AE-201 UDCÄ24

O CN uu Heat Transfer Analogies

A. Bhattacharyya

AKTIEBOLAGET ATOMENERGI STOCKHOLM, SWEDEN 1965

AE-201

HEAT TRANSFER ANALOGIES

Ajay Bhattacharyya

Abstract

This report contains descriptions of various analogues uti- lised to study different steady-state and unsteady-state heat trans- fer problems. The analogues covered are as follows: 1 . Hydraulic a) water flow b) air flow 2. Membrane 3. Geometric Electrical a) Electrolytic-tank b) Conducting sheet 4. Network a) Resistance b) R- C A comparison of the different analogues is presented in the form of a table. The bibliography contains 81 references, (This report was submitted to the Royal Institute of Techno- logy, Stockholm, in partial fulfilment of the preliminary require- ments for the degree of Teknologie Licentiat,,}

Printed and distributed in November 1965 LIST OF CONTENTS

Page

Abstract 1 Introduction 3 I. Hydraulic Analogy 4 I, 1 Application in air-conditioning 8 I. 2 Application in design of regenerators 10 1.3 Steady-state heat exchangers 11 1.4 Transient conditions in parallel and counter flow heat-exchangers 14 1.5 Air-flow analogue 16 1.6 Practical problems 16 II. Membrane analogy 17 III. Geometric electrical analogues 19 III. 1 General description of the electrolytic tank analogue 21 III. 2 Axi- symmetric problems 23 III. 3 Application in refrigeration 23 IIIo 4 Distributed heat generation 25 III. 5 Temperature control 26 III. 6 Sources of error in the electrolytic tank analogue 2? HI. 7 General description of the conducting sheet analogue 28 IV. Network analogues 29 IV. 1 Network analogues for steady-state problems 31 IV. 2 Advantages and disadvantages of resistance network analogues 33 IV. 3 Network analogues for unsteady-state problems ' (RC-network) 34 V. Comparison of analogues 37 Acknowledgement 39 Bibliography 40 - 3 -

Introduction

Two physical phenomena are considered to be analogous when the mathematical equations describing their nature are identical in form. This applies to both algebraic and differential equations. As an example, the case of steady-state heat conduction in two dimen- sions is considered. The differential equation governing the pheno- mena is

where t = temperature, and x, y = coordinates. The vertical deflection of a homogenous membrane which is stretched with uniform tension at the edge and is not subjected to any lateral pressure gives rise to the following differential equation

-^-f = O (2-4) Sy2 where z - vertical deflection of membrane.. It is readily seen that the differential equations governing the two entirely different systems are same or in other words there is complete analogy between the temperature field and the vertical deflection of the membrane. Differential equation governing some heat flow phenomena are unsolvable. In some other cases even if they are solvable analyti- cally the solutions are very tedious, complex and time consuming. In some of the problems belonging to the two categories mentioned above it is possible to find physical phenomena which are mathema- tically analogous and experimentally easier to investigate. It should be specifically mentioned that an analogy does not give any analyti- cal solutions. - 4

Such analogies as mentioned above exist among a large number of natural phenomena. Many problems in elasticity, vibrations, fluid flow and heat transfer can be solved by utilising analogies. In this paper only analogies used in the field of heat transfer will be discussed.

L Hydraulic Analogy

Heat flow and fluid flow under laminar flow conditions are governed by basic equations which are similar. The equations for steady state can be written as

Quantity = Constant. Driving force

For heat flow Q = U A • AT (1-1) where Q = heat flow U = overall heat transfer coeff. A s area of heat flow AT= temp. diff.

For fluid flow Vk = K • AP (1 - 2) where V, = volumetric flow K = constant AP= pressure head

To solve periodic heat flow problems with the help of the above mentioned analogy it is necessary to take into account the effect of storage of heat or fluid. Moor (63) and Bäckström (8) independent of each other designed,,similar hydraulic circuits to study periodic heat flow problems. For a clearer understanding of the basic principles involved, the hydrau- lic circuit called HYDROCAli designed by Moor (63) is discussed below. Ref. 63 gives the use of hydrocal in the solution of a simple heat transfer problem - the heating of a rod from one end, no heat flow being permitted from the other end or from the side area. As shown in the fig. 1, the rod is divided into five incremental blocks. Thermal capacity of each block is arbitrarily lumped at the centre of the block. Six glass tubes called standpipes are mounted as shown in the fig, 2a and 2b. The pipes numbering from 1 to 5 represent the five ttiermal capacities of the five increments. Quantity of water in the standpipe represents stored heat in the rod increment. Quantity of water needed to raise the level of water in the standpipe per unit length, say per centi- meter, represents the increment's heat capacity. The level of water represents the temperature level in the increment assuming that the initial level in the standpipe corresponded to the initial temperature level in the increment. Each tube beginning from 0 to 4 is connected to the next tube on the right hand side» Connecting tubes offer constant re- sistance to water flow because the flow is maintained in the la.- minar region by proper choice of dimensions. This constant re- sistance to water flow represents the resistance to heat flow from the centre of one block to the next. The resistance between the standpipe numbered 0 and 1 is slightly different from other re- sistances because it represents thermal resistance due to one half of a block plus a surface heat transfer resistance. To represent the higher temperature source used for heating the rod the stand- pipe numbered 0 is connected to a water tank at a higher level. The level of water in the standpipes is adjusted to the initial level of the temperature in the rod. Connections are opened and the water levels in the different standpipes start changing. At any time the levels car», be directly interpreted as temperatures of the centres of the various blocks. (See fig. 2b). Bäckström (8) describes various application of the analogy in studying transient heat flow problems connected with cold storage of food. Five examples of increasing complications are described in the ref. 8. One of the problems considered is illustrated in the fig. 3a and fig. 3b. The initial level of water in B represents the initial temperature of the body put in the cold storage room. The level of water in S is kept constant to simulate the constant temperature of the surroun- dings. Similarly the initial level of water in R and C denote the tern- - 6 -

pexature of the room and the cooling coils at the beginning of a transient phenomena. The flow of water from the standpipe C is handregulated in such a way that a valve is opened when the water level exceeds the level corresponding to + Z C. The water flow is stopped when the level falls below the level corresponding to - 8 C. In this way regulation with "pressostat" in a cold storage room is simulated manually in an ana- logue. Differential equations for transient heat transfer and laminar flow are presented together for a better understanding of the principles governing the design of a hydraulic analogue (14). Continuity and Navier-Stokes equation for laminar flow through a circular pipe yield, the following equation for pressure drop over a length dx of a capillary tube of diameter D .

32 ]x V dx dP = pg dh = ^ (1 ~ 3) c where P = pressure drop in the capillary tubing p = density h = difference in hydrostatic pressure in length units \x - viscosity V = velocity x = distance g = acceleration due to gravity D = capillary diameter Let Q = volumetric flow c or, Q = ^ D 2 V (1 - 4) c 4 c c x '

Substituting for V^, from equation (l - 3) in equation (1 - 4} one obtains

IT. p g D dh Q = 2 (1 . 5) 128 u dx - 7 -

IT D 4 P g c Let - = 1 (1 - 6) 128 V- B

1 dh or • (1 - 7) Q c B dx where B is a constant for a particular hydraulic system. Differentiating equation (1 - 7) w. r-, t. x one obtains

dx " B , 2 V *) dx

T o standpipe cross section capillary length

A length of tube Ax over a time increment A 6 is considered here. Storage capacity per unit length in the standpipe for a length Ax of the capillary tube is S A x. If Q- and Q~ are rate of volumetric flow entering and leaving a capillary tubing of length Ax over a time interval of A 8 then

Q_ A9 - Q, A9 = h (S Ax) (1 - 9)

Let Q2 - Qt ~ AQc A Q Ax " ö A0 ^ I

Assuming that the differences are very small one obtains

^ = s|| (1 - 11)

dx dö x '

Combining equation (1-8) and equation (1 - 11) one obtains

^ „ J_ A , s K d8 ~ SB d, x2 ' ' Let ^L= a1

2 dh 1 d h or dW = a T~2 dx The differential equation governing one dimensional heat flow through an infinite slab is as follows

where t = temperature 9 = time

I. 1 Application in air-conditioning

In designing air-conditioning systems one needs detailed informa- tion about the magnitude and the variation of cooling loads» A standard pro- blem in air-conditioning is the evaluation of cooling load for a room which has a sunlit glass or has artificial lighting» This evaluation is very compli- cated because of the interaction of radiation, convection and conduction phe- nomena. The energy input to the space of the room throxtgh the sunlit glass can be classified as follows (52) (59) (60)

i) direct solar radiation ii) sky or diffuse radiation iii) radiant energy from the room surface of the glass to the walls of the room iv) convectjve energy to indoor air from the room surface of the glass

The last of the above mentioned four posts is the only part which immediately affects the cooling load« Radiative heat energy is not directly absorbed by the air in the room. The floor, the ceiling and the- walls absorb radiative heat energy and become hotter» The rate at which the temperature of the surfaces go up is a function of the following factors?

i) thermal capacity of the walls, the floor and the ceiling ii) thermal conductivity of the wall materials iii) coefficient of absorption and emission of the surfaces _ 9 -

As the surfaces get hotter heat is transmitted to the air layer in contact with the walls by convection. The heat transfer phenomena described above is very complex for any analytical solution. A full-scale experiment is impractical. Such an experiment is of doubtful value when the parameters are changed in different design problems, Leopold (52) and Mackey et al. (59) (60) have described some hydraulic analogues designed under some simplifying assumptions for different air conditioning problems. Leopold (52) has described a hydraulic analogue which was used to study the cooling requirements for a room which has indi- rect illumination with load equivalent 36. 6 W/m . Two cases were considered. Case A. An amount of air was introduced in the room which would be sufficient to maintain the temperature of the air constant in the room if there were no ther- mal storage. The air temperature variation with time was studied.

