The thermal response of an enclosure to periodic excitation: The CIBSE approach

Building and Environment, Vol. 29, No. 2, pp. 217-235, 1994

~)

Pergamon

Copyright © 1994 Elsevier Science Ltd. Printed in Great Britain. All rights reserved 0360-1323/94 $6.00+0.00

The Thermal Response of an Enclosure to Periodic Excitation: The CIBSE Approach M. G. DAVIES* In order to relate the temperatures and heat flows at the surface o f a slab when it undergoes sinusoidal excitation, five quantities (conductivity, specific heat, density, thickness and driving period) have to be specified. They can however be reduced to just two groups, ( i ) the characteristic admittance, a, which has the usual units o f thermal transmittance ( W/mZK) but also takes account o f the fact that heat flows and temperatures are not in phase, and (ii) the cyclic thickness, r, a non-dimensional form o f the thickness o f the slab. The slab transmission matrix can be expressed in terms o f a and z, and the overall transmission matrix for a wall, including films, is formed by multiplying the slab matrices. From this, the wall admittance and cyclic transmittance can be found. An example is given. Equations are then developed for the response o f an enclosure when subjected to periodic excitation. It is shown that the response of a wall can be expressed exactly in thermal circuit terms, either as a pair o f lumped elements (units W/K) (to express admittance), or as three pairs o f lumped elements to express two admittances and the cyclic transmittance. The relations between various temperature and heat flows in an enclosure can be shown graphically in the form o f vector diagrams. The special status o f ventilation in estimating room response is mentioned. These considerations serve to support methods currently advanced in the CIBSE Guide to estimate periodic changes in room temperature.

1. INTRODUCTION

medium, in a slab, in a series of slabs including the surface films, and an entire room. It will be shown later how the matrix formulation, which is needed in the earlier part of the presentation, leads to an exact representation of a wall using a thermal circuit, and thus by Fourier synthesis to the transient response.

WEATHER conditions are in continual change, and the activities within a building, both of occupants and plant, vary. The inside temperature field is normally unsteady as a result and it is convenient to classify its behaviour into three time-span categories : (i) A condition averaged over a few days. Over this period, the effects of heat flow into the building fabric due to its thermal capacity and out again largely cancel each other and we are left with the socalled 'steady state' condition. This is the simplest condition to analyse and is indeed the most important since it provides much of the information regarding the energy needed to achieve a given indoor climate. Technical aspects are covered fully in for example various sections of the current CIBSE Guide [1], Book A. (ii) The part of the response that tends to recur day by day. This is of importance where the internal climate is much affected by solar gain on sunny days, or due to daily varying internal heat loads. Technical aspects are given in [1] : tables for the admittance of various structures and their means of calculation are given in [1] Section A3 and their use in Section A8. (iii) The response to any suddenly imposed loads. The Guide does not address this issue.

2. THESEMI-INFINITE SOLID It is convenient to start with one-dimensionalheat flow (in the x direction) in a solid with one exposed surface and which extends indefinitelyin the x direction. Suppose that its surface has imposed upon it a sinusoidally varying temperature Ts cos (2nt/P), where Ts is the amplitude of variation, t is current time and P is the periodic time. We shall initially only be concerned with the daily variation when P = 24 × 3600 seconds. It does not matter for the present at what time of day, t is taken to be zero. It is useful for the development of the theory to use exponential notation. The quantity exp (ju) denotes a vector of unit length making an angle of u radians with the horizontal. When it is included in expressing a physically significant quantity, by convention it is the projection of the vector on the horizontal--cos (u)--that is the observable quantity, exp (ju) is periodic in u: exp ( j ( u + 7r)) = - exp (ju) and exp ( j ( u + 21r)) = exp ( j u ) . In the present case, the variation of surface temperature can be written as Tsexp ( j 2 ~ t / P ) . The surface temperature has the value Ts, 0, - Ts, 0 and Ts, when t = 0, P/4, P/2, 3P/4 and P, etc. This sinusoidal variation is supposed to be superposed upon some steady state condition which is not of current interest. The temperature at distance x within the slab measured

This paper is intended to amplify the theory of periodic heat transfer in the Guide and to show how it can be extended to estimate transient change. The earlier sections deal progressively with heat flow in a semi-infinite *School of Architecture and Building Engineering, The University, Liverpool L69 3BX, U.K. 217

M. G. Davies

218

from the surface, and at time t will be notated as T(x, t). The heat flow across any cross section within the solid is

q(x,t)=-x

, OT(x, t) ~-

(1)

and if this varies over some small section 6x, the quantity (c~[-20T(x,t)/Ox]/Sx).6x represents the difference between the heat flows into and out of the section. It must be balanced by the rate of increase of internal energy, O[pc6x.T(x,t)]/& and we have the Fourier continuity equation, . a 2

z

T(x, t) a T(x, t) ~?x2 = pc at

(2)

The temperature within the slab is given by T(x, t) = T, exp L

L P2 i

(i)

jexp L

-jLWJ

(ii)

J exp LI ~ ]

(iii)

(iv) (3)

This distribution will be found to satisfy the Fourier continuity equation. The term (i) simply represents the amplitude of the imposed variation. (ii) indicates that the amplitude decays exponentially within the slab. (iii) indicates that T(x, t) at fixed time is also periodic in x with a wavelength given by A = [ 4 1 r P 2 ] '2. L pc j

(4)

Terms (iii) and (iv) can be combined so that the expression becomes

T(x, t) =

T exPL- L (i)

j jcos (ii)

27r ~ -

.

(5)

(v)

Term (v) is part of the standard expression for a travelling wave: P denotes the time periodicity and A the space periodicity. The wave proves to be very highly damped however. At a depth of only one wavelength, (x = A), the amplitude term (ii) has the value of e x p ( - 2 ~ ) or 1/535. For brick, for example, with 2 = 0.84 W/mK, p = 1700 kg/m 3 and c = 800 J/kgK, A = 0.82 m, so at this depth, Tvaries throughout the day by only 1/535 of its surface variation ; it is virtually extinct. A depends on the thermal parameters in the form of ;t/pc and since dense materials tend to have high conductivities, 2~pc tends not to vary much from material to material. We are more concerned however with quantities at the surface. The heat flow at x is given by equation (1). From it the ratio of the heat flux at the surface to the temperature at the same time can be found :

T(ol}) =

exp J4 (i)

= a,

say.

(6)

(ii)

This ratio is the characteristic admittance for the slab material, a (written bold since it is a vector quantity). Factor (i) denotes its magnitude. 2pc represents the ability of the materials both to conduct heat into the surface and store it behind the surface. Unlike 2~pc, 2" pc varies strongly from material to material, so that brick will have a much higher admittance than insulating materials. The thermal capacity of a material can only make itself apparent in conditions which change in time. In periodic heat flow, the thermal capacity always appears in association with the periodic time, that is, as the quantity pc/P. Clearly as P becomes large (so approaching steady state conditions), the heat flow associated with unit variation of temperature at the surface tends to zero. The magnitude of the characteristic admittance for brick subjected to daily sinusoidal variation is 9.1 W/m2K. Dense concrete has higher values and lightweight materials lower values. It is convenient to define the magnitude of a as

The factor (ii) in equation (6) expresses the fact that when T and q undergo sinusoidal variation, the times of their maximum values do not coincide. The phase of q is ~/4 radians, 45 °, 1/8 of a cycle or 3 hours in a 24 hour cycle ahead of T. If the slab exercised a purely resistive function, q would be in phase with T. If the slab is purely capacitative q is 90 ahead of T; this is the thermal analogue of the current flow into an electrical capacity, which is 90 ° ahead of the voltage. In crude terms one can argue that heat has to be put into a capacity before the capacity can exhibit a temperature. In the case of the semi-infinite slab, the phase lead is half way between the phase differences of a pure resistance and pure capacity. The characteristic admittance is independent of slab thickness. Heat flow in a slab of finite thickness is discussed in the next section.

3. T H E F I N I T E T H I C K N E S S

SLAB

In the semi-infinite slab of the last section only wave motion in the positive direction is possible. The thermal field in a slab of finite thickness however can be composed of waves in both directions :

ex~l:)2~t/~P) = A + exp

[ ic],J2 .] -

P2

(x +yx)

wa~es moving to righ| amplitude A + gpc

L, 2

(8) waves moving to left amplitude A

It is more convenient to express T(x, t) in terms of the actual temperature amplitudes T~ and T2 at the surfaces

Thermal Response to Periodic Excitation of the slab at x = 0 and x = X. It is also better to use the symmetrical and skew symmetrical hyperbolic functions, cosh (u) = (exp (u) + e x p ( - u ) ) / 2 and

sinh(u) = ( e x p ( u ) - e x p ( - u ) ) / 2 ,

(a)

219 2

magnitude (9a,b) 1

rather than the exponential forms themselves. It then follows after some algebra that the temperatures and heat flows at surfaces 1 and 2 can be related as

T, = cosh (z +jT) " T2 + [sinh (z +jz)/a]" q2

(10a)

q, = [sinh (z+jz) "a]" T2+cosh (z+jz) "q2

(10b)

(b)

0

"

~

I

I

2

"t

I

3

where

[npc]'/2 = L~i I

.x.

