The air-fin cooler is a device in which hot-process fluids, usually liquids, flow inside extended surface tubes and atmospheric air is circulated outside the tubes by forced or induced draf
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For the annulus, the tube diameter is replaced with the equivalent diameter for pres-sure loss:
∆P f =8nhpf ρV2
2
L
d e = 4nhpf ρV2L
The turn loss for both inner pipe and annulus withNhphairpins is
11.6.6 Wall Temperature and Further Remarks
It may be noted that the wall resistance presents the lowest resistance to the flow of heat between the hot and cold fluids Hence, an excellent approximation to the wall temperature may be obtained via the computation ofthe product of
R is and the
heat flux Then, ifthe hot fluid is carried within the inner tube, the wall temperature will be
T w = T b− R is q
where
R is = R io S
R io = r io + r do + r mo
In the event, that the cold fluid is carried in the inner tube, the wall temperature will be
T w = t b+ R is q
11.6.7 Series–Parallel Arrangements
When two streams are arranged for counterflow, the LMTD represents the maximum thermal potential for heat transfer that can be obtained Often, on the industrial scale,
a single process service may entail the use ofmore than a single long hairpin It then follows that it is desirable to connect the hairpins in series on both the annulus and inner pipe sides, as in Fig 11.26 In this configuration, the temperature potential remains the LMTD for counterflow
In some services, there may be a large quantity ofone fluid undergoing a small temperature change and a small quantity ofanother fluid undergoing a large temper-ature change It may not be possible to circulate the large volume offluid through the required number ofhairpins with the pressure drop available Under these circum-stances, the larger volume offluid may be manifolded in the series–parallel arrange-ment shown in Fig 11.27 The inner pipe fluid has been split between the exchangers
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Figure 11.26 Double-pipe heat exchangers in series (From Kraus et al., 2001, with per-mission.)
T1
T2
t2
t1
II
I
Figure 11.27 Double-pipe heat exchangers in series–parallel (From Kraus et al., 2001, with permission.)
designated I and II Both ofthese exchangers are in counterflow relative to each other but not in the same sense as in Fig 11.26 In Fig 11.27, theT ’s refer to the series
streams and thet’s refer to the parallel streams.
Departures from true counterflow and true co-current (parallel) flow can be han-dled by the logarithmic mean temperature difference correction factorF
Kern (1950) presents a derivation for the factorγ to be used in a modification of the heat transfer rate equation
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q = U o Sγ(T1− t1) (11.160) whereT1− t1represents the total temperature potential, the difference in the fluid
stream inlet temperatures, in the exchanger configuration After a laborious and de-tailed derivation, Kern (1950) gives, for one series hot fluid andn parallel cold fluid
streams,
γ =
1−T2− t1
T1− t1
Z − 1 nZ
ln
Z − 1 Z
T
1− t1
T2− t1
1/n
Z
where
Z = T1− T2
n(t2− t1)
For one series cold fluid andn parallel hot fluid streams, Kern (1950) gives
γ =
1−T1− t2
T1− t1
1− Z
nz
ln
(1 − Z)
T
1− t1
T1− t2
1/n + Z
TABLE 11.4 Dimensions of Multitube Double-Pipe Exchangersa
Source: After Saunders (1988).
aFin thicknesses are identical to those listed in Table 11.3 The dimensions shown here are for low-pressure units.
