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When paper reached a certain temperature, auto-ignition occurs, and then there is a process that makes the temperature rise to around 1000 to 1500 Celsius degrees. How can I make a simplified model for this process. the purpose of the model is the temperature change over time. the paper i am using in this paper is a normal A4 office paper. Furthermore you can assume the auto-ignition temperature is constant and that the heat transfer is neglectable.

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    $\begingroup$ Well, if there was no heat transfer, then maybe it could get so hot, but not in real world. $\endgroup$
    – Mithoron
    May 17, 2023 at 18:25
  • $\begingroup$ @Mithoron Maybe he means that the source that gives rise to the heat transfer directs all its energy to the paper, this is, if the source gives $\pu{5 J}$ to the system, then $\pu{0.1 J}$ won't drift away as a loss. $\endgroup$ May 17, 2023 at 18:35
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    $\begingroup$ Too many variables undeclared: rate of heating, surface exposed (Crumpled? on a hot plate? suspended?), air movement (confined? open to still air? wind?)... $\endgroup$ May 17, 2023 at 23:14
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    $\begingroup$ If there are gaseous reactants(O2), products(H2O, CO2) and inerts(N2), the heat transfer is among the major factors that determine the temperature. $\endgroup$
    – Poutnik
    May 18, 2023 at 8:18
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    $\begingroup$ A4 isn't a type of paper: it is a standard for the size of the paper. apiece of asbestos could be "A4". So the designation is useless in understanding the properties of the object. $\endgroup$
    – matt_black
    May 18, 2023 at 10:00

1 Answer 1

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The simplest way to start is to model an unsteady heat transfer in two dimensions. The heat equation is a partial differential equation that, under no flow condition, takes the following form $$ \rho c_\mathrm{p} \frac{\partial T}{\partial t} - \kappa \left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) = h(T' - T) \tag{1} $$ where:

  1. $\rho$ is the density of the paper.
  2. $c_\mathrm{p}$ is the specific heat capacity at constant pressure of the paper.
  3. $\kappa$ is the thermal conductivity of the paper.
  4. $h$ is the volumetric heat transfer coefficient between the source and the paper.
  5. $T'$ is the temperature of the source.

Consider the rectangle defined as the set $D = \{(x,y) \in \mathbb{R}^2 : 0 \leq x \leq L_x \; \wedge 0 \leq y \leq L_y\} $. The initial condition is that $T(x,y,0) = \pu{298.15 K}$ $\forall (x,y)\in D$. The boundary conditions are an important choice, for illustration purposes, I considered the following:

  1. Three Dirichlet's boundary conditions over the sides of a rectangle, with the exception of the lower side: \begin{align} T(x,L_y,t) &= 298.15 \; \pu{K} \hspace{1 cm} x \in [0,L_x] \\ T(0,y,t) &= 298.15 \; \pu{K} \hspace{1 cm} y \in [0,L_y] \\ T(L_x,y,t) &= 298.15 \; \pu{K} \hspace{1 cm} y \in [0,L_y] \\ \end{align}
  2. A Neumann's condition in the lower side, this is, a no flux condition \begin{equation} \underline{\nabla} T(x,0,t) \cdot \hat{\underline{n}} = 0 \hspace{1 cm} x \in [0,L_x] \\ \end{equation} In the following figures, this side is the one where you "enter" into the carvern.

Some recommendations:

  • Consider, when first treating a problem like this, that all the model parameters are constant. Thermodynamic and transport properties will considerably change from ambient temperature to $T \approx 2000 \; \pu{K}$, so I would advise to obtain these magnitudes as a function of temperature. But first, try to obtain some sense of what are the influence of the parameters in the model (some are obvious, of course).
  1. You can model more realistically the source of the ignition, represented in the term $h(T' - T)$, considering it as a function of time. As an example, I will consider that the heat transfer coefficient is a linear function of the time, i.e. $h(t)=h_0t$.

I did some runs using the properties of wood, since it was not easy to find those of paper. I display the size of a real A4 paper; $\pu{210 mm}$ (in the $x$ direction) and $\pu{290 mm}$ (in the $y$ direction):

enter image description here enter image description here enter image description here

Some final points:

  1. The paper of course will burn, and we will lose its mass, so our grid of integration should appear as a hole. You can regard this model as an approximation for the first instants of time, when the paper is still not obliterated from existence.
  2. In Wikipedia I found that the autoignition of paper is around $\pu{220 °C}$ so you would want to double-check your values.
  3. You may use different boundary conditions (I don't know the situation of the paper), specially for the Neumann's condition. Here, the source is explicitly in the partial differential equation, but you can come with another way of reflecting additional behavior changing the ones I have selected.
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