The duct of Example 11.12 is lengthened by 50 %. If the duct...

The duct of Example 11.12 is lengthened by $50 \%$. If the duct discharge pressure is set at a value of $p_{d}=46.2 \mathrm{kPa}(\mathrm{abs}),$ determine the mass flowrate of air through the lengthened duct. The average friction factor for the duct remains constant at a value 0.02

Key Concepts

Impact of Duct Lengthening

Increasing the duct length increases the overall friction losses in the system as the frictional pressure drop is directly proportional to the length of the duct. This enhanced friction loss reduces the effective pressure driving the flow, thereby lowering the mass flow rate for the same set discharge pressure. Understanding this impact is important in designing and analyzing duct systems in various engineering applications.

Duct Flow and Friction Losses

In duct and pipe flow, friction losses occur due to the interaction between the flowing fluid and the duct walls. This friction is a key factor in determining the pressure drop along the duct, which in turn affects the mass flow rate of the fluid. Understanding these friction losses is essential, especially when modifications such as lengthening the duct are made.

Darcy-Weisbach Equation

The Darcy-Weisbach equation is a fundamental relation used to quantify the pressure drop due to friction in a pipe or duct. It relates the pressure loss to factors such as the friction factor, duct length, fluid velocity, and characteristic dimension (typically diameter) of the duct. This equation is critical for analyzing how changes in duct geometry, like an increase in length, affect flow parameters.

Friction Factor

The friction factor is a dimensionless parameter that quantifies the resistance to flow due to friction within the duct. It encapsulates the influence of duct surface roughness and flow regime (laminar or turbulent) on the pressure drop. In problems like the one described, where the friction factor remains constant despite changes in duct length, it simplifies the analysis by isolating length as the primary variable impacting friction losses.

Pressure Drop and Discharge Pressure

Pressure drop is the loss of pressure due to friction and other resistances as the fluid flows along the duct. The discharge pressure, which is set as a boundary condition, determines the available driving pressure for the flow. Changes in the duct's characteristics, such as length, alter the pressure drop and thus influence the mass flow rate through the system.

Mass Flow Rate

Mass flow rate is the measure of the mass of fluid passing through a section of the duct per unit time. It is determined by the balance between the driving pressure (such as the discharge pressure) and the frictional losses along the duct. Calculating the mass flow rate involves evaluating this balance, often using equations like the Darcy-Weisbach equation in conjunction with continuity equations.

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