Erläuterungen zu der D1A-Reihe Analogrechner

Complete English translation of the original German-language document (17 pages).

Explanatory Notes on the DIA Series Analog Computers

[page 1: cover — Telefunken logo and title page: “Explanatory Notes on the DIA Series / ANALOG COMPUTERS”, with a graph illustration and label “FG”]

[page 2: verso of cover — publisher imprint only]

Published by TELEFUNKEN A.G., Department for Technical Publications and Exhibition, BERLIN 12, Ernst-Reuter-Platz 7

Analog Computers

Electronic computing systems can be divided into two groups according to their principal mode of operation:

a) Digital computers or digital calculators
b) Analog computers

In digital computers, the computational quantities are expressed as numerical sequences that represent a characteristic number of magnetic pulses as precisely as possible. The achievable accuracy of results depends, in the case of digital computers, primarily on the number of digits — in other words, on the number of impulses per word — and is independent of the shape and magnitude of the individual impulses.

In analog computers, the computational quantities are represented continuously by the changes in the computational variables proportional to the changes in the quantities being computed. The simplest example is the slide rule, where lengths serve as the analogs of numbers.

The analog computer does not differ from the digital computer in the type of problem representation. Since it is often possible to find a suitable mathematical description for physical processes, it is especially suited, in particular, for control engineering work. With the aid of the analog computer, it is then possible to solve the same mathematical problem on the same machine without the machine recognizing — so to speak — any difference between the equations.

Mechanical and electrical analog computers exist, to which the latter, due to their easier programmability, more convenient operation, and higher computing accuracy, are generally more suitable for general application. In the following, only the electronic analog computer will be discussed.

In electronic analog computers, computational quantities are represented by electrical voltages: 3 V or 10 V; in some cases up to 30 V or 0.1 V can also serve as the machine unit.

With the aid of an electronic network as the physical model of the process under investigation, the variables are measured, i.e., the continuously varying quantities at certain points in this network provide the desired solution.

The analog computer can encompass the following capabilities:

Solution of ordinary linear and nonlinear differential equations and systems of such equations, i.e., those in which differentiation is carried out with respect to only one independent variable.
Solution of general systems of physical equations, by which differential equations as well as purely algebraic equations can be solved.

This already expresses the fact that the analog computer is used primarily for the solution of dynamic problems. These same problems occur in many branches of mathematics, natural science, and technology. This is a significant part of all technical problems.

Important further applications are the use as a simulator for equipment elements, devices and procedures, for the cost reduction of trial and error experiments, or for the verification of approximate calculations before the actual experiment is conducted.

Examples:

Electrical engineering:
Analysis and synthesis of linear and nonlinear networks; calculation of processes in electrical machines, e.g., in transformers, motors, transmission lines, etc.
Mechanical engineering:
Investigation of vibrations and self-excited oscillations; hydraulic and pneumatic processes; gyroscopic processes and pneumatic components.
Aeronautical engineering:
Simulation of aircraft behavior; guidance and control of aircraft; flight calculations.
Nuclear engineering:
Simulation of reactor processes; general kinetics calculations.
Control engineering:
Dynamic processes in control systems of all kinds; design of controllers; stability analysis.
Process engineering:
Simulation of process-engineering sequences; optimization of process-engineering sequences in chemistry; electrochemical technology.
Biology and physiology:
Heart, kidney, and circulatory models; physiological control systems.

The DIA-Series Desk Analog Computer

Electronic analog computers consist of an assembly of computing elements with the aid of which any mathematical description of an investigated process is assembled into a computing circuit.

Computing elements are:

Summing and integrating amplifiers
Potentiometers for setting fixed coefficients
Multipliers
Function generators for generating arbitrarily varying functions
Comparators for comparing two voltages

These computing elements are assembled in a specific number and combination into table-top and rack-mounted computers, together with the necessary power supply units. All computing elements are accessible at a central patch field, where the setting of the computing elements and the programming connections are made using plugs and programming cards.

Figure 1 — RAT 740

In the RAT 740 shown in Figure 1, one can see at the top section on the left the monitoring indicator for the allowable computing range (i.e., the voltage supervision indicator), as well as the power supply unit with display and safety devices for the required supply voltages. The instrument serves simultaneously for the adjustment of all amplifier zero points. Below this is the patch field with the connections of all computing amplifiers. Under this are the controls for all computing amplifiers, i.e., for the coefficient potentiometers. Below the patch field there are also 20 coefficient potentiometers, the 2 function generators with associated switchable dial-type indicator.

