I am working on an analytical solution for a numerical model, I have integrated the formula
dT2/dt = FCp(T1-T2)/M2*Cp
Where
T1 = Temperature in First Reactor
T2 = Temperature in Second Reactor
F = Feed Rate into Reactor 2
Cp = Specific Heat Capacity
M2 = Mass in Reactor 2
I have integrated the equation to obtain
T2 = T1 - k*exp(-Ft/M2)
Where
k = constant
t = time
My initial conditions are
at t = 0
T1 = 30
T2 = 20
So from this I obtain
20 = 30 - k*exp(0)
Therefore k = 10
And
T2 = T1 - 10*exp(-Ft/M2)
However when I use this equation to compare my calculated T2 values to known T2 values from the numerical model, the calculated values are not within a range of negligible difference as desired. I checked my equation for k using other known values of T2 and T1 and each time the value returned is different from the initially calculated value of 10.
For Example
at t = 600
T1= 36.3
T2 = 28.9
k = 20.1
Is someone able to please give me a reason for why this may be happening?