Incropera, Laminar and Turbulent

Description

The flat plate correlation, based on Incropera (ref 1), 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. The HTC can be at a local distance from the plate's leading edge or it can be the average HTC over the plate length. This correlation differs from the other flat plate correlations since it can also include blending with a free convection heat transfer coefficient.

Type: BI_FLAT_PLATE_NU
Subtype: INCP_FLPLATE_BI

Table 1. Inputs List
Index	UI Name (.flo label)	Description
1	Velocity Type (VEL_TYPE)	Method to get a velocity for the Reynolds number. Use Chamber Velocity (convector must be attached to a flow chamber). Use Velocity Input. If AUTO, the chamber velocity will be used if the convector is attached to a flow chamber, and user input velocity will be used if the convector is not attached to a flow chamber.
2	Chamber for Velocity (VEL_CH)	ID for the low chamber that will be used for the velocity. If AUTO, the flow chamber ID attached to the convector will be used.
3	Constant Velocity (FLOW_VEL)	The flow velocity if VEL_TYPE=2 or VEL_TYPE=AUTO, and the convector is not attached to a chamber.
4	Length Scale (LENGTH)	The distance back from the plate's leading edge for the local HTC, or the entire length of the plate for an average HTC.
5	Local or Average HTC (LOC_AVG)	Type of equation to use. Local. Average.
6	Transition Re (RE_TRANS)	The Reynolds Number where the transition from laminar to turbulent flow occurs. If AUTO, RE_TRANS=500,000.
7	HTC Multiplier (HTC_MULT)	A constant multiplier to scale the value of the heat transfer coefficient obtained from the correlation.
8	Free Convection Nu (FREE_HTC)	The equation to use for free convection blending. None (do not calculate free convection HTC). McAdams Vertical Plate. Horizontal Plate. Churchill-Chu Horizontal Cylinder. If AUTO, FREE_HTC=2.
9	Free Mixing Sign (FREE_ASSIST)	The sign of the free and forced HTC blending. Assist (positive). Oppose (negative). If AUTO, FREE_ASSIST=1.
10	Free Length Scale (FREE_LEN)	The length scale for the free convection HTC calculation. If AUTO, FREE_LEN = LENGTH
11	Horizontal Free Surface Dir (FREE_SURF_DIR)	Direction of horizontal plate that is used if FREE_HTC=3 Up or radially out. Down or radially in.

Formulation

This correlation uses a Nusselt Number equation by found in Incropera (ref 1).

For Re < RE_TRANS (~500,000)

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

$N u_{l o c a l_l a m i n a r} = \frac{N u_{a v e r a g e_l a m i n a r}}{2}$

For Re > RE_TRANS (~500,000) with a portion of laminar flow also on the plate

$A = .037 * R E_T R A N S^{0.8} - .664 * R E_T R A N S^{0.5}$

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

For all turbulent flow on the plate (boundary layer tripped near leading edge)

$N u_{a v e r a g e_t u r b u l e n t} = 0.037 * {(\frac{ρ * V * L}{μ})}^{0.8} * P r^{0.333}$

For all turbulent flow at the local point on the plate

$N u_{l o c a l_t u r b u l e n t} = 0.0296 * {(\frac{ρ * V * L}{μ})}^{0.8} * P r^{0.333}$

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, 0.6<Pr<60

Table 2. Output List
Index	.flo label	Description
1	TNET	Thermal network ID which has the convector where this correlation is used.
2	CONV_ID	Convector ID which is using this correlation.
3	VELOCITY	Flow velocity along the plate.
4	LENGTH	Plate length.
5	RE	Reynolds number.
6	REGIME	Laminar or turbulent.
7	TYPE	Local or average.
8	Nu	Nusselt number.
9	HTC	Calculated Heat Transfer Coefficient.

Heat Transfer Correlation References

Incropera, F. and Dewitt, D. Fundamentals of Heat and Mass Transfer, 6th Edition, John Wiley & Sons, 2006.