An improvement to the basic energy balance model for urban thermal environment analysis

Energy and Buildings, 14 (1990) 143 - 152

143

An Improvement to the Basic Energy Balance Model for Urban Thermal Environment Analysis C. P. TSO, B. K. CHAN and M. A. HASHIM

Faculty of Engineering, University of Malaya, 59100 Kuala Lumpur (Malaysia) (Received October 25, 1989)

ABSTRACT

An improvement is made to the basic energy balance model for urban thermal environment analysis by incorporating the important building .mass effect on the surface energy balance. Results show better correlation with typical city temperatures in Malaysla, suggesting that the mass effect should be considered in all advanced urban climatological models.

Keywords: thermal environment, heat island effect, urban climatology, surface energy balance. INTRODUCTION

The theoretical modelling of the urban thermal environment has been developed over the years, beginning from the basic models of Halstead et al. [1], Myrup [2] and Outcalt [3], which consider the timedependent energy balance on the earth's surface within the surface boundary layer (SBL). Such models have been applied by Ackerman [4], Dozier and Outcalt [5], Tapper [6] and others. The model may be refined by including the additional atmospheric boundary layer (ABL) in which changes in wind velocity, temperature, specific hmnidity, and other parameters may be simulated by solving the appropriate time-dependent one~limensional differential conservative equations [ 7 , 8 ] . Extensions to two~limensional models [e.g., 9 - 1 1 ] and three
In most of the above advanced models, the solution in the ABL region provides the upper boundary condition to the SBL. And in the SBL, the energy balance remains essentially the same, namely a balance of the fluxes of radiation, latent and sensible heat, and heat flow into the ground. Since it is generally agreed that buildings play an important role in the urban microclimate, the present authors are incited to investigate the inclusion of a new term to account for the effect of building mass on the above energy balance. It is noteworthy that the building geometry is already incorporated approximately by the use of the surface roughness parameter in the basic model. The canyon model [15] and sky view-factor studies [16] also provide further focusing on the effect of sizes and geometries of building clusters in cities.

THE MODEL

The basic framework of the model is shown in Fig. 1 where the energy balance is made onto a plane on the earth's surface. The assumption is now made that the building mass is reduced to a homogeneous plane which has no volume, but has the ability to store thermal energy at the rate of mcCc dTo/ dr, where mc is the building mass per unit surface area, Cc the specific heat of the building mass, and dTo/dt the rate of change of the air temperature. Thus in this 'point model', at the surface plane (level O in Fig. 1), the soil, the building and the air temperatures are always in thermal equilibrium (at To). This is only a step further from the classical massless case, where the substrate and the air temperatures are assumed Elsevier Sequoia/Printed in The Netherlands

144

Uz

T~

qz LEVEL 2

~t

SURFACE BOUNDARY

R

H

~t

LE

LAYER (SBL)

.

2d

LEVEL

0

LEVEL

s

LEVEL

b

I--

r

"t

M

-

--

NOTATIONS

ENERGY

BALANCE

Fig. 1. Framework of model.

to be in thermal equilibrium at the surface plane. The equation of energy balance is then

M = R--H--LE--S where

M

= heat storage = net radiation flux R = sensible heat flux to the air H = latent heat of water L = evaporation rate (hence LE is the E latent heat flux) S = heat flux into the soil. Following the classical approach, the following finite difference equations are used for the turbulent fluxes,

--PaCak:U H-

To)

[ln(Z2/Zo)]~

LE-

--paLk~U~ [ln(Z~/Zo)]~ ( q : - - q o )

(2) (3)

where = air density =specific heat of air at constant pmssure q o , q2 = specific humidities at levels 0 and 2, respectively (see Fig. 1) = v o n K~rm~n constant k = wind velocity at level 2 = height of the SBL Z2 Pa

ca

= surface roughness Zo To,T2 = temperatures at levels O and 2, respectively. The soil heat flux is expressed by the simple Fourier law as S

(4)

= --(k,/d)(T, -- To)

where = soil thermal conductivity k, = soil temperatures at level s T, = soil depth at level s. • d The finite
T, = T~'+ p,C,d~ (Tb where At p, C, T~'

- -

2T, + To)At

(5)

time step soil density soil specific heat capacity soil temperature at level s, from the previous time step. The finite difference form of the heat storage term is

M =

= = = =

mcCc(To -- T~)/At

(6)

where T~ is the air temperature from the previous time step.

