An Example of Heating and Cool.ing Load Calculation Method

for Air-Conditioning of Building by Digital Computer

Shoiohi Kuramochi1

Taieei Construction Co., Ltd., Tokyo, Japan

Today, many air conditioning engineers recognize that the actual. load for air

conditioning system can hardly be achieved by the conventional heating and cooJ.ing

load ca1oulation method.· In 1966, the author worked out a J.oad calculation system,

in which the numerical calculation method was adopted to solve the problem on heat

transmission, and has finally perfected- this calculation system for actual. use

quite reoentJ.y. The characteristics of this calculation method oan be summarized

as folJ.ows: The air conditioning system in Japan is in majority operated intermit­

tently in a day and quite particular phenomena appear during the early hours after

the commencement of operation and right after the termination of operation. These

phenomena are caused by the transition of heat retained by the building and air

conditioning system as heat capacity. The calculation method worked out has

advantages over the others in representing the phenamina mentioned. To be more

specific, while any physical system doesn't particularly need to be conditioned

linear or invariable for calculating heat transmission, the storage heat load can

be accurately calculated by use of this oalcul.ation method. Moreover, this calcu­

lation system is marked by its reliability in securing an accuracy within the al­

lowable· degree for load calculation. It also permits easy understanding for the

practice engineers in general, hence it will be widely useful for the professional

engineers. Improvement and modification to this calculation system can be also

made without much difficulties. This calculation system is mainly composed of two

parts, namely, 1. the calculation for obtaining the outputs of the heat source equipment

of air conditioning system, and the heat transfer rate of thermal. medium

transportation system and the changes of room temperatures, and

2. the calculation for the heating and cooling load changes in the air­

conditioned rooms.

In the text of this report, the equations adopted for this calculation are

firstly Ulltstrated1 then the formation of the cal.culation system is explained in

detail and lastly the results of the calculation made for the model building in ex­

istence are checked and assured by carrying out the actual measurements and by com­

paring the measured values with the values obtained by the calculation system.

Key Words: Computer, air conditioning, heating load, cooling load,

numerical method, room temperature, medium temperature, simulation,

radiation, measurement, intermittent operation, evaluation.

1. Introduction

The heating and cooling load of air conditioning is a heat that changes temperature and humidity

in the room being air-conditioned, other than produced by an air-conditioning s,yetem. It occurs in the

form of an external heat invading the room or an internal heat produced in the room.. The heat from the

air-conditioner absorbs this heating and cooling load, thereby maintaining the temperature and humidity

of the air in the room at a required level.

The calculation system that will be referred to in this paper is composed of the following two

principal parts:

1 Mechanical Engineer

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(1) Calculation of heat transfer and change in room air temperature relative to heat source equip­

ment output and heat medium transportation system.

(2) Calculation of change in the heating and cooling load in the room being air-conditioned.

This paper will deal with the subject, starting will the details of

the result obtained from the calculation upon application to an existing

the actual measurement data to confirm the exactness of the calculation.

the calculation system will be explained:

the calculation system. Then,

building will be compared with

First of all, the features of

(1) The system gives a relation between cooling and heating load Qlt) and output of air-condition­

ing ~eat sourc7 ~(t)' therewith enabling calculation of the room air temperature 9i(t) under an es­

tabl~shed cond~t~on.

Despite the fact that air-conditioning systems are operated intermittently in many cases in Japan,

thermal changes at the early stage of operation and after the operation have not been much regarded. By

the said calculation system, such changes can be clarified.

(2) A relation between the time t 4 required for room air temperature ~. to reach a predetermined

level after the air-conditioning system starts operation and the capacity of~the air-conditioning heat

source ~ can be obtained, thereby permitting reasonable design of heat medium transportation systems.

(3) The calculation system deals with heat transfer by a numerical method. { 1,2) 1 This elimi­

nates the need for any linearity or invariability of the thermal system and permits calculation inde­

pendent of such changes in heat flow and temperature. Also, the calculation system is easy for practical

designers to understand, and can be applied and improved without difficulty, thus offering wide use by

professional designers.

(4) In making up the calculation system, approximation methods of new concept have been intro­

duced; (a) simulation of the heat medium transportation system in the air-conditioning system, (b) simu­

lation of radiant heat transportation in the room, and (c) simulation of heat exchanging in the air­

conditioning system.

The heating and cooling load in the room being air-conditioned is defined as a heat in a give-and­

take relation with the room air, where the heat causing the room air temperature to increase is expres­

sed as "positive" and that responsible for the decrease as "negative". Other definitions necessary in

the calculation will be given as required.

2. Principal Equations Adopted in Calculation System

2.1 Excitations

Excitations are used as an input factor to achieve the calculation, which can be obtained by means

of necessary relative formulae available from the design condition data. These formulae are prepared as

subrutine. While the design condition involves solar radiation, sky radiation, other effective radia­

tions, atmosphere, humanbody, animals, lights and heaters, their relation with the excitations directly

related thereto has been well known.

2.2 Numerical method of heat transfer

(1) Fundamental equation of steady state heat transfer (2-dimensional)

~j = Kij (If~ - l)i)

)\ ,d -."A

d

••• (1)

••• ( 1) '

where, Q.j: Rate of heat flow from point j

figure 1), fi! ~1 : Temperatures at points i and

to be the temper~tures at points 1, 2, ••• and Q1

to adjacent point i, d: Divided dimensions (see

j, )\: Thermal conductivity. Assuming &-1, (} , •••

to be the rate of a heat flow to point 1, we ~ave,

1 Figures in brackets indicate the literature reference at the end of this paper.

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For the steady-state heat transfer, the left member of the eq (2) is zero.

