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ANALOGUES
CHAPTER 10
ANALOGUES 10.1 General One must distinguish between direct and indirect analogues or analogue computers. In a direct analogue the analogous variable has the same significance everywhere within the analogue system, while in an indirect analogue this need not be so and solutions are obtained with the help of a mathematical model. A direct analogue to one-dimensional heat flow can be an electrical network of capacitances and resistances in which there is a direct analogy between corresponding points in the thermal and in the electrical system. In an example of an indirect analogue the equation of heat conduction and the boundary conditions are transformed into finite difference equations which are the mathematical model to be solved by an analogue. Indirect analogues may range from simple devices for solving particular mathematical problems to general purpose computers. Only some examples of the former type of indirect analogues are dealt with here. Analogues can further be divided into continuous and discrete types; in the first, continua such as metal sheets are used, and in the second type lumped elements such as resistances and capacitances are used. Among the many possible systems, only mechanical and electrical systems have been used to any extent in heat flow problems. Analogue systems become much simpler if the relation between the rates of heat flow and the temperature differences is linear. The scale of analogous quantities can be chosen at will and this is done from certain practical points of view. For direct analogues the similarity laws to be discussed later on in Chapter 11 are automatically satisfied and one need not consider them specifically, except in cases of transient heat flow, where the time as a physical quantity of the original problem is usually retained in the analogue. The time scales in the original problems and in their analogues are in many cases different, but this is of no real significance. 282
ANALOGUES
283
10.2 Steady heat flow There is not much interest in analogues for one-dimensional heat flow, since calculations are easier in these cases. For two-dimensional heat flows, analytical and numerical solutions often become difficult because of the shape of the boundaries. Steady heatflowby conduction is governed by the Laplace equation which is important in many branches of physics and engineering. A well-known mechanical analogue of the continuous type consists of the viscous flow of a liquid between two plates; if the motion is sufficiently slow, the rate of mass flow corresponds to the rate of heat flow and the pressure to the temperature (Ref. 10.4). The streamlines of the flow can be made visible by introducing dyes; lines of constant pressure (corresponding to isothermals) are orthogonal to the streamlines. Boundaries of any shape can be placed in theflowfield.Even heat flow with internal sources can be dealt with by this method, if there are sources of liquid emanating from the walls (Ref. 10.5). In an electrical analogue of the continuous type one uses electrically conducting papers or an electrolytic tank. Electrically conducting papers of constant thickness are available for various values of the specific resistance. The boundaries are obtained by cutting the paper to the desired form and the external heat transfer conditions there can be represented by suitable resistances between the boundaries and the reference points for the temperatures of the external medium. The voltage at any point is analogous to the temperature, and the electrical current density to the density of the heat flow. Lines of constant voltage can be found with the help of a voltmeter and a contact sliding on the surface of the paper. The density of the heat flow is obtained from the voltage gradient in the same direction and from the resistivity R of the sheet, which is the resistance between two parallel edges of a square and is independent of the size of the square, i.e. 1 dV <■ —
*
*
<1
where n is a distance in a direction characterized by the subscript n. Composite bodies can be simulated by using sheets of different thicknesses. Heat sources can be represented by local supplies of electrical current. In an electrolytic tank the metal sheet is replaced by a conducting liquid, otherwise the arrangement is similar except that alternating currents must be used to avoid polarization of the liquid. An electrical analogue of the discrete type consists of networks of resistances where each point corresponds to a point in the heat flow field and the resistances between the mesh-points correspond to the specific resistance of the field.
