Principles of One-Dimensional Fluid Flow

Fluid Mechanics · FE Reference Handbook section

Fluid Mechanics

8 formulas

10 exam-style examples

~60 min

All Fluid Mechanics lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Principles of One-Dimensional Fluid Flow within Fluid Mechanics. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what principles of one-dimensional fluid flow describes physically and when it applies.
State every one of the 8 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: γ = 62.4 lb/ft³ or 9.81 kN/m³; convert psi to feet of head early.

Lecture

Why this section exists. Principles of One-Dimensional Fluid Flow is the part of Fluid Mechanics that lets you connect a pipeline, jet or submerged surface to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as continuity plus energy, with one head-loss or force term. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. γ = 62.4 lb/ft³ or 9.81 kN/m³; convert psi to feet of head early. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Crane lowering a steel plate girder onto bridge bearings while ironworkers guide it. — Photo 1. Where this shows up in practice: principles of one-dimensional fluid flow.
Capstone Studio instructional photograph

Fluid Mechanics — Principles of One-Dimensional Fluid Flow: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a pipeline, jet or submerged surface. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 8 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

Q	Quantity produced by "Q = Av" — read its definition and unit from the handbook line directly above the equation.
mo	Quantity produced by "mo = ρQ = ρAv" — read its definition and unit from the handbook line directly above the equation.
A	Quantity produced by "A = cross-sectional area of flow" — read its definition and unit from the handbook line directly above the equation.
v	Quantity produced by "v = average flow velocity" — read its definition and unit from the handbook line directly above the equation.
ρ	Quantity produced by "ρ = the fluid density" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

The Continuity Equation
So long as the flow Q is continuous, the continuity equation, as applied to one-dimensional flows, states that the flow passing
where
For steady, one-dimensional flow, mo is a constant. If, in addition, the density is constant, then Q is constant.

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

One-dimensional flow: continuity plus Euler's equation along a streamline — Principles of One-Dimensional Fluid Flow

Water flows steadily at 0.150 m³/s through a reducer from 360.0 mm to 100.0 mm diameter; the outlet is 2.0 m above the inlet. The inlet gauge pressure is 207 kPa. Using continuity and Euler's (frictionless) equation along the centre streamline, compute both velocities and the outlet pressure.

Given

Q = 0.150 m³/s
D₁ = 360.0 mm, D₂ = 100.0 mm
p₁ = 207 kPa
Δz = 2.0 m

Find

V₁, V₂ and p₂

Start with the thinking

Continuity fixes the velocities from areas alone — pressure never enters that step.
Euler integrated along a streamline for incompressible flow is Bernoulli: pressure falls where velocity or elevation rises.

Step-by-step solution

Formula
$Q = A_{1} V_{1} = A_{2} V_{2}$
Areas
$A_{1} = π (0.360)^{2} /4 = 0.10179 m^{2}, A_{2} = 0.00785 m^{2}$
Substituting
$V_{1} = 0.150/0.10179 = 1.47 m / s, V_{2} = 0.150/0.00785 = 19.10 m / s$
Formula
$\frac{p _{1}}{ρ} + \frac{V _{1}^{2}}{2} + g z_{1} = \frac{p _{2}}{ρ} + \frac{V _{2}^{2}}{2} + g z_{2}$
Velocity term
$ρ (V_{2}^{2} - V_{1}^{2}) /2 = 1000 (19.1 0^{2} - 1.4 7^{2}) /2 = 181.3 k P a$
Elevation term — ρgΔz = 1000(9.81)(2.0) = 19.6 kPa
Substituting
$p_{2} = 207 - 181.3 - 19.6 = 6.1 k P a$

Answer: V₁ = 1.47 m/s, V₂ = 19.10 m/s, p₂ = 6.1 kPa

Why the other options are there

p₂ = 388.3 kPa (sign of the velocity term reversed)
V₂ = 0.41 m/s (diameter ratio not squared)

Reference: FE Reference Handbook — Fluid Mechanics → Principles of One-Dimensional Fluid Flow

Example 2

One-dimensional flow: continuity plus Euler's equation along a streamline — Principles of One-Dimensional Fluid Flow (2)

Water flows steadily at 0.130 m³/s through a reducer from 360.0 mm to 130.0 mm diameter; the outlet is 1.5 m above the inlet. The inlet gauge pressure is 339 kPa. Using continuity and Euler's (frictionless) equation along the centre streamline, compute both velocities and the outlet pressure.

