Closing the door

The problem of opening and shutting doors plagues the refrigeration and air conditioning businesses. How many times per hour can one allow those pesky customers open a door to an AC area without adding to the heat load too much? Or those people who complain the refrigerator is not cooling effectively, when they keep opening the fridge door wide? But the problem that bothers me is: how much air pollution (PM2.5) is getting in to my living room each time the door is opened to the balcony whose air is not purified? In my friend Chandan’s living room the PM2.5 level immediately responds, going up by easily 10-20%.

A strip of the door of height h and width dx has an area dA (Fig.a):

                                                                

                                                              dA= h dx                                                                                                                                                                         

When the door is being shut it has an angular velocity w and if r is the distance from the pivot point, the velocity v at that point is given by (Fig.b):   

                                               

                                                            v = wr

where r can be calculated from the distance x and the perpendicular distance y (see part b) of above diagram):

r = (x² + y²)^1/2

Let  n_o be the number of particles of PM2.5 in the room initially. Then the rate of change of n is given by:

dn/dt = n_ov dA

But the velocity v is independent of time, and so it is also independent of y. Setting y = 0, r = x:

dn/dt = n_ow òh x dx = n_ow h d²/2

So that, since door area A = hd and taking Dq = wDt:

Dn = A (Dq) d n₀

where x is integrated across the width d of the door, and x  = 0 at the pivot of the door. Of course, this assumes that the time period when the door is being accelerated to velocity v, and when it is decelerated to zero velocity, is negligible. Also, the temperature is being assumed to be the same in both rooms 1 & 2, as is the density. This may not be a very realistic assumption and, indeed, others do not make it (see below).

Take n₀ = 3x10²⁵ m^-3 (1 atm at normal temperature & pressure), h = 2 m. d = 1 m, and    Dq = w(Dt) = p/4 (45° door opening). 

so that  Dn is independent of the time taken, and only dependent on the angle Dq traversed.

So: Dn = 2.36 x 10²⁵ atoms

A room of volume 22.4 m³ contains 6x10²⁶ atoms (1000 mols x Avogadro number, at NTP), so the Dn = 2.36 x 10²⁵ atoms corresponds to 0.88 m³. That is, opening the door by this much allows about 4% of the outside PM2.5 contamination to get in: so the PM2.5 level increases by about 4% of the outside PM2.5 level (for every 45° opening). For a 90° opening, it should be ~8%.

As a reality check: if we assume a 45° angle (which is what we did), the volume pushed out by the door while closing is (pr2/8)(h) = 0.78 m³.

The latter volume is 88% of the former, so a 12% difference can be attributed to the fact that the velocity v is not constant across the door (in the x-direction).

The amount of 880 L is probably an overestimate because the angular velocity w is assumed to be constant while the door is moving, whereas it is more likely that it starts from zero and increases.

But if it is only dependent on the angle Dq traversed then the time variation of the angular velocity will not matter.

This analysis will work if the door is closing: moving CW about its hinge so that air is being forced from Room2 to Room1.

However, if it is opening (moving CCW), the air from room 1 is not being pushed, but pulled (while the air already in room 2 is being pushed CCW).

The analysis for that case is different: so opening and closing the door gives rise to different effects - and I shall not go there.

I found 4 papers in the literature that tackled a similar topic albeit with different assumptions.

Foster et al [1] refers to the case of an air curtain above the door so it is not directly applicable, and it has an analytical solutions as well as a computational fluid dynamics (3D CFD) study.

Lagus [2] considers a pressure difference between both rooms with buoyancy effects driven by the difference in densities in the two rooms. But, for the case of equal density, no buoyancy effects occur, and the volume displaced by the door is given by:

V_p = A(Dq)d/2

which is similar to that derived above for Dn.

Hendiger et al [3] also consider the flow rate due to a pressure difference between the two rooms, and obtain the flow rate Q as:

Q  (m³/s) = 0.067 (Dp)^0.63

Where Dp is in Pa, and the exponent n is usually between 0.6 and 0.7. They also conclude that: “quick door swing causes a greater transfer of contaminants, regardless of the width of the door opening, which shows that it is necessary to open the door more slowly.” [3].

Marshall et al [4] have derived an expression – using very similar logic to that gven above – but specifically trying to calculate the force required to close a door, overcoming air resistance. They show that the force required is pretty low, until the point where it is almost shut, and there it increases sharply for the last few degrees of shutting it:

“When the windows in a small room are shut, the effect of displacing air as the door is pushed makes the door more diffcult to push when it is nearly closed.”

 But they add that air flow under the door will reduce this effect somewhat at small angles.

A more complete analysis would take into account temperature and pressure profiles (as has been done [1,2,3] by other authors. But simply the effect of door swing has also been considered [2.4], although the latter is more rigorous since it takes acceleration into account.

1 1)    “Three-dimensional effects of an air curtain used to restrict cold room infiltration” A.M. Foster et al Applied Mathematical Modelling 31 (2007) 1109–1123

2 2)     “Air inleakage due to door opening and closing” P.L.Lagus (26^th Nuclear Air Cleaning & Treatment Conference, Sep.2000)

3 3)     “Influence of the Pressure Difference and Door Swing on Heavy Contaminants Migration between Rooms” Jacek Hendiger  et al PloS One 11(5): e0155159. doi:10.1371/journal.pone.0155159 (11th May 2016)

4 4)     “Slamming doors due to open windows”, D.A.Marshall et al J.Phys. Special Topics A2.7 (23Nov.2011)