## Chilled-Water-Based Air Cooling Coil[LINK]

The input object Coil:Cooling:Water is simpler than the detailed geometry model. The simple model provides a good prediction of the air and water outlet conditions without requiring the detailed geometric input required for the detailed model. A greatly simplified schematic of enthalpy and temperature conditions in a counter flow cooling/dehumidifying coil is shown in the schematic Figure. The input required to model the coil includes only a set of thermodynamic design inputs, which require no specific manufacturer’s data. The coil simulation model is essentially a modification of one presented by Elmahdy and Mitalas (1977), TRNSYS, 1990 and Threlkeld, J.L. 1970. The model calculates the UA values required for a Dry, Wet and Part Wet & Part Dry Coil and iterates between the Dry and Wet Coil to output the fraction wet. There are two modes of flow operation for this model: Cross Flow, which is widely applicable in HVAC systems and the second being Counter flow mode. The default value in program is set up for Counter Flow. In addition the coil has two modes of analysis: Simple Analysis and Detailed Analysis. The Simple analysis mode operates the coil as either wet or dry while the detailed mode simulates the coil as part wet part-dry. While the detailed mode provides more accurate results, it is significantly slower than the simple model. The simple mode gives good results for an annual simulation but will not be adequate for a time step performance analysis.

### Heat Transfer and Energy Balance[LINK]

The cooling coil may be completely dry, completely wet with condensation, or it may have wet and dry sections. The actual condition of the coil surface depends on the humidity and temperature of the air passing over the coil and the coil surface temperature. The part-dry part-wet case represents the most general scenario for the coil surface conditions. There are subroutines present in the model for both the dry and wet regions of the coil, and a subroutine that iterates between the dry and wet subroutines to calculate the fraction of the coil surface that is wet. For each region the heat transfer rate from air to water may be defined by the rate of enthalpy change in the air and in the water. The rates must balance between each medium for energy to be conserved.

### Model Description[LINK]

The Model has two blocks: 1st = Design Block with the Design Inputs. This block calculates the Design U-Factor Times Area Value (UA) values required by the model. Using these UA values the model simulates the operating conditions. The operating block is the one containing the operating conditions, the conditions at which the coil operates. Following is the list of Design and Operating inputs and subsequently the Design and Operating variables used in the model.

Design Inputs (User Inputs)
DesWaterVolFlowRate: |
Maximum Water Volume Flow Rate |

DesAirVolFlowRate: |
Maximum Air Volume Flow Rate |

DesInletWaterTemp: |
Inlet Water Temperature at Design Condition |

DesInletAirTemp: |
Inlet Air Temperature at Design Condition |

DesOutletAirTemp: |
Outlet Air Temperature at Design Condition |

DesInletAirHumRat: |
Inlet Air Humidity Ratio at Design Conditions |

DesOutletAirHumRat: |
Outlet Air Humidity Ratio at Design Conditions. |

Operating Conditions (From Nodes – not user inputs)
InletWaterMassFlowRate: |
Entering Water Mass Flow Rate at operating condition |

InletWaterTemp: |
Inlet Water Temperature at operating condition |

InletAirMassFlowRate: |
Entering Air Mass Flow Rate at operating condition |

InletAirTemp: |
Inlet Air Temperature at operating condition |

InletAirHumRat: |
Entering air humidity ratio at operating conditions |

The various U-Factor Times Area values (UA) required by this model are calculated from the above inputs, which are explained later in the document. The various UA are:

UA Descriptions of Model
CoilUATotal: |
Overall heat transfer coefficient (W/C) |

CoilUAInternal: |
Overall internal UA (W/C) |

CoilUAExternal: |
Overall external UA (W/C) |

CoilUInternal: |
Internal overall heat transfer coefficient (W/m ∙C) |

CoilUWetExternal: |
Wet part external overall heat transfer coefficient (W/m ∙C) |

CoilUDryExternal: |
Dry part external overall heat transfer coefficient (W/m ∙C) |

The UA values are calculated assuming a wet coil at the design conditions. Following are a few important calculations to understand the working of the model. The model is basically divided into two blocks: the Design Block and the Operating Block.

The Design Block is a one time calculation. The aim of the Design Block is to calculate the Coil UA for use in the operating Block.

### Design Block Calculations:[LINK]

The design block has the code for calculating the six Coil UA values required by the operating block. Reasonable assumptions have been made in the calculations to maintain the simplicity of the model.

Heat transfer in a wet coil model is based on enthalpy rather than temperature to take into account latent effects. While heat transfer rates are commonly expressed as the product of an overall heat transfer coefficient, UA, and a temperature difference, the use of enthalpy-based heat transfer calculations requires an enthalpy-based heat transfer coefficient which we denote as DesUACoilTotalEnth and hence the equation.

Q = DesUACoilTotalEnth * (Hair,mean - Hwater,mean). The value of Q is calculated using product of air mass flow rate and difference in inlet and outlet air enthalpies at design conditions.

The relation between the enthalpy-based UA and the temperature-based UA is DesUACoilTotalEnth = CoilUA / Cp. CoilUA is the conventional heat transfer coefficient and Cp = specific heat of the air.

We need the following quantities for our design calculations. The *Psy* functions are the EnergyPlus built-in psychrometric functions.

˙mair=ρair˙Vair

hair,in=PsyHFnTdbW(Tair,in,wair,in)

hair,out=PsyHFnTdbW(Tair,out,wout)

hw,sat,in=PsyHFnTdbW(Tw,in,PsyWFnTdpPb(Tw,in,Patm))

˙Qcoil=˙mair(hair,in−hair,out)

Tw,out=Tw,in+˙Qcoil/(˙mw,maxCp,w)

hw,sat,out=PsyHFnTdbW(Tw,out,PsyWFnTdpPb(Tw,out,Patm))

We now calculate the design coil bypass factor. The bypass factor is not used in subsequent calculations. It is calculated solely to use as check on the reasonableness of the user-input design inlet and outlet conditions. First we make an initial estimate of the apparatus dew point temperature:

Tair,dp,app=PsyTdpFnWPb(wair,out,Patm)

we also need the “slope” of temperature versus humidity ratio on the psych chart between the inlet and outlet air conditions:

ST,w=(Tair,in−Tair,out)/(wair,in−wair,out)

We now obtain the actual design apparatus dewpoint temperature by iterating over the following two equations:

wair,dp,app=PsyWFnTdpPb(Tair,dp,app,Patm)

Tair,dp,app=Tair,in−ST,w(wair,in−wair,dp,app)

The apparatus dewpoint enthalpy is then:

hair,dp,app=PsyHFnTdbW(Tair,dp,app,wair,dp,app)

and the coil bypass factor is:

Fcoilbypass=(hair,out−hair,dp,app)/(hair,in−hair,dp,app)

If the iterative procedure doesn’t converge, or the coil bypass factor is too large (greater than 0.5), or the apparatus dewpoint enthalpy is less than the saturated air enthalpy at the water inlet temperature, the design outlet air conditions are reset to 90% relative humidity at the same outlet enthalpy. The above design calculations are then repeated.

We are now ready to calculate the design coil UA. This will be accomplished by inverting the simple coil calculation routine *CoolingCoil* using the Regula Falsi method. First we make an initial estimate of the coil UA.

Δhlmd=((hair,in−hw,sat,out)−(hair,out−hw,sat,in))/log((hair,in−hw,sat,out)−(hair,out−hw,sat,in))

UAcoil,enthalpy−based=˙Qcoil/Δhlmd

UAcoil,ext=Cp,airUAcoil,enthalpy−based

We set the internal UA to 3.3 times the external UA (as a typical value for a coil). Then the total UA is:

UAcoil,tot=1(1/UAcoil,int+1/UAcoil,ext)

The next step is to estimate the coil external heat transfer surface area. This is done in the function *EstimateHEXSurfaceArea*:

Areacoil,ext=EstimateHEXSurfaceArea

using the following assumptions:

Tube inside diameter = 0.0122 (m)

Tube side water velocity = 2.0 (m/s)

Inside to outside coil surface area ratio (Ai/Ao) = 0.07 (-)

Fins overall efficiency = 0.92 (-)

Aluminum fins, 12 fins per inch with fins to total outside surface area ratio of 90%.

Airside combined heat and mass transfer coefficient = 140 (W/m2∙°C)

Interior and exterior U values (really UA’s per unit exterior surface area) are calculated by dividing the above UA’s by the area. The resulting Ucoil,ext is assumed to be Ucoil,ext,wet; Ucoil,ext,dry is set equal to Ucoil,ext,wet. We now have all the starting values needed for inverting the simple coil model using the chosen Regula Falsi iterative method. Once the iteration is completed, we have coil UA’s and U’s that yield the design outlet air and water enthalpies given the inlet design conditions and flow rates. Note that the simple coil model can not exactly match the specified design outlet air temperature and humidity ratio. It can only match the design air outlet enthalpy. Generally the simple coil model will yield outlet conditions near the saturation curve if any dehumidification is occuring. Typical outlet relative humidities are around 95%.

The above calculations yield coil UA’s for the design inlet conditions and air and water flow rates. As the flow rates vary during the time step calculations, the UA’s need to be adjusted, since coil UA’s are a rather strong function of air and water side flow rates. Each time step the coil UA’s are modified using the same formulas as are used in the hot water coil model. Refer to that model for the flow dependences.

### Operating Block Calculations:[LINK]

There are two modes of coil analysis in the operating block. They are the Simple analysis mode and the detailed analysis mode. The simple analysis mode assumes the coil to be either all wet or either all dry and execute the model , on the other hand the detailed mode checks for part wet part dry mode of operation and reports surface area wet fraction of coil, however the program execution time in detailed mode is noticeably higher.

The operating block for Detailed Mode Analysis of this coil model is divided into three modes of coil performance. The modes being

Coil is completely dry: There is no moisture condensation on the coil surface and the coil is a dry coil. This is an extreme condition when the entering air has very low humidity ratio or is dry air.

Coil is completely wet: The entire coil is wet due to complete condensation on the surface of the coil.

Part Wet Part Dry Mode: This is the usual/frequent mode of operation of coil, as shown in Figure, where part of the coil at entry of air is dry and as air cools condensation occurs and part of the coil becomes wet.

The Part Wet Part Dry Mode of operation is essentially a function the Coil Completely Dry and Coil Completely Wet mode. This subroutine iterates between the Dry Coil and the Wet Coil to give outputs, a detailed explanation is given later in the document. The operating block requires 5 inputs, which are mentioned earlier in the document. These inputs are automatically generated from the node connections in Energy Plus. The user does not have to input any information to run this coil model.

The option to identify which mode of operation the coil should perform ie, for a given set of inputs would the coil be Dry, Wet or Part Wet Part Dry, is decided by set of conditions described below.

**IF** (**Temperature Dewpoint Air < Water Inlet Temperature**) **THEN** the coil is Dry and we call the Subroutine Coil Completely Dry. In this case outlet temperature of air would be higher than the air dewpoint and hence there would be no condensation.

**IF** (**Temperature Dewpoint Air > Water Inlet Temperature**) **THEN** the coil is completely wet, call subroutine Coil Completely Wet, it is assumed that moisture condensation occurs over completely surface of the coil. However we go ahead and check for the coil being partially wet with the following condition.

**IF (AirDewPointTemp < AirInletCoilSurfTemp) THEN,** the coil is Partially Wet because there is possibility that air temperature will go below its dewpoint and moisture will condense on latter part of the cooling coil.

The Operating Block for Simple Mode Analysis is divided into two modes of coil performance, the two modes being

Coil is completely dry: There is no moisture condensation on the coil surface and the coil is a dry coil.