284
HEAT TRANSFER IN STRUCTURES
10.3 Transient heat flow A mechanical analogue for transient heat flow by conduction which uses liquid flow in tubes and capillaries is known by the name of Hydrocal (Ref. 10.6). This analogue is of the discrete type with vertical tubes representing heat reservoirs and capillaries representing thermal resistances. The rate of fluid flow between two vertical tubes is Qn-l,n = Kn-lAHn-H^)
(10.2)
where Hn-.1 and Hn are the heights of the fluid levels in the tubes n—\ and n and Kn^liH is the reciprocal of the resistance for the fluid flow in the capillary. The equation of motion for a system of tubes as in Fig. 10.1 is Wm
dH, - — Kn,n-l(Hn-l—Hn)+Kn,n+l(Hn+i àt
— Hn)
(10.3)
where the subscript n refers to the nth tube, /£„,„_! is the reciprocal of the capillary resistance between the tube n and n—l and Wn is the cross-section of area of the nth tube. ï I
"h Π
H, T L
/ h-K 12
1LLJ -4*-K -*-| \
1
Hn
U
23
FIG. 10.1. Afluidflowanalogue of the discrete type ("Hydrocal"). Obviously Eq. (10.3) is of the same form as Eq. (8.5.1), if Wn, -£„,„-! and Hn are the quantities analogous to the heat capacities, conductances and temperatures. In the equipment for this mechanical analogue, each tube has a tap so that initially the level of the liquid in the tube can be adjusted to correspond to the initial temperature of the corresponding heat reservoir. An additional tap common to all tubes is often arranged so that the flow of the liquid can be stopped at any moment for reading the instantaneous liquid levels. Boundary conditions corresponding to heat transfer by forced convection at the surfaces can be satisfied by keeping the end tubes at a level corresponding to the temperature of the external medium and connecting these tubes to neighbouring ones by a capillary whose flow resistance is equal to the reciprocal of the external heat transfer coefficient. If the temperature of the external medium is
ANALOGUES
285
constant, the liquid level in the corresponding tube is kept constant by using a tank with a large cross-section instead of a tube. The analogue can be used for complex bodies and for non-linear problems which arise in the case of thermal properties varying with temperature, of radiative heat transfer at the surfaces and so on. Extensions are also possible to two- and three-dimensional heat flow and even to ablation problems, but then the equipment becomes more complicated. An application of this mechanical analogue to a problem of heat flow in structures is given in Ref. 10.7. For linear problems electrical analogues are easier to handle and more flexible than mechanical ones. A direct analogue of the discrete type can be constructed with the help of electrical resistances and capacitances and such an arrangement for one-dimensional heatflowis shown in Fig. 10.2.
FIG. 10.2. An example of an electrical analogue of the direct and discrete type for one-dimensional heat flow with capacitors and resistors.
Heat reservoirs correspond to the capacities and thermal resistances to the electrical resistances as follows by comparison with Fig. 8.10. Further, the voltage at each point corresponds to a temperature and the electrical current between two capacitances to a heat flow, both quantities referring to analogous points in the two systems. If the initial temperature distributions are arbitrary, the points 0,1,..., Ν+1 must initially be connected to electrical sources, so that an analogous initial distribution of the voltages at these points is obtained; at the start of heat transfer these connections are broken unless there are internal heat sources. As mentioned in Section 10.1
286
HEAT TRANSFER IN STRUCTURES
the time scale can be chosen at will and there are two practical possibilities. The time constants can be made large, so that all changes occur slowly and can be read directly on instruments or registered by electromechanical devices (plotters) : or the time constants are small, so that the whole process can be repeated many times per second and may be displayed as a standing pattern on a cathode ray tube. The advantage of the last method lies in the possibility of using smaller capacitances and making all changes of the parameters visible immediately; the disadvantage is the greater complexity of the equipment. The solution of non-linear problems requires more elaborate equipment. For details see Ref. 10.1. An example of an indirect analogue is a method due to Liebmann, Ref. 8.12. The starting point is the implicit finite difference equation Eq. (8.7.1) which re-written with voltages instead of temperatures becomes: Vn.m-V^ =p(Vn_ltm-2Vntm+Vn+lim\ (10.4) In contrast to Chapter 8 this equation is used here in a step-by-step procedure where the voltages at the time t = (m—1) At and at all places x = ηΔχ, (n = 1,..JV) are known and the corresponding voltages at t= mAt are sought. The analogue for solving Eq. (10.4) is the circuit of Fig. 10.3; the balance of the electrical currents at node n corresponds to Eq. (10.4) if all nodes are kept at the voltages indicated in Fig. 10.3 and if RP = pRx (10.5) Rx can be chosen at will. The computer based on this circuit is shown in Fig. 10.4. The voltages VniYn_x already known are adjusted by sliding resistances R'0, R'l9 ... R'n and then the unknown voltages can be read by connecting a voltmeter to the points 1,2, . . . , « , one at a time. The resistances Rol, ... and RPtl··. have to be large compared with the resistances R' so that the voltages V0m_l9 Vi,m-v ··· depend only on the position of the resistances R'. In order to avoid adjustments by hand, servo-mechanisms and automatic recording units have been proposed. As equipment of this type becomes more elaborate, it would become a special-purpose analogue-computer. Vn-,m
n-l
Vnifn
Rx
|
n
Vn +
Rx
I|fn
n+l
iVn.m-l
FIG. 10.3. The basic circuit for an electrical analogue of the indirect and discrete type due to Liebmann (Ref. 8.12).
