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.

When calculus is applied to problems of subjects like physics and economics, it usually leads to equations involving the first and second derivatives of functions, and the task is to recover the original function from these equations. If the derivative is completely known and is continuous, we can use the second fundamental theorem

However, we usually have only a relation between the function and its derivatives rather than a full knowledge of the derivative. For example, we may know that f (x)= f (x)2 for every x. So, we need to find more ways of relating information about f with information about f .We already have two important instances: Fermat's theorem and the monotonicity theorem. In this chapter we will explore several consequences of these results. The payoffs will be new techniques of calculating limits (§6.2), approximation of functions by polynomials (§6.3), use of integration to measure arc length, surface area, and volume (§6.4), and error estimates for numerical calculations of integrals (§6.5). Sections 6.2 and 6.3 are required for the final two chapters on sequences and series, while sections 6.4 and 6.5 are important for the applications of calculus.

Darboux's theorem says that if a function f is differentiable on an interval then f will have the intermediate value property on that interval. Thus, f cannot have a jump discontinuity and it behaves like a continuous function in some ways. Nevertheless, it need not be continuous or even bounded.

As an example, consider the function defined by f (0) = 0 and f (x) = x2 sin(1/x) when x ≠ 0. This function is differentiable at non-zero points by the chain rule. It is also differentiable at zero by a direct calculation:

Thus, f is differentiable at every point. However, f is not continuous at zero: which does not exist.

We can modify the above example slightly to get a function that is differentiable but whose derivative is not bounded. Define g(0) = 0 and g(x) = x3/2 sin(1/x) when x ≠ 0. We have

We say f is continuously differentiable on an interval I if it is differentiable on I and f is continuous on I.

Appendix: Solutions to Odd-Numbered Exercises

Significance of Transport Processes

The conversion of raw materials into useful products in a predictable, efficient, economical and environment-friendly manner is an essential part of many branches of engineering. There are two types of transformations: chemical transformations (involving chemical reactions) and physical transformations (melting, evaporation, filtering, mixing, etc.). Both of these transformations involve the motion of constituents relative to each other, and they often involve the transfer of energy in the form of heat. In operations involving fluid flow and mixing, there are forces exerted on the fluid due to pumps, impellers, etc. (input of mechanical energy), in order to overcome the frictional resistance generated by the flow. The subject of this text is the transport of the components in materials relative to each other, the transport of heat energy and the transport of momentum due to applied forces.

This text is limited to operations carried out in the fluid phase. Although solids transport and mixing does form an important part of material transformation processes, fluid-phase operations are the preferred mode for conversion because the transport is enabled by the two fundamental processes: convection and diffusion. Convection is the transport of mass, momentum and energy along with the flowing fluid. Diffusion is transport due to the fluctuating motion of the molecules in a fluid, which takes place even in the absence of fluid flow. Convection does not take place in solids since they do not flow, and diffusion in solids due to vacancy or interstitial migration is a very slow process, which makes it infeasible to effect material transformations over industrial timescales.

Fluids are of two types: liquids and gases. In liquids, the molecules are closely packed, and the distance between molecules is comparable to the molecular diameter. In contrast, in gases, the distance between molecules is about 10 times larger than the molecular diameter under conditions of standard temperature and pressure (STP). Due to this, the density of a liquid is about 103 times that of a gas. In a gas, the molecules interact through discrete collisions, and the period of a collision is much smaller than the average time between collisions.

The two transport mechanism considered in this text are convection and diffusion. Convection is transport due to the flow. It is directional, and takes place only along the flow streamlines. Transport across streamlines, and transport across surfaces (where there is no fluid velocity perpendicular to the surface) necessarily takes place due to diffusion.

Diffusion is the process by which material is transported by the random thermal motion of the molecules within the fluid, even in the absence of fluid flow. The random velocity fluctuations of the molecules are isotropic, and they have no preferred direction. The characteristic velocity and length for the thermal motion are the molecular velocity and the microscopic length scale, which is the molecular size in a liquid or the mean free path (distance between intermolecular collisions) in a gas. While random molecular motion is always present in fluids, when the concentration/temperature/velocity fields are uniform, there is no net transport due to the random motion. Diffusion takes place only when there is a spatial variation, and transport is along direction of variation.

The molecular mechanisms of mass, momentum and thermal diffusion, are discussed in this chapter. Constitutive relations for the fluxes are derived from a molecular description, and the diffusion coefficients are estimated.

The gas diffusivities are estimated using kinetic theory for an ideal gas made of hard spheres, which undergo instantaneous collisions when the surfaces are in contact, but which do not exert any intermolecular force when not in contact. Real gas molecules do not interact like hard spheres—the interaction force between molecules is repulsive at small separations and attractive at larger separations. Diatomic and polyatomic molecules are also not spherically symmetric, and their interaction depends on the relative orientation of the molecules. The diffusion coefficients in the hard sphere model are proportional to √T, where T is the absolute temperature. For molecules with continuous intermolecular potential, the diffusion coefficients are proportional to a power of the temperature which higher than ½. The pressure-density relationship for real gases is also more complicated than that for an ideal gas, and the virial corrections need to be included for dense gases.

