HW2 Example MatLab Code is posted on the course website Due date October 18

Lecture Objectives: Discuss HW 2 Finish unsteady-state heat conduction Define building system of equations

Homework 2 (Similar to HW1B, but unsteady, and more realistic) Top view Glass Te_i Te_o Tinter_surf ≠ Tair 2.5 m Surface radiation Tair_in 10 m 10 m IDIR Ts_i Idif South East Insulation Ts_o Tair_out Concrete Surface radiation Idif IDIR

Internal node Finite volume method Boundaries of control volume For node “I” - integration through the control volume

Physical approach (finite volume method) For uniform grid

Internal node finite volume method After some math work: Explicit method Implicit method

Unsteady-state conduction Implicit method b1T1 + +c1T2+=f(Tair,T1,T2) a2T1 + b2T2 + +c2T3+=f(T1 ,T2, T3) Air 1 2 3 4 5 6 Air 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

Internal node finite volume method Explicit method Rearranging: Implicit method Rearranging:

Explicit method Accuracy when compared to explicit ? - simple for calculation - but unstable Problem with stability can be fixed with appropriate time step: Accuracy when compared to explicit ?

Implicit method (internal node) kI-1=kI+1=kI AI BI CI FI Internal nodes B1 C1 T1 F1 A2 B2 C2 T2 F2 x = A3 B3 C3 T3 F3 1 2 3 4 5 A4 B4 C4 T4 F4 A5 B5 T5 F5

Implicit method (surface nodes) B0 C0 T0 F0 Surface nodes A1 B1 C1 T1 F1 T O Air T I Air A2 B2 C2 T2 F2 x = A3 B3 C3 T3 F3 1 2 3 4 5 6 external internal A4 B4 C4 T4 F4 A5 B5 C5 T5 F5 A6 B6 Dx T6 F6 For surface nodes: flux coming in = flux going out Surface node: 0 Calculate B0 and C0 Surface node: 6 Calculate A6 and B6

Linearization of radiation equations Surface to surface radiation Equations for internal surfaces - closed envelope Ti Tj Linearized equations: Calculate h based on temperatures from previous time step Or for your HW3

Linearized radiation means linear system of equations Calculated based on temperature values from previous time step B0 C0 T0 F0 A1 B1 C1 T1 F1 A2 B2 C2 T2 F2 These coefficient will have Some radiation convection coefficients x = A3 B3 C3 T3 F3 A4 B4 C4 T4 F4 A5 B5 C5 T5 F5 A6 B6 T6 F6

Accuracy as a function of  and x

Discretization of a non-homogeneous wall structure Section considered in the following discussion

System of equation for more than one element Roof air Left wall Right wall Floor Elements are connected by: Convection – air node Radiation – surface nodes

Energy balance for air unsteady-state heat transfer QHVAC

Example Tair is unknown and it is solved by system of equation :

System of equations (matrix) for single zone (room) 8 elements Three diagonal matrix for each element x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Air equation

System of equations for a building Matrix for the whole building 4 rooms Rom matrixes Connected by common wall elements and airflow in-between room – Airflow simulation program (for example CONTAM) Energy Simulation program “meet” Airflow simulation program