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Conservation of Mass

Objectives How to use the principle of continuity to analyze simple flow systems.

Some of the most important fundamental principles used to analyze engineering systems are the conservation laws. A conservation law is an immutable law of nature declaring that certain physical quantities are conserved. Stated another way, a conservation law states that the total amount of a particular physical quantity is constant during a process. A familiar conservation law is the first law of thermodynamics, which states that energy is conserved. According to the first law of thermodynamics, energy may be converted from one form to another, but the total energy is constant. Another conservation law is Kirchhoff's current law, which states that the algebraic sum of the currents entering a circuit node is zero. Kirchhoff's current law is a statement of the law of conservation of electric charge. Other quantities that are conserved are linear and angular momentum.

In this section, we examine the principal conservation law used in fluid mechanics, the law of conservation of mass. Like the first law of thermodynamics, the law of conservation of mass is an intuitive concept. To introduce the conservation of mass principle, consider the system shown in Figure 10. The system may represent any region in space chosen for analysis. The boundary of the system is the surface that separates the system from the surroundings. We may construct a mathematical representation of the conservation of mass principle by applying a simple physical argument. If an amount of mass, m_in, is supplied to the system, that mass can either leave the system or accumulate within the system, or both. The mass that leaves the system is m_out, and the change in mass within the system is m. Thus, the mass that enters the system equals the mass that leaves the system plus the change in mass within the system. The conservation of mass principle may therefore be expressed mathematically as

 

We see that the conservation of mass law is nothing more than a simple accounting principle that maintains the system's “mass ledger” in balance. In fact, this conservation law is often referred to as a mass balance because that is precisely what it is. Equation 7-21) is more useful when expressed as a rate equation. Dividing each term by a time interval t, we obtain

 

where and are the inlet and outlet mass flow rates, respectively, and mt is the rate at which mass accumulates within the system. The law of conservation of mass is called the continuity principle, and Equation 7-22, or a similar relation, is referred to as the continuity equation.

We now examine a special case of the configuration given in Figure 10. Consider the converging pipe shown in Figure 11. The dashed line outlines the boundary of the flow system defined by the region inside the pipe wall and between sections 1 and 2. A fluid flows at a constant rate from section 1 to section 2. Because fluid does not accumulate between sections 1 and 2, mt=0, and Equation 7-22 becomes

 

where the subscripts 1 and 2 denote the input and output, respectively. Thus, the mass of fluid flowing past section 1 per unit time is the same as the mass of fluid flowing past section 2 per unit time. Because m·=rQ, Equation 7-23 may also be expressed as

. The law of conservation of mass.

 

where and Q denote density and volume flow rate, respectively. Equation 7-23, and its alternative form, Equation 7-24, are valid for liquids and gases. Hence, these relations apply to compressible and incompressible fluids. If the fluid is incompressible, the fluid density is constant, so ₁=₂=. Dividing Equation 7-24 by density, , yields

 

which may be written as

 

where A and V refer to cross-sectional area and average velocity, respectively. Equations 7-25 and 7-26 apply strictly to liquids, but these relations may also be used for gases with little error if the velocities are below approximately 100 m/s.

. Continuity principle for a converging pipe.

The continuity principle can also be used to analyze more complex flow configurations, such as a flow branch. A flow branch is a junction where three or more conduits are connected. Consider the pipe branch shown in Figure 12. A fluid enters a junction from a supply pipe where it splits into two pipe branches. The flow rates in the branching pipes depend on the size of the pipes and other characteristics of the system, but from the continuity principle it is clear that the mass flow rate in the supply pipe must equal the sum of the mass flow rates in the two pipe branches. Thus, we have

 

Junctions in flow branches are analogous to nodes in electrical circuits. Kirchhoff's current law, which is a statement of the law of conservation of electric charge, states that the algebraic sum of the currents entering a node is zero. For a flow branch, the continuity principle states that the algebraic sum of the mass flow rates entering a junction is zero. The mathematical expression for this principle is similar to Kirchhoff's current law, and is written

 

The continuity relation given by Equation 7-27 for the specific case illustrated in Figure 12 is equivalent to the general form of the relation given by Equation 7-28. We have

 

where minus signs are used for mass flow rates and because the fluid in each pipe branch is leaving the junction. The mass flow rate, , is positive because the fluid in the supply pipe is entering the junction.

. A pipe branch.

In the following example, we analyze a basic flow system using the general analysis procedure of: (1) problem statement, (2) diagram, (3) assumptions, (4) governing equations, (5) calculations, (6) solution check, and (7) discussion.

EXAMPLE

Problem Statement A converging duct carries oxygen (=1.320 kg/m3) at a mass flow rate of 110 kg/s. The duct converges from a cross-sectional area of 2 m2 to a cross-sectional area of 1.25 m2. Find the volume flow rate and the average velocities in both duct sections.   Diagram The diagram for this problem is shown in Figure 13. . Converging duct for Example 7.6.   Assumptions . The flow is steady. The fluid is incompressible. There are no leaks in the duct.   Governing Equations Two equations are needed to solve this problem, the relation for mass flow rate and the continuity relation.   Calculations By continuity, the volume flow rate and mass flow rate are equal at sections 1 and 2. The volume flow rate is The average velocity in the large section is and the average velocity in the small section is   Solution check After a careful review of our solution, no errors are found.   Discussion Note that velocity and cross-sectional area are inversely related. The velocity is low in the large portion of the duct and high in the small portion of the duct. The maximum velocity in the duct is below 100 m/s, so the oxygen may be considered an incompressible fluid with little error. Our assumption that the fluid is incompressible is therefore valid.

 

Application: Analyzing a Pipe Branch

Pipe branches are used frequently in piping systems to split a flow into two or more flows. Consider a pipe branch similar to the one shown in Figure 12. Water enters the pipe junction at a volume flow rate of 350 gal/min, and the flow splits into two branches. One pipe branch has an inside diameter of 7 cm, and the other pipe branch has an inside diameter of 4 cm. If the average velocity of the water in the 7-cm branch is 3 m/s, find the mass flow rate and volume flow rate in each branch and the average velocity in the 4-cm branch. A flow schematic with the pertinent information is shown in Figure 14. . Flow schematic for the preceding Example. First, we convert the volume flow rate in the supply pipe to m3/s. The cross-sectional areas of the pipes branches are The volume flow rate in the 7-cm pipe branch is and the mass flow rate is In order to find the flow rates in the other pipe branch, the continuity principle is required. Solving for Q3, we obtain and the corresponding mass flow rate is Finally, the average velocity in the 4-cm branch is As a way of checking for errors, we use the continuity principle to assure that our flow rates are correct. The volume flow rate into the junction must equal the sum of the volume flow rates out of the junction. Thus, The flow rates balance, so our answers are correct. Notice that the flow rates in the pipe branches are nearly equal (11.54 kg/s and 10.54 kg/s), but the velocities are quite different (3 m/s and 8.38 m/s). This is due to the difference in diameters of the pipes. The velocity is nearly three times higher in the 4-cm pipe than in the 7-cm pipe.

 

Practice!

1. Water flows through a converging pipe at a mass flow rate of 25 kg/s. If the inside diameters of the pipes sections are 7 cm and 5 cm, find the volume flow rate and the average velocity in each pipe section.
2. A fluid flows through a pipe whose diameter decreases by a factor of three from section 1 to section 2 in the direction of the flow. If the average velocity at section 1 is 10 ft/s, what is the average velocity at section 2
3. Air enters a junction in a duct at a volume flow rate of 2000 CFM. Two square duct branches, one measuring 12 in×12 in and the other measuring 16 in×16 in, carry the air from the junction. If the average velocity in the small branch is 20 ft/s, find the volume flow rates in each branch and the average velocity in the large branch.
4. Two water streams, a cold stream and a hot stream, enter a mixing chamber where both streams combine and exit through a single tube. The mass flow rate of the hot stream is 5 kg/s, and the inside diameter of the tube carrying the cold stream is 3 cm. Find the mass flow rate of the cold stream required to produce an exit velocity of 8 m/s in a tube with an inside diameter of 4.5 cm.