145

The specific humidity of atmosphere qo in eqn. (3) is given b y qo = E~[3.74 + 2.64(To/10) ~] X 10 -3

APPLICATION TO THE URBAN BOUNDARY LAYER

(7)

The model is n o w applied to an area in the Kuala L u m p u r city in Malaysia. The city is located at latitudinal and longitudinal meridians 3 ° 9' N and 101 ° 44' E, respectively, and has an area o f about 244 km 2 and a population of a b o u t 1.2 million, and the chosen area covers an inner ring of approximately 9.34 × 106 m 2 in the heart of the city as shown in Fig. 2. To enable the solution of the model, further information and assumptions are needed as described below, and as summarized in Table 1. To arrive at a typical building mass per unit land area, a survey is conducted on the specific region, which for convenience is further divided into 24 subregions. The building mass is assumed to consist of only

where E~ is the evaporating fraction, an interpretation of the relative humidity as the fraction of the total area occupied by freely evapotranspiring surfaces [2, 6]. Other assumptions made in formulating this model are that [2] horizontally homogeneity is assumed in all meteorological and soil parameters, the turbulent diffusivities for heat and water vapor are given b y the near-neutral value for m o m e n t u m , the turbulent fluxes of heat and water vapor are assumed to be constant between Z o and Z2, the c a n o p y is uniquely characterized b y the roughness length Zo, and the temperature, wind speed and specific humidity are constant at height Z2. TABLE 1 Input parameters used in present model Item

Parameter

Comment

Source

Air

Specific heat o f air, Ca

1006 J/kg °C

At 27 °C

[ 17 ]

Density of air, Pa

1.175 kg/m 3

At 27 °C

[18]

Thermal conductivity of soil, ks Specific heat of soil, Cs

1.225 W/m °C ~ t 1185 J/kg °C

Average values, between dry and saturated clay soil

[ 18 ]

Density of soil, Ps

1800 kg/m 3

Concrete

Specific heat of concrete, Cc

880 J/kg °C

Water

Latent heat of evaporation, L

2.44 × 104 J/kg

At 27 °C

Miscellaneous

Average mass of concrete, m c

700 kg/m 2

Average value over a defined city area

Roughness length, Z o

5m

An assumption

Wind velocity at 300 m,

5 m/s

Average value reported

[19 ]

25 °C

Average value reported

[19]

Specific humidity at 300 m, q~

0.003

An assumption

Evaporating fraction at surface, Ef

10%

An assumption

Night-time sky temperature, Tsky

--5 °C

Soil temperature at depth 2d (20 cm), T b

25 °C

Soil

[ 17 ] [18]

U2 Air temperature at 300 m,

T2

[19] An assumption

146

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KELANG KG. BAHARU

~bl • ~

j..r ~

,'M

NATIONAL

1 NATIONAL PAL.ACE J

,ee*

0

I

|/~

I

t KM

I

Fig. 2. Surveyed area in Kuala Lumpur city.

1

BOUNDARY OF SURVEYED AREA BUILOING AREA

•

147 concrete, and to facilitate the estimation of numerous buildings a 'concrete volume factor (CVF)' has been defined as the ratio of the concrete volume in a building to the total external volume of a building. Based on detailed calculations using engineering plans o f typical buildings in the survey areas, the CVF of multi-storey buildings is about 0.132, the CVF of shop-houses and residential units is taken to be 0.160. Next, estimation of concrete volume in each building is tediously conducted for all buildings in each subregion, relying on site visits and reference to a six~hain plan which has a scale o f 1:4752. F r o m this finding, the building mass is estimated to consist of a concrete volume of 2.747 × 10 ~ m 3 on a total ground area of 9 . 3 3 3 × l 0 s m 2, giving a final mc of about 700 kg of concrete per m 2. Some details are shown in Table 2. For the net radiation flux, instead of using a functional relation between R and the solar constant, the latitude, solar declination, albedo and transmission coeffiTABLE 2 Composition of land areas and concrete volumes in subregions Subregion

Total land area (m 2)

Estimated concrete volume (m3)

A B

247587 211898

C

304027 35932 145479

141487 127770 21873 99457

176148

48968

D

E F G

H I J K L M N O P

Q R S T U V W

X Tot~

104599

61357 107067 94771 367722 260452 176379

39011 8231 214241 55377 132548

112839 613297 89914

25686 216689 61199

74853 332298 346342 627606 129159 259761 1883359 1945089 729745 9333076

26409 64437 59004 331574 159486 138177 261260 240188 100252 2746623

68699

~OC

0~* 60C

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R • ~ x ~n~t), W/m~ ~ - 7,27 x1~=¢1 t IN SE~NDS,IS ZEROAT 6 A,M, ~6 i I J [ t I i ] i I I 08 I0 12 14 ~ B (6 A.~ {6 ~.) TIME ~ THE DAY (LST) Fig. 3. Idea~zed s o l ~ i n a n i t y

v~ation.

