Question

PROBLEM: Ammonia is used as a refrigerant. It is compressed outdoors, and the ammonia is then liquefied and circulated through the refrigerator's evaporator heat exchanger. The evaporator is indoors. The room containing the evaporator has a net open volume of 1000 m3 and a ventilation rate equal to two air changes per hour. A leak develops in an ammonia line that results in 50 g of ammonia being released per minute. 1. How long will it take for the ammonia concentration to increase to the PEL concentration? 2. How long will it take to reach the IDLH concentration? 3. If the flow of ammonia is stopped at the time the IDLH concentration is reached, how long will it be before the concentration is reduced to the PEL concentration? You may assume that the mixtures of gases are ideal at the low pressures involved. Also assume that the ammonia is mixed with the air in the room completely and instantaneously and that the air- ammonia mixture leaving the room has the same concentration as the mixture in the room. Hint: Use a differential equation and separation of variables to solve.

Answer 1

It's an interesting question. Let's start with noting down the 'Permissible Exposure Limit' (PEL) and 'Immediately Dangerous to Life or Health' (IDLH) for Ammonia#

OSHA PEL for Ammonia = 50 ppm (35 mg/m3)

IDLH for Ammonia = 500 ppm (350 mg/m3)

The problem says that due to the leak NH3 is leaking into the room (of volume 1000 m³) and the ventilation not being high enough, NH3 concentration is building inside the room. As a result, first it will reach the PEL limit and if the leak continues, the NH3 concentration will reach the IDLH limit.

To solve all the parts, we have to first develop the equation relating the NH3 concentration with time. To do this, write the material balance for NH3 in the control volume (the control volume here is the room itself).

F_IN^NH3 - F_OUT^NH3 = Accumulation of NH₃ inside the room..................(1)

............(2) (Here, I've converted the leak rate for NH3 (given in mass) to molar leak rate by dividing it with the molecular mass of NH3 (which is 17))

The ventilation rate has been given as 2 air changes per hour. This basically means that the ventilation system is able to replenish twice the volume of the room in an hour. This basically means the ventilation flow rate is (This is basically 2 multiplied by the volume of the room divided by the time in minutes)

If you need a formula, it would be something like this:

Flow rate in m³/h = Number of air changes * Volume of the room / time

The ventilation rate is therefore,

Therefore, .......(3)

Accumulation is given by the diferential form as follows:

.........(4)

Putting (2), (3) and (4) into (1) yields:

-------------(5)

(5) is a differential equation which can be solved by separation of variables as follows:

Integrating both sides and placing the limits to integration:(The initial concentration is 0 (at t=0) and at time =t is C_NH3)

Separation of variables basically means to separate the two variables by taking each variable to different sides of the equation and integrating both sides.

For Part (a) we are required to find the time when the final concentration (C_NH3) becomes (Converting weight to moles by dividing by molecular mass of NH3 and dividing by 1000 to convert mg to grams)

Putting C_NH3 = 0.002059 mol/m^3 in equation (6):

For part (b) we are required to find the time when the final concentration (C_NH3) becomes (Converting weight to moles by dividing by molecular mass of NH3 and dividing by 1000 to convert mg to grams)

Putting C_NH3 = 0.02059 mol/m^3 in equation (6):

For Part (c), we need to rewrite the mass balance, as the leakage has been arrested, thereby making  

(F^NH3_OUT still remains unchanged)

Accumulation term is the same. The differential equation thus becomes:

Solving by separation of variables:

..........(7)(The limits indicate that we wnat to get to the PEL limit (i.e. 0.002059 mol/m^3) from the IDLH limit (i.e. 0.02059 mol/m^3))

't' is the time that it takes to reduce the concentration from 0.02059 mol/m^3 to 0.002059 mol/m^3

Therefore, it would take 69.0845 min to reduce the concentration from the IDLH level to PEL level

Hope this helps!