10.1.1 Rate Processes in API Unit Operations

Table 10.1 lists the rate processes involved in common unit operations in API synthesis. Characterization of each rate process is often feasible; the apparent complexity of unit operations arises from the combination of several rate processes in a single operation. In many cases, one rate process is limiting and dominates the others.

Other chapters in this book also address several of these rate processes and operations, including “REACTION KINETICS AND CHARACTERIZATION,” “ANALYTICAL METHODS FOR MASS BALANCE AND REACTION KINETIC ANALYSES,” “DESIGN OF DISTILLATION AND EXTRACTION OPERATIONS,” “DESIGN AND SCALE‐UP OF PHARMACEUTICAL CRYSTALLIZATIONS,” “DESIGN OF FILTRATION AND DRYING OPERATIONS,” “KILO LAB AND PILOT PLANT MANUFACTURING,” “QUALITY BY DESIGN APPROACHES FOR API,” and “QbD CASE STUDY: API CASE STUDY.”

10.1.2 Scale Dependence and Scale Independence

The intrinsic rate of a chemical reaction is independent of scale; in other words, if the reaction could be conducted without limitation by any other rate process, it would run at the same rate (per unit volume) at all scales; the reaction time, starting from the same initial composition and held at the same temperature, would be the same at all scales. This is true for reactions whose kinetics are slow compared with other rate processes. The choice of solvent, reagents, and/or catalyst combination for reactions has been largely the domain of development chemists, and these variables are taken here to be fixed already; in any case, those effects, including reagent solubility, are also independent of scale.

All other rate processes listed in Table 10.1 are scale dependent. For example, the rate of heat transfer depends strongly on the ratio of surface area to volume, which reduces on scale‐up. This means that heat removal on scale will be slower than in the laboratory, unless specific measures are taken to provide additional surface area or an increased temperature driving force. Therefore the time required to heat or cool a batch of material tends to increase on scale. Similarly, the rate of mass (or phase) transfer is scale dependent; the agitation conditions inside the vessel determine the time required to reach equilibrium between the phases; it is easier to make this time short in the lab than on scale.

The rates of addition and removal of material also change with scale, normally taking longer at larger scales. For example, the rate of disengagement of gas evolved from a reaction depends again on the surface‐area‐to‐volume ratio, so reactions such as decarboxylation have to be conducted more slowly on scale to avoid partial loss of the reactor contents due to swelling of the batch.

The impact of scale dependence is to lengthen processing times on scale, reducing productivity somewhat but more importantly potentially allowing additional phenomena to occur that were not observed to the same extent in the laboratory, such as impurity‐forming reactions, product degradation, catalyst poisoning or deactivation, unwanted crystal forms, nucleation, agglomeration, or breakage. Fortunately, each of the rate processes that contribute to such problems can be characterized or estimated using classical chemical engineering methods, allowing scale‐up problems to be anticipated and resolved in advance or at least resolved quickly once they occur.

10.1.3 Scale‐Up and Achievement of Similarity

Unit operations need to produce similar results and a similar quality product at each scale; historically the pharmaceutical industry has demonstrated this through process validation at each facility and operation thereafter with rigid limits on process parameters. Recent regulatory guidance [1] encourages adoption of “quality by design” (QbD) and development of a “design space” in which the process may operate, with flexibility to move operating conditions around in this space without needing to obtain additional regulatory approval. The design space is proposed by the applicant and is intended to be applicable to the process at any scale. This implies a greater investment in developing process understanding during the development phase, balanced by flexibility, a reduced regulatory burden, and opportunities for continuous improvement once the process is in full‐scale manufacturing.

There is some debate about how best to apply these principles while also satisfying business, process safety, and environmental requirements, and the underlying benefit of improved process understanding is underlined in this chapter. The operating window in which acceptable or similar results are obtained can be determined, demonstrated, and justified in terms of the chemical engineering rate processes involved in unit operations. For example, the design space for a chemical reaction whose intrinsic kinetics have been shown to be slow (compared with other rate processes involved) can be demonstrated to be scale independent, giving a flexible operating window that can be used at any scale. On the other hand, where specific equipment performance requirements exist, these can be framed in terms of the chemical engineering “rate constants” or time constants required by the process.

In the next section, we will see that these rate constants arise naturally in the chemical engineering rate equations, which provide a first principles basis for defining a scale‐independent design space. The application of this approach is illustrated for several common types of chemical reaction, and the same general principles and methods apply to other API synthesis operations. Examining the rate equations also leads to familiar scale‐up rules.

10.1.4 Characterization of Reaction Systems

Guidance specific to certain classes of reaction is given below, but to avoid repetition for each class, the following general guidelines on reaction characterization are provided here. Guidelines on equipment characterization are given toward the end of the chapter.

Assuming that the solvent and reagents/catalyst have been selected, reactions may be characterized by a sequence of initial screening experiments to identify scale‐dependent physical rate process limitations, followed by a more detailed kinetics study if necessary.

In all experiments aimed at characterizing a reaction, progress should be followed by either taking multiple samples or using an in situ analytical technique, not by relying on a single endpoint sample. If composition is analyzed using HPLC, determination of response factors will become more important as characterization proceeds. Other possibilities include infrared (IR), ultraviolet (UV), and Raman spectroscopy or heat flow (Q_r).

Samples should be analyzed to indicate both reaction progress (e.g. product level, conversion, or current yield) and product quality (e.g. impurity level). If either of these variables is sensitive to physical rates (e.g. agitation), the reaction is not limited by chemical kinetics, at least under typical processing conditions. There may be some scope to reduce the rates of the chemical reactions (relative to mixing) by operating at, e.g. lower concentrations and/or temperatures; otherwise, if the process persists in its current form, the focus of characterization should be the physical capabilities of lab and larger‐scale equipment (see Section 10.6).

Under conditions where there is no effect of physical rate processes, a kinetics study may follow in which temperature, equivalents, and/or other case‐specific variables affecting the scale‐independent chemical rates are varied experimentally. The choice of conditions and the sampling program for these experiments should be informed by the results of the previous experiments and an initial kinetic model [2], and in any event, multiple samples should be taken again to capture when the reactions are occurring as well as the final result.

