Ground Heat Transfer Calculations using Foundation:Kiva[LINK]

Kiva^TM is an open source foundation heat transfer calculation tool developed by Big Ladder Software.

http://bigladdersoftware.com/projects/kiva/

Kiva is the product of Neal Kruis’s dissertation where he demonstrated that accurate foundation heat transfer calculations can be performed quickly (on the order of 5 seconds) without any noticeable loss of accuracy relative to a mesh-independent, fully three-dimensional simulation.

Approach[LINK]

Within EnergyPlus, Kiva is used to perform two-dimensional finite difference heat transfer calculations. Each foundation is represented by a single floor and wall in Kiva, meaning that individual walls in EnergyPlus are mapped to a single representative wall in the two-dimensional context using an area weighted average for any non-uniform boundary conditions among the walls.

Kiva uses the boundary conditions from EnergyPlus:

• weather data,

• solar position, and

• zone temperatures (from previous timestep),

• zone radiation (solar, IR, etc.)

to calculate the resulting convective heat gains and surface temperatures for the floor and wall surfaces associated with a single Foundation:Kiva object. Because Kiva performs multi-dimensional finite difference calculations, the associated surfaces do not use the same HeatBalanceAlgorithm (e.g., Conduction Transfer Functions) as the rest of the model.

Two-dimensional Approximation[LINK]

The two-dimensional approximation method employed by Kiva relies on knowing the footprint shape, area, and exposed perimeter of each instance. The appropriate footprint shape, area, and exposed perimeter for each instance will be defined within the context of the overall geometry of the Foundation surfaces.

The general method is to define the width of the floor (the distance from the symmetry plane to the wall interior),$w$ in the two-dimensional context as:

$$A/P_{exp}$$

where $A$ is the area of the foundation footprint, and $P_{exp}$ is the exposed perimeter of the foundation (See SurfaceProperty:ExposedFoundationPerimeter).

Kiva also has the capability to adjust this width to account for concave foundation footprint shapes (Note: this also relies on detailed input of the exposed foundation perimeter for each segment of the footprint polygon). This adjustment is based on the boundary layer adjustment method described by Kruis and Krarti (2017). The approach adjusts the exposed perimeter to account for interactions in heat flow within concave corners and narrow gaps between two exposed edges.

This approach allows for accurate representation of building foundation heat transfer without performing three-dimensional calculations. Because the two-dimensional context is symmetric, the domain can be divided in half to further reduce the number of calculations.

Numerical Calculations[LINK]

Kiva automatically discretizes the two-dimensional domain into rectangular cells. The size of each cell is defined by the Foundation:Kiva:Settings object’s “Minimum Cell Dimension” and “Maximum Cell Growth Coefficient”. The “Minimum Cell Dimension” defines the smallest possible dimension of a cell within the domain. Cells along a block boundary start at this size and grow geometrically away from the boundary according to the “Maximum Cell Growth Coefficient”. This is evident from Figures 1 and 2 which show the discretization for a single foundation at close and far perspectives, respectively.

The discretized partial differential equations are solved using the Alternating Direction Implicit (ADI) finite difference time stepping scheme. This scheme provides relatively fast calculations with stable results as demonstrated by Kruis and Krarti (2015).

Boundary Conditions[LINK]

Symmetry Plane: Zero heat flux in the horizontal direction.

Wall Top: Zero heat flux in the vertical direction (assumes heat transfer through the exterior wall above is one-dimensional in the horizontal direction).

Deep Ground: Either constant temperature or zero vertical heat flux, depending on user input. Deep ground depth may be automatically calculated based on water table estimates using a method defined by Williams and Williamson (1989).

Far-Field: Zero heat flux in the horizontal direction. If this boundary is sufficiently far from the building, this will result in an undisturbed ground temperature profile.

Interior Surfaces:

• Convection is calculated according to the TARP method (see SurfaceConvectionAlgorithm:Inside).

• Long and short wave radiation is passed from EnergyPlus radiant exchange and interior solar distribution algorithms. Note: Kiva uses area weighted averages to define the radiation incident on walls in the two-dimensional context.

Exterior Surfaces:

• Convection is calculated according to the DOE-2 method (see SurfaceConvectionAlgorithm:Outside). Wind speeds along the exterior grade are calculated at the roughness height.