Case B. Variation of the amount of air supplied to maintain a constant temperature in the room.

Fig. 4 shows a sketch of the system under consideration and the fig. 5 shows the corresponding hydraulic analogue for the case A described above. Mackey et al» (60) describes another case study with their hydraulic analogue of an air-conditioning problem which was si- multaneously solved analytically. Results from the analogue com- pare favourably with the results from the analytical study. This proves the reliability of the analogue systems. Mackey et al. (59) describes another application of the same analogue in which cooling loads and instantaneous rate of heat gain from a sunlit glass and wall were studied over an analogous time period of 24 hours» One singular feature of this experiment was that one had to start with fictitious values of temperature and heat flow in - 10 -

the beginning» After running through the whole cycle of 24 hours which was 24 minutes in the model 4 or 5 times, one obtained an exactly re- producible cyclic variation. The total amount of heat input due to solar radiation should be equal to the heat removed by the air flow over a pe- riod of 24 hours,, The rate of heat input due to solar radiation varies in such a way that it has the same value after 24 hours. In the hydraulic ana- logue this condition is fulfilled by ensuring that after a complete cycle the levels of the fluid in the standpipes of the analogue should be same as they were in the corresponding time in the previous cycle. The heat input due to solar radiation varies cyclically over 24 hours, Fig. 6 shows the corresponding variation in cooling load obtained with the help of the analogue» In the analogue a periodic flow pump is used to obtain a variation in flow to simulate a variation in radiant heat input,

I. 2 Application in design_of regenerators

Duhne (14) describes a simple analogue to study the transient heat transfer phenomena in an infinite slabs The infinite slab is originally at an uniform temperature t-. The temperature of the surface is raised to tC Duhne (14) mensions that the comparison of data from the analogue and a mathematical solution indicated a mean deviation of 0,47 percent and a maximum deviation of - 1.20 percent. Duhne and Garcia (15) have used a similar hydraulic analogue to prepare charts for the solution of problems of heat storage in bodies of parallelopiped form. These charts can be used for designing regenerators in which the temperature rise in the gas is small. Results from the analogue model for one-dimensional heat flow through an infinite slab is used for calculating a 3 dimensional phenomena, A parameter Z is defined as follows: Q - Q Z«-g (1-15) 00 where Q = heat stored by the solid at any time, above its initial tempera- ture t o Q = heat stored by the solid above its initial temperature t when 9=0 seconds oo -li-

lt can be shown that (1 5)

Z = Z Z Z (1-16) x y z \ . / where the suffixes x, y and z denotes the cases where the heat flows one dimensionally in the x, y and z direction respectively. The equa- tion (1 - 16) makes it possible to evaluate Z , Z and Z separately from the charts prepared with the help of the analogue mentioned above.

I, 3 Steady-state heat exchangers

All the hydraulic analogies which have been discussed up till now solved problems connected with periodic heat flow, Juhasz and Hooper (28) have presented a method for studying steady-state heat exchangers using a hydraulic analogue. The analogue was developed only for parallel flow heat exchanger problems. In a parallel flow heat exchanger as shown in the fig. 7 .the temperature difference between the two fluids decrease towards the outlet. In a hydraulic analogue the difference in level between two standpipes connected with a capillary tube decreases with time. In the analogue proposed by Juhasz et al, time in the analogue is equi- valent to relative area of heat exchange.

Let y = relative surface in the heat exchanger, dimensionless Then y = 0 at the inlet and y = j at the outlet to the exchanger Let 8 = time in the model, hours Then y = n 8 y y The following relations are valid for the system described above;

At=nt- A, (1 - 17)

y = ny 8 (1-18)

D4 2 1 7 -

Ai, C 2 p? « St - (1 - 20) m. C 1 Pi

C P? ÄTT where, 2 A = area of parallel flow heat exchanger, m C = volume of liquid flowing through the capillary of the model per -unit time and unit he ad, meter /meter, hour C = specific heat of fluid 1 at constant pressure, Kj/kg °C

C n specific heat of fluid 2 at constant pressure, Kj/kp °C p2 D « inside diameter of capillary tube, meter K = constant, l/meter, hour JL = length of capillary tube, meter m, - mass flow of fluid 1, kg/hour

m9 ~ mass flow of fluid 2, kg/hour n Ä conversion factor between time in the model and relative area in the heat exchanger, hour -1 n = conversion factor between height in the model and the

Is temperature in the heat exchanger, C/meter 2 S, = cross section area of tube 1, meter 2 So " cross section area of tube 2, meter Ca U = overall heat transfer coefficient in the heat exchanger, KJ/meter • hour • C = temperature difference between the fluid 1 and the fluid Z, °C = level difference in the model between levels 1 and 2, meter '13 -

Juhasz et al. (29) discuss certain additional attachments to the analogue described above to take into account the following conditions.

a. variation in specific heat with temperature b. change of phase c. heat loss to the surroundings d. variation in overall coefficient of heat transfer with temperature.

Variation in specific heat affects the rate at which the tempera- ture of the fluids rise. In the analogue it will be reflected as a varia- tion in the rate at which the height, of the liquid columns in the stand- pipes will change for the same rate of flow through the capillary tube. This is achieved by inserting a template in either of the standpipes to vary the internal area of the standpipes with height. The profile of the template corresponds to the variation of specific heat with temperature» Change of phase affects the heat transfer rate in the following ways

1. Temperature of the fluid undergoing change of phase stays • constant irrespective of the rate of heat transfer» It is equivalent to an infinite specific heat. This is arranged in the model by connecting a reservoir of a very large diameter in parallel with the tube corresponding to the fluid undergoing a change of phase» 2. Coefficient of heat transfer changes significantly, A different set of capillary tubes corresponding to different values of U can be connected in parallel.

Heat leakage to the surroundings can be simulated in the following way. The tube representing the fluid from which the leakage takes place is connected to a large reservoir whose level corresponds to the tem- perature of the surroundings. The capillary tube connecting the large reservoir to the tube is selected in accordance with the magnitude of the over-all coefficient of heat transfer between the fluid and the surroundings. - 14

Variation in the over-all coefficient of heat transfer with temperature can be taken into account in the same way as described above in connection with change of phase. Different capillary tubes in parallel can.be used for different levels of the liquid in the tube.

I._4_ __Transient conditions in^garallel and cpunter flow heat-exchangers

In the process industry it is very common to find heat exchangers with automatic control. För an effective design of the control system it is necessary to understand the transient behaviour of such heat exchangers. Analytical solutions describing transient conditions are often very compli- cated. Added complications like longitudinal heat conduction as in liquid metal heat exchangers or like variable specific heats make the analytical solutions more complicated* For an easier understanding of the problem Juhasz and Clark (29) had proposed a hydraulic analogue to study, under some simplifying assumptions, the transient conditions in cotinterflow and parallel flow heat exchangers» The analogue is valid for a co-axial heat exchanger. The inner tube carries the hot fluid and the cold fluid flows through the annulus. The basic principles involved in designing the ana- logue are same as they were in other analogues for periodic heat flow de- scribed previously. The novel feature of this analogue is that the movement of the liquids with respect to the tube and the shell is simulated with the help of rotary valves. The principle behind the use of rotary valves is de- scribed below. For the sake of clarity a more simple example than the one given in ref. 29 is described here, A body A which is divided into 4 incremental blocks is heated with a liquid B. The volume of the liquid B flowing past the body A is also divided into incremental blocks as shown in the fig. 8e At 8 = 8 the whole of the body A is at the same temperature t . The O zK liquid B is also at an uniform temperature t . After a time interval tB from 8 , the block IB is in contact with the o block 1A. Heat flow takes place from the block 1B to the block 1A by con- vection. Subsequently, heat flows from 1A to 2A, 3A and 4A by conduction.» - 15 -

