PDE and heat diffusion

A ground source heat pump can recover 3-4 times more heat than the amount of electricity used to run it. For example a 1kW compressor heat pump can recover 4kW heat. This seems like a great way to get "free" extra heat. However, there are some ifs and buts. One category of concerns regard how the electricity is produced, see the linked article.

Another, afaik not explored concern is that bringing heat to your home means putting cold into the ground. The cited article states that the ground "a few feet down is a constant 6C".

In this exercise you will explore the heat equation and calculate the heat diffusion in the ground. We will study One home sinking cold into an infinite reservoir around it. In these problems consider under what assumption you can model the spatial dimensions with 1,2 or 3 DOF. Hint: Consider different coordinate systems, rectangular, cylindrical, spherical etc.

4xx solve an elliptic stationary formulation.
6xx Solve also the seasonal (time dependent) parabolic problem.

using finite difference solution methods

Heat Equation

The general heat equation can be written

$\begin{displaymath}{\partial T \over \partial t} = \kappa \nabla^2 T + {\kappa h \over \lambda} \end{displaymath}$

Where $\lambda$ is the thermal conductivity, and $\kappa={\lambda \over c_v \rho}$ , is the thermal conduction capacity

= specific heat capacity $\rho$ is material density, and finally

is the heat addition.

For our heat pump problem we have:

: Thermal conductivity of soil in a bore hole $\lambda = 2$ (W/(m C)
: Heat generation of the ground loop
: Bore hole temp
: Earth temperature C
: Surface temperature C

We consider a heat pump installation (see figure above) with one vertical ground loop bored 45m into the ground. For numerical solution, we need to discretize it on a finite domain. A reasonable choice is a domain significantly larger than 45m. We choose a domain extending 90m horizontally, and 100m vertically.

Stationary problem (cmput 4/5/6xx)

After a sufficiently long time ${\partial T \over \partial t} = 0$ and we can write the stationary heat equation as:

$\begin{displaymath}\nabla^2 T = -{h \over \lambda} \end{displaymath}$

For one ground loop, we have circular symmetry about the z-axis, and it is useful to consider cylindrical coordinates (see figure). In cylindrical coordinates $(r,\alpha,z)$ the Laplacian operator is:

$\begin{displaymath}\nabla^2 T = {1 \over r} {\partial T \over \partial r} (r {\... ...l^2 T \over \partial \alpha} + {\partial^2 T \over \partial z} \end{displaymath}$

The main goal here is for you to discretize the stationary heat equation and solve for the heat distribution . There are however some perhaps subtle issues in transforming the real world problem into a computatonal solution.

To make the modeling simple we will solve a pure boundary value problem. In the above specification temperatures are given for the earth soil, surface and bore hole. What boundary condiiton do you need for the line r = 0 below the bore hole? Use these boundary conditions, set up the finite difference equation system and solve for the distribution using stepsizes h_r, h_z = 10m and h_r, h_z = 1m. Plot the solutions and compare.
In this solution we have made some assumptions, in particular we study solve for the temperature in only a finite region, not the whole earth. How can you verify that this region is adequately large?
Energy is withdrawn from the bore hole to heat the house. As mentioned in the specification, bore hole temperature is typically a few degrees below freezing. This is due to the brine being cooled to a few degrees below freezing in the heat pump, and the assumption of a good heat transfer between brine and bore. (In lining and filling the boreholes materials to achieve good heat transfer would be used.) It is also specified that it is desired to withdraw -8000W of energy from the bore hole. Does your current solution guarantee this? Why/why not?
Can you think of an alternative way of formulating the problem to use the desired energy withdrawal in your solution process?

Time dependent problem (cmput 5/6xx)

Here we assume the heat pump is on for half the year, off the other half. Let the initial temperature in our region (before the heat pump is ever turned on)

, and solve for time t = [1,2,3, ..., 48] months, using h=10m and h=1m given:

: Specific heat capacity of (dry) soil (J/(kg C)
: Density of soil $\rho = 1555$ (kg/ $\rm {m}^3$ )
: Heat generation of the ground loop for Oct-Mar, otherwise
: Bore hole temp for Oct-Mar. Neumann otherwise.
: Surface temperature C for Oct-Mar, 15C otherwise

When does the yearly behavior turn repetitive (periodic)?

Further questions (cmput 5/6xx)

The article linked above from Green Edmonton cites the concern with using coal produced electricity. In Japan heat pumps run directly from internal combustion engines are used. What might be a reasonable estimate of system efficiency for a compression ignition engine if both the engine waste heat and heat pump heat is recovered to the house?

What would happen for a block of adjacent homes? Since so few heat pumps are installed little is known about the cumulative effect. Do they each have a detrimental effect on the neighbors? Will we get permafrost? Consider a row along a street and a dense 2D array in a subdivision with typical Edmonton lot sizes. How can you model each of these cases? What symmetries exist? How many dimensions do do you need to solve the PDE?

Will we get permafrost?

Hot off the press, famous British physicist and adviser to the British government on energy policy Prof. David MacKay says YES.

Now these statements always comes with some qualifications. David assumes a relatively dense suburb and that heat pumps will be installed on relatively poorly insulated buildings without upgrades to insulation. Instead he recommends that Britain adopts air source heat pumps. You can find details about his model and reasoning here:Sustainable Energy - without the hot air, p301-306

Are David MacKay's conclusion and model assumption reasonable?
What are densities of Canadian and US suburbs? What would his conclusion be for North America?
His model does not account for the motion of ground water. Why might this be significant? Does it change the conclusions?
Can you find any other aspects of the model that needs detailed scrutiny?

For those who solve the time dependent problem, notice how the wave propagates through the ground. David made a nice illustration of this (right). Assuming a sinusoidal boundary condition at the ground, the temperature at each depth also varies sinusoidally, but phase shifted. Notice how the temp at "level 3" is completely out of phase with the surface, so it is the coldest in the middle of the summer! (Level three is 5-9 meters down depending on the soil type.)

419/675 Computational Differential Equations