Air flows through a 40 ft length of insulated constantarea...

Air flows through a 40 ft length of insulated constantarea duct with $D=2.12 \mathrm{ft}$. The relative roughness is $e / D=0.002 .$ At the duct inlet, $T_{1}=100^{\circ} \mathrm{F}$ and $p_{1}=$ 17.0 psia. At a location downstream, $p_{2}=14.7$ psia, and the flow is subsonic. Is sufficient information given to solve for $M_{1}$ and $M_{2} ?$ Prove your answer graphically. Find the mass flow rate in the duct and $T_{2}$.

Key Concepts

Mass Flow Rate Calculation

The mass flow rate in a duct is determined by the product of the cross-sectional area, the density of the fluid, and the flow velocity. In compressible flows, density and velocity depend on local conditions such as pressure, temperature, and Mach number. Accurate calculation of mass flow rate requires using the appropriate compressible flow relations along with duct geometry and wall roughness effects.

Fanno Flow

Fanno flow describes the behavior of a compressible fluid flowing in a constant?area duct with friction, where the effects of viscous dissipation are balanced by the energy redistribution within the flow. In such ducts, the flow is adiabatic so that no heat transfer occurs, and the friction produces irreversible losses that alter the Mach number, pressure, temperature, and density along the duct. This model is fundamental in analyzing duct flows where friction cannot be neglected.

Mach Number

The Mach number is the ratio of the flow velocity to the local speed of sound in the medium. It is a key parameter in compressible flows, indicating how significant compressibility effects are. In a Fanno flow, the Mach number changes along the duct due to friction, and its determination is central to assessing the state and evolution of the flow conditions.

Friction Factor

The friction factor quantifies the resistance to flow offered by the conduit wall due to roughness and viscous effects. In the context of internal flows, it is usually determined from the Moody diagram or empirical correlations that relate the relative roughness and the Reynolds number. In frictional compressible flow such as Fanno flow, the friction factor plays a crucial role in linking the duct geometry and wall characteristics to the changes in flow properties.

Compressible Flow Relations

These are the thermodynamic and fluid dynamic equations that describe how pressure, temperature, density, and Mach number are related in flows where variations of these quantities are significant. For adiabatic flows such as Fanno flow, special relations that account for friction and the non-isentropic nature of the process are used, and they are often represented graphically by Fanno lines to facilitate analysis.

Graphical Methods in Flow Analysis

Graphical techniques are often used to solve Fanno flow problems by plotting the relationships between relevant flow parameters (for instance, linking the friction parameter with Mach number along a Fanno line). Such graphs help in visually determining the correct solution by matching given operating conditions to theoretical curves, thus serving as a proof method for determining whether the available information is sufficient to uniquely define the flow state.

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