Temperature and heat

A preliminary caution

Before we even get started on our modeling project it is worthwhile considering the meaning of the terms, temperature (T) and heat (Q), to see if we really understand what we are modeling. One of the problems of the science of thermodynamics that it is difficult to give a satisfactory definition of temperature without implying the concept of heat, and that it is equally difficult to give a satisfactory definition of heat without bringing in the concept of temperature. Our course is not a course in thermodynamics, and it is not appropriate to launch ourselves into the subtleties of thermodynamic argument. It will be sufficient to note that it is usual to start with the idea of thermal equilibrium. Two bodies are said to be at the same temperature if no heat (energy) flows (or they remain in thermal equilibrium) when they are brought into contact with each other. Temperatures, or at least scales of temperature, are then defined in terms of certain measurable physical properties such as the volumetric expansion of mercury. It is shown in textbooks of thermodynamics that one particular scale of temperature, the perfect-gas scale, has special significance because it is identical with an absolute thermodynamic scale of temperature based on the second law of thermodynamics. (Unfortunately there is no perfect gas, and if you delve deeply into the temperature literature, you discover that temperature is only defined with respect to physical standards, and that therefore they are arbitrary)

Heat is said to pass from one system to another if the two systems are initially at different temperatures and are then placed in contact with each other (note that this is actually a circular definition with the above paragraph). The quantity of heat passing to a system is measured in terms of the product of mass (m), specific heat (c_p) and temperature rise. Note that heat is only one form of energy in a body, therefore it is difficult to define an absolute heat quantity in a body. Usually we can really only talk about heat transfer and changes in heat quantity. (Again, if you delve into the heat literature, you find that the quantity heat is just as poorly defined as temperature.)

The above comments are, in practice, largely irrelevant, except in experiments where you are worried about temperatures to 3 decimal places. However, the concept that the underlying equations or physics of a process are poorly know or understood at some level should be a warning to us as modelers. At this early stage in our thermal modeling, we really don't fully understand what we are modeling. For thermal problems, this is not much of a problem (we understand well enough for the questions we are asking), but for modeling in general this is a serious warning, since we often do not fully understand the processes. No amount of modeling will improve our understanding of the basic processes (although it will hopefully highlight errors to a perceptive modeler).

Heat Transport

The movement of heat energy from one location to another (when there is no radiation or movement of material) is given by the heat flux (q, energy per area per time). The heat flux is the flow of heat energy into a unit volume of material and therefore by heat conservation,

dQ/dt = -dq/dx (A)

(in one dimension, per unit volume, also to be strictly correct the derivatives need to partials [we will discuss this in class]). Note this is just a statement of conservation, i.e. the amount of heat (Q) at a point varies in time solely on difference between the heat flux (q) coming in one side and going out the other (dq/dx). Heat flux is controlled by another important thermal quantity, the thermal conductivity K. Fourier's Law of Heat Conduction establishes the linear proportionality between heat flow q_i and temperature gradient in a thermally isotropic medium

q_i= -K dT/dx_i (B)

where the minus sign denotes that heat flows down the temperature gradient. As heat flow is a flux of energy through a unit area per unit time, K has dimensions [mlt^-3T^-1], and is measured in W m^-1 K^-1 in the SI system. Its value for silicates is in the range 1-5 W m^-1 K^-1. (The subscripts in the above equation indicate that the Law holds for all directions simultaneously. For anisotropic media the thermal conductivity is a tensor K_ij, which is symmetric because heat flow from an isolated point occurs along straightlines. Isotropy may be safely assumed for polycrystals.)

The change of heat in a region causes a temperature change

m c_p D T = D Q (C)

which actually is used to define the specific heat (c_p). In materials with large thermal expansions there is a significant difference between the temperature change if the system is kept at constant pressure or constant volume during heating. Fortunately, this difference is small in earth materials and we can often get away with using a simple value of specific heat (in geo-problems usually c_p, constant pressure, is appropriate).

The conductive properties of a material are also conveniently expressed by its thermal diffusivity k, defined as

k = K / r c_p

where the dimensions of k are [l²t^-1], and its value is of the order of 10^-6 m²s^-1 in silicates.

A Thermal Model

At this point we have a basic knowledge that heat flows down temperature gradients, and that the temperature of a region rises proportionally to the quantity of heat added. This gives us sufficient information to do back of the envelope calculations and have a general view of thermal conduction problems. To proceed with a model (for whatever our ultimate purpose [to be discussed in class]) we need to chose a basic approach. Thermal problems are simple enough that it is typical to approach them in a direct mathematical simulation. So we proceed to develop a mathematical description on which we can base our model.

