# Simultaneous solutions of coupled thermal airflow problem for natural ventilation in buildings.

IntroductionNatural and hybrid ventilations are important strategies of sustainable building designs. Depending on wind pressure forces and/or buoyancy effects created from indoor and outdoor temperature difference, natural and hybrid ventilations can reduce the usage of mechanical power for ventilation and improve indoor air quality by introducing outdoor air directly to occupied spaces. Natural and hybrid applications include but are not limited to passive cooling ventilation systems, such as night cooling and double skin facades, and large vertical spaces, such as atria and light wells. The design of these systems and spaces includes the determination of the size, location, and number of the ventilation openings, as well as the height and width of the atria and light wells so that the use of buoyancy and wind effects is optimal for ventilation (Allard and Ghiaus 2005). This design process requires the prediction of airflow rates through ventilation openings and air temperatures in each room of a building by solving a coupled thermal and airflow problem.

Numerical solutions of airflow ventilation rates, and temperatures of room air and wall assemblies, are often treated separately. Multi-zone airflow network models, such as CONTAM (Walton and Dols 2008), COMIS (Feustel 1999), and other solution methods (Herrlin and Allard 1992), predict building airflows without solving energy equation. As a result, a known air temperature is required to be provided for each room. On the other hand, building energy analysis software tools are often not designed for inter-zonal airflows as a multi-zone network model does. To solve a coupled thermal airflow problem, many previous efforts have been conducted to integrate a building airflow network model with a building energy analysis tool in the past 20 years. Walton (1982) developed one of the earliest thermal airflow multi-zone models for multi-room analysis. Huang et al. (1999) linked COMIS 3.0 with the EnergyPlus building energy simulation program (Crawley et al. 2001). Axley et al. (2002) developed a coupled thermal and airflow simulation tool and demonstrated it for modeling a naturally ventilated commercial building. McDowell et al. (2003) integrated CONTAM with the TRNSYS building energy simulation software package (Beckman et al. 1994). Gu (2007) added an airflow network model to EnergyPlus and showed some encouraging results. Another popular energy simulation package with its own airflow models is the Environmental Systems Performance, Research version (ESP-r) building simulation program (Clarke 1985). These software tools and studies solve a coupled thermal airflow problem in a similar manner. Either the air temperature in the energy equation or the air pressure in the airflow mass balance equation is solved first with the other parameter kept at a constant. The newly solved parameter is then substituted successively in the other equation. This segregate and successive solution of energy and airflow equations requires relaxation factors to be selected carefully to avoid abrupt changes of room air temperature or pressure in the successive substitution procedure.

For highly coupled thermal airflow problems, the segregate method could cause solution fluctuation or even divergence for poor relaxations (Weber et al. 2003). Ketkar (1993) and Schneider et al. (1995) found that numerical instabilities can be caused when the airflow balance equation and the energy equation are of quite different orders of magnitude. For example, a slight change of air temperature as determined by the energy equation may cause a huge variation of airflow results in the airflow balance equation. In such case, a strong relaxation factor has to be used in the segregate solver, which, however, may slow down the overall convergence of the solution. Weber et al. (2003) developed an integrated thermal airflow simulation tool, TRNFlow, by linking COMIS and TRNSYS. They found that a single relaxation factor might not be possible to guarantee convergence at each time step during the whole simulation period. A method to generate adapted relaxations automatically was proposed and demonstrated by a two-zone case with buoyancy-driven airflow. This method, however, has not been verified by other studies.

Another group of methods for solving highly coupled thermal airflow problems are simultaneous solutions, one of which is to assemble the airflow and energy balance equations for a whole building in a single matrix (Ketkar 1993; Schneider et al. 1995). It was shown that the simultaneous solvers provided a suitable alternative to the segregate methods when numerical instabilities or slow convergence occurs. However, for large problems, simultaneous solvers were found to require more computer power and memory due to increased numbers of equations.

