Transient Analysis of Snow-Melting System Performance

406 ASHRAE Transactions: Research

ABSTRACT

The transient response of snow melting systems for pave-ments, by virtue of their thermal mass and the fact that they areoperated intermittently, has a significant effect on overallsystem performance. Traditional steady-state methods of snowmelting system load calculation have not been able to take intoaccount the thermal history of the system or the transientnature of storm weather. This paper reports on the work of arecent ASHRAE-sponsored project that has sought to assessthe significance of these effects on system performance. Thedevelopment of a numerical transient analysis method thatincludes a boundary condition model able to deal with meltingsnow conditions is described. This numerical method has beenincorporated into a transient analysis tool that can be used bypractitioners to examine transient effects on system perfor-mance. The results of a parametric study that examined tran-sient effects using real storm event data are presented. Effectsof storm structure were found to be most significant. Calcula-tions of back and edge losses under transient conditions arealso presented.

INTRODUCTION

Previous research and design guidelines published byASHRAE for snow-melting systems (e.g., Chapman andKatunich 1956, Ramsey et al. 1999a, 1999b) have been basedon steady-state conditions. Design loads (surface heat fluxes)have been calculated by taking the instantaneous weatherconditions and calculating the flux required at the surface tomeet certain design criteria. These calculations—recentlyupdated following the work of Ramsey et al.—are presented inthe Applications Handbook (ASHRAE 1999, ch. 49) in tabu-lar form for a range of locations. These loads have been calcu-

lated using twelve years of historical weather data andprocessed to find various percentiles of loads not exceeded.

In steady-state calculations of this type, no account istaken of the history of the storm up to the point of interest, andno account is taken of the dynamic response of the heated slab.However, this design heat flux can never be provided at thesurface instantaneously. As current practical snow-meltingsystems employ heated elements – either hydronic or electri-cal – embedded some distance below the surface of a slab, thesystem’s time constant is on the order of hours. Not only doesthe heating system have significant thermal mass, but theweather is also highly transient.

Designers of snow-melting systems are not concernedonly with determining the required surface flux to melt snowat a particular location. A significant part of the design prob-lem is to determine the necessary depth and spacing of theheating elements, along with the operating temperature (orelectrical power) as well as the disposition of insulation, inorder to achieve the design surface flux. It is, furthermore,necessary to consider the losses from the system at the backand edges of the heated slab.

It is possible to use steady-state methods to find the rela-tionship between the heat input at the heating element and theflux at the upper slab surface. However, the fact that snow-melting systems are operated intermittently—depending onweather conditions—and that the time constant of suchsystems is relatively long, must mean that this is not a conser-vative approach. Because of the interest of snow-meltingsystem designers in simultaneously designing the slabconstruction and in the possible significance of transienteffects, TC6.1 initiated a research project, “Development of atwo-dimensional transient model of snow-melting systems,and use of the model for analysis of design alternatives”(1090-RP). This paper reports on the work of this project.

Transient Analysis of Snow-MeltingSystem Performance

Simon J. Rees, Ph.D. Jeffrey D. Spitler, Ph.D., P.E. Xia XiaoMember ASHRAE Member ASHRAE Student Member ASHRAE

Simon J. Rees is a visiting assistant professor, Jeffrey D. Spitler is a professor, and Xia Xiao is a research assistant in the School of MechanicalEngineering, Oklahoma State University, Stillwater, Okla.

4591 (RP-1090)

mphillips

Text Box

© 2002, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions 2002, Vol 108, Part 2. For personal use only. Additional distribution in either paper or digital form is not permitted without ASHRAE’s permission.

ASHRAE Transactions: Research 407

The type of model that has been developed can bedescribed as a transient two-dimensional finite volume modelwith a comprehensive storm boundary condition model. Thedevelopment of the boundary condition model(s) and couplingwith a finite volume conductive heat transfer solver isdiscussed. A research implementation of this model has beenused to calculate back and edge losses from slabs and find theminimum input fluxes required to meet design criteria for arange of design parameters. The model has also been imple-mented as a transient analysis tool, driven by a user interface,and used with storm weather data from a library of storms. Ashort description of the analysis tool is given, along with adiscussion of the resulting fluxes and edge and back lossescalculated in the parametric study.

BACKGROUND

Besides the one-dimensional steady-state approachadopted within ASHRAE, most previously published modelsof snow-melting systems have either modeled steady-statebehavior while accounting for the two-dimensional geometry(e.g., Kilkis 1994a, 1994b; Schnurr and Rogers 1970) or havemodeled transient behavior while only accounting for one-dimensional geometry (e.g., Williamson 1967). Two excep-tions are papers by Leal and Miller (1972) and Schnurr andFalk (1973).

Leal and Miller (1972) reported on a two-dimensionaltransient model that used a “point-matching” technique tosolve for the temperature distribution. It is not clear from thepaper how the “point-matching” technique accommodated themixed geometry (round tube in square slab). The paper did notpresent any results under actual snow-melting conditions.Schnurr and Falk (1973) presented a two-dimensional, tran-sient, apparently explicit finite difference model. It is unclearfrom the paper how the mixed geometry problem was handled.Given the relatively coarse grid, it appears that a fairly simpleapproximation was made.

In the steady-state load calculation procedure developedby Ramsey et al. (1999a, 1999b), the objective is to find theinstantaneous flux required to achieve a given free area ratiocondition. This approach can be thought of as a one-dimen-sional surface heat balance. The steady-state energy balanceequation for the snow-melting surface can be written as

(1)

where

qo = total heat flux per unit area of the surface, Btu/h⋅ft2 (W/m2);

qs = total sensible heat flux, Btu/h⋅ft2 (W/m2);

qm = melting load, Btu/h⋅ft2 (W/m2);

Ar = snow-free area ratio;

qh = sum of the convection and radiation losses, Btu/h⋅ft2 (W/m2);

qe = evaporative losses, Btu/h⋅ft2 (W/m2).

Any buildup of snow on the slab acts as an insulator forthe losses from the snow-melting surface. This model assumesany layer of snow acts as a perfect insulator. For this reason,the convective, radiative, and evaporative losses may beexpressed in terms of the fraction of the snow-free area on thesurface. The snow-free area ratio is defined as the ratio of thearea of snow-free area of the surface to its total area. This isgiven by,

(2)

where

Ar = snow-free area ratio;

Af = equivalent snow-free area, ft2 (m2);

At = total surface area of the slab, ft2 (m2).

Free area ratio is consequently used as a key design crite-rion in the design of snow-melting systems. Details of thecalculation of the terms of Equation 1 are given in Ramsey etal. (1999b) and the 1999 ASHRAE Handbook—Applications(ASHRAE 1999). These heat transfer relationships have beenused in this work where applicable. However, to deal withtransient effects, the coupled mass and heat transfer problemhas to be treated with special boundary conditions in which thetime varying mass of ice and liquid is calculated.

BOUNDARY CONDITIONS

Much of the effort of this project has been in developingdifferent boundary condition models suitable for modeling thevariety of conditions that can occur during winter storms. Theapproach to this aspect of the modeling task has been to treatthe snow layer as quasi one-dimensional. Effectively, eachsurface node of the two-dimensional slab model is coupled toan instance of the surface boundary condition model. Hence,variations in conditions across the slab are modeled, but nolateral heat transfer effects within the snow layer are consid-ered.

The situation of most interest in modeling the heated slabduring the storm is clearly when snow is being melted on theslab. Indeed, this has been the focus of much of the modelingwork in the project. However, unlike the steady-state loadcalculation, in order to model a whole “storm event” itbecomes not only necessary to be able to model melting snowbut also a wide variety of surface and weather conditions thatmay occur. The simplest condition is clearly when the slab isdry. This is likely to be the initial condition during the transientsimulations. The slab surface may also be wet, covered in“slush,” or solid ice. The slab can be wet not only because ofrain but also at the final stages of melting the snow.

Consequently, the approach taken to modeling the surfaceheat transfer has been to identify a number of possible surfaceconditions and develop models for each situation. An algo-rithm has been developed that keeps track of the surface condi-tion (in each cell on the surface grid). The surface condition isdetermined from the surface temperature and the mass of ice

qo qs qm Ar qh qe+( )+ +=

Ar

Af

At-----=


and water on each cell. Which model is applied to calculate thesurface condition at the end of the current time step of thesimulation is decided based on the condition at the end of theprevious time step, the current type of precipitation, and thecurrent surface temperature. A summary of the possiblecurrent conditions and the possible conditions at the end of thetime step are given in Table 1. The surface conditions that havebeen considered are defined as follows:

• Dry: The surface is free of liquid and ice. The surfacetemperature may be above or below freezing.

• Wet: The surface is above freezing and has some liquidretained on it, but no ice.

• Dry Snow: The snow has freshly fallen snow on it but noliquid. The snow can be regarded as a porous matrix ofice. The surface temperature is below freezing so thatsnow is not currently being melted.

• Slush: The surface contains ice in the form of snowcrystals that are fully saturated with water. Water pene-trates the ice matrix to the upper surface. The surfacetemperature is at freezing point.

• Snow and Slush: The surface contains snow that ispartly melted. The lower part of the snow is saturatedwith water and the upper is as dry snow. This is the gen-eral melting snow condition and the surface temperatureis at freezing point.

• Solid Ice: The ice on the surface is in solid form ratherthan porous like snow, i.e., as liquid that has frozensolid. The surface temperature must be below freezing.

• Solid Ice and Water: The surface consists of solid iceand water. This can occur when rain falls on solid ice orwhen the solid ice is being melted. Melting can be frombelow or above. The surface temperature is at freezing.

The primary concern in the boundary condition models isto find the correct relationship between surface temperatureand surface flux. However, dealing with this variety of weatherconditions requires simultaneous consideration of both heatand mass transfer. In addition to finding the heat balance at theslab surface at each time step, it is also necessary to keep trackof mass transfer in both solid and liquid phases. The mass ofboth ice and liquid on each cell of the grid is calculated at eachstep. Calculation of the height of the saturated layer in thesnow also enables the mass of snow (dry snow that is) to becalculated. The mass of solid ice (sheet ice) is also calculatedif this condition arises. Keeping track of the masses of eachphase and type of solid requires the integration of the meltingrate, evaporation rate, and the rate of liquid runoff.

Calculating the masses of snow, ice, and liquid enablesthese data to be used to define the condition of the surface atthe end of each time step. The rules used to define the surfacecondition are as follows.

The surface is assumed dry unless the following condi-tions occur:• If the mass of liquid is greater than zero and the mass of

ice is zero the condition is wet.• If the mass of liquid is greater than zero and the mass of

ice is greater than zero and the mass of snow is greaterthan zero, it is assumed to be snow and slush.

• If the mass of liquid is greater than zero and the mass ofice is greater than zero, but the mass of snow is zero, it isassumed slush.

• If the mass of liquid is zero and the mass of ice is greaterthan zero and the mass of snow is greater than zero, it isassumed to be dry snow.

• If the mass of liquid is zero and the mass of solid ice isgreater than zero, but the mass of snow is zero, it isassumed to be solid ice.

• If the mass of liquid is greater than zero and the mass ofsolid ice is greater than zero, but the mass of snow iszero, it is assumed to be solid ice and water.

The usual method of specifying the boundary conditionsof the finite volume solver can only be used directly for thebasic types of boundary condition – namely, fixed tempera-ture, fixed flux, or a linear mixed condition. As phase changeoccurs at the boundary of the finite volume model, the bound-ary conditions are highly nonlinear. This is because, generally,there is an increase in surface temperature with flux. However,at the melting point, the flux into the snow may increase with-out any change in temperature. Because of this, it is necessaryto have a much more complicated model for the calculation ofthe slab surface temperature that is more loosely coupled to thefinite volume solver.

The finite volume solver is coupled to the boundarycondition models by passing surface temperature informationand heat flux information between the two models. Becausethe temperature becomes fixed at the point of melting, it isnecessary that the finite volume solver pass the surface flux ithas calculated to the boundary condition model. The boundarycondition model is then responsible for calculating the surfacetemperature (given the environmental conditions, currentmass of ice, etc.). This temperature is, in turn, passed back tothe finite volume solver. This process of passing the calculated

TABLE 1 Possible Slab Surface Conditions

vs. Different Initial Conditions

Initial SurfaceCondition

Precipitation Condition

None Rain Snow

1. Dry 1 2,6 1, 3, 4, 5, (2)

2. Wet 1, 2, 6 2, 6, (1) 2, 4, 6

3. Dry snow layer 3, 4, 5, (2) 4, 5 3, 4, 5, (2)

4. Slush layer 2, 4, 6, (1) 2, 4, 6, (1) 2, 4, 5, (7)

5. Snow and slush 4, 5, (6, 7) 4, 5, (6, 7) 4, 5, (6, 7)

6. Solid ice layer 2, 4, 6 2, 4, 6 5, 7, (4)

7. Solid ice and water 7, 6, 2, 1, (4) 7, 2, 6 5, 7, (2, 4)


flux from the finite volume solver to the boundary conditionmodel, and the temperature back again, is iterative. When thecalculation has converged, the heat flux (temperature gradi-ent) calculated by the finite volume solver becomes consistentwith the surface temperature calculated by the boundarycondition model. This iteration process has to be underrelaxedconsiderably because of the highly nonlinear nature of theboundary conditions. This means that considerably morecomputational effort is required to reach a converged state thanwith a simple boundary condition. When calculating steady-state loads for design purposes, it is conservative to assumefull cloud cover and ignore contributions to melting from solarfluxes. However, when modeling the whole storm event (i.e.,including hours before and after precipitation), solar fluxesmay be more significant. However, as no solar weather datawere available, this has not been possible.

The boundary condition model is really a collection ofmodels for each type of surface condition. Of these models,the model of melting snow is probably of most interest andcomplexity. Because of this, and due to space limitations, onlythis model is discussed in detail. The complete boundarycondition model is discussed in a companion paper (Rees andSpitler 2002) as well as in the 1090-RP final report (Spitler etal. 2001).

The Melting Snow Model

First, conceptually, the snow during the melting process isconsidered as a layer of “dry” snow (ice crystals with no liquidwater) and a layer of saturated snow (slush) adjacent to the slabsurface. Both the snow layer and the saturated layer may beconsidered as porous media. The dry snow layer has air in thevoid space between the snow crystals, and the saturated layerhas water in the void space between the snow crystals.

The mass transfers of interest to or from the snow layer areshown in Figure 1. The snowfall rate is determined fromweather data. Snowmelt rates are determined based on anenergy balance, to be discussed below. Sublimation was notfinally included in the model, as it seemed an insignificanteffect. It is also assumed that as melting occurs, the slush-snowline will move so that previously dry snow will become satu-

rated slush. Mass transfers to and from the saturated (slush)layer are shown in Figure 2.

All of the mass transfer processes described above havesome corresponding heat transfer. In addition, convection andradiation from the top surface of the snow layer and conductiveheat transfer to and through the snow and slush layers areimportant. The crux of the modeling problem is how to orga-nize the heat transfer model so that an appropriate accuracy isachieved given the limitations of the weather data available. Anumber of degrees of complexity were examined, rangingfrom a single node at the slab surface to a detailed finitevolume model of the snow layers. The approach finallyadopted employs three nodes—one at the upper surface of thesnow layer, one in the center of the snow layer, and one at thesaturated (slush) layer. This model is represented schemati-cally in Figure 3.

A number of assumptions are made with this model.These include the following:

• Uniform temperature in the slush/liquid layer. • Melting of snow occurs at the lower node only, either at

the interface between the snow and slush layers or in theslush layer.

• Transfer of solid snow from the snow layer to the slushlayer is explicitly accounted for in the mass balance.However, from a heat transfer standpoint, it may beneglected. Because the lower node covers both the slushlayer and the bottom of the snow layer, it makes no dif-

Figure 1 Mass transfer to/from the snow layer.

Figure 2 Mass transfer to/from the slush layer.

Figure 3 Schematic representation of heat transfer in thethree-node snowmelt model.


ference whether the snow melts at the interface or in theslush layer. Therefore, no heat transfer path accountingfor the transfer of solid snow from the snow layer to theslush layer is shown in Figure 3.

• While convection from the upper surface of the snow isaccounted for, convection due to airflow through theporous snow layer is neglected. The model does notnecessitate neglecting this convection, so it may beincluded if further research indicates that it is important.

• Likewise, convection and evaporation from the slushlayer are neglected (when covered with a layer of drysnow).

• Rainfall occurring after a snow layer has formed isaccounted for directly only at the saturated layer.

• The snow melting process is treated as a quasi-one-dimensional process.

