Heat Transfer Coefficients (HTC) Correlations

The following sections describe the available heat transfer correlations (HTCs) in Flow Simulator. Correlation equations and literature references are included. All HTC options appear in the 1D-Thermal Convection Resistor. In addition, the duct flow HTC options appear in the Tube/Pipe flow elements and the Advance Orifice flow element.


Nomenclature:	Subscripts:
Nu: Nusselt Number	lam: Laminar Regime
Re: Reynolds number	turb: Turbulent Regime
Pr: Prandtl multiplier	tran: Transition Regime
: Dynamic Viscosity

Heat Transfer Correlations:

Lapides-Goldstein:

Dittus-Boelter:

Sieder-Tate:

Gnielinski:

Bhatti-Shah:

External Heat Transfer:

The following section lists cross flow convection and free convection configurations that are available in Flow Simulator. These options are primarily used by connecting a 1D Thermal Convection Resistor to an internal chamber in the flow path.

Colburn (Plate in Cross Flow):

Incropera (Plate in Cross Flow)

The flat plate correlation based on Incropera (reference 4) calculates an average Nusselt number for the laminar flow over the entire plate, or the mixed laminar and turbulent flow over the plate. The plate is assumed to have a constant surface temperature.

For Re < ~500,000:

N u_{a v e r a g e_l a m i n a r} = 0.664 * {(\frac{ρ * V * L}{μ})}^{0.5} * P r^{0.333}

For Re > ~500,000:

N u_{a v e r a g e_m i x} = [0.037 * {(\frac{ρ * V * L}{μ})}^{0.8} - 871.0] * P r^{0.333}

Where:

$L = P l a t e l e n g t h i n c r o s s f l o w d i r e c t i o n$

$V = f l u i d c r o s s f l o w v e l o c i t y$

$ρ = f l u i d f i l m d e n s i t y$

$μ = f l u i d f i l m v i s c o s i t y$

P r = f l u i d P r a n d t l N u m b e r

Note: Valid for 10 < Re < 10⁸, Pr > 0.6

Churchill-Bernstein (Cylinder in Cross Flow):

McAdams (Vertical Prism in Free Convection):

Equation for the low Rayleigh number, 10⁴ < Ra < 10⁹

N u = .59 * R a^{.25}

Equation for the high Rayleigh number, 10⁹ < Ra < 10¹³

N u = .13 * R a^{.3333}

Linear interpolation between these two equations for Ra ~ 10⁹.

Where:

Ra: Rayleigh number: Gr * Pr

Gr: Grashof number

G r = H^{3} * {(\frac{ρ}{μ})}^{2} * G R A V * β * a b s (T_{w a l l} - T_{f l u i d})

H: Height of the vertical plate or cylinder

$ρ$ : Fluid bulk density

$μ$ : Fluid film viscosity

GRAV: The Earth's gravity for non-rotating; centripetal acceleration, $ω^{2} * r$ , for rotating

$β$ : Fluid bulk compressibility factor

Horizontal Plate in Free Convection:

“Stable” Configuration

Stationary hot surface facing down.
Stationary cold surface facing up.
Rotating hot surface facing out.
Rotating cold surface facing in.

N u = 0.27 * {(G r * P r)}^{0.25}

Note: Valid for 10⁵ < (Gr*Pr) < 10¹⁰

“UnStable” Configuration

Stationary hot surface facing up.
Stationary cold surface facing down.
Rotating hot surface facing in.
Rotating cold surface facing out.

N u = 0.54 * {(G r * P r)}^{0.25}

Note: Valid for 10⁴ < (Gr*Pr) < 10⁷

N u = 0.15 * {(G r * P r)}^{0.333}

Note: Valid for 10⁷ < (Gr*Pr) < 10¹¹

Linear transition when Gr*Pr is close to 10⁷.

Where:

G r = L^{3} * {(\frac{ρ}{μ})}^{2} * F * β * (T_{w a l l} - T_{f l u i d})

$G r$ = Grashof number

$L$ = Length scale of the flat plate, usually area/perimeter

$ρ$ = Fluid bulk density

$μ$ = Fluid film viscosity

$F$ = Force, gravity force for stationary and centripetal for rotating

$β$ = Fluid bulk compressibility factor

Churchill-Chu(Horizontal Cylinder in Free Convection):

Where:

Gr: Grashof number

D_outer: Cylinder outside diameter

ρ: Fluid bulk density

μ: Fluid film viscosity

GRAV: Earth Gravity

β: Fluid bulk compressibility factor

Note: Valid for 10⁶ < (Gr*Pr) < 10¹²

Turbulent Duct Flow Entrance Effects

The following entrance effects correlations compute an HTC multiplier called Hm. HTC multiplier has limits and must be greater than unity. The Local option resolves heat transfer correctly for all x locations along the axial length of a pipe. The Averaged options use total pipe length L as the input. They correctly predict overall heat transfer, but not local temperature variation.

Abrupt Contraction, Local x

Uniform Bellmouth, Local x

Abrupt Contraction, Averaged L

Uniform Bellmouth, Averaged L

Uniform Blend, Local x

Uniform Blend, Averaged L

Laminar Duct Flow HTC correlations

The following Laminar HTC options are available in Flow Simulator. It is worth noting that Laminar HTCs depend on distance from inlet x, and thus do not need a special Entrance Effects multiplier.

Muzychka-Yovanovich:

Hausen: