Save Energy

Team: 67

School: Los Alamos High

Area of Science: Environmental Science

Interim: • Problem Definition

In the winter, office buildings are normally heated twenty-four hour a day, even though there are no people working after hours. This project will estimate the amount of energy that can be save if the heating system of office buildings is operated differently. For example, the heat is turned off when officers all left for home, and is turned on before officers come to work. For the estimate, we have to calculate how long it will take to warm up an office building during winter for the given conditions, for example, the material of the building, temperatures outside, the way the building is heated, etc. From the result of the calculation, the office building will start to be heated at a certain time before the building starts to be occupied, and the building is already warmed up when the building is occupied.

• Plan for solving the problem computationally

We will numerically solve the heat conduction equation with source (heating) and multi-materials, for example, air, bricks, and drywall with insulation. The source is a function of temperature and space. When the temperature is lower than a certain value, the source is on, and when temperature gets to another certain value, the source will be off. The total time during which the source is on is the cost of heating.

To calculate the saving, we have to simulate two situations, one for keeping the building warm for twenty-four hours a day, and the other for keeping the building warm for eight hours. In the first case, the building starts at warm, but in the second case, it starts at cold.

We will start with a differential equation, the heat conduction equation. From the differential equation, we will obtain an integral form of the heat conduction equation. The integral equation is valid everywhere, although the differential equation is not valid across material interfaces. From the integral equation, we will derive a set of difference equations through certain approximations for heat fluxes at (numerical) cell interfaces. The possible approximations we intent to try are Euler (forward) method, Crank-Nicholson method, and backward method. We will evaluate the advantages and disadvantages of each approximation before using a particular method.

If we use Crank-Nicholson or backward method, what faces us will be a set of linear equations, or a linear system. We will iteratively solve the linear system. Among the possible approaches is Guass-Seidel iteration, red-black iteration, and multi-grid method. We intend to use red-black or multi-grid method to solve the linear system.

• Progress to Date

We have derived the integral form of the heat conduction equation that is valid anywhere including material interfaces. From the integral form, the difference equation of heat conduction with multi-materials has been obtained. The advantages and disadvantages of Euler (forward) method, Crank-Nicholson method, and backward method have been evaluated. From the evaluation, we decide to use Crank-Nicholson method that is stable and second order accurate in time. From the approximation of Crank-Nicholson method, we have derived the difference equation for heat conduction. Up to now, all the formulations allow both two- and three-dimensions. We will decide late whether we will write computer codes for two- or three-dimensional heat conductions. We are currently designing the structure of computer codes.

• Expected Results

We expect that we will be able to answer the original question stated in the project through three-dimensional simulations. The coefficients of heat conduction of each material involved are assumed to be constant. We also expect to be able to show temperature changes in space and time through graphics and visualization.


• Reference:

1. J. Crank, P. Nicolson, A practical method for numerical evaluation of solution of partial differential equations of the heat-conduction type, Proc. Camb. Philos. Soc. 43 (1947) 50-67.
2. M. Jakob, Heat Transfer, Vol. I (Wiley, New York, 1949).
3. S. V. Patankar, Numerical Heat Transfer and Fluid Flow (Hemisphere, New York, 1980).
4. P. Wesseling, Theoretical and practical aspects of a multigrid methods, AIAM J. Sci. Comput. 3 (1982) 387.
5. Y. Jaluria and K. E. Torrance, Computational Heat Transfer ( Springer-Verlag, Berlin, 1986).

Team Members:

  Edward Dai
  Aidan Bradbury

Sponsoring Teacher: Lee Goodwin

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