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Resources Water Engineering - Lecture 3 Prepared by T. Wagener & P Reed T P. Penn State University 1 Wolf Creek Dam, Kentucky 2 Chapter 3 Flow Processes and Hydrostatic Forces l d d 3 Today s Today's Content 1. 1 Control Volume Approach for Hydrosystems 2. 2 Continuity 3. In-class Example 4. Details on the Quiz 4 1 Control Volume Approach for f Hydrosystems d Hydrosystem processes transform the space and time distribution of water in hydrologic and hydraulic systems. The commonality of systems all hydrosystems is the physical laws that define the flow of fluid in these systems. A consistent mechanism f d i t t h i for developing th l i these physical laws is the control volume approach. 5 Example Control Vol.: The Watershed Vol. 6 Example Control Vol.: The Watershed Vol. 7 http://www.catskillcenter.org/atlas/hydrology/hyd_smallwatershed_l.htm Example Control Vol. 1: The Watershed 1 Fluxes? Fl ? 8 Example Control Vol.: The Watershed 9 Following the Definition of Fluid Mechanics A system is a collection of matter of fixed entity (always the same atoms or fluid particles), which may move, flow, and interact with its surroundings. A control volume is a volume in space (a geometric entity, independent of mass) through which fluid may flow. g g g y We need to describe the laws governing fluid motion using both system concepts (consider a given mass of fluid) and control volume concepts (consider a given volume). To do this we need an analytical tool to shift from one representation to the other. The Reynolds Transport Theorem provides this tool. [ [Young et al., 2004, p.122] g , ,p ] 10 Control Volume Approach A system is typically defined from the fluids viewpoint as a given quantity of mass. mass The system has a system boundary. The control surface can be a boundary physical boundary or a hypothetical surface. 11 Figure 3.1.1 (p. 29) Control volume approach. (a) System and surrounding; (b) Control volume as a system. Extensive/Intensive Properties Extensive properties (EP) are related to the total mass of the system (control volume), whereas intensive properties (IP) are independent of the amo nt of mass amount mass. The EPs we will look at are: Mass m Momentum mV (Continuity (Momentum Equation) Equation) Energy E (Energy Equation) IP: mass per unit mass, momentum per unit mass (velocity v), energy per unit mass e For an extensive property B, the corresponding intensive property is defined as the quantity of B per unit mass. 12 dB = dm continued B= sys dm = d sys 13 Generalizing The mass rate of flow out of the control volume is: The rate of flow of an extensive property B is the produce of the mass rate and the intensive property: And, if the velocity varies across the flow section: 14 dm = m = V A dt CS dB = B = V A dt CS B= CS V dA General Control Volume Equation (Reynolds T (R ld Transport Th t Theorem) ) dB sys d = d + V dA dt dt CV CS The total rate of change of extensive property of a flow flo 15 = The t f h Th rate of change of extensive property stored in the control volume + The t t f Th net rate of outflow of extensive property through the control surface Using Reynolds Transport Thm for Continuity The control volume expression for "conservation of is mass" commonly termed the "continuity equation" IN CLASS DERIVATION: d Given : Gi = d + CS V dA dt dt CV Provide a step-by-step derivation of the continuity equation for 1-D steady, incompressible fl i ibl flow dB sys Qin = Qout What is your extensive property? Intensive property? We are showing all of the steps often skipped in CE 360 16 Conservation of Mass, Continuity Equation STEP 1: Define extensive/intensive properties & apply p p pp y mass balance Extensive property is mass, B = m Intensive property, = dB/dm = 1 Law of conservation of mass of a system is constant dBsys/dt = dm/dt = 0 dB sys d Therefore, dt = dt CV d + V dA CS d 0= d + CS V dA dt CV 17 Conservation of Mass, Continuity Equation STEP 2: Define the control volume and control surface terms d mass mass d = vol - time ( vol ) = time dt CV Represents time rate of change of fluid mass contained in CV CS V dA = 2 mass length length ) ( vol time Represents net rate of mass flow through the control volume 18 Conservation of Mass, Continuity Equation STEP 3: Define a discrete input/output form of the control p p surface term CS V dA = m out - min The integral is equivalent to the summation of infinitesimally small mass flow rate terms For discrete mass inputs/outputs passing normal to the control surface we use the finite summations to yield d d + mout - min = 0 where m = AV dt CV 19 Conservation of Mass, Continuity Equation STEP 4: Apply assumptions to g Qin = Qout pp y p get (i) Steady flow d d d = 0 dt CV (ii) 1-D incompressible flow with a singe stream input m out = min out AoutVout = in AinVin (incompressible ) Qout = Qin 20 !!! We will use the general control volume equation (approach) to develop continuity, energy, energy and momentum equations for hydrosystem (hydrologic and hydraulic) processes throughout this class. class Make sure that you understand the derivation of the equations! 21 2 Storage Representation ofContinuity The continuity equation is one of the main equations used in the modeling of hydrologic systems. systems 22 Storage For a constant density unsteady flow, we can define the volume of fluid stored in a control volume, S: S= CV d The temporal change of storage, S, can be computed as: d dS d = dt dt CV The net outflow can be defined as: CS V dA = V dA + outlet inlet V dA = Q (t ) - I (t ) 23 Typical Form Storage Input I 24 Output S Control Volume Q continued 25 In class In-class Example 26 Water Balance Equation 1. What is the continuity equation for a watershed (also called water balance equation)? 2. What are the variables that we can measure? 3. What do you think we are most interested in from an engineering point of view? (S is the amount of water stored within the watershed) 27 Solution 28 4 Details on Quiz 29 The Quiz ALL quizzes are closed book book. All questions will be qualitative in nature! The i Th quiz will t k about 25 minutes. ill take b t i t We will continue the class for about 30 minutes after the quiz. There will be one question straight from q g the book! 30 End of Lecture 3 Remember Surface Tension 31

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