After a time interval A8 from. S + A6 , the block IB which is c. o cooler due to heat loss to 1A in the previous interval is in contact with the block 2A. Similarly, 2B is in contact with 1A. If the longi- tudinal heat conduction in the liquid is not negligible then heat will flow from 2B to 1B« The other heat flow paths are shown with arrow- heads. In this way as the liquid B moves relative to the body A the com- bination of the blocks from the set A and the set B keep changing. , In the hydraulic analogue proposed by Juhasz et al. (29) the blocks are represented by standpipes containing water. For a time internal A8, the standpipe 1A is connected to the standpipe 1B. In the next time interval A 0_ the standpipe IB is connected to the stand- pipe 2A and the standpipe 2B to 1A. These changes in the connections, between the different standpipes at the end of each time interval is conveniently and quickly accomplished by using rotary valves. Since time in the analogue is equivalent to time in the heat exchanger, when the x*otary valves are being reset there should not be any flow between the standpipes belonging to the same set» This is an added complication for the mechanical design of the system» In the co-axial heat exchanger described, by Juhasz et al. (29) one needs 4 sets of standpipes as described below. Set Description A Hot liquid B Inner tube wall C Cold liquid D Outer shell wall

To simulate the heat loss to the surroundings an additional standpipe with constant level is connected to the standpipes represen- ting the outer shell. Variable specific heat of the fluids can be taken into account using templates as described in ref, 28. For a counter- flow heat exchanger the standpipes belonging to the set B and the set C will be connected in opposite order. - 16 -

I._5 Air-flow analogue

All the analogues which have been described so far were based on laminar flow of liquids,, Coyle (11) had designed, an analogue using air as the flow medium* Fig» 9 shows a simple case of heating of an infinite slab . of a constant thickness. Heating is done at the boundaries AA • and BB« ,

OO' is the line % of symmetry for the temperature profile. An air-flow ana- logue for one-half of the slab is also shown in fig. 9» The basic principles of an air-flow analogue is same as that of liquid-flow analogues. The air pressure and mass of air in each standpipe is equivalent to temperature and quantity of heat in the corresponding blocks in the slab» Variation in thermal conductivity was taken into account by using a capillary the bore of which could be partially blocked for a varying pro- portion of its length by a wire sliding within it9 With liquid-flow analogues the problem of air-locks limit the con- venience with which cross-connections in a central exchange panel of a large scale analogue can be made» With air or any other gas, however, this problem does not exist. One could use a gas lighter than air to reduce the end losses in the capillaries,, But the advantage of using air is that one need not have a closed-system.

I. 6 Practi_c_aJ.j3roblems

Moore (63) has discussed the different practical problems associated in constructing hydraulic analogues» The problems can b.e classified as follows:

i) manufacturing tolerances ii) hydraulic phenomena like meniscus effect in standpipes or flow resistance in the rubber tubing joining the flow tubes in series iii) chemical phenomena like aging of rubber or corrosion

.Leopold (52) used a silicone liquid (Dow-Corning Corp. DC 20) of 20 centistokes kinematic viscosity. The change of viscosity with tempera- ture is less than that for water. So a constant temperature bath was not required for the conductances» The chemical problems reported by Moore (63) were absent in the experimental" setup of Leopold (52). - 17 -

Errors: The errors in the system will be caus'ed by the following in- accuracies:

i) inaccuracy due to dividing a continuous system into a system with finite increments. The larger the number of divisions the less would be the error due to this, ii) inaccuracies in physical constants like thermal conductivity etc. iii) errors in the physical dimension of the analogue iv) unaccounted for resistances in the flow tube paths. The errors due to the last two inaccuracies would be of the order of 3 percent, according to Moore (63).

II. Membrane analogy (7?) (81) (76) (24) (64) Small deflections of the surface of an ideal membrane stretched over a closed boundary and dilated by a pressure p satisfy the following differential equation

-4 s(x,y)+il 2(x,y) = -H (2 - ,) 3x, 3y

An ideal membrane is one which has no mass and which is sub- jected to an uniform tension T per unit length of boundary. It is also assumed that the membrane deflections are sufficiently small so that for all points on the membrane the sine of the angle of inclination of the tangent plane from the horizontal may be taken equal to the angle itself, expressed in radians. The Poisson equation for heat conduction with internal heat generation is as follows

2.. .2 »»» i

where q = internal heat generation rate per unit X = thermal conductivity. In the case of the stretched membrane if there is no dilation pressure the governing diffential equation can be written as follows

z(x, y) + » Z(x, y) s o (2-3) Sx^ 3y

Similarly-, if there is no internal heat generation the Poisson equation becomes

= 0 (2-4)

Equations (2-1) and (2 - 3) are similar to the equations (2 - 2) and (2 - 4) respectively. The following scale factors are defined.

i) Geometrical scale factor {3

h A - length in the heat transfer problem length in the membrane problem

ii) Temperature scale factor -y

, temperature 11» where v = *~ T~n—r= ;—: 1 membrane deflection « z '

It can be shown that

,2

Using the above mentioned similarity, problems of 2-dimensional steady-state conduction with internal heat generation can be solved with the help of a membrane analogue. For s.olving a particular problem one constructs the two-dimensional boundaries of the region. A soap film is used as a membrane, The soap film is obtained (81) by using a soap solution which contains sodium oleate, glycerin and water. The boundary tempera- tures are duplicated by building up the edges of the model boundary in scale to the boundary values of the problem. The deflection of the membrane - 19 -

z (x, y) is interpreted as the temperature t at the location x, y. A travelling micrometer gauge is used for measuring the elevations at different locations» Handpowered machine screws with vernier scales are used for determining x and y. According to Wilson and Miles (81) the accuracy with which a measuring point can be located is "t 0. 025 mm. Schneider and Cambel (77) mention that the accuracy of these experimental solutions can be held to within - 3 per cent. It is not possible to measure skin tension 7 per unit length of • f boundary directly. — is indirectly calculated from an auxiliary film stretched over a circular boundary (76). For problems in which there is no internal heat generation this measurement is not ne- cessary. Aluminium is usually used as the material for the models to reduce oxidation. The soap film is very sensitive to vibra.tion, tem- perature variation and contamination. To prolong the life of a film and to obtain consistent results it is necessary to perform the expe- riment in a^ibration-free and temperature-controlled enclosure. Durability of the film is increased by aging the fresh soap solution in a closed vessel for a period of 24 hours. There is an aging effect in the film. In the beginning the thickness of the film is non-uniform. The membrane should be allowed to drain for at least one hour before any measurements are taken. Advantages of a membrane analogue is that it is inexpensive. For • steady-state heat conduction problems in irregular geometries the so- lution can be obtained in a simpler way than geometric electrical analogues.

HE. Geometric electrical analogues

Geometric electrical analogues are mainly of two types

i) Electrolytic tank analogtie ii) Conducting sheet analogue - 20 -

Like the membrane analogues the geometric electrical analogues are also continuous field analogues. For a clearer understanding of the analogy the differential equation governing the steady-state electrical conduction in a sheet of conducting material (either solid or liquid) is presented below. Fig. 10 shows a differential element dx « dy of a con- diicting material whose boundary is subjected to a given voltage distri- bution e, . A distributed current i per unit area is supplied over its sur- face. From the equation of continuity one obtains

\ (V i x +• -~3x- • dx ) •/ dyy - x^\y3y>'i • dy -5- ( i + -r-^~ * dy'y yy } }. dx - iy dx

- i • dx • dy a 0 (3 - 1) where

i = current per unit length of boundary in the x-direction i = current per unit length ooff boundary in the y-direction

From the equation (3-1) one obtains

3x i and i can be rewritten in terms of voltage gradients

1 9e Xx = " R ' IE

y R where R = resistance between two parallel edges of a square. Substituting equations (3 - 3) and (3 - 4) in (3 - 2) one obtains

3. 2 .I® + ±® -. - R i (3-5) 3^ ^ - 21 -

If no distributed current is supplied over the surface of the con ducting material then,

2 2 3 3 e + e „ n /->. _ i\

It is readily seen that a complete analogy exists between the equation (3 - 5) and the equation (2 - 2) which is Poisson equation for heat conduction with internal heat generation. The equation (3 - 6) is the Laplace equation and is analogous to the equation (2 - 3) describing steady-state conduction without generation (32).

