Injection Conditions

In Simcenter STAR-CCM+ In-cylinder, particles enter the fluid continuum through injectors. The injector defines where, in what direction, and how the particles enter.

The following sections describe the injection conditions for a hollow/solid cone injector. For information on the injection conditions for a nozzle injector, see Theory Guide—Huh Model.

Spray Angle and Initial Velocity

For a hollow/solid cone injector, the parcel streams are injected in random directions conforming to a uniform distribution on the surface of the cone.

The velocity magnitude of the parcels injected at nozzle $i$ is calculated as:

Figure 1. EQUATION_DISPLAY

v_{i} = \frac{\dot{m_{i}}}{ε_{i} ρ {(d_{i} / 2)}^{2} π}

(571)

with:

Figure 2. EQUATION_DISPLAY

{\dot{m}}_{i} = \frac{\dot{m} d_{i}^{2}}{\sum_{i = 1}^{n} d_{i}^{2}}

(572)

where:

$ρ$ is the density of the specified Liquid Fuel.
$d_{i}$ is the specified Hydraulic Diameter of nozzle $i$ .
$\dot{m}$ is the specified Mass Flow Rate.
$n$ is the specified number of Nozzle Holes.

For hollow/solid cone injectors, $ε_{i}$ is the specified Area Ratio of nozzle $i$ . For nozzle injectors, this parameter equals 1.

For the Mass Flow Rate, Simcenter STAR-CCM+ In-cylinder requires either the specification of a constant mass flow rate profile with start and end of fuel injection or tabular data that describe the mass flow rate as a function of crank angle or of time. The table must provide data from start of injection (SOI) to end of injection (EOI) only and begin and end at zero mass flow rate. The following table gives an example of a mass flow rate table for multi-pulse injection:

"CA","m_dot (kg/s)"
425,0.0000
427,0.0082
429,0.0000
450,0.0000
452,0.0088
455,0.0088
457,0.0000
475,0.0000
477,0.0094
485,0.0094
487,0.0000

For the remainder of the cycle, Simcenter STAR-CCM+ In-cylinder automatically applies zero mass flow. When necessary, Simcenter STAR-CCM+ In-cylinder automatically applies appropriate offsets to the crank angle or the time data so that multiple cycles can be handled automatically.

If the imported mass flow rate profile does not represent the correct start and end points of injection, you can shift the mass flow rate profile accordingly.

For a table that describes the mass flow rate as a function of crank angle, Simcenter STAR-CCM+ In-cylinder interpolates the table values as:

Figure 3. EQUATION_DISPLAY

{\dot{m}}_{int e r} = int e r p o l a t e T a b l e ({C A}_{c y c l e} - I C A T + I C A A)

(573)

where:

$int e r p o l a t e T a b l e ()$ linearly interpolates the tabular data.
${C A}_{c y c l e}$ is the cycle wrapped crank angle in deg (repeats 0 to cycle length).
$I C A T$ is the specified Injection Crank Angle Target.
$I C A A$ is the specified Injection Crank Angle Anchor.

For a table that describes the mass flow rate as a function of time, Simcenter STAR-CCM+ In-cylinder interpolates the table values as:

Figure 4. EQUATION_DISPLAY

{\dot{m}}_{int e r} = int e r p o l a t e T a b l e (\frac{{C A}_{c y c l e} - I C A T + I T A \cdot R P M}{R P M})

(574)

where:

$I T A$ is the specified Injection Time Anchor.
$R P M$ is the specified engine angular velocity RPM.

The mass flow rate at time-step $t$ is then calculated as:

Figure 5. EQUATION_DISPLAY

\dot{m} (t) = \frac{\int_{t - Δ t}^{t} {\dot{m}}_{int e r} (t) d t}{Δ t}

(575)

For information on the specified values, see Engine Reference—Injector Dialog.

