ASHRAE Journal - July 2010 - 21

The method presented here is strictly applicable only when these values are used. The amplitudes of these pulses are determined from the building load profile and the coefficient of performance of the heat pumps. The evaluations of the three effective ground thermal resistances and of the temperature penalty are not straightforward. This article proposes correlations, based on multiple calculations, to calculate these values. Furthermore, a simple way to evaluate the equivalent borehole thermal resistance, based on the work of Hellström4 is proposed. With these correlations, a simple spreadsheet-based calculation can be carried out to obtain borehole length estimates. In the next section, the methods used to generate the correlations are presented. This is followed by a brief description of the methodology used to evaluate Rb. Finally, two sizing examples, for single and multiple boreholes systems, are presented to demonstrate the ease of use of the proposed method. A Microsoft Excel spreadsheet accompanies this article (see link in the box above). The procedure presented here and the accompanying spreadsheet are not intended to replace commercially available borehole sizing software (which cover a larger spectrum of conditions) but to provide designers with a simple tool to guide them through ASHRAE Handbook calculations. Correlations for R6h, R1m and R10y The effective ground thermal resistances account for transient heat transfer from the borehole wall to the far-field undisturbed ground temperature. Several ways exist to evaluate thermal resistances in the ground. In this work, the approach proposed by Kavanaugh and Rafferty1 and contained in the ASHRAE Handbook2 is used. It is based on the cylindrical heat source solution originally proposed by Carslaw and Jaeger5 used in conjunction with temporal superposition as proposed by Ingersoll and Plass6 and reviewed by Bernier.7 The effective thermal resistances are expressed as follows:

Ground Load, W (Btu/h)

The sizing calculation spreadsheet is available for free at www.ashrae.org/ borehole.xls.

qh

qm

qy

Hour of the Year 10 Years 1 Month 6 Hours

Figure 1: Three consecutive ground load pulses.

After a time period equivalent to H²/(90a), where H is the borehole depth, Eskilson8 has shown that axial effects start to be significant. The error introduced when using the cylindrical heat source has been calculated by Philippe, et al.9 Based on these results, it appears that the axial effects are only significant for the R10y term and that the error remains below 5% for typical values of thermal diffusivities. More accurate solutions, such as the two-dimensional finite line source model (Eskilson8) could be used. These solutions are more complex to solve and the gain in accuracy in the context of an engineering approximation does not warrant their use here. In the present work, the Gfunction is calculated precisely based on the work of Baudoin.10 Alternatively, readers can use the graphical values presented by Kavanaugh and Rafferty.1 For each of the three effective ground resistances, R6h, R1m and R10y, a total of 48 calculations are performed over the following range of typical operating conditions:

0.05 m ≤ rbore ≤ 0.1 m 0.025 m / day ≤ α ≤ 0.2 m / day
2 2

(3)

Then, to avoid complicated calculations of the G-function, these results were curve-fitted in the following form:
R= 1 k f ( α ,rbore )

R1m = R10y =

1 2 2 G αt10y+1m+6h / rbore − G αt1m+6h / rbore   k

1 2 2 G αt1m+6h / rbore − G αt6h / rbore   k

1 2 R6h = G αt6h / rbore k

(

) )
(2)

(

)

(

2 f = a0 + a1rbore + a2 rbore + a3 α+a4 α 2 +a5ln (α ) +

(4)

(

) (

)

a6ln (α ) + a7 rbore α+a8 rbore ln (α ) + a9 αln (α )

2

where G-function represents the cylindrical heat source solution, k is the ground thermal conductivity, a is the ground thermal diffusivity, and rbore is the borehole radius (Figure 3). The cylindrical heat source solution is strictly valid for onedimensional (in the radial direction) transient heat transfer.
July	2010	

As shown, the resulting correlation function f depends only on two dimensional parameters, a and rbore, given in m²/day and in m, respectively. The correlation coefficients for f6h, f1m and f10y are given in Table 1. The correlated values are compared with the calculated values in Figure 2. The results are plotted in terms of effective thermal resistances for a particular value of ground thermal conductivity, k=1 W/(m·K) [0.58 Btu/h·ft·°F]. The bottom and left axes present results in SI units while I-P units are used in the top and right
ASHRAE	Journal	 21



