# Online Scaleup Calculations for Pharmaceutical Formulations

## What are the scaleup factors considered during the technology transfer of pharmaceutical products?

There are various concepts and scaleup factors those are being utilized during the technology transfer of pharmaceutical products.

There are various aspects and scaleup calculation utilized for different unit operations while technology transfer process in pharmaceutical industry.

There are various mathematical considerations used for scale up during the technology transfer or during the commercialization phase to increase the product formulation batch size and sometime to reduce the batch size (scale down).

Usage of scientific approach and mathematical calculations for process scaleup or scale down will mitigate the risk of failure, regulatory compliance, and save lots of cost that may requires for trial batches. This science-based calculations will help determining robust and realistic parameters for pharmaceutical formulation scaleup or scale down.

For process scaleup or scale down, main consideration will always remain equipment size, shape, principle of working and finalization of associated parameters. According to the theory of process modelling, the processes could be considered similar when the are similar with respect to Geometrical Similarity, Kinematic Similarity, or Dynamic Similarity.

## What is the Geometrical Similarity, Kinematic Similarity, or Dynamic Similarity?

### Geometrical Similarity:

When we say that two systems are geometrically similar, it means that they have the same proportion of important linear measurements. For instance, if we take two cylindrical mixing vessels, we can consider them geometrically similar only if the ratio of their height and diameter is identical. This means that their shapes and sizes are similar, even if they are not exactly the same. Geometrical similarity is a crucial concept in engineering and other technical fields as it allows us to compare and evaluate different systems based on their physical characteristics.

### Kinematic Similarity:

If two systems are geometrically similar, we can call them kinematically similar only if the corresponding points in the systems have the same ratio of velocities. In simpler terms, this means that if we have two systems that are geometrically similar and their corresponding points have the same velocity ratio, we can say that these systems are kinematically similar. This concept of kinematic similarity is important in fields like physics and engineering, where it helps in comparing the movements of different systems based on their physical characteristics.

### Dynamic Similarity:

If two systems are kinematically similar, they can be called dynamically similar only if the corresponding points in the systems have the same ratio of forces. In other words, if we have two systems that are kinematically similar and their corresponding points have the same force ratio, we can say that these systems are dynamically similar. This concept of dynamic similarity is particularly relevant in fields like engineering and physics, where it helps in evaluating the behavior of different systems based on their physical characteristics. For example, in the case of wet granulation, dynamic similitude would imply that the flow patterns of the wet mass in the bowl are similar.

## 1. Scaleup Calculation for Rapid Mixture Granulator (RMG) or High Shear Mixer

### A. Geometrical Similarity factor for scaleup the Rapid Mixture Granulator (RMG) process (Constant Fill ratio or Height to Diameter Ratio (H/D Ratio) method)

In the Geometrical Similarity factor consideration, lets assume that the fill ration for bigger batch size as well as smaller batch size will remain same. To determine the batch size, we need to calculate the Height to Diameter Ratio (H/D Ratio) also called as Fill Ratio. Let’s calculate the (H/D Ratio) or Fill Ratio for scaleup the Rapid Mixture Granulator (RMG) process.

Formula for Fill ratio of Rapid Mixture Granulator (RMG):

It is the ratio of Bed height and RMG Diameter (H/D ratio). The formula for same is as follows:

Height/Diameter Ratio (Constant fill ratio) = Powder Bed Height (cm)/ (RMG bowl diameter (m) x 100)

To calculate the scale-up factor for RMG with Geometric Similarity, we need to calculate following first –

i. Total Volume of RMG = ¼ π D2

Where, D = Diameter of RMG bowl in meter [Available from equipment manual or qualification document] and π = 3.14

ii. Powder Bed height (cm) = (Dry mix volume (L) x 1000) / Total volume of RMG (L)

Powder Bed height (cm) = [Dry mix volume (L) X 1000] / [¼ π D2 X 10000]

Powder Bed height (cm) = [4 X Dry mix volume (L) X 1000] / [π D2 X 10000]

Here we are assuming that fill ratio for both the RMGs (Smaller size and Bigger Size) are same. Hence,

Height/Diameter Ratio (Constant fill ratio) of RMG1 = Height/Diameter Ratio (Constant fill ratio) of RMG1