Given

Q = 0.130 m³/s
D₁ = 360.0 mm, D₂ = 130.0 mm
p₁ = 339 kPa
Δz = 1.5 m

Find

V₁, V₂ and p₂

Start with the thinking

Continuity fixes the velocities from areas alone — pressure never enters that step.
Euler integrated along a streamline for incompressible flow is Bernoulli: pressure falls where velocity or elevation rises.

Step-by-step solution

Formula
$Q = A_{1} V_{1} = A_{2} V_{2}$
Areas
$A_{1} = π (0.360)^{2} /4 = 0.10179 m^{2}, A_{2} = 0.01327 m^{2}$
Substituting
$V_{1} = 0.130/0.10179 = 1.28 m / s, V_{2} = 0.130/0.01327 = 9.79 m / s$
Formula
$\frac{p _{1}}{ρ} + \frac{V _{1}^{2}}{2} + g z_{1} = \frac{p _{2}}{ρ} + \frac{V _{2}^{2}}{2} + g z_{2}$
Velocity term
$ρ (V_{2}^{2} - V_{1}^{2}) /2 = 1000 (9.7 9^{2} - 1.2 8^{2}) /2 = 47.1 k P a$
Elevation term — ρgΔz = 1000(9.81)(1.5) = 14.7 kPa
Substituting
$p_{2} = 339 - 47.1 - 14.7 = 277.1 k P a$

Answer: V₁ = 1.28 m/s, V₂ = 9.79 m/s, p₂ = 277.1 kPa

Why the other options are there

p₂ = 386.1 kPa (sign of the velocity term reversed)
V₂ = 0.46 m/s (diameter ratio not squared)

Reference: FE Reference Handbook — Fluid Mechanics → Principles of One-Dimensional Fluid Flow

Example 3

One-dimensional flow: continuity plus Euler's equation along a streamline — Principles of One-Dimensional Fluid Flow (3)

Water flows steadily at 0.065 m³/s through a reducer from 370.0 mm to 140.0 mm diameter; the outlet is 1.0 m above the inlet. The inlet gauge pressure is 390 kPa. Using continuity and Euler's (frictionless) equation along the centre streamline, compute both velocities and the outlet pressure.

Given

Q = 0.065 m³/s
D₁ = 370.0 mm, D₂ = 140.0 mm
p₁ = 390 kPa
Δz = 1.0 m

Find

V₁, V₂ and p₂

Start with the thinking

Continuity fixes the velocities from areas alone — pressure never enters that step.
Euler integrated along a streamline for incompressible flow is Bernoulli: pressure falls where velocity or elevation rises.

Step-by-step solution

Formula
$Q = A_{1} V_{1} = A_{2} V_{2}$
Areas
$A_{1} = π (0.370)^{2} /4 = 0.10752 m^{2}, A_{2} = 0.01539 m^{2}$
Substituting
$V_{1} = 0.065/0.10752 = 0.60 m / s, V_{2} = 0.065/0.01539 = 4.22 m / s$
Formula
$\frac{p _{1}}{ρ} + \frac{V _{1}^{2}}{2} + g z_{1} = \frac{p _{2}}{ρ} + \frac{V _{2}^{2}}{2} + g z_{2}$
Velocity term
$ρ (V_{2}^{2} - V_{1}^{2}) /2 = 1000 (4.2 2^{2} - 0.6 0^{2}) /2 = 8.7 k P a$
Elevation term — ρgΔz = 1000(9.81)(1.0) = 9.8 kPa
Substituting
$p_{2} = 390 - 8.7 - 9.8 = 371.5 k P a$