Initial Temperature

For the initial temperature of the injected fuel, you can either set a constant value or import a table that describes the temperature as a function of time or of crank angle. The table must provide data from SOI to EOI only. Multiple cycles are handled automatically.

The following table gives an example of a temperature table for multi-pulse injection:

"CA","T (K)"
425,354.45
427,365.15
429,354.45
450,354.45
452,365.15
455,365.15
457,354.45
475,354.45
477,365.15
485,365.15
487,354.45

Initial Droplet Diameter

For a hollow/solid cone injector, you specify a droplet diameter distribution. The following distribution functions are supported:

Cumulative distribution function (CDF) in tabular form
Probability density function (PDF) in tabular form
Rosin-Rammler distribution function

If no distribution functions are available, you can set a constant droplet diameter instead.

CDF Table

A CDF table has a column of droplet diameter values and a column of CDF values, $(D, F (D))$ .

The droplet diameter must be in m. The column header for the CDF values must be CDF and the CDF values must be strictly increasing—duplicate values are not allowed.

Example:

CDF Table Plot of CDF Table

CDF Table	Plot of CDF Table
"D(m)","CDF" 0.00000000e+00,0.00000000e+00 5.00000000e-07,3.12499951e-07 7.50000000e-07,2.37304406e-06 1.00000000e-06,9.99995000e-06 1.25000000e-06,3.05171125e-05 1.50000000e-06,7.59346168e-05 1.75000000e-06,1.64117391e-04 2.00000000e-06,3.19948805e-04 2.25000000e-06,5.76484160e-04 ... ... ... 1.60000000e-05,9.99972069e-01 1.62500000e-05,9.99988004e-01 1.65000000e-05,9.99995117e-01 1.67500000e-05,9.99998121e-01 1.70000000e-05,9.99999318e-01 1.72500000e-05,9.99999767e-01 1.75000000e-05,9.99999926e-01 1.77500000e-05,9.99999978e-01 2.00000000e-05,1.00000000e+00

"D(m)","CDF"
0.00000000e+00,0.00000000e+00
5.00000000e-07,3.12499951e-07
7.50000000e-07,2.37304406e-06
1.00000000e-06,9.99995000e-06
1.25000000e-06,3.05171125e-05
1.50000000e-06,7.59346168e-05
1.75000000e-06,1.64117391e-04
2.00000000e-06,3.19948805e-04
2.25000000e-06,5.76484160e-04
             ...
             ...		   
             ... 
1.60000000e-05,9.99972069e-01
1.62500000e-05,9.99988004e-01
1.65000000e-05,9.99995117e-01
1.67500000e-05,9.99998121e-01
1.70000000e-05,9.99999318e-01
1.72500000e-05,9.99999767e-01
1.75000000e-05,9.99999926e-01
1.77500000e-05,9.99999978e-01
2.00000000e-05,1.00000000e+00

The cumulative distribution function gives the fraction of the injector mass flow rate with a droplet diameter smaller than $D$ . Hence $F (D_{M}) = 0.5$ defines $D_{M}$ as the mass-median droplet diameter.

This definition provides three essential properties of a cumulative distribution function:

$\lim_{D \to 0} F (D) = 0$

$\lim_{D \to \infty} F (D) = 1$

$\frac{d F (D)}{d} \geq 0$

PDF Table

A PDF table has a column of droplet diameter and a column of PDF values,

(D, P (D))

. The droplet diameter must be in m and the PDF values must be strictly increasing or decreasing—duplicate values are not allowed.

