CH ENGR 101B Study Guide - Quiz Guide: Thermal Conductivity, Air1, Abet
9 Jan 2019
School
Department
Course
Professor
Homework # 2
101B
UCLA, Winter 2018
Tsur
Ambient air
T∞, h
Electrical cable
Insulation
Ts,1
r1
r2
Ts,2
Chemical and Biomolecular Engineering
Heat Transfer
UCLA, Winter 2018
Due: January 24, 2018
Read Chapter 2; Review Example 2.4 in Chapter 2 with IHT
1. 2.7
2. [ABET] An electric cable of radius r1 and
thermal conductivity kc is enclosed by an
insulating sleeve whose outer surface is of
radius r2 and experiences convection heat
transfer and radiation exchange with the
adjoining air and large surroundings,
respectively. When electric current passes
through the cable, thermal energy is
generated within the cable at a volumetric
rate q
̇.
a. Write the steady-state forms of the heat diffusion equation for the insulation and the cable. Verify that
these equations are satisfied by the following temperature distributions:
Insulation:
2
,2 ,1 ,2
1
2
ln
( ) ( )
ln
s s s
r
r
T r T T T r
r
; Cable:
22
1
,1 2
1
( ) 1
4
s
c
qr r
T r T kr
Sketch the temperature distribution, T(r), in the cable and the sleeve, labeling key features.
b. Applying Fourier’s law, show that the rate of conduction through the sleeve may be expressed as
,1 ,2
'
2
1
2 ( )
ln
s s s
r
k T T
qr
r
Applying an energy balance to a control surface placed around the cable, obtain an alternative
expression for q'r, expressing your result in terms of q
̇ and r1.
c. Applying an energy balance to a control surface placed around the outer surface of the sleeve, obtain
an expression from which Ts,2 may be determined as a function of q
̇, r1, h, T∞, ε, and Tsur.
d. Consider conditions for which 250 A are passing through a cable having an electric resistance per unit
length of Re' = 0.005 Ω/m, a radius of r1 = 15 mm, and a thermal conductivity of kc = 200 W/m·K. For
ks = 0.15 W/m·K, r2 = 15.5 mm, h = 25 W/m2·K, ε = 0.9, T∞ = 25°C, and Tsur = 35°C, evaluate the
surface temperatures Ts,1 and Ts,2, as well as the temperature T0 at the centerline of the cable.
e. With all other conditions remaining the same, compute and plot T0, Ts,1, and Ts,2 as a function of r2 for
15.5 ≤ r2 ≤ 20 mm.
3. 2.52
Document Summary
When electric current passes through is generated within the cable at a volumetric rate q . thermal energy the cable, Ts,2: write the steady-state forms of the heat diffusion equation for the insulation and the cable. Verify that these equations are satisfied by the following temperature distributions: Sketch the temperature distribution, t(r), in the cable and the sleeve, labeling key features: applying fourier"s law, show that the rate of conduction through the sleeve may be expressed as. ,2 q r s k t s ln. 15. 5 r2 20 mm: 2. 52.