Coil is completely wet: The entire coil is wet due to complete condensation on the surface of the coil.

The option to identify which mode of operation the Simple mode analysis should perform ie, for a given set of inputs would the coil be Dry or Wet is decided by set of conditions described below.

**IF** (**Temperature Dewpoint Air < Water Inlet Temperature**) **THEN** the coil is Dry and we call the Subroutine Coil Completely Dry. In this case outlet temperature of air would be higher than the air dewpoint and hence there would be no condensation.

**IF** (**Temperature Dewpoint Air > Water Inlet Temperature**) **THEN** the coil is completely wet, call subroutine Coil Completely Wet, it is assumed that moisture condensation occurs over completely surface of the coil. However we go ahead and check for the coil being partially wet with the following condition.

The above is a simple mode of analysis and the results are very slightly different from the detailed mode of analysis. The algorithms used in Simple mode and the Detailed mode are identically similar. The surface area wet fraction in the coil is reported as 1.0 or 0.0 for wet or dry coil respectively. The program defaults to simple mode of analysis for enabling higher execution speed.

### Effectiveness Equations:[LINK]

There are two modes of flow for the coil, Counter Flow mode or the Cross Flow mode, default set up is as cross flow since most air condition applications have cross flow heat exchangers. According to the mode of flow the following NTU - Effectiveness relationships are used to calculate coil effectiveness, which is used later by all the three modes (Dry, Wet, Part Wet) for calculating air outlet conditions and heat transfer.

Following are the relations used for calculating effectiveness equation for the Heat exchangers.

Counter Flow Heat Exchanger: Effectiveness Equation:

ηCounterFlow=(1−Exp(−NTU×(1−RatioStreamCapacity)))1−RatioStreamCapacity×Exp(−NTU×(1−RatioStreamCapacity))

In Equation the variable Ratio_StreamCapacity is defined as below

RatioStreamCapacity=MinCapacityStreamMaxCapacityStream

In equation capacity of stream is defined as below in equation

(Min,Max)CapacityStream=(MassFlowRate×Cp)air,water

NTU in equation , is defined as the Number of Transfer Units, it is a function of Coil UA and the Minimum Capacity of Stream. The Coil UA is a variable in this equation and depends on which mode of the coil operation (Dry, Wet, Part Wet) is calling upon equation , i.e., if it is Coil Completely Dry calling upon the effectiveness equation with the value of Dry UA total, which in our case is defined as CoilUA_total. Equation gives definition for NTU.

NTU=CoilUAMinStreamCapacity

Cross Flow Heat Exchanger: Effectiveness Equation:

ηCrossFlow=1−EXP{Exp(−NTU×RatioStreamCapacity×NTU−0.22)−1RatioStreamCapacity×NTU−0.22}

The variables in the above equation have already been defined earlier. Depending on the mode of operation of the coil model the cross or the counter flow equations are used to calculate the effectiveness.

### Coil Outlet Conditions:[LINK]

Calculating the Outlet Stream Conditions using the effectiveness value from equation or depending on the mode of flow. The energy difference between the outlet and inlet stream conditions gives the amount of heat transfer that has actually take place. Temperature of air and water at outlet to the coil is given as in following equations

TempAirOut=TempAirinlet−ηcross,counter×MaxHeatTransferStreamCapacityAir

TempWaterOut=TempWaterInlet+ηCross,counter×MaxHeatTransferStreamCapacityWater

In the above equations and the maximum heat transfer is calculated as shown in the following equation

MaxHeatTransfer=MinStreamCapacity×(TempAirInlet−TempWaterInlet)

### Coil Completely Dry Calculations: (operating block)[LINK]

Since the coil is dry, the sensible load is equal to total load and the same with the humidity ratios at inlet and outlet, as in equations and .

QSensibleDryCoil=QTotalDryCoil

HumRatioInlet=HumRatioOutlet

Total Heat Transfer in dry coil is as follows:

QTotalDryCoil=CapacityAir×(AirTempIn−AirTempOutlet)

The variables in the above equation are calculated earlier in equations and to give the total cooling load on the coil.

### Coil Completely Wet Calculations: (operating block)[LINK]

In wet coil we need to account for latent heat transfer, hence calculations are done using enthalpy of air and water instead of stream temperatures Hence we need to define coil UA for the wet coil based on enthalpy of the operating streams and not design streams.

Similar to equations and we calculate the air outlet enthalpy and water outlet enthalpy ie by replacing temperature with enthalpy of the respective streams. The input variable for Coil UA in equation for calculating NTU, in this case it would be enthalpy based and is given as shown in equation

CoilUAEnthalpyBased=1(CpSatIntermediateCoilUAInternal+CpAirCoilUAExternal)

Total Coil Load in case of Wet Coil is the product of mass flow rate of air and enthalpy difference between the inlet and outlet streams as given in the following equation

QTotal=∗Mair×(EnthAirInlet−EnthAirOutlet)

Once the enthalpy is known the outlet temperatures and outlet humidity ratios of the wet coil are calculated as in equations below.

**IF (TempCondensation < PsyTdpFnWPb(InletAirHumRat ,Patm)) THEN**

AirTempOut=AirTempinlet−(AirTempinlet−CondensationTemp)×η

and

**OutletAirHumdityRatio = PsyWFnTdbH(OutletAirTemp,EnthAirOutlet)**

**ELSE**

There is no condensation and hence the inlet and outlet Hum Ratios are equal , and outlet temperature is a function of outlet air enthalpy as below

**OutletAirTemp = PsyTdbFnHW (EnthalpyAirOutlet, OutletAirHumRat)**

**and**

**OutletAirHumRat = InletAirHumRat**

**ENDIF**

Effectiveness η used in equation is defined in equation and Condensation Temperature is calculated using psychrometric function as in equation .

η=1−Exp{−CoilUAExternalCapacitanceAir}

CondensationTemp=PsyTsatFnHPb(EnthAirCondensateTemp,Patm)

EnthAirCondensateTemp=EnthAirInlet−(EnthAirInlet−EnthAirOutlet)η

Once the air outlet temperature are known, then sensible load is calculated as a product of capacitance of air and temperature difference at inlet and outlet, as in equation

QSensible=CapacitanceAir×(AirTempInlet−AirTempOutlet)

### Coil Part Wet Part Dry Calculations: (operating block)[LINK]

The Coil would perform under part wet part dry conditions when Air Dewpoint Temperature is less than Coil surface temperature at inlet to air. In this case part of the coil used value of Dry UA for heat transfer and part the coil used Wet UA value for heat transfer.

This problem is solved utilizing the fact that the Exit conditions from the Dry Part of the Coil would become the inlet conditions to the wet part of the coil (see Figure) and the coil model determines by iteration what fraction of the coil is wet and based on that it calculates the areas and subsequently the UA values of that dry and wet part, based on the area of the dry and wet part respectively. Explained below are the steps followed to the estimating the wet dry behavior of the coil.

- Iterate between the Dry Coil and the Wet Coil. First calculate Coil Completely Dry performance by estimating the wet dry interface water temperature using equation and inputting this variable as the water inlet temperature to dry Coil.

WetDryInterfaceWaterTemp=WaterTempInlet+AreaWetFraction∗(WaterTempOutlet−WaterTempInlet)

The value of Surface Area Wet fraction is estimated initially as follows

AreaWetFractionEstimate=AirDewPtTemp−InletWaterTempOutletWaterTemp−InletWaterTemp

For the above mentioned iteration the value of Coil UA for Wet and Dry part need to be varied according to the new respective area of the wet and dry parts. This estimate of Wet and Dry area is a product of the estimated Surface Area Fraction and total coil external area, which keeps varying as will be explained further in the document.

UA value for Dry part of the Coil is estimated as below.

CoilUADryExternal=SurfAreaDry1CoilUDryExternal+1CoilUInternal

Where Surface Area Dry = (Total Coil Area – Wet Part Area), where the Wet part area is the product of Surface fraction Wet and Total Coil Area.

UA value for the Wet part of the Coil requires Wet UA external and Wet UA Internal, which are calculated as below.

WetPartUAExternal=CoilUWetExternal×SurfaceAreaWet

WetPartUAInternal=CoilUInternal×SurfaceAreaWet

It is essential to remember that the mode of calculation for the coils remains the same as in completely wet and completely dry mode, only the UA values and water, air outlet and inlet values change.

Now Iterate between the Dry Coil and wet Coil with the above respective UA, and usual operating inputs except the variable water inlet temperature for dry Coil is replaced with Wet Dry Interface Water temperature, and in the Wet Coil the Outlet Air Temperature from dry Coil is the inlet air temperature to Wet Coil. The iteration proceeds till the Outlet Water Temperature from Wet Coil equals the Wet Dry Interface Water Temp, which is the input to Dry Coil.

Dry Part Inputs: (changed operating inputs) :Iteration Case 1: Explained In Programming Fashion:

**CALL CoilCompletelyDry (WetDryInterfcWaterTemp, InletAirTemp, DryCoilUA, &**

** OutletWaterTemp, WetDryInterfcAirTemp, WetDryInterfcHumRat, &**

**DryCoilHeatTranfer).**

Input the calculated values calculated by Dry Coil above into Wet Coil below. The variables have been highlighted in color red and blue.

**CALL**CoilCompletelyWet (InletWaterTemp, **WetDryInterfcAirTemp**, **WetDryInterfcHumRat**

WetPartUAInternal,WetPartUAExternal, &

**EstimateWetDryInterfcWaterTemp**, OutletAirTemp, OutletAirHumRat, &

WetCoilTotalHeatTransfer, WetCoilSensibleHeatTransfer, &

EstimateSurfAreaWetFraction, WetDryInterfcSurfTemp)

Iterate Between the above two Wet and Dry Coil calls until the two variables in blue ie **WetDryInterfcWaterTemp = EstimateWetDryInterfcWaterTemp.**The key is to have the difference between the variables (WetDryInterfcWaterTemp – OutletWaterTemp) in Dry Coil equal to (InletWaterTemp-EstimatedWetDryInterfcWaterTemp) in Wet Coil. This equality quantized the relative part of coil that is dry and part that is wet on the basis of heat transfer that has occurred.

After the above convergence check for the coil being dry otherwise iterate to calculate surface fraction area wet.

**IF**

{(AreaFractionWet≤0.0)and(WetDryInterfaceSurfTemp>AirDewPt)}

**THEN CoilCompletelyDry**

If equation is satisfied then Coil is Dry and simply output the value for Dry Coil calculated else the coil is partially wet and then iterate to find the surface fraction area wet. Start with the initially guess value of surface area fraction (equation wet and iterate on the entire loop starting from until the Wet Dry Interface Temperature equals the Air Dewpoint Temperature. The value of Surface Area fraction wet at which the interface air temperature equals is dewpoint is the transition point from wet to dry and gives the % of coil that is dry and % that is wet.

Graphs Showing the Performance of the coil model at optimum operating conditions are shown below. All values of variable used have been normalized.

IBPSA BuildSim-2004. 2004. Colarado Boulder: An Improvement of Ashrae Secondary HVAC toolkit Simple Cooling Coil Model for Building Simulation, Rahul J Chillar, Richard J Liesen M&IE ,UIUC.

Stoecker, W.F. <dates unspecified> Design of Thermal Systems,: ME 423 Class Notes , M& IE Dept UIUC.

Brandemeuhl, M. J. 1993. HVAC2 Toolkit: Algorithms and Subroutines for Secondary HVAC Systems Energy Calculations, ASHRAE.

Elmahdy, A.H. and Mitalas, G.P. 1977. “A Simple Model for Cooling and Dehumidifying Coils for Use In Calculating Energy Requirements for Buildings ASHRAE Transactions, Vol.83 Part 2, pp. 103-117.

