Thermodynamics

220 | Thermodynamics2 kgH 216 kgO 2FIGURE 5–1Mass is conserved even duringchemical reactions.dA cFIGURE 5–2Control surfaceINTERACTIVETUTORIALSEE TUTORIAL CH. 5, SEC. 1 ON THE DVD.→→V n18 kgH 2 OThe normal velocity V n for a surface isthe component of velocityperpendicular to the surface.nV5–1 ■ CONSERVATION OF MASSConservation of mass is one of the most fundamental principles in nature.We are all familiar with this principle, and it is not difficult to understand.As the saying goes, You cannot have your cake and eat it too! A person doesnot have to be a scientist to figure out how much vinegar-and-oil dressing isobtained by mixing 100 g of oil with 25 g of vinegar. Even chemical equationsare balanced on the basis of the conservation of mass principle. When16 kg of oxygen reacts with 2 kg of hydrogen, 18 kg of water is formed(Fig. 5–1). In an electrolysis process, the water separates back to 2 kg ofhydrogen and 16 kg of oxygen.Mass, like energy, is a conserved property, and it cannot be created ordestroyed during a process. However, mass m and energy E can be convertedto each other according to the well-known formula proposed by Albert Einstein(1879–1955):E mc 2(5–1)where c is the speed of light in a vacuum, which is c 2.9979 10 8 m/s.This equation suggests that the mass of a system changes when its energychanges. However, for all energy interactions encountered in practice, withthe exception of nuclear reactions, the change in mass is extremely smalland cannot be detected by even the most sensitive devices. For example,when 1 kg of water is formed from oxygen and hydrogen, the amount ofenergy released is 15,879 kJ, which corresponds to a mass of 1.76 10 10kg. A mass of this magnitude is beyond the accuracy required by practicallyall engineering calculations and thus can be disregarded.For closed systems, the conservation of mass principle is implicitly used byrequiring that the mass of the system remain constant during a process. Forcontrol volumes, however, mass can cross the boundaries, and so we mustkeep track of the amount of mass entering and leaving the control volume.Mass and Volume Flow RatesThe amount of mass flowing through a cross section per unit time is calledthe mass flow rate and is denoted by ṁ. The dot over a symbol is used toindicate time rate of change, as explained in Chap. 2.A fluid usually flows into or out of a control volume through pipes orducts. The differential mass flow rate of fluid flowing across a small areaelement dA c on a flow cross section is proportional to dA c itself, the fluiddensity r, and the component of the flow velocity normal to dA c , which wedenote as V n , and is expressed as (Fig. 5–2)(5–2)Note that both d and d are used to indicate differential quantities, but d istypically used for quantities (such as heat, work, and mass transfer) that arepath functions and have inexact differentials, while d is used for quantities(such as properties) that are point functions and have exact differentials. Forflow through an annulus of inner radius r 1 and outer radius r 2 , for example, 21dm # rV n dA cdA c A c2 A c1 p 1r 2 2 r 2 1 2 butdm # m # total (total mass flow ratethrough the annulus), not ṁ 2 ṁ 1 . For specified values of r 1 and r 2 , thevalue of the integral of dA c is fixed (thus the names point function and exact 21