{\partial \ r} = 0 \qquad \text{at } r = 0 \tag{9}$$

and

$$-k\frac{\partial}{\partial r}\frac{T}{r} = h(T\_a - T\_{sr}) + q\_{nz} \quad \text{at } r = \text{R} \tag{10}$$

where, qnz is the transferred heat conduction to the droplet through the glass nozzle. It can be calculated as:

$$q\_{nz} = \frac{4\,k\_{nz}\,h\_{nz}}{d\_{nz}})^{0.5} \,\left(T\_a - T\_{sr}\right) \tag{11}$$

The heat transfer coefficient, hnz, can be correlated from Thomas (1999) as:

Three types of liquids were selected for the drying process experiments. The first type was sodium sulphate decahydrate solution (60 wt % solid). The second type was a concentrated fruit juice (60 wt % fruit juice powder of apple, peach and blueberry, MTC product). The third type was an organic paste (20% sodium chloride, 25% dispersal pigment; Goteks prod‐ uct) used for adhesive and coating applications. Table 2 shows the physical properties of the sample materials. Droplets ranging from 9 mm to 14 mm diameter were subjected to the drying process. However, the actual size of the droplets in typical spray drying applications is much smaller. The droplets with small sizes require developing a more accurate technique to measure both core and surface temperatures. Therefore, the current research assumes that

**ρd (Kg/m3) cpd (kJ/kgK) k (W/mK) kd (W/mK) Deff (m²/s) Dv(m²/s)**

decahydrate solution <sup>3110</sup> 1.1 0.180 0.246 1.14x10-5 3.45x10-5 Fruit juice 1650 1.7 0.126 0.188 1.10x10-5 3.40x10-5 Organic paste 3400 2.4 0.251 0.422 1.52x10-5 3.50x10-5

In the drying process, the droplet is first heated by the hot air flow with significant evapora‐ tion from the surface. The temperature of the surface increases and approaches the wet-bulb temperature, indicating the constant drying rate period. In order to propose a mathematical model, the droplet was assumed to have a fixed size with no change during this period. Al‐ so, the droplet was assumed to have a uniform initial temperature and moisture content.

> 2 2 <sup>2</sup> ( ) *T T <sup>T</sup> t rr r*

0 0 *<sup>T</sup> <sup>k</sup> at r*

( ) *a sr nz <sup>T</sup> k h T T q at r R*


Equation 8 was solved with the following boundary conditions using explicit finite differen‐

¶

¶ (8)

¶ -= = ¶ (9)

¶ (10)

 ¶

¶

= +

a

*r*

*r* ¶

**Table 2.** Physical properties of sodium sulphate decahydrate solution, fruit juice and organic paste.

the mechanisms of the drying large and small droplets are similar.

182 Wind Tunnel Designs and Their Diverse Engineering Applications

Temperature distribution within the droplet can be represented as

¶

¶

Sodium sulphate

ces,

and

**3.1. Mathematical model**

$$h\_{nz} = 0.26 \, + \, \text{Re}^{0.6} \, \text{Pr}^{0.33} \left( k\_{air} / d\_{nz} \right) \tag{12}$$

During the falling rate period, the formation of a receding evaporation front divides the droplet into two regions, a dry crust at the outer surface and a wet region inside the core. Therefore, heat transfer equations are formulated for each region. The physical model and the coordinate system for analysis are shown in Figure 14.

**Figure 14.** Physical model and the coordinate system of the droplet cross-section.

Energy balance for the wet core, 0 < r < z, can be represented as follows:

$$\frac{\partial^{\circ}\mathbf{T}\_{w}}{\partial\;t} = \alpha\_{w} (\frac{\partial^{\circ}\mathbf{T}\_{w}}{\partial\;r^{2}} + \frac{2}{r}\frac{\partial\mathbf{T}\_{w}}{\partial r})\tag{13}$$

Heat is transferred through the crust into the wet core where evaporation occurs at the inter‐ face between the core and the crust. Vapor then diffuses through pores of the crust into the drying medium. Thus, moisture is transferred mainly by vapor flow. Consequently, vapor diffusion must be taken into account in formulating the equations for the dry (crust) region. The energy balance for the crust region, z < r < R, can be represented as follows:

$$\frac{\partial}{\partial t} \frac{T\_d}{t} = \alpha\_d (\frac{\partial^2 T\_d}{\partial \ r^2} + \frac{2}{r} \frac{\partial T\_d}{\partial r}) + \frac{\partial}{\partial r} (\frac{M}{RT} \frac{\partial P\_v}{\partial r}) \frac{\Delta h\_v}{\rho\_d c p\_d} D\_v \tag{14}$$

#### **3.2. Boundary conditions**

At the center of the sphere, r = 0

$$-k\_w \frac{\partial^\circ \mathcal{T}\_w}{\partial^\circ r} = 0\tag{15}$$

At the surface, r = R:

$$-k\_d \frac{\partial T\_d}{\partial \ r} = h(T\_a - T\_{sr}) + q\_{nz} \tag{16}$$

At the receding evaporation front (r = z), the moving boundary conditions are

$$-k\_d \frac{\partial T\_d}{\partial \, r} + k\_w \frac{\partial}{\partial \, r} = \stackrel{\bullet}{\mathbf{m}} \, \Delta \mathbf{h} \mathbf{v} \tag{17}$$

The drying rate, m • , was defined in Eqs. (5 & 6). However, in this case, *z* " , in Eq. (6) repre‐ sents the distance from the droplet surface (r = R) to the receding evaporation front (r = z).

Heat and mass transfer coefficients, h and *kc* ' , can be determined by the correlations found in [28,29] as follows:

$$\Delta \mathbf{N} \mu = \mathbf{2} + \Phi \mathbf{R} e^{0.5} \mathbf{P} r^{0.33} \tag{18}$$

and

0.5 0.33 *Sh Re Sc* = + 2 b(19)

where Φ and β are constants ranging from 0.6-0.7 for Re (500 -17000).

Thermal conductivity of the wet region can be evaluated according to [30] as:

$$k\_w = k\_d + k\_v X$$

where *<sup>X</sup>*¯ is an average moisture content and *kv* is

$$k\_v = \frac{D\_v M}{RT} \cdot \frac{dp}{dT} \Delta h\_v \tag{21}$$

The non-linear Eqs. (13&14) with the boundary conditions as in Eqs. (15-17) were solved by a finite difference method. The proposed equations were solved in a program using Turbo-Pascal V.6, and the computational results compared with the experimental results.