(2) Fundamental equation of unsteady state heat transfer (2-dimensional)

4t.~ = C(~1.dt- 19-1)

191.dt p(~2 + 19-3 + ~4 + li-5 + (~- 4l&;)

d. c(

N=--A

••• (2)

••• (3)

••• (4)

••• (5)

••• (6)

••• (?)

where ~1 At1 Temperature at point 1 after time interval of At, a: Thermal diffusivity, capacity of element volume, c: Specific heat, (: Specific weight, .1t: Time interval, 0(:

conductivity.

0: Heat Heat

The various equations of heat transfer to be produced by the above fundamental equation and the traditional equations will be illustrated in appendix.

(3) Temperature variation in thermal medium transportation system ( 5)

With a tube as shown in a figure 2 (1) considered to be a model of the thermal medium transporta­tion tube, it is assumed that the surface of the tube is insulated, and the temperatures of the tube and thermal medium are equal at any contacting point and time. Next, changes in temperature with time at respective points as shown in figure 2 (2) where the medium temperature is changed in step order as much as 1:.. &-1 in

the transportation system maintained at a constant temperature are replaced by linear changes as shown in figure 2 (3), and it is assumed that & changes at a uniform rate during the period of 2m1.t:1t to re~ch a final level, ~2 corresponding to ~1 th~t freely changes can be obtained by the following equatJ.on.

2m1 2m1-m+1 t9: + :E 1(n • .1t-2m1.~t) --- i! IJ; ((n-2m1+1ll-1 Mtl

m=1 2m1

f m1

c.r.v.llt

••• (8)

••• (q)

where, &1 ( at): Temperature of medium fluid at inlet of system at time n.At, ~2( t): Temperature of ~edJ.um fluid at outlet of system at time n • .1.t, LlE;1{ At): Temperature dirlerence at inlet during.t..t at time n.llt, n, m: Number showing time by multtitte 6f4t, f: Total heat capacity of transportation system, p, v: Flow rate of medium fluid, c: Specific heat of medium fluid, {: Specific weight of medium liquid.

When m1 is under a relation of 0.5~m1 f-1, it is regarded as 1, and if under a relation of m1 ~1, an integer closet to the actual value is employed. Especially when m1 < 0.5 1 G-1 is always considered equal to ~?~ This means that the effect of the heat capacity of the system is to be taken into account when it is~arge, and such effect may be ignored when small.

3. Change in Room Air Temperature and Heat Medium Transportation System

The calculation system dealt with in this Section takes the heating and cooling-load Q as the input

data, thereby computing the output of heat source equipment Qas and the change of room air temperature

9-i• Calculation system concerning Q will be explained in Section 4. Between_ the two calculation systems for ~i and Q, input and output must be exchanged to carry out the calculation. The heat produced

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by the heat source equipment is carried into the air-conditioning room by the transportation system 1 and

the heating or cooling load is carried into the heat source equipment by the same systema These heat

transportation phenomena are under the influence of: (1) Heat capacity of the transportation system

plus heat medium, (2) Thermal load caused by transportation equipment power, and (3) Heat from outside

of the transportation system, whereby the thermal characteristics of the transportation system will be

formeda In addition, the air-conditioning system is provided with automatic control unite With these

factors, an air-conditioning process system will be made up under restrictions by the design require­

ments.

In this Section, assuming a simple basic model as shown in figure 3 further explanation will be

made.

3o1 Conditions for design calculation

Principal control values for the air-conditioning system include (a) room temperature ~imt (b) chil­

led or hot water temperature ~w1mand (c) conditioning air temperature ~f1m• As restrictive conditions,

there are (a) capacity of heat source GRBm, (b) flow rate of chillied or hot water p, (c) flow rate of

air-conditioning air v1, (d) volume of outdoor air induced v0 , (e) time required for room air tempera­

ture to reach a stable level at early stage of operation t4, (f) time required for heat source output to

reach a stable level at early stage of operation t 1 and (g) time required till complete elimination of

heat source output after operation stop t 5 •

Figure 4 shows the changes in QL:rn, Q and t}i at early stage of operation, and figure 5 indicates

their trend of subsquent changes after operation is over. The control values and restrictive conditions

are appl.ied as design requirements together with outdoor and indoor conditions.

