Chapter 2: The Law of Conservation of Mass
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1 Chapter 2: The Law of Conservation of Mass Table of Contents The Law of Conservation of Mass... 2 Assumption: Steady State / Steady Flow (SSSF)... 2 Mass Flow Rate... 2 Assumption: 1D Flow... 3 Assumption: Uniform Flow... 3 Example... 4 Example... 5 Example... 6 Mass Flow Contributions to the 1 st Law... 7 Flow Work... 8 Enthalpy... 9 Example Example
2 The Law of Conservation of Mass The Law of Conservation of Mass (COM) is: In words, the rate of change of mass within a system is equal to the rate at which mass enters the system minus the rate at which mass leaves the system. The mass flow rate ( ) has units of kg/s and is the mass of fluid flow per unit of time. Assumption: Steady State / Steady Flow (SSSF) In many cases we can assume steady state/steady flow (SSSF) when analyzing systems. Steady state means the fluid properties within the system do not vary with time (this includes properties at the system boundaries, plus at the inlet and outlet). Steady flow means the mass flow rates into and out of the system do not vary with time. Applying the SSSF assumption to the COM gives: We can further simplify the COM for SSSF systems having a single inlet and a single outlet: Mass Flow Rate Consider a system having a single inlet and a single outlet. The inlet has cross sectional area and the outlet has cross sectional area. Fluid flowing into the system has average velocity and density. Similarly, fluid flowing out of the system has average velocity and density. Divide the cross sectional area of the inlet into infinitesimally small areas (). The volume of fluid flowing through during time is the product of the length of fluid normal to the surface ( ) and the area : 2
3 The mass of fluid flowing through during time is the product of volume and fluid density : Rearranging gives an expression for, the mass flow rate: We will make two key assumptions regarding fluid flow in this course: one dimensional flow (1D flow) and uniform flow. Assumption: 1D Flow Assuming 1D flow means we assume the fluid velocity is always parallel to, where is the unit vector normal to the cross sectional area. This gives. Assumption: Uniform Flow Assuming uniform flow means the state of the fluid is the same everywhere on cross section. Fluid properties, including fluid velocity and fluid density, are constant with respect to and can therefore be pulled out of the integral expression for : The numerator of the mass flow rate,, is the volumetric flow rate. Volumetric flow rate has units of m 3 /s. 3
4 Example The swimming pool used for the 2016 U.S. Olympic Trials was a temporary structure, assembled and disassembled within a few weeks. A similar temporary pool was used for the 2008 and 2012 Olympic Trials. You can watch a time lapse video of the pool construction here: a. Estimate the volume of water in the filled pool (gallons). b. Estimate how long would it take to fill the pool using a garden hose. What is the fluid velocity leaving the garden hose? c. The pool was actually filled with a hook and ladder fire truck ( trials pool begins filling with a fire truck/). Estimate how long it took to fill the pool. 4
5 Example The exhaust hood in your kitchen has a flow rate of m 3 /min. Your kitchen has a ceiling height of 2.5 m and a floor that measures 5 x 6 m. a. How long would it take the exhaust hood to remove all of the air in your kitchen? Report your answer in minutes. b. Assume kitchen air is 30 C and 1 bar. Calculate the mass flow rate through the exhaust fan. Report your answer in kg/s. c. If the fan inlet has rectangular dimensions of 60 x 120 cm and a circular outlet that is 30 cm in diameter, what are the inlet and outlet air velocities? Report your answer in m/s. 5
6 Example This image is an angiogram of a diseased superficial femoral artery (SFA). The angiogram is created by injecting a contrast dye in the artery, and then taking X ray images to visualize the blood vessel. A lesion, or a segment of diseased artery, can be treated a number of ways, including with stents, or with balloon angioplasty. Physicians also use ultrasound to determine if an intervention is necessary. The ultrasound measures blood velocity and the blood velocity measurements are used to calculate a PSVR (peak systolic velocity ratio, or the velocity of blood through the lesion divided by the velocity of blood through the healthy artery). Physicians usually consider treatment when the cross sectional area of the lesion is half that of the healthy artery. Some medical literature suggests the threshold for treatment should be a PSVR of 2. Do you agree with the recommendation for a PSVR of 2? 6
7 Mass Flow Contributions to the 1 st Law Recall the 1 st Law from Chapter 1:,, The rates of energy added as mass enters the system and energy lost as mass leaves the system can be expressed in terms of the mass flow rate ( ) and the specific kinetic, potential, and internal energies of the mass crossing the system boundary: Specific kinetic energy is: Specific potential energy is: 1 2 Specific internal energy is: Substituting these expressions for specific kinetic energy and specific potential energy into the 1 st Law gives:
8 Flow Work Flow work is the energy needed to push fluid into or out of a system. We must therefore include flow work in the 1 st Law for every system inlet and every system outlet. Recall from Chapter 1 that work is the product of force and displacement. To calculate flow work, force is the product of the fluid pressure and the cross sectional area of the inlet (or outlet) perpendicular to the flow direction. Displacement is the fluid displacement, the product of fluid velocity and time. Assuming 1D, uniform flow gives: Substituting the expressions for force and displacement into the expression for work gives: Rearranging this equation gives an expression for the rate of flow work: It is reasonable to include flow work in the terms in the 1 st Law, since it is considered at every system inlet and every system outlet. To do so, we need to first express flow work in terms of the mass flow rate, where : The 1 st Law becomes:
9 Enthalpy Enthalpy is a property that combines internal energy and flow work into a single term: Specific enthalpy is: Incorporating specific enthalpy into the 1 st Law gives: 1 2 Of note, the left hand side of this expression will be zero when we assume SSSF
10 Example Pump performance curves for Pentair WhisperFlo pool pumps are shown below. For model WFE12 (the uppermost curve), determine the following. a. The pressure rise for a volumetric flow rate of 10 m 3 /hr. Report your answer in m. b. The rate of flow work. Report your answer in W. c. The efficiency of converting electric power into flow work, if the pump requires 1570 W. Report your answer in %. Commented [Office1]: y.com/p 6862 pentair whisperflo pump.aspx 10
11 Example The main air duct for the Combustion Building at the Maurice J. Zucrow Laboratories has a rectangular cross section with width 1.5 m and height 1 m. The blower that feeds the duct must provide an air velocity of 5 m/s and a pressure differential between the inlet and outlet of 1.2 kpa. The building air is 25 C and 1 bar. a. Calculate the mass flow rate. Report your answer in kg/s. b. Calculate the rate of flow work. Report your answer in kw. 11
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