In the previous chapter, a Cartesian co-ordinate system was used to analyse the transport between surfaces of constant co-ordinate, and the boundary conditions were specified at a fixed value of the co-ordinate z. For configurations with curved boundaries, such as a cylindrical pipe or a spherical particle, the boundaries are not surfaces of constant co-ordinate in a Cartesian system. It is necessary to apply boundary conditions at, for example, x2 + y2 + z2 = R2 for the diffusion around a spherical particle of radius R. It is simpler to use a co-ordinate system where one of the co-ordinates is a constant on the boundary, so that the boundary condition can be applied at a fixed value of the co-ordinate. Such co-ordinate systems, where one or more of the co-ordinates is a constant on a curved surface, are called curvilinear co-ordinate systems.

The procedure for deriving balance laws for a Cartesian co-ordinate system can be easily extended to a curvilinear co-ordinate system. First, we identify the differential volume or ‘shell’ between surfaces of constant co-ordinate separated by an infinitesimal distance along the co-ordinate. The balance equation is written for the change in mass/momentum/energy in this differential volume in a small time interval Δt. The balance equation is divided by the volume and Δt to derive the differential equation for the field variable. The balance equations for the cylindrical and spherical co-ordinate system are derived in this chapter, and the solution procedures discussed in Chapter 4 are applied to curvilinear co-ordinate systems.

Cylindrical Co-ordinates

Conservation Equation

A cylindrical surface is characterised by a constant distance from an axis, which is the x axis in Fig. 5.1. It is natural to define one of the co-ordinates r as the distance from the axis, and a second co-ordinate x as the distance along the axis. The third co-ordinate ϕ, which is the angle around the x axis, is considered later in Chapter 7. For unidirectional transport, we consider a variation of concentration, temperature, or velocity only in the r direction and in time, and there is no dependence on ϕ and x.

An anecdote about Prof. P. K. Kelkar, founding director of IIT Kanpur and former director of IIT Bombay, was narrated to me by Prof. M. S. Ananth, my teacher and former director of IIT Madras. A distraught young assistant professor at IIT Kanpur approached the director and complained that ‘the syllabus for the course is too long, and I am will not be able to cover everything’. Prof. Kelkar replied, ‘You do not have to cover everything, you should try to uncover a few things.’ In this book, my objective is to uncover a few things regarding transport processes.

The classic books on transport processes, notably the standard text Transport Phenomena by Bird, Stewart and Lightfoot written about 60 years ago, provided a comprehensive overview of the subject organised into different subject areas. At that time, engineers were required to do design calculations and modeling for different unit operations, and for the sequencing of these operations in process design. This required expertise in laboratory and pilot scale experiments on unit operations and scaling up of these operations using correlations. Proficiency in developing, understanding and using design handbooks and correlations was also needed. In this context, the study of transport processes at the microscopic level, and its implications for design for unit operations, was a pioneering advance that has since become an essential part of the chemical engineering curriculum.

In the last half century, sophisticated computational tools have been developed for detailed flow modeling within unit operations, and for the selection and concatenation of unit operations for achieving the required material transformations. The ease of search for information and data today was inconceivable half a century ago. Routine calculations have been automated, and there is little need for routine tasks such as unit conversion, graphical construction and interpreting engineering tables. There is now a greater need for understanding physical phenomena and processes and their mathematical description.

Using a rigorous understanding of transport processes, an engineer usually contributes to process design in one of two ways. The first is the development and enhancement of models and computational tools for modeling of flows and transformations in unit operations; these result in higher resolution, better representation of the essential physics and inclusion of new phenomena.

Calculus is a magical subject. A first encounter in school leads to a radical revision of one's ideas of what is mathematics. We are transported from a rather staid enterprise of counting and measuring to an adventure encompassing change, fluctuation, and a vastly increased ability to understand and predict the workings of the world. At the same time, the student encounters “magic” with both its connotations: awe and wonder on the one hand, mystery and a sense of trickery on the other. Calculus can appear to be a bag of tricks that are immensely useful, provided the apprentice wizard can perfectly remember the spells. As the student pursues mathematics further at university, her instructors may use courses in analysis to persuade her that calculus is a science rather than a mystical art. Alas, all too often the student perceives the new instruction as mere hair-splitting which gives no new powers and may even undermine her previous attainments. The first analysis course is for many an experience that makes them regret taking up higher mathematics.

