Introduction

Diafiltration (DF) is the removal of low molecular weight solutes, such as salts, from a solution of a high molecular weight solute, such as a protein. The basis of the separation is that the high molecular weight solute (‘macrosolute’), or particle, does not pass through the membrane but the low molecular weight ‘microsolute’ does. It is somewhat similar to dialysis but, in the latter, a dialysing fluid flows counter-currently on the permeate side, providing a mass transfer driving force that enhances removal of the microsolute. Dialysis is discussed briefly towards the end of this chapter.

There are many ways to perform DF and in this chapter a number of different approaches are examined. Throughout the analyses presented, the solution is assumed to be composed of two solutes; a macrosolute whose concentration is denoted by cA and a microsolute whose concentration is denoted by cB. Complete rejection of the macrosolute is assumed throughout and the microsolute is assumed to have a rejection coefficient of zero. An additional and important assumption is that the flux is determined by the macrosolute concentration only. Furthermore, the flux is assumed to be at the limiting value as given by the limiting flux model. The one exception to this is in Section 6.4.3, where a more complex, essentially empirical, model for the flux is used. The most notable feature of that model is that the flux depends on the concentrations of both the macrosolute and the microsolute.

Introduction

In Chapter 2 it was seen that dead-end filtration is characterised by the flow of suspension normal to the filter. In crossflow microfiltration (CFMF), the suspension flows parallel (or tangential) to the membrane. The rationale behind the crossflow mode of operation is that the ‘scouring’ action of the flow parallel to the membrane inhibits the growth of filter cake, thus creating the potential for high filtrate fluxes and steady state operation. CFMF is normally considered when the solids in a suspension prove difficult to separate in a dead-end filter or a centrifuge. Dead-end filtration is difficult when the particles are ‘small’ and have the tendency to form highly resistant and compressible filter cakes. Problems in centrifuging suspensions arise when the particles are small and have densities that are close to that of the suspending fluid. All of the above characteristics are typical of microbial suspensions. Consequently, crossflow systems have the potential to be economically viable for bacterial cells (as compared to centrifugation) and larger filamentous cells (as compared to pre-coat rotary vacuum filtration). CFMF is also useful for separating shear sensitive animal cells, for clarifying beverages such as beer and fruit juice, in separation of blood cells from plasma (plasmapheresis) and in sterile filtration of pharmaceuticals. Use of crossflow microfiltration as a key component of submerged membrane bioreactors (MBRs) is now commonplace. These devices combine biological waste treatment with membrane filtration, the performance of the latter being improved by the presence of the air bubbles required by the biological reactions. This use of air sparging to improve membrane filtration is discussed later in this chapter.

It is very important to remember that comparing dead-end and crossflow microfiltration is not really comparing like with like. In dead-end processes, a filter cake is recovered. An example of this is use of rotary vacuum filters in production of cakes of baker's yeast used in the bread making industry. In CFMF, however, no substantial cake is formed and the best that can be achieved is to concentrate a suspension. Both dead-end and crossflow techniques can, of course, be used for soluble product recovery.

Introduction

Membrane fouling refers to a deterioration in membrane performance due to interactions between the feed and the membrane. This typically involves an increase in membrane resistance and an increase in solute rejection. Cake formation and concentration polarisation are technically not considered to be membrane fouling processes, at least in this book. Cake formation is viewed as forming an additional resistance. Concentration polarisation is seen as reducing the driving force for filtration.

There is a vast literature on the subject of membrane fouling but despite this, membrane fouling still remains a largely empirical science. This is not surprising given the huge variation in membrane and feed characteristics that is encountered in membrane filtration processes. Furthermore, the underlying physical processes are complicated, involving solute–solute, particle–particle, solute–membrane and particle–membrane interactions. Identifying which interactions are important is usually very difficult indeed, especially when complex mixtures are involved.

Introduction

In this chapter, mathematical models for analysis, design and optimisation of a variety of continuous, batch and fed-batch UF systems are developed. In all cases the system is assumed to be operating under limiting flux conditions, i.e., Eq. 4.22 applies. It should be recalled that this equation implies complete rejection of the solute. It is seen that, despite the relative simplicity of the models, the presence of the logarithmic term in the flux expression requires one to routinely use numerical techniques to solve the model equations. In addition, it is shown how a graphical method can be useful for certain types of calculations and also it is demonstrated that by employing special functions, numerical solution can be avoided in solving certain batch and continuous problems.

