———-‘ n‘v n v _4 a '- h3~'( g .‘ z~. ' ‘. v 2.2; x,.-, ‘ A ,J.- 91. LIN-M43. cum § § 7%.. ‘3: '3 :t "270 3“. 'E ' a. v -. 'J. ‘ a 3% 5:1; “'35 -..: autumn}; ' . 1 xx» 1.... ‘ 1"?qu W :vu'nr anvr’w-“1run "'{W 35,; ' 'ri 3. ‘r-ra ‘ w ‘U';‘_l-‘~ ‘ 1' ( a. "5 2n . '4 .v , g, t. - - v - - u; v v . ' ‘h " "” "*t‘ ' ' z ' ' ‘ . ' - . t »:x - , ”Qantas , . . , . :— r r ..1 a. “WV“. 57.6: . waw-y f . a " "4' MIN: Ar» _ , A ..., .r "133:. 1. . ‘ a' . L ‘ H 3 ‘ ,.. . . . y . - . .w , . I. ' " ' ' " A .. ‘,,..;.'...,...f ' M " v- w -... .- , .....1 .v. ..A,. I ) v0:i¢m’:.1.urr-rn r I” yr ' ‘ — . , -. r '1 - ‘. ‘ nyl‘ ' l . ’ ' v Van . ‘ M — - . ,..‘ r" . l . . . i '9' . lt’""'-‘":""'F’" - ‘I. I . r“ , ‘:..‘.. 71-. 4M»- '{firfr .. . ;_ .9 .. . .‘o J. . « . ...... .~ - ”‘ ”—- 2155:. ... ,2. ~- - - -;- u". . .5 ' . . ‘ ' , , , . . . ‘ . h , . ., ‘ I I . - l w" ‘ ‘l v .m— glvvfiiulv- H" x 7 ~ .. --— 33.1-4.6! . '0: W: q . .r.:. r,‘ .1?” ' v . v. ... . » . wm Viplvfi ..3. m .:.'-.‘,'.r.;’.:. 1,; " .. .. m2 and It is the solution to Eqs. (4.11), (4.12) and (4.13). Next, & is obtained in a similar manner. The result is d8] P g _ - 1, d1 (4.24) d, f0 (L4) 6, A. being the solution to Eqs. (4.16), (4.17) and (4.12). Also, Edi-2. is obtained as d8; P 68) 4 with 1. being the solution to the problem described in Eqs. (4.20), (4.21) and (4.12). The sensitivity of f to (o for various values of a) and ¢ for a 1% perturbation in a) is listed in Table 25. Similarly, the sensitivity of f to 4) for different values of a) and 4, for a 1% perturbation in 4, is listed in Table 26. Actual changes in f and the change predicted by the associated design sensitivity analysis method are also 45 tabulated. The relative error between the actual and approximate change is also tabulated in Tables 27 and 28. The sensitivity of 3, and 32 to a) for various values of r.) and 4) for a 1% perturbation in r.) are shown in Table 29 and Table 30. Table 33 and Table 34 Show the sensitivity of 3, and 32 to 4, for a 1% perturbation in 4, . In this case the heaters were assumed to be on during the previous day. Tables 31-34 also Show the % error in the computation of the sensitivities of g, and 32 with respect to the variables a) and ¢ using the adjoint variable approach (Exact) and the finite difference approach (Approximate, based on a 1% step size). Tables 27 and 28 also list the % error in the computation of the sensitivities of f with respect to to and 4; . The results for the sensitivity of 32 obtained with the design sensitivity analysis are very accurate. However, the results of the sensitivity of f and 3, are not as accurate due to the very small changes in f and 3,, leading to inaccuracy in the difference between ,1!) and ft?) , 8i” and g?) values in calculating the approximate change. The design sensitivity analysis method will be used on a daily basis to determine the required sensitivities. For each new value of a) and 4) the heat conduction equation is used to solve for the heat-flux at the Slab surface, q,, the maximum slab surface temperature, and the maximum heating element temperature. At the beginning of each day, the previous day’s temperature distribution is used as the initial conditions for the current day. The sensitivities are then calculated and a new 46 value for a) and 4) can be determined. The procedure is then repeated for the next day. However, with this model there is no guarantee that f will approach zero at optimality. Moreover, the cost of electric usage is not minimized. A model that would insure minimum cost of electric usage, while meeting the required loads is now formulated. The problem is to find to and 4, that will minimize the cost of electric usage subject to the system constraints. In mathematical form the problem is to find a) and 4) that minimize f the cost of electric usage, or f - K [G's-(mm where K = cost of electricity (S/KWH) g = heat source power (KW) subject to 813 821 833 In this case 1 P Siam-q, dtse WM)” 5 85°F (T )m s 130°F - :11 f- KLPIODSe ml")8(x-L)dxdt (4.26) (4.27) (4.28) (4.29) (4.30) 47 ——--0’Kf job 5 “(7’) )(t- 4TH);- L) M, (4.31) _ fo’. f” Se ’1”IL:3)[‘;4’]8(x- -L)dxdt (4-32) A controller that utilizes the information from the load prediction algorithm and the results of the optimization algorithm can be used to operate the heating mats. Figure 38 is an illustration of the controller functions in block diagram form. Hourly dry bulb temperatures for January for Chicago, Minnesota, Washington DC, and Las Vegas are shown in Figures 39-42. load profiles for different days in January were calculated for the four representative cities. The results are presented in Figures 43-58. These figures illustrate that some days are characterized by high load profile fluctuations, while others are characterized by nearly uniform load profiles. This suggests that no single strategy for operating the heating system over a period of one year will work. The load profiles change on an hourly, daily, and monthly basis. This fact makes it difficult to control the temperature of the air in the heated space. Hence, a controller such as the one described in this study is necessary to maintain a comfortable environment. For this reason such a heating system may not be suitable for heating residential or office buildings. However, it may be suitable for heating storage and maintenance facilities. An expensive control system may be economically unfeasible, however, if used in maintenance or storage facilities. CHAPTER V RESULTS, DISCUSSION AND CONCLUSIONS 5.1 Results and Discussion In Section 3.2.1 the results of varying the heater placement were presented. The results indicate that higher slab surface temperatures were achieved when the heaters were installed closer to the concrete slab than when they were installed at a deeper location. There was a 50% decrease in the maximum heat-flux delivered by the slab when the mat depth was increased from 7 to 13 inches. However, the corresponding slab surface temperature decreased by only 5%. Installing the mats even deeper into the ground resulted in lower slab temperatures and thus lower heat-flux at the Slab surface for the same energy input. The time lag between when the heater energy was input and when it was output at the slab surface was also affected by the depth of placement of the heaters. Even with the heaters installed at 2 inches beneath the slab, a 3 hour time lag was observed. However, time lags of up to 50 hours were recorded for mat depths of up to 2 feet. The reason for that is two-fold. First, with deeper mats, there is an increase in the storage capacity. Second, there is an increase in the thermal resistance between the heaters and the slab surface. This 48 49 results in the operation of the heaters at higher temperatures to meet the loads. Figure 59 shows that the deeper mats operated at higher temperatures. Hence, the slower diffusion of heat is a result of the combined effect of the larger storage volume being heated to a higher temperature. The performance of the heating system is also affected by the amount of energy input to the mats. Figure 60 shows the maximum slab surface temperature as a function of mat depth for several energy inputs. The results indicate that the maximum allowable slab surface temperature was not exceeded at any mat depth even when the mats were operated at full power for 5 hours. Figure 61, however, shows that the maximum allowable temprature of the heating element was exceeded when the mats were energized for more than 3 hours. Determination of the optimum mat depth was discussed in Chapter 111. It was also found that the response of the system depends on the sand properties. The thermal properties of the sand vary with moisture content. Figure 5 shows the thermal properties of sand for different sand moisture contents. In the next section the response of the system for different sand properties is discussed. 5.1.2 Effoct of Moisture Content on Sand Thormal Proportios In this section the importance of soil moisture content and its effect on the estimation of the heat loss through the building foundation is discussed. The information presented here is given in the Feasioilig Stody for Qollefl'ng Sit; Soil Chgacterization Thermal Propet'ty Data for Residential Construction (Salomone, 1988). 50 The impact of moisture in and around foundations on heat transfer is not completely understood. This has resulted in poor estimation of foundation heat loss. Computer models have generated results that differ by factors of two to three from measured data. The largest uncertainty in modeling heat and mass transfer around building foundations is the soil condition. To better estimate the seasonal energy loss through the foundation, some guidance in selecting the soil thermal properties must be provided to the building analyst. Information on where the moisture content is significant and should be included in the energy loss estimation, and where it can be ignored and its impact on heat transfer is also needed. The thermal resistivity of soils is determined by the type and distribution of the soil components: minerals, organic matter, moisture and air. The thermal resistivity of soils usually depends on composition (percent sand, silt, clay and organic matter), moisture content and dry density. Figure 62 shows the variation of sand thermal resistivity with moisture content. The figure shows that generally below the critical moisture content there is a large increase in the thermal resistivity for a small decrease in moisture content. In this region the thermal resistivity is considered to be unstable. Above the critical moisture content, the thermal resistivity is fairly constant and this region is called the "stable region". Hence, from this figure the thermal resistivity for a given moisture content can be determined. For soils with a minimum moisture content greater than the critical moisture content, the soil can be considered stable, and a constant value of thermal resistivity can be used. However, if the minimum moisture content is less than the critical 51 moisture content, the appropriate value for thermal resistivity would fall between the value of thermal resistivity established for the stable region and the thermal resistivity of the soil in the dry state. As moisture is added to the soil around the soil particles or wedges of water at the contacts, a path for the flow of heat bridges the air gap between the soil particles. By increasing the effective contact area between particles, the wedges of water greatly increase the thermal conductivity of the soil. As the moisture content increases further, the effective contact area no longer increases. Consequently, a significant increase in the thermal conductivity is not evident when more moisture is added to fill the pore space. The critical moisture content is then the moisture content at which