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SCIENCE CHINA Technological Sciences, Volume 61 , Issue 1 : 98-109(2018) https://doi.org/10.1007/s11431-017-9120-5

New friction factor and Nusselt number equations for laminar forced convection of liquid with variable properties

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  • ReceivedJun 20, 2017
  • AcceptedAug 11, 2017
  • PublishedSep 15, 2017

Abstract

Friction factor and heat transfer coefficient of liquid flow with variable properties can significantly differ from that with constant properties. Existing equations obtained by regression analysis of experimental data use correction factors to account for variable property effect. They are limited to specific kind of fluid and low or medium temperature differences. The correction factors of the equations for heating and cooling conditions are different. New explicit friction factor and Nusselt number equations for laminar forced convection of liquid with variable properties are derived with a first order approximation of dynamic viscosity-temperature variation. The new equations are applicable to all kinds of liquids and can be used for large temperature differences. Governing equations of laminar forced convection of water and ethanol are numerically solved using computational fluid dynamics (CFD) method and the results are used to verify the derived equations. The derived equations show good predictions of friction factors and Nusselt numbers for both heating and cooling conditions and show more accurate predictions than the existing equations. A dimensionless number is also introduced based on theoretical analysis to evaluate property variation effects on friction factors and heat transfer coefficients.


Funded by

National Natural Science Foundation of China(51576103)

National S&T Major Project(ZX06901)


Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant No. 51576103), and the National S&T Major Project (Grant No. ZX06901).


References

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  • Figure 1

    (Color online) Normalized physical property distributions in the radial direction.

  • Figure 2

    Schematic of the physical model and grids.

  • Figure 3

    (Color online) Velocity and temperature distribution in the cross section of case 1 (water-heating) (a) and cross section of case 2 (ethanol-heating) (b).

  • Figure 4

    (Color online) Velocity and temperature distribution in the cross section of case 3 (water-cooling) (a) and cross section of case 4 (ethanol-cooling) (b).

  • Figure 5

    (Color online) Friction factor variation with bulk temperature of case 1 (water-heating) (a) and of case 2 (ethanol-heating) (b).

  • Figure 6

    (Color online) Friction factor variation with bulk temperature of case 3 (water-cooling) (a) and of case 4 (ethanol-cooling) (b).

  • Figure 7

    (Color online) Friction factor variation with bulk temperature of case 2 (16 kW/m2) and case 5 (8 kW/m2).

  • Figure 8

    (Color online) Dimensionless parameter variations with bulk temperature for cases 1, 2, 3 and 4.

  • Table 1   Cases with different boundary conditions

    Cases

    Liquid

    Tube length (mm)

    Mass flux (kg/(m2 s))

    Inlet temperature (K)

    Heat flux (kW/m2)

    1

    Water

    1300

    75

    275

    40

    2

    Ethanol

    2000

    75

    255

    16

    3

    Water

    1300

    50

    500

    −25

    4

    Ethanol

    4000

    60

    450

    −6

    5

    Ethanol

    2000

    75

    255

    8

  • Table 2   Constants of dynamic viscosity-temperature function for selected liquids ()

    Liquids

    A

    B

    C

    D

    Temperature range (K)

    Water

    −24.71

    4209

    0.04527

    −3.376×10−5

    273.15–643.15

    Ethanol

    −6.21

    1614

    0.00618

    −1.132×10−5

    168.15–516.15

    Chloroform

    −4.172

    915

    0.00270

    −4.108×10−6

    201.15–536.15

    Methanol

    −39.350

    4826

    0.10910

    −1.127×10−4

    233.15–512.15

    Cyclopropane

    −3.074

    268

    0.00026

    −8.830×10−8

    146.15–397.15

    Propane

    −7.764

    722

    0.02381

    −4.660×10−5

    86.15–369.15

    Isoprene

    −2.228

    636

    −0.00732

    7.665×10−6

    127.15–486.15

    Chlorobenzene

    −4.573

    1196

    0.00137

    −1.378×10−6

    228.15–623.15

    Chloroprene

    −3.583

    709

    0.00138

    −1.841×10−6

    143.15–533.15

    Tolune

    −5.878

    1287

    0.00458

    −4.499×10−6

    233.15–588.15

  • Table 3   Friction factor variation with bulk temperature of case 2 (ethanol-heating)

    T b

    ΔT

    f CFD

    Eq. (13)

    Eq. (27)

    Eq. (30)

    Eq. (31)

    301.17

    71.33

    0.121

    87.95%

    0.01%

    −2.18%

    1.95%

    329.71

    77.26

    0.076

    81.75%

    1.03%

    −2.34%

    1.75%

    344.91

    79.57

    0.061

    78.24%

    0.83%

    −2.76%

    1.34%

    366.33

    82.51

    0.046

    74.99%

    0.86%

    −3.39%

    0.86%

    379.78

    84.33

    0.038

    74.22%

    1.26%

    −3.93%

    0.55%

    392.64

    86.11

    0.032

    74.53%

    2.04%

    −4.81%

    0.01%

    398.86

    86.97

    0.030

    75.16%

    2.62%

    −5.50%

    −0.43%

  • Table 4   Nusselt number variation with bulk temperature of case 1 (water-heating) and case 2 (ethanol-heating)

