CHAPTER 2: MULTI-PHASE MATHEMATICAL MODELING OF
2.6 NUMERICAL PROCEDURE
CO2 exits the nozzle at the sublimation point, the point where a mixture of dry ice
particles and vapor coexist, of 195 K and atmospheric pressure. Both vapor and solid particles are accelerated by the air flow from the Coanda nozzle. Prior to initiating CFD calculations, it was essential to compute the fraction of CO2 that is solid, represented by
the expression, 1-x2, at the nozzle outlet. This fraction is determined by a steady state
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kinetic energy of the flowing CO2. The expression x2represents the mass fraction of vapor.
The resulting equation representing energy balance is: 𝐻2= 𝐻1 + 1 2 𝑣12 − 1 2[( 𝑚̇ 𝐴) ∗ ( 𝑥2 𝜌𝑣+ 1−𝑥2 𝜌𝑠 )] 2 𝐻2 = 𝑥2 𝐻2𝑣 + (1 − 𝑥2)𝐻2𝑠(15)
The outlet properties are fixed by virtue of the CO2 being at sublimation conditions,
where the enthalpy of dry ice, denoted as 𝐻2𝑠 is 152.1 kJ/kg and the enthalpy of vapor state, denoted as 𝐻2𝑣 is 723.1 kJ/kg. The density of the solid state, denoted as 𝜌𝑠 is 1,562 kg/m3 and the vapor density, denoted as 𝜌𝑣 is 2.82 kg/m3.(44)
The inlet enthalpy, H1,of liquid CO2 depends on temperature T1. For instance, at
30°C (45) H1 is 602.5 kJ/kg. By specifying T1, equation 15 has only one remaining
unknown variable, x2. We can then determine H2 of the dry ice and gaseous CO2 from the
energy balance equation (38). Prior to simulation, the energy balance equation was used to determine the mass fraction of dry ice (1-x2). These values along with the outlet
velocities (v2) were then used as inputs into the CFD simulation.
In addition, three particle sizes were selected as inputs into the CFD simulation to illustrate closed and open nozzle behavior. Since we were unsure as to what an exact dry ice particle size coming out of the nozzle would be, we selected particle sizes at 10µm, 100µm and 1000µm for observation purposes. The full set of CFD input and boundary values are shown in Table 2.2.
Table 2.2:Operating conditions and parameters
Process Variables Value
Density of CO2 vapor 2.819 kg/m3
Density of CO2 solid 1,562 kg/m3
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Temperature 194.65 K
Pressure 101.325 kPa
Fraction of CO2 vapor at an inlet temperature of 0°C 0.598
Fraction of CO2 solid at an inlet temperature of 0°C 0.402
Mass flow rate of CO2 vapor at an inlet temperature of
0°C
0.00075 kg/s Mass flow rate of CO2 solid at an inlet temperature of
0°C
0.00051 kg/s Velocity of CO2 vapor at an inlet temperature of 0°C 53.9 m/s
Velocity of CO2 solid at an inlet temperature of 0°C 0.066 m/s
Fraction of CO2 vapor at an inlet temperature of 10°C 0.642
Fraction of CO2 solid at an inlet temperature of 10°C 0.358
Mass flow rate of CO2 vapor at an inlet temperature of
10°C
0.00081 kg/s Mass flow rate of CO2 solid at an inlet temperature of
10°C
0.00045 kg/s Velocity of CO2 vapor at an inlet temperature of 10°C 57.9 m/s
Velocity of CO2 solid at an inlet temperature of 10°C 0.058 m/s
Fraction of CO2 vapor at an inlet temperature of 20°C 0.694
Fraction of CO2 solid at an inlet temperature of 20°C 0.306
Mass flow rate of CO2 vapor at an inlet temperature of
20°C
0.00087 kg/s Mass flow rate of CO2 solid at an inlet temperature of
20°C
0.00039 kg/s Velocity of CO2 vapor at an inlet temperature of 20°C 62.6 m/s
Velocity of CO2 solid at an inlet temperature of 20°C 0.050 m/s
Fraction of CO2 vapor at an inlet temperature of 30°C 0.784
Fraction of CO2 solid at an inlet temperature of 30°C 0.216
Mass flow rate of CO2 vapor at an inlet temperature of
30°C
0.000987 kg/s Mass flow rate of CO2 solid at an inlet temperature of
30°C
0.00027 kg/s Velocity of CO2 vapor at an inlet temperature of 30° C 71.0 m/s
Velocity of CO2 solid at an inlet temperature of 30° C 0.035 m/s
Particle size diameters 10µm, 100µm, 1000µm Air-CO2 diffusivity 0.000016 m2/s
Fluid 1 inlet species mass fraction (Air = 0; CO2 = 1)
Fluid 2 inlet species mass fraction (Air = 1; CO2 = 0)
The two cases examined are as follows.
Case A - Pressure outlet: A pressure is specified at the outlet. This refers to the case where the nozzle is open to the environment.
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Case B - Wall: An adiabatic wall has been modeled at constant temperature. Simulated dry ice particle velocities with varying sizes were compared for both open and closed boundary nozzles at each inlet temperature
A polyhedral mesh and surface re-mesher were used for the spray-nozzle, as shown in the grid geometry in Figure 2.6. The surface re-mesher was chosen to improve the overall quality of surface mesh and to optimize the surface mesh for generating volume mesh. The polyhedral mesher was chosen to fill the volume inside the surface mesh for which the solver equations work through. Particle data were extracted from the simulation post- processing. Three particle sizes (10µm, 100µm, and 1000µm) were selected to illustrate behavior. For convergence, the residual was set to 1 x 10-4m.
The following model assumptions were made in this work. 1. the flow through the Coanda nozzle is representative of only one phase, namely, dry ice. There is no phase change through the nozzle itself, 2. we assumed and modeled the dry ice particle sizes coming out of the nozzle to be 10μm, 100μm and 1000μm respectively without any variations of size.