4.4.11. Correlation Heat Transfer Coefficient

A convective flux boundary condition q_n = h\left(T - T_{fluid}\right) can be used to model problems where a full conjugate heat transfer simulation is deemed impractical. To facilitate specification of these models, Aria includes a number of heat transfer coefficient correlations [40] that cover laminar and turbulent flows of forced and free convection of gas and liquids (including liquid metals) for internal and external flow geometries.

All correlations define a Nusselt number \mathrm{Nu} from which the heat transfer coefficient h is computed as

(4.26)h = \dfrac{k \mathrm{Nu}}{D_h} ,

where k is the scalar thermal conductivity and D_h is a characteristic length for the problem. As shown in the example below, correlations are defined in heat transfer correlation coefficient command blocks and are referenced in the convective flux boundary condition command block.

Begin Sierra myJob
  ...
  Begin Aria Material my_fluid
    ...
    # Define k, rho, cp, mu models
    thermal conductivity = constant value = 1.0
    specific heat capacity = constant value = 1000.0
    density = constant value = 10.0
    viscosity = constant value = 0.01
  End
  ...

  Begin aria region my_region
    ...
    Begin equation system energy
      ...

      # Define conv flux BC using specified correlation
      Begin convective flux boundary condition bar_coupling
        add surface surface_1
        use correlation convection model pipe_flow
        use advective bar my_bar
      End

      # Equivalent line cmd
      # BC Bar_Flux for energy on surface_1 = generalized_nat_conv bar = my_bar hcorr = pipe_flow

      # Define h
      Begin heat transfer correlation coefficient pipe_flow
        laminar correlation = type 2 # Nu = 3.666 analytic laminar pipe flow
        turbulent correlation = type 72 # Gnielisnki correlation w/ Nu = func(Re, Pr, f)

        # Define Reynolds number above which the turbulent correlation will be used
        transition Reynolds Number = 3000

        # Define additional inputs needed to compute Nu
        compute characteristic length model = hydraulic diameter
        compute friction factor model = smooth tube
        compute fluid velocity model = advection_velocity
        compute reynolds number
        compute prandtl number
      End

      # Postprocess average htc
      postprocess average of expression heat_transfer_coefficient on surface_1 as avg_htc
    End
  End
End

In this example, laminar and turbulent correlations are used to model the conjugate heat transfer of internal flow past a pipe. To place emphasis on the correlation, details concerning the advective bar fluid model or pipe heat conduction model have been omitted.

To start, all heat transfer correlation models must define the thermal conductivity k and characteristic length D_h. The thermal conductivity is always specified as a material property, whereas the characteristic length is specified via a line command. In this example, the hydraulic diameter model (as specified in the advective bar block) would be used for the characteristic length.

In the above example, the type 72 Gnielinski correlation has been used. For this correlation, the Nusselt number is a function of the Reynolds number \mathrm{Re} = \dfrac{U D_h}{\nu}, Prandtl number \mathrm{Pr} = \dfrac{\nu}{\alpha}, and non-dimensional friction factor. Correspondingly, line commands need to be specified to compute each of these non-dimensional parameters. Here, for the friction factor computation, the smooth tube model has been specified whereas the Reynolds number and Prandtl number will be computed from material properties defined in aria material command block and applied to the volume bar block. Note a characteristic speed also needs to be specified to compute the Reynolds number. Here, the compute fluid velocity line command specifies to use the advection_velocity expression.

While this example has highlighted how to define and use a forced convection correlation with T_{fluid}\left(\mathbf{x},t\right) modeled using a 1D advective bar, in a similar manner free convection correlations can be defined with the bulk fluid temperature T_{fluid}\left(t\right) modeled using a bulk node.

Below is an example of using an external free convection correlation with a bulk node. To start, we see the use advective bar line command has been replaced with the use bulk element line command, where the specified name now refers to a bulk node. Next, for the correlation, \mathrm{Nu} is now a function of \mathrm{Pr} and the \mathrm{Ra} Rayleigh number. As this free convection correlation does not depend on a friction factor, or Reynolds number, there is no need to specify them. This includes any unused prerequisites (e.g. characteristic speed). Unlike before, the Rayleigh number is automatically computed for the given correlation from its prerequisites and so no line command is needed. However, the Rayleigh number prerequisites do need to be specified. Note that material properties are now pulled from the material assigned to the bulk node.