Below these elements, the multipliers and comparators appear on the computing panel along with a choice of various computing and testing states with the associated switches. At the bottom of the computer is an input connector strip for auxiliary equipment such as oscilloscopes and XY recorders. This connector strip also serves for exact setting of the coefficient potentiometers by the coefficient-setting procedure, and for precise installation according to the repetitive computing mode. The potentiometer in the lower section enables a continuous variation of the repetition rate between 1/10 sec and 100 sec.

Figure 2 — Computing Amplifier Assembly Unit

One can recognize in the internal construction of the computer an assembly unit with magazines for plug-in cards with a computing module. In the present case these are 45-amplifier computing modules. The cards are easily interchangeable.

[page 5]

Figure 3 — Operational Amplifiers

The figure shows a computing amplifier (more precisely, an operational amplifier), consisting of a so-called main amplifier and an auxiliary amplifier on a plug-in unit. Between them is a chopper relay used for stabilization of the amplifier. This is one of the more important roles of the auxiliary amplifier.

All other computing elements are similarly built on plug-in cards and stored in magazines.

Figure 4 — Precision Analog Computer System RA 800 HYBRID

Since in analog computer technology the computing precision per unit per equation or per system of equations is proportional to the number of computing elements employed, the computing precision of the available problem is particularly important in aeronautical and space engineering, but also in all other applications where the number of computing elements is no longer adequate using table-top computers. The programming effort also becomes very significant when greater precision is required. Large computer systems require a higher number of individual computing elements; the precision of the assembly (i.e., the accuracy of the individual components) must also be higher; the technical error limits of the individual computing components must be maintained more precisely. For large analog computing systems in combination, from 150 computing amplifiers, 50 potentiometers, and 30 multipliers, one achieves error limits that are somewhat smaller by a factor of 10 compared to table-top systems. A logical extension of the capabilities is the so-called Hybrid combination; the ratio of analog and digital computing elements is approximately 1:1. In the two latter cases a control of the computer procedure by means of punch-tape or card readers for service purposes can be provided, and digital-to-analog-logic elements are also available.

[page 5 continued]

In both latter cases, a Precision Analog-to-Digital system can be included, and the control of the computer process is carried out through punch cards or card readers for the maintenance and servicing of computing elements. Additional digital-to-analog logic elements are also available.

[page 6]

Figure 5 — Precision Analog Computer Systems

Because the computation precision for complex problems increases with the number of precision computing elements and is often enhanced by substitution through table-top instruments, several table-top analog computers can be interconnected into a Precision Analog Computer.

The actual basis for complex computing construction and the simulation of problems of a higher order of magnitude is provided by the higher-level analog computer.

Symbolic Representation of Analog Computer Elements

For the systematic representation of computing circuits of the analog computer, which is required for all practical operations in particular, a standardized symbolic notation for the computing elements has been established, and the usual mathematical operations are assigned to these standard symbols. The connection between individual computing elements is indicated by lines.

1. Summing amplifier

For inversion, a general symbol is:
A symbol with triangle or amplifier block.
For summation of multiple input signals, the inputs are joined at the summing junction.

2. Integrator

For integration, a general symbol with an integrator block.
For summation (multiple inputs), the integration is carried out on the algebraic sum of the input signals.

3. Offset (bias)

For general use, the symbol shows a fixed reference voltage applied to the input.
Notation: a symbol indicating a constant value.

4. Coefficient Potentiometer

For setting fixed gains; symbols show the potentiometer configuration.
For x gain or general symbol.

5. Multiplier

For the multiplication of two variable quantities; symbol as shown.

6. Comparator

For comparing two voltages; symbol as indicated.

[page 6: Figure 6 — Symbols table showing computing element symbols and their operators]

The Operational Amplifier

The figure shows the circuit principle of the operational amplifier (inverting). It consists essentially of a high-gain amplifier with very high input impedance with input resistor R₁ and feedback resistor R₂.

