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

2. 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)

3. 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.

4. 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. 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.

6. 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

7. 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.

8. 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:

9. 5.2 Conservation of Mass Principle (III)
Alternative derivation ;
Reynolds Transport Theorem for fixed CV
Mass conservation principle

10. 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.

11. 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).

12. 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.

13. 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

14. 5.3 Overall Efficiency : turbine-gen & pump-motor
For turbine-generator combination ;
For pump-motor combination ;
In general, efficiency : ×, loss : +

15. 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

16. 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.

17. 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

18. 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.)

19. 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)

20. 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

21. 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

22. 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.

23. 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 = ?

24. 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

25. 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:

26. 5.6 Conservation of Energy Principle
Alternative derivation ;
Reynolds Transport Theorem for fixed CV
Energy conservation principle

27. 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

28. 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.

29. 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)

30. 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

31. 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.

32. 5.7 Energy Analysis of Steady Flows
For single-stream devices, mass flow rate is constant.

33. 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

34. 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.

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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