no significant increase in thermal conductivity is observed. It has been shown by Salomone ( 1988) that the critical moisture content depends on density and soil type. The critical moisture content increases as density decreases and texture becomes finer. In addition to the importance of moisture contents in soils, the concept of moisture migration also plays a role in the estimation of foundation heat-flux. The concept of moisture migration has been explained by Radhakrishna (1980) as follows: "Heat flow through a wet soil occurs mainly by conduction. At high moisture levels, liquid fills the gaps between soil particles and provides a continuous medium, making the soil-water mass an efficient thermal conductor. When a significant heat-flux is emitted from a source within a soil mass, liquids tend to vaporize near the heat source and to condense in cooler regions away from it. On the other hand, capillary suction causes a liquid return flow that tends to maintain 52 a constant moisture distribution throughout the soil (promoting thermal stability). Once the amount of liquid in the soil drops below the critical moisture content, the opposing heat flow causes the liquid thermal bridges between the soil particles to break down more rapidly than the capillary suction can replace them. This in turn increases the thermal gradients in the soil, causing more moisture migration and resulting in a significant increase in the thermal resistivity around the heat source. This condition is termed thermal instability or thermal runaway." Therefore, in a purely diffusive field, the soil moisture content is usually lower in the higher temperature region and higher in the lower temperature region. Since the temperature distribution depends on the soil moisture flow and the thermal conductivity, which, in turn, is a function of the thermally driven moisture distribution, then the phenomenon of coupled heat and mass transfer exists in the ground. Results of computer simulations of a basement wall backfilled with a sandy soil have shown that the coupled effect increases the wall heat loss by 9% during the winter (Labs et al., 1988). Whereas for clay soil the coupled effect results in no appreciable differences (Shen, 1986). No coupled heat and mass transfer was included in this study. Despite the influence of the above factor on thermal conductivity of soils, it has been reported by Salomone (1988) that a majority of existing computer models that are used to calculate foundation heat-fluxes do not account for variation in soil thermal conductivity. But rather, constant values of soil thermal properties are often used. Because of the uncertainty in modeling heat and mass transfer around 53 uninsulated building foundations, more representative values of soil thermal properties are required. The uninsulated case is needed in order to optimize the foundation insulation levels. Knowledge of the energy savings of going from no insulation to the first increment of insulation is needed to determine optimum insulation levels. 5.1.3 Effect of Sand Moisture Content on Heat loss To gain a better understanding of the influence of sand moisture content on the heat loss calculations, numerous computer simulations were performed for various sand properties with different moisture contents. Figures 63-68 show the heat-flux at the slab surface for various moisture contents. The results clearly indicate the dependency of heat-flux values on sand moisture content. Figure 63 indicates that when sand properties with zero percent moisture content were used, the slab heat- flux was lower than when sand properties with 10% or 20% moisture content were used. However, when sand with 40% moisture content was used the heat-flux was lower than that for dry sand. Figures 64 and 65 suggest that when the heating elements are installed deeper in the ground and hence a thicker sand layer is used, no definite relation can be made between percent moisture content and heat loss. However, the results in these figures Show that moisture content also influences the heat diffusion rate. In all cases, when properties of dry sand were used in the computer simulations, the heat diffusion rate was the smallest. Figure 69 shows the time delay between when the energy was input to the heating elements, and when 54 it was output at the slab surface for various mat depths and heat inputs. As in the case of moist sand, the time delay was nearly independent of the energy input. However, it was found to be dependent on mat depth. When the heating mats were installed closest to the concrete slab the time delay was a minimum. When they were installed further away from the slab the time lag increased. With the heating mats installed about 6 inches from the slab surface the peak heat-flux occurred about 2 hours after the heating mats were turned off. However, for a mat depth of 21 inches about 56 hours elapsed between the time of the peak heat-flux and the time the energy input was terminated. Figure 70 Shows the peak heat-flux versus mat depth for various energy inputs. The results shown in Figure 71 indicate that the maximum allowable slab temperature of 85° F was exceeded at mat depths less than 9 inches when the mats were energized for more than 3 hours. Whereas, when sand properties with 20% moisture content were used, the maximum allowable slab surface temperature was exceeded at a mat depth of 7 inches when the mats were energized for 5 hours, as shown in Figure 60. Because of the influence of sand moisture content on heat-flux calculations, computer simulations with thermal properties of dry sand were used to determine the optimal insulation levels for the four representative US. cities selected for these studies. The results are shown in Figures 72, 73 and 74. In this case, when dry sand was used, the heat lost to the ground with no insulation was on the average 48% of the energy input. When an R-5 insulation layer was added the heat loss was 35%. When R-40 insulation was simulated the 55 heat lost to the ground was 12%, that is a 75% decrease from the case of no insulation. Tables 33-36 list the results of the life-cycle cost analysis for each of the representative cities. For all four cities, R-20 insulation was the optimum insulation level, based on life-cycle costs, to be used. In Chapter 111 different insulation levels were selected as Optimum for the different cities, when sand with a 20% moisture content was used. The results clearly indicate the effect of sand thermal properties on the optimum insulation levels to be selected for each city. The effect of sand moisture content on the maximum allowable heating elements temperature was also investigated. Figure 75 shows the maximum allowable heating element temperature for various energy inputs, when dry sand was used. The results indicate that the maximum allowable temperature of 130°F was exceeded when the heaters were turned on for more than one hour. Whereas, Figure 61 shows that the heaters could be safely turned on as long as three hours. However, the maximum allowable temperature was exceeded when the heaters were energized for more than 3 hours. The results presented in Figures 76-84 strongly emphasize the importance of using the appropriate soil thermal properties for foundation heat-flux calculations. Sand with a thermal diffusivity of 0.044 ftz/hr was used to generate these results. Figures 76-78 show the slab heat-flux history for various mat depths. Figures 79-81 represent the variation of the heat-flux for different sand bed thicknesses. The effect of varying insulation levels is presented in Figures 82-84. In this case, the results 56 suggest that some insulation must be added to reduce the heat loss to the ground. In addition, the results indicate that the first increment of insulation was sufficient. However, additional insulation was not economically feasible. Compared to the previous cases when dry sand and sand with 20% moisture content were used, the present case clearly demonstrates the effect of sand thermal properties on the heat- flux calculations, and hence on the performance of the heating system. The maximum heat-flux values were about 25% higher than for moist sand, and 1.6 times larger for dry sand. The heat diffusion rate was about 30% higher for moist sand, and 3.5 times that of dry sand. 5.1.4 Selection of Insulation Produog In general, insulation material can be placed on the outside or inside of the foundation walls or under the slab. In this study the insulation was placed horizontally below the heaters. In all cases the insulation is exposed to the soil or sand. Insulation materials, under these conditions, must not degrade or lose their thermal resistance when exposed to moisture. Acceptable materials are extruded polystyrene boards (XEPS) under any condition and molded expanded polystyrene boards (MEPS) for vertical applications when porous backfill and adequate drainage are provided (Labs et al., 1988). Exterior insulation can be installed on the outside surface of basement, crawl space, and slab-on-grade foundation walls or extend horizontally away from them. Exterior placement is referred to whenever the insulation is in contact with the soil. 57 When insulation materials are placed outside the foundation structure, the following considerations are applicable: 1. When placed vertically on a foundation wall, the insulation must have sufficient compressive strength to withstand the lateral pressures generated by the backfill soil without excessively deforming the insulation layer (and, therefore, changing the insulation’s thermal resistance). At a depth of 7 feet below ground, the lateral pressure is estimated to be between 200 psf and 450 psf. For a one-story basement depth or less, most rigid or semi-rigid insulations are strong enough to be serviceable (Labs et al., 1988). When placed horizontally, the insulation must be strong enough to resist the vertical compressive stresses in the soil. These stresses increase at approximately 120 psf per foot depth below the ground surface (labs et al., 1988). It must also resist the stresses and displacements imposed on it during backfilling and any subsequent settlement. The surface should be carefully prepared and compacted beneath the insulation to provide an even support to the insulation layer and backfilled above to prevent direct damage. labs et al. (1988) report that since waterproofing the foundation outside an external insulation is not recommended because of the problems of tracing a water leak, the insulation will usually be exposed to ground moisture conditions outside the waterproofing layer. The severity of this exposure can be mitigated by a good drainage 58 system around the foundation and by a polyethylene sheet to separate the insulation from the soil. An exterior insulation placed below grade is considered to be in an undesirable environment because of moisture gain of the insulation. Extruded polystyrene boards are acceptable under any condition. They have relatively low water absorption by total immersion (0.3%), their water vapor permeability is 1.1 perm-in, flexural strength is 50 psi, and compressive resistance of 25 psi. It loses some R-value with time but is very stable after an initial "aging" process. The thermal conductivity is usually given as a five year aged value. Insulation sheet thickness is usually limited to less than 4 inches. Another type of insulation that is also suitable for external placement is MEPS insulation. It has similar flexural and compressive Strengths as XEPS insulation, but it has a higher permeability to water vapor, and less resistance to moisture absorption. However, MEPS insulation is available in thicker sections (up to 32 inches) than most types of foam insulations, and is less expensive than XEPS insulation (labs et al., 1988). 