A large water tank has an inlet pipe and an outlet pipe. The...

A large water tank has an inlet pipe and an outlet pipe. The inlet pipe has a diameter of $2.00 \mathrm{~cm}$ and is $1.00 \mathrm{~m}$ above the bottom of the tank. The outlet pipe has a diameter of $5.00 \mathrm{~cm}$ and is $6.00 \mathrm{~m}$ above the bottom of the tank. A volume of $0.300 \mathrm{~m}^{3}$ of water enters the tank every minute at a gauge pressure of 1.00 am. a) What is the velocity of the water in the outlet pipe? b) What is the gauge pressure in the outlet pipe?

A large water tank has an inlet pipe and an outlet pipe. The inlet pipe has a diameter of $2.00 \mathrm{~cm}$ and is $1.00 \mathrm{~m}$ above the bottom of the tank. The outlet pipe has a diameter of $5.00 \mathrm{~cm}$ and is $6.00 \mathrm{~m}$ above the bottom of the tank. A volume of $0.300 \mathrm{~m}^{3}$ of water enters the tank every minute at a gauge pressure of 1.00 am.
a) What is the velocity of the water in the outlet pipe?
b) What is the gauge pressure in the outlet pipe?

Key Concepts

Gauge Pressure

Gauge pressure represents the pressure measured relative to the ambient atmospheric pressure. It is key in such problems since the pressure readings provided are above or below atmospheric pressure, requiring adjustments when applying fluid dynamics and Bernoulli’s equation.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the gravitational force acting on it. It is directly proportional to the depth of the fluid and is an important factor when considering differences in elevation in problems involving liquid tanks or reservoirs.

Bernoulli’s Principle

Bernoulli’s principle is a fundamental concept in fluid mechanics that states that in a streamline flow, the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant. This principle is crucial for analyzing how pressure and velocity change along the flow in pipes, especially when changes in diameter and elevation occur.

Continuity Equation

The continuity equation for incompressible fluids asserts that the mass flow rate remains constant across any cross-section of a closed system. This means that the product of the cross-sectional area and the velocity of the fluid remains constant, which is essential for relating velocities in pipes of different diameters.

Transcript

00:01 A large water tank that has inlet and outlet pipe and the diameter for the inlet pipe di i it is equals to 2 .00 cm and the height from the bottom of tank it is h i equals to 1 .00 meter for the inlet pipe and the outlet pipe has diameter so d o it is equals to 5 .0 centimeter and it is at the height h o equals to 6 .00 meter and volume of the tank is delta v equals to 0 .3 mm cube in time delta t equals to one minute that is 60 second.

00:37 So for the part a of the question and we have gaze pressure equals to 1 .00 atm also.

00:46 So for the part a of the question, we have to determine the velocity of the water at the outlet.

00:51 So we have to calculate here v not.

00:55 So we can apply the velocity of outlet in the pipe can be determined.

01:00 Determined by a not v not this will be equals to delta v by delta t this is the rate of flow okay and a not this will be the area of the outlet pipe so it can be written as pi by four multiplied by d not square multiplied by v0 this will be equals to delta v by delta t okay so from here after rearranging we will get that v0 it will be equals to velocity at the outlet it will be equals to delta v by delta t multiplied by four divided by pi d not square okay so substituting values here so we will get that delta v it is equals to 0 .3 double 0 divided by delta t which is 60 second and 4 divided by pi multiplied by the outer the outlet which is 5 cm so 5 into 10 to the power minus 2 meter so whole square and from here after solving velocity at the outlet v0 is equal to 2 point it will be equal to 2 .546 meter per second.

02:03 So this becomes the answer for this part a of the question.

02:07 Now, moving to the part b in which we have to calculate the gauge pressure in the outlet pipe.

02:13 So first of all, determine the velocity of the inlet...

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