SME
Transport Phenomena
Heat transport
Use this for “A wall of 0.15 m brick (k = 0.7 W/(m·K)) and 0.08 m foam insulation (k = 0.03) has films of 8 and 30 W/(m²·K). Inside is 20 °C, outside −10 °C. Find the heat flux.”
Diagram
R″
=
1
hᵢ
+
L₁
k₁
+
L₂
k₂
+
1
hₒ
R″ = 0.1 + 0.2857 + 1.25 + 0.04
R″ = 1.676 m²·K/W
q″
=
Tᵢₙ − Tₒᵤₜ
R
″
=
27
1.676
q″
=
16.11
W/
m
2
Each
drop
is
q″
×
R:
1.61
K,
4.6
K,
20.1
K,
0.645
K;
the
steepest
line
is
through
the
insulation,
the
layer
with
the
most
L
k
.
This is an example. Type your own values to start a new problem.
R″ (m²·K/W)
=
1
hᵢₙ (W/(m²·K))
+
L₁ (m)
k₁ (W/(m·K))
+
L₂ (m)
k₂ (W/(m·K))
+
1
hₒᵤₜ (W/(m²·K))
q″ (W/m²)
=
Tᵢₙ (°C)
−
(
Tₒᵤₜ (°C)
)
R″ (m²·K/W)
Formulas
R″
=
1
hᵢₙ
+
L₁
k₁
+
L₂
k₂
+
1
hₒᵤₜ
1.6757
=
1
10
+
0.2
0.7
+
0.05
0.04
+
1
25
q″
=
Tᵢₙ − Tₒᵤₜ
R
″
16.1125
=
22 − (−5)
1.6757
Assumptions
•
Steady, one-dimensional conduction through flat layers, with perfect contact between them.
•
Per square metre of wall, the films and layers are resistances in series, like resistors.
•
Heat flows from Tᵢₙ to Tₒᵤₜ; q″ comes out negative when the outside is warmer.
Step-by-step
WE KNOW
Tᵢₙ = 22 °C, hᵢₙ = 10 W/(m²·K), L₁ = 0.2 m, k₁ = 0.7 W/(m·K), L₂ = 0.05 m, k₂ = 0.04 W/(m·K), hₒᵤₜ = 25 W/(m²·K), Tₒᵤₜ = −5 °C
FIND
total resistance (R″), heat flux (q″)
Step 1 · Find total resistance (R″)
Use R″ = 1 ÷ hᵢₙ + L₁ ÷ k₁ + L₂ ÷ k₂ + 1 ÷ hₒᵤₜ
Resistances
in
series
add,
like
resistors:
each
film
is
1
h
and
each
layer
L
k
.
R″
=
1
10
+
0.2
0.7
+
0.05
0.04
+
1
25
R″ = 0.1 + 0.28571 + 1.25 + 0.04
R″ = 1.675714 m²·K/W
Step 2 · Find heat flux (q″)
Use q″ = (Tᵢₙ − Tₒᵤₜ) ÷ R″
Heat flows like current: the temperature difference over the total resistance.
q″
=
22 − (−5)
1.675714
q″
=
27
1.6757
q″
=
16.1125
W/
m
2
Check
Put the numbers back in. Both sides should match.
1.675714
=
1
10
+
0.2
0.7
+
0.05
0.04
+
1
25
✓
16.1125
=
22 − (−5)
1.675714
✓
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A liquid film falling down a wall
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Heat lost from an insulated pipe; the critical radius
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
Heat lost from an insulated pipe; the critical radius
Use this for “A steam line of outer radius 0.03 m at 120 °C gets 0.02 m of insulation (k = 0.05 W/(m·K)) in 25 °C air with h = 8 W/(m²·K). Find the heat lost per metre.”
Centre temperature of a wire with heat generation
Use this for “A 2 mm-radius rod with k = 20 W/(m·K) makes 1 × 10⁸ W/m³ of heat, and its surface is at 90 °C. How hot is its centre?”