Water from a pipe is coming at a rate of 100 liters per minute. If the radius of the pipe is 5 cm, t

Q: Why do we use diameter instead of radius in the Reynolds number formula?

It is a standard convention in fluid mechanics for flow through circular pipes. The characteristic length scale for a pipe is taken as its diameter $D$. If you use radius, the critical values for laminar/turbulent transitions would be different.

Q: How do I convert liters per minute to cubic meters per second?

Remember that $1 \text{ Liter} = 10^{-3} \text{ m}^3$ and $1 \text{ minute} = 60 \text{ seconds}$. So, multiply the value by $10^{-3}$ and divide by $60$.

Q: What does 'order of' mean in the question?

It refers to the closest power of 10 when the number is expressed in scientific notation ($a \times 10^b$). Here, $2.12 \times 10^4$ has an order of magnitude of $10^4$.

Q: What does a Reynolds number of $10^4$ physically mean?

The Reynolds number is the ratio of inertial forces to viscous forces. A value of $10^4$ (which is greater than $4000$) means inertial forces dominate, and the flow is highly turbulent and chaotic.

Q: Can I just use CGS units for everything instead of SI?

Yes, you can. However, SI is generally safer to avoid confusion, especially with the unit of viscosity (Poise in CGS vs Pa s in SI). $1 \text{ Pa s} = 10 \text{ Poise}$.

Question

Water from a pipe is coming at a rate of $100$ liters per minute. If the radius of the pipe is $5 	ext{ cm}$, the Reynolds number for the flow is of the order of (density of water = $1000 	ext{ kg} / 	ext{m}^3$, coefficient of viscosity of water = $1 	ext{ mPa s}$)

Options:

(A) $10^3$

(B) $10^4$

(C) $10^2$

(D) $10^6$

Sigma Percentile · Accepted Answer

### Analyzing the Setup
Imagine a pipe with water gushing through it. We are given the volume flow rate $Q = 100 	ext{ L/min}$, the radius of the pipe $r = 5 	ext{ cm}$, and the properties of water like density $ho = 1000 	ext{ kg/m}^3$ and viscosity $\eta = 1 	ext{ mPa s}$. Our goal is to find the order of magnitude of the Reynolds number to understand the nature of this flow.

### The Master Equation
To find the Reynolds number, we use its standard formula:
$$R_e = \frac{ho v D}{\eta}$$

But wait, we don't have the velocity $v$ directly. We have the volume flow rate $Q$. We know that the volume flow rate is the cross-sectional area times the velocity ($Q = A \cdot v$). So, velocity is $Q$ divided by the area, which is $\pi r^2$. And the diameter $D$ is simply twice the radius ($2r$).

Let's substitute these into our Reynolds number formula:
$$R_e = \frac{ho \left(\frac{Q}{\pi r^2}ight) (2r)}{\eta}$$

The $r$ in the numerator cancels one $r$ in the denominator. We get a neat, modified formula:
$$R_e = \frac{2 ho Q}{\pi r \eta}$$

### The Trap of Units
Before we plug in the numbers, we must ensure all units are in the standard SI system. This is where silly mistakes happen! Let's convert them carefully:
*   **Flow Rate:** $Q = 100 	ext{ L/min} = \frac{100 	imes 10^{-3} 	ext{ m}^3}{60 	ext{ s}} = \frac{10^{-2}}{6} 	ext{ m}^3/	ext{s}$
*   **Radius:** $r = 5 	ext{ cm} = 5 	imes 10^{-2} 	ext{ m}$
*   **Viscosity:** $\eta = 1 	ext{ mPa s} = 10^{-3} 	ext{ Pa s}$

### Final Calculation
Now, let's carefully substitute all these SI values into our modified formula:
$$R_e = \frac{2 	imes 1000 	imes \left(\frac{10^{-2}}{6}ight)}{\pi 	imes (5 	imes 10^{-2}) 	imes 10^{-3}}$$

Let's crunch the numbers. The powers of ten combine nicely:
$$R_e = \frac{2000 	imes 10^{-2}}{6 	imes \pi 	imes 5 	imes 10^{-5}}$$
$$R_e = \frac{20}{30 \pi 	imes 10^{-5}} = \frac{2}{3 \pi 	imes 10^{-5}}$$
$$R_e = \frac{2 	imes 10^5}{3 \pi}$$

Since $\pi \approx 3.14$, $3\pi \approx 9.42$. Dividing $200,000$ by $9.42$ gives us approximately $21,231$.

In scientific notation, this is $2.12 	imes 10^4$. Therefore, the **order of magnitude** of the Reynolds number is **$10^4$**. Since $R_e > 4000$, this indicates the flow is highly turbulent!

Select Answer:

Visualized Solution

\text{Visualizing the Flow}

\text{Reynolds Number Formula}

\text{Velocity from Flow Rate}

\text{Modified Formula}

\text{Unit Conversions}

\text{Substituting Values}

\text{Calculation}

\text{Final Answer}

The Sigma Insight: Flow of Fluid

Analyzing the Setup Imagine a pipe with water gushing through it

The Master Equation

The Trap of Units Before we plug in the numbers, we must ensure all units are in the standard SI system

Final Calculation

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