The model is formed by five primary equations—a massbalance for the solid ice, a mass balance for liquid water, anda heat balance on each node. The mass balance on the ice isgiven by

(3)

where

mice = mass of snow per unit area in the snow layer, lbm/ft2 or kg/m2;

θ = time, h or s;

= snowfall rate in mass per unit area, lbm/(h⋅ft2) or kg/(s⋅m2);

= rate of snow that is transferred to the slush in solid form, lbm/(h⋅ft2) or kg/(s⋅m2).

The mass balance on the liquid is given by

(4)

ml = mass of liquid water per unit area in the slush layer, lbm/ft2 or kg/m2;

= rainfall rate in mass per unit area, lbm/(h⋅ft2) or kg/(s⋅m2);

= snowmelt rate in mass per unit area, lbm/(h⋅ft2) or kg/(s⋅m2);

= rate of runoff in mass per unit area, lbm/(h⋅ft2) or kg/(s⋅m2).

A simple heuristic approach has been taken to estimatethe amount of runoff. In order to approximate the effect ofwater being retained in the snow due to capillary action, therunoff is limited to 10% of the melt rate until the saturatedlayer is 2 in. thick. (Conceptually, there is some height atwhich the capillary forces are in balance with the gravitationalforces on the retained liquid). The runoff rate is increased tothe melt rate after this point in order to prevent more waterbeing retained.

In order to calculate the heat balances on the snow andsaturated layers, it is necessary to work out the total mass ofthese two layers. This can be done by assuming an effectiveporosity (or relative density) and calculating the thickness ofthese layers. The total height of the snow and slush layers canbe found from the mass of ice by

(5)

where

htotal = the total thickness of the snow and saturated layers, ft or m;

neff = the effective porosity of the ice matrix (applies to both layers), dimensionless;

ρice = the density of ice, lbm/ft3 or kg/m3.

Similarly, the height of the saturated layer can be calcu-lated from the mass of liquid,

(6)

and the height of the snow layer can be found by subtracting,hsnow=htotal – hsat. Having worked out the height of the respec-tive layers, the mass of the dry snow layer can be found:

(7)

The mass balance equations are coupled to the energybalance equations by the melt rate. The energy balance on thesnow layer is given conceptually as

(8)

However, each of the various terms must be defined inadditional detail. The conductive heat flux from the slush layerto the snow layer is given by

(9)

where

ksnow = thermal conductivity of the snow, Btu/(h⋅ft⋅°F) or W/(m⋅K);

tsat = temperature of the slush layer, °F or °C;

tsnow = temperature of the snow node, °F or °C.

The heat flux due to snowfall is given as

(10)

The convective heat flux is given by

(11)

The radiative heat flux is given by

dmice

dθ-------------- m· snowfall

″m· melt

″–=

m· snowfall″

m· melt″

dml

dθ--------- m· melt

″m· rain

″m· runoff

″–+=

m· rain″

m· melt″

m· runoff″

htotal

mice

ρice 1 neff–( )--------------------------------=

hsat

ml

ρlneff------------- ,=

msnow ρicehsnow 1 neff–( )=

msnowcp

dtsnowdθ

------------------- qconduction, snow″

qsnowfall″

– qconvection″

– qradiation″ .–=

qconduction, snow″ ksnow

0.5hsnow--------------------- tslush tsnow–( ),=

qsnowfall″

m· snowfall″

cp, ice tsnow ta–( ).=

qconvection″

hc tsurface ta–( ).=


(12)

The snow surface temperature is found from a heat balanceon the surface node:

(13)

As this surface temperature appears in Equations 11 and12, it has to be determined iteratively. The energy balance at theslush node presumes that the liquid/ice mixture is in thermody-namic equilibrium and, therefore, the temperature is uniform atthe melting point. Then, the energy balance is given by

(14)

Assuming rainwater will be at the air temperature, theheat flux due to rainfall is given by

(15)

The mean radiant temperature and convection coefficientare calculated in the same manner as in Ramsey et al. (1999b).

HEAT TRANSFER IN THE PAVEMENT

The finite volume method has been used in the project tosolve the heat conduction equation that describes the temper-ature distribution within the slab and around the pipe. Thecode uses block structured boundary fitted grids to deal withcomplex geometries.

The FVM starts from the integral form of the partialdifferential equation so that in the case of Fourier’s equationfor heat conduction, we start with

(16)

where φ is the temperature and Γ is the thermal diffusivity, Vis the volume and S is the surface of a control volume, and nis a vector normal to the surface. The left-hand term of theequation is the temporal term and the right-hand term repre-sents the diffusion fluxes.

A physical space approach for dealing with complexgeometries can be derived from the vector form of the equationabove. We will consider the diffusion fluxes and temporal termin turn. The approach taken here is discussed in Ferziger andPeri (1996).

A second-order approximation is to assume that the valueof the variable on a particular face is well represented by thevalue at the centroid of the cell face. If we consider the diffu-sion flux at the east face of a cell, we can write

(17)

where Se is the area of the east face. Our main difficulty is incalculating the gradient of the variable at each cell face.

Referring to Figure 4, we can define local coordinates at thecell face. In the direction normal to the face at its centroid, wedefine the coordinate n, and on the line between neighboringcentroids, we define the coordinate ξ, which passes throughthe face at point e'.

In order to calculate the gradient of the variable at the cellface, we would like to use the values of the variable at the cellcentroid as we are calculating these implicitly. We couldcalculate the gradient using the values at øP and øE and thedistance between these points, LP,E. In this case,

, which is only accurate if the grid isorthogonal. What we would really like is to preserve second-order accuracy by making the calculation of the gradient alongthe normal to the face and at the centroid of the face by usingthe values of the variable at points P' and E'. However, we arenot calculating the values of the variable at these points implic-itly. We use a “deferred correction” approach to calculating theflux as follows:

(18)

where we use central differencing to get the gradients. Theterms in the square brackets on the right of Equation 18 arecalculated explicitly, i.e., using the previous values of the vari-able. As the solution approaches convergence, the terms

and cancel out, leaving as wedesired. In order to calculate explicitly from thecentral difference , it is necessary to use inter-polation to get and , but this is easily implemented.

In formulating a finite volume solution, we need to inte-grate the partial differential equation with respect to time. Weuse a first-order backward differencing approach in a fullyimplicit formulation. The fully implicit approach results in thefollowing discretized equation:

(19)

qradiation″ σεs Tsurface

4TMR

4–( ).=

tsurface tsnow

0.5hsnow

ksnow--------------------- qconvection

″qradiation

″+( )–=

m· melt″

hif qconduction, slab″

qrainfall″

qconduction, snow″

.–+=

qrainfall″

m· rainfall″

cp, water ta tslush–( ).=

∂∂t---- φ Vd

v∫ Γ∇φn S,d

s∫=

FeD Γ∇φndS Γ∇φ n⋅( )eSe,≈

Se

∫=

∇φ( )

Figure 4 The local coordinate system at the east face of atypical finite volume cell.

FeD ΓeSe ∂φ ∂ξ⁄( )

e′≈

FeD ΓeSe

∂φ∂ξ------

e′ΓeSe

∂φ∂n------

e

∂φ∂ξ------

e′–

old,+=

∂φ ∂ξ⁄( )e′ ∂φ ∂ξ⁄( )

e′old ∂φ ∂n⁄( )n

∂φ ∂n⁄( )nφ

P′ φE′–( ) L

P′E′⁄φ

P′ φE′

ρ∆V φPn 1+ φP

n–( ) Fn

DFs

DFw

DFe

D+ + +[ ]

n 1+t,∆=


where variable values at the previous time level have super-script n and those at the current time level have superscriptn+1. The discretized equation can then be said to be first-orderaccurate in time and second-order accurate in space. Thisscheme is unconditionally stable. After integrating the p.d.e.and applying the discretization procedures we have discussed,we arrive at an algebraic equation for each control volume ofthe form,

(20)

where aP for one control volume becomes one of anb for thenext cell. For a two-dimensional model, this results in a penta-diagonal matrix equation that can be solved convenientlyusing the Strongly Implicit Method (Stone 1968).