%%%

(11) 2

is non-dimensional but is in effect in radians. In the present context it is best thought of as the non-dimensionalized slab thickness, the cyclic thickness. A value of of about 0.1 or less represents a 'thin' wall in the sense that there can be little temperature difference across it. If is larger than about 3, excitation at surface 1 has little effect at surface 2: the wave has decayed. For a single thickness of brick (X = 0.1 m) for daily excitation, z is around 0.8, and the wall is neither thick nor thin. The single slab is not of interest in itself but it serves to illustrate certain properties which are wanted for more realistic constructions. A transmission characteristic describes the heat flow at surface 2 due to excitation at surface 1 (or vice versa), and a storage characteristic describes the response at surface 1 due to excitation at surface 1 itself. (A similar quantity can be defined at surface 2, but for the single slab the two are the same.)

u = Iq~]

r, = o

=

%'N%%.% \

3

Fig. 1. The cyclic transmittance u = q2/T, of a slab subjected to a sinusoidal variation of temperature on its front surface, and isothermal at its rear surface as a function of its cyclic thickness z = x/(npc/P2)'X. (a) Magnitude of u for a = 1 W/m2K, (b) phase lag in radians of q2 behind T,.

difference between them is zero as must be the case. When is large (larger than about 3), q~ lags behind T, by a value which tends to z itself, less ~/4. The ~/4 comes from the admittance term, the remainder from the thickness. The wave crest velocity is given as A/P which is around 34 mm an hour for brick under diurnal excitation. It is easy to show that when the slab is thin (so that cos (2z) -~ 1 - (2z) 2/2 and tan (z) ~ z),

3.1. Thermal transmittance If the slab is excited at surface 1, the heat flow q2 will depend upon what further construction follows surface 2. If surface 2 is adiabatic, q2 = 0 by definition. The only other condition that can be considered in isolation is that where surface 2 is isothermal, so T2 = 0. The cyclic transmittance u is

~

phase

2 u = ~ e x p ( j x 0)

(13)

which is the steady state transmittance (with of course no phase lag). It will be noted that (14)

a -sinh (r +j~)

(12a) that is, the effect at surface 1 due excitation at surface 2 is the same as the effect at surface 2 due to excitation at surface 1.

a

~/[(cosh (2~) - cos (23))/21

(i) "exp [ j I 4 -

arctan [ tan (z) ] ] ]

L

JJ/

(12b)

(ii) (The manipulation from complex to polar form is presented in the Appendix.) Factor (i) is the magnitude of u and is shown in Fig. l(a) for a = 1 W/mZK as a function of the non-dimensional thickness z. Its form is rather similar to a rectangular hyperbola though it falls away more rapidly than this at high r values. (The steady state transmittance 2/X against thickness X is necessarily a rectangular hyperbola.) Factor (ii) gives the phase difference between q2 and T, and it is shown in Fig. 1(b). When r is small the BAE 29:2-F

3.2. Thermal storage The ratio of q~ to T, depends upon whether surface 2 is taken to be adiabatic (q2 = 0) or isothermal (T2 = 0). The adiabatic case is important in connection with room interior walls : either surface of a partition wall equally excited from both sides behaves as though the wall has a perfectly insulating surface at its mid-plane. The isothermal case leads to the discussion of an exterior wall. An exterior wall has both a transmission parameter (discussed above) and a storage parameter as shown below. Taking the adiabatic case first, the surface admittance Ysa = I q ~ ]

= a" tanh (z +jz) q2~ 0

(15a)

M. G. Davies

220

extinct. This corresponds to the double path when z is equal to about 3, as Fig. 2(a) shows. Consider now the case where instead the temperature at surface 2 is held at z e r o - - t h e isothermal case. Then T, = 0 and the isothermal surface admittance y~, is

[-cosh (2~) --cos (2"C)] ''2

= a.L

j

(i)

(ii)

• exp /"

~+arctan|sinh(2~)|||.

(15b) Ysi =

(iii) Factor (i) is simply the magnitude of the characteristic admittance. Term (ii) demonstrates how the magnitude of y~, is affected by the thickness of the slab ; its form is shown in Fig. 2(a). Factor (iii) indicates how far the phase of heat flow is ahead of the temperature (Fig. 2(b)). When the slab is thin, y~ tends to the value 3,~=

P -'exp

=j"

~.

I} -

-I/ (b)

phase

~

(17a)

[cosh (2~) + cos (2z)7" : = a" Lcos h (20 - c o s ( 2 r ) j (i)

(ii)

• exp /

-arctan

lsinh(2})//f

(17b)

Off)

(16)

The conductivity is effectively infinite. The slab acts as a pure capacity and the heat flow is 9 0 ahead of temperature, as in the case of the electrical capacity. When the slab is thick, y~,, tends to the value for the characteristic admittance, as must be the case. Factor (ii) and so y~, however has a maximum magn i t u d e - - l . 1 4 5 a for z = 1.18--to be explained as follows: Suppose that T, is the independent variable. When the slab is thin (and adiabatic), it has little thermal capacity and so q, must be small ; clearly q, must increase with z. q, however consists of two c o m p o n e n t s - - o n e due to a wave travelling to the right generated "directly' by T,, and the other due to a wave moving to the left after reflection at the adiabatic surface. If these waves were similar to mechanical waves with little attenuation, q~ would increase from zero to a large value when the slab thickness X was equal to a quarter wavelength, that is, = 2n/4 or about 1.6, and would decrease again to zero when r = 2n/2. Since the thermal wave is so strongly damped however, the reflected wave has progressively less effect at the surface as X or T increases, q~ has a maximum value at a value o f z (1.18) which is less than 2~z/4 and with further increase in thickness eventually tends toward its semi-infinite value. It slightly undershoots, and has a minimum corresponding to the zero mentioned above. As noted earlier, when the path length corresponds to the wavelength A the wave is virtually

(a)

I q7"1' ] =7"2~a 0/ t a n h ( T + ] ~ )

The effect of thickness is exactly the reciprocal in magnitude and the negative in phase of its effect in tlie adiabatic case. When the slab is thin, 2 it Y~i = X" exp ( j x 0) = X

(18)

which is the steady state transmittance, q, now equals q2, Y~i = u, and the thermal capacity has no effect. 3.3. Comparison qf'values A flux q~ at surface 1 can be generated by TL with T2 = 0, and expressed by the y~, parameter, or it can be generated by Tz with T, = 0, expressed by the u parameter. It will be apparent from the above analysis that v,~ >

2/X > u.

(19)

This must be so : if T t drives heat into the slab periodically, some will be lost from surface 2 as in steady conditions but some more will be temporarily stored in the slab itself, so q l must be greater than its steady state value ; ),~, > 2IX. Similarly, if T2 drives heat periodically, some will be stored in the slab, so the q~ that results is less than its steady state value ; u < 2IX. 3.4. Discussion For future development it is convenient to present the relations between the surface values in matrix form :

-

ksinh(z+#)-a

I¢ 0 [

I

I

I

2

i I 0

i 1

i 2

I

~

3

3

i

Fig. 2. The surface admittance .~,~ = q~/T, of a slab adiabatic at its rear surface, as a function of z. (a) Magnitude of y,~,, (b) phase ofy~,.

cosh(z+R)

Jkq2J"

(20)

The transmission matrix will be seen to be a function of two parameters only, r the cyclic thickness and a the characteristic admittance. These seem eminently suited to the purpose in hand. a, including the thermal properties of the material as it" pc and independent of thickness, has the units, W/m2K, of the building parameters y and u that we wish to find. r, including material properties as Z/pc, provides a direct and easily interpreted measure of how y and u are affected by the finite thickness of the slab. The matrix can be abbreviated as

Thermal Response to Periodic Excitation

Le21

e,qr q. e22JLq2J

e~'e22-e~2-e2t=unity,

andthat

ql It will be noted that

e~t=e22. (22a,b)

If the slab is thin, so that z2 can be neglected in comparison with 1, the matrix has the form 1

J" 2~pcX

•

(22)

P Attention should be drawn however to other schemes of notation. A matrix formulation for thermal purposes was first put forward by van Gorcum in 1951 [2] who expressed the elements as

ell=e22=cos(?X),

and

e2~ = 27"sin(TX)

(23a,b,c)

where

pc

y2=-~-j~o

and

2~

co= p .

it is in some ways misleading: a slab is a distributed resistance/capacitance system, not a lumped one. Only if it is 'thin' in the sense used above can it be said to behave as a lumped element. Much the best single period (P = 24 hours) application of this extensively explored theory to buildings was developed by workers at the U.K. Building Research Station and now presented in [1]. Tables of admittances and transmittances were published relating heat flows to the room index and ambient temperatures (not the slab surface temperatures used above); see [4] and [5]. The technique they developed readily enables designers to evaluate the risk of overheating of buildings at the design stage. The aim of the present paper is in part to underpin the evaluation of wall parameters and to extend their use to find temperature swings in a building. Overall parameters are discussed in the next section.

4. T H E M U L T I - L A Y E R E D C O N S T R U C T I O N

sin (TX) )Q,

el2-

221

(24a,b)

A great a m o u n t of work was conducted worldwide in this area during the next two decades, much of it in essentially this notation. It is easy to show that the ?, X, 2 formulation is exactly equivalent to the a, z formulation but it is arguably not as suitable. (i) It appears to suggest that the transmission matrix is a three parameter operator rather than one of two parameters only. (ii) A term of form cos (u) suggests a quantity which oscillates with increasing u. Since 7 is complex, cos (7X) in fact goes to infinity with increasing X, but its effect is better described by a hyperbolic function. The same applies to the other trigonometric functions. (iii) It is better t o work in terms of z+jr than 7X since the z form makes it clear that we have to do with a complex quantity, and r itself is readily interpretable while 7 is not. In some of the literature the z grouping was recognized as a group though its significance was not widely understood. (The present author has reviewed this literature. See [3].) (iv) Since 2 is already included in the definition of ~, 27 seems an inept construct. It clearly merits a single symbol such as a or a. Its significance was for the most part not recognized, though it was hinted at in a few papers of eastern European origin. (v) The j(o notation is suitable in electrical and acoustical work where high frequencies are the rule. For building applications with a basic frequency of once per day it seems better to use period P rather than angular frequency (o. Some writers have drafted their work in terms of the total slab resistance R = X/2 and thermal capacity C = pcX. This is a perfectly possible formulation but

