單級單吸清水離心泵設(shè)計

單級單吸清水離心泵設(shè)計,單級單吸清水離心泵設(shè)計,單級單吸,清水,離心泵,設(shè)計 Varennes, al, C.P Keywords: are vari , shor effects. In one example considered, it is shown that the borehole length is 15% shorter when axial C211 2009 Elsevier Ltd. All rights reserved. coupled increasingl is presente ively rejecter or extractor is used at peak conditions to reduce the length of the ground heat exchanger. they necessitate a high level of expertise. Furthermore, it is not easily possible to obtain ground temperature distributions like the ones shown later in this paper. In this paper hourly simulations are performed using the so-called finite and infinite line source approximations where the borehole is approximated by a line with a constant heat transfer rate per unit length. These approximations present, in a convenient analytical form, the solution to the tran- sient 2-D heat conduction problem. Despite their advantages, hourly simulations based on the line source approximation are * Corresponding author. Departement des genies civil, geologique et des mines, E cole Polytechnique de Montreal, C.P. 6079 Succ. Centre-ville, Montreal, (Qc), H3C 3A7 Canada. Tel.: 1 514 340 4711x4620; fax: 1 514 340 3970. Contents lists available Renewable journal homepage: www.else Renewable Energy 35 (2010) 763770 E-mail address: denis.marcottepolymtl.ca (D. Marcotte). lates a heat transfer fluid in a closed circuit from the GLHE to a heat pump (or a series of heat pumps). Typically, GLHE consists of boreholes that are 100150 m deep and have a diameter of 1015 cm. The number of boreholes in the borefield can range from one, for a residence, to several dozens, in commercial applications. Furthermore, several borehole configurations (square, rectangular, L-shaped) are possible. Typically, a borehole consists of two pipes forming a U-tube (Fig.1). The volume between these pipes and the borehole wall is usually filled with grout to enhance heat transfer from the fluid to the ground. In some situations it is advantageous to design so-called hybrid systems in which a supplementary heat tion and minimum/maximum heat pump entering water temper- ature 8,3. There are design software programs that perform these calculations. Some use the concept of the g-functions developed by Eskilson 5. The g-functions are derived from a numerical model that, by construction, includes the axial effects. The other approach is to perform hourly simulation. This last approach is essential for design of hybrid systems in which supplemental heat rejection/ injection is used. There are several software packages that can perform hourly borehole simulations. For example, TRNSYS 9 and EnergyPlus 4 use the DST 6 and the short-time step model 5, respectively. Even though these packages account for axial effects, Hybrid systems Underground water freezing 1. Introduction Geothermal systems using ground- exchangers (GLHE) are becoming growing energy costs. Such a system The operation of the system is relat 0960-1481/$ see front matter C211 2009 Elsevier Ltd. doi:10.1016/j.renene.2009.09.015 closed-loop heat y popular due to d in Fig. 1. simple: a pump circu- Given the relatively high cost of GLHE, it is important to design them properly. Among the number of parameters that can be varied, the length and configuration of the borefield are important. There are basically two ways to design a borefield. The first method involves using successive thermal pulses (typically 10-years1 month6 h) to determine the length based on a given configura- Finite line source Ground loop heat exchangers are considered. Infinite line source conduction effects are considered. In another example dealing with underground water freezing, the amount of energy that has to be removed to freeze the ground is three times higher when axial effects The importance of axial effects for borehole heat-pump systems D. Marcotte a,b,c, * , P. Pasquier a , F. Sheriff b , M. Bernier a Golder Associates, 9200 lAcadie, Montreal, (Qc), H4N 2T2 Canada b CANMET Energy Technology Centre-Varennes, 1615 Lionel-Boulet Blvd., P.O. Box 4800, c Departement des genies civil, Geologique et des mines, E cole Polytechnique de Montre article info Article history: Received 13 May 2008 Accepted 18 September 2009 Available online 23 October 2009 abstract This paper studies the effects systems. The axial effects line source methods. Using important. Unsurprisingly All rights reserved. design of geothermal c (QC), J3X 1S6 Canada . 