Initial Conditions

In a transient calculation, the initial conditions applied tothe calculation can be just as significant as the boundary condi-tions. In a transient snow-melting calculation, the initial slaband ground temperatures have to be initialized according tolocation and weather conditions. In order to take account ofground heat transfer, it is necessary, not only to specify thetemperature at the bottom boundary (as described above), butalso to calculate the temperature profile through the calcula-tion domain. The approach taken involves using a one-dimen-sional model of the slab and ground. The ground temperatureis specified at the lower boundary, using an analytical solutiongiven by Kusuda and Achenbach (1965), and a two-weekperiod is simulated using the same boundary condition modelsdescribed above, using this one-dimensional model. To initial-ize the two-dimensional model, the final temperature profilecalculated using the one-dimensional model is taken and usedto define the initial temperatures (according to cell depth) overthe two-dimensional grid. For each storm, the two-dimen-sional calculation is made with six hours of weather databefore the start of any snow precipitation and twenty-fourhours afterward.

THE TRANSIENT ANALYSIS TOOL

A tool for practitioners that is able to carry out transientanalysis of snow-melting systems has been one of theoutcomes of this work. The tool is designed to allow the simu-lation of the operation of hydronic, constant wattage electricand self-regulating electric snow-melting systems. The pave-ment can be either on-grade or suspended, and its constructioncan be defined in up to six layers. The geometry of the slab andembedded heating elements is modeled in two-dimensions,using a section of the slab that includes half of the heatingelement and a section of the slab that is one-half of the pipe/cable spacing wide. This makes maximum use of the geomet-ric symmetry and means that the thermal boundary conditionsare representative of the central part of the slab. Back lossescan be considered using this approach but not edge losses (thegeometric model is different in the edge loss study discussed

later). It is not possible to model a larger part of the slab in theanalysis tool, as the analysis would require excessive run time.

The key geometric parameters—pipe spacing, pipe depth,and slab thickness—can be entered in the main interface. Thetransient analysis tool includes interfaces to a number ofcomponent databases. These allow the selection of the mate-rials that are used in the construction of the pavement layersand the type of cable or pipe used as a heating element. Theuser is able to select specific storms from 46 cities to be usedas input data along with different ground conditions andsystem controls. An illustration of the user interface is shownin Figure 5. Output from the tool includes mass of ice, heightof snow, surface flux, and temperature. These data arepresented in the form of plots of hourly averaged data and thedistribution over the surface at each hour, along with thenumerical results in tabular form.

The transient analysis tool includes two other compo-nents besides the conductive heat transfer solver. These are theuser interface and the grid generation software. The organiza-tion of these three elements is shown in Figure 6. The graph-ical user interface allows the user to input the basic designparameters and view the results of the calculations. Behind thescenes, however, a detailed description of the slab geometryhas to be set up for every calculation. This is done by code inthe user interface and by the grid generator.

aPφP a∑ nbφnb b,+=

Figure 5 The transient analysis tool user interface.


Grid Generation

The solver that is used to calculate the conductive heattransfer from the heating elements through the slab requires anumerical grid to be supplied that defines the geometry of eachdesign to be analyzed. The geometry of a flat slab with embed-ded circular pipes or cables is not easy to represent as a compu-tational grid, using either rectangular or polar coordinatesystems (it is a mixed geometry problem). The approach takenhere has been to use an existing finite volume solver that is ableto deal with a multiblock boundary fitted grid system. In thistype of grid system, the cells are arranged in a structuredmanner but are deformed where necessary to allow the geom-etry of the domain boundaries to be followed exactly.

The purpose of the grid generator is to generate the coor-dinates of every grid vertex from a description of the geometricboundaries. The geometry in the problem of concern in thisproject is always similar (i.e., a pipe or cable in a slab),although exact dimensions vary for each case. This allows theuse of a parametric grid generator rather than one that requiresinteractive user input. The problem geometry can be deducedfrom a few design parameters, such as pipe size, slab thick-ness, pipe depth, and pipe spacing. The grid outline can thenbe defined in terms of a number of lines and arcs, along withthe number of grid cells and their spacing along the boundaryof the slab and pipe/cable. This information is then passed tothe grid generator module. The grid generator module inter-prets the description of the grid outline to generate thecomplete grid coordinates. The numerical method uses amultiblock approach. This means that the geometry is brokendown into a number of subdomains or blocks. Within each ofthese blocks, the grid cells are arranged in a regular row andcolumn manner and each of the blocks is effectively “glued”to one or more others at the block edges. This enables more

complex geometries to be defined and allows better controlover the grid cell distribution. A simple algebraic grid gener-ation algorithm (Gordon and Hall 1973) is applied to calculatethe cell vertex positions in each block.

The most important factors that define a good grid are thenumber of cells and their distribution. Numerical errors areminimized when the grid is dense, smooth, and orthogonal.One way of optimizing the grid design it is to have small cellsin the region of the grid where greater temperature gradientsare expected and to increase the size of the cells in otherregions. Therefore, since the cells close to the pipe have moretemperature gradient as compared to the cells far away fromthe pipe/cable, the size of the cell should increase with itsdistance from the pipe. While doing this, special care has to betaken to retain reasonable smoothness. An algorithm wasdeveloped to control the distribution of cells along the blockedges for the whole range of geometric parameters expected inthis application. The final topological arrangement adoptedwas to use four blocks for the representation of the half slaband pipe geometry. This arrangement of blocks and an exam-ple grid is shown in Figure 7.

Storm Selection

A large amount of storm weather data have been collectedby the previous project (926-RP) for the calculation of steady-state design loads—12 years of data for 46 cities. The transientcalculation method is significantly more computationallyintensive than a steady-state calculation and so it is only feasi-ble to make calculations for specific storm events (about 50-60 hours of data). This means that storm events had to beselected from the original data for use in both the transientanalysis tool and for use in the parametric study discussedlater. There are a number of possible indicators that could beused as criteria for selecting storms from the twelve years ofdata for each city. It is important that the selected stormsinclude some of the “severest” storms at each cite (i.e., thosethat might represent a system design condition for a class IIIsystem1), as well as some that might represent design condi-tions for lower classification systems. It was also thoughtimportant that the storms be categorized in a way that is mean-ingful to the average user of the software.

It is possible to conceive of a number of measures thatcould be used to rank different storm events. The criteriaexamined were

• correlation with storms reported in the news media,• total precipitation in the storm,• hours above a certain percentile steady-state snow melt-

ing load,• total snow-melting energy consumption (from the

steady-state calculation), and• total storm duration (hours of snow precipitation).

Figure 6 The design tool software architecture. 1. A class III system is defined as one that will keep the surfacesnow-free during snowfall (Chapman and Katunich 1956).


It was found that the weather data did not correlate wellwith storms reported in the media. The media tends to giveemphasis to depth of snow (including drift effects), whereasthe peak steady-state loads often occur when there is strongwind and low temperature. Of the other criteria, those that useintegrated data (total precipitation, energy consumption,storm duration) give too much emphasis to longer storms thatoften do not include the severest steady-state loads. The crite-ria we decided upon was to select on the basis of number ofhours the steady-state load was in a band about a given percen-tile (the 75th, 90th, 95th, and 99th percentiles). This oftengives many more storms than required for the storm libraryand so this list was sorted again according to total energyconsumption. This resulted in approximately 15 storms beingselected for each of the 46 cities and the data being formulatedinto a database for use in the transient analysis tool.

THE PARAMETRIC STUDY

The objective of the parametric study undertaken in thiswork was to analyze the effects of two phenomena on thesnow-melting load not previously considered in the handbookpresentation: back and edge losses and transient design condi-tions/operation of the snow melting system. To that end, twoparametric studies were undertaken where the parametersvaried include configuration, location, and storm. In the firststudy (the center zone cases), a model with a width equal tohalf of the pipe/cable spacing (as in the transient analysis tooldiscussed above) was used to examine back losses and theeffects of transient operation. In the second study (the edgezone cases), a geometric model corresponding to the two pipesclosest to the slab edge and including a width of soil of 18 in.(450 mm) was used to examine the effect of edge losses. In

both studies, soil to the depth of 18 in. (450 mm) below the slabsurface was included in the model.

For each combination of parameters studied, multiplesimulations were performed with different heat flux levels. Aniterative scheme was used to determine the heat flux level,resulting in a minimum free area ratio of zero (Ar=0, see Equa-tion 2), and the minimum heat flux level, resulting in a mini-mum free area ratio of one (with the heat flux level fixedthroughout the storm). Since the original steady-state loadswere based on nonexceedance percentiles, fluxes were deter-mined that would maintain either

• a free area ratio of one, for all hours of the storm, excepta number of hours equivalent to the number of hours forwhich the steady-state 99% nonexceedance conditionswere exceeded,

• or, for the free area ratio of zero case, the minimum fluxwas found that only allowed the snow height to increasefor a number of hours equivalent to the number of hoursfor which the steady-state 99% nonexceedance condi-tions were exceeded.

