# RoomAir Models[LINK]

The group of models described in this section is used to account for non-uniform room air temperatures that may occur within the interior air volume of a zone. These models are accessed using the RoomAirModelType input object. RoomAir modeling was added to EnergyPlus starting with Version 1.2. Although there are many types of analyses (comfort, indoor air quality, etc) that might benefit from localized modeling of how room air varies across space, most of the room air models in EnergyPlus only address the distribution of temperature within the zone. This allows surface heat transfer and air system heat balance calculations to be made taking into account natural thermal stratification of air and different types of intentional air distribution designs such as under-floor and side-wall displacement ventilation that purport to extract room air at higher-than-mean temperatures. The exception is the RoomAirflowNetwork model, which integrates the AirflowNetwork model and applies the nodal airflow model within zones. Note that EnergyPlus does not have completely general methods of modeling room air that are applicable to every conceivable type of airflow that might occur in a zone. Such models (e.g. RANS-CFD) are too computationally expensive to use with EnergyPlus for the foreseeable future. The models that are available in EnergyPlus offer only limited modeling capabilities for select room airflow configurations. Also note that because the complete mixing model for room air has long been the standard in building energy simulation, there is not currently a consensus on how to best model non-uniform air temperatures in buildings. Therefore, it is up to the user to have a good understanding of when, where, and how to apply the room air models available in EnergyPlus. The rest of this section provides some guidance in the way of examples and further discussion of the models available in EnergyPlus.

EnergyPlus offers the different types of air models listed in the table below along with the input objects associated with the use of that model.

Summary of room air models available in EnergyPlus
Air model name Applicability Input Objects Required
Well-Mixed All zones None, default
User Defined Any zone where the user has prior knowledge of the temperature pattern ‘RoomAirModelType’, ‘RoomAir:TemperaturePattern:UserDefined’, ‘RoomAir:TemperaturePattern: xx’
One-Node Displacement Ventilation (Mundt) displacement ventilation in typical office-type zones ‘RoomAirModelType’, ‘RoomAirSettings:OneNodeDisplacementVentilation’, ‘RoomAir:Node’’
Three-Node Displacement Ventilation (UCSD) displacement ventilation ‘RoomAirModelType’, ‘RoomAirSettings:ThreeNodeDisplacementVentilation’
Under-Floor Air Distribution Interior Model (UCSD) Interior zones served by a UFAD system ‘RoomAirModelType’, ‘RoomAirSettings:UnderFloorAirDistributionInterior’
Under-Floor Air Distribution Exterior Model (UCSD) Exterior zones served by a UFAD system ‘RoomAirModelType’, ‘RoomAirSettings:UnderFloorAirDistributionExterior’
UCSD Cross Ventilation cross ventilation ‘RoomAirModelType’, ‘RoomAirSettings:CrossVentilation’
RoomAirflowNetwork Room air model using AirflowNetwork ‘RoomAirModelType’, ‘RoomAirSettings:AirflowNetwork’ ‘RoomAirflowNetwork:Node’, ‘RoomAirflowNetwork:Node:AdjacentSurfaceList’, ‘RoomAirflowNetwork:Node:InternalGains’, ‘RoomAirflowNetwork:Node:InternalGains’, ‘AirflowNetwork:IntraZone:Node’, ‘AirflowNetwork:IntraZone:Linkage’

The room air models are coupled to the heat balance routines using the framework described by Griffith and Chen (2004). Their framework was modified to include features needed for a comprehensive program for annual energy modeling rather than one for hourly load calculations. The formulation is largely shifted from being based on the setpoint temperature to one based on the current mean air temperature. This is necessary to allow for floating temperatures and dual setpoint control where there may be times that the mean zone temperatures are inside the dead band. The coupling framework was also extended to allow for exhaust air flows (e.g. bathroom exhaust fans) in addition to air system return flows.

The inside face temperature calculation is modified by rewriting the zone air temperature, Ta, with an additional subscript, i, for the surface index (TajTai,j or TaTai ). The inside face heat balance is solved for its surface temperature using,

Tsi,j=Tsoi,jYi,o+nzk=1Tsoi,jkYi,knzk=1Tsi,jkZi,k+nqk=1Φi,kq′′kii,jk+Tai,jhci,j+q′′LWS+q′′LWX+q′′SW+q′′solZi,o+hci,j

where:

Ts is the inside face temperature

i is the subscript indicates individual surfaces

j is the subscript indicates current time step

k is the subscript indicates time history steps

Tso is the outside face temperature

Yi are the cross CTF coefficients

Zi are the inside CTF coefficients

φi are the flux CTF coefficients

q′′ki is the conduction heat flux through the surface

hci is the surface convection heat transfer coefficient

Ta is the near-surface air temperature

q′′LWS is the longwave radiation heat flux from equipment in zone

q′′LWX is the net long wavelength radiation flux exchange between zone surfaces

q′′SW is the net short wavelength radiation flux to surface from lights

q′′sol is the absorbed direct and diffuse solar (short wavelength) radiation.