This study investigates simultaneous solvers for modeling coupled thermal airflow problems. In a fully simultaneous solver, air temperatures and pressures for all rooms of a building are solved simultaneously by a single Jacobian matrix. A semi-simultaneous method is also proposed in which a Jacobian matrix for the unknown air temperature and pressure of one room is solved when keeping air temperatures and pressures of other rooms as constants. Then the same procedure is repeated for each room of a building, one after another. As a comparison, the segregate solvers with fixed and adapted relaxations are also studied and verified. All four solvers are compared for the simulation of a two-zone building with buoyancy-driven lows. The fully simultaneous and semi-simultaneous solvers are then validated by an experimental study of natural ventilation under combined wind and buoyancy forces in a light well. The simultaneous solvers developed in this article are not limited to coupled thermal airflow problems in natural ventilation but can be applied to other multi-zone problems with coupled equations.

[FIGURE 1 OMITTED]

Theory

The formulation of a coupled thermal airflow problem of natural ventilation can be shown by the two-zone building in Figure 1, which is based on the case from Weber et al. (2003). The air is assumed to be well mixed and static, so the unknown air parameters of each zone are temperature (T) and pressure (P). The air pressure is also assumed to vary hydrostatically relative to the zone pressure, which is located at the floor of each room.

The airflow from zone i to zone j through path ij([F.sub.ij]) is often modeled by a power-law function of the pressure drop across the path, [DELTA][P.sub.ij]. Assuming [F.sub.ij] is positive for [DELTA][P.sub.ij] > 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

[DELTA][P.sub.ij] = [P.sub.ij,w] + [P.sub.ij,s] + [P.sub.i] - [P.sub.j], (2)

The stack pressure for path ij is

[P.sub.ij,s] = [g / 2] [([[rho].sub.i] + [[rho].sub.j])([z.sub.i] - [z.sub.j]) + ([[rho].sub.j] + [[rho].sub.i])([h.sub.1] + [h.sub.2])]. (3)

Equation 3 is obtained as follows. When [z.sub.1] [not equal to] [z.sub.2] (e.g., path ij is an inclined airflow passage with nonzero length), and the low direction inside the path ij is often unknown, the air density in path ij is often calculated by

[p.sub.ij] = [1 / 2] ([[rho].sub.i] + [[rho].sub.j]). (4)

The pressure difference across path ij is calculated by

[DELTA][P.sub.ij] = [P.sub.i] - [[rho].sub.i]g[h.sub.1] + [[[[rho].sub.i] + [[rho].sub.j]] / 2] g([z.sub.1] - [z.sub.2]) - ([P.sub.j] - [[rho].sub.j]g[h.sub.2]) + [P.sub.ij,w]. (5)

Since [z.sub.1] = [z.sub.i] + [h.sub.1] and [z.sub.2] = [z.sub.j] + [h.sub.2],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Comparing Equations 2 and 6, Equation 3 can be obtained. Based on Fourier's law of conduction, the heat flux from surface thermal node [k.sub.1] to [k.sub.2] of the wall separating zone i and j is

q = -k [[partial derivative]T / [partial derivative]y] (7)

For zone i, the air mass and energy conservation equations can be written as

d[m.sub.i] / dt = [summation over (j)] [F.sub.ji] - [summation over (j)] [F.sub.ij] + [F.sub.i], (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

By applying the Euler method for transient terms and assuming the ideal gas law to obtain air density, Equations 8 and 9 can be discretized, respectively, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Equations 10 and 11 can be combined to a matrix formation for zone i, where a vector for the zone state variables is defined as [X.sub.i] = [([P.sub.i],[T.sub.i]).sup.T]:

f([X.sub.i]) = 0. (12)

Equation 12 is nonlinear due to the power-law terms. Applying the Newton-Raphson method, the equations can be linearized to obtain a Jacobian matrix:

J([X.sup.(n).sub.i])([DELTA][X.sup.(n).sub.i]) = J([X.sup.(n).sub.i])([X.sup.(n).sub.i] - [X.sup.(n+1).sub.i]) = f([X.sup.(n).sub.i]), (13)

where [J.sup.(n)] = [([partial derivative]f/[partial derivative]x).sup.(n)]. The elements of the Jacobian matrix can be found by applying the derivative of Equations 10 and 11 over either air temperature or pressure.

