Transient Heat Transfer from a Buried Pipe

Figure 1. Computational domain and parameters

An accurate estimate of the heat transfer from a buried pipe to the surrounding ground is essential for the design of the ground loop portion of a ground-source heat pump. Exact analytical solutions to this problem are complicated by the fact that heat pump systems rarely operate continuously. Complete numerical simulations of system designs can be carried out, but these are unwieldy and difficult to justify for initial scoping calculations, or for preliminary performance estimates.  It was desirable to  develop  simple algebraic correlations that could be used to approximate the intermittent overall heat transfer between a fluid flowing in an isolated buried pipe and the surrounding ground.

Figure 2. Sample heat flow results

A finite difference model of this transient problem was developed in a SINDA-like numerical package.  Figure 1 shows the computational domain and system and material parameters that were used in the numerical model.  The model was exercised over a wide range of ground and fluid properties and operating conditions.  Figure 2 presents one sample set of results showing  dimensionless heat flow as a function of time at the pipe wall.  Algebraic correlations of the results were developed in order to provide readily accessible and simple design equations.  Figure 3 demonstrates the agreement between the dimensionless average heat transfer and the algebraic correlation as a function of dimensionless time.

Figure 3. Sample average heat flow results with algebraic correlation

References

J.W. Stevens, 2002, “Coupled Conduction and Intermittent Convective Heat Transfer From a Buried Pipe,”  Heat Transfer Engineering, Vol 23, n. 4, pp. 34-43.

J. W. Stevens, 2000, “Intermittent Convective Heat Transfer for Ground-source Heat Pump Design,”  Proceedings of the ASME Advanced Energy Systems Division – 2000, AES-Vol. 40, pp. 147-152.

J.W. Stevens, 1998, "Transient Heat Transfer Approximations for Ground-source Heat Pump Design,"  Proceedings of the ASME Advanced Energy Systems Division – 1998, AES-Vol. 38, pp. 415-424.

The Second Law of Thermodynamics (part 3)

A Third Statement of the Second Law

            A third way of stating the second law is to say that it is impossible to create a refrigerator that uses no power. This is equivalent to saying that, by itself, heat always flows from a warmer place to a cooler place. Recall that the purpose of a refrigerator (or air conditioner) is to take heat out of a cool place and move it to a warm place. According to this statement of the second law, this won’t happen spontaneously. Therefore, a common way for a refrigerator to function is to establish a region that is even colder than the space to be cooled, and a separate region that is even hotter than the spot where the heat is to go. However, in doing this, we have added something to our original setup, i.e. the refrigerator along with its associated input of power. 
Now, if it were possible to build a refrigerator that did its job without any input of power, we could take such a refrigerator, enclose it in a box, and use it to make heat flow spontaneously from a cooler place to a warmer place. Such a device has never been demonstrated and would violate the second law. Thus, saying that heat always flows down a temperature gradient is equivalent to saying that it is impossible to build a refrigerator that requires no power. 
            In a similar way, this statement of the second law implies, and is implied by the first statement that we used, i.e. that it is impossible to convert all heat into work. Recall that a heat engine is a device that operates between a hot reservoir and a cold reservoir and produces work. But if it were possible to build a refrigerator that did not require any work to operate, we could imagine a composite device consisting of a normal heat engine combined with a refrigerator that uses no work. 
The net result of the composite device would be a heat engine that changed all heat into work which would violate the second law. 
Alternately, we could imagine a composite device where a normal refrigerator was connected to a device that converts all heat into work. The combined device would violate our third statement of the second law since it would move heat from a cool place to a warm place without any input of work.








Summary
          The Second Law of Thermodynamics is based on the experience of many years of observations and is as solidly grounded as the First Law of Thermodynamics. It can be expresed in many different ways but all the expressions imply one another. The second law establishes values for different forms of energy, allowable directions for processes, and theoretical limits on all heat engine efficiencies.

Second Law, part 1

Second Law, part 2