For the center zone parametric study, the tube spacing,tube depth, insulation, soil conductivity, location, and stormtype were the parameters varied (see Table 2). The soilconductivity was only varied when the insulation level at theslab bottom was 2 in. (50 mm); hence, the number “1.5”. Thiscorresponds to 360 different cases, although a substantiallylarger number of simulations were required in order to find thecorrect minimum flux.

The edge zone parametric study, summarized in Table 3,was organized in a way similar to that of the center zone.However, because the edge zone in many cases cannot be kept

Figure 7 Block definitions of the numerical grid for a slab containing a pipe and anexample grid.


snow-free (without idling) without unreasonably high heatfluxes, it makes little sense to search for the heat flux thatmaintains the Ar=1 or Ar=0 condition. Furthermore, theamount of computational time required for an edge zone ismuch higher. Therefore, it was decided to just simulate theedge zones for the minimum flux found by the center zonestudy.

PARAMETRIC STUDY RESULTS

The results for the center-zone parametric studies, withAr=1, are summarized in Table 4. Table 4 shows the ratio of theminimum required heat fluxes to maintain a free area ratio of1 for the entire storm (excepting the number of hours where thesteady-state design loads exceeded the 99% nonexceedancelevel) to the steady-state 99% nonexceedance loads (shown inthe second row of the table). The minimum heat fluxes are thenpresented for each combination of the parameters (with andwithout insulation, two soil conductivities, two different tubedepths, and three different spacings.) The ratio of the mini-mum required heat fluxes to the 99% nonexceedance loadsshows the relationship between the steady-state design loads(without back losses) and the actual transient loads when backlosses are included. Results for the Ar=0 cases are shown inTable 5.

Looking at the results shown in Tables 4 and 5, the follow-ing observations may be made:

• Many of the fluxes exceed, by significant margins, alevel that is feasible to obtain in practice. The very highfluxes required indicate that, in many situations, it isimpractical to meet the design goal of maintaining a freearea ratio of one at the same statistical level assumed inthe steady-state analysis without idling the system.

• More than any other factor, the results are most sensitiveto the storm itself. This will be discussed in more detailbelow. Briefly, it has been noted that storms that start offwith relatively low loads, perhaps even ceasing to snowfor a few hours, then increasing in intensity, will havemuch lower ratios than storms that start off with highloads.

• After the storm, the results are most sensitive to thespacing. The farther apart the tubes, the more difficult itis to maintain a completely snow-free surface. Also, asthe tubes are placed deeper, the effect of spacing is lessimportant.

• The results are somewhat less sensitive to the depth ofthe tubing. Generally, the depth is more important forthe storms that are intense early on. With storms that areless intense in the early hours, the depth makes rela-tively little difference. Furthermore, there are a signifi-cant number of scenarios where increasing the depthdecreases the flux requirement. This will be discussedbelow.

• The results are almost completely insensitive to the soilconductivity and whether or not insulation has beeninstalled.

The fact that insulation is relatively unimportant heredoes not mean that insulation might not be important in energyconsumption or some other facet of operation. Also, unlessfurther research proves otherwise, we would not assume thatthe storms are typical of the location. We would not draw theconclusion that systems in, say Minneapolis, can be sized onsteady-state considerations only, whereas systems in Spokanemust be sized significantly higher than what might be drawnfrom the steady-state analyses.

TABLE 2 Center Zone Parametric Study

Number Parameter Levels

3 Tube spacing (6,8,12 in.; 150, 200, 300 mm)

2 Tube depths (2,4 in.; 50, 100 mm)

2 Insulation levels at the slab bottom (none, 2 in. [50 mm])

1.5 Soil conductivities (for noninsulated case, two values corresponding to saturated clay and dry light soil; for

insulated case, just use saturated clay)

10 Locations (Spokane, Reno, SLC, Colorado Springs, Chicago, OKC, Minneapolis, Buffalo, Boston, and

Philadelphia)

2 Storms (one 99% storm each for free area ratio one or zero)

Varies The number of simulations required to find either the minimum flux for Ar=1 or Ar=0 varies

TABLE 3 Edge Zone Parametric Study

Number Parameter Levels

3 Spacing (6,8,12 in.)

2 Depths (2,4 in.)

2 Insulation levels at the slab bottom (none, 2 in. expanded polystyrene)

1.5 Soil conductivities (for noninsulated case, two values corresponding to saturated clay and dry light soil; for

insulated case, just use saturated clay)

10 Locations (Spokane, Reno, SLC, Colorado Springs, Chicago, OKC, Minneapolis, Buffalo, Boston, and

Philadelphia)

2 Storms (one 99% storm each for free area ratio one or zero)

1 Heat flux (the same as the minimum flux found by center zone study).


TAB

LE

4

Nor

mal

ized

Min

imum

Req

uire

d H

eat F

luxe

s fo

r Tr

ansi

ent C

ondi

tions

with

Bac

k Lo

sses

(Fre

e A

rea

Rat

io =

1)

Cit

ySp

okan

eR

eno

SLC

Col

. Spr

.C

hica

goO

KC

Min

neap

olis

Buf

falo

Bos

ton

Phi

lade

lphi

a

99%

Ste

ady

Loa

d, B

tu/h

⋅ft2 (

W/m

2 )15

9(5

02)

137

(432

)12

0(3

79)

219

(691

)23

5(7

41)

260

(820

)25

4(8

01)

330

(104

1)22

9(7

22)

246

(776

)

No insulation layer

Soil conductivity,0.50 Btu/h⋅ft2.°F(0.87 W/m⋅K)

Pipe Depth2 in. (50 mm)

Spac

ing

6 in

.(1

50 m

m)

2.3

1.7

1.2

1.7

1.5

1.6

1.4

1.3

1.5

1.2

Spac

ing

8 in

.(2

00 m

m)

2.7

1.9

1.3

2.0

1.7

2.0

1.5

1.5

1.6

1.4

Spac

ing

12 in

.(3

00 m

m)

4.6

2.7

1.6

3.2

2.6

3.3

2.1

2.1

2.2

1.9

Pipe depth4 in. (100 mm)

Spac

ing

6 in

.(1

50 m

m)

3.3

2.2

1.4

2.6

2.3

2.4

1.7

1.6

1.6

1.4

Spac

ing

8 in

.(2

00 m

m)

3.8

2.2

1.4

2.8

2.4

2.6

1.8

1.6

1.7

1.5

Spac

ing

12 in

.(3

00 m

m)

5.4

2.6

1.5

3.8

3.2

-2.

21.

92.

01.

8



Spac

ing

6 in

.(1

50 m

m)

2.3

1.6

1.2

1.6

1.4

1.6

1.3

1.3

1.5

1.3

Spac

ing

8 in

.(2

00 m

m)

2.7

1.8

1.3

1.9

1.5

1.9

1.5

1.5

1.7

1.4

Spac

ing

12 in

.(3

00 m

m)

5.0

2.7

1.6

3.1

2.4

3.1

2.1

2.1

2.3

1.9


Spac

ing

6 in

.(1

50 m

m)

3.4

2.2

1.5

2.5

2.1

2.3

1.7

1.6

1.7

1.4

Spac

ing

8 in

.(2

00 m

m)

3.7

2.2

1.4

2.7

2.2

2.5

1.8

1.6

1.7

1.5

Spac

ing

12 in

.(3

00 m

m)

5.4

2.6

1.6

3.7

2.9

-2.

22.

02.

11.