After Equation 13 is solved, the zone state variables can be corrected by Equation 14:

[X.sup.(n+1).sub.i] = [X.sub.(n).sub.i] - [DELTA][X.sup.(n).sub.i] (14)

Due to the nonlinearity of the problem, the Jacobian matrix needs to be updated after the correction, so iterations are often needed.

In a fully simultaneous solver, Equation 13 is applied to all zones of a building to assemble a single linear matrix:

J([X.sup.(n)])([DELTA][X.sup.(n)]) = J([X.sup.(n)])([X.sup.(n)] - [X.sup.(n+1)]) = f([X.sup.(n)]), (15)

where X = [([X.sub.1]...,[X.sub.i]...,[X.sub.N]).sup.T] for a building with N zones.

The size of the matrix J([X.sup.(n)]) is proportional to j the product of the number of zones by the number of zone state variables. For the example in Figure 1, given two zones and two unknown zone state variables and P, the matrix of the fully simultaneous method will be 4 x 4. A building with N zones will create a matrix of 2N x 2N. Since all parameters are solved simultaneously at the iteration of n and n + 1, huge computer memory is required for storing the information. The calculation of the Jacobian matrix also needs extensive data processing efforts. The matrix can be solved directly by a direct matrix solver such as the Lu factorization method.

As a comparison, a segregate solver calculates Equation 10 for air pressures for all zones or Equation 11 for all air temperatures separately when keeping the other parameters as constants at every iteration. Then the solved air parameters in one equation are substituted successively in the other equation. The resultant matrix will be with the size of N x N for a building with N zones, and it can be solved iteratively so that less computer storage and processing are needed than for a simultaneous solver. However, relaxation factors are required to avoid abrupt change of air parameters during the correction and substitution process. In such a case, Equation 14 for zone i becomes

[x.sup.(n+1).sub.i] = [x.sup.(n).sub.i] - [r.sup.(n).sub.i] x [DELTA][x.sub.(n).sub.i], (16)

where [x.sub.i] can be the air temperature and pressure. For example, a segregate solver can start with the input air temperatures [T.sup.(n).sub.i] and solve the air mass balance equation for all zones to obtain air pressures and airflows. Then, these solved airflows are used in the energy balance equation to calculate the correction term [DELTA][T.sup.(n).sub.i], which updates [T.sup.(n+1).sub.i] based on Equation 16. The updated air temperature is then used as new input for the air mass balance equation at the next iteration n + 1. It was found that a fixed relaxation factor, which ensures convergence at some time step, may slow down the iteration at other time steps where a relaxation is unnecessary. A poorly selected relaxation factor may cause numerical instabilities for highly coupled thermal airflow problems. Weber et al. (2003) suggested an adapted relaxation method to avoid numerical instabilities:

[r.sup.(n).sub.i] = 0.5[r.sup.(n-1).sub.i] when [DELTA][T.sup.(n).sub.i] [DELTA][T.sup.(n-1).sub.i] [less than or equal to] 0; (17)

[r.sup.(n).sub.i] = 1.5[r.sup.(n-1).sub.i] when [DELTA][T.sup.(n).sub.i] [DELTA][T.sup.(n-1).sub.i] [greater than or equal to] 0. (18)

This study proposes a semi-simultaneous solver, which has some characteristics between fully simultaneous and segregate solvers. The semi-simultaneous method solves air temperature and pressure of one zone, e.g., [X.sup.(n).sub.i], simultaneously by Equation 13 while air temperatures and pressures of other zones are kept as constants. [X.sup.(n).sub.i] is then substituted in Equation 13 for the next zone, e.g., zone j, to calculate [X.sup.(n).sub.j]. The same procedure is repeated for each of the rest zones until a full sweep of iterations for all zones is completed. More sweeps are necessary before Equation 13 converges for all zones. Furthermore, an external iteration loop is needed to consider the impact of heat transfer through building walls on the zone thermal airflow model. In this method, the Jacobian matrix is only 2 x 2, which is significantly less than the fully simultaneous method. As a simultaneous solution, relaxation factor is also avoided in this solver.