## User Defined RoomAir Temperatures[LINK]

The input object RoomAir:TemperaturePattern:UserDefined provides a capabity for users to define the sort of air temperature pattern he or she expects in the zone. With these models, the pattern is generally set beforehand and does not respond to conditions that evolve during the simulation. (Exception: the pattern available through the RoomAir:TemperaturePattern:TwoGradient object will switch between two different pre-defined vertical gradients depending on the current value of certain temperatures or thermal loads. )

The user-defined patterns obtain the mean air temperature, TMAT, from the heat balance domain and then produce modified values for:

• Tai is the adjacent air temperature which is then used in the calculation of inside face surface temperature during the heat balance calculations.

• Tleaving is the temperature of air leaving the zone and entering the air system returns.

• Texhaust is the temperature of air leaving the zone and entering the exhaust.

• Tstat is the temperature of air “sensed” at the thermostat (not currently used in air system control because air system flows use load-based control).

The user defined room air models used indirect coupling so that the patterns provide values for, or ways to calculate, how specific temperatures differ from TMAT. The various ΔT values determined from the model are applied to TMAT as follows:

Tai=ΔTai+TMAT

(where “*i’s“* represents each surface in the zone that is affected by the model)

Tleaving=ΔTreturn+TMAT

Texhaust=ΔTexhaust+TMAT

Tstat=ΔTstat+TMAT

The patterns defined by the object ‘RoomAir:TemperaturePattern:SurfaceMapping’ are fairly straightforward. The user directly inputs values for ΔTai for each surface. The pattern “maps” specific surfaces, identified by name, to ΔTai values. This provides completely general control (but in practice may be cumbersome to use). The other patterns focus on temperature changes in the vertical direction. Surfaces do not need to be identified, but all the surfaces with the same height will be assigned the same ΔTai values.

The patterns defined by the object ‘RoomAir:TemperaturePattern:NondimensonalHeight’ apply a temperature profile based on a non-dimensionalized height, Z. The height of each surface is defined to be the z-coordinate of the surface’s centroid relative to the average z-coordinate of the floor surfaces. The zone ceiling height is used as the length scale to non-dimensionalize each surface’s height so that,

Zi=(zi,centroidzfloorAvg)/CeilingHeight

(where “*i’s“* represents each surface in the zone that is affected by the model)

The values for Zi are constrained to be between 0.01 and 0.99 because the value is meant to describe the air layer near the surface (say approximate 0.1 m from the surface) rather than the surface itself.

The user-defined profile is treated as a look up table or piecewise linear model. The values for ΔTai are determined by searching the Z values in the user-defined profile and performing linear interpolation on the associated ΔTa values.

The patterns defined by the object ‘RoomAir:TemperaturePattern:ConstantGradient’ apply a constant temperature gradient in the vertical direction. The model assumes that TMAT occurs at the mid-plane so that ZTMAT=0.5 (by definition). The surface Zi values are compared to ZTMAT and then scaled with zone ceiling height to obtain values for the change in height (in units of meters), Δz. The user defined gradient, grad, (units of C/m) is then used to determine ΔTai values using:

ΔTai=Δzgrad

The patterns defined by the object ‘RoomAir:TemperaturePattern:TwoGradient’ are very similar to the constant gradient pattern above but the value of grad used at any given time is selected by interpolating between two user-defined values for grad. Five options are available, three based on temperatures and two based on thermal loads – see the Input Output Reference. The user provides upper and lower bounding values. If the current value of the “sensing” variable lies between the upper and lower bounds, then grad is determined using linear interpolation. If the designated value is above the upper bound then the upper value for grad is used (no extrapolation). Similarly, if the designated value is below the lower bound, then the lower value for grad is used. Note that “upper” and “lower” indicate the temperature and heat rate bounds and that the values for grad do not have to follow in the same way; the grad value for the lower bound could be higher than the grad value for the upper bound (providing a something of a reverse control scheme). Rather than directly using ΔT values from the user, the temperatures for the return air, exhaust and thermostat are determined based on user-entered heights (in units of meters from the floor) and applying the current value for grad.

## One-Node Displacement Ventilation RoomAir Model[LINK]

The input object RoomAirSettings:OneNodeDisplacementVentilation provides a simple model for displacement ventilation. Mundt (1996) points out that a floor air heat balance provides a simple and reasonably accurate method of modeling the temperature near the floor surface. The slope of a linear temperature gradient can then be obtained by adding a second upper air temperature value that comes from the usual overall air system cooling load heat balance. The figure below diagrams the temperature distribution versus height being calculated by the model. Mundt’s floor air heat balance is extended to include convection heat gain from equipment and by ventilation or infiltration that may be introduced near the floor in order to maintain all the terms in the air heat balance of the Heat Balance Model. This yields the following heat balance for a floor air node,

ρcp˙V(TAirFloorTSupply)=hcFloorAFloor(TFloorTAirFloor)+QConvSourceFloor+QInfilFloor

where:

ρ is the air density

cp is the air specific heat at constant pressure

V is the air system flow rate

Tsupply is the air system’s supply air drybulb temperature

hcFloor is the convection heat transfer coefficient for the floor

Afloor is the surface area of the floor

Tfloor is the surface temperature of the floor

QconvSourceFloor is the convection from internal sources near the floor (< 0.2 m)

QInfilFloor is the heat gain (or loss) from infiltration or ventilation near the floor.