Begin Sierra myJob
  ...

  # Define k, rho, cp, mu models
  Begin Aria Material my_bulk_material
    viscosity            = user_function name = air_viscosity X = temperature
    density              = user_function name = air_density   X = temperature
    specific heat        = user_function name = air_specheat  X = temperature
    thermal conductivity = user_function name = air_cond      X = temperature
  End
  ...

  Begin aria region my_region
    ...

    # Define bulk node
    Begin bulk fluid element my_bulk_node
      ...
      material = my_bulk_material
      Initial Temperature = 600.0

      # Setting a large volume here will in effect result in an infinite reservoir
      bulk element volume = constant v = 1.0e10
      bulk eq energy for temperature using p0 with mass src
    End

    Begin equation system energy
      ...

      # Define conv flux BC using specified correlation
      Begin convective flux boundary condition bulk_node_coupling
        add surface surface_1
        use correlation convection model external_free_conv
        use bulk element my_bulk_node
      End

      # Define h
      Begin heat transfer correlation coefficient external_free_conv
         LAMINAR CORRELATION = type 64
         TURBULENT CORRELATION = type 64 # Churchill-Chu free convection

         # Define D_h as a constant instead of using an expression
         characteristic length = 1.0

         # Define additional inputs needed to compute Nu(Ra, Pr)
         gravitational constant = 9.8 # g
         wall length = 1.0 # L
         compute prandtl number # Pr
         compute wall temperature # T_w
         compute fluid temperature # T_f
      End

      # Postprocess average htc
      postprocess average of expression heat_transfer_coefficient on surface_1 as avg_htc
    End
  End
End

Note

In some cases, it is desirable to use one of the built in heat transfer correlations with a fixed reference temperature. The current work around is to couple the surface to a bulk node with a large volume assigned in order to mimic an infinite reservoir. The reference temperature will fluctuate near the initial temperature conditions applied to the bulk node.

Warning

Both laminar and turbulent correlations are always required to be specified. When both correlations are free convection correlations, default constant expressions of one are set for the transitional Reynolds number, characteristic speed, and Reynolds number (if not already provided). This behavior will be fixed in the future so that specification of both correlations is not always required. For now, setting both correlations to the same type can be used as a work around.

As shown in these examples, depending on the correlation, additional line commands will need to be specified. The supported line commands can be found here. If a correlation prerequisite is not met, an error will be thrown indicating the prerequisite expressions that the correlation needs. Supported correlations are listed next.

4.4.11.1. Correlations Listing

The following parameters will be referred to in the various correlation definitions:

Table 4.1 Parameter Definitions

Symbol

Name

f

friction factor

\mathrm{Re}

flow Reynolds number

\mathrm{Pr}

flow Prandtl number

\mathrm{Ra}

flow Rayleigh number

\mathrm{Pr}_w

flow Prandtl number evaluated with wall conditions

T

fluid temperature

T_w

wall temperature

L

wall length

g

gravitational constant

r

ratio of annulus inner diameter to outer diameter

\rho

fluid density

\mu

dynamic viscosity

\nu

kinematic viscosity

\beta

expansion coefficient

\theta

wall angle (in radians)

D_e

entrance length

4.4.11.2. Forced Convection (Internal)

Type 1

This correlation applies to cases of fully-developed internal laminar flow with a constant wall flux or constant wall temperature, for which a Nusselt number may be determined analytically.

(4.27)\mathrm{Nu} = 4.36

Regime

Laminar

Phase

Gas|Liquid

Range

\mathrm{Pr} > 0.6

Type 2

This correlation applies to cases of fully-developed internal laminar flow with a constant wall flux or constant wall temperature, for which a Nusselt number may be determined analytically.

(4.28)\mathrm{Nu} = 3.666

Regime

Laminar

Phase

Gas|Liquid

Range

\mathrm{Pr} > 0.6

Type 3

This correlation by Hausen applies to cases of thermally fully-developed laminar flow with entrance effects.

