Suppose my room is x degrees C. The air in my AC ductwork is a constant y degrees C and is blowing at z m/s. We assume there is no heat transfer from the outside. How can I figure out how long it will take for my room to reach the desired temperature?

  • 3
    $\begingroup$ That depends on what's in your room. You don't just have to cool the air down, but also the walls and whatever else is in there. If there are a lot of people in the room, you have to assume that each of them is a 100W heat source. Then there are the lights and other electric and electronic equipment. $\endgroup$ – CuriousOne Jun 11 '16 at 6:44
  • $\begingroup$ Is the room closed with respect to airflow going out? There is a sense in which the room is basically a tank of air with an inlet pipe (and possibly fixed wattage heaters if there are people present) and could potentially be modelled as such, given your assumption of perfect insulation. $\endgroup$ – Robert de Graaf Jun 11 '16 at 12:20
  • $\begingroup$ Too many missing details. Is this really a problem which you are struggling to solve? Or is it a passing curiosity? $\endgroup$ – sammy gerbil Jun 11 '16 at 22:13
  • $\begingroup$ @RobertdeGraaf Yes. $\endgroup$ – moonman239 Jun 13 '16 at 21:58

You are missing some key params, like volume of the room, humidity or energy consumption.

Simplifying the assumptions:

Steady state conditions
1D heat transfer from AC to room
Fluid properties are uniform
Adiabatic system
0 thermal radiation (Shades drawn)
Cooling happens everywhere simultaneously
No new energy enters or leaves system

The last one is the least likely, since the AC is bringing in energy to power it, and heat will be lost to surroundings heating the room.

You need Newton's law of cooling, since you say temps are constant and adiabatic system.

Since you give z in m/s, I would propose assuming an area, A, that the air flows through. You then need to assume that the system expels mass at an equal rate.

Then you can say:

q/A = h * deltaT

Where q is the convective heat flux and h is convection heat transfer coefficient (assuming air). I don't know how you would get q without statements about the energy the AC needs, but this is solvable if you determine a relationship between h and z, through either AC manufacturer or estimate (text gives h = 10.9 W * s^0.8/m^2.8 * K * z^0.8 for a similar problem…determining that can be involved). Then you can determine q, the energy needed.

Using q would be more accurate, since you're looking for change in internal energy over time, dU/dt. Since U is only dependent on the volume, density (rho), specific heat (c) and temperature of the solution (assuming air), you get…

dT/dt = q/(rho * V * c)

Then you use newton's law of cooling,

T(t) = Ti + (T_AC - Ti)exp(-kt)

Solve for t.

This again is my assuming a number of key params: Volume of room, quality of air/humidity, properties of fluid (assuming air), and a relationship with velocity from your manufacturer as well as the AC's opening area.

There are complicated ways to go about driving at this accurately, but even those require simulations.

  • $\begingroup$ While this would be a useful comment, I'm not sure it qualifies as an answer $\endgroup$ – John Rennie Jun 11 '16 at 7:16
  • $\begingroup$ Really not finding your comment qualified @JohnRennie $\endgroup$ – double0darbo Jun 17 '16 at 22:34

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