HI. 1 _ j^eneral_descrip_tion_of_ the electrolytic J;ank_analogue_

An electrolytic tank analogue (1) (24) (2) (4) (58) consist of a tank filled with an electrolyte. The geometric form o£ the tank is similar to that of the heat transfer region to be investigated. The tank should be large enough so that models of convenient size can be used in it. It is easier to eliminate edge-effects in a large tank. If there is no control system to maintain a constant temperature, a large tank tends to stabilize rapid room-temperature variations. Fig, 11 reproduced from the ref, 74 shows a sketch of an electrolytic tank analogue. A short description of an analogue designed by Langmuir, Adams and Meikle (58), who were pioneers in this field, follows. Langmuir et al. utilised the electrolytic analogue to study the heat flow pheno- menon through a thick corner. The model consisted of an electrolytic bath in a shallow tank» The hot and cold surfaces of the thermal cor- ner were represented by metallic sides. The inner and the outer surfaces i.e., the metallic sides were connected to a power supply to establish a potential difference across the liquid electrolyte repre- senting the homogeneous wall material. The potential difference between the inner and the outer surfaces was analogous to the temperature difr ference between the two wall surfaces in the heat flow problem. - 22 -

In the mathematical analogy described by the equations (3 - 5) and (5 - 6) it is assumed that the conducting medium is purely resistive. Be- cause of electrolysis effect, it is not possible to use direct current in this analogy. With alternating current because of the polarization effect it is difficult to obtain a purely resistive conducting medium. The junction of the electrode and the electrolyte constitutes a capacitive impedance. The effect of polarization is considered to be the inclusion of a reactive and resistive impedance in parallel, in the immediate vicinity of an electrode surface, which is in series with the electrolyte resistance. The magnitude of the polarization impedance is determined by the electrolyte-electrode combination and the frequency of the tank voltage,> A figure of merit F is used for comparing different electrolyte-electrode combinations. It is defined as,

(3-7) where r = resistance per cubic centimeter of the electrolyte s. = surface impedance of one square centimeter of electrode surface

The reciprocal of F is the equivalent extension of the length of the electrolyte required to give the observed impedance. The higher the value of F the better is the combination. The following table gives some results from an experiment described by Amort (1),

Table 3 - 1 Combination F in l/cm

Tap water + Cu-electrode 7 Distilled water + polished brass 1 00 Distilled water + HC1 cleaned bradd 300

The frequency of the tank voltage is also to be taken into account. - 23 -

Polarization effects are more pronounced at low frequencies. If the frequency is too high, the effects of stray capacitances create trouble. Amort (1) recommended frequencies from about 400 cps to 1500 cps. Anthony and Fridensohn (3) (4) and McNall, Jr., and Janssen (65) used 60 cps. A curve is reproduced (see fig. 12) from the ref. 1 to show how resistance and capacitance components of polarization for distilled water and polished brass electrodes vary with frequency.

Nukiyama and Tanasawa (68) describes the use of an electroly- tic analogue to study an axi- symmetric case of radial heat flow from a finite hollow cylinder. The bases and the inner surface of the cy- linder was at 0 C. The inner surface was kept at T °C. The percen- tage error in calculating the heat flow by assuming the cylinder to be infinite was evaluated with the help of the analogue. Since it was an axi-symmetric case the tank was made in the form of a cylindrical wedge cut by two radial planes both passing through the cylinder a%iso When the cylinder was assumed to be finite the copper electrodes si- mulating the bases and the inner surface of the cylinder was kept at the same potential. To simulate the case of an infinite cylinder the copper electrodes representing the two bases were replaced with in- sulating plates. In an infinite cylinder with an axi-symmetric tempe- rature distribution there cannot be any axial heat flow. By maintaining the same potential difference in both the cases the variation in current was measured to evaluate the error in neglecting end effects.

Pierre (74) used an electrolytic analogue to study the influence of metallic frames on the insulating value of, and the temperature distribution in the insulation of cold-storage chambers on board ship. One of the problems investigated and its model is shown in fig. 13, Pierre (74) had investigated different conducting mediums but had found ordinary tap water to be best suited for the purpose. - 24 -

The width of the tank between the copper electrodes representing the external and the internal surface of the wall was increased to account for the film resistances according to the relation given below.

\. X. 6, 8 = 6. + —i-+ —- +\. S^ (3-8) int ext b where

5 = distance between the copper plates 5. = thickness of insulation i V = thermal conductivity of insulation a. = thermal conductance of the film between the air in the mt cold storage chamber and the wall of the chamber a , = thermal conductance of the film between the ambient ext fluid and the outer wall surface 6, = thickness of bearing structures forming part of the wall of the cold storage chamber X, = thermal conductivity of material forming bearing structure

in the experiment a constant voltage was applied across the copper terminals. Decrease in the insulating value due to the presence of metal frames was evaluated by measuring the current flows in the presence and absence of the frame models in the bath. The following equation gave the percentage reduction in the insulating value

ir R. y - i - R w 7/ vfr where

i*. = the current flowing through the tank in the presence of the frame model i. = the current flowing through the tank in the absence of the frame model R. _ = the ohmic resistance of the model in the absence of the frame ins R, = the ohmic resistance of the model in the presence of the frame - 25 -

P-Ji éPL^Iikuted *? ? §•* B^P^I ation The electrolytic tank analogues which have been described so far did not take into account distributed heat generation. The distributed heat generation can be approximated by introducing current into the electrolyte through a grid of input electrodes. The current to different electrodes in a grid can be controlled and measured independently of each other to simulate non-uniformly distributed heat generation. McNall, Jr., and Janssen (65) described an electrolytic analogue to study the problem of heat conduction within;, a transistor. In a tran- sistor the heat generated at the interface between the collector and the base flows through several different materials of varying thermal conductivity to an infinite heat sink e9 g, atmosphere. Fig. 14 shows one of the configurations simulated in the analogue. Each piece of material was represented by a geometrically similar shallow rectangu- lar lucite tray containing a solution of potassium chloride in distilled water. Gold plated copper strips were used to form the equipotential lines in the boundary between the two different materials. Electrical connection between adjacent trays were accomplished by soldering the tips of the gold-plated copper strips in corresponding trays. The in- put current to simulate the interface heat generation was fed at the corresponding copper strips in the model» Interface resistance at the clamped joint was represented by having a resistor between the two trays representing the copper support and the aluminium chassis (see fig. 14). Each strip in the interface, where heat generation was simu- • lated, had an independent circuit which contained a 25,000 ohm resistor,, Such a high value of resistance ensured that the effect of changing the input current in one circuit on currents in the other circuits would be nagligible. The arrangement of the analogue was very flexible. Changes in configuration were obtained conveniently by merely soldering and unsoldering some of the strips in the.interface» The size and shape of the lucite trays were easily changed by inserting insulating baffles0 The error due to lumping in the surface of discontinuties in the form. of strips is discussed in detail in ref, 65. The error was of the order -26-

o.f 0,5 C, The experimental accuracy was of the order of 6 - 8 percent. It is mentioned that the accuracy of the analogue could have been improved by increasing the frequency to 1500 cps from 60 cps and by using graphi- tiaied. or platinized electrodes. Malavard and Miroux (66) had -»jtilised an electrolytic bath with a sloping bottom to study heat transfer problems with rotational symmetry. Fig. 15 shov/3 sketches of such a problem and its analogue. The mercury was contained in a heat ir.sulated glass tube with a hemispherical bottom. The heat a our so v?ae at the centre of the surface at the position of the cathode spot ox the arc which was assumed to be stationary. The heat was dissipated by evaporation of mercury at the free surface, The heat transfer condition at es-ch point of this surface caxi be expressed by tiie relation,

-\g£+.A. *(T)«0 (3- 10) where

"k :s thermal conductivity of mercury A »a constant related to the heat of evaporation f (T) « function defining the pressure of saturated vapour, as given by an experimental diagram»

In the analogue, the above equation gives a relation between the elec- trode current and the potential. Correct settings were obtained by successive approximation, starting from a plausible temperature distribution.

•P?i §.. - The electrolyte tank analogues which have been described so far did net contain any special arrangements for temperature control of the electro- lyte. The conductivity of most of the electrolytes is a strong function of tem- pevatare. For example» the resistivity of a weak solution of potassium chlo- ride in distilled watsr decreases about 1,8 per cent for each 1 °C tempera- ture increase (65). McNall and Jans sen (65) allowed a maximum temperature rise of 1,7 °C in their experiment. Anthony and Fridensohn (3) (4) had de- signed an electrolytic tank with 7, 920 electrode positions. A hydraulic system was incorporated to continuously circulate the electrolyte. The purpose of circulation was to maintain a constant temperature and to reduce the effects of contamination and evaporation. The hydraulic system also permitted a very - 27 -

accurate control over the electrolyte level in the problem. Sodium chromate was used as the electrolyte. The conductivity of sodium chrornate solution changes 2. 5 percent per C. To keep the con- ductivity of the solution constant within 0. 5 % it was necessary to maintain the temperature within 0. 2 C. The analogue was designed to handle heat transfer problems encountered in nuclear reactor design. Distributed heat generation due to fission or neutron absorption is common in such problems. In the analogue the distributed heat generation was simulated by introducing current into the electrolyte through a grid of input elec- trodes. Each electrode had a resistance in series which was 100 times larger than the other resistances in the system. That made the current in each electrode practically independent of all other adjustments made in the system while the problem was being set up. The sum of the currents fed into the system were collected in sinks representing coolant passages in a nuclear reactor core. Maximum number of sinks available were 20. The experimental accuracy, verified against an analytical solution of a standard pro- blem, was of the order of 3 per cent.