(4.29)\begin{split} \mathrm{Nu} &= 3.66 + \dfrac{0.0668 I}{1 + 0.4 I^{0.66667}} \\ I &= \dfrac{D_{h}}{D_{e}}\mathrm{Re}\mathrm{Pr} \end{split}

Regime

Laminar

Phase

Gas|Liquid

Range

\mathrm{Pr} \gg 1 or unheated entrance

Type 4

This correlation by Nusselt applies to cases of thermally fully-developed turbulent flow with entrance effects.

(4.30)\mathrm{Nu} = 0.036\mathrm{Re}^{0.8}\mathrm{Pr}^{0.3333}\big{(}D_h / D_e\big{)}^{0.05556}

Regime

Turbulent

Phase

Gas|Liquid

Type 14

This correlation by Azer-Chao applies to cases of constant surface temperature liquid metals in tubes.

(4.31)\mathrm{Nu} = 5.0 + 0.05\left(\mathrm{Re}\mathrm{Pr}\right)^{0.8}\mathrm{Pr}^{0.25}

Regime

Turbulent

Phase

Liquid Metal

Type 15

This correlation by Sliecher applies to cases of constant surface temperature liquid metals in tubes.

(4.32)\mathrm{Nu} = 4.8 + 0.0156\left(\mathrm{Re}\mathrm{Pr}\right)^{0.85}\mathrm{Pr}^{0.08}

Regime

Laminar|Turbulent

Phase

Liquid Metal

Range

0.004 < \mathrm{Pr} < 0.1 and \mathrm{Re} < 500000

Type 23

This correlation by Petukhov applies to cases of turbulent gas flow with friction factor correlation for smooth surfaces.

(4.33)\mathrm{Nu}=\dfrac{0.125fRePr}{1.07 + 12.7\sqrt{0.125f}\big(\mathrm{Pr}^{0.6667}-1\big)}

Regime

Turbulent

Phase

Gas

Range

3000 < \mathrm{Re} < 5000000 and 0.5 < \mathrm{Pr} < 2000

Type 25

This correlation by Dittus-Boelter applies to cases of turbulent flow.

(4.34)\begin{split} \mathrm{Nu}&=0.023{\mathrm{Re}}^{0.8}{\mathrm{Pr}}^{e} \\ e&=\begin{cases} \begin{array}{c} 0.3\\ 0.4 \end{array} \begin{array}{c} T_{w}<T\\ \mbox{else} \end{array} \end{cases} \end{split}

Regime

Turbulent

Phase

Gas|Liquid

Range

10000 < \mathrm{Re} < 1000000 and 0.7 < \mathrm{Pr} < 160

Type 26

This correlation by Sieder-Tate applies to cases of turbulent gas flow.

(4.35)\mathrm{Nu}=0.027\mathrm{Re}^{0.8}\mathrm{Pr}^{0.3333}\bigg(\frac{T}{T_{w}}\bigg)^{0.098}

Regime

Turbulent

Phase

gas

Range

10000 < \mathrm{Re} < 1000000 and 0.7 < \mathrm{Pr} < 160

Type 39

This correlation by Sieder-Tate applies to cases of turbulent liquid flow.

(4.36)\begin{split} \mathrm{Nu}&=0.027{\mathrm{Re}}^{0.8}{\mathrm{Pr}}^{0.3333}\bigg(\frac{\mu}{\mu_{w}}\bigg)^{0.14} \\ \mu_{w}&=\Bigg(\mu^{-0.2661}+\frac{T_{w}-T}{37.073}\Bigg)^{-3.758} \end{split}

Regime

Turbulent

Phase

Liquid

Range

10000 < \mathrm{Re} < 1000000 and 0.7 < \mathrm{Pr} < 160

Type 72

This correlation by Gnielinski applies to cases of turbulent gas flow in a circular tube with friction factor correlation for smooth surfaces.

(4.37)\mathrm{Nu}=\frac{\frac{f}{8}{(\mathrm{Re}-1000)\mathrm{Pr}}}{1+12.7\sqrt{\frac{f}{8}}\big({\mathrm{Pr}}^{0.6667}-1\big)}

Regime

Turbulent

Phase

Gas

Range

3000 < \mathrm{Re} < 5000000 and 0.5 < \mathrm{Pr} < 2000

Type 73

This modified correlation by Gnielinski applies to cases of turbulent flow in an annulus with friction factor correlation for annular surfaces where the inner wall is heated and the outer wall is insulated.