Threlkeld, J.L. 1970. Thermal Environmental Engineering, 2nd Edition, Englewood Cliffs: Prentice-Hall,Inc. pp. 254-270.

ASHRAE Secondary HVAC Toolkit TRNSYS. 1990. A Transient System Simulation Program: Reference Manual. Solar Energy Laboratory, Univ. Wisconsin-Madison, pp. 4.6.8-1 - 4.6.8-12.

Kays, W.M. and A.L. London. 1964. Compact Heat Exchangers, 2nd Edition, New York: McGraw-Hill.

Clark, D.R.. 1985. HVACSIM+ Building Systems and Equipment Simulation Program Reference Manual, Pub. No. NBSIR 84-2996, National Bureau of Standards, U.S. Department of Commerce, January, 1985

Elmahdy, A.H. 1975. Analytical and Experimental Multi-Row Finned-Tube Heat Exchanger Performance During Cooling and Dehumidifying Processes, Ph.D. Thesis, Carleton University, Ottawa, Canada, December, 1975.

Elmahdy, A.H., and Mitalas, G.P. 1977. “A Simple Model for Cooling and Dehumidifying Coils for Use in Calculating Energy Requirements for Buildings,” ASHRAE Transactions, Vol. 83, Part 2, pp. 103-117.

## Chilled-Water-Based Detailed Geometry Air Cooling Coil[LINK]

The input object Coil:Cooling:Water:DetailedGeometry provides a coil model that predicts changes in air and water flow variables across the coil based on the coil geometry. A greatly simplified schematic of enthalpy and temperature conditions in a counterflow cooling/dehumidifying coil is shown in the following schematic figure. In addition, the variables required to model a cooling/dehumidifying coils and their definitions are extensively listed in “Table. Coil Geometry and Flow Variables for Coils”. The input required to model the coil includes a complete geometric description that, in most cases, should be derivable from specific manufacturer’s data. The coil simulation model is essentially the one presented by Elmahdy and Mitalas (1977) and implemented in HVACSIM+ (Clark 1985), a modular program also designed for energy analysis of building systems. The model solves the equations for the dry and wet sections of the coil using log mean temperature and log mean enthalpy differences between the liquid and the air streams. Elmahdy and Mitalas state that crossflow counterflow coils with at four rows or more are approximated well by this model. This does not constitute a major limitation since cooling and dehumidifying coils typically have more than four rows.

### Heat Transfer and Energy Balance[LINK]

The cooling coil may be completely dry, completely wet with condensation, or it may have wet and dry sections. The actual condition of the coil surface depends on the humidity and temperature of the air passing over the coil and the coil surface temperature. The partly wet-partly dry case represents the most general scenario for the coil surface conditions. The all dry and all wet cases can be considered as limiting solutions of the wet or dry areas respectively going to zero. In the general case, equations are written for both the dry and wet regions of the coil. For each region the heat transfer rate from air to water may be defined by the rate of enthalpy change in the air and in the water. The rates must balance between each medium for energy to be conserved. Equations through express the energy balance between the water and the air for the case of dry and wet coils respectively. Equations and represent the heat transfer rate between water and air based on the actual performance of the coil. The UA parameter can be calculated from the parameters in the following table.

Coil Geometry and Flow Variables for Coils
A |
air, air side |
LMTD |
log mean temperature difference |

aa, bb |
coeff. in enthalpy approximation |
˙m |
mass flow rate |

C1, C2 |
coeff. in air side film coeff. |
mf |
metal and fouling |

Cp |
specific heat |
μ |
viscosity |

D |
diameter, effective diameter |
o |
outside (air side) |

Dhdr |
hydraulic diameter on air side |
Pr |
Prandtl number |

D |
dry region |
˙Q |
heat transfer rate |

δ |
thickness |
R |
overall thermal resistance |

Δ |
spacing |
Re |
Reynolds number |

F |
heat transfer film coefficient |
ρ |
ratio of diameters |

Fai |
variable in fin eff. calculation |
s |
surface, outside of metal |

fin, fins |
air side fin geometry |
St |
Stanton number |

H |
enthalpy |
T |
temperature |

η |
efficiency |
tube |
water tube |

I0() |
mod Bessel fn, 1st kind, ord 0 |
UAdry |
dry heat xfer coeff. * dry area |

I1() |
mod Bessel fn, 1st kind, ord 1 |
UcAw |
wet heat xfer coeff. * wet area |

K0() |
mod Bessel fn, 2nd kind, ord 0 |
ub, ue |
variables in fin eff. calculation |

K1() |
mod Bessel fn, 2nd kind, ord 1 |
V |
average velocity |

I |
inside (water side) |
w |
water, water side, or wet region |

K1 |
variable in sol’n form of eq. |
wa |
humidity ratio |

K |
thermal conductivity |
Z |
variables in sol’n form of eq. |

L |
length |
1, 2, 3 |
positions (see diagram) |

Equations through represent two sets of three equations with 7 unknowns: ˙Qd , Ta,1, Ta,2, Tw,2, Tw,3, ˙ma , ˙mw . However, normally at least four of these variables are specified, for example: inlet water temperature, outlet air temperature, water flow rate, air flow rate, so that the system of equations is effectively closed.

˙Qd=maCpa(Ta,1−Ta,2)

˙Qd=mwCpw(Tw,3−Tw,2)

˙Qd=(UAdry)(LMTD)

˙Qw=ma(Ha,2−Ha,3)

˙Qw=mwCpw(Tw,2−Tw,1)

˙Qw=(UcAw)(LMHD)

In order to manipulate these equations, the log mean temperature and enthalpy differences are expanded as shown in Equations and . Finally, a linear approximation of the enthalpy of saturated air over the range of surface temperature is made using Equation . Note that in Equation Hw refers to the enthalpy of saturated air at the water temperature.

LMTD=(Ta,1−Tw,3)−(Ta,2−Tw,2)lnTa,1−Tw,3Ta,2−Tw,2

LMHD=(Ha,2−Hw,2)−(Ha,3−Hw,1)lnHa,2−Hw,2Ha,3−Hw,1

Hw=aa+bbTw

Equation is derived from the above equations and is used to solve for the coil conditions when all of the inlet conditions are given as input. Operating in this manner, the coil does not have a controlled outlet air temperature.

Tw,2=(1−Z)(Ha,1−aa−K1CpaTa,1)+ZTw,1(bb−mwCpwma)bb−ZmwCpwma−(1−Z)K1Cpa

An alternative solution method is to define the coil leaving air temperature as an input with a variable water flow rate. In this case Equations and are more convenient. Equations through define terms that are used to simplify Equations , and .

Tw,2=(1−Z)(Ha,3−aa)+Tw,1(mwCpwma−bbZ)mwCpwma−bb

Tw,2=(Zd−1)Ta1Cpa+Tw,3(Cpa−ZdmwCpwma)Zd(Cpa−mwCpwma)

Z=exp(UcAw(1ma−bbmwCpw))

K1=Zd−1Zd−maCpamwCpw

Zd=exp(UcAdry(1maCpa−1mwCpw))

### Underlying Correlations, Properties, and Assumptions[LINK]

Overall heat transfer coefficients are calculated from the specified coil geometry and by using empirical correlations from fluid mechanics and heat transfer. For the water side, Equation gives the film heat transfer coefficient in SI units:

fi=1.429(1+0.0146Tw)V0.8wD−0.2i

This is valid for Reynolds numbers greater than 3100 based on water flow velocity and pipe inside diameter and is given in Elmahdy and Mitalas(1977) as recommended in the standard issued by the Air-Conditioning and Refrigeration Institute (1972) for air-cooling coils. The definition of overall inside thermal resistance follows directly as shown in Equation.

Ri=1fiAi

Equation gives the film coefficient for the air side. Another form of the same equation is Equation , which is familiar from the data presented in Kays and London (1984). For coil sections that have a wet surface due to condensation, the air side film coefficient is modified according to Equation . The correction term, a function of air Reynolds number, is valid for Reynolds numbers between 400 and 1500. The coefficients in Equation and are calculated by Equations and that are functions of the coil geometry. Elmahdy (1977)explains the modifier for the wet surface and coefficients for the film coefficient. Equations through show definitions and values of common parameters and properties.

fo=C1ReC2amaAaminflowCp,aPr2/3a

C1ReC2a=StaPr2/3a

fo,w=fo(1.425−5.1×10−4Rea+2.63×10−7Re2a)

C1=0.159(δfinDhdr)−0.065(δfinLfin)0.141

C2=−0.323(ΔfinsLfin)0.049(DfinΔtuberows)0.549(δfinΔfins)−0.028

Dhdr=4AaminflowδcoilAstotal

Rea=4δcoil(1+wa)maAstotalμa

Pra=0.733

μa=1.846×10−5

The film coefficients above act on the extended surface of the air side, that is the area of the fins and the tubes. Therefore, the fin efficiency must also be considered in calculating the overall thermal resistance on the outside. Gardner(1945) gives the derivation of Equation , used as a curve fit to find the fin efficiency as a function of film coefficient. This equation is based on circular fins of constant thickness. To model a coil with flat fins, an effective diameter – that of circular fins with the same fin area – is used. Equations through define variables used in Equation . The overall efficiency of the surface is shown by Equation . Note that the efficiency is found by the same equations for the wet surface using the wet surface film coefficient.

ηfin=−2ρfai(1+ρ)[I1(ub)K1(ue)−K1(ub)I1(ue)I0(ub)K1(ue)+K0(ub)I1(ue)]

fai=(Dfin−Dtube)2√2fokfinδfin

ρ=DtubeDfin

ue=fai1−ρ

ub=ueρ

ηo=1−(1−ηfin)AfinsAstotal

The definition of overall outside thermal resistance is given in Equation as a function of fin efficiency and film coefficient. For a wet coil surface the resistance must be defined differently because the heat transfer equations are based on enthalpy rather than temperature differences, as shown in Equation .

Ro=1foηoAs,total

Ro,w=Cp,a/bbfo,wηo,wAs,total

Equation gives the last two overall components of thermal resistance. They represent the metal tube wall and internal fouling. The fouling factor, due to deposits of dirt and corrosion of the tube inside surfaces, is assumed to be 5x10-5 m2·K/W. All components of thermal resistance are added in series to produce the overall heat transfer coefficients shown in Equations and .

Rmf=δtubektubeAi+FlAi

UAdry=AdryAs,total[1Ri+Rmf+Ro]

UcAw=AwAs,total[1/1bbbbRi+Rmf+Ro,w]

### Solution Method of Model[LINK]

The complicated equations derived above were implemented in a successive substitution solution procedure to calculate the coil performance based on the input parameters. The MODSIM implementation of a cooling coil, the TYPE12 subroutine, was the motivation for this approach; the method used there has been retained with modifications for the uncontrolled coil model. Clark (1985) contains notes about the MODSIM routine.

In the general case, the cooling coil is only partially wet. For an uncontrolled coil, Equation is used to find the water temperature at the boundary. Several simple equations in the loop adjust the boundary point until the dry surface temperature at the boundary is equal to the dewpoint of the inlet air. For the controlled coil, Equations and give two calculations of the boundary temperature, and the water flow rate and boundary position are adjusted until the two equations agree.

Special cases occur when the coil is all wet or all dry. The coil is solved as if it were all wet before the general case is attempted. If the wet surface temperatures at the coil inlet and outlet are both below the dewpoint, no further solution is required. However, to ensure a continuous solution as flow variables are changed, when the surface is all dry or when it is wet with only the dry surface equations yielding a surface temperature below the dewpoint at the water outlet, the general solution is used to calculate the unknowns. In the solution of the controlled coil the outlet air enthalpy, given some resulting dehumidification, must correspond to the enthalpy at the specified outlet air temperature.