Where possible, the parameters in Eq. (10.3) such as rate constant and activation energy should be fitted using a kinetic model to reliable results of experiments. Model development in this manner tends to be iterative, and the proposed reaction scheme may change a number of times before the model fits the data to an acceptable degree. Chemical knowledge that certain intermediate species is unlikely to exist for long may allow the rate constants of certain reactions to be set arbitrarily high so that the remaining parameters can be fitted with greater confidence. Model development should take place at the same time as experimentation and the design of remaining experiments may change as a result of indications from the model.

It is difficult to overemphasize the importance of conducting a few well‐designed, well‐monitored experiments versus a large set of experiments with scant data recording. Isothermal experiments are preferred, though not absolutely necessary as long as the temperature versus time is recorded during the heat‐up and the location of “time zero” is clear. HPLC remains the most powerful method to follow product and impurity levels, and it is preferable to convert directly from HPLC area to moles rather than HPLC area percent. When starting materials are sufficiently pure, enough peaks are captured by HPLC, several response factors are known, and (preferably) an internal standard is also used, assumptions about the reaction scheme proposal can be checked directly using HPLC area. This can reveal the existence of undetected intermediates or by‐products and allow the reaction scheme to be revised accordingly. An excellent illustration of the procedure is available [3].

At the end of this phase, such a model may be used to find optimum conditions inside or outside the experimental region and to assist with definition of a scale‐independent design space. Some excellent examples of the success of this approach are available [4–8]. Further experimentation should be focused on model verification, especially at the most forcing conditions over which it may be used.

10.2.1 Rate Equations

If, for example, the reaction involved is simply A + B → P, the rate equations for this system may be written as follows. The rate of change of concentration of A or B is

(10.1)

where the rate expression on the right‐hand side is of this or similar form:

(10.2)

and the kinetic rate constant follows the Arrhenius relationship

(10.3)

More generally than (Eq. 10.1), and when several reactions are occurring in the same mixture, the rate of change of concentration of any species in the mixture is given by

(10.4)

From Eqs. (10.1) to (10.4), the rate constant depends on temperature, and the rates of reaction depend on concentrations. This is the chemical engineering basis for the classical approach by chemists to study temperature and “equivalents” as two of the primary variables affecting the outcome of a reaction. In the above example, with 1 mol of A reacting with 1 mol of B, 1 equiv. of A would mean 1 mol of A/mol of B, 2 equiv. of A would be 2 mol of A/mol of B, etc.

Integrating Eq. (10.4) in time, the final outcome (composition) of reaction will depend on the concentrations and temperature profiles followed during reaction and the time allowed for reaction. Temperature is often (but not always) held constant or nearly constant, in which case final concentrations at completion depend only on initial concentrations and initial temperature. Example Problem 10.4 follows this behavior.

Temperature is an important influence and can change the relative rates of the reactions, depending on their activation energies. From a safety standpoint, temperature must be controlled to avoid thermal runaway, and the batch reactor is not ideal for this purpose. The rate of change of reactor temperature is given by

(10.5)

In which the log mean temperature difference is

(10.6)

The approximation on the right of Eq. (10.6) applies when the heat transfer fluid flow rate is high, i.e. the heat transfer fluid temperature is almost constant between the jacket inlet and outlet.

The rate of heat evolution summed over all chemical reactions in the mixture is

(10.7)

Note that heat of reaction in Eq. (10.7) is the heat released per mole of reactant consumed when the stoichiometric coefficient of that reactant is 1 and is negative when the reaction is exothermic. Equation (10.5) shows that temperature can be controlled by balancing scale‐dependent heat removal with scale‐independent chemical kinetics. The former can be enhanced by increasing the surface area and/or increasing the temperature driving force, normally reducing the jacket inlet temperature. These measures can allow the reaction to run at the same temperature and concentration on scale, in the same reaction time as in the laboratory. There is an upper limit on the scale at which this will be feasible. On the other hand, reaction rate can be reduced to balance reduced heat removal by operating at lower concentration or lower temperature; the former reduces productivity; the latter could change the balance between desired and undesired reactions, lengthens reaction times, and also reduces the available temperature driving force for cooling.

For the common case where the temperature is held constant during reaction and the initial concentrations are the same at each scale, Eq. (10.5) with the left‐hand side set to zero leads to the scale‐up basis

(10.8)

Equations (10.4) and (10.8) taken together indicate that for scaling batch homogeneous reactions that are limited by chemical kinetics, a scale‐independent design space can be expressed using temperature and equivalents as factors, as long as there is sufficient assurance that adequate heat transfer to maintain temperature control will be available at all scales. The latter condition amounts to a statement about the equipment capability at each location, specifically the heat removal capacity (in W/l) of the equipment operating at the required temperature.

All of the above effects can be calculated using mechanistic models in which Eqs. (10.2)–(10.7) are solved by integration in time. In practice, batch homogeneous reactions may not be allowed to run to completion, because stopping the reaction earlier produces higher yield (yield peaks during the batch) and less impurity than waiting until the end. Similarly, batch homogeneous reactions may be run using temperature profiles, e.g. charge at low temperature, heat to reaction temperature, then hold, and possibly heat again. In these cases, the temperature profile followed by the reaction may be scale dependent for the reasons given above, with, e.g. longer heating ramps on scale; this will impact the relative rates of the reactions and could affect quality as well as rate.

If Eq. (10.3) is known for each reaction, these effects can be predicted easily, and operating conditions can be optimized to a high degree as shown in Example Problem 10.4; good examples of this approach covering a variety of reaction types are available [9–13].

10.2.2 Characterization Tests for Batch Homogeneous Reactions

The general guidelines described in Introduction should be followed with regard to experimental design.

The reaction should first be run in 2–3 otherwise identical (replicate) experiments in which stirrer speed is varied between 200 and 1000 rpm; four or more samples should be taken, biased toward the beginning of the reaction, e.g. after 10, 30, 60 minutes, and at the end.

Under conditions where there is no effect of physical rate processes, a kinetics study may follow in which temperature and equivalents are varied experimentally and the parameters in Eq. (10.3) fitted.

If agitation conditions affected the result of the initial screening experiments, the result of the reaction is influenced by the rate of mixing or more specifically the rate of bulk blending of the vessel contents, known as “macromixing.” This is unusual for a batch homogeneous reaction, but if it occurs, the scale‐dependent macromixing time constant will be important as the outcome of reaction is influenced by this characteristic and not just chemical kinetics; see the section below on equipment characterization. A reaction such as this might be better run in fed‐batch mode, with the fed reagent maintained at a low concentration. Alternatively, there may be some scope to reduce the rates of the chemical reactions (relative to mixing) by operating at lower concentrations and/or temperatures.

