Question

Real gas pseudo pressure is used to derive the diffusivity equation for a highly compressible fluid, for which viscosity \(\mu\) and gas compressibility factor \(Z\) are functions of pressure \(p\). Which of the following represents the partial derivative of the real gas pseudo pressure with respect to time \(t\)?

• A. \( \left(\dfrac{2p}{\mu Z}\right)\dfrac{\partial p}{\partial t} \)
• B. \( \left(\dfrac{p^{2}}{2\mu Z}\right)\dfrac{\partial p}{\partial t} \)
• C. \( \left(\dfrac{\mu Z}{2p}\right)\dfrac{\partial p}{\partial t} \)
• D. \( \left(\dfrac{2\mu Z}{p^{2}}\right)\dfrac{\partial p}{\partial t} \)

Hint:

Differentiate the pseudo pressure integral using the Leibniz rule to get \(\partial m/\partial p\), then multiply by \(\partial p/\partial t\) using the chain rule.

Answer 1

The Correct Option is A

Step 1: Write down the definition of the real gas pseudo pressure:
To handle the strong pressure dependence of viscosity and compressibility factor for a highly compressible real gas, the diffusivity equation is linearised using the real gas pseudo pressure, defined as \[ m(p) = 2\int_{p_{0}}^{p} \frac{p'}{\mu(p')Z(p')}\, dp' \] where \(p_{0}\) is an arbitrary fixed reference pressure and \(p'\) is a dummy integration variable that ranges up to the actual pressure \(p\), which itself is a function of both position and time, \(p = p(x,t)\).
Step 2: Differentiate the pseudo pressure with respect to pressure:
Because the reference pressure \(p_{0}\) is a constant, the Leibniz rule for differentiating an integral with a variable upper limit gives \[ \frac{\partial m}{\partial p} = 2\cdot \frac{p}{\mu(p) Z(p)} = \frac{2p}{\mu Z} \] This is simply the integrand evaluated at the upper limit \(p\), since the lower limit is fixed.
Step 3: Apply the chain rule to bring in time:
Since \(m\) depends on \(t\) only through \(p(x,t)\), the chain rule gives \[ \frac{\partial m}{\partial t} = \frac{\partial m}{\partial p}\cdot \frac{\partial p}{\partial t} = \frac{2p}{\mu Z}\cdot \frac{\partial p}{\partial t} \]
Step 4: Compare with the given options:
The expression obtained, \(\left(\dfrac{2p}{\mu Z}\right)\dfrac{\partial p}{\partial t}\), matches option (A) exactly. The other options either invert the ratio, use the wrong power of \(p\), or drop the factor of 2 that comes from the definition of \(m(p)\).
Final Answer:
\[ \boxed{\dfrac{\partial m}{\partial t} = \left(\dfrac{2p}{\mu Z}\right)\dfrac{\partial p}{\partial t}\ \text{(Option A)}} \]