### Application of Cooling Coil Model to Heating Coils[LINK]

The implementation of detailed heating coil models in IBLAST was another important aspect of the system/plant integration. The same kind of loops exist to provide hot water to the heating coils from the boilers as exist to supply the cooling coils with chilled water from the chillers. Some simplifications can be made, however, since the enthalpy change of the air flowing over a heating coil is entirely sensible. There is no condensation in a heating coil. In order to allow heating and cooling coils to be specified using the same geometric parameters, a heating coil simulation was developed from the cooling coil model described above by eliminating the wet surface analysis.

In addition, it was concluded that, since much simpler and less computationally expensive heating coil simulations are possible, an option was provided in IBLAST for a heating coil design using only the UA value of the coil, the product of heat transfer coefficient and coil area. This model was largely based on the TYPE10 subroutine implemented in MODSIM. The equations used to model the performance of the TYPE10 heating coil are as follows:

Ta,out=Ta,in+(Tw,in−Tain)ε(min(Cp,a˙ma,Cp,w˙mw)Cp,a˙ma)Tw,out=Tw,in−(Ta,out−Tain)(Cp,a˙maCp,w˙mw)

where the coil effectiveness is given by:

ε=1−exp⎛⎜
⎜
⎜
⎜⎝{exp[−(min{Cp,a˙ma,Cp,w˙mw}max{Cp,a˙ma,Cp,w˙mw}){NTU}0.78]−1}(min{Cp,a˙ma,Cp,w˙mw}max{Cp,a˙ma,Cp,w˙mw}){NTU}−.22⎞⎟
⎟
⎟
⎟⎠

The parameter NTU is the number of transfer units and is defined as a function of the UA value of the coil as follows:

NTU=UAmin(Cp,a˙ma,Cp,w˙mw)

## Hot-Water-Based Air Heating Coil[LINK]

The input object Coil:Heating:Water provides a model that uses an NTU–effectiveness model of a static heat exchanger. The model is an inlet – outlet model: given the inlet conditions and flow rates and the UA, the effectiveness is calculated using the formula for the effectiveness of a cross-flow heat exchanger with both fluid streams unmixed. The effectiveness then allows the calculation of the outlet conditions from the inlet conditions.

The inputs to the model are: (1) the current inlet temperatures and flow rates of the air and water fluid streams and (2) the UA of the coil. Note that the UA is fixed in this model and is not a function of the flow rates.

There are 2 alternative user inputs for the component: the user may input the design water volumetric flow rate and the UA directly; or the user may choose to input the more familiar design heating capacity plus design inlet & outlet temperatures and let the program calculate the design UA. These alternative user inputs are fully described in the EnergyPlus Input Output Reference document.

### Model Description[LINK]

The air and water capacitance flows are defined as:

˙Cair=cp,air⋅˙mair

˙Cwater=cp,water⋅˙mwater

The minimum and maximum capacity flows are then:

˙Cmin=min(˙Cair,˙Cwater)

˙Cmax=max(˙Cair,˙Cwater)

The capacitance flow ratio is defined as:

Z=˙Cmin/˙Cmax

The number of transfer units (*NTU*) is:

NTU=UA/˙Cmin

The effectiveness is:

ε=1−exp(e−NTU⋅Z⋅η−1Z⋅η)

Where η=NTU−0.22 .

The outlet conditions are then:

Tair,out=Tair,in+ε⋅˙Cmin⋅(Twater,in−Tair,in)/˙Cair

Twater,out=Twater,in−˙Cair⋅(Tair,out−Tair,in)/˙Cwater

The output of the coil in watts is:

˙Qcoil=˙Cwater⋅(Twater,in−Twater,out)

The UA value is recalculated for each timestep. A nominal UA, *UA0*, at the rating point is calculated by the program using the input for rated conditions and a search routine called regula falsi.

User input for the ratio of convective heat transfers at the nominal or rated operating point, “*r*,” is used in the model. This ratio is defined as

r=ηf(hA)air(hA)water

where,

ηf is the fin efficiency, (dimensionless)

*h* is the surface convection heat transfer coefficient

* A* is the surface area

The value calculated for *UA*0 is used with the input for *r* to characterize the convective heat transfer on the water sides at the nominal rating operation point using

((hA)w)0=UA0(r+1r)

and on the air side at the nominal rating point using

(ηf(hA)a)0=r(hA)w,0

Then the following equations are used to calculate a new UA as a function of the flow rates and inlet temperatures at each timestep.

xa=1+4.769⋅10−3(Tair,in−Tair,in,0)

ηf(hA)a=xa(˙ma˙ma,0)0.8(ηf(hA)a)0

xw=1+(0.0141+0.014Twater,in,0)(Twater,in−Twater,in,0)

(hA)w=xw(˙mw˙mw,0)0.85(hA)w,0

UA=(1(hA)w+1ηf(hA)a)−1

The above formulas are from the following reference, along with further references. The equation for xw was modified from that published in Wetter (1999) to correct a small error.

Wetter, M. 1999. Simulation Model: Finned Water-to-Air Coil Without Condensation. LBNL-42355. This document can be downloaded from .

## Single-Speed Electric DX Air Cooling Coil[LINK]

This model (object names Coil:Cooling:DX:SingleSpeed and

Coil:Cooling:DX:TwoStageWithHumidityControlMode, with CoilPerformance:DX:Cooling) simulates the performance of an air-cooled or evaporative-cooled direct expansion (DX) air conditioner. The model uses performance information at rated conditions along with curve fits for variations in total capacity, energy input ratio and part-load fraction to determine the performance of the unit at part-load conditions (Henderson et al. 1992, ASHRAE 1993). Sensible/latent capacity splits are determined by the rated sensible heat ratio (SHR) and the apparatus dewpoint (ADP)/bypass factor (BF) approach. This approach is analogous to the NTU-effectiveness calculations used for sensible-only heat exchanger calculations, extended to a cooling and dehumidifying coil.

This model simulates the thermal performance of the DX cooling coil and the power consumption of the outdoor condensing unit (compressor, fan, crankcase heater and evap condenser water pump). The total amount of heat rejected by the condenser is also calculated and stored for use by other waste heat recovery models (e.g., Coil:Heating:Desuperheater). The performance of the indoor supply air fan varies widely from system to system depending on control strategy (e.g., constant fan vs. AUTO fan, constant air volume vs. variable air volume, etc.), fan type, fan motor efficiency and pressure losses through the air distribution system. Therefore, this DX system model does not account for the thermal effects or electric power consumption of the indoor supply air fan. EnergyPlus contains separate models for simulating the performance of various indoor fan configurations, and these models can be easily linked with the DX system model described here to simulate the entire DX air conditioner being considered (e.g., see AirLoopHVAC:Unitary:Furnace:HeatCool, AirLoopHVAC:UnitaryHeatCool, ZoneHVAC:WindowAirConditioner or AirLoopHVAC:UnitaryHeatPump:AirToAir).

### Model Description[LINK]

The user must input the total cooling capacity, sensible heat ratio (SHR), coefficient of performance (COP) and the volumetric air flow rate across the cooling coil at rated conditions. The capacity, SHR and COP inputs should be “gross” values, excluding any thermal or energy impacts due to the indoor supply air fan. The rated conditions are considered to be air entering the cooling coil at 26.7°C drybulb/19.4°C wetbulb and air entering the outdoor condenser coil at 35°C drybulb/23.9°C wetbulb. The rated volumetric air flow should be between 0.00004027 m3/s and 0.00006041 m3/s per watt of rated total cooling capacity (300 – 450 cfm/ton). The rated volumetric air flow to total cooling capacity ratio for 100% dedicated outdoor air (DOAS) application DX cooling coils should be between 0.00001677 (m3/s)/W (125 cfm/ton) and 0.00003355 (m3/s)/W (250 cfm/ton).

The user must also input five performance curves or performance tables that describe the change in total cooling capacity and efficiency at part-load conditions:

1) Total cooling capacity modifier curve or table (function of temperature)

2) Total cooling capacity modifier curve or table (function of flow fraction)

3) Energy input ratio (EIR) modifier curve or table (function of temperature)

4) Energy input ratio (EIR) modifier curve or table (function of flow fraction)

5) Part load fraction correlation curve or table (function of part load ratio)

- The total cooling capacity modifier curve (function of temperature) is a curve with two independent variables: wet-bulb temperature of the air entering the cooling coil, and dry-bulb temperature of the air entering the air-cooled condenser coil (wet-bulb temperature if modeling an evaporative-cooled condenser). The output of this curve is multiplied by the rated total cooling capacity to give the total cooling capacity at the specific entering air temperatures at which the DX coil unit is operating (i.e., at temperatures different from the rating point temperatures). This curve is typically a biquadratic but any curve or table with two independent variables can be used.

Note: The data used to develop the total cooling capacity modifier curve (function of temperature) should represent performance when the cooling coil is ‘wet’ (i.e., coil providing sensible cooling and at least some dehumidification). Performance data when the cooling coil is ‘dry’ (i.e., not providing any dehumidification) should **not** be included when developing this modifier curve. This model automatically detects and adjusts for ‘dry coil’ conditions (see section “Dry Coil Conditions” below).

TotCapTempModFac=Func(Twb,i,Tc,i)

where

Twb,i = x values = wet-bulb temperature of the air entering the cooling coil, °C

Tc,i = y values = dry-bulb temperature of the air entering an air-cooled condenser or wet-bulb temperature of the air entering an evaporative-cooled condenser, °C

* The total cooling capacity modifier curve (function of flow fraction) is a curve with one independent variable being the ratio of the actual air flow rate across the cooling coil to the rated air flow rate (i.e., fraction of full load flow). The output of this curve is multiplied by the rated total cooling capacity and the total cooling capacity modifier curve (function of temperature) to give the total cooling capacity at the specific temperature and air flow conditions at which the DX unit is operating. This curve is typically a quadratic but any curve or table with one independent variable can be used.

TotCapFlowModFac=Func(ff)

where

ff=flow fraction=(Actual air mass flow rateRated air mass flow rate)=x value

**Note:** The actual volumetric air flow rate through the cooling coil for any simulation time step where the DX unit is operating must be between 0.00002684 m3/s and .00006713 m3/s per watt of rated total cooling capacity (200 - 500 cfm/ton). The simulation will issue a warning message if this air flow range is exceeded.

- The energy input ratio (EIR) modifier curve (function of temperature) is a curve with two independent variables: wet-bulb temperature of the air entering the cooling coil, and dry-bulb temperature of the air entering the air-cooled condenser coil (wet-bulb temperature if modeling an evaporative-cooled condenser). The output of this curve is multiplied by the rated EIR (inverse of the rated COP) to give the EIR at the specific entering air temperatures at which the DX coil unit is operating (i.e., at temperatures different from the rating point temperatures). This curve is typically a biquadratic but any curve or table with two independent variables can be used.

Note: The data used to develop the energy input ratio (EIR) modifier curve (function of temperature) should represent performance when the cooling coil is ‘wet’ (i.e., coil providing sensible cooling and at least some dehumidification). Performance data when the cooling coil is ‘dry’ (i.e., not providing any dehumidification) should **not** be included when developing this modifier curve. This model automatically detects and adjusts for ‘dry coil’ conditions (see section “Dry Coil Conditions” below).

