Two galvanized iron pipes of diameter D are connected to a...

Two galvanized iron pipes of diameter $D$ are connected to a large water reservoir, as shown. Pipe $A$ has length $L$ and pipe $B$ has length $2 L .$ Both pipes discharge to atmosphere. Which pipe will pass the larger flow rate? Justify (without calculating the flow rate in cach pipe). Compute the flow rates if $H=10 \mathrm{m}, D=50 \mathrm{mm},$ and $L=10 \mathrm{m}$

Key Concepts

Pipe Flow Resistance

Pipe flow resistance is a comprehensive concept that encapsulates the cumulative effects of frictional losses, pipe geometry, and flow characteristics on the fluid movement. It helps explain why, under the same pressure conditions, a shorter pipe offers less resistance and therefore allows a higher flow rate compared to a longer pipe.

Hagen-Poiseuille Law

The Hagen-Poiseuille law governs laminar flow through a cylindrical pipe, establishing a precise relation between the volumetric flow rate and the pressure drop. It shows that the flow rate is highly sensitive to the pipe diameter (proportional to the fourth power) and inversely proportional to the length, which highlights the significant impact of pipe dimensions on flow efficiency in laminar regimes.

Frictional Losses in Pipe Flow

Frictional losses arise from the interaction between the moving fluid and the pipe walls, causing a drop in pressure along the length of the pipe. These losses are directly influenced by factors such as pipe length, diameter, the roughness of the pipe's interior surface, and the flow regime (laminar or turbulent), making them a critical factor in determining the overall flow rate.

Head Loss

Head loss is the reduction in the total energy per unit weight of a fluid as it flows through a pipe, primarily due to friction and turbulence. This concept explains how the driving pressure or elevation head is diminished along the length of the pipe, which in turn affects the flow rate, especially when comparing pipes of different lengths.

Darcy-Weisbach Equation

The Darcy-Weisbach equation is a fundamental relation used in fluid mechanics to quantify the head loss due to friction in a pipe. It relates the pressure drop to factors such as pipe length, diameter, fluid velocity, and a friction factor, and is particularly important for turbulent flow conditions. This equation helps in understanding how longer pipes, with higher frictional resistance, lead to reduced flow rates.

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