    T b

    ΔT

    NuCFD

    Eq. (14)

    Eq. (29)

    Eq. (32)

    Eq. (33)

    Eq. (34)

    case 1 (water-heating)

    332.50

    49.95

    4.88

    −10.63%

    0.38%

    −2.73%

    −4.48%

    −4.36%

    351.64

    50.45

    4.72

    −7.60%

    −1.09%

    −0.66%

    −2.19%

    −2.51%

    370.71

    50.70

    4.63

    −5.85%

    −1.52%

    0.21%

    −1.12%

    −1.74%

    383.37

    50.88

    4.59

    −5.01%

    −1.63%

    0.55%

    −0.67%

    −1.43%

    414.71

    51.64

    4.52

    −3.37%

    −1.95%

    1.20%

    0.20%

    −0.82%

    427.13

    52.09

    4.49

    −2.78%

    −2.19%

    1.53%

    0.59%

    −0.53%

    case 2 (ethanol-heating)

    301.17

    71.33

    5.350

    −22.61%

    −2.55%

    −4.51%

    −6.76%

    −7.69%

    317.79

    75.14

    5.191

    −18.97%

    −0.31%

    −2.02%

    −5.29%

    −5.19%

    329.71

    77.26

    5.128

    −17.52%

    0.47%

    −1.14%

    −5.13%

    −4.27%

    344.91

    79.57

    5.079

    −16.39%

    0.95%

    −0.55%

    −5.41%

    −3.62%

    366.33

    82.51

    5.040

    −15.50%

    1.21%

    −0.07%

    −6.03%

    −3.09%

    386.28

    85.22

    5.015

    −14.94%

    1.38%

    0.54%

    −6.19%

    −2.53%

    398.86

    86.97

    5.003

    −14.66%

    1.50%

    1.23%

    −5.53%

    −1.95%

  • Table 5   Friction factor variation with bulk temperature of case 4 (ethanol-cooling)

    T b

    ΔT

    f CFD

    Eq. (13)

    Eq. (27)

    Eq. (30)

    Eq. (31)

    381.21

    −38.78

    0.11

    −25.69%

    1.94%

    −2.25%

    −0.09%

    368.60

    −38.55

    0.13

    −26.66%

    1.28%

    −2.31%

    0.01%

    355.40

    −38.28

    0.16

    −27.84%

    0.57%

    −2.43%

    0.07%

    341.53

    −37.99

    0.20

    −29.29%

    −0.21%

    −2.65%

    0.04%

    334.35

    −37.84

    0.23

    −30.12%

    −0.63%

    −2.81%

    −0.01%

    326.97

    −37.68

    0.26

    −31.04%

    −1.05%

    −3.00%

    −0.10%

    319.41

    −37.51

    0.30

    −32.04%

    −1.48%

    −3.23%

    −0.20%

  • Table 6   Nusselt number variation with bulk temperature of case 3 (water-cooling) and case 4 (ethanol-cooling)

    Tb

    ΔT

    NuCFD

    Eq. (14)

    Eq. (29)

    Eq. (32)

    Eq. (33)

    Eq. (34)

    case 3 (water-cooling)

    455.51

    −34.37

    4.31

    1.31%

    1.67%

    −1.81%

    −0.58%

    −2.43%

    444.14

    −34.60

    4.25

    2.66%

    0.49%

    −0.72%

    0.52%

    −2.59%

    432.67

    −34.69

    4.22

    3.41%

    −0.06%

    −0.33%

    0.93%

    −2.85%

    420.97

    −34.76

    4.20

    3.98%

    −0.37%

    −0.15%

    1.13%

    −3.13%

    409.32

    −34.90

    4.17

    4.56%

    −0.63%

    −0.03%

    1.25%

    −3.46%

    397.61

    −35.14

    4.15

    5.24%

    −0.91%

    0.22%

    1.47%

    −3.77%

    373.89

    −36.01

    4.07

    7.16%

    −1.72%

    0.78%

    1.93%

    −4.71%

    case 4 (ethanol-cooling)

    381.21

    −38.78

    4.12

    5.50%

    −1.93%

    −2.00%

    −0.11%

    −0.37%

    368.60

    −38.55

    4.08

    6.60%

    −0.90%

    −1.19%

    0.39%

    0.53%

    355.40

    −38.28

    4.03

    7.63%

    0.02%

    −0.51%

    0.73%

    1.30%

    341.53

    −37.99

    3.99

    8.62%

    0.88%

    0.07%

    0.97%

    2.01%

    334.35

    −37.84

    3.97

    9.13%

    1.30%

    0.33%

    1.07%

    2.34%

    326.97

    −37.68

    3.94

    9.63%

    1.71%

    0.58%

    1.16%

    2.66%

    319.41

    −37.51

    3.92

    10.13%

    2.09%

    0.79%

    1.25%

    2.95%

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