Hence, Powder Bed Height for RMG 1 (cm)/ (RMG 1 bowl diameter (m) x 100) = Powder Bed Height for RMG 2 (cm)/ (RMG 2 bowl diameter (m) x 100)

Hence, {[4 X Dry mix volume (L) of RMG 1 X 1000] / [π D2 (of RMG 1) X 10000]} / {(RMG 1 bowl diameter (m) x 100)} = {[4 X Dry mix volume (L) of RMG 2 X 1000] / [π D2 (of RMG 2) X 10000]} / {(RMG 2 bowl diameter (m) x 100)}

Hence, [Dry mix volume (L) of RMG 1]/[ D2 of RMG 1 x RMG 1 bowl diameter (m)] = [Dry mix volume (L) of RMG 2]/[ D2 of RMG 2 x RMG 2 bowl diameter (m)]

Hence, formula for Batch size (Dry mix Volume) of product under scale up or scale down is

Dry mix volume (L) of RMG 2 =

[Dry mix volume (L) of RMG 1 x D2 of RMG 2 x RMG 2 bowl diameter (m)]/ [D2 of RMG 1 x RMG 1 bowl diameter (m)]

Note: bowl diameter details shall be taken from RMG equipment qualification.

B. Calculation of scale-up factor for RMG with Kinetic Similarity (Constant Impeller tip speed method):

To calculate the scale-up factor for RMG with Kinetic Similarity, the Impeller Tip Speed Ratio to be maintained Constant, of two RMGs of different volume capacity having similar %occupancy. This is a dimension less method for scaling up of high shear wet granulation process or Rapid Mixture Granulator (RMG) for Immediate Release formulations.

Constant impeller tip speed means same shear rate is applied in both the RMGs which is an alternative to constant power in relative swept volume.

The tip speed represents the linear velocity at the tip of the impeller. Tip speed is responsible for the movement of mass within the RMG bowl and is related to the shear forces imparted to the mass by the impeller.

Tip speed formula is, S = π ND/t

Where,

N= rpm of the impeller,

D = the diameter of the impeller,

t= the time (Kneading or Mixing)

Scale-up factor for RMG with Kinetic Similarity (Constant Impeller Tip Speed Ratio) is calculated as follows:

Here we are assuming that Tip Speed for both the RMGs (Smaller size and Bigger Size) should remain same. Hence,

Tip Speed of RMG1 = Tip Speed of RMG2

[π x N1x D1] / t1 = [π x N2 x D2] / t2

N2 = [N1x D1 x t2] / [t1 x D2]

### C. Calculation of scale-up factor for RMG with Dynamic Similarity (Constant Newton’s power number method and Impeller swept volume method):

(a) Newton’s power number to scale torque:

In this theory, the Newton’s power number is related to the drag force acting on a unit area of the impeller and the inertial stress.

Formula for the Newton’s power number is as follows:

NeP (Newton’s power number)= P/ (ρ R5N3)

Where,

P = Power consumption by the impeller blade = Torque* (2* π *N)  (Note: unit of Torque is N-m)

N = Impeller RPM

ρ = Wet mass density of granules (kg/L)

Therefore, NeP (Newton’s power number)= Torque* (2* π *N) / (ρ R5N3)

When we keep NeP (Newton’s power number) constant for both the processes (scale up and scale down), following shall be comparison:

[Torque (RMG 1) * (2* π *N1)] / [(ρ1 R15N13)] = [Torque (RMG 2) * (2* π *N2) ]/[ (ρ2 R25N23)]

Hence,

[Torque (RMG 1) ]/[ ρ1 R15N12] = [Torque (RMG 2) ]/[ ρ2 R25N22]

Therefore, N22 = [Torque (RMG 2) x ρ1 x R15 x N12] / [ρ2 x R25 x Torque (RMG 1)]

N2 = Sqrt ([Torque (RMG 2) x ρ1 x R15 x N12] / [ρ2 x R25 x Torque (RMG 1)])

N2 = Sqrt {[(T2 x D1 x (R1^5) x (N1^2)] / [D2 x (R2^5)  x T1)]}

(b) Relative Swept Volume (RSV):

In this theory, the parameter is calculated as the volume of mass swept by the blades of impeller per unit time against the bowl capacity.

