# Bulk Modulus Application

1. Jan 21, 2016

### reddawg

1. The problem statement, all variables and given/known data
See image.

2. Relevant equations
pressure = density*gravity*depth

3. The attempt at a solution
The pressure at 5000 ft according to the book is 322,000 psf. This makes sense because density*gravity*depth = 2*32.2*5000 = 322,000 psf. How do I apply the bulk modulus equation to find the pressure at 5000 ft factoring a variable density with depth (The book says its 323,200 psf which makes sense because density increases with depth, although very slightly)?

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2. Jan 21, 2016

### Staff: Mentor

What is the hydrostatic equation, expressed in terms of the derivative of pressure with respect to depth?

3. Jan 21, 2016

### reddawg

That would be just [density*gravity]. So how do I apply that?

4. Jan 21, 2016

### Staff: Mentor

Right. $$\frac{dp}{dz}=\rho g$$
Now, from the bulk modulus equation, if the density approaches $\rho_0$ at low pressures, what is the density at pressure p?

5. Jan 21, 2016

### reddawg

That's where I have trouble. I always end up with just [density*gravity*depth]. Is it just (1/g)*(dp/dz) ?

6. Jan 21, 2016

### Staff: Mentor

The relationship between density and pressure does not involve z. It's strictly a physical property relationship (sort of like the ideal gas law, except for a liquid).

Chet

7. Jan 21, 2016

### reddawg

So, solving for density using the Bulk Modulus equation:

ρ = B*(Δρ/p)

8. Jan 21, 2016

### Staff: Mentor

Actually, the equation is $$\frac{1}{\rho}\frac{d\rho}{dp}=\frac{d(\ln \rho)}{dp}=\frac{1}{B}$$ What do you get if you integrate that, subject to the initial condition $\rho=\rho_0$ at p --> 0?

9. Jan 21, 2016

### reddawg

I get:

(1/B)*p = ln(ρ/ρ0) when factoring in the initial conditions.

10. Jan 21, 2016

### Staff: Mentor

Good. Now solve for $\rho$ in terms of p. What do you get?

11. Jan 22, 2016

### reddawg

ρ = ρ0ep/B

12. Jan 22, 2016

### Staff: Mentor

OK. Now substitute that into the hydrostatic equation in post #4. What do you get? Can you integrate that from z =0?

13. Jan 22, 2016

### reddawg

(dp/dz) = ρ0gep/B

How do I rearrange that to integrate from z=0 to h?

14. Jan 22, 2016

### Staff: Mentor

Cmon man.

$$e^{-\frac{p}{B}}dp=\rho_0 g dz$$