III. 6 Sources of error in the electrolytic tank analogue

A chart based on discussion of errors, presented by Amort (1) is given below. For a more detailed information ref, 1 7 should be consulted.

Type of error Precautions to Established minimize error magnitude of error Presence of probes Experimental deter- Very small with finite spacing mination of best combination of elec- trodes and electro- lyte Meniscus, probe Less than 1 per cent penetration and capillary action Variation of probe Levelling of probe penetration depths carriage - 28 -

Tank-edge meniscus Fill tank exactly to up- per edge of the model: add wetting agent Electrolyte Constant circulation Depends on ambient evaporation of electrolyte humidity and time for problem solution Variation in Levelling of tank Can be made in- electrolyte depth significant Combined tank Large figure of me- Can be held to 0. 2 per errors rit for electrod- cent (see ref» 17) electrolyte combi- nation, sufficient tank size for accu- rate models, tank levelling Mechanical errors Minimize backlash and friction. Struc- ture must be rigid

III. 7 General description of the conducting sheet analogue

The basic equations governing the potential distribution as a re- sult of current flow in an electrically conducting sheet are same as those governing current flow through an electrolyte. In a conducting sheet ana- logue the sheet is given the same shape as the prototype thermal system (75). An isothermal boundary is simulated by an equipotential boundary by using a highly conducting material. An adiabatic boundary is represen- ted by an electrically insulating material. Fig. 16 shows a sketch of a simple conducting sheet analogue. In a conducting sheet analogue the power supply can be either d-c or a-c. In a conducting sheet analogue there is no problem, of electrolysis as in the case of electrolytic tank analogue. The conducting material most commonly used is Teledeltos facsimile paper (27). Stainless steel sheets were used by Fitts et al. (21). Kayan (3 8) (39) (40) (42) has done an extensive amount of work in the development of analogues using conducting sheets,, It was even pro- posed that analogues of this type should be called a "Kayanalogger". In a conducting sheet analogue, using a uniform sheet, the dimen- sions are obtained in the same way as in an electrolytic tank analogue. "In using uniform sheet, linear distances o.n the sheet are commensurate 29 -

of resistance to heat transfer between the boundary fluid, at its pre- vailing temperature, and the wall itself (45). " In ref. 39 conducting sheets were used to study problems similar to the ones studied in ref. 74 using the electrolytic tank analogue, Kayan (45) has utilised the conducting sheet analogue to study a great variety of problems. Some of the more important variations are listed below.

1. Problem of an embedded structural element within a wall (39) 2. Problem of heat transfer pipes embedded in a slab (42) 3. Problem of wall of a multiple-component material (40)

Fig. 1 7 is a description of a problem in refrigeration (40). Fig. 18 is a description of the conducting sheet analogue for the same problem. The aim was to obtain the steady-state internal temperature lines, in the analogue the solid wall was considered to be the basic material in the problem. The lengths of the actual heat flow paths were directly represented on the conducting sheet. The insulating wall, however, has higher resistance. To account for that, the electrical characteristics of the sheet representing the insulation was modified. By making perforations (see fig. 1 8) in the conducting sheet the effective electrical unit resistance of a given section was altered. This alteration was made in progressive steps during an investigation. This enabled one to make a progressive study of the effect of different values of thermal resistance in insulation. Uniformity of the sheet electrical conductivity in all directions is an important condition for accurate results. The resistance ratio bet- ween the solid sheet and the perforated sheet should be the same in all directions. This is dependent largely upon the accuracy of cutting and duplicating the mesh. Kayan (40) obtained an apparent deviation of less than 5 percent.

IV» Network analogues

Network analogues (32) are the most important form of analogues because of their flexibility and ease of measurement. The mathematical - 30 -

basis for direct network analogue simulators lies in the calculus of finite differences. The electrical analogue model is constructed by recognizing the formal similarity between the finite difference equation applying to any node point in a discretized system and Kirchhoff 's current-law equation for a node formed by passive electrical elements. A discretized system in the form of line segments is shown in the fig. 19a. The nodes are numbered as Z. 0 and 1. The transient or steady- state values of the dependent variable y at these three adjacent node points are indicated by y?, y^, y.. It can be shown that 4^^ cbc h2 hT where h = Ax. Considering a second-order differential equation 40 (4 " 2> dx Substituting from, the equation (4-1) the equation (4 - Z) can be rewritten as

(4-3) 2 h h Fig. 19b shows a node formed in an electrical circuit by the junction of two resistors. From Kirchhoff's current law, one obtains

i1+i2=0 or ei " e0 . e2 " e0 —R- + —R— - 31 -

A comparison of the equations (4 - 3) and (4 - 4) indicate that a typical node of fig, . 19b is analogous to a typical node of the finite difference grid if

The voltage e« at the node 0 is then proportional to y^, the dependent variable. Network analogues fox steady-state problems consist of only resistors» Unsteady-state problems may be represented by either pure resistance networks or resistance-capacitance networks. iy^l Network analogues j£pj_ steady ^ A network analogue describing a steady-state heat transfer phenomenon consists only of resistors (24) (75) (18) (21), Uniformly or non-uniformly distributed heat generation can be simulated by the introduction of flux at the network junctions,, Design of a simple analogue is described below (18). A fin attached to a wall is shown in fig. 20a» It is assumed that there is no heat transfer from the free end of the fin. The fin receives heat from the hot gas around it. The gas is kept at t C. The wall o *? temperature is t C. The resistance R~, representing the gas to surface heat transfer for the four sides of each fin segment, is given by •R ~ __JL_ (A _ K\ K { b) o" *opfh where

a~ ~ heat transfer coefficient from the gas to fin Pj. = perimeter of the fin segment receiving heat h = width of a fin segment receiving heat

The lumped thermal resistance R, between each node point in the fin is given by - 32 -

where

X.£ = thermal conductivity of fin material A,. = cross-sectional area of thin fin

From the equations (4 - 5) and (4 - 6) one obtains

Rn t \,A, u x x _ ', c :- g (gay) (4 _ 7}

In an experims ntal set up either Rn or TX. can be arbitrarily chosen. The value of the other resistance is given by equation (4 - 7). The value of B is determined by the geometrical and boundary con- ditions of the problem. The superimposed voltage from the voltage source will be directly proportional to the difference in temperature between the gas and the wall. Fig. 20b shows the network analogue for the problem shown in fig0 20a« Network analogues have been used very extensively to study steady-state heat transfer problems. Two interesting examples are cited below. Ellerbrock, Jr., Schum and Nachtigall (18) utilised the network analogue to study the temperature distribution in a cooled turbine blade. Three different network analogues were designed for the following blades;

1. 13-fin shell-supported air-cooled blade 2. strut supported air-cooled blade 3. liquid cooled blade.

The local values of temperatures obtained from the analogue were not in complete agreement with the values obtained by ana- lytical calculations. The differences, however, were not anything serious. In the midchord region experimental values obtained from an actual heat transfer experiment indicated that the analogue values were nearer truth. Lumping of resistances did not introduce any - 33 -

serious error in the result. Reducing the number of subdivisions in the blade-analogue from 55 to 31 elements changed the local temperature differences by about - 1 per cent. Even though the local temperatures obtained from the analogue and mathematical analysis showed some difference, the average temperature of the blade obtained by the two methods showed very good agreement, Fitts, Flemons and Rogers (21) designed a combined network and conducting sheet analogue to measure sheath temperatures of finned fuel elements,, Stainless steel sheet was used as the conducting medium. Lumped resistances were utilised to simulate the boundary film. Resistive elements were attached to the boundary of the conduc- ting sheet. Fitts et al, (21) mention that mechanical and material tolerances introduced an error of about 4 per cent» Error due to thermoelectric voltage was maximum 0. 1 7 C. Another possible source of error was due to variation of the resistance of the conductors with changing temperature. It was found to be negligible.