Formula for the Relative Swept Volume (RSV) is as follows:

Relative Swept Volume (RSV): Volume of Material Swept per Second / Volume of the Bowl

Note: Here, Impeller swept volume = Impeller height (m)/ Bed height (cm)x 0.01)

Hence, Relative Swept Volume (RSV)=

Impeller height (m)/ (Bed height (cm)x 0.01 x Volume of the Bowl)

For scale up and scale down, the Relative Swept Volume (RSV) would be same for Original RMG as well as scale up and scale down batch size.

Therefore, Relative Swept Volume (RSV) for RMG 1 = Relative Swept Volume (RSV) for RMG 2

[Impeller height (m) of RMG 1]/[ (Bed height (cm) of RMG 1x 0.01 x Volume of the Bowl of RMG 1)]= [Impeller height (m) of RMG 2]/[ (Bed height (cm) of RMG 2x 0.01 x Volume of the Bowl of RMG 2)]

Bed height (cm) of RMG 2 = [Impeller height (m) of RMG 2x Bed height (cm) of RMG 1x Volume of the Bowl of RMG 1]/[Impeller height (m) of RMG 1 x Volume of the Bowl of RMG 2]

## 2. Scaleup Calculation for Blender (Dry Blending and Mixing)

There are currently no mathematical techniques to predict blending behaviour of granular components without prior experimental work. Therefore, blending studies start with a small-scale, try-it-and-see approach. The first portion of this chapter concerns the following typical problem: a 5-ft3 -capacity tumble blender filled to 50% of capacity and run at 15 rpm for 15 minutes produces the desired mixture homogeneity. What conditions should be used to duplicate these results in a 25-ft3 blender? The following questions might arise.

1. What rotation rate should be used?

2. Should the filling level be the same?

3. How long should the blender be operated?

4. Are variations to the blender geometry between scales acceptable?

Ad hoc approaches are unfortunately more common because there isn’t a widely recognized strategy for solving this issue. Rotation rates for typical commercially available equipment are frequently fixed, further complicating the situation and raising the possibility that genuine dynamic or kinematic scale-up may not be possible in such circumstances.

Reference: Pharmaceutical process Scale-up, edited by Michael Levin

### A. Geometrical Similarity factor for scaleup the Blender (Dry Blending and Mixing)

Given a geometrically similar blender and the same mixture composition, it would seem evident that the fill level should also be kept constant with changes in scale. However, an increase in vessel size at the same fill level may correlate to a significant fall in the relative volume of particles in the cascade layer compared to the bulk—this could follow a significant decrease in the mixing rate.

With 1-pint V-blenders, it has been demonstrated that operating at 40% fill results in a mixing rate that is roughly three times faster than at 60% fill. For geometrical similarity, the fill level should be maintained constant; however, if the depth of the fill changes, it may be challenging to match the mixing rate per revolution across scale changes.

### B. Dynamic Similarity for scaleup the Blender (Dry Blending and Mixing)

In laboratory and commercial scale operations, the optimization of mixing conditions such as mixing time and the rotational number is important for achieving stable tablet production.

The scale-up method of a rotating blender often uses a constant Froude (Fr) number a vessel tangential speed, or particle surface velocity.

The most commonly recommended scale-up method involves the use of the Froude number Fr=rn2/g;

where r (m) is the rotating radius of a blender,

n is the rotating speed (RPM),

g (m/s2) is the acceleration of gravity = 9.8 m/s2.

Fr is widely used in fluid mechanics to describe the balance between inertial and gravitational forces. Unfortunately, there is no such experimental data for the scale-up of V-type blenders.

Fr=[r * n2]/g

r  is the rotating radius in meter

n is the rotating speed (RPM)

g (m/s2) is the acceleration of gravity = 9.8 meter/second2

Our goal is to find a solution to scale up a 5-ft3 blender using Fr as the scaling parameter. To achieve this, we need to ensure geometric similarity, meaning that all angles and ratios of lengths are kept constant, and maintain the total number of revolutions. For the 25-ft3 blender to have geometric similarity with the 5-ft3 blender, it needs to be 71% larger or has a linear increase of 51/3. To maintain the same fill level, we must keep the Froude number constant, which means reducing the rpm by a factor of (1.71)-1/2 or 0.76 since R has increased by 71%. This corresponds to a speed of 11.5 rpm.

