11.4.4 Pressure Loss Data Tube Side The pressure loss inside tubes ofcircular cross section in a shell-and-tube heat exchanger is the sum ofthe friction loss within the shell-and-tubes a
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where
r c= A bp
ζ = N ss
where withX Las the longitudinal tube pitch andN sstaken as the number ofsealing strip pairs,
N cc =D s − 2l c
X L
C =
1.35 for Res≤ 100
1.25 for Res > 100 (11.91c)
Here
A bp = L bc (D s − D o + 0.5N P w P )
is the crossflow area for the bypass, whereN P is the number ofbypass divider lanes that are parallel to the crossflow streamB, w Pis the width ofthe bypass divider lane (m), andL bcis the central baffle spacing
J Sis the correction factor that accounts for variations in baffle spacing at the inlet and outlet sections as compared to the central baffle spacing:
J S= N b− 1 +
L∗
i
(1−n)
−L∗(1−n)
N b− 1 +L∗
i
(1−n) +L∗(1−n) (11.92)
whereN bis the number ofbaffles and
L∗
i = L bi
L∗
o= L L bo
n =
3
5 for turbulent flow 1
HereL biis the baffle spacing at the inlet (m),L bois the baffle spacing at the outlet
(m), andL bcis the central baffle spacing (m)
J R is the correction factor that accounts for the temperature gradient when the
shell-side fluid is in laminar flow:
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J R =
10
N r,c
0.18
For 20 < Re s < 100, a linear interpolation should be performed between the two
extreme values In eq (11.81), Resis the shell-side Reynolds number andNr,cis the number ofeffective tube rows crossed through one crossflow section
11.4.4 Pressure Loss Data
Tube Side The pressure loss inside tubes ofcircular cross section in a shell-and-tube heat exchanger is the sum ofthe friction loss within the shell-and-tubes and the turn losses between the passes ofthe exchanger The friction loss inside the tubes is given by
∆P f =4f ρu2
2
L
whereu is the linear velocity ofthe fluid in the tubes, or
∆P f = 4f G2
2ρ
L
whereG is the mass velocity ofthe fluid in the tubes In eqs (11.95), f is the friction
factor
The fluid will undergo an additional pressure loss due to contractions and expan-sions that occur during fluid turnaround between tube passes Kern (1950) and Kern and Kraus (1972) have proposed that this loss be given by one velocity head per turn:
∆P t= 4ρu2
In an exchanger with a single pass,
∆P t =4ρu2
and in an exchanger withn p passes, there will ben p− 1 turns Hence
∆P t = 2(n p − 1)ρu2 (11.98) Friction factors may be obtained from Fig 11.12, which plots the friction factor
as a function of the Reynolds number inside the tube and the relative roughness,/d i.
The figure is due to Moody (1944), and it may be noted that when the flow is laminar,
f = 16
Trang 3Laminar flow
Laminar flow
Critical zone Transitionzone Fully rough zone
Smooth pipes
0.09 0.08
0.03
0.008 0.006 0.004
0.015
0.0008 0.0006 0.0004 0.0002 0.0001
0.000,01
e
0.009 0.008
0.1
f=64
Re —
103 2 3 4 5 6 8104 2 3 4 5 6 8105 2 3 4 5 6 8106 2 3 4 5 6 8107 2 3 4 5 6 8108
Recr
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Many investigators have developed friction factor relationships as a function of the Reynolds number The use ofthe friction factor give by eq (11.77) in the Petukhov and Gnielinski correlations ofeqs (11.76) and (11.78) has been noted Other func-tions can be fitted to the curves in Fig 11.10 Two ofthem for smooth tubes are
f =
0.046
Re0.20
3× 104≤ Re ≤ 106
(11.100)
0.079
