# 3.3.4. Verification of Particle Evolution Equations

It is generally necessary to check the implementation of the equations described in Particle Transport Model to insure that they produce the expected effect. This process is referred to as verification. There are several stages of verification, many of which are focused on software design details, but many of which are intimately linked with the physics model implementation. In this section, a series of verification tests is presented that provides a test for the correct implementation of the models described in the previous sections. For the particle transport models developed here, this is accomplished by taking certain asymptotic limits of the evolution equations for which an analytical solution exists and insuring that the particle evolution approaches that solution as the particle properties approach the appropriate limiting values. For example, in the limit of zero Reynolds number, certain behavior is expected and the particle should approach that behavior as the diameter approaches zero. The limiting behavior is based on the limiting behavior of the model equations and is not based on matching particular experimental data, although many of the limiting behaviors correspond to well known phenomena. The key objective of the verification process here is to insure that the equations are satisfied, so that when numerical examples are given, nominal values for material and transport properties are employed.

## 3.3.4.1. Verification of Particle Momentum and Trajectories

In this section the solutions of Equations (3.622) through (3.628) are tested. For all of these tests, it is presumed that the source terms indicated in (3.694) through Effect of the particle phase on the gas phase are set to zero so that the gas velocities and other properties are fixed. Setting the gas-phase source terms to zero is referred to as one-way coupling because the gas phase affects the particular phase, but not vice versa.

### 3.3.4.1.1. Terminal Velocity

Falling particle will reach a terminal velocity if the gas velocity is held fixed that is given by

(3.711)

Because this terminal velocity includes the Reynolds number dependence of the drag coefficient, the full range of the drag coefficient can be tested by changing, for example, the particle diameter. Note that the slip velocity magnitude does appear in the drag coefficient for a range of values through the Reynolds number. For these Reynolds numbers (), the drag coefficient can be replaced by the first relationship in Eqn. (3.623) to give

(3.712)

For , using the second relationship from Eqn. (3.623) in Eqn. (3.711) yields simply

(3.713)

The full range of terminal velocities obtained by varying is shown in Fig The Fuego-computed particle velocity and trajectory are compared with predictions from eq-Theory-Particles-smallReynoldsGasVel and eq-Theory-Particles-smallReynoldsPos (left) and error (right). Parameters from Table %s with d_p = 3\cdot 10^{-2}. Note that while the velocities are continuous at , there is a discontinuity in the rate of change of the velocity with diameter of Reynolds number at this point.

### 3.3.4.1.2. Small Reynolds number

In the limit of small Reynolds number the solution of Eqn. (3.622) approaches that given by Eqn. (3.626) and (3.627) where simplifies to so that approaches a constant value even as the velocity changes. In this case, the analytic solution of Eqn. (3.622), where the gas velocity is fixed, is

(3.714)

where the superscript indicates the initial conditions. This limit can be tested by approaching zero Reynolds numbers with successively smaller diameters. The solution to Eqn. (3.628) is obtained is similarly obtainable by integration of (3.714) once and is

(3.715)

Using the values indicates in table Input parameters related to the particle trajectory verification, and with no heat and mass transfer, particles were evolved from rest using Fuego. The resulting particle trajectories are compared with those predicted in Equations (3.714) and (3.715), and the velocities and positions are plotted in Figure 3.8.

Fig. 3.8 The Fuego-computed particle velocity and trajectory are compared with predictions from (3.714) and (3.715) (left) and error (right). Parameters from Table 3.14 with

Fig. 3.9 The Fuego-computed particle velocity and trajectory are compared with predictions from (3.714) and (3.715) (left) and error (right). Parameters from Table 3.14 with

Fig. 3.10 The Fuego-computed particle velocity and trajectory are compared with predictions from (3.714) and (3.715) (left) and error (right). Parameters from Table 3.14 with

Table 3.14 Input parameters related to the particle trajectory verification

### 3.3.4.1.3. Turbulent dispersion

Verification of particle dispersion is hampered by the fact that it is a stochastic and not deterministic process. Therefore, verification is conducted by checking that the asymptotic behavior is correct for large numbers of samples. In an isotropic homogeneous turbulent field, the turbulent dispersion of a large number of particles should result in the mean square displacement of particles in proportion to the effective diffusion coefficient and the evolution time. The diffusion coefficient in the limit where and not determines the eddy interaction time is proportional to [127]. For verification purposes, the mean gas velocity is set to zero so that the effective diffusion coefficient is proportional to . Note that the early time behavior is different, but that the early time behavior decays over the time scale . The long-time dispersion behavior averaged over a sufficient statistical sample should be verified to follow [101]

(3.716)

Such behavior should hold true in a statistical sense for both the dispersion of the mean parcel locations and for the change in the extent of the parcel itself. Because the dispersion is driven by a Gaussian process, it can be expected that statistical differences are reduced in proportion to the inverse of the square root of the number of samples.