ASHRAE Journal - July 2010

Table of Contents for the Digital Edition of ASHRAE Journal - July 2010

ASHRAE Journal - July 2010
Table of Contents
Commentary
Industry News
Letters
Meetings and Shows
Sizing Calculation Spreadsheet: Vertical Geothermal Borefields
HVAC for Prisons
Control Strategies for Variable Speed Pumps In Super High-Rise Building
Maintaining Green
Sustainable Products Capabilities
Emerging Technologies
Technical Topics
Special Products
People
Products
Classified Advertising
Advertisers Index
ASHRAE Journal - July 2010 - Intro
ASHRAE Journal - July 2010 - ASHRAE Journal - July 2010
ASHRAE Journal - July 2010 - Cover2
ASHRAE Journal - July 2010 - 1
ASHRAE Journal - July 2010 - 2
ASHRAE Journal - July 2010 - Table of Contents
ASHRAE Journal - July 2010 - Commentary
ASHRAE Journal - July 2010 - 5
ASHRAE Journal - July 2010 - Industry News
ASHRAE Journal - July 2010 - 7
ASHRAE Journal - July 2010 - 8
ASHRAE Journal - July 2010 - 9
ASHRAE Journal - July 2010 - 10
ASHRAE Journal - July 2010 - 11
ASHRAE Journal - July 2010 - Letters
ASHRAE Journal - July 2010 - 13
ASHRAE Journal - July 2010 - 14
ASHRAE Journal - July 2010 - 15
ASHRAE Journal - July 2010 - 16
ASHRAE Journal - July 2010 - 17
ASHRAE Journal - July 2010 - Meetings and Shows
ASHRAE Journal - July 2010 - 19
ASHRAE Journal - July 2010 - Sizing Calculation Spreadsheet: Vertical Geothermal Borefields
ASHRAE Journal - July 2010 - 21
ASHRAE Journal - July 2010 - 22
ASHRAE Journal - July 2010 - 23
ASHRAE Journal - July 2010 - 24
ASHRAE Journal - July 2010 - 25
ASHRAE Journal - July 2010 - 26
ASHRAE Journal - July 2010 - 27
ASHRAE Journal - July 2010 - 28
ASHRAE Journal - July 2010 - 29
ASHRAE Journal - July 2010 - HVAC for Prisons
ASHRAE Journal - July 2010 - 31
ASHRAE Journal - July 2010 - 32
ASHRAE Journal - July 2010 - InsertA
ASHRAE Journal - July 2010 - InsertB
ASHRAE Journal - July 2010 - 33
ASHRAE Journal - July 2010 - 34
ASHRAE Journal - July 2010 - 35
ASHRAE Journal - July 2010 - Control Strategies for Variable Speed Pumps In Super High-Rise Building
ASHRAE Journal - July 2010 - 37
ASHRAE Journal - July 2010 - 38
ASHRAE Journal - July 2010 - 39
ASHRAE Journal - July 2010 - 40
ASHRAE Journal - July 2010 - 41
ASHRAE Journal - July 2010 - 42
ASHRAE Journal - July 2010 - 43
ASHRAE Journal - July 2010 - Maintaining Green
ASHRAE Journal - July 2010 - 45
ASHRAE Journal - July 2010 - 46
ASHRAE Journal - July 2010 - 47
ASHRAE Journal - July 2010 - 48
ASHRAE Journal - July 2010 - 49
ASHRAE Journal - July 2010 - 50
ASHRAE Journal - July 2010 - Sustainable Products Capabilities
ASHRAE Journal - July 2010 - 52
ASHRAE Journal - July 2010 - 53
ASHRAE Journal - July 2010 - 54
ASHRAE Journal - July 2010 - 55
ASHRAE Journal - July 2010 - 56
ASHRAE Journal - July 2010 - 57
ASHRAE Journal - July 2010 - 58
ASHRAE Journal - July 2010 - 59
ASHRAE Journal - July 2010 - 60
ASHRAE Journal - July 2010 - 61
ASHRAE Journal - July 2010 - 62
ASHRAE Journal - July 2010 - 63
ASHRAE Journal - July 2010 - 64
ASHRAE Journal - July 2010 - 65
ASHRAE Journal - July 2010 - 66
ASHRAE Journal - July 2010 - 67
ASHRAE Journal - July 2010 - 68
ASHRAE Journal - July 2010 - 69
ASHRAE Journal - July 2010 - 70
ASHRAE Journal - July 2010 - 71
ASHRAE Journal - July 2010 - 72
ASHRAE Journal - July 2010 - 73
ASHRAE Journal - July 2010 - 74
ASHRAE Journal - July 2010 - 75
ASHRAE Journal - July 2010 - 76
ASHRAE Journal - July 2010 - 77
ASHRAE Journal - July 2010 - 78
ASHRAE Journal - July 2010 - 79
ASHRAE Journal - July 2010 - 80
ASHRAE Journal - July 2010 - 81
ASHRAE Journal - July 2010 - 82
ASHRAE Journal - July 2010 - 83
ASHRAE Journal - July 2010 - 84
ASHRAE Journal - July 2010 - 85
ASHRAE Journal - July 2010 - 86
ASHRAE Journal - July 2010 - 87
ASHRAE Journal - July 2010 - 88
ASHRAE Journal - July 2010 - 89
ASHRAE Journal - July 2010 - 90
ASHRAE Journal - July 2010 - 91
ASHRAE Journal - July 2010 - 92
ASHRAE Journal - July 2010 - 93
ASHRAE Journal - July 2010 - 94
ASHRAE Journal - July 2010 - 95
ASHRAE Journal - July 2010 - Emerging Technologies
ASHRAE Journal - July 2010 - 97
ASHRAE Journal - July 2010 - 98
ASHRAE Journal - July 2010 - 99
ASHRAE Journal - July 2010 - Technical Topics
ASHRAE Journal - July 2010 - 101
ASHRAE Journal - July 2010 - 102
ASHRAE Journal - July 2010 - 103
ASHRAE Journal - July 2010 - Special Products
ASHRAE Journal - July 2010 - 105
ASHRAE Journal - July 2010 - 106
ASHRAE Journal - July 2010 - 107
ASHRAE Journal - July 2010 - People
ASHRAE Journal - July 2010 - Products
ASHRAE Journal - July 2010 - Classified Advertising
ASHRAE Journal - July 2010 - 111
ASHRAE Journal - July 2010 - Advertisers Index
ASHRAE Journal - July 2010 - Cover3
ASHRAE Journal - July 2010 - Cover4
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