Transfer function method of calculating cooling loads, heat extraction and space temperature

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Publisher’s version / Version de l'éditeur:

ASHRAE Journal, 14, 12, pp. 54-56, 1973-01-01

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Transfer function method of calculating cooling loads, heat extraction

and space temperature

Mitalas, G. P.

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NATIONAL

CONSEIL NATl

Price 10 cents

Transfer Function Method of Calculating

Cooling Loads, Heat Extraction And

Space Temperature

P%W by Gintas P. Mitalas Reprinted from ASHRAE JOURNAL Vol. 14, No. 12, December 1972

p. 54-56

By permission of the American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Technical Paper No. 383

of the

Division of Building Research

OTTAWA January 1973

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LA METHODE DE FONCTION DE TRANSFERT ET LE CALCUL DE L'EFFORT DE REFROIDISSEMENT, DE

L'ENLEVEMENT DE LA CHALEUR ET DE LA TEMPERATURE AMBIANTE

SOMMAI RE

Une nouvelle methode de calcul de I'effort de refroidissement, l a methode de fonction de transfert, a kt6 introduite dans I'Bdition de 1972 du Handbook of Funda~nentals. Le present article est une breve etude des modalitis de calcul de cette methode, mais il s'agit surtout de montrer que celleci possede un champ d'application plus vaste que 1"'ancienne" methode. On compare 1'"ancienne" methode et la methode elargie quant au calcul des profils d'effort de refroidissement. Les rksultats d'un calcul- Cchantillon montrent graphiquement les applications elargies de cette methode relativement

i

un programme d'exploitation variable, au calcul de I'enlevement de la chaleur a une temperature ambiante variable de I'air et aux caracteristiques d1en18ve- rnent de la chaleur d'une borne de conditionnement de I'air ( y

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Transfer Function Method

of

Calculating

Cooling Loads,

Heat Extraction

&

Space

Temperature

GINTAS P. MITALAS Member ASHRAE

A new cooling load calculation procedure, the transfer func- The ASHRAE Task Group on Energy ~ e q u i r e m e n t s after

tion method, has been incorporated rn the 1972 ASHRAE five year's work, developed a computer-oriented method

HANDBOOK OF FUNDAMENTALS, 7his presents a that has now been incorporated in the 1972 AsHRAE HAND-

short discussion of the calculation procedure based on the BOOK OF FUNDAMENTALS. This new method has been desk-

transfer f u n c t i o n method, and demonsfrates ,he extended nated as the transfer function method because it utilizes the

capability o f this new, computer-orlenfed, method. transfer function concept to relate cooling load to heat

gain, and heat extraction rate to cooling load and room air temperature.

A transfer function is a set of coefficients that relate

T

HE growing concern about e n e r g consumption and an output function at some Specific time to the value of one the consequent urge to analyze more thoroughly the o r more driving functions at that tinle and to previous values design of alternative systems for the heating and cooling of of both the input and output functions- For example, a buildings has led to a demand for improved methods of component of cooling load a t time T, QT, can be related to

predicting the performance of air-conditioning systems. the corresponding component of the heat gain q T by an expression of the form

7 ,

I I I I I I I 1 I I I I 4 0 QI = v0qe.r

+

v l q e , ~ - A

+

V2qs.r-2A f

. .

= wlQr-,i

-

w ~ Q , - z A

-

w3Ir-3*

-

.

b - _/-\

-

l o _{Values of the v,, V,}

. .

_{and the w,,}

_w,

.

_,

.

_coefficients

S O L - A I R T C M P C R A T U R I

'

'\ depend on heat storage characteristics of the building and on

A1 OUlSlDE SUBFACE

-

s -

.

& the nature of the particular heat gain component. Table I

5

(Table 44 in Chapter 22) lists values of these coefficients for

3 4 -

-

various components of heat gain and for three classes of

building, viz., heavy, medium and light-weight construction.

2 2 - The new computer-oriented procedure is an extension

IPANSFER F U N C I I O N

'.'

-...

IfXAMPLf 16 )

Q

...

and development of the old familiar methods. It uses the

2

-

"...