• Exterior long wave radiation is calculated using the same algorithms used for other EnergyPlus surfaces. Note: there is no explicit radiant exchange between the ground and building surfaces.

• Exterior solar incidence is uniform along the exterior grade surfaces. No shading is taken into account. Solar incidence along the wall exterior

Multiple Kiva Instances[LINK]

In some cases, a single Foundation boundary condition might require multiple Kiva instances:

Multiple Zones: If the surfaces in several zones reference the same Foundation:Kiva object, each zone will be calculated using separate Kiva instances.

Walk-Out Basements: Walkout basements are defined by using walls of different heights all referencing the same Foundation:Kiva object.

A separate Kiva instance will be run for any walls with different heights associated with the same Foundation:Kiva object. Figure 4 shows how the grouping of walls by height based on the basement in Figure 5, including the portion that is only a slab.

The resulting five two-dimensional contexts will look like Figures 6 - 10.

Each Kiva instance with a different wall height will calculate different heat fluxes, convective coefficients and surface temperatures for both the wall and the floor. The heat flux through the associated floor will be weighted according to the fraction of the total exposed perimeter, $P_{exp,tot}$, represented by each segment of different height. The total heat flux through the walkout basement floor is:

$$\dot{q}=\sum _i^{N_{segs}}\frac{P_{exp,i}}{P_{exp,tot}}\cdot h_i(T_{\infty }-T_{floor,i})$$

The weighted average convective coefficient for the walkout basement floor surface is:

$$\overline{h}=\sum _i^{N_{wall,segs}}\frac{P_{exp,i}}{P_{exp,tot}}\cdot h_i$$

The weighted average temperature for the floor surface is:

$${\overline{T}}_{floor}=T_{\infty }-\dot{q}/\overline{h}$$

Multiple Floor Surfaces: If a floor has multiple constructions (e.g., carpeted and bare) each surface must reference a separate Foundation:Kiva object, or be combined into a single equivalent construction.

Core Zone Slabs[LINK]

Because core zones have no exposed perimeter, they are assumed to exchange heat only with the deep ground boundary condition. This is calculated using a one-dimensional finite difference formulation. The associated Kiva instance will use only the description of the slab and the deep ground boundary condition to define the heat flux through the surface.

Warm-Up[LINK]

The traditional “warm-up” period in EnergyPlus (of repeating a single day) presents several challenges for foundation heat transfer calculations:

• As the ground can have time constants on the order of years, a single day is simply not long enough to adequately capture the thermal history of the ground.

• Any repetition of a single day would erase any pre-calculated thermal history and likely take much longer to converge.

Instead, Kiva instances are initialized independently from the rest of the simulation using the accelerated initialization method developed by Kruis (2015). This method looks back in the weather file and simulates long timestep (on the order of weeks or months) calculations using an implicit numerical scheme. These long timesteps allow Kiva to capture a long term history of the ground without running the entire building model.

The initialization of the ground relies on assumptions of indoor air temperatures (as they are not yet calculated by EnergyPlus). When a thermostat is assigned to a zone with Kiva foundation surfaces, the assumed temperature is equal to the setpoint (or a weighted average of heating and cooling setpoints depending on outdoor temperature). For zones without thermostats, a constant 22 ^oC indoor temperature is assumed.

Validation[LINK]

Kiva has been tested against the BESTEST Ground coupled cases with accuracy within 3% of the reference solutions (Kruis and Krarti, 2015).

References[LINK]

[1] N. Kruis and M. Krarti, “Kiva^TM: A Numerical Framework for Improving Foundation Heat Transfer Calculations,” Journal of Building Performance Simulation, vol. 8, no. 6, pp. 449-468, 2015.

[2] N. Kruis, “Development and Application of a Numerical Framework for Improving Building Foundation Heat Transfer Calculations,” Ph.D. Dissertation. University of Colorado, 2015.

[3] N. Kruis and M. Krarti, “Three-dimensional accuracy with two-dimensional computation speed: using the Kiva^TM numerical framework to improve foundation heat transfer calculations,” Journal of Building Performance Simulation, vol. 10, no. 2, pp. 161?182, 2017.

[4] T. Williams and A. Williamson, “Estimating Water-Table Altitudes for Regional Ground-Water Flow Modeling, U.S. Gulf Coast,” Ground Water, vol. 27, no. 3, pp. 333-340, 1989.