CHE 378: Heat and Mass Transfer Spring 2008 Homework 6 Due: Friday, February 22, 2008 at the beginning of lecture
1. Middleman 2.1 2. Middleman 2.11 Clarification: The units of K given in the problem are: K [=]
2 cm 3 of H 2 at STP cm of glass [=] cm s at
CHE 378: Heat and Mass Transfer Spring 2008 Homework 7 Due: Friday, February 29, 2008 at the beginning of lecture
1. Middleman 2.23 2. Middleman 3.28 3. Middleman 3.35 4. Middleman 3.36 5. Middleman 3.56
CHE 378: Heat and Mass Transfer Spring 2008 Homework 2 Due: Friday, January 25, 2008 at the beginning of lecture
1. A spherical shell with inner radius r1 and outer radius r2 has surface temperatures T1 and T2, respectively, where T1>T2. Determine the tem
CHE 378: Heat and Mass Transfer Spring 2008 Homework 11 Due: Friday, April 18, 2008 at the beginning of lecture
1. Middleman 6.54 2. In a textbook on unit operations, the following statement was made: For a given mass-transfer operation, KL, was experimen
CHE 378: Heat and Mass Transfer Spring 2008 Homework 1 Due: Friday, January 18, 2008 at the beginning of lecture
1. An aluminum pan whose thermal conductivity is 237 W/(m*C) has a flat bottom with diameter 20 cm and thickness 0.4 cm. Heat is transferred s
CHE 378: Heat and Mass Transfer Spring 2008 Homework 4 Due: Friday, February 8, 2008 at the beginning of lecture
1. A rear window defroster in an automobile consists of uniformly distributed highresistance wires molded into the glass. When power is applie
Mechanisms of Heat Transfer
Thermodynamics allows us to determine the internal energy change and net
heat transferred for a given process, or the steady-state net rate of heat transfer.
Additional information is needed, however, if we are interested in de
One-Dimensional Transient Heat Conduction
(bounded solids)
planar slab
A n cos n exp n2 X Fo
An
n 1
x/L
4sin n
2 n sin 2 n
T Ta
To Ta
n tan n Bi
Bi
X Fo t / L
2
hL
k
Jn x
cylinder
A n J o n exp n2 X Fo
n 1
r/R
X Fo t / R 2
2Bi
An
J o ( n ) n2 Bi
Transient Heat Transfer by Conduction
For a non-steady-state process, the temperature in a conducting material will, in general, be
a function of position and time. Due to the objects finite thermal conductivity k, there will
be an internal resistance to
Straight rectangular fin
T Ta cosh N (1 s )
Tw Ta
cosh N
1/ 2
1/ 2
Rcond
hL2
N
kB
Rconv
z
s
L
Tw
N=0
local convective
driving force: (T Ta)
T
maximum possible
local convective
driving force: (Tw Ta)
Ta
N
0
L
z
What is the maximum possible rate
Heat Conduction Through a Cylindrical Solid
one-dimensional, steady-state, no energy generation
v
q k T
L
conduction in the radial direction only
r2
r1
T2
T1
qr k
dT
dr
heat flux in radial or r-direction
no heat loss along the two
ends of the annulus
L
r
Heat Transfer From Extended Surfaces
Cooling Fins
h, T
Qconvection
Qconduction
Qconvection
Qconvection
Ts , A
Sometimes our goal is to promote heat loss
from a hot surface. But h may be small,
such as for free convection of gases, and/or
it may be insuffi
steady-state, one-dimensional
L
internal energy generation within inner cylinder only at
a uniform rate per unit volume of S;
inner cylinder is surrounded by insulation (outer layer)
R2
R1
ki S
h, Ta
ko
What is the temperature profile across this
composit
Microscopic equation of change for the internal energy
(conduction + convection)
We derive the general partial differential equation that describes how the temperature
at each location in an incompressible material varies with time. We include energy
tran
Heat Conduction Through a Plane Wall
one-dimensional, steady-state
L
no heat loss along
top/bottom surfaces
A (cross-sectional area in direction of heat transfer)
x
conduction in
x-direction only
T1
T2
A
Qin
x
Qout
x +x
x
fixed shell of volume Ax
Qin, Qou
cylinder
dV 2rdr 2R 2 d
1
d
0
1
d
T
1
TdV
dV
2 d
0
0
1 A1J o 1 exp 12 X Fo
sphere
dV 4r 2dr 4R 3 2d
2A1
long time
Q tot VC P T To
1
d
VC P To Ta 1
2
BiJ o 1
2
exp
X Fo
1
2
1
0
1
2
d
1
3 2 d
0
0
sin 1
1 A1
exp 12 X Fo
1
long time
3A1
Bi si
Diffusion with a Homogeneous Chemical Reaction
(steady-state, one-dimensional)
gas A
(well-mixed)
pA or cA,g
z=0
z
z
z + z
liquid B
Gas A with constant pressure (or concentration) dissolves in liquid
B in dilute amounts and diffuses isothermally throughou
Heat Transfer: Thermodynamics and Fundamental Concepts
1st Law of Thermodynamics (closed system or constant mass)
U = heat + work added to/removed from the system (no PE,KE changes)
Both heat and work are forms of energy transfer between the system and
th
The Composite Solid
L
steady-state; no energy generation
k1
T1
A1
k2
A2
k3
A3
no heat loss along top/bottom
Now consider the following planar
composite solid, in which the
temperature drop is the same across
each material.
T2
While the heat transfer acros
Steady-State Heat Conduction With Internal Energy Generation
one-dimensional
In various cases, heat conduction occurs in a medium within which internal energy is also generated
by some process. Examples include resistive heating in electrical conductors (
Diffusion of Naphthalene in a Narrow Tube (Ex. 2.1.1)
L
solid A
xA1
z=0
z
A
cross-sectional area S
B
stream of A,B (ideal gas)
xA2 < xA1
constant T,P
B: nitrogen
(insoluble in A)
A: naphthalene
There is a significant vapor pressure of solid A at the given
Two tanks of equal volumes are kept at the same T and P, one filled with just N 2
(gas) and the other filled with just CO2 (gas). At some point, the line connecting
the two tanks is opened, and both species begin to diffuse from one tank to the
other thro
Sustained Release Hollow Fiber System (2.3 M)
cA,out 0
Rout
gas mixture A,B
rapidly moving stream with no A
CA,in
Rin
A represents some active chemical agent (e.g., insecticide) that has
been encapsulated inside a hollow fiber membrane for controlled rele
Diffusion Across a Thin Barrier Separating Two Fluids
steady-state, one-dimensional, dilute limit
L
well-mixed fluid mixture with
species A of small concentration
well-mixed fluid mixture with
species A of small concentration
c1o
uniform concentration of
At this time, T To beyond this distance,
and so the system has not yet sensed
the presence of a heat flux at the surface.
To
t1
T
Lets consider the T response of a
semi-infinite medium (with Bi > 1).
t2
t3
increasing time
Ta
0
y
erf erf
1/ 2
4
t
Fundamentals of Mass Transfer
Mass transfer: the transport or movement of matter from one location to another
(the material of interest is described by a net velocity)
one of several key differences with heat transfer
(we cant assign a velocity to energy