-

_{standard approach to calculate most of the components of}

heat gain: heat gain from lights is the power supplied to the

I

-

2--

<

lights, etc. The heat gain through opaque wall and roof ele-

ments is the only exception in that it is not calculated using

1

1 ],

1 ,

j0

,,

Total Equivalent Temperature Difference (TETD), but

T ~ M C , H O U R rather by using the appropriate sol-air temperature data

Fig. I Heat fluxes at inside surface of outside wall calc~ilated and combining them with a wall heat transfer function, i.e.,

by rrmrsfer f~cnctiotz & T E T D _{this component of heat gain is found by using a transfer}

function approach. Transfer function coefficients for many

700

1 I I I 1 I I I I I I widely used types of walls and roofs are listed in Tables 39 and 40 of the 1972 HANDBOOK.

boo

-

_{The heat fluxes at the inside surface of the outside}

S O L A l HEAT G A I N

-4

'\ wall, given in Fig. 1, show the results of the two calculation

SOU

-

/ \

-

methods i.e., transfer function and TETD. The heat fluxes

&

-

1 1 calculated by the two methods agree reasonably well.

x

3 4 0 C - / 1 LY 6 X O U l

-

The transfer function approach to converting a com-

ponent of heat gain into the associated component of cool-

i

2,oe

-

ing load is, in principle, very similar to the time-averaging

L.

-

approach that was recommended in the 1967 ASHRAE

5

I Z O C

-

HANDBOOK. The new technique differs in that it applies a

IV T ~ A N S F ~ I F U N C ~ I O N F O R _{different weighting factor to each of the preceding values of}

I oc heat gain rather than giving exactly the same weight to sev-

eral values of heat gain. The calculated values of cooling

1 V,,L

-1

load are more accurate when the new factors are used.

0 2 4 b 8 10 1 1 14 16 I 8 20 12 I 4

I I M t , HOUt

Fig. 2 Coolitzg load profiles calcitlated by transfer function & G.P. Mitalas is Assistant Research Oficer, National Research

nvet aging methods Council, Div. o f Building Research, Ottawa, Ont., Canada.

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Table I. Coefficients of Room Transfer Functionsa

(Table 44 of Chapter 22, 1972 ASHRAE HANDBOOK OF FUNDAMENTALS)

--.

-Heat Gain Component V I

I

v2

Dimensionless

- - -. - -. ...

Solar heat gain through glassC with no interior shading device L 0.2727 -0.3400

M 0.2217 -0.3354

H 0.2155 -0.3712

Conduction heat gain through roofs, exterior walls, partitions & doors, & windows L 0.6982 - 1.2017

with blinds or drapes M 0.7108 - 1.4456

H 0.7055 -1.5668

Heat generated b y the lightsd

Heat generated by equipment & people & dissipated b y radiation

Heat generated b y equipment & people & dissipated by convection

and

Sensible heat gain b y ventilation 8 infiltration

These heat gain components appear os cooling load on air-

conditioning unit without delay. Thus v,= 1 .O.

~ ~ = ~ ~ = ~ ~ = w ~ = w ) = w ~ = O . O The w coefficientso

&'The coefficients o f the transfer function given in this table were calculated b y procedures outlined in Ref. 44.These coefficients are applicable for the case where all the heat gain energy appears eventually or cooling load.The computer program used for these calculations was developed ot the National Research Council of Canada, Division of Building Research.

The letters L, M and H denote the following room construction features used to calculate the above room transfer functions:

L = Light construction: frome exterior wall, 2 in. concrete floor slob. approximotely 3 0 I b of material per sq ft of floor area.

M = Medium construction: 4 in. concrete e ~ t e r i o r wall, 4 in. conmete Roor slab, approximately 70 Ib o f building material per rq. ft. of floor orea.

H = Heavy construction: 6 in. concrete exterior wall, 6 in. concrete floor slab, approximotely 1 3 0 I b of building materials per sq f t of floor area. For all carer, the room was assumed t o be furnished.

The coefficients of the transfer function that relote room cooling load to the solar heot gain through gloss depend on where the solar energy is absorbed. I f the window is shoded b y an inside blind or curtain, most of the solar energy is absorbed b y the shade and is tronsferred to the room b y convection and long-wove radiation in about the same proportion as the heat goin through wells and raofs.Thur the heat gain through windows with inside shading devices can b e combined with the wall and roof heat gain and converted to cooling load using the transfer function for heat gain through walls.

If theventilating air i s exhausted through the space above the ceiling, it remover some of the heat from the lights before it enters the room. This heat ir o load on the air-conditioning plant if the air is recirculated, even lhough i t is not part of the heat gain o f the room. The percent of heat gain appearing in the room depends on the type of lighting fixture, its mounting, and the exhaust air flow.

The wi coefficients (of the denominator) apply to all components of input except for the convection prrrt of heot gain from and equipment.

Table II. Normalized Coefficients of Room Air Transfer FunctionRJ' (Table 46 of Chapter 22,1972 ASHRAE HANDBOOK OF FUNDAMENTALS)

---pp

Conshuction

/

I

"*

Btu/(hr) (sq ft) ( F deg) dimensionless _{i = O}

---

--

.-I

Light

/

1.73

1

-3.50

1

2.22

1

-0.45

/

1.0000

1 -

1.8260

1

1.0697

1

-0.2005

1

0.0432

Medium -4.22

1

3.08 - 0.74 -2.1092

1

1.4606 -C.3331

1

0.0183

Heavy

/

::::

I

-4.55

I

3.61

1

-0.95

1

1.0000 -2.2908

1

1.7252

I

_-0.4277

,

_0.0067

-

a A simplifled procedure for calculating room air transfer function coefficients i s given in Ref. 45.

The g* coefficients given in this table ore for a room with zero heat conductonce to surrounding spaces and are normalized to unit floor oreo. To get the g coefficients for a room with a total conductance K between room air and surroundings, ventilation rate V, and infiltration rate VI, it i s necessary to multiply eoch g;* value by room floor oreo and then o d d lKS(V,+YI,)1.08lZi-03pi to the resulting govolue (whereV,andVl,areincfm).

Note that go value changer with the changes o f V, and VI, values.

Daily profiles of cooling load calculated using transfer function and averaging methods as well as the heat gain used for these calculations are shown in Fig. 2. These .'

curves show that the averaging method gives a substantially different cooling load profile as compared to the profile calculated by the transfer function method.

The main advantage of the new procedure, however, is that it takes the calculation a step beyond the determina- tion of cooling load and evalwates the rate at which heat will be removed from a space and the temperature of the space when a specific size and type of cooling unit is used. This phase of the analysis also allows the designer to evaluate the effects of different schedules of operation. This was

not possible with the old methods. By enabling the designer to evaluate the deviations of room air temperature from the nominal design value it is possible for him to exploit the finite width of the comfort zone and select equipment that can maintain conditions within the zone, albeit not always at the center of the zone.

Results of a calculation are given in Fig. 3 to illustrate the flexibility of this method. In one case, room air temperature control is set at 77 F all the time; in the other case the thermostat set point is at 77 F only during the occu- pancy period; the rest of the time it is set at 85 F. The transfer function coefficients used for this calculation are given in Table I1 (Table 46 in Chapter 22).

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Fig. 3 Roorn air temperatrcres

& herif rxfractiot~ rates for con- rir~ituirs & i?itertnittet~f ol~eraliot~

0 1 4 b B 10 I 2 14 I6 18 2 0 11 1 4

T I M E , HOUR

Fig. 4 Sample of results f o r E.\-ample 19 (se6 Table 48, Chap- ter 2 2 , ASHRAE HANDBOOK OF

FUNDAMENTALS, 1972)

0 2 4 6 8 10 1 2 4 16 18 2 0 22 2 4

TIME, HOUR

Example 19 (from Chapter 22) and the curves in Fig. 4 indicate the sort of result that can be obtained using the new methods. It is interesting to note that in this example the peak heat gain is 197 MBh,* but that the peak cooling load is only 158 MBh. Furthermore, if the cooling unit has a maximum capability of 150 kBtu/hr, it wil be able to keep the room temperature between 75.4 and 77 F. Results also show the consequences of using different throttling ranges for the control system and the conditions that would occur if the cooling system were shut down during the night. Attention is drawn to this example because it illus- trates the kind of analysis that the new method makes possible. The actual results are of little intrinsic interest as it is

not intended to imply that the same ratio of heat extraction to heat gain would obtain'for other situations. Every case should be examined on its own.

CONCLUSION

This is the first time that the transfer function method has been presented in this form for practical design calculation. It can be anticipated, therefore, that some minor difficulties may arise in its application. It is hoped that difficulties, if there are any, will not deter the majority of ASHRAE HAND-

BOOK users from trying the method.

:"000's Btuh.

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