t
Lk
m hL
2
07
21
1
2
0 25 0 17
ll
l
Ste l
. . . Ste
h L/ kl
t
ck
h
TtT
TTt
TT
TTt
f
a
af
a
l l
l
2l
1 18
0
00
12
1 83 2
Ste
. Ste
,
,
.
Heat and Mass Transfer 4-117
1999 by CRC Press LLC
Incropera, F.P. and Viskanta, R. 1992. Effects of convection on t
T T C C T T c,o c,i c h,i c,i
m /m c h / , , m c m c c pc h ph
T T T m,Cmin m,Cmax lm
min
4-150 Section 4
1999 by CRC Press LLC
3. Calculate the Reynolds number Re = GDh/and/or any other pertinent dimensionless groups
(from the basic definitions) neede
about +/1% of level, although melting standards are available to +/0.5C.
The phase-change materials melt at well-defined temperatures, yielding easily discernible evidence
that their event temperature has been exceeded. When more than one phase-change pai
(4.4.72)
with R(t) calculated from the transcendental equation
(4.4.73)
The solutions for inward melting of a cylinder, where heating is applied at the outer radius ro, are the
same as the above-described ones for the outward-melting cylinder, if the repl
finned tubes, as shown in Figures 4.5.6 and 4.5.5a; (2) flat or continuous (plain, wavy, or interrupted)
external fins on an array of tubes, as shown in Figures 4.5.7 and 4.5.5b; (3) longitudinal fins on individual
tubes. The exchanger having flat (contin
9(1), 5260.
Wen, C.Y. and Yu, Y.H. 1966. A generalized method for predicting the minimum fluidization velocity,
AIChE J., 12(2), 610612
Wen, C.Y. and MIller, E.N. 1961. Ind. Eng. Chem., 53, 5153.
Nu Pr susp
dp
cp
g
pp
pg p
t
p
g
hd
k
C
C
V
gd
0.3 0.21
4-1
solid surfaces immersed in the liquid, and as freezing progesses the crystals grow in the form of intricately
shaped fingers, called dendrites. This complicates the geometry significantly and makes mathematical
modeling of the process very difficult. An i
h
Heat and Mass Transfer 4-115
1999 by CRC Press LLC
By using Table 4.4.2, the geometric coefficients for the cylindrical shape of the fish are P = 1/2 and
R = 1/16, while d is the cylinder diameter, = 0.1 m. Substituting these values into Equation (4.4.
gD
G
g
gL
g
p
cicemoic
m
c
15
2
4
2
1 1 1 . 222
Heat and Mass Transfer 4-143
1999 by CRC Press LLC
0.7); the exception is liquid metals (Pr < 0.03). Hence, there is generally no need to identify the thermal
boundary condition in turbulent flow for all fl
(4.5.49)
Gray and Webb (see Webb, 1994) hypothesized the friction factor consisting of two components: one
associated with the fins and the other associated with the tubes as follows.
(4.5.50)
where
(4.5.51)
and ft (defined the same way as f) is the Fanni
Txt
t
Txt
x
lxXtt
l
l
2
, ,
2 in
0 for 0
T xt T x t l f , in t0, a 0
T t T t l0 0 0 ,for
T X t T t l f for 0
k
T
x
h
dX t
dt
t
Xt
ls
l
l
for 0
4-110 Section 4
1999 by CRC Press LLC
The analytical solution of this problem yields the temperature distribu
With different materials for different ranges, thermocouples have been used from cryogenic temperatures
(a few Kelvin) to over 3000 K. In the moderate temperature range, ambient to 1200C, manufacturers
quoted calibration accuracy can be as good as 3/8% of
h
dX t
dt
ts
s
XtXt
s
l
l
l
for 0
TxtTTT
x
t
iif
s
l
l
l
,
erfc
erfc
2
TxtTTT
x
t
sf
s
,
00
2
erfc
erfc
ek
k
TT
TT
e
s
sif
fss
s
2 2
0 erf erfc Ste
l
ll
l
Stes
sf
s
cTT
h
0
l
X t t s 2 1 2
Heat and Mass Transfer 4-109
1999 by CRC Press LLC
(4.4.41)
F
and Liquid-Vapor Phase Change Phenomena by V.P. Carey (Taylor and Francis, Washington, D.C., 1992)
provide an introduction to the physics of boiling and condensation processes. The text by J.G. Collier,
Convective Boiling and Condensation (2nd ed., McGraw
(4.5.77)
so that
(4.5.78)
where 1 and 2 are generally specified for the surface or can be computed for plate-fin surfaces
from Shah (1981) and Kays and London (1984):
(4.5.79)
Now compute the fluid flow lengths on each side (see Figure 4.5.17) from the de
Influence of Operating and Design Variables.