5.2 Conclusions The following conclusions are supported by the results in this study: 1. The results indicate that the response of the system was strongly sensitive to the mat depth. Placing the mats close to the slab surface allows for quick response to changing loads, but it has the disadvantage of allowing the inside air temperature to drop rapidly when the mats are turned off. Deeper mats cannot respond quickly but the storage 59 capacity is greater which is needed in order to use off-peak electric power. The effect of varying the thickness of the sand bed beneath the mats on the response of the system was found to be insignificant. Since the purpose of providing a sand layer beneath the mats is to protect the heating elements from stones and hard objects that can cut into the mats and cause them to fail, a 2 inch sand layer was found to be sufficient. Determining the optimal insulation thicknesses based on life-cycle costs depends on fuel prices, insulation cost, economic parameters, and climate. The optimal insulation thickness determined in this study may change if different fuel prices, insulation costs, and economic parameters are used. The mathematical model for determining the optimal insulation thickness developed in Section 3.2, is more expensive (regarding computer time) than the parametric study. The optimal solution given by this model depends on the terminal time P and the grid size. The optimal solution must be obtained for P equal to at least one year. And to achieve accurate results the grid size must be small. For foundation heat-flux calculations the appropriate soil thermal properties must be selected. The response of the system is greatly influenced by the soil thermal properties. The thermal resistivity of soil changes with moisture content. It increases with moisture content 10. 60 up to the critical moisture content. Beyond that, the thermal resistivity tends to remain constant. Because of the difficulty to control the temperature of the air in the heated space, this heating system may not be suitable for heating residential or office buildings. However, it may be suitable for heating storage and maintenance facilities. When the heating mats are installed a predetermined distance below the concrete slab, there is a considerable time lag between thermostat demand and heat delivery to the space. Hence, controls that can detect load changes early must be provided for this type of slab- heating. A load prediction algorithm is needed to predict load changes 24 hours in advance in order to determine the Start time of energizing the mats, the period of charging, and whether a continuous or intermittent heating strategy should be deployed. The increased below-grade heat loss for slab heating results in a higher energy usage than for direct systems. Nevertheless, since off-peak power is used exclusively, the energy cost for slab heating could be less. Slab-heating systems share the advantage of other radiant heating methods in that the air temperature can be lower than for convective- type systems. In addition, they are quiet and clean. 61 5.3 Rooommondotiops for Fptore Work 1. Analysis of earth contact systems that (1) realistically couple the earth contact facets of a building to the other building elements including above grade roofs and walls, (2) model the effect of changes in the internal loads due to people, lights, equipment, and direct solar gain, on earth contact heat transfer, and (3) account for the interaction between various passive design strategies are needed. To make more useful predictions of the energy performance of common earth contact configurations (basements, crawl spaces, floor slabs) more attention is needed on modeling three-dimensional heat flow from the floor, from corners, and from the building as a whole. Detailed measurements of foundation heat transfer in buildings is needed. Because soil thermal properties are strongly influenced by moisture content, soil moisture retension characteristics, water table movement, and coupled heat and mass transfer which affect the moisture distribution around foundations, and associated heat transfer, more understanding of the above factors is needed. Soil thermal conductivity is the primary variable that influences heat loss or gain from earth contact surfaces and underground electric cables. Also, the impact of moisture in and around foundations on heat transfer is not completely understood. The biggest uncertainty in modeling heat and mass transfer around building foundations is the soil condition. Some systematic testing of soils is needed to better 62 select the soil thermal properties for foundation heat-flux calculations. In addition, studies of the influence of moisture on the heat transfer are also needed to determine when the moisture content is significant and should be included in the heat loss calculation, and when it can be ignored. Because optimization methods using economic analysis depend on fuel prices, insulation costs, and economic parameters, knowledge of fuel prices, insulation costs, and interest rates in the different cities of the United States is needed to determine optimal insulation levels for buildings in each city. TABLES 63 TABLE 1 List of Climate Cities City, State HDD Min. Max. (based on 65F) Temp (F) Temp (F) Minneapolis, MN 8,400 -22.5 92.5 Chicago, IL 6,600 -2.5 92.5 Washington DC 4,200 12.5 87.5 Las Vegas, NV 2,700 17.5 112.5 64 TABLE 2 Description of Prototype Building in Washington, DC Building Type Building gross floor area Building net conditioned area Number of zones Building Location North latitude West longitude Time Zone Number Daylight Savings Time Typical Weekday Operating Schedule Occupancy start hour Operating hours/day Summer Thermostat Schedule Beginning month Ending month Typical Occupied Schedule Weekdays .................. from Saturdays .................. from Sundays .................... from Thermostat Set Point Temperatures Summer occupied temperature Winter occupied temperature Winter unoccupied temperature Daylighting analysis Lighting system type Percent light heat to space Wall U-Factor (Btu/hr-ftz- F) Wall construction group Color correction Roof U-Factor (Btu/hr-ftz- F) Roof construction code Color correction Shading coefficient Window U-Factor (Btu/hr-ftz- F) Space mass code Leakage coefficient Occupied air change rate Warehouse 100,000 rt2 100.000 {12 5 38.5 deg 77 deg 5 No 8 10 May September 800 to 1700 900 to 1600 0 to 0 80 deg 70 deg 70 deg ’TI’TI'TI : No : Sus-Fluor : 100 : 0.087 : E : Medium : 0.056 : 9 Light : 0.55 : 0.68 : Medium : 3 : 0.5 air changes per hour 65 TABLE 3 Description of Prototype Building in Chicago, IL Building Type Building gross floor area Building net conditioned area Number of zones Building Location North latitude West longitude Time Zone Number Daylight Savings Time Typical Weekday Operating Schedule Occupancy start hour Operating hours/day Summer Thermostat Schedule Beginning month Ending month Typical Occupied Schedule Weekdays .................. from Saturdays .................. from Sundays .................... from Thermostat Set Point Temperatures Summer occupied temperature Winter occupied temperature Winter unoccupied temperature Daylighting analysis Lighting system type Percent light heat to space Wall U-Factor (Btu/hr-ftz- F) Wall construction group Color correction Roof U-Factor (Btu/hr-ftz- F) Roof construction code Color correction Shading coefficient Window U-Factor (Btu/hr-ftz- F) Space mass code Leakage coefficient Occupied air change rate : 6 : 70 deg Warehouse 100,000 rt2 100,000 rt2 5 : 42 deg 87.5 deg No 8 10 May September 800 to 1700 E 900 to 1600 Oto 0 80 deg '11’1'1'11 70 deg : No : Sus-Fluor : 100 : 0.087 : E : Medium : 0.056 : 9 : Light : 0.55 : 0.68 : Medium : 3 : 0.5 air changes per hour 66 TABLE 4 Description of Protorype Building in Minneapolis, MN Building Type Building gross floor area Building net conditioned area Number of zones Building Location North latitude West longitude Time Zone Number Daylight Savings Time Typical Weekday Operating Schedule Occupancy start hour Operating hours/day Summer Thermostat Schedule Beginning month Ending month Typical Occupied Schedule Weekdays .................. from Saturdays .................. from Sundays .................... from Thermostat Set Point Temperatures Summer occupied temperature Winter occupied temperature Winter unoccupied temperature Daylighting analysis Lighting system type Percent light heat to space Wall U-Factor (Btu/hr-ftz- F) Wall construction group Color correction Roof U-Factor (Btu/hr-ftz- F) Roof construction code Color correction Shading coefficient Window U-Factor (Btu/hr-ftz- F) Space mass code Leakage coefficient Occupied air change rate :No Warehouse 100,000 n2 100,000 it2 :No : E 5 : 44.5 deg : 93.1 deg 6 8 10 : June August :800 to 1700 '900 to 1600 0 to 0 80 deg F 70 deg F 70 deg F : Sus-Fluor : 100 : 0.07 : Medium : 0.047 ' 9 Light : 0.55 : 0.45 : Medium : 3 : 0.5 air changes per hour 67 TABLE 5 Description of Prototype Building in Las Vegas, NV Building Type Building gross floor area Building net conditioned area Number of zones Building Location North latitude West longitude Time Zone Number Daylight Savings Time Typical Weekday Operating Schedule Occupancy start hour Operating hours/day Summer Thermostat Schedule Beginning month Ending month Typical Occupied Schedule Weekdays .................. from Saturdays .................. from Sundays .................... from Thermostat Set Point Temperatures Summer occupied temperature Winter occupied temperature Winter unoccupied temperature Daylighting analysis Lighting system type Percent light heat to space Wall U-Factor (Btu/hr-ftz- F) Wall construction group Color correction Roof U-Factor (Btu/hr-ftz- F) Roof construction code Color correction Shading coefficient Window U-Factor (Btu/hr-ftz- F) Space mass code Leakage coefficient