EIRTempModFac=Func(Twb,i,Tc,i)

where

Twb,i = x values = wet-bulb temperature of the air entering the cooling coil, °C

Tc,i = y values = dry-bulb temperature of the air entering an air-cooled condenser or wet-bulb temperature of the air entering an evaporative-cooled condenser, °C

- The energy input ratio (EIR) modifier curve (function of flow fraction) is a curve with one independent variable being the ratio of the actual air flow rate across the cooling coil to the rated air flow rate (i.e., fraction of full load flow). The output of this curve is multiplied by the rated EIR (inverse of the rated COP) and the EIR modifier curve (function of temperature) to give the EIR at the specific temperature and air flow conditions at which the DX unit is operating. This curve is typically a quadratic but any curve or table with one independent variable can be used.

EIRFlowModFrac=Func(ff)

where

ff=flow fraction=(Actual air mass flow rateRated air mass flow rate)=x value

- The part load fraction correlation (function of part load ratio) is a curve with one independent variable being part load ratio (sensible cooling load / steady-state sensible cooling capacity). The output of this curve is used in combination with the rated EIR and EIR modifier curves to give the “effective” EIR for a given simulation time step. The part load fraction (PLF) correlation accounts for efficiency losses due to compressor cycling. This curve is typically a linear, quadratic, or cubic but any curve or table with one independent variable can be used.

PartLoadFrac=PLF=Func(PLR)

where

PLR=part load ratio=(actual sensible cooling loadsteady-state sensible cooling load)=x values

The part-load fraction correlation should be normalized to a value of 1.0 when the part load ratio equals 1.0 (i.e., no efficiency losses when the compressor(s) run continuously for the simulation time step). For PLR values between 0 and 1 (0 < = PLR < 1), the following rules apply:

PLF≥0.7PLF≥PLR

If PLF < 0.7 a warning message is issued, the program resets the PLF value to 0.7, and the simulation proceeds. The runtime fraction of the coil is defined as PLR/PLF. If PLF < PLR, then a warning message is issued and the runtime fraction of the coil is limited to 1.0.

A typical part load fraction correlation for a conventional, single-speed DX cooling coil (e.g., residential or small commercial unit) would be:

PLF = 0.85 + 0.15(PLR)

All five part-load curves are accessed through EnergyPlus’ built-in performance curve equation manager (curve: quadratic, curve:cubic and curve:biquadratic). It is not imperative that the user utilize all coefficients shown in equations (449) through (453) if their performance equation has fewer terms (e.g., if the user’s PartLoadFrac performance curve is linear instead of quadratic, simply enter the values for a and b, and set coefficient c equal to zero).

For any simulation time step, the total (gross) cooling capacity of the DX unit is calculated as follows:

## Coils[LINK]

## Chilled-Water-Based Air Cooling Coil[LINK]

The input object Coil:Cooling:Water is simpler than the detailed geometry model. The simple model provides a good prediction of the air and water outlet conditions without requiring the detailed geometric input required for the detailed model. A greatly simplified schematic of enthalpy and temperature conditions in a counter flow cooling/dehumidifying coil is shown in the schematic Figure. The input required to model the coil includes only a set of thermodynamic design inputs, which require no specific manufacturer’s data. The coil simulation model is essentially a modification of one presented by Elmahdy and Mitalas (1977), TRNSYS, 1990 and Threlkeld, J.L. 1970. The model calculates the UA values required for a Dry, Wet and Part Wet & Part Dry Coil and iterates between the Dry and Wet Coil to output the fraction wet. There are two modes of flow operation for this model: Cross Flow, which is widely applicable in HVAC systems and the second being Counter flow mode. The default value in program is set up for Counter Flow. In addition the coil has two modes of analysis: Simple Analysis and Detailed Analysis. The Simple analysis mode operates the coil as either wet or dry while the detailed mode simulates the coil as part wet part-dry. While the detailed mode provides more accurate results, it is significantly slower than the simple model. The simple mode gives good results for an annual simulation but will not be adequate for a time step performance analysis.

Simplified Schematic of Cooling/Dehumidifying Coil

## Heat Transfer and Energy Balance[LINK]

The cooling coil may be completely dry, completely wet with condensation, or it may have wet and dry sections. The actual condition of the coil surface depends on the humidity and temperature of the air passing over the coil and the coil surface temperature. The part-dry part-wet case represents the most general scenario for the coil surface conditions. There are subroutines present in the model for both the dry and wet regions of the coil, and a subroutine that iterates between the dry and wet subroutines to calculate the fraction of the coil surface that is wet. For each region the heat transfer rate from air to water may be defined by the rate of enthalpy change in the air and in the water. The rates must balance between each medium for energy to be conserved.

## Model Description[LINK]

The Model has two blocks: 1st = Design Block with the Design Inputs. This block calculates the Design U-Factor Times Area Value (UA) values required by the model. Using these UA values the model simulates the operating conditions. The operating block is the one containing the operating conditions, the conditions at which the coil operates. Following is the list of Design and Operating inputs and subsequently the Design and Operating variables used in the model.

## Intermediate calculated U-Factor Times Area Values: The Crux of the Model[LINK]

The various U-Factor Times Area values (UA) required by this model are calculated from the above inputs, which are explained later in the document. The various UA are:

The UA values are calculated assuming a wet coil at the design conditions. Following are a few important calculations to understand the working of the model. The model is basically divided into two blocks: the Design Block and the Operating Block.

The Design Block is a one time calculation. The aim of the Design Block is to calculate the Coil UA for use in the operating Block.

## Design Block Calculations:[LINK]

The design block has the code for calculating the six Coil UA values required by the operating block. Reasonable assumptions have been made in the calculations to maintain the simplicity of the model.

Heat transfer in a wet coil model is based on enthalpy rather than temperature to take into account latent effects. While heat transfer rates are commonly expressed as the product of an overall heat transfer coefficient, UA, and a temperature difference, the use of enthalpy-based heat transfer calculations requires an enthalpy-based heat transfer coefficient which we denote as DesUACoilTotalEnth and hence the equation.

Q = DesUACoilTotalEnth * (Hair,mean - Hwater,mean). The value of Q is calculated using product of air mass flow rate and difference in inlet and outlet air enthalpies at design conditions.

The relation between the enthalpy-based UA and the temperature-based UA is DesUACoilTotalEnth = CoilUA / Cp. CoilUA is the conventional heat transfer coefficient and Cp = specific heat of the air.

We need the following quantities for our design calculations. The

Psyfunctions are the EnergyPlus built-in psychrometric functions.˙mair=ρair˙Vair

hair,in=PsyHFnTdbW(Tair,in,wair,in)

hair,out=PsyHFnTdbW(Tair,out,wout)

hw,sat,in=PsyHFnTdbW(Tw,in,PsyWFnTdpPb(Tw,in,Patm))

˙Qcoil=˙mair(hair,in−hair,out)

Tw,out=Tw,in+˙Qcoil/(˙mw,maxCp,w)

hw,sat,out=PsyHFnTdbW(Tw,out,PsyWFnTdpPb(Tw,out,Patm))

We now calculate the design coil bypass factor. The bypass factor is not used in subsequent calculations. It is calculated solely to use as check on the reasonableness of the user-input design inlet and outlet conditions. First we make an initial estimate of the apparatus dew point temperature:

Tair,dp,app=PsyTdpFnWPb(wair,out,Patm)

we also need the “slope” of temperature versus humidity ratio on the psych chart between the inlet and outlet air conditions:

ST,w=(Tair,in−Tair,out)/(wair,in−wair,out)

We now obtain the actual design apparatus dewpoint temperature by iterating over the following two equations:

wair,dp,app=PsyWFnTdpPb(Tair,dp,app,Patm)

Tair,dp,app=Tair,in−ST,w(wair,in−wair,dp,app)

The apparatus dewpoint enthalpy is then:

hair,dp,app=PsyHFnTdbW(Tair,dp,app,wair,dp,app)

and the coil bypass factor is:

Fcoilbypass=(hair,out−hair,dp,app)/(hair,in−hair,dp,app)

If the iterative procedure doesn’t converge, or the coil bypass factor is too large (greater than 0.5), or the apparatus dewpoint enthalpy is less than the saturated air enthalpy at the water inlet temperature, the design outlet air conditions are reset to 90% relative humidity at the same outlet enthalpy. The above design calculations are then repeated.

We are now ready to calculate the design coil UA. This will be accomplished by inverting the simple coil calculation routine

CoolingCoilusing the Regula Falsi method. First we make an initial estimate of the coil UA.Δhlmd=((hair,in−hw,sat,out)−(hair,out−hw,sat,in))/log((hair,in−hw,sat,out)−(hair,out−hw,sat,in))

UAcoil,enthalpy−based=˙Qcoil/Δhlmd

UAcoil,ext=Cp,airUAcoil,enthalpy−based

We set the internal UA to 3.3 times the external UA (as a typical value for a coil). Then the total UA is:

UAcoil,tot=1(1/UAcoil,int+1/UAcoil,ext)

The next step is to estimate the coil external heat transfer surface area. This is done in the function

EstimateHEXSurfaceArea:Areacoil,ext=EstimateHEXSurfaceArea

using the following assumptions:

Tube inside diameter = 0.0122 (m)

Tube side water velocity = 2.0 (m/s)

Inside to outside coil surface area ratio (Ai/Ao) = 0.07 (-)

Fins overall efficiency = 0.92 (-)

Aluminum fins, 12 fins per inch with fins to total outside surface area ratio of 90%.

Airside combined heat and mass transfer coefficient = 140 (W/m2∙°C)

Interior and exterior U values (really UA’s per unit exterior surface area) are calculated by dividing the above UA’s by the area. The resulting Ucoil,ext is assumed to be Ucoil,ext,wet; Ucoil,ext,dry is set equal to Ucoil,ext,wet. We now have all the starting values needed for inverting the simple coil model using the chosen Regula Falsi iterative method. Once the iteration is completed, we have coil UA’s and U’s that yield the design outlet air and water enthalpies given the inlet design conditions and flow rates. Note that the simple coil model can not exactly match the specified design outlet air temperature and humidity ratio. It can only match the design air outlet enthalpy. Generally the simple coil model will yield outlet conditions near the saturation curve if any dehumidification is occuring. Typical outlet relative humidities are around 95%.

## Variable UA[LINK]

The above calculations yield coil UA’s for the design inlet conditions and air and water flow rates. As the flow rates vary during the time step calculations, the UA’s need to be adjusted, since coil UA’s are a rather strong function of air and water side flow rates. Each time step the coil UA’s are modified using the same formulas as are used in the hot water coil model. Refer to that model for the flow dependences.

## Operating Block Calculations:[LINK]

There are two modes of coil analysis in the operating block. They are the Simple analysis mode and the detailed analysis mode. The simple analysis mode assumes the coil to be either all wet or either all dry and execute the model , on the other hand the detailed mode checks for part wet part dry mode of operation and reports surface area wet fraction of coil, however the program execution time in detailed mode is noticeably higher.

The operating block for Detailed Mode Analysis of this coil model is divided into three modes of coil performance. The modes being

Coil is completely dry: There is no moisture condensation on the coil surface and the coil is a dry coil. This is an extreme condition when the entering air has very low humidity ratio or is dry air.

Coil is completely wet: The entire coil is wet due to complete condensation on the surface of the coil.

Part Wet Part Dry Mode: This is the usual/frequent mode of operation of coil, as shown in Figure, where part of the coil at entry of air is dry and as air cools condensation occurs and part of the coil becomes wet.

The Part Wet Part Dry Mode of operation is essentially a function the Coil Completely Dry and Coil Completely Wet mode. This subroutine iterates between the Dry Coil and the Wet Coil to give outputs, a detailed explanation is given later in the document. The operating block requires 5 inputs, which are mentioned earlier in the document. These inputs are automatically generated from the node connections in Energy Plus. The user does not have to input any information to run this coil model.