In the next section, continuous feed-and-bleed systems are examined, where the basic computational problem is to solve systems of non-linear algebraic equations (NLAEs). First, process analysis calculations are examined where the goal is to calculate the exit concentration for a given system. In process design calculations, the required area for a given process specification is computed. In process optimisation calculations, the problem of how to minimise the required area in multi-stage systems is examined. Subsequent sections examine batch, fed-batch and single pass continuous systems.

Introduction to reverse osmosis

Reverse osmosis (RO) is the membrane filtration of solutions of low molecular weight solutes, especially salts [1]. At this small scale, it is appropriate to discard the idea of the membrane having distinct pores but instead to view it as a matrix through which both the solvent and solute diffuse. Diffusion occurs through the space between the chains of the polymer from which the membrane is constructed. However, the membrane may contain a small number of defects resulting in the presence of some pores.

The ability of a species to diffuse through an RO membrane is typically controlled by charge effects rather than size and, consequently, RO is used extensively for desalination processes. It is also used for water softening, i.e., removal of cations, especially calcium, and in production of high purity water such as that required in electronics industries. Smaller scale RO units are often used to produce high purity water for laboratory purposes.

Just like ultrafiltration, the flux of pure water through RO membranes is partly limited by the osmotic pressure of the solution. Osmotic pressure is a colligative property, meaning that at a fixed mass concentration, it increases as the molecular weight of the solute decreases. In other words, the osmotic pressure depends on the number of solute molecules per unit volume. Thus, the osmotic pressures of salt solutions are much greater than those of macromolecular solutions at the same molar concentration. This contributes to the need for the very high pressures, up to 100 bar, that are used in RO.

Introduction

Modelling, analysis and design of chemical engineering systems generally proceed on two fronts. First, the mass and/or energy balances for the system are formulated. On their own, these balances are rarely sufficient to derive equations with which practical calculations can be performed. This is because the balances typically contain some unknown term, or terms, which quantify the rate at which some key process is occurring. Quantifying this rate is the second step in the analysis. A classic example would be the kinetic term in a model of a batch reactor. Alternatively, an equilibrium relation might provide a crucial relationship between compositions in the liquid and vapour streams of a distillation process, thus providing the missing information required to solve the complete model equations. In this chapter the underlying rate equations for UF design and analysis are established. A number of different equations, of varying complexity, are presented and their main features outlined. The rate equation in this case is simply an equation that relates the permeate flux to measurable process variables, notably the crossflow velocity, the solute concentration and the trans-membrane pressure. In subsequent chapters, each of the rate equations is combined with appropriate mass balances for batch and continuous ultrafiltration and diafiltration systems. As will be seen, the combination of the rate equation and the mass balances leads to a variety of computational challenges. However, with modern software tools, these are easily overcome.

In addition to reviewing the basic theory of UF, this chapter provides the reader with a very brief overview of methods for predicting mass transfer coefficients, viscosities and osmotic pressures for a range of solutions relevant to UF practice. This is by no means an exhaustive review but is intended to give the reader a sense of the issues involved in actually trying to apply the various UF theories. The data provided should also prove useful for performing practice calculations.

It is often said, quite correctly, that to really learn a foreign language one needs to immerse oneself in it. For me, getting to grips with chemical and process engineering requires the same degree of immersion. It means formulating lots of problems using the core engineering skills of material and energy balancing. It also means using key mathematical and computational techniques on a regular basis. Having taught on an interdisciplinary degree programme for more than 20 years, it is clear to me that very few students can cope with doing a relatively small amount of engineering while at the same time studying lots of biology, or some other, less quantitative, subject. For most people, constant practice of engineering skills is required. The late and great golfer, Seve Ballesteros, once said: To give yourself the best possible chance of playing to your potential, you must prepare for every eventuality. That means practice. Engineering is similar to golf; the more time students spend solving practice problems, the better they become at doing engineering.

This book is all about immersing the reader in the field of membrane filtration. To use a football term, it is about encouraging the reader to ‘get stuck into’ quantitative aspects of membrane process design and analysis. As a consequence, the book is somewhat ‘equation heavy’ but that is the nature of engineering. In any event, none of the mathematics used is beyond what an undergraduate engineering student should be able to cope with.