6079 Succ. Centre-ville, Montreal, (Qc), H3C 3A7 Canada of axial heat conduction in boreholes used in geothermal heat pump examined by comparing the results obtained using the finite and infinite ous practical design problems, it is shown that axial effects are relatively t boreholes and unbalanced yearly ground loads lead to stronger axial at ScienceDirect Energy rarely used in routine design due to the perceived computational axial effects on the GLHE design. Our main finding is that for models is the change in temperature felt at a given location and Nomenclature a Thermal diffusivity (m 2 s C01 ) A, B, C, D Synthetic load model parameters (kW) b r/H C s Ground volumetric heat capacity (Jm C03 K C01 ) erfc (x) Complementary error function (erfcx 1 2 p p R N x e C0t 2 dt EWT Temperature of fluid entering the heat pump (K or C14 C) F o Fourier number, F o at/r 2 k s Volumetric ground thermal conductivity (Wm C01 K C01 ) H Borehole length (m) HP Heat Pump q 0 Radial heat transfer rate (W) q Radial heat transfer rate per unit length (Wm C01 ) S Borehole spacing (m) r Distance to borehole (m) r b Borehole radius (m) R b Borehole effective thermal resistance (KmW C01 ) t Time DT (r, t) Ground temperature variation at time t and distance r from the borehole (K or C14 C) T f Fluid temperature (K or C14 C) T g Undisturbed ground temperature (K or C14 C) T w Temperature at borehole wall (K or C14 C) u H 2 at p x, y Spatial coordinates (m) z Elevation (m) D. Marcotte et al. / Renewable Energy 35 (2010) 763770764 burden. The major difference between the finite and infinite line source lies in the treatment of axial conduction (at the bottom and top of the borehole) which is only accounted for in the former. The theoretical basis of the finite line source, although more involved than for the infinite line source, was first established by Ingersoll et al. 7. It has been rediscovered recently by Zeng et al. 15 who improved the model by imposing a constant temperature at the ground surface. Lamarche and Beauchamp 11 have made a useful contribution to speed up the computation of Zengs model. Finally, Sheriff 13 extended Zengs model by permitting the borehole top to be located at some distance below the ground surface. She also did a detailed comparison of the finite and infinite line source responses, but did not examine the repercussion on borefield design. At first glance, the axial heat-diffusion is likely to decrease (increase) the borehole wall temperature in cooling (heating) modes respectively. Therefore, designing without considering axial effects appears to provide a safety factor for the design. But, is it really always the case? Moreover, are the borehole designs incorporating axial effects significantly different from those neglecting it? Under which circumstances are we expected to have significant design differences? These are the main questions we seek to answer. The main contribution of this research is to describe, using synthetic case studies, the impact of considering Fig. 1. Sketch of a time due tothe effectof a constant pointsourcereleasing q 0 units of heat per second 7: DTr;t q 0 4pk s r erfc C18 r 2 at p C19 (1) where erfc is the complementary error function, r the distance to the point heat source, and a is the ground thermal diffusivity. The line is then represented as a series of points equally spaced. In the limit, when the distance between point sources goes to zero, many realistic circumstances the axial effects cannot be neglec- ted. Therefore, design practices should be revised accordingly to include the axial effects. We first review briefly the theory for infinite and finite line source models. Then, we present three different design situations. The first two situations involve the sizing of geothermal systems with and without the hybrid option, under three different hourly ground load scenarios. The last design problem examines the energy required and ground temperature evolution in the context of ground freezing for environmental purposes. 