Heat Ex Changers Formulas

8/3/2019 Heat Ex Changers Formulas

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Energy Technology Heat Exchangers Formulas, Tables and Figures

LESSON 1. HEAT EXCHANGERS

FORMULAS

Thermal resistance for conduction in a plane wall:kA

LR condt =, .

Thermal resistance for convection:hA

R convt1

, = .

Overall heat transfer coefficient, U:UA

RR ttot1 == .

Thermal resistance for conduction in a cylindrical wall: Lk

rr

R condt 2

)/ln( 12, = .

Thermal resistance for convection in a cylindrical wall:rLhAh

R convt2

11, == .

The overall heat transfer coefficient in a cylindrical wall depends on the surface area it isexpressed: ( ) 1332211 ...

===== tii RAUAUAUAU .


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The Reynolds number for internal flow is based on the internal diameter of the tube and the

mean fluid velocity over the tube cross section:

intmintmD

DuDuRe == . As cmAum =& , for

a circular tube the Reynolds number can be expressed as: int

DD

mRe

&4= .

The Reynolds number for external cross flow over a cylinder is based on the external diameter

of the tube and the free flow velocity, u or V:

extext

D

VDDuRe == .

Nusselt number:fluid

Dk

hDNu = .

Prandtl number:

=Pr .

Thermal diffusivity: pck / = . Zhukauskas correlation for flow across a bank of tubes:

4/1

36.0

=

s

m

D,maxDPr

PrPrCReNu . With

20N L


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Dittus-Boelter correlation for internal turbulent fully developed flow:

n

DD PrReNu

5/4

023.0= .Where n = 0.4 for heating (Ts > T), n = 0.3 for cooling (Ts < T), fluid properties evaluated at

2/)( salent TTT += , and

10)/(

000,10

1607.0

DL

Re

Pr

D .

Sieder and Tate correlation for internal turbulent fully developed flow with great variation ofthe properties:

14,0

3/15/4027.0

=

s

DD PrReNu

. Properties evaluated at 2/)( salent TTT += ,

except s at Ts, and

10)/(

000,10

700,167.0

DL

Re

Pr

D .

For concentric tubes, the hydraulic diameter is used for the flow between the tubes. Thehydraulic diameter is the quotient between four times the cross-sectional area and the wetted

perimeter:wet

c

hP

AD

4= . For circular tubes: intext

intext

intext

h DDDD

DDD =

+

=)(

4/)(4 22

.


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For the case when the heat capacity rates of both fluids are equals, ChCc, the temperaturedifference is a constant throughout the heat exchanger, T1 = T2 = Tlm.

Total heat transfer in a shell-and-tube or in a cross-flow heat exchanger:CFmlTUAFq ,= ,

where Fis the correction factor represented graphically in Figures 2 to 5, based on the factors

P and R where t lowercase is assigned to the temperature of the tube-side fluid, not matter

whether it is the hot or the cold.

inin

inout

tT

ttP

= ;inout

outin

tt

TTR

= . tlowercase is assigned to the temperature of the tube-side fluid,

not matter whether it is the hot or the cold.

If the temperature change of one fluid is negligible (phase change), P orR is zero and Fis 1,hence, heat exchanger behaviour is independent of the specific configuration.

The maximum heat transfer of a heat exchanger is the product of the smaller heat capacityrate multiplied by the maximum possible temperature difference: )( ,, incinhminmax TTCq = .

The effectiveness of a heat exchanger, , is the ratio of the actual heat transfer rate of a heatexchanger to the maximum possible heat transfer rate (0 1):


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For the cross-flow heat exchangers with both fluids unmixed, the correspondingequation is only valid for Cr = 1, however it may be used to a good approximation for

all values ofCr.

The explicit relation for NTU for the case of shell-andtube heat exchangers provides theNTU per shell pass, so this result is multiplied by the number o passes, n, to obtaien the

NTU for the entire exchanger. For NTU 0.25 all heat exchangers have approximately the same effectiveness. More generally, for Cr> 0 and NTU 0.25, the counter-flow heat exchanger is the most

effective.

For any exchanger, maximum and minimum values of the effectiveness are associatedwith Cr= 0 y Cr= 1, respectively.

For compact heat exchangers, heat transfer and flow characteristics are typically presented inthe format of figures (Figures 12 and 13).