9

2 in. (50 mm) insulation layer

Soil conductivity,0.50 Btu/h⋅ft2°.F(0.87 W/m⋅K)


Spac

ing

6 in

.(1

50 m

m)

2.3

1.6

1.1

1.7

1.5

1.7

1.4

1.2

1.3

1.1

Spac

ing

8 in

.(2

00 m

m)

2.7

1.8

1.1

2.0

1.7

2.0

1.5

1.4

1.4

1.2

Spac

ing

12 in

.(3

00 m

m)

4.3

2.3

1.3

2.9

2.7

3.3

2.0

1.8

1.8

1.7


Spac

ing

6 in

.(1

50 m

m)

3.1

1.8

1.1

2.3

2.2

2.3

1.6

1.3

1.3

1.2

Spac

ing

8 in

.(2

00 m

m)

3.3

1.8

1.1

2.5

2.3

2.5

1.6

1.3

1.3

1.2

Spac

ing

12 in

.(3

00 m

m)

4.5

2.1

1.2

3.2

3.1

3.4

2.0

1.5

1.5

1.4


TAB

LE

5

Nor

mal

ized

Min

imum

Req

uire

d H

eat F

luxe

s fo

r Tr

ansi

ent C

ondi

tions

with

Bac

k Lo

sses

(Fre

e A

rea

Rat

io =

0)

Cit

ySp

okan

eR

eno

SLC

Col

. Spr

.C

hica

goO

KC

Min

neap

olis

Buf

falo

Bos

ton

Phi

lade

lphi

a

99%

Ste

ady

Loa

d, B

tu/h

⋅ft2 (

W/m

2 )67

(211

)11

3(3

57)

104

(328

)11

2(3

53)

83(2

62)

113

(357

)11

3(3

57)

112

(353

)17

2(5

43)

150

(473

)

No insulation layer



Spac

ing

6 in

.(1

50 m

m)

2.4

1.3

1.2

0.7

2.2

1.6

2.4

1.5

1.6

2.7

Spac

ing

8 in

.(2

00 m

m)

2.9

1.4

1.4

0.7

2.5

2.1

2.7

1.7

1.7

3.2

Spac

ing

12 in

. (3

00 m

m)

4.8

1.9

1.9

0.8

3.4

4.3

3.6

3.2

2.3

5.4


Spac

ing

6 in

.(1

50 m

m)

4.7

1.9

1.7

1.3

2.8

3.2

2.9

2.3

2.0

4.3

Spac

ing

8 in

.(2

00 m

m)

4.7

1.9

1.7

1.2

2.9

3.7

2.9

2.6

2.0

4.6

Spac

ing

12 in

.(3

00 m

m)

4.8

2.3

2.5

1.2

3.5

5.2

3.4

3.3

2.2

-

Soil conductivity,1.4 Btu/h⋅ft2⋅°F

(2.4 W/m⋅K)


Spac

ing

6 in

.(1

50 m

m)

2.2

1.3

1.3

0.7

2.2

1.3

2.3

1.5

1.6

2.6

Spac

ing

8 in

.(2

00 m

m)

2.4

1.5

1.4

0.6

2.5

1.8

2.6

1.7

1.7

3.1

Spac

ing

12 in

.(3

00 m

m)

4.1

2.1

2.0

0.7

3.5

3.5

3.5

2.9

2.3

5.3


Spac

ing

6 in

.(1

50 m

m)

4.1

1.7

1.7

1.2

2.9

2.6

2.9

2.0

2.0

4.2

Spac

ing

8 in

.(2

00 m

m)

4.0

1.8

1.8

1.2

3.0

3.0

2.8

2.3

2.0

4.6

Spac

ing

12 in

.(3

00 m

m)

4.1

2.1

2.6

1.2

3.6

5.0

3.3

3.4

2.3

-


Soil conductivity,0.50 Btu/h⋅ft2⋅°F

(0.87 W/m⋅K)


Spac

ing

6 in

.(1

50 m

m)

2.7

1.3

1.2

0.7

2.1

1.9

2.2

1.6

1.5

2.7

Spac

ing

8 in

.(2

00 m

m)

3.0

1.4

1.3

0.6

2.3

2.5

2.5

2.1

1.6

3.2

Spac

ing

12 in

.(3

00 m

m)

5.2

1.8

1.7

0.7

2.9

4.4

3.3

3.2

2.1

5.3


Spac

ing

6 in

.(1

50 m

m)

5.0

1.7

1.4

1.1

2.4

3.5

2.6

2.4

1.9

4.1

Spac

ing

8 in

.(2

00 m

m)

5.0

1.7

1.5

1.1

2.5

3.7

2.5

2.5

1.8

4.4

Spac

ing

12 in

.(3

00 m

m)

5.1

2.0

2.0

1.1

3.1

4.9

2.7

2.9

1.9

-


Sensitivity to the Storm

As noted above, the most important factor in the ratiobetween the minimum heat flux for transient conditions andthe 99% steady-state load is the storm itself. To illustrate this,we might first consider a storm where the ratio is fairly high,in Spokane. Weather data for this storm are plotted in Figure8. For this storm, the high intensity loads occur early. As analternative, consider Salt Lake City, which has relatively lowratios between the transient requirements and the 99% steady-state loads. Weather conditions and loads for the Salt LakeCity storm are shown in Figure 9.

For this storm, what is immediately obvious is that thereis a 75th percentile load occurring well before the rest of thestorm. In this case, the system will come on and warm the slabbefore the rest of the storm hits. This will be analogous toidling the slab. Therefore, a relatively lower heat flux appliedto the slab for some time before the high intensities occur isvery effective. To further demonstrate this phenomenon, thestorm was artificially modified by moving the second batch ofprecipitation forward in time. This artificially modified stormis shown in Figure 10.

If we compare the case with no insulation, soil conduc-tivity of 0.5 Btu/h⋅ft⋅°F (0.87 W/m⋅K), spacing of 12 in. (300mm), and depth of 2 in. (50 mm), the minimum heat fluxrequired with the original storm is 191 Btu/h⋅ft2 (603 W/m2);with the artificially modified storm, it increases to 229 Btu/h⋅ft2 (723 W/m2). This represents an increase in the ratio from1.6 to 1.9.

More examples can be added, but what this shows is thata chance event—the actual snowfall pattern over time—canhave a significant influence on the ratio required heat flux fortransient conditions. This makes it difficult to come up withsimple correlations to relate the transient requirements to thesteady-state loads. Ultimately, if a free area ratio of one isabsolutely required, idling at some level will probably berequired. On the other hand, if some time lag in clearing off thesnow is allowable, then the steady-state heat fluxes or moder-ately increased heat fluxes might be acceptable.

Effect of Depth on the Minimum Flux Requirement

It might be expected that increasing the depth of the tubein the slab would increase the heat flux requirement. However,there are a large number of cases where increasing the deptheither decreases the heat flux requirement or has a negligibleinfluence. Generally speaking, for the storms that had rela-tively high ratios, as discussed above, increasing the depthtended to increase the heat flux requirement. Presumably, thisis due to the high intensity hours early in the storm, placing apremium on the quickness of the response. Conversely, for thestorms with relatively low ratios, increasing the depth eitherdecreased the heat flux requirements or had little effect. Thisappears to be due to the fact that, for the lower ratio storms, thelatter hours of the storm are more important, and, therefore, theadditional time delay due to putting the tubing deeper is notimportant. Instead, the more uniform heat flux allows moreuniform melting.

Figure 8 Weather conditions and loads for Spokane, Ar = 1. Figure 9 Weather conditions and loads for Salt Lake City,Ar = 1.

Figure 10 Artificially modified storm in Salt Lake City,Ar = 1.


Within any given storm, the wider the spacing, the morelikely it is that increasing the depth of the tubing will decreasethe minimum heat flux requirement. Again, the more uniformheat flux yielded by the deeper tubing is more important whenthe tube spacing is higher. This can be demonstrated byconsidering that, for any given storm, there is one hour thattends to control the required heat flux (i.e., decreasing the heatflux slightly will cause the free area ratio to drop below one forthat hour). For the Salt Lake City storm, Figure 11 shows thesurface heat flux for an input heat flux of 165 Btu/h⋅ft2 (521 W/m2), 12 in. (300 mm) spacing, and 2 in. (50 mm) and 4 in. (100mm) depths, at the controlling hour. In Figure 11, the normal-ized distance “0.0” represents the location above the tube; andthe distance “0.5” would be midway between the tubes. Whatwe can see is that for this hour, a higher surface heat flux isachieved at the position between the pipes (0.5 position) withthe deeper pipe spacing. With the shallower pipe, a “stripe” ofsnow would be seen at the position between the pipes.

Transient Load Less Than the Steady-State Load?

