Semantics making Bernoulli's Eq KILLER see if you can make sense

[missnola2a]

**semantics making Bernoulli's Eq KILLER! see if you can make sense

Homework Statement

a pipe that carries a fluid ρ = 873 kg/m3 lies along the slope of a smooth hill which is 140 m high. The speed of the liquid at the bottom of the hill is 6.5 m/s. The diameter of the crude oil pipe line at the top of the hill reduces by a factor 5. Please answer the following:
Hint
(a) By what factor the area of the oil pipe line reduces at the top of the hill?

(b) What is the ratio of speeds of the oil from the bottom of the hill to the top of the hill?

(c) How much potential energy per unit volume the oil gains as it moves from the bottom to the top of the hill?
J/m3
(d) What is the change in pressure in the pipe line during this motion along the hill?

Homework Equations

The Attempt at a Solution

OK ... so I first of all am confused by this because I can't tell if the pipe starts at the bottom of the hill and goes up or vice versa.

This may not matter mathematically, but I am a conceputal person.

That said,
a1v1=a2v2

(pie drops out)
r1^2 v1= r2^2 v2

this is where i am a little stuck, should it be 5r1=r2
OR r2=r1/5

this changes the equation a lot. after I get v2 (or v1) I put it into origional

P1 + 1/2 rho V1^2 + rho*g*Y1 = P2 + 1/2 rho V2^2

*** assuming that PE for 2 is 0, but again, i am confused as to where in the system Y=0

thanks for your help

 

[minger]

Well, first you could assume that the pipe starts at the top and then see if there is enough kinetic energy to make it to the top. Also, it says that the diameter of the crude oil pipe line at the top of the hill reduces by a factor 5

So, the diameter at the top of the hill is 5 times smaller than at the bottom.

 

[tomcue]

!

First of all, semantics refers to the meaning and interpretation of words and phrases. Bernoulli's equation is a fundamental principle in fluid mechanics that relates the pressure, velocity, and elevation of a fluid in a closed system. Therefore, the phrase "Semantics making Bernoulli's Eq KILLER" does not make sense in this context.

Moving on to the problem, it seems that the pipe starts at the bottom of the hill and goes up. This means that the oil is gaining potential energy as it moves up the hill, and the pressure will decrease as the velocity increases.

To answer the questions:

(a) The area of the pipe line reduces by a factor of 25 (5^2) at the top of the hill. This is because the diameter is reduced by a factor of 5, which means the area is reduced by a factor of 25.

(b) The ratio of speeds will be the inverse of the ratio of diameters, since the volume flow rate must remain constant. So the ratio of speeds will be 1/5.

(c) The potential energy gained per unit volume can be calculated using the formula PE = mgh, where m is the mass per unit volume (rho), g is the acceleration due to gravity, and h is the height gained. So the potential energy gained per unit volume will be (873 kg/m^3)(9.8 m/s^2)(140 m) = 1,235,320 J/m^3.

(d) To calculate the change in pressure, we can use the Bernoulli's equation P1 + 1/2 rho V1^2 + rho*g*Y1 = P2 + 1/2 rho V2^2. Since the elevation is increasing, Y1 < Y2 and thus the pressure at the top of the hill (P2) will be lower than at the bottom (P1). The change in pressure will be negative, indicating a decrease in pressure.

I hope this helps clarify the problem for you. Remember to always pay attention to the given information and units, and to think about the physical meaning behind the equations. Good luck with your studies!