What is the nature of steady state for a temperature field?

Question

Suppose we define a temperature field that varies as a function over space.. now, the steady-state is the state where the temporal variations of this function are zero, that is the function only varies as we move across space.

So, once we are in steady-state there is a 'set' spatial distribution of temperature. My question is what are the general properties of this spatial distribution?

As in, how should be the temperature be distributed such that the field doesn't evolve with time but has different values at each point in space? In a way, my question is related to what is really motivating the evolution of the temperature field to steady-state.

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Though the question may look broad, I seek the general qualitative principles which tell us how real-life temperature distributions evolve. I have some exposure to the Fourier law and Laplace equation, however, I am asking the general physical principles involve in driving the evolution of the system.

I am not asking for a mathematical answer but rather the physical principles which the mathematics tries to capture. In simple words, rules of thumbs with the reasoning of how they came about. The picture that I have in mind is a substance that is at rest and simply transferring heat by the mechanisms of conduction, convection, and radiation.

Finally, I am not really thinking of a kind of system which involves nuclear/ chemical reactions. Simply general principles about physically evolving temperature fields.

Answer 1

If there is a non zero static vetorial field of temperature gradient, there is some source of heat. An example is a boiler full of water, with an electrical resistance inside.

It is like a constant water flow of a river, that requires continuous supply from glaciers or underground reservoirs.

If there is no source, the static flow is zero, what means for the case of temperature, no gradient.

Answer 2

how should be the temperature be distributed such that the field doesn't evolve with time but has different values at each point in space?

It is easy in one dimension (for a uniform bar or a plate): a constant gradient, with heat reservoirs at both sides.

In more dimensions, lines or surfaces that are isothermal will be unavoidable, I think.

(image source)

Answer 3

So, once we are in steady-state there is a 'set' spatial distribution of temperature. My question is what are the general properties of this spatial distribution?

One of the general properties of the steady-state spatial distribution of temperature is that it can be represented as a scalar field, meaning there is specific value of temperature at a given location. In cartesian coordinates such a scalar field is described as

$$T=T(x,y,z)$$

Another general property is obtained if one takes the gradient of this scalar field.

$$\nabla T=\biggr(\frac{\partial T}{\partial x},\frac{\partial T}{\partial y},\frac{\partial T}{\partial z}\biggl )$$

The gradient gives you a physical (vector) quantity describing the direction of the temperature change, and the rate at which the temperature changes the most rapidly, around a particular location (x,y,z). The units are degrees per unit length (degrees Kelvin per meter in SI units).

Finally, in the case of heat conduction in a homogenous isotropic material, one can then relate the temperature gradient to the local heat flux density $q$ , or

$$q=-k\nabla T$$

Where

$q$ is the local heat flux density [$W.m^{-2}$]

$k$ is the material's thermal conductivity [$W.m^{-1}.K^{-1}$]

$\nabla T$ is the temperature gradient [$K.m^{-1}]$

These are some of the general properties I'm aware of. Others with more knowledge than I (e.g. @Chet Miller) can probably give you a lot more.

Anyway, hope this helps.