To solve this problem, we first need to understand how the flow in the pipes behaves under the given conditions. We are dealing with two distinct scenarios: one with a single larger pipe of radius R1 and another with two smaller pipes each of radius R2. The goal is to maintain the same total mass flux Q0 while ensuring the pressure difference per unit length remains constant in both setups.

Understanding Mass Flux and Flow Rate

Mass flux (Q) is defined as the mass flow rate per unit area. In a cylindrical pipe, the relationship between mass flux, pressure difference, and pipe radius is governed by the principles of fluid dynamics, particularly the Hagen-Poiseuille equation for laminar flow.

Flow Rate in a Single Pipe

For a single cylindrical pipe with radius R1, the flow rate Q can be expressed as:

• Q = (πR1^4ΔP) / (8μL)

Here, ΔP is the pressure difference per unit length, μ is the dynamic viscosity of the fluid, and L is the length of the pipe. This equation shows that the flow rate is proportional to the fourth power of the radius, which is a crucial aspect of our analysis.

Flow Rate in Two Smaller Pipes

Now, for two smaller pipes, each with radius R2, the total mass flux (Q) through both pipes can be expressed as:

• Q = 2 * (πR2^4ΔP) / (8μL)

This equation indicates that the flow rate through each smaller pipe also depends on the fourth power of the radius, but now we have two pipes contributing to the total flow.

Setting Up the Equation

Since the total mass flux must remain constant, we can set the two equations equal to each other:

• (πR1^4ΔP) / (8μL) = 2 * (πR2^4ΔP) / (8μL)

We can simplify this equation by canceling out the common factors (π, ΔP, 8μ, and L) from both sides:

• R1^4 = 2R2^4

Finding the Ratio R1/R2

To find the ratio R1/R2, we can take the fourth root of both sides:

• (R1 / R2)^4 = 2

Taking the fourth root gives us:

• R1 / R2 = 2^(1/4) = √(√2) ≈ 1.1892

Conclusion

Therefore, the ratio of the radius of the larger pipe to the radius of each smaller pipe is approximately 1.1892. This means that for the flow characteristics to remain consistent while adapting the system from one larger pipe to two smaller pipes, the size of the smaller pipes must be about 84% of the radius of the larger pipe.