10.2.3 Achieving Similarity on Scale‐Up

Achieving similarity on scale‐up relies on having characterized both the reaction and the intended scale‐up facility. Equations (10.2)–(10.7) provide the basis for a scale‐independent design space.

When kinetics are rate limiting, other than agitation that is necessary to promote heat transfer, no particular additional agitation requirement exists for homogeneous reactions. As noted above, there may be an optimum temperature profile to adopt during reaction and/or an optimum time at which to stop the reaction. These may be determined from a kinetic model, as in Example Problem 10.4. A scale‐independent design space may include factors such as equivalents, temperature, and reaction time.

When mixing is rate limiting, unusual for batch homogeneous reactions, agitation influences the outcome, and similar macromixing time constants will be required at each scale; if the process persists in this form, the macromixing time constant should be a factor in design space definition.

10.3.1 Rate Equations

Although relatively complex at first sight, this multiphase, chemically reacting, time‐dependent problem can be broken into classical chemical engineering elements and rate processes just like the batch homogeneous reaction above. The rate equation for chemical species concentration in the liquid phase is given by combination of chemical kinetics with the film theory of scale‐dependent mass transfer:

(10.9)

Only the equations for components that transfer between phases contain the second term; for dissolving gases, solubility is given by Henry's law:

(10.10)

In other heterogeneous reactions, the solubility expression differs; for liquid–liquid (aqueous–organic) systems,

(10.11)

where S_i is the partition coefficient for component i between the phases. For solid–liquid systems,

(10.12)

where f() is a function determined from phase equilibrium fundamentals and/or by experimental measurement (e.g. using NRTL equations or similar).

When the transferring component is a solvent (e.g. in a reactive distillation or solvent swap), the equilibrium is described using the Antoine or similar vapor pressure equation [14].

Examination of Eq. (10.9) indicates that when chemical kinetics are slow relative to mass transfer, the concentration of hydrogen (or any dissolving, reacting solute) will tend to saturation with modest k_La values. On the other hand, if in Eq. (10.9), the kinetics of reactions consuming hydrogen are rapid compared to mass transfer, the concentration of hydrogen will tend toward zero during reaction. These two extremes of behavior change the outcome of reaction; slow mass transfer extends reaction time, may lead to increased impurities, and can cause reaction to stall. The outcome of a batch multiphase reaction at constant temperature therefore depends on temperature, concentration, equivalents (including catalyst loading), pressure (if the solute is in the gas phase), and potentially k_La.

In reactions with three phases present, such as the additional solid catalyst phase in hydrogenation, that phase should be dispersed, and its surface area made available for mass transfer in a similar way to the gas phase in the present example. Complete suspension of a solid phase is usually sufficient to prevent mass transfer to or from that phase being rate limiting; when the solid particles have the same characteristics at both scales, complete suspension is approximately equivalent to maintaining constant k_La for those particles (see the Section 10.6).

For multiphase reactions, the rate equation for solution phase temperature is

(10.13)

Equation (10.13) contains one additional term compared with Eq. (10.5) to account for any heat effect associated with mass transfer of dissolving material between phases; this effect is often negligible but can be significant when solvent is vaporized (in distillation) or when a large quantity of solute is transferred quickly (e.g. when crystals come out of solution suddenly). The heat flow signature from the chemical reactions will be determined by kinetics when mass transfer is rapid and by mass transfer when kinetics are rapid. Neglecting the final term in Eq. (10.13), the scale‐up relationship for maintaining constant temperature when kinetics limit is identical to the corresponding equation for homogeneous reactions:

(10.14)

Equation (10.14) when applied to pressure reactions assumes constant pressure, i.e. constant solubility. The scale‐up relationship when kinetics are rapid relative to mass transfer can be obtained by noting that in this case, the rate of dissolution of the limiting solute dominates the rates of the main reactions:

(10.15)

Substituting Eq. (10.15) into Eq. (10.13) leads to

(10.16)

The above result may also be obtained by writing Eq. (10.13) in dimensionless form using t^* = k_La·t and T^* = T/T₀, neglecting the last term and substituting Eq. (10.15) for Q_r:

(10.17)

The second term on the right‐hand side is constant when pressure and initial temperature are fixed, and when the left‐hand side is set equal to zero (constant temperature), Eq. (10.16) is obtained.

Equations (10.16) and (10.17) indicate that when the solution concentration of solute is close to zero, the required rate of heat removal is directly proportional to the rate of mass transfer, i.e. higher k_La implies the need for greater heat removal.

An additional constraint exists for pressure reactions, such as hydrogenation, which cannot be taken for granted in practice. The ability to maintain a desired or constant pressure in the headspace depends on the balance between supply of fresh gas to the headspace and removal of gas by mass transfer:

(10.18)

Equation (10.18) shows that rapid mass transfer may cause the headspace pressure to reduce, while a temperature rise may cause it to increase. To maintain a constant pressure on scale‐up requires adequate temperature control and adequate gas supply:

(10.19)

The right‐hand side of Eq. (10.19) requires knowledge of the solution concentration of the dissolved gas. A conservative estimate may be made by setting the dissolved concentration to zero, leading to

(10.20)

A crude estimate of the required flow rate to maintain constant pressure may also be made from the number of moles of gas required for reaction and the intended reaction time, i.e.

(10.21)

Similar to homogeneous batch reactions, Eqs. (10.9)–(10.13) and (10.18) can be easily incorporated into dynamic mechanistic models for multiphase reactions and the unknown parameters (e.g. rate constants, activation energies) regressed against experimental data. The effects of changing concentrations and equivalents (including catalyst loading), temperatures, and pressures and the effects of mass transfer, heat transfer, and gas supply limitations can then be predicted. In many hydrogenations, for example, neither pressure nor temperature is maintained constant, and the profiles they follow may be scale dependent; even the sequencing of nitrogen and hydrogen purges before reaction can affect the result. The impact of these changes can be predicted using classical chemical engineering rate equations with appropriate reaction and equipment characterization. Example Problems 10.1 and 10.2 illustrate this approach.

10.3.2 Characterization Tests for Multiphase Reactions

The general guidelines described in Introduction should be followed with regard to experimental design.

