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A diffuser for an incompressible, inviscid fluid of density \(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}\) consists of a diverging section of pipe. At the inlet the diameter is \(D_{i}=0.25 \mathrm{m},\) and at the outlet the diameter is \(D_{o}=0.75 \mathrm{m} .\) The diffuser length is \(L=1 \mathrm{m},\) and the diameter increases linearly with distance \(x\) along the diffuser. Derive and plot the acceleration of a fluid particle, assuming uniform flow at each section, if the speed at the inlet is \(V_{i}=5 \mathrm{m} / \mathrm{s}\). Plot the pressure gradient through the diffuser, and find its maximum value. If the pressure gradient must be no greater than \(25 \mathrm{kPa} / \mathrm{m}\), how long would the diffuser have to be?

Step by step solution

01

Setup

To begin, consider the continuity equation for incompressible flow, which is \(A_{i} V_{i} = A_{o} V_{o}\), where \(V_{i}\) and \(V_{o}\) are speeds at the inlet and outlet respectively and \(A_{i}\) and \(A_{o}\) are cross sectional areas at the inlet and outlet. The area \(A\) at a distance \(x\) from the inlet can be given as \(A=\pi((D_{i}+x(D_{o}-D_{i})/L)/2)^2\). So, by the continuity equation, the speed \(V\) can be written as \(V= A_{i}V_{i}/A\).

02

Find Acceleration

The acceleration \(a\) of the fluid can be obtained by differentiating the velocity \(V\) with respect to time \(t\). That is \(a=dV/dt= (dV/dx) (dx/dt) = V (dV/dx)\).

03

Use Bernoulli's Equation

Next, apply Bernoulli's equation between inlet and any point on the diffuser. Bernoulli's equation can be written as \(\rho V_{i}^{2}/2 + p_{i} = \rho V^{2}/2 + p\), where \(p_{i}\) is the inlet pressure, and \(p\) is the pressure at a distance \(x\). Rearranging for pressure gives \(p = p_{i} + \rho (V_{i}^2 - V^2 )/2\).

04

Determine Pressure Gradient

The pressure gradient \(dp/dx\) can be find by differentiating pressure \(p\) with respect to \(x\).

05

Find Maximum Pressure Gradient

The maximum pressure gradient can be determined by equating the derivative of the pressure gradient with respect to \(x\) to zero and solving for \(x\).

06

Adjust Length of Diffuser

If the pressure gradient cannot exceed \(25 \mathrm{kPa} / \mathrm{m}\), the length of the diffuser needs to be adjusted such that the maximum pressure gradient occurs when \(x = L\). Solve for \(L\) when the pressure gradient equals \(25 \mathrm{kPa} / \mathrm{m}\).

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