Suppose a building has a time constant of one hour. Assume the outside temperature remains at a constant of 50 degrees F. Further suppose that there are no people, machines, etc., causing an increase in temperature, and that the heating system is heating the room at a rate of U(t) = e^t in units degrees F per hour starting at time t =0. Find an exact expression for the time it takes for the building temperature to increase from an initial value of 50 degrees F to 75 degrees F.
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You're very welcome.
Heat differences tend to experience exponential decay. The inverse of the decay rate is the time constant. See Wikipedia's article on exponential decay. Since it's one hour, if you express t in hours, you don't have to worry about it, which is nice.
Call the temperature in the building H(t).
From the time constant information and knowing that heat differences experience an exponential decay, I can say
dH/dt = -(H(t) - 50)
The second thing I need to address is that the building is being heated. That equation is relatively straightforward.
dH/dt = U(t) = e^t
Since both of these processes are happening at the same time, I superimpose the two differential equations and get
dH/dt = e^t - (H(t) - 50)
It's first-order, so you should only need one boundary condition. That's the initial condition, H(0) = 50.
Once you've got a closed-form solution, you can set it equal to 75 and solve for t.
You could also solve this in terms of the difference between the building temperature and outside temperature, but I don't think it would be any easier, with the heating term.
Heat differences tend to experience exponential decay. The inverse of the decay rate is the time constant. See Wikipedia's article on exponential decay. Since it's one hour, if you express t in hours, you don't have to worry about it, which is nice.
Call the temperature in the building H(t).
From the time constant information and knowing that heat differences experience an exponential decay, I can say
dH/dt = -(H(t) - 50)
The second thing I need to address is that the building is being heated. That equation is relatively straightforward.
dH/dt = U(t) = e^t
Since both of these processes are happening at the same time, I superimpose the two differential equations and get
dH/dt = e^t - (H(t) - 50)
It's first-order, so you should only need one boundary condition. That's the initial condition, H(0) = 50.
Once you've got a closed-form solution, you can set it equal to 75 and solve for t.
You could also solve this in terms of the difference between the building temperature and outside temperature, but I don't think it would be any easier, with the heating term.