http://people.csail.mit.edu/jaffer/Work/thermal | |

## Scheme for Thermal Simulation |

For systems with a few time constants, spreadsheet programs can
provide quick and easy simulation of heat and temperature over short
durations. As systems grow more complex or if simulation of a
**typical meteorological year**
is required, spreadsheets become unwieldy and brittle.

Scheme for Linear System Analysis shows how to solve linear systems in the frequency domain; this page details how to solve them in the time domain.

Lumped element thermal simulations can be modeled as electrical networks composed of grounded capacitors (thermal mass) and resistors (thermal resistance). With all their poles lying in the left half-plane, such systems are unconditionally stable; and quantization errors are attenuated by the integrations associated with the capacitors.

Each resistor can be modeled as a switched-capacitor operating at the
sampling frequency
*f _{c}* = 1 /

Resistor (in Circuit) | Equivalent Switched-Capacitor |
---|---|

Ei Ri,k Ek o---+---/\/\---+---o | >-Ii,k-> | | | _|_ _|_ Ci --- --- Ck | | _|_ _|_ /// /// | _ / \_/ >--+------+ fc | ___|___ ___O___ | |_ Ei | | | | Ek o---+--+ +--+-||--+-+---o | >-Ii,k-> Ci,k | | | _|_ _|_ Ci --- --- Ck | | _|_ _|_ /// /// |

I = (_{i,k}E - _{k}E) / _{i}R
_{i,k} | I = (_{i,k}E - _{k}E) · _{i}C
· _{i,k}f
_{c} |

Equating the I currents at sampling
frequency _{i,k}f gives:
_{c} | C =
_{i,k} | 1R · _{i,k}f
_{c} | = t · _{c}G
_{i,k} |

This technique requires storage for (scalars) charge
*Q _{i}* and voltage

- Calculate each node's voltage from its charge and capacitance:
*E*_{i}= Q_{i}/ C_{i} - For each pair of connected nodes indexed by
*i*and*k*, where*i < k*:- calculate the transfer charge
*Q*= (_{t}*E*-_{k}*E*) ·_{i}*C*= (_{i,k}*E*-_{k}*E*) ·_{i}*t*·_{c}*G*_{i,k} - add
*Q*to_{t}*Q*(the charge accumulating in_{i}*C*)_{i} - subtract
*Q*from_{t}*Q*(the charge accumulating in_{k}*C*)_{k}

- calculate the transfer charge
- Repeat this procedure
*T*·*f*times, where_{c}*T*is the desired length of the simulation in seconds.

I am a guest and not a member of the MIT Computer Science and Artificial Intelligence Laboratory.
My actions and comments do not reflect in any way on MIT. | ||

SCM for Engineering | ||

agj @ alum.mit.edu | Go Figure! |