To verify the scaling of the particle dispersion, a large number of particles were evolved using specified fluid variables. All of the mean fluid velocities are set to zero and only the turbulent kinetic energy, , combined with time and length-scale information from the turbulent energy dissipation, , acted on the particles. In this case, the only force on the particle is associated with as defined in Equations (3.629) and (3.630). The parameters related to particle dispersion used in the test are given in Input parameters related to the particle dispersion verification. The flow was evolved for a time of (nearly ) and simulations were carried out for , , and particles. The results are shown in The mean square displacement of particles is shown as a function of time. Different curves show the statistical noise associated with different numbers of particles considered. where the linear dependence of the mean-square displacement with time is evidence for large particle samples. The large number of samples required to get convergence is noteworthy.

Fig. 3.11 The mean square displacement of particles is shown as a function of time. Different curves show the statistical noise associated with different numbers of particles considered.

Table 3.15 Input parameters related to the particle dispersion verification
Fluid properties | Particle properties
--- | ---

## 3.3.4.2. Verification of particle heat and mass transfer

There are a number of terms describing the heat and mass transfer to and from particles that must be tested. In this section, the terms that describe the particle evolution in a constant gas-phase environment will be tested. To accomplish this, the source terms indicated in Eqn. (3.694) through (3.698) are set to zero so that the gas velocities and other properties are fixed (one-way coupling).

### 3.3.4.2.1. Droplet heating and cooling

If the particles have sufficiently high boiling points and enthalpies of vaporization, the evaporation rate in Eqn. (3.667) will be zero which implies . In this case, the droplet heating and cooling are governed by the equations presented in Conserved scalars and transfer numbers for various applications. There are several limits that can be tested. The most common situation that is encountered is conduction/convection dominated heating or cooling, and this can be split into high and low Reynolds number regimes. When radiant heating and cooling are the most significant processes, the system evolves according to the radiative heat flux terms resulting in different behavior.

To verify conduction dominated heating or cooling, the particle emissivities are set to zero, eliminating the radiative interaction terms. This leads to a simplification of Eqn. (3.690)

(3.717)

for which an analytic solution

(3.718)

is obtained if , , and are constant and is the initial particle temperature. To verify radiant heat flux dominated heating or cooling, the gas-phase diffusion coefficient, , can be set to zero and non-evaporating particles can be initialized with appropriately high temperatures for cooling or with an appropriate incident flux. Neglecting gas-phase conduction, Eqn. (3.690) is written

(3.719)

where the film temperature is equal to the particle temperature for . In this case, the analytic solution of Eqn. (3.719) is given in implicit form

(3.720)

where is the temperature seen by the particle.

Table 3.16 Input parameters related to the particle heating and cooling verification
Fluid properties | Particle properties
--- | ---

### 3.3.4.2.2. The wet bulb temperature

The wet bulb state is the state at which droplet heating is zero that occurs when the heat transfer to the droplet is perfectly balanced by the enthalpy of vaporization and no droplet heating occurs. The droplet temperature will tend to approach the wet-bulb temperature, either by being heated by excess enthalpy conducted through the surface or by being cooled when more enthalpy is used in vaporization than is conducted to the surface. The system of Equations (3.690), (3.667), and (3.670) with in Eqn. (3.652) determines the wet bulb state. Droplets with initial temperatures set to the wet bulb temperature should evolve with no change in the droplet temperature. It is verified in the following sections that particles initially at the wet bulb temperature do not change temperature, and that droplets approach the wet bulb temperature from other initial droplet temperatures.

