Advanced Tube Element

Advanced Tube Description and Quick Guide

The advanced tube (AT) is a compressible flow element. It has many of the same inputs as the Incompressible Tube element but obviously the flow solution is different due to compressibility effects. The AT element could be used instead of the standard compressible tube element in models using a gas.

Figure 1.

Some of the benefits of AT element over the standard compressible tube include the ability to divide the element walls into multiple circumferential segments and to use more than 20 axial segments. Each axial and circumferential segment of the AT can have unique friction and heat transfer inputs. The AT element can also calculate fluid swirl and windage changes for rotating annular passages or cylinders with the tube axis the same as the rotation axis.

The advanced tube element includes rotation (or gravity), wall friction, area change, and heat transfer effects. The advanced tube has a length and number of stations inputs like a standard tube. The wall friction and heat transfer effects use the length to determine the loss and temperature change. The losses due to sudden area change do not account for separated flow in this element. Use an expansion/contraction element to get losses when flow may separate due to the area change.

Flow in the advanced tube element is limited to subsonic. The maximum Mach number will be 1.0 and occur at the element exit station for a constant area duct. The maximum Mach number will occur at the minimum area for a duct with changing area. Use the Supersonic Tube element for converging-diverging geometries when the Mach number can exceed 1.

Advanced Tube Element Inputs

Table of the inputs for the advanced tube element.

Element Specific Input Variables
Index UI Name (.flo label) Description
1 Cross-Sectional Shape

(CS_SHAPE)

Specifies method for defining AT geometry parameters: hydraulic diameter (DHI), wetted perimeter (PWI), and flow area (AFI). The cross-section shape is also used to find an effective hydraulic diameter.

The following typical settings are set automatically by the GUI based on the selections made for Geometric Input Type and Size.

1: Circular AT with uniform area along the length of the AT. A single value for area is set in STATION_CS_AREAS

2: Circular AT with uniform diameter along the length of the AT. A single value for diameter is set in STATION_HYDDIAMS_OR_PERIMS

6: Arbitrary cross-sectional shape with uniform area and hydraulic diameter along the length of the AT. A single value for area is set on STATION_CS_AREAS and a single value for hydraulic diameter is set on STATION_HYDDIAMS_OR_PERIMS

7: Arbitrary cross-sectional shape with uniform area and wetted perimeter along the length of the AT. A single value for area is set on STATION_CS_AREAS and a single value for wetted perimeter is set on STATION_HYDDIAMS_OR_PERIMS

11: Tapered circular AT with area specified at inlet and exit. Assumes linear tapering of area. Inlet and exit area values are set on STATION_CS_AREAS

12: Tapered circular AT with diameter specified at inlet and exit. Assumes linear tapering of diameter. Inlet and exit diameter values are set on STATION_HYDDIAMS_OR_PERIMS

16: Tapered arbitrary cross-section shaped AT with area and hydraulic diameter specified at inlet and exit. Assumes linear tapering of area and hydraulic diameter. Inlet and exit area values are set on STATION_CS_AREAS. Inlet and exit hydraulic diameter values are set on STATION_HYDDIAMS_OR_PERIMS.

17: Tapered arbitrary cross-section shaped AT with area and wetted perimeter specified at inlet and exit. Assumes linear tapering of area and wetted perimeter. Inlet and exit area values are set on STATION_CS_AREAS. Inlet and exit wetted perimeter values are set on STATION_HYDDIAMS_OR_PERIMS.

21: Circular AT with area specified each station. Assumes linear tapering of area between stations. NUM_STATIONS number of area values are set on STATION_CS_AREAS

22: Circular AT with diameter specified each station. Assumes linear tapering of diameter between stations. NUM_STATIONS number of diameter values are set on STATION_HYDDIAMS_OR_PERIMS

26: Arbitrary cross-section shaped AT with area and hydraulic diameter specified each station. Assumes linear tapering of area and hydraulic diameter between stations. NUM_STATIONS number of area values are set on STATION_CS_AREAS. NUM_STATIONS number of hydraulic diameter values are set on STATION_HYDDIAMS_OR_PERIMS

27: Arbitrary cross-section shaped AT with area and wetted perimeter specified each station. Assumes linear tapering of area and wetted perimeter between stations. NUM_STATIONS number of area values are set on.

There are additional settings for triangle, rectangle, elliptical and annular shapes.

2 Number of Stations

(NUM_STATIONS)

Number of stations in the AT.

Station 1 is at the inlet plane; station NUM_STATIONS is at the exit plane. NSTA can range from 2 to unlimited. During the solution process, the AT will be discretized into NUM_STATIONS minus one segments with average temperatures, pressures, and Reynolds Numbers for each segment.

An advanced tube should be modelled with at least 2 stations. The number of stations can have a big impact on the convergence of attached chambers. If a chamber attached to this element is having difficulty converging, try increasing the number of stations. Of course, analysis speed may decrease as the number of stations increase.

3 Number of AT wall sides

(NUM_WALL_SIDES)

  • Number of circumferential wall segments around the AT. Wall segments can have their own loss mode, heat transfer mode, roughness definition, and turbulator definitions (heat transfer, friction, geometry).
4 Number of bends

(NUM_BENDS)

Number of bends in the AT. Each bend is defined by a bend radius, bend angle, location in the AT (either distance from start of AT, or a specified straight segment length between bends), loss multiplier and combination angle.
5 Total Length

(LENGTH)

Length of the AT in inches. Do not include length within bends unless the bend losses are not otherwise accounted.
6 Station Length Fraction

(STATION_MODE)

Flag specifying station location definitions.

0: Stations are uniformly distributed along the length of the AT

1: Station location is defined as a percentage of length of the AT. User specifies percentage for each station in the STATION_LOCATIONS array. Valid values are 0-100.

2: Station location is defined as distance from the start of the AT. User specifies distance from start for each station in the STATION_LOCATIONS array. Valid values are 0-LENGTH

7 Turbulent Friction Relation

(FRIC_RELATION)

A flag that specifies which friction relation is used.

0.0: Smooth Wall Power law (Abuaf)

1.0: Swamee-Jain (approx. to Colebrook-White)

Swamee-Jain (1.0) is recommended for non-zero roughness.

8 Roughness type

(ROUGH_TYPE)

Flag specifying measurement method of user-input ROUGHNESS value. Roughness values will be converted to sand-grain roughness equivalent. For more information see Friction Correlations section in General Functions and Routines.

0: Equivalent sand-grain roughness

1: Average absolute roughness

2: Root mean square roughness

3: Peak-to-valley roughness

9 HTC Relation

(HTC_RELATION)

The “Duct Flow” Nu correlation used for turbulent flow.

See the “HTC Correlations” in the “General Functions and Routines” section for the equations.

-2) User Input Nu

-1) User Input HTC

1) Lapides-Goldstein

2) Dittus-Boelter

3) Sieder-Tate Combo

4) Gnielinski Combo

5) Bhatti-Shah

7) Sieder-Tate Turbulent Only

8) Gnielinski-Turbulent Only

9) Rotating Parallel Duct (Morris)

41) Rotating Shaft (Seghir-Ouali and Gai)

10 Heat Transfer Inlet Effects

(HT_INLET_EFF)

Flag specifying heat transfer inlet effects applied for the AT.

See the Heat Transfer Coefficients (HTC) section in General Functions and Routines.

0: No inlet effects

1: Abrupt local or uniform average inlet effects

3: Abrupt average inlet effects

4: Uniform local inlet effects

5: Between uniform average and local inlet effects

6: Between abrupt average and local inlet effects

11 Portion of Ustrm Cham. Dyn. Head Lost

(DQ_IN)

Inlet dynamic head loss. Valid range is 0.0 to 1.0 inclusive. An entry outside this range will cause a warning message and the value used will be 0 or 1 (whichever value is closest to the entry).

If DQ_IN > 0 and the upstream chamber has a positive component of relative velocity aligned with the centerline of the tube, the driving pressure will be reduced by the equation:

P i n = P s u p s t r e a m + ( 1.0 D Q I N ) * ( P i n n o l o s s P s u p s t r e a m ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbGaamyAaiaad6gacqGH9aqpcaWGqbGaam4Ca8aadaWgaaWc baWdbiaadwhacaWGWbGaam4CaiaadshacaWGYbGaamyzaiaadggaca WGTbaapaqabaGcpeGaey4kaSYaaeWaa8aabaWdbiaaigdacaGGUaGa aGimaiabgkHiTiaadseacaWGrbWdamaaBaaaleaapeGaamysaiaad6 eaa8aabeaaaOWdbiaawIcacaGLPaaacaGGQaWaaeWaa8aabaWdbiaa dcfacaWGPbGaamOBa8aadaWgaaWcbaWdbiaad6gacaWGVbGaeyOeI0 IaamiBaiaad+gacaWGZbGaam4CaaWdaeqaaOWdbiabgkHiTiaadcfa caWGZbWdamaaBaaaleaapeGaamyDaiaadchacaWGZbGaamiDaiaadk hacaWGLbGaamyyaiaad2gaa8aabeaaaOWdbiaawIcacaGLPaaaaaa@63F1@

(Default value = 0.)