Occupied air change rate Warehouse 100,000 r12 100,000 it2 5 36.1 deg 115.1 deg 8 No 8 10 May October 800 to 1700 900 to 1600 0 to 0 80 deg 70 deg 70 deg 'TI'TI'TI : No : Sus-Fluor : 100 : 0.22 : E : Medium : 0.053 9 : Light : 0.55 : 0.81 : Medium : 3 : 0.5 air changes per hour 68 TABLE 6 Characteristics of Eq. (2.8) for three Values of the Weighting Factor 7 7 Number of Solution Truncation Name of Stability unknowns Method error Method 0 l Explicit 0141, (Ax)2,(Ay)2] Forward-time. Conditional central Space (FI‘CS), Euler Method. in 5 Implicit O[(At)2, (A102, (Ay)2] Midpoint-time, Unconditional cenu'al space, (MTCS); Crank- Nicolson. ‘ 1 5 Implicit O[At, (Ax)2,(Ay)2] Backward-time, Unconditional central space, (BTCS); laasonen Method. TABLE 7 Properties of Materials Used Material Thermal conductiviy pc Moisture content (Btu/hr-ft-F) (Btu/ft3-F) (Percent) Concrete .54 22.0 Sand .16 18.9 0 .36 22.0 5 .59 25.1 10 .86 31.4 20 .91 37.6 30 .92 43.8 40 Soil .50 20.0 Insulation .017 2.1 69 TABLE 8 Peak load Summary for Prototype Building in Washington, DC COOLING HEATING Time of Peak Jul hour = 17 Jan hour = 6 Outside Temp 87.5 deg F 12.5 deg F Sensible Latent Sensible (Btu/hr) (Btu/hr) (Btu/hr) Glass Solar 105,532 0 Glass Conduction 37,128 -121,992 Wall Conduction 19,001 -62,431 Roof Conduction 98,000 -322,000 Opaque Solar 84,737 0 Door Conduction 0 0 Misc Conduction 0 0 Occupants 153,491 217,368 0 Lights 94,690 0 Equipment 30,711 0 Misc Sensible 0 0 Infiltration 77,880 -277,956 Total 701,169 -7 84,379 Total Load/Area 7.0 (Btu/hr-ftz) -7.0 70 TABLE 9 Peak load Summary for Prototype Building in Chicago, IL COOLING HEATING Time of Peak Apr hour = 17 Feb hour = 5 Outside Temp 87.5 deg F -2.5 deg F Sensible Latent Sensible (Btu/hr) (Btu/hr) (Btu/hr) Glass Solar 106,387 0 Glass Conduction 37,128 -153,816 Wall Conduction 19,001 -78,718 Roof Conduction 98,000 -406,000 Opaque Solar 86,288 0 Door Conduction 0 0 Misc Conduction 0 0 Occupants 153,491 217,368 0 Lights 94,690 0 Equipment 30,71 1 0 Misc Sensible O 0 Infiltration 86,589 -364,306 Total 712,284 -1,002,839 Total Load/Area 7.1 (Btu/hr-ftz) - 10.0 71 TABLE 10 Peak Load Summary for Prototype Building in Minneapolis, MN COOLING HEATING Time of Peak Jul hour = 17 Jan hour = 6 Outside Temp 92.5 deg F -22.5 deg F Sensible Latent Sensible (Btu/hr) (Btu/hr) (Btu/hr) Glass Solar 108,335 0 Glass Conduction 31,590 -129,870 Wall Conduction 19,656 -80,808 Roof Conduction 105,750 -434,750 Opaque Solar 63,218 0 Door Conduction 0 0 Misc Conduction 0 0 Occupants 153,491 217,368 0 Lights 94,690 0 Equipment 30,711 0 Misc Sensible O O Infiltration 117,465 -401,941 Total 724,905 -1,047,369 Total Load/Area 7.2 (Btu/hr-ftz) -10.5 72 TABLE 11 Peak load Summary for Prototype Building in las Vegas, NV COOLING HEATING Time of Peak Jul hour = 17 Jan hour = 5 Outside Temp 112.5 deg F 17.5 deg F Sensible Latent Sensible (Btu/hr) (Btu/hr) (Btu/hr) I Glass Solar 89,853 0 Glass Conduction 82,134 -132,678 Wall Conduction 89,232 -l44, 144 Roof Conduction 172,250 -278,250 Opaque Solar 126,253 0 Door Conduction 0 0 Misc Conduction 0 0 Occupants 153,491 217,368 0 Lights 94,690 0 Equipment 30,71 1 0 Misc Sensible 0 0 Infiltration 166,471 -232,7 1 3 Total 1,005,086 -787,785 Total Load/Area 10.1 (Btu/hr-ftz) -7.9 73 TABLE 12 Building Energy Consumption for Various Insulation Levels Insulation Annual Energy Consumption (KWH) (hr-ftZ-F/Btu) Las Vegas Washington DC Chicago Minneapolis R-0 411,460 575.225 851.132 856.519 R-5 356,790 498,796 738,044 742,715 R-lO 333,771 466,616 690,429 694,798 R-20 313,630 438,458 648,765 652,871 R-30 304,998 426,390 630,910 634,902 R-40 302,121 422,368 624,957 628,912 TABLE 13 Fuel Cost Savings Insulation Annual Fuel Cost Savings ($/lineal ft) (hr-ft2-F/Btu) Las Vegas Washington DC Chicago Minneapolis R-0 - - - - R-5 1.49 2.08 3.07 3.09 R- 10 2.16 3.02 4.47 4.50 R-20 2.67 3.73 5.51 5.55 R-30 2.98 4.16 6.15 6.19 R-40 3.01 4.20 6.21 6.25 74 TABLE 14 Insulation Costs Inulation Level Total Cost (R-value) (S/Lineal Foot) (F-ftz-hr/Btu) 5 15.4 10 26.2 20 40.0 30 66.2 40 92.3 TABLE 15 Economic Parameters Used in the Analysis Inflation Rate (1) Fuel Price Inflation Rate (ip) After Tax Discount Rate (d) Finance Rate (m) State and Federal Tax Bracket (I) Down Payment Percentage (DP) Property Tax Rate (t,) Analysis Period in Years (N E) Mortgage Period in Years (NL) O&M Fraction (M s) Resale Value (R,) 5% 7% 10% 12% 30% 10% 1% 30 30 0% 0% 75 $383528:va DO coamcEmagnsmaB owaoEUuoEU mmwo> m3u>§ .88 Rue: eon mus—on 5 one mace =< omv w; Sm 3N m: 5: 3K mém mam own CNN hm _ on mam man an VNN o: o: :2. awn mam hmm EN ma Om Em a? mom vow X: X: 3:. _ .on Cam omm omm #43 ON Em mwm mom com new crew wén 0.vo Sm onm New m2 2 8..o Se 2N mom fiwm 55m odm oHN gm mwm 8N 0E m. - - - - - - - - m3 m3 mom EN c :52 020 :33 >23 552 020 :33 >8:— :EE 0:5 :33 >84 amended 32?-5 .23 3.00 296-85 :28. me>mm 360 296-85 35 $00 295-85 33m 8:235 wax—22‘ .80 296-83 2 mAmE. 76 TABLE 17 Annual Energy End-Use of Prototype Building in Washington, DC Electric Site (KWH) (MBtu) Heating Energy Electric Resistance 402,255 1,372.89 I Cooling Energy 0 0 Domestic Hot Water Energy 0 0 Building Miscellaneous Lights 77,589 264.81 Equipment 3,524 12.03 Consumption Totals 483,367 Unit Cost $0.036 Dollar Cost $17,401 $17,401 Site Energy (MBtu) 1,649.7 1,649.7 Source Energy (MBtu) 5,607.1 5,607 .1 77 TABLE 18 Annual Energy End-Use for Prototype Building in Chicago, IL Electric Site (KWH) (MBtu) Heating Energy Electric Resistance 595,197 2,031.41 Cooling Energy 0 0 Domestic Hot Water Energy 0 0 Building Miscellaneous Lights 77,589 264.81 Equipment 3,524 12.03 Consumption Totals 676,309 Unit Cost $0.036 Dollar Cost $24,347 $24,347 Site Energy (MBtu) 2,308.2 2,308.2 Source Energy (MBtu) 7,845.2 7,845.2 78 TABLE 19 Annual Energy End-Use of Prototype Building in Minneapolis, MN Electric Site (KWH) (MBtu) Heating Energy Electric Resistance 598,964 2,044.26 Cooling Energy 0 0 Domestic Hot Water Energy 0 0 Building Miscellaneous Lights 77,589 264.81 Equipment 3,524 12.03 Consumption Totals 680,077 Unit Cost $0.036 Dollar Cost $24,483 $24,483 Site Energy (MBtu) 2,321.1 2,321.1 Source Energy (MBtu) 7,888.9 7,888.9 79 TABLE 20 Annual Energy End-Use of Prototype Building in Las Vegas, NV Electric Site (KWH) (MBtu) Heating Energy Electric Resistance 287,734 982.03 Cooling Energy 0 0 Domestic Hot Water Energy 0 0 Building Miscellaneous Lights 77,589 264.81 Equipment 3,524 12.03 Consumption Totals 368,846 Unit Cost $0.036 Dollar Cost $13,278 $13,278 Site Energy (MBtu) 1,258.9 1,258.9 Source Energy (MBtu) 4,278.6 4,278.6 80 TABLE 21 Monthly Energy Consumption for Prototype Building in Washington, DC Month Electricity Torals Through (KWH) Month (KWH) Jan 89,631 89,631 Feb 72,638 162,269 Mar 58,313 220,581 Apr 30,946 251,528 May 28,905 280,433 Jun 11,264 291,696 Jul 8,960 300,656 Aug 9,064 309,720 Sep 18,969 328,689 Oct 25,050 353,739 Nov 49,381 403,120 Dec 80,247 483,367 TABLE 22 Monthly Energy Consumption for Prototype Building in Chicago, IL Month Elecrricity Totals Through (KWH) Month (KWH) Jan 114,188 114,188 Feb 98,583 212,771 Mar 83,523 296,294 Apr 47,996 344,290 May 42,668 386,958 Jun 20,695 407,653 Jul 12,437 420,090 Aug 13,215 433,304 Sep 29,669 462,973 Oct 35,034 498,007 Nov 71,662 569,669 Dec 106,641 676,309 81 TABLE 23 Monthly Energy Consumption for Prototype Building in Minneapolis, MN Month Electricity Totals Through (KWH) Month (KWH) Jan 125,159 125,159 Feb 100,846 226,005 Mar 82,925 308,930 Apr 46,151 355,081 May 23,205 378,286 Jun 19,941 398,227 Jul 13,291 411,518 Aug 16,499 428,018 Sep 20,763 448,780 Oct 40,828 489,608 Nov 77,199 566,807 Dec 113,269 680,077 TABLE 24 Monthly Energy Consumption for Prototype Building in Las Vegas, NV ' Month Electricity Totals Through (KWH) Month (KWH) Jan 69,095 69,095 Feb 46,498 115,593 Mar 35,546 151,139 Apr 18,655 169,794 May 18,785 188,579 Jun 9,488 198,067 Jul 6,937 205,004 Aug 7,139 212,144 Sep 11,042 223,185 Oct 34,618 257,803 Nov 44,675 302,478 Dec 66,368 368,846 82 TABLE 25 Sensitivity of Objective Function to Design Variable 0) for Various Values of 0) and (l) ‘° 0.2 0.4 0.8 1.2 1.6 4 2 -39.56509 -39.58174 -8.530717 6.603876 14.13012 4 -33.46498 2479936 —7.683368 8.939914 23.37762 8 -22.19187 -16.70038 -5.917057 4.476792 14.37560 TABLE 26 . Sensitivity of Objective Function to Design Variable 9 for Various Values of 00 and o 9 1 2 4 6 8 (0 0.2 0.651563 0.646303 0.632301 0.612745 0.585630 0.4 0.455344 1.052629 1.032117 1.003699 0.964971 0.8 -1.7l3885 0.940619 1.141854 1.128383 1.112934 83 TABLE 27 Sensitivity“ of Objecrive Function to Design Variable (r) (D flail-Am) flw—Aw) Approximate Exact E335); 0.2 17.883556 18.000122 -29. 141500 -39.565090 35.768 0.4 12.764373 12.938846 -21.809125 -39.581740 81.492 0.8 7.023667 7.136330 -7.041438 -8.530717 21.150 1.2 6.759366 6.661518 4.077000 6.603876 61.980 1.6 9.789231 9.471698 9.922906 14.130120 42.399 *¢ = 2 TABLE 28 Sensitivity* of Objective Function to Design Variable (I) (p flirt-Alp) flip—A6) Approximate Exact E333); 1 12.256743 12.252656 2.043500 0.455344 -77.718 2 12.864738 12.838117 0.665525 1.052629 58.165 4 14.229255 14.174500 0.684438 1.0321 17 50.798 6 15.629166 15.545157 0.700075 1.003699 43.371 8 17.056112 16.942379 0.710831 0.964971 35.753 0) 0.4 84 TABLE 29 Sensitivity of Maximum Slab-Surface Temperature to Design Variable (l) for Various Values of 0) and q) ‘9 0.2 0.4 0.8 1.2 1.6 8.047215 8.047573 7.552066 6.026809 4.840493 8.041915 8.037479 7.999564 7.871912 7.361530 8.047216 8.042663 7.994562 7.923996 7.826599 TABLE 30 Sensitivity of Maximum Heating-Element Temperature to Design Variable (l) for Various Values of (l) and o ‘0 0.2 0.4 0.8 1.2 1.6 43.39164 32.73029 25.38326 18.78799 14.32776 43.39164 31.72807 26.87 849 24.34707 21.54462 43.39162 31.72807 26.87870 24.48542 22.73963 N .- 85 TABLE 31 Sensitivity* of Maximum Slab-Surface Temperature to Design Variable (I) - Relative (l) g, (OH-ACO) g, ((1)—Ace) Approxrmate Exact Error % 0.2 71.349380 71.322769 6.652500 8.047215 20.966 0.4 72.696212 72.642937 6.659125 8.047573 20.850 0.8 75.336128 75.236053 6.254688 7.552066 20.742 1.2 77.618996 77.496712 5.095167 6.026809 18.284 1.6 79.475533 79.340271 4.226938 4.840493 14.515 3k 9 = 2 TABLE 32 Sensitivity* of Maximum Heating-Element Temperature to Design Variable (I) (l) g 2((1H'ACD) 82((D+A(D) Approximate Exact E233); 0.2 85.88414 85.71233 42.95250 43.39164 1.022 0.4 93.17790 92.92178 32.01463 32.73029 2.236 0.8 104.65557 104.25001 25.34769 25.38326 0.140 1.2 113.42435 112.97691 18.64338 18.78799 0.776 1.6 119.93759 119.47914 14.32659 14.32776 0.008 86 TABLE 33 Sensitivity* of Maximum Slab-Surface Temperature to Design Variable (l) _ . Relative (p 3 ,(¢)+Aq)) 3 ,(4) A0) Approxrmate Exact Error % 1 73.250633 73.248657 0.098800 0.121303 22.776 2 73.242897 73.243866 0024225 -0.029616 22.254 4 73.199142 73.199509 -0.018350 -0.021529 17.326 6 73.158585 73.158928 -0.017150 -0.020670 20.522 8 73.121185 73.121658 —0.023650 -0.028735 21.499 *m=04 TABLE 34 . Sensitivity“ of Maximum Heating-Element Temperature to Design Variable (I) _ ' Relative q) 32(q)+A¢)) 32(4) Ad) Approxrmate Exact Error % 1 95.224998 95.213409 0.579450 0.609339 5.158 2 95.222610 95.226059 -0.086225 -0.0897 10 4.042 , 4 95.057800 95.059402 -0.080100 -0.080896 0.994 6 94.905006 94.906502 -0.074800 -0.07 6326 2.040 8 94.764351 94.765762 -0.070550 -0.074035 4.940 0. 