### 3.3.4.2.3. -Law Evaporation and Condensation

For particles at the wet bulb temperature with no radiative losses there will be no particle heating so that and are independent of the radius. Then, if the slip velocity is negligible the particle will evaporate at a rate proportional to the diameter squared, following the well-known -law for particle evaporation

(3.721)

To test this behavior, droplet heating should be prevented by setting the droplet temperature to the wet bulb temperature, the radiative heat transfer should be inhibited by setting the emissivity to zero, and the particle Reynolds number should approach zero in the sense that errors on the order of are introduced by finite slip velocity correlations. In this case, plotting the droplet diameter squared versus time should yield a linear line with a slope given by

(3.722)

To verify the -law for particle evaporation along with the wet-bulb temperature, three water droplets are evolved in an atmosphere of humid air. The wet-bulb temperature is computed separately for those conditions to be , and particles are selected at that temperature and above and below that temperature. The basic fluid and particle properties employed in the simulation are provided in Input parameters related to the verification of droplet evaporation. and the relationship between the initial particle diameter and temperature are indicated in Particle initial conditions for verification of droplet evaporation.. There is no flow and gravitational acceleration is not present so that there is no slip velocity maintaining the zero-Reynolds-number limit. For the conditions given, . Using these values the evaporation times for the three particles are , , for particles labeled 1, 2, and, 3 in Particle initial conditions for verification of droplet evaporation., respectively. The evaporation times predicted in Fuego are , , in agreement with the predictions. The predictions assume that the deviation from the wet bulb temperature are insignificant. The evolution of for the particles indicated in Particle initial conditions for verification of droplet evaporation. is shown in d^2 (left) and T_p (right) as a function of time for evaporating water droplets; the trajectories are close to linear as expected and a linear curve fit for each profile gives an coefficient of unit indicating a high degree of correlation. Also shown in d^2 (left) and T_p (right) as a function of time for evaporating water droplets is the early evolution of the particle temperature. The temperature is shown to converge to the computed wet bulb temperature and to maintain itself at that temperature.

Fig. 3.12 (left) and (right) as a function of time for evaporating water droplets

Table 3.17 Input parameters related to the verification of droplet evaporation.
Fluid properties | Particle properties | Evaporation properties
--- | --- | ---

Table 3.18 Particle initial conditions for verification of droplet evaporation.
| 1 | 2 | 3
--- | --- | --- | ---

### 3.3.4.2.4. -law for Fast Moving Droplets

For particle droplets with large , the available Nusselt and Sherwood number correlations indicate that the evaporation rate should increase with for . In these cases, setting the particles to the wet-bulb temperature with no radiative losses should lead to

(3.723)

with

(3.724)

being the constant rate of evaporation. Note that as the particle diameter approaches zero its Reynolds number must also approach zero so that the evaporation rate should transition to a -law behavior as it nears the fully evaporated state.

## 3.3.4.3. Verification of Lagrangian-Eulerian Coupling

The primary objective of this section is to verify that the source terms seen by the particle are appropriately reflected by equivalent (to the degrees appropriate) source terms in the gas-phase conservation equations. This primarily tests the source terms indicated in Equations (3.694) through (3.698). Because the particles affect the gas phase in the same way that the gas phase affects the particles, this is referred to as two-way coupling. The verification problems are formulated as one-dimensional problems (though the one dimension need not be aligned with the x, y, or z axis) to the maximum extent possible by employing symmetry boundary conditions on the four sides normal to the flow direction and imposing an initial uniform flow in the remaining direction. The general configuration employed is a cylindrical channel as indicated in The general configuration for verification of two-way coupling. The domain should be of sufficient length that the particles equilibrate with the gas-phase flow.. This avoids any wall effects and provides a means of identifying the transfer from inlet conditions to outlet conditions. Further, in all of these tests gravitational acceleration should be set to zero and the effects of radiation negated by setting the particle emissivity to zero.

