G8
Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
G8 Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers Edward S. Gaddis1 . Volker Gnielinski2 1 2
Technische Universita¨t Clausthal, Clausthal-Zellerfeld, Germany Karlsruher Institut fu¨r Technologie (KIT), Karlsruhe, Germany
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731
2
Required Geometrical Data . . . . . . . . . . . . . . . . . . . . . . . . 731
3 3.1
Mean Shell-Side Heat Transfer Coefficient . . . . . . . . 732 Definition of Mean Shell-Side Heat Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 Determination of Mean Shell-Side Heat Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732
4.3.1 4.3.2 4.3.3
3.2
4 4.1 4.2 4.3
1
Mean Shell-Side Nusselt Number . . . . . . . . . . . . . . . . . . 732 Mean Nusselt Number of a Tube Bundle. . . . . . . . . . . . 732 Mean Nusselt Number of an Ideal Tube Bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 Correction Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733
Introduction
The method described in this chapter for calculating the mean heat transfer coefficient on the shell-side of a baffled shell-and-tube heat exchanger is based on the corresponding method for calculating the mean heat transfer coefficient in a tube bundle with a cross flow. However, the flow configuration on the shell-side of a baffled shell-and-tube heat exchanger leads to a number of deviations from the case of a flow across a tube bundle. The geometry of the baffles in the heat exchanger shell – as shown in Fig. 1 – generates a main stream through the heat exchanger tube bundle, which is partly across and partly parallel to the tubes. Unavoidable clearances between the outer surface of the tubes and the holes in the heat exchanger baffles, as well as between the baffles and the inside shell surface lead to leakage streams, which participate in heat transfer, but not to the same extent as the main stream. Since the tubes in the tube bundle cannot be brought uniformly and very close to the shell, bypass streams occur in the gaps between the outer tubes of the bundle and the inside surface of the heat exchanger shell; such bypass streams do not participate effectively in heat transfer. These geometrical factors dictated by the design of a shell-and-tube heat exchanger lead to deviations between the heat transfer in a tube bundle and that on the shell-side of a baffled heat exchanger; they can be taken into account by means of correction factors. The correction factors described hereafter are based on data given by Bell [1]. The presented method of calculation was checked by Gnielinski and
VDI-GVC (ed.), VDI Heat Atlas, DOI 10.1007/978-3-540-77877-6_30, Springer-Verlag Berlin Heidelberg 2010
#
4.4 4.5
Correction Factor for Number of Tube Rows . . . . . . . 733 Correction Factor for Temperature Dependence of Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 Correction Factor for Shell-Side Flow Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 End Effects in Baffled Shell-and-Tube Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 Limitations of the Proposed Calculation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738
5
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738
6
Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740
7
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741
Gaddis [2] through a large number of experimental measurements available in the open literature.
2
Required Geometrical Data
The following geometrical parameters are required for the calculation of the mean shell-side heat transfer coefficient: Di D1 DB do dB H LE
Shell inside diameter Baffle diameter Tube bundle diameter Outer diameter of tubes Diameter of holes in baffles Height of baffle cut Sum of the shortest connections e and e1 (see Fig. 3, LE ¼ 2e1 þ Se) Total number of tubes in heat exchanger including blind nT and support tubes nW Number of tubes in both upper and lower windows (baffle cuts) Number of pairs of sealing strips nS nMR Number of main resistances in cross flow between adjacent baffles (needed only if nS 6¼ 0, see Fig. 6 for the determination of nMR) S Baffle spacing (assumed constant, otherwise see Sect. 4.4) Transverse pitch s1 Longitudinal pitch s2 Tube arrangement: in-line or staggered.
732
G8
Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
G8. Fig. 1. Fluid flow on the shell-side of a baffled shell-and-tube heat exchanger. SM Main stream (partly across and partly parallel to the tubes). SL Leakage stream. SB Bypass stream.