The option to identify which mode of operation the coil should perform ie, for a given set of inputs would the coil be Dry, Wet or Part Wet Part Dry, is decided by set of conditions described below.

IF(Temperature Dewpoint Air < Water Inlet Temperature)THENthe coil is Dry and we call the Subroutine Coil Completely Dry. In this case outlet temperature of air would be higher than the air dewpoint and hence there would be no condensation.IF(Temperature Dewpoint Air > Water Inlet Temperature)THENthe coil is completely wet, call subroutine Coil Completely Wet, it is assumed that moisture condensation occurs over completely surface of the coil. However we go ahead and check for the coil being partially wet with the following condition.IF (AirDewPointTemp < AirInletCoilSurfTemp) THEN,the coil is Partially Wet because there is possibility that air temperature will go below its dewpoint and moisture will condense on latter part of the cooling coil.The Operating Block for Simple Mode Analysis is divided into two modes of coil performance, the two modes being

Coil is completely dry: There is no moisture condensation on the coil surface and the coil is a dry coil.

Coil is completely wet: The entire coil is wet due to complete condensation on the surface of the coil.

The option to identify which mode of operation the Simple mode analysis should perform ie, for a given set of inputs would the coil be Dry or Wet is decided by set of conditions described below.

IF(Temperature Dewpoint Air < Water Inlet Temperature)THENthe coil is Dry and we call the Subroutine Coil Completely Dry. In this case outlet temperature of air would be higher than the air dewpoint and hence there would be no condensation.IF(Temperature Dewpoint Air > Water Inlet Temperature)THENthe coil is completely wet, call subroutine Coil Completely Wet, it is assumed that moisture condensation occurs over completely surface of the coil. However we go ahead and check for the coil being partially wet with the following condition.The above is a simple mode of analysis and the results are very slightly different from the detailed mode of analysis. The algorithms used in Simple mode and the Detailed mode are identically similar. The surface area wet fraction in the coil is reported as 1.0 or 0.0 for wet or dry coil respectively. The program defaults to simple mode of analysis for enabling higher execution speed.

## Effectiveness Equations:[LINK]

There are two modes of flow for the coil, Counter Flow mode or the Cross Flow mode, default set up is as cross flow since most air condition applications have cross flow heat exchangers. According to the mode of flow the following NTU - Effectiveness relationships are used to calculate coil effectiveness, which is used later by all the three modes (Dry, Wet, Part Wet) for calculating air outlet conditions and heat transfer.

Following are the relations used for calculating effectiveness equation for the Heat exchangers.

Counter Flow Heat Exchanger: Effectiveness Equation:

ηCounterFlow=(1−Exp(−NTU×(1−RatioStreamCapacity)))1−RatioStreamCapacity×Exp(−NTU×(1−RatioStreamCapacity))

In Equation the variable Ratio_StreamCapacity is defined as below

RatioStreamCapacity=MinCapacityStreamMaxCapacityStream

In equation capacity of stream is defined as below in equation

(Min,Max)CapacityStream=(MassFlowRate×Cp)air,water

NTU in equation , is defined as the Number of Transfer Units, it is a function of Coil UA and the Minimum Capacity of Stream. The Coil UA is a variable in this equation and depends on which mode of the coil operation (Dry, Wet, Part Wet) is calling upon equation , i.e., if it is Coil Completely Dry calling upon the effectiveness equation with the value of Dry UA total, which in our case is defined as CoilUA_total. Equation gives definition for NTU.

NTU=CoilUAMinStreamCapacity

Cross Flow Heat Exchanger: Effectiveness Equation:

ηCrossFlow=1−EXP{Exp(−NTU×RatioStreamCapacity×NTU−0.22)−1RatioStreamCapacity×NTU−0.22}

The variables in the above equation have already been defined earlier. Depending on the mode of operation of the coil model the cross or the counter flow equations are used to calculate the effectiveness.

## Coil Outlet Conditions:[LINK]

Calculating the Outlet Stream Conditions using the effectiveness value from equation or depending on the mode of flow. The energy difference between the outlet and inlet stream conditions gives the amount of heat transfer that has actually take place. Temperature of air and water at outlet to the coil is given as in following equations

TempAirOut=TempAirinlet−ηcross,counter×MaxHeatTransferStreamCapacityAir

TempWaterOut=TempWaterInlet+ηCross,counter×MaxHeatTransferStreamCapacityWater

In the above equations and the maximum heat transfer is calculated as shown in the following equation

MaxHeatTransfer=MinStreamCapacity×(TempAirInlet−TempWaterInlet)

## Coil Completely Dry Calculations: (operating block)[LINK]

Since the coil is dry, the sensible load is equal to total load and the same with the humidity ratios at inlet and outlet, as in equations and .

QSensibleDryCoil=QTotalDryCoil

HumRatioInlet=HumRatioOutlet

Total Heat Transfer in dry coil is as follows:

QTotalDryCoil=CapacityAir×(AirTempIn−AirTempOutlet)

The variables in the above equation are calculated earlier in equations and to give the total cooling load on the coil.

## Coil Completely Wet Calculations: (operating block)[LINK]

In wet coil we need to account for latent heat transfer, hence calculations are done using enthalpy of air and water instead of stream temperatures Hence we need to define coil UA for the wet coil based on enthalpy of the operating streams and not design streams.

Similar to equations and we calculate the air outlet enthalpy and water outlet enthalpy ie by replacing temperature with enthalpy of the respective streams. The input variable for Coil UA in equation for calculating NTU, in this case it would be enthalpy based and is given as shown in equation

CoilUAEnthalpyBased=1(CpSatIntermediateCoilUAInternal+CpAirCoilUAExternal)

Total Coil Load in case of Wet Coil is the product of mass flow rate of air and enthalpy difference between the inlet and outlet streams as given in the following equation

QTotal=∗Mair×(EnthAirInlet−EnthAirOutlet)

Once the enthalpy is known the outlet temperatures and outlet humidity ratios of the wet coil are calculated as in equations below.

IF (TempCondensation < PsyTdpFnWPb(InletAirHumRat ,Patm)) THENAirTempOut=AirTempinlet−(AirTempinlet−CondensationTemp)×η

and

OutletAirHumdityRatio = PsyWFnTdbH(OutletAirTemp,EnthAirOutlet)ELSEThere is no condensation and hence the inlet and outlet Hum Ratios are equal , and outlet temperature is a function of outlet air enthalpy as below

OutletAirTemp = PsyTdbFnHW (EnthalpyAirOutlet, OutletAirHumRat)andOutletAirHumRat = InletAirHumRatENDIFEffectiveness η used in equation is defined in equation and Condensation Temperature is calculated using psychrometric function as in equation .

η=1−Exp{−CoilUAExternalCapacitanceAir}

CondensationTemp=PsyTsatFnHPb(EnthAirCondensateTemp,Patm)

EnthAirCondensateTemp=EnthAirInlet−(EnthAirInlet−EnthAirOutlet)η

Once the air outlet temperature are known, then sensible load is calculated as a product of capacitance of air and temperature difference at inlet and outlet, as in equation

QSensible=CapacitanceAir×(AirTempInlet−AirTempOutlet)

## Coil Part Wet Part Dry Calculations: (operating block)[LINK]

The Coil would perform under part wet part dry conditions when Air Dewpoint Temperature is less than Coil surface temperature at inlet to air. In this case part of the coil used value of Dry UA for heat transfer and part the coil used Wet UA value for heat transfer.

This problem is solved utilizing the fact that the Exit conditions from the Dry Part of the Coil would become the inlet conditions to the wet part of the coil (see Figure) and the coil model determines by iteration what fraction of the coil is wet and based on that it calculates the areas and subsequently the UA values of that dry and wet part, based on the area of the dry and wet part respectively. Explained below are the steps followed to the estimating the wet dry behavior of the coil.

WetDryInterfaceWaterTemp=WaterTempInlet+AreaWetFraction∗(WaterTempOutlet−WaterTempInlet)

The value of Surface Area Wet fraction is estimated initially as follows

AreaWetFractionEstimate=AirDewPtTemp−InletWaterTempOutletWaterTemp−InletWaterTemp

For the above mentioned iteration the value of Coil UA for Wet and Dry part need to be varied according to the new respective area of the wet and dry parts. This estimate of Wet and Dry area is a product of the estimated Surface Area Fraction and total coil external area, which keeps varying as will be explained further in the document.

UA value for Dry part of the Coil is estimated as below.

CoilUADryExternal=SurfAreaDry1CoilUDryExternal+1CoilUInternal

Where Surface Area Dry = (Total Coil Area – Wet Part Area), where the Wet part area is the product of Surface fraction Wet and Total Coil Area.

UA value for the Wet part of the Coil requires Wet UA external and Wet UA Internal, which are calculated as below.

WetPartUAExternal=CoilUWetExternal×SurfaceAreaWet

WetPartUAInternal=CoilUInternal×SurfaceAreaWet

It is essential to remember that the mode of calculation for the coils remains the same as in completely wet and completely dry mode, only the UA values and water, air outlet and inlet values change.

Now Iterate between the Dry Coil and wet Coil with the above respective UA, and usual operating inputs except the variable water inlet temperature for dry Coil is replaced with Wet Dry Interface Water temperature, and in the Wet Coil the Outlet Air Temperature from dry Coil is the inlet air temperature to Wet Coil. The iteration proceeds till the Outlet Water Temperature from Wet Coil equals the Wet Dry Interface Water Temp, which is the input to Dry Coil.

Dry Part Inputs: (changed operating inputs) :Iteration Case 1: Explained In Programming Fashion:

CALL CoilCompletelyDry (WetDryInterfcWaterTemp, InletAirTemp, DryCoilUA, &OutletWaterTemp, WetDryInterfcAirTemp, WetDryInterfcHumRat, &DryCoilHeatTranfer).Input the calculated values calculated by Dry Coil above into Wet Coil below. The variables have been highlighted in color red and blue.

CALLCoilCompletelyWet (InletWaterTemp,WetDryInterfcAirTemp,WetDryInterfcHumRatWetPartUAInternal,WetPartUAExternal, &

EstimateWetDryInterfcWaterTemp, OutletAirTemp, OutletAirHumRat, &WetCoilTotalHeatTransfer, WetCoilSensibleHeatTransfer, &

EstimateSurfAreaWetFraction, WetDryInterfcSurfTemp)

Iterate Between the above two Wet and Dry Coil calls until the two variables in blue ie

WetDryInterfcWaterTemp = EstimateWetDryInterfcWaterTemp.The key is to have the difference between the variables (WetDryInterfcWaterTemp – OutletWaterTemp) in Dry Coil equal to (InletWaterTemp-EstimatedWetDryInterfcWaterTemp) in Wet Coil. This equality quantized the relative part of coil that is dry and part that is wet on the basis of heat transfer that has occurred.After the above convergence check for the coil being dry otherwise iterate to calculate surface fraction area wet.

IF{(AreaFractionWet≤0.0)and(WetDryInterfaceSurfTemp>AirDewPt)}

THEN CoilCompletelyDryIf equation is satisfied then Coil is Dry and simply output the value for Dry Coil calculated else the coil is partially wet and then iterate to find the surface fraction area wet. Start with the initially guess value of surface area fraction (equation wet and iterate on the entire loop starting from until the Wet Dry Interface Temperature equals the Air Dewpoint Temperature. The value of Surface Area fraction wet at which the interface air temperature equals is dewpoint is the transition point from wet to dry and gives the % of coil that is dry and % that is wet.