Walls and roofs usually consist of more than one block or layer. A construction of brick plastered on one side constitutes two slabs in close contact and it is clear that the plaster and brick temperatures at the interface must be the same. The heat flow must similarly be continuous. If the plaster lies between surfaces 1 and 2 and the brick between 2 and 3, we can write

e,qm ql

Le2~ e22JLqaA and

[T2]=~f,, f, zIFT31 (25) q2

Lf2, f22]Lq3J

where ejk denotes the complex elements for the plaster layer and.~'k those for the brick. Clearly the temperatures and flows at the bounding surfaces can be expressed as

[T,]=~el] ql

e,2lFf,,

Le2~ ez2ALfzl

f,2l[-v31 fz21Lq3J

(26)

and this can be generalized to any n u m b e r of layers of any solid material. A wall construction may contain a cavity across which heat is transferred by convection and radiation. These two processes are normally combined and expressed in terms of a cavity resistance r .... which has a typical value of 0.18 m2K/W. There will be a temperature difference across the cavity, but the heat flow at the two bounding surfaces is the same since any storage in the cavity air is negligible. The transmission matrix for a cavity is accordingly

['0 The matrices relating conditions (i) between the room index temperature T] and the internal wall surfaces, and (ii) between the wall external surface (surface n) and ambient at To can similarly be written as

r,V, ljLq,j

and

rolr o-i 1jLq, j. (27a,b)

M. G. Davies

222

r~ is the internal film resistance, often taken to be around 0.12 m2K/W, and ro is the outside film resistance, which depends strongly on wind conditions but is often given a value of 0.05 m2K/W. The overall quantities can be related by multiplication of a sequence of such matrices to form the product or wall matrix with elements e~k: [~]

= [e'~,

e'~2]rTo]"

(28)

e'21 e'22AkqoA The following points may be noted regarding the product matrix. (i) The sequence of matrices must be assembled before multiplication in the order of the physical wall, for example, [inner film] [inner brick] [cavity] [outer brick] [outer film]. The values of the overall wall elements e~k change if the positions of two of these matrices are exchanged. (ii) In steady state conditions, e'~, = e ' 2 2 = 1, e2t = 0 and e'L2 is formed as the sum of the thermal resistances such as r~ and )(/2 through the wall. The conventional U value is given as l/e'~ 2. This is the only element which carries any information. In the steady state case, the ordering of the matrices does not matter. (iii) It turns out that the determinant of the wall matrix is unity :

e'l, "e~2-e~, "e'12 = 1.

(29a)

Since the e;k elements are complex, this represents two relations : real part of (e't ~"e'22-e2, "e'~2) = 1

(29b)

imaginary part o f ( e ' ~ , ' e 2 z - e ~ , "e'12) = 0.

(29c)

(iv) The wall matrix consists of 8 real numbers. Since there are two inherent relations between them, the matrix has only 6 degrees of freedom. (v) A film matrix has one parameter and a slab matrix has two parameters. The wall matrix indicated in (i) thus has 1 + 2 + 1 + 2 + 1 or 7 parameters ; there is no limit to the number of input parameters for a multilayer wall. (vi) Thus the behaviour of a wall of arbitrary construction, driven at some fixed period P, has to be summarized by just 6 real or 3 complex values. (vii) These lead to three output parameters--two admittances and one transmittance : inside admittance

IqT~l

= e%, :7~,= 0

outside admittance

[ ol q°

4. I. Thermal parameters jor an external wall As an illustration of these methods, we will find the wall parameters for an exterior wall of lightweight concrete with a thin layer of plaster on its inside surface. Details are given in Table 1. The steady transmittance or U value is 1/1.259 or 0.794 W/m 2K. The plaster has a small r value and thus behaves virtually as a lumped capacity. The concrete has a much larger value of r so that we can expect considerable attenuation of a wave as it traverses the wall. Computation will be reported to four decimal places though it will be recognized that none of the input quantities, with possible exception of the thicknesses, is actually known accurately. Some details of the algebra used are given in the Appendix. We use the quantities A = cosh (z).cos (T),

B = sinh (z)-sin (z),

(33a,b)

C = cosh (,)" sin (z) + s i n h (~) "cos (~) `/2 D =

cosh (r)- sin (,) - sinh (r)" cos (r)

,/2

(33c,d)

The primary transmission matrix has the structure

A +jB (-D+jC).a

(C+jD)/a] A +jB J"

Thus the plaster matrix is 1.0000 + j x 0.0160

0.0260 + j × 0.00014 m2K/W

- 0 . 0 0 6 5 + j x 1.2290

1.0000 + j × 0.0160

(30)

e'l 2

_

e,~ t

(31)

el2

Ti=O

7-1 = 0

(as was the case for the homogeneous slab), so the two admittances are not equal. The transmittance however is the same in either direction. The negative signs indicate the phase of q. If the wall interior is taken to be to the left and outside to the right, and x is taken to be positive to the right, an increase in Ti must clearly lead to an increase in q~ in the positive x direction (though they will not be exactly in phase). An increase in To however must clearly lead to an increase in qo in the negative x direction. Similarly, an increase in To will lead to an increase in q~ in the negative x direction, at any rate for a thin wall. (For a thick wall, a much attenuated q~ may be more nearly in phase with the T,, that generated it ; the arithmetic leads to this conclusion. The scheme as a whole is entirely serf-consistent.) There is no useful purpose served in writing down detailed algebraic formulae for wall parameters. The procedure will be illustrated by examples for an external and an internal wall.

T o = 11

e

I2

(32) The admittance involving qo is not used in the CIBSE Guide applications. In general, e22 is not equal to e'~

W/m2K It will be seen that the element e,2~ is, to this accuracy, equal to the slab resistance in Table 1. Further, the quantity 27tpcX/P is 1.2290 W/m2K, equal to this accuracy to the element e212. The element e ~ , = e22 I is nearly unity. The other element in each pair mentioned is relatively small. All this is as expected for a thermally thin slab. The concrete matrix is

T h e r m a l R e s p o n s e to P e r i o d i c E x c i t a t i o n

223

Table 1. Details of wall construction X

2

p

c

Element

(m)

(W/mK)

(kg/m 3)

(J/kgK)

(m2K/W)

Inner film Plaster Concrete Outer film

. 0.013 0.200 .

1000 1000

0.120 0.026 1.053 0.060

.

. 0.50 0.19 .

.

Resistance

. 1300 600 .

a

z

(W/m2K)

--

--6.8753 0.1264 2.8793 2.1431 ---

1.259

Table 2. Matrix factors

Plaster Concrete

- - 4 . 6 7 8 7 + j × 3.5272

A

B

C

D

1.0000 -2.3406

0.0160 3.5344

0.1788 0.9588

0.0010 4.1790

- - 0 . 5 3 4 2 + j x 2.2274, m2K/W

- 16.4045+jx (-0.3314)

-5.1671+jx

3.8767

W / m 2K The wall parameters are found from these elements. We recall the structure of this array :

- - 2 . 3 4 0 6 + j x 3.5344 -- 12.0326+j x 2.7606

0 . 3 3 3 0 + j x 1.4514 m2K/W

Ti = e'l l " To + e'12 " qo

(35a)

-- 2 . 3 4 0 6 + j x 3.5344

qi "= e'21" To +e'22" qo

(35b)

W/m2K This slab is thermally fairly thick and it is no longer possible to interpret individual elements. The real and imaginary parts of the determinant of each matrix will be found to be 1 and 0 within rounding error. The product matrix [plaster matrix] × [concrete matrix] follows by routine multiplication : - - 2 . 7 1 0 2 + j x 3.5669

0 . 2 4 8 4 + j x 1.5482

--

e'l t" e ~ 2 - e~l" e'~2 = 1.

[qo]

r~=o

e',2 '

--0.5342 '+ j x 2.2274 (37)

The inside admittance y~ is

4.1829 + j x 3.8966

W/m2K This has now to be post-multiplied by the outside film matrix

['0 00607,w]

[-q i ] e'22 Y i = LTiJ ro = 0 = - e', - =2

-5.1671 + j x 3.8767 - 0.5342 + j × 2.2274

= 2.82 W/m2K, 2.64 hours.

(38)

The outside admittance yo is ]-qo-]

giving

r, = 0 - 2 . 7 1 0 2 + j x 3.5669

(36)

The cyclic transmittance u is

= 0.437 W / m : K , 5.10 hours.

m2K/W -- 16.4045 + j x (--0.3314)

where subscripts i and 0 denote inside and outside conditions. Also

0.0858 + j x 1.7622

e'l,

- 4 . 6 7 8 7 + j × 3.5272

e'] 2

- 0.5342 + j x 2.2274

= 2.56 W/m2K, 2.63 hours.

(39)

m2K/W - 16.4045 + j x ( - 0 . 3 3 1 4 )

-5.1671 + j x 3.8767

W/m2K It is convenient to note the surface admittance at the inner surface for the plaster/concrete/outer film in Cartesian form : y(surface) =

- 5 . 1 6 7 1 + j × 3.8767

(34)

Finally this has to be premultiplied by the inner film matrix

giving

decrement factor = cyclic transmittance U value 0.437 = 0.79~' 5.10 hours

0.0858 + j x 1.7622

= ( 2 . 0 5 2 2 + j x 3.0321) W/mEK.

It will be noted that either admittance, 2.82 or 2.56 > U value 0.794 > cyclic transmittance 0.437. The CIBSE Guide expresses the cyclic transmittance as a 'decrement factor' :

= 0.550, 5.1 hrs.

(40)

The Guide also lists 'surface factors' F for each construction. Any heat flow qt which acts at the exposed surface o f the plaster (at T]) has a heating effect within the enclosure. Its heating effect is the same as that of a reduced heat flow F ' q l , acting at the r o o m index temperature T~, where

M. G. Davies

224 film transmittance F - film transmittance + surface ad--mittance

(41 a)

the wall thicker than this is hardly significant, but the thickness cannot be justified on thermal grounds.