In calculus, we mainly study continuous change. However, there are situations where discrete changes have to be considered. For example, when we try to describe a number such as _ by its decimal representation, we actually create an iterative process of successively better approximations: 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, and so on. A similar situation arises when we work with the Taylor polynomials of a function— we successively approximate a function by polynomials of increasing degree. What is common to the two examples is that there is a first stage, a second stage, a third stage, and we are interested in what happens as we keep going. Clearly, we need to develop a theory of limits for this context.We shall do so in this chapter. Further, we shall work out in detail the situation when discrete changes accumulate and we are interested in the total. This will have many similarities as well as a direct relation with integration.

As an example, let us consider a geometry problem that leads to an iterative method for approximating square roots by fractions. It is named Heron's method after a Greek mathematician, but the evidence is strong that this kind of reasoning was carried out earlier in ancient Iraq and India, three to four thousand years ago. The statement of the problem is: “Given a rectangle, construct a square with the same area.” Now, if the rectangle has sides a and b, the square needs to have side √ab. To us, this may be a triviality, but what if the only numbers you know are the fractions? Then the problem will, in general, have only approximate solutions. How do we find good fractional approximations to √ab? Consider the following steps.

The final square is obviously a bit too big. Nevertheless, its side of (a + b)/2 is visibly better than the initial sides of a and b. If we have ab = N, we can repeat the process with a rectangle whose sides are a = (a + b)/2 and b0 = N/a. This will lead to a new and further improved square with side (a + b)/2.

This chapter explores two key concepts in contract law. First, it identifies the parties to a contract and the rules that help in that process. In particular, the doctrines of privity and agency, which assist with determining who incurs rights and obligations under a contract, are discussed. Second, the chapter considers the terms of a contract, including how to identify, incorporate and interpret them. Specific attention is paid to the various types of contract terms and how they should be interpreted.

This chapter considers how intangible assets that businesses develop, such as inventions, designs and brands, as well as business ideas and information, can be protected pursuant to Australian intellectual property (IP) laws. It identifies that various IP rights are protected by statute, while others are protected pursuant to common law. It explains that some forms of intellectual property require businesses to apply and register for protection before an IP right can be claimed, such as designs, patents and trade marks, while others, such as copyright, dont require any application or registration. The chapter highlights that an array of IP rights can be used to protect different aspects of a business’s goods and/or services. For example, a business’s product might have a patent that attaches to it regarding how it works, a registered trade mark protecting its brand name, and a design for its appearance. The more IP rights that can be used to protect a particular good or service, the more defiant the good or service will be to imitation and competition.

There are many taxes in Australia that operate at the federal, state and local levels. Taxpayers must understand their tax obligations by identifying which taxes apply to them and understanding the requirements they must meet in order to comply with the tax laws. Federal tax laws are administered by Australia’s federal revenue authority, the Australian Taxation Office (ATO), and state taxes are administered by state-based revenue authorities. This chapter will cover two types of federal taxes: income tax and the goods and services tax (GST).

In an ideal world, people who enter into contracts would choose to enter the contract, and agree to its terms, because they have accurate and comprehensive information on which to base their decision. This chapter will first explain when a contract may be invalid, because one or both parties entered into the contract under some sort of misapprehension, or on the basis of misinformation. We will look at mistake (mutual mistake, unilateral mistake and common mistake) and misrepresentation. We will also briefly explain the old action ‘non est factum’ (‘not my deed’) and the remedy of rectification. Second, the chapter will explore when a contract may be invalid because of ‘unfair’ conduct by one of the parties – for example, where one party (Party A) enters an agreement because another party (Party B) subjected A to undue pressure (duress or undue influence); or because the conduct of B is so ‘unconscientious’ or the contract is able to be set aside in equity for unconscionable conduct.

This chapter examines the law of sale of goods. The statutory regime across the states and territories is explained before the specific concept of contracts for the sale of goods is discussed. The chapter then considers the various implied terms that become a part of such contracts and the consequences for violation of these terms. A brief discussion of the various rules pertaining to delivery follows, before the chapter concludes with an outline of the various remedies available to aggrieved parties when sale of goods contracts are breached.

This chapter will introduce the idea of ‘ethics’, and then the subset of ‘business ethics’. You will read real-world examples of (often poor) behaviour from companies, and understand how that behaviour can be considered through the lenses of business ethics, the ethical director, corporate governance with a focus on corporate social responsibility, and ethical marketing and advertising. By the end of this chapter, you will have a broad understanding of how these concepts fit together, and how they interact with the legal regulations around companies and businesses discussed in other parts of this text.