“Floor splits” are the fraction of total convective or infiltration loads that are dispersed so as to add heat to the air located near the floor. The user prescribes values for floor splits as input. No guidance is known to be available to use in recommending floor splits, but the user could for example account for equipment known to be near the floor, such as tower computer cases, or supplementary ventilation designed to enter along the floor. The equation above can be solved directly for TAirFloor and is used in the form of the equation below,

TAirFloor=ρcp˙VTSupply+hcFloorAFloorTFloor+˙QConvSourceFloor+˙QInfilFloorρcp˙V+hcFloorAFloor

The upper air node temperature is obtained by solving the overall air heat balance for the entire thermal zone for the temperature of the air leaving the zone and going into the air system return, Tleaving.

TLeaving=˙Qsyspcp˙V+TSupply

where ˙Qsys is the air system heat load with negative values indicating a positive cooling load. Values for ˙Qsys are computed by the load calculation routines and passed to the air model. The vertical temperature gradient or slope, dT/dz, is obtained from,

dTdz=TLeavingTAirFloorHreturn

where Hreturn is the distance between the air system return and the floor air node assumed to be 0.1 m from the floor and z is the vertical distance.

The constant slope allows obtaining temperatures at any vertical location using,

Tai=TleavingdTdz(zleavingzi)

So for example the temperatures near the ceiling can easily be determined. Accounting for the location of the thermostat inside the zone (e.g. 1.1 m) is accomplished by returning the temperature for the appropriate height to the appropriate air node used for control. If the walls are subdivided in the vertical direction as shown in the figure above, then the air model can provide individual values for each surface based on the height and slope. However, no additional heat balances are necessarily made (in the air domain) at these points as all the surface convection is passed to the model in the totaled value for ˙Qsys.

## Three-Node Displacement Ventilation RoomAir Model[LINK]

### Overview[LINK]

The input object RoomAirSettings:ThreeNodeDisplacementVentilation provides a simple model for heat transfer and vertical temperature profile prediction in displacement ventilation. The fully-mixed room air approximation that is currently used in most whole building analysis tools is extended to a three node approach, with the purpose of obtaining a first order precision model for vertical temperature profiles in displacement ventilation systems. The use of three nodes allows for greatly improved prediction of thermal comfort and overall building energy performance in low energy cooling strategies that make use of unmixed stratified ventilation flows.

The UCSD Displacement Ventilation Model is one of the non-uniform zone models provided through the Room Air Manager in EnergyPlus. The intent is to provide a selection of useful non-uniform zone air models to enable the evaluation of air-conditioning techniques that use stratified or partially stratified room air. Such techniques include displacement ventilation (DV) and underfloor air distribution (UFAD) systems. The methodology can also include, in principle, natural displacement ventilation and also wind-driven cross-ventilation (CV).

### Displacement Ventilation[LINK]

A DV system is a complete contrast to a conventional forced air system. In a conventional system conditioned air is delivered at ceiling level and the intent is to create a fully mixed space with uniform conditions. In a DV system conditioned air is delivered at floor level and low velocity in order to minimize mixing and to establish a vertical temperature gradient. The incoming air “displaces” the air above it which, in turn, is exhausted through ceiling level vents. In DV a noticeable interface occurs between the occupied zone of the room and a mixed hot layer near the ceiling of the room(Dominique & Guitton, 1997). Maintaining the lower boundary of this warm layer above the occupied zone is one of the many unique challenges of displacement ventilation design. Often DV systems use 100% outside air. The vertical displacement air movement means that convective heat gains introduced near the ceiling will be removed without affecting the occupied region of the room. Also a fraction of the heat gains that occur in the occupied zones rise as plumes into the upper part of the space, thereby reducing the cooling load. Similarly the fresh air will be used more effectively than with a fully mixed system: the fresh air won’t be “wasted” in the upper, unoccupied region of the room. Finally, the vertical temperature gradient means that the average room temperature can be higher for a DV conditioned room than with a conventionally conditioned room: the occupants feel the lower temperature in the lower region of the room and are unaffected by the higher temperature near the ceiling. However, whenever the outside air temperature is above $19^{}$C this advantage is mostly lost: the internal loads must be removed from the space independently of the airflow pattern (during the warmer hours buildings tend to be almost closed to the outside, operating in closed loop). The inflow temperature advantage is then only useful for the minimum outside air that must always be provided (in most cases this remaining advantage is negligible).

DV systems have limitations. In order to avoid chilling the occupants the supply air temperature used for DV is considerably higher than that used in conventional forced-air systems. This can lead to problems in removing both sensible and latent loads. Exterior spaces may have conditions that are not conducive to establishing a vertical temperature gradient. DV systems seem to be best suited to interior spaces with only moderate loads.

