## Description

A duct
is made of drawn tubing that has an inside diameter of 8 inches. Air
flows through the duct at a rate of 500 cfm. The duct is **X**
ft long and lays horizontally. At the duct inlet, the dynamic loss
associated with flow from the room into the duct is K = 0.35. At the
duct outlet, the dynamic loss associated with an attached vent is K = 7 *
**Y** - 26.

The density and kinematic viscosity of the air are 0.0753 lbm/ft^3 and 1.62E-4 ft^2/s. The surface roughness of the drawn tubing is 0.0000015 ft.

- Determine the pressure drop (inches of air) along the duct length.
- Plot the pressure drop (inches of air) as a function of volumetric flow rate (ft^3/min), from 100 to 1000 cfm, in increments of 100 cfm.
- Plot the frictional head loss (inches of air) as a function of volumetric flow rate (ft^3/min), from 100 to 1000 cfm, in increments of 100 cfm.
- Plot the dynamic head loss (inches of air) as a function of volumetric flow rate (ft^3/min), from 100 to 1000 cfm, in increments of 100 cfm.

*All plots must be computer-generated*

The number of letters in your first name multiplied by 10 = **X**

The number of letters in your last name = **Y**

My name if 5 letters

X = 50

y - 5

## Explanation & Answer

Attached. Please let me know if you have any questions or need revisions.

A duct is made of drawn tubing that has an inside diameter of 8 inches. Air flows through the duct

at a rate of 500 cfm. The duct is X ft long and lays horizontally. At the duct inlet, the dynamic loss

associated with flow from the room into the duct is K = 0.35. At the duct outlet, the dynamic loss

associated with an attached vent is K = 7 * Y - 26.

The density and kinematic viscosity of the air are 0.0753 lbm/ft^3 and 1.62E-4 ft^2/s. The surface

roughness of the drawn tubing is 0.0000015 ft.

1. Determine the pressure drop (inches of air) along the duct length.

2. Plot the pressure drop (inches of air) as a function of volumetric flow rate (ft^3/min), from

100 to 1000 cfm, in increments of 100 cfm.

3. Plot the frictional head loss (inches of air) as a function of volumetric flow rate (ft^3/min),

from 100 to 1000 cfm, in increments of 100 cfm.

4. Plot the dynamic head loss (inches of air) as a function of volumetric flow rate (ft^3/min),

from 100 to 1000 cfm, in increments of 100 cfm.

*All plots must be computer-generated*

The number of letters in your first name multiplied by 10 = X

The number of letters in your last name = Y

Given:

𝐷 = 8 𝑖𝑛 =

2

3

𝑓𝑡

𝜈 = 1.62 𝑥 10−4

𝑞 = 500

𝑓𝑡 2

𝑠

𝑘1 = 0.35

𝐿 = 𝑋 𝑓𝑡 = 50 𝑓𝑡

𝐾2 = 7𝑌 − 26 = 30

𝑓𝑡 3

𝑚𝑖𝑛

=

25 𝑓𝑡 3

3 𝑠

𝑙𝑏

𝜌 = 0.0753 𝑓𝑡 3

∈= 0.0000015 𝑓𝑡 = 1.5 𝑥 10−6 𝑓𝑡

(𝑆𝑖𝑛𝑐𝑒 𝐼 ℎ𝑎𝑣𝑒 5 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝑖𝑛 𝑚𝑦 𝑛𝑎𝑚𝑒, 𝑠𝑜 𝑋 = 10(5) = 50 𝑓𝑡)

(𝑆𝑖𝑛𝑐𝑒 𝐼 ℎ𝑎𝑣𝑒 8 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝑖𝑛 𝑚𝑦 𝑠𝑢𝑟𝑛𝑎𝑚𝑒, 𝑠𝑜 𝐾2 = 7𝑌 − 26 = 7(8) − 26 = 30)

Solution:

1. Determine the pressure drop (inches of air) along the duct length.

From mechanical energy balance,

𝑊′ =

∆𝑃

+ ∆𝐾𝐸 + ∆𝑃𝐸 + ∑ 𝐹𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑜𝑠𝑠

𝜌

Since there is no pump involved and there is no change in elevation while the diameter of the pipe

is uniform throughout, change in potential energy can be ignored and total work is zero. Therefore,

the equation will be:

∆𝑃

+ ∆𝐾𝐸 + ∑ 𝐹𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑜𝑠𝑠 = 0

𝜌

The pressure drop can be calculated as:

−∆𝑃 = 𝜌 (∆𝐾𝐸 + ∑ 𝐹𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑜𝑠𝑠 )

where ∑ 𝐹𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑎𝑛𝑎𝑙 𝑙𝑜𝑠𝑠 = 𝐹𝑙𝑜𝑠𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 + 𝐹𝑙𝑜𝑠𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 + 𝐹ℎ𝑒𝑎𝑑 𝑙𝑜𝑠𝑠 +

𝐹𝑙𝑜𝑠𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑣𝑎𝑙𝑣𝑒𝑠/𝑓𝑖𝑡𝑡𝑖𝑛𝑔𝑠 + 𝐹𝑙𝑜𝑠𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑚𝑒𝑡𝑒𝑟𝑖𝑛𝑔 𝑑𝑒𝑣𝑖𝑐𝑒

Since only head loss is present,

−∆𝑃 = 𝜌(𝐹ℎ𝑒𝑎𝑑 𝑙𝑜𝑠𝑠 )

𝑤ℎ𝑒𝑟𝑒, 𝐹ℎ𝑒𝑎𝑑 𝑙𝑜𝑠𝑠 = 𝐹𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 ℎ𝑒𝑎𝑑 𝑙𝑜𝑠𝑠 + 𝐹𝑑𝑦𝑛𝑎𝑚𝑖𝑐 ℎ𝑒𝑎𝑑 𝑙𝑜𝑠𝑠

Therefore, the final equation will be:

−∆𝑃 = 𝜌(∆𝐾𝐸 + 𝐹𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 ℎ𝑒𝑎𝑑 𝑙𝑜𝑠𝑠 + 𝐹𝑑𝑦𝑛𝑎𝑚𝑖𝑐 ℎ𝑒𝑎𝑑 𝑙𝑜𝑠𝑠 )

Solving f...