2. Theoretical background The basic building block of both infinite and finite line source GLHE system. the combined effect felt at distance r from the source is obtained by integration along the line. 2.1. Infinite line source In an infinite medium, the line-integration gives the so-called (infinite) line source model 7: DTr;t q 4pk s Z N r 2 =4at e C0u u du (2) 2.2. Finite line source In the case of a finite line source, the upper boundary is considered at constant temperature, taken as the undisturbed ground temperature 15. This condition is represented by adding a mirror image finite line source with the same load, but opposite sign, as the real finite line. Then, integrating between the limits of the real and image line, one obtains 15,13: DTr;t;z q Z H 0 erfc C16 du 2 at p C17 C0 erfc C16 d 0 u 2 at p C17 0 1 A du (3) In hourly simulations, the fluid temperature (T f in Fig. 1)is required. This necessitates knowledge of the borehole thermal resistance R b (i.e. from the fluid to the borehole wall), and of the borehole wall temperature (T w in Fig. 1) 2. The average borehole wall temperature it obtained by integrating Equation (3) along z. However, this is computationally intensive due to the double integration. Lamarche and Beauchamp 11 have shown, using an appropriate change of variables, how to simplify Equation (3) to a single integration. Accounting for small typos in 11 and 15 as noted by Sheriff 13, the average temperature difference, between a point located at distance r from the borehole and the undisturbed ground temperature, is given by: DTr;t q 2pk s 0 B B B Z b 2 1 p b erfcuz z 2 C0b 2 q dzC0D A C0 Z b 2 4 p b 2 1 p erfcuz z 2 C0b 2 q dzC0D B 1 C C C A (4) D. Marcotte et al. / Renewable Energy 35 (2010) 763770 765 4pk s 0 du d u where du r 2 zC0u 2 q and d 0 u r 2 zu 2 q , z is the elevation of the point where the computation is done. The left part of the integrand in Equation (3) represents the contribution by the real finite line, the right part, the contribution of the image line. Fig. 2 shows the vertical temperature profile obtained with Equation (3) at radial distance r2 m, after 200 days, and at r1 m, after 2000 days of heat injection. The corresponding infinite lines-source temperature is indicated as a reference. In this example,theboreholeis50 mlong,thegroundthermalparameters are k s 2.1 Wm C01 K C01 and C s 2e06 Jm C03 K C01 . The ground is inti- tially at 10 o C. The applied load is 60 W per m for a total heating power of 3000 W. As expected, the importance of axial effects and the discrepancy between infinite and finite models increases with the Fourier number (at/r 2 4.54 and 181.4 for these two cases). 10 12 14 16 18 20 22 24 0 10 20 30 40 50 60 Temperature ( o C) Depth (m) Vertical temperature profile Infline, r=2, t=200 d Fline, r=2, t=200 d Fline average, r=2, t=200 d Infline, r=1, t=2000 d Fline, r=1, t=2000 d Fline average, r=1, t=2000 d Fig. 2. Vertical ground temperature profile at radial distances r1mandr2mafter respectively 2000 days and 200 days, F o (r1, t2000)181.4 and F o (r2, C01 C01 t200)4.54.Constantheatinjectionof3000 W.Thermalparameters:k s 2.1 Wm K , C s 2e06 Jm C03 K C01 . where br/H, r is the radial distance from the borehole center, u H 2 at p and D A , and D B are given by: D A b 2 1 q erfc C18 u b 2 1 q C19 C0b erfcub C0 e C0u 2 b 2 1 C0e C0u 2 b 2 u p p ! and D B b 2 1 q erfc C18 u b 2 1 q C19 C00:5 C18 b erfcub b 2 4 q erfc C18 u b 2 4 q C19C19 C0 e C0u 2 C0 b 2 1 C1 C00:5 C16 e C0u 2 b 2 e C0u 2 C0 b 2 4 C1 C17 u p p ! 