### Non-uniform zone models[LINK]

Several types of models have been proposed as suitable for inclusion in building energy simulation (BES) programs. These models must be simple enough not to impose an undue computational burden on a BES program, yet provide enough predictive capability to produce useful comparisons between conventional and stratified zone operation strategies. ASHRAE RP-1222 (Chen & Griffith 2002) divides the candidate models into two categories: nodal and zonal. Nodal models describe the zone air as a network of nodes connected by flow paths; each node couples convectively to one or more surfaces. Zonal models are coarse–grained finite volume models. ASHRAE RP-1222 provides a short history (and examples) of each type of model. In terms of nodal models for displacement ventilation we mention the Mundt model (Mundt 1996), since it is implemented in EnergyPlus, and the Rees-Haves model (Rees & Haves 2001) since it is a well developed nodal-type model and is implemented in the RP-1222 toolkit. The Rees-Haves model, while successful in predicting the flow and temperature field for geometries similar to those used in its development, can suffer from lack of flexibility and clarity in the modeling approximations. When dealing with diverse geometries it is not clear that the flow coefficients used in the model are applicable or why they can be used since plumes, the fundamental driving mechanisms of the displacement flow, are not explicitly modeled. This is the main difference between the DV models implemented in theRP-1222 toolkit and the model that is described here.

The UCSD DV model is closer to a nodal model than to a zonal model. However, it is best to classify it in a separate category: plume equation based multi-layer models (Linden et al. 1990, Morton et al. 1956). These models assume that the dominant mechanism is plume-driven flow from discrete internal sources and that other effects (such as buoyancy driven flow at walls or windows) may be neglected. Alternatively, these heat sources also produce plumes that can be included in the model. The result is a zone divided vertically into two or more well separated regions – each region characterized by a single temperature or temperature profile. This characterization allows the physics of the heat gains and the ventilation flow to be represented in a realistic manner, without the introduction of ad hoc assumptions.

### Single Plume Two Layer Model[LINK]

The simplest form of the plume equation based models is the case of a single plume in an adiabatic box with constant supply air flow. For this configuration two layers form in the room: a lower layer with similar density and temperature as the inflow air and a mixed upper layer with the same density / temperature as the outflow air. The main assumption of this model, successfully validated against scaled model experiments (Linden et al. 1990), is that the interface between the two layers occurs at the height (h) where the vertical buoyancy driven plume flow rate is the same as the inflow rate. For a point source of buoyancy in a non-stratified environment (a plume) the airflow rate increases with vertical distance from the source according to:

˙V=CB1/3z5/3

where:

˙V = plume volume flux [m3/s]

B = buoyancy flux [m4/s3]

z = vertical distance above source [m]

C=65α(910α)1/3π2/3

α = plume entrainment constant; a value of 0.127 is used, suitable for top-hat profiles for density and velocity across the plumes.

For an ideal gas:

Δρ/ρ=ΔT/T

resulting in the following relation between heat input rate and buoyancy flux:

B=g˙QρCpT

where

ρ = density of air [kg/m3]

T = air temperature [K]

g = acceleration of gravity [m/s2]

˙Q = heat input rate [W]

Cp = specific heat capacity of air [J/kgK]

Since the plume volume flow rate increases with height with exponent 5/3, for any room inflow rate (F, (m3/s)) there will always be a height (h,(m)) where the plume driven flow rate matches the inflow rate. This height is obtained by setting (1.1) equal to F and solving for z = h:

h=F3/5/(C3/5B1/5)

Substituting in and introducing air properties at 20C gives:

h=24.55F3/5/˙Q1/5

### Multiple Plumes and Wall Heat Transfer[LINK]

Of course, it would be rare for a real world case to consist of a single point-source plume originating on the floor, unaffected by heat gains from walls and windows. For multiple plumes of equal strength a straight-forward extension of the single is possible. N plumes of unequal strength result in the formation of n vertical layers. This case is much more complex but if we are satisfied with a first order precision model the equal strength model can be used by averaging the plume strengths (Carrilho da Graça, 2003). Even in a case where all plumes are of equal strength, nearby plumes may coalesce. Plumes that are less than 0.5 meters apart at their source will coalesce within 2 meters (Kaye & Linden,2004).

As the complexity of the physical systems modeled increases some limitations must be imposed. In particular, the biggest challenge remains the interaction between wall driven boundary layers (positively and negatively buoyant) and displacement flows. For this reason, the model that is developed below is not applicable when:

1. Downward moving buoyancy driven airflow rate is of the same order of magnitude as plume driven flow (these airflow currents are typically generated on lateral surfaces or in the ceiling whenever these surfaces are much cooler than the room air).

2. Upward moving wall or floor generated buoyancy flux in the lower layer is of the same order of magnitude as plume driven flow.

Although these limitations are significant it is important to note that even in the presence of dominant convection from the floor surface, a buoyancy, two layer flow can be established whenever the plume buoyancy flux is more than 1/7 of the horizontal flux (Hunt et al. 2002). A two layer structure can also originate when the only heat source is a heated portion of the room floor, as long as the heated area does not exceed 15% of the room floor (Holford et al. 2002).

For the case of multiple non-coalescing plumes (n), with equal strength, the total vertical airflow for a given height is:

˙V=CnB1/3z5/3

resulting in a mixed layer height of:

h=24.55F3/5n3˙Q1/5

### Implementation[LINK]

The model predicts three temperatures that characterize the three main levels in the stratification of the room:

1. A floor level temperature Tfloor to account for the heat transfer from the floor into the supply air.

2. An occupied subzone temperature Toc representing the temperature of the occupied region.

3. An upper level temperature Tmx representing the temperature of the upper, mixed region and the outflow temperature.

We assume that the model for multiple, equal strength plumes (equations and will be adequate for our calculations. The supply air flow rate ˙V is obtained by summing all the air flows entering the zone: supply air, infiltration, ventilation, and inter-zone flow. The heat gain ˙Q is estimated by summing all the convective internal gains located in the occupied subzone – task lights, people, equipment – and dividing this power equally among the n plumes. With these assumptions we can describe the implementation.