87 TABLE 35 Life-Cycle Cost Analysis for Prototype Building in Washington ,DC Insulation Fuel Life Fuel Life Total Life Level (R-value) Cycle Cost Cycle Cost Cycle Cost (F-ftz-hr/Btu) Savings 403 - 403 5 322 81 337 10 295 108 321 20 262 141 301 30 249 153 314 40 238 165 329 All costs are in dollars per lineal foot. TABLE 36 Life-Cycle Cost Analysis for Prototype Building in Chicago, IL Insulation Fuel Life Fuel Life Total Life Level (R-value) Cycle Cost Cycle Cost Cycle Cost (F-ftz-hr/Btu) Savings 596 - 596 5 477 119 492 10 436 159 562 20 387 209 426 30 369 227 434 40 352 244 443 All costs are in dollars per lineal foot. 88 TABLE 37 Life-Cycle Cost Analysis for Prototype Building in Minneapolis, MN Insulation Fuel Life Fuel Life Total Life Level (R-value) Cycle Cost Cycle Cost Cycle Cost (F-ftz-hr/Btu) Savings 600 - 600 5 480 120 495 10 440 160 465 20 390 210 429 30 371 228 435 40 354 245 445 ' All costs are in dollars per lineal foot. TABLE 38 Life-Cycle Cost Analysis for Prototype Building in Las Vegas, NV Insulation Fuel Life Fuel Life Total Life Level (R-value) Cycle Cost Cycle Cost Cycle Cost (F-ftz-hr/Btu) Savings 288 - 288 5 230 58 246 10 211 77 237 20 187 101 224 30 178 110 243 40 170 118 261 All costs are in dollars per lineal foot. FIGURES 2 _ _ _ _ _ L 9a é/g///////////////////////Z é/é/v \\\\\§\\\\\\\ \\\\\\\\\\ \\\\\\\\\\\\Q\\\\\\\\\ 89 ZZZ/£6, \§\§\\\\\\\\> ////////// §§§ 96/ \ % 1998 1989 ’A & 60 «We _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 6 4 3 2 300qu Hydro. Oil 8: Gas Nuclear Cool Electrical energy production by fuel in the US. Non—Utility Figure l. 91 Ambient Air Inside Air Concrete Slab INSULATION Figure 3. The deepheat concept. 92 __i__ 0 Tim Inside Air T1 ,,,,,,,,,,,,,,, A Concrete 0 T2 Slab Heating , Mats . Sand . Insulation 0 ith Control _,'."Ti """"""" Volume 'Tm """"" “ Soil 0 O . ................... Figure 4. The one-dimensional grid. A"i- 1' Axl ¢——M——+ —-k P i-l 1 i- 1/2 Figure 5. The interface conductivity. 93 thermal resistivity 4.4 Thermal Resistivity (h r—ft—OF/Btu) Moisture Content (Percent) Figure 6. Variation of sand thermal properties with moisture content. 0.0 40 (Ju/ZU!) ‘Kiwsnilia IowJaul Qslab (Btu/hr-ftz) Qslob (Btu/hr—ftz) 94 T ' I I I - 2—4 a.m. d -2.5" .= II 15_ s d, 2.5 _ .. ti=2n - 12 Sand with 20% moisture content 9— .. 6* _ 3— .. ~ \ O I I ' I ' O 24 48 72 96 Time (hours) Figure 7. Slab heat—flux as a function of mot depth. 18 ' I ' I ' I ' T F r ' I 4 2—4 a.m. _ 15__ “"2'5" di=2.5" _ d ti=2.' - Sand with 20% moisture content 12~ — 9.. 6.. 3.. O ' I ' I ' I ' I O 24 48 72 96 120 144 168 Time (hours) Figure 8. Slab heat—flux as a function of mat depth. 95 40 I I i fl 4 1 A—A 5—hr. Step ‘ . O—O 4—hr. Step ‘ A di=2.5 'n- G—O 3—hr. Step ‘3’: 32‘ ti-2 in. B—El 2-hr. Ste " l E ‘ Heat source strength 200 Btu/hr—ft2 « } Sand with 20% moisture content a 24— _ x q q 3 E ,L 16— _ O a) I -l d ‘6 a 8— - d 1 O I I T I I r 5 10 15 20 25 Mat Depth (inches) 30 Figure 9. Peak slab heat—flux versus mat depth for various energy inputs. 70 T L 1 I I T , I r G—é) 2—Hour Step ‘ B—El 55-Hour Step a A—A 4—Hour Step 56" 0-0 5—Hour Step —- g ‘ a .8 42_ Heat source strength 200 Btu/hr—ft2 .1 .3: Sand with 207. moisture content 8 4 . 0.. di=2.5" B 28‘ ti-2" _ (I) .§ . . t_. 14— _ I I I I I 15 20 25 Mat Depth (inches) 30 Figure 10. Time of peak heat—flux as a function of mat depth. 96 25 l . I I I I I 2—4 a.m. ‘ til-0.5” 'l d3=2.5" 20- ti=2u -‘ < \-\ . ._\ _ \\\ ‘ . \‘ ._\. 3.5—w \K;-\.._\ ‘ \\>"‘\..\ - ._ 0.0 1 ' I 'i | ' l ‘ l O 1 2 3 4 5 b Time (days) Figure 24. Variation of slab heat-flux for various insulation levels. QSIOb (Btu/hr—ftz) Qslab (Btu/hr—ftz) 103 10.0 v I I l I l I L 2—6 a.m. — R"0 - 25 — R-5 . rm. d-- . " ’\\ I —‘ R—iO /“ \"3. d =12 5" ------- R—20 _ xx 3 ' 7.5 \\.§ Figure 25. Variation of slab heat—flux for various insulation Time (days) levels. 8'0 ' l l 77 T T ' f 2—6 a m — R-O J — R—5 I d;=2.5" _ R—1O 6.4% ,A. \ ds=16.5” ------ R-ZO ‘ " — — R—4O ‘ ./\. 3.5 . 4 8- \ '-\_ Sand with 207. moisture content Figure 26. Variation of slab heat—flux for various insulation Time (days) levels. oslob (Btu/hr—ftz) QSIOb (Btu/hr—ftz) 104 I v I I I I 2—6 a.m. R—O J —— R—5 di‘Z-s” —- R—lO ds=20.5” ------- R—20 ‘ — — R—4O Time (days) Figure 27. Variation of slab heat-flux for various insulation levels. .p—I l T I 5 8 10 12 14 Time (days) Figure 28. Variation of slab heat—flux for various insulation levels. Total Life—Cycle Cost ($/Iineal ft) Total Life-Cycle Cost ($/lineal ft) 105 350 330~ 310— 290— 270-4 250 I 10 l 20 T 30 Insulation R—value (hr—ftZ-OF/Btu) Figure 29. Life—Cycle cost of insulation options for Washington 40 DC. I 10 Insulation R—value (hr-ftZ—oF/Btu) Figure 30. Life—cycle cost of insulation options for Chicago, I 20 l 30 40 IL. 106 Total Life—Cycle Cost ($/lineal ft) T T I l I 1 O 10 20 3O 4O Insulation R—value (hr—ftZ-OF/Btu) Figure 31. Life—cycle cost of insulation options for Minneapolis, MN. 260 I I I I ‘ I 240- Total Life—Cycle Cost ($/linea| ft) I I I I a I I O 10 20 30 4O Insulation R-value (hr—ftZ—oF/Btu) Figure 32. Life—cycle cost of insulation options for Las Vegas, NV. ‘107- 20 I I I I I I I T 50 H Peak - G—Q Time of Peak -50 (g: 15_ 2—4 a.m. IT dI=2.5" 4O é ‘ tI=2" ' 3 3 10— —30 .a 2 am ‘ x -20 8 5— l- 0. ~10 O T l ' T ' I I T O 5 IO 15 20 25 30 Mat Depth (inches) Figure 33. Variation of slab heat-flux with mat depth. (smou) need Io emu 108 16 I 1 I f I ' l ' I ' I ' I f I ’ - — 3hr. inter. q ------- 4hr. inter. .. -—-— 5hr.inter. 12— ,,.—.—.:..-.-..r.“...‘f..'.‘...’.'..f ...... ‘ A ...................... N . t: I -I di=2.5" " L { ti=2u .3 8‘ n _ m ds=8.5 v _Q ‘ Sand with 207. moisture content q C 7) O 4— -l O I V I I r fi I I I v I fi— 0 12 1 6 2O 24 28 32 36 Time (hours) Figure 354. Slab heat—flux with intermittent heating. 64 Sand with 20% moisture content 4 ... -.— ‘—_- —' ——-— — ................ .................... A N ’r‘ E t~-2” \ I 3 4‘ .. (13 dI=2.5 v _Q .. d8=8.5" 2 (D O 2.. . —— 7hr. inter. ------- 6hr. inter. - - 5hr. inter. r T I I I I r T I I I j T r O 4 8 12 16 20 24 28 32 Time (hours) Figure 35. Slab heat—flux with intermittent heating. 36 109 30 f I I I I T ' I ' I ....... Energy Input dI=2.5" . —— ds=12.5” .. ‘ —- ds=8.5” ”'2 24" —- (15:25" Sand with 20% moisture content - <37 ‘ ‘ L 18— _ .C 3 4 ' N e / 12— ' — :8 / U) ' _——-—-I o I .......... / ....«.--——---—-"“" ‘ O T I t I T 2: I i: I .. I i: ' ;. O 4 8 12 16 20 24 Time (hours) Figure 36. Slab heat—flux with intermittent heating. Qslab (Btu/hr-ftz) Time (hours) Figure 37. Slab heat—flux for various energy inputs. llO Houny and Hourly Ambient Ambient Temperature Temperature Measurements History Temperature Pre- Hourly Load Houfly Load diction Algorithm History Measurements Solution to 2 Solution of Adjoint Load Profile Pre- State Equations Variable Problem diction Algoritm Optimization Algorithm Slab Surface Mat Temperature Temperature f I Set Point Set Point __:=I Controller Slab Surface 1 1 Mat Temperature Thermostat — Thermostat Power Switch Figure 38. Block diagram of controller functions. Dry Bulb Temperature (0F) Dry Bulb Temperature (OF) 111 4'0 I ' I I I I f I I I I I 30- - .. .I 20- J 10— — O- ._ —10— Minneapolis, MN - _2O ' I T I ' I ' I ' I ' I ' I ' O 4 8 12 16 20 24 28 32 Time (days) Figure 39. Outside air temperature fluctuations in January. 50 I I I I I a If? I I I I I I 40— - ~ -I 30-I * 20- a 10- .. ‘ I 0- Chicago, IL A ‘10 I I I I I I fl T I I I I I I I O 4 8 12 16 20 24 28 32 Time (days) Figure 40. Outside air temeprature fluctuations in January. 3 4. 3 2 1 . _. «I. O O O O O O _ 6 5 4 3 2 1 Coy 830.353 325 be 55V SBanEE 92m \CQ Dry Bulb Temperature (0F) Dry Bulb Temperature (0F) 60 112 I ' 7 ' I r ' I I ' I r _ Washington. DC _1 a Ia I ' I T I ' I ' I fii I ' O 4 8 12 16 20 24 28 32 Time (days) Figure 41. Outside air temperature fluctuations in January. 50— 40— T I ' I I I ' I I I ' I ' ' I Las Vegas, NV . I I I I I I O 4 8 12 16 20 24 28 32 l I I I I T 1' Time (days) Figure 42. Outside air temperature fluctuations in January. 10‘ 6 4 ANOTEEEV 83 AV TC Cg €75}me ES #4 Load (Btu/hr—ftz) Load (Btu/hr—ftz) 113 IO 2e _ "‘ 'I O I I I I I I I I I I I I I I I I I I I f I 1 4 7 1O 13 16 19 22 25 Time (hours) Figure 43. Load profile for day 3 in January; Chicago, IL. 10 I I r I T I I I I T 89 _ 6—1/—v——/—/_/—-/—f — 4'— —I 2- _ O f I I I I I I I V I. I I I I I I I I I I I I 1 4 7 1O 13 16 19 22 25 Time (hours) Figure 44. Load profile for day 9 in January; Chicago, IL. 6 4 Aluiisnv 83 12 1IO 8 6 4 «Elisa» 33 «4 10 114 Load (Btu/hr—IIZ) . I , I 1 4 7 1O 13 16 19 22 25 I I I I I I I T I I I I T T j’ r r I f Time (hours) Figure 45. Load profile for day 19 in January; Chicago, IL. Load (Btu/hr—ftz) m I I I I I I if I I I r I I I T I t I T 4 7 1O 13 16 19 22 25 Time (hours) Figure 46. Load profile for-day 29 in January; Chicago, IL. 6 m AN:I:<33 83 2 4 2 0 «Elisa: 83 I‘]( Load (Btu/hr—ftz) Load (Btu/hr—ftz) 115 I fiI I I I 10 13 Time (hours) Figure 47. Load profile for day 3 in January; Las 16 I I r I 19 22 25 Vegas, NV. 41..,. 1013 Time (hours) p \1—1 16 I r I T 19 I Figure 48. Load profile for day IO in January; Las Vegas, NV. Amziiimv n93 Fi . 6 . 4v «melzsé 83 2 2 ‘ Load (Btu/hr-ft2) Load (Btu/hr—ftz) 8 I I I I fir I T I I I I r I I I fit 1 - 6— a 4— .4 2_. _ O I I I m I I I I I I I I I I I I I I I I I 1 4 7 1O 13 16 19 22 25 Time (hours) Figure 49. Load profile for day 19 in January; Las Vegas, NV. 8 I I I I ' I I I I fl T 6- _ 4—4 .— 2— 4 O I I I I I I rfi I I I I If fiI I I I r I 1 4 7 1O 13 16 19 22 25 Time (hours) Figure 50. Load profile for day 29 in January; Las Vegas, NV. 10‘ A1..-E\33 83 CI 10- @cliES Ex: Load (Btu/hr-ftz) Load (Btu/hr—ftz) 10 f I I I I I I T I I I I I I I If fi «1 6_ —-4 4d -— 2.. .. -I d O I I I I I I I I I I I r I r I II I I I I 1 4 7 IO 13 16 19 22 25 Time (hours) Figure 51. Load profile for day 3 in Jonuory; Minneapolis, MN. 10 I I I fifi I I I I I I I 1 -4 8A ._ 6— _ T -I 4— _ 2" 3 O T I I I I I I r I I I I I I I I I I T r I 4 7 IO 13 16 19 22 25 Time (hours) Figure 52. Load profile for day IO in January; Minneapolis, MN. RV PC 4 Aisliamv 2:: my IO 9» 6 4 @:L<3$ 83 Load (Btu/hr—ftz) Load (Btu/hr—ftz) 10 I I I I I I I I I I I I I I I I I I I I I I ‘ 1 89 .. 6— _ -——/—\\ 1 \—\_ 4_ _ 2— .. O T T Ifi I I I I I I I I I I I I I I I I I I 1 4 7 1O 13 16 19 22 25 Time (hours) Figure 53. Load profile for day 17 in January; Minneapolis, MN. 10 ' ' T . ' l I I l ' I I I I I I I I ' ' I 8-— u 6— ‘ 4.4 _. 2- _ .J . 0 fl I r I I I r I I I I I I I I I I I I I I I 1 4 7 1O 13 16 19 22 25 Time. (hours) Figure 54. Load profile for day 24 in January; Minneapolis, MN. 6 A.» €155sz coo... 2 6 4 $:L:\2E 83 2 ‘ Load (Btu/hr—ftz) Load (Btu/hr—ftz) 119 8 I T l I l I I T I I f T 6- _ 4— _ 2- _ O I I I I I I I I I I I I I I I I I I I I I I r 1 4 7 1O 13 16 19 22 25 Time (hours) Figure 55. Load profile for day 3 in January; Washington DC. 8 ' ' I I I I I I l I I ' I I I / V 6- _ ‘ I 4— _ s w 2-1 _. .1 . O ' ' I ' ' T I ' I ' I I ' ' I I ' I ' I I ' ' 1 4 7 1O 13 16 19 22 25 Time (hours) Figure 56. Load profile for day 9 in January; Washington DC. 6 4 $7538 83 2 6 ¢ «Elias BS 2 ‘ Load (Btu/hr—ftz) Load (Btu/hr—ftz) 120 8 I I I I r I I f ' T I I T f 6— _ 4— u 2— -1 O I I I I I I I I I I I I I I I I I I I I I I I 1 4 7 1O 13 16 19 22 25 Time (hours) Figure 57. Load profile for day 19 in January; Washington DC. 8 T I I I I I I F I f I ' I ' r 6— _ \ q .l 4— — l d 2-4 .1 O I I I I I I I I I I I I I I I I I I I I I I I 1 4 7 1O 13 16 19 22 25 Time (hours) Figure 58. Load profile for day 29 in January; Washington DC. A¢OV OuszcuvaCCQh 121 1 10 T I T I I I I I I m f I H dS=.2 ft e—e ds=1 ft f: H ds=2 ft 0 V 100- — at.) {Izzu 3 E di=2.5" <1) 0- Heot source on (2—4 a.m.) E 90- ,2 Sand with 207. moisture content 0) 8 1 3 O (f) +4 80“ "‘ O \- Q) I .\: 4 70 I I I I 1 I I Ill ‘ r; 1; f" w. 0 24 48 72 96 120 144 168 Time (hours) Figure 59. Variation of heating element temperature with mat depth. Maximum Slab Temperature (OF) Max. Heating Element Temperature (OF) 122 90 I I Y I r I I I T I G—O 2-Hr. Step ‘ B—E] 3—Hr. Step 1 A—A 4—Hr. Step 86d di-2.5 in. 0‘0 5—Hr. Step .— ‘ ti=2 in. . 82‘— 202 moisture content _ 78— d 74— _ 7o . , , r T , j , I , j , 5 8 1 1 14 17 20 23 26 Figure 60. Maximum slab surface temperature Mat Depth (inches) as a function of mat depth and energy input. 170 I T I T I T I T 1 Sand with 20% moisture content ds=8.5" 150—4 d;-2.5" — .. ”=2" —. 130— — -I -4 1 Ice — g . 90- — 4 . 70 I I I I I ' I I I I O 1 2 3 4 5 6 Hours Mats Energized Figure 61. Maximum heating element temperature as a function of energy input. 123 . Thermal Resistivity Critical moisture content Unstable Region Stable Region Figure 62. Variation of soil thermal resistivity with moisture content. Moisture Content Qslab (Btu/hr—ftz) 12 l I l ’\ LI 0:7 T // \ \ — — 207: _ 9'— ; (”is _ 40% _. \\ ds=4 5 _ If \ \'\.\\\ ti=2" .. / ~\\\\ d,=2.5 6— ll- {\2 O—Zam T l‘ \ka- .. a l i i 4 ' 5 Time (days) Figure 63. Effect of soil moisture content on slab heat—flux. Qslab (Btu/hr—ftz) l l I 1 ....... 0% —— 10% _ \ - — 20% I/ ‘ \ —“ 40% _ / \\ ds=8.5“ \ \ \ t;=2" - ///\\_\x d;=2.5" / \\\ ‘ i "‘s‘-\\. — l/ ‘I‘Q-r-Tfff \4Q I I I l O 1 2 3 4 5.) Time (days) Figure 64. Effect of soil moisture content on slab heat—flux. 5.00 I I l I I I I j ....... 0% .. A\ —' 10% - — 207. 3.75— N: I/ Q\ IL - I’ / .\. \\\ t'=2 ‘ { II/ . \\\ g d,=2 5" :5: 2.50- I7/ ............ _ ..{\\ o_2 om _ .0 _ l' \ .. \t-nf\. _ ..,_:‘\\ \T‘ 1.25— I/ 0.00 1’ , fl , . , . 2 3 4 Time (days) Effect of soil moisture content on slab heat—flux. U: Figure 65. 4.0 V I T I I I r *I II II T ....... 0% _ —- 10% fl - — 207. L — 407. 3 O— \ \ " \‘ ds=16.5" .4 ll \\ .\. \\ .. \_ d;=2.5 Qslab (Btu/hr-ftz) m o l {7“ \ // o I {\f 0.0 I l . I I F . I I O 1 2 3 4 5 6 7 Time (days) Figure 66. Effect of soil moisture content on slab heat—flux. QSIOb (Btu/hr—ftz) QSIOb (Btu/hr-ftz) 2.60 I | v 1 r i I I T I Y I 1 ....... 0% / _ \ —- 10% . /l,/ \t\ 0—2 O-m- — — 20% 4 § {2.5“ -——- 40% 1.95"“ /,// \\ dl d -—205 — /l/ A \ \\ i s— J I, / \.\\ tl—2 ‘ I/ / '\\‘\\ 1.30-« " \ \ _ ’I/ / ................... \\ d [I / ....... \\\\\\\ q I """""" ~- ”‘5: h/ i. """" -\..‘ 065* I . — I 4 ,J ~ I i/ . 0.00 I’m T ‘ i r I ' I i I r I 0 1 2 3 4 5 6 7 Time (doys) Figure 67. Effect of soil moisture content on slob heat—flux. 2.0 w I . I T I 1 , 1 .. E 0% * 6P0 —- 10% . /// \\ - - 20% l I \ —- 407 1 5" // \ o _ ’1’ /'A \. \\§~ .1 / / \.\.\\\\ d It . \_§\\ 1 0— ’/ / .\\\ _ II \\ x/ / ................................................... s I’/' ./ tl=2 ._ ........................ 4 0 5% I) / (15:24 5 _ ‘ I”/ 0-2 0 m ‘ I . d,=2.5 / OO 4"} ___A....[' t I Y r I I r I f O 1 2 3 4 5 6 7 Time (days) Figure 68. Effect of soil moisture content on slob heat-flux. 127 Time of Peak (hours) j ' I 1 1 V i T I ' i ‘ T ' I 4 6 8 10 12 14 16 18 20 22 Mot Depth finches) Figure 69. Time of peak heat—flux cs 0 functim of mot depth and energyinput. 60 1 l ' i l * r l ' I ' I v r r G—e 2—Hour Step ‘ G—EJ 3-Hour Step‘ A 50.0 A—A 4-Hour Step- 2: d;-2.5" O—O 5—Hour Step | d - t-=2" { 40- ' — 3 Dry sond El . . g 30— — L: d -1 E x q .. C (D ‘1 104 — '1 , -i O i T T l ‘ i ' J ‘ i i ' I f 4 6 8 1O 12 14 16 18 20 22 Mot Depth (inches) Figure 70. Peak heat—flux as 0 function of mot depth input and energy 128 100 I I I I . I v G—O 2—Hr. Step A ‘ G—B S—Hr. Step ‘ 3- A—A 4—Hr. Step V 94‘ d;=2.5" o—e 5—Hr. Step “ Q) E d ti-2H q 8 Dry sand 3 88— - E r a) q q f— .0 2 _ m E q 3 .§ x _. O 2 70 f fl I I ‘ I l 5 8 1 1 14- 17 20 23 Mat Depth (inches) Figure 71. Maximum slab surface temperature as a function of mat depth and energy input. 1O ' f I ' I ' I I ds=4.s~ —— R—o "‘ .\ d 2 5” ....... R_5 -4 i" ' — R-1O 8‘ \\ 0—2 a.m. —- R—20 ‘ ‘1; q "-_'\\ ‘ T 6— ":\\\\\ Sand with 0% moisture content _ .C E . \\\ . g -_.'I\\\ 4- ' \\ .— 3 '-\ \ \ U) \ 0 ‘ \\\ d 2 ."-~.\‘\ ‘ \ .-."-..\‘. \ \ \ . ...\.‘\\: ‘ ~ .. ........... -. ""“'M O I t I I I r I .r O 1 2 3 4- 5 Time (days) Figure 72. Variation of slab heat—flux with insulation R—value. Qslob (Btu/hr—ftz) 2 3 Time (days) Figure 73. Variation of slab heat—flux with insulation R—value. f I ' ' ds=8.5" — R—O _ F ....... R_5 ' ff‘\ \ s \ “i=2-5" —- R— 1 0 4m ' ~ \\ \ 0—2 a.m. —' R—ZO - “ — — R—4O 6F - '~..\\\‘.\ _ -o—l \ ' T 3 '._\ .\ Sand with 0% moisture content E \ 3 .. q—l co V .o 2— 2 (n O . 1 _ O ' I ' I I l 4 V 1 V l I [ Sand with 0% moisture content 05:12.5“ R_O 4 ....... R_5 _ di=2.5” __ R_1O 0—2 a m —- R—20 _ 3 — — R—4O 2- 1 __ O ' I ' I I I _ 0 1 2 3 4 5) Time (days) Figure 74. Variation of slab heat—flux with insulation R—value. 130 Max. Heating Element Temperature (0F) 7o . l . I . I . r - I . O 1 2 3 4 5 6 Hours Mats Energized Figure 75. Maximum heating element temperature as a function of energy input for dry sand. 24 I F I I I I I 4 2—4 a.m. q 20_ ds=2'5” di=2.5“ _ .. t,=2" ‘1‘; 16— - L . . i 3 12— -* m J 8.5 a .D O B B—l \J O 12 « ‘\fil 4d — O I I Y I I I I I I I T O 4 8 12 16 20 24 Time (hours) Figure 76. Variation of slab heat—flux with mat depth for “sand=o'044 ftZ/hr. 131 2—4 a.m. .. d -2.5" 20- ' di=2.5" 03km Btu/hr—ftz) Time (hours) Figure 77. Variation of slab heat—flux with mat depth for asond=o.044 ftZ/hr. 2-4 a.m. 05km Btu/hr—ftz) Time (hours) Figure 78. Variation of slab heat—flux with mat depth for “sand=O-O44 rtZ/hr. oslob Btu/hr—ftz) 03km Btu/hr—ftz) 132 2—4 a.m. . 20_ (la-0.5" ds=4.5" 72 96 Time (hours) Figure 79. Variation of slab heat—flux for various sand layer thicknesses for “sand=O-O44 ftZ/hr. 2-4 a.m. I T I ' I I I O 24 48 Time (hours) Figure 80. Variation of slab heat—flux for various sand layer thicknesses for asond=0.044 ftz/hr. QSlOb Btu/hr-ftz) Qslob Btu/hr—ftz) 10.0 .“ 01 9‘ o 5" 01 0.0 20 133 l I ‘ T 2-4 a.m. 7 d =12" * d.-o.s-' 3 t;=2” . J I I I I T l I O 24 48 72 96 Time (hours) Figure 81. Variation of slab heat—flux for various sand layer thicknesses for asond=0.044 ftZ/hr. l ‘ l t T — — w/o insulation — w/ insulation 2—4 a.m. ds'4-5” di=2.5" Time (hours) Figure 82. Variation of slab heat—flux for various insulation levels for asond=0.044 ftz/hr. Qslob Btu/hr—ftz) Qslob Btu/hr—ftz) 134 12 v 1 v I I — - w/o insulation ‘ —— w/ insulation . 2-4 a.m. g— 6-1 3_ O O 24 48 72 Time (hours) Figure 83. Variation of slab heat—flux for various insulation levels for asond=0.044 ftZ/hr. 10.0 I I I r r f -- - w/o insulation ‘ -— w/ insulation q 2-4 a.m. d =12" 7.5~ 3 — 5.0— ~ 2.5“ —l 0.0 v I I I I I T O 24 48 Time (hours). 96 Figure 84. Variation of slab heat—flux for various insulation levels for asond=0.044 ft2/hr. REFERENCES REFERENCES ASHRAE 1985. ASHRAE handbook - 1985 fitndamentals. American Society of Heating, Refrigerating, and Air-Conditioning Engineers, Atlanta, GA. ASHRAE 1987. ASHRAE handbook - HVAC systems and applications. American Society of Heating, Refrigeration, and Air-Conditioning Engineers, Atlanta, GA. Brandemuehl, MJ. and Beckman, WA. 1979. "Economic evaluation and optimization of solar heating systems." Solar Energy, Vol. 23. Christian, J .E. and Strzepek, W.R. 1987. "Procedure of determining the optimum foundation insulation levels for new low-rise residential buildings." ASHRAE Transactions, Vol. 93, Part 1. Class notes, 1990. Applied Energy Conversion, University of Michigan, Ann Arbor, MI. DOEZ Engineers Manual Version 2. IA 1982. Energy and Environmental Division, Building Energy Simulation Group, Lawrence Berkeley Laboratory, Berkeley, CA. Green, MA. 1982. Solar Cells, Operating Principles, Technology, and System Applications. Prentice Hall, Inc. Englewood Cliffs, NJ. Harrison, E. 1959. "The intermittent heating of buildings by off-peak electricity supplies." Journal of the Institute of Heating and Ventilating Engineers, Vol. 27. [ES 1984. Lighting Handbook Reference Volume. Illuminating Engineering Society. Kedl, RJ. 1983. "Assessment of energy storage technology." Oak Ridge N ationa] laboratory Report ORNL/TM - 8997. Labs, K., Carmody, J ., Sterling, R., Shen, L., Huang, YJ., and Parker, D. 1988. Building foundation design handbook. Oak Ridge National laboratory Report ORNL/ Sub 86-72143/ 1. 135 136 MacArthur, J .W., Mathur, A., and Zhao, J. 1989. "On-Line recursive estimation for load profile prediction." ASHRAE Transactions, Vol. 95, Part 1. NBS. Comprehensive Guide for Least Cost Energy Decisions (Special publication 709) US. Government Printing Office, Washington DC. Nevins, R.G., Michaels, KB, and Feyerherm, A.M. 1964. "The effect of floor surface temperature on comfort: Part 1, college age males." ASHRAE Transactions, Vol. 70. Papalambros, P.Y. and Wilde, DJ. 1988 Principles of Optimal Design: Modeling and Computation. New York: Cambridge University Press. Parker, D.S. and Carmody, J .C. 1988. "Economic optimization of building foundation insulation levels." ASHRAE Transactions, Vol. 94, Part 2. Patankar, S.V. 