IV. 2 Advantages and disadvantages of resistance network analogues (55)

Advantages; 1. Any change in conductance can be easily accounted for by changing the particular resistors 2. All the measuring difficulties inherent in the electrolytic tank analogue are eliminated. There is no polarization effect 3. Accuracy is very high. A large number of resistor compo- nents gives rise to a statistical cancellation of errors 4. The problem, can be set up quickly 5. A great variety of problems can be tackled

Disadvantages: 1. The technique is cumbersome when applied to very compli- cated geometries 2, An error is introduced because of lumping. This disadvantage is not serious if the network is large enough - 34 -

IV. 3 Network analojgue_s_for_ unsteady-state_probl^ms_[RC>ne_two_rk)

In solving an unsteady state heat transfer problem account rmxst be taken of both thermal resistance and capacity. In a corresponding network analogue the thermal resistances and capacities are simulated by electrical resistors and capacitors respectively. The following table indicates the different quantities which are analogous in the two systems;

Table 4 - 1

Variable Thermal Electrical

Potential temperature voltage Flux rate of heat transfer current Resistance resistance to heat flow ohmic resistance Capacitance thermal capacity capacitors Time time time

The corresponding equations are of the following form:

-. . , potential difference across a resistance Resistance = *- r; flux _. .. net flux into a capacitance Capacitance = —. r—-?—r *—? 1—TT—T- time rate of change of potential In the electrical system it is necessary that the resistance be inde- pendent of the current flow through the resistor and that the capacitance be the same for all capacitor voltage levels. There should not be any cur- rent leakage from the capacitors. The network is not suitable for systems where thermal conductance and capacitance vary with temperature. Writing the equations governing the electrical and thermal problem in the fig. 21

(4-8)

de 2(e2 + e, - eQ) = ReC -^ (4 - 9) e - 35 -

where t = temperature e := voltage R= resistance C = capacitance T = time

Subscripts: 2. 0, 1 = node designations e = electrical t = thermal

Let T = D.7 (4 - tO)

t = A] e + Bj

q.. = cti where q» *s= heat flux i » current D,,, A-, B, and C. = scale factors

It can be readily shown that

R x Cz R C e e

(4 - 14)

The potential factors A. and B., can be chosen independently. The requirement of similarity places no restriction on these factors. All the quantities are chosen to get a suitable value of D., the scale factor for time. Practical values of C are less than 1 00 • 10 farad. Certain e amount of leakage of current to the ground throxxgh the capacitors is unavoidable. To make the ratio of the current through the analogue to the leakage current as large as possible the maximum value of R 36 -

and C make the transient in the analogue very fast for any manual re cor- ding. In almost all cases it is necessary to record the current flow auto- matically. Amperege through the analogue is very very low because of the high circuit resistance. Fast transients and low amperege necessitates the use of complicared measuring instruments. Paschkis (70) (71) (72) (73) has done an extensive amount of pioneering work on network analysers. Using a network analyser Paschkis (72) studied how the heat flow from the inside surface of a building wall changed when the outside surface temperature changed periodically and the inside air temperature was held constant. In many transient heat transfer problems the direction of heat flow does not change in the time period being considered. In this particular problem the direction of heat flow changed in each cycle. The temperature - time curve for the outer-surface swung periodically around a zero line, e.g. the constant room temperature. Fig. 22a is a sketch of the heat transfer problem. The network analogue for the problem is shown in the fig, 22b. The potential factors were chosen in such a way that the daily cycle of 24 hours was represented by 768 seconds in the ana- logue. The voltage corresponding to the outside surface temperature was changed stepwise after every 4 seconds in the analogue, 4 seconds in the analogue represented 15 minutes in the actual problem. The advantages of using this analogue were:

1. Elimination of building of actual walls 2. Considerably shorter time for experiments 3. Elimination of the dependence on weather 4. Possibility of maintaining constant air temperatures 5. Possibility of investigating separately the influence of the different variables (e.g. film conductance, etc.),

Paschkis and Baker (73) utilised the network analogue to study the temperature distribution in the insulation round a steam pipe line as a func- tion of time for a given time cycle of turning on and off the steam flow. Com- parisons with actual experimental data indicated an excellent agreement, - 37 -

Paschkis and Baker (73) also described a large permanent electric model for general use. It consisted of 525 condensers di- vided into 15 equal groups. The condensers ranged in size ffom 0.1 uf to 20 jif. All condensers were accurate to within .1 1 per cent. All condensers had at least } 5, 000 megohms per microfarad insu- lation resistance* The resistor boards contained four decade re- sistors each, which together yielded any resistance value from 100 ohms to 1, 111, 000 ohms in steps of 100 ohms.

V. Comparison of analogues

Type of analogue Advantages Dis advantage s Hydraulic Cheap equipment, Limited accuracy, easy technique, very tedious calibration suitable from heuris- of capillaries tic point of view* apply to transient conditions

Membrane Cheap equipment, Limited accuracy, applicable to comp- measuring difficulties, licated geometries aging of membrane, very sensitive to am- bient conditions

Electrolytic tank Applicable to compli- Difficult measuring cated geometries technique, limited accuracy

Conducting sheet Cheap equipment, Limited accuracy, easy technique, app- scale distortion, licable to complica- 2 dimension only ted geometries, not sensitive to ambient conditions - 38 -

Type of analogue Advantages Disadvantages

Resistance network High accuracy, easy Cumbersome when technique, applicable applied to complicated to mathematically geometries and un- more complicated steady-state problems problems

RC-networks Apply to transient Limited accuracy, spe- conditions, highly cialized applications, flexible require simple geometries, expensive components and measuring instruments

Note: In the above mentioned table the analogue computers have not been con- sidered. Only direct simulators have been duscussed. Unlike RC-networks analogue computers can steer a system as in an on-line analogue computer. Components lika operational amplifiers in analogue computers do not have physical counterparts. Analogue computers are extremely useful and flexible tools for studying steady and transient phenomena. - 39 -

Acknowledgement

The author takes this opportunity to thank Professor Bo Pierre of the Royal Institute of Technology, Stockholm, for his kind help and keen interest in this work. The author is also indebted to the post- graduate students of the Refrigeration and Applied Thermodynamics Department for their kind help in procuring some of the references» The support given by the Reactor Technology Section of AB Atomenergi, Stockholm, is also gratefully acknowledged. - 40 -

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75. ROHSENOW, W M and CHOI, H Y Heat, mass and momentum transfer Englew. Cliffs N. J. Prentice-Hall 1961, 444-68.

76. SCHNEIDER, P J The Prandtl membrane analogy for temperature fields with permanent heat sources or sinks J. Aeronaut. Sci. 19 (1952) 644.

77. SCHNEIDER, P J and CAMBEL, A B Membrane apparatus for analogic experiments Rev. Sci. Instr. 24 (1953) 513-14.

78. SCHWARTZ, C H, GOLDBERG, S A and ORNING, A A The use of visible light models for the study of radiant heat transfer in furnaces Trans. ASME, Ser. A, J. Eng. Power 84 (1962) 358-64.

79. STEPHENSON, D G Methods of determining non- steady- state heat flow through walls and roofs of buildings J. Inst. Heating Ventilating Eng. 30 (1962) 64-73. - 47 -

80. SHORTLEY, G H and WELLER, R The numerical solution of Laplace 's equation Appl. Phys. 9 (1938) 334-48,

81. WILSON, LH and MILES, A J Application of the membrane analogy to the solution of heat-conduction problems J. Appl. Phys. 21 (1950) 532-35. direction of insulation heat flow

Fig. 1 HEATING OF A ROD

initial water level

Fig. 2 a. STAND PIPES WITH INITIAL LEVEL OF WATER

X Fig. 2,b. WATER LEVEL IN STANDPIPES AFTER A TIME INTERVAL

Fig. 3 a. COLD STORAGE ROOM

B Body placed in a cold storage room R Cold storage room C Cooling coils S Surroundings B start

R -8 C stop C

iF

-35°C

FIg. 3 b. HYDRAULIC MODEL FOR A COLD STORAGE ROOM

Code for fig. 3 b

B Body placed in a cold storage room R Cold "storage room C Cooling coils S Surroundings t. Initial temperature of the body t Initial temperature of the room t-, Temperature of the cooling coil

The required resistances in the capillary tubes joining the different standpipes are calculated by using the following relation,

R i Ki c a Wk k k k wherej time factor between the actual periodic phenomena and ,, , , seconds - actual the model, seconds - model Z area of a standpipe, m wk thermal capacity of a body , j/ C K k total heat release from a body per C, j/S C (H) 20 % total source energy (test constant) heats ceiling surface substantially in- (F) Thermal storage dependent of surface temperature. \

'(C) Convec Ö (J) Air supply vol. tion ceiling (K) Air supply tem^r to air Radiation leaving lamp o (Ö fl (A) 32, 9 % total source energy to air (D) Convection by convection and absorption of floor to r.-adiation by CO- and water vapour. air

V (B ) 47. 1 % total source energy heats (G) Thermal floor surface substantially in- storage dependent of surface temperature.

Fig. 4 ANALYSIS OF ROOM LOAD K

Fig. 5. DIAGRAMMATIC SKETCH OF ANALOGUE SIMULATING L CONDITIONS OF FIG. 4. Instantaneous rate of heat gain .