This book is written to support students in this transition from the expectations of school to those of university. It is intended for students who are pursuing undergraduate studies in mathematics or in disciplines like physics and economics where formal mathematics plays a significant role. Its proper use is in a “calculus with proofs” course taught during the first year of university. The goal is to demonstrate to the student that attention to basic concepts and definitions is an investment that pays off in multiple ways. Old calculations can be done again with a fresh understanding that can not only be stimulating but also protects against error. More importantly, one begins to learn how knowledge can be extended to new domains by first questioning it in familiar terrain. For students majoring in mathematics, this book can serve as bridge to real analysis. For others, it can serve as a base from where they can make expeditions to various applications.

A calculus text written at this point of time can make its plea for existence on the novelty of its exposition and choice of content, rather than any originality in its mathematics. Let me state the case for this book. These explanations are primarily aimed at teachers, but students may also gain some perspective from them if they peruse them after going through at least the first two chapters.

The book aims to give a complete logical framework for calculus, with the proofs reaching the same levels of rigour as a text on real analysis. At the same time it eschews those aspects of analysis that are not essential to a presentation of calculus techniques. So it has the completeness axiom for real numbers, but not Cauchy sequences or the theorems of Heine–Borel and Bolzano–Weierstrass. At the other end of the spectrum, it omits the case for the importance of calculus through its applications to the natural sciences or to economics and finance.

The first chapter provides a description of real numbers and their properties, followed by functions and their graphs. For the most part, the material of this chapter would be known to students, but not in such an organized way. I typically use the first class of the course to ask students to share their thoughts on various issues. What are rational numbers? What are numbers? Which properties of numbers are theorems and which are axioms? What is the definition of a point or a line? What do we mean by a tangent line to a curve? These flow from one to another and from students’ responses. By the end of the hour, with many students firm in their beliefs but finding them opposed just as firmly by others, I have an opportunity to propose that we must carefully put down our axioms and ways of reasoning so that future discussions may be fruitful. One may still ask whether the abstract approach is overdone; is it necessary to introduce general concepts like field and ordered field? The reason for doing so is that it provides a context within which simple questions can be posed and the student can practice creating and writing small proofs as a warm-up to harder tasks that wait ahead.

Integration can be seen as accumulation, the summing up of local changes to get a global result. In the previous chapter, we set up the general process for achieving this. We also saw how a specific kind of information—monotonicity—could be used to obtain global results. With its help we were able to formally define the natural logarithm and exponential functions, which are usually taken for granted in school mathematics.

Further progress requires a closer look at the local behavior of functions. The more we know of the local behavior, the better our chances of extracting global information. These considerations underlie our development of the notions of limit and continuity in this chapter. As applications, we will rigorously develop angles and their radian measures, followed by the trigonometric functions and their properties.

Limits

You have seen in school, the notation lim x!p f (x)= L, which is read as “the limit of f (x) at p is L” and is interpreted as “the values of f (x) approach L as the values of x approach p.” We need a clear definition of what we mean by “approaches.”

Example 3.1.1

Consider f (x) = 2x + 5. What happens if we take values of x that approach 0? Here are some calculations:

We see that as x gets closer to 0, f (x) appears to be getting closer to 5. Can we control this? Can we get the output f (x) close to 5 within any required accuracy level, simply by making the input x appropriately close to 0?

The mass/energy conservation laws are derived for two commonly used co-ordinate systems—the Cartesian co-ordinate system in Section 7.1 and the spherical co-ordinate system in Section 7.2. For unidirectional transport, we have seen that the conservation equation has different forms in different co-ordinate systems. Here, conservation equations are first derived using shell balance in three dimensions for the Cartesian and spherical co-ordinate systems. The conservation equations have a common form when expressed in terms of vector differential operators, the gradient, divergence, and Laplacian operators; the expressions for these operators are different in different co-ordinate systems. The conservation equation derived using shell balance is used to identify the differential operators in the the Cartesian and spherical co-ordinate system, and the procedure for deriving these in a general orthogonal co-ordinate system is explained.

Since the conservation equation is universal when expressed using vector differential operators, it is not necessary to go through the shell balance procedure for each individual problem; it is sufficient to substitute the appropriate vector differential operators in the conservation equation expressed in vector form. It is important to note that the derivation here is restricted to orthogonal co-ordinate systems, where the three co-ordinate directions are perpendicular to each other at all locations.

The discussion in Section 7.1 and 7.2 is restricted to mass/energy transfer. The constitutive relation (Newton's law) for momentum transfer for general three-dimensional flows is more complicated than that for mass/heat transfer. Mass and heat are scalars, and the flux of mass/heat is a vector along the direction of decreasing concentration/temperature. Since momentum is a vector, the flux of momentum has two directions associated with it: the direction of the momentum vector and the direction in which the momentum is transported. Due to this, the stress or momentum flux is a ‘second order tensor’ with two physical directions—the direction of momentum and the orientation of the perpendicular to the surface across which momentum is transported.

