Fundamental Principles of Nuclear Engineering(核工程基本原理）

1Fundamentals of Mathematics and Physics

In engineering field, some practical problems cannot be adequately solved using arithmetic and algebra only. Advanced mathematical tools such as calculus and integral are needed to understand physical process used in nuclear engineering.

1.1Calculus

Arithmetic involves the fixed values of numbers. Algebra involves both literal and arithmetic numbers,which still has fixed values in a given calculation although the literal numbers in algebraic problems can change during calculation.

Here some examples are given. When a weight is dropped and allowed to fall freely, its velocity changes continually. The electric current in an alternating current circuit changes continually. Both of these quantities have a different value at successive instants of time. Physical systems that involve quantities that change continually are called dynamic systems. The solution of problems which involving dynamic systems often need different mathematical techniques from  arithmetic and algebra. Calculus involves all the same mathematical techniques involved in arithmetic and algebra, such as addition, subtraction, multiplication, division, equations, and functions, but it also involves several other techniques. These techniques are not difficult to understand because they can be developed using familiar physical systems, but they do involve new ideas and terminology.

There are many dynamic systems encountered in nuclear engineering field. The decay of radioactive materials, the startup of a reactor, and the  power change of a turbine generator all involve quantities change with time. An analysis of these dynamic systems involves calculus. Calculus is the most helpful tools to understand certain of the basic ideas and terminology which is involved in nuclear facility field, though detailed understanding of calculus is not required for the operational aspect.

These ideas and terminology are encountered frequently, and a brief introduction to the basic ideas and terminology of the mathematics of dynamic systems is discussed in this chapter.

1.1.1Differential and Derivative

In mathematics, differential is a tool to describe the local characteristic of a function using linear techniques. Suppose a function is defined in a region. x0 and x0+Δx are two points (value) in this region. Then the incremental change of the function can be expressed as1: 

Δy=f(x0+Δx)－f(x0)(11)

Using local linear technique, it can be expressed as:  

Δy=A·Δx+o(Δx)(12)

where, A is a constant number independent with Δx, o(Δx) is a higher order infinite small of Δx. We call the function y=f(x) is derivable near the point of x0 and A·Δx is called as the differential of the function y =f(x) at point x0 corresponding to Δx (the incremental change of argument x). It is denoted as dy. The incremental change of argument x is the differential of x. It is denoted as dx. So we get: 

dy=Adx(13)

Here we use an example in physics to explain the concept of differential. One of the most commonly encountered mathematical applications of the dynamic system is the relationship of  position and time of a moving object. Figure 11 represents an object moving in a straight line from position P1 to position P2. The distance to P1 from a fixed reference point, point O, along the line of travel is represented by S1;  the distance to P2 from point O by S2. 

Figure 11Motion Between Two Points

If the time recorded by a timer, when the object is at position P1 is t1, and if the time when the object is at position P2 is t2, then the average velocity of the object between points P1 and P2 equals the distance traveled, divided by the time elapsed as Equation (14).

vav=S2－S1t2－t1(14)

俞冀阳，清华大学工程物理系核能科学与工程管理研究所，副教授，博导。1994年毕业于清华大学工程物理系，1999年获清华大学工学博士后在清华大学工程物理系任教。主要从事核反应堆工程与安全方面的人才培养和科学研究工作。已出版的教材与专著有：《反应堆热工水力学》、《热工流体数值计算》、《核电厂事故分析》、《核心理学》、《核动力装置设计与优化原理》、《核工程基本原理》等。英文版专著有国际原子能机构出版的IAEA-TECDOC-1395《Comparison of Heavy Water Reactor Thermalhydraulic Code Predictions with Small Break LOCA Experimental Data》。

本书着力于核工程所涉及领域的基本原理，打通各个领域的壁垒，使核工程所涉及到的各个领域的基本原理融会贯通，使读者能够掌握全面的知识体系。