12 Rotor Index

(RPMSEL)

Element rotational speed pointer.

-1: Rotates with air.

0.0: Specifies a stationary element.

1.0: Rotor 1, RPM = general data ELERPM(1).

2.0: Rotor 2, RPM = general data ELERPM(2).

3.0: Rotor 3, RPM = general data ELERPM(3).

13 Gravity Multiplier

(GRAV_MULT)

Multiplier on the constant for acceleration due to gravity, Gc. Gc is nominally equal to 32.17405 lbm-ft/lbt-sec2. Gravity effects only available when AT is stationary.
14 Element Inlet Orientation: Tangential Angle

(THETA)

Angle (deg) between the element centerline at the entrance of the element and the reference direction.

If the element is rotating or directly connected to one or more rotating elements, the reference direction is defined as parallel to the engine centerline and the angle is the projected angle in the tangential direction. Otherwise, the reference direction is arbitrary but assumed to be the same as the reference direction for all other elements attached to the upstream chamber.

Theta for an element downstream of a plenum chamber has no impact on the solution except to set the default value of THETA_EX.

(See also THETA_EX)

15 Element Inlet Orientation: Radial Angle

(PHI)

Angle (deg) between the element centerline at the entrance of the element and the THETA direction. (spherical coordinate system)

Phi for an element downstream of a plenum chamber has no impact on the solution except to set the default value of PHI_EX.

(See also PHI_EX)

16

17

18

Exit K Loss:

Axial (K_EXIT_Z)

Tangential (K_EXIT_U)

Radial (K_EXIT_R)

Head loss factors in the Z, U, and R directions based on the spherical coordinate system of theta and phi.

Z = the axial direction. (theta=0 and phi=0)

U = the tangential direction. (theta=90 and phi=0)

R = the radial direction. (theta=0 and phi=90)

Valid values of K_EXIT_i (i = Z, U, R) range from zero (default) to one.

The three loss factors reduce the corresponding three components of velocity exiting the element.

V a c t u a l   e x i t   i d i r = V n o l o s s   e x i t   i d i r * 1 K _ E X I T _ i d i r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGwbWdamaaBaaaleaapeGaamyyaiaadogacaWG0bGaamyDaiaa dggacaWGSbGaaiiOaiaadwgacaWG4bGaamyAaiaadshacaGGGcGaam yAaiaadsgacaWGPbGaamOCaaWdaeqaaOWdbiabg2da9iaadAfapaWa aSbaaSqaa8qacaWGUbGaam4BaiaadYgacaWGVbGaam4Caiaadohaca GGGcGaamyzaiaadIhacaWGPbGaamiDaiaacckacaWGPbGaamizaiaa dMgacaWGYbaapaqabaGcpeGaaiOkamaakaaapaqaa8qacaaIXaGaey OeI0Iaam4saiaac+facaWGfbGaamiwaiaadMeacaWGubGaai4xaiaa dMgacaWGKbGaamyAaiaadkhaaSqabaaaaa@64E5@

(Default value provides no loss, K_EXIT_i=0)

19 Element Exit Orientation: Tangential Angle

(THETA_EX)

Angle (deg) between the element exit centerline and the reference direction.

THETA_EX is an optional variable to be used if the orientation of the element exit differs from that of the element inlet.

The default value (THETA_EX = -999) will result in the assumption that THETA_EX = THETA.

Other values will be interpreted in the manner presented in the description of THETA.

20 Element Exit Orientation: Radial Angle

(PHI_EX)

Angle (deg) between the element exit centerline and the THETA_EX direction.

PHI_EX is an optional variable to be used if the orientation of the element exit differs from that of the element inlet.

The default value (PHI_EX = -999) will result in the assumption that PHI_EX = PHI.

Other values will be interpreted in the manner presented in the description of PHI.

21 Nusselt Number for Laminar Flow

(NU_LAM)

Nusselt number used in the laminar flow region (defaults to 4.36)
22 (RE_POW) Not Used
23 (MDOT_REF) Not Used
24 Pressure Tolerance Value

(PRESSURE_TOL)

User defined inlet total pressure convergence tolerance.

Caution should be used when increasing this very much above the default value.

The total pressure calculated from the station marching from exit to inlet must match the total pressure of the upstream chamber minus any losses (PTIN)

Defaults to 0.000001 psia.

25 Mach Number Tolerance Value

(MACH_TOL)

User defined convergence tolerance for a station mass flows. Mass flow continuity is achieved by changing station Mach numbers until convergence criteria is met.

Defaults to 0.000001 lbm/sec

26 Aspect Ratio

(ASPECT_RATIO)

The aspect ratio of the AT cross section for an Arbitrary-Shape cross section. The aspect ratio should be between 0 and 1.
27 Laminar Friction Effects

(LAMR_FRIC_RLTN)

Laminar friction effects to use at the duct inlet. See the “Friction Correlations” in the “General Functions and Routines” section for the equations.

0) Off, assume fully developed laminar flow.

1) Muzychka-Yovanovich

28 Laminar HTC Relation

(NU_LAM_METHOD)

The “Duct Flow” Nu correlation used for laminar flow.

See the “HTC Correlations” in the “General Functions and Routines” section for the equations.

0) User Input Nu

1,4,5) Muzychka-Yovanovich

2) Hausen

29 No GUI Input

(PROPS_METHOD)

Temperature to be used for fluid properties retrieval.
  • Properties at the local temperature (default)
  • Properties at average bulk fluid temperature (T_inlet+T_outlet)/2
31 Inner Diameter (ID) Rotor Index

(RPMSEL_INNER)

The rotation of the inner diameter wall of an annulus.

Used only for ROTATION_METH=1, Rotating Annulus.

0.0: Specifies a stationary inner annulus wall.

1.0: Rotor 1, RPM = general data ELERPM(1).

2.0: Rotor 2, RPM = general data ELERPM(2).

3.0: Rotor 3, RPM = general data ELERPM(3).

32 Inlet Head Loss Type

(K_IN_METHOD)

The type of inlet losses.

K – Incompr Loss Coef

Kin = inlet head loss / dynamic head,

or K i n = P T , s u p p l y P T , a f t e r   l o s s 0.5   ρ   V 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGlbWdamaaBaaaleaapeGaamyAaiaad6gaa8aabeaak8qacqGH 9aqpdaWcaaWdaeaapeGaamiua8aadaWgaaWcbaWdbiaadsfacaGGSa Gaam4CaiaadwhacaWGWbGaamiCaiaadYgacaWG5baapaqabaGcpeGa eyOeI0Iaamiua8aadaWgaaWcbaWdbiaadsfacaGGSaGaamyyaiaadA gacaWG0bGaamyzaiaadkhacaGGGcGaamiBaiaad+gacaWGZbGaam4C aaWdaeqaaaGcbaWdbiaaicdacaGGUaGaaGynaiaacckacqaHbpGCca GGGcGaamOva8aadaahaaWcbeqaa8qacaaIYaaaaaaaaaa@5889@

1) Cd – Compr Loss Coef

K i n = ( 1.0 C d ) 2 1.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGlbWdamaaBaaaleaapeGaamyAaiaad6gaa8aabeaak8qacqGH 9aqpdaqadaWdaeaapeWaaSaaa8aabaWdbiaaigdacaGGUaGaaGimaa WdaeaapeGaam4qaiaadsgaaaaacaGLOaGaayzkaaWdamaaCaaaleqa baWdbiaaikdaaaGccqGHsislcaaIXaGaaiOlaiaaicdaaaa@4438@

2) Upstream Cross Flow

The inlet loss is assumed to be due to a cross flow velocity at the inlet and is calculated according to Ref. 3.

The data used is for a tube with L/D > 2.83, where the flow has recovered from the inlet effects and is at right angles to the cross flow.

The method is valid for both upstream momentum and upstream inertial chambers. It is not valid for upstream plenum chambers.

33-35 (FUTURE) Reserved for future development
36 Bend Input Mode

(BEND_INPUT_MODE)

Flag indicating how the locations of bends are defined.

0: Bend location is distance from start of the AT in inches.

1: Bend location is defined by the distance of straight AT length between the end of the previous bend (or inlet, if it is the first bend) and the beginning of the current bend

37 Laminar-to-Transition Reynolds Number

(RE_LAM)

Reynolds number below which flow is assumed to be laminar
38 Transition-to-Turbulent Reynolds Number

(RE_TURB)

Reynolds number above which flow is assumed to be turbulent. Flow at Reynolds numbers between RE_LAM and RE_TURB are assumed to be in the transition region.
39 Friction Type

(FRIC_TYPE)

Friction factor output type.

1: Darcy friction factor

2: Fanning friction factor

40 Starting Length

(STLEN)

Starting length (in) used for the HTC inlet multiplier.

Used to modify X in calculating hx/ho:

X = X m e a s + S T L E N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGybGaeyypa0Jaamiwa8aadaWgaaWcbaWdbiaad2gacaWGLbGa amyyaiaadohaa8aabeaak8qacqGHRaWkcaWGtbGaamivaiaadYeaca WGfbGaamOtaaaa@4208@

where Xmeas = Distance from AT inlet (station 1).