3
Mean Shell-Side Heat Transfer Coefficient
3.1
Definition of Mean Shell-Side Heat Transfer Coefficient
p l ¼ do : 2
Heat transfer between the shell-side flow and the outer surface of the tubes of a tube bundle in a baffled shell-and-tube heat exchanger is given by
Equations for calculating the mean shell-side Nusselt number are presented hereafter. The mean shell-side heat transfer coefficient is thus calculated from
Q_ ¼ aADTLM :
The characteristic length l in the mean shell-side Nusselt number in Eq. (3) is the length of a stream line over the tube surface (half tube circumference) and is given by
ð1Þ
DTLM in Eq. (1) is the logarithmic mean temperature difference. For a constant wall temperature boundary condition, the logarithmic mean temperature difference is calculated from the fluid inlet temperature Tin , the fluid outlet temperature Tout , and the wall temperature Tw by DTLM ¼
ðTin # Tw Þ # ðTout # Tw Þ Tin # Tout ! !: ¼ Tin # Tw Tin # Tw ln ln Tout # Tw Tout # Tw
ð2Þ
Equations (1) and (2) define the mean shell-side heat transfer coefficient a.
3.2
The mean shell-side heat transfer coefficient a is calculated from dimensionless correlations determined experimentally. The dimensionless number that comprises the mean shell-side heat transfer coefficient is the mean shell-side Nusselt number defined by al : l
4
Nushell l : l
ð3Þ
ð5Þ
Mean Shell-Side Nusselt Number
The mean shell-side Nusselt number Nushell of a baffled shelland-tube heat exchanger can be calculated from the mean Nusselt number Nubundle of a tube bundle with a cross flow; its numerical value differs from that of the tube bundle because of the flow configuration on the shell-side. To allow for the deviations caused by this flow configuration, a correction factor fW is introduced as follows: Nushell ¼ fW Nubundle :
Determination of Mean Shell-Side Heat Transfer Coefficient
Nushell ¼
a¼
ð4Þ
ð6Þ
A procedure for calculating Nubundle is given hereafter (see also > Chap. G7). Equations for calculating the correction factor fW are presented in Sect. 4.3.3.
4.1
Mean Nusselt Number of a Tube Bundle
The mean Nusselt number Nubundle of a tube bundle under real operating conditions can be calculated from the mean Nusselt
Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
number Nu0;bundle of an ideal tube bundle by introducing correction factors as follows: Nubundle ¼ fN fP Nu0;bundle :
ð7Þ
The correction factor fN takes into consideration the influence of the number of tube rows in the tube bundle and the correction factor fP the influence of the change in the physical properties of the fluid in the thermal boundary layer near the tube surface due to temperature changes during heating or cooling. A procedure for calculating Nu0;bundle is given hereafter. Equations for calculating fN and fP are given in Sects. 4.3.1 and 4.3.2 respectively.
4.2
from that of the central channels see Sect. 4.4. According to > Chap. G7, the void fraction c in Eq. (12) is a function of the transverse pitch ratio a ¼ s1 =do and the longitudinal pitch ratio b ¼ s2 =do and is given by p for b # 1 ð16Þ c¼1& 4a and c¼1&
Nu0;bundle ¼ fA Nul;0
fA ¼ 1 þ
ð8Þ
with Nul;0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:3 þ Nu2l;lam þ Nu2l;turb ;
ð9Þ
Nul;lam
pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi ¼ 0:664 Rec;l 3 Pr
Nul;turb ¼
0:037Re0:8 c;l Pr # $ &0:1 1 þ 2:443Rec;l Pr2=3 & 1
ð11Þ
wl ðReynolds numberÞ cn
ð12Þ
n ðPrandtl numberÞ: a
ð13Þ
with Rec;l ¼ and Pr ¼
V_ : Af
ð14Þ
The cross sectional area Af required to calculate the characteristic velocity w from the fluid flow rate V_ is calculated from Af ¼ Di S:
ð15Þ
S is the baffle spacing assumed constant in all heat exchanger channels; if the baffle spacing of the end channels differs
ð17Þ
ð18Þ
2 : 3b
ð19Þ
The factor fA for in-line and for staggered tube arrangements in dependence on a and b or on b respectively can also be evaluated from diagrams presented in > Chap. G7. The ranges of the pitch ratios, which were examined for developing the above equations, are given in > Chap. G7.
4.3
Correction Factors
Equations for evaluating the correction factors fN , fP , and fW are given hereafter.
