The longitudinal fin double-pipe exchanger is used advantageously where an ap-preciable inequity appears in the composite thermal resistance ofa pair offluids in a plain double-pipe exchan
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Ratio of free flow to frontal area on one side, , dimensionless
⫺0.8
⫺1.0
⫺0.7
⫺0.9
⫺0.6
⫺0.5
⫺0.4
⫺0.3
⫺0.2
⫺0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
KK ce
Re = ⬁
Re = 2000
Re = 10,000
Re = 3000
Re = 5000
Re = 5000
Re = 3000
Re = 10,000
Re = 2000
Re = ⬁
K e
K c
Laminar
Laminar
(From Kays and London, 1984)
St= Nu
Re· Pr =
hd e /k (d e G/µ)(cµ/k) =
h
The fluid properties in eqs (11.129) and (11.130) are evaluated at the bulk tem-perature
T b =1
t b =1
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Ratio of free flow to frontal area on one side, , dimensionless
⫺0.8
⫺0.7
⫺0.6
⫺0.5
⫺0.4
⫺0.3
⫺0.2
⫺0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
KK ce
Re = ⬁
Re = 2000
Re = 10,000
Re = 3000
Re = 5000
Re = 5000
Re = 3000
Re = 10,000
Re = 2000
Re = ⬁
K e
K c
Laminar
Laminar
Kays and London, 1984.)
Flow Friction Data Kays and London (1984) suggest that the pressure drop∆P
in a compact heat exchanger be computed from the equation
∆P
P1 =
G2v1
2g c P1 (Φ1+ Φ2+ Φ3− Φ4) (11.132) where
Φ1= 1 + K c− σ2
Φ2= 2
ν2
ν1
− 1
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Φ3= f S
A
νm
ν1
Φ4 =1− σ2− K e ν2
ν1 Friction factors are correlated on an individual surface basis and are usually plotted
as a function of the Reynolds number The entrance and exit loss coefficients differ for the various types of passages and are plotted as functions of the parameterσ and
the Reynolds number
Four terms may be noted within the parentheses in eq (11.132) These terms de-note, respectively, the entrance or contraction loss as the fluid approaches the ex-changer at line velocity and changes to the exex-changer entrance velocity, acceleration loss, or acceleration gain as the fluid expands or contracts during its passage through the exchanger, flow friction loss, and exit loss
Kays and London (1984) have presented heat transfer and flow friction data for approximately 120 surfaces described by the foregoing Some typical examples are shown in Figs 11.17 through 11.20 Entrance and exit loss coefficients for plate fin cores and rectangular passages are plotted in Figs 11.21 and 11.22
Because the pitch ofthe fins is small, the height ofthe fin is approximately equal to
b, the distance between the separation plates The fin efficiency for the parallel-plate
heat exchanger may be taken as
whereb/2 is halfofthe separation plate distance.
11.6 LONGITUDINAL FINNED DOUBLE-PIPE EXCHANGERS 11.6.1 Introduction
The double-pipe exchanger consists ofa pair ofconcentric tubes or pipes One process stream flows through the inner pipe, and the other flows, either in counter- or co-current (parallel) flow in the annular region between the two pipes The inner pipe may be bare or it may contain as many as 48 longitudinal fins equally spaced around its periphery
Consider the plain double-pipe exchanger shown in Fig 11.23 It usually consists
of two pairs ofconcentric pipes with a return bend and a return head made leaktight
by packing glands The packing glands and bends returning outside rather than inside the return head are used only where the annulus has low fluid pressure Ifthere is no problem with differential thermal expansion, the glands may be omitted and the outer and inner pipes may be welded together to provide a leaktight construction
Two pairs ofconcentric pipes are used to form a hairpin because ofthe
conve-nience the hairpin affords for manifolding streams and the natural loop it can provide
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Return bend
Tee
Gland
Return head
for differential thermal expansion between the inner and outer pipes The hairpin brings all inlets and outlets close together at one end, which is particularly important when multiple hairpins are connected in batteries Moreover, the hairpins need not have the same length An additional merit ofthe double-pipe heat exchanger is the ease in which it usually can be disassembled for inspection and cleaning or reused in another service whenever a process becomes obsolete
The longitudinal fin double-pipe exchanger is used advantageously where an ap-preciable inequity appears in the composite thermal resistance ofa pair offluids in a plain double-pipe exchanger Because heat transfer equipment is usually purchased
