Equation solvers

1. Equation solvers • Matlab • Free versions / open source codes: • Scilab http://www.scilab.org/ • MathCad: • Mathematica: http://www.wolfram.com/mathematica/ • LabView: http://www.ni.com/labview/ • EES: http://www.fchart.com/ees/ • Modelica: https://www.modelica.org/ • ….

2. Lecture Objectives: • Review the previous lectures • Prepare for the exam – solve some example problems • Discuss the final project assignment

3. Review • Heat transfer • Thermal analysis of building elements • External and internal boundary conditions • Weather data for boundary conditions • Modeling procedures • Numerical methods for solving of model equations

4. Review of heat transfer How to model: • Convection at surfaces • Radiation between surfaces • Conduction through building elements Steady state or unsteady state

5. Building elements

6. Weather data (TMY2 database) Use them for External boundary conditions Convection Long-wave Radiation Solar radiation • Direct • Diffuse • Reflected (diffuse)

7. Discretization

8. Discretization for conduction • T – temperature [C] • ρ – density [kg/m3] • cp – specific capacity [J/kgK] • k- conductivity [W/mK] • time [sec] x distance [m] Section considered in the following discussion Discretization in space Discretization in time

9. Finite volume (difference) method Boundaries of control volume Fir each node conservation of energy: explicit implicit

10. Implicit methods - example After rearranging: 2 Equations with 2 unknowns!  =0 To Tw Ti  =36 system of equation Tw Ti  =72 system of equation Tw Ti

11. Unsteady-state conductionImplicit method with linearization b1T1 + +c1T2+=f(Tair,T1,T2) a2T1+b2T2 + +c2T3+=f(T1 ,T2, T3) Air 1 4 3 2 5 Air 6 a3T2+b3T3+ +c3T4+=f(T2 ,T3 , T4) ……………………………….. a6T5+b6T6+ =f(T5 ,T6 , Tair) Matrix equation M × T = F for each time step M × T = F

12. Numerical methods PROBLEM Unsteady-state Steady-state System of equations for unsteady state process (nonlinear) System of equations for steady state process (nonlinear) Explicit Implicit Implicit Linearization (Matrix solver) Nonlinear (Newton-Raphson method) For each time step

13. Modeling steps • Define the domain • Analyze the most important phenomena and define the most important elements • Discretize the elements and define the connection • Write the energy and mass balance equations • Solve the equations (use numeric methods or solver) • Present the result

14. concrete Lconcrete insulation Linsulation Plenum (air) Lplenum acoustic tile Ltile QHVAC mS ,TS Room (air) TR Walmart store (L>>H, D>>H) TF Floor D=100m H-5m door L=200 m Practice for the ExamExample #1

15. Practice for the ExamExample #2 You are considering using the ventilated windows for ventilation of your new building and a sales person claims that it will reduce your annual energy bill by 10%. To check this claims you decided to model the performance of this window for your climate condition. QHVAC TRA Building fan creates under pressure in the room Building fan creates pressure in the room TRA QHVAC Air cavity open to outdoor air at the top, and to indoor air at the bottom

16. Heat transfer for double glazed window

17. Final project: Graduate students – individual - detailed modeling: your model or advance use of software Undergraduate students – group of 2 students - Energy analysis of buildings form Integrated design course or anyother building or building system I can provide several ideas but your ideas are encouraged… Send me a ½ page proposal with your ideas till Wednesday, November 7 Will meet with each student in my office and define/finalize the project assignment