F-12102 Rules of Calculation

Rules of Calcul Calcul ation Contents - Ventilati on Theory

Contents

Ventilation Theory............................................................... 1 Symbols and and Units.............................................................. 2 Rules of Calculation............................................................ Calculation............................................................ 3 Determination Determination of Total Fan Pressure (Appendix A)............ 6 Loss in Discharge elbow elbow (Appendix (Appendix B)............................... 7 Loss in Transition Transition Duct (Appendix (Appendix C)................................. 8 Loss in Connection Piece Piece (Appendix (Appendix D) ........................... 10

Ventilation Theory 1.

Always calculate in total pressure when determining fan and duct system.

2.

The total pressure is the the sum of static static and and dynamic pressure p t = ps + pd

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3.

From sectionx  to sectiony  the equation of Bernoulli is used: p tx = pty + ploss x-y Hence, pressure loss is always the difference in the total pressure.

F-12102 1/9

Rules of Calcul ation Symbols and Units

 Atmospheric pressure at fan altitude... p B 

(PA)

Bearing loss in bearing assembly........ PB 

(W)

 Absolute static inlet pressure .............. p1 

(Pa)

Rotational speed..................................N

 Absolute static outlet pressure ............ p2 

(Pa)

Fan inlet area.......................................A1

(m2)

 Absolute inlet total pressure ................pa1 

(Pa)

Fan outlet area.....................................A2

(m2)

 Absolute outlet total pressure.............. pa2 

(Pa)

Inlet velocity .........................................v1 

(m/s)

Inlet differential static pressure............ps1 

(Pa)

Outlet velocity ......................................v2 

(m/s)

Outlet differential static pressure.........p s2 

(Pa)

Tip speed .............................................Ut 

(m/s)

Inlet dynamic pressure ........................ pd1 

(Pa)

Hub speed............................................Un 

(m/s)

Outlet dynamic pressure...................... pd2 

(Pa)

Impeller power fan total efficiency ....... ηR

Inlet differential total pressure ............. pt1 

(Pa)

Isentropic exponent .............................κ

Outlet differential total pressure .......... pt2 

(Pa)

Compressibility factor ..........................K

(min-1)

Mass flow rate ..................................... qm  (kg/s)

Kinetic energy factor, inlet ...................α A1

Inlet standard volume flow rate ........... q0 (Nm3/h)

Kinetic energy factor, outlet .................α A2

Inlet volume flow rate...........................qv1 (m3/s)

Mol mass..............................................M

Outlet volume flow rate........................qv2 (m3/s)

Mach number through inlet.................. Ma1

Inlet density .........................................ρ1  (kg/m)

Mach number through outlet................M a2

Outlet density.......................................ρ2  (kg/m)

Peripheral mach number .....................M U

Inlet temperature ................................. Tt1 

(K)

Total fan work ......................................Yt 

Outlet temperature............................... Tt2 

(K)

Universal constant of gas ....................R0(J/mol K)

Static inlet temperature........................Ts1 

(K)

Mass constant of gas...........................R (J/kg K)

Static outlet temperature ..................... Ts2 

(K)

Impeller diameter................................. D

(m)

Fan air power.......................................PF 

(W)

Hub diameter .......................................d

(m)

Fan impeller power.............................. PR 

(W)

Fan shaft power...................................P A 

(W)

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(g/mol)

/J/kg)

F-12102 2/9

Rules of Calcul ation Purpose - Definitions

1.

Purpose

The purpose of this document is to define the principal quantities required for the expression of the fan performance characteristics for the Howden Power axial flow fan program for power stations and will in principle follow the EUROVENT ¼ “Terminology of Fan Air Performance”. The exceptions will be that the term “stagnations” change into “total” so that the index for static pressure can be maintained as index s. The atmospheric pressure will be changed from pa0 to pB. 2.

 Absolu te stat ic in let pressure ..........p1 

(Pa)

Inlet dif ferential static pressure .......ps1 

(Pa)

given by ps1 = p1 - pB where pB = atmospheric pressure at fan altitude Inlet density ........................................ρ1  (kg/m3) given by ρ1 = where R =

Definitions

 All sizes on the inlet side of the fan have been given the subscript 1, and all sizes on the outlet side of the fan subscript 2.

