Taylor Rotating Gap

Description

Flow Simulator uses a variety of heat transfer correlations to model the heat exchange phenomenon in flow and thermal networks. The correlation types available with Flow Simulator to model heat transfer in thin rotating gaps are discussed below.

Convection through Thin Gaps

(i) Taylor Rotating Gap: Used to model heat exchange through the thin rotating fluid gap between a stationary and a rotating cylinder.

Type: Thin Gap Nu
Subtype: Taylor Rotating Gap

Table 1. Input List
Index	UI Name (.flo label)	Description	Mandatory/Not Mandatory
1	Rotor Conductor (ROT_COND)	The conductor ID that represents the rotor. On specifying a valid rotor ID, Flow Simulator automatically extracts the rotor dimensions and gap thickness.	Not mandatory. In case a valid rotor ID is not known, this input can be left as 0, and the rotor dimensions and gap thickness can be entered as other input for this correlation.
2	Stator Conductor (ROT_COND)	The conductor ID that represents the stator. On specifying a valid rotor ID, Flow Simulator automatically extracts the rotor dimensions and gap thickness.	Not mandatory. In case a valid stator ID is not known, this input can be left as 0, and the rotor dimensions and gap thickness can be entered as other input for this correlation.
3	Laminar-Transition Taylor No. (TA_LAM)	Taylor number where the laminar regime of flow ends, and the transitional regime starts.	Not mandatory. In AUTO mode, Flow Simulator chooses this value directly from the references.
4	Transition-Turbulent Taylor No. (TA_TUR)	Taylor number where the transitional regime of flow ends and the turbulent regime starts.	Not mandatory. In AUTO mode, Flow Simulator chooses this value directly from the references.
5	Fluid Total Pressure (TOT_PRESS)	Pressure of the fluid in the gap.	Not mandatory. In Auto mode, Flow Simulator uses the pressure of the fluid chamber attached to the convector. If the convector is not attached to a fluid chamber, the pressure will be 14.7 PSIA.
6	Gap Thickness (GAP_THK)	Thickness of the gap between the rotor and stator.	Mandatory. If you have provided a valid rotor and stator conductor ID, which can be used to retrieve and calculate the gap thickness, this can be left as AUTO. If you have left the rotor or stator conductor input as 0, then you must enter a gap thickness.
7	Rotor Radius (ROT_RADIUS)	Rotor outer radius.	Mandatory. If you have provided a valid rotor conductor ID, which can be used to retrieve the rotor outer radius, this can be left as AUTO. If you have left the rotor conductor input as 0, then you must enter a rotor outer radius.
8	HTC Multiplier (HTC_MULT)	A constant multiplier to scale the value of the heat transfer coefficient obtained from the correlation.	Not mandatory. Default value is 1.0.
9	MODE (MODE)	1 = Solid to Solid, convector is connected to two thermal nodes that represent solid surfaces. 2 = Solid to Fluid, convector is connected to one thermal node representing the fluid in the gap or one fluid chamber.	Not mandatory. In Auto mode, Flow Simulator determines the MODE based on the thermal nodes or fluid chamber attached to the convector.
10	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.	Not mandatory. If AUTO, FREE_HTC = 2.
11	Free Mixing Sign (FREE_ASSIST)	The sign of the free and forced HTC blending. Assist (positive). Oppose (negative).	Not mandatory. If AUTO, FREE_ASSIST = 1.
12	Free Length Scale (FREE_LEN)	The length scale for the free convection HTC calculation.	Not mandatory. If AUTO, FREE_LEN = LENGTH.
13	Horizontal Free Surface Dir (FREE_SURF_DIR)	Direction of the horizontal plate that is used if FREE_HTC=3. Up or radially out. Down or radially in.	Mandatory.
14	Rotor Slots Number (ROT_SLT_NUM)	Number of slots on the rotating cylindrical surface.	Mandatory. Enter 0 for no slots on the rotor.
15	Rotor Slots Width at Gap (ROT_SLT_WDTH)	The arc length of the slot at the gap.	Mandatory if ROT_SLT_NUM > 0.
16	Rotor Slots Depth (ROT_SLT_DPTH)	The depth of the slot in the radial direction.	Mandatory if ROT_SLT_NUM > 0.
17	Stator Slots Number (STAT_SLT_NUM)	Number of slots on the stator cylindrical surface.	Mandatory. Enter 0 for no slots on the stator.
18	Stator Slots Width at Gap (STAT _SLT_WDTH)	The arc length of the slot at the gap.	Mandatory if STAT_SLT_NUM > 0
19	Stator Slots Depth (STAT _SLT_DPTH)	The depth of the slot in the radial direction.	Mandatory if STAT_SLT_NUM > 0.