Re0.25
4× 103≤ Re ≤ 105
(11.101)
Shell Side Tinker (1951) also suggested a flow stream model for the determination ofshell-side pressure loss However, the lack ofadequate data caused him to make rather gross simplifications in arriving at the effects to be attributed to the various flow streams Willis and Johnston (1984) developed a simpler method which extends Tinker’s scheme to include end-space pressure losses and includes a simple method for nozzle pressure drop developed by Grant (1980)
The flow streams in the Willis and Johnston method are shown in Fig 11.13 For each ofthe streams, a coefficientn is defined so that
n i = ∆p ˙m i
i (i = b, c, s, t, w) (11.102) where the∆p i’s and the ˙m i’s are the pressure drops and mass flow rates for theith
stream, respectively
The crossflow stream contains the actual crossflow path (path c) and the bypass path (path b) These paths merge into the window stream (path w), and continuity and compatability for these three paths give
where
A
w
s t
B
method
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and the pressure loss between points A and B will be
because∆p cr = ∆p b = ∆p c It can be shown that
∆p cr =
1
n b
1/2 +
1
n c
1/22
˙m2
and in similar fashion, for the parallel combination of the shell-to-baffle leakage path (path s) and the tube-to-baffle leakage path (path t),
∆p l=
1
n s
1/2 +
1
n t
1/22
˙m2
where the leakage flow rate is
With the total flow rate given by
˙m T = ˙m s + ˙m t + ˙m w (11.107)
a combination ofeqs (11.102) and (11.104)–(11.107) gives
1+
n −1/2 c + n −1/2 b −2+ n w
n −1/2 s + n −1/2 t −2
and it is observed that the procedure depends on the values ofthen i’s
Forn c, Butterworth (1979) has proposed that
n c = C c F −b
with
C c = 4a d o2d V
(p − d o )3
˙m
T d o
µA c
−b δ
ov
2ρd o A2
c
(11.109b)
where for square or rotated square pitch,a = 0.061, b = 0.088, and F c = 1.00;
and for equilateral triangular pitch,a = 0.45, b = 0.267, and F c = 0.50 In eq.
(11.109b),
D V =ap2− d o2
d o
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witha = 1.273 for square and rotated square pitch and a = 1.103 for equilateral
triangular pitch In addition,
δov = (1 − 0.02p b )D s
is the height ofthe baffle overlap region,p bis the baffle cut, and
A c=πD o2
2 − 2D o2
2
θ3
2 − sinθ3
2 cos
θ3 2
L
cb
δov − N p zLcb wheren pis the number ofpass partitions andz is the path partition width in line with
the flow
Forn b,
n b= 0.316(δ ov /X L )( ˙m T D e /µA bp ) −0.25 + 2N s
2ρA2
bp
(11.110)
where
A bp = (2w + N p z)Lbc
2(w + L bc ) + N p (z + L bc )
Forn s
n s =4 [0.0035 + 0.264(2 ˙m Tδsb /µA sb )] −0.42 + (δ b /2δ sb ) + 2.03(δ b /2δ sb ) −0.177
2ρA2
sb
(11.111) where
A sb = π(D s− δsb )δ sb
Forn t,
n t = 4 [0.0035 + 0.264(2 ˙m Tδtb /µA tb )] −0.42 + (δ b /2δ tb ) + 2.03(δ b /2δ tb ) −0.177
2ρA2
tb
(11.112) where
A tb = nπ(d o− δtb )δ tb
Forn w,
n w= 1.90e0.6856A w /A CL
2ρA2
w
(11.113)
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whereA w is the window flow area with n tw taken as the number oftubes in the window:
A w = A w1−πd o2n tw
4
A w1= d s 4
θ1
2 − sinθ1
2 cos
θ1 2
and where for square and rotated square layouts,
A CL = (D s − N CL d o )L bc
and for equilateral triangular layouts,
A CL = 2(N CL − 1)(p − D o ) + 2w
Here, to the nearest integer,
N CL=D o − d o
P y
withP y = p for square pitch, P y = 1.414p for rotated square pitch, and P y =
1.732p for equilateral triangular pitch.
