Thermal and Mechanical Processing

Materials Science · FE Reference Handbook section

Materials Science

1 formulas

10 exam-style examples

~47 min

All Materials Science lectures

Learning objectives

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

This chapter section covers Thermal and Mechanical Processing within Materials Science. 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 thermal and mechanical processing describes physically and when it applies.
State every one of the 1 relation 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: percent versus fraction in composition and strain.

Lecture

Why this section exists. Thermal and Mechanical Processing is the part of Materials Science that lets you connect a steel, concrete or polymer specimen under test 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 a definition, a phase-diagram read, or a one-line property calculation. 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. percent versus fraction in composition and strain. 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.

Concrete cylinder under axial load in a compression testing machine. — Photo 1. Where this shows up in practice: thermal and mechanical processing.
Wikimedia Commons, public domain

Materials Science — Thermal and Mechanical Processing: 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 steel, concrete or polymer specimen under test. 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 1 relation 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.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

Cold working (plastically deforming) a metal increases strength and lowers ductility.
Raising temperature causes (1) recovery (stress relief), (2) recrystallization, and (3) grain growth. Hot working allows these
processes to occur simultaneously with deformation.
Quenching is rapid cooling from elevated temperature, preventing the formation of equilibrium phases.
cementite (iron carbide).

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

Steady heat flow through a wall — Thermal and Mechanical Processing

A 0.8 ft thick concrete wall of area 369 ft² has k = 0.6 Btu/(hr·ft·°F) and a 49°F temperature difference. What is the heat flow rate?

Given

k = 0.6 Btu/hr·ft·°F
A = 369 ft²
ΔT = 49°F
t = 0.8 ft

Find

Heat flow q

Start with the thinking

Fourier's law for a plane wall is linear in ΔT.
Thickness divides, area multiplies.

Step-by-step solution

Fourier — q = kAΔT/t
Substituting
$q = 0.6 (369) (49) /0.8$
Evaluate
$q = 13, 561 B t u / h r$

Answer: q ≈ 13,561 Btu/hr

Why the other options are there

8,679 Btu/hr (thickness multiplied)
29.4 Btu/hr (area omitted)

Reference: FE Reference Handbook — Materials Science → Thermal and Mechanical Processing

Example 2

Steady heat flow through a wall — Thermal and Mechanical Processing (2)

A 1.0 ft thick concrete wall of area 65 ft² has k = 1.7 Btu/(hr·ft·°F) and a 39°F temperature difference. What is the heat flow rate?

Given

k = 1.7 Btu/hr·ft·°F
A = 65 ft²
ΔT = 39°F
t = 1.0 ft

Find

Heat flow q

Start with the thinking

Fourier's law for a plane wall is linear in ΔT.
Thickness divides, area multiplies.

Step-by-step solution

Fourier — q = kAΔT/t
Substituting
$q = 1.7 (65) (39) /1.0$
Evaluate
$q = 4, 310 B t u / h r$

Answer: q ≈ 4,310 Btu/hr

Why the other options are there

4,310 Btu/hr (thickness multiplied)
66.3 Btu/hr (area omitted)

Reference: FE Reference Handbook — Materials Science → Thermal and Mechanical Processing

Example 3

Steady heat flow through a wall — Thermal and Mechanical Processing (3)

A 0.7 ft thick concrete wall of area 330 ft² has k = 1.7 Btu/(hr·ft·°F) and a 23°F temperature difference. What is the heat flow rate?

Given

k = 1.7 Btu/hr·ft·°F
A = 330 ft²
ΔT = 23°F
t = 0.7 ft

Find

Heat flow q

Start with the thinking

Fourier's law for a plane wall is linear in ΔT.
Thickness divides, area multiplies.

Step-by-step solution

Fourier — q = kAΔT/t
Substituting
$q = 1.7 (330) (23) /0.7$
Evaluate
$q = 18, 433 B t u / h r$

Answer: q ≈ 18,433 Btu/hr

Why the other options are there

9,032 Btu/hr (thickness multiplied)
39.1 Btu/hr (area omitted)

Reference: FE Reference Handbook — Materials Science → Thermal and Mechanical Processing

Example 4

Steady heat flow through a wall — Thermal and Mechanical Processing (4)

A 0.6 ft thick concrete wall of area 278 ft² has k = 2.1 Btu/(hr·ft·°F) and a 27°F temperature difference. What is the heat flow rate?

Given

k = 2.1 Btu/hr·ft·°F
A = 278 ft²
ΔT = 27°F
t = 0.6 ft

Find

Heat flow q

Start with the thinking

Fourier's law for a plane wall is linear in ΔT.
Thickness divides, area multiplies.

Step-by-step solution

Fourier — q = kAΔT/t
Substituting
$q = 2.1 (278) (27) /0.6$
Evaluate
$q = 26, 271 B t u / h r$

Answer: q ≈ 26,271 Btu/hr

Why the other options are there

9,458 Btu/hr (thickness multiplied)
56.7 Btu/hr (area omitted)

Reference: FE Reference Handbook — Materials Science → Thermal and Mechanical Processing

Example 5

Steady heat flow through a wall — Thermal and Mechanical Processing (5)

A 0.8 ft thick concrete wall of area 280 ft² has k = 0.9 Btu/(hr·ft·°F) and a 81°F temperature difference. What is the heat flow rate?

Given

k = 0.9 Btu/hr·ft·°F
A = 280 ft²
ΔT = 81°F
t = 0.8 ft

Find

Heat flow q

Start with the thinking

Fourier's law for a plane wall is linear in ΔT.
Thickness divides, area multiplies.

