Integration of exponential function - wash in curve: (dP_circ)/dt =

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snacks_au

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The following equation gives the rate of change of the partial pressure of a specific gas (Pcirc) in a circuit of plastic tubing which is being flushed with fresh gas of different partial pressure (Pdel):

dPcirc/dt = (FGF/V) x (Pcirc - Pdel)

FGF = fresh gas flow into the circuit (L/min),
V = volume of the circuit (L),
Pdel = partial pressure of gas in the fresh gas flow (cmH2O),
Pcirc = partial pressure of gas in the circuit (cmH2​O)

Could anyone show me how to integrate the rate equation above to give Pcirc as a function of time. I already have the answer but I can't figure out how to get there:

Pcirc(t) = Pcirc(0) + (Pdel - Pcirc(0)) x (1 - e-t/[V/FGF])

Thanks in advance!
 
snacks_au said:
The following equation gives the rate of change of the partial pressure of a specific gas (Pcirc) in a circuit of plastic tubing which is being flushed with fresh gas of different partial pressure (Pdel):

dPcirc/dt = (FGF/V) x (Pcirc - Pdel)

FGF = fresh gas flow into the circuit (L/min),
V = volume of the circuit (L),
Pdel = partial pressure of gas in the fresh gas flow (cmH2O),
Pcirc = partial pressure of gas in the circuit (cmH2​O)

Could anyone show me how to integrate the rate equation above to give Pcirc as a function of time. I already have the answer but I can't figure out how to get there:

Pcirc(t) = Pcirc(0) + (Pdel - Pcirc(0)) x (1 - e-t/[V/FGF])

Thanks in advance!
Click to expand...

can you integrate:

\(\displaystyle \displaystyle{\int \dfrac{dx}{x-a}}\)
 
Actually I think I am a step closer after you posted that but still not right:


(1) dPcirc/dt = (FGF/V) x (Pcirc - Pdel)

(1/(Pcirc - Pdel)) x dPcirc = (FGF/V) x dt

ln((Pcirc - Pdel) = FGF/V x t + C

Pcirc - Pdel = Pcirc(0) e[(FGF/V)t]

Pcirc = Pcirc(0) e[(FGF/V)t] + Pdel

then if 1 is modified to give the formula for wash-out (i.e. dP/Dt = -m*P), and wash-in = 1- washout

Pcirc = Pcirc(0) (1- e-[(FGF/V)t]) + Pdel


still not quite right. Any ideas?
 
snacks_au said:
Actually I think I am a step closer after you posted that but still not right:


(1) dPcirc/dt = (FGF/V) x (Pcirc - Pdel)

(1/(Pcirc - Pdel)) x dPcirc = (FGF/V) x dt

ln((Pcirc - Pdel) = FGF/V x t + C

Pcirc - Pdel = Pcirc(0) e[(FGF/V)t]

Pcirc = Pcirc(0) e[(FGF/V)t] + Pdel

then if 1 is modified to give the formula for wash-out (i.e. dP/Dt = -m*P), and wash-in = 1- washout

Pcirc = Pcirc(0) (1- e-[(FGF/V)t]) + Pdel


still not quite right. Any ideas?
Click to expand...
How did you get that?

Please show intermediate steps - carefully.
 
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