cient, a simple empirical idealized flux is used, i.e., R = 750 sin(~t)

(W/m 2)

(8)

where ~ is specified as 7.27 × 10 -s s-1. This is based on curve-fitting upon a typical measured diurnal solar intensity on a warm day in Malaysia [20], as illustrated in Fig. 3. At night, the flux is taken as the constant --148.7 W/m 2, estimated from radiative losses between a night sky temperature of --5 °C and a ground temperature of 24 °C. For the length parameters, following ref. 2 the height of the SBL Z~ is taken as 300 m, the soil depth 2d is taken as 20 cm, and the urban surface roughness Zo is assumed to be 5 m. The wind velocity U2, temperature T~ and specific h u m i d i t y q2 at level 2 are taken to be 5 m/s, 25 °C, and 0.003, respectively. The evaporating fraction E~ is assumed to be 10%. The soil temperature Tb at level b is constant at 25 °C. The model equations are conveniently solved by the software Eureka: The Solver [21], using an IBM-AT compatible personal computer. It takes typically about one hour to complete one set o f calculations.

RESULTS Figure 4 shows the diurnal variation o f air and soil temperatures predicted using the present new model. Also shown are the

148

reported air temperatures [19] which are based on the mean of the mean hourly temperatures over twelve months for the year 1983. For comparison with the old model, Fig. 5 shows the predicted temperatures using the same input data but without the building mass effect (mc = 0). More precise comparisons for all air temperatures are

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presented in Table 3. Further results are generated to show the diurnal variation of energy components, in both the new and old model, as shown in Fig. 6 and Fig. 7 respectively. Finally results for the sensitivity of the prediction towards variations in six parameters are presented in Fig. 8, Fig. 9, and Table 4.

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TIME OF THE DAY ( L S T )

TIME OF THE DAY (LST)

Fig. 4. Predicted diurnal variation of temperatures with new model.

1

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PREDICTED AIR TEMPERATURE, TO .............

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Fig. 5. Predicted diurnal variation of temperatures with old model.

TABLE 3 Comparison o f predicted and reported air temperatures Time o f day

Reported city temperature (°C)

Predicted temp. with old model (°C)

Deviation from reported values (%)

Predicted temp. with new model (°C)

Deviation from reported values (%)

0000 0100 0200 0300 0400 0500 0600 0700 0800 0900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

25.6 25.3 25.0 24.8 24.5 24.3 24.2 24.3 25.4 27.7 29.6 31.1 32.0 32.4 32.5 31.9 31.0 29.9 28.7 27.7 27.1 26.6 26.3 25.9 25.6

24.84 24.83 24.82 24.82 24.81 24.81 26.52 28.76 30.85 32.66 34.06 34.96 35.32 35.11 34.34 33.05 31.32 29.25 26.97 25.14 25.02 24.94 24.89 24.86 24.84

--2.97 --1.87 --0.72 0.07 1.28 2.11 9.57 18.3 21.5 17.9 15.1 12.4 10.4 8.35 5.65 3.61 1.04 --2.16 --6.03 --9.23 --7.69 --6.25 --5.37 --4.02 --2.97

25.44 25.26 25.13 25.04 24.97 24.93 25.47 26.58 28.03 29.60 31.10 32.40 33.37 33.93 34.03 33.66 32.83 31.60 30.04 28.43 27.33 26.58 26.06 25.70 25.44

--0.63 --0.16 0.52 0.96 1.93 2.58 5.25 9.40 10.40 6.85 5.08 4.19 4.29 4.73 4.72 5.52 5.91 5.69 4.68 2.62 0.85 --0.08 --0.92 --0.79 --0.63

149

mean of 27.7 °C (calculated from Table 3). The reported annual mean maximum and minimum temperatures for that year are 33.7 °C and 24.0 °C respectively [19].