### 3.3.4.3.1. Mass conservation

The net mass flux through the system should be constant at steady state. Particles subject to evaporation will be iso-kinetically injected into the flow near the inlet (just downstream to ensure no effect of evaporation on the inlet boundary condition). These particles will be allowed to completely evaporate while they flow with the gas-phase through the domain, and the flow will be allowed to come to steady state as can be indicated by the constant exit mass flux. At steady-state, the inlet and outlet mass fluxes should be related according to the integral conservation relation

(3.725)

where is the mass rate of particle injection. The subscripts and indicate the state of the gas-phase fluid at the inlet to the first control volume and the exit from the last control volume. The integral across the inlet and outlet areas will average out any spatial fluctuations. To facilitate reaching steady state flow with a reasonable number of particles, it is recommended that particles with a uniform size be employed. Since particle evaporation rates typically follow the -law behavior, it is preferable to have a relatively large number of particles injected over the evaporative lifetime of a given particle. The error in Eqn. (3.725) is expected to decrease as the frequency of particle injections is increased.

### 3.3.4.3.2. Species conservation

To test the species conservation, the same verification test is employed, but the mass of the individual species is tracked.

(3.726)

The mass source term is written to also account for droplet combustion as relevant. Because the total mass associated with the system changes, the mass fraction of species that do not evaporate from the particle or participate in combustion would tend to decrease; that is, the total mass flux increases while the mass flux of non-evaporating species does not increase. This can act as a second species verification test.

### 3.3.4.3.3. Energy conservation

The configuration for energy conservation is the same as for mass and species conservation: one-dimensional flow with regular iso-kinetic particle injection and allowed to reach steady state. to test the coupled energy conservation, three tests are recommended to cover the range of particle behaviors. These tests would cover non-evaporating particles, evaporating but not combusting particles, and combusting particles.

The equation describing energy conservation for non-evaporating particles, assuming that the temperature of the particles equilibrates with the gas-phase before the outlet plane, is best written

(3.727)

where represents the gas-phase mixture enthalpy. The equilibrium temperature on the right-hand side of Eqn. (3.727) is obtained through an iterative solution of that nonlinear equation with the initial particle and gas temperatures prescribed as and , respectively. Here the outflow gas-phase void fraction, , appears in the first term on the right hand side; it is implicitly included in the other equations in this section, but is unit there based on the state assumptions. Satisfaction of the equality in Eqn. (3.727) provides a test of the first term on the right-hand side of Eqn. Effect of the particle phase on the gas phase. Note that the approach to the equilibrium temperature follows an exponential decay of the form so that extending the domain to double the residence time from to will tend to cause particle and gas temperatures at the outlet to be accordingly closer to a factor of . The thermal relaxation time constant for the non-evaporating particle is

(3.728)

This provides a additional check on the transient behavior of the system and can be used to evaluate time step control provided by Eqn. (3.707). Regardless of the time step, the enthalpy transfers should be conservative at equilibrium.

for the evaporating or combusting particles, the verification test is set up so that the evaporation of the particles is completed within the domain and that the flow reaches steady state prior to evaluation. In this sense it is the same arrangement as indicated above for mass and species conservation. These tests verify the last two terms in Eqn. Effect of the particle phase on the gas phase. With the particles entirely evaporated, an energy balance gives

(3.729)

The initial particle enthalpy is defined

(3.730)

based on the gas-phase enthalpy of the particle species denoted by the subscript . Again, a nonlinear solution of this equation is required to determine because of the nonlinear dependence of the enthalpy on temperature. To satisfy this equation, it is necessary that the specific heat for the condense and gaseous phase of the participating species be identical; otherwise particle cooling during evaporation followed by the warming of products to the equilibrium temperature will result in energy differences that would have to be accounted for by tracking each droplet temperature in time through the domain. Further, the critical temperature, in Eqn. (3.645), should be set essentially to an essentially infinite value of force .

### 3.3.4.3.4. Momentum conservation

As particles with excess momentum transfer their momentum to the gas-phase flow, the net flow rate will increase. The momentum verification test described in the present section differs from the other tests in this section in the sense that a single particle or group of particles is injected at on instant and there is no continuous injection. The injected particle(s) is(are) allowed to equilibrate with the gas-phase flow and the net change in momentum is measured. For this purpose, there must be no net pressure change across the domain boundaries. Integral momentum conservation then gives the final equilibrium velocity based on

(3.731)

Presuming that the velocity equilibrates across the domain, can be brought outside of the integral and summation for an explicit expression. As for the temperature equilibration test in Eqn. (3.727), there is an exponential approach to the equilibrium value that can be used to test transient behavior if spatial fluctuations in the gas-phase velocity are not too strong. In that case, the equilibrium velocity is approached with the exponential time constant given by Eqn. (3.626) and (3.627) so that doubling the duration of the test from to should bring the particle and gas velocities closer by a factor of . Note that this transient behavior can be used to evaluate the time step criteria provided in Eqn. (3.707) by assessing the error associated with liberal time steps. Regardless of the time step, though, the momentum transfer should be conservative at equilibrium.