The changes in Qm.[ at early stage of operation and after the operation is over which are given as

the characteristic of the heat source equipment, can be regarded as linear changes in the case of

ordinary refrigerators and boilers, and thus are treated as a certain restrictive factor. Assuming

n1 = t1/4t, and Qru3m = maximum heat source output, we have Ll~ = Qzmuin1 = constant at early stage of

operation. If n is between 0 and n1, this equation can be rewritten as

QRB(n.llt) ••• ( 10)

be Also, assuming that the time to stop operation is t = 0, and n

5 obtained if n is between 0 and n

5

ts(~t, the fol.lowing relation can

n.<lruJ(o)

"rm<n.<lt) = n5

... (11)

In addition-~ variables necessary for the calculation, such as; W1: Total heat capacity of supply

system of cooling or heating water, W2: Total heat capacity of return system of cooling or heating

water, £1: Total heat capacity of supply air duct system, V: Volume of room, c.r. V(l+~): Total ef­

fective heat capacity of room air, e: Additional coefficient of heat capacity of room air must be as­

sumed for calculationo

Initial conditions are related to the thermal state of the air-conditioning system at the time of

commencement of the calculation, and may be assumed through engineering judgements guided by location of

building, data, time, thermal characteristics of building and system, and statistical data.

heat from

Now, assuming that !9-0 is outdoor air temperature, ~"i is return air

exchanger, and ~f2 is mixed air temperature on the side of the heat

the following equation.

temperature at exchanger, t;. fZ

the inl.et of can be obtained

Slight errors in assumed values

are correct, because the calculation

is advance.

for the initial condition are system itself has convergence

3.2 Variation in room temperature

JoAq as, negligible~equent

and if the time to start

Variation in room temperature is obtainable by the following equation.

718

••• ( 12)

input data calculation

••• (13)

where, QRB(n.At): Rate of heat transported from air conditioning system to room at time n.~t, Q(n.dt): Heating or cooling load in room at time n.dt,4~i(n.dt): Difference in room temperature between 4t at time n.At. And if the supply air temperature at diffuser at time n.~t is &"rl(n.At),

'lru!(n.<lt) ••• (14)

According to the equation (14), it is noted that the room temperature at time (n.dt +dt) is ob­tainable from respective values at time n.At. This fact is of great significance for the subsequent calculation. Apparent variation in retained heat of room air ~ relative to changes in room temperature at the earlY period operation from 0 to t4 is given as,

••• ( 15)

where n4 = t4jdt. The decipive factor for the change in room temperature where the absolute value of ~ is smaller than that of JlflQ}mdt is a balancing relation between Q and Q"'RB• With relative

increase in ~~ however, ita ef?ect on t4 can no longer be ignored, and must be given due consideration in selecting the value of QRBm• At the early stage of usual intermittent air-conditioning operation,

the maximum output of the heat source equipment is required, and ~ is often determined by setting t4 at a required value. The relationship between t4 and ~ can be obtained. Today, there is no better way than a trial and error method to find the relationship between t4 and ~·

3.3 Heat exchanger

Air must be used as final thermal medium when heat produced by the air-conditioning system is transmitted into the room. Heat exchangers of the air-conditioning system in general use today employ water as primary thermal medium and air as secondary thermal medium, excepting heat exchanger for heat source equipment. The capacity and service condition of the heat exchanger are subject to the related design specifications. In order to quantitatively find the characteristics of the heat exchanger under other service conditions, it is necessary to intorporate complicated characteristic relationship as it is into the calculation system. This, however, is so intricate to deal with that a practical simple method will be employed here.

General usage and performance of the heat exchanger most used recently are as follows, with the flow rate of air v1 passing through the heat exchanger considered invariable, chilled or hot water at an almost constant inlet temperature ~'w1m is supplied by controlling its flow rate p by means of a two-w~y or three-way valve. Assuming the return water temperature to be 9'w2 ' the rate of exchanged heat is ex­

pressed as P (&'w1 - ~'w2). This formula is equal, whether two-way or three-way valve is involved.

If the air temperatures at the the corresponding enthalpies are i 2 exchanged heat is had as follows,

inlet and outlet of the heat exchanger are ~£2 and ~L1 ' respectively, and i 11 and the latent heat load to be eliminated is~L' the rate of

... ( 16)

While the values of &1 2

tion, ~· 1 and & 1 serve ~s

w m f m that these centro values may

and &f are establiShed spontaneously during the air-conditioning opera­contra! valuesa Design values p and v1, therefore, must be adjusted so not be exceeded even under the maximum air-conditioning load.

Taking for example a cooling operation to simulate the action of this heat exchanger, the mutual relation with temperatures is shown in figure 6. In figure 6 (1), a relationship under maximum load is given, where the value of c/a is assumed to be invariable under medium load Shown in figure 6 (2).

Under this relation, a = &f2 - 6l-'w1, b = &•w2 - &'w1, b' = ( G-'w2) - G-'w1 and c = 8-f2 - 61-f1. The differ­

ence between ~·w2 and (~'w~ is to correspond to the latent heat load Qt• The relation between c and b is at given by the equation (16). The value of a becomes maximum when the load comes to the maximum, at which point, it is necessary not to overestimate the value of~1•

The values of ~·w1 and ~·w2 obtained from this simulation are not equal tO those actually measured. But what is actually needed is the difference in temperature (&'w2 - 9'w1), and thus if near actual values are desired, they can be obtained by comparing the value of ~·w1 with ~·w1m and adjusting~w2 as

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much as the resultant difference.

Simulation of the automatic control performance at the early stage of operation is achieved as fol­

lows: The val.ue 9-f1 iS'changed toward the predetermined value of B-f1m, andQR5 is increased till ()).i

reaches its predetermined value. With QRM reaching~ RBmt this state is main tamed. After the room

temperature reaches the pre-determined level, the difference in~i corresponding to the load variation is

feed back to the heat source side with the medium temperature &f1 and ~·w29 thereby controlling the out­

put «Jm•

The thermal relations in the heating operation can also be established merely by reversing the

symbols for temperatures and heat given above.

