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Abstract
A model for the simulation of geothermal systems with parallel- and series-connected boreholes is presented. Mass and heat balance problems are formulated for each component in the system and are assembled into system-level problems. A third problem is formulated to account for heat transfer in the bore field, using the finite line source solution. This third problem is coupled to the system-level heat balance problem by an analytical solution of the heat transfer inside boreholes with multiple U-tubes. The simulation model allows for any number of independent fluid loops within the bore field or within individual boreholes and allows for combinations of specified inlet fluid temperatures and heat extraction rates in independent fluid loops. The model accounts for the axial variation of the fluid and borehole wall temperatures and heat extraction rates. The capabilities of the model are demonstrated through three example simulations.
Nomenclature
Variables
| αs | = | Soil thermal diffusivity |
| Δt | = | Simulation time step |
| Δtp | = | Size of aggregation cell p |
| ΔTb | = | Borehole wall temperature drop |
| cp | = | Specific heat capacity |
| COP | = | Heat pump coefficient of performance |
| d | = | Distance between boreholes |
| D | = | Borehole or borehole segment buried depth |
| = | Absolute error tolerance on borehole wall temperatures | |
| = | Absolute error tolerance on fluid temperatures | |
| = | Relative error tolerance on fluid mass flow rates | |
| G | = | Captured solar radiation |
| h | = | Segment-to-segment thermal response factor |
| H | = | Borehole or borehole segment length |
| ks | = | Ground thermal conductivity |
| = | Inlet fluid mass flow rate | |
| = | Outlet fluid mass flow rate | |
| η | = | Solar collector efficiency |
| Nagg | = | Total number of load aggregation cells |
| Nb | = | Total number of boreholes in bore field |
| Nc | = | Total number of system components |
| nin | = | Number of inlets in component |
| Nin | = | Total number of inlets in system |
| nout | = | Number of outlets in component |
| Nout | = | Total number of outlets in system |
| np | = | Number of U-tubes in borehole |
| Np | = | Total number of U-tubes in bore field |
| nq | = | Number of borehole segments in borehole |
| Nq | = | Total number of borehole segments in bore field |
| = | Component heat transfer rate | |
| = | Borehole heat extraction rate | |
| = | Building load | |
| = | Pipe heat transfer rate | |
| = | Borehole radius | |
| = | Delta-circuit thermal resistance | |
| = | Effective borehole thermal resistance | |
| = | Fluid to outer pipe wall thermal resistance | |
| = | Pipe inner radius | |
| = | Pipe outer radius | |
| t | = | Time |
| Ta | = | Ambient temperature |
| Tb | = | Borehole wall temperature |
| Tf | = | Fluid temperature in borehole pipes |
| Tf,in | = | Inlet fluid temperature |
| Tf,out | = | Outlet fluid temperature |
| Tg | = | Undisturbed ground temperature |
| (x,y) | = | Coordinates of boreholes or pipes |
| z | = | Depth |
Matrices and vectors
| A | = | Coefficient matrix of the system of differential equations for fluid temperatures in boreholes |
| = | Coefficient matrix for the component-level mass balance problem | |
| = | Coefficient matrix for the system-level mass balance problem | |
| = | Coefficient matrix for the borehole heat transfer problem | |
| = | Coefficient matrix for the bore field heat transfer problem | |
| = | Coefficient matrix for the component-level heat balance problem | |
| = | Coefficient matrix for the system-level heat balance problem | |
| = | Coefficient vector for the component-level mass balance problem | |
| = | Coefficient vector for the system-level mass balance problem | |
| = | Coefficient vector for the borehole heat transfer problem | |
| = | Coefficient vector for the bore field heat transfer problem | |
| = | Coefficient vector for the component-level heat balance problem | |
| = | Coefficient vector for the system-level heat balance problem | |
| C | = | Component connectivity matrix |
| E | = | Matrix exponential of Az |
| Ein, Eout, Eb | = | Coefficient matrices for boundary condition at z = H |
| H | = | Matrix of segment-to-segment thermal response factors |
| I | = | Identity matrix |
| L | = | Diagonal matrix of eigenvalues of A |
| = | Vector of inlet fluid mass flow rates | |
| = | Vector of outlet fluid mass flow rates | |
| = | Vector of heat extraction rate per unit length of borehole segments | |
| Tb | = | Vector of average borehole segment temperatures |
| Tb,0 | = | Vector of average borehole segment temperatures assuming no heat extraction during current time step |
| Tf,in | = | Vector of inlet fluid temperatures |
| Tf,out | = | Vector of outlet fluid temperatures |
| Tf | = | Vector of fluid temperatures in borehole pipes |
| Tg | = | Vector of undisturbed ground temperatures |
| V | = | Matrix of eigenvectors of A |
| y | = | Vector of mass flow rate fractions |
Indices
| ib, jb | = | Borehole indices |
| ic, jc | = | Component indices |
| ip, jp | = | Pipe indices |
| k | = | Time index |
| specified | = | Specified value of the variable |
| tot | = | Total value in the bore field |
| u,v | = | Borehole segment indices |