I/0.120

1/0.120+(2.0522+jx 3.0321) = 0.770, 1.085 hours. (41b) This value of Fis independent of all heat transfer paths between the room index temperature (environmental or rad-air temperature) and the exterior, other than the one under discussion the plastered concrete wall in the example given. It is a handy simplification but the value of Tj as estimated by taking F" q~, to act at T~ is less than the correct value (found when qt is properly taken to act at T~) by an amount q~ (film transmittance+ surface admittance). 4.2. Admittance Jor an internal wall Although the steady state heat losses from a room are normally associated with outside walls only, the periodic exchange of heat involves all surfaces. If we can assume that the daily pattern of temperature in an adjoining room is the same as that in the room of interest, the heat flows into and out of the two surfaces of the partition wall are always the same. Thus although the wall midplane undergoes swings in temperature, no heat actually flows across it and it can be treated as an adiabatic surface. We are now concerned with only one parameter for the wall--its admittance. This parameter could be found by determining first the transmission matrix for the wall semi-thickness ; the surface admittance .v~, is then e~,/e'~. To illustrate a rather different (but exactly equivalent) approach, we will use equation (15b) for the surface admittance for a slab which is adiabatic on its rear surface. We choose a lightweight concrete wall (2 = 0.19 W/inK, p = 600 kg/m 3 and c = 1000 J/kgK as before so a = 2.8793 W/meK), and of full thickness 100 mm. Thus r, based on the semi-thickness, is 0.5358. Using equation (15b) it is readily found that 1

= 0 . 0 8 7 5 - j x 0.4617 m 2K/W.

(42)

Ysa

Now the admittance y~ of the wall seen from the room index node is related to the surface admittance as 1 Yi

=

1

+ rl

5. T E M P E R A T U R E V A R I A T I O N IN A S I M P L Y VENTILATED ENCLOSURE

We have to see how the parameters u and 3'~ found in the last section are to be used to estimate the thermal response of an enclosure when it is subjected to periodic, and ideally, sinusoidal, excitation. Consider the very simple enclosure illustrated in Fig. 3. It consists of a massive outer wall of area A and five adiabatic surfaces. The inside surfaces simply delimit a space so that we can talk about ventilation rates, but the surfaces play no part in the thermal response. The response to be considered is that to the first harmonic 7",, of the daily variation of ambient temperature, together with the first harmonic of the sunshine :~I absorbed on the outer wall. These can be combined as sol-air temperature 7",,

T,,, = r,, + :zl" ro.

The matrix equation for heat flow through the wall is then

[~]=

,

ell [e'_,,

•A

as follows from continuity. With r~ = 0.12 m2K/W as before, Yi = 1.975 W/m2K, 4.39 hours, or 2 W/m2K, 4.4 hours. A heavier and thicker wall will of course have a y~ value greater than this, but even though y+, itself is very large, y~ cannot exceed 1/ri, or 8.3 W/m2K in this case. It was remarked earlier that a wall with r = 1.18 provided the largest value of y,, ( = 1.14 a). Considerations of thermally 'optimally thick' walls have sometimes entered the design of passive solar buildings. It should be pointed out that the optimal value of y~ is obtained with a smaller value o f t than 1.18. For example, i f a = 12 W/m2K, and r i = 0.12 m2K/W, the optimal r is about 0.8. (See Table 7 of [6].) The thermal penalty of making

i

e,,/A][ T~, 1

(45,

e',_+ JLq~,'AI"

(Q is the heat flow (first harmonic component) into the wall as a whole.) We suppose that this heat is removed by ventilation and by storage in the massive wall. The air to air ventilation conductance V is computed as the throughput of air, ( m 3 / s ) , and the volumetric specific heat of air in j/m3K. According to the environmental temperature model and the tad-air model (7), there is a conductance X (4.5£A in the CIBSE Guide) between room air temperature 7",,, and T+ and the effect of V and X in series will be denoted here by V'. The matrix formulation of this process is

[~,]=Ul ) I,,'V'][T,]I JLQJ'

(46)

When Q and q+~,are eliminated from these equations, we have an expression for T+:

(43)

Ysa

(44)

AT,. +e't2V'To T+ =

Ae':2+e',2 V;

(47)

and dividing by e', 2 and using the relations in equations \1/

Fig. 3. A simple ventilated enclosure with one massive wall and live adiabatic surfaces, excited by variation in ambient and solair temperatures.

225

T h e r m a l R e s p o n s e to Periodic E x c i t a t i o n

(37) and (38), we have T, =

(a) A'u'T~+V'To Ayi + V'

A~

T1

X

T2

(b)

_12

T~ is a weighted mean of the two driving temperatures, T,,, and 7",,. The relation can be written in the form T~- To =

A" uo~Iro -- A ( y i - u ) " To A y i + V'

(49)

This makes clear the delayed warming action of solar radiation absorbed at an outer surface and the delayed and attenuated action of variation in ambient temperature. Equation (48) is very similar to the CIBSE Guide expression for temperature swings. See [1] equations (A8), (10), (12), (13) and (14). Although a matrix approach to calculating wall parameters is virtually unavoidable, an extension to enclosure response on the lines indicated above is clumsy: a matrix can indeed be written down for the joint effect of transmission through the above wall together with a window in it (two paths in parallel), for example, but it becomes cumbersome. It is better to proceed by expressing the effects of all mechanisms--conduction, storage and ventilation--in a thermal circuit formulation. Structural heat gains of the type AuT~, are usually small compared with solar gains transmitted with little loss and no phase lag through windows, or possibly with internal gains due to occupants and equipment. In this case, it is easy to calculate T~ both magnitude and phase using a 2-element representation of the admittance. This will be illustrated in the next section and amplified using vector notation. If structural gains have to be included (by use of the cyclic transmittance u), a thermal circuit formulation is still possible but a wall now has to be represented by 6 elements, as will be shown later. 5.1. Circuit notation A slab of area A, thickness X and conductivity 2 has a conductance of

Q TI - T2

A2 X

(51)

The reference temperature T(reference) is zero by definition. This form makes it clear that the capacity must be directly connected to the reference temperature• In sinusoidal flow, d/dt =- j x 2~t/P so Q =j x

27rpcAX P " (Ti -- T(reference))

(52)

and the slab has a reactive conductance of Q

2~pcAX

T, =.ix ~

AXpcm

reference temperatu re

(c)

T

2AX

~

2AX

'r2

(d) TI

T2

TTT Fig. 4. Conventional thermal circuit symbols. (a) A resistive conductance which might denote convective, radiative or steady conductive heat transfer links. (b) A thermal capacity providing a reactive conductance. (c) Combination of (a) and (b) to represent the behaviour of a thin slab. (d) An assembly of several units of type (c) to represent the behaviour of a thick wall.

These two quantities, referred to unit area, have already been seen in the transmission matrix for a thin slab (expression (22)). A thin slab can accordingly be modelled as shown in Fig. 4(c), where the capacity is shown located at the mid point of the resistive conductance. A thick slab can be modelled as a chain of thin slabs (Fig. 4(d)). Electrical analogues of walls can be built up in this way, and a chain of this kind provides the basis for finite difference calculations for heat flow through a wall which has a distributed resistance and capacity. However, a conceptually quite different form of circuit representation is available when the slab undergoes sinusoidal excitation. Suppose that the surface (area A) of a semi-infinite slab of material undergoes a sinusoidal variation of temperature of amplitude T1. A heat flow, Q, into the slab results and we can write the complex conductance as Q i = Aa" exp

x

=

+j ×

(50)

(units, W/K) and its action in steady state conditions can be represented graphically as the thermal circuit element shown in Fig. 4(a). If it has thermal capacity pc per unit volume, and infinite conductivity, it can be represented as in Fig. 4(b). The thermal capacity is pcAX. The heat flow Q into the capacity is dT, d(T~ - T(reference)) Q = pcAX. d~ = pcAX. dt

TlIt ~-

(48)

(53)

]

=

(i)

+j xA[~]

.

(54)

(ii)

Thus the complex conductance corresponding to an infinite distributed resistive/capacitative medium can be expressed by two lumped elements--a pure resistive conductance (i) and a pure reactive conductance (ii). They act in parallel ; see Fig. 5(a). These conductances are purely equivalent constructs and have no physical reality. The resistive conductance includes a measure of thermal capacity, pc, and the reactive conductance includes a measure of thermal conductivity or resistivity, 2. Factors (i) and (ii) are here numerically equal since the slab is infinitely thick• A wall however can be modelled in this way. In Section 4.1 we discussed the response of a wall made up of plaster, concrete and an outer film. For an area of 10 m z for example, the wall can, according to equation (34), be modelled by a complex conductance F~ between the surface and the reference temperature of

226

M . G. Davies (a)

~1

(b)

i

f'l

LI

Ic)

'~li

L @

Fig. 5. The exact but artificial means of representing the admittance property of a wall when excited at some fixed frequency. (a) Values for a semi-infinite wall, when the resistive and reactive components are equal. (b) Values for the wall discussed in the text. (c) Formal representation of the unit.

Ft = (20.522 + j x 30.321) W/K.

(55)

See Fig. 5(b). The conductance Ft can be handled algebraically as a unit ; it does not have to be split into real and imaginary parts until the time for numerical evaluation. See Fig.

5(c).

W, at t = 6 hours it is zero and at t = 12 hours it is - 500 W. It is most convenient to think of this as a varying input in addition to a steady component, so that the total input does not become negative. The steady state component of the response is not of current interest.) Then 500x ( 1 6 2 + ( 1 1 0 . 7 + j x 163.62))

T.v- T,,

5.2. Response qf an elementary enclosure As an illustration, consider the very simple enclosure where all the walls have a uniform construction and have the temperature T~. The air temperature is 7~v and the convective conductance between T,,. and Tt is C. The ventilation conductance is V. See Fig. 6. If a heat flow Q, is input at Z,v, elementary analysis gives values for the enclosure temperatures : Q.(C+F,) T , , - To = VC+CFI +Ft V'

(56a)

T t - To =

(56b)

Q.C

VC+CFI +FI V"

Suppose the enclosure is cubic, of side 3 m. The area A = 6 x 32 = 54 m 2. Suppose the convective heat transfer coefficient is 3 W/m2K. Then C=54x3=

162W/K.