12 4 8 10 12 2 4 6 8 10 12 AM Time of day PM

Fig. 6. RESULTS FROM AN ANALOGUE 1 Btu/hr. sq. ft. = 3. 15 2 Tube 1 steam Saturation temperature Starting level 1 level reservoir Sat n temp, level Tube 2 Room temp, ilevel Template

Starting level 2 Capillary Capillary e = o 9 = 9 Time "(Area) (y) = 1 Total (y)=0 water out Capillary Steam inl

water in steam out

Fig. 7 DIAGRAMMATIC LAYOUT OF A PARALLEL FLOW ANALOGUE. At e = e i A 2 A 3 A H4 A

3 B 2 B 1 B direction of motion

After 6 =

1 A K2 A 3 A 5 B 4 B Hh B 2 B 1 B After e =

T1 2 A 3 A 4 A B 1 5 B 4 B 3 B 2 lH I B

After e =Ae3+Ae2+Ae1+e0-

1 A 2 A 3 A 4 A

5 B 4 B 3 B 2 B

Fig. 8 BLOCK DIAGRAM FOR TRANSIENT HEATING OF A BODY A WITH A LIQUID B FLOWING OVER BODY A. A v\ liquid level h applied pressure change datum pressure p

overflow

A B

r T | " ! -8- i T r • i • 1 • 1 . 1 1 i . 1 . i . i • 1 1 i § I | i 1 A pumps

Fig. 9 A HEAT FLOW PROBLEM AND ITS AIR-FLOW ANALOGUE. 9i

X

Fig. 10. CURRENT BALANCE ON A CONDUCTING ELEMENT

B

FIG. 11. ELECTROLYTIC TANK ANALOGUE

A. Laminated wood board. B. Wooden laths. C. Electrolyte (water). D. Copper bars. E. Wooden laths with slots for copper bar D. F, Copper frame model. G. Avometer, 3,2 800

600

• r-i O ni & O 200

0 250 500 750 1000 1250 Frequency, cps

FIG. 12. RESISTANCE AND CAPACITANCE COMPONENTS OF POLARIZATION FOR DISTILLED WATER AND POLISHED BRASS ELECTRODES copper bar

ext cold storage chamber wall insulation metal electrolyte frame 6. i

copper bar

plating on the ship's side

Fig. 13. SKETCHES OF A LONGITUDINAL SECTION OF WALL OF COLD STORAGE CHAMBER ON BOARD SHIP AND ITS TEST MODEL Emitter (INDIUM)

Base (GERMANIUM)

A Collector (INDIUM)

Support (COPPER) Clamped joint with interface resistance Chassis (ALUMINIUM)

Fig. 14 a. A TRANSISTOR CONFIGURATION (Heat generation along interface AA , and heat sink along BB )

Lucite tray Electrolyte

Gold-plated copper strips

Electrolyte

Fig. 14 b. ANALOGUE INTERFACE '100 Volts AC

Glass tube z-axis

Electrodes Cathode spot

Insulating wall

Surface of Mercyry

Sloping bottom

Axis of Mercury revolution

Fig. 15. SKETCH OF A MERCURY VAPOUR RECTIFIER AND ITS ELECTROLYTIC ANALOGUE

Boundary Electrodes

DUCT

T Probe

Voltage Divider

Fig. 16. ELECTRICAL FIELD ANALOGUE OF HEAT TRANSFER THROUGH CORNER OF THICK-WALLED DUCT line of symmetry

cement plaster facing

concrete

insulation

concrete

17. CROSS SECTION OF BUILDING STRUCTURE line of symmetry

esh (insulation)

-AV. (air boundary) , (concrete floor)

(concrete slab) (air boundary)

e probe

Fig. 18. CONDUCTING SHEET ANALOGUE h h

R R e.

TT I , . ,_ 2 0 1

h (a) (b) Fig. 19. A DISCRETIZED SYSTEM AND ITS NETWORK ANALOGUE

Gas temp, t C

s s < i adiabatic r -T surface ]

Wall temp. t °C \ / w

(a) Fin illustration

-~LI Voltage source

Gas temp. t,.oC

<=fe^ ..,. ,i R measuring probe

Wall temp. t °c w S LHi?_J.

h (b) Analogue wiring diagram

Fig. 20. ANALOGUE WIRING DIAGRAM FOR SIMPLE CASE OF FIN ATTACHED TO WALL R/2 R/2

i

Fig. 21. A GENERAL NETWORK ELEMENT

P. S. = power supply S. S. = selector switch taking any voltage from + 100 % to zero So = switch taking voltage for constant room air tem- perature M = model, consisting of R resistors, and C condensers

Fig. 22 b. CIRCUIT DIAGRAM

concrete

plaster Outside surface room (constant temp, ) (periodic \ variation \ in temp.)