3.4 Calculation of thermal relation in air conditioning system

Figure 7 shows a thermal relation in the air conditioning system

in figure 3, and the calculation procedure to obtain various values.

the calculation~ some assumptions are set as follows:

(a) Time required to transport heat medium is ignored.

'

based on a simulation model shown To simplify and smoothly accomplish

(b) External heat load invading the heat medium transportation system is proportional to the heat­

ing or cooling load (this is obtainable by calculation as required).

(c) This calculation system includes no exact calculation method for relative humidities. Thus,

the relative humidity of the air in the room is considered always maintained at 50%.

(d) Room air temperatures within one calculation unit (will be referred to later) at same time are

all equal.

(e) Calculation is carried out at each step of time interval t& t. The calculation at t = n • .dt +.at

subsequent to the completion of the cycle of calculation at desired time t = n.dt is started with room

air temperature 9i(n.At + at)•

(f) As shown in figure 7, calculation is carried out in the order of room temperature, return air

duct system, return water pipe system, supply water pipe system and supply air duct system, followed by

calculation of room air temperature with time advanced as much as~t.

(g) ported by

Although the heat flow of the cooling or heating load is reversible, the flow of heat trans­

the heat medium transportation system is inreversible~

(h) The temperatures at various points of the air-conditioning system are calculated by the use of

known related temperatures obtained when time is most adyanced, as a general rule.

Today, these assumptions are within the permissible range of the overall calculation accuracy.

The following will outline the transfer of heat in the air conditioning system step by step.

(1) Return air system

The return air temperature changes with changing room air temperature.

air temperature between times n.dt and n.tlt + 4t to be A.9' 1(n.L]t), it can be Assuming change in return expressed as f~llows:

Suppose that the return air inlet temperature 9-'i(n.l]t +At) is changed to 9 11i(n.llt + 4 t) at the

outlet. This temperature change can be caused to occur by external heat qft invading the system and the

heat ca:pacity of the return air system. A relation between &1 i and &"i can be obtained by the following

equation, using the equations (8) and (9), with qf~ token into account.

The

n.ll (;' ~ i(n.4t~t) = i(n.4t+4t-2m1.At)

ml = f,j(c.a.Vz·At) temperature of a mixture of this return

(2) Return water system

2m1 2m1-m+1 t

+ m'f1

2m1 Lli!l i{(n-2m1-tm)tltj qf2

+ <-~-)4t ••• (18) Ceu•V

2 ... (tq}

air and outdoor air is obtainable from the equation(12).

According to the given order of calculation in this paper, the calculation now comes to the return

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water system at time (n.6t +~t). As shown in figure 3 1 the system contains a water pump, where the

flow rate of water is p, water temperature at the outlet of the beat exchanger is &1w2 , and water temper­

ature at the inlet of beat source equipment is &w2• It is also assumed that all the thermal changes

caused by the air at the inlet of the beat exchanger will be taken over by the return water system, which

will join the effects dependent on the rate of invading heat qwt• power load of the pump %WP and heat

capacity of the system in the return water system.

The effects of the mixed air will be taken over by the equation (16), and by applying equations (8) and (9) to the above changes, the water temperature at the inlet of the heat source equipment

&w2(n.dt + ~t) can be obtained as follows:

2m2 2m2-m+1 , + m~1 2m2 lf&-w2{(n-2m2-!m)~t}

m2=~ P•tlt

'\.2+ K~ + ( p -t ... (20)

••• (21)

where &-f2 and ~·w2 , and t9w1 and &f1 used in equation (16) are values expressed at times (n.Llt + 4t)

and n.4t, respectively.

(3) Supply water system

Thermal changes in the return waters system are carried over into the supply water system. The

heat source equipment is located between the two systems and is subjected to the change in temperature

depending on the output Qas• Assuming that the water temperatures at the inlet and outlet of the heat

source equipment are 9 2 and()- 1, the relation ofQRBwith them isQRB = p(&: 1 - 8-' 2 ). With O.Wt as­

sumed to be the water !emperat~e at the inlet of heat exchanger anu-temper~ture c~ange caused by heat

invading the system qw1 taken into consideration, ~·w1 can be expressed by the following equation in the

same manner as above.

2m3 "" 2113-m-1 &

+ ~1 2m3 ~ w1(n-2m3-~mldt + ••• (22)

3 w1

m = p.l\t ••• (23)

(4) Supply air system

Thermal changes in the supply air system are taken Over by the supply air system via heat exchanger.

Values ~or ~f2 ! ~·w1 and~~ 2 are all as measured at time (n.dt +dt), as well as ~f1 • In the supply air

system ~s prov~ded a fan. l§suming that the air temperature at the outlet of the heat exchanger is ~£ 1 , average tempQ.ratur·e of the diffused air is 19'11f 1, power load from the fan isQKWP' and heat invading tlie

system is ~l , we can obtain &"f1 by the following equation in the same manner as above.

2m4 4 &" & ,_. 2m -m+1 l\&

f1(n,4t+At)= f1(n.At+dt-2m4,~t) + m;'1

2m4 f1[(n-2m4-~m)llt) + ••• (24)

... (25)

The four systems mentioned above are connected by (a) air-conditioning room, (b) the heat e~changer

for air-conditioning and (c) heat source equipment. The performance of these connected systems and heat

medium transportation system will serve as various basic components of an air-conditioning system. Also,

the air-conditioning system is available in other types using boost heater, dual duct system, three or

four piping system, etc. These cases can be handled in the theoretically same manner as Q and Q , where subrutine is prepared for the calculation. By the combination of various basic mode~ and au~ routine as required, various types of air-conditioning systems and associated heat medium transportation

systems can_ be made up.