(4.38)\begin{split} \mathrm{Nu}&=cF\frac{\frac{f}{8}{\mathrm{Re}\mathrm{Pr}}}{b+12.7\sqrt{\frac{f}{8}}\big({\mathrm{Pr}}^{0.6667}-1\big)}\\ b&=1.07+\frac{900}{\mathrm{Re}}-\frac{0.63}{1+10Pr}\\ F&=0.75r^{-0.17} \\ c&=\begin{cases} \begin{array}{c} 1\\ \big(\frac{T}{T_w}\big)^{0.45}\\ \big(\frac{\mathrm{Pr}}{\mathrm{Pr}_w}\big)^{0.11} \end{array} \begin{array}{c} \mbox{GAS and } T > T_w\\ \mbox{GAS and } T < T_w\\ \mbox{LIQUID} \end{array} \end{cases} \end{split}

Regime

Turbulent

Phase

Gas|Liquid

Range

\mathrm{Re} > 10000

Type 74

This modified correlation by Gnielinski applies to cases of turbulent flow in an annulus with friction factor correlation for annular surfaces where the inner wall is insulated and the outer wall is heated.

(4.39)\begin{split} \mathrm{Nu}&=cF\frac{\frac{f}{8}{\mathrm{Re}\mathrm{Pr}}}{b+12.7\sqrt{\frac{f}{8}}\big({\mathrm{Pr}}^{0.6667}-1\big)}\\ b&=1.07+\frac{900}{\mathrm{Re}}-\frac{0.63}{1+10Pr}\\ F&=0.9-0.15r^{0.6}\\ c&=\begin{cases} \begin{array}{c} 1\\ \big(\frac{T}{T_w}\big)^{0.45}\\ \big(\frac{\mathrm{Pr}}{\mathrm{Pr}_w}\big)^{0.11} \end{array} \begin{array}{c} \mbox{GAS and } T > T_w\\ \mbox{GAS and } T < T_w\\ \mbox{LIQUID} \end{array} \end{cases} \end{split}

Regime

Turbulent

Phase

Gas|Liquid

Range

\mathrm{Re} > 10000

4.4.11.3. Forced Convection (External)

Type 5

This modified Ranz-Marshall correlation applies to cases of thermally fully-developed flow with entrance effects.

(4.40)\mathrm{Nu} = 2 + 0.6 \sqrt{\mathrm{Re}} \mathrm{Pr}^{0.3333} 25(D_{e}/D_{h})^{-0.7}

Region

Laminar|Turbulent

Phase

Liquid

Type 18

This correlation applies to cases of flow over a sphere.

(4.41)\mathrm{Nu} = 0.37\mathrm{Re}^{0.6} 1.126 \mathrm{Pr}^{0.3333}

Regime

Laminar|Turbulent

Phase

Gas

Range

17 < \mathrm{Re} < 700000

Type 21

This correlation by Zhulkauskas applies to cases of aligned cylindrical tube bank in cross flow.

(4.42)\begin{split} \mathrm{Nu}&=C_{1}{\mathrm{Re}}^{C_{2}}{\mathrm{Pr}}^{0.36} \\ (C_{1},C_{2})&=\begin{cases} \begin{array}{c} (0.27,0.63)\\ (0.021,0.84) \end{array} \begin{array}{c} \mathrm{Re}<200000\\ \mbox{else} \end{array} \end{cases} \end{split}

Regime

Laminar|Turbulent

Phase

Gas

Range

1 < \mathrm{Re} < 1000000 and 0.7 < \mathrm{Pr} < 500

Type 27

This correlation by Whitaker applies to cases of gas flow over a sphere.

(4.43)\mathrm{Nu}=2+(0.4\sqrt{\mathrm{Re}}+0.06\mathrm{Re}^{0.6667})\mathrm{Pr}^{0.4}\bigg(\frac{T}{T_{w}}\bigg)^{0.175}

Regime

Laminar|Turbulent

Phase

Gas

Range

3.5 < \mathrm{Re} < 76000 and 0.71 < \mathrm{Pr} < 380

Type 28

This correlation by Hilpert applies to cases of cylinder in cross flow.