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[868], (72)
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where
Z = n(T1− T2)
t2− t1
11.6.8 Multiple Finned Double-Pipe Exchangers
There are numerous applications for longitudinal fin pipes and tubes Closest to the double-pipe exchanger is the hairpin with multiple longitudinal-fin pipes A vari-ety ofpipes and tubes are available with longitudinal fins whose numbers, heights, thicknesses, and materials differ Data for some of these configurations are shown in Table 11.4
The procedure for the design and analysis of the multiple-tube exchanger differs little for that used for the single-tube exchanger
11.7 TRANSVERSE HIGH-FIN EXCHANGERS 11.7.1 Introduction
Pipes, tubes, and cast tubular sections with external transverse high fins have been used extensively for heating, cooling, and dehumidifying air and other gases The
fins are preferably called transverse rather than radial because they need not be
circular, as the latter term implies, and are often helical The air-fin cooler is a device
in which hot-process fluids, usually liquids, flow inside extended surface tubes and atmospheric air is circulated outside the tubes by forced or induced draft over the extended surface
Unlike liquids, gases are compressible, and it is usually necessary to allocate very small pressure drops for their circulation through industrial equipment or the cost ofthe compression work may entail a substantial operating charge Except for hydrogen and helium, which have relatively high thermal conductivities, the low thermal conductivities ofgases coupled with small allowable pressure drops tend toward low-external-convection heat transfer coefficients
In the discussion oflongitudinal high-fin tubes in Section 11.6.1, it was noted that
a steel fin 1.27 cm high and 0.0889 cm thick could be used advantageously with a fluid producing a heat transfer coefficient as high as 250 W/m2· K Aluminum and copper have thermal conductivities much higher than steel, 200 and 380 versus 45 W/m· K It would appear that thin high fins made ofaluminum or copper would have excellent fin efficiencies when exposed to various heating and cooling applications ofair and other gases at or near atmospheric pressure
In air-fin cooler services, the allowable pressure drop is measured in centimeters
or inches ofwater and air can be circulated over a few rows ofhigh-fin tubes with large transverse fin surfaces and, at the same time, require a very small pressure drop
Transverse high-fin tubular elements are found in such diverse places as economizers ofsteam power boilers, cooling towers, air-conditioning coils, indirect-fired heaters, waste-heat recovery systems for gas turbines and catalytic reactors, gas-cooled nu-clear reactors, convectors for home heating, and air-fin coolers
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In the services cited involving high temperatures, hot gases flow over the fins and water or steam flows inside the tubes The extended surface element usually consists ofa chromium steel tube whose chromium content is increased with higher anticipated service temperatures A ribbon, similar in composition to the tube, is helically wound and continuously welded to the tube The higher and thicker the fins, the fewer the maximum number offins per centimeter oftube which can be arc-welded because the fin spacing must also accommodate the welding electrode High-temperature high-fin tubes on a closer spacing are fabricated by electrical resistance welding ofthe fins to the tube
High-fin tubes can also be extruded directly from the tube-wall metal, as in the case ofintegral low-fin tubing However, it becomes increasingly difficult to extrude
a high fin from ferrous alloys as hard as those required for high-temperature services, which are often amenable to work hardening while the fin is being formed Whether fins are attached by arc welding or resistance welding, the fin-to-tube attachment for all practical design considerations introduces a neglible bond or contact resistance
High-fin tubes are used in increasing numbers in devices such as the air-fin cooler,
in which a hot fluid flows within the tubes, and atmospheric air, serving as the cooling medium, is circulated over the fins by fans Several high-fin tubes for air-fin cooler service are shown in Fig 11.28 Typea can be made by inserting the tubes through
sheet metal strips with stamped or drilled holes and then expanding the tubes slightly
to cause pressure at the tube-to-strip contacts The tubes and strips may then be brazed Ifthe tubes are only expanded into the plates to produce an “interference fit,”
some bond or contact resistance must be anticipated For practical purposes, when the tubes and strips are brazed together, the joint may be considered a metallurgical bond and the bond resistance can be neglected
In Fig 11.28, tubesb through e are made by winding a metal ribbon in tension
around the tube These types are not metallurgically bonded and rely entirely upon the tension in the ribbon to provide good contact Typef combines tension winding with
brazing, and for the combination of a steel tube and an aluminum fin, the common tin–lead solder is not compatible and a zinc solder is used Typeg employs a tube as
a liner, and high fins are extruded from aluminum, which, like copper, is a metal that can be manipulated to a considerable fin height Typesd, e, and f employ
aluminum for the fins and are arranged to protect the tube from the weather because air-fin coolers are installed outdoors Typeg, sometimes called a muff-type high-fin
tube, has its contact resistance between the inside ofthe integral finned tube and the liner or plain tube Typeh has a mechanical bond which can closely match a
metallurgical bond for contact resistance Typei, an elliptical tube with rectangular
fins, may employ galvanized steel fins When tube ends are circular, they are rolled into headers
Consider a typical air-fin cooler application with a hot fluid inside the tubes
In many instances, carbon steel meets the corrosion-resistance requirements ofthe tube-side fluid From the standpoint ofhigh thermal conductivity and cost, aluminum ribbon is very suitable for tension-wound fins However, aluminum has twice the thermal coefficient ofexpansion ofsteel, and the higher the operating temperature ofthe fluid inside the tubes, the greater the tendency ofthe fins to elongate away from their room-temperature tension-wound contact with the tube, and the greater is
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Figure 11.28 Various types ofhigh-fin tubing (From Kraus et al., 2001, with permission.)
