This extended section can be found on the website www.cambridge.org/kleinandnellis. Maple is an application that can be used to analytically solve algebraic and differential equations. The capability to differentiate, integrate and algebraically manipulate mathematical expressions in symbolic form can be a very powerful aid in solving many types of engineering problems, including some thermodynamics problems. Maple also provides a very convenient mathematical reference; if, for example, you've forgotten that the derivative of sine is cosine, it is easy to use Maple to quickly provide this information. Therefore, Maple can replace the numerous mathematical reference books that might otherwise be required to carry out all of the integration, differentiation, simplification, etc. that is required to solve many engineering problems. Maple and EES can be used effectively together; Maple can determine the analytical solution to a problem and these symbolic expressions can subsequently be copied (almost directly) into EES for convenient numerical evaluation and manipulation in the context of a specific application. This appendix summarizes the commands that are the most useful for thermodynamics problems.

In this chapter, we discuss (1) time averaging in relation to local volume averaging and (2) proper order of time-volume averaging versus volume-time averaging.

Time averaging in relation to local volume averaging

The averaging procedure in multiphase mechanics must be related and can be understood by considering the basis of experimental observation. The relative magnitudes of three quantities determine the method and meaning of averaging. They are the size of the dispersed phase, the spacing between the elements of the dispersed phase, and the volume observed. When applied to a two-phase boiling system, they become the size of bubbles, the mean spacing between bubbles, and the size of observation “window” (or any probe of finite size). For a one-dimensional system, these quantities are bubble size D, bubble spacing S, and observation window or slit width L.

To provide a physical interpretation of Eq. (B.1), which is Eq. (2.4.9) with γv = 1, we consider a dispersed system and an averaging volume in the shape of a rectangular parallelepiped ΔxΔyΔz with its centroid located at (x, y, z), as illustrated in a. Its top view is shown b.

Clearly, for those elements of the dispersed phase k that are completely inside the averaging volume, where δAk is the closed surface of the element. Such an element, labeled ⓐ in b, may be a bubble or a droplet, spherical or nonspherical. Next, we consider those elements of the dispersed phase that are intersected by the boundary surface ΔAx + (Δx/2). One such element is labeled ⓑ in b. The unit outdrawn normal vector nk can be represented by where i, j, and k are unit vectors pointing in the positive directions of x, y, and z-axis, respectively, and e1, e2, and e3 are the direction cosines of nk. If we denote the portion of the interfacial area of element ⓑ that is inside the averaging volume v by δAk, [x + (Δx/2)], and its area of intersection with the surface ΔAx + (Δx/2) by δAk, x + (Δx/2), then