R0 M

p1 R • Ts1

 is mass constant of gas on inlet side

R0 = universal constant of gas M

= mol mass

Inlet st andard volum e flow rate ........q0 (Nm3/h) at barometric pressure 1013 mbar and 0 °C Inlet volum e flow rate ........................qv1 (m3/s) given by qv1 =

qm ρ1

Inlet velocit y .......................................v1  is given as a mean velocity v 1 = Mass f low rate ....................................qm  (kg/s) Fan inlet area .....................................A1

(m2)

Inlet temperature ...............................Tt1 

(K)

Static inlet t emperature.....................Ts1 

(K)

given by Ts1 = Tt1 (1 +

κ −1 2

• α A1 • Ma12 ) -1

where

qm ρ1 • A1

Inlet dynamic pressure......................pd1  given by pd1 = 0.5

(m/s)

(Pa)

• ρ1 • v12

 Absolu te inl et tot al pressu re ............pa1 

(Pa)

given by pa1 = p1 + pd1 Inlet di fferential t otal pressure .........pt1 

(Pa)

given by pt1 = ps1 + pd1

κ

= Isentropic exponent

Ma1

= Mach number through inlet

α A1

= Kinetic energy factor

By convention α A1 is considered equal to 1

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Fan ou tlet area ...................................A2

(m2)

Outlet temperature.............................Tt2 

(K)

Static outlet t emperature ..................Ts2 

(K)

F-12102 3/9

Rules of Calcul ation Definitions

given by Ts2 = Tt2 (1 +

κ − 1 • α 2

 A 2

• Ma22 )-1

Total fan w ork ....................................Y t 

where

given by YtN =

κ

= Isentropic exponent

Ma2

= Mach number through outlet

α A2

= Kinetic energy factor

YtV = R•Ts1

By convention α A2 is considered equal to 1  Abso lute stat ic outl et pressu re ........p2 

(Pa)

Outlet differential static pressure ....ps2 

(Pa)

Outlet density .....................................ρ2  (kg/m3) p2 R • Ts2

given by ρ2 = where R =

R0 M

Outlet volume flow rate.....................qv2 (m3/s) given by q v2 =

−

v12 2

 or

•

=

κ−1

κ

=

cp cp − R

κ −1 p ( 2) κ − p1 p2 −1 p1

1

Fan air p ower  .....................................P F 

(W)

given by PF = qm • Yt

Fan impeller power ............................P R 

(W)

PF

ηR

where ηR = impeller power fan total efficiency

Fan shaft power .................................P A 

(W)

given by P A = PR + PB

ρ2

(m/s)

qm

(Pa)

given by pd2 = 0.5 • ρ2 • v22

 Abso lute outlet to tal press ure  ........pa2 

given by pt2 = ps2 + pd2

given by Tt2 = Tt1 +

(Pa)

Tip speed ............................................U t  given by Ut =

(Pa)

(min -1) (m/s)

π•D•N 60

Hub sp eed ...........................................U n  given by UN =

(K)

PR c p • qm

Rotational speed ................................N

given by pa2 = p2 + pd2

Outlet differential total pressure ......pt2 

where PB = bearing loss in bearing assembly

Total outlet temperature ....................T t2 

ρ2 •  A 2

Outlet dynamic pressure ..................p d2 

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v 22 2

qm

Outlet velocit y ....................................v2  given by v2 =

κ

where K

given by PR =

 is mass constant of gas on inlet side

•K +

κ − 1  ⎞ ⎛  ⎜ ⎛ p 2  ⎞ κ ⎟ v 2 v 2 • κ ⎜ ⎜⎜ ⎟⎟ − 1⎟ + 2 − 1 κ − 1 ⎜ ⎝ p1  ⎠ 2 2 ⎟⎟ ⎜ ⎝   ⎠

given by p s2 = p2 – pB where pB = atmospheric pressure at fan altitude

− ps1 ρ1

p s2

(J/kg)

(m/s)

π • d • N 60

F-12102 4/9

Rules of Calcul ation Definitions

Peripheral mach number ..................MU MU =

U κ • R • Ts1

Mach number thro ugh inlet ..............M a1 Ma1 =

v1 κ • R • Ts1

Mach number thro ugh outlet ............Ma2 Ma2 =

v2 κ • R • Ts2

Dimensionless factors psiN

=

phiN

=

lamN

=

where AL

=

psiV

=

phiV

=

lamV

=

where AD

=

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2 • YtN 2

Un

q v1 Un •  A L

• phiN ηR

psiN

π

( D2 − d2 ) 4 2 • YtV Ut 2 q v1 U t •  A D psi V

π 4

• phi V ηR

D2

F-12102 5/9

Rules of Calcul ation Determinati on of Total Fan Pressure p tF

The fan pressure is determined from the following formula p t2  – p t1  = p t2’  + p loss 2’-2 – p t1’  + p loss 1’-1 where pt2 = ps2 + pd2 pt1 = ps1 + pd1 pt2’ = ps2 + pd2’ ps2’  is stated by the customer and shall be related to an area in m 2. In case the customer does not state an area it is determined by Howden Power. ploss 2’-2

consist in this case og losses in silencer, transition piece, duct piece and exhaust loss.

pt1’ = ps1’ + pd1’ ps1’ is stated by the customer and shall be related to an area in m 2. In case the customer does not state an area it is determined by Howden Power ploss 1–1

consist of losses in silencer and transition piece.

