8.13 Calculating Losses Along a Duct Using the Integral Method
339
L: m, and
θ: No dimensions.
Exercise 8.9 For the conditions described in Exercise 8.8 determine:
(a) The airflow rate provided by the fan, and
(b) The pressure supplied by the fan.
Assume duct discharge pressure, P f = 0.
Solution
(a)
Step 1. Calculate the constant C 1 with Eq. 8.50, using the assumption that P f =
0:
C 1 =
2θ
r
P
3
2
f − Q
3
f = 0−1.7
3
= −4.913
Step 2. Calculate z at duct outlet.
Given that P f = 0, then z → ∞.
Step 3. Calculate the integration constant C 2 :
C 2 = −
π
√
3
− r
2θ
r
2
3
L
I (z) =
π
√
3
= 0.2 ·
2 · 0.5 × 10
−5
0.2
2
3
· 400 + C 2
C 2 = −1.9224
Step 4. Calculate z at the fan location (x = 0) using Eq. 8.54:
I (z) =
1
3
ln
(z − 1)
2
z 2 + z + 1
−
2
√
3
arctan
2z + 1
√
3
Given that for x = 0, Eq. 8.55 gives:
I(z) = r
2θ
r
2
3
x + C 2
I (z) = C 2
Thus:
339
L: m, and
θ: No dimensions.
Exercise 8.9 For the conditions described in Exercise 8.8 determine:
(a) The airflow rate provided by the fan, and
(b) The pressure supplied by the fan.
Assume duct discharge pressure, P f = 0.
Solution
(a)
Step 1. Calculate the constant C 1 with Eq. 8.50, using the assumption that P f =
0:
C 1 =
2θ
r
P
3
2
f − Q
3
f = 0−1.7
3
= −4.913
Step 2. Calculate z at duct outlet.
Given that P f = 0, then z → ∞.
Step 3. Calculate the integration constant C 2 :
C 2 = −
π
√
3
− r
2θ
r
2
3
L
I (z) =
π
√
3
= 0.2 ·
2 · 0.5 × 10
−5
0.2
2
3
· 400 + C 2
C 2 = −1.9224
Step 4. Calculate z at the fan location (x = 0) using Eq. 8.54:
I (z) =
1
3
ln
(z − 1)
2
z 2 + z + 1
−
2
√
3
arctan
2z + 1
√
3
Given that for x = 0, Eq. 8.55 gives:
I(z) = r
2θ
r
2
3
x + C 2
I (z) = C 2
Thus: