I am dumping some equations here I need now and then! The sections about 3-dimensional temperature waves summarize what is described at length in the second part of this post.

**Temperature waves** are interesting for simulating yearly and daily oscillations in the temperature below the surface of the earth or near wall/floor of our ice/water tank. **Stationary solutions** are interesting to assess the heat transport between pipes and the medium they are immersed in, like the tubes making up the heat exchanger in the tank or the solar/air collector.

## Contents

- Heat equation – conservation of energy
*Solutions for oscillating sources – temperature waves*

1D solution

Helpers’ for the 3D case (spherical)

‘Helpers’ for the 2D case (cylindrical)

3D solution

Comparison of surface energy densities: 1D versus 3D*Stationary solutions*

1D – plane

3D – sphere

2D – cylinder: pipe.

Comparison of overall heat flow: 1D versus 2D

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## Heat equation – conservation of energy [Top]

Energy is conserved locally. It cannot be destroyed or created, but it it is also not possible to remove energy in one spot and make it reappear in a different spot. The energy density η in a volume element can only change because energy flows out of this volume, at a flow density j (energy per area and time).

In case of heat energy, the sensible heat energy ‘contained’ in a volume element is the volume times mass density ρ [kg/m^{3}] times specific heat c [J/kgK] times the temperature difference in K (from a ‘zero point’). The flow of heat energy is proportional to the temperature gradient (with constant λ – heat conductivity [J/mK], and heat flows from hot to colder spots.

Re-arranging and assuming that the three properties ρ, c, and λ are constant in space and time, they can be combined into a single property called thermal diffusivity D

In one dimensions – e.g. heat conduction to/from an infinite plane – the equation is

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## 1D solution – temperature waves in one dimension [Top]

I covered it already here in detail. I’m using complex solutions as some manipulations are easier to do with the exponential functions than with trigonometric functions, keeping in mind we are finally interested in the real part.

Boundary condition – oscillating temperature at the surface; e.g. surface temperature of the earth in a year. Angular frequency ω is 2π over period T (e.g.: one year)

Ansatz: Temperature wave, temperature oscillating with ω in time and with to-be-determined complex β in space.

Plugging into 1D heat equation, you get β as a function of ω and the properties of the material:

The temperature should better decay with increasing x – only the solution with a negative sense makes sense, then . The temperature well below the surface, e.g. deep in the earth, is the same as the yearly average of the air temperature (neglecting the true geothermal energy and related energy flow and linear temperature gradient).

Solution – temperature as function of space and time:

Introducing parameter k:

Concise version of the solution function:

Strip off the real part:

Relations connecting the important wave parameters:

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## ‘Helpers’ for the 3D case (spherical) [Top]

Basic stuff

Inserting, to obtain a nicely looking Laplacian in spherical symmetry

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## ‘Helpers’ for the 2D case (cylindrical) [Top]

Basic stuff

latex \frac{\partial}{\partial y})\sqrt{x^2 + y^2 } = \frac{\vec{r}}{r}$

Inserting, to obtain a nicely looking Laplacian in cylindrical symmetry

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## 3D solution – temperature waves in three dimensions [Top]

Boundary condition – oscillating temperature at the surface of a sphere with radius R

Ansatz – a wave with amplitude decrease as 1/r. Why try 1/r? Because energy flow density is the gradient of temperature, and energy flow density would better decrease as 1/m^{2} .

Plugging in, getting β

Same β as in 1D case, using the decaying solution

Inserting boundary condition

The ‘amplitude’ A is complex as β is complex. Getting the real part – this is what you would compare with measurements:

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## Comparison of surface energy densities: 1D versus 3D temperature waves [Top]

This is to estimate the magnitude of the error you introduce when solving an actually 3D problem in only one dimension; replacing the curved (spherical) surface by a plane.

One dimension – energy flow density is just a number:

Real part of this, at the surface (x=0)

How should this be compared to the 3D case? The time average (e.g. yearly) average is zero, to one could compare the average value for half period, when the cosine is positive or negative (‘summer’ or ‘winter’ average). But then, you can as well compare the amplitudes.

Introducing new parameters

3D case: Energy flow density is a vector

The vector points radially of course, its absolute value is

At the surface of the sphere the ‘ugly part’ is zero as

Real part:

Here, I was playing with somewhat realistic parameters for the properties of the conducting material. If the sphere has a radius of a few meters, you can ‘compensate for the curvature’ by tweaking parameters and obtain a 1D solution in the same order of magnitude.

Temporal change – there is a ‘base’ phase different between temperature and energy flow of (about) π/4 which is also changed by introducing curvature. I varied ρ,c, and λ with the goal to make the j curves overlap as much as possible. It is sufficient and most effective to change specific heat only. If the surface is curved, energy ‘spreads out more’. So to make it ‘as fast as’ the 3D wave you need to compensate by a giving it a higher D.

I did not bother to shift the temperature to, say, 10°C as a yearly average. But this is just a linear shift tat will not change anything else – 0°C is arbitrary.

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## 1D stationary solution – plane [Top]

Stationary means, that nothing changes with time. The time derivative is zero, and so is the (spatial) curvature:

The solution is a straight line, and you need to know the temperature at two different points. Indicating the surface x=0 again with 0 and the endpoint x_E with E, and using the definition of j in terms of temperature gradient and distance from the surface (x_E – 0 = Δx).

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## 3D stationary solution- sphere [Top]

The time derivative is zero, so the Laplacian is zero:

Ansatz, guessing something simple

Boundary conditions, as for the 1D case:

Plugging in – getting functions for all r:

At the surface:

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## 2D stationary solution – cylinder, pipe [Top]

Cylindrical Laplacian is zero

Same boundary conditions, plugging in

Solutions for temperature and energy flow at any r:

Expressing r in terms of distance from the surface,

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## Comparison of overall heat flow: 1D versus 2D [Top]

j is the energy flow per area, and the area traversed by the flow depends on geometry. in the 1D case the area is always the same area, equal to the area of the plane. For a cylinder, the area increases with r.

The integrated energy flow J for a plate with area F is

If the two temperatures are given, J decreases linearly with increasing thickness of the cylindrical ‘shell’, e.g. a growing layer of ice.

For a cylinder of length l the energy flow J is…

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Factor r has been cancelled, and the for given temperatures J is only decreasing linearly with increasing outer radius . That’s why vendors of plate heat exchangers (in vessels with phase change material) worry more about a growing layer of sold material than user for e.g. ‘ice on coil’ I quoted a related research paper on ‘ice storage powered’ heat pump system in this post – they make exactly this point and provide some data. In addition to conduction also convection at both sides of the heat exchanger should be taken into account, too, in a ‘serial connection’ of heat transferring components.