I'm new to this forum as I have been stuck on this problem for a while, and didn't know where else to turn. Any help is greatly appreciated!
I'm attempting to mathematically model thermal energy storage within a packed bed sensible heat storage system. The key equation is the following second order PDE, which captures the change in temperature (T) with respect to both time (t) and displacement through the tank (x):
$\left(\rho C_p\right)_m\frac{{\partial T}_m}{\partial t}+G.C_{p,f}\frac{{\partial T}_m}{\partial x}=\frac{\partial}{\partial x}\left(k_m\frac{{\partial T}_m}{\partial x}\right)$
Which may be better written as:
$A\frac{\partial T}{\partial t}+B\frac{\partial T}{\partial x}=C\frac{\partial^2T}{{\partial x}^2}$
Where:
$A = 1717686.55\ \frac{J}{m^3K}$
$B = 236\frac{J}{m^2sK}$
$C = 1.155\frac{W}{mk}$
With the following intial condition:
$T(x,0)=20°C$
And the following boundary conditions:
$T(0,t)=550°C$
$\frac{\partial T}{\partial x}(0,t)=0$
$\frac{\partial T}{\partial x}(L,t)=0$
I have spent the last week attempting to solve this through the separation of variables method with eigenvalues and eigenfunctions, in the hope of getting a solution in the form of:
$T_n\left(x,t\right)=B_nsin\left(\frac{n\pi x}{L}\right)e^{-k\left(\frac{n\pi}{L}\right)^2t\ }$
Where L relates to a length in meters, of the storage tank and $Bn$ represents the various constants of integration obtained throughout the process, solvable via the Fourier Sine Series. However, I am simply unable to get the equation to work.
I have since obtained some research results from literature where the charging process of the packed bed system was as follows:
t = 1 Hour Graph 1
Polynomial solution: $y=13728x^6-50447x^5+67135x^4-37308x^3+6810x^2-445.52x+554.58$
t = 1.5 Hours Graph 2
Polynomial solution: $y=-2710.5x^6+4928.9x^5+2182.7x^4-7837.1x^3+3379.5x^2-480.39x+566.51$
t = 2 Hours Graph 2
Polynomial solution: $y=-5493.2x^6+21095x^5-29059x^4+17302x^3-4758.7x^2+535.14x+533.75$
t = 2.5 Hours Graph 4
Polynomial solution: $y=1090.7x^6-2259.7x^5+742.42x^4+496.34x^3-319.33x^2+50.762x+548.15$
t = 3 Hours Graph 5
Polynomial solution: $y=872.72x^6-2893.5x^5+3281x^4-1720.8x^3+435.16x^2-46.906x+551.42$
I was hoping there may be some way to combine these polynomial equations and approximate them using a sum of sine waves, perhaps through Fourier Transform so that I can build a mathematical model to predict this charging time performance?? I have become completely stuck with this so any help is gratefully received.
Thank you in advance.