Water flows through a 2 -in.-diameter tube that suddenly contracts to 1 in. diameter. The pressu

Water flows through a 2 -in.-diameter tube that suddenly...

Key Concepts

Continuity Equation

The continuity equation is a fundamental principle in fluid mechanics stating that the mass flow rate must remain constant in a steady, incompressible flow. This implies that as the cross-sectional area of a tube changes, the velocity of the fluid must adjust in order to maintain a constant flow rate, making it an essential tool for linking changes in geometry to changes in velocity.

Bernoulli's Principle

Bernoulli's principle relates the pressure, velocity, and elevation in a flowing fluid, asserting that in a streamline flow, an increase in fluid velocity occurs simultaneously with a decrease in pressure or potential energy. This principle is crucial when analyzing pressure drops in systems where the flow velocity changes due to variations in cross-sectional areas.

Minor Losses in Fluid Flow

Minor losses refer to the additional energy losses in a fluid system due to changes in flow direction, geometry, or sudden contractions and expansions. These losses are characterized by loss coefficients and are important in calculating the actual pressure drop and flow rate, especially in systems that involve rapid changes in cross-sectional area.

Volume Flow Rate Calculation

Calculating the volume flow rate involves determining the product of the cross-sectional area of the tube and the fluid velocity. By using relationships derived from the continuity equation and Bernoulli's principle (including any adjustments for minor losses), it is possible to compute the flow rate through a system with varying geometry.

Transcript

00:01 Writing the energy equation for the sudden contraction, energy equation for sudden contraction, sudden contraction, contraction can be written as p1 divided by row plus alpha 1, v1 square divided by 2 plus zz 1 is minus p2.

00:31 P2 is equals to p2 divided by row plus alpha 2 multiplied by v2 squared divided by 2 plus g z 2 plus g z plus head loss due to sudden contraction.

00:45 Plus head loss due to sudden contraction.

00:55 Now we can also write this as p1 divided by row plus alpha 1 multiplied by b1 squared by 2 plus z z1 is equal to is equals to p2 divided by ro plus alpha 2 multiplied by v2 squared divided by 2 plus z z z 2 now the head laws due to sudden contraction can be written as kv squared divided by 2 kv2 squared divided by 2 here p1 divided by row is pressure energy v1 squared divided by 2 is kinetic energy.

01:33 So since, since pipe is horizontal, since pipe is horizontal, hence z1 will be equals to z2.

01:48 Also assume the kinetic kinetic energy correction factor, that is alpha 1 will be equals to alpha 2 will be equals to 1.

01:56 Now we will put all this value, all this value, in the above equation, we get p1 minus p2, by row plus b1 squared divided by 2 is equal to k multiplied by v2 divided by 2 plus v2 squared divided by 2 2 let this is equation number 1 now we will find the k value by using the area ratio the area ratio area ratio area ratio area ratio area ratio area ratio area ratio will be equal to will be equals to a 2 divided by a 1 so a 2 can be written as pi divided by 4 d2 square pi divided by 4 d2 square divided by a 1 can be written as pi divided by 4 d1 square so from here we get area ratio a 2 divided by a 1 is equal to the diameter square 2 square divided by d1 square so we will put the value of the value of d1 and d2 from here we get 1 divided by 2 whole square or we get we get a 2 divided by a 1 is equal to 0 .25 so now we will find the contraction loss coefficient at ar is equal to 0 .25.

03:21 This is at ar is equal to 0 .25 the contra, the from the figure 8 .15 the loss coefficient for flow through sudden area change in the textbook is given as k is equal to 0 .4.

03:40 From continitive equation, from continitive equation, continuity equation, we can write a1 v1 is equal to a2 v2 or we can write v1 is equal to a2 v2 divided by a1 or we can write a2 a2, v2, a2 divided by a1 is, or we have already found as 2 .25 multiplied by v2 is v2...

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