For gas–liquid, gas–liquid–solid, liquid–liquid, and liquid–liquid–solid systems, the reaction should first be run in 2–3 otherwise identical (replicate) experiments in which stirrer speed is varied between 200 and 1000 rpm; 4 or more samples of the reacting phase should be taken during reaction, biased toward the beginning of the reaction, e.g. after 10, 30, 60 minutes, and at the end. With some reactions, additional profiling may be possible by following variables such as hydrogen uptake, pressure, and both pot and jacket temperature. The purpose of changing stirrer speed is to change the mass transfer contact area between the phases. In each of these experiments, any solids present should be well suspended.

To check for mass transfer limitation due to dissolving solids, experiments should be run at an impeller speed, which guarantees suspension, but either the particle size or the mass of solid reagent should be varied.

Under conditions where there is no effect of physical rate processes, a kinetics study may follow in which any of temperature, equivalents, pressure, catalyst loading, and/or other case‐specific variables affecting the scale‐independent chemical rates are varied experimentally.

If agitation conditions affected the result of the initial screening experiments, the outcome of the reaction is influenced by the rate of mass transfer. This is frequently the case for heterogeneous reactions, and characterization of the scale‐dependent equipment characteristics will be important. There may be some scope to reduce the rates of the chemical reactions (relative to mixing) by operating at lower concentrations, pressures, catalyst loadings, and/or temperatures, but this will also reduce volumetric productivity.

10.3.3 Achieving Similarity on Scale‐Up

For batch multiphase reactions, achieving similarity on scale‐up relies on having characterized both the reaction and the intended scale‐up facility. Equations (10.9)–(10.13) and (10.18) provide the basis for a scale‐independent design space.

When kinetics are slow (relative to mass transfer), adequate agitation is necessary to create a dispersion and to promote heat transfer; adequate gas supply is required for pressure reactions. Justifying the scale independence of the design space in this case reduces to demonstrating that pressure and temperature can be maintained in the same range on scale as in the laboratory and that agitation is sufficient to disperse the phases.

When kinetics are fast, k_La should be included explicitly to make the design space scale independent.

For hydrogenations and other catalyzed pressure reactions, a design space could therefore contain concentrations, equivalents, temperature, pressure, catalyst loading, and time as factors; when kinetics are fast, k_La should also be a factor.

As noted above, there may be an optimum temperature or pressure profile to adopt during the reaction and/or an optimum time at which to stop the reaction. These may be determined from a kinetic model, as used in the example described below [15]. Many examples of kinetic modeling of multiphase reactions are available, including a methanethiol reaction [16] and for other hydrogenations [17].

When kinetic models are used in reverse, to work backward from a desired end result to determine the possible operating conditions (or factors) that would produce this result, multiple acceptable combinations of factor settings may be found. For example, very poor mass transfer (low k_La) can be partially compensated for by operating at higher pressure [18]; the effects of low k_La may also be partially mitigated by operating at lower concentrations of starting materials or catalyst. These combinations may be difficult to express in a design space definition but can be found easily using a mechanistic model and justified if the model has been properly verified against experimental data. This has led to discussion of the use of models to more flexibly capture the definition of a design space and verify that a given set of operating conditions lie within it [19].

10.4.1 Rate Equations

The rate of change of concentration of each species in a homogeneous fed‐batch reaction is

(10.22)

Compared with Eq. (10.4), the additional term on the right‐hand side represents addition of feed and the accompanying dilution effect.

When the kinetic rates of the reactions are slow compared with the rate of addition, reaction takes place after the addition, and the result is very similar to a homogeneous batch reaction, described above. When the kinetic rates are comparable with the addition rate, a significant amount of reaction occurs during the feed, which normally changes the outcome of reaction compared with the batch case. This is because the concentration of the fed reactant remains low during the addition.

When the kinetic rates are much faster than the addition rate, the concentration of the fed reactant tends to zero, and in this case the reaction is highly localized around the feed region. Equation (10.22) in this case no longer fully describes the rates of change, and a more detailed analysis involving phenomena known as mesomixing and micromixing is required [20, 21] for an accurate mathematical description. A process scheme for this situation is shown in Figure 10.7, with the notable addition of a “feed zone” or reaction zone in which most of the reaction takes place. The size of the feed zone and its composition are determined by micromixing and mesomixing rates. The outcome of the reaction is predominantly determined by these scale‐dependent rates rather than by chemical kinetics.

The rates of mesomixing and micromixing can be thought of as somewhat analogous to the rate of mass transfer between two phases, which arises with multiphase systems as described above; for this reason, fed‐batch reactions with rapid chemical kinetics are often referred to as “pseudo‐homogeneous,” implying that the feed zone is like a separate phase.

The characteristics in Figure 10.7 arise in a significant fraction of applications, as reactions are often deliberately engineered to run with fast kinetics (e.g. by operating at high concentration, catalyst loading, and temperatures) because the heat output from such systems can then be halted in the event of a cooling failure by stopping the addition. Reactions such as these are also known as “feed controlled” and “dosing controlled.” The existence of the feed zone does not necessarily change the outcome of the reaction significantly; this only occurs if the reaction scheme and kinetics are such that the concentration and temperature “hotspot” that exists in this zone causes the reactions to follow a different path compared with what they would follow if diluted fully throughout the bulk of the vessel. In a significant minority of cases, the feed zone has a major bearing on product quality and yield. Appropriate characterization experiments as described below can be used to determine whether a given reaction is subject to these effects.

Considering Eq. (10.22), when reaction kinetics are slow, such that all or almost all of the reaction takes place after the addition is complete, the solution tends toward that for a batch reaction with the same volume, as described by Eq. (10.4). Integrating Eq. (10.22) in time, the final outcome (composition) of the reaction will then depend on the concentrations and temperature profiles followed during the reaction and the time allowed for the reaction. Temperature is often (but not always) held constant or nearly constant, so final concentrations at completion depend only on initial concentrations and initial temperature. Example Problem 10.4 follows this behavior.

When a significant amount, but not all, of the reaction occurs during feeding, this indicates that some of the chemical kinetic rates are comparable with the addition rate. Equation (10.22) is more informative for this case when examined in dimensionless form; for a second‐order reaction between A and B, using  =C_i/C_A0 and t^* = kC_A0·t, the rate equation for the dimensionless A or B concentration is, from Eq. (10.22),

(10.23)

Or, equivalently,

(10.24)

Defining the initial volume ratio of bulk solution to feed solution as

(10.25)

and noting that feed time is given by

(10.26)

then, at the start of the addition,¹

(10.27)

Substituting Eq. (10.27) into Eq. (10.24) leads to

(10.28)

Equation (10.28) shows that the outcome (or dimensionless concentration profile) of fed‐batch reaction when a significant amount of reaction occurs during the addition depends on the initial concentrations in the bulk and feed vessels, the rate constant (temperature dependent), the addition time, and also the volume ratio of the reagents. This latter point is important as the volume ratio is independent of the number of equivalents, i.e. reactions run at the same equivalents, temperature, and addition time may produce different results when the volume ratios mixed are different.