The input voltage u_e is applied via R₁ to the inverting input (summing junction S) of the amplifier. The output voltage u_a is fed back via R₂ to the same summing junction.

Since the open-loop gain V of the amplifier is very large (V → ∞), the voltage at the summing junction u_S ≈ 0 for all practical purposes. From this it follows:

Current through R₁: i₁ = u_e / R₁
Current through R₂: i₂ = −u_a / R₂

Since (with ideal amplifier) i₁ + i₂ = 0:

u_e / R₁ + (−u_a) / R₂ = 0

∴ u_a = −(R₂/R₁) · u_e

For the case R₁ = R₂, the output voltage is equal and opposite to the input voltage: u_a = −u_e (inverter).

For multiple inputs (summation) with input resistors R₁, R₂, … Rₙ:

u_a = −R_f · (u₁/R₁ + u₂/R₂ + … + uₙ/Rₙ)

This expression is valid for all input voltages simultaneously; the amplifier produces the algebraic sum of the weighted input signals, hence the name summing amplifier. The weighting factor k_i = R_f / R_i can be adjusted by appropriate choice of the input and feedback resistors.

Figure 7 — Operational Amplifier Circuit Principle

The amplifier circuit shown has the following properties:

The potentiometer in the feedback determines the relationship between input and output signals.
In the feedback path (R_f) the sum of all input currents flows.
The output is an inverted representation of the input sum.

The impedance at the summing point S is practically zero; the impedance at the output is very low.

Figure 8 — Summing-Point Principle of Summation

The summation by means of voltage dividers demonstrates the following: Since u_S = 0, the sum of currents at the summing point is zero:

∑ (u_n / R_n) + u_a / R_f = 0

from which:

u_a = −R_f · ∑ (u_n / R_n)

The summation multiplies each input quantity u_n by the corresponding gain factor k_n = R_f / R_n. A gain factor greater than 1 is obtained by making R_n less than R_f, and a factor less than 1 by making R_n greater than R_f.

The analog computer uses normalized resistances so that the gain factors take the values: k_n = 1, 1/3 (for some machines).

The Integrator

Figure 9 — Circuit Principle of the Integrator

The integrator is obtained from the operational amplifier by replacing the feedback resistor R_f with a capacitor C. Using the summing-point principle one obtains:

i₁ = u_e / R
i_C = C · (du_a / dt)

Since i₁ + i_C = 0:

u_e / R + C · (du_a / dt) = 0

∴ u_a(t) = −(1/RC) · ∫ u_e dt + u_a(0)

The output voltage is thus proportional to the time integral of the input voltage. The constant RC is the integration time constant; u_a(0) is the initial condition (the voltage on the capacitor at t = 0).

Figure 9 — Circuit Principle of the Integrator

When the output voltage u_a is to be written as:

u_a(t) = −(1/T) · ∫₀ᵗ u_e dt + u_a(0)

where T = RC is the integration time constant.

The integrating formula given above is exact. The starting formula:

u_a = −(1/T) [u_{e1} · t₁ + u_{e2} · t₂ + … + u_{en} · tₙ]

The integrator adds (accumulates) the input values continuously with respect to time; the area under the u_e curve is formed. The example diagram shows a triangular waveform obtained from the integration of a square wave.

Figure 10 — Functional Diagram of the Integrator

The diagram at left shows a step-response from an integrator: when the input is a constant value u_e, the output rises linearly (integration of a constant produces a ramp). The drawing to the right shows the timing diagram of a double-cycle operational integrator characteristic, whereby the Chopper relay plays an important role: Through the Chopper stabilization, offset drift is practically eliminated. The amplifier operates as follows:

During the first half-cycle the amplifier amplifies normally.
During the second half-cycle the offset is measured and corrected.

A quasi-DC-stable operational amplifier is thereby obtained, and the offset is reduced to the μV range.

The Multiplier

Figure 11 — Multiplier Networks

Multiplication can be achieved in various ways:

Via logarithm tables: If two voltages u_x and u_y are converted to their logarithms, added, and then reconverted from the logarithm, the product u_x · u_y is obtained. This method is limited to positive values.
Via a so-called quarter-squares circuit: Using the identity:

u_x · u_y = ¼ [(u_x + u_y)² − (u_x − u_y)²]

both squared terms can be computed from a single squaring circuit by alternately using u_x + u_y and u_x − u_y.