Heat transfer results are correlated in terms of the Colburn j factor, 3/2H StPrj = and theReynolds number, where both the Stanton ( pGchSt /= ) and Reynolds ( /hGDRe = )

b b d th i l it (it i fl l it


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TABLES AND FIGURES

Figure 1. Tube arrangements in a bank: a) aligned; b) Staggered.

Table 1. Coefficients of Zhukauskas correlation for the tube bank in cross flow.

Configuration ReD,max C m

Ali d 10 102

0 80 0 40


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Table 3. Thermophysical properties of motor oil at atmospheric pressure.

T

(K)

(kg/m3)

cp

(J/kgK)102

(Ns/m2)

106

(m2/s)

k103

(W/mK)107(m

2/s)

Pr 103(K

-1)

273 899.1 1796 385 4.28 147 0.910 47,000 0.70

280 895.3 1827 217 2.43 144 0.880 27,500 0.70290 890.0 1868 99.9 1.12 145 0.872 12,900 0.70

300 884.1 1909 48.6 550 145 0.859 6400 0.70

310 877.9 1951 25.3 288 145 0.847 3400 0.70

320 871.8 1993 14.1 161 143 0.823 1965 0.70

330 865.8 2035 8.36 96.6 141 0.800 1205 0.70

340 859.9 2076 5.31 61.7 139 0.779 793 0.70

350 853.9 2118 3.56 41.7 138 0.763 546 0.70

360 847.8 2161 2.52 29.7 138 0.753 395 0.70

370 841.8 2206 1.86 22.0 137 0.738 300 0.70

380 836.0 2250 1.41 16.9 136 0.723 233 0.70

390 830.6 2294 1.10 13.3 135 0.709 187 0.70

400 825.1 2337 0.874 10.6 134 0.695 152 0.70

410 818.9 2381 0.698 8.52 133 0.682 125 0.70

420 812.1 2427 0.564 6.94 133 0.675 103 0.70

430 806 5 2471 0 470 5 83 132 0 662 88 0 70


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Table 4. Thermophysical properties of air at atmospheric pressure.

T

(K)

(kg/m3)

cp

(J/kgK)107

(Ns/m2)

106

(m2/s)

k103

(W/mK)106(m

2/s)

Pr

100 3.5562 1032 71.1 2.00 9.34 2.54 0.786

150 2.3364 1012 103.4 4.43 13.8 5.84 0.758

200 1.7548 1007 132.5 7.59 18.1 10.3 0.737

250 1.3947 1006 159.6 11.44 22.3 15.9 0.720

300 1.1614 1007 184.6 15.89 26.3 22.5 0.707

350 0.9950 1009 208.2 20.92 30.0 29.9 0.700

400 0.8711 1014 230.1 26.41 33.8 38.3 0.690

450 0.7740 1021 250.7 32.39 37.3 47.2 0.686

500 0.6964 1030 270.1 38.79 40.7 56.7 0.684

550 0.6329 1040 288.4 45.57 43.9 66.7 0.683600 0.5804 1051 305.8 52.69 46.9 76.9 0.685

650 0.5356 1063 322.5 60.21 49.7 87.3 0.690

700 0.4975 1075 338.8 68.10 52.4 98.0 0.695

750 0.4643 1087 354.6 76.37 54.9 109 0.702

800 0.4354 1099 369.8 84.93 57.3 120 0.709

850 0.4097 1110 384.3 93.80 59.6 131 0.716

900 0 3868 1121 398 1 102 9 62 0 143 0 720


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Table 5. Thermophysical properties of saturated water.

T

(K)

P

(bar)

(kg/m3)

hfg

(kJ/kg)

cp

(J/kgK)106

(Ns/m2)

k103

(W/mK)

Pr 106(K

-1)

273.15 0.00611 1000 2502 4217 1750 569 12.99 -68.05

275 0.00697 1000 2497 4211 1652 574 12.22 -32.74

280 0.00990 1000 2485 4198 1422 582 10.26 46.04

285 0.01387 1000 2473 4189 1225 590 8.81 114.1

290 0.01917 999.0 2461 4184 1080 598 7.56 174.0

295 0.02617 998.0 2449 4181 959 606 6.62 227.5

300 0.03531 997.0 2438 4179 855 613 5.83 276.1

305 0.04712 995.0 2426 4178 769 620 5.20 320.6

310 0.06221 993.0 2414 4178 695 628 4.62 361.9

315 0.08132 991.1 2402 4179 631 634 4.16 400.4320 0.1053 989.1 2390 4180 577 640 3.77 436.7