The UCSD DV model is controlled by the subroutine ManageUCSDDVModel which is called from the RoomAirModelManager. The RoomAirModelManager selects which zone model will be used for each zone.

The calculation is done in subroutine CalcUCSDDV. First we calculate the convective heat gain going into the upper and lower regions.

˙Qocz=˙Qoc,conv+˙Qtl,conv+˙Qeleq,conv+˙Qgaseq,conv+˙Qotheq,conv+˙Qhw,conv+˙Qstmeq,conv+˙Qbb,conv

˙Qmxz=˙Qgl,conv+˙Qltp+˙Qhtrad,conv

˙Qtot,conv=˙Qocz+˙Qmxz

Next we sum up the inlet air flows in the form of MCP (mass flow rate times the air specific heat capacity) and MCPT (mass flow rate times Cp times air temperature).

MCPzone=MCPi+MCPvent+MCPmix

MCPsys=inlets˙miCp,i

MCPtot=MCPzone+MCPsys

MCPTzone=MCPTi+MCPTvent+MCPTmix

MCPTsys=inlets˙miCp,iTi

MCPTtot=MCPTzone+MCPTsys

The number of plumes per occupant Nplumesperpers is a user input. The total number of plumes in the zone is:

Nplumes=NoccNplumesperperson

The gains fraction Frgains is a user input via a schedule. It is the fraction of the convective gains in the occupied subzone that remain in that subzone. Using this we calculate the total power in the plumes and the power per plume.

˙Qplumes=(1Frgains)˙Qtot,conv

˙Qperplume=˙Qplumes/Nplumes

We now make an initial estimate of the height fraction Frhb (height of the boundary layer divided by the total zone height).

Frhb=(24.55/24.55HceilHceil)0.000833MCPtotNplumes˙Q1/3perplume3/5

where 0.000833 = 1/(ρaircp,air) converts MCPtot to a volumetric flow rate. Next we iterate over the following 3 steps.

### Iterative procedure[LINK]

1. Call subroutine HcUCSDDV to calculate a convective heat transfer coefficient for each surface in the zone, an effective air temperature for each surface, and HAmx, HATmx, HAoc, HAToc, HAfl, and HATfl. Here HA is surfaceshc,iAi for a region and HAT is surfaceshc,iAiTi for a region. The sum is over all the surfaces bounding the region; hc,i is the convective heat transfer coefficient for surface i, Ai is the area of surface i, and Ti is the surface temperature of surface i.

2. Recalculate Frhb using the Equation [eq:RoomAirHeightBoundaryLayer].

3. Calculate the three subzone temperatures: Tfl, Toc and Tmx.

The hc’s calculated in step 1 depend on the subzone temperatures and the boundary layer height. In turn the subzone temperatures depend on the HA and HAT’s calculated in step 1. Hence the need for iteration

Next we describe Steps 1 and 3 in more detail.

### Step 1[LINK]

Subroutine HcUCSDDV is quite straightforward. It loops through all the surfaces in each zone and decides whether the surface is located in the upper, mixed subzone or the lower, occupied subzone, or if the surface is in both subzones. If entirely in one subzone the subzone temperature is stored in the surface effective temperature variable TempEffBulkAir(SurfNum) and hc for the surface is calculated by a call to subroutine CalcDetailedHcInForDVModel. This routine uses the “detailed” natural convection coefficient calculation that depends on surface tilt and ΔT1/3. This calculation is appropriate for situations with low air velocity.

For surfaces that bound 2 subzones, the subroutine calculates hc for each subzone and then averages them, weighting by the amount of surface in each subzone.

During the surface loop, once the hc for a surface is calculated, the appropriate subzone HA and HAT sums are incremented. If a surface is in 2 subzones the HA and HAT for each subzone are incremented based on the area of the surface in each subzone.

### Step 3[LINK]

The calculation of subzone temperatures follows the method used in the ZoneTempPredictorCorrector module and described in the section Basis for the System and Zone Integration. Namely a third order finite difference expansion of the temperature time derivative is used in updating the subzone temperatures. Otherwise the subzone temperatures are obtained straightforwardly by solving an energy balance equation for each subzone.

Tfl=(Cair,fl(3T1,fl(3/2)T2,fl+(1/3)T3,fl)+HATfl+MCPTtot+0.6TocMCPtot)/((11/6)Cair,fl+HAfl+1.6MCPtot)

Toc=(Cair,oc(3T1,oc(3/2)T2,oc+(1/3)T3,oc)+˙QoczFrgains+HAToc+1.6TflMCPtot)/((11/6)Cair,oc+HAoc+1.6MCPtot)

Tmx=(Cair,mx(3T1,mx(3/2)T2,mx+(1/3)T3,mx)+˙Qocz(1Frgains)+˙Qmxz+HATmx+TocMCPtot)/((11/6)Cair,mx+HAmx+MCPtot)

Here Cair,fl, Cair,oc, and Cair,mx are the heat capacities of the air volume in each subzone. Cair,mx is calculated by:

Rair,mx=Vmx(Δzmx/zceil)ρair,mxcp,air,mxMulcap/(Δtz3600)

Cair,mx=Rair,mxΔtz/Δtsys

The other subzone air heat capacities are calculated in the same manner.