Apart from volume ratio, a further degree of freedom exists in fed‐batch reactions that is not present in the batch case: the order of addition; when the order of addition is reversed, this can radically change the solution concentrations compared with the forward addition. Reverse addition is worth considering and testing when feasible and safe. From a quality point of view, order of addition should follow from the reaction scheme (expressed in Eq. (10.4) using the stoichiometric matrix ν_ij). For example, in the following case, to maximize product it is more favorable to add A to B:

• A + B → product
• A + product → impurity

In the next case, it is more favorable to add B to A:

• A + B → product
• B + product → impurity

When essentially the entire reaction occurs during the addition, Eqs. (10.22) and (10.28) are no longer directly applicable, as the reaction zone will be localized and some reactions will run to completion before the feed has been diluted into the bulk contents. In this case, the dominant effect on the composition of the reaction mixture is the rate of localized mixing near the addition point, a physical phenomenon somewhat analogous to mass transfer in the multiphase example above. The degree to which the reaction exhibits this characteristic may depend on the order of addition. The final outcome of this type of reaction will depend on the time constants for mesomixing or micromixing, whichever is slower and therefore rate determining; these influence the size and residence time of the feed zone, which in turn determines how much of the chemistry takes place in the concentration and temperature hotspot created by the feed. Both meso‐ and micromixing time constants are affected by the addition rate, the addition location, the diameter of the addition nozzle, the agitator configuration, and the agitator rotational speed [14]. Factors directly affecting the intrinsic kinetics have less influence in this scenario, such as reaction temperature.

The rate of change of temperature in fed‐batch reaction systems is given by

(10.29)

Compared with Eq. (10.5) for batch reactions, the two additional terms on the right‐hand side represent contribution of sensible heat by the fed material (e.g. when warmer than the bulk) and any associated thermodynamic heat of mixing that also accompanies the addition. The heat flow signature from the chemical reactions will be determined by kinetics (kinetically controlled) when reaction occurs after the addition, by the rate of addition when the reactions occur entirely during the addition, and by both kinetics and the rate of addition when a significant amount of reaction, but not all, occurs during the addition.

From Eq. (10.29) with Q_f = 0, the scale‐up relationship for maintaining constant temperature when kinetics limit is identical to the corresponding equation for homogeneous reactions:

(10.30)

At the other extreme, when kinetics are instantaneous relative to addition, the rate of heat output is primarily dependent on the rate of addition of limiting fed reactant:

(10.31)

Substitution of Eq. (10.31) into Eq. (10.29) and setting the right‐hand side equal to zero leads to the widely used scale‐up relationship

(10.32)

Noting that the addition volumetric flow rate is the addition volume divided by the addition time, Eq. (10.32) may be rearranged in terms of feed time as

(10.33)

A more complete version of Eq. (10.33) may be obtained by writing Eq. (10.29) in dimensionless form using T^* = T/T₀ and t^* = UAt/ρVC_p and then substituting Eq. (10.31) for Q_r and Eq. (10.27) for Q_f, giving at the start of the addition

(10.34)

Rearranging Eq. (10.34) with the right‐hand side set equal to zero leads to

(10.35)

From Eq. (10.35), a longer feed time implies that a lower UA or ΔT may be tolerated; a higher volume ratio is most likely to be combined with a more concentrated feed, and these effects cancel out.

From Eqs. (10.33) and (10.35), it is clear that the addition time on larger scale will often need to be longer than on smaller scale in order to operate the reaction at the same constant temperature as at lab scale; the magnitude of the increase in addition time will depend on the degree to which the temperature driving force can be increased on scale‐up.

Similarly to homogeneous batch reactions, the equations above can be easily incorporated into dynamic mechanistic models for fed‐batch reactions and unknown parameters (e.g. rate constants, activation energies) regressed against experimental data. The effects of changing order of addition, concentrations (including catalyst loading), temperatures, volume ratio, and addition time and the effects of mixing and heat transfer can be predicted. In some fed‐batch reactions, for example, temperature is deliberately profiled (e.g. using a heating ramp to drive reaction to completion), and the profile may be scale dependent. The impact of these changes can be predicted using classical chemical engineering rate equations. There may be an optimum time at which to stop the reactions, e.g. after product concentration has peaked and before impurities are able to form. To understand these effects and take advantage of the opportunities, a detailed mechanistic model of the reaction is valuable. Several examples illustrating this approach are given below, and more are included in the Refs. [7, 11].

10.4.2 Characterization Tests for Fed‐Batch Reactions

For fed‐batch homogeneous reactions, if feasible the reaction should first be run using forward and reverse additions. If the desired process involves addition of A to B, then run this case followed by an otherwise identical case with B added to A. If the results of these experiments differ significantly, reaction occurs during the addition and is at least in the intermediate regime described above. Temperature or heat flow profiles when monitored during these experiments may indicate the degree of addition‐controlled behavior and the amount of reaction happening during the feed versus after the feed. Likewise, in situ or sample data following the concentration of the bulk reactant both during and after the addition will be informative in this regard.

When a dosing‐controlled reaction is suspected, a further 2 replicate experiments in which only stirrer speed is varied between 200 and 1000 rpm will further indicate the influence of agitation on the rate and outcome of reaction.

In each of the above initial screening experiments, four or more samples should be taken, e.g. after 25, 50, 75, and 100% of the feed have been added. With some reactions, additional profiling may be possible by following variables such as the rate of gas evolution and both pot and jacket temperatures.

Under conditions where there is no effect of agitation on rate or quality, a kinetics study may follow in which addition time, volume ratio, temperature, equivalents, and catalyst loading are varied experimentally. The choice of conditions and the sampling program for these experiments should be informed by the results of the previous experiments and an initial kinetic model [2], and in any event, multiple samples should be taken again to capture when the reactions are occurring as well as the final result.

