Water flows steadily from a large tank through a length of smooth plastic tubing, then discharge

step 1 Let us write the head loss equation using Bernoulli equation. So we can write p1 divided by row

Water flows steadily from a large tank through a length of...

Key Concepts

Laminar Flow

Laminar flow is a flow regime in which a fluid moves in smooth, parallel layers with little to no mixing between them. This behavior is typically seen at lower flow velocities and is characterized by a predictable and linear relationship between the shear stress and the velocity gradient, making it ideal for analytical solutions such as Poiseuille's law.

Reynolds Number

The Reynolds number is a dimensionless parameter used to predict flow regimes in fluid dynamics. It quantifies the ratio of inertial forces to viscous forces in a fluid and is critical in determining whether the flow will be laminar or turbulent. For flow in pipes, a Reynolds number below a certain threshold (typically around 2000) indicates a laminar flow regime.

Hagen-Poiseuille Equation

The Hagen-Poiseuille equation describes the volumetric flow rate of a viscous fluid through a cylindrical pipe under laminar flow conditions. This law relates the pressure drop along the pipe, the viscosity of the fluid, and the geometric dimensions of the pipe (such as its length and diameter) to the flow rate, allowing for the calculation of maximum flow rate before the onset of turbulence.

Hydrostatic Head

Hydrostatic head refers to the pressure exerted by a fluid due to its weight when at rest, which is directly related to the height of the fluid column. In systems where gravity drives the flow, such as a water tank discharging through tubing, the hydrostatic head is a key factor because it provides the driving pressure needed to overcome frictional and entrance losses in the tubing.

Minor (Entrance) Losses

Minor losses refer to additional pressure losses that occur at points of abrupt changes in the flow geometry, such as at the entrance or exit of a pipe. These losses are not accounted for by the simple frictional (or viscous) losses in a long, straight pipe and are often characterized by an entrance loss coefficient. They must be included in calculations for system pressure drops to accurately predict flow behavior, especially near transitions between different flow regimes.

Transcript

00:01 Let us write the head loss equation using bernoulli equation.

00:04 So we can write p1 divided by row plus alpha 1 multiplied by b1 squared divided by 2 plus gzz1, this term minus p2 divided by row plus alpha 2 multiplied by v2 squared divided over 2 plus gz2, this will be equals to the frictional head loss and the major head loss and the minor head loss.

00:31 Both will be, this will be the head loss due to friction plus minor head loss.

00:41 This will be minor head loss.

00:43 So let this is equation number one.

00:45 Here, the hl, the frictional head loss can be written as fl v.

00:51 2, fl, v2, squared, divided 2.

00:53 And the minor head loss, hlm, can be written as k nozzle, k -nozzle, multiplied by v2, v2, v2 squared divided by 2, v2, 2, v2 squared, so let us assume assume assume v1 is equal to 0 v1 equals to 0 z2 equals to 0 p1 z2 equals to 0 p1 z2 equals to 0 p1 is equal to p atm is equal to 0 so from equation one from equation one we can write we can write gz1 minus alpha 2 v2 squared divided over 2 is equals to is equals to k entrance k entrance multiplied by v2 squared divided by 2 v2 squared divided 2 plus f l v2 squared divided by 2 d 2d here it will be d 2d so from here we get z1 is equals to v2 squared divided by 2g bracket close alpha 2 plus k entrance k entrance plus fl divided by d let this equation be equation number 2 now for laminar flow for laminar flow laminar flow friction factor f is equal to 64 divided by reynolds number re so it will be 64 divided by 2300 so it will come out to be 0 .078 278 hence we know that now reynolds number can be written as row vd divided by mu or we have we have we have we have we have vd divided by new should be less than equals to 2300 for liminar flow or from here we can write v is equal to 2300 new divided by d from table property of water at at temperature is equal to 20 degrees centigrade we have kinematic viscosity new is equal to 1 .01 1 .01 multiplied by 10 to of minus 6 meters per second 6 meters square per second so we will put this value and we will get the value of velocity v is equal to 2300 multiplied by 1 .01 multiplied by 10 2d power of minus 6 divided by 3 .18 multiplied by 10 to the power of minus 3 so from here we get the velocity as 0 .73 meter per second now from continuity equation from continuity equation continuity equation we can write the discharge q is equal to area multiplied by velocity is equal to velocity 0 .73 multiplied by area is pi divided by four multiplied by diameter square which is equals to 3 .18 multiplied by 10 2d power of minus 3 square so from here we get the discharge q is equal to 5 .8 into 10 to the power of minus 6 meter cube per second 5 .8 multiplied by 10 to the power of minus 3 meter cube per second.

04:57 We will convert it into liter per minute...

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