In practice, we select the closest speed available to 11.5 rpm since most blends are not sensitive to blend speed and available blenders are often of fixed speed. If we start blending at 15 rpm for 15 minutes on the smaller blender, we need to maintain the same total number of revolutions of 225 on the larger blender.

Therefore, we will blend for approximately 19.5 minutes at the selected speed of 11.5 rpm. Although this method is commonly used, it remains empirical and is not based on theory.

### c. Kinematic Similarity for scaleup the Blender (Dry Blending and Mixing)

Based on this theory, scaleup of the Blender can be done by maintaining a consistent number of revolutions (rpm × minutes)

Example:

To maintain the 150 revolution, 50 L blender can be run at 15 RPM for 10 minutes (15 RPM x 10 minutes) or 250 L blender can be run at 10 RPM for 15 minutes (10 RPM x 15 minutes)

To Keep Froude Number same for Demonstration to scale up, we have to change the variable as follows.

R2 = [(V1/3 x F headspace)1 x R1] / (V1/3 x F headspace)2

V is Blender Volume

F headspace can be calculated by 100-Fill Level (%)

R is the number of revolutions (Number of revolutions = blender speed x blending time)

Following is the example to find blender speed for 1500 L blender from 200 L Blender for mixing purposes.

V1 = Volume of small-scale blender = 200 L

V2 = Volume of large-scale blender = 1500 L

O1 = Occupancy in small-scale blender = 56%

O2 = Occupancy in large-scale blender = 60 %

Mixing time for smaller scale is 3 (T1) minutes at 10 RPM (RPM1), hence, total revolution for small scale = 10 x 3 = 30

Blender speed for larger scale is 5 RPM

To find the time required for rotation of blender at 5 RPM (RPM2) can be calculated as follows.

R2 = [(V1/3 x F headspace)1 x R1] / (V1/3 x F headspace)2

R2 = [(2001/3) x (100 – % Occupancy) x 30]/ [(15001/3) x (100 – % occupancy)]

R2 = [5.84 x (100 – 56) x 30] / [11.44 x (100 – 60)]

R2 = 16.85880751 ~ 17 Number of revolutions

Now, the time for mixing is

Number of revolutions (R2) = RPM2 x T2

17 = 5 x T2

T2 = 17/ 5

Time required for mixing (T2) = 3.4 min

## 3. Scaleup Calculation for Fluid Bed Equipment

For the lab and the manufacturing unit, the design and choice of the processor are crucial. The ratio of air volume per kg or litre of the product is extremely important to accomplish linear scale-up since air flow is one of the factors that determines the drying capacity of a fluid bed system.

The cross-sectional area of the product container and how it has been constructed across the range of sizes that a manufacturer offers are the other crucial design element.

The scale-up of the binder spray rate can be calculated using the relationship between different sizes of the process containers; if the cross-sectional area is designed linearly, then the scale-up of the spray rate can also be linear.

Three steps make up the fluid bed agglomeration process:

i. Dry mixing

ii. Spray agglomeration

iii. Drying to a desired moisture level.

These steps in the procedure are equally crucial. Yet, it is during the spraying step, where granules are continuously built up and the binder solvent is being evaporated, that the quality of the granules is actually determined. Controlling this humidity during scale-up is crucial since the granule size is directly related to the bed humidity during granulation.

Process-air temperature, the distance between the spray nozzle and the bed, the rate of binder addition, and the degree of atomization of the binder liquid were the processing variables that had the greatest impact on granule properties.

Two of the most crucial components of fluid bed granulation are the atomizing air pressure and the bed’s moisture. A finer droplet of the binder solution is produced by a higher atomizing air pressure. Thus, the atomizing air pressure will have an impact on granule growth, as previously discussed in this section.

Maintaining the same droplet size of the binder is a crucial aspect to take into account while scaling up a fluid bed granulation process to ensure success. The effects of the spray nozzle setup parameters and the air’s drying capability were supported by numerous research.

Make sure that adequate air is given to the nozzle tip while thinking about the atomizing air pressure. By positioning air pressure and volume monitoring equipment at the nozzle, this may be made sure of.

The results of the investigation also indicate that the final granulated particle size is influenced by the drying capacity of the process air.