At station 1, X equals STLEN. If one physical AT is modelled as two or more elements strung together, STLEN should include the cumulative length of all AT elements leading into the current element unless something physical re-establishes the boundary layer.

41 Inlet Head Loss

(K_INLET)

Inlet head loss.

This is either a K or Cd depending on K_IN_METHOD.

42-45 (FUTURE) Reserved for future development
46 Rotation Method

(ROTATION_METH)

Method for applying element rotation effects. Option 0 is the traditional method where all element walls are rotating at the same speed and the fluid is rotating at same speed as the walls. Options 1 and 2 calculate swirl and windage change along the length of the tube.

0: Stationary or Forced to Rotor Speed

1: Rotating Annulus

2: Rotating Coaxial Shaft

A1 Bend Radius

(BEND_RADIUS)

(array of 20 values in 4 lines of 5 each, NUM_BENDS values used)

Radius (in) of bend along AT centerline.

(See ‘ r ’ from graphic for COMBINATION_ANGLE below)

Each bend counted in NUM_BENDS must have a specified bend radius. Enter bends in order based on distance from beginning of AT (DIST_FR_STRT).

A2 Bend Angle

(BEND_ANGLE)

(array of 20 values in 4 lines of 5 each, NUM_BENDS values used)

Angle (deg) between the entering and exiting lengths of the bend.

Each bend must have a specified angle.

The maximum bend angle is 180°.

A3 Distance from Start

(DISTANCE)

(array of 20 values in 4 lines of 5 each, NUM_BENDS values used)

Array of values that indicate bend location in the AT. The interpretation of the values in the array depends on the value of BEND_INPUT_MODE:

BEND_INPUT_MODE = 0: DISTANCE is cumulative straight AT segment length (in.) to the start of the current bend,

BEND_INPUT_MODE = 1: DISTANCE is straight segment length (in.) from end of the previous bend (or inlet if it is the first bend in the AT) to the start of the current bend.

A4 Loss Multiplier

(LOSS_MULT)

(array of 20 values in 4 lines of 5 each, NUM_BENDS values used)

Loss multiplier for each bend (Default = 1.0).
A5 Combination Angle

(COMBINATION_ANGLE)

(array of 20 values in 4 lines of 5 each, NUM_BENDS minus one values used)

Relative angle (deg) between two bends in series.

The number of COMBINATION_ANGLE entries will be NUM_BENDS – 1. The first entry will be the combination angle between bends 1 and 2.

A COMBINATION_ANGLE of 0 degrees defines an ‘S’ shaped bend and 180 degrees defines a ‘U’ shaped bend.

Figure 2. Notation for combination of two 90° bends (Ref 5, pg. 228)

The allowable range is 0 to 180 degrees.

A6 STATION_LOCATIONS

(Dynamic array of NUM_STATIONS values)

Dynamic array of NUM_STATIONS station locations, specified as a percentage of length, a fraction of length, or a distance from start of AT, depending on the value of STATION_MODE. For STATION_MODE = 0 (uniformly distributed), this array will not be used.
A7 STATION_RADII

(Dynamic array of NUM_STATIONS values)

Dynamic array of NUM_STATIONS station radii in inches. Only applicable for rotating ATs, that is, RPMSEL not equal to 0.
A8 STATION_HEIGHTS

(Dynamic array of NUM_STATIONS values)

Dynamic array of NUM_STATIONS station heights in inches. Only applicable for stationary ATs, that is, RPMSEL = 0, and when gravitational effects are enabled. Station height is defined as distance above a datum and is used to determine gravitational effects on the fluid.
A9 Station cross-section areas

(STATION_CS_AREAS, dynamic array, length depends on value of CS_SHAPE)

Dynamic array of station cross-section areas. Length of the array is determined by the value of CS_SHAPE. STATION_CS_AREAS is only used for CS_SHAPE = 1, 6, 7, 11, 16, 17, 21, 26, or 27
CS_SHAPE Description Form of CS_AREAS
1 Circular AT with uniform area 1 value = AT area
6 Arbitrary cross-section AT with uniform area and hydraulic diameter 1 value = AT area
7 Arbitrary cross-section AT with uniform area and wetted perimeter 1 value = AT area
11 Tapered circular AT with area specified at inlet and exit 2 values = inlet and exit area
16 Tapered arbitrary cross-section shaped AT with area and hydraulic diameter specified at inlet and exit 2 values = inlet and exit area
17 Tapered arbitrary cross-section shaped AT with area and wetted perimeter specified at inlet and exit 2 values = inlet and exit area
21 Circular AT with area specified at each station NUM_STATIONS values = station area
26 Arbitrary cross-section shaped AT with area and hydraulic diameter specified at each station NUM_STATIONS values = station area
27 Arbitrary cross-section shaped AT with area and wetted perimeter specified at each station NUM_STATIONS values = station area
A10 Station hydraulic diameter or wetted perimeter

(STATION_HYDDIAMS_OR_PERIMS, dynamic array, length and meaning of value depends on value of CS_SHAPE)

Dynamic array of station hydraulic diameter or perimeter. Length of and meaning of value in the array is determined by the value of CS_SHAPE. STATION_HYDDIAMS_OR_PERIMS is only used for CS_SHAPE = 2, 6, 7, 12, 16, 17, 22, 26, or 27
CS_ SHAPE Description Form of CS_HYDIAMS_ OR_PERIMS
1 Circular AT with uniform area 1 value = 0
2 Circular AT with uniform diameter 1 value = Dh
6 Arbitrary cross-section AT with uniform area and hydraulic diameter 1 value = = Dh
7 Arbitrary cross-section AT with uniform area and wetted perimeter 1 value = Pw
11 Tapered circular AT with area specified at inlet and exit 1 values = 0
12 Tapered circular AT with diameter specified at inlet and exit 2 values = Din, Dexit
16 Tapered arbitrary cross-section shaped AT with area and hydraulic diameter specified at inlet and exit 2 values = Dh,in,Dh,exit
17 Tapered arbitrary cross-section shaped AT with area and wetted perimeter specified at inlet and exit 2 values = Pw,in,Pw,exit
21 Circular AT with area specified at each station 1 value = 0
22 Circular AT with diameter specified at each station NUM_STATIONS values = Dh,station
26 Arbitrary cross-section shaped AT with area and hydraulic diameter specified at each station NUM_STATIONS values = Dh,station
27 Arbitrary cross-section shaped AT with area and wetted perimeter specified at each station NUM_STATIONS values = Pw,station
A11 WALL_SIDE_FRACTIONS

(Dynamic array of length NUM_WALL_SIDES)

Array of length NUM_WALL_SIDES specifying the portion of the AT perimeter represented by each wall side (segment). WALL_SIDE_FRACTIONS can be input as either a fraction or a percentage. If input as a fraction:

i = 1 N U M _ W A L L _ S I D E S W A L L _ S I D E _ F R A C T I O N S ( i ) = 1.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGfWbqabSWdaeaapeGaamyAaiabg2da9iaaigdaa8aabaWdbiaa d6eacaWGvbGaamytaiaac+facaWGxbGaamyqaiaadYeacaWGmbGaai 4xaiaadofacaWGjbGaamiraiaadweacaWGtbaan8aabaWdbiabggHi LdaakiaadEfacaWGbbGaamitaiaadYeacaGGFbGaam4uaiaadMeaca WGebGaamyraiaac+facaWGgbGaamOuaiaadgeacaWGdbGaamivaiaa dMeacaWGpbGaamOtaiaadofadaqadaWdaeaapeGaamyAaaGaayjkai aawMcaaiabg2da9iaaigdacaGGUaGaaGimaaaa@5C6F@

And if input as a percentage:

i = 1 N U M _ W A L L _ S I D E S W A L L _ S I D E _ F R A C T I O N S ( i ) = 100 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGfWbqabSWdaeaapeGaamyAaiabg2da9iaaigdaa8aabaWdbiaa d6eacaWGvbGaamytaiaac+facaWGxbGaamyqaiaadYeacaWGmbGaai 4xaiaadofacaWGjbGaamiraiaadweacaWGtbaan8aabaWdbiabggHi LdaakiaadEfacaWGbbGaamitaiaadYeacaGGFbGaam4uaiaadMeaca WGebGaamyraiaac+facaWGgbGaamOuaiaadgeacaWGdbGaamivaiaa dMeacaWGpbGaamOtaiaadofadaqadaWdaeaapeGaamyAaaGaayjkai aawMcaaiabg2da9iaaigdacaaIWaGaaGimaiaacwcaaaa@5D20@

Wall sides cover the length of the AT and thus apply at every station.

A12 WALL_SIDE_TYPES

(Dynamic array of length NUM_WALL_SIDES)

Array of length NUM_WALL_SIDES specifying the surface type of each wall side (segment).

0: Smooth surface

1: Rough surface

2: Turbulated surface

For types 1 and 2, additional inputs are required to define roughness and/or turbulator geometry.

