Water is flowing in a pipe as depicted in the figure. As shown, p = 117 kPa, d = 4.67 cm, d' = 3.06 cm, v = 4.36 m/s, and h = 1.11 m. What pressure is indicated on the upper pressure gauge? Take y = 0 at the bottom of the left pipe so that the center of the left pipe is y1 = d/2.
Without the figure, I have no idea what the two diameters are, nor where the uppoer pressure gauge is located.
Use the continuity equation
V+d^2 = constant
to relate velocity to pipe diamter.
Then use the Bernoulli equation
(1/2)*(rho)^V^2 + P +(rho)*g*h = 0
to relate pressure (P) to velocity and height. rho is the density of water.
To find the pressure indicated on the upper pressure gauge, we can use Bernoulli's equation, which states that the sum of the pressure, kinetic energy, and potential energy at any two points in the flow of a fluid is constant.
Let's name the two points where we want to find the pressure: point 1 is at the bottom of the left pipe and point 2 is at the top of the left pipe.
Using Bernoulli's equation, we have:
P1 + (1/2) * ρ * v1^2 + ρ * g * y1 = P2 + (1/2) * ρ * v2^2 + ρ * g * y2
Where:
P1 = Pressure at point 1 (bottom of the left pipe)
v1 = Velocity of water at point 1
y1 = Height of point 1
P2 = Pressure at point 2 (top of the left pipe)
v2 = Velocity of water at point 2
y2 = Height of point 2
ρ = Density of water
g = Acceleration due to gravity
In this case, the height difference y2 - y1 is equal to the total height of the left pipe h.
Substituting the given values into the equation:
117 kPa + (1/2) * ρ * (4.36 m/s)^2 + ρ * 9.8 m/s^2 * (d/2) = P2 + (1/2) * ρ * 0^2 + ρ * 9.8 m/s^2 * (d/2 + h)
Now let's calculate the height difference h2 = d/2 + h:
h2 = (4.67 cm) / 2 + 1.11 m = 2.335 cm + 1.11 m = 1.144 m
Substituting the value back into the equation:
117 kPa + (1/2) * ρ * (4.36 m/s)^2 + ρ * 9.8 m/s^2 * (d/2) = P2 + ρ * 9.8 m/s^2 * 1.144 m
Now we need to convert the given pressure from kilopascals to pascals (1 kPa = 1000 Pa):
117 kPa = 117,000 Pa
Substituting the values, we get:
117,000 Pa + (1/2) * ρ * (4.36 m/s)^2 + ρ * 9.8 m/s^2 * (4.67 cm / 2) = P2 + ρ * 9.8 m/s^2 * 1.144 m
Now, we need to consider that the density of water ρ is approximately 1000 kg/m^3.
Substituting the value into the equation:
117,000 Pa + (1/2) * 1000 kg/m^3 * (4.36 m/s)^2 + 1000 kg/m^3 * 9.8 m/s^2 * (4.67 cm / 200) = P2 + 1000 kg/m^3 * 9.8 m/s^2 * 1.144 m
Simplifying the equation:
117,000 Pa + 9,132.96 Pa + 226.16 Pa = P2 + 10,752.96 Pa
Now, let's solve for P2:
117,000 Pa + 9,132.96 Pa + 226.16 Pa = P2 + 10,752.96 Pa
Subtracting 10,752.96 Pa from both sides:
117,000 Pa + 9,132.96 Pa + 226.16 Pa - 10,752.96 Pa = P2
Now, let's calculate:
115,606.16 Pa = P2
Therefore, the pressure indicated on the upper pressure gauge is 115,606.16 Pa.
To find the pressure indicated on the upper pressure gauge, we can use the Bernoulli's equation, which relates the pressure, velocity, and elevation of a fluid flow.
The Bernoulli's equation for an incompressible fluid flow is:
P + (1/2)ρv^2 + ρgh = constant
where:
P is the pressure of the fluid
ρ is the density of the fluid
v is the velocity of the fluid
g is the acceleration due to gravity
h is the elevation of the fluid
In this case, we are given the values of p, d, d', v, and h. We need to convert the given values to appropriate units before using the equation.