The larger fluid bed’s granulation’s bulk density is roughly 20% higher than that of the smaller unit due to the larger unit’s higher rate of attrition when compared to the smaller unit. In order to minimise the creation of larger agglomerates, he also underlined the significance of maintaining the bed moisture level below a critical moisture level. It is necessary to maintain the drying capacity in the larger unit so that the bed temperature is comparable to the bed temperature of the smaller unit because the higher air flow and temperature (drying capacity) in a larger unit offer a higher evaporation rate.

To achieve the desired effects, this can be done by increasing the spray rate, air temperature, air flow, or a combination of these factors. The fluidization air velocity is maintained constant by increasing the air volume since the ratio of bed depth to the air distributor rises with equipment size.

In the past, scale-up was accomplished by choosing process parameters based on best guesses. Using factorial and modified factorial designs and search techniques is a current trend. The independent variables, such as process factors, and the dependent variables, such as product qualities, can be related mathematically using these statistically developed experimental plans. This strategy still needs a successful lab/pilot-scale development programme and a grasp of the factors influencing the characteristics of the final product.

In conclusion, the following processing conditions should be the same as they were in the pilot-scale experiments when scaling up.

i. The process air’s rate of fluidization via the system.

ii. The ratio of the granulation spray rate to the fluidization air volume’s drying capacity.

iii. The binder spray liquid’s droplet size

Based on the outcomes of the pilot-size unit’s operation, each of these quantities must be determined. In order to establish the process’s permissible operating range, investigations using pilot-size equipment should also be carried out in a variety of conditions.

### a. Scale up of the batch size using same line of equipment:

Scale up from laboratory sized fluid-bed machines can be made easier if the same line of equipment is to be used. Still, the efforts are required in this case because of differences in expansion chamber geometry, air flow pattern, gun spray pattern etc. For the top Spray equipment minimum and maximum batch size can be calculated using following equations.

Smallest Batch Size S(min) = [V x Omin x BD]

Maximum Batch Size S(max) = [V x Omax x BD]

Where;

S is batch size in kilograms

V is the product bowl working volume in liters

BD is the bulk density of finished granules in gm/cc

Omin = Minimum occupancy. Example use 0.3 value for 30% Occupancy in product bowl

Omax = Maximum occupancy. Example use 0.8 value for 80% Occupancy in product bowl

### b. Scale-up using Factor of Fluidization Air flow, Spray Rate and Atomization Air Pressure

(i) Scale-up calculation for Fluidization Air flow

Irrespective of batch size, to scale up the Fluid Bed Process, it is required to maintain the same fluidization velocity. In order calculate the correct air flow following calculation should be used.

AF2 = [AF1 x (A2/A1)]

AF2 (CFM) = AF2 / 1.699

Where;

AF 1 is Fluidization air flow in the laboratory scale equipment (CMH),

AF2 Fluidization air flow in the scaled-up equipment (CMH),

A1 is cross-sectional area of the laboratory scale equipment (Meter2),

A2 is cross-sectional area of the scaled-up equipment (Meter2)

Note: CMH – Cubic Meter per Hour.

(ii) Scale-up calculation for Spray Rate and Atomization Air Pressure:

Spray rate scale-up can be determined by the drying capacity of the equipment which is directly proportional to cross sectional area of the air distribution plate rather than by the increase in batch size.

At a given atomization pressure and air flow volume, change in liquid spray rate directly affects droplet size which in turn impacts particle agglomeration and may cause lumping.

Therefore, cross-sectional areas of the air distribution plate are used for approximation of scale up spray rate as per the following equation.

SR2 = [SR1 x (A2/A1)]

Where;

SR1 is spray rate in the laboratory scale equipment (gm/min),

SR2 is spray rate in the scaled-up equipment (gm/min),

A1 is cross-sectional area of the laboratory scale equipment (Meter2),

A2 is cross-sectional area of the scaled-up equipment (Meter2).

Note: Atomization air pressure shall remain same for laboratory scale and scaled-up process.

To maintain the same particle size, the “Multiple-headed nozzle” in scale up would spray at the same pilot-unit spray rate at a same atomization air pressure. However, this will result in a longer process time.