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, McGraw-Hill, N .Y. Radhakrishna, H.S., Chur, F.Y., and Boggs, A.A. 1980. ”Thermal instability and its prediction in cable backfill soils." IEEE Transactions on Power Apparatus and Systems, Vol. PAS-99, No. 3. Salomone, L. R. 1988. F easibilily study for collecting site soil characten'zation thermal property data for residential construction. Oak Ridge National Laboratory ORNL/ Sub / 86-04923 / 1. General References Ackerman, M. and Dale, J .D. 1987. "Measurement and prediction of insulated and uninsulated basement wall heat losses in a heating climate." ASHRAE Transactions, Vol. 93, Part 1. Beck, J .V. and Arnold, KJ. 1977. Parameter Estimation in Engineering and Science. John Wiley & Sons, New York. Beck, J .V., McLain, H.A., Karnitz, M.A., Shonder, J.A., and Segan, E.G. 1988. "Heat losses form underground steam pipelines." Transactions of the ASME, Vol. 110. Bonacina, C. and Comini, G. 1973. "On the solution of the nonlinear heat conduction equations by numerical methods." International Journal of Heat and Mass Transfer, Vol. 16. 137 Borresen, BA. 1990. "Controllability - Back to Basics." ASHRAE Transactions, Vol. 96, Part 2. Braun, J .E. 1990. "Reducing energy costs and peak electrical demand through optimal control of building thermal storage." ASHRAE Transactions, Vol. 96, Part 2. ’ Brian, P.L.T. 1961. "A finite difference method of high-order accuracy for the solution of three-dimensional transient heat conduction problems." A.I.Ch.E. Journal, Vol. 7. Brodrick, J .R. 1989. "The equipment R&D benefits of characterizing the energy requirements of office buildings and multifamily housing." ASHRAE Transactions, Vol. 95. Campbell, J. 1990. "Calculation of heat requirements with intermittent heating." ASHRAE Transactions, Vol. 96, Part 1. Ceylan, TH. and Myers, GE. 1980. "Long-time solutions to heat conduction transients with time-dependant inputs." Journal of Heat Transfer, Vol. 96, Part 2. Christian, J. E. 1988. "Foundation futures: energy saving opportunities offered by ASHRAE Standard 90.2P." ASHRAE Transactions, Vol. 96, Part 2. Cleaveland, J .P. and Akridge, J .M. 1990. "Slab-on-grade thermal loss in hot climates." ASHRAE Transactions, Vol. 96, Part 1. ‘ Crawley, DB. and Huang, Y.J. 1989. "Using the office building and multifamily data bases in the assessment of HVAC equipment performance." ASHRAE Transactions, Vol. 95, Part 1. Douglas, J. and Rachford, H.H. 1956. "On the numerical solution of heat conduction problems in two and three space variables." Trans. Am. Math. Soc, 82. Huang, Y.J., Ritschard, KL, and Fay, J. M. 1989. "DOE-2.1D data base of building loads for prototypical multifamily buildings." ASHRAE Transactions, Vol. 95, Part 1. Huang, Y.J., Shen, L.S., Bull, J., and Goldberg, L. 1988. "Whole-house simulation of foundation heat-flows using the DOE-2.1C program." ASHRAE Transactions, Vol. 94, Part 2. J aluria, Y. and Torrance, KE. 1986. Computational Heat Transfer. Hemisphere Publishing Corporation, A subsidiary of Harper & Row, Publishers, Inc. 138 Krarti, M. and Claridge, DE. 1988. "Analytical calculation procedure for underground heat losses." ASHRAE Transactions, Vol. 94, Part 2. Kusuda, T. and Achenbach, RR. 1965. "Earth temperature and thermal diffusivity at selected stations in the United States." ASHRAE Transactions, Vol.96, Part 2. ' MacCluer, CR. 1989. "Temperature variations of flux-modulated radiant slab systems." ASHRAE Transactions, Vol. 95, Part 1. MacCluer, CR. 1990. "Analysis and simulation of outdoor reset control of radiant slab heating systems." ASHRAE Transactions, Vol. 96, Part 1. MacCluer, C.R., Miklavcic, M., and Chait, Y. 1989. "The temperature stability of a radiant slab-on-grade." ASHRAE Transactions, Vol.95, Part 1. Meixel, G. D. Jr. and Bligh, T.P. 1983. Earth Contact Systems: Final Report University of Minnesota, Underground Space Center DOE/SF/11508-TS. Mitalas, GP. 1987. "Calculation of below-grade residential heat loss: low-rise residential building." ASHRAE Transactions, Vol. 93, Part 1. Ozisik, MN. 1980. Heat Conduction. A Wiley - Interscience Publication, John Wiley & Sons. ' Parker, D.S. 1987. "F-factor correlations for determining earth contact heat loads." ASHRAE Transactions, Vol. 93, Part 1. Perry, E. H., Cunningham, GT. and Scesa, S. 1985. "Analysis of heat losses through residential floor slabs." ASHRAE Transactions, Vol. 91, Part 2A. Rosenfeld, A. and Moriniere, 0. de la, 1985. "The high cost-effectiveness of cool storage in new commercial buildings." ASHRAE Transactions, Vol. 91, Part 2. Smith-Gates Easyheat 1990. The foundation of a reliable heating system, Farmington, CT. Sterling, R., Meixel, G., Shen, L., Labs, K., and Bligh, T. 1985. "Assessment of the energy savings potential of building foundation research." ORNL/Sub/84- 002401 / 1, Oak Ridge National Laboratory. Thiyagarajan, R and Yovanovich, MM. 1974. "Thermal resistance of a buried cylinder with constant flux boundary condition." ASME Journal of Heat Transfer, Vol. 96. 139 Walton, G. N. 1987. "Estimating 3-D heat loss from rectangular basements and slabs using 2-D calculations." ASHRAE Transactions, Vol. 93, Part 1. APPENDICES APPENDIX A 140 APPENIHXI\ Sample Output of ASEAM2.1 (System Energy Requirement for building in Chicago, IL) Report System Cycle Month Bin Bin Oper Baseboard Temp Hours Hours Load. SB 1 1 1 42.5 1.7 1.7 219,632 SB 1 1 1 37.5 35.5 35.5 283,996 SB 1 1 1 32.5 95.2 95.2 348,359 33 ' 1 1 1 27.5 39.8 39.8 412,723 SB 1 1 1 22.5 20.6 20.6 477,087 SB 1 1 1 17.5 15.5 15.5 541,450 SB 1 1 1 12.5 19.4 19.4 605,814 SB 1 1 1 7.5 1.7 1.7 670,178 SB 1 1 1 2.5 0.8 0.8 734,541 SB 1 1 2 57.5 2.8 2.8 34,303 SB 1 1 2 52.5 11.1 11.1 78,466 SB 1 1 2 47.5 11.3 11.3 143,165 SB 1 1 2 42.5 13.4 13.4 207,864 SB 1 1 2 37.5 32.6 32.6 272,563 SB 1 1 2 32.5 35.4 35.4 337,262 SB 1 1 2 27.5 35.3 35.3 401,961 SB 1 1 2 22.5 30.0 30.0 466,660 SB 1 1 2 17.5 9.0 9.0 531,359 SB 1 1 2 12.5 10.0 10.0 596,058 SB 1 1 2 7.5 8.1 8.1 660,757 SB 1 1 2. 2.5 8.3 8.3 725,456 SB 1 1 2 -2.5 0.8 0.8 790,155 SB 1 1 3 67.5 5.3 5.3 0 SB 1 1 3 62.5 5.1 5.1 5,680 SB 1 1 3 57.5 6.2 6.2 24,988 SB 1 1 3 52.5 15.8 15.8 65,439 SB 1 1 3 47.5 16.4 16.4 127,734 SB 1 1 3 42.5 23.7 23.7 192,385 SB 1 1 3 37.5 75.6 75.6 257,036 SB 1 1 3 32.5 61.1 61.1 321,687 SB 1 1 3 27.5 18.2 18.2 386,337 SB 1 1 3 22.5 2.8 2.8 450,988 141 .0 9.5 0 4 82.5 11.9 11.9 .0 9.5 0 3 17.5 4 87.5 SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB 1 7 7 6.7 4 77.5 6.7 4 67.5, 14.2 '14.2 4 72.5 1 1 2,880 19,596 61,837 118,066 8.7 .0 24.0 4 52.5 23.6 23.6 8.7 4 47.5 47.4 47.4 4 62.5 4 57.5 24 1 .1 182,572 4 42.5 44.1 44 4 37.5 22.9 22.9 247,079 1 311,586 .0 .0 2 2.0 4 32.5 4 27.5 5 87.5 0 0 1 0 5 82.5 21.5 21.5 0. 1.7 1 5 77.5 41.6 41.6 5 72.5 39.7 39.7 14,543 53,992 103,227 5 67.5 20.5 20.5 1 1 1 5 62.5 11.9 11.9 5 57.5 29.0 29.0 167,123 5 52.5 26.9 26.9 5 47.5 30.3 30.3 231,018 294,913 9 0.4 .9 24.9 6. 6 87.5 31.1 31.1 6.9 0.4 5 42.5 358,809 5 37.5 1 1 1 1 0 6 92.5 24 6 82.5 42.6 42.6 6 77.5 28.4 28.4 6 72.5 42.1 42.1 10,057 45,753 91,710 6 67.5 21.1 21.1 1 6 62.5 23.5 23.5 6 57.5 9.1 0 0 9.1 1 1 0 .0 .0 6.4 7 87.5 58.3 58.3 7 82.5 74.5 74.5 7 77.5 37.2 37.2 7 72.5 40.7 40.7 7 67.5 11.7 11.7 7 62.5 7 57.5 0.0 6 52.5 6 47.5 0.0 6.4 7 92.5 1 1 1 11,011 42,554 1 1 1 .6 0.0 .6 1 0.0 0.0 1 0 0.0 7 52.5 142 8 92.5 16.5 16.5 8 87.5 43.4 43.4 1 SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB 6 82.5 43.6 43.6 a 77.5 62.7 62.7 a 72.5 43.0 43.0 8 67.5 19.2 ’19.2 1 13,024 40,913 .8 .0 .0 .8 1 0.0 0.0 1 8 62.5 8 57.5 1 0 0 0 9 92.5 22.6 22.6 8 52.5 9 87.5 13.4 13.4 9 82.5 20.5 20.5 1 1 9 77.5 21.5 21.5 9 72.5 35.5 35.5 18,630 47,127 106,368 9 67.5 26.5 26.5 9 62.5 47.8 47.8 1 1 9 57.5 30.5 30.5 169,764 .2 4 0.4 .2 4 0.4 9 52.5 9 47.5 9 42.5 10 82.5 233,160 0 0.0 6.0 0.0 6.0 1 10 77.5 16.2 16.2 10 72.5 21.9 21.9 10 67.5 23.8 23.8 10 62.5 33.0 33.0 10 57.5 61.8 61.8 10 52.5 35.9 35.9 10 47.5 21.0 21.0 10 42.5 10 37.5 10 32.5 11 62.5 1 1 1 5,556 24,568 57,875 120,609 SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB 1 184,366 248,124 9.0 9.0 1 1 1 1 .7 .7 0 11,017 41,474 91,573 155,842 0 6.0 0. 0.0 1 1 6.0 11 57.5 17.0 17.0, 11 52.5 45.8 45.8 11 47.5 41.9 41.9 11 42.5 38.1 38.1 11 37.5 33.6 33.6 11 32.5 26.0 26.0 11 27.5 11 22.5 11 17.5 11 12.5 12 57.5 12 52.5 1 1 220,111 284,380 1 348,649 1 1 1 412,918 .2 .1 .3 .0 7 4 7.2 477,187 .1 4 3.3 541,456 3 1 0 46,897 99,819 0 0.0 6.7 6.7 1 1 3.3 3.3 SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB Hlflrikifi|Hlflr‘P‘HlHI‘I‘I‘P‘P‘F‘H‘HIHP‘P‘H'HIHFJP‘HlHidF‘F‘H|Hldr‘h‘H'HI‘E‘P‘H'H MNNNNNNNNNNMNNNNNNNMNNNNNNNNNNNNNNHHHHHHHHHH HD‘F‘F‘H‘HI‘I‘I‘ hahohohahahiNlNHN ...: fiUUWUUWh’UUWh’NNNNNNNNNNNNNHHHHHHHHHN 143 UIUIUIUIUIUIUIMUMUIUIUIUIUIUIUUIUIMUIUIUIUIUIUIUIUIUIUIUIUIUIUIUIUIUIMUMU‘IUIUIUI 191. 200. MONOC‘UQUQOQ”GOOOOOQ‘GdOwNwGMfiUOUUDODNOQDNNwl-i 191. 200. MONOG‘UOUO’OQNODOOOOG‘MQOUNmebUQMWOUNQODNNWI-J 164,041 228,263 292,484 356,706 420,928 485,150 549,372 613,593 677,815 742,037 319,101 383,465 447,828 512,192 576,556 640,919 705,283 769,647 834,010 116,025 180,724 245,423 310,123 374,822 439,521 504,220 568,919 633,618 698,317 763,016 827,715 892,414 2,791 39,536 104,187 168,838 233,489 298,139 362,790 427,441 492,092 556,742 621,393 0 144 7 7 .1 .2 7 7 4 72.5 16.3 16.3 4 67.5 32.8 32.8 4 62.5 23.3 23.3 4 57.5 27.0 127.0 4 52.5 33.4 33.4 4 47.5 60.6 60.6 4 42.5 137.9 137.9 4 82.5 4 77.5 2 SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB 1,134 31,292 95,799 160,306 2 224,812 289,319 .1 353,826 4 32.5 45.0 45.0 .1 94 4 27.5 10.0 10.0 4 37.5 94 418,332 482,839 5 87.5 0.3 0.3 0 5 82.5 2 .5 5 77.5 25.4 25.4 5 72.5 39.3 39.3 5 67.5 69.5 69.5 5 62.5 74 5 5.5 84,079 147,974 2 2 .1 74.1 211,870 5 57.5 90.0 90.0 5 52.5 89.1 89.1 5 47.5 77.8 77.8 5 42.5 36.1 36.1 275,765 339,661 403,556 6.6 467,451 6 6 92.5 13.1 13.1 6. 5 37.5 2 0 6 87.5 12.9 12.9 6 82.5 34.4 34.4 2 2 6 77.5 48.6 48.6 6 72.5 84.9 84.9 6 67.5 98.9 98.9 6 62.5 81.5 81.5 2 75,424 138,909 2 2 202,394 6 57.5 93.9 93.9 6 52.5 25.0 25.0 6 47.5 265,880 329,365 .0 4 2 .0 .6 4 2 O 7 92.5 2 2 2 .7 7 87.5 24.7 24 7 82.5 50.5 50.5 7 77.5 83.8 83.8 7 72.5 171.3 171.3 7 67.5 111.3 111.3 7 62.5 55.4 55.4 73,846 ' 136,977 2 2 200,107 7 57.5 12.0 12.0 7 52.5 8 92.5 2 2 263,238 2.0 4 2.0 4 O .5 2 2 8 87.5 17.6 17.6 SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB SB F‘F‘k‘h‘h‘h‘h‘F‘P‘F‘F‘F‘F‘F‘F‘F‘F‘P‘F‘F‘h‘h‘F‘P‘F‘P‘P‘h‘F‘F‘hih‘klh‘F‘P‘P‘h‘k‘h‘h‘h‘k‘h‘ NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN UUD‘O‘DD‘ODDDDGGQQOGG 82. 77. 72. 67. 62. 57. 52. 92. 87. 82. 77. 72. 67. 62. 57. 52. 47. 42. 82. 77. 72. 67. 62. 57. 52. 47. 42. 37. 32. 62. 57. 52. 47. 42. 37. 32. 27. 22. 17. 12. 57. 52. 47. 42. 145 UIU'IUIUIUIUIUIUIUIUIUIUIMMMMMUIUIUIUIUIUIMUIMUIUIUIU'IUIUIUIUIUIUIUIUIUIUIUIUIUIUI 33.4 92.3 57.2 9.0 MWQQN UIUIOI§O 0: 86.5 G 0'! 0'! H H N N 84.5 (db ONO) WOOOUOOHNOUHOOOMO wmmnmmwu mmwwomwmmmewI-I H 33. 92. 166.0 166. 131.8 131. ’57. w 86. 126. 112. U'I'DOIQNIO dDdWOQDGOfi‘OI—‘UOOOUOOHMOUHQOOGQU‘INMUUMQ‘OONQOU" 0 0 0 80,822 143,693 206,564 269,435 0 0 0 0 0 95,109 158,504 221,900 285,296 348,692 412,087 0 0 0 4,094 46,102 109,859 173,617 237,374 301,131 364,889 428,646 63,929 128,198 192,467 256,736 321,005 385,274 449,543 513,812 578,081 642,350 706,619 131,938 196,159 260,381 324,603 SB SB SB SB SB SB SB SB HIHrdrah-Hidid NNNNNMNN 12 12 12 12 12 12 12 12 37. 32. 27. 22. 17. 12. UIUIUIUIUIUIUIUI 146 91. 108. 77. 64. 44. 29. 14. 