Chapter Review Guide for Fluid Mechanics, by Franzini et al.  You can use this page to test your understanding of each chapter.  The focus is identifying important equations and major concepts.  However, don't use this as an excuse for not reading the book!  If you find any mistakes (typos, etc.), let Dr. E. know as soon as possible!!

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Table of Contents

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Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8a Chapter 8b

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Chapter 1
Approach to problem solving
Systems of units (BG and SI)
Dimensions (F, M, L, T)

Chapter 2
Concepts

Ideal fluid
Real fluid
Surface tension and capillarity
Vapor pressure

Density, r
Specific weight, g
Specific volume, u
Bulk modulus (also know as volume modulus), E_u
Specific gravity, s
viscosity (absolute and kinematic), m and n
surface tension, s
wetting angle, q

g = rg
u = 1 / r
E_u = -u dp/du
Du / u y - DP / E_u
s = r/r_ref = g/g_ref    (where the reference substance is often water)
pu = RT
t= F/A = m U/Y = m du/dy
n = m / r
h = (2s cos q)/ gr

Chapter 3
Concepts

Pressure at a point in a fluid is the same in all directions
Pressure varies as a function of depth and specific weight in fluids
Barometer
Pressure measurement

Bourdon gage, Pressure Transducer, Piezometer Column, Simple and Differential Manometer

Forces and locations of forces on submerged plane and curved areas
Buoyancy and Stability
Fluid masses subjected to acceleration

Pressure head, h
Atmospheric, absolute and gage pressures, p_atm, p_abs and p_gage
Centroid of area, h_c, y_c
Centroid of Pressure, h_p, y_p
Moment of Inertia (about an arbitrary axis and about a centroidal axis), I_O and I_c
Weight Force, W
Buoyancy force, F_B
System subscript, S
Control volume subscript, CV

DP/dz = -  g
p = gh
h = p / g
p_abs = p_{atm +} p_gage
Be able to derive pressure relationships for piezometers and manometers
F = xp dA = gh_cA
y_p = I_O / y_cA = y_c + IC / y_cA
W = specific weight of object times its volume
F_B = specific weight of water times volume of water displaced by object

W =  F_B   (for an object that is not rising or falling in a fluid)

Chapter 4
Concepts

Ideal vs. real
Incompressible vs. compressible
Steady
Uniform
Laminar
Turbulent
Path line
Streamline
Flowrate (volume, mass, weight)
Mean velocity
System (S) and control volume (CV)
Continuity
Flow net, streamlines and equipotential lines
Velocity and acceleration in steady flow
Velocity and acceleration in unsteady flow

Volumetric flowrate, Q
Mass flowrate, m with dot on top (I'll just use m' here)
Weight flowrate, G
Velocity, V
Acceleration, a
Steady flow subscript, st
Bolding indicates a vector

Q = VA
V = Q / A
m' = rQ    (for constant density flow)
G = gQ    (for constant density flow)
System - Control Volume Equation:

In words: change of X in the moving system equals change of X within the control volume plus the flow of X out of the control volume minus flow of X into the control volume.

A₁V₁ = A₂V₂ (incompressible flow, steady flow)
r₁A₁V₁ = r₂A₂V₂ (steady flow)
g₁A₁V₁ = g₂A₂V₂ (constant gravity, steady flow)
a_st = V cV/cs   (steady flow, s is direction tangent to streamline)
a = V CV/Cs + CV/ct    (unsteady flow, s is direction tangent to streamline)

Chapter 5
Concepts

Energy (kinetic, potential, pressure, internal,...)
General energy equation
Head (pressure, elevation, velocity, total, piezometric, static,...)
Power
Efficiency
Cavitation
Energy grade line
Hydraulic grade line
Head loss

a = Kinetic energy correction factor
I = internal energy per unit weight of fluid
p/g = flow work, pressure acting on flow
h_M = shaft work, put into fluid by machine per unit weight of fluid
Q_H = energy put into fluid by external heat source per unit weight of fluid
h_l = head loss

Energy = Force time distance
Energy/unit weight has units of length
General energy equation (5.8):
Bernoulli's equation (5.13):

h_l = (I₂ - I₁) - Q_H
Power = hgQ  = energy per time
Efficiency = power output / power input
Cavitation (5.36):

Chapter 6
Concepts

Impulse-momentum principle
Navier Stokes equations
Force exerted on pressure conduits
Force exerted on stationary and moving vanes or blades
Absolute and relative velocities
Jets and Jet Propulsion
Pumps and Turbines, water
Head equivalent of h_M
Reaction and rotating jets
Fans and Propellers, gases