(57)

Suppose the wall has the construction of that discussed earlier. Then F~ = 54 x ( 2 . 0 5 + i x 3.03) = ( l l 0 . 7 + j x

163.62) W/K. (58)

With a volume of 27 m 3 and an air change rate of 5 air changes an hour, V = 27 x 1200 x 5/3600 = 45 W/K.

(59)

Suppose that a sinusoidal input of 500 W of period 24 hours is input into the air. (Thus at t = 0 say, Q~, = 500

TO Fig. 6. Thermal circuit for a simple enclosure excited internally.

J'[45 x 162+ 162 x ( 1 1 0 . 7 + j x 163.62)] + ( 1 1 0 . 7 + j x 163.62) x45] J g

(60a) 136350+j x 81810 = 3.504 K, - 1.154 hours 30204.9 + j x 33869.3 (60b) and similarly, T I - T , , = 1.785 K , - 3 . 2 1 8 hours.

(61)

These results are qualitatively as expected: since the heat flow is input directly into the air, the swing in air temperature is larger than that for the surfaces. The temperatures must lag behind the heat input, but the lag in T,,. is smaller than that for T~. The heat flow into the wall itself lags (24/2~) × arctan(3.03/2.05) or 3.73 hours behind the phase of heat input. These temperatures and their attendant heat flows can be illustrated using vector diagrams as will be shown immediately. 5.3. Vector notation If the air temperature for example in a room varies sinusoidally from T,~vto - T~, and back again during 24 hours, it can be represented as a vector of length T~ which rotates, conventionally anti-clockwise, once in 24 hours. It is taken to be horizontal and pointing to the right at the time when it has its maximum value (e.g. at 1500 hours). The actual magnitude of T,, at some later time is the projection of T,, on the horizontal. The projection in this case at 2100 hours is zero, so T,,. is zero (that is, it has its steady state value), and has its minimum value at 0300 in the morning. Suppose that air undergoing sinusoidal variation in temperature is in contact with an infinitely thick slab whose surface temperature T, varies sinusoidally as a result. Ifh~ is the convective film coefficient, the heat flow vector q~ from the air to the surface is q~ = (T..,v-T~)'hc = f~'a = T~'aexp(jTz/4). (i) (ii) (iii)

(62)

Thermal Response to Periodic Excitation (a)

/¢ 1 " /

(c)

(b) 1

227

/¢

1

/

L q (conducted)

Fig. 7. Elementary vector diagrams. (a) Relation between air and surface temperatures with the air as the driving temperature. (b) Relation between the air impedance and the surface impedance and convective film resistance. (c) Relation between a heat flow incident upon a surface and the losses by convection and conduction.

Term (i) expresses the flow through the boundary layer, where it is proportional to h~. Term (ii) expresses it as into the slab, when it is proportional to the characteristic admittance a. Term (iii) elaborates (ii) by making it clear that qs is not only just proportional to a (that is, to ~/(2~2pc/P)) but that the phase of qs is ~/4 radians, 45 ° or 3 hours in 24 hours ahead of T~. The relationship between these quantities is shown in Fig. 7(a). Ta~ being the 'cause' must on physical grounds be greater in magnitude and ahead in phase of the 'effect'--the variation T~--it produces, q~ must be in phase with the difference between them, T a ~ - T , and must be 45 ° ahead of Ts itself. The figure is displayed at some arbitrary time and is supposed to make one anticlockwise revolution in 24 hours. When the difference vector Tav- T~ is vertical, both Ta~ and T~ have momentarily the same value and the heat flow between them is zero. From the above equations,

ray q~- 1

1

h~

a

(63)

and the admittance y~ say as seen from T~,v is q~ Y " - T.~

(64)

SO 1

y~

1

1

= -- + . hc a

(65)

(This equation is similar in structure to equation (43). hc is similar to 1/r~and a is similar to y~.) These quantities too can be expressed in vector form as Fig. 7(b). It is identical in shape to Fig. 7(a), but its units are different-m2K/W as compared with K - - a n d since it does not represent a time-varying relationship, it is not supposed to rotate. A third vector diagram is possible. Suppose that the air temperature T~ remains at zero at all times, but that a flux qs is absorbed at T~. (This could be supposed due to solar gain or some sort of periodic electrical heating.) Then q~ = ( T , - 0 ) - h ~ + T ~ ' a . (i) (ii)

qs as the driving mechanism must be larger in size than any of the individual flows that result from it, as will be seen. q(convected) must be in phase with T~ and q(conducted) must be 45 ° ahead of qs. In the case of a heat flow, an 'effect' can be ahead of a 'cause' in phase. In applying vector ideas to the simply ventilated enclosure of Section 5.2, we note first that all vectors have to be referred to the phase of the convective heat input, which is the imposed variable. This will be taken arbitrarily to be horizontal so that the temperature and heat flow configurations are positioned at the time of maximum heat input. To = 0, so T,v is 3.504 K in magnitude (see equation (60b)) and lags 1.154 hours or about 17° behind the reference phase and TI is 1.785 K with a lag of 3.218 hours or 48 °. These vectors are shown in Fig. 8(a). The phase of the difference, Taw- T~, is a little ahead of reference. (It has a magnitude of 2.176 K and a phase of 0.510 hours or +8°.) Indeed, since all the heat which is convected to the slab under the action of Tav-T~ is conducted into the slab and the slab heat flow is itself 3.728 hours or about 56 ° ahead of Ts, the Tar-- TI vector itself must make an angle of 56 ° with T~, as shown. The phase angles of 17° and 48 ° depend upon the choice of ventilation rate and convection coefficient (leading here to V = 45 W / K and C = 162 W/K) ; the phase angle of 56 ° on the other hand depends upon the choice of wall parameters alone.

48° or 3.218 hr

phase of heat input

. . . . . . . . . .

"N~

~ T a v ~

h

3.504K Tav-Tl= 2.176K

r x

(a) convective exchange

= flow into wall 353W /.._......~.---~ation

loss 158W

(66) convective input 500W amplitude

Term (i) represents the heat q(convected) lost to the air and (ii) the heat q(conducted) which passes into the solid slab. Their vector relationship is shown in Fig. 7(c) which is identical in shape with the other two but has differing u n i t s - - W / m 2. It is supposed to rotate.

(b)

Fig. 8. Vector diagrams illustrating the behaviour of the enclosure of Fig. 6. (a) Relation between temperatures. (b) Relation between heat flows.

M . G. Davies

228

A heat flow diagram can be shown similarly. See Fig. 8(b). By definition, the heat input of 500 W is a horizontal vector. The ventilation loss Qv = 45 x 3.504 or 158 W ; it is in phase with T,, and so its vector is parallel to the 72,, vector. The heat exchange between the air and the walls is 162 x 2.176 or 353 W. The enclosure can be excited in two other ways. (i) Suppose that the heat is input instead at the surface itself. In this case, T~ is the higher temperature ; T~,~ is in phase with it, so that the T~ and T~,,.vectors (and hence the difference Tt-T~,~) coincide in direction and the temperature configuration collapses onto a line. The incident flow splits into a conducted and a convected component with a phase difference between them, but the flows through C and V are the same. (ii) The enclosure can be driven too by a supposedly sinusoidal variation in ambient temperature To. In this case, the heat flow driven by T,, through V, C and into the walls is the same and now the heat flow configuration collapses onto a line. There are differences in the magnitudes and phases of the temperatures however: T, is smaller than T,, and lags behind it; T,~ lies between To and T, in magnitude and phase. The difference vectors To-T~,,. and T:,~- T~ however have the same direction or phasethe same phase as the heat flow vector. 5.4. An enclosure including radiant exchange In the enclosure discussed above, all surfaces were supposed to be at the same temperature and so there could be no radiant exchange. In order that a radiant transfer can be included, at least one surface must be at a different temperature from the rest. Indeed, in order to set up a satisfactory model for radiant exchange gener-

(a)

h

//

//

//

//I

ally, we have to consider an enclosure where at least three (composite) surfaces have separately specified temperatures ; it is preferable that all six surfaces of a rectangular enclosure should be individually specified. Given that we have a satisfactory model in the first place however, the most convenient situation to discuss in the present connection is one where five of the surfaces have the common temperature T~ and a further surface has another temperature T2. A cubic enclosure will again be selected, of side A. Five of the surfaces are taken to be at the common temperature T~. The surface is massive and has a complex conductance F~ as shown in Fig. 9(a). The other surface at T2 is massless. Its conductance to reference temperature is F2 and this could be supposed to be made up entirely of the outside film resistance ro. The air temperature T~,vexchanges heat with these surfaces through the convective conductances

6'1 = 5Ah~.~, and

I 1 - e, i 5/6 S, : 5A(;Jlr ~- 5Ahr

(c)

%v

//

1

X

Tra

~:~

5/6 (68b)

Reference [6], pp. 313-315, shows vector diagrams of temperature and heat flow for an enclosure of this kind.

(b)

xI Tar

~A/V~

//

1 -

s~ = A~:,h~ + Ahr"

Si+ Ci

//

(68a)

and

P //

(67a,b)

The ventilation Conductance V is present as before. There will now be a radiative exchange between T~ and T 2. Following the rad-air model of [7], p. 167, these will be handled as separate exchanges with the radiant star node T,~ which has radiant conductances St and S~ with T~ and T2:

Tar

T~s

C2 = Ah~2.

%.

S2+ C z

'VV~

f v. (d)

Ll

Fig. 9. Thermal circuit of an enclosure to include radiant exchange. (a) A two star model, which handles radiant and convective exchanges separately. (b) Radiant and convective exchanges merged to form the rad-air model. (c) The internal and external conductances combined to form separate overall conductances between rad-air temperature and ambient. (d) The overall conductances themselves combined into a global loss conductance L.