4. Heating and Cooling Load in Room

When the material of wall surrounding the room, dimensions, room, atmospheric condition and initial condition are given, and designated, the heating and cooling load tends to be determined.

721

position and service condition of room temperature and humidity are In this Section, the calculation

the

system whereby the transmitting course and thermal rate of the heating and cooling load can be obtained will. be explained.

4.1 Preparation for calcUlation s,yatem

(1) Calculation unit and area element

Unline the conventional method to calculate the load per each room, the calculation system being referred to in this Section is such that a plurality of rooms that little differ in temperature and are considered equal in temperature variation during the calculation are dealt with in one cal.culation unit. In case one room has two or more typical temperatures, as many calculation units ae the number of typical temperatures are provided in the building. Thus, while the calculation unit has something related to the concept of zoning, it still depends on the typical room temperatures. Practicall.y, rooms where the temperature difference is always within ± 1o5°C may be regarded as one calculation unit.

Heating or cooling load in the room is obtained by the calculation of trans:la:l.t heat transfer pheno­mena, except convection heat transfer, draft and latent heat that are directly obtainable. When the room air absorbs the heating or cooling load, its temperature changes at a rate dependent on ita heat capaci­ty. Ae the heat capacity of this room air, an apparent heat capacity c.r.V(1+p) is used. Walls, floor and ceiling forming a room play a leading role in the transfer of heat, radiant heat that has invaded the room cannot become a load without having been absorbed by the solid objects in the room.

The transmission course of the heating or cooling lead is formed as a thermal system chiefly by the surrounding structural objects, and its thermal characteristics depend largely upon the construction and composition of the structual surroundings, the load phenomenon occurs in largely different manner depend­ing on whether the glass window is wide or narrow, or whether a curtain wall or concrete wall. is used.

A calculation of heat transfer does not necessarily require simulation exactly to the thermal com .. position of the room, in this calculation system, the areas of structural objects equal. in heat transfer phenoemonea are totall.ed for calculation and the results are proportionally divided per area.. These area are called area elements for calculation.

(2) Thick wall and thin wall

Disturbances with large changes are caused by the external atmospheric condition, and response de­pends on the properies of the outer wall. To estimate the rate of heat transfer from the wall body, thermal transmittance or thermal conductivity is used. The rate of temperature changes inside the solid can be determined according to the thermal diffusivity. The wall bodies are available in many types, such as concrete wall having a heat capacity with property of heat insulation, metal panels having a low heat capacity with poor heat insulating power, a combination of metal panels with heat retaining materi­al that is low in heat capacity yet has proper heat insulation, and glass plates that let radiant heat penetrate through.

These wall bodies are classified by heat capacity per unit area, according to which those high in heat capacity are call.ed a thick wall and those low in heat capacity are called a thin wall. The rela­tion expressed by equation (6) is referred to for practical classification, whereby the divided measure­ment 11d11 relative to divided time .4 t is determinedo Accordingly, the wall whose thickness is divided into two or leas portions is called a thin wall, and that with three or more divisions of its thickness is called a thick wall. For the calculation on the thin wall, equation (1) or (48) is applied combined with such an expedient as including part of the heat capacity of the wall body in that of room air ..

~.2 Course of heating or cooling load

The calculation of heating or cooling load in the room is aimed at clarification of the quantity of heat transfer through the analysis of the transfer course of the heating or cooling load. The flow rate and direction of heat always involves the temperature gradient on the course according to Fourier's low. With a temperature change of the heat medium on the course, the rate of its retained heat increases or decreases, and this behavior of the heat has an important bearing on the load. Kink of disturbance and intensity of the heat flow are given as the design considerations, and various equations shown in the a.}l}len.diX are applied for the calculation of heat transfer inside the building. The application of these equations is very simple, and will be outl.ined below in connection with the pointe particularly considered in this calculation method.

(1) Heat transfer on interior structural bodies

The room bas many pillars and beams in addition to the walls,. While these structural bodies do not serve very much as the course of the external heat, they release or absorb the retained heat with changes in room temperatureo Also the surrounding wall has a rugged surface, which has a similar thermal action .. To compensate for these thermal actions, subrutines by the application of equations (5) and (36) through

722

(38) are prepared.

(2) Radiant heat transfer in the room

Let us assume a film at the boundary dividing the exterior and interior structural bodies. This assumed film represents a surface condition of the interior structual body, and its surface temperature (MRT) is &'Im = ji Am X f;-r.nJ ~ Ain• The effective radiant heat transfer between the external surface and the film can be obtained by equation (42). If there is no marked temperature difference over the entire exterior structure, its average temperature (MRT) is B'om = Y: AonX G-or/fAon• The radiant heat transfer­red to the assumed film from the exterior can safely be considered to be received evenly by the whole surface of the interior structure.

(3) Radiation from interior heating bodies

Interior heating bodies are available locally and in the objects evenly scattered on the floor, such as human bodies and lighting equipment. In the latter case, radiant heat is emitted evenly to the ceiling and floor.