Wall sides cover the length of the AT and thus apply at every station.

A13 LOSS_MODE_ON_EACH_SIDE

(Dynamic array of length NUM_WALL_SIDES)

Array of length NUM_WALL_SIDES specifying the momentum loss calculation method for each wall side (segment).

0: No momentum loss

1: Specified roughness, uniform along AT length

2: Specified friction coefficient, uniform along AT length

3: Specified Kloss, uniform along AT length

4: Specified friction multiplier applied to calculated friction, uniform along AT length

12: Linearly tapered friction coefficient, specified at inlet and exit

14: Linearly tapered friction multiplier applied to calculated friction, specified at inlet and exit

21: Specified roughness for each AT segment (between stations)

22: Specified friction coefficient for each AT segment (between stations)

23: Specified Kloss for each AT segment (between stations)

24: Specified friction multiplier applied to calculated friction for each AT segment (between stations)

The choice of LOSS_MODE_ON_EACH_SIDE will determine the length of and interpretation of the values in the LOSS_QUANTITIES_ON_SIDE_X arrays

Wall sides cover the length of the AT and thus apply at every station.

A14 HEAT_MODE_ON_EACH_SIDE

(Dynamic array of length NUM_WALL_SIDES)

Array of length NUM_WALL_SIDES specifying the heat transfer calculation method for each wall side (segment).

0: Adiabatic

1: Uniform heat load (Btu/s)

2: Uniform heat load (Btu/lbm)

3: Uniformly distributed delta T (Not implemented)

4: Fixed fluid total temperature at AT exit (Not implemented)

5: Calculated inner wall convection

6: Inner wall convection with constant Nusselt number

7: Inner wall convection with constant heat transfer coefficient

8: Calculated inner wall convection with constant Hmult

16: Inner wall convection with linearly tapered Nusselt number, specified at inlet and exit.

17: Inner wall convection with linearly tapered HTC, specified at inlet and exit.

18: Calculated inner wall convection with linearly tapered Hmult, specified at inlet and exit.

21: Specified heat load (Btu/s) at each AT segment (between stations)

22: Specified heat load (Btu/lbm) at each AT segment (between stations)

23: Specified delta T across each AT segment (between stations) (Not implemented)

24: Specified total temperature at each AT station (Not implemented)

26: Inner wall convection with Nusselt number specified for each AT segment (between stations)

27: Inner wall convection with HTC specified for each AT segment (between stations)

28: Calculated inner wall convection with Hmult specified for each AT segment (between stations)

The choice of HEAT_MODE_ON_EACH_SIDE will determine the length of and interpretation of the values in the HEAT_QUANTITIES_ON_SIDE_X arrays

Wall sides cover the length of the AT and thus apply at every station.

A15 LOSS_QUANTITIES_ON_SIDE_X These arrays hold the loss quantities on side number X. There will be NUM_SIDES number of these arrays in the database and their length and interpretation of contained values is determined by the value of LOSS_MODE_ON_EACH_SIDE as follows:
LOSS_ MODE_ON_ EACH_SIDE Description Form of LOSS_QUANTITIES_ON_SIDE_X
0 No momentum loss 1 value = 0 (not used)
1 Specified uniform roughness 1 value = roughness
2 Specified uniform friction coefficient 1 value = friction coefficient
3 Specified uniform Kloss 1 value = Kloss
4 Specified uniform friction multiplier 1 value = friction multiplier
12 Linearly tapered friction coefficient, specified at inlet and exit 1 value = friction coefficient
14 Linearly tapered friction multiplier, specified at inlet and exit 1 value = friction multiplier
21 Specified roughness for each AT segment (between stations) (NUM_STATIONS – 1) values = segment roughness
22 Specified friction coefficient for each AT segment (between stations) (NUM_STATIONS – 1) values = segment friction coefficient
23 Specified Kloss for each AT segment (between stations) (NUM_STATIONS – 1) values = segment Kloss
24 Specified friction multiplier for each AT segment (between stations) (NUM_STATIONS – 1) values = segment friction multiplier

NOTE: Interpretation of roughness values specified for LOSS_QUANTITIES_ON_SIDE_X will depend on the value of the ROUGH_TYPE input parameter.

A15 HEAT_QUANTITIES_ON_SIDE_X These arrays hold the heat transfer quantities on side number X. There will be NUM_SIDES number of these arrays in the database and their length and interpretation of contained values is determined by the value of LOSS_MODE_ON_EACH_SIDE as follows:
HEAT_ MODE_ON_ SIDE_X Description Form of LOSS_QUANTITIES_ON_SIDE_X
0 Adiabatic 1 value = 0 (not used)
1 Uniform heat load (BTU/s) 1 value = Heat load in BTU/s
2 Uniform heat load (BTU/lbm) 1 value = Heat load in BTU/lbm
3 Uniformly distributed ΔT 1 value = ΔT across the whole AT in oF
4 Fixed fluid total temperature at AT exit 1 value = TTexit­ (oF)
5 Calculated inner wall convection 1 value = 0 (not used)
6 Inner wall convection with constant Nusselt number 1 value = Nu
7 Inner wall convection with constant HTC number 1 value = HTC (Btu/hr-ft2-oF)
8 Calculated inner wall convection with constant Hmult 1 value = Hmult
16 Inner wall convection with linearly tapered Nusselt no. 2 values = Nuin, Nuexit
17 Inner wall convection with linearly tapered HTC 2 values = HTCin, HTCexit
18 Calculated inner wall convection with linearly tapered Hmult 2 values = Hmultin, Hmultout
21 Specified heat load (BTU/s) at each AT segment (between segments) (NUM_STATIONS – 1) values = heat load (BTU/s)
22 Specified heat load (BTU/lbm) at each AT segment (between segments) (NUM_STATIONS – 1) values = heat load (BTU/lbm)
A15 HEAT_QUANTITIES_ON_SIDE_X

(Continued)

HEAT_ MODE_ON_ SIDE_X Description Form of LOSS_QUANTITIES_ON_SIDE_X
23 Specified ΔT across each AT segment (between stations) (NUM_STATIONS – 1) values = ΔTsegment (oF)
24 Specified fluid total temperature at each station NUM_STATIONS values = TTstation
26 Inner wall convection with Nusselt number specified for each segment (between stations) (NUM_STATIONS – 1) values = Nusegment
27 Inner wall convection with HTC specified for each AT segment (between stations) (NUM_STATIONS – 1) values = HTCsegment
28 Calculated inner wall convection with Hmult specified for each AT segment (between stations) (NUM_STATIONS – 1) values = Hmultsegment

NOTE: For Heat Modes using calculated inner wall convection, the method of heat transfer calculation is determined by the value of the HTC_RELATION input parameter.

A16 WALL_TEMPERATURE_ON_SIDE_X

(Dynamic array, number of values is input by the user)

Surface temperature of the AT wall for use in heat transfer and fluid calculations. The user has three options for number of values to input for wall temperature. The size of the array determines how the values in the array are determined:
Size of Array Interpretation of Values
1 Value Uniform wall temperature (oF) along the whole AT
2 Values Inlet and exit wall temperature (oF) linearly interpolated along the length of the AT
NUM_STATIONS Values Wall temperature (oF) specified at every station along the AT

Advanced Tube Element Theory Manual

The advanced tube element routine simulates compressible gas flow through a passage where friction is a significant pressure loss mechanism. Both laminar and turbulent flows are accommodated by the routine as is heat transfer with either internal turbulators or with conduction/convection/radiation using thermal networks.

The length of the AT is divided into segments of arbitrary length, the number of segments, typically (but not limited to) between 5 and 15, being that specified in the input file. The beginning and end of each segment is represented as a “station” so there is one more station than the number of segments used for the AT. The geometry of the AT flow passage is defined by a combination of two of the three input variables, flow area, hydraulic diameter, and wetted perimeter, specified for each of the AT stations, the remaining variable being calculated using the equation:

D h y d = 4 * A r e a P e r i m e t e r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGebWdamaaBaaaleaapeGaamiAaiaadMhacaWGKbaapaqabaGc peGaeyypa0ZaaSaaa8aabaWdbiaaisdacaGGQaGaamyqaiaadkhaca WGLbGaamyyaaWdaeaapeGaamiuaiaadwgacaWGYbGaamyAaiaad2ga caWGLbGaamiDaiaadwgacaWGYbaaaaaa@48D1@

The AT routine is divided into two main sections: a flow direction calculation and a flow iteration loop. In the flow direction section, the procedures described in the paragraphs on computing the element flow inlet and outlet conditions are employed to define the inlet driving pressure (PTS), the inlet temperature, the secondary fluid mass fraction, and the exit back pressure (PSEB). If the AT is rotating and its inlet and outlet are at different radii, an estimate of the pumping effect due to rotation is used to compute an effective inlet pressure, PTSM. The procedure used to calculate the pressure ratio, PTSM / PTS, is identical to that for a forced vortex turning at the specified element RPM. If PTS (or PTSM) is greater than PSEB, these pressures are employed with a simple overall flow coefficient, based on inlet pressure drop and estimated friction effect, to estimate the fluid velocity at the AT exit plane. If PTS (or PTSM) is less than PSEB,, the calculation is repeated with the flow direction reversed.