Graphs Showing the Performance of the coil model at optimum operating conditions are shown below. All values of variable used have been normalized.

Air Outlet Temperature Vs Air Mass Flow Rate

Sensible Load variations Vs Air mass Flow Rate

Total and Sensible Load variations Vs Air Mass Flow Rate

Surface Area Fraction Wet Vs Air Mass Flow Rate

## References[LINK]

IBPSA BuildSim-2004. 2004. Colarado Boulder: An Improvement of Ashrae Secondary HVAC toolkit Simple Cooling Coil Model for Building Simulation, Rahul J Chillar, Richard J Liesen M&IE ,UIUC.

Stoecker, W.F. <dates unspecified> Design of Thermal Systems,: ME 423 Class Notes , M& IE Dept UIUC.

Brandemeuhl, M. J. 1993. HVAC2 Toolkit: Algorithms and Subroutines for Secondary HVAC Systems Energy Calculations, ASHRAE.

Elmahdy, A.H. and Mitalas, G.P. 1977. “A Simple Model for Cooling and Dehumidifying Coils for Use In Calculating Energy Requirements for Buildings ASHRAE Transactions, Vol.83 Part 2, pp. 103-117.

Threlkeld, J.L. 1970. Thermal Environmental Engineering, 2nd Edition, Englewood Cliffs: Prentice-Hall,Inc. pp. 254-270.

ASHRAE Secondary HVAC Toolkit TRNSYS. 1990. A Transient System Simulation Program: Reference Manual. Solar Energy Laboratory, Univ. Wisconsin-Madison, pp. 4.6.8-1 - 4.6.8-12.

Kays, W.M. and A.L. London. 1964. Compact Heat Exchangers, 2nd Edition, New York: McGraw-Hill.

Clark, D.R.. 1985. HVACSIM+ Building Systems and Equipment Simulation Program Reference Manual, Pub. No. NBSIR 84-2996, National Bureau of Standards, U.S. Department of Commerce, January, 1985

Elmahdy, A.H. 1975. Analytical and Experimental Multi-Row Finned-Tube Heat Exchanger Performance During Cooling and Dehumidifying Processes, Ph.D. Thesis, Carleton University, Ottawa, Canada, December, 1975.

Elmahdy, A.H., and Mitalas, G.P. 1977. “A Simple Model for Cooling and Dehumidifying Coils for Use in Calculating Energy Requirements for Buildings,” ASHRAE Transactions, Vol. 83, Part 2, pp. 103-117.

## Chilled-Water-Based Detailed Geometry Air Cooling Coil[LINK]

The input object Coil:Cooling:Water:DetailedGeometry provides a coil model that predicts changes in air and water flow variables across the coil based on the coil geometry. A greatly simplified schematic of enthalpy and temperature conditions in a counterflow cooling/dehumidifying coil is shown in the following schematic figure. In addition, the variables required to model a cooling/dehumidifying coils and their definitions are extensively listed in “Table. Coil Geometry and Flow Variables for Coils”. The input required to model the coil includes a complete geometric description that, in most cases, should be derivable from specific manufacturer’s data. The coil simulation model is essentially the one presented by Elmahdy and Mitalas (1977) and implemented in HVACSIM+ (Clark 1985), a modular program also designed for energy analysis of building systems. The model solves the equations for the dry and wet sections of the coil using log mean temperature and log mean enthalpy differences between the liquid and the air streams. Elmahdy and Mitalas state that crossflow counterflow coils with at four rows or more are approximated well by this model. This does not constitute a major limitation since cooling and dehumidifying coils typically have more than four rows.

Simplified Schematic of Cooling/Dehumidifying Coil

## Heat Transfer and Energy Balance[LINK]

The cooling coil may be completely dry, completely wet with condensation, or it may have wet and dry sections. The actual condition of the coil surface depends on the humidity and temperature of the air passing over the coil and the coil surface temperature. The partly wet-partly dry case represents the most general scenario for the coil surface conditions. The all dry and all wet cases can be considered as limiting solutions of the wet or dry areas respectively going to zero. In the general case, equations are written for both the dry and wet regions of the coil. For each region the heat transfer rate from air to water may be defined by the rate of enthalpy change in the air and in the water. The rates must balance between each medium for energy to be conserved. Equations through express the energy balance between the water and the air for the case of dry and wet coils respectively. Equations and represent the heat transfer rate between water and air based on the actual performance of the coil. The UA parameter can be calculated from the parameters in the following table.

Equations through represent two sets of three equations with 7 unknowns: ˙Qd , Ta,1, Ta,2, Tw,2, Tw,3, ˙ma , ˙mw . However, normally at least four of these variables are specified, for example: inlet water temperature, outlet air temperature, water flow rate, air flow rate, so that the system of equations is effectively closed.

˙Qd=maCpa(Ta,1−Ta,2)

˙Qd=mwCpw(Tw,3−Tw,2)

˙Qd=(UAdry)(LMTD)

˙Qw=ma(Ha,2−Ha,3)

˙Qw=mwCpw(Tw,2−Tw,1)

˙Qw=(UcAw)(LMHD)

In order to manipulate these equations, the log mean temperature and enthalpy differences are expanded as shown in Equations and . Finally, a linear approximation of the enthalpy of saturated air over the range of surface temperature is made using Equation . Note that in Equation Hw refers to the enthalpy of saturated air at the water temperature.

LMTD=(Ta,1−Tw,3)−(Ta,2−Tw,2)lnTa,1−Tw,3Ta,2−Tw,2

LMHD=(Ha,2−Hw,2)−(Ha,3−Hw,1)lnHa,2−Hw,2Ha,3−Hw,1

Hw=aa+bbTw

Equation is derived from the above equations and is used to solve for the coil conditions when all of the inlet conditions are given as input. Operating in this manner, the coil does not have a controlled outlet air temperature.

Tw,2=(1−Z)(Ha,1−aa−K1CpaTa,1)+ZTw,1(bb−mwCpwma)bb−ZmwCpwma−(1−Z)K1Cpa

An alternative solution method is to define the coil leaving air temperature as an input with a variable water flow rate. In this case Equations and are more convenient. Equations through define terms that are used to simplify Equations , and .

Tw,2=(1−Z)(Ha,3−aa)+Tw,1(mwCpwma−bbZ)mwCpwma−bb

Tw,2=(Zd−1)Ta1Cpa+Tw,3(Cpa−ZdmwCpwma)Zd(Cpa−mwCpwma)

Z=exp(UcAw(1ma−bbmwCpw))

K1=Zd−1Zd−maCpamwCpw

Zd=exp(UcAdry(1maCpa−1mwCpw))

## Underlying Correlations, Properties, and Assumptions[LINK]

Overall heat transfer coefficients are calculated from the specified coil geometry and by using empirical correlations from fluid mechanics and heat transfer. For the water side, Equation gives the film heat transfer coefficient in SI units:

fi=1.429(1+0.0146Tw)V0.8wD−0.2i

This is valid for Reynolds numbers greater than 3100 based on water flow velocity and pipe inside diameter and is given in Elmahdy and Mitalas(1977) as recommended in the standard issued by the Air-Conditioning and Refrigeration Institute (1972) for air-cooling coils. The definition of overall inside thermal resistance follows directly as shown in Equation.

Ri=1fiAi

Equation gives the film coefficient for the air side. Another form of the same equation is Equation , which is familiar from the data presented in Kays and London (1984). For coil sections that have a wet surface due to condensation, the air side film coefficient is modified according to Equation . The correction term, a function of air Reynolds number, is valid for Reynolds numbers between 400 and 1500. The coefficients in Equation and are calculated by Equations and that are functions of the coil geometry. Elmahdy (1977)explains the modifier for the wet surface and coefficients for the film coefficient. Equations through show definitions and values of common parameters and properties.

fo=C1ReC2amaAaminflowCp,aPr2/3a

C1ReC2a=StaPr2/3a

fo,w=fo(1.425−5.1×10−4Rea+2.63×10−7Re2a)

C1=0.159(δfinDhdr)−0.065(δfinLfin)0.141

C2=−0.323(ΔfinsLfin)0.049(DfinΔtuberows)0.549(δfinΔfins)−0.028

Dhdr=4AaminflowδcoilAstotal

Rea=4δcoil(1+wa)maAstotalμa

Pra=0.733

μa=1.846×10−5

The film coefficients above act on the extended surface of the air side, that is the area of the fins and the tubes. Therefore, the fin efficiency must also be considered in calculating the overall thermal resistance on the outside. Gardner(1945) gives the derivation of Equation , used as a curve fit to find the fin efficiency as a function of film coefficient. This equation is based on circular fins of constant thickness. To model a coil with flat fins, an effective diameter – that of circular fins with the same fin area – is used. Equations through define variables used in Equation . The overall efficiency of the surface is shown by Equation . Note that the efficiency is found by the same equations for the wet surface using the wet surface film coefficient.

ηfin=−2ρfai(1+ρ)[I1(ub)K1(ue)−K1(ub)I1(ue)I0(ub)K1(ue)+K0(ub)I1(ue)]

fai=(Dfin−Dtube)2√2fokfinδfin

ρ=DtubeDfin

ue=fai1−ρ

ub=ueρ

ηo=1−(1−ηfin)AfinsAstotal

The definition of overall outside thermal resistance is given in Equation as a function of fin efficiency and film coefficient. For a wet coil surface the resistance must be defined differently because the heat transfer equations are based on enthalpy rather than temperature differences, as shown in Equation .

Ro=1foηoAs,total

Ro,w=Cp,a/bbfo,wηo,wAs,total

Equation gives the last two overall components of thermal resistance. They represent the metal tube wall and internal fouling. The fouling factor, due to deposits of dirt and corrosion of the tube inside surfaces, is assumed to be 5x10-5 m2·K/W. All components of thermal resistance are added in series to produce the overall heat transfer coefficients shown in Equations and .

Rmf=δtubektubeAi+FlAi

UAdry=AdryAs,total[1Ri+Rmf+Ro]

UcAw=AwAs,total[1/1bbbbRi+Rmf+Ro,w]

## Solution Method of Model[LINK]

The complicated equations derived above were implemented in a successive substitution solution procedure to calculate the coil performance based on the input parameters. The MODSIM implementation of a cooling coil, the TYPE12 subroutine, was the motivation for this approach; the method used there has been retained with modifications for the uncontrolled coil model. Clark (1985) contains notes about the MODSIM routine.

In the general case, the cooling coil is only partially wet. For an uncontrolled coil, Equation is used to find the water temperature at the boundary. Several simple equations in the loop adjust the boundary point until the dry surface temperature at the boundary is equal to the dewpoint of the inlet air. For the controlled coil, Equations and give two calculations of the boundary temperature, and the water flow rate and boundary position are adjusted until the two equations agree.

Special cases occur when the coil is all wet or all dry. The coil is solved as if it were all wet before the general case is attempted. If the wet surface temperatures at the coil inlet and outlet are both below the dewpoint, no further solution is required. However, to ensure a continuous solution as flow variables are changed, when the surface is all dry or when it is wet with only the dry surface equations yielding a surface temperature below the dewpoint at the water outlet, the general solution is used to calculate the unknowns. In the solution of the controlled coil the outlet air enthalpy, given some resulting dehumidification, must correspond to the enthalpy at the specified outlet air temperature.

## Application of Cooling Coil Model to Heating Coils[LINK]

The implementation of detailed heating coil models in IBLAST was another important aspect of the system/plant integration. The same kind of loops exist to provide hot water to the heating coils from the boilers as exist to supply the cooling coils with chilled water from the chillers. Some simplifications can be made, however, since the enthalpy change of the air flowing over a heating coil is entirely sensible. There is no condensation in a heating coil. In order to allow heating and cooling coils to be specified using the same geometric parameters, a heating coil simulation was developed from the cooling coil model described above by eliminating the wet surface analysis.