(4.44)\begin{split} \mathrm{Nu} &=C_{1}\mathrm{Re}_f^{C_{2}}\mathrm{Pr}^{0.3333} \\ \mathrm{Re}_f &= \mathrm{Re}\Big(\frac{T}{T_{f}}\Big)^{0.7}\\ (C_{1},C_{2}) &= \begin{cases} \begin{array}{c} (0.989,0.33)\\ (0.911,0.385)\\ (0.683,0.466)\\ (0.193,0.618)\\ (0.027,0.805) \end{array} \begin{array}{c} \mathrm{Re}_f <4\\ \mathrm{Re}_f <40\\ \mathrm{Re}_f <4000\\ \mathrm{Re}_f <40000\\ \mbox{else} \end{array} \end{cases} \end{split}

Regime

Laminar|Turbulent

Phase

Gas

Range

0.4 < \mathrm{Re} < 400000

Type 34

This correlation by Zhulkauskas applies to cases of cylinder in cross flow

(4.45)\begin{split} \mathrm{Nu}&=C_{1}{\mathrm{Re}_f}^{C_{2}}\mathrm{Pr}^{C_{3}} \\ \mathrm{Re}_f&=\mathrm{Re}\Big(\frac{T}{T_{f}}\Big)^{0.7}\\ T_{f}&=\frac{T+T_{w}}{2} \\ (C_{1},C_{2})&=\begin{cases} \begin{array}{c} (0.989,0.33)\\ (0.911,0.385)\\ (0.683,0.466)\\ (0.027,0.805) \end{array} \begin{array}{c} \mathrm{Re}_f<4\\ \mathrm{Re}_f<40\\ \mathrm{Re}_f<4000\\ \mbox{else} \end{array} \end{cases} \\ C_{3}&=\begin{cases} \begin{array}{c} 0.36\\ 0.37 \end{array} \begin{array}{c} \mathrm{Pr} > 10\\ \mbox{else} \end{array}\end{cases} \end{split}

Regime

Laminar|Turbulent

Phase

Gas

Range

1 < \mathrm{Re} < 1000000 and 0.7 < \mathrm{Pr} < 500

Type 35

This correlation by Churchill-Bernstein applies to cases of cylinder in cross flow.

(4.46)\begin{split} \mathrm{Nu}&=0.3+\frac{0.62\sqrt{\mathrm{Re}_f}{\mathrm{Pr}}^{0.3333}}{(1+(0.4/{\mathrm{Pr}})^{0.6667})^{0.25}}(1+({\mathrm{Re}_f}/282000)^{0.625})^{0.8} \\ \mathrm{Re}_f&=\mathrm{Re}\Big(\frac{T}{T_{f}}\Big)^{0.7} \\ T_{f}&=\frac{T+T_{w}}{2} \end{split}

Regime

Laminar|Turbulent

Phase

Gas

Range

\mathrm{Re}\mathrm{Pr} > 0.2

Type 36

This correlation applies to cases of flow over a flat plate.

(4.47)\begin{split} \mathrm{Nu}&=0.332\sqrt{\mathrm{Re}_f}\mathrm{Pr}^{0.3333} \\ \mathrm{Re}_f&=\mathrm{Re}(T/T_{f})^{0.7} \\ T_{f}&=\frac{T+T_{w}}{2} \end{split}

Regime

Laminar

Phase

Gas

Range

\mathrm{Pr} > 0.6

Type 37

This correlation applies to cases of flow over a flat plate.

(4.48)\begin{split} \mathrm{Nu}&=0.664\sqrt{\mathrm{Re}_f}\mathrm{Pr}^{0.3333} \\ \mathrm{Re}_f&=\mathrm{Re}(T/T_{f})^{0.7} \\ T_{f}&=\frac{T+T_{w}}{2} \end{split}

Regime

Laminar

Phase

Gas

Range

\mathrm{Pr} > 0.6

Type 38

This correlation applies to cases of flow over a flat plate.

