Question

Water $\left(\mu=9.0 \times 10^{-4} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}, \rho=1000 \mathrm{~kg} / \mathrm{m}^{3}\right)$ enters a \(4-\mathrm{cm}\)-diameter and 3 -m-long tube whose walls are maintained at \(100^{\circ} \mathrm{C}\). The water enters this tube with a bulk temperature of \(25^{\circ} \mathrm{C}\) and a volume flow rate of $3 \mathrm{~m}^{3} / \mathrm{h}$. The Reynolds number for this internal flow is (a) 29,500 (b) 38,200 (c) 72,500 (d) 118,100 (e) 122,9000

Short answer

(a) 29,500 (b) 35,000 (c) 42,200 (d) 50,000

Step-by-step solution

Convert the diameter to meters

The diameter of the tube is given in centimeters, so we need to convert it to meters. Using 1 meter = 100 centimeters, the diameter in meters is: \(d = 4\,\text{cm} \times \frac{1 \,\text{m}}{100 \,\text{cm}} = 0.04\,\text{m}\)

Calculate the area of the tube

Using the diameter, we can find the area of the tube using the formula for the area of a circle: \(A = \pi(\frac{d}{2})^2\) Substituting the diameter \(d\): \(A = \pi(\frac{0.04 \,\text{m}}{2})^2 = 0.001256 \,\text{m}^{2}\)

Calculate the average flow velocity

We are given the volume flow rate and can use that to find the average flow velocity using the formula: \(v = \frac{Q}{A}\) Substituting the volume flow rate \(Q = 3 \,\text{m}^{3}/\text{h}\) and the area \(A\): \(v = \frac{3 \,\text{m}^{3}/\text{h}}{0.001256\,\text{m}^{2}} \times \frac{1 \,\text{h}}{3600 \,\text{s}} = 0.705\,\text{m/s}\)

Calculate the Reynolds number

Now, we can use the formula for Reynolds number: \(\text{Re}=\frac{\rho u d}{\mu}\) Substitute the given values for dynamic viscosity \(\mu = 9.0 \times 10^{-4} \,\text{kg}/\text{m}\cdot \text{s}\), density \(\rho = 1000 \,\text{kg}/\text{m}^{3}\), average flow velocity \(v\), and diameter \(d\): \(\text{Re}=\frac{1000 \,\text{kg}/\text{m}^{3} \times 0.705\,\text{m/s} \times 0.04\,\text{m}}{9.0 \times 10^{-4}\,\text{kg}/\text{m}\cdot\text{s}} = 31,333.33\) Rounding the value of the Reynolds number to the nearest hundred, we get: \(\text{Re} = 31,300\) Comparing this result with the given options, the closest one is: (a) 29,500 So, the Reynolds number for this internal flow is approximately 29,500.