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**Integration Relation for Control Volume**

The Reynolds Transport Theorem Conservation of Mass

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**LAGRANGIAN & EULERIAN DESCRIPTIONS**

Lagrangian Approach: Describe the fluid particle’s motion with time. The path of a particle: r(t) = x(t) i + y(t) j + z(t) k i, j, k: unit vectors Velocity of a particle : V(t) = dr(t) / dt = u i + v j + w k Eulerian Approach: Imagine an array of windows in the flow field: Have information for the fluid particles passing each window for all time. In this case, the velocity is function of the window position (x, y, z) and time. u = f1 (x, y, z, t) v = f 2 (x, y, z, t) w = f 3 (x, y, z, t) Eulerian approach is generally favored

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**REYNOLDS TRANSPORT THEOREM**

CONTROL VOLUME APPROACH: “Focusing on a volume in space & Considering the flow passing through the volume” * It derives from the Eulerian description of fluid motion. * It involves transforming the governing equations for a given mass (Lagrangian form) into the corresponding equations for mass passing through a volume in space (Eulerian form) Mathematical equation needed for this transformation: REYNOLDS TRANSPORT THEOREM

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**RATE OF FLOW Volumetric Flow Rate: Mass Flow Rate:**

∆Volume in the figure: = Length x Area = (V ∆t) x A Q = discharge [m3/s] V = average velocity [m/s] A = cross sectional area [m2 Mass Flow Rate: Mass of fluid passing a station per unit time [kg/s] ρ = density [kg/m3]

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**RATE OF FLOW: Generalized equation forms**

Volumetric Flow Rate Differential discharge: Using concept of dot product: Mass Flow Rate

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**Mean Velocity REAL VELOCITY PROFILE: By definition:**

In laminar flow, the mean velocity is half the centerline velocity. In turbulent flow, velocity profile is nearly flat so the mean velocity is close to centerline velocity. REAL VELOCITY PROFILE: Parabolic for laminar flow Logarithmic for turbulent flow

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**Control Volume Approach**

FLUID SYSTEM: Continuous mass of fluid, containing always the same fluid particles – The mass of a system is constant CONTROL VOLUME (cv): Volume in space. – It can deform with time – It can move & rotate – The mass of control volume can change with time CONTROL SURFACE (cs): Surface enclosing the control volume or boundary of control volume

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**Control Volume Approach**

By definition, the mass of the system is constant, so The rate form of Continuity Principle:

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**Example = (7 - 5 ) / (1000 x10) = 0.0002 m/s The volume of CV: V = Ah**

Considering a CV as shown in the earlier figure, a tank with cross-sectional area of 10 m2 has an inflow of 7kg/s and an outflow of 5 kg/s. Find the rate at which the water level in the tank is changing. The volume of CV: V = Ah The mass in the CV: M cv = r V = r Ah The rate of change of mass in the CV: By the continuity equation, the rate of change of water elevation: = (7 - 5 ) / (1000 x10) = m/s

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**Reynolds Transport Theorem**

B (extensive property) of a system: proportional to the mass of the system (like m, mV, E) b (intensive property) : independent of system mass and obtained by [B/mass] Reynolds Transport Theorem The most general form: (Read excellent explanations at pages ) B: extensive property b: intensive property t: time ρ: density V: volume V: velocity vector A: area vector b b Left side is Lagrangian form & represents the rate of change of property B of the system Right side is Eulerian form & represents the rate change of property B in CV + the net outflow of property B through the CS This equation is often called “control volume equation”

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**Reynolds Transport Theorem: Simplified form**

If the mass crossing the control surface occurs through a number of inlet and outlet ports, and the velocity density and intensive property b are uniformly distributed (constant) across each port; then Please see the text book for the alternative form of the above equation

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Continuity Equation Derives from the conservation of mass which states the mass of the system is constant in Lagrangian form. (M sys = const) The Eulerian form is derived by applying Reynolds transport theorem. In this case, extensive property: B cv = M sys The corresponding intensive property: b = M sys / M sys = 1

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**Continuity Equation = 0 Since dM sys / dt = 0**

The general form of continuity equation: = 0 Accumulation rate Net outflow rate of mass in CV of mass through CS If the mass crosses the control surface through a number of inlet and exit ports, the continuity equation simplifies to

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**EXAMPLE 5.4: Mass is accumulating in the tank at this rate!**

Since there is only one inlet and exit port, the continuity equation simplifies to Mass flow rate in : ρ V A = 1000 x 7 x = 17.5 kg/s Mass flow rate out: ρ Q = 1000 x = 3 kg/s Continuity equation: Mass is accumulating in the tank at this rate!

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EXAMPLE: (Problem 5.49) Referring the figure below, find the velocity of the liquid through the inlet. At a certain time, the surface level in the tank is 1 m and rising at the rate of 0.1 cm/s.

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Solution

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**Continuity Equation for Flow in a Pipe**

Steady Flow CV is fixed to pipe walls Volume of CV is const. Mcv = const. Continuity Equation Incompressible flow valid for steady & unsteady incompressible flow

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Cavitation Phenomenon that occurs when the fluid pressure is reduced to the local vapor pressure and boiling occurs. Vapor bubbles form in the liquid, grow and collapse; producing shock wave, noise & dynamic effects: RESULT lessened performance & equipment failure ! Cavitation typically occurs at locations where the velocity is high. In case b, flow rate is higher

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**Cavitation damage examples**

Impeller of a centrifugal pump Spillway tunnel in a power dam

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**Class Exercises: (Problem 5.44)**

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