Time to Reach equilibrium temperature
Time to Reach equilibrium temperature
(OP)
if anyone could provide guidance, it would be appreciated.
here's the problem:
a hot gas of known temperature is exhausting at a known rate into a room. the room is being ventilated at a known rate. i need to estimate the time it takes for the room to reach an equilibrium temperature 20 deg higher than the original temperature. any basic formula/method for estimating this?
here's the problem:
a hot gas of known temperature is exhausting at a known rate into a room. the room is being ventilated at a known rate. i need to estimate the time it takes for the room to reach an equilibrium temperature 20 deg higher than the original temperature. any basic formula/method for estimating this?





RE: Time to Reach equilibrium temperature
Basic method would be to perform an energy balance.
RE: Time to Reach equilibrium temperature
but if the gas mixes and the exiting gas is at some partial mixture temperature, that's one heck of a tough problem.... maybe impossible to calculate.
you have to make some assumptions about the temperature of the exiting gas as a function of time. its not reasonable to say that the exiting gas is always the gas that was initially in the room.
not much help, i know.
magicme
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"not all that glitters is gold"
RE: Time to Reach equilibrium temperature
The exhausting air would be made up of:
The hot gas
The ventilating air
Air that was already in the room
A few initial combinations should suffice to bound the problem and get an idea of the solution's sensitivity to the assumptions.
RE: Time to Reach equilibrium temperature
RE: Time to Reach equilibrium temperature
That said, the time to equilibrium will be be anyway influenced by the heat capacity of anything that's in the room (machinery, presumably 50% of wall thickness or 100% if they are externally insulated).
As you see you should better specify your goals and your constraints before going to any equation set up.
prex
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RE: Time to Reach equilibrium temperature
Google around for sweep-through purging and you'll probably learn of the problems involved.
RE: Time to Reach equilibrium temperature
RE: Time to Reach equilibrium temperature
Best regards
Morten
RE: Time to Reach equilibrium temperature
Then when you say the the room is ventilated, do you mean that you are introducing a cold air stream into the room and exhausting that air back out?
When I read this last week, I thought you were just dumping hot air into the room and exhausting out a mixture of that hot air and the room air. Now I'm not so sure what you are asking???
Also, you need to know the volume of the room and if the air flows are different, you will have a pressure gradient that will cause in(ex)filtration. I am going to assume that the air flows are constant, but please provide more detail.
RE: Time to Reach equilibrium temperature
It could be a situation where a piece of equipment is giving off an exhaust gas into the room (hopefully nothing dangerous) and there is also an exhaust fan pulling the air from the room (giving rise to a ventilation rate.) Or maybe it's just the heat from a large piece of equipment.
I ran across a situation not too long ago where the ventilation setup was intake air dampers in small room with large diesel engine (combustion exhaust air vented to outside). Small room connected to another room via some wall openings. Second room had exhaust fans of (supposedly) known ventilation flow rate. So from the view of the larger room, there would be hot air "exhausting" into the larger room. However, in that case, what was being questioned was the equilibrium value, not the time it took to get to equilibrium.
Don't know if this helps your calculation any...
Patricia Lougheed
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RE: Time to Reach equilibrium temperature
mdot is the mass flow rate thru the room (kg / s)
MASS is the constant mass (kg) of the gas in the room
T is the absolue temp (K) in the room (changing with time)
Tinlet is the temp of the entering gas
Texit = T
time = time ins seconds
net heat into the room is
Qdot = mdot * Cp * (Tinlet - T)
In terms of the MASS in the room, that net heat into the room also equals (MASS * T) / time
so....
mdot * (Tinlet - T ) = (MASS * T) / time
and the room temperature is ...
T = (mdot * Tinlet) /(MASS/time + mdot)
???
I'll have to make some numbers before I really believe this.
regards,
magicme
------------------------------------
there's no place like gnome.
RE: Time to Reach equilibrium temperature
Tzero = initial room temperature, then...
T = (mdot * Tinlet + {MASS/time}*Tzero}) /(MASS/time + mdot)
or alternately....
T = { (time/MASS)*mdot*Tinlet +Tzero } / { 1 + mdot *time / MASS }
i think that's correct...
magicme
------------------------------------
there's no place like gnome.
RE: Time to Reach equilibrium temperature
A tank has 10 gal of brine in which 5 Lb of salt are dissolved. Brine consisting of 3 Lbs of salt per gallon enters the tank at 2 GPM and a well-stirred mixture leaved at the same rate.
Let A be the number of Lbs of salt in the tank at time t.
dA/dt = rate gained - rate lost
Since 2 GPM of brine at 3 Lbs salt per gallon enter, we have 6 Lbs per Minute entering. This is the rate gained.
Since there are always 10 gallons in the tank and since there are A Lbs of salt at time t, the rate lost is:
(A Lb/10 gal)*(2 GPM) = (A Lbs/5 Min)
So, dA/dt = 6 - (A/5)
Initial conditions are A=5 at t=0
Solving the Diff Eq, you get:
-ln(30-A) = (t/5) + c
plugging in the initial conditions, c = -ln(25)
So, the solution is:
A = 30 - 25e^(-t/5)
Hope this helps point you in the right direction.