Short-Term Behavior and Steady-State Value of BHE Thermal Resistance

Antonella Priarone1, Stefano Lazzari*2

1. University of Genova, DIME-TEC, Via all’Opera Pia 15/a, Genova, Italy, I-16145.2. University of Bologna, DIN, Viale Risorgimento 2, Bologna, Italy, I-40136.

Topic: A transient parametrical study of the grout thermal resistance of the BHE, i.e., the resistance between the tubes and the ground, is performed through COMSOL Multiphysics®.

Numerical Model: Dimensionless transient heattransfer by conduction in BHE and surroundingground.Governing Eqs. Dimensionless quantities

B.C. and I.C.

Parameter values:k* = 0.4, 0.7, 1.0; (ρc)* = 0.4, 0.7, 1.0

The average dimensionless temperature at theinterface tubes-grout (g-function ggt) and that atthe interface grout-ground (g-function gb) areevaluated. Their difference gRgt = ggt − gb isrepresentative of the dimensionless groutthermal resistance per unit length.

Results: For a given value of k*, in the short-term period as the value of (ρc)* increases thevalue of ggt decreases. On the contrary, as FoDincreases, the curves approach and tend to acommon value, which is a decreasing functionof k* (Fig.3, left).

For given values of k*, there exists athreshold value of FoD such that the curvesbecome independent of both (ρc)* and FoD:the corresponding value of gRgt represents thedimensionless steady-state grout thermalresistance per unit length (Fig.3, right).The evaluation of all the contributions to thesteady-state BHE thermal resistance showsthat case I could perform better than case II.

As the distance between the tubes increases,the dimensionless grout thermal resistancefor unit length of a single U-tube BHEdecreases (Fig.4).A comparison with Paul’s correlation

shows a sufficient agreement.Conclusions: the pipe spacing and the heatcapacity ratio play an important role in thetransient behavior of the BHE internalresistance, whereas the pipe spacing and theconductivity ratio play an important role inthe steady value of the BHE thermalresistance.

Figure 3. Plot of gRgt and gRgt as a function of FoD.

Figure 1. Double and single U-tube BHEs (lengths are in mm).

Excerpt from the Proceedings of the 2014 COMSOL Conference in Cambridge

I II III IV V

( )* *

*2 ** *

∂ = ∇∂T k T

cτ ρ

**2 *

*

∂ = ∇∂T Tτ

( ) *

* ** *

14

− ∇ ⋅ =nS

t

Tk Dπ

( )* *

20

,τ

τρ

−= = =g g

g Dg

T T kT k Fo

Q c D

( )( )( )

** ,= =gt gt

g g

ckk c

k c

ρρ

ρ

* 0=T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.00E-04 1.00E-02 1.00E+00 1.00E+02

ggt

FoD

0.4, 0.40.4, 0.70.4, 10.7, 0.40.7, 0.70.7, 11, 0.41, 0.71, 1

k*, (ρc)*

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

1.00E-04 1.00E-02 1.00E+00 1.00E+02

gRgt

FoD

0.4, 0.40.4, 0.70.4, 10.7, 0.40.7, 0.70.7, 11, 0.41, 0.71, 1

k*, (ρc)*case I

case I

00.10.20.30.40.50.60.70.80.9

11.1

1.00E-04 1.00E-02 1.00E+00 1.00E+02

ggt

FoD

0.7, 0.4 (case III)0.7, 0.7 (case III)0.7, 1 (case III)0.7, 0.4 (case IV)0.7, 0.7 (case IV)0.7, 1 (case IV)0.7, 0.4 (case V)0.7, 0.7 (case V)

k*, (ρc)*

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1.00E-04 1.00E-02 1.00E+00 1.00E+02

gRgt

FoD

0.7, 0.4 (case III)0.7, 0.7 (case III)0.7, 1 (case III)0.7, 0.4 (case IV)0.7, 0.7 (case IV)0.7, 1 (case IV)0.7, 0.4 (case V)0.7, 0.7 (case V)0.7, 1 (case V)

k*, (ρc)*

Figure 4. Plot of gRgt and gRgt as a function of FoD.

single U-tube BHE

single U-tube BHE

1*

*0

1

1Rgt gt g

t

g R kk D

′= ⋅ = ⋅

β

β