(4.49)\begin{split} \mathrm{Nu}&=0.0296{\mathrm{Re}_f}^{0.8}\mathrm{Pr}^{0.3333} \\ \mathrm{Re}_f&=\mathrm{Re}(T/T_{f})^{0.7} \\ T_{f}&=\frac{T+T_{w}}{2} \end{split}

Regime

Turbulent

Phase

Gas

Range

0.6 < \mathrm{Pr} < 60

4.4.11.4. Free Convection (External)

Type 58

This correlation by Churchill-Chu applies to free convection from a vertical surface, or non-vertical surface if angle is less than 60^{\circ} (measured from vertical).

(4.50)\begin{split} \mathrm{Nu}&=0.68+0.67\frac{\mathrm{Ra}^{0.25}}{(1+(0.492/\mathrm{Pr})^{0.565})^{0.4444}} \\ \mathrm{Ra}&= \mathrm{Gr}\mathrm{Pr} \\ \mathrm{Gr}&=g cos(\theta) L^3(\rho_{f})^{2}\frac{|T_{w}-T|}{T_{f}\mu_{f}^{2}} \\ \mu_{f}&=\mu(T_{f}/T)^{0.7} \\ \rho_{f}&=\rho\frac{T}{T_{f}} \\ T_{f}&=\frac{T+T_{w}}{2} \end{split}

Regime

Laminar

Phase

Gas

Range

\mathrm{Ra} < 1e^{9} and \theta < {60}^{\circ}

Type 59

This correlation by Churchill-Chu applies to free convection from a vertical surface, or non-vertical surface if angle is less than 60^{\circ} (measured from vertical).

(4.51)\begin{split} \mathrm{Nu}& = 0.68+0.67\frac{\mathrm{Ra}^{0.25}}{(1+(0.492/\mathrm{Pr}_f)^{0.565})^{0.4444}} \\ \mathrm{Ra}& = \mathrm{Gr}\mathrm{Pr}_f \\ \mathrm{Gr}& = g L^3 (\rho)^{2}\frac{|T_{w}-T|}{T_{f}\mu_{f}^{2}} \\ \mathrm{Pr}_f& = \mathrm{Pr}\dfrac{\mu_{f}}{\mu} \\ \mu_{f}& = \Bigg(\mu^{-0.2661}+\frac{T_{f}-T}{37.073}\Bigg)^{-3.758} \\ T_{f}& = \frac{T+T_{w}}{2} \end{split}

Regime

Laminar

Phase

Liquid

Range

\mathrm{Ra} < 1e^{9} and \theta < {60}^{\circ}

Type 60

This correlation by Churchill-Chu applies to free convection from a vertical surface.

(4.52)\begin{split} \mathrm{Nu}&=\Big(0.825+\frac{0.387\mathrm{Ra}^{0.16667}}{(1+(0.492/\mathrm{Pr})^{0.5625})^{0.2963}}\Big)^{2} \\ \mathrm{Ra}&= \mathrm{Gr}\mathrm{Pr} \\ \mathrm{Gr}&=gL^3(\rho_{f})^{2}\frac{|T_{w}-T|}{T_{f}\mu_{f}^{2}} \\ \mu_{f}&=\mu(T_{f}/T)^{0.7} \\ \rho_{f}&=\rho\frac{T}{T_{f}} \\ T_{f}&=\frac{T+T_{w}}{2} \\ \end{split}

Regime

Laminar|Turbulent

Phase

Gas

Type 61

This correlation by McAdams applies to free convection from a horizontal surface; either the upper surface of a heated plate or the lower surface of a cooled plate.

(4.53)\begin{split} \mathrm{Nu}&=\begin{cases} \begin{array}{c} 0.54\mathrm{Ra}^{0.25}\\ 0.15\mathrm{Ra}^{0.3333} \end{array} \begin{array}{c} \mathrm{Ra}<1e^{7}\\ \mbox{else} \end{array} \end{cases} \\ \mathrm{Ra}&=\mathrm{Gr}\mathrm{Pr} \\ \mathrm{Gr}&=gL^3(\rho_{f})^{2}\frac{|T_{w}-T|}{T_{f}\mu_{f}^{2}} \\ \mu_{f}&=\mu(T_{f}/T)^{0.7} \\ \rho_{f}&=\rho\frac{T}{T_{f}} \\ T_{f}&=\frac{T+T_{w}}{2} \\ \end{split}

Regime

Laminar|Turbulent

Phase

Gas

Range

1e^{4} < \mathrm{Ra} < 1e^{11}

Type 62

This correlation by McAdams applies to free convection from a horizontal surface; either the upper surface of a cooled plate or the lower surface of a heated plate.