ANALOGUES
287
It depends on the circumstances whether a direct analogue, an analogue computer or a digital computer is preferable for solving a given problem. For steady heat flow problems with linear relations between the rate of
Generotor FIG. 10.4. A computer based on the basic circuit of Fig. 10.3.
heat flow and the temperature differences, direct analogues can be easily constructed, often largely with standard laboratory equipment. For moderate accuracies these analogues are useful and inexpensive, particularly for two-dimensional cases. Analogues are valuable for transient problems, when parameters in a design have to be optimized, and when a large number of temperature distributions of a similar type have to be found. If available, the general purpose digital computer is best where moderate or small amounts of work of varying types are involved. It also has great advantages whenever "non-linear" boundary conditions or similar complications occur. References General : 10.1 KARPLUS, W. J., Analog Simulation, MacGraw-Hill, New York (1958). 10.2 KARPLUS, W. J. and SOROKA, W. W., Analog Methods—Computation and Simulation, MacGraw-Hill, New York (1959). 10.3 SOROKA, W. W., Simulation in science and technology. Appl. Mech. Reviews 13, 9, 621-623 (1960). Particular: 10.4 HELE-SHAW, H. S., The flow of water, Nature 58, 34-36 (1898), The motion of a perfect liquid, Proc. Roy. Inst. 16, 49-64 (1899). 10.5 MOORE, A. D., Fields from fluid flow mappers, /. Appl. Phys. 20, 790-804 (1949). 10.6 MOORE, A. D., The hydrocal, Ind. Engng. Chem. 28, 704-708 (1936). 10.7 KNUTH, E. L. and KUMM, E. L., Application of hydraulic analog method to one-dimensional transient heat flow, Jet Prop. 26, 649-654, 659 (1956).
ANALOGUES 10.1 General One must distinguish between direct and indirect analogues or analogue computers. In a direct analogue the analogous variable has the same significance everywhere within the analogue system, while in an indirect analogue this need not be so and solutions are obtained with the help of a mathematical model. A direct analogue to one-dimensional heat flow can be an electrical network of capacitances and resistances in which there is a direct analogy between corresponding points in the thermal and in the electrical system. In an example of an indirect analogue the equation of heat conduction and the boundary conditions are transformed into finite difference equations which are the mathematical model to be solved by an analogue. Indirect analogues may range from simple devices for solving particular mathematical problems to general purpose computers. Only some examples of the former type of indirect analogues are dealt with here. Analogues can further be divided into continuous and discrete types; in the first, continua such as metal sheets are used, and in the second type lumped elements such as resistances and capacitances are used. Among the many possible systems, only mechanical and electrical systems have been used to any extent in heat flow problems. Analogue systems become much simpler if the relation between the rates of heat flow and the temperature differences is linear. The scale of analogous quantities can be chosen at will and this is done from certain practical points of view. For direct analogues the similarity laws to be discussed later on in Chapter 11 are automatically satisfied and one need not consider them specifically, except in cases of transient heat flow, where the time as a physical quantity of the original problem is usually retained in the analogue. The time scales in the original problems and in their analogues are in many cases different, but this is of no real significance. 282
ANALOGUES
283
10.2 Steady heat flow There is not much interest in analogues for one-dimensional heat flow, since calculations are easier in these cases. For two-dimensional heat flows, analytical and numerical solutions often become difficult because of the shape of the boundaries. Steady heatflowby conduction is governed by the Laplace equation which is important in many branches of physics and engineering. A well-known mechanical analogue of the continuous type consists of the viscous flow of a liquid between two plates; if the motion is sufficiently slow, the rate of mass flow corresponds to the rate of heat flow and the pressure to the temperature (Ref. 10.4). The streamlines of the flow can be made visible by introducing dyes; lines of constant pressure (corresponding to isothermals) are orthogonal to the streamlines. Boundaries of any shape can be placed in theflowfield.Even heat flow with internal sources can be dealt with by this method, if there are sources