In detail, the semi-simultaneous solver follows the following procedure.

1. Initialize zone pressures, temperatures, and zone mass and heat capacity.

2. Calculate [P.sub.ij,w] and [P.sub.ji,w] for airflow paths.

3. Start internal iteration for a single zone.

a. Calculate [P.sub.ij,s] and [P.sub.ji,s] for airflow paths.

b. Calculate [DELTA][P.sub.ji], [DELTA][P.sub.ij], [partial derivative][F.sub.ji] / [partial derivative][P.sub.i], and [partial derivative][F.sub.ij] / [partial derivative][P.sub.i].

c. Fill [J.sub.(n)] and calculate [f.sub.(n)].

d. Solve [J.sub.(n)][DELTA][X.sup.(n)] = [f.sup.(n)] directly by the LU factorization method.

e. Correct [X.sup.(n+1).sub.i] = [X.sup.(n).sub.i] - [DELTA][X.sup.(n).sub.i].

f. Update zone properties, e.g., density and viscosity, and zone mass.

g. Check convergence of internal iteration; if it is not convergent or the total number of internal iteration is less than the maximum iteration, repeat Steps a-f; if the internal iteration is convergent, move to the next zone and repeat Step 3.

4. After calculating thermal airflows for all zones, solve 1D heat conduction equations for all walls in the building.

5. Check overall convergence; the solution is convergent if [f.sup.(n+1)] is less than a certain criterion or the maximum number of external iterations is reached; if it is not convergent, repeat Steps 3 and 4; if it is convergent, move to the next time step.

Comparison and verification

The fully simultaneous, semi-simultaneous, and segregate solvers with fixed and adapted relaxations are implemented in CONTAM97R, a multi-zone airflow network model with energy analysis capabilities in the CONTAM family. To compare these four methods, a two-zone building with buoyancy-driven airflows from Weber et al. 2003) is used. Figure 2 shows the plan view of the two-zone building in CONTAM97R. The room size is 45 [m.sup.3] (1589.2 [ft.sup.3]) each. The air temperature is maintained as constant in room 2 at 20[degrees]C (68[degrees]F) and initially in room 1. The outdoor wind speed is zero with a constant temperature of 10[degrees]C (50[degrees]F). The height of airflow path (af) is 1 m (3.3 ft) for af 1, 5 m (16.4 ft) for af 2, and 3 m (9.8 ft) for af 3, which are relative to the floor of the corresponding room. The flow coefficient and exponent is 1.0 kg/s/[Pa.sup.n] (2.2 [lb.sub.m]/s/[(1.5 x [10.sup.-4] psi).sup.n]) and 0.6 (1.3 [lb.sub.m]/s/[(1.5 x [10.sup.-4] psi).sup.n]), respectively, for all airflow paths. The wall thickness is 152 mm (0.5 ft) with a conductivity of 0.047 W/(m x K) (0.027 BTU/(h x ft x [degrees]F)) for all the heat transfer path (ht). A transient simulation is conducted for a period of 24 h with a time step of 5 min.

[FIGURE 2 OMITTED]

