# Free convection to enclosed bodies (ht.conv_free_enclosed)¶

ht.conv_free_enclosed.Nu_Nusselt_Rayleigh_Hollands(Pr, Gr, buoyancy=True, Rac=1708)[source]

Calculates the Nusselt number for natural convection between two theoretical flat horizontal plates using the Hollands [1] correlation recommended in [2]. This correlation supports different aspect ratios, so the plates can be real, finite objects and have their heat transfer accurately modeled. The influence comes from the Rac term, which should be calculated separately, using Rac_Nusselt_Rayleigh or Rac_Nusselt_Rayleigh_disk.

$\text{Nu} = 1 + \left[1 - \frac{1708}{\text{Ra}} \right]^* \left[k_1 + 2 \left(\frac{\text{Ra}^{1/3}}{k_2} \right)^{1 - \ln({\text{Ra}}^{1/5}/k_2)} \right]^* + \left[\left(\frac{\text{Ra}}{5803}\right)^{1/3} - 1\right]^*$
$k_1 = \frac{1.44}{1 + 0.018/{Pr} + 0.00136/{Pr}^2}$
$k_2 = 75\exp(1.5\text{Pr}^{-0.5})$
Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - plate temperature difference [-]

buoyancybool, optional

Whether or not the plate’s free convection is buoyancy assisted (hot plate) or not, [-]

Racfloat, optional

Critical Rayleigh number, [-]

Returns
Nufloat

Nusselt number with respect to height between the two plates, [-]

Notes

For $$Ra < {Ra}_c$$, Nu = 1; for cases not assited by buoyancy, Nu is also 1.

References

1

Hollands, K. G. T. “Multi-Prandtl Number Correlation Equations for Natural Convection in Layers and Enclosures.” International Journal of Heat and Mass Transfer 27, no. 3 (March 1, 1984): 466-68. https://doi.org/10.1016/0017-9310(84)90295-3.

2

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd ed. 2010 edition. Berlin ; New York: Springer, 2010.

Examples

>>> Nu_Nusselt_Rayleigh_Hollands(5.54, 3.21e8, buoyancy=True)
69.02668649510


Plates - 1 m height, 2 m long, 0.2 m long vs a 1 m^3 cube

>>> Nu_Nusselt_Rayleigh_Hollands(.7, 3.21e6, buoyancy=True, Rac=Rac_Nusselt_Rayleigh(H=1, L=2, W=.2, insulated=False))
4.666249131876

>>> Nu_Nusselt_Rayleigh_Hollands(.7, 3.21e6, buoyancy=True, Rac=Rac_Nusselt_Rayleigh(H=1, L=1, W=1, insulated=False))
8.786362614129

ht.conv_free_enclosed.Nu_Nusselt_Rayleigh_Holling_Herwig(Pr, Gr, buoyancy=True)[source]

Calculates the Nusselt number for natural convection between two theoretical flat horizontal plates. The height between the plates is infinite, and one of the other dimensions of the plates is much larger than the other.

This correlation is for the horizontal plate Rayleigh-Benard classic heat transfer problem, not for real finite geometry plates.

This model is a non-linear equation which is solved numerically. The model can calculate Nu for Ra ranges between 350 and larger numbers; [1] recommends $$10^{5} < Ra < 10^{15}$$.

$\text{Nu} = \frac{{Ra}^{1/3}}{[0.05\ln(\frac{0.078}{16}{Ra}^{1.323}) + 2D]^{4/3}}$
$D = -\frac{14.94}{{Ra}^{0.25}} + 3.43$
Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - plate temperature difference [-]

buoyancybool, optional

Whether or not the plate’s free convection is buoyancy assisted (hot plate) or not, [-]

Returns
Nufloat

Nusselt number with respect to height between the two plates, [-]

Notes

A range of calculated values are provided in [1]; they all match the results of this function. This model is recommended in [2].

For $$Ra < 1708$$, Nu = 1; for cases not assited by buoyancy, Nu is also 1.

No success has been found finding an analytical solution in the major CAS packages, but the nonlinear function is in fact a function of one variable; this means a pade or chebyshev expansion could be performed.

References

1(1,2)

Hölling, M., and H. Herwig. “Asymptotic Analysis of Heat Transfer in Turbulent Rayleigh–Bénard Convection.” International Journal of Heat and Mass Transfer 49, no. 5 (March 1, 2006): 1129-36. https://doi.org/10.1016/j.ijheatmasstransfer.2005.09.002.

2

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd ed. 2010 edition. Berlin ; New York: Springer, 2010.

