Gas Storage Tank Pressure Thresholds Solution

In industrial mechanical and chemical plant engineering, pressurized gas containment vessels operate under automated digital monitoring to prevent explosive rupture or structural implosion. Under high-temperature thermodynamic regimes, non-ideal gases deviate from linear equations of state, exhibiting asymptotic pressure surges when thermal and volumetric inputs approach acoustic resonance or critical condensation limits.

A UNLV mechanical testing lab monitors the internal pressure $P$ (in megapascals, $\text{MPa}$) of an experimental carbon dioxide storage vessel using the rational multivariable model:

where $x$ represents normalized core gas temperature (in hundreds of Kelvin, $x > 0$) and $y$ represents a dimensionless chamber volume expansion ratio ($y > 0$).

During a dynamic test cycle, a control valve malfunction attempts to set the operational parameters to $(x, y) = (2, 2)$.

1. State the algebraic domain condition for $g(x, y)$ and write the mathematical domain $D$ in set-builder notation. Describe the geometric curves in the $xy$-plane that must be excluded from the operational domain.
2. Determine the mathematical range $R$ of the pressure function across its unrestricted domain and across the physical positive quadrant ($x > 0, y > 0$).
3. Evaluate the pressure function at the target settings $(x, y) = (2, 2)$. Explain what happens as operating conditions approach this point, and discuss how automated safety control logic must respond.

Step-by-Step Solution:

1. Identify the Denominator Restriction and Domain: State the Non-Zero Denominator Condition for Rational Functions:

   $$x^2 - y^2 \ne 0$$
   Factoring as a difference of squares:
   $$(x - y)(x + y) \ne 0 \implies y \ne x \quad \text{and} \quad y \ne -x$$
   • Set-Builder Notation:
     $$D = \{(x, y) \in \mathbb{R}^2 \mid y \ne \pm x\}$$
   • Geometric Description: The domain excludes the two perpendicular diagonal lines $y = x$ and $y = -x$ intersecting at the origin. Within the physical operating quadrant where temperature $x > 0$ and expansion ratio $y > 0$, the single excluded operational boundary is the diagonal line $y = x$. (Defined in Domain and Range of Multivariable Functions.)

2. Determine the Pressure Range: State the Range Analysis for Rational Inversion Functions: Let $u = x^2 - y^2$.

   • In the first quadrant ($x > 0, y > 0$), when temperature dominates ($x > y$), $u \in (0, \infty)$, yielding positive pressures $P = \frac{10}{u} \in (0, \infty)$.
   • When volume expansion dominates ($y > x$), $u \in (-\infty, 0)$, yielding negative gauge pressures $P \in (-\infty, 0)$ (representing vacuum conditions).
   • Because a fraction with a fixed non-zero numerator ($10$) can never equal zero ($10/u \ne 0$ for any finite $u$), zero is excluded from the range.
   • Range:
     $$R = (-\infty, 0) \cup (0, \infty)$$

3. Evaluate at $(2, 2)$ and Asymptotic Limit Analysis: Substitute $x = 2$ and $y = 2$ into the denominator:

   $$x^2 - y^2 = 2^2 - 2^2 = 4 - 4 = 0$$
   Evaluating $g(2, 2) = \frac{10}{0}$, which is undefined.
   • As operating conditions approach $(2, 2)$ from the region where $x > y$ (temperature higher than expansion ratio), the denominator approaches $0$ through positive values:
     $$\lim_{(x, y) \to (2, 2), x > y} \frac{10}{x^2 - y^2} = +\infty\text{ MPa}$$
   • As operating conditions approach $(2, 2)$ from the region where $x < y$, the denominator approaches $0$ through negative values:
     $$\lim_{(x, y) \to (2, 2), x < y} \frac{10}{x^2 - y^2} = -\infty\text{ MPa}$$

Meaning of the Answer & Real-Life Application:

The line $y = x$ represents an infinite pressure singularity in the empirical state equation. In physical reality, attempting to operate near $x = y$ drives the containment vessel toward infinite hoop stress, guaranteeing explosive over-pressure rupture if $x \to y^+$, or severe vacuum collapse if $x \to y^-$. Control systems in UNLV energy facilities utilize software interlocks that define an exclusion buffer around $y = x$ (e.g., enforcing $|x - y| \ge 0.25$), triggering immediate emergency pressure-relief venting and burner cut-offs whenever operational telemetry approaches the singular diagonal line.