### 3.3.4.3.5. Parallel implementation

All of these tests described in the present section are implemented on both one and four processors with domains that cross multiple processors to test the passing of particle information across domain boundaries.

### 3.3.4.3.6. Verification Tests for Lagrangian-Eulerian Coupling

In this section, the verification tests suggested in the previous sections are described. Verification tests for non-reacting iso-kinetic particle flows (energy conservation only), for isothermal and non-iso-kinetic particle flows (momentum conservation only) and for reacting/evaporating particle flows (mass and energy conservation) are all included. Together these test all of Eqn. (3.694) through Effect of the particle phase on the gas phase. Not yet covered with documented verification problems is the radiation coupling in (3.698).

#### Energy conservation verification

Energy conservation for non-evaporating particles was verified in Fuego using nominal parameter values for both the particles and the fluid. For the Eulerian phase, the mass, momentum, and energy equations were evolved, but not the species equations. Under these conditions, fluid properties are manually specified. The fluid properties listed in table Input parameters related to the verification of energy conservation without evaporation were specified as constants. Since the fluid viscosity, specific heat, and Prandtl numbers are specified, the thermal conductivity is computed from these quantities. The constant specific heat leads to a linear dependence of enthalpy on temperature, and without loss of generality the enthalpy is set to the temperature , corresponding to an enthalpy reference temperature of . The simulation was carried out in the `cylindricalchannel.g` configuration with an inlet/outlet area of 3.10583 (with unit radius, this is nominally , but low resolution of the circular cross section leads to a smaller area) and a length of 20. Particles are inject iso-kinetically at the downstream location 1 unit from the inlet and flow 19 units to the outlet in 19 time units. With iso-kinetic flow, and with zero particle viscosity from Eqn. (3.675). The particle thermal response time from Eqn. (3.728) is and is sufficiently small that particle thermally equilibrate while traveling the length of the channel.

There are two ways to carry out the energy conservation verification: using Eqn. (3.727) during a period in which the flow has reached steady state and integrating Eqn. (3.727) in time over the entire simulation. Both of these approaches are employed here. The surface integrals specified in the conservation equation were carried out within Ensight. To compute the flux of the particles and associated enthalpy out of the domain (last terms in (3.727)), the particle deposition tracking in Fuego was employed (keywords: enthalpy_deposition_density, enthalpy_deposition_rate, etc.) and these quantities were also integrated over the outlet surface. The balance of the steady-state flux is described first. for the two terms on the left hand side of (3.727), the boundary conditions provided the values of for the fluid inlet enthalpy flux and for the particle inlet enthalpy. The compute equilibrium temperature is from (3.727). The outlet gas temperature was in the range of 883.4 to 834.8 with a mean of 834.1 and the exiting particles are between 832 and 835 . to supplement the stead-state enthalpy flux, the integrated enthalpy flux is computed and plotted in The enthalpy over time in nonreacting channel flow with hot particles injected. The net input enthalpy includes the initial domain enthalpy and the enthalpy of all of the injected particles over time. Squares show the enthalpy associated with particles in the domain. Circles show the enthalpy of the particles that have left the domain. Diamonds show the enthalpy associated with fluid in the domain. Triangles show the excess enthalpy of fluid that has left the domain (difference between the outlet enthalpy and inlet that was accounted for in the net input category). The sum of the categories indicated by symbols is shown to agree with the net input enthalpy indicating overall conservation of enthalpy.. The net input enthalpy includes the initial domain enthalpy and the enthalpy of all of the injected particles over time can be compared with the enthalpy in the domain and that which has left the domain. These two quantities agree to within , which is taken to be suitable (the integration of quantities within Ensight does not use the same algorithms as employed in Fuego, leading to some error). Also examined in this test is the particle mass deposition rate at the outlet. Because there is no evaporation in this scenario, the complete particle mass injected should be deposited (or pass through) the outlet. Using the keywords mass_deposition_density and mass_deposition_rate and integrating these over the outlet surface, The mass deposition rate integrated over the outlet is shown as a function of time. The corresponding particle inlet mass flux is 0.1. shows that the mass tracked as crossing the outlet plane matches the input particle mass, 0.1, to within statistical fluctuations.