325 0.1351 987.2 2378 4182 528 645 3.42 471.2

330 0.1719 984.3 2366 4184 489 650 3.15 504.0

335 0.2167 982.3 2354 4186 453 656 2.88 535.5

340 0.2713 979.4 2342 4188 420 660 2.66 566.0

345 0.3372 976.6 2329 4191 389 668 2.45 595.4

350 0 4163 973 7 2317 4195 365 668 2 29 624 2


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10

Figure 2. Correction factor for a shell-and-tube heat exchanger with one

shell and any multiple of two tube passes (2, 4, 6,).

0.50

0.60

0.70

0.80

0.90

1.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P

F

R=0.1

R=0.2

R=0.4

R=0.6

R=0.8R=1

R=1.2

R=1.5

R=2

R=3

R=4

R=6R=10


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11

Figure 3. Correction factor for a shell-and-tube heat exchanger with two shell

passes and any multiple of four tube passes (4, 8,).

0.50

0.60

0.70

0.80

0.90

1.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P

F

R=0.2

R=0.4

R=0.6

R=0.8

R=0.9

R=1.2R=1.5

R=2

R=3

R=4

R=6

R=10


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12

Figure 4. Correction factor for a single-pass, cross-flow heat exchanger with

both fluids unmixed.

0.50

0.60

0.70

0.80

0.90

1.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P

F

R=0.2

R=0.4

R=0.6

R=0.8

R=1.2

R=1.5

R=2

R=3

R=4


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13

Figure 5. Correction factor for a single-pass, cross-flow heat exchanger with

one fluid mixed and the other unmixed.

0.50

0.60

0.70

0.80

0.90

1.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P

F

R=0.1

R=0.2

R=0.4

R=0.6

R=0.8

R=1.2R=1.5

R=2

R=3

R=4

R=6

R=8


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14

Figure 6. Effectiveness of a parallel flow heat exchanger.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1 2 3 4 5 6

NTU

E

Cr = 0

Cr = 0.2

Cr = 0.4Cr = 0.6

Cr = 0.8

Cr = 1


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16

Figure 8. Effectiveness of a shell-and-tube heat exchanger with one shell

and any multiple of tow tube passes (2, 4, 6,).

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1 2 3 4 5 6

NTU

E

Cr = 0

Cr = 0.2

Cr = 0.4Cr = 0.6

Cr = 0.8

Cr = 1


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Figure 9. Effectiveness of a shell-and-tube heat exchanger with two shell

passes and any multiple of four tube passes (4, 8,).

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1 2 3 4 5 6

NTU

E

Cr = 0

Cr = 0,2

Cr = 0,4Cr = 0,6

Cr = 0,8

Cr = 1


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18

Figure 10. Effectiveness of a single-pass, cross-flow heat exchanger with

both fluids unmixed.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1 2 3 4 5 6

NTU

E

Cr = 0

Cr = 0.2

Cr = 0.4Cr = 0.6

Cr = 0.8

Cr = 1


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Figure 11 A. Effectiveness of a single-pass, cross-flow heat exchanger with

one fluid mixed (Cmin) and the other unmixed (Cmax).

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1 2 3 4 5 6

NTU

E

Cmix/Cunmix = 0

Cmix/Cunmix = 0.2

Cmix/Cunmix = 0.4Cmix/Cunmix = 0.6

Cmix/Cunmix = 0.8

Cmix/Cunmix = 1


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Figure 11 B. Effectiveness of a single-pass, cross-flow heat exchanger with

one fluid mixed (Cmax) and the other unmixed (Cmin).

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1 2 3 4 5 6

NTU

E

Cmix/Cunmix = 100

Cmix/Cunmix = 8

Cmix/Cunmix = 4Cmix/Cunmix = 2

Cmix/Cunmix = 1.5

Cmix/Cunmix = 1


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Figure 12. Heat transfer and friction factor for a circular tube- circular fins heat exchanger,

surface CF-7.0-5/8J from Kays and London.


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Figure 13. Heat transfer and friction factor for a circular tube-continuous fin heat

exchanger, surface 8.0-3/8T from Kays and London.

Documents

Heat Ex Changers Formulas