Question

Water is traveling through a horizontal pipe with a speed of 1.7 m/s and at a pressure of 205 kPa. This pipe is reduced to a new pipe which has a diameter half that of the first section of pipe. Determine the speed and pressure of the water in the new, reduced in size pipe.

Answer 1

Answer:

The velocity is  $v_2 = 6.8 \ m/s$

The pressure is  $P_2 = 204978 Pa$

Explanation:

From the question we are told that

 The speed at which water is travelling through is  $v = 1.7 \ m/s$

  The pressure is  $P_1 = 205 k Pa = 205 *10^{3} \ Pa$

   The diameter of the new pipe is $d = \frac{D}{2}$

Where D is the diameter of first pipe

   

According to the principal of continuity we have that

       $A_1 v_1 = A_2 v_2$    

Now  $A_1$ is the area of the first pipe which is mathematically represented as

       $A_1 = \pi \frac{D^2}{4}$

and  $A_2$ is the area of the second pipe which is mathematically represented as  

       $A_2 = \pi \frac{d^2}{4}$

Recall   $d = \frac{D}{2}$

        $A_2 = \pi \frac{[ D^2]}{4 *4}$

        $A_2 = \frac{A_1}{4}$

So    $A_1 v_1 = \frac{A_1}{4} v_2$

substituting value

        $1.7 = \frac{1}{4} * v_2$    

        $v_2 = 4 * 1.7$    

       $v_2 = 6.8 \ m/s$

   

According to Bernoulli's equation  we have that

     $P_1 + \rho \frac{v_1 ^2}{2} = P_2 + \rho \frac{v_2 ^2}{2}$

substituting values

     $205 *10^{3 }+ \frac{1.7 ^2}{2} = P_2 + \frac{6.8 ^2}{2}$

     $P_2 = 204978 Pa$