second semester ay 2019-2020 · 2020-01-22 · thickness 3. constant wall area and uniform...
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Second Semester AY 2019-2020
Department of Chemical EngineeringUniversity of the Philippines Diliman 1
ChemE 131Thermal Systems
2
Metals: added energy increases the kinetic energy of free electrons
Atomic/
molecular
motion
excited by
the added
energy
3
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
Transfer of energy of motion
between adjacent molecules
4
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
OUTSIDE INSIDE
0T 0T
1T
→t
=heating rate q
5
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
0T
1T
0 Lx
T
thickness L 0 L= − =
surface area A=
Aheat flux
q=
q dE
A dx −
P
From thermodynamics:
dE C dT=
P
P
C dTq k
A C dx
= −
dE
dx= −
thermal diffusivity =
P
k
C= q dT
kA dx= −
Fourier's Law
6
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
Quantity SI unit English unit
heat flow rate, q J/s or W Btu/h, Btu/min
heat flux, q/A W/m2 Btu/ft2h
temperature, T ºC, K ºF, ºR
thermal conductivity, k W/mK Btu/ftºF
direction/dimension mm, m in, ft
q dTk
A dx= −
7
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
xq
yqzq
q T T Tk
A x y z
= − + +
qk T
A= −
If conduction occurs
within the wall in
all directions:
8
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
9
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
10
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
Find the steady-
state temperature
profile across the
given plane wall as
a result of heat
conduction.
11
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
12
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
Assumption Implication
steady state conditions q = constant
constant transfer area q/A may be constant
isotropic material with constant properties
material properties do not vary significantly within the
temperature range
wall thickness is significantly smaller than the width and height
one-dimensional (1D) heat conduction along the wall
thickness
heat convection from the bulk surroundings occurs at a significantly higher rate than the heat conduction
Twall = Tsurroundings
k = constant
13
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
Fourier's law of conduction
q dTk
A dx= −
s ,2
s ,1
TL
0 T
q dx k dTA
→ = −
s ,2 s ,1T
L q kA
T−
= −
→
x q k
T
dA
d→ = −
Calculate the heat loss per m2 ofsurface area through a 25.4-mm thickinsulating wall made of fiber insulatingboard (k = 0.048 W/m·K), where theinside temperature is 352.7 K and theoutside temperature is 297.1 K.
14
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
15
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
OUTx 0.0254 m=
q dTk
A dx= −
INSIDE OUTSIDE
INT 352.7 K= OUTT 297.1 K=
INx 0 m=
OUT OUT
IN IN
x T
x T
qdx k dT
A
= −
OUT IN
OUT IN
T Tqk
A x x
−= − −
Wk 0.048
m K=
2
q W105.1
A m=
Calculate the heat loss per m2 of surface areafor an insulating wall composed of a 25.4-mmthick fiber insulating board (k = 0.048 W/mK),where the inside wall temperature is 403.5 Kand the outside temperature is 276.5 K.If the insulation is used to cover a 50-ft2 wall,how thick must the fiber board be to avoid anincrease in the heat loss flux?
16
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
17
inner wall
Ti = 352.7 K
fiber
insula
ting b
oard
ambient
surroundings
To = 276.5 K
heat transfer
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
Assumptions:
1. steady state heat conduction
2. heat transfer is only along the direction of wall thickness
3. constant wall area and uniform thickness
4. fiber insulating board wall is isotropic
5. fiber insulating board has constant properties (not temperature dependent)
6. no convective heat transfer resistance near the surface of the outer wall
7. perfect contact between inner wall and the insulating wall
18
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
k0.048
W/mK
Ti
403.5 K
To
276.5 K
xi = 0 xo = 0.0254 m
2basis: 1 m wall area
q dTk
A dx= − constant=
Tk
x
= −
o i
o i
T Tk
x x
−= −
−
19
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
( )( )( )
2o i
o i
276.5 403.5 KT T Wq kA 0.048 1 m
x x m K 0.0254 0 m
−− = − = −
− −
2q 240 W (for 1 m wall surface area)=
( )
2
2
2
for a 50-ft wall:
276.5 403.5 K240 W W 0.048
m K x0.3048 m50 ft
1 ft
− = −
x 0.1178 m (or 117.8) mm wall thickness =
A furnace wall is insulated by a material withthermal conductivity expressed as
During operation, the temperature inside thefurnace reaches 1800F, releasing heat at 3500Btu/hr·ft2. Determine the insulation thicknesssufficient to bring the temperature outside thewall to 300F.
20
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
( )F
Btuk 0.08 1 0.003T
hr ft F
= +
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
x
INSIDE OUTSIDE
INT 1800 F= OUTT 300 F=
( )Tk 0.08 1 0.003= +
Assumptions
steady state conditions
constant transfer area
isotropic material
material properties do not vary significantly within the temperature range
wall thickness is significantly smaller than
the width and height
heat convection from the bulk surroundings occurs at
a significantly higher rate
2
Btuq 3500
hr ft=
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
q dTk
A dx= − ( )
OUT OUT
IN IN
x T
x T
q dx k T dT
A
→ = −
( ) ( )OUT OUT
IN IN
x T
x T 3500 dx 0.08 1 0.003T dT→ = − +
( ) ( )3002
1800 3500 x 0.08 T 0.0015T→ = − +
x 0.1423 ft (1.71 in) =
Determine the heat transfer rate across the frustum shown at the right, whose thermal conductivity is measured to be 27 Btu/hrftF.
23
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
24
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
x b 4 3 r x
2r 6 b 4
+ = → =
+
b 4 b b 8 in
6 4
+= → =
1r x
4=
25
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
q dTk
A dx= −
dx q k dT
A→ = −
( )( )
o
o
12 in 200 F
21
8 in 400 F4
dxq 27 dT
x
= −
( )o
o
12 ft
12 200 F
400 F8 ft
12
16 1 q 27T
x
→ − = −
26
What is heat conduction?
Heat conduction flux
Thermal conductivity
1D heat conduction
( ) o
o
Btu27 200 400 F
hr ft Fq
16 12 12 1
12 8 ft
− −
=
− −
Btuq 2120.58
hr=
ChemE 131Thermal Systems
27
Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies linearly with temperature according to the equation
28
o2 o F
Btuk 0.2 0.005T
h ft F
= +
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