If rate or quality is sensitive to agitation, the reaction is not limited by chemical kinetics at current conditions but by localized mixing, at least under typical agitation conditions. The main focus of process design and characterization should be the mixing capabilities of lab and larger‐scale equipment (see Section 10.6). There may be some scope to reduce the rates of the chemical reactions (relative to mixing) by operating at lower concentrations and/or temperatures.

10.4.3 Achieving Similarity on Scale‐Up

Achieving similarity on scale‐up relies on having characterized both the reaction and the intended scale‐up facility. Equations (10.22) and (10.29) provide the foundation for a scale‐independent design space.

When kinetics are rate limiting, other than agitation that is necessary to promote heat transfer, no particular additional agitation requirement exists. As noted above, there may be an optimum temperature profile to adopt during reaction and/or an optimum time at which to stop the reaction. These may be determined from a kinetic model, as in Example Problem 10.4. A scale‐independent design space may include factors such as equivalents, temperature, and reaction time.

When mixing is rate limiting (reaction is dosing controlled and mixing affects quality), agitation influences the outcome, and similar mesomixing or micromixing time constants will be required at each scale; in order to achieve a scale‐independent design space, the mixing time constants should be factors in design space definition. Additional factors may include order of addition, addition time, volume ratio, equivalents, and temperature.

In the intermediate regime where a significant portion but not all of reaction occurs during the addition, reaction will tend to become more dosing controlled on scale‐up, as the addition time is increased to compensate for lower heat transfer area per unit volume. A scale‐independent design space may include factors such as order of addition, addition time, volume ratio, equivalents, temperature (profile), and reaction time.

The original publication in 2007 [18] was based on 30 virtual simulations to which a polynomial equation was fitted for plotting; the above figures were reproduced in 2009 by running 440 virtual experiments in the same factor space with no interpolation.

Because kinetics limit this reaction at or near current operating conditions and because most of the reaction occurs after the addition, a scale‐independent design space for these responses may be constructed using two variables: temperature and equivalents of acid. When all other input material quantities are held constant and adequate temperature control is maintained on scale, the result of this reaction at any point in this space would be independent of scale.

Figure 10.15 is quite typical in that there appear to be two “edges of failure” when the responses are overlapped [27]. One response relates to the quality attribute (amount of alkene in this case), and another relates in this case to a business attribute (reaction time). In a strictly QbD context, Figure 10.15 therefore has only one edge of failure: if the quench window is too short, the alkene level will exceed its limit; if the reaction time is too long, this will not directly impact quality but will slow productivity.

The design space shown in Figure 10.15 is approximately 10 mmol wide and 5° high. However if a rectangular region of proven acceptable ranges had to be defined inside this space, the maximum width would be 7 mmol, and the maximum height 2–3°. This illustrates how the design space has the potential to offer greater flexibility than rigidly defined proven acceptable ranges.

On the other hand, Figure 10.15 remains quite restrictive, in that all other factors (such as substrate concentration) have to be held constant for it to apply. This is one of the reasons why a more dynamic design space definition based on the full mechanistic model, rather than one set of response surfaces at otherwise fixed conditions, has been advocated by industry [13]. The model verification statistics associated with this example are discussed at the end of this chapter.

10.5.1 Plug Flow

Equations presented above for batch reactions may be applied directly to plug flow reactions after noting that the independent time variable now applies to position along the reactor [28]:

(10.36)

Here V is the cumulative volume of the reactor since the material entered at time zero and Q is the volume flow rate. When a plug flow reactor is operated at the same temperature and with the same residence time as a batch reactor with that cycle time, the end results are the same. This convenient result means that data collected in batch mode may be used for design in continuous mode and vice versa.

Plug flow reactors may be used for homogeneous systems, liquid–liquid and gas–liquid systems, with appropriate attention to ensuring that the phases are dispersed and separated when required and that both phases travel along the tube without accumulation.

Reaction characterization experiments follow the same logic as the corresponding problems in batch systems described above [14]. Equipment performance characterization in terms of heat transfer is always required; mixing and mass transfer characterization will be required when kinetics are rapid and in multiphase systems [14, 29]. Design spaces are expressed by taking into account similar factors to batch reactors.

Plug flow reactors with multiple addition points along the reactor length are analogous to fed‐batch systems with multiple sequential additions and likewise may be characterized and predicted using a similar approach to fed‐batch systems.

Useful additional information on application of continuous flow systems for API is available in the Refs. [30–32].

10.5.2 CSTRs in Series

Plug flow behavior may be approximated using a cascade of stirred reactors in series [21]; this can also ease the problem of running slurry reactions in continuous mode. The approach to reaction characterization and design for these systems is described above. Equipment characterization in terms of heat transfer is always required, and batch characterization tests may be used for this purpose (see below).

Models developed for process prediction in batch or fed‐batch systems may be reused to design continuous stirred tank reactor systems by adding appropriate feeding and removal between reactors in the cascade [34]. When the number of reactors is large, a simpler plug flow model will give almost equivalent results.

10.6.1 Heat Transfer

The product of overall heat transfer coefficient U and wetted area A appears throughout this chapter as equipment characteristics required for process scale‐up.

The product UA is best evaluated using a solvent test in the intended process vessel, to which solvent is charged and the fill level and agitator speed are set to those of the intended process. The batch is heated and/or cooled over the range of pot and jacket temperatures required by the process. Pot and jacket inlet temperatures are monitored continuously; jacket outlet temperature is monitored if available. In the absence of reaction and any other heat effects, Eq. (10.5) may be used to fit UA for both heating and cooling.

UA may also be estimated from chemical engineering correlations developed from measurements in similar types of equipment [35]. The coefficients in such heat transfer correlations may vary depending on the precise configuration, and in some cases, new coefficients may be required to describe unusual vessel configurations. In mature applications of equipment characterization, those coefficients are stored with the equipment configuration data in an equipment database for future reuse.

10.6.2 Mass Transfer

The product of mass transfer coefficient k_L and interfacial area a (per unit volume) appears in this chapter whenever interphase transfer is involved.

The product k_La is best evaluated using a test in the intended process vessel, in which mass transfer either is the only phenomenon occurring or is the rate‐limiting phenomenon.

