Theoretical burst pressure and working pressure are two critical parameters in the design and operation of pressure vessels, but they serve different purposes and have distinct definitions:

### Theoretical Burst Pressure:

**Definition:**The theoretical burst pressure is the calculated pressure at which a pressure vessel is expected to fail or rupture under ideal conditions.**Calculation:**It is determined using material properties (such as tensile strength), geometric dimensions (such as wall thickness and inner diameter), and assumptions about stress distribution. For thin-walled cylinders, the formula is: [ P_b = \frac{2 \cdot t \cdot \sigma_t}{D} ] where ( P_b ) is the burst pressure, ( t ) is the wall thickness, ( \sigma_t ) is the tensile strength of the material, and ( D ) is the inner diameter.**Purpose:**It provides an upper limit of the pressure the vessel can withstand before failure. It is used for design and safety assessments to ensure the vessel is sufficiently strong.

### Working Pressure (or Design Pressure):

**Definition:**The working pressure, also known as the design pressure, is the maximum pressure that the pressure vessel is designed to operate under during normal conditions.**Establishment:**It is set by engineers and designers based on the intended use of the vessel, including factors such as process requirements, safety margins, and regulatory standards.**Safety Margin:**The working pressure is always set lower than the theoretical burst pressure to incorporate a safety factor. This safety factor accounts for uncertainties in material properties, manufacturing tolerances, and variations in operating conditions.

### Key Differences:

**Purpose:**

**Theoretical Burst Pressure:**Indicates the pressure at which the vessel is likely to fail.**Working Pressure:**Indicates the maximum safe operating pressure during normal use.

**Safety Margin:**

**Theoretical Burst Pressure:**No inherent safety margin; it is a theoretical limit.**Working Pressure:**Includes a safety margin to ensure safe operation under varying conditions.

**Usage:**

**Theoretical Burst Pressure:**Used for design validation and safety assessments.**Working Pressure:**Used for specifying operational limits and ensuring regulatory compliance.

**Calculation:**

**Theoretical Burst Pressure:**Calculated based on material strength and geometry.**Working Pressure:**Determined by design criteria, including safety factors and operational requirements.

### Practical Example:

Suppose a pressure vessel has a theoretical burst pressure of 100 MPa. The engineers might set the working pressure at 25 MPa, incorporating a safety factor of 4. This means the vessel is designed to operate safely at pressures up to 25 MPa, well below the 100 MPa at which it would theoretically fail. This approach ensures safety and reliability in practical use.