### Mixed calculation[LINK]

The above iterative procedure assumed that displacement ventilation was taking place: i.e., conditions were favorable temperature stratification in the zone. Now that this calculation is complete and the subzone temperatures and depths calculated, we check to see if this assumption was justified. If not, zone conditions must be recalculated assuming a well-mixed zone.

If Tmx<Toc or MCPtot0 or HfrHceil<Hfl,top+Δzocc,min then the following mixed calculation will replace the displacement ventilation calculation.

Note: Δzocc,min is the minimum thickness of occupied subzone. It is set to 0.2 meters. Hfl,top is the height of the top of the floor subzone. It is defined to be 0.2 meters; that is, the floor subzone is always 0.2 meters thick and Tfl is the temperature at 0.1 meter above the floor surface.

The mixed calculation iteratively calculates surface convection coefficients and room temperature just like the displacement ventilation calculation described above. In the mixed case however, only one zone temperature Tavg is calculated. The 3 subzone temperatures are then set equal to Tavg.

First, Frhb is set equal to zero.

Then the code iterates over these steps.

1. Calculate Tavg using:

Tavg=(Cair,z(3T1,z(3/2)T2,z+(1/3)T3,z)+˙Qtot,conv+HAToc+HATmx+HATfl+MCPTtot)/((11/6)Cair,z+HAoc+HAmx+HAfl+MCPtot)

Tmx=Tavg

Toc=Tavg

Tfl=Tavg

2. Call HcUCSDDV to calculate the hc’s.

3. Repeat step 1

### Final calculations[LINK]

The displacement ventilation calculation finishes by calculating some report variables. Using Equation [eq:RoomAirHeightBoundaryLayer], setting the boundary height to 1.5 meters and solving for the flow, we calculate a minimum flow fraction:

˙Vmin=(1.5/24.55)5/3Nplumes˙Q1/3perplume

Frminflow=.000833MCPtot/˙Vmin

We define heights:

Htrans=FrhbHceil

Hmxavg=(Hceil+Htrans)/2

Hocavg=(Hfltop+Htrans)/2

Hflavg=Hfltop/2

Using the user defined comfort height we calculate the comfort temperature.

If mixing:

Tcomf=Tavg

If displacement ventilation:

If Hcomf < Hflavg:

Tcomf=Tfl

Else if HcomfHflavg and Hcomf<Hocavg:

Tcomf=(Tfl(HocavgHcomf)+Tmx(HcomfHflavg)/(HocavgHflavg)

Else if HcomfHocavg and Hcomf<Hmxavg:

Tcomf=(Toc(HmxavgHcomf)+Tmx(HcomfHocavg)/(HmxavgHocavg)

Else if HcomfHmxavg and Hcomf<Hceil:

Tcomf=Tmx

Using the user defined thermostat height we calculate the temperature at the thermostat.

If mixing:

Tstat=Tavg

If displacement ventilation:

If Hstat < Hflavg:

Tstat=Tfl

Else if HstatHflavg and Hstat<Hocavg:

Tstat=(Tfl(HocavgHstat)+Tmx(HstatHflavg)/(HocavgHflavg)

Else if HstatHocavg and Hstat<Hmxavg:

Tstat=(Toc(HmxavgHstat)+Tmx(HstatHocavg)/(HmxavgHocavg)

Else if HstatHmxavg and Hstat<Hceil:

Tstat=Tmx

The average temperature gradient is:

If HmxavgHflavg>0.1:

GradTavg=(TmxTfl)/(HmxavgHflavg)

else GradTavg=9.999

The maximum temperature gradient is:

If HocavgHflavg>0.1:

GradTmax,1=(TocTfl)/(HocavgHflavg)

else GradTmax,1=9.999

If HmxavgHocavg>0.1:

GradTmax,2=(TmxToc)/(HmxavgHocavg)

else GradTmax,2=9.999 and

GradTmax=max(GradTmax,1,GradTmax,2)

For reporting purposes, if the zone is deemed to be mixed, the displacement ventilation report variables are set to flag values.

If Tmx<Toc or MCPtot0 or HfrHceil<Hfl,top+Δzocc,min or TmxToc<ΔTCritRep

GradTavg=9.999

GradTmax=9.999

FRmin,flow=1.0

Htrans=9.999

Finally, the zone node temperature is set to Tmx.

## Under-Floor Air Distribution Interior Zone Model[LINK]

### Overview[LINK]

The input object RoomAirSettings:UnderFloorAirDistributionInterior provides a simple model for heat transfer and nonuniform vertical temperature profile for interior zones of a UFAD system. These zones are expected to be dominated by internal loads, a portion of which (such as occupants and workstations) will act to generate plumes. The plumes act to potentially create room air stratification, depending on the type & number of diffusers, the amount and type of load, and the system flowrate. In order to better model this situation the fully-mixed room air approximation that is currently used in most whole building analysis tools is extended to a two node approach, with the purpose of obtaining a first order precision model for vertical temperature profiles for the interior zones of UFAD systems. The use of 2 nodes allows for greatly improved prediction of thermal comfort and overall building energy performance for the increasingly popular UFAD systems.