In one such test for headspace fed gas–liquid reactions, typically hydrogenations [36], solvent is charged, and the fill level set to that of the intended process. The headspace is evacuated and then pressurized with gas, and the agitator is turned on, with the speed increasing quickly to the intended process speed. Headspace pressure and both headspace and liquid temperature are monitored continuously. In the absence of reaction and any other phase transfer effects, Eq. (10.17) may be used to fit both gas–liquid (surface gassing) k_La and solubility. Alternatively, a reaction that consumes the dissolving gas may be run under conditions of high catalyst loading, such that mass transfer is rate limiting; in this case, Eq. (10.9) may be used to fit k_La to an indicator of reaction progress, such as hydrogen uptake. Note that (i) the k_La for surface gassing is very sensitive to the submergence of the top impeller [37] and (ii) sparging gas may be of little value in a “dead‐end” system (i.e. unless gas is continuously removed from the headspace) and mass transfer due to surface gassing is more important.

The depressurization test described above must be done carefully in order to provide useful data; for example, the headspace and liquid must be at the same temperatures before the agitator is turned on, to avoid a pressure recovery due to heating of the gas by the liquid.

For solid–liquid and liquid–liquid reactions, analogous characterization tests in which the uptake of solute is monitored (e.g. by sampling the liquid) may be used, or a known fast chemical reaction may be run under conditions in which dissolution of solute is rate limiting. Equation (10.9) again provides the basis for fitting k_La to the monitored profiles.

Alternatively, empirical estimates may be made using chemical engineering correlations; for these, the molecular diffusion coefficient of the solute in the solvent is required, which limits applicability. A feature that makes solid–liquid systems somewhat easier to predict is that the wetted area does not change with stirrer speed once the solids are suspended, that is, unless the particles are broken as a result of agitation. For liquid–liquid systems, the effect of minor components on the droplet size and the resulting surface area can be very significant, making accurate estimates difficult.

In all operations involving contact between multiple phases, a certain minimum level of agitation is required (even if kinetics are slow) in order to ensure that a dispersion of one phase exists in the other. In sold–liquid and liquid–liquid systems, the minimum stirrer speeds at which such a dispersion is created may be estimated with greater certainty than k_La.

For solid–liquid systems, this level of agitation (N_JS – the agitator speed that just suspends the solid particles) exposes the full surface area, and k_La on scale may be taken as approximately equivalent to that which applied in a lab experiment using the same raw materials at the same conditions in which all of the solids were suspended; therefore similar k_La can be achieved even if neither lab or plant k_La is known. N_JS may be estimated for a variety of impeller and tank configurations [38].

For liquid–liquid systems, a balance must be struck between mass transfer rate (favored by small droplets) and subsequent quick phase separation (favored by large droplets). Operating at N_JD – the agitator speed that just disperses the liquid droplets of one phase in the other – exposes significant surface area while reducing the likelihood of forming a stable emulsion; once again the k_La on scale may be taken as similar to that which applied in a lab experiment with the same raw material at the same conditions in which the liquids were just dispersed; therefore similar k_La can be achieved even if neither lab or plant k_La is known. N_JD may be estimated for a variety of impeller and tank configurations [34, 39].

If solid–liquid or liquid k_La is a factor in definition of a design space, the proximity of agitator speed to N_JS or N_JD, respectively, is a reasonable surrogate variable for k_La; for example, a dimensionless agitator speed that is scale independent could be defined as a factor:

(10.37)

N^* in Eq. (10.37) might, for example, in a particular application need to be above 1.0 in order to avoid mass transfer limitations.

10.6.3 Liquid Mixing

In batch homogeneous reactions there is the possibility of reaction rate limitation by macromixing, at very low agitation rates. A macromixing time constant 30–60 seconds should eliminate this dependence, and correlations are available to estimate this factor [40]. The liquid mixing characteristics relevant for fed‐batch reactions are meso‐ and micromixing time constants; these appear in equations that describe the rate of localized mixing near the addition point in fed‐batch and continuous reactors and may be rate determining when kinetics are fast, i.e. for pseudo‐homogeneous, dosing‐controlled reactions as described above.

The time constants for meso‐ and micromixing may be characterized by running tests reactions with known kinetics in the target equipment [14, 41]. The outcome of these reactions is mixing sensitive under certain conditions and varies with factors such as impeller speed, addition rate, and addition location. When the product distribution or selectivity from each experiment is combined with a mathematical model based on Figure 10.7, the time constants for meso‐ or micromixing for each set of conditions may be obtained.

Alternatively, formulas are available for these time constants [14], and if the spatial distribution of the local rate of energy dissipation, ε, is known, the time constants may be obtained from these. Specialized tools such as computational fluid dynamics or laser anemometry may be used to estimate the spatial distribution of ε. As a general rule, ε is a multiple (e.g. 10–100) of the vessel‐averaged power input per unit mass (W/kg) near the impeller and a fraction (e.g. 1–10%) of the average near the liquid surface.

In continuous plug flow reactor systems, the extent of “axial dispersion” or longitudinal mixing along the axis of the flow may be characterized using pulse or step response experiments. Given the small scale that often applies in continuous pharmaceutical production/flow chemistry, the flow regime is not always turbulent (e.g. small diameter tubes with concentrated materials flowing at low velocities). In general, long, thin tubes, whether coiled or straight, lead to a low level of axial dispersion, and inserts such as static mixers further enhance the degree of plug flow and reduce axial dispersion to the point where assuming idealized plug flow behavior is a good approximation. An extensive discussion on this topic is available [42] containing further detail of measurement techniques and results to support process development and scale‐up. Application of PFRs in “flow chemistry”/continuous manufacturing also allows operation at higher temperature and pressure than typical for batch reactors, and in this case, fluid properties may differ significantly from those at ambient conditions, resulting in typically lower residence times for a given mass flow rate, due to thermal expansion.

10.6.4 Phase Separation

Phase separation times, e.g. after a liquid–liquid reaction or extraction, are longer on scale that in the laboratory [43]. As a rough indicator, the separation time scales with the liquid depth, so a separation time of one minute in the laboratory can easily extend to one hour on scale. Excessive separation times can be avoided by avoiding the formation of very fine droplets or stable emulsions; operating a liquid–liquid reaction at or near N_JD as described above represents a good balance between high mass transfer area and short phase separation time.

10.6.5 Gas Disengagement

When reactions evolve gas, the rate of gas evolution scales with the reaction volume, but the ability of the liquid surface to allow the gas to escape scales with the cross‐sectional area of the vessel. The maximum velocity of gas escape through the liquid surface is approximately 0.1 m/s [34], allowing the maximum volume flow rate of gas evolution to be estimated:

(10.38)

This can be converted into a molar rate using the ideal gas law

(10.39)

If the rate of reaction exceeds the maximum rate of gas evolution, significant foaming will occur, and in some cases material will be lost from the reaction vessel. This problem becomes more likely on scale, as the maximum rate of gas evolution per unit volume reduces:

(10.40)

Equation (10.40) indicates that the volumetric rate of reactions evolving gases may need to be reduced on scale (e.g. by slower feeding or otherwise) in order not to exceed the limitations imposed by the equipment.