Answer: V₁ = 0.60 m/s, V₂ = 4.22 m/s, p₂ = 371.5 kPa

Why the other options are there

p₂ = 398.7 kPa (sign of the velocity term reversed)
V₂ = 0.23 m/s (diameter ratio not squared)

Reference: FE Reference Handbook — Fluid Mechanics → Principles of One-Dimensional Fluid Flow

Example 4

One-dimensional flow: continuity plus Euler's equation along a streamline — Principles of One-Dimensional Fluid Flow (4)

Water flows steadily at 0.075 m³/s through a reducer from 180.0 mm to 110.0 mm diameter; the outlet is 1.5 m above the inlet. The inlet gauge pressure is 516 kPa. Using continuity and Euler's (frictionless) equation along the centre streamline, compute both velocities and the outlet pressure.

Given

Q = 0.075 m³/s
D₁ = 180.0 mm, D₂ = 110.0 mm
p₁ = 516 kPa
Δz = 1.5 m

Find

V₁, V₂ and p₂

Start with the thinking

Continuity fixes the velocities from areas alone — pressure never enters that step.
Euler integrated along a streamline for incompressible flow is Bernoulli: pressure falls where velocity or elevation rises.

Step-by-step solution

Formula
$Q = A_{1} V_{1} = A_{2} V_{2}$
Areas
$A_{1} = π (0.180)^{2} /4 = 0.02545 m^{2}, A_{2} = 0.00950 m^{2}$
Substituting
$V_{1} = 0.075/0.02545 = 2.95 m / s, V_{2} = 0.075/0.00950 = 7.89 m / s$
Formula
$\frac{p _{1}}{ρ} + \frac{V _{1}^{2}}{2} + g z_{1} = \frac{p _{2}}{ρ} + \frac{V _{2}^{2}}{2} + g z_{2}$
Velocity term
$ρ (V_{2}^{2} - V_{1}^{2}) /2 = 1000 (7.8 9^{2} - 2.9 5^{2}) /2 = 26.8 k P a$
Elevation term — ρgΔz = 1000(9.81)(1.5) = 14.7 kPa
Substituting
$p_{2} = 516 - 26.8 - 14.7 = 474.5 k P a$

Answer: V₁ = 2.95 m/s, V₂ = 7.89 m/s, p₂ = 474.5 kPa

Why the other options are there

p₂ = 542.8 kPa (sign of the velocity term reversed)
V₂ = 1.80 m/s (diameter ratio not squared)

Reference: FE Reference Handbook — Fluid Mechanics → Principles of One-Dimensional Fluid Flow

Example 5

One-dimensional flow: continuity plus Euler's equation along a streamline — Principles of One-Dimensional Fluid Flow (5)

Water flows steadily at 0.030 m³/s through a reducer from 340.0 mm to 90 mm diameter; the outlet is 1.5 m above the inlet. The inlet gauge pressure is 383 kPa. Using continuity and Euler's (frictionless) equation along the centre streamline, compute both velocities and the outlet pressure.

Given

Q = 0.030 m³/s
D₁ = 340.0 mm, D₂ = 90 mm
p₁ = 383 kPa
Δz = 1.5 m

Find

V₁, V₂ and p₂

Start with the thinking

Continuity fixes the velocities from areas alone — pressure never enters that step.
Euler integrated along a streamline for incompressible flow is Bernoulli: pressure falls where velocity or elevation rises.

Step-by-step solution

Formula
$Q = A_{1} V_{1} = A_{2} V_{2}$
Areas
$A_{1} = π (0.340)^{2} /4 = 0.09079 m^{2}, A_{2} = 0.00636 m^{2}$
Substituting
$V_{1} = 0.030/0.09079 = 0.33 m / s, V_{2} = 0.030/0.00636 = 4.72 m / s$
Formula
$\frac{p _{1}}{ρ} + \frac{V _{1}^{2}}{2} + g z_{1} = \frac{p _{2}}{ρ} + \frac{V _{2}^{2}}{2} + g z_{2}$
Velocity term
$ρ (V_{2}^{2} - V_{1}^{2}) /2 = 1000 (4.7 2^{2} - 0.3 3^{2}) /2 = 11.1 k P a$
Elevation term — ρgΔz = 1000(9.81)(1.5) = 14.7 kPa
Substituting
$p_{2} = 383 - 11.1 - 14.7 = 357.2 k P a$

