Numerical Scheme
Finite difference methods provide the mathematical framework for simulating transient heat or mass transfer in porous media. In paper and board manufacturing, the crank nicolson algorithm resolves the second-order partial differential equations that describe how moisture spreads through a web during drying. This implicit method averages the explicit and implicit estimates of the state of the web at consecutive time steps, creating a tridiagonal matrix that is solved efficiently.
By discretizing both the temporal and spatial domains, the algorithm approximates the moisture concentration at each depth of the multi-ply board. The resulting system of algebraic equations is solved at each time step, yielding a continuous and stable profile of the moisture concentration.
Solution Stability
Numerical stability dictates the choice of time steps in dynamic simulation models. Unlike explicit schemes that restrict the time increment to prevent mathematical oscillation, the crank nicolson algorithm is unconditionally stable. It allows engineers to simulate long drying sections without encountering numerical blowups.
The precision of the simulation remains high because the truncation error is second-order in both time and space.
Board Converting
Moisture profile control during high-speed converting depends on real-time predictions of coat weight drying rates. When applying aqueous barriers to paperboard, the crank nicolson algorithm calculates the liquid diffusion into the cellulosic network before the hot air knives evaporate the carrier fluid. Plant operators configure the drying tunnels based on these transient calculations.
Accurate predictions prevent the board from curling and preserve the structural integrity of the barrier layer.