(4.54)\begin{split} \mathrm{Nu}&=0.27\mathrm{Ra}^{0.25} \\ \mathrm{Ra}&=\mathrm{Gr}\mathrm{Pr} \\ \mathrm{Gr}&=gL^3(\rho_{f})^{2}\frac{|T_{w}-T|}{T_{f}\mu_{f}^{2}} \\ \mu_{f}&=\mu(T_{f}/T)^{0.7} \\ \rho_{f}&=\rho\frac{T}{T_{f}} \\ T_{f}&=\frac{T+T_{w}}{2} \\ \end{split}

Regime

Laminar|Turbulent

Phase

Gas

Range

1e^{5} < \mathrm{Ra} < 1e^{10}

Type 63

This correlation by Morgan applies to free convection from a horizontal cylinder.

(4.55)\begin{split} \mathrm{Nu}&=C_1 \mathrm{Ra}^{C_2} \\ \mathrm{Ra}&=\mathrm{Gr}\mathrm{Pr} \\ \mathrm{Gr}&=gL^3(\rho_{f})^{2}\frac{|T_{w}-T|}{T_{f}\mu_{f}^{2}} \\ \mu_{f}&=\mu(T_{f}/T)^{0.7} \\ \rho_{f}&=\rho\frac{T}{T_{f}} \\ T_{f}&=\frac{T+T_{w}}{2} \\ (C_{1},C_{2})&=\begin{cases} \begin{array}{c} (0.675,0.058)\\ (1.02,0.148)\\ (0.85,0.188)\\ (0.48,0.25)\\ (0.125,0.333) \end{array} \begin{array}{c} {\mathrm{Ra}}<1e^{-2}\\ {\mathrm{Ra}}<100\\ {\mathrm{Ra}}<10000\\ {\mathrm{Ra}}<1e^{7}\\ \mbox{else} \end{array} \end{cases} \end{split}

Regime

Laminar|Turbulent

Phase

Gas

Range

1e^{-10} < \mathrm{Ra} < 1e^{12}

Type 64

This correlation by Churchill-Chu applies to free convection from a horizontal cylinder.

(4.56)\begin{split} \mathrm{Nu}&=\Big(0.6+\frac{0.387\mathrm{Ra}^{0.16667}}{(1+(0.559/\mathrm{Pr})^{0.5625})^{0.2963}}\Big)^{2} \\ \mathrm{Ra}&=\mathrm{Gr}\mathrm{Pr} \\ \mathrm{Gr}&=gL^3(\rho_{f})^{2}\frac{|T_{w}-T|}{T_{f}\mu_{f}^{2}} \\ \mu_{f}&=\mu(T_{f}/T)^{0.7} \\ \rho_{f}&=\rho\frac{T}{T_{f}} \\ T_{f}&=\frac{T+T_{w}}{2} \\ \end{split}

Regime

Laminar|Turbulent

Phase

Gas

Range

1e^{-5} < \mathrm{Ra} < 1e^{12}

Type 65

This correlation by Churchill-Chu applies to free convection from a sphere.

(4.57)\begin{split} \mathrm{Nu}&=2+\frac{0.589\mathrm{Ra}^{0.25}}{(1+(0.469/\mathrm{Pr})^{0.5625})^{0.4444}} \\ \mathrm{Ra}&=\mathrm{Gr}\mathrm{Pr} \\ \mathrm{Gr}&=gL^3(\rho_{f})^{2}\frac{|T_{w}-T|}{T_{f}\mu_{f}^{2}} \\ \mu_{f}&=\mu(T_{f}/T)^{0.7} \\ \rho_{f}&=\rho\frac{T}{T_{f}} \\ T_{f}&=\frac{T+T_{w}}{2} \\ \end{split}

Regime

Laminar|Turbulent

Phase

Gas

Range

\mathrm{Ra} > 1e^{11} and \mathrm{Pr} > 0.7