Examples

>>> Nu_Nusselt_Rayleigh_Holling_Herwig(5.54, 3.21e8, buoyancy=True)
77.54656801896913

ht.conv_free_enclosed.Nu_Nusselt_Rayleigh_Probert(Pr, Gr, buoyancy=True)[source]

Calculates the Nusselt number for natural convection between two theoretical flat plates. The height between the plates is infinite, and one of the other dimensions of the plates is much larger than the other.

This correlation is for the horizontal plate Rayleigh-Benard classic heat transfer problem, not for real finite geometry plates.

Two sets of equations are used.

For the laminar regime $$1708 < \text{Ra} \le 2.2\times 10^{4}$$:

$\text{Nu} = 0.208(\text{Ra})^{0.25}$

For the turbulent regime $$2.2\times 10^{4} < \text{Ra}$$:

$\text{Nu} = 0.092(\text{Ra})^{1/3}$
Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - plate temperature difference [-]

buoyancybool, optional

Whether or not the plate’s free convection is buoyancy assisted (hot plate) or not, [-]

Returns
Nufloat

Nusselt number with respect to height between the two plates, [-]

Notes

This model is recommended in [2] as a rough model.

For $$Ra < 1708$$, Nu = 1; for cases not assited by buoyancy, Nu is also 1.

References

1

Probert, SD, RG Brooks, and M Dixon. “Heat Transfer across Rectangular Cavities.” CHEMICAL AND PROCESS ENGINEERING, 1970, 35.

2

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd ed. 2010 edition. Berlin ; New York: Springer, 2010.

Examples

>>> Nu_Nusselt_Rayleigh_Probert(5.54, 3.21e8, buoyancy=True)
111.46181048289132

ht.conv_free_enclosed.Nu_Nusselt_vertical_Thess(Pr, Gr, H=None, L=None)[source]

Calculates the Nusselt number for natural convection between two theoretical vertical flat plates using the correlation by Thess [1] in [1]. This is a variant on the horizontal Rayleigh-Benard classic heat transfer problem. This correlation supports different aspect ratios, so the plates can be real, finite objects and have their heat transfer accurately modeled. The recommended range of the correlation is H/L < 80.

For 1e4 < Ra < 1e7:

$\text{Nu} = 0.42{Pr}^{0.012} {Ra}^{0.25} \left(\frac{H}{L}\right)^{-0.25}$

For 1e7 < Ra > 1e9 (or when geometry is unknown):

$\text{Nu} = 0.049{Ra}^{0.33}$
Parameters
Prfloat

Prandtl number with respect to fluid properties [-]

Grfloat

Grashof number with respect to fluid properties and plate - plate temperature difference [-]

Hfloat, optional

Height of vertical plate, [m]

Lfloat, optional

Length of vertical plate, [m]

Returns
Nufloat

Nusselt number with respect to distance between the two plates, [-]

References

1

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd ed. 2010 edition. Berlin ; New York: Springer, 2010.

Examples

>>> Nu_Nusselt_vertical_Thess(.7, 3.21e6)
6.112587569602785

>>> Nu_Nusselt_vertical_Thess(.7, 3.21e6, L=10, H=1)
28.79328626041646

ht.conv_free_enclosed.Nu_vertical_helical_coil_Ali(Pr, Gr)[source]

Calculates Nusselt number for natural convection around a vertical helical coil inside a tank or other vessel according to the Ali [1] correlation.

$Nu_L = 0.555Gr_L^{0.301} Pr^{0.314}$
Parameters
Prfloat

Prandtl number of the fluid surrounding the coil with properties evaluated at bulk conditions or as described in the notes [-]

Grfloat

Prandtl number of the fluid surrounding the coil with properties evaluated at bulk conditions or as described in the notes (for the two temperatures, use the average coil fluid temperature and the temperature of the fluid outside the coil) [-]

Returns
Nufloat

Nusselt number with respect to the total length of the helical coil (and bulk thermal conductivity), [-]

Notes

In [1], the temperature at which the fluid surrounding the coil’s properties were evaluated at was calculated in an unusual fashion. The average temperature of the fluid inside the coil $$(T_{in} + T_{out})/2$$ is averaged with the fluid outside the coil’s temperature.

The correlation is valid for Prandtl numbers between 4.4 and 345, and tank diameter/coil outer diameter ratios between 10 and 30.

References

1(1,2)

Ali, Mohamed E. “Natural Convection Heat Transfer from Vertical Helical Coils in Oil.” Heat Transfer Engineering 27, no. 3 (April 1, 2006): 79-85.