Fig. 3.13 The enthalpy over time in nonreacting channel flow with hot particles injected. The net input enthalpy includes the initial domain enthalpy and the enthalpy of all of the injected particles over time. Squares show the enthalpy associated with particles in the domain. Circles show the enthalpy of the particles that have left the domain. Diamonds show the enthalpy associated with fluid in the domain. Triangles show the excess enthalpy of fluid that has left the domain (difference between the outlet enthalpy and inlet that was accounted for in the net input category). The sum of the categories indicated by symbols is shown to agree with the net input enthalpy indicating overall conservation of enthalpy.

Fig. 3.14 The mass deposition rate integrated over the outlet is shown as a function of time. The corresponding particle inlet mass flux is 0.1.

Table 3.19 Input parameters related to the verification of energy conservation without evaporation
Fluid properties | Particle properties
--- | ---

#### Momentum conservation verification

To verify the momentum transfer within Fuego, simulations were carried out with ten particles injected into a cylindrical channel. Momentum conservation for non-evaporating particles was verified using nominal parameter values for both the particle and the fluid. For the Eulerian phase, the mass and momentum were evolved, but not the species and energy equations. Under these conditions, fluid properties are manually specified. The fluid and particle properties listed in table Input parameters related to the verification of energy conservation without evaporation were specified as constants. The simulation was carried out in the cylindrical_channel.g configuration with an inlet/outlet area of 3.10583 (with unit radius, this is nominally , but low resolution of the circular cross-section leads to a smaller area), and a length of 20. The channel had symmetry boundary conditions on the side to prevent drag from affecting the momentum field. Particles are injected with an initial velocity of 10 units into a stationary fluid phase at the initial time. The particles were injected at the time origin with a diameter of 0.12, a density of 10, and a velocity of 10. The injection occurred as a line of ten particles across the channel (in the narrow direction). The particle flow direction was oriented lengthwise down the channel and the point of injection was the midpoint of the channel, 10 units from the inlet and outlet. Zero pressure boundary conditions were specified on the inlet and outlet that allowed the flow to continue in the absence of any forces (ie. particle drag).

Based on the values in Input parameters related to the verification of energy conservation without evaporation, the initial momentum imparted by the particles is 0.904779 and the computed total momentum at the end of the Fuego simulation is 0.893059. That is, the final combined momentum is 1.3 % less than the initial momentum. The reason for this discrepancy is unclear. The final particle and fluid momentum are 0.001299 and 0.89176, respectively. The temporal evolution of the fluid, particle, and total momentum is shown in The momentum associated with the particle phase, the fluid phase, and the combined momentum shown. The right-hand panel shows the initial period in greater detail.. There the fluid momentum has been shifted back one time step (0.01 units) to account for the staggered time-stepping algorithm: the fluid evolves based on the previous time steps’ particle momentum transfer. The simulation was evolved for 30 time units, but no changes for the single precision arithmetic was observed after 21 units.

Fig. 3.15 The momentum associated with the particle phase, the fluid phase, and the combined momentum shown. The right-hand panel shows the initial period in greater detail.

The velocity after the equilibration is computed to be based on Eqn. (3.731). Carrying out the Fuego simulation results in an equilibration average velocity of , which is only 0.1% below the computed value. The predicted resulting momentum and velocity were computed with differing methods within Ensight: the momentum was computed using the volume integral while the velocity was computed using the spatial mean frequency. It is not clear what level of numerical error is attributable to the Ensight algorithms.

The average gas velocity had equilibrated with within 99% of its final velocity within 1.06 time units. The value of for this system is from (3.627) and the momentum and velocity equilibration times are somewhat longer than the suggested response time near the particle equilibration time of a few since the particles were not injected uniformly through the domain. The result is substantial gas-phase velocity inhomogeneities that take much longer than to dissipate.