Answer: V₁ = 0.33 m/s, V₂ = 4.72 m/s, p₂ = 357.2 kPa

Why the other options are there

p₂ = 394.1 kPa (sign of the velocity term reversed)
V₂ = 0.09 m/s (diameter ratio not squared)

Reference: FE Reference Handbook — Fluid Mechanics → Principles of One-Dimensional Fluid Flow

Example 6

One-dimensional flow: continuity plus Euler's equation along a streamline — Principles of One-Dimensional Fluid Flow (6)

Water flows steadily at 0.045 m³/s through a reducer from 310.0 mm to 90 mm diameter; the outlet is 3.5 m above the inlet. The inlet gauge pressure is 457 kPa. Using continuity and Euler's (frictionless) equation along the centre streamline, compute both velocities and the outlet pressure.

Given

Q = 0.045 m³/s
D₁ = 310.0 mm, D₂ = 90 mm
p₁ = 457 kPa
Δz = 3.5 m

Find

V₁, V₂ and p₂

Start with the thinking

Continuity fixes the velocities from areas alone — pressure never enters that step.
Euler integrated along a streamline for incompressible flow is Bernoulli: pressure falls where velocity or elevation rises.

Step-by-step solution

Formula
$Q = A_{1} V_{1} = A_{2} V_{2}$
Areas
$A_{1} = π (0.310)^{2} /4 = 0.07548 m^{2}, A_{2} = 0.00636 m^{2}$
Substituting
$V_{1} = 0.045/0.07548 = 0.60 m / s, V_{2} = 0.045/0.00636 = 7.07 m / s$
Formula
$\frac{p _{1}}{ρ} + \frac{V _{1}^{2}}{2} + g z_{1} = \frac{p _{2}}{ρ} + \frac{V _{2}^{2}}{2} + g z_{2}$
Velocity term
$ρ (V_{2}^{2} - V_{1}^{2}) /2 = 1000 (7.0 7^{2} - 0.6 0^{2}) /2 = 24.8 k P a$
Elevation term — ρgΔz = 1000(9.81)(3.5) = 34.3 kPa
Substituting
$p_{2} = 457 - 24.8 - 34.3 = 397.8 k P a$

Answer: V₁ = 0.60 m/s, V₂ = 7.07 m/s, p₂ = 397.8 kPa

Why the other options are there

p₂ = 481.8 kPa (sign of the velocity term reversed)
V₂ = 0.17 m/s (diameter ratio not squared)

Reference: FE Reference Handbook — Fluid Mechanics → Principles of One-Dimensional Fluid Flow

Example 7

One-dimensional flow: continuity plus Euler's equation along a streamline — Principles of One-Dimensional Fluid Flow (7)

Water flows steadily at 0.130 m³/s through a reducer from 380.0 mm to 90 mm diameter; the outlet is 3.5 m above the inlet. The inlet gauge pressure is 380 kPa. Using continuity and Euler's (frictionless) equation along the centre streamline, compute both velocities and the outlet pressure.

Given

Q = 0.130 m³/s
D₁ = 380.0 mm, D₂ = 90 mm
p₁ = 380 kPa
Δz = 3.5 m

Find

V₁, V₂ and p₂

Start with the thinking

Continuity fixes the velocities from areas alone — pressure never enters that step.
Euler integrated along a streamline for incompressible flow is Bernoulli: pressure falls where velocity or elevation rises.