Correction Factor for Number of Tube Rows
A tube bundle in a cross flow with a number of tube rows nR Chap. G7. However, the flow structure in the first few rows in a tube bundle between adjacent segmental baffles in a shell-and-tube heat exchanger with a large number of segmental baffles – because of the frequent change in the flow direction – differs from that in a tube bundle with a cross flow. For that reason, the correction factor fN is ignored. Thus, in Eq. (7)
The characteristic length l in the Reynolds number in Eq. (12) is defined by Eq. (4) and the characteristic velocity w is given by w¼
0:7ððb=aÞ & 0:3Þ : c1:5 ððb=aÞ þ 0:7Þ2
fA ¼ 1 þ
ð10Þ
and
for b < 1
For staggered tube arrangement:
4.3.1
where
p 4ab
The tube arrangement factor fA in Eq. (8) is calculated from the following equations: For in-line tube arrangement:
Mean Nusselt Number of an Ideal Tube Bundle
An ideal tube bundle is defined arbitrarily as follows: Number of tube rows is nR # 10, number of tubes per row is # 10, ratio of tube length to tube diameter is # 10, physical properties of fluid is independent of temperature, fluid velocity in the free cross section at the inlet of the tube bundle is uniform and perpendicular to the free cross section and smooth tube surface. Large deviations from the idealized situation can be accounted for by introducing correction factors. The Nusselt number Nu0;bundle of an ideal tube bundle is based on the procedure given in > Chap. G7 and is calculated from
G8
fN ¼ 1
ð20Þ
is substituted irrespective of the number of tube rows in the tube bundle.
4.3.2
Correction Factor for Temperature Dependence of Physical Properties
Heating or cooling the fluid through the tube bundle influences the fluid temperature in the thermal boundary layer near the
733
734
G8
Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
surface of the tubes. To account for the change in the physical properties of the fluid in that layer due to change of temperature a correction factor fP is introduced as follows: For heating of liquids [ðPr=Prw Þ > 1]: ! Pr 0:25 : ð21Þ fP ¼ Prw For cooling of liquids [ðPr=Prw Þ < 1]: ! Pr 0:11 : fP ¼ Prw
ð22Þ
For gases approximately: fP ¼
Tm Tw
!nP
:
ð23Þ
The exponent nP in Eq. (23) depends on the type of gas used; very limited data about nP are available in the literature (nP # 0 for cooling of air and # 0:12 for cooling of nitrogen; see > Chap. G7). To account for the change of the fluid temperature in the flow direction, the physical properties in the presented equations are to be evaluated at the mean fluid temperature Tm given by Tm ¼
4.3.3
Tin þ Tout : 2
ð24Þ
Correction Factor for Shell-Side Flow Configuration
The correction factor fW for the shell-side flow configuration is formed as a multiplication of the geometry correction factor fG, the leakage correction factor fL, and the bypass correction factor fB , or
fW ¼ fG fL fB :
ð25Þ
Equations for evaluating the correction factors fG , fL , and fB are given hereafter. Geometry Correction Factor
The geometry correction factor fG takes in consideration the deviation of the mean Nusselt number for a baffled shell-andtube heat exchanger from that of a real tube bundle described in Sect. 4.1. This deviation is due to the specific shell-side flow, which is partly across the tube bundle in the space between adjacent baffles and partly parallel to the tubes in the upper and lower windows (baffle cuts) of the heat exchanger. According to K. J. Bell, V. Gnielinski and E. S. Gaddis [1, 2], fG can be calculated from fG ¼ 1 % RG þ 0:524RG0:32 ;
ð26Þ
where RG ¼
nW : nT
ð27Þ
In Eq. (27), nW is the number of tubes in both upper and lower windows; tubes that lie on the edges of the baffles and are thus partly in the cross flow between the baffles and partly in the parallel flow in the windows count as half tubes. nT is the total number of tubes in the heat exchanger. Figure 2 may be used to calculate the geometry correction factor fG. Leakage Correction Factor
The leakage correction factor fL is calculated from ! AGTB AGTB fL ¼ 0:4 þ 1 % 0:4 expð%1:5RL Þ; ASG ASG
ð28Þ
where ASG is the sum of the areas of all gaps between the tubes and the holes in a baffle and between the shell and a baffle, or
G8. Fig. 2. Geometry correction factor fG as a function of the parameter RG .
Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
ASG ¼ AGTB þ AGSB :
ð29Þ
The area AGTB of all gaps between the tubes and the holes in a baffle is given by " # nW ! p dB2 $ do2 ð30Þ AGTB ¼ nT $ 4 2 and the area AGSB of the gap between the shell and a baffle is given by # 360 $ g p" AGSB ¼ Di2 $ D12 ; ð31Þ 4 360 where g is the central angle of a baffle cut (see Fig. 9) measured in degrees and is given by $ % 2H g ¼ 2cos$1 1 $ : ð32Þ D1 The ratio RL is calculated from RL ¼
ASG ; AE
ð33Þ
where AE is the area for the cross flow between two baffles measured in the row of tubes on or near the diameter of the shell that is parallel to the edge of the windows, or AE ¼ SLE :
G8. Fig. 3. Illustration of the shortest connection LE .
contribute to heat transfer, the leakage streams that flow through the gaps between the shell and the baffles do not participate in heat transfer; therefore the gap area AGSB should be kept as small as possible. Figure 4 may be used to calculate the leakage correction factor fL. Bypass Correction Factor
Bypass streams between the inner surface of the shell and the outermost tubes of the tube bundle do not participate effectively to heat transfer. Sealing strips – as shown in Fig. 5 – can be fixed in the tube bundle to reduce the level of bypass streams. This is particularly important in shell-and-tube heat exchangers with floating-heads, where the gaps responsible for the bypass streams are relatively large. The bypass correction factor fB is given by rffiffiffiffiffiffiffiffi%) ( $ nMR 3 2nS ð35Þ fB ¼ exp $bRB 1 $ for nS % nMR 2 and fB ¼ 1 for nS >
nMR ; 2
ð36Þ
where b ¼ 1:5 for laminar flow ðRec;1 < 100Þ
ð34Þ
LE is the sum of the shortest connections e between adjacent tubes and e1 between the outermost tubes in the bundle and the shell, measured in the row of tubes on or near the diameter of the shell that is parallel to the edge of the windows (LE ¼ 2e1 þ Se). Figure 3 illustrates the meaning of LE , e and e1 ; equations for the calculation of e for different tube bundle geometries are also given in the figure. The baffle spacing S is assumed constant, otherwise see Sect. 4.4. While the leakage streams that flow through the gaps between the tubes and the holes in the baffles
G8
and b ¼ 1:35 for transition region and turbulent flow ðRec;1 & 100Þ The ratio RB is given by RB ¼
AB : AE
ð37Þ
In Eq. (35), nS is the number of pairs of sealing strips (in Fig. 5, nS ¼ 2) and nMR is the number of main resistances in cross flow and thus the number of the shortest connections, which the flow
735
736
G8
Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
G8. Fig. 4. Leakage correction factor fL as a function of RL with the ratio AGTB =ASG as a parameter.
4.4
G8. Fig. 5. Reduction of bypass streams by means of sealing strips.
End Effects in Baffled Shell-and-Tube Heat Exchangers
In many cases, the dimensions of the heat exchanger nozzles necessitate larger inlet and outlet baffle spacing (spacing between heat exchanger sheets and first or last baffle) compared with the central baffle spacing. This leads to a different mean heat transfer coefficient in the end channels (inlet and outlet channels) compared with the central channels. The mean heat transfer coefficient aE in the end channels and the mean heat transfer coefficient aC in the central channels can be calculated separately from the previous equations using the baffle spacing SE for the end channels and the baffle spacing SC for the central channels. The mean heat transfer coefficient a for the whole heat exchanger can be calculated from a¼
crosses on its way between the upper and the lower edges of adjacent baffles. Figure 6 illustrates the determination of nMR . The area AE in Eq. (37) is calculated from Eq. (34) and the area AB is the cross sectional area that is responsible for the bypass streams and is given by AB ¼ SðDi ! DB ! e Þ for e < ðDi ! DB Þ
ð38Þ
AB ¼ 0 for e $ ðDi ! DB Þ:
ð39Þ
and
The tube bundle diameter DB is the diameter of a circle, which touches the outermost tubes in the space between the upper and lower edges of adjacent baffles (see Fig. 9). Figure 7 may be used to calculate the bypass correction factor fB.
2SE aE þ ðL ! 2SE ÞaC ; L
ð40Þ
where L is the total tube length. Other factors that influence the mean heat transfer coefficient in end channels are ● Velocity distribution at inlet of inlet channel and outlet of outlet channel dictated by the construction of the nozzles particularly the inlet nozzle ● Absence of parallel flow at inlet of inlet channel and at outlet of outlet channel ● Absence of leakage streams through the heat exchanger sheets However, these deviations are relatively small in a baffled shell-and-tube heat exchanger with a large number of baffles (L & 2SE ).
Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
G8. Fig. 6. Determination of the number on main resistances nMR in cross flow.
G8. Fig. 7. Bypass correction factor fB in dependence on RB and the ratio nS =nMR .
G8. Fig. 8. Longitudinal section of the heat exchanger in the example (dimensions in mm).
G8
737
738
G8
Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
theoretical predictions. In this range of operation, as well as with gases, care must be taken (i.e., higher factors of safety). The above procedure of calculation is valid for heat exchangers with geometrical parameters that lie within the following ranges: RG ! 0:8 RL ! 0:8 RB ! 0:5 0:2 ! S=Di ! 1 Heat exchangers with geometrical ratios outside the above ranges, led to high deviations between experimental measurements and theoretical predictions. Also, heat exchanger geometries leading simultaneously to small numerical values of the correction factors fG , fL and fB (i.e., a very small numerical value for fW from Eq. (25)) showed high deviations. Thus, in addition to the above limitations, the calculation procedure should be used only for heat exchangers that have a correction factor for the shell-side flow configuration fW , which lies within the following range: fW % 0:3: G8. Fig. 9. Cross-section of the heat exchanger in the example (dimensions in mm).
4.5
Limitations of the Proposed Calculation Procedure
A large number of experimental measurements obtained from shell-and-tube heat exchangers with segmental baffles and different geometries are available in the literature and were considered to check the calculation procedure presented in this chapter. The tested heat exchangers had staggered tube arrangement with different transverse and longitudinal pitch ratios as follows: ● Staggered square: with a pitch ratio t=do ¼ 1:2, corresponding to a transverse pitch ratio a ¼ 1:697 and a longitudinal pitch ratio b ¼ 0:849 ● Equilateral triangle: with pitch ratios in the range 1:2 ! ðt=do Þ ! 2:2 corresponding to transverse pitch ratios in the range 1:2 ! a ! 2:2 and longitudinal pitch ratios in the range 1:039 ! b ! 1:905, the ratioðb=aÞ is constant and is equal to 0.866 Measurements with in-line tube arrangement were not available. The experimental measurements were made with different flow rates, different physical properties and during heating or cooling the shell-side fluid; the fluids used were water and oil with different viscosities. Measurements with gases were not available. The operating conditions during the experiments covered the following ranges: ● Reynolds number: 10 < Rec;l < 105 ● Prandtl number: 3 < Pr < 103 Measurements obtained at low Reynolds numbers (Rec;l < 102 ) with a high Prandtl number ( Pr $ 103 ) showed high deviations between experimental measurements and
5
Example
The shell-and-tube heat exchanger shown in Figs. 8 and 9 has two tube passes and four baffles with equal baffle spacing. Water with a volumetric flow rate of V_ ¼ 20 m3 h&1 is cooled in the shell-side from an inlet temperature Tin ¼ 63:3 oC to an outlet temperature Tout ¼ 56:7 oC. The wall temperature of the tubes is constant at Tw ¼ 50 oC. The heat exchanger has the following geometrical data: Shell inside diameter Di = 310 mm Baffle diameter D1 = 307 mm Tube bundle diameter DB = 285 mm Outer diameter of tubes do = 25 mm Hole diameter in baffles to accommodate dB = 26 mm the tubes Baffle cut H = 76 mm Total number of tubes nT = 66 Number of tubes in upper and lower nW = 25 windows (Tubes that lie on the edges of the baffles count as half tubes) Number of pairs of sealing strips nS = 0 Baffle spacing S = 184 mm Transverse pitch s1 = 32 mm Longitudinal pitch s2 = 27.7 mm Tube arrangement: staggered Other dimensions are given in Figs. 8 and 9. It is required to calculate the mean shell-side heat transfer coefficient. Solution: Mean water temperature: T þT ¼ 60 ( C Equation (24): Tm ¼ in 2 out ¼ 63:3þ56:7 2 Physical properties of water: At mean temperature Tm ¼ 60 oC: Kinematic viscosity n = 0:471 ) 10&6 m2 s&1 Thermal conductivity l = 654 ) 10&3 W m&1 K&1 Prandtl number Pr = 2.96
Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
At wall temperature Tw ¼ 50 oC: Prandtl number Prw = 3.54 Calculation of the mean Nusselt number Nu0;bundle for an ideal tube bundle: a¼
b¼
s1 32 ¼ ¼ 1:28 do 25
s2 27:7 ¼ ¼ 1:11 do 25
p p ¼ 1 ! 4"1:28 ¼ 0:386 (since b > 1) Equation (16): c ¼ 1 ! 4a p p Equation (4): l ¼ 2 do ¼ 2 " ð25 " 10!3 Þ ¼ 0:03927 m Equation (15): Af ¼ Di S ¼ ð310 " 10!3 Þ " ð184 " 10!3 Þ ¼ 0:05704 m2 _ !1 Equation (14): w ¼ AVf ¼ ð20=3600Þ 0:05704 ¼ 0:0974 m s 0:0974 " 0:03927 wl ¼ 21038 ¼ Equation (12): Rec;l ¼ cn 0:386 " " 10!6 Þ pffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffip pð0:471 Equation (10): Nul;lam ¼ 0:664 Rec;l 3 Pr ¼ 0:664 21038 p ffiffiffiffiffiffiffiffiffi 3 2:96 ¼ 138:3 0:037Re0:8 c;l Pr " #¼ Equation (11): Nul;turb ¼ !0:1 1 þ 2:443Rec;l Pr2=3 ! 1
0:037 " 210380:8 " 2:96 " # ¼ 160:7 1 þ 2:443 " 21038!0:1 2:962=3 ! 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Equation (9): Nul;0 ¼ 0:3 þ Nu2l;lam þ Nu2l;turb ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:3 þ 138:32 þ 160:72 ¼ 212:3 Calculation of the tube arrangement factor fA : 2 ¼ 1:6 (staggered tube Equation (19): fA ¼ 1 þ 3b2 ¼ 1 þ 3"1:11 arrangement) Equation (8): Nu0;bundle ¼ fA Nul;0 ¼ 1:6 " 212:3 ¼ 339:7 Calculation of the mean Nusselt number Nubundle (real tube bundle): Calculation of the correction factor fN for the number of tube rows: The correction factor fN is ignored (see Sect. 4.3.1), and therefore Equation (20): fN ¼ 1 is used. Calculation of the correction factor fP for temperature dependence of physical properties: % &0:11 % & Pr 2:96 0:11 Equation (22): fP ¼ ¼ ¼ 0:981 (cooling Prw 3:54 of liquid) Equation (7): Nubundle ¼ fN fP Nu0;bundle ¼ 1:0 " 0:981 " 339:7 ¼ 333:2 Calculation of the mean shell-side Nusselt number Nushell : Calculation of the geometry correction factor fG : Equation (27): RG ¼ nnWT ¼ 25 66 ¼ 0:379 Equation (26): fG ¼ 1 ! RG þ 0:524RG0:32 ¼ 1 ! 0:379 þ 0:524 " 0:3790:32 ¼ 1:005 Calculation of the leakage correction factor fL : " # pð262 !252 Þ " # pðdB2 !do2 Þ ¼ 66 ! 25 Equation (30): AGTB ¼ nT ! n2W 4 4 2 ¼ 2143 mm2 ' ( " # ' !1 1 ! 2"76 Equation (32): g ¼ 2cos!1 1" ! 2H D1 ¼#2cos 307 ¼ 119:4 360!g Equation (31): AGSB ¼ p4 Di2 ! D12 360 ¼ p4 ð3102 ! 3072 Þ 360!119:4 ¼ 972 mm2 360 Equation (29): ASG ¼ AGTB þ AGSB ¼ 2143 þ 972 ¼ 3115 mm2
G8
Calculation of the sum of the shortest connections LE after Fig. 3: pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Check: b ( 12 2a þ 1 ? → 1:11mm ( 12 2 " 1:28 þ 1 ¼ 0:943mm → yes According to Fig. 3: e ¼ s1 ! do ¼ 32 ! 25 ¼ 7 mm e1 can be determined from the drawings; in this example e1 can be calculated from the following equation: e1 ¼
Di ! DB 310 ! 285 ¼ ¼ 12:5 mm 2 2
According to Fig. 9: number of shortest connections e between the tubes in tube bundle = 8 LE ¼ 2e1 þ Se ¼ 2 " 12:5 þ 8 " 7 ¼ 81 mm Equation (34): AE ¼ SLE ¼ 184 " 81 ¼ 14904 mm2 3115 ¼ 0:209 Equation (33): RL ¼ AASGE ¼ 14904 ' ( AGTB Equation (28): fL ¼ 0:4 ASG þ 1 ! 0:4 AAGTB expð!1:5RL Þ SG % & 2143 2143 ¼ 0:4 þ 1 ! 0:4 expð!1:5 " 0:209Þ 3115 3115 ¼ 0:805
