# Effect of Rotating Cylinder on Mixing Performance in a Cylindrical Double-Ribbon Mixer
## 2 Overview of Mixing Indices and Subdomain Mixing Index (SMI)  
Mixing quality plays an important role in all the major mixing industries, especially those related to pharmaceutical, chemical, and food manufacturing. This motivated several researchers worldwide to develop a wide range of mixing indices to evaluate the quality of mixing. Of these, the Lacey mixing index (LMI) [26; 27], generalized mean mixing index (GMMI) [28], and modified generalized mean mixing index (MGMMI) [29] are the most commonly used mixing indices. LMI is a sample-based mixing index that yields statistical analysis considering the standard deviation, variation, and the coefficient of variation. Sample-based mixing indices may not produce accurate results, as they purely depend on the sampling size and conditions. GMMI and MGMMI are not sample-based, and hence, these methods consider all the particles in a computational domain. The major issue of over-predicting the mixing degree (index higher value greater than 1) by GMMI was addressed by MGMMI, which can only have values between 0 and 1. However, a disadvantage of the MGMMI is that it overestimates the mixing value, even in segregated conditions. To circumvent the above-mentioned issues, a new non-sampling mixing index, called a subdomain-based mixing index (SMI), was introduced by [PERSON] et al. [5]. The advantages of the SMI over the other mentioned indices were illustrated in their study. According to their study, SMI can be explained as follows. When a computational domain \(Z\) is divided into \(n_{w}\),\(n_{h}\),and \(n_{d}\) elements for the width, height, and depth, respectively, then the total number of subdomains (\(Q\)) is given as  
\[Q=n_{w}\times n_{h}\times n_{d}. \tag{1}\]  
Assuming that the domain contains \(M\) types of particles, the number of particles of type \(k\) (\(1\leq k\leq M\)) in the _ith_ subdomain (\(S_{i}\)) is denoted as \(n_{ki}\). Hence the total number of particles of all types in \(S_{i}\) corresponds to \(\sum_{k=1}^{M}n_{ki}\). Thus, the total number of particles (\(N\)) for the entire domain \(Z\) is given as follows:  
\[N=\sum\
olimits_{i=1}^{Q}\sum\
olimits_{k=1}^{M}n_{ki}. \tag{2}\]  
If all the particle types have the same particle counts, then the fraction of \(k\)-type particles with respect to the maximum number of particles among the various types in subdomain \(S_{i}\) is given as  
\[P_{ki}=\frac{n_{ki}}{\max(n_{1i};n_{2i},n_{3i},\ldots n_{Mi})}\leq 1,1\leq k \leq M. \tag{3}\]  
If two or more types of particles possess the same maximum number of particles in a subdomain, only one of these types of particles is selected to determine the denominator in Equation (3). Finally,the SMI for \(S_{i}\) can simply be obtained from the arithmetic mean of the individual fractions of each particle type \(k\), with the exception of the fraction of the majority type, as follows:  
\[SMI(S_{i})=\frac{1}{M-1}(\sum\
olimits_{k=1}^{M}P_{ki}-1). \tag{4}\]  
If all the particle types have the same particle counts in the subdomain, then the value of \(SMI(S_{i})\) is 1. The \(SMI(S_{i})\) is 0 when only one type of particle exists in the subdomain. Cases where there are no particles in the subdomain are not considered, and \(SMI(S_{i})\) is not defined. Hence, the value of SMI is always bound between 0 (segregation condition) and 1 (perfectly mixed condition). SMI values close to 1 generally represent randomly mixed conditions.