The mass, momentum, and energy conservation must be maintained along the length of the AT. The conservation equations solved in the advanced tube are the same as those solved for the compressible tube although the solution method is different. The equations and methods used are described here.

Mass Equation

The continuity equation is given as:

m ˙ = ρ A v = c o n s t a n t   f o r   e a c h   t u b e   s t a t i o n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGTbWdayaacaWdbiabg2da9iabeg8aYjaadgeacaWG2bGaeyyp a0Jaam4yaiaad+gacaWGUbGaam4CaiaadshacaWGHbGaamOBaiaads hacaGGGcGaamOzaiaad+gacaWGYbGaaiiOaiaadwgacaWGHbGaam4y aiaadIgacaGGGcGaamiDaiaadwhacaWGIbGaamyzaiaacckacaWGZb GaamiDaiaadggacaWG0bGaamyAaiaad+gacaWGUbaaaa@59C6@

Momentum Equation

The momentum equation is given as (ref 2):

ρ A V d V = A d p + d F f + d F b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHRiI8cqaHbpGCcaWGbbGaamOvaiaadsgacaWGwbGaeyypa0Ja ey4kIiVaeyOeI0IaamyqaiaadsgacaWGWbGaey4kaSIaey4kIiVaam izaiaadAeapaWaaSbaaSqaa8qacaWGMbaapaqabaGcpeGaey4kaSIa ey4kIiVaamizaiaadAeapaWaaSbaaSqaa8qacaWGIbaapaqabaaaaa@4ECD@

Where:

d F f = f r i c t i o n a l   f o r c e s = f ρ V 2 2 D h d x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGKbGaamOra8aadaWgaaWcbaWdbiaadAgaa8aabeaak8qacqGH 9aqpcaWGMbGaamOCaiaadMgacaWGJbGaamiDaiaadMgacaWGVbGaam OBaiaadggacaWGSbGaaiiOaiaadAgacaWGVbGaamOCaiaadogacaWG LbGaam4Caiabg2da9maalaaapaqaa8qacaWGMbGaeqyWdiNaamOva8 aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqaa8qacaaIYaGaamira8aa daWgaaWcbaWdbiaadIgaa8aabeaaaaGcpeGaamizaiaadIhaaaa@550B@

d F b = c e n t r i f u g a l   f o r c e s = A ρ ω 2 r d x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGKbGaamOra8aadaWgaaWcbaWdbiaadkgaa8aabeaak8qacqGH 9aqpcaWGJbGaamyzaiaad6gacaWG0bGaamOCaiaadMgacaWGMbGaam yDaiaadEgacaWGHbGaamiBaiaacckacaWGMbGaam4BaiaadkhacaWG JbGaamyzaiaadohacqGH9aqpcaWGbbWaaSaaa8aabaWdbiabeg8aYj abeM8a39aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqaa8qacaWGYbaa aiaadsgacaWG4baaaa@54D3@

Combined Equation

The mass and momentum equations can be combined to give (ref 2):

M x M x + Δ x ( 1 M 2 ) M ( 1 + γ 1 2 M 2 ) d M = x x + Δ x [ γ M 2 f 2   D h ] d x + x x + Δ x [ 1 + γ M 2 2 ] d T T T T x x + Δ x [ ω 2 2 R T s ] d ( r 2 ) = I m     MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGfWbqabSWdaeaapeGaamyta8aadaWgaaadbaWdbiaadIhaa8aa beaaaSqaa8qacaWGnbWdamaaBaaameaapeGaamiEaiabgUcaRiabfs 5aejaadIhaa8aabeaaa0qaa8qacqGHRiI8aaGcdaWcaaWdaeaapeWa aeWaa8aabaWdbiaaigdacqGHsislcaWGnbWdamaaCaaaleqabaWdbi aaikdaaaaakiaawIcacaGLPaaaa8aabaWdbiaad2eadaqadaWdaeaa peGaaGymaiabgUcaRmaalaaapaqaa8qacqaHZoWzcqGHsislcaaIXa aapaqaa8qacaaIYaaaaiaad2eapaWaaWbaaSqabeaapeGaaGOmaaaa aOGaayjkaiaawMcaaaaacaWGKbGaamytaiabg2da9maawahabeWcpa qaa8qacaWG4baapaqaa8qacaWG4bGaey4kaSIaeuiLdqKaamiEaaqd paqaa8qacqGHRiI8aaGcdaWadaWdaeaapeWaaSaaa8aabaWdbiabeo 7aNjaad2eapaWaaWbaaSqabeaapeGaaGOmaaaakiaadAgaa8aabaWd biaaikdacaGGGcGaamira8aadaWgaaWcbaWdbiaadIgaa8aabeaaaa aak8qacaGLBbGaayzxaaGaamizaiaadIhacqGHRaWkdaGfWbqabSWd aeaapeGaamiEaaWdaeaapeGaamiEaiabgUcaRiabfs5aejaadIhaa0 WdaeaapeGaey4kIipaaOWaamWaa8aabaWdbmaalaaapaqaa8qacaaI XaGaey4kaSIaeq4SdCMaamyta8aadaahaaWcbeqaa8qacaaIYaaaaa Gcpaqaa8qacaaIYaaaaaGaay5waiaaw2faamaalaaapaqaa8qacaWG KbGaamiva8aadaWgaaWcbaWdbiaadsfaa8aabeaaaOqaa8qacaWGub WdamaaBaaaleaapeGaamivaaWdaeqaaaaak8qacqGHsisldaGfWbqa bSWdaeaapeGaamiEaaWdaeaapeGaamiEaiabgUcaRiabfs5aejaadI haa0WdaeaapeGaey4kIipaaOWaamWaa8aabaWdbmaalaaapaqaa8qa cqaHjpWDpaWaaWbaaSqabeaapeGaaGOmaaaaaOWdaeaapeGaaGOmai aadkfacaWGubWdamaaBaaaleaapeGaam4CaaWdaeqaaaaaaOWdbiaa wUfacaGLDbaacaWGKbWaaeWaa8aabaWdbiaadkhapaWaaWbaaSqabe aapeGaaGOmaaaaaOGaayjkaiaawMcaaiabg2da9iaadMeapaWaaSba aSqaa8qacaWGTbaapaqabaGcpeGaaiiOaiaacckaaaa@9AC0@

The total pressure ratio across a segment can be related to the integral ( I m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabe aaqaaaaaaaaaWdbiaadMeapaWaaSbaaWqaa8qacaWGTbaapaqabaaa aaaa@3861@ ) by:

P t   x + Δ x P t = e I m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaamiua8aadaWgaaWcbaWdbiaadshacaGGGcGa amiEaiabgUcaRiabgs5aejaadIhaa8aabeaaaOqaa8qacaWGqbWdam aaBaaaleaapeGaamiDaaWdaeqaaaaak8qacqGH9aqpcaWGLbWdamaa CaaaleqabaWdbiaadMeapaWaaSbaaWqaa8qacaWGTbaapaqabaaaaa aa@448A@

Inlet Head Loss

Having calculated (or guessed) a Mach number just inside the element inlet, inlet head loss computations are made to determine the total pressure, P t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbGaamiDa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@392F@ , at this location. The simplest loss function is a constant K-loss input value (K) that is used in the following equation to calculate P t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbGaamiDa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@392F@ :

P t i = P t s 1 + K * q / P t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbGaamiDa8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGH 9aqpdaWcaaWdaeaapeGaamiuaiaadshapaWaaSbaaSqaa8qacaWGZb aapaqabaaakeaapeGaaGymaiabgUcaRiaadUeacaGGQaGaamyCaiaa c+cacaWGqbGaamiDa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaaaaa@4592@

q = d y n a m i c   p r e s s u r e = 1 2 * ρ * V 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGXbGaeyypa0JaamizaiaadMhacaWGUbGaamyyaiaad2gacaWG PbGaam4yaiaacckacaWGWbGaamOCaiaadwgacaWGZbGaam4Caiaadw hacaWGYbGaamyzaiabg2da9maalaaapaqaa8qacaaIXaaapaqaa8qa caaIYaaaaiaacQcacqaHbpGCcaGGQaGaamOva8aadaahaaWcbeqaa8 qacaaIYaaaaaaa@4F2C@

P t s = u p s t r e a m   s o u r c e   p r e s s u r e   ( b e f o r e   K   l o s s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbGaamiDa8aadaWgaaWcbaWdbiaadohaa8aabeaak8qacqGH 9aqpcaWG1bGaamiCaiaadohacaWG0bGaamOCaiaadwgacaWGHbGaam yBaiaacckacaWGZbGaam4BaiaadwhacaWGYbGaam4yaiaadwgacaGG GcGaamiCaiaadkhacaWGLbGaam4CaiaadohacaWG1bGaamOCaiaadw gacaGGGcWaaeWaa8aabaWdbiaadkgacaWGLbGaamOzaiaad+gacaWG YbGaamyzaiaacckacaWGlbGaaiiOaiaadYgacaWGVbGaam4Caiaado haaiaawIcacaGLPaaaaaa@60D4@