One interesting factor for the Ar=0 cases (Table 5) is thatfor Colorado Springs, the minimum required heat flux is actu-ally lower for some parametric combinations than the 99%steady-state load. While, at first glance, this may seem ratherunlikely, reviewing the actual storm weather data will demon-strate how this is possible. Figure 12 shows the dry-bulbtemperature and steady-state snow-melting loads, starting 18hours before the storm. As can be seen, the air temperature isquite warm prior to the storm. Therefore, it is expected that theslab will be above freezing when the snowfall begins, and itwill melt some of the snow using the thermal energy stored inthe slab. Therefore, in this case, the transient effects help,rather than hinder, the performance.

Back and Edge Losses

So far, the required heat fluxes under transient conditions(with back losses) have been compared to those required understeady-state conditions (without back losses). It is also inter-esting to review the actual back losses for the various cases.The back losses vary every hour of the simulation, but gener-ally they are near zero when the system is switched on, thenclimb to a moderately stable value as long as the systemremains on. For the sake of simplicity, we report the maximumpercentage of back losses, which should be representative ofthe moderately stable value. Table 6 gives the maximumpercentage back loss and Table 7 gives the maximum percent-age edge loss for the Ar=1 cases. The results for the Ar=0 casesare not shown, as space is limited and the results are very simi-lar.

The edge zone simulations were calculated using theminimum required heat flux inputs determined for the centerzone simulations. The percentage edge loss is defined here asthe ratio between the heat transfer through the edge of the slaband the heat input in the adjacent pipe. As noted above, themodel geometry is extended to include two whole pipes andsoil beyond the edge. Where insulation was included, this was2 in. (50 mm) thick and extended the full height of the slabedge. In brief, they show trends and orders of magnitude simi-lar to the maximum percentage back losses.

The following trends regarding the back and edge lossesmay be observed:• Back and edge losses vary significantly from storm to

storm.• Insulation is quite successful in reducing the back and

edge loss, often giving an order-of-magnitude reduction.• Compared to whether or not insulation is used, soil con-

ductivity has a small impact on the losses.• Spacing also has a relatively minor impact. As the tubes

are spaced farther apart, they run at higher temperaturesand, hence, have higher back losses.

Figure 11 Distribution of the surface heat flux. Figure 12 Colorado Springs storm with warm conditionsprior to snowfall.


TAB

LE

6

Max

imum

Per

cent

age

Bac

k Lo

ss fo

r Tr

ansi

ent C

ondi

tions

with

Bac

k Lo

sses

(Fre

e A

rea

Rat

io =

1)

Cit

ySp

okan

eR

eno

SLC

Col

. Spr

.C

hica

goO

KC

Min

neap

olis

Buf

falo

Bos

ton

Phi

lade

lphi

a

no insulation layer

Soil Conductivity,0.50 Btu/h⋅ft2⋅°F

(0.87 W/m⋅K)

Pipe Depth2 in.

(50 mm)

Spac

ing

6 in

. (1

50 m

m)

16.4

18.1

16.6

12.8

917

.511

.112

.212

.113

.313

.5

Spac

ing

8 in

.(2

00 m

m)

17.1

18.7

16.8

13.8

618

.012

.213

.112

.613

.714

.1

Spac

ing

12 in

. (3

00 m

m)

16.6

19.0

16.7

15.3

218

.113

.514

.313

.318

.614

.5

Pipe Depth4 in.

(100 mm)

Spac

ing

6 in

. (1

50 m

m)

22.5

23.9

24.3

21.8

225

.220

.620

.720

.922

.021

.8

Spac

ing

8 in

.(2

00m

m)

21.6

24.4

24.6

22.5

025

.621

.321

.321

.322

.422

.4

Spac

ing

12 in

.(3

00 m

m)

21.0

24.6

24.6

23.2

225

.7-

21.9

21.6

22.4

22.6


(2.4 W/m⋅K)

Pipe Depth2 in.

(50 mm)

Spac

ing

6 in

. (1

50 m

m)

17.3

20.3

19.1

14.1

518

.011

.613

.513

.915

.015

.0

Spac

ing

8 in

.(2

00m

m)

18.4

21.2

19.4

15.5

618

.713

.114

.714

.315

.515

.7

Spac

ing

12 in

.(3

00 m

m)

16.9

22.0

19.4

17.3

719

.815

.216

.414

.922

.116

.5

Pipe Depth4 in.

(100 mm)

Spac

ing

6 in

. (1

50 m

m)

24.5

27.8

28.4

24.7

627

.723

.223

.823

.524

.824

.8

Spac

ing

8 in

.(2

00 m

m)

24.1

28.4

28.8

25.5

828

.324

.024

.424

.125

.325

.4

Spac

ing

12 in

. (3

00 m

m)

23.6

28.8

28.7

26.7

428

.8-

25.5

24.6

25.5

26.0



(0.87 W/m⋅K)

Pipe Depth2 in.

(50 mm)

Spac

ing

6 in

. (1

50 m

m)

2.3

2.4

2.2

1.97

1.5

1.0

1.6

2.7

2.1

1.5

Spac

ing

8 in

.(2

00 m

m)

2.5

2.4

2.2

2.13

1.7

1.1

1.7

2.6

2.5

1.6

Spac

ing

12 in

.(3

00 m

m)

2.5

2.4

2.4

2.29

1.9

1.3

2.0

1.9

2.0

1.8

Pipe Depth4 in.

(100 mm)

Spac

ing

6 in

. (1

50 m

m)

3.6

3.3

3.2

3.23

2.8

2.2

2.8

1.9

3.0

2.7

Spac

ing

8 in

.(2

00 m

m)

3.5

3.3

3.3

3.34

2.9

2.2

3.0

2.0

3.5

2.7

Spac

ing

12 in

. (3

00 m

m)

3.6

3.5

3.4

3.42

3.0

2.3

3.1

3.0

3.1

3


TAB

LE

7

Max

imum

Per

cent

age

Edg

e Lo

sses

(Fre

e A

rea

Rat

io =

1.0

)

Cit

ySp

okan

eR

eno

SLC

Col

. Spr

.C

hica

goO

KC

Min

neap

olis

Buf

falo

Bos

ton

Phi

lade

lphi

a

no insulation layer


(0.87 W/m⋅K)

Pipe Depth2 in.

(50 mm)

Spac

ing

6 in

. (1

50m

m)

23.7

27.2

26.8

20.8

21.6

17.3

18.7

20.4

20.9

19.6

2

Spac

ing

8 in

.(2

00 m

m)

23.1

26.4

25.6

20.2

20.7

16.4

17.8

19.4

19.6

18.0

1

Spac

ing

12 in

.(3

00 m

m)

19.6

22.7

22.8

17.7

17.2

13.8

15.6

16.4

16.4

14.8

4

Pipe Depth4 in.

(100 mm)

Spac

ing

6 in

.(1

50 m

m)

27.2

29.4

20.2

25.8

24.2

22.3

23.1

24.7

25.3

23.7

9

Spac

ing

8 in

.(2

00 m

m)

24.9

28.7

29.3

24.5

22.9

21.1

22.2

23.6

24.2

22.8

5

Spac

ing

12 in

. (3

00 m

m)

21.8

26.1

26.6

21.9

19.8

-20

.020

.821

.219

.73


(2.4 W/m⋅K)

Pipe Depth2 in.

(50 mm)

Spac

ing

6 in

. (1

50 m

m)

27.3

33.0

34.3

27.6

28.1

23.4

24.3

26.3

27.0

25.0

7

Spac

ing

8 in

.(2

00 m

m)

26.9

31.9

32.7

26.8

26.2

21.9

22.8

24.9

25.0

22.9

4

Spac

ing

12 in

. (3

00 m

m)

20.9

28.1

28.4

23.1

23.1

17.9

19.2

20.9

20.4

18.6

6

Pipe Depth4 in.

(100 mm)

Spac

ing

6 in

. (1

50 m

m)

30.5

36.0

36.8

32.6

32.0

28.9

29.1

31.2

31.6

29.5

0

Spac

ing

8 in

.(2

00 m

m)

28.5

35.0

35.7

31.3

31.0

26.7

27.6

29.9

30.0

27.9

9

Spac

ing

12 in

. (3

00 m

m)

24.5

31.1

31.7

27.0

23.3

-24

.325

.825

.424

.03

2 in.(50 mm) insulation layer


(0.87 W/m⋅K)

Pipe Depth2 in.

