PROBLEM 3.10
KNOWN: Construction and dimensions of a device to measure the temperature dependence of a liquid’s thermal conductivity.
FIND: (a) Overall height of the apparatus using bakelite, (b) Overall height of the apparatus using aerogel, (c) Required heater area and electrical power to minimize heat losses for bakelite and aerogel.
SCHEMATIC: t t l l
= 15 W/m · K k
∆ l l
= 0.8 W/m
T l l
· K t t t = 1 mm
Stainless Steel
Low Thermal Conductivity
Stainless Steel t t t
ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional heat transfer.
PROPERTIES: Table A.3, bakelite (300 K): k = 1.4 W/m
⋅
K.
ANALYSIS: The heat flux through the device is constant and is evaluated using Eq. 3.5 q = k l
∆ T l
=
⋅ t l
For each stainless steel sheet,
= 800 W/m
2
′′ ss
∆ T t ss ss
or
∆
T = k ss ss
=
-3
= 0.053°C
The temperature difference from top to bottom is
∆ T = ∆ T ss
× × ∆ T l
×
N + ∆ T lcm
×
(N - 1) or
∆
T lcm
=
1
(N - 1)
[
∆
T -
∆
T ss
× × ∆
T l
×
] 1
4
[ ]
For the low thermal conductivity material, lcm
∆ T lcm t lcm
or t lcm
= k lcm
∆ T lcm q
′′
.
(a) For bakelite, t lcm
=
800 W/m
2
Continued…
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The total height is
H = 2Nt + Nt + (N - 1)t lcm
PROBLEM 3.10 (Cont.)
-3 -3
= 177 × 10
-3
m = 177 mm
(b) For aerogel, t lcm
=
800 W/m
2
The total height is
× lcm
<
-3 -6
= 20.7 × 10
-3
m = 20.7 mm
(c) For side area, A s
= 4HL and heater area, A h
= L
2
. The heater-to-side area ratio is
10 =
L
2
L
= or L = 40 H
4HL 4H
For bakelite, L = 40 × 177 mm = 7.1 m and
<
2 3 h
For aerogel, L = 40 × 20.7 mm = 830 mm = 0.83 m and
<
2 < h
COMMENTS: (1) It may be expected that the small device utilizing the aerogel low thermal conductivity material will reach steady-state faster than the large device using the bakelite plates.
(2) The stainless steel sheets are isothermal to within 0.053 degrees Celsius. Precise placement of the thermocouple beads on the stainless steel sheets is not required. (3) The device constructed of bakelite is large. The device constructed of the nanostructured aerogel material reasonably sized.
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted.
Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976
United States Copyright Act without the permission of the copyright owner is unlawful.