DISCUSSION

From Fig. 4 it is seen that the predicted air temperature rises to a maximum of 34.0 °C at 13:30, and then declines more gradually to a minimum of 24.9°C at 05:00. The reported air temperature rises to a maximum of about 32.5°C, also at around 13:30, and declines similarly asymmetrically down to a minimum of 24.2 °C at 06:00. Hourly comparisons from Table 3 indicate that the largest deviation of the predicted air temperature of about 10% occurs at 08:00, while the deviation is within 1%, at night from 20:00 to 03:00. And apart from underprediction of less than 1% between 21:00 and 01:00, the model generally over-predicts the air temperature. This is also reflected in the mean of the diurnal temperatures being 28.7 °C, 3.4% higher than the reported

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T I M E OF T H E DAY { L S T )

Fig. 7. Predicted energy balance components with old model.

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T~E OF THE D~Y (LST)

T~MEO~T.E DAY¢'ST~

Fig. 6. ~ d i c ~ d new model.

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~ i ¢ . 8. ~ f f e c t ~mpemtu~s.

componenN with

of bufldi~

m~

~m

o~ dlu~

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TABLE 4 Sensitivity o f parameters to m a x i m u m temperatures Parameter

Reference quantity

Variation o f 50% on reference

Effect on maximum T O

Effect on maximum T s

Wind speed, U2

5 m/s

2.5 - 7.5 m/s

37.8 to 32.1 °C (11.0 to --5.6%)

31.0 to 28.3 °C (6.0 to --3.0%)

Roughness length, Zo

5 m

2.5 - 7.5 m

35.7 to 33.0 °C (4.9 to --3.1%)

29.5 to 28.7 °C (1.2 to --1.6%)

Building mass, mc

700 kg/m 2

350 - 1050 kg/m ~

34.8 to 33.3 °C (2.2 t o - - 2 . 1 % )

29.5 to 28.9 °C (1.1 t o - - 1 . 1 % )

Temperature at 300 m, T~

25 °C

12.5 - 37.5 °C

25.5 to 42.0 °C (--25.0 to 23.3%)

24.9 to 33.2 °C (--14.7 to 13.6%)

Specific humidity at 300 m, q~

0.003

0.0015 - 0.0045

31.6 to 36.4 °C (--7.1 to 6.9%)

28.0 to 30.4 ~C (--4.1 to 4.1%)

Evaporating fraction, Ef

10%

5

36.9 to 31.8 °C (8.4 to --6.7%)

30.6 to 28.1 °C (4.8 to --3.8%)

-

15%

At reference conditions, m a x i m u m T O and T s are 34.03 °C and 29.20 °C,.respectively.

150

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~ ~o

REFERENCE

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BUILDING M A S S

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Fig. 9. Analysis for the effect of varying six input parameters on maximum and minimum air temperatures.

It is admitted that the excellent comparison so presented m a y be fortuitous, since the model is rather basic and many parameters, including R, E~, U2, Zo and m, have been approximated or assumed, based on a survey of other works. Nevertheless, the singular feature of importance is the ability to predict the lag in the m a x i m u m temperature, with accompaniment of an asymmetric profile. This is borne out by comparison with prediction from the old model, in Fig. 5 and Table 3, where in sharp contrast the air temperature is symmetrical about the noon, which is also the point of symmetry for the input solar radiation. O n the whole, prediction with the old model shows a maxim u m of 35.3 °C at 12:00, a m i n i m u m of 24.8°C at 04:30, and deviations from reported values by as much as 20% (at 08:00). The soil temperature, however, displays a lag characteristic for both the new and old model, with a greater lag in the new

model. In the absence of diurnal soil temperature data these will not be discussed further. The role of the building mass is also highlighted in the energy components in the new model in Fig. 6, as compared to those in the old model in Fig. 7. The additional building mass stores energy up to a maxim u m at around 09:00, declines to zero at around 14:00, and begins to emanate energy until the minimum point at about 18:30 is reached. The skew symmetric form of the M component influences all other components, except for the externally added R component, by way of a shift along the time axis. The m a x i m u m air temperature occurs at the time when the sum of the outgoing and storage energies exceeds those of the incoming solar energy. Turning now to sensitivity analysis, Fig. 8 shows the effect of the building mass on the air temperature. As the mass increases,