Calculation of the bypass correction factor fB : nS nMR ¼ 0 (no sealing strips) Check: e ðDi ! DB Þ ? → [e ¼ 7 mm, ðDi ! DB Þ ¼ ð310 ! 285Þ ¼ 25 mm] → yes Equation (38): AB ¼ SðDi ! DB ! e Þ ¼ 184ð310 ! 285 ! 7Þ ¼ 3312 mm2 3312 Equation (37): RB ¼ AABE ¼ 14904 ¼ 0:222 Check: (Rec,1 ( 100)? → yes → b ¼ 1:35 h ' qffiffiffiffiffiffi(i Equation (35): fB ¼ exp !bRB 1 ! 3 n2nMRS ¼ exp½!1:35" " pffiffiffiffiffiffiffiffiffiffiffi# 0:222 1 ! 3 2 " 0 * ¼ 0:741 (since nS < nMR 2 ) Calculation of the factor fW for shell-side flow configuration: Equation (25): fW ¼ fG fL fB ¼ 1:005 " 0:805 " 0:741 ¼ 0:599 The mean shell-side Nusselt number Nushell can be calculated from Equation (6): Nushell ¼ fW Nubundle ¼ 0:599 " 333:2 ¼ 199:6 Calculation of the mean shell-side heat transfer coefficient a: !3 l ¼ 3324 W m!2 K!1 ¼ 199:6"654"10 Equation (5): a ¼ Nushell l 0:03927 The procedure of evaluating the thermal performance on the shell-side of an existing shell-and-tube heat exchanger is based usually on the knowledge of the volumetric flow rate V_ of the shell-side fluid (or the mass flow rate), the fluid inlet temperature Tin , the wall temperature Tw and the total heat transfer area A as well as the other dimensions of the heat exchanger. The heat transfer calculations yield in this case the fluid outlet temperature Tout . On the other hand, dimensioning a new heat exchanger to fulfill the requirements of a particular process is based usually on the knowledge of V_ , Tin , Tout , and Tw . The heat transfer calculations yield in this case the total heat transfer area A required to fulfill the process requirements on the shell-side. Moreover, in each case the wall temperature Tw can be determined only in conjunction with the thermal and fluid dynamic conditions of the fluid inside the heat exchanger tubes. In both cases (performance evaluation of an existing heat exchanger or dimensioning of a new heat exchanger for a particular job), one of the parameters used in the example (Tout or A) is not known a priori; it has to be assumed, checked,
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Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
and corrected if necessary. Thus, the calculations are based most probably on iteration. The given example represents only a single step in the iteration procedure.
l Nu0;bundle Nubundle
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Symbols
Latin Letters A Total surface ara of all tubes in a shell-and-tube head exchanger (m2(mm2)). Area responsible for bypass streams (m2 (mm2)) AB Smallest area for cross flow between two baffles AE measured in the row of tubes on or near the diameter of the shell that is parallel to the edge of the baffles (m2 (mm2)) Cross sectional area defined by Eq. (15) Af (m2 (mm2)) Area of gap between the shell and a baffle AGSB (m2 (mm2)) Area of all gaps between the tubes and the holes in a AGTB baffle (m2 (mm2)) sum of the areas of all gaps between the tubes and ASG the holes in a baffle and between the shell and a baffle (m2 (mm2)) a Thermal diffusivity (in the definition of Prandtl number, Eq. (13)) (m2 s 1) a Transverse pitch ratio (¼ s1 =do ) (l) b Longitudinal pitch ratio (¼ s2 =do ) (l) Baffle diameter (m (mm)) D1 Tube bundle diameter (m (mm)) DB Shell inside diameter (m (mm)) Di Diameter of holes in baffles to accommodate the dB tubes (m (mm)) Outer diameter of tubes (m (mm)) do e Shortest connection between adjacent tubes in the same tube row or in adjacent tube rows (see Fig. 3) (m (mm)) Shortest connection between the outermost tube in e1 the bundle and the shell measured in the tube row on or near the diameter of the shell that is parallel to the edge of the baffles (see Fig. 3) (m (mm)) Tube arrangement factor (l) fA Bypass correction factor (l) fB Geometry correction factor (l) fG Leakage correction factor (l) fL Correction factor for number of tube rows (l) fN Correction factor for change in physical properties fP in the thermal boundary layer near the surface of the tubes (l) Correction factor for shell-side flow configuration fW (l) H Height of baffle cut (m (mm)) L Total length of heat exchanger tubes (m (mm)) Sum of the shortest connections e and e1 measured LE in the row of tubes on or near the diameter of the shell that is parallel to the edge of the baffles (see Fig. 3) (m (mm))