4.3 How to set up load variation calculation system

Tb achieve calculation by computer, the design conditions and area element per room are applied as input data. Of the results from the calculation, necessary data are taken as output, and expressed with a proper time intenal. The calculation system must be ready to take all kinds of possible input date conditions. If any special conditions occur, and the system is not prepared to permit their application thereto, an approximation method is emplyed in a form close to the existing method. If the effect of the approximation is not permissible, then the calculation system itself must be improved for higher ac­curacy. Such instances often occur in respect to multi-layer wall, or when the air-conditioning room is adjacent to a room under special condition.

5. Example of Calculation

The calculation system being dealt with in this paper is based on the simulation of a air-condition­ing system, and is a computer program intended to obtain design data. The details of the setup, however, are the questions directly related to the prograrmning, and thus are omitted in this paper. Already, this calculation method has provide to be able to provide permissible calculation data upon application to several existing buildings and comparison with actual. values. The .following is an example of the practi­cal application.

5.1 Application of calculation method

(1) Preparation for calculation

Prior to calculation, ·the air-conditioning system is determined. Then, input data are prepared from the related materials. Since the Capacities of the associated equipment are unknown prior to the designing, they are estimated from the actual statistic values. The capacity of heat source equipment .for intermittent air conditioning ~ is obtainable from ita relation with the time required to stabi­lize the room air temperature t4, and accordingly values for p and v1 can also be determined. For the operation of the calculation system, engineering decisions, such as initial conditions, are required. Various values estimated at first are ·corrected to proper values upon investigation of the calculation results, and are recalculated if' necessary. For example, the temperature change in room air temperature in the adjacent part that will not be calculated is assumed and if this assumption is found far different from the calculation result obtained later, the calculation is repeated from the beginning.

(2) Application to building ( 6)

Location: Tokyo. Name: 0 Building. Purpose of use: to provide office spaces. Construction: SRC (curtain waJ.ls used as outer walls at the south and east sides). Scale: Nine storied, 3 basement floors and 3 penthouses. Total area: 7,260m2. Air-conditioning area: 3,400m2. Operation: Intermit­tent (8:30 to 17:30).

The application of this calculation method to the air conditioning of this building was attempted. In this application, actual values were emplyed, instead of exterior and interior design conditions. For the air-conditioning system, the model shown in figure 3 could be applied as it is, and the calcula­tion was carried out in one calculation unit.

723

(3) Results of calculation

By this calculation method, the temperatures and heat flows in the building and at the respective

components of the air-conditioning system 9 could be calculated in detail over the given calculation

perior. However, recording of all the data on the temperature and heat flow is tremendous and practical­

ly meaningless. Thus 9 what was considered necessary for the design was selected from among these data

as the output, from which values for ~9 &i and only were picked up and diagrammed as given in figure

8 (2) with full lines and bracken line. The changes of the heat medium temperatures &f1, &-r2, &"r1,

&'w1 and &•w2 were as shown in figure 9 with full lines.. From these calculation results, the following

were found out.

(a) To bring the room air temperature at the early stage of operation to a predetermined level,

the heat source equipment is to be operated with full output for a time., According to the conventional

design method, however, the flow rate of air in the supply air system becomes short. Thus, the output

declines prior to arrival. at the predetermined room air temperature level. As a result of a trial. cal­

culation with increased supply air rate to avoid the shortage, it was found that full output operation

has to be continued till the room air temperature reaches the predetermined level, as shown in figure

10.

(b) The part indicated with A in figure 8 (2) at the initial stage of operation represents what is

spent for the increase in the retained heat of the equipment (negative heat in the cooling operation)~

In contrast, the part B appearing after the operation is finished results from the discharge of this

heat. Both rates of heat are not always equal, however.

(c) By the application of this calculation method to the air conditioning design, adequate value

for Qrum, can be discovered. There are other advantages that rational heat medium transportation system

suited to ~ can be designed and that a proper air-conditioning plan can be set up.

In this example, calculation was made using known values for QRBm, p and v1 of the system set up

by the conventional design method, thus given other results than those for rational equipment.

5·2 Comparison with measured values and evaluation

Changes of ~ and &-i obtained from the actual. air conditioning measurement in this building are

given in figure 8-12) with dotted lines together with exterior conditions as figure 8 (1). Changes of

heat medium temperatures are shown in figure 9 with dotted lines~ The corresponding calculation results

are as given in figure 8 (2) and 9 with full lines and bracken line. A comparison between both results

snows that the change of Qru3, Q and &i with time shows similar values and trends.. Although the heat

medium temperatures themselves are a little different because of difference in control method, the neces­

sary temperature difference are in near agreement,.

What calls for specific attention here is the fact that the existing air-conditioning system is the

product of the conventional design method and not of a rational method. Also, when it comes to the

actual construction, there were unexpectedly many causes of heating and cooling load, such as that from

the heat medium transportation system exposed on the roof. With these points taken into consideration,

a difference by 10% or so between the calculated and actual values for ~ is upavoidable ..

For reference, the time and labor required for this calculation method are such that IBM ,360 com­

puter took 20 to 130 seconds with6t at 300 to 60 seconds per one calculation day, and a little more

labor than for conventional method was required for input data preparation.

Judging from the above, we believe the practical value of this calculation method by degital com­

puter is very high.