The UCSD UFAD Interior Zone Model is one of the non-uniform zone models provided through the Room Air Manager in EnergyPlus. The intent is to provide a selection of useful non-uniform zone air models to enable the evaluation of air-conditioning techniques that use stratified or partially stratified room air. Such techniques include displacement ventilation (DV) and underfloor air distribution (UFAD) systems. The methodology can also include natural displacement ventilation and also wind-driven cross-ventilation (CV).

### Underfloor air distribution systems[LINK]

UFAD systems represent, in terms of room air stratification, an intermediate condition between a well-mixed zone and displacement ventilation. Air is supplied through an underfloor plenum at low pressure through diffusers in the raised floor. The diffusers can be of various types: e.g., swirl, variable-area, displacement, and produce more or less mixing in the zone. UFAD systems are promoted as saving energy due to: higher supply air temperature; low static pressure; cooler conditions in the occupied subzone than in the upper subzone; and sweeping of some portion of the convective load (from ceiling lights, for instance) into the return air without interaction with the occupied region of the zone.

Modeling a UFAD system is quite complex and involves considerably more than just a non-uniform zone model. The zones’ coupling to the supply and return plenums must be modeled accurately (particularly radiative transfer from a warm ceiling to a cool floor and on into the supply plenum by conduction). The supply plenum must be accurately modeled, giving a good estimate of the supply air temperature and conduction heat transfer between supply & return plenums through the slab. The HVAC system must be modeled including return air bypass and various types of fan powered terminal units.

The UCSD UFAD interior zone model is similar to the UCSD DV model. The most obvious difference is that the UFAD model has no separate near-floor subzone. Like the UCSD DV model it is a plume equation based multi-layer model (2 layers in this case). The zone is modeled as being divided into 2 well separated subzones which we denote as “occupied” and “upper”. Each subzone is treated as having a single temperature. The boundary between the 2 subzones moves up & down each time step as a function of zone loads and supply air flow rate. Thus at each HVAC time step, the height of the boundary above the floor must be calculated, portions of surfaces assigned to each subzone, and a separate convective heat balance performed on each subzone.

### Model Description[LINK]

The UFAD interior zone model is based upon non-dimensional analysis of the system and using the non-dimensional description to make a comparison between full-scale UCB test chamber data & small-scale UCSD salt tank measurements.

In order to do the non-dimensional comparisons, we need to define two dimensionless parameters. One is Γ, and the other is ϕ. Lin & Linden (Lin & Linden, 2005) showed that in a UFAD system, the buoyancy flux of the heat source (B) and the momentum flux of the cooling jets (M) are the controlling parameters on the stratification. Since [B]=L4T3 and[M]=L4T2 , we can have a length scale as M3/4/B1/2 .

Definition of Γ for the single-plume single-diffuser basic model

We observed, in our small-scale experiments, that the total room height does not affect the interface position, or the height of the occupied zone. In other words, H might not be the critical length scale for the stratification. Therefore, we started to use A as the reference length. Then Γ is defined as:

Γ=M3/4B1/2A1/2=(Q2/A)3/4B1/2A1/2=Q3/2A5/4B1/2

Definition for multi-diffuser and multi-source cases

We only considered single-diffuser, single-source cases in above analysis. Suppose there are n equal diffusers and m equal heat sources in a UFAD room. We shall divide the number of diffusers up into a number of separate heat sources so that each subsection with n’ = n/m diffusers per heat source will have the same stratification as other subsections. Further, the air flow and the heat load into the subsection Q’ and B’ will be Q=Q/m B=B/m respectively, where Q’ and B’ are the total air flow and the total heat load for the entire UFAD space. Then the momentum flux each diffuser per heat source carries is Md=(1nQ)2/A. Equation [eq:GammaSinglePlumeSingleDiffuser] will be modified as:

Γ=(nMd)3/4B1/2nA=Q3/2(nA)5/4B1/2=Q3/2m(nA)5/4B1/2

Full-scale cases

Because B is the buoyancy flux of the heat sources and M is the momentum flux of the cooling jets, in a real full-scale room, we shall consider the total room net heat load (plume heat input, minus the room losses) and the total net flow rate coming from the diffusers (input room air flow, minus the room leakage). Further, if the diffuser is swirl type, the vertical momentum flux should be used.

Γ=(Qcosθ)3/2m(nA)5/4(0.0281W)1/2

where Q is the net flow rate coming out from all diffusers (m3/s); W is the total net heat load (kW); A is the effective area of each diffuser (m2); n’ is the number of diffusers per heat source; qθ is the angle between the diffuser slots and the vertical direction and m is the number of heat sources.

Definition of ϕ

In our theoretical model, two-layer stratification forms at steady state, provided that each diffuser carries the same momentum flux, and each heat source has the same heat load. We could define a dimensionless parameter ϕ, which indicates the strength of stratification.

Small-scale cases

In our salt-water tank experiments, fluid density r is measured. Define that:

ϕ=ρuρlρuρo

where ρu and ρl are the fluid density of the upper layer and lower layer, separately; and ρo is the source density at the diffusers.