Examples

>>> Nu_vertical_helical_coil_Ali(4.4, 1E11)
1808.5774997297106

ht.conv_free_enclosed.Nu_vertical_helical_coil_Prabhanjan_Rennie_Raghavan(Pr, Gr)[source]

Calculates Nusselt number for natural convection around a vertical helical coil inside a tank or other vessel according to the Prabhanjan, Rennie, and Raghavan [1] correlation.

$Nu_H = 0.0749\text{Ra}_H^{0.3421}$

The range of Rayleigh numbers is as follows:

$9 \times 10^{9} < \text{Ra} < 4 \times 10^{11}$
Parameters
Prfloat

Prandtl number calculated with the film temperature - wall and temperature very far from the coil average, [-]

Grfloat

Grashof number calculated with the film temperature - wall and temperature very far from the coil average, and using the total height of the coil [-]

Returns
Nufloat

Nusselt number using the total height of the coil and the film temperature, [-]

Notes

[1] also has several other equations using different characteristic lengths.

References

1(1,2)

Prabhanjan, Devanahalli G., Timothy J. Rennie, and G. S. Vijaya Raghavan. “Natural Convection Heat Transfer from Helical Coiled Tubes.” International Journal of Thermal Sciences 43, no. 4 (April 1, 2004): 359-65.

Examples

>>> Nu_vertical_helical_coil_Prabhanjan_Rennie_Raghavan(4.4, 1E11)
720.6211067718227

ht.conv_free_enclosed.Rac_Nusselt_Rayleigh(H, L, W, insulated=True)[source]

Calculates the critical Rayleigh number for free convection to begin in the Nusselt-Rayleigh parallel horizontal plate scenario. There are actually two cases - one for the top plate to be insulated (adiabatic) and the other where it has infinite thermal conductivity/is infinitely thin or not present (perfectly conducting). All real cases will lie between the two.

Parameters
Hfloat

Distance between the two plates, [m]

Lfloat

Length of the plates, [m]

Wfloat

Width of the plates, [m]

insulatedbool, optional

Whether the top plate is insulated or uninsulated, [-]

Returns
Racfloat

Critical Rayleigh number, [-]

Notes

Splines have been fit to data in [1] for the uninsulated case and [2] for the insulated case. The data is presented in the original papers and in [3].

References

1

Catton, Ivan. “Effect of Wall Conduction on the Stability of a Fluid in a Rectangular Region Heated from Below.” Journal of Heat Transfer 94, no. 4 (November 1, 1972): 446-52. https://doi.org/10.1115/1.3449966.

2

Catton, Ivan. “Convection in a Closed Rectangular Region: The Onset of Motion.” Journal of Heat Transfer 92, no. 1 (February 1, 1970): 186-88. https://doi.org/10.1115/1.3449626.

3

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

Examples

>>> Rac_Nusselt_Rayleigh(1, .5, 2, False)
2530.500000000005
>>> Rac_Nusselt_Rayleigh(1, .5, 2, True)
2071.0089443385655

ht.conv_free_enclosed.Rac_Nusselt_Rayleigh_disk(H, D, insulated=True)[source]

Calculates the critical Rayleigh number for free convection to begin in the parallel horizontal disk scenario. There are actually two cases - one for the top plate to be insulated (adiabatic) and the other where it has infinite thermal conductivity/is infinitely thin or not present (perfectly conducting). All real cases will lie between the two.

Parameters
Hfloat

Distance between the two disks, [m]

Dfloat

Diameter of the two disks, [m]

insulatedbool, optional

Whether the top plate is insulated or uninsulated, [-]

Returns
Racfloat

Critical Rayleigh number, [-]

Notes

The range of data covered by this function is D/H from 0.4 to infinity. As inifinity is not well suited to polynomial form, the upper limit is 6 in actuality. Values outside that range are rounded to the limits.

This function provides 17-coefficient polynomial fits to interpolate in the table of values in [1]. The source of the coefficients is cited as being from [2].

References

1

Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

2

Buell, J. C., and I. Catton. “The Effect of Wall Conduction on the Stability of a Fluid in a Right Circular Cylinder Heated From Below.” Journal of Heat Transfer 105, no. 2 (May 1, 1983): 255-60. https://doi.org/10.1115/1.3245571.

Examples

>>> Rac_Nusselt_Rayleigh_disk(H=1, D=.4, insulated=False)
151199.9999999945

>>> Rac_Nusselt_Rayleigh_disk(H=1, D=4, insulated=False)
1891.520931853363

>>> Rac_Nusselt_Rayleigh_disk(2, 1, True)
24347.31479211917