the bond or contact resistance In one variation oftyped in Fig 11.28, the ribbon
is wound withJ rather than L feet, with the J ’s pressing against each other and the
tube at room temperature As the feet become heated during operation, they expand against each other
11.7.2 Bond or Contact Resistance of High-Fin Tubes
The bond resistance ofseveral types ofinterference-fit high-fin tubes shown in Fig
11.28 has been studied by Gardner and Carnavos (1960), Shlykov and Ganin (1964), and Yovanovich (1981) Gardner and Carnavos pointed out that in its most general
sense, the term interference fit implies the absence ofa metallurgical bond, as opposed
to the extrusion ofa fin from a tube wall, and the welding, soldering, or brazing ofa
fin to the tube The interference fit is produced by mechanically developing contact pressure through elastic deformation either by winding a ribbon under tension about
a tube, as in typesb through e in Fig 11.28, or by expanding a tube against the fins
as in typea, or a combination ofpressing the root tube against the liner or the liner
against the root tube, as in typeg.
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11.7.3 Fin Efficiency Approximation
The fin efficiency ofthe radial fin ofrectangular profile was given in Chapter 3:
mr2
a − r2
b
I1(mr a )K1(mr b ) − K1(mr a )I1(mr b )
I0(mr b )K1(mr a ) + I1(mr a )I o (mr b )
where
m =
2h kδ
1/2
ρ = r b
r a φ = (r a − r b )3/2
2h
kA p
1/2
R a= 1
1− ρ R b =
ρ
1− ρ The modified Bessel functions in the radial fin efficiency expressions are obtained from tables or from software, and their employment to obtain the efficiency involves a somewhat laborious procedure An alternative has been provided by McQuiston and Tree (1972), who suggest the approximation
where
m =
2h kδ
1/2
ψ = r b1− ρ
ρ
1+ 0.35 ln1ρ
(11.164) whereρ is the radius ratio,
ρ = r b
r a
11.7.4 Air-Fin Coolers
The air-fin cooler consists ofone or more horizontal rows oftubes constituting a
section through which air is circulated upward by mechanical draft The fan that
moves the air may be above the section providing an induced draft or it may be below the section, providing a forced draft.
