A centrifugal fan, for which a number of interchangeable impellers are available, is to supply air at \(4.5 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) to a ventilating duct at a head of \(100 \mathrm{~mm}\) water gauge. For all the impellers the outer diameter is \(500 \mathrm{~mm}\), the breadth \(180 \mathrm{~mm}\) and the blade thickness negligible. The fan runs at \(188.5 \mathrm{rad} \cdot \mathrm{s}^{-1}\) ( \(\left.30 \mathrm{rev} / \mathrm{s}\right)\). Assuming that the conversion of velocity head to pressure head in the volute is counterbalanced by the friction losses there and in the impeller, that there is no whirl at inlet and that the air density is constant at \(1.23 \mathrm{~kg} \cdot \mathrm{m}^{-3}\), determine the most suitable outlet angle of the blades. (Neglect whirl slip.)

Step by step solution

01

Calculate the flow velocity

First, determine the flow velocity (peripheral velocity) at the outer diameter of the impeller. Use the formula: \[ U = \frac{\text{Outer diameter}}{2} \times \text{angular velocity} \]Given the outer diameter, D = 0.5 m, and angular velocity, \(\omega = 188.5 \text{ rad} \cdot \text{s}^{-1} \), the flow velocity is: \[ U = \frac{0.5}{2} \times 188.5 = 47.125 \text{ m} \cdot \text{s}^{-1} \]

02

Calculate the volume flow rate

Next, use the given volume flow rate, \( Q = 4.5 \text{ m}^3 \cdot \text{s}^{-1} \), to find the radial velocity at the outlet. The radial velocity, \(V_r\), can be calculated using the formula: \[ Q = A \cdot V_r \]where A is the area at the impeller outlet. Since the breadth (b) is 0.18 m and the diameter (D) is 0.5 m, the area is: \[ A = b \cdot D = 0.18 \cdot 0.5 = 0.09 \text{ m}^2 \]The radial velocity, \(V_r\), is then:\[ V_r = \frac{Q}{A} = \frac{4.5}{0.09} = 50 \text{ m} \cdot \text{s}^{-1} \]

03

Apply Bernoulli's equation for the flow head

Using the head (pressure head), convert the given water gauge head to a velocity head. Given head, \( h = 100 \text{ mm} = 0.1 \text{ m} \). Using the density of air, \( \rho = 1.23 \text{ kg} \cdot \text{m}^{-3} \), the velocity head can be found using Bernoulli’s equation:\[ P = \rho \cdot g \cdot h \]\[ P = 1.23 \cdot 9.81 \cdot 0.1 \approx 1.20663 \, \text{ Pa} \]

04

Relate velocity head to velocities

To find the relationship between the tangential and radial velocities at the outlet, recall that the head is related to these velocities by:\[ P = \rho \left( \frac{V_{t}^2}{2} \right) \]Since the pressure head must be counterbalanced by the conversion of velocity head to pressure head, solve for the tangential velocity \( V_t \):\[ 1.20663 = 1.23 \left( \frac{V_{t}^2}{2} \right) \]\[ V_{t}^2 = \frac{1.20663 \times 2}{1.23} \approx 1.962 \] \[ V_{t} \approx \sqrt{1.962} \approx 1.4 \text{ m} \cdot \text{s}^{-1} \]

05

Determine the outlet blade angle

Finally, calculate the angle of the outlet blades, \(\beta\), using the tangential and radial components of velocity. The blade angle at the outlet can be determined from the relation:\[ \tan(\beta) = \frac{V_r}{V_t} \]\[ \tan(\beta) = \frac{50}{1.4} \approx 35.714 \]\[ \beta = \arctan(35.714) \approx 88.41^\circ \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Flow Velocity

When dealing with centrifugal fans, the flow velocity at the impeller’s outer diameter is key. It measures how fast the air moves due to the fan’s rotation.
To find this, use the formula: \[ U = \frac{\text{Outer diameter}}{2} \times \text{angular velocity} \]
Here, the given outer diameter (0.5 m) and angular velocity (188.5 rad/s) yield a flow velocity of 47.125 m/s. This tells us how quickly the air is rotated by the fan blades.

Radial Velocity

Radial velocity (Vr) shows the speed of air moving from the center to the edge of the fan.
This is crucial because it affects the volume of air moved by the fan.
With a volume flow rate (Q) of 4.5 m³/s and an outlet area (A) of 0.09 m², we find Vr using: \[ V_r = \frac{Q}{A} = \frac{4.5}{0.09} = 50 \text{ m} \cdot \text{s}^{-1} \]
This calculation helps determine how efficiently the fan moves air outward from the impeller.

Bernoulli's Equation

Bernoulli's equation links pressure, velocity, and height in fluid dynamics.
In this fan problem, we convert water gauge head to velocity head.
Given a head (h) of 0.1 m and air density (ρ) of 1.23 kg/m³, Bernoulli’s equation is: \[ P = \rho \cdot g \cdot h \]
Substituting values, we get: \[ P = 1.23 \cdot 9.81 \cdot 0.1 \approx 1.20663 \text{ Pa} \]
This pressure difference is key to understanding the energy transformation within the fan.

Outlet Blade Angle

The outlet blade angle (β) impacts how air exits the fan.
Using the tangential (Vt) and radial velocities (Vr), we apply: \[ \tan(\beta) = \frac{V_r}{V_t} \]
Given Vr of 50 m/s and Vt of 1.4 m/s, we find β: \[ \tan(\beta) = \frac{50}{1.4} \approx 35.714 \]
Solving, β ≈ 88.41°.
This steep angle ensures that air leaves the fan blades effectively, maximizing the fan's performance.

Air Density

Air density (ρ) influences how much air mass is moved by the fan.
In this problem, ρ is given as 1.23 kg/m³ and used in calculations involving pressure and velocity relationships.
For instance, ρ was critical in Bernoulli’s equation to convert the water gauge head to velocity head.
This density impacts the fan’s ability to generate pressure and thus efficiently move air through the duct system.