### 5.8.1. Model Derivation

#### 5.8.1.1. Population Balance Equation

The starting point for the MUSIG model is the population balance equation. Let represent the number density of particles of mass at time . Then the population balance equation is:

| (5–137)

where , , , and respectively represent the birth rate due to breakup of larger particles, the death rate due to breakup into smaller particles, the birth rate due to coalescence of smaller particles, and the death rate due to coalescence with other particles. These rates may further be expressed as:

| (5–138)

| (5–139)

| (5–140)

| (5–141)

where represents the specific breakup rate (the rate at which particles of mass break into particles of mass and ) and represents the specific coalescence rate (the rate at which particles of mass coalesce with particles of mass to form particles of mass .

#### 5.8.1.2. Size Fraction Equations

The next step of the MUSIG model is to discretize Equation 5–137 into size groups, or bins. Let represent the number density of size group :

| (5–142)

Also define the mass and volume fraction of size group to be and , respectively, and recognize that . Now integrate Equation 5–137 over the bin size dimension and multiply by to give:

| (5–143)

or:

| (5–144)

Defining the size fraction , this equation may also be written as:

| (5–145)

which is the size fraction equation used by the MUSIG model. A further simplification is to assume that all size groups share the same density and velocity yielding the homogeneous MUSIG model:

| (5–146)

#### 5.8.1.3. Source Terms

The contribution of the birth rate due to breakup of larger particles to the source term in Equation 5–146 is:

| (5–147)

Similarly, the contribution of the death rate due to breakup into smaller particles is:

| (5–148)

Note that the total source due to breakup is zero when summed over all size groups:

| (5–149)

For the discretized coalescence sources, you must define the coalescence mass matrix as the fraction of mass due to coalescence between groups at , which goes into group :

| (5–150)

The contribution of the birth rate due to coalescence of smaller particles to the source term in Equation 5–146 is:

| (5–151)

Similarly, the contribution of the death rate due to coalescence into larger groups is:

| (5–152)

Note that this formulation for the coalescence source terms guarantees that the total source to coalescence is zero when summed over all size groups:

| (5–153)

This follows from the requirement that together with the following property of the mass matrix for all :

| (5–154)

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