# Re: [EnergyPlus_Support] energy balance equation

Hello,
This section of the engineering documentation is an excerpt from published papers:
Taylor, R. D., Pedersen, C. O., Fisher, D. F., Lawrie, L., and Liesen, R. J., ?Impact of Simultaneous Simulation of Buildings and Mechanical Systems in a Heat Balance Based energy Analysis Programs on System Response and Control?, Proceedings of Building Simulation ?91, International Building Performance Simulation Association, August 20-22, 1991, Sophia-Antipolis, Nice France.

Taylor, R. D., Pedersen, C. O., and Lawrie, L., ?Simultaneous Simulation of Buildings and Mechanical Systems in Heat Balanced Based energy Analysis Programs?, Proceedings of Building Simulation ?90, International Building Performance Simulation Association, December 3-5, 1990, Liege, Belgium.

These papers can take you through the derivation in even more excruciating detail.

To answer your question  Equation 1.8 is the Euler Formulation and is not what was finally chosen to be used.  The Third Order substitution for the temperature was found to be stable, therefore equation 1.11 is what is currently implemented as the air heat balance equation.  Then all the variables are used from the current time step except for 3 zone air temperature from 3 previous time steps that are weighted in according to the third order formulation.

• heating/cooling load at time step t is calculated on the basis of set point temperature at time step t and heat transfer (internal surfaces, air mixing with other zones, infiltration from outside, internal convective gains etc..) at time step (t-dt)
This would be at the current time step t, not (t-dt).
• if the system is able to provide such load (as by definition for a Purchased Air system) the zone temperature at time t matches exactly the set point scheduled for time t
Correct.
• if the system is not able to provide all the required load (as for a real system), zone temperature is updated on the basis of the difference between the cooling/heating load calculated at time step (t-dt) and the system power at time t
If there is not enough capacity then Tz at t is updated with the current conditions for infiltration, loads, etc. and the current system capacity.  The history is preserved in the old zone air temps and the capacitance from the zone surfaces.

My interest is in the meaning of "Zone/Sys Sensible Cooling (Heating) Rate" output at time t for a Purchased Air system.
Is it the cooling (heating) load calculated at time t or (t-dt) or what is the time shift?

That would be at time t.

Thanks Richard Liesen

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