Therefore, ρl = ρo gives ϕ = 1, which means the largest stratification (displacement ventilation case); ρl = ρu leads to ϕ = 0, in which case there is no stratification (mixed ventilation case).

Full-scale cases

Similarly, we can define ϕ for full-scale cases by using temperature.

ϕ=TrTozTrTs

where Tr, Toz, and Ts are the return air temperature, the occupied zone temperature and the supply temperature, respectively (K). Again ϕ = 1 occurs in displacement ventilation; while ϕ = 0 happens in mixed ventilation.

Comparisons between full-scale UCB data and small-scale UCSD data

The figures (Figure 2. Data comparisons in the non-dimensional (a) regular Γ-ϕ plot and Figure 3. (b) log-log Γ-ϕ plot.} show the comparisons between UCB’s data and the UCSD salt tank data in the Γ-ϕ plot. As seen in the figures, the full-scale and small-scale data are on the same trend curve. This provides the evidence that the salt-tank experiments have included most characteristics of a UFAD system. Note that big Γ (>20) of UCB’s experiments all have large DDR (from 1.19 to 2.18). The largest DDR (2.18) even gives a negative (TrToz)/(TrTs), which is NOT shown in the figures.)

We could work out the occupied zone temperature by using the least-square fitting line suggested in Figure 3. Hence the interface height is needed to determine a entire two-layer stratification. Figure 4 shows the dimensionless interface height(h/nA) of the UCSD small-scale experiments plotted against Γ. Note that for the experiments with elevated heat source, the interface heights have been modified byh=h12hs where hs is the vertical position of the elevated heat source. All data then are located along a line in Figure 4. Since the salt-tank experiments are concluded to represent important characteristics of a full-scale UFAD room, this figure provides some guidelines for estimate the interface position in a real UFAD room.

Formulas for EnergyPlus based on the dimensionless parameter Γ

If we have input including the supply temperature Ts (K); the number of diffusers n; the number of heat sources m; the vertical position of the heat sources hs (m); the heat load W (kW); the effective area of a diffuser A (m2); and the total supply air flow rate Q (m3/s) then the output will be;

Tr=0.0281WQgTs+Ts

Toz=Tr1.6Γ0.76(TrTs)

h=nmA(7.43ln(Γ)1.35)+12hs

where Tr is the return temperature (K); Toz is the occupied subzone temperature (K); h is the interface height (m); and Γ is defined above.

### Implementation[LINK]

The implementation closely follows the procedure described in the displacement ventilation zone model. The model predicts two temperatures that characterize the two main levels in the stratification of the room:

1. An occupied subzone temperature Toc representing the temperature of the occupied region;

2. An upper level temperature Tmx representing the temperature of the upper, mixed region and the outflow temperature.

We will use Equation [eq:InterfaceHeightEquation] to calculate the interface height and do a heat balance calculation on each subzone. Γ is given by Equation [eq:GammaFullScaleCases]. The supply air flow rate ˙V is obtained by summing all the air flows entering the zone: supply air, infiltration, ventilation, and inter-zone flow. The heat gain ˙Q is estimated by summing all the convective internal gains located in the occupied subzone – task lights, people, equipment – and dividing this power equally among the n plumes. With these assumptions we can describe the implementation.

The UCSD UFI model is controlled by the subroutine ManageUCSDUFModels which is called from the RoomAirModelManager. The RoomAirModelManager selects which zone model will be used for each zone.

The calculation is done in subroutine CalcUCSDUI. First we calculate the convective heat gain going into the upper and lower regions.

˙Qocz=˙Qoc,conv+˙Qtl,conv+˙Qeleq,conv+˙Qgaseq,conv+˙Qotheq,conv+˙Qhw,conv+˙Qstmeq,conv+˙Qbb,conv+˙Qwaterheater,conv+˙Qfuelcell,conv

˙Qmxz=˙Qgl,conv+˙Qltp+˙Qhtrad,conv

˙Qtot,conv=˙Qocz+˙Qmxz

Next we sum up the inlet air flows in the form of MCP (mass flow rate times the air specific heat capacity) and MCPT (mass flow rate times Cp times air temperature).

MCPzone=MCPi+MCPvent+MCPmix

MCPsys=inlets˙miCp,i

˙msystot=inlets˙mi

MCPtot=MCPzone+MCPsys

MCPTzone=MCPTi+MCPTvent+MCPTmix

MCPTsys=inlets˙miCp,iTi

MCPTtot=MCPTzone+MCPTsys

The number of plumes per occupant Nplumesperpers is a user input. The total number of plumes in the zone is:

Nplumes=NoccNplumesperperson

Using this we calculate the total power in the plumes and the power per plume.

˙Qplumes=˙Qocz

˙Qperplume=˙Qplumes/Nplumes

The number of diffusers per plumes is also a user input. To obtain the number of diffusers in the zone:

Ndiffusers=NplumesNdiffusersperplume

The area Adiff is also a user input. For swirl diffusers and for displacement diffusers this area is used as input. For the variable area diffusers, though, we calculate the area. We assume 400 ft/min velocity at the diffuser and a design flow rate per diffuser is 150 cfm (.0708 m