How Does thermalmodel.com Work?
The model used by thermalmodel.com is a linear state space model based off conservation of heat flow:
is the sum of all heat transfer in or out, in Watts is the sum of all heat generated or consumed, in Watts
The class of model is referred to as a Lumped-Element model, which produces accurate results when each "lump", or in the terminology of this website, "node", can be assumed to have uniform temperature. A node can be assumed to be uniform temperature when the diffusion of heat within it is much greater than the transfer of heat into or out of it. This is quantified by a small Biot number.
What follows is the definition and manipulation of fundamental heat transfer equations into the numerical form thermalmodel.com uses to calculate node temperatures when it runs a model.
Fourier's Law and Thermal Conductance
Fourier's Law in 1 dimension states:
is heat transfer, in Watts is Thermal Conductance or Thermal Transmittance, in Watts per degree Kelvin is the temperature difference, in degrees
Thermal Resistance
The reciprocal of Thermal Conductance is Thermal Resistance ,
is heat transfer, in Watts is the temperature difference, in degrees is Thermal Resistance, in degrees Kelvin per Watt
Thermal Capacitance
Thermal Capacitance (different than Thermal Conductance above) can be thought of as the amount of heat energy absorbed by an object to increase its temperature by 1 degree:
is Thermal Capacitance , or Heat Capacity, in Joules per degree Kelvin is heat transfer, in Joules is the temperature difference, in degrees is the rate of change of heat energy with respect to temperature, in Joules per degree
Rearranging and differentiating the equation above with respect to time, we obtain the following:
is heat transfer, in Watts (i.e. Joules per second) is Thermal Capacitance , or Heat Capacity, in Joules per degree Kelvin is the derivative of temperature with respect to time, in degrees per second
Model Equation
We now have two equations for
is Thermal Capacitance , or Heat Capacity, in Joules per degree Kelvin is the derivative of temperature with respect to time, in degrees per second is the temperature difference, in degrees is Thermal Resistance, in degrees Kelvin per Watt is the net heat generated or consumed, in Watts
This partial differential equation can be simplified to a numerical equation. The results of the equation are reasonable when the time step size between discrete states is small. You'll notice that if you rerun a model with too large a time step, unstable results appear - at high time step sizes, errors compound and blow up the temperature differences to unreasonable numbers. If you see instability, decrease the time step size and rerun your model.
Assuming small time steps, this equation holds for each node in the model:
is the change in temperature of the node of interest at that time step, in degrees is the time step, in seconds is Thermal Capacitance , or Heat Capacity, of the node of interest, in Joules per degree Kelvin is the current temperature of a node that is thermally connected with the node of interest, in degrees is the current temperature the node of interest, in degrees is the Thermal Resistance between the two thermally connected nodes, in degrees Kelvin per Watt if conduction or convection, or in degrees Kelvin to the fourth per Watt if radiation is the net heat generated or consumed by the node of interest, in Watts
Under the hood, thermalmodel.com solves the above equation for each node at each timestep, resulting in a set of temperature values for each node at each timestep over the duration of the model.
Fixed temperature, i.e. boundary condition nodes do not change temperature over time. They influence nodes that are thermally connected to them, but their calculated temperature difference at each time step is discarded rather than added to their previous temperature.
The source code for the model can be seen here as a standalone Javascript package on npm.
