Chapter 5: Mass, Bernoulli, and Energy Equations Hyun Sik Yoon (Inwon Lee) Department of Naval Architecture and Ocean Engineering Pusan National University Spring 2011

5. Introduction This chapter deals with 3 equations commonly used in fluid mechanics The mass equation : conservation of mass principle. The Bernoulli equation : conservation of kinetic, potential, and flow energies of a fluid stream and their conversion to each other. The energy equation : conservation of energy principle. (Bernoulli + Heat & Work Input)

5. Objectives After completing this chapter, you should be able to Apply the mass equation to balance the incoming and outgoing flow rates in a flow system. Recognize various forms of mechanical energy, and work with energy conversion efficiencies. Understand the use and limitations of the Bernoulli equation, and apply it to solve a variety of fluid flow problems. Work with the energy equation expressed in terms of heads, and use it to determine turbine power output and pumping power requirements.

5.1 Conservation of Mass One of the most fundamental principles in nature. Mass : cannot be created or destroyed during a process. For closed systems : mass conservation is implicit (automatically satisfied) since the mass of the system remains constant during a process. For control volumes : mass can cross the boundaries  we must keep track of the amount of mass entering and leaving the control volume.

5.2 Mass and Volume Flow Rates Mass flow rate, : the amount of mass flowing through a control surface per unit time. The dot over a symbol : time rate of change. Flow rate across the entire cross-sectional area of a pipe or duct is obtained by integration While this expression for is exact, it is not always convenient for engineering analyses.

5.2 Average Velocity and Volume Flow Rate Integral in  average values of r and Vn For many flows, variation of r is very small: Volume flow rate is given by Note: many textbooks use Q instead of for volume flow rate. Mass and volume flow rates are related by

5.2 Conservation of Mass Principle (I) The conservation of mass principle can be expressed as Where and are the total rates of mass flow into and out of the CV, and dmCV/dt is the rate of change of mass within the CV.

5.2 Conservation of Mass Principle (II) For CV of arbitrary shape, rate of change of mass within the CV net mass flow rate (mass flux) - : outflow (+) - : inflow (-) Conservation of mass for CV is:

5.2 Conservation of Mass Principle (III) Alternative derivation ; Reynolds Transport Theorem for fixed CV Mass conservation principle

5.2 Steady Flow Processes For steady flow, the total amount of mass contained in CV is constant. Net mass outflow = 0 ; For incompressible flows, Ex) Two inlets, one outlet, r = const.

5.3 Mechanical Energy Mechanical energy : the form of energy converted to mechanical work completely and directly by an ideal mechanical device such as an ideal turbine. (≠ heat) Three forms : P/r (flow), V2/2 (kinetic), and gz (potential) Flow (pressure) work, P/r : work done on CS during outflow/inflow Mechanical energy change of a fluid during incompressible flow becomes For zero losses, Demech represents the work supplied to the fluid (Demech>0) or extracted from the fluid (Demech<0).

5.3 Efficiency Transfer of emech is usually accomplished by a rotating shaft : shaft work Compressor (flow), Fan (kinetic), Pump (potential) : motor  shaft work  emech Turbine : emech  shaft work. Mechanical efficiency of a device or process : Efficiency = (output)/(input) If hmech < 100%, losses have occurred during conversion.

5.3 Pump and Turbine Efficiencies In fluid systems, we are usually interested in increasing the pressure, velocity, and/or elevation of a fluid. In these cases, efficiency is better defined as the ratio of (supplied or extracted work) vs. rate of increase in mechanical energy

5.3 Overall Efficiency : turbine-gen & pump-motor For turbine-generator combination ; For pump-motor combination ; In general, efficiency : ×, loss : +

5.3 Ex 5-3 : hydraulic turbine-generator Water depth = 50m, Water supply = 5000 kg/s Electrical output = 1862 kW Generator efficiency = 95% Overall efficiency, Turbine efficiency, Shaft power of turbine At the inlet and outlet of the turbine, mechanical energy decrease are given as Overall efficiency, turbine efficiency and shaft power

5.4 The Bernoulli Equation The Bernoulli equation is an approximate relation between pressure, velocity, and elevation and is valid in regions of steady, incompressible flow where net frictional forces are negligible. Equation is useful in flow regions outside of boundary layers and wakes.

5.4 The Bernoulli Equation Acceleration of a fluid particle Decomposition of Motion  distance along the streamline, s + radius of curvature Alternative derivation in x-y

5.4 The Bernoulli Equation Derivation Force balance on the fluid particle in s-dir. Left-Hand-Side : Resultant force in s-dir. Both sides  For incompressible flow (r = const.)

