Air is delivered through a constant-diameter duct by a fan. The air is inviscid, so the fluid velocity profile is "flat" across each cross section. During the fan start-up, the following velocities were measured at the time \(t\) and axial positions \(x\) : $$\begin{array}{llll} & x=0 & x=10 \mathrm{m} & x=20 \mathrm{m} \\ t=0 \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} \\\t=1.0 \mathrm{s} & V=1.00 \mathrm{m} / \mathrm{s} & V=1.20 \mathrm{m} / \mathrm{s} & V=1.40 \mathrm{m} / \mathrm{s} \\\t=2.0 \mathrm{s} & V=1.70 \mathrm{m} / \mathrm{s} & V=1.80 \mathrm{m} / \mathrm{s} & V=1.90 \mathrm{m} / \mathrm{s} \\\t=3.0 \mathrm{s} & V=2.10 \mathrm{m} / \mathrm{s} & V=2.15 \mathrm{m} / \mathrm{s} & V=2.20 \mathrm{m} / \mathrm{s}\end{array}$$ Estimate the local acceleration, the convective acceleration, and the total acceleration at \(t=1.0 \mathrm{s}\) and \(x=10 \mathrm{m} .\) What is the local acceleration after the fan has reached a steady air flow rate?

Step by step solution

01

Calculate the Local Acceleration

This can be obtained by differentiating the velocity \(V\) with respect to time \(t\) at the specific point \(x=10m\). At \(t=1.0s\), \(V=1.2m/s\), and at \(t=0s\) \(V=0m/s\), hence the local acceleration \(a_{local} = (V_{t=1s} - V_{t=0s}) / (t_{1s} - t_{0s}) = 1.2m/s^2\)

02

Calculate the Convective Acceleration

This can be obtained by differentiating velocity with respect to position \(x\) while keeping \(t\) constant at the specific moment \(t=1s\). This can be calculated as follows: \(a_{conv} = (V_{x=20m} - V_{x=0m}) / (x_{20m} - x_{0m}) = 0.04m/s^2\)

03

Calculate the Total Acceleration

This is the sum of local and convective acceleration. Thus, total acceleration \(a_{total} = a_{local} + a_{conv} = 1.24m/s^2\)

04

Calculate the Local Acceleration after constant Flow Rate

Once a constant flow rate has been reached, the velocity no longer changes with time, therefore, the local acceleration will be zero; that is, \(a_{local} = dV/dt = 0m/s^2\) where V is the constant velocity.

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