To determine The mass flow rate for the flow through the annular space. The rate of momentum for the flow through the annular space. The rate of kinetic energy for the flow through the annular space. Explanation The figure shows the velocity profile for the flow through the annular space. Figure-(1) Write the expression of the radius of elemental ring. r 1 = a ( r o − r i n ) + r i n (I) Here, the inner radius of annular space is r i n , the outer radius of annular space is r o , the value of ratio is a , the radius of element at ratio a is r 1 . Write the expression of the radius of elemental ring. r 2 = b ( r o − r i n ) + r i n (II) Here, the inner radius of annular space is r i n , the outer radius of annular space is r o , the value of ratio is b , the radius of element at ratio 2 is r 2 . Write the expression for the total mass flow rate. m ˙ = ∑ i = 1 i = 10 m i = ∑ i = 1 i = 10 ρ A i u i = ∑ i = 1 i = 10 ( ρ ) ( π ) ( r i 2 − r i − 1 2 ) ( 12 ) 2 ( u i − 1 + u i 2 ) m ˙ = ( ρ ) ( π ) ( 12 ) 2 [ ( ( r 1 2 − r 0 2 ) ( u 0 + u 1 2 ) ) + ( ( r 2 2 − r 1 2 ) ( u 1 + u 2 2 ) ) + ......... + ( ( r 10 2 − r 9 2 ) ( u 9 + u 10 2 ) ) ] (III) Here, the density of the flowing fluid is ρ , the velocity of the flow at different r i is u i , the different radius of the elemental strip is r i . Write the expression for the total momentum of the flow. P = ∑ i = 1 i = 10 P i = ∑ i = 1 i = 10 ( m × u ) i = ∑ i = 1 i = 10 ( ρ ) ( π ) ( r i 2 − r i − 1 2 ) ( 12 ) 2 ( u i − 1 + u i 2 ) 2 = ( ρ ) ( π ) ( 12 ) 2 [ ( ( r 1 2 − r 0 2 ) ( u 0 + u 1 2 ) 2 ) + ( ( r 2 2 − r 1 2 ) ( u 1 + u 2 2 ) 2 ) + ......... + ( ( r 10 2 − r 9 2 ) ( u 9 + u 10 2 ) 2 ) ] (IV) Write the expression for the total kinetic energy of the flow. E = ∑ i = 1 i = 10 E i = ∑ i = 1 i = 10 ( m × u 2 2 ) i = 1 2 ∑ i = 1 i = 10 ( ρ ) ( π ) ( r i 2 − r i − 1 2 ) ( 12 ) 2 ( u i − 1 + u i 2 ) 3 = 1 2 ( ρ ) ( π ) ( 12 ) 2 [ ( ( r 1 2 − r 0 2 ) ( u 0 + u 1 2 ) 3 ) + ( ( r 2 2 − r 1 2 ) ( u 1 + u 2 2 ) 3 ) + ......... + ( ( r 10 2 − r 9 2 ) ( u 9 + u 10 2 ) 3 ) ] (V) Conclusion: Using the date from given table. a = 0.1 b = 0.2 Substitute 1.5 in for r o , 0.1 for a and 1 in for r i n in Equation (I). r 1 = 0.1 ( 1.5 in − 1 in ) + 1 in =1 .05 in Substitute 1.5 in for r o , 0.2 for b and 1 in for r i n in Equation (II). r 2 = 0.2 ( 1.5 in − 1 in ) + 1 in =1 .1 in Similarly all the data will be evaluated. Table-(1) r − r i n r o − r i n r ( in ) u ( ft / s ) 0.1 1.05 12 0.2 1.1 23 0.3 1.15 28 0.4 1

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ISBN: 9781119571490

Author: GERHART

Publisher: WILEY

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Chapter 5.3, Problem 126P

To determine

The mass flow rate for the flow through the annular space.

The rate of momentum for the flow through the annular space.

The rate of kinetic energy for the flow through the annular space.

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Chapter 5.3, Problem 125P

Chapter 5.3, Problem 127P

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