Introduction to dead-end filtration

In Chapter 1 the various membrane filtration processes were classified and it was seen that filtration and microfiltration involve the separation of particulate matter from liquids. Filtration and microfiltration are essentially the same process but the term microfiltration is usually reserved for filtration of particles with diameters in the range 0–10 µm. Many biological suspensions, such as suspensions of bacteria or yeast, fall within this range. Typically, synthetic membranes (‘microfilters’) are used with pore diameters less than 1 µm in diameter and many companies manufacture membranes with pore diameters of 0.22 µm and 0.45 µm. Filtration is a technology that has been used extensively in more traditional chemical industries where separation of larger particles such as crystals may be required. For these bigger particles, synthetic membranes are not necessary and filter cloths with considerably large pore sizes are employed. Often, suspensions of smaller particles, such as yeast cells employed in brewing processes, are filtered with filter cloths rather than synthetic membranes. However, this requires use of filter aids as discussed later.

The important point to note throughout this chapter is that filtration and microfiltration operate on the same principle and the underlying theory covers both processes. Both terms are used at various times throughout this chapter.

Introduction

This book is largely concerned with solving process problems in the membrane filtration of liquids. In that sense, it is more a chemical engineering book than a membrane science book. There are many fine books available which provide much more information on membrane synthesis and structure, module design, transport processes in membranes and applications of membrane technologies in industrial and medical processes [1–4].

Nonetheless, a small amount of background on membrane separations is needed before the business of process modelling, analysis and design can begin. So, in this chapter, some general terminology is defined, membranes and modules are described and some of the main characteristics of the various membrane filtration techniques are outlined.

A membrane is simply a physical barrier through which pure solvent can pass while other molecules or particles are retained. In the case of ultrafiltration (UF) and microfiltration (MF) this semi-permeability is largely a result of the relative sizes of the solute/particles and the membrane pores. Solute retention is a little different in reverse osmosis (RO) and is largely determined by charge effects. The RO membrane can be thought of as a matrix through which solvent and solute diffuse at different rates. Nanofiltration (NF) occupies a transition zone between UF and RO and solute retention is complex, involving both size and charge effects.

Introduction

A key part of the modelling and analysis of chemical and bioprocess engineering systems is the formulation of mass and/or energy balances. A large part of the skill of an engineer is his or her ability not only to state these balances in words, but more importantly, to express them in mathematical form. Once this has been done, the remainder of the problem is essentially one of applying well-established mathematical and/or computational techniques. In steady state processes, the system balances are in the form of systems of algebraic equations, usually non-linear in nature. An example is multi-stage continuous feed-and-bleed ultrafiltration that is covered in Chapter 5. In a batch process, the balances are usually formulated as systems of ordinary differential equations. An example is batch crossflow microfiltration that is covered in Chapter 3.

This book is aimed at a wide audience, from chemical engineers with a strong background in mathematics and computing, to students studying interdisciplinary subjects like biotechnology or bioprocessing who might be a little less mathematically literate. The aim of this appendix is to provide a brief summary of the main mathematical and computational techniques that are employed throughout the remainder of the text. Coverage of the mathematical topics is brief but provides the student with sufficient information that he or she can review the key topics in more depth using appropriate sections of their preferred mathematics text if desired. Coverage of the computational topics provides just enough background so that those who do not have a strong background in numerical methods will have some appreciation of how a computational tool such as MATLAB actually solves a system of differential or algebraic equations. Again, the student can consult a numerical methods text to obtain a deeper understanding of these topics if needed. However, this should not be necessary for the problems encountered in this book.

In this chapter, we work with objects that possess a magnitude and a direction. These objects are known as physical vectors or simply vectors. There are two types of vectors: bound vectors, which are fixed to a specified point in the space, and free vectors, which are allowed to move around in the space. Ironically, free vectors are often used when working in rigid domains, whereas bound vectors are often used when working in flowing or flexible domains. We mostly deal with bound vectors.

We denote vectors with underlined bold letters, such as v, and we denote scalars with nonunderlined letters, such as a, unless otherwise noted. Familiar examples of vectors are velocity, acceleration, and forces. For these vectors, the concept of direction and magnitude are natural and easy to grasp. However, it is important to note that vectors can be built depending on the user's interpretation and objectives. As long as a magnitude and direction can be attached to a physical property, then vector analysis can be used. For instance, for angular velocities of a rigid body, one needs to describe how fast the rotation is, whether the rotation is counterclockwise or clockwise, and where the axis of rotation is. By attaching an arrow whose direction is along the axis of rotation, whose length determines how fast the rotation is, and pointing in the direction consistent with a counterclockwise or clockwise convention, the angular velocity becomes a vector. In our case, we adapt the right-hand screw convention to represent the counterclockwise direction as a positive direction (see Figure 4.1).