Formulation

Expression for the Taylor number, for the flow between two concentric cylinders, is given by (Ref 1):

T a = \frac{v_{r}}{µ} \sqrt{\frac{l_{a g}^{3}}{r_{o r}}}

Where,

Expression for the Nusselt number, based on the Taylor number, is provided as (Ref. 1):

Table 2.
$Nu = 2$	$Ta < 41,$ laminar…
$Nu = 0.212 {Ta}^{0.63} {Pr}^{0.27}$	$100 \geq Ta \geq 41$ vortex…
$Nu = 0.386 {Ta}^{0.5} {Pr}^{0.27}$	$100 < Ta$ turbulent…

H T C = \frac{N u * k}{2 * l_{a g}} w h e r e k = f l u i d c o n d u c t i v i t y a t f i l m t e m p e r a t u r e

µ

= dynamic viscosity of air

$v_{r}$ = peripheral speed of the rotor

$r_{or}$ = rotor outer radius

$l_{ag}$ = thickness of the air gap

If MODE=2, Solid to Fluid, the Nu is doubled to account for two convectors that are needed to simulate the heat transfer between the two solid surfaces.

Slots

If the rotor and/or stator surface is not smooth (slots or saliency), then the HTC can be affected. In-house CFD studies, as well as trends in other references, were used to determine a simple multiplier to the smooth the surface HTC. The multiplier is based on qualitative rather than quantitative evidence due to the lack of public references and the complexity of the flow field. If this correlation does not correspond to your experience, enter 0 for the slot number and use an HTC multiplier input.

The HTC multiplier for slots is based on:

Taylor number.
Smooth surface versus slotted surface perimeter ratio (smooth ratio = 1 for no slots, minimum of 0.5).
Slot aspect ratio (depth/width, maximum of 1).

The HTC multiplier calculation steps:

Calculate the Taylor number as usual.
Get the HTC multiplier for smooth ratio = 0.5.
Linearly scale the HTC multiplier for the actual smoothness fraction.
Linearly scale again for the slot aspect ratio. HTC multiplier approaches 1 as the aspect ratio approaches 0.

Perform steps 1-4 if the rotator and stator slots. The final HTC is the HTC multiplier that is furthest from 1.

This graph shows the data used for the HTC multiplier. Notice that the multiplier is less than 1 for laminar flow and greater than 1 for turbulent flow. The multiplier for slots on the rotor is twice as large as the multiplier for slots on the stator.

Table 3. 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	TA_FLOW	Solver-calculated Taylor number for the model.
4	PRESSURE	User-defined fluid pressure.
5	GAP_THK	Solver calculated/auto-retrieved gap thickness for the model.
6	ROT_RADIUS	Solver calculated/auto-retrieved rotor radius for the model.
7	NU	Calculated Nusselt number.
8	HTC	Heat transfer coefficient calculated as per the Taylor Rotating Gap correlation (Ref.1).

Heat Transfer Correlation References

Dave Staton, Aldo Boglietti, Andrea Cavagnino, "Solving the More Difficult Aspects of Electric Motor Thermal Analysis", IEEE Transactions on Energy Conversion, vol. 20, no. 3, pp. 620-628, Sept. 2005, doi: 10.1109/TEC.2005.847979. Eqn. 14-17.
Gazley, C.: "Heat Transfer Characteristics of rotating and axial flow between concentric cylinders", Trans ASME, Jan 1958, pp.79-89.