Question

(II)Estimate the air pressure inside a category 5 hurricane, where the wind speed is 300 km/h (Fig. 10-52).

Short answer

The air pressure inside a category 5 hurricane is \(96.78\;{\rm{kPa}}\).

Step-by-step solution

Understanding the Bernoulli’s equation

The air pressure inside a hurricane can be estimated by utilizing the reduced form of Bernoulli’s equation.

Given data

The wind speed inside the hurricane is \({v_1} = 300\;{\rm{km/h}}\).

The standard value for the density of air is \(\rho = 1.3\;{\rm{kg/}}{{\rm{m}}^3}\), and the pressure outside the hurricane is equal to atmospheric pressure \({P_1} = {P_{atm}} = 101.3 \times {10^3}\;{\rm{Pa}}\).

Evaluating the air pressure inside the hurricane

The air pressure inside a hurricane is calculated below:

\(\begin{array}{c}{P_1} - {P_2} = \frac{1}{2}\rho \left( {{{\left( {{v_2}} \right)}^2} - {{\left( {{v_1}} \right)}^2}} \right)\\{P_1} - {P_2} = \frac{1}{2}\rho \left( {{{\left( 0 \right)}^2} - {{\left( {{v_1}} \right)}^2}} \right)\\{P_1} - {P_2} = \frac{1}{2}\rho {\left( {{v_1}} \right)^2}\\{P_2} = {P_1} - \frac{1}{2}\rho {\left( {{v_1}} \right)^2}\end{array}\)

Here, \({v_2}\) is the wind speed outside the hurricane.

Substitute the values in the above equation.

\(\begin{array}{c} = 101.3 \times {10^3}\;{\rm{Pa}} - \left( {\frac{1}{2}\left( {1.3\;{\rm{kg/}}{{\rm{m}}^3}} \right){{\left( {300\;{\rm{km/h}} \times \frac{{1000\;{\rm{m}}}}{{1\;{\rm{km}}}} \times \frac{{1\;{\rm{h}}}}{{3600\;{\rm{s}}}}} \right)}^2}} \right)\\ = 96786.47\;{\rm{Pa}} \times \left( {\frac{{1\;{\rm{kPa}}}}{{1000\;{\rm{Pa}}}}} \right)\\ = 96.78\;{\rm{kPa}}\end{array}\)

Hence, the air pressure inside a hurricane is \(96.78\;{\rm{kPa}}\).