Step-by-step solution

Formula
$Q = A_{1} V_{1} = A_{2} V_{2}$
Areas
$A_{1} = π (0.380)^{2} /4 = 0.11341 m^{2}, A_{2} = 0.00636 m^{2}$
Substituting
$V_{1} = 0.130/0.11341 = 1.15 m / s, V_{2} = 0.130/0.00636 = 20.43 m / s$
Formula
$\frac{p _{1}}{ρ} + \frac{V _{1}^{2}}{2} + g z_{1} = \frac{p _{2}}{ρ} + \frac{V _{2}^{2}}{2} + g z_{2}$
Velocity term
$ρ (V_{2}^{2} - V_{1}^{2}) /2 = 1000 (20.4 3^{2} - 1.1 5^{2}) /2 = 208.1 k P a$
Elevation term — ρgΔz = 1000(9.81)(3.5) = 34.3 kPa
Substituting
$p_{2} = 380 - 208.1 - 34.3 = 137.5 k P a$

Answer: V₁ = 1.15 m/s, V₂ = 20.43 m/s, p₂ = 137.5 kPa

Why the other options are there

p₂ = 588.1 kPa (sign of the velocity term reversed)
V₂ = 0.27 m/s (diameter ratio not squared)

Reference: FE Reference Handbook — Fluid Mechanics → Principles of One-Dimensional Fluid Flow

Example 8

One-dimensional flow: continuity plus Euler's equation along a streamline — Principles of One-Dimensional Fluid Flow (8)

Water flows steadily at 0.055 m³/s through a reducer from 390.0 mm to 90 mm diameter; the outlet is 0.5 m above the inlet. The inlet gauge pressure is 398 kPa. Using continuity and Euler's (frictionless) equation along the centre streamline, compute both velocities and the outlet pressure.

Given

Q = 0.055 m³/s
D₁ = 390.0 mm, D₂ = 90 mm
p₁ = 398 kPa
Δz = 0.5 m

Find

V₁, V₂ and p₂

Start with the thinking

Continuity fixes the velocities from areas alone — pressure never enters that step.
Euler integrated along a streamline for incompressible flow is Bernoulli: pressure falls where velocity or elevation rises.

Step-by-step solution

Formula
$Q = A_{1} V_{1} = A_{2} V_{2}$
Areas
$A_{1} = π (0.390)^{2} /4 = 0.11946 m^{2}, A_{2} = 0.00636 m^{2}$
Substituting
$V_{1} = 0.055/0.11946 = 0.46 m / s, V_{2} = 0.055/0.00636 = 8.65 m / s$
Formula
$\frac{p _{1}}{ρ} + \frac{V _{1}^{2}}{2} + g z_{1} = \frac{p _{2}}{ρ} + \frac{V _{2}^{2}}{2} + g z_{2}$
Velocity term
$ρ (V_{2}^{2} - V_{1}^{2}) /2 = 1000 (8.6 5^{2} - 0.4 6^{2}) /2 = 37.3 k P a$
Elevation term — ρgΔz = 1000(9.81)(0.5) = 4.9 kPa
Substituting
$p_{2} = 398 - 37.3 - 4.9 = 355.8 k P a$

Answer: V₁ = 0.46 m/s, V₂ = 8.65 m/s, p₂ = 355.8 kPa

Why the other options are there

p₂ = 435.3 kPa (sign of the velocity term reversed)
V₂ = 0.11 m/s (diameter ratio not squared)

Reference: FE Reference Handbook — Fluid Mechanics → Principles of One-Dimensional Fluid Flow

Example 9

One-dimensional flow: continuity plus Euler's equation along a streamline — Principles of One-Dimensional Fluid Flow (9)

Water flows steadily at 0.105 m³/s through a reducer from 180.0 mm to 80 mm diameter; the outlet is 0.0 m above the inlet. The inlet gauge pressure is 408 kPa. Using continuity and Euler's (frictionless) equation along the centre streamline, compute both velocities and the outlet pressure.

Given

Q = 0.105 m³/s
D₁ = 180.0 mm, D₂ = 80 mm
p₁ = 408 kPa
Δz = 0.0 m

Find

V₁, V₂ and p₂

Start with the thinking

Continuity fixes the velocities from areas alone — pressure never enters that step.
Euler integrated along a streamline for incompressible flow is Bernoulli: pressure falls where velocity or elevation rises.