Equation (11.108) establishes the window mass flow as a function of the total mass flow, and a simple computation then determines the total baffle-space pressure loss via
where
∆p cr = n c ˙m c or ∆p cr = n b ˙m b
∆p w = n w ˙m w
The total pressure loss contains components due to the baffle-space pressure loss established by the foregoing procedure, the end-space pressure loss, and the nozzle pressure loss The end-space pressure loss is taken as
∆p e = N e ˙m2
e+1
2n we ˙m2
where
n e = n cr D s+ δov
2δov
L bc
L be
2
(11.115a)
with
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n cr =
1
n c
1/2
+
1
n c
1/2−2
(11.115b)
and where
n we= 1.9e0.6856(A w L bc /A CL L)
2ρA2
w
(11.115c)
The average end-space flow rate is
˙m e= ˙m t + ˙m w
2 Grant (1980) gives the pressure drop in the inlet nozzle as
∆p n1= G2A1
ρ1A2
A 1
A2 − 1
(11.116)
whereG1is the entry mass velocity,G1 = ρ1u1, A1is the inlet nozzle area, andA2
is the bundle entry area For the outlet-nozzle pressure loss,
∆p n2= G22
2ρ2
1−
A3
A4
2 +
1
c− 1
2
(11.117)
whereG2is the exit mass velocity,G2= ρ2u2, A3is the outlet nozzle area, andA4is the bundle exit area The recommended value ofthe contraction coefficient isc =2
3 The total shell-side pressure loss will be
∆P T = ∆p n1 + (F T + 1) ∆p e + (N b − 1) ∆p AB ∆p n2 (11.118)
Equation (11.118) has assumed that the pressure losses in the end spaces at inlet and outlet are identical The factorF T is the transitional correction factor and is based on the crossflow Reynolds number
Rec= ˙m c d o
µA c
where for Rec < 300, the entire method is not valid; 0 ≤ Re c < 1000, F T =
3.646e −0.1934; and Re
c ≥ 1000, F T = 1
11.5.1 Introduction
One variation ofthe fundamental compact exchanger element, the core, is shown
in Fig 11.14 The core consists ofa pair ofparallel plates with connecting metal
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corrugated fins stamped from a continuous strip of metal By spraying braze powder on the plates, the entire assembly ofplates, fins, and bars can be thermally bonded in a single furnace operation (From Kraus et al., 2001, with permission.)
members that are bonded to the plates The arrangement ofplates and bonded mem-bers provides both a fluid-flow channel and prime and extended surface It is observed that ifa plane were drawn midway between the two plates, each halfofthe connecting metal members could be considered as longitudinal fins
Two or more identical cores can be connected by separation or splitter plates, and this arrangement is called a stack or sandwich Heat can enter a stack through
either or both end plates However, the heat is removed from the successive separating plates and fins by a fluid flowing in parallel through the entire network with a single average convection heat transfer coefficient For this reason, the stack may be treated
as a finned passage rather than a fluid–fluid heat exchanger, and, ofcourse, due consideration must be given to the fact that as more and more fins are placed in a core, the equivalent or hydraulic diameter ofthe core is lowered while the pressure loss is increased significantly
Next, consider a pair ofcores arranged as components ofa two-fluid exchanger in crossflow as shown in Fig 11.15 Fluids enter alternate cores from separate headers
at right angles to each other and leave through separate headers at opposite ends of the exchanger The separation plate spacing need not be the same for both fluids, nor need the cores for both fluids contain the same numbers or kinds of fins These are dictated by the allowable pressure drops for both fluids and the resulting heat transfer coefficients When one coefficient is quite large compared with the other, it is entirely permissible to have no extended surface in the alternate cores through which the fluid with the higher coefficient travels An exchanger built up with plates and fins as in
Fig 11.16 is a plate fin heat exchanger.
The discussion ofplate fin exchangers has concentrated thus far on geometries involving two or more fluids that enter the body ofthe compact heat exchanger by
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2001, with permission.)
means ofheaders In many instances, one ofthe fluids may be merely air, which is used as a cooling medium on a once-through basis Typical examples include the
air-fin cooler and the radiators associated with various types ofinternal combustion
engines Similarly, there are examples in which the compact heat exchanger is a coil that is inserted into a duct, as in air-conditioning applications A small selection of compact heat exchanger elements available is shown in Fig 11.16
11.5.2 Classification of Compact Heat Exchangers
Compact heat exchangers may be classified by the kinds ofcompact elements that they employ The compact elements usually fall into five classes:
1 Circular and flattened circular tubes These are the simplest form of compact
heat exchanger surface The designation ST indicates flow inside straight tubes (ex-ample: ST-1), FT indicates flow inside straight flattened tubes (ex(ex-ample: FT-1) and FTD indicates flow inside straight flattened dimpled tubes Dimpling interrupts the boundary layer, which tends to increase the heat transfer coefficient without increas-ing the flow velocity
2 Tubular surfaces These are arrays oftubes ofsmall diameter, from 0.9535
cm down to 0.635 cm, used in service where the ruggedness and cleanability ofthe conventional shell-and-tube exchanger are not required Usually, tubesheets are com-paratively thin, and soldering or brazing a tube to a tubesheet provides an adequate seal against interleakage and differential thermal expansion