The number of independent parameters in a problem is reduced when the dependent and independent parameters are expressed in dimensionless form. In the problem of the settling sphere in Section 1.6.1 and the flow through a pipe in Section 1.6.3, the original problem contained one dependent and four independent dimensional quantities. Using dimensional analysis, this was reduced to one independent and one dependent dimensionless groups. The mass transfer problem in Section 1.6.2 contained one dependent and six independent quantities. The problem was reduced to a relationship between one dependent and two independent dimensionless groups, using dimensional analysis and the assumption that the solute mass and total mass can be considered as different dimensions. In the heat transfer problem in Section 1.6.3, there were one dependent and eight independent dimensional quantities. This was reduced to a relationship between one dependent and three independent dimensionless groups, using dimensional analysis and the assumption that the thermal and mechanical energy can be considered as different dimensions. Thus, dimensional analysis has significantly reduced the number of parameters in the problem.

It is not possible to further simplify the problem using dimensional analysis. In order to progress further, experiments can be carried out to obtain empirical correlations between the dimensionless groups. Another option, pursued in this text, is to do analytical calculations based on a mathematical description of transport processes. Before proceeding to develop the methodology for the analytical calculations, a physical interpretation of the different dimensionless groups is provided in this chapter.

In dimensional analysis, there is ambiguity in the selection of the dimensional parameters for forming the dimensionless groups. This ambiguity is reduced by a physical understanding of the dimensionless groups as the ratio of different types of forces. Here, a broad framework is established for understanding the different dimensionless groups and the relations between them. The forms of the correlations depend on several factors, such as the flow regime, flow patterns and the boundary conditions.

It is important to note that the correlations listed here are indicative, but not exhaustive. Some commonly used correlations are presented to obtain a physical understanding of the terms in the correlation, and to illustrate their application. More accurate correlations applicable in specific domains can be found in specialised handbooks/technical reports.

In this chapter, we take a closer look at the idea that local information about functions should help us resolve integration problems.We already saw that continuity guarantees integrability. Another application of continuity combined with integration was in enabling the definition of the trigonometric functions. On the other hand, continuity did not give us new tools to calculate integrals. In this chapter we shall study the stronger property of differentiability, which will eventually give us techniques for calculating integrals. It also has its own significance, independent of integration. We shall use it for a better understanding of the shapes of graphs of functions.

Among the continuous functions, the ones that are easiest to integrate are the “piecewise linear” ones. Their graphs consist of line segments, such as in the example below:

This suggests that we try to locally approximate functions by straight line segments. If we can get good approximations of this type, we can use them to assess the integral. We shall give the name “differentiable” to functions that can be locally approximated by straight lines. Most of this chapter is devoted to identifying these functions and to calculating the corresponding straight lines. Then we make the first connection between the processes of differentiation and integration, the so-called first fundamental theorem of calculus. Finally, we see that differentiation has a life of its own, and we use it to explore the problems of finding the extreme values and sketching the graph of a function.

Derivative of a Function

Let us consider what happens if we zoom in for a closer look at the graph of a function such as y = x2, near a point such as (1,1).

We see that the graph of the function y = x2 looks more like the line y = x2 - 1 as we zoom in towards (1,1), and at some stage becomes indistinguishable from it.

This can happen even for functions with rapid oscillations. Let us look at the function defined by y = x2 cos(1/x) if x ≠ 0, and y = 0 if x = 0, near the origin.

No matter how much we zoom in, the function has infinitely many oscillations. Nevertheless, their amplitudes decrease and in that sense the function becomes closer to the line y = 0 as we zoom in.

Transport of heat/mass is enhanced by an externally generated flow past an object or a surface in ‘forced convection’. Here, the flow is specified, and it is not affected by the change in temperature/concentration due to the heat/mass transfer. The known fluid velocity field is substituted into the convection–diffusion equation in order to determine the temperature/concentration field and the transport rate.

In the previous chapter, we examined the limit of low Peclet number, where transport due to convection is small compared that due to diffusion. There, the approach was to neglect convection altogether, and solve the diffusion equation. In the limit of high Peclet number, an equivalent approach would be to neglect diffusion altogether, and solve the convection equation to obtain the concentration/temperature fields. This approach is not correct for the following mathematical and physical reasons.