If the heat flux is steady in both space and time, then Fourier's law of heat conduction can be combined with the concept of heat conservation (or continuity of heat assuming no internal heat production), to allow direct calculations of steady state temperature gradients or heat flows. For example, the heat flux near the surface of the earth is generally close to one heat flow unit (hfu), which is defined as 10^-6 calories per square centimetre per second (41.84 mW m^-2). This is a very small quantity compared to the solar heat flux, so it has no effect on the climate of the earth. It is insufficient to prevent the development of permafrost near the poles, though it does produce a measurable warming of the water in the lowest meter or two of the ocean basins. However, it is simple to directly calculate the approximate background temperature gradient that this flux implies in the near surface crust (order 30 K/km).

Geoscientists are often more interested in the unsteady case. For example, how will the temperature vary with both time and distance from the intrusive contact when a thick sill of hot magma is emplaced in a flat-lying set of strata? This problem is an example of a one-dimensional heat flow problem. In its simplest form, suppose that one side of a uniform layer of material (of density r ) is suddenly raised in temperature by an increment D T. Heat starts to diffuse into the region of lower temperature, but at first the temperature gradient is much larger close to the interface than further away from it. So long as a variable temperature gradient persists, more heat flows into a volume element from one side than leaves from the other: this heat is used to increase the temperature of the element, and the amount used per unit volume depends on the density and specific heat c_p. (c_p is defined as the heat necessary to raise the temperature of one kilogram by one kelvin). For the moment, we need not be concerned whether this is the specific heat at constant volume, c_v, or at constant pressure, c_p).

By considering the conservation of energy (heat) in equation A, we can see that a space (or time) varying temperature gradient in equation B leads to a time varying variation in the local heat quantity (Q) and therefore a time variation in the local temperature (T) by equation C. Taking equation A and substituting B (actually it's space derivative) into the right hand side and C (actually it's per unit time per unit volume form) into the left hand side gives

r c_p ¶ T/¶ t = dQ/dt = -dq/dx = K ¶ ²T/¶ x²

(again in one dimension, x, and using partial derivative symbols in B and C). This is usually expressed as

¶ T/¶ t = k ¶ ²T/¶ x² (D)

where k is the thermal diffusivity. This equation, called the one-dimensional heat flow equation states that the rate at which the temperature at a point is changing with time is proportional to the rate at which the temperature gradient at that point is changing in the direction of heat flow. So, if the heat flow is steady (i.e., if the left-hand side is equal to zero) then the temperature gradient must be a constant: the temperature must vary linearly in the direction of heat flow.

The internal temperature gradient will be constant, if the 1D problem is steady state in time, unless there are internal heat sources. Internal heat sources or sinks are easily included by adding a source term to the right-hand side

¶ T/¶ t = k ¶ ²T/¶ x² + q

Where q is the heat source or sink in terms of Watts per cubic meter.

If the temperature gradient is not initially linear (as in the example of the intrusion of a sill), then as heat diffuses into the overlying layer of thickness d, the temperature gradient will eventually become almost linear and the heat flow almost uniform from one side of the layer to the other (assuming the magma doesn't cool and the surface stays at a constant temperature).

It is very useful when dealing with problems to be able to get an idea of the space and time scales involved. An example would involve answering the question: how long do we need to model, seconds, years or millions of years. In thermal problems this question of scaling can often be approached with the thermal relaxation time. This is a characteristic time for heat to diffuse through a layer and for the layer to approach a linear temperature gradient. It may be obtained by dividing d² by k:

T_r = d²/k

For a more complete discussion of heat flow problems of geological interest see Turcotte and Schubert (1982, Chapter 4).

More Than One Dimension

So far we have mainly talked about problems with one spatial dimension. For a homogeneous and isotropic material it is relatively straightforward to write down the multi-dimensional form of equation D (although there are several logical and theoretical difficulties, most of which stem from the heterogeneity of any real system). The multi-dimensional form of D just takes the sum of the spatial temperature curvatures in orthogonal directions,

¶ T/¶ t = k Ñ ²T (E)

(which I will explain in class), where the Ñ ² operator stands for taking the second spatial derivative in each direction and summing the result.

Towards a Model

Equation E represents the distillation of many of our ideas about heat flow, and will form the basis of our thermal model. Remember, however, that there are many things that we have already thrown out in constructing E (and we haven't even started modeling). We demand isotropy and homogeneity. We disallow convection, radiation or any heat sources or sinks. We haven't talked about it, but in fact we have disallowed phase changes or crystal habit changes. We have even disallowed dimensional changes, expansions, and an almost infinite list of other possible variations (many of which we may not even know about!).

Well, at least equation E represents an approximate mathematical description of the process that we want to model. We will talk in class about how to go about turning E into a model.

A final note, or problem

Equation E is often used as the starting point of thermal modeling. In sloppy work, equation E is used even when k varies in space, and modeling is carried out by varying k in different regions of the model, using equation E. Note however that you actually get a different equation than E if you decide at the outset to let k be a variable. Can you derive the correct equation for variable k?