Based on operational experience and research over the last several decades, many variables have been
identified that have a significant influence on fouling. The most important variables are summarized next.
Fl
For the same melting problem but with the boundary condition of an imposed time-dependent heat
flux q0(t),
(4.4.59)
the quasi-static approximate solution is
(4.4.60)
Xt
k
h
T t T dt t
s
t
f
2
0
0
0
12
l
for
l
T xt T t T t T
x
Xt
x X t t l f ,
qxtk
dT x
equal to the heat transfer coefficient which would have existed at those conditions. The usual recommendation
is to adjust for the Prandtl number of Schmidt number using a relation of the form:
(4.6.18)
based on laminar results. That recommendation has no
TTTT
TT
TT
hicihoco
hici
hoco
,
,
,
ln
MTD = LMTD countercurrent F
4-172 Section 4
1999 by CRC Press LLC
where (LMTD)countercurrent is calculated from Equation (4.5.89) and F is the configuration correction
factor for the flow configuration involved. F
p
gL
g
m
c
p
p
G
gp
Kf
L
r
K
icii
c
h
i
m
i
o
e
i
o
2
22
2
1
1
1
211
1
2
1
2
11
m
m
io
io
v
vv
1
m
R
p
T
ave
lm
R
p
fLG
g D chm
4
2
21
,m / A0
Heat and Mass Transfer 4-141
1999 by CRC Press LLC
FIGURE 4.5.14 Entrance and exit pressure loss coeffi
conventional PHEs. A double-wall PHE is used to avoid mixing of the two fluids. A wide-gap PHE is
used for fluids having high fiber content or coarse particles. A graphite PHE is used for highly corrosive
fluids. A flow-flex exchanger has plain fins on on
tubes because it is easier to clean the tube side. If this is not possible or desirable, then a removable
bundle with a rotated square tube layout should be chosen to facilitate cleaning.
Pressure Drop. Tube-side pressure drop in plain tubes is discussed
and constructions/surface features. It increases maintenance costs: (1) cleaning techniques, (2) chemical
additives, and (3) troubleshooting. It may cause a loss of production: (1) reduced capacity and (2)
shutdown. It increases energy losses: (1) reduced
is 1.2 1.2 6 m (4 4 20 ft). Fouling is one of the major potential problems in many compact
exchangers except for the plate heat exchangers. With a large frontal area exchanger, flow maldistribution
could be another problem. Because of short transient time
C
C
C
C
CC
CC
*
1*
NTU NTU
NTU
NTU
for
1 for
min
1
1 min
1 max
C
CC
CC
*C C
F
C
C
C
NTU
NTU NTU 1
1
1 NTU 1
cf 1
*1
*
ln
F
*
1
NTU 1 1
1
1 NTU 1 1 1
1
11
1
R
RP
PR
P
P
ln
11
Heat and Mass Transfer 4-133
1999 by CRC Press LLC
The overall heat transfer c
individually finned tubes to
(4.5.35)
Regenerators. For regenerator matrices having cylindrical passages, the pressure drop is computed
using Equation (4.5.29) with appropriate values of f, Kc, and Ke. For regenerator matrices made up of
any porous materi
the point of saving money by reducing shell diameter and number of tubes and may require excessive
clear way for pulling the bundle; the bundles may be springy and difficult to handle during maintenance.
Most heat exchangers fall into the 6:1 to 10:1 rang
1122
121
2
1212
123
123
1
1
22
j
e
b
e
ml
a ml r
r
a r ml r
b
r
r
r
r
e
tanh
.
.
.
. . ln
*.
*.
* . * exp . .
*
*
*
*
for
for
for
for
0 6 2 257
0 6 2 257
0 9107 0 0893
0 9706 0 17125
2
2
0 445
0 445
0 246 0 13 1 3863
m
h
k
llr
d
d ff
ej
fe
o
2
2
12
*
j