There is also an option to calculate an inlet K loss based on a built-in correlation based on data reported in Ref 3, Fig 8. Converting Cd from the plot to head loss using:

K i n = ( 1.0 C d ) 2 1.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGlbWdamaaBaaaleaapeGaamyAaiaad6gaa8aabeaak8qacqGH 9aqpdaqadaWdaeaapeWaaSaaa8aabaWdbiaaigdacaGGUaGaaGimaa WdaeaapeGaam4qaiaadsgaaaaacaGLOaGaayzkaaWdamaaCaaaleqa baWdbiaaikdaaaGccqGHsislcaaIXaGaaiOlaiaaicdaaaa@4438@

V e l   H e a d   R a t i o = ( P t s P s t u b e _ i n l e t ) ( P t s P s u p s t r e a m   c h a m b e r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGwbGaamyzaiaadYgacaGGGcGaamisaiaadwgacaWGHbGaamiz aiaacckacaWGsbGaamyyaiaadshacaWGPbGaam4Baiabg2da9maala aapaqaa8qadaqadaWdaeaapeGaamiuaiaadshapaWaaSbaaSqaa8qa caWGZbaapaqabaGcpeGaeyOeI0IaamiuaiaadohapaWaaSbaaSqaa8 qacaWG0bGaamyDaiaadkgacaWGLbGaai4xaiaadMgacaWGUbGaamiB aiaadwgacaWG0baapaqabaaak8qacaGLOaGaayzkaaaapaqaa8qada qadaWdaeaapeGaamiuaiaadshapaWaaSbaaSqaa8qacaWGZbaapaqa baGcpeGaeyOeI0IaamiuaiaadohapaWaaSbaaSqaa8qacaWG1bGaam iCaiaadohacaWG0bGaamOCaiaadwgacaWGHbGaamyBaiaacckacaWG JbGaamiAaiaadggacaWGTbGaamOyaiaadwgacaWGYbaapaqabaaak8 qacaGLOaGaayzkaaaaaaaa@6D37@

In addition to the inlet losses, pressure losses due to bends can also be included in the AT element. There are 2 ways to model bend losses in Flow Simulator: 1) include the bend in the AT element, 2) Use a separate bend element with the AT element only accounting for the straight length. Each method has advantages and disadvantages. Option 2 may lead to more accurate results and allows for the pressures upstream and downstream of the bends to be visible in the GUI. Option 1 is faster, and accuracy is sufficient for engineering calculations. If option 1 is used, the bend K loss is calculated the same way as the bend element K loss (see the bend element for calculation details).

Energy Equation

Start with the differential form of the steady state energy balance equation:

d H d x = m ˙ C p d T d x = h C ( T w T ) + m ˙ ( ω 2 r d r d x g c d z d x ) + q i n ' ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaamizaiaadIeaa8aabaWdbiaadsgacaWG4baa aiabg2da9iqad2gapaGbaiaapeGaam4qa8aadaWgaaWcbaWdbiaadc haa8aabeaak8qadaWcaaWdaeaapeGaamizaiaadsfaa8aabaWdbiaa dsgacaWG4baaaiabg2da9iaadIgacaWGdbWaaeWaa8aabaWdbiaads fapaWaaSbaaSqaa8qacaWG3baapaqabaGcpeGaeyOeI0IaamivaaGa ayjkaiaawMcaaiabgUcaRiqad2gapaGbaiaapeWaaeWaa8aabaWdbi abeM8a39aadaahaaWcbeqaa8qacaaIYaaaaOGaamOCamaalaaapaqa a8qacaWGKbGaamOCaaWdaeaapeGaamizaiaadIhaaaGaeyOeI0Iaam 4za8aadaWgaaWcbaWdbiaadogaa8aabeaak8qadaWcaaWdaeaapeGa amizaiaadQhaa8aabaWdbiaadsgacaWG4baaaaGaayjkaiaawMcaai abgUcaRiaadghapaWaa0baaSqaa8qacaWGPbGaamOBaaWdaeaapeGa ai4jaaaakmaabmaapaqaa8qacaWG4baacaGLOaGaayzkaaaaaa@6519@

The AT uses the following energy integral [ref 4 section 1.6]:

These integrals are calculated in a semi-analytic way.

Integral I H 1 , x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGjbWdamaaBaaaleaapeGaamisaiaaigdacaGGSaGaamiEaaWd aeqaaaaa@3A76@ is calculated as:

Integral I H 21 , x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGjbWdamaaBaaaleaapeGaamisaiaaikdacaaIXaGaaiilaiaa dIhaa8aabeaaaaa@3B32@ is calculated as:

Where:

Integral I H 22 , x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGjbWdamaaBaaaleaapeGaamisaiaaikdacaaIYaGaaiilaiaa dIhaa8aabeaaaaa@3B32@ is calculated as:

Finally, temperature at location x is calculated as:

Coupling with the Thermal Network Solver

The total amount of heat added to the fluid is given by:

Heat added, Δ Q a d v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHuoarcaWGrbWdamaaBaaaleaapeGaamyyaiaadsgacaWG2baa paqabaaaaa@3B7A@ , is a result of convection from the AT wall:

Wall temperature along a segment is constant and we assume that there is a fluid temperature that satisfies:

The fluid temperature can be calculated as:

Rotating Annulus and Coaxial Shaft Angular Momentum Balance

If the rotation method is “Rotating Annulus” or “Rotating Coaxial Shaft”, an angular momentum balance equation is solved to calculate the change of fluid swirl and fluid temperature due to windage. These rotation methods require that the element is attached to inertial chambers at each end. These rotation methods can be used instead of cavities for long annular or coaxial cylinders. For example, the Rotating Annulus method is used between the red and blue cylinders while the Rotating Coaxial Shaft is used inside the red cylinder in this figure.
Figure 3.


The angular momentum equation that is solved:

T q r T q s = m ˙ * ω * R o u t * X K o u t R i n * X K i n MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubGaamyCa8aadaWgaaWcbaWdbiaadkhaa8aabeaak8qacqGH sislcaWGubGaamyCa8aadaWgaaWcbaWdbiaadohaa8aabeaak8qacq GH9aqpceWGTbWdayaacaWdbiaacQcacqaHjpWDcaGGQaWaaeWaa8aa baWdbiaadkfapaWaaSbaaSqaa8qacaWGVbGaamyDaiaadshaa8aabe aak8qacaGGQaGaamiwaiaadUeapaWaaSbaaSqaa8qacaWGVbGaamyD aiaadshaa8aabeaak8qacqGHsislcaWGsbWdamaaBaaaleaapeGaam yAaiaad6gaa8aabeaak8qacaGGQaGaamiwaiaadUeapaWaaSbaaSqa a8qacaWGPbGaamOBaaWdaeqaaaGcpeGaayjkaiaawMcaaaaa@5718@

T q r = t o r q u e   f r o m   t h e   r o t o r   s u r f a c e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubGaamyCa8aadaWgaaWcbaWdbiaadkhaa8aabeaak8qacqGH 9aqpcaWG0bGaam4BaiaadkhacaWGXbGaamyDaiaadwgacaGGGcGaam OzaiaadkhacaWGVbGaamyBaiaacckacaWG0bGaamiAaiaadwgacaGG GcGaamOCaiaad+gacaWG0bGaam4BaiaadkhacaGGGcGaam4Caiaadw hacaWGYbGaamOzaiaadggacaWGJbGaamyzaaaa@5695@

T q s = t o r q u e   f r o m   t h e   s t a t o r   s u r f a c e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubGaamyCa8aadaWgaaWcbaWdbiaadohaa8aabeaak8qacqGH 9aqpcaWG0bGaam4BaiaadkhacaWGXbGaamyDaiaadwgacaGGGcGaam OzaiaadkhacaWGVbGaamyBaiaacckacaWG0bGaamiAaiaadwgacaGG GcGaam4CaiaadshacaWGHbGaamiDaiaad+gacaWGYbGaaiiOaiaado hacaWG1bGaamOCaiaadAgacaWGHbGaam4yaiaadwgaaaa@5782@

m ˙ = m a s s   f l o w r a t e   t h r o u g h   t h e   t u b e   MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGTbWdayaacaWdbiabg2da9iaad2gacaWGHbGaam4Caiaadoha caGGGcGaamOzaiaadYgacaWGVbGaam4DaiaadkhacaWGHbGaamiDai aadwgacaGGGcGaamiDaiaadIgacaWGYbGaam4BaiaadwhacaWGNbGa amiAaiaacckacaWG0bGaamiAaiaadwgacaGGGcGaamiDaiaadwhaca WGIbGaamyzaiaacckaaaa@5674@