on the basis of its performance in the fouled condition, the composite thermal resis-tance is the sum ofthe convective film resisresis-tances and the fouling resisresis-tances The advantage of the finned annulus lies in its ability to offset the effects of poorer heat transfer in one fluid by exposing more surface to it than the other Indeed, even if the composite resistances ofboth fluids are low, as discussed subsequently, there may still be an advantage in the use ofthe finned inner pipe
Fins are usually 0.089 cm thick (0.035 in and 20 BWG) A steel fin with a thermal conductivity of50 W/m· K and a height of1.27 cm (1
2in.) on exposure to a composite resistance of0.004 m2 · K/W (corresponding to a film coefficient of 250 W/m2·
K) has a fin efficiency ofabout 0.65 Exposed to a composite resistance of0.002
m2· K/W, the efficiency drops to about 0.5 Hence, the high fin has its limitations,
although metals ofhigher thermal conductivity extend the range ofapplication Fin surface is inexpensive compared with prime surface, but its usefulness diminishes significantly below a composite resistance of0.002 m2· K/W For the case where
both composite resistances are very large, any improvement in the surface exposed
to the higher resistance may save considerable linear meters ofexchanger Moreover, inner pipes are available with fins on the inside as well as the outside ofthe pipe, and the inner pipes are also available with continuous twisted longitudinal fins, which cause some mixing in the annulus As a class, however, these show a small increase
in heat transfer coefficient for a large expenditure of pressure loss, and for viscous fluids, the mixing and its effects decay rapidly
The disposition ofthe fins about the pipe is shown clearly in Fig 11.24 They form
a radial array ofchannels, with each channel composed oftwo fins Channels may
be attached by continuously spot-welding them to the outside ofthe inner pipe or
by other brazing or welding procedures It should be noted that contact between the
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perime-ter, and surfaces for the annular region in the double-pipe heat exchanger: (a) extruded fins;
(b) welded U-fins; (c) detail oftwo welded U-fins (From Kraus et al., 2001, with permission.)
channels and the other pipe should be continuous over the entire channel length but need not be very wide In another method ofattaching longitudinal fins, grooves are plowed in the outside diameter ofthe inner pipe Metal ribbon is then inserted into the grooves as fins and the plowed-up metal is peened back to form a tight bond between fins and the inner pipe In the laminar or transition flow regimes, fins are sometimes offset every 30 to 100 cm The common double-pipe exchanger units available are summarized in Table 11.3
Source: After Saunders (1988).
aOne outer tube–one inner tube: standard units The fin thickness for extruded or soldered fins is 0.5 mm for fins up to 12.7 mm high and 0.8 mm high for greater heights Fin thickness for welded fins is 0.89 mm for fin heights up to 25.4 mm The dimensions shown here are for low-pressure units.
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11.6.2 Physical Data for Annuli
Extruded Fins For the finned annular region between the inner and outer pipes
shown in Fig 11.24a, the cross-sectional area for n t identically finned inner pipes
each havingn f extruded fins will be
A = π
4D2
i −π
4d2
o − n f b fδfn t (11.134) There are two wetted perimeters One of them is for heat transfer:
P Wh=πd o + n f (2b f − δf )n t (11.135)
where the tips ofthe fins are presumed adiabatic The other is for pressure loss:
P Wf = πD i + P Wh
or
P Wf = πD i+πd o + n f (2b f− δf )n t (11.136) The equivalent diameter for heat transfer will be
d e= 4A
P Wh =
(π/4)D2
i −(π/4)d2
o − n f b fδfn t (πd o − n fδf + 2n f b f )n t (11.137)
and the equivalent diameter for pressure drop will be
d e= P4A
Wf = (π/4)D2i −
(π/4)d2
o − n f b fδfn t
πD i + (πd o − n fδf + 2n f b f )n t (11.138)
The surface area per unit length per tube will be
S = S = S b + S f
whereS bis the unfinned surface of the inner tube per unit length,
Then, per unit length, with
the surface area on the annulus side of the inner pipe per unit length is
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Welded U-Fins The configuration for the annular region that accommodates
welded U-fins is shown in Fig 11.24b, and detail for a pair of the fins is shown
