Determine the mass flow rate of air under adiabatic conditions through a \(50 \mathrm{~mm}\) diameter pipe \(85 \mathrm{~m}\) long from a large reservoir in which the pressure and temperature are \(300 \mathrm{kPa}\) and \(15^{\circ} \mathrm{C}\). The pressure at the outlet of the pipe is \(120 \mathrm{kPa}\). Assume a negligible pressure drop at the pipe inlet and a mean value of \(0.006\) for \(f\). To what value would the inlet pressure need to be raised to increase the mass flow rate by \(50 \%\) ?

Adiabatic flow refers to a process where there is no heat exchange between the fluid and its surroundings. This concept often applies to fluid mechanics, especially in situations where heat insulation is significant. For the given problem, since the flow is adiabatic, we assume no heat is transferred to or from the air as it moves through the pipe.

Adiabatic processes are prevalent in thermodynamics and fluid mechanics problems. When working with adiabatic flow, it's crucial to remember that changes in the fluid’s internal energy are entirely due to work done by or on the fluid. This simplifies some calculations, as we don't need to account for heat transfer, just the work related to pressure and volume changes.

Understanding this concept helps in accurately using equations like the continuity equation and the Darcy-Weisbach equation, which assume no heat loss or gain during the flow. This is critical for calculating mass flow rate and other properties of the flow in a system with well-insulated boundaries.

The Darcy-Weisbach equation is fundamental in fluid mechanics for determining pressure loss due to friction in a pipe. It links the pressure drop (\text{ΔP}) to the flow conditions and pipe characteristics. The form used in the problem is:

\textbf{Darcy-Weisbach Equation:}
\[ \frac{dp}{\text{dp}} = \frac{fl}{d}\frac{v^2}{2} \]

Where:
\textbf{f} = Friction factor
\textbf{l} = Length of the pipe
\textbf{d} = Diameter of the pipe
\textbf{v} = Flow velocity

This equation assumes a steady and incompressible flow. The friction factor (f) depends on the Reynolds number and the relative roughness of the pipe, and for a given problem, it may either be provided or need to be calculated.

For our problem, the value of f is given as 0.006. Once we have the pressure values and dimensions of the pipe, we plug them into the equation to find the velocity or mass flow rate. Combining with the continuity equation helps solve for various fluid flow properties under the given conditions.

The continuity equation is another cornerstone concept in fluid mechanics, stating that the mass flow rate of a fluid must remain constant from one cross-section of a pipe to another, assuming there is no addition or loss of mass along the way. It's expressed mathematically as:

\[ Q = A \times v \]

Where:
\textbf{Q} = Volumetric flow rate
\textbf{A} = Cross-sectional area of the pipe
\textbf{v} = Flow velocity

For compressible flows like air, the mass flow rate (ṁ) is given by integrating density (ρ), area, and velocity:

\[ \text{mass flow rate} = \rho \times A \times v \]

This relationship is crucial in solving our main problem since it allows us to relate physical dimensions of the pipe and flow properties to the mass flow rate. For our scenario with given diameters and lengths, the continuity equation simplifies solving for the required mass flow rate, ensuring we correctly model how much air moves through the system per unit time.

Pressure drop in fluid mechanics is the reduction in pressure from one point in a pipe to another. It is caused by friction, changes in elevation, or pipe fittings. In the context of the Darcy-Weisbach equation, pressure drop (\text{ΔP}) is a function of the flow velocity, pipe characteristics, and the friction factor. The equation becomes:

\[ \frac{dP}{\text{dx}} = \frac{4fL}{d}\frac{mass \text{ flow rate}^2}{2 \rho A^2} \]

This relationship helps in determining the pressure drop along the length of the pipe due to friction. For our problem, assuming negligible pressure drop at the inlet, knowing the inlet and outlet pressures helps us compute the mass flow rate using given pipe dimensions and flow properties.

Analyzing pressure drop is critical in practical applications as it influences system design, energy requirements, and operational efficiency. Paying attention to factors like pipe roughness, flow rates, and lengths will ensure accurate calculations and optimal performance of fluid systems.