Figure 3 compares the predicted airflow rates through path 1 for the four solvers. Given that the airflow from the ambient to room 1 is positive, the airflow through af 1 is an inflow to room 1 in the beginning of the simulation, which causes the temperature of room 1 to decrease. The airflow from the ambient into the house continues to drop with the decrease of the room temperature until it reaches around 14.914[degrees]C (58.845[degrees]F). At this point, the stack effect at af 1 and af 3 is balanced, so the airflow through the building is zero. However, the heat loss through the external walls (ht 1, ht 2, and ht 4) keeps driving the temperature of room 1 down so that the airflow of af 1 reverses to outflow after this point. Figure 3 shows that a slight variation of air temperature of room 1 (in the order of 1 x [10.sup.-3][degrees]C [3.4 x [10.sup.-4][degrees]F]) changes the airflow characteristics significantly (from inflow to outflow) near this critical point. Note that this strong dependence of airflow direction on temperature variation was only caused by the sensitivity of the numerical method used. In reality, it often takes several degrees of temperature variation for the low to change direction. For such a highly coupled thermal airflow problem, the segregate solver with fixed relaxation of 0.1 causes numerical instabilities and eventually produced erroneous results, as shown in Figure 3. By using the adapted relaxation method in Equation 17, the segregate solver reaches a convergent result, although there is a slight fluctuation near the critical point. In comparison, both simultaneous solvers produce identical and convergent results without using any relaxation factor.

[FIGURE 3 OMITTED]

The minor fluctuation of the adapted relaxation method can be explained in Figure 4. Near the critical point, the maximum iteration, which is set to be 100 in this case, is reached even for a relaxation of 0.013. An increase of the maximum iteration number will not improve the situation. However, this minor fluctuation does not affect the overall convergence. Figure 4 also shows that heavy relaxations and many iterations are needed for highly coupled thermal airflows, for which the simultaneous solvers provide good alternative methods without using any relaxation. In the meanwhile, this case verifies the adapted relaxation method of Weber et al. (2003). Note that due to incompleteness of data from their studies, this case is not set up exactly as the one from Weber et al.'s study. However, similar trend of results are observed in their analysis.

[FIGURE 4 OMITTED]

A comparison of simulation time and computational storage is shown in Table 1. Because of numerical instability, the iterative solver with fixed relaxation requires the most simulation time, which is many times greater than other methods, especially the semi-simultaneous solver. Moreover, the semi-simultaneous solver requires the fewest computational storage to construct the matrix equation, although the difference of all methods is minimal for the two-zone problem. The benefits of simultaneous solvers can be illustrated further by a validation study in the next section.

Validation

The case of the two-zone building only provides pure simulation results for comparison purposes. This study selects another case with experimental data to validate the simultaneous solvers. Figure 5 shows the experimental study of a light well in a wind tunnel with airflows driven by combined wind and buoyancy forces (Kotani et al. 2003). The size of the light well is 144 mm x 144 mm x 480 mm (0.47 ft x 0.47 ft x 1.57 ft) with an internal void of 72 mm x 72 mm x 480 mm (0.24 ft x 0.24 ft x 1.57 ft), which corresponds to a 41-story building for a scale of 1/250. The size of the opening is 12 mm x 72 mm (0.04 ft x 0.24 ft) for the lower one and 72 mm x 72 mm (0.24 ft x 0.24 ft) for the top. The heat in the void is generated by Nichrome wires for a range of 10 W to 40 W (34.1 BTU/h to 136.5 BTU/h). The wind speed is adjusted to be from 0 to 1.5 m/s (4.9 ft/s) with the wind direction perpendicular to the lower opening. The air in the wind tunnel is kept at 12[degrees]C (53.6[degrees]F). In this study, the light well is divided into eight vertical zones with same size, as shown in Figure 5. The experiment neglects heat transfer through the light well structures, so this study does not consider heat conduction in walls. An orifice airflow equation (Equation 18) is used for the lower, inter-zonal, and top openings with a flow exponent of 0.5 and a discharge coefficient provided from the original study:

[F.sub.ij] = [C.sub.d][A.sub.ij][square root of 2[rho][DELTA][P.sup.ij]]. (19)

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Figure 6 compares the predicted airflow rates through the light well at steady state by two previous simulation studies (Kotani et al. 2003; Zhai et al. 2009) and the current study with the measured data. Both simultaneous solvers predicted almost identical results, which agree well with the measured airflow rates. It seems the current study predicts slightly better results than the other studies, which should not be conclusive without further confirmation by more validation studies. Figure 6 also shows that all predictions overestimate the airflow rates, which was attributed to the inaccurate measurement of heat generation rate used in the simulations (Kotani et al. 2003).