Step-by-step solution

Fourier — q = kAΔT/t
Substituting
$q = 0.9 (280) (81) /0.8$
Evaluate
$q = 25, 515 B t u / h r$

Answer: q ≈ 25,515 Btu/hr

Why the other options are there

16,330 Btu/hr (thickness multiplied)
72.9 Btu/hr (area omitted)

Reference: FE Reference Handbook — Materials Science → Thermal and Mechanical Processing

Example 6

Steady heat flow through a wall — Thermal and Mechanical Processing (6)

A 1.5 ft thick concrete wall of area 236 ft² has k = 2.5 Btu/(hr·ft·°F) and a 48°F temperature difference. What is the heat flow rate?

Given

k = 2.5 Btu/hr·ft·°F
A = 236 ft²
ΔT = 48°F
t = 1.5 ft

Find

Heat flow q

Start with the thinking

Fourier's law for a plane wall is linear in ΔT.
Thickness divides, area multiplies.

Step-by-step solution

Fourier — q = kAΔT/t
Substituting
$q = 2.5 (236) (48) /1.5$
Evaluate
$q = 18, 880 B t u / h r$

Answer: q ≈ 18,880 Btu/hr

Why the other options are there

42,480 Btu/hr (thickness multiplied)
120.0 Btu/hr (area omitted)

Reference: FE Reference Handbook — Materials Science → Thermal and Mechanical Processing

Example 7

Steady heat flow through a wall — Thermal and Mechanical Processing (7)

A 1.2 ft thick concrete wall of area 80 ft² has k = 2.0 Btu/(hr·ft·°F) and a 33°F temperature difference. What is the heat flow rate?

Given

k = 2.0 Btu/hr·ft·°F
A = 80 ft²
ΔT = 33°F
t = 1.2 ft

Find

Heat flow q

Start with the thinking

Fourier's law for a plane wall is linear in ΔT.
Thickness divides, area multiplies.

Step-by-step solution

Fourier — q = kAΔT/t
Substituting
$q = 2.0 (80) (33) /1.2$
Evaluate
$q = 4, 400 B t u / h r$

Answer: q ≈ 4,400 Btu/hr

Why the other options are there

6,336 Btu/hr (thickness multiplied)
66.0 Btu/hr (area omitted)

Reference: FE Reference Handbook — Materials Science → Thermal and Mechanical Processing

Example 8

Steady heat flow through a wall — Thermal and Mechanical Processing (8)

A 0.7 ft thick concrete wall of area 270 ft² has k = 2.4 Btu/(hr·ft·°F) and a 39°F temperature difference. What is the heat flow rate?

Given

k = 2.4 Btu/hr·ft·°F
A = 270 ft²
ΔT = 39°F
t = 0.7 ft

Find

Heat flow q

Start with the thinking

Fourier's law for a plane wall is linear in ΔT.
Thickness divides, area multiplies.

Step-by-step solution

Fourier — q = kAΔT/t
Substituting
$q = 2.4 (270) (39) /0.7$
Evaluate
$q = 36, 103 B t u / h r$

Answer: q ≈ 36,103 Btu/hr

Why the other options are there

17,690 Btu/hr (thickness multiplied)
93.6 Btu/hr (area omitted)

Reference: FE Reference Handbook — Materials Science → Thermal and Mechanical Processing

Example 9

Steady heat flow through a wall — Thermal and Mechanical Processing (9)

A 0.9 ft thick concrete wall of area 324 ft² has k = 1.7 Btu/(hr·ft·°F) and a 88°F temperature difference. What is the heat flow rate?

Given

k = 1.7 Btu/hr·ft·°F
A = 324 ft²
ΔT = 88°F
t = 0.9 ft

Find

Heat flow q

Start with the thinking

Fourier's law for a plane wall is linear in ΔT.
Thickness divides, area multiplies.

Step-by-step solution

Fourier — q = kAΔT/t
Substituting
$q = 1.7 (324) (88) /0.9$
Evaluate
$q = 53, 856 B t u / h r$

Answer: q ≈ 53,856 Btu/hr

Why the other options are there

43,623 Btu/hr (thickness multiplied)
149.6 Btu/hr (area omitted)

Reference: FE Reference Handbook — Materials Science → Thermal and Mechanical Processing

Example 10

Steady heat flow through a wall — Thermal and Mechanical Processing (10)

A 0.8 ft thick concrete wall of area 371 ft² has k = 0.7 Btu/(hr·ft·°F) and a 26°F temperature difference. What is the heat flow rate?

Given

k = 0.7 Btu/hr·ft·°F
A = 371 ft²
ΔT = 26°F
t = 0.8 ft

Find

Heat flow q

Start with the thinking

Fourier's law for a plane wall is linear in ΔT.
Thickness divides, area multiplies.

Step-by-step solution

Fourier — q = kAΔT/t
Substituting
$q = 0.7 (371) (26) /0.8$
Evaluate
$q = 8, 440 B t u / h r$

Answer: q ≈ 8,440 Btu/hr

Why the other options are there

5,402 Btu/hr (thickness multiplied)
18.2 Btu/hr (area omitted)

Reference: FE Reference Handbook — Materials Science → Thermal and Mechanical Processing

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 steel, concrete or polymer specimen under test, 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

Thermal and Mechanical Processing contains 1 relation; you must be able to find this page in under 15 seconds.
Exam style: a definition, a phase-diagram read, or a one-line property calculation.
Unit rule: percent versus fraction in composition and strain.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

percent versus fraction in composition and strain
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.