of liquid emanating from the walls (Ref. 10.5). In an electrical analogue of the continuous type one uses electrically conducting papers or an electrolytic tank. Electrically conducting papers of constant thickness are available for various values of the specific resistance. The boundaries are obtained by cutting the paper to the desired form and the external heat transfer conditions there can be represented by suitable resistances between the boundaries and the reference points for the temperatures of the external medium. The voltage at any point is analogous to the temperature, and the electrical current density to the density of the heat flow. Lines of constant voltage can be found with the help of a voltmeter and a contact sliding on the surface of the paper. The density of the heat flow is obtained from the voltage gradient in the same direction and from the resistivity R of the sheet, which is the resistance between two parallel edges of a square and is independent of the size of the square, i.e. 1 dV <■ —
*
*
<1
where n is a distance in a direction characterized by the subscript n. Composite bodies can be simulated by using sheets of different thicknesses. Heat sources can be represented by local supplies of electrical current. In an electrolytic tank the metal sheet is replaced by a conducting liquid, otherwise the arrangement is similar except that alternating currents must be used to avoid polarization of the liquid. An electrical analogue of the discrete type consists of networks of resistances where each point corresponds to a point in the heat flow field and the resistances between the mesh-points correspond to the specific resistance of the field.
284
HEAT TRANSFER IN STRUCTURES
10.3 Transient heat flow A mechanical analogue for transient heat flow by conduction which uses liquid flow in tubes and capillaries is known by the name of Hydrocal (Ref. 10.6). This analogue is of the discrete type with vertical tubes representing heat reservoirs and capillaries representing thermal resistances. The rate of fluid flow between two vertical tubes is Qn-l,n = Kn-lAHn-H^)
(10.2)
where Hn-.1 and Hn are the heights of the fluid levels in the tubes n—\ and n and Kn^liH is the reciprocal of the resistance for the fluid flow in the capillary. The equation of motion for a system of tubes as in Fig. 10.1 is Wm
dH, - — Kn,n-l(Hn-l—Hn)+Kn,n+l(Hn+i àt
— Hn)
(10.3)
where the subscript n refers to the nth tube, /£„,„_! is the reciprocal of the capillary resistance between the tube n and n—l and Wn is the cross-section of area of the nth tube. ï I
"h Π
H, T L
/ h-K 12
1LLJ -4*-K -*-| \
1
Hn
U
23
FIG. 10.1. Afluidflowanalogue of the discrete type ("Hydrocal"). Obviously Eq. (10.3) is of the same form as Eq. (8.5.1), if Wn, -£„,„-! and Hn are the quantities analogous to the heat capacities, conductances and temperatures. In the equipment for this mechanical analogue, each tube has a tap so that initially the level of the liquid in the tube can be adjusted to correspond to the initial temperature of the corresponding heat reservoir. An additional tap common to all tubes is often arranged so that the flow of the liquid can be stopped at any moment for reading the instantaneous liquid levels. Boundary conditions corresponding to heat transfer by forced convection at the surfaces can be satisfied by keeping the end tubes at a level corresponding to the temperature of the external medium and connecting these tubes to neighbouring ones by a capillary whose flow resistance is equal to the reciprocal of the external heat transfer coefficient. If the temperature of the external medium is
ANALOGUES
285
constant, the liquid level in the corresponding tube is kept constant by using a tank with a large cross-section instead of a tube. The analogue can be used for complex bodies and for non-linear problems which arise in the case of thermal properties varying with temperature, of radiative heat transfer at the surfaces and so on. Extensions are also possible to two- and three-dimensional heat flow and even to ablation problems, but then the equipment becomes more complicated. An application of this mechanical analogue to a problem of heat flow in structures is given in Ref. 10.7. For linear problems electrical analogues are easier to handle and more flexible than mechanical ones. A direct analogue of the discrete type can be constructed with the help of electrical resistances and capacitances and such an arrangement for one-dimensional heatflowis shown in Fig. 10.2.
FIG. 10.2. An example of an electrical analogue of the direct and discrete type for one-dimensional heat flow with capacitors and resistors.