Table 3.20 Input parameters related to the verification of momentum conservation
Fluid properties | Particle properties
--- | ---

#### Mass and energy conservation verification

Mass and energy conservation for evaporating and reacting particles was verified in Fuego using fluid-phase parameter values taken from the thermodynamic databases and evaluated with Cantera. For the eulerian phase, the mass, momentum, energy, and species equations were evolved. The fluid and particle properties listed in table Input parameters related to the verification of momentum conservation were specified.’ Note that the critical temperature for water was set to a large value so that the enthalpy of the particles could be compute directly from as in Eqn. (3.730); this initial value of the particle enthalpy is also given in table Input parameters related to the verification of energy conservation with evaporation. using the value of computed in Fuego, . Since the fluid composition and enthalpy are specified as boundary conditions or evolved within Fuego, the viscosity specific heat, and thermal conductivity are computed from these quantities. The simulation was carried out in the cylindrical_channel.g configuration with an inlet/outlet area of 3.10583 (with unit radius, this is nominally , but low resolution of the circular cross-section leads to a smaller area) and a length of 20. Particles are injected iso-kinetically at a downstream location 1 unit from the inlet flow 19 units to the outlet in 19 time units.

Net mass conservation is evaluated using Eqn. (3.725). The fluid-phase mass flux at the inlet, using the Cantera computed density of , is . The particle mass flux is and combining the inlet mass fluxes gives an expected outlet mass flux of . The value computed from the outlet using Ensight is ; there is a statistical variation in this quantity with a standard deviation of . In a similar manner, mass conservation for individual species is evaluated using Eqn. (3.726). For water, the inlet mass flux is and the mass injected is . The value computed at the outlet plane is which is approximately 1.5% below the expected value of . The outlet water mass flux has a statistical variation associated with it characterized by a standard deviation of . While there is no source term for species like , the inlet and outlet mass fluxes can be computed. The inlet flux of is and that for the outlet is computed to be , with a standard deviation of . Enthalpy conservation is evaluated using (3.729). The enthalpy flux into the domain associated with the fluid phase is while that associated with the particle phase is . The difference between these, , is the expected enthalpy flux at the outlet, the right hand side of (3.729). Using Ensight to evaluate the enthalpy flux at the outlet gives ; this is a 2% discrepancy, the source of which is uncertain at this point. There is a statistical variation in the outlet enthalpy flux characterized by a standard deviation of , approximately 0.04% of the total enthalpy change.

Table 3.21 Input parameters related to the verification of energy conservation with evaporation.
Fluid properties | Particle properties | Evaporation properties
--- | --- | ---

## 3.3.4.4. Verification summary

To summarize the verification process, the following list shows the equations that are covered by verification tests described in the various sections in this document. The equations that are not fully covered are also indicated. The lack of verification coverage is only significant for the radiative terms.

> - Turbulent dispersion

>   - (3.622), (3.628), (3.629), (3.630), (3.631)

> - Droplet heating and cooling

>   - (3.675), (3.690), (3.691)

> - The wet bulb temperature, d^2-Law Evaporation and Condensation

>   - (3.650), (3.652), (3.643), (3.667), (3.668), (3.670), (3.671)

> - d^{1.5}-law for Fast Moving Droplets

>   - (3.660) , (3.661), (3.662), (3.664)

> - Mass conservation, Species conservation, Energy conservation

>   - (3.694), (3.695), Effect of the particle phase on the gas phase

> - Momentum conservation

>   - (3.696)

> - Equations employed but not yet fully covered in verification tests

>   - (3.623), (3.632), (3.646), (3.651), (3.669), (3.683), (3.698), (3.699)

Fig. 3.16 Condensed-phase conduction is approximated based on the difference between the film and mean droplet temperatures and on an estimated heat transfer coefficient that describes a boundary layer thickness. Over this boundary layer thickness, the temperature difference is presumed to act.

Fig. 3.17 Terminal velocities for particles as a function of diameter and particle Reynolds numbers determined from (3.712)

Fig. 3.18 Water droplet evaporation and condensation with initial temperatures set to the wet bulb temperature. Left plot exhibits the linear -law behavior while the right hand plot shows the droplet temperatures as constant (no heating).

Fig. 3.19 Aluminum particle evaporation with and without combustion with initial temperatures set to the wet bulb temperature showing the linear -law behavior.

Fig. 3.20 The general configuration for verification of two-way coupling. The domain should be of sufficient length that the particles equilibrate with the gas-phase flow.