If section 2’ coincides with section 2/or section 1’ coincides with section 1 p t2’ = pt2  and/or pt1’ = pt1. ploss

occurs in the various components, which are built in the arrangement to be quoted.

a) silencer loss is indicated in tables and contains friction and Carnot loss. b) Loss in connection piece on suction side after silencer is calculated according to appendix D ( F-12102 9/9) c) Loss in connection piece on pressure side for silencer is calculated according to appendix C (F-12102 8/9) d) Loss in elbow on pressure side is calculated according to appendix B (F-12102 7/9) e) Carnot loss from fan is calculated as pcarn = 0.5 • ρ2 (v 2 – va)2 see appendix C (F12102 8/9) f) Loss in a straight duct piece below 3 m can as a rule be neglected. g) The density is stated by the customer or calculated from given information about barometric pressure, temperature, humidity and the composition of the air.

 At free suction fan ps1 = 0 and the velocity is also zero which means that the links pt1’ and ploss 1’-1 is zero. The latter, however, depends on the design of the inlet cone.

 Appendix A

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F-12102 6/9

Rules of Calcul ation Loss in Discharge Elbow

If the discharge elbow is located more than 3 • Da from the fan outlet the latter will not affect the pressure loss. For a discharge elbow designed as Howden Power rules ζb = 0.25. For other designs please refer to literature on pressure losses in elbows. If the distance L is shorter than 3 • Da  the pressure losses will be increased when the impeller/hub ratio (D/d) is smaller than 2.4. The pressure loss in the elbow is calculated on the basis of the following formula (the loss in the ducting between fan and discharge elbow has not been taken into account):

ρ • 0.5

Ploss 2-a

=

Ploss a-2’

= ζb

where If

D d

• (v 2 − v Da ) 2

• 0.5 • ρ a • v a 2

D ⎞ ⎛  ζ b = 0.25 + ⎜ 2.4 − ⎟ • ζ d ⎠ ⎝ 

> 2.4 ζ b = 0.25

 Appendix B

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F-12102 7/9

Rules of Calcul ation Loss i n Transition Duct for Silencer, Pressure Side

If a silencer with transition duct is installed downstream, the losses in the system from 2 to 2’ shall be calculated as follows (friction losses in the duct [0.02 • L/Dh • pda] between fan and transition duct as well as friction losses in the duct after the silencer are not included):

ploss a-b = loss in transition duct

ploss 2-2’ = (ploss 2-a)+(ploss a-b)+ploss b-2’)+ (ploss a-2’)

ploss 2-a = carnot loss

where

* This loss only occur if L < 3 • D a. The standard pressure losses in a silencer in which the baffles are installed in a straight duct without transition duct are shown in table 2.

ploss b-2’ = loss in silencer corrected to case conditions (table 2)

= 0.5 • ρa • (va – vb)2 ploss a-2’* = system loss =

ζ • 0.5 • ρa • va2 (table 1)

= 0.5 ρ2 • (v2 – va)2

losses in silencer v = 12 m/s

Tabel 1

 Air density 1.2 kg/m3

L (casing) mm

YBA-2

YCA

PA

Pa

1150 1650 2150 2650 3150 3650

130 140 150 160 170 180

170 180 190

Table 2

 Appendix C

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F-12102 8/9

Rules of Calcul ation Loss i n Connection Piece, Suction Side

If the connection piece is situated between suction side silencer and fan the pressure loss is calculated on the basis of the following formulas:

The graph is only valid if -

the inlet of the connection piece is a squared section

-

the axis of silencer or duct is in line with the axis of the inlet box

ploss 1-1’(conn) = ζconn • pd1 where pd1 =

⎛ q  ⎞ 0.5 • ρ1 • ⎜ v1 ⎟ ⎝  A 1  ⎠

2

 A1 = area of inlet box

ζconn to be tdetermined from the below graph 4 •  A DH =

-

the ration of the side in the inlet box is 2:1, 4 which gives DH1 = b and D H1' = a. 3

O

 A = the area of section 1’ or 1 O = the circumference of section 1’ or 1

 Appendix D

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F-12102 9/9