The conservation of mass principle is one of the most fundamental principles in nature. We are al... more The conservation of mass principle is one of the most fundamental principles in nature. We are all familiar with this principle, and it is not difficult to understand. A person does not have to be a rocket scientist to figure out how much vinegar-and-oil dressing will be obtained by mixing 100 g of oil with 25 g of vinegar. Even chemical equations are balanced on the basis of the conservation of mass principle. When 16 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 of hydrogen and 16 kg of oxygen. Technically, mass is not exactly conserved. It turns out that mass m and energy E can be converted to each other according to the well-known formula proposed by Albert Einstein (1879-1955): E = m c 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 there is equivalence between mass and energy. All physical and chemical systems exhibit energy interactions with their surroundings , but the amount of energy involved is equivalent to an extremely small mass compared to the system's total mass. For example, when 1 kg of liquid water is formed from oxygen and hydrogen at normal atmospheric conditions, the amount of energy released is 15.8 MJ, which corresponds to a mass of only 1.76 × 10-10 kg. However, even in nuclear reactions, the mass equivalence of the amount of energy interacted is a very small fraction of the total mass involved. Therefore, in most engineering analyses, we consider both mass and energy as conserved quantities. For closed systems, the conservation of mass principle is implicitly used by requiring that the mass of the system remain constant during a process. For control volumes, however, mass can cross the boundaries, and so we must keep track of the amount of mass entering and leaving the control volume. Mass and Volume Flow Rates The amount of mass flowing through a cross section per unit time is called the mass flow rate and is denoted by m ˙. The dot over a symbol is used to indicate time rate of change. A fluid flows into or out of a control volume, usually through pipes or ducts. The differential mass flow rate of fluid flowing across a small area element dA c in a cross section of a pipe is proportional to dA c itself, the fluid density ρ, and the component of the flow velocity normal to dA c , which we denote as V n , and is expressed as (Fig. 5-2) δ m ˙ = ρ V n d A c (5-2) Note that both δ and d are used to indicate differential quantities, but δ is typically used for quantities (such as heat, work, and mass transfer) that are path functions and have inexact differentials, while d is used for quantities (such as properties) that are point functions and have exact differentials. For flow through an annulus of inner radius r 1 and outer radius r 2 , for example, FIGURE 5-1 Mass is conserved even during chemical reactions. 2 kg H 2 16 kg O 2 18 kg H 2 O + FIGURE 5-2 The normal velocity V n for a surface is the component of velocity perpendicular to the surface. dA c V n
The conservation of mass principle is one of the most fundamental principles in nature. We are al... more The conservation of mass principle is one of the most fundamental principles in nature. We are all familiar with this principle, and it is not difficult to understand. A person does not have to be a rocket scientist to figure out how much vinegar-and-oil dressing will be obtained by mixing 100 g of oil with 25 g of vinegar. Even chemical equations are balanced on the basis of the conservation of mass principle. When 16 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 of hydrogen and 16 kg of oxygen. Technically, mass is not exactly conserved. It turns out that mass m and energy E can be converted to each other according to the well-known formula proposed by Albert Einstein (1879-1955): E = m c 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 there is equivalence between mass and energy. All physical and chemical systems exhibit energy interactions with their surroundings , but the amount of energy involved is equivalent to an extremely small mass compared to the system's total mass. For example, when 1 kg of liquid water is formed from oxygen and hydrogen at normal atmospheric conditions, the amount of energy released is 15.8 MJ, which corresponds to a mass of only 1.76 × 10-10 kg. However, even in nuclear reactions, the mass equivalence of the amount of energy interacted is a very small fraction of the total mass involved. Therefore, in most engineering analyses, we consider both mass and energy as conserved quantities. For closed systems, the conservation of mass principle is implicitly used by requiring that the mass of the system remain constant during a process. For control volumes, however, mass can cross the boundaries, and so we must keep track of the amount of mass entering and leaving the control volume. Mass and Volume Flow Rates The amount of mass flowing through a cross section per unit time is called the mass flow rate and is denoted by m ˙. The dot over a symbol is used to indicate time rate of change. A fluid flows into or out of a control volume, usually through pipes or ducts. The differential mass flow rate of fluid flowing across a small area element dA c in a cross section of a pipe is proportional to dA c itself, the fluid density ρ, and the component of the flow velocity normal to dA c , which we denote as V n , and is expressed as (Fig. 5-2) δ m ˙ = ρ V n d A c (5-2) Note that both δ and d are used to indicate differential quantities, but δ is typically used for quantities (such as heat, work, and mass transfer) that are path functions and have inexact differentials, while d is used for quantities (such as properties) that are point functions and have exact differentials. For flow through an annulus of inner radius r 1 and outer radius r 2 , for example, FIGURE 5-1 Mass is conserved even during chemical reactions. 2 kg H 2 16 kg O 2 18 kg H 2 O + FIGURE 5-2 The normal velocity V n for a surface is the component of velocity perpendicular to the surface. dA c V n
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