# How to solve temperature evolution PDE for a water heater?

I am trying to understand a PDE model for a heater.

Please kindly guide in the correct direction so that I can understand and possibly solve this equation?

The below is a heat equation for a water heater with two heating element and sensors.

\eqalignno{&{\partial T\over \partial t}+V{\partial T\over \partial x} =({D_{c}+D_{a}}){\partial^{2}T\over \partial x^{2}} +{Q_{\rm ele}\over \rho c_{v}}+{U\cdot A_{s}\over AH\rho c_{v}}({T_{\rm amb} -T})\cr &&\hbox{(1)} }

$$D_{a} = ({\theta d})^{2}\sqrt {-g\beta {\partial T\over\partial x}} {\hbox{(3)}}$$ where θ=0.46 is the Von Kármán constant. g=9.8 m/s2 is the acceleration due to gravity and β=207×10−6/K is the thermal expansion coefficient of water. The minus sign in (3) is due to the fact that natural convection only occurs when the instantaneous temperature gradient is negative, i.e., (∂T)/(∂x)<0. If the water tank has a diameter of d=0.5 m, (3) can be further simplified to $$D_{a} =\gamma \sqrt {-{\partial T\over\partial x}} \eqno{\hbox{(4)}}$$