ω = r o t a t i o n a l   s p e e d   r a d / s MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDcqGH9aqpcaWGYbGaam4BaiaadshacaWGHbGaamiDaiaa dMgacaWGVbGaamOBaiaadggacaWGSbGaaiiOaiaadohacaWGWbGaam yzaiaadwgacaWGKbGaaiiOamaabmaapaqaa8qacaWGYbGaamyyaiaa dsgacaGGVaGaam4CaaGaayjkaiaawMcaaaaa@4F61@

X K i n ,   o u t = R a t i o   o f   t h e   f l u i d   a b s o l u t e   r o t a t i o n a l   s p e e d   t o   t h e   r o t a t i o n a l                                             r e f e r e n c e   f r a m e   a t   i n l e t   a n d   e x i t   o f   t h e   t u b e   s e g m e n t MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaadIfacaWGlbWdamaaBaaaleaapeGaamyAaiaad6gacaGG SaGaaiiOaiaad+gacaWG1bGaamiDaaWdaeqaaOWdbiabg2da9iaadk facaWGHbGaamiDaiaadMgacaWGVbGaaiiOaiaad+gacaWGMbGaaiiO aiaadshacaWGObGaamyzaiaacckacaWGMbGaamiBaiaadwhacaWGPb GaamizaiaacckacaWGHbGaamOyaiaadohacaWGVbGaamiBaiaadwha caWG0bGaamyzaiaacckacaWGYbGaam4BaiaadshacaWGHbGaamiDai aadMgacaWGVbGaamOBaiaadggacaWGSbGaaiiOaiaadohacaWGWbGa amyzaiaadwgacaWGKbGaaiiOaiaadshacaWGVbGaaiiOaiaadshaca WGObGaamyzaiaacckacaWGYbGaam4BaiaadshacaWGHbGaamiDaiaa dMgacaWGVbGaamOBaiaadggacaWGSbaabaGaaiiOaiaacckacaGGGc GaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaadkhacaWGLbGaamOzaiaadwgacaWGYbGa amyzaiaad6gacaWGJbGaamyzaiaacckacaWGMbGaamOCaiaadggaca WGTbGaamyzaiaacckacaWGHbGaamiDaiaacckacaWGPbGaamOBaiaa dYgacaWGLbGaamiDaiaacckacaWGHbGaamOBaiaadsgacaGGGcGaam yzaiaadIhacaWGPbGaamiDaiaacckacaWGVbGaamOzaiaacckacaWG 0bGaamiAaiaadwgacaGGGcGaamiDaiaadwhacaWGIbGaamyzaiaacc kacaWGZbGaamyzaiaadEgacaWGTbGaamyzaiaad6gacaWG0baaaaa@C86B@

R i n , o u t = F l o w   r a d i u s   a t   i n l e t   a n d   e x i t   o f   t h e   t u b e   s e g m e n t MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbWdamaaBaaaleaapeGaamyAaiaad6gacaGGSaGaam4Baiaa dwhacaWG0baapaqabaGcpeGaeyypa0JaamOraiaadYgacaWGVbGaam 4DaiaacckacaWGYbGaamyyaiaadsgacaWGPbGaamyDaiaadohacaGG GcGaamyyaiaadshacaGGGcGaamyAaiaad6gacaWGSbGaamyzaiaads hacaGGGcGaamyyaiaad6gacaWGKbGaaiiOaiaadwgacaWG4bGaamyA aiaadshacaGGGcGaam4BaiaadAgacaGGGcGaamiDaiaadIgacaWGLb GaaiiOaiaadshacaWG1bGaamOyaiaadwgacaGGGcGaam4Caiaadwga caWGNbGaamyBaiaadwgacaWGUbGaamiDaaaa@6DA0@

The flow radius is always the center of an annulus, but the flow radius is not well defined for a Rotating Coaxial Shaft. You must supply the flow radius for the Rotating Coaxial Shaft method. If the flow radius is 0, the XK=1 and the fluid is rotating at the same speed as the shaft. A flow radius of 0.25 times the shaft diameter is a good value to use unless experience indicates a different value.

The torque equations use the same skin friction coefficient equations used for cylindrical surfaces in cavities.

The Reynolds number will use an effective velocity based on axial and rotational velocities. This effective Reynolds number is used in the heat transfer coefficient and friction calculations. Furthermore, the friction uses an effective length to account for the spiral path of the flow through the tube. See reference 6 by Gazley.

V e f f = V a x i a l 2 + V r e l a t i v e _ t a n g e n t i a l 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGwbWdamaaBaaaleaapeGaamyzaiaadAgacaWGMbaapaqabaGc peGaeyypa0ZaaOaaa8aabaWdbiaadAfapaWaa0baaSqaa8qacaWGHb GaamiEaiaadMgacaWGHbGaamiBaaWdaeaapeGaaGOmaaaakiabgUca RiaadAfapaWaa0baaSqaa8qacaWGYbGaamyzaiaadYgacaWGHbGaam iDaiaadMgacaWG2bGaamyzaiaac+facaWG0bGaamyyaiaad6gacaWG NbGaamyzaiaad6gacaWG0bGaamyAaiaadggacaWGSbaapaqaa8qaca aIYaaaaaqabaaaaa@56B0@

R e e f f = ρ   V e f f   D h   μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbGaamyza8aadaWgaaWcbaWdbiaadwgacaWGMbGaamOzaaWd aeqaaOWdbiabg2da9maalaaapaqaa8qacqaHbpGCcaGGGcGaamOva8 aadaWgaaWcbaWdbiaadwgacaWGMbGaamOzaaWdaeqaaOWdbiaaccka caWGebWdamaaBaaaleaapeGaamiAaaWdaeqaaOWdbiaacckaa8aaba WdbiabeY7aTbaaaaa@4978@

A table of results is written to the .res file for these rotation methods. This table contains swirls, relative tangential velocities, and torques at each tube station.

Solution Method

The main steps in the solution method are:

  1. Guess an exit station Mach number.
  2. Calculate an exit station flowrate using the exit station Mach number.
  3. Loop through each AT station from the exit to the inlet.
    • Calculate a station temperature, Mach number, and pressure that satisfies the mass, momentum, and energy equations.
  4. Compare the inlet station total pressure with the total pressure of the upstream chamber minus any losses (PTI).
  5. If the inlet total pressures do not match within the convergence tolerance, adjust the exit station Mach number, and repeat steps 2-4.

Advanced Tube Element Outputs

The following listing provides details about the elements output variables.

Name Description Units
CROSS-SECTION Shape of AT cross-section. (None)
LENGTH Length of the AT. Inch, m
NUM_STATIONS Number of stations in the AT. (None)
NUM_CIRCUMF_WALL_SEGS Number of circumferential wall segments around the AT. (None)
RI AT inlet radius. Inch, m
RE AT exit radius. Inch, m
K_INLET Inlet head loss (user input). (None)
FRICTION_TYPE Friction factor calculation used in the solution (DARCY, FANNING, or N/A). (None)
TURB_FRIC Turbulent friction relation used for solution (ABAUF (Smooth Wall Power Law), SWAMEE (Colebrook White), or OFF). (None)
LAM_FRIC Laminar friction relation used for the solution. (None)
INPUT_ROUGHNESS_ TYPE Roughness input type used for solution (SAND_GRAIN, AVERAGE_ABSOLUTE, ROOT_MEAN_SQUARE, or PEAK_TO_VALLEY). (None)
K_CONTRAC_RESULT Back-calculated K loss. (unitless)
CD_RESULT Result calculated from actual mass flow rate divided by ideal mass flow rate. The ideal mass flow rate assumes K=0. (unitless)
QTOTAL Total heat change over the entire AT. Btu/s, W
PTS Driving pressure relative to the rotational reference frame (that is, rotor) at the AT inlet. psia, MPa
PTIN Total pressure relative to the rotational reference frame (that is, rotor) at the AT inlet, includes inlet losses. psia, MPa
PSIN Static pressure relative to the rotational reference frame (that is, rotor) at the AT inlet.

Limited by critical pressure ratio for supersonic flows when inlet area is smaller than exit area.

psia, MPa
PTEX Total pressure relative to the rotational reference frame (that is, rotor) at the AT exit including supersonic effects. psia, MPa
PSEX Static pressure relative to the rotational reference frame (that is, rotor) at the AT exit.