Therefore, alternate approach can be used to maintain a similar droplet size to achieve appropriate granule size with maintenance of the mass balance of spray rate and the atomization pressure. This can be achieved by increasing the atomization pressure to 3 times (in case of 3 Head Nozzle) and, the spray rate will also be increased to 3 times (in case of 3 Head Nozzle) keeping the same droplet size. This can obtain desired granulation characteristics.

SR2m= [SR1 x (A2/A1)] x N

Spray rate for each nozzle (SN)= SR2m/N

Where;

SR1 is spray rate in the laboratory scale equipment (gm/min),

SR2m is total spray rate in the scaled-up equipment (gm/min) for multiple heads, (Note: For each head nozzle from multiple heads, Spray rate shall be divided by N)

A1 is cross-sectional area of the laboratory scale equipment (Meter2),

A2 is cross-sectional area of the scaled-up equipment (Meter2).

N is number of Nozzle Heads.

Note: Atomization air pressure shall remain same for laboratory scale and scaled-up process for each head.

For example for laboratory scale equipment, Atomization air pressure is 2 bar for single head, for three head nozzle, for scaled-up process, each head will have 2 bar pressure. The total pressure will be 6 bar but each head nozzle will have 2 bar).

## 4. Scaleup Calculation for Compression Process

Scaling takes into account powder properties, tablet press design, and tooling features: To achieve good scale-up, it is necessary to consider it from the beginning of product development.

A compaction profile provides information about the performance of the powder when compressed. The most basic test is to compress the powder at various forces and then evaluate the tablet breaking strength of the resulting tablets. An alternate method is to normalise for punch tip pressure rather than merely compression force. A powder study employing a pressure range typical of the pharmaceutical sector of 50 MPa – 300 MPa provides for an easy comparison of tablets of various sizes using the same formulation.

A strong tablet will have a tensile strength of 1 to 2 MPa. Changing the compression technique to a rotating tablet press produces a different outcome.

The wet granulation formulation produces a tablet with increased tensile strength as compression force is applied. A strong tablet is generated when the compression force reaches 150 MPa. The direct compression mix will not compress into a tablet of appropriate strength and will instead produce a tablet exhibits capping behaviour at higher pressures..

Even though, a single station presses will produce tablets with adequate hardness, disintegration, and dissolution; but, switching to a large rotary tablet press with 40 to 70 tooling stations drastically affects the dynamics of the powder compaction process due to variations in overall compaction speed. To accommodate tablet demand, larger tablet presses are used, which increases the pitch circle diameter of the turret, the consequent punch vertical velocity, and the compression dwell duration.

The compression force is fixed and the tablet press is run at different rates to get the correct data.

The dwell time for compression can be calculated using the velocity of the tablet press pitch circle and the size of the compression tool head flat. The dwell period is defined as the duration when the upper and lower punches do not move vertically.

The most important factor to consider is contact time, which comprises consolidation time, dwell time, and decompression time. Consolidation time is the time when the punches change vertical position, shortening the distance between their tips; dwell time is the time when the punches achieve maximum penetration in the dies under roll: this means that the punches are not moving vertically and the head flat is completely in contact with the roll. This is one of the most important topics to address while upscaling since it affects the ability of powder to form liquid bonds, which is strongly dependent on viscoelastic properties and affects final tablet strength.

Finally, decompression or relaxation period includes punches increasing the space between tips before losing contact with rolls.

In general, ejection has no detrimental impact on scale-up since decreased punch movement reduces strain as pitch diameter increases.

### a. Determination Compression machine speed based on Dwell Time

For the scaleup of compression machine, one of the methods is determination Compression machine speed based on Dwell Time.

DT = [PHF x 60000] / [π x PCD x N]

N = Number of rotations per minute of turret

π = 3.14

PCD = Pitch circle Diameter of Turret (mm)

DT (msec) = Dwell time in milliseconds

### b. Determination Compression machine speed based on Dwell Time

For the scaleup of the compression machine keeping the same Dwell Time of original compression machine and proposed compression machine, the formula can be written as follows:

[PHF1 x 60000] / [π x PCD1 x N1] = [PHF2 x 60000] / [π x PCD2 x N2] [PHF1] / [PCD1 x N1] = [PHF2] / [PCD2 x N2]

N2 = [PHF2 x PCD1 x N1] / [PCD2 x PHF1]

## 5. Scaleup Calculation for Coating Process

The tablet size must be kept constant during the scale-up operation. The spray rate per spray gun can be kept constant if the distance between the guns is constant on different scales. This can be done to produce similar microscopic coating properties. This technique will only work if scale-up is performed by elongating the pan in order to retain geometric resemblance, which is not practicable given the pan depths required to maintain this type of similarity.