6. Conclusions

In the future, calculations of the type mentioned above will have to depend on computers by all

means. As the calculation methods for solid heat transfer, there are weight function method, response

factor method and analog method, in addition to the numerical method employed in our system.. All these

methods, however, pertain to the problem on less than 3096 of the heating and cooling load, and as to

remaining part of the load, they have little to differ. The gist of the heating and cooling load calcu­

lation should lie, not in the methodology of the solid heat transfer calculation, but in the systemati­

zation of the calculation method altogether ..

Our calculation method introduced in this paper has been systematized since several years ago, -in­

dependently of the progress in other methods. With the numerical method, too 9 it is possible to reduce

the calculation time and memory capacity if proper consideration is given, and satisfactory results

724

coul.d be !l'btained in respect to accuracy~

Improvement of this calculation method itself and interchange with other methods are the problems remaining to be solved,_

This paper limited clarification to the design problems ahead of the heat source equipment outputi and has not dealt with the air conditioning energy. It, however, can be easily obtained using, the handling of the heat medium transportation system as a guide, provided the performance of the system components is already known.

WhUe it is still pre,ature to draw a conclusion because of insufficient data, the physical value for the wall body, a ( = ~ ) is considered different between cooling and heating period. For such phenomenon, this calculation method is more advantageously applicable than other methods as in the case of intermittent air conditioning operation.

To conclude this report, the author expresses his profound appreciation to Dr. Kenichi Hiraga under whom he works for the opportunity of compiling this paper and very helpful advicea and guidance.

7• Appendix

7.1 Various equations related to unsteady state heat transfer

Temperature in one dimensional object, see figure 11 (1)

... (26)

One dimensional, surface temperature of object in contact with fluid with temperature of, see figure 11 (2)

&1.6t = 2P(&2 + N.~f + <zi. - N - 1) &;) N.\9-f+2G;

N + 2 see figure 11 (3)

... (27)

••• (28)

One dimensional, surface temperature of object subjected to heat flux q, see figure 11 (4)

... (29)

Heat produced at uniform rate inside object Q, see figure 11 (5)

.... (30)

One dimensional, surface temperature of object with heat produced at uniform rate inside Q, see figure 11 (6)

.... (31)

One dimensional, heat conduction between two objectst aes figure 11 (7)

- 1) e;) ••• (32)

725

where, t\I: Thermal conductivity of object with point 1, Au: Thermal conductivity of object without point 1.

Two dimensional, irregular shape surface temperature of object

In case of figure 11 (8):

••• (33)

In case of figure 11 (9):

••• (34)

In ease of figure 11 (10):

($-2 + ~ 1 ')

&1.Jt = 4P 2 + N.er + (lfP- N- 1)$-~ ••• (35)

Value for 0( used in equations (33) to (35) is a corrected heat transfer coefficient inversely pro­portionate to increase or decrease in area of the object surface.

7o2 Values of P and N

Because of the condition where multiplying coeffecient for ~1 in the right member of the equations for G-1•(lt will not become minus, 4t has a permissible maximwn limit. For example, from equations (26) and (27) given in the latter paragraphs, we have

p ~ 1/2 inside object ••• (36)

1 p 52( 1-INJ on object surface ••• (37)

Also, the permissible range of P relative to any value of N is as follows: P ~1/3 ••• (38)

7o3 Surface heat transfer ( 3)

Q = .w. -11'1) ••• (39)

where, Q: Rate of heat transfer at surface, D<: Total heat transfer coefficient, CX 0 : Convection heat transfer coefficient, ~r: Radiation heat transfer coefficient.

7.4 Radiation (solar) temperature &8

.... (41)

where, I: intensity of radiation, a: absorption rate.

7o5 Radiation ( 3)

••• (42)

726

••• (43)

••• (44)

••• (45)

••• (46)

where, Q12: Net rate of radiation from surface A1 to A2, A1, A2: Areas of surfaces exchanging heat by radiation opposite to each other, ~12 : Total shape factor between A1 and A2, T1, T2: Surface temperatures of A1 _end A2, dA1, ~: Small element areas in surfaces A and Az, ¢1 , ¢>2 : Angles be­tween normals on dA1 and dA2 and li.iie connecting dA1 and ~' ·a1, a2 : lbaorptJ.on factors of A1 and A2 , f,, ~: Emissitivities of X1 and A2•

7o6 Equivalent out door temperature &8

and equivalent temperature difference ~~e ( 4)

110

= 11.,. + t(l18

('(") -Item)

tll1o = &.,. + r( Irs( 't) - i9'sm) - 1)-1

1 23 & sm = 21+ 1: lf's(t)

t=o

••• (47)

••• (48)

where, &1 : Room air temperature, f: Decrement factor, \: Time lag, &-a(L): Value of 9'8 as much as before time being calculated.

B. References

( 1) H. s. Carslow and J. J. Jarger, Conduction of heat in Solid, P.467-478.

( 2) Y. Kudo, Dennetsu Gairon, P.394~416.

( 3) K. Watanabe, others, Kenchiku Ke~ Genron, P.29, 60.

( 4) H. Uchida, Kuuki Chyosei no Kihon Keikaku, P.90, 91

727

( 5) s. Kuramochi 1 An Calculation Method of ~Conditioning Heating and Cooling Load, SHASHJ, Vol.43, No.12, Vo1.44 9 No.1

( 6) s. Kuramochi, Investigation and Measure­ments of Air-Conditioning on Buildings (1) (3), AHASEJ, Vol.44, No.2, No.3.

