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Professional Engineeing Publication

The prediction of cold water temperature at the off-design points (wet bulb temperature and cooling range other than design conditions) is to find an approach satisfying the cooling tower characteristic value at the design condition. In finding an approach at the off-design points, the L/G at the off-design points must be first assumed. There are three methods in assuming L/G ratio at the off-design point.

Constant Fan BHP (BHP off = BHP dsn)
Constant Fan Pitch (VOL off = VOL dsn)
Constant Air Mass Flow Rate (GAS off = GAS dsn)

1) Relationship between Design & Off-Design L/G Ratio and Design & Off-Design BHP @ Constant Fan BHP

First, let's see the fan bhp formula.

Fan BHP

=VOL x TP / (6356 x Fan Effi.)
= VOL x (VP + SP) / (6356 x Fan Effi.)
= VOL x (1/2g x Density x VEL² + K x 1/2g x DEN x Vel²) / (6356 x Fan Effi.)
= VOL x DEN x VEL² x (1 + K) /1/2g / (6356 x Fan Effi.)
= VOL x DEN x VEL² x (Area² /Area²) x (1 + K) /1/2g / (6356 x Fan Effi.)
= VOL x DEN x VOL² x 1 / Area² x (1 + K) /1/2g / (6356 x Fan Effi.)
= VOL³ x DEN x 1 / Area² x (1 + K) /1/2g / (6357 x Fan Effi.)
(The term of 1 / Area2 x (1 + K) /1/2g / (6357 x Fan Effi.) could be considered as a constant under the assumption that the fan efficiency at the design conditions is equal to the fan efficiency at the off-design conditions.)
= Constant x VOL3 x Density

Where,

VOL = Air Volume @ Fan (ACFM)
TP = Total Pressure @ Fan (Inch Aq.)
VP = Velocity Pressure @ Fan (Inch Aq.)
SP = Static Pressure (Inch Aq.)
g = Acceleration Gravity (ft/min²)
DEN = Air Density @ Fan (Lb/ft³)
VEL = Air Velocity @ Fan (FPM)
K = Overall Pressure Drop Coefficient
Area = Net Fan Area (ft²)

The relation of BHP off = BHP dsn is established under the assumption of constant fan BHP, which means that the fan bhp at the off-design is always equal to the fan bhp at the design regardless the off-design conditions.

BHP off = Const. x VOL off³ x DEN off
BHP dsn = Const. x VOL dsn³ x DEN dsn
BHP off = BHP dsn ------------------------------------------------------ Eq. 16-1

Since BHP off = BHP dsn, the Eq. 16-1 presenting the relation with VOL off and VOL dsn can be written in the following form.

Const. X VOL off³ x DEN off = Const. X VOL dsn³ x DEN dsn
VOL off = VOL dsn x (DEN dsn / DEN off)^1/3 --------------------------- Eq. 16-2
VOL dsn / SV dsn = G dsn (SV = Specific Volume of Air at Fan)
VOL dsn = G dsn x SV dsn
= L dsn / L/G dsn x SV dsn (L = Water Flow in Pound)--------- Eq. 16-3

Substitute VOL dsn in the right side of Eq. 16-2 by Eq. 16-3. Then, the below form is obtained.

VOL off = L dsn x (1 / L/G dsn) x SV dsn x (DEN dsn / DEN off)^1/3------ Eq. 16-4
L/G off = L off / G off = L off / (VOL off / SV off) ------------------------ Eq. 16-5

Substitute VOL off in the denominator of right side of Eq. 16-5 by Eq. 16-4.

L/G off = L off / [(L dsn x (1 / L/G dsn) x SV dsn x (DEN dsn / DEN off)^1/3) / SV off]
= L/G dsn x (L off / L dsn) x (DEN off / DEN dsn)^1/3 x (SV off / SV dsn) --- Eq. 16-6

Therefore, L/G at off-design point can be obtained from Eq. 16-6.

2) Relationship between Design & Off-Design L/G Ratio Design & Off-Design BHP @ Constant Fan Pitch

The relation of VOL off = VOL dsn is established under the assumption of constant fan pitch, which means that the air volume at the off-design is always same as the air volume at the design regardless the off-design conditions.