Q' = rate of fluid striking a moving body, function of relative velocities between water and body
V = absolute velocity of water
u = relative velocity of water (to some moving body)
u = absolute velocity of moving body

nF = d(mV)_s / dt
NF = rQ (DV), for steady flow and stationary bodies
NF = rQ' (Du), for steady flow and a single moving vane (only a portion of fluid momentum is transfered)
NF = rQ (Du), for steady flow and a series of moving vanes (all fluid momentum is transfered)
h_M = (u₁ V₁ COs a₁ - u₂ V₂ COs a₂) / g

Chapter 7
Concepts

Simlitude and Similarity
Reynolds Number
Froude Number
Mach Number
Weber Number
Scale Ratios
Dimensional Analysis and the Pi Theorem

Subscripts:  p = prototype, m = model
Scale ratio, L_r
Velocity ratio, V_r
Force ratio, F_r
Reynolds number, R
Froude Number, F
Mach Number, M
c = sonic velocity in medium in question
Weber Number, W
 s = surface tension

R = LVr/m
F = V (gL)^-0.5
M = V / c
W = V (s/rL)^-0.5

Chapter 8a (sections 1 - 16)
Concepts

Laminar and Turbulent Flow
Critical Reynolds number
        Laminar Flow,  R < 2000
Friction in circular conduits
Friction factor
Pipe Roughness (hydraulically smooth, transitionally rough, fully rough)
Moody Diagram
Pipe flow Problems (head-loss, discharge, and sizing problems)
Solution of pipe flow problems using the Moody Chart
Solution of pipe flow problems using explicit equations
Empirical equations for pipe flow

Reynolds number, R
Hydraulic radius, Rh
Cross-section area of flowing fluid, A
Wetted Perimeter, P
Friction factor, f
Equivalent height of roughness projections, e
relative roughness, e/D
Energy Gradient, S (= hL / L, where L = length of pipe)
roughness coefficients,  CHW and n

Rh = A / P
hl = f (L/D) (V²/2g)
f = 64/R, for laminar flow in circular pipes
,  for turbulent flow
Various derived and "fitted" equations, see especially pages 289-291.
,  BG units, formula is not unit-consistent, limited applicability
,  SI units, formula is not unit-consistent, limited applicability
,  BG units, formula is not unit-consistent, limited applicability
,  SI units, formula is not unit-consistent, limited applicability

Type 1:  solved directly.  Type 2, and 3 problems: solved iteratively using Moody's diagram, iteratively using equations, or solved using equation solvers, see pages 294-302.
Empirical equations, see page 304-305.

Chapter 8b (sections 17 - 28)
Concepts

Minor Losses
When to include minor losses

When a fitting creates a major loss (such as a valve that is almost closed)
For short pipe lengths (a good rule of thumb is to include minor losses if the pipe length is < 1000 D)

Solution of various pipe problems

loss coefficient, k
equivalent length, L/D or N (multiply by pipe diameter, D, to estimate equivalent length of straight pipe that would give same head loss as source of minor loss)

Entrances:  h'e = ke V²/2g, where ke ranges ~0 to ~ 0.8, see page 309.
Submerged discharges: h'd = V²/2g, see page 310.
Contractions:  h'c = kc V²/2g, where kc is a function of D2/D1, where D2 = diameter after contraction, D1 = diameter before contraction, kc ranges from 0 to 0.5,  see page 312.
Expansions, sudden:  h'x = (V1 - V2)²/2g, where V2 = velocity after contraction, V1 = velocity before expansion
Expansions, gradual:  h' = k' (V1 - V2)²/2g, where V2 = velocity after contraction, V1 = velocity before expansion, and k' is a function of cone angle, see page 316.
Pipe fittings: h = k V²/2g, where k depends on the fitting, see page 317.  Alternatively, one can use the equivalent length concept (L/D from Table 8.3 times the pipe diameter).  Add the equivalent length of the fitting to the pipe length.  This ensures that the head loss caused by the fitting is taken into account, but as pipe friction.
Bends:  hb = kb V²/2g, where kb is a function of relative roughness and the ratio of the bend radius to the pipe diameter, see page 320.

Single-pipe flow with minor losses:  Total head loss is sum of pipe friction loss and minor loss, see page 321.
Pipeline with pump or turbine:  Total head provided by a pump or supplied to a turbine is a function of energy (change in elevation and velocity, usually) and the pipe friction and minor losses, see page 327.  Minor losses are often ignored.
Branching pipes: Need to use continuity and energy equations, see page 331.  Minor losses and even velocity heads are often ignored.
Pipes in series:  Continuity shows that the flow through each pipe is the same, the total head loss is the sum of the head loss in each pipe.  See page 338.
Pipes in parallel:    Continuity shows that the flow through the system is the sum of the flows in the individual pipes.  The head loss to get from point 1 to point 2, through any of the pipes, is the same, therefore the head loss of each pipe is the same, see page 341.

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