Relation Between Mean Fluid Temperature and Outlet Temperature for Single U-Tube Boreholes

Round 1

Reviewer 1 Report

This paper aims to provide a relationship to be used in simulation of ground heat exchangers that will give the outlet fluid temperature when the mean fluid temperature has already been determined.  For many years, the simple assumption that the mean fluid temperature is equal to the simple average of the entering and exiting fluid temperatures, combined with an energy balance on the fluid, has sufficed to predict the exiting fluid temperature from the ground heat exchanger.  This approximation works reasonably well as long as the time step is long enough and the amount of short-circuiting is not too high.  In my experience, this approximation works pretty well for typical North American boreholes of around 100 m or less depth, using time steps that correspond to multiple fluid passes through the borehole.  (Compare this to Figure 10 in the paper, where the outlet temperature predicted with a conventional model coupled with the effective borehole thermal resistance has maximum error below 12 minutes; at one hour has an error of about 0.3°C, and is essentially zero at 2 hours.)  Nevertheless, for cases where the time step is low relative to the transit time or with significant short-circuiting, a better model may be required, and contributions in this area are of interest.

This is a generally well-written paper and appears to have a sound basis.  However, I have some concern that the paper promises more than it delivers, misrepresents the range of applicability in the conclusions, and, as the paper stands now, provides a nice solution for a problem that is of not very much interest.  At the least, its contribution should be clarified, or perhaps the authors the authors can extend their presentation to demonstrate this generally useful contribution.

It might also be pointed out at the beginning that when we speak of ground heat exchanger simulation, there are at least three scenarios of interest:

Analysis of thermal response tests, where an approximately fixed load is imposed on a ground heat exchanger for a short duration. Many simulations of this scenario assume that the load is fixed and that the temperatures evolve based on the time from the beginning.  As thermal response tests are usually conducted for 48 hours or more and the early response is usually neglected, the usefulness of the model provided by this paper is limited for this application. Simulation of the ground heat exchanger as part of a design. In this case, the loads change with time, but are generally known a priori and are treated as inputs.  If a rectangular pulse approximation is used, the load will stay constant over the pulse, but then durations of less than two hours are seldom of interest.  If an actual hourly or shorter time step were used, then a model that can improve the prediction of fluid temperature would be very useful.  But it would need to do so under conditions when the load is changing.  It’s also possible that the flow rate would be changing, but for design purposes this might be neglected. Simulation of the ground heat exchanger as part of a whole building energy simulation. In this case, the loads are often not known in advance, and the entering fluid temperature and flow rate are inputs.  The exiting fluid temperature and heat rejection/extraction rate are outputs at each time step.  Again, the load in this case changes with every time step and, in many systems, the flow rate will be changing too.  And, in the reviewer’s opinion, here is where a simple model for predicting the outlet fluid temperature is needed.

So, my critique of the paper can now be summarized as follows.  As far as I can tell, the authors only provide a solution for Scenario 1, and while of academic interest, this has little practical application.  A model like the authors present, that uses a relatively simple formulation to better predict the exiting fluid temperature under Scenario 2 and 3 conditions when the heat transfer rate and possibly the fluid flow rate are changing would be very useful.  But, as presented, I don’t see how this can actually be done with the authors’ model.  If it can be, great!  I would then suggest the authors present that as an additional section and I would consider them to have made a significant contribution. 

What would be necessary is that the authors explain:

How to treat t* under long-term operating conditions when the heat transfer rate and fluid flow rate are changing. Or, if it is only applicable when the fluid rate does not change, I think this would still be a suitable contribution, but that would need to be clearly stated. Demonstration that this procedure works by comparison to their detailed FEM with time-varying heat transfer rates and (possibly) flow rates.

If the authors cannot extend their work to cover this scenario, then, in my view, the paper is of limited interest and I would not recommend publication in a top journal.  If it were to be published, it needs to be revised to more clearly state the limited applicability of the work.  These revisions would include:

The introduction would need to be revised – lines 55-56 and following suggest that the purpose of CARM is only to model TRTs. I disagree.  A model such as CARM or the others should be capable of modeling Scenarios 2 & 3, albeit slowly.  (Actually, this should be revised even if the authors can demonstrate that their work can handle Scenarios 2 & 3.) The authors cannot state “The obtained expressions of φ hold for … and any working condition, both in quasi-stationary and in transient regime.” (Lines 408-411)  Likewise, the statement “Thus, the obtained expressions of φ are a useful complement of the BHE models that yield Tfm.” is incorrect or only correct if it is limited to Scenario 1.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

Dear authors of the manuscript entitled “Relation between mean fluid temperature and outlet temperature for single U-tube boreholes”; the paper presents valuable and interesting information for the low enthalpy geothermal field. A lot of simulations results are included and validated to finally valuate the BHE thermal resistance. The quality of the paper makes it suitable to be considered for the publication in this journal after a minor revision.

Section 1

The introduction section thoroughly describes the state of the art of the field in which the paper is focused. Numerous references and works are cited and the problematic is accordingly described. However, this section should be improved by better explaining the purpose of the present paper. Please shortly explain how the new correlations for the dimensionless coefficient will be determined.

Section 2

Some more specific information must be given regarding the COMSOL model simulated. Why a working period of only 100 h was performed in the simulations?

Section 3

This section is well written and accordingly describes how the correlations were obtained for each of the regimes.

Section 4

Briefly explain the foundation of the analytical BHE model proposed by Man et al. It will help the reader to better understand the validation of the code.

Please provide quantitative values of the discrepancies (shown in Figure 6) between the 3D numerical simulation and the analytical model.

Sections 5, 6 and 7

These sections are also quite complete and well structured.

Section 8

Conclusions are poor in relation to all the information presented in the work. Try to improve this section and highlight the contributions of the paper.

 

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

The authors have improved the paper by extending the work.  Kudos to the authors for this improvement; it is now a much more useful contribution.  My only comment now is that the explanation given on lines 451-465 is crucial, but it lacks clarity. 

Specifically, this text should be clarified:

The time interval t needed by the fluid to go from the inlet to the outlet is evaluated: 14 minutes for the case considered. For each hour, the dimensionless time t* of Eq. (12) starts from zero at the beginning of the hour, and the application of the correlations for  starts after time interval t from the beginning of the hour. During the time interval t of the first hour, Tout is kept constant with the same value as in the initial instant, as is physically consistent and confirmed by the 3D simulation. During the time interval t of any other hour, Tout is considered as linearly varying with time from the last time instant of the preceding hour to the first time instant after t of that hour.

The main item that is unclear is related to the 14 minutes – is this the transit time of the fluid, assuming plug flow?  In a real system, the non-uniform velocity distribution causes the effect of a step change at the inlet to be smeared at the outlet.  If this is the case, that’s fine, but please be clear about it.  If this is addressed, the paper will be acceptable for publication.

However, I would suggest to the authors that this is such an important contribution that it may be better to break out the explanation in its own paragraph.

Author Response

We thank Reviewer 1 for the positive evaluation of our manuscript and for his final suggestion.

We have specified that the time interval Δt mentioned in the comment by Reviewer 1 is evaluated by assuming plug flow.

The first lines of the paragraph, in the new text, are as follows:

"The time interval Δt needed by the fluid to go from the inlet to the outlet is evaluated by assuming plug flow, i.e. a uniform velocity in each cross section: 14 minutes for the case considered."