5.4 The Bernoulli Equation Limitations on the use of the Bernoulli Equation Steady flow : d/dt = 0 Frictionless flow No shaft work : wpump= wturbine=0 Incompressible flow : r = constant No heat transfer : qnet,in=0 Applied along a streamline (except for irrotational flow)

5.4 The Bernoulli Equation Interpretation of Bernoulli Equation Energy point of view : Conservation of mechanical energy Pressure point of view : Conversion between static – dynamic – hydrostatic pressure Head point of view : Conversion between pressure – velocity – elevation head

5.4 The Bernoulli Equation Static / Dynamic / Hydrostatic pressure Static : thermodynamic pressure Dynamic : pressure rise with flow being stopped isentropically (no loss) Hydrostatic : effect of depth origin (actually not real pressure) Total pressure = static + dynamic + hydrostatic Stagnation pressure = static + dynamic Pitot tube : measure Pstat - P  velocity V Manometer : pressure difference

5.5 Ex. 5-6 : Application of the Bernoulli Eq. (I) Water discharge from a large tank, V2=? Smooth and rounded outlet Free surface (open top)  ①, outlet  ② Discussion : sharp outlet  nonzero loss  Bernouill equation does not hold.

5.5 Ex. 5-7 : Application of the Bernoulli Eq. (II) Siphoning out gasoline Min. time to withdraw 4 liter of gasoline Bernouill eq. (no loss)  max. V2  min. time P3 = ?

5.6 General Energy Equation One of the most fundamental laws in nature is the 1st law of thermodynamics, which is also known as the conservation of energy principle. It states that energy can be neither created nor destroyed during a process; it can only change forms Falling rock, picks up speed as PE is converted to KE. If air resistance is neglected, PE + KE = constant

5.6 General Energy Equation The energy content of a closed system can be changed by two mechanisms: heat transfer Q and work transfer W. Conservation of energy for a closed system can be expressed in rate form as Net rate of heat transfer to the system: Net power input to the system:

5.6 Conservation of Energy Principle Alternative derivation ; Reynolds Transport Theorem for fixed CV Energy conservation principle

5.6 General Energy Equation Recall general RTT “Derive” energy equation using B=E and b=e Break power into rate of shaft and pressure work

5.6 General Energy Equation Derivation of pressure work When piston moves down ds under the influence of F=PA, the work done on the system is dWboundary=PAds. If we divide both sides by dt, we have For generalized control volumes: Note sign conventions: is outward pointing normal Negative sign ensures that work done is positive when is done on the system.

5.6 General Energy Equation Moving integral for rate of pressure work to RHS of energy equation results in: Recall that P/r is the flow energy, which is the work associated with pushing a fluid into or out of a CV per unit mass. (net energy input) = (energy change inside CV) + (net energy flux)

5.6 General Energy Equation As with the mass equation, practical analysis is often facilitated as averages across inlets and exits Since e=u+ke+pe = u+V2/2+gz

5.7 Energy Analysis of Steady Flows For steady flow, time rate of change of the energy content of the CV is zero. Net rate of energy transfer to a CV by heat and work transfers during steady flow is equal to the difference between the rates of outgoing and incoming energy flows with mass.

5.7 Energy Analysis of Steady Flows For single-stream devices, mass flow rate is constant.

5.7 Energy Analysis of Steady Flows Divide by g to get each term in units of length  equivalent column height of fluid, i.e., Head

5.7 Energy Analysis : Special Case (Loss = 0) If we neglect piping losses, and have a system without pumps or turbines  Bernoulli equation Bernoulli equation : special case of the energy equation.

5.7 Energy Analysis : Correction Factor (I) Kinetic energy correction factor, α Recall, during the simplification of energy flux term This is not correct for the kinetic energy term, because Kinetic energy correction factor α should be introduced Then the energy equation becomes

5.7 Energy Analysis : Correction Factor (II) Calculation of α For fully-developed laminar flow cf) For turbulent flow, α  1.06 If uniform flow V(r)=const is assumed, then α = 1.0

5.7 Ex. 5-13 : Hydroelectric Power Hydroelectric power generation from a dam Conditions Flow rate : 100 m3/s Elevation : z1 ─ z2=120m Total irreversible head loss from piping : hL = 35m Overall efficiency of turbine-gen unit = 80% Electric power output ? Mass flow rate of water through the turbine Energy equation : P1≈P2 ≈Patm, V1≈V2 ≈0

5.7 Ex. 5-13 : Hydroelectric Power Hydroelectric power generation from a dam (cont’d) Electric power output ? Extracted turbine head & corresponding turbine power (ideal case) Note that this is the power output from a perfect turbine-generator Electrical power from the actual turbine-generator

5.7 Ex. 5-14 : Application of energy equation Fan selection for computer cooling Conditions Computer case : 12cm×40cm×40cm (half filled) Fan : to replace the air in the void space for 1 sec Fan installation hole : d=5cm Efficiency of fan-motor unit = 0.3, rair=1.2 kg/m3 Wattage (Input power) of fan ? Volume of the case and mass flow rate of air Average velocity

5.7 Ex. 5-14 : Application of energy equation Fan selection for computer cooling (cont’d) Wattage (Input power) of fan ? Set CV : very small velocity at inlet ①

5.7 Ex. 5-14 : Application of energy equation Fan selection for computer cooling (cont’d) Pressure difference across the fan Energy equation between ③ and ④