Example:

PDF Table Plot of CDF Table

PDF Table	Plot of CDF Table
"D(m)","PDF" 0.00000000e+00,0.00000000e+00 2.50000000e-07,1.95312498e-01 5.00000000e-07,3.12499902e+00 7.50000000e-07,1.58202750e+01 1.00000000e-06,4.99995000e+01 1.25000000e-06,1.22066587e+02 1.50000000e-06,2.53105779e+02 1.75000000e-06,4.68868350e+02 2.00000000e-06,7.99744041e+02 ... ... ... 1.80000000e-05,3.26415271e-02 1.82500000e-05,8.94951042e-03 1.85000000e-05,2.27248660e-03 1.87500000e-05,5.32734238e-04 1.90000000e-05,1.14927268e-04 1.92500000e-05,2.27403021e-05 1.95000000e-05,4.11290355e-06 1.97500000e-05,6.77577482e-07 2.00000000e-05,1.01313324e-07

"D(m)","PDF"
0.00000000e+00,0.00000000e+00 
2.50000000e-07,1.95312498e-01 
5.00000000e-07,3.12499902e+00 
7.50000000e-07,1.58202750e+01 
1.00000000e-06,4.99995000e+01 
1.25000000e-06,1.22066587e+02 
1.50000000e-06,2.53105779e+02 
1.75000000e-06,4.68868350e+02 
2.00000000e-06,7.99744041e+02  
             ...
             ...		   
             ... 
1.80000000e-05,3.26415271e-02 
1.82500000e-05,8.94951042e-03 
1.85000000e-05,2.27248660e-03 
1.87500000e-05,5.32734238e-04 
1.90000000e-05,1.14927268e-04 
1.92500000e-05,2.27403021e-05 
1.95000000e-05,4.11290355e-06 
1.97500000e-05,6.77577482e-07 
2.00000000e-05,1.01313324e-07

The probability density function gives the fraction of the injector mass flow rate with a droplet diameter inside a certain diameter range. This fraction is calculated from the PDF as the integral over that range:

$P (D_{1} \leq D \leq D_{2}) = \int_{D_{1}}^{D_{2}} P (D) d D$

Hence, any probability density function satisfies the following conditions:

$\lim_{D \to 0} P (D) = 0$

$\lim_{D \to \infty} P (D) = 0$

$P (- \infty < D < \infty) = \int_{- \infty}^{\infty} P (D) d D = 1$

The mass-median droplet diameter $D_{M}$ is given by:

$\int_{- \infty}^{D_{M}} P (D) d D = \int_{D_{M}}^{\infty} P (D) d D = 0.5$

Simcenter STAR-CCM+ In-cylinder internally converts the PDF table into a CDF table. The probability density function is related to the cumulative distribution function $F (D)$ as:

$F (D_{0}) = P (D \leq D_{0}) {= \int}_{- \infty}^{D_{0}} P (D) d D$

Note

For the table conversion, Simcenter STAR-CCM+ In-cylinder uses trapezoidal rule integration. For this reason, the accuracy and consistency of the generated CDF table strongly depends on the discretization of the input PDF—the more data points you use to the describe the PDF, the smaller is the error of the trapezoidal rule applied to the data points.

Rosin-Rammler

The Rosin-Rammler distribution quantifies the particle size using the following cumulative distribution function:

Figure 6. EQUATION_DISPLAY

F (D) = 1 - \exp [- {(\frac{D}{D_{ref}})}^{q}]

(576)

with:

Figure 7. EQUATION_DISPLAY

D_{ref} = Γ (1 - \frac{1}{q}) \cdot SMD

(577)

Figure 8. EQUATION_DISPLAY

Γ (z) = \int_{0}^{\infty} D^{z - 1} e^{- D} dD

(578)

where:

$q$ is the specified Exponent.
$SMD$ is the specified Sauter Mean Diameter.

For information on the specified values, see Engine Reference—Injector Dialog.

Droplet Count

At each time-step, Simcenter STAR-CCM+ In-cylinder injects $5.0 E 6 \cdot Δt$ parcels per nozzle, where $Δt$ is the time-step size. The injection time for the parcels is chosen randomly within the time-step. The diameter of the droplets represented by a parcel is stochastically sampled from the specified distribution. The droplet count for each parcel is then adjusted to ensure that the parcel meets the given fuel mass for the nozzle at that time-step.