Final answer:
Hydrostatic test pressure when design temperature is above test temperature is calculated using the hydrostatic pressure relationship P = ρgh, with temperature adjustments.
Step-by-step explanation:
The correct answer is option P₂ = P₁ + ρgh₁. The formula for calculating hydrostatic test pressure at a higher design temperature compared to the test temperature involves adjusting the pressure to account for the effect of temperature difference on pressure. When design temperature is above test temperature, you need to consider the expansion of the fluid and the vessel material leading to a lower density at design temperature. This usually involves using hydrostatic pressure formulas combined with temperature correction factors. For a static fluid, the hydrostatic pressure increases with depth according to the relationship P = ρgh, where ρ is the density at test temperature, g is the acceleration due to gravity, and h is the height of the fluid column.
Furthermore, when taking temperature into consideration, a common practical approach is to increase the pressure proportionally to the temperature increase using the ideal gas law, although this would be more appropriate for gases rather than liquids. The correct determination often depends on having the precise material properties at both test and design temperatures. The specific parameters in the equation such as the density (ρ) and the height (h) would be measured or given at the test temperature, but adjustments might be needed for the design conditions.