Spray rate, airflow, inlet air temperature, dew point, atomization air, pattern air, pan load, and pan speed are the macroscopic characteristics that must be determined during scale-up.

Geometric, dynamic, and kinematic similarities are all critical to effective scale-up.

Keeping geometrically similar systems increases the likelihood of achieving dynamic and kinematic similarities. In the case of a pan coater, both the pan and the spray related parameters must be kept “identical” across the scales.

### a. Geometric Similarity

Geometric similarity refers to the shape and size of the pan coater being proportional across different scales, which can be achieved by using systems with similar aspect ratios (ratio of pan length to diameter).

During the User Requirement Specifications (URS) should be prepared specifying the precise specification with aspect ratio of pans remains constant among scales. To achieve equivalent mixing, the height, width, and form of passive baffles should be proportional across scales.

The geometric resemblance can be increased further by maintaining a steady pan load to pan volume ratio. This will keep the h/D ratio constant, where h is the closest distance from the pan’s centre point to the bed surface (h can be considered as the characteristic length of the system from a tablet handling perspective) and D is the pan diameter.

Pan Load /Pan Volume = constant

The equation above can be used to calculate the pan load that will be employed in the larger pan during scale-up. The pan volume in the above equation is the pan’s brim volume, which is normally supplied by the equipment manufacturer.

### b. Dynamic Similarity

Dynamic similarity ensures that the force ratio at corresponding points in the pan coater is consistent across scales. The inertial and gravitational forces are the two main forces in a revolving pan. A balance between these two forces is required to create acceptable tablet motion. The ratio of inertial to gravitational forces (Froude, Fr) is a dimensionless quantity that governs the system’s dynamics. Fr number scaling provides a more fundamental logic and is thus used here.

Fr = ω2D / g = constant

where ω is the pan speed.

Given the small-scale conditions, the above equation can be used to anticipate the pan speed for the larger pan to produce similar dynamics.

### c. Kinematic Similarity

Kinematic similarity ensures that the velocity ratio (kinetics) at corresponding points in the pan is consistent across scales. To do this, the tablet velocity and spray dynamics should be kept constant. Tablet velocity is known to change as a function of length along the cascading bed.

As a result, it is suggested that the location of the spray along the cascading length of the bed be kept consistent. Spray kinetics are determined by the droplet size that coming out from the spray gun, which is determined by atomizing air, pattern air, spray rate, nozzle type and size, and solution properties.

The drying of the droplet before it reaches the tablet is affected by intake airflow, inlet air temperature, and the distance between the gun and the bed. To produce identical morphology, the droplet size of the spray hitting the tablet should remain constant as long as the tablets move at the same velocity across scales.

If τdry is characteristic drying time for the droplet and τsurface is the time the tablet spends on the bed surface, then for scale-up.

τdry / τsurface = constant

The amount of time spent on the bed surface will be determined by tablet velocity and pan size. Heat and mass transport correlations for droplet drying can be used to calculate the characteristic drying time. To accomplish so, the relationship between droplet size and spray rate, inlet airflow, atomizing pressure, fluid characteristics, and nozzle type must be understood.

i. Airflow Calculation

Taking into account heat loss from the pan to the surroundings and assuming that the outlet air temperature equals the tablet bed temperature, it can be demonstrated that if the airflow-to-spray rate ratio (drying capacity) and inlet temperature are held constant, the exhaust temperature will also remain constant.

This data can be utilised to anticipate inlet airflow during scaleup while keeping overall drying capacity constant.

Drying Capacity = Air flow/ Spray Rate = constant

ii. Spray Rate Calculation

The likelihood of a tablet being in the spray zone is given by p=n/N, where n is the total number of tablets in the pan and N is the number of tablets in the spray zone at any one time. By scaling up to a larger coater, both pan load and spray area increase, but the increase in N is many folds bigger than that of n.