In addition, it was concluded that, since much simpler and less computationally expensive heating coil simulations are possible, an option was provided in IBLAST for a heating coil design using only the UA value of the coil, the product of heat transfer coefficient and coil area. This model was largely based on the TYPE10 subroutine implemented in MODSIM. The equations used to model the performance of the TYPE10 heating coil are as follows:

Ta,out=Ta,in+(Tw,in−Tain)ε(min(Cp,a˙ma,Cp,w˙mw)Cp,a˙ma)Tw,out=Tw,in−(Ta,out−Tain)(Cp,a˙maCp,w˙mw)

where the coil effectiveness is given by:

ε=1−exp⎛⎜ ⎜ ⎜ ⎜⎝{exp[−(min{Cp,a˙ma,Cp,w˙mw}max{Cp,a˙ma,Cp,w˙mw}){NTU}0.78]−1}(min{Cp,a˙ma,Cp,w˙mw}max{Cp,a˙ma,Cp,w˙mw}){NTU}−.22⎞⎟ ⎟ ⎟ ⎟⎠

The parameter NTU is the number of transfer units and is defined as a function of the UA value of the coil as follows:

NTU=UAmin(Cp,a˙ma,Cp,w˙mw)

## Hot-Water-Based Air Heating Coil[LINK]

## Overview[LINK]

The input object Coil:Heating:Water provides a model that uses an NTU–effectiveness model of a static heat exchanger. The model is an inlet – outlet model: given the inlet conditions and flow rates and the UA, the effectiveness is calculated using the formula for the effectiveness of a cross-flow heat exchanger with both fluid streams unmixed. The effectiveness then allows the calculation of the outlet conditions from the inlet conditions.

The inputs to the model are: (1) the current inlet temperatures and flow rates of the air and water fluid streams and (2) the UA of the coil. Note that the UA is fixed in this model and is not a function of the flow rates.

There are 2 alternative user inputs for the component: the user may input the design water volumetric flow rate and the UA directly; or the user may choose to input the more familiar design heating capacity plus design inlet & outlet temperatures and let the program calculate the design UA. These alternative user inputs are fully described in the EnergyPlus Input Output Reference document.

## Model Description[LINK]

The air and water capacitance flows are defined as:

˙Cair=cp,air⋅˙mair

˙Cwater=cp,water⋅˙mwater

The minimum and maximum capacity flows are then:

˙Cmin=min(˙Cair,˙Cwater)

˙Cmax=max(˙Cair,˙Cwater)

The capacitance flow ratio is defined as:

Z=˙Cmin/˙Cmax

The number of transfer units (

NTU) is:NTU=UA/˙Cmin

The effectiveness is:

ε=1−exp(e−NTU⋅Z⋅η−1Z⋅η)

Where η=NTU−0.22 .

The outlet conditions are then:

Tair,out=Tair,in+ε⋅˙Cmin⋅(Twater,in−Tair,in)/˙Cair

Twater,out=Twater,in−˙Cair⋅(Tair,out−Tair,in)/˙Cwater

The output of the coil in watts is:

˙Qcoil=˙Cwater⋅(Twater,in−Twater,out)

The UA value is recalculated for each timestep. A nominal UA,

UA0, at the rating point is calculated by the program using the input for rated conditions and a search routine called regula falsi.User input for the ratio of convective heat transfers at the nominal or rated operating point, “

r,” is used in the model. This ratio is defined asr=ηf(hA)air(hA)water

where,

ηf is the fin efficiency, (dimensionless)

his the surface convection heat transfer coefficientAis the surface areaThe value calculated for

UA0 is used with the input forrto characterize the convective heat transfer on the water sides at the nominal rating operation point using((hA)w)0=UA0(r+1r)

and on the air side at the nominal rating point using

(ηf(hA)a)0=r(hA)w,0

Then the following equations are used to calculate a new UA as a function of the flow rates and inlet temperatures at each timestep.

xa=1+4.769⋅10−3(Tair,in−Tair,in,0)

ηf(hA)a=xa(˙ma˙ma,0)0.8(ηf(hA)a)0

xw=1+(0.0141+0.014Twater,in,0)(Twater,in−Twater,in,0)

(hA)w=xw(˙mw˙mw,0)0.85(hA)w,0

UA=(1(hA)w+1ηf(hA)a)−1

The above formulas are from the following reference, along with further references. The equation for xw was modified from that published in Wetter (1999) to correct a small error.

## References[LINK]

Wetter, M. 1999. Simulation Model: Finned Water-to-Air Coil Without Condensation. LBNL-42355. This document can be downloaded from .

## Single-Speed Electric DX Air Cooling Coil[LINK]

## Overview[LINK]

This model (object names Coil:Cooling:DX:SingleSpeed and

Coil:Cooling:DX:TwoStageWithHumidityControlMode, with CoilPerformance:DX:Cooling) simulates the performance of an air-cooled or evaporative-cooled direct expansion (DX) air conditioner. The model uses performance information at rated conditions along with curve fits for variations in total capacity, energy input ratio and part-load fraction to determine the performance of the unit at part-load conditions (Henderson et al. 1992, ASHRAE 1993). Sensible/latent capacity splits are determined by the rated sensible heat ratio (SHR) and the apparatus dewpoint (ADP)/bypass factor (BF) approach. This approach is analogous to the NTU-effectiveness calculations used for sensible-only heat exchanger calculations, extended to a cooling and dehumidifying coil.

This model simulates the thermal performance of the DX cooling coil and the power consumption of the outdoor condensing unit (compressor, fan, crankcase heater and evap condenser water pump). The total amount of heat rejected by the condenser is also calculated and stored for use by other waste heat recovery models (e.g., Coil:Heating:Desuperheater). The performance of the indoor supply air fan varies widely from system to system depending on control strategy (e.g., constant fan vs. AUTO fan, constant air volume vs. variable air volume, etc.), fan type, fan motor efficiency and pressure losses through the air distribution system. Therefore, this DX system model does not account for the thermal effects or electric power consumption of the indoor supply air fan. EnergyPlus contains separate models for simulating the performance of various indoor fan configurations, and these models can be easily linked with the DX system model described here to simulate the entire DX air conditioner being considered (e.g., see AirLoopHVAC:Unitary:Furnace:HeatCool, AirLoopHVAC:UnitaryHeatCool, ZoneHVAC:WindowAirConditioner or AirLoopHVAC:UnitaryHeatPump:AirToAir).

## Model Description[LINK]

The user must input the total cooling capacity, sensible heat ratio (SHR), coefficient of performance (COP) and the volumetric air flow rate across the cooling coil at rated conditions. The capacity, SHR and COP inputs should be “gross” values, excluding any thermal or energy impacts due to the indoor supply air fan. The rated conditions are considered to be air entering the cooling coil at 26.7°C drybulb/19.4°C wetbulb and air entering the outdoor condenser coil at 35°C drybulb/23.9°C wetbulb. The rated volumetric air flow should be between 0.00004027 m3/s and 0.00006041 m3/s per watt of rated total cooling capacity (300 – 450 cfm/ton). The rated volumetric air flow to total cooling capacity ratio for 100% dedicated outdoor air (DOAS) application DX cooling coils should be between 0.00001677 (m3/s)/W (125 cfm/ton) and 0.00003355 (m3/s)/W (250 cfm/ton).

The user must also input five performance curves or performance tables that describe the change in total cooling capacity and efficiency at part-load conditions:

1) Total cooling capacity modifier curve or table (function of temperature)

2) Total cooling capacity modifier curve or table (function of flow fraction)

3) Energy input ratio (EIR) modifier curve or table (function of temperature)

4) Energy input ratio (EIR) modifier curve or table (function of flow fraction)

5) Part load fraction correlation curve or table (function of part load ratio)

Note: The data used to develop the total cooling capacity modifier curve (function of temperature) should represent performance when the cooling coil is ‘wet’ (i.e., coil providing sensible cooling and at least some dehumidification). Performance data when the cooling coil is ‘dry’ (i.e., not providing any dehumidification) should

notbe included when developing this modifier curve. This model automatically detects and adjusts for ‘dry coil’ conditions (see section “Dry Coil Conditions” below).TotCapTempModFac=Func(Twb,i,Tc,i)

where

Twb,i = x values = wet-bulb temperature of the air entering the cooling coil, °C

Tc,i = y values = dry-bulb temperature of the air entering an air-cooled condenser or wet-bulb temperature of the air entering an evaporative-cooled condenser, °C

* The total cooling capacity modifier curve (function of flow fraction) is a curve with one independent variable being the ratio of the actual air flow rate across the cooling coil to the rated air flow rate (i.e., fraction of full load flow). The output of this curve is multiplied by the rated total cooling capacity and the total cooling capacity modifier curve (function of temperature) to give the total cooling capacity at the specific temperature and air flow conditions at which the DX unit is operating. This curve is typically a quadratic but any curve or table with one independent variable can be used.

TotCapFlowModFac=Func(ff)

where

ff=flow fraction=(Actual air mass flow rateRated air mass flow rate)=x value

Note:The actual volumetric air flow rate through the cooling coil for any simulation time step where the DX unit is operating must be between 0.00002684 m3/s and .00006713 m3/s per watt of rated total cooling capacity (200 - 500 cfm/ton). The simulation will issue a warning message if this air flow range is exceeded.Note: The data used to develop the energy input ratio (EIR) modifier curve (function of temperature) should represent performance when the cooling coil is ‘wet’ (i.e., coil providing sensible cooling and at least some dehumidification). Performance data when the cooling coil is ‘dry’ (i.e., not providing any dehumidification) should

notbe included when developing this modifier curve. This model automatically detects and adjusts for ‘dry coil’ conditions (see section “Dry Coil Conditions” below).EIRTempModFac=Func(Twb,i,Tc,i)

where

Twb,i = x values = wet-bulb temperature of the air entering the cooling coil, °C

Tc,i = y values = dry-bulb temperature of the air entering an air-cooled condenser or wet-bulb temperature of the air entering an evaporative-cooled condenser, °C

EIRFlowModFrac=Func(ff)

where

ff=flow fraction=(Actual air mass flow rateRated air mass flow rate)=x value

PartLoadFrac=PLF=Func(PLR)

where

PLR=part load ratio=(actual sensible cooling loadsteady-state sensible cooling load)=x values

The part-load fraction correlation should be normalized to a value of 1.0 when the part load ratio equals 1.0 (i.e., no efficiency losses when the compressor(s) run continuously for the simulation time step). For PLR values between 0 and 1 (0 < = PLR < 1), the following rules apply:

PLF≥0.7PLF≥PLR

If PLF < 0.7 a warning message is issued, the program resets the PLF value to 0.7, and the simulation proceeds. The runtime fraction of the coil is defined as PLR/PLF. If PLF < PLR, then a warning message is issued and the runtime fraction of the coil is limited to 1.0.

A typical part load fraction correlation for a conventional, single-speed DX cooling coil (e.g., residential or small commercial unit) would be:

PLF = 0.85 + 0.15(PLR)

All five part-load curves are accessed through EnergyPlus’ built-in performance curve equation manager (curve: quadratic, curve:cubic and curve:biquadratic). It is not imperative that the user utilize all coefficients shown in equations (449) through (453) if their performance equation has fewer terms (e.g., if the user’s PartLoadFrac performance curve is linear instead of quadratic, simply enter the values for a and b, and set coefficient c equal to zero).

For any simulation time step, the total (gross) cooling capacity of the DX unit is calculated as follows: