Question

The variation of pressure with density in a thick gas layer is given by \(P=C \rho^{n}\), where \(C\) and \(n\) are constants. Noting that the pressure change across a differential fluid layer of thickness \(d z\) in the vertical \(z\) -direction is given as \(d P=-\rho g d z,\) obtain a relation for pressure as a function of elevation \(z .\) Take the pressure and density at \(z=0\) to be \(P_{0}\) and \(\rho_{0},\) respectively.

Short answer

Answer: The expression for pressure as a function of elevation is \(P(z) = \left[P_{0}^{(1-n)/n} - \frac{(1-n)gz}{nC^{1/n}}\right]^{n/(1-n)}\), where \(P_0\) is the pressure at \(z = 0\), \(g\) is the acceleration due to gravity, and \(C\) and \(n\) are constants.

Step-by-step solution

Express density as a function of pressure

Given the relation \(P = C\rho^n\), we can express the density \(\rho\) as a function of pressure \(P\) by solving for \(\rho\): $$\rho = \left(\frac{P}{C}\right)^{1/n}.$$

Substitute the density expression into the pressure change equation

Now, let's substitute the expression for \(\rho\) in terms of \(P\) into the pressure change equation, \(dP = -\rho g dz\): $$dP = -\left(\frac{P}{C}\right)^{1/n} g dz.$$

Rearrange the equation to separate variables

We can rearrange the above equation to separate the variables \(P\) and \(z\) to one side each as follows: $$\frac{dP}{P^{1/n}} = -\frac{g}{C^{1/n}} dz.$$

Integrate both sides of the equation

Now we can integrate both sides of the equation. For the integration boundaries, use \(P = P_0\) when \(z = 0\) and \(P = P(z)\) when \(z = z\): $$\int_{P_0}^{P(z)} \frac{dP}{P^{1/n}} = -\frac{g}{C^{1/n}} \int_0^z dz.$$ The left integral results in: $$n\left[\frac{P^{(1-n)/n}}{1-n}\right]_{P_0}^{P(z)}.$$ The right integral results in: $$-\frac{g}{C^{1/n}}z.$$

Solve for P(z)

Now we simply need to solve for \(P(z)\): $$n\left[\frac{P^{(1-n)/n}}{1-n} - \frac{P_{0}^{(1-n)/n}}{1-n}\right] = -\frac{g}{C^{1/n}}z.$$ The expression for \(P(z)\) will be: $$P(z) = \left[P_{0}^{(1-n)/n} - \frac{(1-n)gz}{nC^{1/n}}\right]^{n/(1-n)}.$$