Thermal Response to Periodic Excitation They relate to sinusoidal excitation by ambient temperature, a purely convective source, and a source inputting at T~, T2 and Z,v. It is probably of more significance however to consider the room index temperature itself, since this is the quantity most closely related to the comfort temperature. This can be done by developing the model progressively as shown in the following figures. The model of Fig. 9(a) based on the separate convective and radiant star temperatures 72,vand T~ respectively, can be transformed to the rad-air model of Fig. 9(b). 72,~ and the external conductances F~, F2 and V remain, but T~ is excluded and is replaced by the rad-air temperature Tr, with its links of S~ and C~ to T~ and similarly to T2, and the link X with T,~, where

X-

( S + C)" C

S

'

S=S~+S2

and

C=C~+C2. (69a,b,c)

This in turn can be expressed in terms of overall conductances between the rad-air temperature and reference as in Fig. 9(c) : 1

1

1

L -SI+C

+Ft

and

Lzsimilarly.

(70)

Finally, the two wall conductances can be lumped as

L = L~+Lz

(71)

to form Fig. 9(d). This is formally similar to Fig. 6 and the variations in 72,~and T~, due to a sinusoidally varying heat input Q. at T~v are

Q,(X+L) T.v- VX+XL+LV'

L.

-

Q,X VX+XL+LV" (72a,b)

Heat can be input at locations other than T,,v. If heat is input at a bounding surface, by electrical resistance heating or by embedded hot water pipes, it is to be modelled as input at T~ or Tz, though that would be unrealistic for this particular model. The longwave radiation Qr from an internal hot-body heat source can be taken with sufficient accuracy to act at the radiant star node Try. According to the rad-air model, this is equivalent to an input of (I + ~)Qr at T~a and a withdrawal of eQr, from T,~, where e = C/S and is near enough 0.5. Thus if the hot body source had a total output Q, made up of 2/3 radiant and 1/3 convective outputs (i.e. Q~ = 2/3Qt, Q, = 1/3Q,), the net input at T,v is (1/3)Qt minus 0.5 x (2/3)Qt, or zero. In this case the whole input of Qt can be treated as though input at T~a. Then Qt(X+ v) Tra-VX+XL+LV

Qt or

L+(1/V+I/X)

1'

(73)

This expression is formally identical in structure with the expression A8.12 in the 1970 IHVE Guide [8] for the peak-to-mean deviation t,~ in environmental temperature tel with symbols as shown :

Qt

EA Y+ Cv

(74)

area x admittance ventilation conductance exchange conductance

rad-air notation L = ELi ~ ~Ajy~ V X

(l/v+ I/x)

229 environmental temperature notation ZAY Cpv EAh~ C~(= (I/Cp~,+I/Y.Ah~)

~)

(V and Cpv are identical in meaning. L and EA Y only differ in relation to the form for the inside film resistance r~. The lines of reasoning leading to X and to EAh, however are fundamentally different.) If Qr and Qa do not have these special values, the form for Tr~ is a little more involved and leads to a rather lower value of Tr,. There is little point however in making such adjustments because equation (74) as it stands can only provide a crude estimate of the daily variation about steady-state conditions. L (or EA Y) is a measure of the response of the fabric of an enclosure when it is subjected to sinusoidal excitation of a period of 24 hours. If the variation in heat input really were sinusoidal with a period of 24 hours and amplitude Q,, equation (74) would be expected to provide a reliable estimate of the amplitude of variation of Try. But the heat input is not of this form. Solar gains through windows--the most probable cause for overheating--are zero by night and peak sometime during the day. If the pattern of heating is taken to be the same over a few consecutive days, it can be resolved into a time-averaged or steady state component and first, second, third etc. Fourier harmonic components (based on P = 24, 12, 8 . . . hours). The real response is to be found as the sum of terms Tr~ formed from the component excitations and the corresponding admittance values. A number of studies report such an exercise, including one by the present author ([91, p. 215), based on the sum of 10 Fourier components. This procedure is clearly far outside the scope of a manual calculation which can take account of the first harmonic alone. For manual calculations, an estimate of Qt is provided as

Qt ~- (peak value of gain) - (daily mean gain). This will not give a badly misleading value when the gain is smoothly spread out over several hours, as is the case for solar gains. It would be seriously in error if the gain resulted from a large input of short duration. The Guide method is not suited to estimating the response due to such transient excitation. (Solar radiation falling upon opaque walls and roofs too has its effect on inside temperatures. With a single node, Tb to describe the heat exchanges at five of the room's surfaces, Fig. 9 is too crude to model the effect of such solar gain (the absorbed fraction of the incident intensity). Figure 9 however will serve to explain one way of handling it. First, a node is needed to represent the actual position where the solar gain is received, that is, on the external surface of the wall or roof. Such a node is not made explicit in Fig. 9(a) but could be by insertion of a node T~ say at the outside of the wall opposite to T~. Since the complex conductance F~ includes both the effect of the wall and the outside film, insertion of T] implies separate expression of these mechanisms. Once this is done, the effect inside the room of solar gains Q]

230

M. G. Davies

physically acting at T~ can be computed quite readily by supposing instead that a (much) reduced flow Q'; acts at T~,. The relation Q'[/Q~ depends on the various thermal conductances making up the overall conductance L~ in Fig. 9(c), but not upon the remaining conductances V and L2. This scaling procedure is explained in Appendix

3 of [101.) A further limitation to the validity of these estimates is imposed by assuming that the ventilation conductance V is constant. This is unrealistic : people open windows when it gets too hot. In fact this may not much affect the magnitude of temperature swings, since V is normally numerically fairly small compared with ZA Y. V however is of the same order and may well exceed the steady state conduction loss ZA U: this leads to unreliability in the steady state elevation. The Guide method is again not well suited to handling the response to an enclosure with time-varying parameters. 5.5. Response with time lug The Guide [1] pp. A3-45 gives an account of how to find the admittance of a layered construction including time lags; Tables A3.16 to 21 give the admittances (magnitude and lag) of a large number of constructions. Section A8 p. 13 provides a worked example of finding the magnitude of swing in environmental temperature but does not in fact show what use may be made of the time lag information. This is implicit in equations (73) or (74) and to illustrate how it can be done, we extend the Guide example a little. Construction details follow : Outside wall, 105 mm brickwork, 50 mm mineral fibre, 105 mm brickwork, 13 mm lightweight plaster. For the thermal parameters of this construction, see Table A3.17, construction 3b. Partitions, 105 mm brickwork, lightweight plaster on each side. Table A3.20, construction I b. Floor, 150 mm cast concrete, 50 mm screed, linoleum tiles. Table A3.21, construction 1a. Ceiling, bare concrete, Table A3.21, construction la. Window, single glazed, alum/n/urn frame with thermal break occupying 10% of glazed area. A quoted time lag of 2 hours for example implies a phase lag o f a = 3 0 . The real and imaginary components of the quoted Ys are then Y" cos (a) and Y" sin (a). These are to be summed as shown.

outside wafl partitions floor ceiling window

W,m2K t.7 3.6 4.3 6.0 5.7

hours 2 3 2 2 0

A Y'co~(a) 16.02 106.91 74.4g 103.92 39.90

A Y's i nI a ) 9,25 106,91 43.00 60.00 000

341.23

219.16

The Guide's value of C,. = 20 W/K, and the amplitude of solar gain is 1098 W so the amplitude of variation of td is estimated as 1098

6. AN EXACT C I R C U I T M O D E L FOR A WALL Figure 5(b) presented a two-element circuit to represent the thermal behaviour of a wall when subjected to sinusoidal excitation. Its purpose was to model the admittance parameter of the wall--the response of its inner surface when excitation took place at or adjacent to its inner surface and it is sufficient to do this. However, we need to know in addition the response of a wall at its inner surface when excited at its outer surface and the circuit of Fig. 5(b) does not permit this. A circuit with more parameters is needed, as is shown below. First a comment. Figure 10(a) shows a pair of elements similar to Fig. 5(b), in which A ~is a resistive conductance with units W/K, and B~ (units J/K) provides a reactive c o n d u c t a n c e / x (2n/P) x B~ W/K. Together they provide a complex conductance Qi

2n

T. = A~+/x p B~.

(75)

The same conductance can be provided by the series pair of Fig. 10(b), if A 2 and B2 are so chosen that I

1

A~ +

1

2n

.ix pB2

=

2n

A,+jx ~B,

'

(76)

Our choice of a parallel or series form is in principle quite arbitrary. However, thermal quantities are usually expressed as ' W per.. ' not "... per W', etc. Thus thermal capacity is J/kgK, not kgK/J. Thermal admittance is expressed as 'W/m2K ' and since it is complex, when multiplied by the appropriate surface area, can be written in the form,

C+jx D with units W/K

Y area 5 42 2(1 20 7

elements have lags of 2 hours ; one is larger and one is zero. The Guide suggests a lag of only one hour, seemingly at variance with this value. However, when account is taken of the effect of the higher harmonics of solar irradiance, the phase lag will be less than the value of 2.1 hours which derives from the first harmonic alone. (The Guide goes on to include the effect of timevarying structural gains (negligible), casual gains due to occupants and the variation due to swings in ambient temperature.)

1098

t¢, = ( 3 4 1 . 2 3 + j x 219]16)+20 = 36i.23-Fjx 2i9]i6" (74) So te~ = 2.6 K, 2.1 hours. Thus the internal temperature should peak about 2 hours after the time of maximum solar gain. This is obvious, since most of the construction

which leads naturally to the parallel expression. For this reason, a complex unit of parallel form will be adopted here.

y

l' 1

A,t L7

TI

BI

(a) (b) Fig. I0. To illustrate that the complex conductance associated with T~ can be modelled by elements either in parallel or in series.