In Figure 11, the general block diagram of a multiplier network is shown schematically. The input voltages can be both positive and negative (four-quadrant operation).

Figure 12 — Multiplier Block Diagram

Figure 12 shows the multiplier circuit, which comprises a Summing amplifier, Differencing amplifier, and nonlinear function units. The Parabolic multiplier uses the following principle: The algebraic expression is computed, so that the precise algebraic expression is applicable:

u_a = u_x · u_y

The Parabolic multiplier gives the best output accuracy of the various multiplier variants; this result is achieved simultaneously with linear or nonlinear operations.

Das Parabolic multiplier delivers the same input accuracy. One important thing: it works in all four quadrants, which means the inputs can be both positive and negative. The technical implementation is achieved as follows:

Figure 13 — Circuit Principle of the Time-Division Multiplier

Another multiplier method is the so-called Modulation or Time-Division Multiplier. It is a method known as Pulse-Width-Pulse-Height modulation, where the pulse width of one variable is proportional to x and the pulse height of the other variable is proportional to y. The impulse-width is thereby — in each case — the machine variable. The implementation principle is shown in the figure.

The pulse width is determined by the comparator: if the ramp voltage from the comparator integration exceeds u_x, the comparator changes its switching state and the pulse ends. Thus for a given ramp slope, the pulse width t_i is proportional to x. During this pulse the amplitude of the current is proportional to y. The mean value of the output of the switch multiplied by integration gives:

u_a ∼ x · y

It is equally possible to make the multiplication by varying the variable y. The division is therefore also possible by varying x.

Figure 14 — Circuit Principle of the Servo Multiplier

The third multiplier method is performed with the aid of a so-called Servo Multiplier. This proceeds from the observation that a potentiometer can realize a position-dependent value of the output voltage; it works when, for a given potentiometer wiper position, a continuously varying voltage proportional to the wiper position can be tracked. The servo multiplier uses a motor-driven potentiometer; when a reference voltage is applied to the potentiometer,the output is:

u_a(t) = −(x/E) · y = −(x · y) / E

The difference voltage at the input of the servo amplifier drives the motor; the motor drives the potentiometer slider, and the difference voltage approaches zero. In steady state:

x − E = 0, i.e., x ≈ 0

For small changes of x the motor drives the slider of the potentiometer (in practice this sits directly at the motor-shaft); the position of the slider is proportional to x/E. The output voltage y_a = x · y/E² gives the product of both input variables. Multiplication is possible in all four quadrants.

[page 11]

Figure 15 — Functional Diagram of the Function Generator

Function generators permit the approximation of an arbitrarily given function in general. Especially helpful is the fact that the same circuit can achieve different approximated functions through simple parameter change.

The technical realization of the function generator is essentially through the use of diode networks (diode-resistor function generators). This type of function generator generates a piecewise-linear approximation of the desired function. The function is approximated by a number of straight-line segments. The breakpoints and slopes of these segments are adjustable.

In practice it is technically expedient to use about 10 to 30 diode segments. The device has approximately 10 adjustment potentiometers. The function can be set up visually by comparison with the desired graph, and through variable breakpoint setting.

Figure 16 — Trimming of a Function Generator

The adjustment of a function generator is performed by means of an oscilloscope or an XY recorder, where the generated curve and the desired curve are displayed simultaneously. The comparison of the two curves allows adjustment of the breakpoints and slopes until the function is correctly approximated. This adjustment is made continuously and with reference to an accurately specified function.

The Feedback System

Figure 17 — Feedback Model of an Automobile Suspension

A simplified mechanical model for the description of an automobile suspension can be set up as a system having 2 degrees of freedom of mechanical vibration. The Differential equation system for this can be read directly from the computing circuit of the analog computer. The variables from the model of Figure 16 describe the computing variables: The movements of the front and rear points of the automobile body in relation to the road surface can be expressed as functions of positions of computing amplifiers as well as through the road-surface excitation. By means of a minor modification, one can also have available the full computing circuit while taking into account also the variable springs and shock absorbers. By means of systematic analog calculation, one can then survey the behavior of a system both qualitatively and quantitatively in the shortest possible time; in addition, short or repetitive model computing operations and actual experimental tests can be conducted.