151 the temperature profile tends to be more flattened, with the m a x i m u m temperature being reduced and occurring later in time. However, the nocturnal air temperature is higher, as a larger building mass requires more time to cool down. At 2000 kg/m ~ the m a x i m u m of 31.9 °C occurs at 15:00, which has a lag of three hours after noon. Figure 9 displays the m a x i m u m and minim u m temperatures in response to separate variations in six input parameters; as one parameter is varied, the other parameters are held at their reference values. The trend of the variations agrees qualitatively with those predicted by Tapper et al. [6] for Christchurch, New Zealand. A more precise effect on the m a x i m u m temperatures is presented in Table 4, which shows that for a +50% variation on the reference value of a parameter, the building mass is the least sensitive parameter, as it brings about only +2% change in the m a x i m u m air temperature. In order of increasing sensitivity are the roughness length, the specific humidity at 300 m level, the evaporating fraction, the wind speed and the temperature at 300 m level; the last parameter causing a change of about +25% in the m a x i m u m air temperature. The same order of sensitivity is maintained in the case of m i n i m u m temperatures. It is n o t e w o r t h y that in this paper, the 'old model' is n o t identical to Myrup's model in at least two main ways. Firstly, a curvefitted solar radiation for a typical warm day is applied here, but this has the same characteristic that the intensity is symmetrical about the noon position. Secondly, in M y m p ' s case, the city substrate is taken to be 20 cm of concrete, whereas here the substrate is taken to be soil. Nevertheless in both cases the energy storage effect is absent. It is reiterated here that the present investigation focuses on the role of the building mass in urban climate, and hence the modelling has been kept at the basic level. Perhaps the results are encouraging enough to warrant the recommendation that a building mass term be included in more refined models whenever surface energy balance is involved. Moreover, in the present model, the assumption of an equilibrium surface temperature among the soil, the building

and the air, raises the possibility of a ramification in energy balancing, where multi-regions with different temperatures participate in energy exchanges. Work in this direction is now in progress.

CONCLUDING REMARK By including a time-dependent heat storage t e r m which arises from the presence of building mass, the energy balance model has improved significantly with respect to its ability to predict diurnal temperature variation in the city. The building mass does n o t seem to affect the m a x i m u m air temperature to as great an e x t e n t as some other parameters in the model, though it has a definite influence on the lag of the m a x i m u m air temperature. This basic energy balance model has worked satisfactorily for the present city but further studies in modelling and in more precise evaluation of input parameters should be pursued.

ACKNOWLEDGEMENT Part of the present study is supported through the ECOVILLE project under the Institute of Advanced Studies, University of Malaya.

LIST OF SYMBOLS Ca Cc Cs d E E~ H k ks L M mc qo q~ R 8

specific heat of air at constant pressure (J/kg °C) specific heat of concrete (J/kg °C) specific heat capacity of soil (J/kg °C) soil depth of level s (m) evaporation rate of water (kg/m 2 s) evaporating fraction (%) sensible heat flux to the air (W/m 2) von K~rm~n constant soil thermal conductivity (W/m °C) latent heat of water (J/kg) heat storage term (W/m 2) building/concrete mass per unit land area (kg/m 2) specific humidity at level O (g/g) specific humidity at level 2 (g/g) net radiation flux (W/m ~) heat flux into the soil (W/m ~)

152

t At To T~ T2 Tb T~ Ts' Us Zo Z2 ~0

time (s) time step (s) temperature at level O (°C) temperature at level O at previous time step (°C) temperature at level 2 (°C) soft temperature at level b (°C) soft temperature at level s (°C) soil temperature at level s at previous time step (°C) wind velocity at level 2 (m/s) surface roughness (m) the height of SBL (m) parameter in idealized equation for R

8

9

10

11

(s -I) Pa ps

air density (kg/m 3) soil density (kg/m 3)

12

13 REFERENCES 1 M. H. Halstead, R. L. Richman, W. Covey and J. D. Merryman, A preliminary report on the design of a computer for micrometeorology, J. Meteor., 14 (1957) 308 - 325. 2 L. O. Myrup, A numerical model of the urban heat island, J. Appl. Meteor., 8 (6) (1969) 908 -

918. 3 S. I. Outcalt, The development and application of a simple digital surface~limate simulator, J. Appl. Meteor., 11 (1972) 629 - 636. 4 T. P. Ackerman, A model of the effect of aerosois on urban climates with particular applications to the Los Angeles Basin, J. Atmos. Sc~, 34 (1977) 531 - 547. 5 J. Dozier and S. I. Outcalt, An approach toward energy balance simulation over rugged terrain, Geograph. Anal., 11 (1) (1979) 65 - 85. 6 N. J. Tapper, P. D. Tyson, I. F. Owen and W. J. Hastie, Modelling of the winter urban heat island over Christchurch, New Zealand, J. Appl. Meteor., 20 (1981) 365 - 376. 7 T. N. Carlson and F. E. Boland, Analysis of urban-rural canopy using a surface heat flux/

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