Vorlesung 5

1. Cases to Replace States by Averages:

   • No variation of states in a direction (zero gradients).

   • No significant transport or resistance in a direction.

   • When detailed resolution is unnecessary, and averaging does not lose critical information.

2. Method:

   • Use cross-sectional or volumetric averaging to represent the state with a single average value.

   • Example: In a tubular reactor with turbulent flow, replace the detailed temperature profile T(z,r,θ,t) with an average T(z,t).

1. Definition: Integral balances are mathematical expressions that account for the total accumulation, inflow, outflow, and production/consumption of a property over a control volume.

2. Types of Integral Balances:

   • Mass Balance: Accounts for the mass entering and leaving a system.

   • Energy Balance: Considers energy changes within a system.

   • Momentum Balance: Focuses on momentum changes due to forces and flows.

Applications:

• When modeling systems where detailed spatial resolution is unnecessary or impractical.

• In well-mixed systems where spatial variations are negligible.

• For systems with large control volumes where averaging over the entire volume is appropriate.

• Used to simplify complex systems into manageable calculations.

1. Definition: Lumped models assume that system properties (e.g., temperature, concentration) are uniform throughout the entire control volume.

2. Characteristics:

   • No spatial variation; only temporal changes are considered.

   • Simplifies systems to 0D models by assuming homogeneous properties.

   • Often used for systems with rapid mixing or where spatial effects are negligible.

1. Homogeneous Domains:

   • Assumes uniform properties throughout but may consider spatial gradients.

   • Can account for variations across different regions within a control volume.

2. Lumped Models:

   • Ignores spatial gradients entirely.

   • Assumes instantaneous mixing leading to uniformity.

   • Focuses solely on time-dependent changes.

1. Relationship:

   • Differential Balances: Provide detailed, point-wise descriptions of changes at specific locations within a system.

   • Integral Balances: Offer an overall perspective by integrating differential balances over a control volume.

2. Use Cases:

   • Differential Balances: Used for detailed, localized analysis.

   • Integral Balances: Used for system-wide or averaged analysis, simplifying complex systems.