1. Convert p from kPa to Pa:
p = 117 kPa = 117,000 Pa
2. Convert d and d' from cm to m:
d = 4.67 cm = 0.0467 m
d' = 3.06 cm = 0.0306 m
3. Convert v from m/s to the square of m/s:
v = 4.36 m/s
4. Convert h from m to the square of m:
h = 1.11 m
Now, let's calculate the pressure indicated on the upper pressure gauge by using the Bernoulli's equation.
The constant term in the equation can be obtained by substituting the values at point 1 (bottom of the left pipe):
P1 + (1/2)ρv1^2 + ρgh1 = constant
Since the pressure is measured relative to atmospheric pressure, we can assume P1 is equal to atmospheric pressure (close to 0). Therefore, the equation becomes:
(1/2)ρv1^2 + ρgh1 = constant (equation 1)
Now, we need to calculate the value of the constant term using the given values at point 2 (upper pressure gauge):
P2 + (1/2)ρv2^2 + ρgh2 = constant (equation 2)
Since we are asked to find the pressure indicated on the upper pressure gauge, P2 is the unknown.
To solve for P2, we first need to find the values of v1, v2, and h2.
v1 = v (velocity at the bottom of the left pipe) = 4.36 m/s
To find v2, we can use the principle of conservation of mass, which states that the mass flow rate is constant in an incompressible fluid (assuming no leaks or sources). Therefore:
A1v1 = A2v2
where A1 and A2 are the cross-sectional areas of the corresponding pipe sections.
The cross-sectional area A1 can be calculated using the diameter d1:
A1 = π(d1/2)^2 = π(0.0467/2)^2
Similarly, the cross-sectional area A2 can be calculated using the diameter d2:
A2 = π(d2/2)^2 = π(0.0306/2)^2
Now, we can solve for v2:
A1v1 = A2v2
v2 = (A1v1) / A2
Finally, we need to find the value of h2. Since the pressure is measured relative to a specific point (y=0 at the bottom of the left pipe in this case), the elevation h2 is equal to the difference in heights between the two points. In this case, h2 is the height difference between the bottom of the pipe (y=0) and the upper pressure gauge (y=h). Therefore, h2 = h.
Now, we have all the values required to calculate the pressure indicated on the upper pressure gauge.
Substituting the values into equation 2:
P2 + (1/2)ρv2^2 + ρgh2 = constant
Since ρ and the constant term cancel out, the equation becomes:
P2 + (1/2)ρv2^2 + ρgh2 = (1/2)ρv1^2 + ρgh1 (equation 3)
Substituting the known values:
P2 + (1/2)ρv2^2 + ρgh2 = (1/2)ρv1^2 + ρgh1
P2 + (1/2)(ρv2^2) + (ρgh) = (1/2)(ρv1^2) + (ρgh1)
Since P1 is close to atmospheric pressure (close to 0), we can assume P1 and P2 are approximately equal. Therefore, we can simplify the equation as follows:
(1/2)(ρv2^2) + (ρgh) = (1/2)(ρv1^2) + (ρgh1)
Now, follow the steps below to calculate the pressure indicated on the upper pressure gauge:
1. Calculate ρ (density of water) using the known values. The density of water is approximately 1000 kg/m^3.
2. Calculate v2 using the previously calculated values for A1, A2, and v1.
3. Substitute the known values of ρ, v2, v1, h, and h1 into the equation above:
(1/2)(ρv2^2) + (ρgh) = (1/2)(ρv1^2) + (ρgh1)
4. Finally, solve for P2 by subtracting the other terms from both sides of the equation, then convert the units to kPa if necessary:
P2 = (1/2)(ρv1^2) + (ρgh1) - (1/2)(ρv2^2) - (ρgh)
By following these steps, you should be able to calculate the pressure indicated on the upper pressure gauge.