# Integration Relation for Control Volume

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Integration Relation for Control Volume
The Reynolds Transport Theorem Conservation of Mass

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

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

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]

RATE OF FLOW: Generalized equation forms
Volumetric Flow Rate Differential discharge: Using concept of dot product: Mass Flow Rate

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

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

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

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

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”

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

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

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

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!

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.

Solution

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

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

Cavitation damage examples
Impeller of a centrifugal pump Spillway tunnel in a power dam

Class Exercises: (Problem 5.44)