231

T h e r m a l R e s p o n s e to Periodic E x c i t a t i o n

TIC ~Q1 ~

Q~ OT2

Fig. l 1. An assembly of three complex units so as to represent exactly the thermal behaviour of a wall when excited sinusoidally.

Consider then the circuit of Fig. 11 which consists of three complex units Y,, Yb and Yc arranged in T-section. The relation between the input and output quantities can be shown to be

[TI I

1,

Va }Qa"~-yb"t- YaYb

=

Q,

y~

Yc l+yb

T2 J

l

Yc = - 164.045+jx ( - 3 . 3 1 4 ) W / K as above,

(81a)

Ya = 2 0 . 5 8 4 + j x 13.369 W/K,

(81b)

Yb = 1 8 . 8 2 4 + j × 12.370 W / K as above.

(81c)

It will be seen that both the real and imaginary parts of Y~are negative. For an isolated thin slab the imaginary part is positive, but the real part is negative. Now negative electrical resistances do not exist so it is not possible to model a wall/exactly using an electrical analogue computer, as was fashionable before the advent of digital computationl Unlike the two element unit of Fig. 5(b), this network can model the wall transmission characteristics. It will be readily checked from Fig. 11 that uA, the value of Q~/T2 when Tt is kept at zero, is given as uA -

v..v. v.+ v~+ vc

(82)

and that y , A , the value of Q,/T~ when 7"2 is kept at zero, is .

(77)

Q2

y,A -

Now the transmission matrix for a wall section of area A can be written as

Y~" (Yb + Yc) y~ + Yb + Yc "

(83)

,%,

The following correspondences may be noted :

(The values of u and of y~ found from the Y values in equations (81a), (81b) and (81c) and these expressions agree with the values found in Section 4.1.) Although the equivalence of the six lumped elements of Fig. ! 1 and the distributed resistance and capacity of a multi-layered wall is exact, the circuit does not provide a physical representation. For :

(i) We have already seen that the wall matrix has 6 degrees of freedom ; the T circuit is determined by the choice of 6 elements. (ii) The wall matrix has a determinant of unity; the T circuit matrix too has this property.

(i) two of its capacities are not linked to the reference temperature as they must be in any physical analogue of thermal capacity, and (ii) the values of the elements depend upon the frequency or period at which the wall response is required.

We can therefore represent the overall behaviour of the wall by a thermal circuit consisting of three complex elements in T formation. The circuit elements can be found by identifying them with the elements of the wall matrix. Thus

Nevertheless, the formalism has two potential applications which may have not been much exploited.

I l:Iell Q'

I_e'2,'A

e'22J

Q2 "

Yc =- e~l" A

(79a)

and

L-

e'2, .A e'l, -- 1

and

Vb =

e'21 "A e ~ 2 - - 1"

(79b,c)

To illustrate this, consider the plastered concrete wall discussed in Section 4.1. Suppose that Tt is to be identified as the exposed plaster surface and T2 with ambient temperature To. Thermal circuit elements must be based on conductances (W/K) rather than transmittances (W/m2K), so we will take an area A = 10 m 2. Then Y~ = -- 164.045 + j x ( - 3.314) W/K,

(80a)

1I, = 22.53 ! + j x 22.554 W / K (coincidentally almost equal),

(80b)

Yb = 18.824 + j X 12.370 W/K.

(80C)

If instead, T~ is to be identified with the room index temperature (the rad-air or environmental temperature), we have

6.1. Application to a multi-cell enclosure A complete building can be regarded as an assembly of rooms (1,2, 3 . . . ) , each with its own index temperature (T~, T~2. . . . ) together with ambient temperature and reference temperature. A thermal network can be set up with a T section network linking the temperature nodes between which heat may flow directly. Ventilation conductances V~, V2 . . . can be introduced linking them to ambient temperature. Inter-room ventilation however introduces a new feature. To see this, consider the very simple pair of rooms of Fig. 12. Their common wall is supposed to be massive

v

Q3

Fig. 12. Thermal circuit of two rooms separated by a massive wall and with inter-room ventilation.

232

M . G. D a v i e s

and is accordingly modelled as the three complex conductances shown. The elements include the surface resistances. (These resistances can be modelled separately but such complication is not needed here.) The rooms will be designated 1 and 3. The other surfaces are taken to be adiabatic so they do not affect the response. Heat inputs Q, and Q3 are taken for convenience to act at T,~ and T,~. Air infiltrates from ambient to room 1 so there is a ventilation conductance V as shown. (Strictly speaking, V is the link between the room air temperature T~,~ and ambient and the link X between T,,,.n and T,~ should be included. X however is numerically very large and the link between T~ itself and ambient is normally near enough V.) We suppose that the air then moves to room 3 from which it is entirely lost to ambient. Thus there is no corresponding ventilation link between room 3 and ambient. The heat flow to T~3 however includes the infiltration gain, V ( T , - T~3). (In the simple situation of Fig. 12, we imagine T~.~to be greater than T~ since room 3 is heated, but if there were substantial conduction losses from room 3 taken to be zero here--it might well be lower.) This heat gain (or loss), though proportional to a temperature difference, is of a quite different kind from the gain or loss by conduction through the partition wall. In the case of conduction, the loss from room 1 is a gain to room 3. in the elementary theory for finding room heat needs, no account is taken of the heat lost by ventilation from a room : the need V(T~- To) is calculated on the basis of the temperature of the air at entry To. If this flow passes on to a second room as supposed here, it acts as a quasi pure heat source--a source that will necessarily input V ( T , ~ - T~3) to room 3 without acting as a loss to room 1. The inter-room ventilation is shown as a quasi-pure source in Fig. 12. We have to make use of the fictitious wall node T, as shown. Heat continuity at each of the three nodes in Fig. 12 can then be expressed as the following matrix :

-

Y.

-V

Y~, + YI, + Y~

Yh

-

Y~

Yb+V

2

LT,3]

LQ,J (84)

The elements Ejk of this matrix have the following structure : (i) The elements E~ ~, E22 and E33 (the principal diagonal) are formed as the sums of the conductances which are directly,linked to the nodes 1, 2 and 3 respectively. (ii) E,, = E~, and is the negative of the conductance Y,, linking nodes 1 and 2. A similar relation holds for nodes 2 and 3. Nodes 1 and 3 are not directly linked by a conductance and accordingly E~ 3 is zero. (iii) Properties (i) and (ii) form the atandard structure of the continuity equations matrix. The matrix is symmetrical, that is, Ek.i =Eik- However, this matrix shows a non-standard feature: E3n ¢ E~3. E3, is the negative of the ventilation conductance V. In setting up a continuity matrix for an assembly of rooms, account must be taken of the direction of air flow

between rooms, it introduces the asymmetry to the continuity matrix as indicated above. Apart [¥om this complication, the above equations can be solved using a computer routine to handle linear equations with complex coefficients. The phases of the heat inputs can be included if this is necessary and the solution gives the magnitudes and phases of the temperatures which result. The so-called 'wall temperatures' (T~ above) are needed for the solution but are not likely to be of substantive interest. (The steady state case is a simple version of the scheme sketched above. Wall nodes are not needed, but the ventilation term remains.) 6.2. The response to transient excitation It was remarked above that the response to solar gains after a succession of days when the pattern of irradiance was substantially the same can be synthesized by the addition of a number of Fourier components representing the steady state and periods of 24, 12, 8, 6 hours etc. of the incident radiation. A synthesis along these lines in fact enables us to lk~rm for example a pattern of excitation which has a value of near zero during the time 0 to P/2 and a value near unity between P/2 and P. The more harmonics used, the closer the values of zero and unity will be approximated and the steeper the rise and fall between these values. Any value of P can be chosen 10 days for example. The wall admittances for the surfaces of some room for periods P, P/2, P / 3 . . . Pill can be found as indicated earlier and then by superposition of the responses for periods P, P/2 . . . the response of the room to some chosen form of excitation can be found. The switching on of lighting for example implies an abrupt or step change fi'om zero to some steadily maintained value. A hot water heating system leads to a more ramp-like excitation. If we simply wish to know the response to a step change in input, the basic period P has to be chosen to be long enough l\3r the enclosure to have come sufficiently near to its new steady state value and this may be a matter of hours or days, according to the amount of storage in the building. The more sudden the form of excitation, the larger the number of harmonics that have to be summed to provide adequate accuracy, so this approach to calculating transient response is not efficient from the computational standpoint. It is however fairly simple and run times with today's computers are unlikely to impose much penalty.

7. C O N C L U S I O N S The behaviour of a wall, a layered structure, when subjected to time-varying excitation depends upon the impedance of the layers just as it does in steady conditions but is very much modified by the thermal capacity of the layers. If the excitation is periodic, due for example to diurnal weather variation, the behaviour can be expressed in terms of temperature waves moving in both directions through the wall. This implies the use of trigonometric functions of position in the wall, and since the waves are also highly damped, hyperbolic functions of position also enter the analysis.

Thermal Response to Periodic Excitation Steady state U values have the units W / m 2 K and the dynamic parameters associated with periodic flow also have these units. It is convenient therefore to define a basic material property with these units, the characteristic admittance, which depends on the conductivity and thermal capacity of the material, together with the driving period :

Since the heat flow is 1/8 of a cycle or n/4 radians ahead of the temperature in a semi-infinite solid excited at its exposed surface, this should be included in the definition of characteristic admittance, so we have a = a exp (jn/4). The finite thickness X of a slab is best expressed as the cyclic thickness r :

[npc] '/2 = L~ j • X.

