Fluid kinematics Chapter 3 Description and visualization of fluid’s motion (velocity, acceleration).

Fluid Flow An ideal fluid is a fluid that has no internal friction or viscosity and is incompressible. The ideal fluid model simplifies fluid-flow analysis No real fluid has all the properties of an ideal fluid, it helps to explain the properties of real fluids.

LAGRANGIAN AND EULARIAN METHODS OF STUDY OF FLUID FLOW In the Lagrangian method a single particle is followed over the flow field, the co-ordinate system following the particle. The flow description is particle based and not space based. A moving coordinate system has to be used. In the Eularian method, the description of flow is on fixed coordinate system based and the description of the velocity etc. are with reference to location and time i.e., V = V (x, y, z, t)

Principles of Fluid Flow 1/2 The continuity equation results from conservation of mass: mass in=mass out ρQ (in)= ρQ (out) Continuity equation: A1V1 = A2V2 Area  speed in region 1 = area  speed in region 2 For liquid, same density

Principles of Fluid Flow 2/2 The speed of fluid flow depends on cross-sectional area. Bernoulli’s principle states that the pressure in a fluid decreases as the fluid’s velocity increases. P1>P2 & V1<V2 A1V1 = A2V2

Flow Rate Volume rate of flow Mass flow rate Constant velocity over cross-section Variable velocity Mass flow rate

Examples Example. 1 Discharge in a 25cm pipe is 0.03 m3/s. What is the average velocity? Example. 2 A pipe whose diameter is 8 cm transports air with a temp. of 20oC and pressure of 200 kPa abs. At 20 m/s, what is the mass flow rate?

Flow Rate Only x-direction component of velocity (u) contributes to flow through cross-section

Example: Find the ratio of velocities:

Example

Example

a) b)

c)

Example Find inlet velocity, if head increases by 0.1cm/s Continuity equation

Example: The open tank in the figure contains water at 20°C. For incompressible flow, derive an analytic expression for dh/dt in terms of (Q1, Q2, Q3). (b) If h is constant, determine V2 for the given data if: V1 = 3 m/s and Q3 = 0.01 m3/s. Volume=Atank.h =0

The Energy Balance (First Law of Thermodynamics)

Conservation of energy - 1st law of thermodynamics “when heat or work are transferred to a control mass this will result in change of energy stored in it” Q heat added to the system W work done by the system

The Energy Balance Reminder: The general balance equation is: - Rate of Rate of Rate of Rate of Rate of Creation – Destruction + Flow in – Flow out = Accumulation In nature energy can be neither created nor destroyed. With respect to a control volume (CV): Rate of energy flow into CV Rate of energy efflux from CV Rate of accumulation of energy into CV - = (1) Units [Energy/time]

The Energy Balance Rate of Energy Flow into CV: Rate of Energy Flow out of CV: Rate of Energy Accumulation: e [=J/kg] is the specific energy: e (internal) + e (kinetic)+ e (potential=gravity) e = u + V2/ 2+ g z

Landau Potential (Grand potential) Name Symbol Formula Natural variables Internal energy                                                       Helmholtz free energy           Enthalpy                         Gibbs free energy                   Landau Potential (Grand potential) where T = temperature, S = entropy, p = pressure, V = volume. The Helmholtz free energy is often denoted by the symbol F, but the use of A is preferred by IUPAC [2]. Ni is the number of particles of type i in the system and μi is the chemical potential for an i-type particle

The Energy Balance Substituting in equation (1) : Q and W are positive when transferred from surroundings to system. MODERN Convention Note: 2nd edition of textbook uses old convention: Q is positive when transferred from surroundings to system. W is positive when transferred from system to surroundings

The Energy Balance The energy balance reduces to: (2) The work term includes: Shaft work, Wshaft Work due to pressure (injection work)

pressure

The Energy Balance The net work is: Where specific volume u = 1/  [m3/kg] Jule (j)= N.m From (2): (3) This is the Energy equation

The Energy Balance We may use the enthalpy as: h = u + P/ρ For multiple inlets – outlets the energy equation becomes: (4) We may use the enthalpy as: h = u + P/ρ

Simplification of Energy Equation For single inlet (1) – outlet (2) and steady-state conditions (ie no accumulation of energy and mass) , The energy equation becomes: (5)

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