in Fig 11.24c Observe that z is the fin root width and thickness, which is usually
taken as 2δf The free area for flow forn f fins andn tinner tubes withδf = z/2 is
A =π
4D2
i −π
4d2
o + n fδfb f+ z
2
Ifd o z, the wetted perimeter for heat flow will be the circumference of the inner
tube less the thicknesses ofthen f fins plus twice the heights ofthen f fins
P Wh=πd o + n f (2b f− δf )n t (11.141) Here, too, the tips are presumed to be adiabatic The wetted perimeter for pressure loss is
P Wf = πD i + P Wh
or
P Wf = πD i+πd o + n f (2b f − δf )n t (11.142) Then the two equivalent diameters are, for heat transfer,
d e=P4A
Wh =(π/4)D2i −
(π/4)d2
o + n fδf (b f + z/2)n t
πd o + n f (2b f− δf )n t (11.143)
and for pressure drop,
d e=(π/4)D2i −(π/4)d2
o + n fδf (b f + z/2)n t
πD i+πd o + n f (2b f− δf )n t (11.144)
The surface areas,S b , S f , S, and the surface area per unit length S will be the same
as those for the extruded fin configuration and are given by eqs (11.139)
11.6.3 Overall Heat Transfer Coefficient Revisited
Kern (1950), Kern and Kraus (1972), and Kraus et al (2001) all report on a method originally proposed by Kern for evaluation of the overall heat transfer coefficient when it has a component that involves fouling in the presence of fins The equation for an overall heat transfer coefficient is a complicated expression because of the annulus fouling and the fin efficiency It can be developed from a series summation ofseveral thermal resistances that are identified in Fig 11.25 These resistances are given in m2· K/W
After both inside and outside heat transfer coefficients,h i andh o, have been
de-termined and after both fouling resistances,r di andr do, have been specified (either
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r t o f,
r⬙ o w
r t o f,
t f
t fw
r o
r do
r do
r do
r i,OD
r i (at 10)
r di (at 10) r di,OD
Dirt
Dirt Tube
t c
ex-changer The thermal resistances are based on gross fin and outer pipe surface, and the tip of the fin is considered adiabatic
one or both can be zero), the steps can be arranged in systematic order The detailed procedure that follows is based on a finned annular passage and an internally un-finned tube
1 Withh io = h i (d i /d o ), form the inside film resistance:
r io= 1
2 The inner pipe fouling resistancer dimust be referred to the outer tube surface.
Hence,
r dio= d d o
3 The pipe wall resistance referred to the outside of the inner pipe is
r mo= d oln(d o /d i )
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However, when the diameter ratiod i /d o ≥ 0.75, r mocan be computed with an error ofless than 1% from the arithmetic mean diameter:
r mo= d o − d i
2k m
2πd o π(d o + d i ) =
d o (d o − d i )
k m (d o + d i ) (m2· K/W) (11.147)
4 At this point, the sum ofthe internal resistances referred to the outside ofthe inner pipe is
R io = r io + r dio + r mo (m2· K/W)
and this resistance must be referred to the gross outside surface of each inner pipe per meter:
Thus,
R is= R io S
5 The annulus heat transfer coefficient ish o, so that
r o = 1
6 The annulus fouling resistancer do must be combined with h o to obtain the value ofthe annulus coefficient working on the fin and prime outer surface Let this resistance be designated asr o, so that
r o= h1
o + r do = r o + r do (m2· K/W)
h o= r 1
o + r do
The fin efficiency will be given by
ηf o= tanhmb
where
m =
2h o
k mδf
1/2
Then, with the weighted fin efficiency defined by eq (11.9b),
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ηov,o= 1 −S f o
S o (1 − η f o ) (11.9b) or
ηov,o= ηf o S S f o + S bo
f o + S bo
the value of the heat transfer coefficient to the finned and prime surface corrected for the weighted fin efficiency and based on the outside surface of the inner pipe will be
h oη = h oηo
so that the resistance is
r oη= 1
7 The overall resistance is the sum ofeqs (11.149) and (11.152) Thus,
1
U o = R is + r oη
or
U o = 1
The overall heat transfer coefficient given by eq (11.153) is the coefficient to be
used in the rate equation:
11.6.4 Heat Transfer Coefficients in Pipes and Annuli
Heat transfer coefficients for both the inside of the tubes and the annular region containing the fins have been presented in Section 11.4.3
11.6.5 Pressure Loss in Pipes and Annuli
The friction relationships ofeqs (11.95) and the turn loss relationship ofeq (11.96) also pertain to the double-pipe heat exchanger However, when hairpins are consid-ered, the total friction loss in the inner pipe will be
∆P f =8nhpf ρV2
2
L
d = 4nhpf ρV2
L