(50 mm)

Spac

ing

6 in

. (1

50 m

m)

4.1

4.0

4.0

3.1

2.9

2.2

3.0

3.0

2.8

2.48

Spac

ing

8 in

.(2

00m

m)

4.1

3.8

3.8

3.0

2.8

2.1

3.0

2.9

2.7

2.37

Spac

ing

12 in

. (3

00 m

m)

3.6

3.6

3.6

2.9

2.5

1.8

2.7

2.6

2.4

2.14

Pipe Depth4 in.

(100 mm)

Spac

ing

6 in

. (1

50 m

m)

5.0

4.8

4.8

4.1

3.8

2.9

4.0

3.8

3.7

3.30

Spac

ing

8 in

.(2

00 m

m)

4.8

4.6

4.7

4.0

3.6

2.8

3.9

3.7

3.6

3.18

Spac

ing

12 in

. (3

00 m

m)

4.3

4.4

4.4

3.7

3.3

2.5

3.6

3.4

3.3

2.91


The depth is relatively important—the deeper tubing has,as expected, higher back losses. Naturally, this is more signif-icant for uninsulated slabs and higher conductivity soil.

CONCLUSIONS

A new computer tool for transient analysis of snow-melt-ing systems has been developed. The key features include thefollowing:

• In contrast to previously reported simulations, the toolperforms transient simulations and accounts for backlosses. Furthermore, it uses a boundary-fitted grid toaccurately represent the geometry of the hydronic tubingor embedded cable.

• A boundary conditions model accounts for accumula-tion and melting of snow and ice.

• A library of design storms, compiled from 12 years ofweather data, allows the program user to select samplestorms from a wide range of U.S. locations.

• The program is capable of modeling control systemswhere the system is automatically turned on when snowis detected, as well as scheduled operation, which couldbe used to model a system that is idled.

• The program has a user-friendly interface with librariesfor pavement properties, tube types, etc.

Besides the development of the software, a parametricstudy of snow-melting systems that investigated the effects oftransient conditions, back losses, edge losses, tube spacing,tube depth, insulation, and soil conductivity on system perfor-mance was conducted. Key findings for cases with a free arearatio of one include the following:

• The 99% steady-state nonexceedance loads are notclosely correlated to the performance of the systemunder transient conditions with back losses. In otherwords, the heat flux required to maintain the pavementsnow-free for the number of hours with steady-stateloads less than the 99% nonexceedance loads may beanywhere from one to five times as high as the steady-state 99% nonexceedance load.

• Many of the calculated fluxes exceed, by significantmargins, a level that is feasible to obtain in practice. Thevery high fluxes required indicate that, in many situa-tions, it is impractical to meet the design goal of main-taining a free area ratio of one at the same statisticallevel assumed in the steady-state analysis without idlingthe system.

• More than any other factor, the results are most sensitiveto the storm itself. Storms that start off with relativelylow loads, perhaps even ceasing to snow for a few hours,then increasing in intensity, will have much lower ratios(of heat flux required under transient conditions withback losses to the 99% nonexceedance load) than stormsthat start off with high loads.

• After the storm, the results are most sensitive to the

spacing. The farther apart the tubes, the more difficult itis to maintain a completely snow-free surface. Also, asthe tubes are placed deeper, the effect of spacing is lessimportant.

• The required design heat fluxes under transient condi-tions with back losses are almost completely insensitiveto the soil conductivity and whether or not insulation hasbeen installed. Insulation is, however, useful for reduc-ing back losses, particularly as the snow event increasesin time.

For a free area ratio of zero, the results are much moredifficult to interpret. With respect to the importance of thestorm, the tube spacing, and the tube depth, the results aresimilar to those for a free area ratio of one. However, withrespect to the ratio between the heat flux required under tran-sient conditions with back losses to the 99% nonexceedanceload, the results depend highly on how the concept of a freearea ratio of zero is mapped to transient conditions.

As to recommendations for future work, we found thereis almost no published literature that describes any experimen-tal measurements of snow melting. Experimental validationwould be worthwhile. Two kinds of experimental validationmight be considered: first, laboratory scale testing of snowmelting under highly controlled conditions. This would beuseful to characterize the transport of water through the snowand runoff. A second type of experimental validation wouldinvolve testing of sample sections of snow-melting systemsunder actual snow conditions.

There is some question about how conservative the resultsin this work are, due to the fact that solar radiation wasassumed to be zero. While this is probably a very reasonableassumption during the storm event (and in steady-state calcu-lations), assuming this to be the case for weeks beforehandprobably causes the temperatures in the slab at the beginningof the storm to be somewhat low, compared to reality. There-fore, it might be prudent to examine the possibility of obtain-ing the solar irradiation data, and then modifying thesimulation program developed in this work to include theeffects of solar irradiation.

ACKNOWLEDGMENTS

The research reported in this paper was funded underASHRAE research contract 1090-RP and sponsored by TC6.1, Hydronic and Steam Heating Equipment and Systems.The authors are grateful for the guidance provided by theProject Monitoring Subcommittee during the study.

REFERENCES

ASHRAE. 1999. 1999 ASHRAE Handbook—Applications.Atlanta: American Society of Heating, Refrigeratingand Air-Conditioning Engineers, Inc.

Chapman, W.P., and S. Katunich. 1956. Heat requirements ofsnow melting systems. ASHAE Transactions 62:359-372.


Ferziger, J.H., and M. Peri. 1996. Computational methodsfor fluid dynamics. Berlin: Springer-Verlag, pp. 364.

Gordon, W.J., and C.A. Hall. 1973. Construction of curvilin-ear coordinate systems and applications to mesh genera-tion. Int. Journal of Numerical Methods in Engineering.7: 461-477.

Kusuda, T., and P.R. Achenbach. 1965. Earth temperatureand thermal diffusivities at selected stations in theUnited States. ASHRAE Transactions. Atlanta: Ameri-can Society of Heating, Refrigerating and Air-Condi-tioning Engineers, Inc.

Kilkis, I.B. 1994a. Design of embedded snow-melting sys-tems: Part 1, Heat requirements—An overall assessmentand recommendations. ASHRAE Transactions 100(1):423-433.

Kilkis, I.B. 1994b. Design of embedded snow-melting sys-tems: Part 2, Heat transfer in the slab—A simplifiedmodel. ASHRAE Transactions 100(1): 434-441.

Leal, M.R.L.V., and P.L. Miller. 1972. An analysis of thetransient temperature distribution in pavement heatinginstallations. ASHRAE Transactions 78(2): 61-66.

Ramsey, J.W., M.J. Hewett, T.H. Kuen, S.D. Petersen, T.J.Spielman, and A. Briefer. 1999. Development of Snowmelting load design algorithms and data for locationsaround the world (ASHRAE 926-RP), Final Report.

Atlanta: American Society of Heating, Refrigeratingand Air-Conditioning Engineers, Inc.

Ramsey, J.W., M.J. Hewett, T.H. Kuen, and S.D. Petersen.1999. Updated design guidelines for snow melting sys-tems. ASHRAE Transactions 105(1): 1055-1065.

Rees, S.J., and J.D. Spitler. 2002. A storm weather boundarycondition model for horizontal building and pavementsurfaces. To be submitted.

Schnurr, N.M., and M.W. Falk. 1973. Heat transfer designdata for optimization of snow melting systems.ASHRAE Transactions 76(2): 257-263.

Schnurr, N.M., and D.B. Rogers. 1970. Transient analysis ofsnow melting systems. ASHRAE Transactions 77(2):159-166.

Spitler, J.D. S.J. Rees, X. Xiao, and M. Chulliparambil.2001. Development of a two-dimensional transientmodel of snow-melting systems, and use of the model foranalysis of design alternatives, Final Report. Atlanta:American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Stone, H.L. 1968. Iterative solution of implicit approxima-tions of multidimensional partial differential equations.SIAM Journal of Numerical Analysis 5(3): 530-558.

Williamson, P.J. 1967. The estimation of heat outputs forroad heating installations. Transport Road ResearchLaboratory (U.K.: Report LR77), p. 63.

This paper has been downloaded from the Building and Environmental Thermal Systems Research Group at Oklahoma State University (www.hvac.okstate.edu) The correct citation for the paper is: Rees, S.J., J.D. Spitler and X. Xiao. 2002. Transient Analysis of Snow-melting System Performance. ASHRAE Transactions. 108(2): 406-423. Reprinted by permission from ASHRAE Transactions (Vol. #108, Part 2, pp. 406-423). © 2002 American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

http://www.hvac.okstate.edu/

Transient Analysis of Snow-Melting System Performance

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