Another reason for the airflow overestimation can be the use of eight zones to model the light well. The actual temperature gradient may not be predicted appropriately, as illustrated in Figure 7. For different scenarios of wind speeds and heat generation rates, the temperature predictions by the simultaneous solvers are generally greater than the measured data, which contributes to the overestimation of airflow rates.

[FIGURE 7 OMITTED]

On the other hand, the overall trend of the prediction agrees reasonably well with the measured temperatures with some major exceptions in the lower zones of the light well, where the simultaneous solvers underestimate the temperatures. Similar problems are observed in the calculations of Kotani et al. (2003). In the current study, the light well was divided into eight zones, since the original studies by Kotani et al. (2003) also used eight zones in their simulations. For clarity purposes, their simulation results were not shown in Figure 7. However, their study did not model the light well by more than eight zones, and therefore, there are no resistance models available if more zones need to be simulated. Moreover, it is still a challenging topic of multi-zone models to simulate a large space by dividing the space into subzones. To model a large space is not the focus of this study. Therefore, for validation purposes, the light well was not modeled by more than eight zones. It is possible that a better estimation can be obtained by other simulation models, such as computational fluid dynamics (CFD) with more accurate inputs of the heat generation rates. However, as computationally economic methods, the simultaneous solvers provide predictions of airflow rates and temperatures in the light well with certain accuracy. Note that both simultaneous solvers calculate almost identical results in Figure 7, so only one simulation line is shown for each scenario.

The computational storage for the matrix construction is found to be 550 bytes for the iterative solver with adapted relaxation, 100 bytes for the semi-simultaneous solver, and 2560 bytes for the fully simultaneous solver. Figure 8 compares the simulation time for the three methods. The iterative solver with adapted relaxation is about four times slower than the semi-simultaneous method in some cases, e.g., when the internal heat is 10 W (34.1 BTU/h) and the wind is 0.5 m/s (1.6 ft/s). For the rest of the cases, the simulation time is close for all three methods. When the simulations were run on an Intel[R] Xeon[R] 2.67-GHz CPU, the average simulation time is 0.25 s for the iterative method, 0.15 s for the semi-simultaneous method, and 0.16s for the fully simultaneous method. Once again, the semi-simultaneous solver uses the fewest computational storage and provides the fastest solution in general. Further tests of the methods on more complicated problems are necessary to generalize this conclusion. Note that the semi-simultaneous solver took less computational storage than the two-zone building case, which is because the heat conduction through walls was not solved in the light well case.

Conclusion

This study investigates simultaneous solutions of coupled thermal airflow problems in natural ventilation applications by comparing them to the segregate solvers for solving the same problems. Two simultaneous solvers are proposed and studied. The fully simultaneous method solves a single Jacobian matrix for air temperatures and pressures of all zones simultaneously, which requires more computer memory than other methods. The semi-simultaneous method solves a matrix of air temperature and pressure for each zone, one after another, and requires less computer storage. The following conclusions are reached.

[FIGURE 8 OMITTED]

1. It is shown that the segregate iterative solver with fixed relaxations can cause numerical instabilities for highly coupled problems. This study verified that the adapted relaxation method from a previous study is able to avoid some numerical convergence problems, although there may be minor fluctuation near the critical point, where the airflow direction is reversed in a two-zone building case.

2. The common benefit of the proposed simultaneous solvers is that solution convergence can be obtained without using any relaxation, even for highly coupled thermal airflow problems.

3. Both methods are also validated for the predictions of airflow rates and temperatures in a light well with combined wind and buoyancy effects. It is shown that the simultaneous solvers predict the results reasonably well, considering their computational cost is much lower than more advanced models, such as CFD.

Further studies of the simultaneous solvers are necessary for more general or complicated applications of natural and hybrid ventilations. With the fast development of computer power and memory storage, the simultaneous solvers may become an important and popular technique for solving multiply coupled building physics. This study is conducted with the hope to look into this potential for the near future.