Nul;0 Nul;lam Nul;turb Nushell nMR nP nR nS nT nW Pr Prw Q_ RB RG RL Rec;l S SB SC SE SL SM s1 s2 T Tw t w V_
Half circumference of tube (m (mm)) Mean Nusselt number of an ideal tube bundle (see Eq. (8)) (l) Mean Nusselt number of a tube bundle under real operating conditions (see Eq. (7)) (l) Mean Nusselt number for a single tube (l) Mean Nusselt number for a single tube with laminar flow (l) Mean Nusselt number for a single tube with turbulent flow (l) Mean shell-side Nusselt number (¼ al=l) (l) Number of main resistances in cross flow between adjacent baffles (see Fig. 6) (l) Exponent of temperature ratio in Eq. (23) (l) Number of tube rows in cross flow in a tube bundle between adjacent baffles (l) Number of pairs of sealing strips (l) Total number of tubes in heat exchanger including blind and support tubes (l) Number of tubes in both upper and lower windows (baffle cuts) (l) Prandtl number (¼ n=a) (l) Prandtl number at wall temperature Tw (l) Heat flow rate between the shell-side fluid and the outer surface of the tubes (W) Ratio AB =AE (see Eq. (37)) (l) Ratio nW =nT (see Eq. (27)) (l) Ratio ASG =AE (see Eq. (33)) (l) Reynolds number for a tube bundle (¼ wl=cn) (l) Baffle spacing (in the case of same baffle spacing in all channels) (m (mm)) Bypass stream (see Fig. 1) Baffle spacing in central channels (m (mm)) Baffle spacing in end channels (inlet and outlet channels) (m (mm)) Leakage stream (see Fig. 1) Main stream (see Fig. 1) Transverse pitch (m (mm)) Longitudinal pitch (m (mm)) Absolute temperature (T ¼ T þ 273:1) (K) Absolute wall temperature (Tw ¼ Tw þ 273:1) (K) Pitch for an equilateral triangle or a staggered square tube arrangement (m (mm)) Characteristic velocity in the definition of the Reynolds number (m s"1) Fluid flow rate (m3 s"1 (m3 h"1))
Greek Letters a Mean shell-side heat transfer coefficient in heat exchanger (W m"2 K"1) Mean shell-side heat transfer coefficient in central aC channels (W m"2 K"1) Mean shell-side heat transfer coefficient in end aE channels (inlet and outlet channels) (W m"2 K"1) b Constant in Eq. (35) (l) g Central angle of a baffle cut (# ) T Fluid temperature (# C)
Shell-Side Heat Transfer in Baffled Shell-and-Tube Heat Exchangers
Tw DTLM l n c
Wall temperature of tubes ( C) Logarithmic mean temperature difference ( C) Thermal conductivity (W m!1 K!1) Kinematic viscosity (m2s!1) Void fraction (see Eqs.(16) and (17))(1)
Subscripts in At inlet m Mean value out At outlet Notice: The units between brackets (mm, mm2, and m3 h!1) are not consistent with the M.K.S. units system; they are used in the example for convenience.
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Bibliography Bell KJ (1963) Final report of the cooperative research program on shell and tube heat exchangers. University of Delaware, Engineering Experimental Station, Bulletin No. 5, Newark, Delaware Gnielinski V, Gaddis ES (1978) Berechnung des mittleren Wa¨rmeu¨bergangskoeffizienten im Außenraum von Rohrbu¨ndelwa¨rmeaustauschern mit Segment-Umlenkblechen. vt verfahrenstechnik 12(4):211–217
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