Step-by-step solution

Formula
$Q = A_{1} V_{1} = A_{2} V_{2}$
Areas
$A_{1} = π (0.180)^{2} /4 = 0.02545 m^{2}, A_{2} = 0.00503 m^{2}$
Substituting
$V_{1} = 0.105/0.02545 = 4.13 m / s, V_{2} = 0.105/0.00503 = 20.89 m / s$
Formula
$\frac{p _{1}}{ρ} + \frac{V _{1}^{2}}{2} + g z_{1} = \frac{p _{2}}{ρ} + \frac{V _{2}^{2}}{2} + g z_{2}$
Velocity term
$ρ (V_{2}^{2} - V_{1}^{2}) /2 = 1000 (20.8 9^{2} - 4.1 3^{2}) /2 = 209.7 k P a$
Elevation term — ρgΔz = 1000(9.81)(0.0) = 0.0 kPa
Substituting
$p_{2} = 408 - 209.7 - 0.0 = 198.3 k P a$

Answer: V₁ = 4.13 m/s, V₂ = 20.89 m/s, p₂ = 198.3 kPa

Why the other options are there

p₂ = 617.7 kPa (sign of the velocity term reversed)
V₂ = 1.83 m/s (diameter ratio not squared)

Reference: FE Reference Handbook — Fluid Mechanics → Principles of One-Dimensional Fluid Flow

Example 10

One-dimensional flow: continuity plus Euler's equation along a streamline — Principles of One-Dimensional Fluid Flow (10)

Water flows steadily at 0.090 m³/s through a reducer from 320.0 mm to 60 mm diameter; the outlet is 4.0 m above the inlet. The inlet gauge pressure is 324 kPa. Using continuity and Euler's (frictionless) equation along the centre streamline, compute both velocities and the outlet pressure.

Given

Q = 0.090 m³/s
D₁ = 320.0 mm, D₂ = 60 mm
p₁ = 324 kPa
Δz = 4.0 m

Find

V₁, V₂ and p₂

Start with the thinking

Continuity fixes the velocities from areas alone — pressure never enters that step.
Euler integrated along a streamline for incompressible flow is Bernoulli: pressure falls where velocity or elevation rises.

Step-by-step solution

Formula
$Q = A_{1} V_{1} = A_{2} V_{2}$
Areas
$A_{1} = π (0.320)^{2} /4 = 0.08042 m^{2}, A_{2} = 0.00283 m^{2}$
Substituting
$V_{1} = 0.090/0.08042 = 1.12 m / s, V_{2} = 0.090/0.00283 = 31.83 m / s$
Formula
$\frac{p _{1}}{ρ} + \frac{V _{1}^{2}}{2} + g z_{1} = \frac{p _{2}}{ρ} + \frac{V _{2}^{2}}{2} + g z_{2}$
Velocity term
$ρ (V_{2}^{2} - V_{1}^{2}) /2 = 1000 (31.8 3^{2} - 1.1 2^{2}) /2 = 506.0 k P a$
Elevation term — ρgΔz = 1000(9.81)(4.0) = 39.2 kPa
Substituting
$p_{2} = 324 - 506.0 - 39.2 = - 221.2 k P a$

Answer: V₁ = 1.12 m/s, V₂ = 31.83 m/s, p₂ = -221.2 kPa

Why the other options are there

p₂ = 830.0 kPa (sign of the velocity term reversed)
V₂ = 0.21 m/s (diameter ratio not squared)

Reference: FE Reference Handbook — Fluid Mechanics → Principles of One-Dimensional Fluid Flow

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a pipeline, jet or submerged surface, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Principles of One-Dimensional Fluid Flow contains 8 relations; you must be able to find this page in under 15 seconds.
Exam style: continuity plus energy, with one head-loss or force term.
Unit rule: γ = 62.4 lb/ft³ or 9.81 kN/m³; convert psi to feet of head early.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

γ = 62.4 lb/ft³ or 9.81 kN/m³; convert psi to feet of head early
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.