For example, in a vessel of 2 L nominal volume with diameter 0.115 m, filled to a level of 1 L at 20 °C, the maximum volumetric rate of a reaction evolving 1 mol of gas (e.g. CO₂) per mole of product without batch swelling is 0.1 × (1/0.104) × (1.013 25 × 10⁵/8.314 × (273.15 + 20)) = 39.97 mol/m³·s ≈ 0.04 mol/l·s. The same process running in a 2000 L vessel with diameter 1.5 m at 1600 L batch size is limited to a rate of 0.1 × (1/1)(1.013 25 × 10⁵/8.314 × (273.15 + 20)) = 4.16 mol/m³·s ≈ 0.004 mol/l·s, i.e. 10 times slower, to avoid batch swelling.

10.7.1 Parameter Uncertainty

All models or equations with parameters that are derived originally from experimental data have a degree of uncertainty associated with their calculations or predictions. This is true for calibration curves, regression lines, response surfaces generated by design of experiments, chemical engineering correlations for estimating equipment performance, models based on ordinary differential equations (such as many of the examples shown above), and those using partial differential equations (such as computational fluid dynamics, in which turbulence models, “wall laws,” and other model parameters are ultimately based on experimental data). Unless these latter models involving differential equations are integrated over a sufficiently fine “grid,” their uncertainty is further increased by lack of precision.

Once accepted as useful, models are often used without taking this uncertainty into account, but experienced practitioners will always allow a factor of safety to compensate for a margin of error, even if the size of that error is not known. When the original data and the model are both available, it is possible to use statistical methods to quantify the uncertainty level. This makes the potential deficiencies of the model more evident and explicit and may also focus further experimentation on reducing those uncertainties.

More details about how to calculate uncertainty are given elsewhere [44], and Figure 10.17 illustrates the typical situation for a linear model in which the slope and intercept have been fitted.

Confidence bands define an envelope within which there is a certain confidence level (typically 95%) of the true location of the best‐fit line. Prediction bands (or intervals) define a wider envelope within which there is a certain confidence level (e.g. 95%) that all datapoints/observations will lie. The width of prediction bands relative to the model prediction indicates the likely relative error of the model predictions; this also reflects underlying variability or lack of reproducibility in the experimental data. For typical linear models such as that in Figure 10.17, the band widths are at a minimum at the center of the experimental data and increase in either direction from the center. This tallies with the belief that models are most applicable near the conditions where the experiments were run and become less reliable at more extreme conditions.

Any such model should in the first instance be based on reliable, reproducible data; there should also be sufficient data to avoid “overfitting,” i.e. the number of observations should be significantly greater than the number of model parameters fitted. The degree to which such a model can be said to be verified depends on the width of its prediction bands compared with the accuracy needed in the intended application.

For example, if a model predicts an impurity level of 0.1%, with 95% prediction band widths of 0.05%, one could state with a high degree of confidence that the impurity level will be less than 1%; more formally, the probability of impurity exceeding 1% is almost zero, p(Impurity > 1%) ≈ 0. The same model is not sufficiently accurate to state with the same degree of confidence that the impurity level will be less than 0.2%, i.e. p(Impurity > 0.2%) > 0. However if the 95% prediction band width was 0.01%, the model would be suitable for more confident predictions at lower impurity levels. There is therefore an element of fitness for purpose when judging whether a model has been verified sufficiently to apply it in a given situation.

Similarly, if the correlation used to calculate the stirrer speed required to suspend solids has a stated accuracy of 20% and the predicted N_JS = 80 rpm, this level of verification is sufficient to say that 40 rpm is too little – p(Suspended) ≈ 0‐ and 120 rpm is too much – p(Suspended) ≈ 1, but not whether 75 rpm is sufficient – 0 < p(Suspended) < 1.

Returning to Example Problem 10.4, confidence and prediction bands may be calculated for a dynamic model based on chemical engineering rate equations, and typical results are shown in Figures 10.18 and 10.19.

Comparing Figures 10.18 and 10.19, the relative width of prediction bands is greater for alkene than for product. This reflects greater uncertainty in the alkene measurements and predictions.

In QbD work, the criticality of factors is sometimes judged according to their proximity to factor settings that define the “edge of failure,” e.g. crossing an impurity limit. Prediction bands should be taken into account when judging criticality in this way, as these will show that failure will occasionally occur at factor settings that are nearer to intended operating conditions.

10.7.2 Implications for Design Space Definition

The existence of uncertainty means that formal probability statements may be required to properly define a design space; while this implies some additional work, it has the benefit of quantifying the degree of assurance (or risk) that exists in relation to how the process will perform. In general, this approach will lead to design spaces that are smaller and more conservative than when uncertainty is ignored and that maximize the probability that the product quality will be in specification. Most popular statistical software packages at present do not take uncertainty into account in this way [45].

The relevant probability can be calculated in a variety of ways, and Figure 10.20 illustrates application to Example Problem 10.4. Recall from Figure 10.19 that the relative uncertainty of alkene is greater than that for product in this case; this leads to a relatively broader region of conditions in which the impurity has a significant probability of failing to meet specification. On the other hand, uncertainty in product predictions is low, leading to a narrow region separating success and failure. To produce Figure 10.20, the relative uncertainty in quench window was taken to be proportional to that of alkene and the relative uncertainty of ProductMaxTime proportional to that of product.

Figure 10.20 illustrates the regions of the factor space with the highest probability of success; these are the most favorable regions in which to operate the process. In the green area of Figure 10.20, the probability of success exceeds 80%. Figure 10.21 compares the design space defined on this basis (p > 80%) with that obtained by overlapping the average responses; as expected, taking account of uncertainty reduces the size of the design space.

The above results show that for QbD purposes, uncertainty in data or responses predicted by any model should be explicitly taken into account in defining the design space; probability is the natural way to do this. Uncertainty will tend to shrink the design space away from the edges of the area where average responses overlap, in line with good engineering practice. When the peak probability is far from 100%, this highlights the need to obtain greater process understanding, or an improved process, before proceeding.