0 1000 2000 3000 4000 5000 0 2 4 6 8 10 12 Days T ( o C) Infinite Finite FEM Fig. 3. Comparison of Finite and Infinite line source model with finite element model (FEM) for a 30 m borehole. Average temperature variation computed at 0.5 m from the borehole axis, over the borehole length. Constant heat transfer rate of 1000 W. Thermal parameters: k s 2.1 Wm C01 K C01 , C s 2e06 Jm C03 K C01 . and 50 m radius cylinder. The borehole is represented by a 30 m 0.001 0.01 0.1 1 10 100 1000 10 20 30 40 50 60 70 80 r=0.075 m r=1 m Ground temperature Time (y) Temperature ( o C) Fig. 4. Comparison of Finite (solid) and Infinite (broken) line source model, computed at distance 1 m and 0.075 m from the borehole. Constant heat transfer rate per unit 1 2 3 4 5 6 100 0 100 Cooling (+) Heating () load Time (h) Load (kw) 1 2 3 4 5 6 200 100 0 100 Load decomposition Load (kw) Fig. 6. Principle of temporal superposition for variable loads. D. Marcotte et al. / Renewable Energy 35 (2010) 763770766 long and 0.075 m radius cylinder delivering 1000 W. The axis of revolution is located at the borehole center and constitutes The particular case rr b in Equation (4) gives the borehole wall temperature. 2.3. Numerical validation Fig. 3 compares the variation in temperature over time computed with finite and infinite line source to the numerical results of a finite element model (FEM) constructed within COMSOL C211 . The finite element model is 2-D with axial symmetry around the borehole axis. The ground is represented bya 50 m long length of 100 W/m. Thermal parameters: k s 2.1 Wm C01 K C01 , C s 2e06 Jm C03 K C01 . a thermal insulation boundary whereas all external boundaries are set to the undisturbed ground temperature. Over 6000 triangular 0 100 200 300 400 500 600 700 800 900 1000 12 12.5 13 13.5 Borehole length (m) Temperature ( o C) Average temperature vs borehole length Infinite linesource Finite linesource Fig. 5. Infinite vs finite line source average temperature along a vertical profile. The load is 20 W/m, thermal parameters: k s 2.1 Wm C01 K C01 , C s 2e06 Jm C03 K C01 . Temperature computed after one year at r1 m from the borehole. elements equipped with quadratic interpolating functions are used to discretize the model. The agreement between the FEM model and the finite line source is almost perfect, the maximum absolute difference in temperature over the 5000 days period being only 0.019 o C. Fig. 4 compares the temperature obtained with the infinite and finite line source models, at r1m and r0.075 m (a typical value for r b ), with the thermal parameters specified above. A 1 o C temperature difference between the infinite and finite models is obtained after 2.5 y and 2 y,at1mand0.075m respectively. Note that the temperature reaches a plateau for the finite line source model indicating that a steady-state condition has been reached. In contrast, the infinite line source model exhibits a linear behavior. Fig. 5 shows the ground temperature, computed at a distance of 1 m from the borehole, for increasing values of the borehole length. As expected, the finite line source solution reaches the infinite line source solution for long boreholes. 0 5 10 15 20 25 30 35 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 COP vs EWT EWT COP Cooling Heating Fig. 7. COP as a function of EWT. 0 10 20 30 40 1 2 3 4 5 6 7 8 9 11 12 13 14 17 18 19 20 21 24 25 27 28 29 30 33 34 35 36 37 42 43 44 45 47 48 49 50 53 54 55 56 57 60 61 66 67 68 69 75 76 77 78 79 80 81 82 85 86 87 88 89 94 95 96 97 100 101 106 107 108 109 111 112 113 114 117 118 119 120 121 126 127 128 129 134 135 136 137 142 143 144 145 147 148 149 150 155 156 157 158 159 160 161 166 167 168 169 174 175 176 177 182 183 184 185 190 191 192 193 198 199 200 201 204 205 210 211 212 213 218 219 220 221 224 225 Borehole location and priority number D. Marcotte et al. / Renewable Energy 35 (2010) 763770 767 3. Design of complete geothermal systems In this section we compare the design length of borefields obtained