(86)

is dimensionless but is in effect in radians. Thus although five quantities (2, p, c, X and P) are initially needed to define the behaviour of the slab, the relations between temperatures and heat flows at its two surfaces involve just the two parameters a and x. They appear in complex trigonometric and hyperbolic functions and are conveniently expressed as a slab transmission matrix. The overall properties of a layered structure, including inside and outside films and any cavity, can be computed by multiplication of the series of slab matrices. The product matrix has 8 numerically distinct elements, but since its determinant is necessarily unity, it has only 6 degrees of freedom, no matter how many parameters (the a and z values for each layer, together with film resistances) are

233

needed to specify the construction. This allows computation of 3 quasi independent wall characteristics-two admittances and a cyclic transmittance, each of which has a magnitude and time lead or lag. The 1986 CIBSE Guide [1] provides tables of values for one of these admittances and for the cyclic transmittance (expressed in terms of the steady state transmittance). It is possible to express the behaviour of a wall subjected to sinusoidal excitation exactly in terms of a thermal circuit. If the wall admittance alone is of interest, a thermal element of just two real elements is sufficient. If the transmittance properties too are required, three complex elements (6 real elements) are needed which lead of course to the three wall parameters. In estimating the thermal response of an enclosure, the philosophy adopted is that the total response can be decomposed into a steady state component, and components for harmonics of period 24, 12, 8 . . . hours. Thus the form of excitation (e.g. due to solar gain or step in heat input) has to be resolved into components, the admittances for each period and so the temperature components are computed, and the temperatures are finally summed. If expressed in thermal circuit terms, different values for the elements are needed for each period. However, much of the response is given by including the steady state and first harmonic only. This can be performed manually and is the procedure advanced in the CIBSE Guide. The circuit formulation appears to make readily possible a simple estimate of the response of an assembly of rooms though care must be exercised over the direction of air movement between rooms. The various relations between air and surface temperatures in periodically varying conditions--their magnitudes and phase relations---can be illustrated through construction of a vector diagram. The heat flows can be similarly expressed.

REFERENCES

1. CIBSE Guide Book A, Chartered Institution of Building Services Engineers, London (1986). 2. A . H . van Gomum, Theoretical considerations on the conduction of fluctuating heat flow, Applied Science Research, Vol. A2, pp. 272-280, Martinus Nijhoff, The Hague (1951). 3. M.G. Davies, Transmission and storage characteristics of sinusoidally excited walls--a review, Appl. Energy 15, 167 231 (1983). 4. A.G. Loudon, Summertime temperatures in buildings without air conditioning, J. Inst. Heat. Vent. Engrs 37, 280-292 (1970). (Originally published as Conference Proceedings in February 1968.) 5. N . O . Milbank and J. Harrington-Lynn, Thermal response and the admittance procedure, Build. Serv. Engrs 42, 38-51 (1974). 6. M . G . Davies, Transmission and storage characteristics of walls experiencing sinusoidal excitation, Appl. Energy 12, 26%316 (1982). 7. M. G. Davies, The basis for a room global temperature, Phil. Trans. Roy. Soc. London A339, 153-191 (1992). 8. IHVE Guide Book A, Institution of Heating and Ventilating Engineers, London (1971). 9. M.G. Davies, The passive solar heated school in Wallasey III : Model studies of the thermal response of a passive school building, Energy Res. 10, 203-234 (1986). 10. M.G. Davies, Rad-air temperature : The global temperature in an enclosure, Build. Serv. Eng. Res. Technol. 10, 89-104 (1989).

APPENDIX

COMPUTATIONAL FORMULAE In a steady state analysis of heat flow through a wall, we ignore the thermal capacity of the wall materials and simply have to add together the thermal resistances of each layer; there is only one basic quantity to be found--the U value. In dealing with periodic flow, thermal storage is important and its effect is

to introduce a phase difference between the temperature and the heat flow at some point in the wall ; this makes necessary the use of complex algebra, which can be expressed in vector form. In addition, the high damping of the temperature waves makes necessary the use of hyperbolic functions. Finally there are three quasi-independent quantities to be found--two admittances and a transmittance. At a fundamental level, the procedure for periodic flow is the same as for steady flow--we add the imped-

M . G. D a v i e s

234

cosh (z + # ) ~ cosh (r)" cos (r) + j " sinh (r)" sin (r) (Cartesian form)

(ASa)

(polar form)

(ASb)

- [(cosh (20 + c o s (2r))/2] 12 x exp [j" arctan (tan (r)" tanh (r))]

sinh (r + j r ) --- sinh (r)" cos (r) + j " cosh (r)" sin (z) Fig. Al. Vector diagram to illustrate the hyperbolic functions of r +jr.

(Cartesian form)

(A9a)

(polar form).

(A9b)

= [(cosh (2r) - c o s (2r))/2] / -~ x exp [j" arctan (tan (r)/tanh (r))]

ances of the layers. The attendant algebra however is more involved than simple addition, and it is convenient to summarize the computational formulae which are needed. Admittance--the ratio of heat flow q to temperature T at a point--to take some specific quantity, can be written in Cartesian form as a+jb. This is convenient for addition and subtraction. For multiplication and division, it is better seen in polar form :

a+jb ~ ,v"(a'-+b2)'exp(j'arctan(b/a)).

(AI)

The polar form is the better form too for interpretation. ,](aZ+b 2) is the magnitude of the admittance. The quantity arctan (b/a) (in radians) is the angle by which q is ahead of T. For excitation of period 24 hours, q is ahead of T by arctan (b/a) x (24/2g) hours. The product

(a+jb)(c+jd) =- ( a c - b d ) + j ( a d + b c ) .

The transmission matrix uses cosh (r + j r ) as it stands in Cartesian form and we will write A = cosh (~)'cos (r) sinh (r +.jr) however sinh (r +jr)" a. Since

and

B - sinh (r)" sin (r). (A l Oa,b)

appears

as

a -- a(1/.¢'2+/'

sinh (r + j r ) / a

l/x/2),

and (All)

it is convenient to write

C=

sinb (r)' cos (r) + cosh (r)" sin (r)

\ ,/9 -

(AI2)

and D .

(A2)

sinh (z)- cos (r) - c o s h (r)" sin (r) . . . . . . V/2

(Al3)

Then This is used in multiplying out the elements in wall layers. The polar form is not needed. The quotient

[T,] = F cosh(r +.jr) sinh(r+,r)/qFTq q,

(a+jb) _ ac+bd bc-ad (c+jd) c2+d 2 + j x c2q_d2

=

Cartesian Ibrm Va~+b-q ''~ V D,c-aa77 --= Lc2_}_d2] .expL/.arctanLac~dJJ. (A3)

=(T

D 2 - C 2 + 2AB = O.

A + I,B2 - 1

+ T + ) c o s h ( r + j r ) + ( T - T~)sinh(r+jr).

(AI5)

(AI6)

140A - (394/30 + 28A/90)

x x/(2520A+41580)-B2+8050/3=O.)

2

exp ( r + # ) - e x p ( - r - j r ) ,~

(Al4a,b)

and expanding to r ~ (which has a coefficient of 1/40320) we have

• exp (r +jr) + exp ( - r - jr) +T+) : : --T+)

(C + j D ) / a ] [ T 2 7 A+jB JLq2A

A third is found by eliminating r between A and B. This cannot be done exactly but expanding up to r 4, we have

(A4)

This can be written as

+(T

A+jB L(-D+jC).a

A ~- B ~ + 2 C D - I,

The space variation of temperature in a solid undergoing sinusoidal excitation can be written as the sum of two travelling waves

T(x) = T+" exp (-- r - - j r ) + T • exp (z +jr). movingto right movingto left

F

cosh (r + j r ) J L q 2 l

(Since A, B, C and D are all functions of r alone, there must be three relations between them. Two of these are

polar form

T(.'c) = ( T

Lsinh (~ + j r ) • a

(A5a)

(AI7)

The calculation of the phase angles in equations (12b) and (15b) should be mentioned. When r is small, tan (r) ~ tanh (r) -~ r,

(A5b)

(AI8)

SO It is more convenient to use hyperbolic functions since cosh (u) is symmetrical in u and sinh(u) is skew-symmetrical, while exp (u) and exp ( - u) do not have symmetry properties. We have however to express the complex hyperbolic functions in Cartesian and polar forms. This can be done by expanding cosh (r + j r ) for example, noting that cosh(/r) : - c o s ( r )

and

sinh (.#) =- /" sin (r)

(A6)

and using the above relations. It can also be done geometrically. We note that cosh (r +./r) ~ ~ e " # ' + ~ e ~'e j~

(A7)

and can thus be represented as a vector of length {e~making an angle of + r with the horizontal, plus a vector of length ~e making an angle of r. See Fig. A1. sinh (r +.jr) is similarly the difference between these two vectors. Either approach leads readily to the relations

- a r c t a n (tan (r)/tanh (r)) ~- - a r c t a n (r/r) = - arctan (1) = - ~z/4.

(A 19)

So phase (u) - ( ~ / 4 - 7z/4) 0 lbr small values of r, as Fig. 1(b) shows. When r is large, tanh (r) ~ 1, and arctan (tan (r)/tanh (r)) tends to arctan (tan (r)) or - r . So phase (u) tends to n / 4 - r . Clearly, the thicker the wall, the greater phase lag between maxima at the two surfaces. In the case of the admittance y~,, of a slab with an adiabatic rear surface, the arctan component in term (iii) of equation 15(b) tends to ~/4 and zero for small and large values o f t respectively, so the phase ofy~,, tends to n/2 and ~/4 for these respective limits, again as illustrated• As far as the effect of thickness goes, the admittance 3',, of a

Thermal Response to Periodic Excitation slab whose rear surface is isothermal is the reciprocal in magnitude o f that for y~. Phase (ysi) = ~ / 2 - phase (y~) and is zero and ~/4 at the low and high limits of z. The overall response of two layers of solid material in contact is found by multiplication of the two transmission matrices. Each of the eight elements e,,,, eta2, e , ~ . . . of the product matrix is formed as the sum of four components. None of the

6A( 29:2-6

235

elements in general have the same value, but the product matrix continues to have a determinant of unity. Multiplication by a film matrix, five of whose elements are zero and only one actually carries information, only changes four of the elements of the product matrix it multiplies. The two admittances and the transmittance are functions of these elements, as illustrated in the main text.