I_J T ~r----r-- -,

' i 3 ! i

d

l I I I 1 L __ ---f--- __j_ __ f---~ I I r I 4 i 1 I 2 !

i I l I

1------- __ L_ ~--L __ I I I I --1

I I 5 I I y

I I -r

i I I I L ____ L_ __ _ _j ___ _ _j

'---X

Fig-1 Heat Conduction in Solid, Two-Dimension

728

~--------------~--------------~

(}I (t>O)

B'

l 0

.... Q:)

t (h (0)

81 (t

()

I

10 11 L',tl

12/ltl

~x

Llt1

llt 2 ilt

Fig-2(1)-(3) Simulation of Thermal Medium Transportation

System

729

F p pl Pz T1

fan pump

v 0 -wv--j

8i

vl H.X.

control Panel.

pump thermostat

flwl flwz p v4 v3 ewl

Bwz Ref

valve for heating

" If

II for cooling II "

three way Valve

Fig-3 A Model of Air-conditioning System

730

operation start

B I

' Bi

Fig-4 Change of Thermal Medium Temp. just after

Operation Start

Qn.v._ __ Q Rs"'

Q

B (see Fi

Ql l

operation stop

-t

Fig-5 Change of Value of QRB, Q"RB and Q just after

Operation Stop

731

CD

I

~( 1 )case -c::: . ....

..... atQm~

..... ::::1 0

~ ( 2) -c::: .... .....

case at Q ~ .....

::::1 0

Fig-6(1),(2) Simulation of Heat-exchanger Performance

732

~

"' "'

out door air

rl® return air system : r/ ® return water system J

leD air conditioning room I l (]) heat exchanger l

I l G) heat souce equipment 1

Y @ supply air system }-®-- Y ® supply water system J

Fig-7 Calculation Procedure Diagram

~ :r

oc D.B. W.B.

1

total sky .rc'J .

c4c'Jt. loiJ .

~

1967- 8 -10

bulb temP·

wind velocity W.V. \ --...______:..

10 -h

Fig-8 (1) Conditions of Weather

5

3

2

1

9

N I s .

-+-'

It ~ ~~

~ w '-"

-+-" -+-" cd ~ X tn

QO ,~

operation stqp

'

6 -~- .. . . 0 ... .. 0...... . ......... . • • •• 0.

5

4

3

calculation A -----------[-------------,....,. -------/ .i.Q fresh airload

I -------- --- -----------------, ' r---t.. I /

I /

calculation 2 Bi .......................... 7···· ....... ..

measurement

0 0 0 0 0 •• 0

1 /

B .... 8 9 10 11 12 13 14 15 16 17 18 h

Fig-8 Cont. (2) Values of QRB' Q and ei obtained by Calculation and Measurement

OC Bi

29 1 28 27 26 25

~ w ~

oc 30~peration start

- , fJr2 - ,_ ----

operation stop~ ------~- ~~

B 25t

l20r 11 _____ ;

--1 --·---------15r 11 __ ... ______ ........ -- fJ{l . I ----11 f) I ------ W2

lOr I\ f) ----- - -- --1 ---- w 1 ----

51-

08

-- -I -------.. .-symbol --calculation

---measurement J I I_ L

9 10 11 12 13 14 15 16 17 18 -h

Fig-9 Thermal Medium Temp. obtained by Calculation

and Measurement

6

5

4

3

2

1

f-

f-

f-

.-

8

operation start

Q RB

ei -"'An.

I I I

9 10 11-time

-

-

-

-

-

oc 29 e 28 1 27 26 25 24

Fig-10 Results of Calculation in case of enough Air

supply

737

I 3 1 2 I I I I I I I I I

Ld d d----i

( 1 )

f 1 2 I

lJ ( 2 )

Fig-11 (1 and 2) Variation of Heat Transfer to be adopted

738

~ V> ~

I /; I

I / I ;:; I :;; ;:; I

I ---1

f s~ 11 1 r1 I I

~· I I I I I I I

/ ·~d

I d--1 One dimensionlal

( 3 )

4

3

1

I I I d~-

2

//A/#/@

Two dimensional

Fig-11 Cont. (3) Variation of Heat Transfer to be adopted

Q-.....

2

I

--~-<--d__,J

( 4 )

Q ' I I I I I

I 3 1 2 I I I I I I I I I

I I

Ld d d_J

( 5 ) Fig-11 Cont. (4 and 5) Variation of Heat Transfer to be adopted

740

f 1

I I I I I

l 3T

I I I

-d

Q

2 I I I I

d-l----d~ 2

( 6 )

II I I I I I I

.L I .l.

1T I

2T I

I I I I

d d~

( 7 )

Fig-11 Cont. (6 and 7) Variation of Heat Transfer to be adopted

741

~

"' "'

r~d-l

----~

f f 5

( 8 ) ( 9 )

Fig-11 Cont. (8 and 9) Variation of Heat Transfer to be adopted

I

I I

-~--

1

I _L __

-I 2 I

' f I 1 I __ j__ --~-

1 I I

I I

(10)

I I --r--1

I 'i:l

_j

Fig-11 Cont. {10) Variation of Heat Transfer to be adopted

743