with the finite and infinite line source models for given hourly ground load scenarios. These calculations imply that single borehole solutions will need to be superimposed spatially. We have already seen an instance of this principle of superposition while computing the line source solution from a series of constant point sources along a line 7, see Equations (1 and 2). The additivity of 40 30 20 10 40 30 20 10 16 22 32 38 40 52 58 62 64 71 73 84 90 92 98 102 104 116 122 124 130 132 138 140 152 154 162 164 170 172 178 180 186 188 194 196 202 206 208 214 216 222 Coord. x (m) Coord. y (m) Fig. 8. Borehole grid and priority number. Number indicates effects (variation in temperature) stems from the linear relation between q and DT, and the fact that energy is an extensive and additive variable. The temporal superposition also followsthe same general principle of addition of effects as described by Yavuzturk and Spitler 14 and illustrated by Fig. 6. When the load is varying hourly, a new pulse is applied each hour. It is simply the difference between the load for two consecutive hours. More formally, for the infinite line source as an example, with a single borehole, we have: DTr;t X i; t i C20t q C3 i 4pk Z N r 2 =4atC0t i e C0u u du (5) where: q* 1 q 1 , and q* i q i C0q iC01 , i2.I, t I C20t, is the incremental load between two successive hours. With multiple boreholes, Table 1 Number of boreholes required, complete geothermal system. Constant T assumes a constant ground surface temperature of 10 o C, Periodic T assumes a periodic ground surface temperature with an amplitude of C620 o C in phase with the heat load. Scenario Borehole length Infinite line Finite line Constant T Periodic T Balanced (AC017) 100 m 33 33 34 Balanced 50 m 76 74 80 Cooling dominant (A17) 100 m 39 36 37 Cooling dominant 50 m 93 79 81 Heating dominant (AC030) 100 m 57 53 56 Heating dominant 50 m 134 115 124 DTx 0 ;t X n j1 X i; t i C20t q 0 i 4pk Z N kx j C0 x 0 k 2 =4atC0t i e C0u u du (6) where: n is the number of boreholes, x j and x 0 are the coordinate vectors of borehole j and point where temperature is computed, respectively. Note that for long simulation periods, the computa- tional burden becomes important. 0 10 20 30 40 10 15 23 26 31 39 41 46 51 59 63 65 70 72 74 83 91 93 99 103 105 110 115 123 125 131 133 139 141 146 151 153 163 165 171 173 179 181 187 189 195 197 203 207 209 215 217 223 order of inclusion in the design when required. In the test cases that follow we assume that all of the building heating and cooling loads are to be provided by the GLHE system, i.e. there is no supplementary heat rejection/injection. Synthetic building loads are used to enhance the reproducibility of our results. These building loads are simulated using: QtAC0B cos C18 t 8760 2p C19 C0C cos C18 t 24 2p C19 C0D cos C18 t 24 2p C19 cos C18 2t 8760 2p C19 (7) In Equation (7), t is in hours, A controls the annual load unbalance, B the half-amplitude of annual load variation, C and D Table 2 Number of boreholes required, hybrid system. HP capacity represents 40% of maximum building load. The last two column represent the percentage of the building load supplied by the HP for each mode. Scenario Borehole length Number of boreholes % Energy Infinite Finite Cooling Inf. (Fin.) Heating Inf. (Fin.) Balanced 100 m 19 19 69 (69) 77 (78) Balanced 50 m 37 37 67 (67) 72 (73) Cooling dominant 100 m 24 24 69 (69) 86 (86) Cooling dominant 50 m 41 39 70 (69) 90 (88) Heating dominant 100 m 37 37 67 (67) 83 (86) Heating dominant 50 m 55 53 72 (70) 70 (73) 5 10 15 20 25 30 6 Time (year) Rock temperature at x=(13,5,12.0) Infinite linesource Finite linesource 2.0). Energy 35 (2010) 763770 the half-amplitude of daily load fluctuations. D/C controls the relative importance of the damped component used to simulate larger daily fluctuations in winter and summer. Coefficients A to D are in kW. We consider three different load scenarios, each with B100, C50, and D25. One is approximately balanced (AC017),