In the induced-draft air-fin cooler, the heated air is thrown upward to a good height
by its high exit velocity A relatively small amount ofthe heated air is sucked back
to reenter the air intake below the section and thereby cut down the temperature difference available between the ambient air and the process fluid In a forced-draft unit, the air leaves at a low velocity at a point not far from the high entrance velocity
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ofthe air to the fan below the section Hot air is more apt to be sucked back into
the fan intake, causing recirculation Following a trend in cooling towers that started
some years ago, induced-draft units now appear to be preferred Usually, the section has cross bracing and baffles to increase rigidity and reduce vibration
The design and analysis of air-fin coolers differs only in a few respects from the longitudinal fin exchangers in Section 11.6 The principal difference is in the air side, where air competes with other fluids as a coolant Because air is incompressible and liquids are not, only a small pressure drop can be expended for air circulation across the finned tubes, lest the cost ofair-compression work become prohibitive In most applications, the allowable air-side pressure drop is only about 1.25 cm (12 in.) of water The air passes over the finned tubing in crossflow, and this merely requires the use of the proper heat transfer and flow friction data The temperature excursion ofthe air usually cannot be computed at the start ofthe calculations because the air volume, and hence the air temperature rise, are dependent on the air pressure drop and flow area ofthe cooler
Most widely used are the integral-fin muff-type tube (Fig 11.28g), the L-footed
tension-wound tube (Fig 11.28g), and the grooved and peened tension-wound tube
(Fig 11.28h) These tubes usually employ nine or eleven fins per inch Numerous
other tubes are manufactured in accordance with the types shown in Fig 11.28b, c,
andf Other tubes have serrated or discontinuous fins The latter tubes are fabricated
to their own standards by manufacturers of air-fin coolers
Physical Data As indicated in Fig 11.29, tubes may be arranged in either triangu-lar or in-line arrangements Observe that the pitch in these arrangements is designated
byP t , P l, orP d, whereP t = transverse pitch (m), P l = longitudinal pitch (m), and
p t
p t
p d
p d
Figure 11.29 Tube arrangements: (a) triangular; (b) in-line (From Kraus et al., 2001, with
permission.)
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P d = diagonal pitch (m) The diagonal pitch is related to the transverse and longitu-dinal pitch by
P d =
P
t
2
2
+ P2
l
1/2
(11.165) and in the case ofan equilateral triangular arrangement,
P d = P t
When there aren tubes in a row andn r rows, the total number oftubes will be
Letz be the clear space between the tubes, which are L meters long The fins are
b meters high:
b = d a − d b
2 whered a andd b are, respectively, the outer and inner diameters ofthe fin The fins areδfthick and the minimum flow areaA = Aminwill depend on the transverse pitch
P t For
P t > 2P d − d b−z + δ2zδ f
f
A = Amin = n L
P t − d b− 2zδ f
z + δ f
(11.167) and for
P t < 2P d − d b− 2zδ f
z + δ f
A = Amin= 2n L
P d − d b−z + δ2zδ f
f
(11.168) The surface area of the tube (between the fins) will be
S b= πn t Ld b z
and the surface of the fins, which accounts for the heat transfer from the tips of the fins, will be
S f =z + δ πn t L
f
1 2
d2
a − d2
b
+ d aδf
(11.170)
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This makes the total surface
S = S b + S f (11.171a) the finned surface per total surface,
S f
S =
S f
and the surface per unit length per tube,
friction data in tube bundles containing high-fin tubes have been reported by Jameson (1945), Kutateladze and Borishaniskii (1966), and Schmidt (1963) The correlation ofBriggs and Young (1963) is based on a wide range ofdata Their general equation for tube banks containing six rows of tubes on equilateral triangular pitch is
Nu=hd b
k = 0.134Re0.681· Pr1/3
2(P f− δf )
d a − d b
0.20
P f − δf δf
0.1134
(11.172) where
Re= d b G µ and where the range ofparameters is
1000< Re < 18,000 0.33 mm < δ f < 20.02 mm
11.13 mm < d b < 40.89 mm 1.30 mm < P f < 4.06 mm
11.42 mm < b = d a−d b
2 < 16.57 mm 24.99 mm < P t < 111 mm
Vampola (1966) proposed a correlation based on an extensive study ofdifferent finned tubes For more than three tube rows,
Nu= hd e
k = 0.251Re0.67
P t − d b
d b
−0.20
×
P
t − d b
P f− δf + 1
−0.20P
t − d b
P d − d b
0.40
(11.173) where
Re= d e G µ