13. OHQU’NHQQ 91. 108. 77. 64. 44. 29. 14. 13. cumulus-mm 388,825 453,047 517,268 581,490 645,712 709,934 774,156 838,378 APPENDIX B 147 APPENDIX B Computer Program Listing C************************************************************** C THIS PROGRAM SOLVES 1D, UNSTEADY HEAT EQUATION BY EMPLOYING C IMPLICIT CRANK-NICOLSON SCHEME FOR A COMPOSITE MEDIUM C THE OUTPUT WILL BE IN CN.OUT C************************************************************** C DESCRIPTION OF INPUT PARAMETERS C C TSLAB C NSLAB C DLS geese 000000000 "I 068 max Duo 0 0 (:52 O 2 000000000 SLAB THICKNESS. NO. OF CV S IN SLAB. POSITION OF LINE SOURCE W.R.T SLAB. NO. OF CV S IN REGION BETWEEN SLAB AND LINE SOURCE. DISTANCE FROM LINE SOURCE TO INSULATION. NO. OF CVS IN DIN. INSULATION THICKNESS. NO. OF CV 8 IN INSULATION. POSTION OF TGRD BELOW INSULATION. NO OF CVS IN DGRD. GROUND TEMPERATURE. T INFINITY. CONVECTIVE HEAT TRANSFER COEFFICIENT. K FOR CONCRETE. K FOR SAND. K FOR EARTH. K FOR INSULATION. RHO*C FOR CONCRETE. RHO‘C FOR SAND. RHO‘C FOR EARTH. RHO*C FOR INSULATION. TOTAL NO. OF GRID POINTS TIME STEP. GRID SIZE. THE MAXIMUM TIME. THE NUMBER OF TIME STEPS AFTER WHIC PRINTOUT OCCURS. THE SOLUTION VARIABLE AT THE NTH TIME STEP. THE SOLUTION VARIABLE AT THE N-lTH TIME STEP. ********************************************************************** DIMENSION T(3000),TOL(3000) DIMENSION A(30(X)).B(3000),C(3000),R(30(X)),SOLN(3000) REAL 148 TSLAB ,TIN,TINF,TGRD,HCON,KCONC,KSAND,MTIME,'I'IME ,PI,TINF1 , + KEARTHKINSLRHOCCRHOCSRHOCERHOCIRNIN ,KEQ1,KEQ2,KEQ3,L INTEGER NSLAB,NLS,NIN,NTIN,NGRD PRINT’l‘,’ENTER SLAB THICKNESS, TSLAB=’ READ(*,*)TSLAB ~ PRWT‘,’ENTER NO. OF CVS IN SLAB, NSLAB=’ READ(*,*)NSLAB PRIN'I“,’EN'I'ER LINE SOURCE POSTION, DLS=’ READ(*,*)DLS PRW,’ENTER NO. OF CVS IN DLS, NLS=’ READ(*,*)NLS PRIN'PJENTER DIN=’ READ(*,*)DIN PRIN'P,’ENTER NO. OF CVS IN DIN, NINR=’ READ(*,*)RNIN PRINT‘,’ENTER INSULATION THICKNESS, TIN=’ READ(*,“')TIN PRIN'I‘,’EN’I'ER NO OF CVS IN TIN, NTIN=’ READ(*,*)NTIN PRINT‘JENTER DGRD=’ READ(*,*)DGRD PW,’EN1ER NGRD=’ READ(*,"‘)NGRD PRIN'P,’DT=' READ(*,*)DT PRIN'I‘JENTER MAXIMUM NO. OF TIME STEPS, MTIME=’ READ(*,*)MTIME PRINT‘JPRINTOUT STEP, NSTEP=' READ(*,*)NSTEP PRIN'I‘JENTER HEAT GENERATION TERM=’ READ(*,*)GH PRIN'I‘h’TIME HEAT SOURCE ON?’ READ(*,*)TIME1 PRIvi'I'IME HEAT SOURCE OFF?’ READ("',*)’I'IMI~32 PRINT‘,’INITIAL CONDITIONS=?’ READ(*,*)TINT OPEN(10,STATUS=’OLD’,FILE=’CN.DAT’) READ(10,5)TINF,TGRD,HCON,KCONC,KSAND,KEARTH,KINSL,RHOCC,RHOCS, + RHOCERHOCI 5 FORMAT(7(1X,F9.4)/4(1X.F9.4)) NIN=INT(.5+RNIN) TSLAB=TSLAB/(12.) DLS=DLS/(12.) TIN=TIN/(12.) DIN=DIN/(12.) DGRD=DGRD/(12.) C 149 DX l=TSLAB/(NSLAB+.5) DX3=DIN/RNIN DX2=(DLS-.5*DX3)/NLS DX4=TININTIN DX5=DGRD/NGRD ALFAC=KCONCIRHOCC ALFAS=KSANDIRHOCS ALFAE=KEARTHIRHOCE ALFAI=KINSLIRHOCI KEQl=KCONC*KSAND*(DX1+DX2)/(KCONC* DX1+KSAND*DX2) KEQ2=KSAND*KINSL*(DX3+DX4)/(KSAND" DX3+KINSL“DX4) KEQ3=KINSL*KEARTH*(DX4+DX5)/(KINSL*DX4+KEARTH*DXS) ALFAQ1=KEQIIRHOCC ALFAQZ=KEQ1IRHOCS ALFAQ3=KEQ2/RHOCS ALFAQ4=KEQ2/RHOCI ALFAQ5=KEQ3/RHOCI ALFAQ6=KEQ3IRHOCE FOC=ALFAC*DT/(2."DX1**2) FOSl=ALFAS*DT/(2.*DX2"2) FOSZ=ALFAS*DT/(2.*DX3**2) FOI=ALFAI*DT/(2.*DX4**2) FOE=ALFAE*DT/(2.*DX5"*2) FOEQ =2.*ALI=AS*DT/(Dx2+nx3)**2 FOEQl=2.*ALFAQ1"DT/(DX1+DX2)“2 FOEQ2=2.*ALFAQ2"DT/(DX1+DX2)**2 FOEQ3=2.*ALFAQ3*DT/(DX4+DX3)"2 FOEQ4=2.*ALFAQ4*DT/(DX4+DX3)**2 FOEQ5=2.*ALFAQ5*DT/(DX5+DX4)**2 FOEQ6=2.*ALFAQ6'DT/(DXS+DX4)**2 BIOT=.5*HCON*DXl/KCONC NTOT=NSLAB+NLS+NIN+NTIN+NGRD C OPEN THE OUTPUT FILE THEN WRITE THE BASIC INPUT DATA C C OPEN(20,STATUS=’OLD’, FILE=’CN.OUT’) OPEN(50,STATUS='OLD’. FILE=‘TEMP.OUT’) WRITE(20,100) WRITE(20,101)TSLAB,NSLAB,DLS,NLS,DIN,NIN,TIN,NTIN,DGRD,NGRD WRITE(20,102)KCONC,RHOCC,KSAND.RHOCS,KEARTHRHOCE, + KINSLRHOCI,TINF,TGRD,HCON,DT,NTOT,GH,TIME1,TIME2 - C WRITE CALCULATED PARAMETERS ONTO OUTPUT FILE C WRITE(20,200)DX1 ,DX2,DX3 ,DX4,DX5 ,ALFAC ,ALFAS ,ALFAI,ALFAE, 150 + KEQ1,KEQ2,ICEQ3,ALFAQ1,ALFAQZ,ALFAQ3,ALFAQ4.ALFAQ5.ALFAQ6, + FOC,FOS l .FOSZ.FOI,FOE,FOEQ,FOEQ1,FOEQ2,FOEQ3,FOEQ4,FOEQ5, + FOEQ6,G,BIOT ISTEP1=0 ISTEP2=0 ISTEP3=0 TIME=0. C C SET THE INITIAL CONDITION C DO 90 1:1, NTOT+1 T(I)=TINT TOL(I)=TINT 90 CONTINUE C C SOLVE FOR T ON INTERIOR POINTS AT (N+l)TI-I TIME STEP C INCREMENT THE ITERATION COUNTERS AND CHECK THE MAXIMUM LIMIT C OF ITERATIONS C 20 ISTEP1=ISTEP1+1 ISTEP2=ISTEP2+ 1 IS'I'EP3=ISTEP3+1 'I'IME=TIME+DT G=0. IFOST'EPI.GT.MTIME)GO TO 40 IF(IS'I'EP3.GE.(TI1\{E1/DT))'IHEN IF(ISTEP3.LE.('ITME2/DT))'IHEN G=GH*DT/(RHOCS"DX3) ENDIF EN DIF C C FORM THE 'I'RIDIAGONAL SYSTEM OF EQUATIONS C CALL FMTDIG(NSLAB,NLS,NIN,NTIN,NGRD,N’I‘OT,G,BIOT,TGRD, + FOC,F OS 1 ,FOSZ,FOI,FOE,FOEQ,FOEQ1,FOEQ2,FOEQ3,FOEQ4,FOEQ5, + FOEQ6,TINF,A,B,C,R,T,'IOL,RHOCC,DXl,DT) C C INVER'I‘ THE TRIDIAGONAL SYSTEM OF EQUATIONS C CALL TDIG(A,B,C,R,SOLN,QI,QOHCON,T'WF,NTOT,NGRD,KINSL,DX4) C C SAVE THE OLD SOLUTION C DO 25 I=1,N'I‘OT+1 T(I)=SOLN(I) TOL(I)=T(I) 25 CONTINUE C C WRITE THE TEMPERATURE FIELD ON OUTPUT FILE C 151 IFOST'EPZ.EQ.NSTEP)THEN WRITE(50,99)TIME,T(1),T(NSLA8+NLS+2) C WRITE THE HEAT FLUX ON OUTPUT FILE QT.OUT C OPEN(30,STATUS=’OLD’.FILE=’Q1.0U'I‘ OPEN(40,STATUS=’OLD’,FILE=’Q0.0UT’) ' WRI'I'E(30,77)(TIME/24),QI WRITE(40,77)TIME,QO 77 FORMAT(1X,F9.3 ,5X,F20.6) ISTEP2=0 GO TO 20 ENDIF GO TO 20 88 FORMATU,1X,’AT T = ’,F7.3,1X,’TEMPERATURE FIELD IS:’) 99 FORMAT(1X,3(F8.4,2X)) 100 FORMATQOX ’ttttttttttttfitw*IlutmtltttttttttttttttttttInt-*4!“-’/ 9 + 32X,‘OUTPUT FOR CRANKNC’lZOX, + ’***************#*l#**************.**********’//2X 9 + ’"BASIC INPUT DATA FOR CRANKNCz’I) 101 FORMAT(1X,’SLAB THICKNESS’,10X,F9.4,12X,’NSLAB’5X,I4/ + 1X,’LINE SOURCE POSITION’,4X,F9.4,12X,’NLS’,7X,I4/ + 1X,’INSULATION POSITION ’,5X.F9.4,12X,’NIN’,7X,I4/ + 1X,'INSULATION THICKNESS’,4X,F9.4,12X,’NTIN’,6X,I4/ + 1X,’POSITION OF WATER TABLE’,lX,F9.4,12X,’NGRD’,6X,I4) 102 FORMATU 1X,’K CONCRETE’,4X.F3.4,12X,’RHOC CONCRETE’,4X,F8.4/ + 1X,’K SAND',8X,F8.4,12X,'RHOC SAND’,8X,F8.4/ + 1X,'K EARTH’,7X,F8.4,12X,’RHOC EARTH’,7X,F8.4/ + 1X,’K INSULATION’JX,F8.4,12X,’RHOC INSULATION’,2X,F8.4// + 1X,’T INFINITY =’.2X.F6.2,3X,’T GROUND =',2X,F6.2,3X, + ’h CONVECTION =’2X.F6.2,3X/1X,’DT =’,1X,F6.2,5X,’NTOT’,IS, + 3X,’HEAT SOURCE =’1X,F6.2,2X,’TIME1 =’,F4.l,2X,’TIME2 =’.F4.l) 200 FORMATU/IX,’DX1=’,F8.4,3X,’DX2=’,F8.4,3X,’DX3=’58.4, + 3X,’DX4=’,F8.4,3X,'DX5=’,F8.4//1X,’ALFAC=’,F8.4,3X, + ’ALFAS=’,F8.4,3X,’ALFAI=’,F8.4,3X,’ALFAE=’,F8.4//1X, + ’KEQ1=’,F8.4,3X,’KEQ2:’,F8.4,3X,’KEQ3=’,F8.4,3X,’ALFAQ1=’, + F8.4//1X,’ALFAQ2=’.F8.4,3X,’ALFAQ3=',F8.4,3X,’ALFAQ4=’,F8.4,3X, + ’ALFAQ5=',F8.4//lX,'ALFAQ6=’,F8.4,3X,’FOC=’,F8.4,3X,’FOS l=’, + F8.4,3X,’F082=',F8.4//1X,’FOI=’,F8.4,3X,’FO&’,F8.4,3X,’FOEQ=’, + F8.4,3X,’FOEQ1=’,F8.4,3X,’FOEQ2=’,F8.4//1X,’FOEQ3=’,F8.4,3X, + ’FOEQ4=’,F8.4,3X,’FOEQS=’,F8.4,3X,’FOEQ6=’,F8.4//1X,’G=’,F8.4,3X, + ’BIOT=’,F8.4//) 40 STOP END 152 Ck******t*************************‘k****************************‘k***tit SUBROUTINE FMTDIG(NSLAB,NLS ,NIN,NTIN,NGRD,NTOT,G,BIOT, + TGRD,FOC,FOSI,FOSZ,FOI,FOE,FOEQ,FOEQ1,FOEQ2,FOEQ3 ,FOEQ4, + FOEQS,FOEQ6,TI1\IF,A,B,C,R,T,TOL.RHOCC.DXl.DT) C , C THIS SUBROUTINE FORMS THE TRIDIAGONAL MATRIX FOR TI-E C CRANK-NICOLSON NETHOD. THE GENERIC FORM OF THE EQUATIONIS: A*T(I-1) + B‘TG) + C‘T(I+1) = R 0000 DIMENSION T(3000),TOL(3000),A(3000).B(3000),C(3000),R(3000) Cl=3.E-09 C C FIRST BOUNDARY CONDITION AT X=0. C B(1)=l.+2.*FOC+4.*FOC*BIOT+2.*Cl*DT“'(TlNF**3.)/CRHOCC*DX1) C(l)=-2.*FOC R1=2.*FOC*TOL(2)+(1.-2.*FOC-4.*FOC*BIOT)*TOL(1) R2=8.*FOC*BIOT"‘TINF R3=C1*DT"(2.*(TII~IF"*4)-3.*(TINF**2)*(TOL(l)-TTNF)**2)/(RHOCC*DX 1) R(1)=R1+R2+R3 C C INTERIOR NODES FOR FIRST LAYER C IF(NSLAB.EQ.1)THEN A(2)=-FOC*2. 8(2): 1 .+FOEQ1+2.*FOC C(2)=-FOEQ1 ' R(2)=FOEQ1*TOL(3)+2.*FOC*TOL(1)+(1-FOEQl-FOC*2.)*TOL(2) ENDIF IF(NSLAB.GE.2)THEN A(2)=-FOC*2. B(2)=l.+3.*FOC C(2)=-FOC R(2)=FOC*(TOL(3)+TOL(1)*2.)+(1-3.* FOC)*TOL(2) DO 10 1:2, NSLAB A(I)=-FOC B(I)=1.+2.*FOC C(I)=-FOC R(I)=FOC*(TOL(I+1)+TOL(I-1))+(l-2.*FOC)*TOL(I) 10 CONTINUE A(NSLAB+1)=-FOC B(NSLAB+1)=1.+FOC+FOEQ1 C(NSLAB+1)=-FOEQ1 R(N SLAB+1)=FOEQ1*TOL(NSLAB+2)+FOC*TOL(NSLAB)+ + (l.-FOC-FOEQ1)*TOL(NSLAB+1) ENDIF 153 P=NSLAB+2 C C INTERIOR NODES FOR SECOND LAYER C IF(NLS .EQ.1)THEN A(P)=-FOEQ2 B(P)=] .+FOEQ+FOEQ2 C(P)=-FOEQ R(P)=FOEQ*TOL(P+1)+FOEQ2*TOL(P-1)+(l .-FOEQ-FOEQ2)*TOL(P) ENDIF IF(NLS .GE.2)THEN A(P)=-FOEQ2 B(P)= 1 .+FOS 1+FOEQ2 C(P)=-FOS 1 R(P)=FOS 1*TOL(P+1)+FOEQ2*TOL(P- l)+(1 .-FOS 1 -FOEQ2)*TOL (P) P=NSLAB+NLS DO 20 I=NSLAB+3, P A(I)=-FOS 1 B(I)=1.+2.*FOS l C(I)=-FOS l R(1)=FOS1*(TOL(I+1)+TOL(I-1))+(l.-2.*FOS1)*TOL(I) 20 CONTINUE A(P+ l)=-FOS l B(P+1)=l .+FOS 1+FOEQ C(P+ l )=-FOEQ R(P+ l )=FOEQ*TOL(P+2)+FOS 1 *TOL(P)+(1 .-FOEQ-FOS 1)* TOL (P+ 1 ) ENDIF P=NSLAB+NLS+1 INTERIOR NODES FOR THIRD LAYER 000 0 IF(NIN.EQ. 1)THEN A(P+l)=-FOEQ B(P+l)=l .+FOEQ3+FOEQ C(P+l)=-FOEQ3 R(P+ 1 )=FOEQ3*TOL(P+2)+FOEQ*TOL(P)+(l .-FOEQ-FOEQ3)*TOL(P+1 )+G ENDIF IF(NIN.GE.2)THEN A(P+1)=-FOEQ B(P+l)=1.+FOEQ+FOSZ C(P+ l )=-FOSZ R(P+ l )=FOS2*TOL(P+2)+FOEQ*TOL(P)+( 1 .-F OS 2-FOEQ)*TOL(P+ 1 )+G DO 30 I=P+2, P+NIN-l ‘ A(I)=-FOSZ B(I)=1.+2.*FOSZ C(I)=-F082 R(I)=FOSZ* (T OL(I+ l)+TOL(I- l ))+( 1 .-2.*FOSZ)*TOL(I) 30 CONTINUE 154 P=P+NIN A(P)=-FOSZ B(P)=l.+FOS2+FOEQ3 C(P)=-FOEQ3 R(P)=FOEQ3*TOL(P+1)+FOSZ*TOL(P- l)+(1 .-F082-FOEQ3)*TOL(P) ENDIF ‘ C . P=NSLAB+NLS+NIN+1 C C INTERIOR N ODES FOR FOURTH LAYER C IF(NTTN.EQ.1)THEN A(P+1)=-FOEQ4 B(P+ 1 )=1 .+FOEQ4+FOEQ5 C(P+1)=-FOEQ5 R(P+l)=FOEQ5*TOL(P+2)+FOEQ4*TOL(P)+(1 .-FOEQ4 -FOEQ5)*TOL(P+ 1) ENDIF IF(NTIN.GE.2)THEN A(P+1)=-FOEQ4 B(P+1)=1.+FOEQ4+FOI C(P+l)=-FOI R(P+1)=FOI*TOL(P+2)+FOEQ4*TOL(P)+(1.-FOI-FOEQ4)*TOL(P+ 1) DO 40 I=P+2, P+NTIN-l A(I)=-FOI B(I)= l.+2.*FOI C(I)=-FOI R(1)=FOI*(TOL(I+ l)+TOL(I-1))+(l .-2.*FOI)*TOL(I) 40 CONTINUE P=P+NTIN A(P)=-FOI B(P)=l .+FOI+FOEQ5 C(P)=-FOEQ5 R(P)=FOEQ5*TOL(P+1)+FOI*TOL(P-1)+(1.-FOEQ5-FOI)*TOL(P) ENDIF P=NTOT-NGRD+1 C C INTERIOR NODES FOR FIFTH LAYER A(P+l)=-FOEQ6 B(P+l)=1.+FOEQ6+FOE C(P+1)=-FOE R(P+1)=FOE"'TOL(P+2)+FOEQ6*TOL(P)+(l.-FOE-FOEQ6)*TOL(P+l) DO 50 I=P+2, NTOT A(I)=-FOE B(I)=1.+2.*FOE C(I)=-FOE R(I)=FOE*(TOL(I+1)+TOL(I-1))+(1.-2.*FOE)*TOL(I) 50 CONTINUE 155 C BOUNDARY NODE AT X=L. C C A(NTOT+ l)=-FOE B(NTOT+1)=1.+3.*FOE C(NTOT+1)=—FOE*2. R(NTOT+l)=FOE*(TOL(NTOT)+4.*TGRD)+(1.-3.*FOE)*TOL(NTOT+1) RETURN END C*******************‘k‘k‘kt‘k‘kt**************************************ak‘ki‘k‘k'k SUBROUTINE TDIG(A,B,C,R,SOLN,QI,QO,HCON,TINF,NTOT,NGRD, + KINSLDX4) C REAL KINSL C THIS SUBROUTINE USES GAUSS-ELIMINATION THEN BACK SUBSTITUTION C TO GIVE THE SOLUTION OF THE SYSTEM OF EQUATIONS C 10 20 30 DIMENSION A(3000),B(3000),C(3000),BN(3000).R(3000),SOLN(3000) DO 10 I=1.NTOT+1 BN(I)=B(I) CONTINUE DO 20 I=2.NTOT+1 D=A(I)/B N(I- 1) BN(I)=BN(I)-C(I-1)*D R(1)=R(I)-R(I-l)*D CONTINUE SOLN(NTOT+1)=R(NTOT+1)/BN(NTOT+1) N=NTOT+1 D0 30 I=1,NTOT J =N-I SOLNU )=(R(J )-C(J )* SOLN (J + l))/BN(J ) CONTINUE N =NTOT-NGRD QI=(SOLN( l )-TINF)"HCON+. l Se-08*(SOLN(1)**4-TTNF**4) QO=KINSL*(SOLN(N)-SOLN(N+1))/DX4 RETURN END MICHIGAN stars UNIV. LIBRQRIES lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 31293008987509