1 Balances

1.1 The balance: recipe and form

The field of transport phenomena covers the transport of the three most important quantities – mass, energy, and momentum – in any (physical or chemical) process. The addition of the words ‘in any process’ in particular is an indication of one of the most important features of the field: transport phenomena is, above all, an engineering field with a wide range of applications.

Nonetheless, the field is also fundamental, given that it forms the basis for many other chemical engineering disciplines such as reactor engineering, separation technology, and fluid mechanics. This makes transport phenomena a must for any chemical engineer. A good knowledge of the subject is also very useful to those in other professions such as mechanical, mining, civil, and building engineers, physicists, chemists, and materials scientists.

The area covered by the field of transport phenomena and the discipline of chemical engineering is considerable. There are, for example, all kinds of processes in the chemical and petro-chemical industry, the flow of one or more phases through a pipeline, the behaviour of bubbles in a bioreactor, or the filling of a casting mould with liquid metal. At the other end of the scale, the field is also very important to everyday matters, such as the heat emission of a radiator and the associated air flows in the room, and transport of oxygen by blood flow. Fortunately, these very different processes can be clearly understood and described with a limited number of rules.

Flow phenomena and heat and mass transfer are described in this field in terms of continuum properties, with only occasional references to molecular processes. This is how the basis is laid for chemical engineering: the expertise of designing and improving processes in which substances are transported, transformed, processed, or shaped. It is important here to fully understand the essence of a process – that is, to identify the essential stages in the transport of mass, heat, and/or momentum. The transport of these three quantities can, as it happens, be described in exactly the same way. Transport phenomena lays the basis for physical technology and provides the necessary tools. This textbook is about these tools.

First and foremost, transport phenomenon is a subject of balances and concepts by which physical processes and phenomena can be described. In many cases, the subject is about deviations from a state of equilibrium and the subsequently occurring resistances to heat and mass transport. It frequently concerns a quantitative description of cause and effect. With the help of these still somewhat vague terms, it is possible to gain an outlined, but also very detailed, understanding and description of the aforementioned and countless other processes. This chapter will discuss the term balance in extensive detail.

For the description of the transport of any quantity, such as the transport of oxygen from bubbles to the liquid phase in a fermenter or the transport of heat through the wall of a furnace, the balance is an essential tool. The basic principle of the balance is the bookkeeping of a selected physical quantity. This concept is of particular importance when working with what are known as conserved quantities; these are quantities (like mass and energy) that are not lost during a process, but conserved.

The field of transport phenomena deals with steady-state or transient (time-dependent) processes in which mass, energy, and momentum are exchanged between domains as a result of driving forces (differences in concentrations of mass, energy and momentum, and/or in pressure). Transport phenomena is therefore primarily about the ‘bookkeeping’ of the three physical quantities: mass, energy, and momentum.

This bookkeeping is done by drawing up balances over control volumes. A control volume is a domain (or system) with boundaries and can be closed or open, usually allowing for transport or exchanges across the boundaries. Such bookkeeping can refer to large control volumes, which involve macro-balances; however, balances can also be drawn up in relation to very small control volumes – these are known as micro-balances, which provide information at a local scale. In almost all cases, solving problems, such as about transport or transfer rates or about changes in concentrations or temperatures, starts with drawing up one or more of such balances.

The next step is to derive from such balances proper equations, in many cases differential equations, the latter requiring initial and/or boundary conditions of course. The final step is about solving these (differential) equations to find the answer to the problem under consideration. In this approach, it is essential to denote all quantities with symbols!

The general recipe for drawing up a balance and solving the problem can be summarised as follows:

1. Make a sketch of the situation. Use symbols rather than numerical values to indicate quantities.

2. Select the quantity G that is being transported or transferred in the process under consideration.

3. Select the control volume V about which information is to be obtained.

4. Find out whether and if so, how, the quantity of G in the control volume V changes during a brief period of time ∆t. Draw up the balance (using symbols).

5. Solve the (differential) equation resulting from the balance.

The quantity of G in V can change in all kinds of ways. These should be examined systematically and, if applicable, included in the balance. For example, during ∆t, G can flow into V from outside. As a result, the quantity of G inside V increases. It is also possible for G to flow outwards, from inside V. In this case, the quantity of G in V falls. We refer to inflow and outflow, respectively. Of course, it is also possible for the production of G to occur inside V during period ∆t: as a result, the total quantity of G in V increases. Negative production (= destruction, consumption, annihilation) is also possible, for example, if G stands for the mass of a reagent that is being transformed into another substance in a chemical process.

Bear in mind that G may not necessarily be the quantity in which you are interested. In order to calculate temperature T, for example, a thermal energy balance has to be drawn up, and the thermal energy U must be selected for G rather than T because thermal energy (not temperature) can be transported and transferred.

The general structure for a balance is now as follows (see also Figure 1.1):

The change of G in V during Δt = G at time (t + Δt) in V — G at time t in V

   = quantity of G that flows from outside into V during Δt +

   — quantity of G that flows outside from inside V during Δt +

   + net quantity of G that is produced in V during Δt

From now on, the symbol ϕ will be used to denote a transport (rate), with the dimension ‘quantity of G per unit of time’. Instead of transport rate, the term flow rate is used. The letter P stands for net production per unit of time or net production rate. With the help of this notation, the quantity of G that flows ‘inwards’ (= from the outside to the inside) during the period of time ∆t can, if ∆t is very short, be written as the product of the flow rate ‘in’ at time t and the period of time ∆t:

The same applies to the flow of G from the inside to outside and to the net production during ∆t:

This means the balance is

Dividing both sides of the equation (1.1) by ∆t and taking the limit ∆t → 0 produces

Equation (1.2) is the basic form of the balance and is called the balance equation. The left-hand side therefore stands for the incremental change of the total quantity of G in V, while the three ways in which the total quantity of G in V can change are given on the right-hand side. The left-hand side is also known as ‘unsteady-state term’. All terms in equation (1.2) have the same dimension – should have the same dimension: quantity of G per unit of time.

If, for a given quantity G, the net production...