BHP off = Const. x VOL off³ x DEN off
VOL off³ = BHP off / (Const. x DEN off)
VOL off = BHP off^1/3 / (Const. x DEN off)^1/3 ------------------------------ Eq. 16-7

BHP dsn = Const. x VOL dsn³ x DEN dsn
VOL dsn³ = BHP dsn / (Const. x DEN dsn)
VOL dsn = BHP dsn^1/3 / (Const. x DEN dsn)^1/3 ---------------------------- Eq. 16-8

From the assumption of constant fan pitch, the relation of VOL off = VOL dsn is established and the following forms are obtained.

BHP off^1/3 / (Const. X DEN off)^1/3 = BHP dsn^1/3 / (Const. X DEN dsn)^1/3
BHP off = BHP dsn x (DEN off / DEN dsn) --------------------------------- Eq. 16-9

L/G dsn = L dsn / G dsn = L dsn / (VOL dsn / SV dsn) ---------------------- Eq. 16-10

Solve Eq. 16-10 for VOL dsn and rewrite.

VOL dsn = ( 1 / L/G dsn) x L dsn x SV dsn ---------------------------------- Eq. 16-11

VOL off = VOL dsn = ( 1 / L/G dsn) x L dsn x SV dsn ---------------------- Eq. 16-12

L/G off = L off / G off = L off / (VOL off / SV off) --------------------------- Eq. 16-13

Substitute VOL off in the denominator of right side of Eq. 16-13 by Eq. 16-12 and rewrite.

L/G off = L off / [((1 / L/G dsn) x L dsn x SV dsn ) / SV off]
= L/G dsn x (L off / L dsn) x (SV off / SV dsn) ------------------------ Eq. 16-14

3) Relationship between Design & Off-Design L/G Ratio Design & Off-Design BHP @ Constant Gas

The relation of GAS off = GAS dsn is established under the assumption of constant gas mass flow rate, which means that the air mass flow rate at the off-design is always equal to the air mass flow rate at the design regardless the off-design conditions.

BHP off = Const. x VOL off³ x DEN off
VOL off = GAS off x SV off
BHP off = Const. x (GAS off x SV off)³ x DEN off
GAS off³ = BHP off / (Const. x DEN off x SV off³)
GAS off = BHP off^1/3 / (Const. x DEN off^1/3 x SV off) ----------------------- Eq. 16-15

BHP dsn = Const. x VOL dsn³ x DEN dsn
VOL dsn = GAS dsn x SV dsn
BHP dsn = Const. x (GAS dsn x SV dsn)³ x DEN dsn
GAS dsn³ = BHP dsn / (Const. x DEN dsn x SV dsn³)
GAS dsn = BHP dsn^1/3 / (Const. x DEN dsn^1/3 x SV dsn) -------------------- Eq. 16-16

From the assumption of constant fan pitch, the relation of GAS off = GAS dsn is established and the following forms are obtained.

BHP off^1/3 / (Const. x DEN off^1/3 x SV off) = BHP dsn^1/3 / (Const. x DEN dsn^1/3 x SV dsn)

Therefore, BHP off = BHP dsn x (DEN off / DEN dsn) x (SV off / SV dsn)³ --- Eq. 16-17

L/G dsn = L dsn / G dsn -------------------------------------------------------- Eq. 16-18
L/G off = L off / G off ----------------------------------------------------------- Eq. 16-19

Eq. 16-19 can be written as Eq. 16-20 using the relation of G dsn = G off
L/G dsn = L dsn / G off -------------------------------------------------------- Eq. 16-20
Also, Eq. 16-20 can be solved for G off as Eq. 16-21.
G off = (1 / L/G dsn) x L dsn --------------------------------------------------- Eq. 16-21

Substitute the G off of right side of Eq. 16-19 by Eq. 16-21 and rewrite it as below.

L/G off = L off / [(1 / L/G dsn) x L dsn] = L/G dsn x (L off / L dsn) ------------- Eq. 16-22