brick

Fig» 22 a. DIAGRAM OF A WALL

LIST OF PUBLISHED AE-REPORTS 169. Concentration of 17 elements in subcellutar fractions of beef heart tissue determined by neutron activation analysis. By P. O. Wester. 1964. 29 p. 1—130. (See the back cover earlier reports.) Sw. cr. 8:—. 131. Measurements of hydrodynamic instabilities, flow oscillations and bur 170. Formation of nitrogen-13, fluorine-17, and fluorine-18 in reac'or-irradiated nout in a natural cirkulation loop. By K. M. Becker, R. P. Mathisen, O. H2O and D2O and applications to activation analysis and fast neutron Eklind and B. Norman. 1964. 21 p. Sw. cr. 8:—. flux monitoring. By L. Hammar and S. Forsen. 1964. 25 p. Sw. cr. 8:—. 132. A neutron rem counter. By 1. O. Andersson and J. Braun. 1964. 14 p. 171. Measurements on background and fall-out radioactivity in samples from Sw. cr. 8:—. the Baltic bay of Tvären, 1957—1963. By P. O. Agnedal. 1965. 48 p. Sw. 133. Studies of water by scattering of slow neutrons. By K. Sköld, E. Pilcher cr. 8:—. and K. E. Larsson. 1964. 17 p. Sw. cr. 8:—. 172. Recoil reactions in neutron-activation analysis. By D. Brune. 1965. 24 p. 134. The amounts of As, Au, Br, Cu, Fe, Mo, Se, and Zn in normal and urae- Sw. cr. 8:—. mic human whole blood. A comparison by means neutron activation 173. A parametric study of a constanl-Mach-number MHD generator with analysis. By D. Brune, K. Samsahl and P. O. Wester. 1964. 10 p. Sw. cr. nuclear ionization. By J. Braun. 1965. 23 p. Sw. cr. 8:—. 8:—. 174. Improvements in applied gamma-ray spectrometry with germanium semi- 135. A Monte Carlo method för the analysis of gamma radiation transport conductor dector. By D. Brune, J. Dubois and S. Hellström. 1965. 17 p. from distributed sources in laminated shields. By M. Leimdörfer. 1964. Sw. cr. 8:—. 28 p. Sw. cr. 8:—. 136. Election of uranium atoms from UO2 by fission fragments. By G. Nilsson. 175. Analysis of linear MHD power generators. By E. A. Witalis. 1965. 37 p. 1964. 38 p. Sw. cr. 8:—. Sw. cr. 8:—. 137. Personell neutron monitoring at AB Atomenergi. By S. Hagsgård and 176. Effect of buoyancy on forced convection heat transfer in vertical chann- C.-O. Widell. 1964. 11 p. Sw. cr. 8:—. els — a literature survey. By A. Bhattacharyya. 1965. 27 p. Sw. cr. 8:—. 138. Radiation induced precipitation in iron. By B. Solly. 1964. 8 p. Sw. cr. 177. Burnout data for flow of boiling water in vertical round ducts, annuli 8:—. and rod clusters. By K. M. Becker, G. Hernborg, M. Bode and O. Erik- 139. Angular distributions of neutrons from (p, n)-reactions in some mirror son. 1965. 109 p. Sw. cr. 8:—. nuclei. By L. G. Strömberg, T. Wiedling and B. Holmqvist. 1964. 28 p. 178. An analytical and experimental study of burnout conditions in vertical Sw. cr. 8:—. round ducts. By K. M. Becker. 1965. 161 p. Sw. cr. 8:—. 179. Hindered El transitions in Eu'« and Tb'". By S. G. Malmskog. 1965. 19 p. 140. An extended Greuling-Goertzel approximation with a Pn -approximation in the angular dependence. By R. Håkansson. 1964. 21 p. Sw. cr. 8:—. Sw. cr. 8:—. v 141. Heat transfer and pressure drop with rough surfaces, a literature survey. 180. Photomultiplier tubes for low level Cerenkov detectors. By O. Strinde- By A. Bhattachayya. 1964. 78 p. Sw. cr. 8:—. hag. 1965. 25 p. Sw. cr. 8:—. 142. Radiolysis of aqueous benzene solutions. By H. Christensen. 1964. 50 p. 181. Studies of the fission integrals of U235 and Pu239 with cadmium and Sw. cr. 8:—. boron filters. By E. Hellstrand. 1965. 32 p. Sw. cr. 8:—. 143. Cross section measurements for some elements suited as thermal spect- 182. The handling of liquid waste at the research station of Studsvik,Sweden. rum indicators: Cd, Sm, Gd and Lu. By E. Sokolowski, H. Pekarek and By S. Lindhe and P. Linder. 1965. 18 p. Sw. cr. 8:—. E. Jonsson. 1964. 27 p. Sw. cr. 8:—. 183. Mechanical and instrumental experiences from the erection, commis- 144. A direction sensitive fast neutron monitor. By B. Antolkovic, B. Holm- sioning and operation of a small pilot plant for development work on qvist and T. Wiedling. 1964. 14 p. Sw. cr. 8:—. aqueous reprocessing of nuclear fuels. By K. Jönsson. 1965. 21 p. Sw. 145. A user's manual for the NRN shield design method. By L. Hiärne. 1964. cr. 8:—. 107 p. Sw. cr. 10:—. 184. Energy dependent removal cross-sections in fast neutron shielding 146. Concentration of 24 trace elements in human heart tissue determined theory. By H. Grönroos. 1965. 75 p. Sw. cr. 8:—. by neutron activation analysis. By P. O. Wester. 1964. 33 p. Sw. cr. 8:—. 185. A new method for predicting the penetration and slowing-down of 147. Report on the personnel Dosimetry at AB Atomenergi during 1963. By neutrons in reactor shields. By L. Hjärne and M. Leimdörfer. 1965. 21 p. K.-A. Edvardsson and S. Hagsgård. 1964. 16 p. Sw. cr. 8:—. Sw. cr. 8:—. 148. A calculation of the angular moments of the kernel for a monatomic gas 186. An electron microscope study of the thermal neutron induced loss in scatterer. By R. Håkansson. 1964. 16 p. Sw. cr. 8:—. high temperature tensile ductility of Nb stabilized austenitic steels. 149. An anion-exchange method for the separation of P-32 activity in neu- By R. B. Roy. 1965. 15 p. Sw. cr. 8:—. tron-irradited biological material. By K. Samsahl. 1964. 10 p. Sw. cr 187. The non-destructive determination of burn-up means of the Pr-144 2.18 8:—. MeV gamma activity. By R. S. Forsylh and W. H. Blackadder. 1965. 150. Inelastic neutron scattering cross sections of Cu'" and Cu*5 in the energy 22 p. Sw. cr. 8:—. region 0.7 to 1.4 MeV. By B. Holmqvist and T. Wiedling. 1964. 30 p. 188. Trace elements in human myocardial infarction determined by neutron Sw. cr. 8:—. activation analysis. By P. O. Wester. 1965. 34 p. Sw. cr. 8:—. 151. Determination of magnesium in needle biopsy samples of muscle tissue 189. An electromagnet for precession of the polarization of fast-neutrons. by means of neutron activation analysis. By D. Brune and H. E. Sjöberg. By O. Aspelund, J. Bäckman and G. Trumpy. 1965. 28 p. Sw. cr. 8:—. 1964. 8 p. Sw. cr. 8:—. 190. On the use of importance sampling in particle transport problems. By 152. Absolute El transition probabilities in the dofermed nuclei Yb"; and B. Eriksson. 1965. 27 p. Sw. cr. 8:—. Hfi". By Sven G. Malmskog. 1964. 21 p. Sw. cr. 8:—. 191. Trace elements in the conductive tissue of beef heart determined by 153. Measurements of burnout conditions for flow of boiling water in vertical neutron activation analysis. By P. O. Wester. 1965. 19 p. Sw. cr. 8:—. 3-rod and 7-rod clusters. By K. M Becker, G. Hernborg and J. E. Flinta. 192. Radiolysis of aqueous benzene solutions in the presence of inorganic 1964. 54 p. Sw. cr. 8:—. oxides. By H. Christensen. 12 p. 1965. Sw. cr. 8r-. 154. Integral parameters of the thermal neutron scattering law. By S. N. 193. Radiolysis of aqueous benzene solutions at higher temperatures. By PuroTiit. 1964. 48 p. Sw. cr. 8:—. H. Christensen. 1965. 14 p. Sw. cr. 8:—. 155. Tests of neutron spectrum calculations with the help of foil measurments 194. Theoretical work for the fast zero-power reactor FR-0. By H. Häggblom. 1965. 46 p. Sw. cr. 8:—. in a D2O and in an H^O-moderated reactor and in reactor shields of concrete and iron. By R. Nilsson and E. Aalto. 1964. 23 p. Sw. cr. 8:—. 195. Experimental studies on assemblies 1 and 2 of the fast reactor FRO. 156. Hydrodynamic instability and dynamic burnout in natural circulation Part 1. By T. L. Andersson, E. Hellstrand, S-O. Londen and L. I. Tirén. two-phase flow. An experimental and theoretical study. By K. M. Beck- 1965. 45 p. Sw. cr. 8:—. er. 5. Jahnberg, I. Haga, P. T. Hansson and R. P. Mathisen. 1964. 41 p. 196. Measured and predicted variations in fast neutron spectrum when pene- Sw. cr. 8s—. trating laminated Fe-D2O. By E. Aalto, R. Sandlin and R. Fräkr. 1965. 157. Measurements of neutron and gamma attenuation in massive laminated 20 p. Sw. cr. 8:—. shields of concrete and a study of the accuracy of some methods of 197. Measured and predicted variations in fast neutron spectrum in massive calculation. By E. Aalto and R. Nilsson. 1964. 110 p. Sw. cr. 10:—. shields of water and concrete. By E. Aalto, R. Fräki and R. Sandlin. 1965. 158. A study of the angular distributions of neutrons from the Be* (p,n) B' 27 p. Sw. cr. 8:—. reaction at low proton energies. By. B. Antolkovic', B. Holmqvist and 198. Measured and predicted neutron fluxes in, and leakage through, a con- T. Wiedling. 1964. 19 p. Sw. cr. 8:—. figuration of perforated Fe plates in D2O. By E. Aalto. 1965. 23 p. Sw. 159. A simple apparatus for fast ion exchange separations. By K. Samsahl. cr. 8:—. 1964. 15 p. Sw. cr. 8r- 199. Mixed convection heat transfer on the outside of a vertical cylinder. 160. Measurements of the Fes< (n, p) Mn5* reaction cross section in the neutron By A. Bhattacharyya. 1965. 42 p. Sw. cr. 8:—. energy range 2.3—3.8 MeV. By A. Lauber and S. Malmskog. 1964. 13 p. 200. An experimental study of natural circulation in a loop with parallel Sw. cr. 8:—. flow test sections. By R. P. Mathisen and O. Eklind. 1965. 47 p. Sw. 161. Comparisons of measured and calculated neutron fluxes in laminated cr. 8:—. iron and heavy water. By. E. Aalto. 1964. 15. p. Sw. cr. 8:—. 201. Heat transfer analogies. By A. Bhattacharyya. 1965. 55 p. Sw. cr. 8:—. 162. A needle-type p-i-n junction semiconductor detector for in-vivo measure- ment of beta tracer activity. By A. Lauber and B. Rosencrantz. 1964. 12 p. Sw. er. 8:—. Förteckning över publicerade AES-rapporter 163. Flame spectro photometric determination of strontium in water and biological material. By G. Jönsson. 1964. 12 p. Sw. cr. 8:—. 1. Analys medelst gamma-spektrometri. Av D. Brune. 1961. 10 s. Kr 6:—. 164. The solution of a velocity-dependent slowing-down problem using case's 2. Beslrålningsförändringar och neutronatmosfär i reaktortrycktankar — eigenfunction expansion. By A. Claesson. 1964. 16 p. Sw. cr. 8:—. några synpunkter. Av M. Grounes. 1962. 33 s. Kr 6:—. 165. Measurements of the effects of spacers on the burnout conditions for 3. Studium av sträckgränsen i mjukt stål. Av G. Ostberg och R. Attermo. flow of boiling water in a vertical annulus and a vertical 7-rod cluster. 1963. 17 s. Kr 6:—. By K. M. Becker and G. Hemberg. 1964. 15 p. Sw. cr. 8:—. 4. Teknisk upphandling inom reaktorområdet. Av Erik Jonson. 1963. 64 s. 166. The transmission of thermal and fast neutrons in air filled annular ducts Kr 8:—. through slabs of iron and heavy water. By J. Nilsson and R. Sandlin. 1964. 33 p. Sw. cr. 8:—. 5. Ågesta Kraftvårmeverk. Sammanställning av tekniska data, beskrivningar 167. The radio-thermoluminescense of CaSO^: Sm and its use in dosimetry. m. m. för reaktordelen. Av B. Lilliehöök. 1964. 336 s. Kr 15:—. By B. Bjärngard. 1964. 31 p. Sw. cr. 8:—. 168. A fast radiochemical method for the determination of some essential Additional copies available at the library of AB Atomenergi, Studsvik, trace elements in biology and medicine. By K. Samsahl. 1964. 12 p. Sw. Nyköping, Sweden. Transparent microcards of the reports are obtainable cr. 8:—. through the International Documentation Center, Tumba, Sweden.

EOS-tryckerierna, Stockholm 1955