I am trying to mathematically model the temperature distribution and evolution in a greenhouse, which conserves heat due to the greenhouse effect. Here is a transverse schematic ($H$ is the height of the box, $l$ its width and $L$ is length but is not represented in this 2D schema) of the situation I'm trying to solve:

Scheme of the situation There are several radiative energy flows (visible $F_d$ from the sun, infrared $F_i$ from the atmosphere, infrared $F_s=\sigma T_s^4$ from the black soil and infrared $F_p=\sigma T_p^4$ from the plastic/glass wall). The plastic/glass absorbs infrared but not visible light, it is why $F_d$ goes directly into the black soil in the schema. There are also convective flows inside the greenhouse due to the airflow $Q$ but also due to natural convection between the two plates of different temperature ($T_s > T_p$). Adimensionnal numbers and correlations can be used to deduce the convective factor $h$.

For the moment, I made the hypothesis that $F_d$ and $F_i$ were constant. I also have considered a stationary situation where the energy does not accumulate, the outgoing energy is thus equal to the incoming energy. By doing such energy balance locally (air and soil) and globally, I have obtained the following algebraic system of non-linear equation, with three unknows $P$, $T_s$ and $T_p$, where $P$ is the heat given to the air inside the greenhouse:

\begin{array}{r c l} P = h(T_p-T) + h(T_s-T) \\ F_d + F_i = P + \sigma T_p^4 \\ F_d + \sigma T_p^4 = \sigma T_s^4 + h(T_s-T) \end{array}

Using the Newton-Raphson method, I was able to solve this non-linear system, but I would now like to refine my mathematical model. So here is my two problems:

$F_d(t)=A\sin(\pi (t-t_r)/(t_s-t_r))$ is not constant (where $t_r$ and $t_s$ are respectively the sunrise and sunset time). The hypothesis of a stationary problem is therefore erroneous. I tried to modify my equation system but I'm stuck because I don't know where to put the time derivatives and the appropriate constants (heat capacity, volumic mass ?). Once I have the differential equation system, I think I know how to proceed, using Crank-Nicolson discretization method and returning to an algebraic system that can be solved by Newton-Raphson.

In addition to being able to have the temporal evolution of temperatures $T$ (air), $T_s$ (black soil) and $T_p$ (plastic/glass plate), I would also like to have their spatial distribution. What I mean is the value of $T(z,t)$ for all position $z$ and for all time $t$, knowing the initial condition $T(z,t_0)$ for all $z$, and assuming a constant air flow $Q$. I guess because of the airflow $Q$, there will be a air transport equation involved, but I'm not sure how to include these new equations.

I will be infinitely grateful to anyone who can help. It is a personal project that I have set myself for educational purposes, I am not an expert but I try as best I can to carry out this project.

  • $\begingroup$ I think I know how to proceed, using Crank-Nicolson discretization method and returning to an algebraic system that can be solved by Newton-Raphson. Newton-Raphson method is used to find the root of a function while Crank-Nicolson is a method of solving PDEs, so I think you have some misunderstandings to clear up. $\endgroup$
    – Kyle Kanos
    Jan 9 '19 at 11:00
  • $\begingroup$ Crank-Nicolson discretizes a differential system that becomes an algebraic system with unknowns $x(t+\delta t)$. If linear, LU factorization can be used to solve it. In our case, it is not because $F(x)=0$ is a non-linear system, where $x(t+\delta t)$ is a vector of $3$ unknowns. It can be solved using several successive applications of the recurrent equation: $x_{n+1} = x_n + J_F(x_n)^{-1} F(x_n)$, where $J_F$ is the Jacobian matrix of the function $F$, will converge towards the exact solution $x(t+\delta t)$. It is the well-known Newton-Raphson method. $\endgroup$
    – A. Reagan
    Jan 9 '19 at 12:11
  • $\begingroup$ It's also well-known that the "Thomas algorithm" solves the problem without resorting to root-finding... $\endgroup$
    – Kyle Kanos
    Jan 9 '19 at 12:13
  • $\begingroup$ Thomas algorithm can solve linear system when the matrix is tridiagonal, the system I want to solve is non-linear due to the unknown temperature $T$ that appears in the system with an exponent 4 due to Stefan-Boltzman law. $\endgroup$
    – A. Reagan
    Jan 9 '19 at 12:18

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