Lecture Objectives: Review discretization methods for advection diffusion equation –Accuracy –Numerical Stability Unsteady-state CFD –Explicit vs. Implicit method Boundary conditions and Airpak –Example

Advection diffusion equation 1-D, steady-state e w P E W xx xx xx Example: Equation for temperature of water flowing through hot pipe Assume that diffusion in y direction ins negligible: This is incorrect assumption introduced jut to simplify example to 1-D problem! y x Q Q Temperature is changing along x T1 T2 T3 T4 Tn … Vx model

N N+1 N-1 xx xx xx General equation Advection diffusion equation 1-D, steady-state Different notation:

Advection equation 1-D, steady-state P E W xx xx xx 1) Upwind scheme: 2) Central differencing scheme: Higher order differencing scheme: Quadratic upwind differencing Scheme (QUICK) N N+1 N-1 N+2 N-2 We need to find coefficients a P, a W, a E, a WW, a EE, WW W P E EE Vx<0 Vx>0 3) Hybrid of upwind and central differencing scheme

Quadratic upwind differencing Scheme (QUICK) For advection only: Advection coefficient: Diffusion coefficients : Coefficients: Source:

General Transport Equation unsteady-state W E N S H L P Equation in the algebraic format: We have to solve the system matrix for each time step ! Unsteady-state 1-D Transient term: Are these values for step  or  +  ? If: -  - explicit method -  +  - implicit method

General Transport Equation unsteady-state 1-D Fully explicit method: Implicit method: Value form previous time step (known value) Make the difference between - Calculation for different time step - Calculation in iteration step

Boundary conditions in CFD application in indoor airflow Real geometry Model geometry Where are the boundary Conditions?

CFD ACCURACY Depends on airflow in the vicinity of Boundary conditions 1) At air supply device 2) In the vicinity of occupant 3) At room surfaces Detailed modeling - limited by computer power

Surface boundaries Wall surface W use wall functions to model the micro-flow in the vicinity of surface Using relatively large mesh (cell) size mm for forced convection thickness

Airflow at air supply devices Complex geometry - Δ~10 -4 m We can spend all our computing power for one small detail momentum sources

Diffuser jet properties High Aspiration diffuser D L D L How small cells do you need? We need simplified models for diffusers

Peter V. Nielsen Simulation of airflow in the vicinity of occupants How detailed should we make the geometry?

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