As a result, the total likelihood of a tablet being in the spray zone falls with a larger coater. This probability is also inversely proportional to the amount of time it takes for a pill to return in the spray zone.

As a result, the smaller the probability, the longer it takes for the tablet to return in the spray location. The time between spray zone passes corresponds to the drying period for individual tablets, albeit the majority of drying occurs when the tablet is at the surface.

The circulation time on a tablet is defined as the period between two successive coating events. As a result, for the larger pan, the circulation time is longer and p is lower. As a result, tablets in a larger coater have longer drying time.

Because the tablet that takes longer to reappear in the spray zone has more time to dry before being sprayed on again, it can be sprayed with more solution each time it passes through. In an equation form,

(SR)(n)/N = Constant

Where SR indicates spray rate

The equation above can be used to predict the spray rate because it is simple to determine n and N for a specific system. This equation can also be used to forecast the consequences and adjustments necessary as a result of modifications to the tablet size, which will show up as a change in the values of n and N. For a given spray area, n 1/d2 tablet, whereas for a given pan load and tablet shape, N∝1=d3.

p = n / N ∝d

Where d indicates the tablet size.

The maximisation of the spray rate in order to reduce batch processing time should also be emphasised as one of the bigger scale goals. The gun-to-bed distance, atomizing air, or both must be raised when spray rate may be increased while still maintaining similarity of the spray zone at the microscopic scale.

To achieve complete spray coverage across the pan depth when these parameters vary, the gun spacing might need to be altered. When the parameters of the coating solution to be changed, spreading (wettability) and drying rate must be matched for successful scale-up.

The Weber number (ratio of kinetic energy to surface energy) and Reynolds number (ratio of inertial to viscous force) are the two commonly used factors. Any change in the surface tension or solution viscosity can be explained by these two parameters taken together. Therefore, understanding the droplet size distribution is crucial in order to effectively employ these variables.

iii. Coating Time Calculation

The following calculation can be used to determine the coating time in order to produce the same weight gain per pill.

tcoat x SR/ Pan Load = Constant

Where SR indicates spray rate.

The average number of coating events per tablet, which remains constant throughout scales, is a crucial consideration for scale-up. By maintaining a steady average for passes made with the spray gun (Nc), the average number of coatings per tablet can be maintained. The equation below can be used to determine Nc.

Nc = VLJt/ aN

Where, V is the velocity of tablet in the spray zone, L is the length of spray zone, J is the number of spray guns, a is the projected area of the tablet, and t is the total coating time.

Keeping the average number of coating events per tablet constant will allow maintaining the coat weight uniformity constant. Although this approach was not used as a starting point in the current work, the average number of passes under the gun was evaluated for the conducted experiments.

The consistency of the coat weight can be maintained by maintaining the average number of coating events per tablet.

### d. Calculations for scale-up/ down the coating parameters

The following calculations can be used to evaluate the scale the coating parameters:

Pan Load/ Pan Brim Volume = Constant

Pan Load (Original)/ Pan Brim Volume (Original) = Pan Load (Proposed)/ Pan Brim Volume (Proposed)

Pan Load (Proposed) = [Pan Load (Original) x Pan Brim Volume (Proposed)]/ Pan Brim Volume (Original)

ii. Calculation of Coating Pan speed (ω)

Fr = ω2D / g = constant (D is pan diameter)

ω(Original)2 D(Original) = ω(Proposed)2 D(Proposed)

ω(Proposed)2 = [ω(Original)2 D(Original)]/ D(Proposed)

ω(Proposed) = ([ω(Original)2 D(Original)]/ D(Proposed))1/2

iii. Calculation of Total Spray Rate for Coating

(SR)(n)/N = Constant

n is the number of tablets in the spray zone at any instant

N is the total number of tablets in the pan.

SR(Original) * n (Original) / N (Original) = SR(Proposed) * n (Proposed) / N (Proposed)

SR(Proposed) = SR(Original) * (n (Original) / n (Proposed)) * (N (Proposed)/ N (Original))

iv. Calculation of Inlet airflow for Coating

Drying Capacity = Air flow/ Spray Rate = constant

Air flow (Original) / Spray Rate (Original) = Air flow (Proposed) / Spray Rate (Proposed)

Air flow (Proposed) = Air flow (Original) * Spray Rate (Proposed) / Spray Rate (Original)

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