Nomenclature

[A.sub.ij] = area of path ij, [m.sup.2] ([ft.sup.2])

[C.sub.d] = discharge coefficient

[C.sub.i] = specific heat of zone i, J/(kg x [degrees]C)(BTU/ ([lb.sub.m] x [degrees]F))

[C.sub.ij] = flow coefficient of path ij, kg/(s x [Pa.sup.n])([lb.sub.m] /(s x [psi.sup.n]))

f = air mass or energy balance function

[F.sub.i] = air mass source per unit time in zone i, kg/s ([lb.sub.m]/s)

[F.sub.ij] = airflow rate of airflow path ij, kg/s ([lb.sub.m]/s)

g = acceleration of gravity, m/[s.sup.2](ft/[s.sup.2])

h = relative height of airflow path

[h.sub.k] = overall heat transfer coefficient at the wall surface node k, W/[degrees]C (BTU/(h x [degrees]F))

J = Jacobian matrix

k = thermal conductivity, W/(m x [degrees]C)(BTU/(h x ft x [degrees]F))

[m.sub.i] = air mass of zone i, kg ([lb.sub.m])

n = flow exponent; the nth iteration

N = number of zones in a building

[n.sub.ij] = low exponent of path ij

[P.sub.i] = pressure of zone i, Pa (psi)

[P.sub.j] = pressure of zone j, Pa (psi)

[P.sub.ij,s] = stack pressure at path ij, Pa (psi)

[P.sub.ij,w] = wind pressure at path j,Pa(psi)

[DELTA][P.sub.ij] = pressure difference across airflow path ij, Pa (psi)

q" = heat flux for wall conduction, W/[m.sup.2] (BTU/(h x [ft.sup.2]))

[Q.sub.i] = heat source in zone i, W (BTU/h)

[r.sub.i] = relaxation factor of zone i

[T.sub.i] = air temperature of zone i, [degrees]C ([degrees]F)

[V.sub.i] = volume of zone i, [m.sup.3] ([ft.sup.3])

[x.sup.i] = unknown air parameter (temperature or pressure) of zone i

[X.sub.i] = vector of zone state variables

[z.sub.i] = floor elevation of zone i, m (ft)

[z.sub.j] = floor elevation of zone j, m (ft)

Greek letters

[DELTA] = difference

[rho] = air density, kg/[m.sup.3] ([lb.sub.m]/[ft.sup.3])

DOI: 10.1080/10789669.2011.591258

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Received December 15, 2010; accepted May 11, 2011

Liangzhu (Leon) Wang, PhD, is Member ASHRAE. W. Stuart Dols is Member ASHRAE. Steven J. Emmerich is Member ASHRAE.

Liangzhu (Leon) Wang, (1) * W. Stuart Dols, (2) and Steven J. Emmerich (2)

(1) Department of Building, Civil and Environmental Engineering, Concordia University, 1455 De Maisonneuve Blvd. West Montreal, Quebec, Canada H3G1M8

(2) Indoor Air Quality and Ventilation Group, Building Environment Division, Engineering Laboratory, National Institute of Standards and Technology, 100 Bureau Dr., Stop 8633, Gaithersburg, MD 20899-8633, USA

* Corresponding author e-mail: leonwang@bcee.concordia.ca

Table 1. Comparison of simulation time and computational storage among different solvers for the buoyancy-driven low in a two-zone building. Iterative solver Iterative solver with fixed with adapted Methods relaxation relaxation Simulation time (s) 16.5 0.6 Computational storage 320 320 required (bytes) Semi-simultaneous Fully simultaneous Methods solver solver Simulation time (s) 0.04 0.09 Computational storage 250 325 required (bytes)

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Author: | Wang, Liangzhu "Leon"; Dols, W. Stuart; Emmerich, Steven J. |
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Publication: | HVAC & R Research |

Article Type: | Report |

Geographic Code: | 1CANA |

Date: | Jan 1, 2012 |

Words: | 5501 |

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