Limited by critical pressure ratio for supersonic flows.

psia, MPa
PSEB Effective sink (static) pressure downstream of the AT. psia, MPa
TTS Total temperature of fluid relative to the rotational reference frame (that is, rotor) at the AT inlet. degF, K
TSIN Static temperature of fluid relative to the rotational reference frame (that is, rotor) at the AT inlet. degF, K
INVEL Velocity of fluid relative to the rotational reference frame (that is, rotor) at the transition inlet. ft/s, m/s
TTEX Total temperature of fluid relative to the rotational reference frame (that is, rotor) at the AT exit. degF, K
TSEX Static temperature of fluid relative to the rotational reference frame (that is, rotor) at the AT exit. degF, K
EXVEL Velocity of fluid relative to the rotational reference frame (that is, rotor) at the transition exit. ft/s, m/s
Station Geometry Table of AT geometry. NONE
STA Column of stations. Station 1 is listed as “Inlet” and station NUM_STATIONS is listed as “Exit”. NONE
X Station location as a distance from the inlet. Inch, m
RADIUS Station radius from engine center line. Inch, m
HEIGHT Station height from some datum, used in gravitational effects calculations. Inch, m
DH Station hydraulic diameter. If not user input, calculated from relation: Dh = 4*A/P. Inch, m
PERIM Station wetted perimeter. If not user input, calculated from relation: P = 4*A/Dh. Inch, m
AREA Station cross-sectional area. If not user input, calculated from relation: A=Dh*P/4. in2, m2
Station Bulk Data Station-by-station fluid information. NONE
PT Fluid total pressure at station location. psia, MPa
PS Fluid static pressure at station location. psia, MPa
TT Fluid total temperature at station location. degF, K
TS Fluid static temperature at station location. degF, K
VEL Fluid velocity at station location. ft/s, m/s
THETA Fluid theta angle at station location. Will only change if there is a bend in the AT, otherwise it is the same as at “Inlet”. deg
PHI Fluid phi angle at station location. Will only change if there is a bend in the AT, otherwise it is the same as at “Inlet”. deg
REYF

Fluid Reynolds number used in the friction calculation at the AT station.

R e y f = m ˙   D h A   μ f i l m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbGaamyza8aadaWgaaWcbaWdbiaadMhacaWGMbaapaqabaGc peGaeyypa0ZaaSaaa8aabaWdbiqad2gapaGbaiaapeGaaiiOaiaads eapaWaaSbaaSqaa8qacaWGObaapaqabaaakeaapeGaamyqaiaaccka cqaH8oqBpaWaaSbaaSqaa8qacaWGMbGaamyAaiaadYgacaWGTbaapa qabaaaaaaa@478A@

m ˙ = t u b e   m a s s   f l o w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGTbWdayaacaWdbiabg2da9iaadshacaWG1bGaamOyaiaadwga caGGGcGaamyBaiaadggacaWGZbGaam4CaiaacckacaWGMbGaamiBai aad+gacaWG3baaaa@45D9@

D h = t u b e   h y d r a l i c   d i a m e t e r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGebWdamaaBaaaleaapeGaamiAaaWdaeqaaOWdbiabg2da9iaa dshacaWG1bGaamOyaiaadwgacaGGGcGaamiAaiaadMhacaWGKbGaam OCaiaadggacaWGSbGaamyAaiaadogacaGGGcGaamizaiaadMgacaWG HbGaamyBaiaadwgacaWG0bGaamyzaiaadkhaaaa@4E40@

A = t u b e   a r e a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGbbGaeyypa0JaamiDaiaadwhacaWGIbGaamyzaiaacckacaWG HbGaamOCaiaadwgacaWGHbaaaa@407A@

μ f i l m = f l u i d   v i s c o s i t y   b a s e d   o n   T f i l m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH8oqBpaWaaSbaaSqaa8qacaWGMbGaamyAaiaadYgacaWGTbaa paqabaGcpeGaeyypa0JaamOzaiaadYgacaWG1bGaamyAaiaadsgaca GGGcGaamODaiaadMgacaWGZbGaam4yaiaad+gacaWGZbGaamyAaiaa dshacaWG5bGaaiiOaiaadkgacaWGHbGaam4CaiaadwgacaWGKbGaai iOaiaad+gacaWGUbGaaiiOaiaadsfapaWaaSbaaSqaa8qacaWGMbGa amyAaiaadYgacaWGTbaapaqabaaaaa@5A4A@

T f i l m = T s + T w a l l 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubWdamaaBaaaleaapeGaamOzaiaadMgacaWGSbGaamyBaaWd aeqaaOWdbiabg2da9maalaaapaqaa8qacaWGubWdamaaBaaaleaape Gaam4CaaWdaeqaaOWdbiabgUcaRiaadsfapaWaaSbaaSqaa8qacaWG 3bGaamyyaiaadYgacaWGSbaapaqabaaakeaapeGaaGOmaaaaaaa@454B@

Unitless
REGIME Flow regime at current station (TURB, LAM, or TRAN). NONE
RHO Fluid density at station location. lbm/ft3, kg/m3
CP Fluid specific heat at station location.

Btu/lbm-oF,

J/kg-oK

Segment Bulk Data Segment-by-segment heat addition/temperature rise results.
KSEG Head loss (K-loss) across the current segment. NONE
Q_CONV Heat added to fluid across segment due to convection. Btu/s, W
Q_FLUX Heat added to fluid across segment due to heat flux. Btu/s, W
Q_ROTA Heat added to fluid across segment due pumping. Btu/s, W
Q_GRAV Heat added to fluid across segment due to buoyancy (gravitational effects). Btu/s, W
Q_TOT Total heat added to fluid across segment. Btu/s, W
DTCONV Fluid temperature rise (or fall) across segment due to convection. degF, K
DTFLUX Fluid temperature rise (or fall) across segment due to heat flux. degF, K
DTROTA Fluid temperature rise (or fall) across segment due to pumping. degF, K
DTGRAV Fluid temperature rise (or fall) across segment due to buoyancy (gravitational effects). degF, K
DTTOT Total fluid temperature rise (or fall) across segment. degF, K
Station Data for Circumferential Wall Segment X Station-by-station data for each circumferential wall segment in the model. NONE
WFRAC Fraction of the AT circumference modeled by the current wall side segment. NONE
ARC Arc length of AT modeled by the current wall side segment. Inch, m
TWALL User defined wall temperature. degF, K
TFILM Fluid temperature used to determine fluid properties for heat transfer calculations (Cp, and so on). degF, K
TWADIAB Adiabatic wall temperature. degF, K
MU_WALL Dynamic viscosity of fluid at wall temperature, TWALL. lbm/Hr-Ft, kg/sec-m
MU_FILM Dynamic viscosity of fluid at film temperature, TFILM. lbm/Hr-Ft, kg/sec-m
COND_FILM Conductivity of the fluid at film temperature, TFILM.

Btu/hr-ft-degF,

W/m-degK

PR_FILM Prandtl number at film temperature. Unitless
RECOV Recovery factor for the TWADIAB calculation. NONE
REYN Average fluid Reynolds number used in the HTC calculation for the AT segment. Same as REYN for an incompressible fluid. Unitless
Segment Data for Circumferential Wall Segment X Segment-by-segment (between stations) data for each circumferential wall segment in the model. NONE
SURFAREA Segment surface area =Segment Arc Length * Segment Length`. in2, m2
KSEGW Head loss (k-loss) across current segment and wall side. NONE
QCONV Heat added across segment and wall side due to convection heat transfer. Btu/s, W
QFLUX Heat added across segment and wall side due to heat flux. Btu/s, W
FR_EQ Friction equation type (MOODY, ABAUF, or OFF). NONE
SGROUGH Segment roughness, interpreted by solver according to ROUGHNESS_TYPE value. Inch, m
FMULT Friction multiplier for each segment. NONE
FRIC Friction factor value for each segment and wall side, either Fanning or Darcy depending on FRICTION_TYPE. NONE
HT_EQ Heat transfer equation used for the solution (OFF, DITBOELT (Dittus-Boelter), SIEDTATE (Sieder-Tate), GNIELINSKI , BHATSHAH(Bhatti-Shah), TURBULAT, FIX_HTC, FIX_NUSS, FIX_HEAT_FIX_DTT, FIX_TTEX, FIX_TT). NONE
HINMT HTC multiplier at inlet to the segment. NONE
HMULT HTC multiplier for the segment. NONE
NUSSLT Segment Nusselt number. UNITLESS
HTC Final calculated heat transfer coefficient for each segment.

Btu/hr-ft2-oF,

W/m2-oK

FLUID_SWIRL RPM of the fluid/SWIRL_REF_RPM. None
VEL_TAN_REL Tangential velocity relative to the surface. ft/s, m/s
VEL_EFF Effective velocity. Includes axial and tangential components. ft/s, m/s
TORQUE Torque between the surface and fluid. ft-lb, N-m

References

  1. Hubbartt, J. E., H. O. Sloan and V. L. Arne, Method for Rapid Determination of Pressure Change for One-dimensional Flow with Heat Transfer, Friction, Rotation, and Area Change, NACA TN 3150, June 1954.
  2. Prabhudharwadkar D., Dweik Z., Murali Krishnan R., One Dimensional Model for Rotating Channels in the Turbine system. Part 1: Formulation and Validation for Single Phase Flow, ASME Power Conference, Power 2013-98126, 2013.
  3. Rhode, J. E., H. T. Richards and G. W. Metger, Discharge Coefficients for Thick Plate Orifices with Approach Flow Perpendicular and Inclined to the Axis, NASA TN D-5467, October 1969.
  4. Kreyszig, E., Advanced Engineering Mathematics, 8th Ed., John Wiley & Sons, 1999.
  5. Miller, D, Internal Flow Systems, Miller Innovations, 1990.
  6. Gazley, C.: 'Heat Transfer Characteristics of rotating and axial flow between concentric cylinders', Trans ASME, Jan 1958, pp.79-89.