Are there any commonly accepted algebraic models for vacuum tube operation (triode, tetrode, and pentode)? In the same way that BJTs have a Gummel-Poon or Ebers-Moll model, and (macroscale) MOSFETs have a cutoff/linear/saturation algebraic model, is there a similar model for vacuum tubes? A DC-accurate model plus some dynamic components (dominant capacitances) would be excellent, but I'm having trouble finding any references. Higher order effects (equivalent to something like the Early effect in BJTs) would be good to know about as well, especially if they tend to impact practical designs.
The general small-signal model of a tube is pretty much outlined at the link Madmanguruman posted: a current source in parallel with a plate resistance and the associated inter-electrode capacitances.
For non-linear time domain analysis, the situation is more complicated as outlined here. As the article states, it's possible to derive a mathematical model of a triode or pentode based on the Langmuir-Child law, but that model doesn't accurately represent the real behavior of a tube in certain areas of its operation. The best models are "phenomenological" i.e. designed to fit a tube's actual performance curves as closely as possible without regard to underlying physics.
There is a program I've used available that will let you take published tube curves, fit the model to the curves, and then spit out a SPICE subcircuit. It works well for triodes, I don't know if it can be used for pentodes though. There are also many, many ready-to-run SPICE models of varying quality for a variety of different tubes on the web.