Mathematically, when the diffusion term is neglected, the convection–diffusion equation is reduced from a second order to a zeroth order differential equation in the cross-stream co-ordinate. The second order differential for the concentration/temperature field is well posed only if two boundary conditions are specified in each co-ordinate. When diffusion is neglected, the resulting zeroth order equation cannot satisfy both boundary conditions in the cross-stream co-ordinate specified for the original problem. Physically, when diffusion is neglected, there is transport due to convection only along fluid streamlines, and there is no transport across the streamlines. The concentration/temperature is a constant along streamlines in the flow. At bounding surfaces (the pipe surface in a heat exchanger, or particle surfaces in the case of suspended particles), there is no flow perpendicular to the surface. When we neglect diffusion, there is no flux across the surface. Therefore, we obtain the unphysical result that there is no mass/heat transfer across the surface.

A more sophisticated approach is required to obtain solutions for transport in strong convection, based on the following physical picture. In the limit of high Peclet number, mass or heat diffusing from a surface gets rapidly swept downstream due to the strong convection, and so the concentration/temperature variations are restricted to a thin ‘boundary layer’ close to the surface.

Calculus has two parts: differential and integral. Integral calculus owes its origins to fundamental problems of measurement in geometry: length, area, and volume. It is by far the older branch. Nevertheless, it depends on differential calculus for its more difficult calculations, and so nowadays we typically teach differentiation before integration.

We shall revert to the historical sequence and begin our journey with integration. Our first reason is that it provides a direct application of the completeness axiom without needing the concept of limits. The second is that important functions such as the trigonometric, exponential, and logarithmic functions are most conveniently constructed through integration. Finally, the student should become aware that integration is not just an application of differentiation or a set of techniques of calculation.

Suppose we wish to find the area of a shape in the Cartesian plane.We can, at least, estimate it by comparing the shape with a standard area, that of a square.We cover the shape with a grid of unit squares and count how many squares touch it, and also how many squares are completely contained in it. This gives an upper and a lower estimate for the area.We can obtain better estimates by taking finer grids with smaller squares. The figures given immediately below illustrate this process of iteratively improving the estimates.

We have said that we are estimating area. But what is our definition of area? In school books you will find descriptions such as “Area is the measure of the part of a plane or region enclosed by the figure.” It should be evident that this is not a very useful prescription. It means nothing without a description of the measuring process. In fact, the estimation process described above could become the basis for a meaningful definition of area, by requiring it to be a number that lies above all the lower estimates produced by the process, and below all the upper estimates. Its existence would be guaranteed by the completeness axiom. This is a promising start, but the sceptic can raise various objections that would have to be answered:

• 1. Could there be a figure for which multiple numbers satisfy the definition of its area?

• 2. If we slightly shifted or rotated the grids, could that change our calculation? That is, could moving a figure change its area?

In the analysis of transport at high Peclet number in Chapter 9, it was assumed that the fluid velocity field is specified, and is not affected by the concentration or temperature variations. There are situations, especially in the case of heat transfer, where variations in temperature cause small variations in density, which results in flow in a gravitational field due to buoyancy. Examples of these flows range from circulation in the atmosphere to cooking by heating over a flame. In the former, air heated by the earth's surface rises and cold air higher up in the atmosphere descends due to buoyancy; in the latter, hotter and lighter fluid at the bottom rises due to buoyancy and is replaced by colder and heavier fluid at the top, resulting in significantly enhanced heat transfer.

The heat transfer due to natural convection from heated objects is considered here, and correlations are derived for the Nusselt number as a function of the Prandtl number and the Grashof number. The Prandtl number is the ratio of momentum and thermal diffusion. The Grashof number, defined in Section 2.4 (Chapter 2), is the square of the Reynolds number based on the characteristic fluid velocity generated by buoyancy. In order to determine the heat transfer rate, it is necessary to solve the coupled momentum and energy equations, the former for the velocity field due to temperature variations and the latter for the temperature field. The equations are too complex to solve analytically, and attention is restricted to scaling the equations to determine the relative magnitudes of convection, diffusion and buoyancy. We examine how the dimensionless groups emerge when the momentum and energy equations are scaled, and how these lead to correlations for the Nusselt number. The numerical coefficients in these correlations are not calculated here.

Boussinesq Equations

Consider a heated object with surface temperature T0, in a ambient fluid with temperature T∞ far from the object, as shown in Fig. 10.1 The fluid density is ρ∞ far from the object, but the temperature variation causes a variation in the density near the object. This density variation results in a buoyancy force, which drives the flow.