Heat reservoirs correspond to the capacities and thermal resistances to the electrical resistances as follows by comparison with Fig. 8.10. Further, the voltage at each point corresponds to a temperature and the electrical current between two capacitances to a heat flow, both quantities referring to analogous points in the two systems. If the initial temperature distributions are arbitrary, the points 0,1,..., Ν+1 must initially be connected to electrical sources, so that an analogous initial distribution of the voltages at these points is obtained; at the start of heat transfer these connections are broken unless there are internal heat sources. As mentioned in Section 10.1
286
HEAT TRANSFER IN STRUCTURES
the time scale can be chosen at will and there are two practical possibilities. The time constants can be made large, so that all changes occur slowly and can be read directly on instruments or registered by electromechanical devices (plotters) : or the time constants are small, so that the whole process can be repeated many times per second and may be displayed as a standing pattern on a cathode ray tube. The advantage of the last method lies in the possibility of using smaller capacitances and making all changes of the parameters visible immediately; the disadvantage is the greater complexity of the equipment. The solution of non-linear problems requires more elaborate equipment. For details see Ref. 10.1. An example of an indirect analogue is a method due to Liebmann, Ref. 8.12. The starting point is the implicit finite difference equation Eq. (8.7.1) which re-written with voltages instead of temperatures becomes: Vn.m-V^ =p(Vn_ltm-2Vntm+Vn+lim\ (10.4) In contrast to Chapter 8 this equation is used here in a step-by-step procedure where the voltages at the time t = (m—1) At and at all places x = ηΔχ, (n = 1,..JV) are known and the corresponding voltages at t= mAt are sought. The analogue for solving Eq. (10.4) is the circuit of Fig. 10.3; the balance of the electrical currents at node n corresponds to Eq. (10.4) if all nodes are kept at the voltages indicated in Fig. 10.3 and if RP = pRx (10.5) Rx can be chosen at will. The computer based on this circuit is shown in Fig. 10.4. The voltages VniYn_x already known are adjusted by sliding resistances R'0, R'l9 ... R'n and then the unknown voltages can be read by connecting a voltmeter to the points 1,2, . . . , « , one at a time. The resistances Rol, ... and RPtl··. have to be large compared with the resistances R' so that the voltages V0m_l9 Vi,m-v ··· depend only on the position of the resistances R'. In order to avoid adjustments by hand, servo-mechanisms and automatic recording units have been proposed. As equipment of this type becomes more elaborate, it would become a special-purpose analogue-computer. Vn-,m
n-l
Vnifn
Rx
|
n
Vn +
Rx
I|fn
n+l
iVn.m-l
FIG. 10.3. The basic circuit for an electrical analogue of the indirect and discrete type due to Liebmann (Ref. 8.12).
ANALOGUES
287
It depends on the circumstances whether a direct analogue, an analogue computer or a digital computer is preferable for solving a given problem. For steady heat flow problems with linear relations between the rate of
Generotor FIG. 10.4. A computer based on the basic circuit of Fig. 10.3.
heat flow and the temperature differences, direct analogues can be easily constructed, often largely with standard laboratory equipment. For moderate accuracies these analogues are useful and inexpensive, particularly for two-dimensional cases. Analogues are valuable for transient problems, when parameters in a design have to be optimized, and when a large number of temperature distributions of a similar type have to be found. If available, the general purpose digital computer is best where moderate or small amounts of work of varying types are involved. It also has great advantages whenever "non-linear" boundary conditions or similar complications occur. References General : 10.1 KARPLUS, W. J., Analog Simulation, MacGraw-Hill, New York (1958). 10.2 KARPLUS, W. J. and SOROKA, W. W., Analog Methods—Computation and Simulation, MacGraw-Hill, New York (1959). 10.3 SOROKA, W. W., Simulation in science and technology. Appl. Mech. Reviews 13, 9, 621-623 (1960). Particular: 10.4 HELE-SHAW, H. S., The flow of water, Nature 58, 34-36 (1898), The motion of a perfect liquid, Proc. Roy. Inst. 16, 49-64 (1899). 10.5 MOORE, A. D., Fields from fluid flow mappers, /. Appl. Phys. 20, 790-804 (1949). 10.6 MOORE, A. D., The hydrocal, Ind. Engng. Chem. 28, 704-708 (1936). 10.7 KNUTH, E. L. and KUMM, E. L., Application of hydraulic analog method to one-dimensional transient heat flow, Jet Prop. 26, 649-654, 659 (1956).