Figure 18 — Single-Shock Behavior of a Passenger Vehicle

The figure shows solution curves for the described automobile suspension system, which is also used in practice for structural and design purposes. The differential equations of the analog computer were set up with an oscilloscope and recorded with an XY recorder. The system receives the excitation of a road step at one wheel from a certain height, while the vehicle is occupied by one person. As the disturbance at time t = 0, a spring displacement of −0.15 m is applied to the rear wheel. This excitation shows the rear of the automobile above the axle. The dashed curves show the behavior at 4 persons. The curves give all essential information for assessment of the spring characteristics of a standard passenger vehicle. The variation of parameters (change of spring and damper characteristics and the effect of external disturbances) is extraordinarily simple.

[page 13]

Figure 19 — Differential Equations of a Transformer

An application from electrical engineering is the calculation of a transformer’s behavior, where the equivalent circuit of the transformer is taken as the starting point. The equations are the differential equations of the primary and secondary currents and the magnetic flux; in the case of ohmic load this is the differential equation of the secondary current. R₁ and R₂ are the winding resistances, L₁ and L₂ are the leakage inductances, and M is the mutual inductance.

Figure 20 — Computing Circuit of a Transformer

The figure shows the appropriate computing circuit from which the solution structure for both differential equations and the coupling between them can be derived. When the computation of the transformer is performed by introducing the magnetization characteristic as a function, this opens the way to examining the effect of variable measuring conditions, e.g., by simple modification of the potentiometers. In this manner, arbitrary parameter variation in the transformer characteristics becomes possible in the simplest manner, so that any desired voltage or current excitation conditions can be observed and the appropriate differential equations can be studied. The corresponding differential equations in the actual computing circuit can be observed and also for the computation of the Amplitude-Normalization which corresponds to the computing method and also for the determination of Amplitude magnitudes. All voltages are then ≤ 1 (Machine unit); the voltage values can exceed ±10 V (machine unit), and then the individual values can be accumulated. Together with a Simulation (e.g., core losses, reactive current, etc.) and the influence of saturation, the main task of the programmer next to the setup of the actual problem equations is the Amplitude Normalization.

Figure 21 — Representation of Control System Elements

One of the most important application fields of analog computer technology is control engineering. The reason for this lies in the fact that in the practical case each common control element of each type is described as an analog computing element or a combination of the same.

When using standard — i.e., in analog computers included for general purposes — computing elements, the transfer function, as required by the control element, is obtained not only by combining several computing elements, but also can be realized with a specific network.

Another principle underlying the simulation of D-link elements is that in practice it can be accomplished only by a Differentiation of a limited measured variable, which not only improves the noise characteristics of the differentiation but also reduces the amplitude of higher harmonics.

[page 14 — Table: Representation of Control System Elements]

The table shows the control element types with their corresponding transfer functions and analog computing circuit implementations:

No.	Control Element	Transfer Function	Implementation with Standard Computing Elements	Implementation with Passive Networks
4a	P-element	V	Potentiometer, inverting amplifier	Voltage divider
4b	I-element	1/pT	Integrator	RC network (capacitor in feedback)
4c	PI-element with PT₁ member	(1 + pT₁)/(pT · (1 + pT₁))	PI-controller with lag element	RC network
4d	Proportional-Integral-Derivative element with delay	(1 + pT₁)/(1 + pT₂)	Summation of P and I outputs	RC network with additional elements
4e	D-element	pT_D	Differentiator (limited)	Differentiating network
4f	PD-element with PT₁	(1 + pT_D)/(1 + pT₁)	Series combination	RC lead-lag network

Variables themselves, as well as the integration of a D-element through an integrator can allow the formation of the input quantities. In general, analog computing elements for control elements are preferred because all parameters can be varied independently without mutual coupling, and thus the optimal set of parameters can be found. The reduced circuit complexity with passive networks, however, results in less flexibility with the lower cost of switching elements.

Freshwater-Temperature Control

A fresh-water temperature control is applicable in many cases without complex computing operations such as Addition, Multiplication, Integration. The standard block diagram of this represents a Freshwater-Temperature regulation.

Modern steam turbines are equipped with a very high freshwater temperature. An essential component of Freshwater-Temperature regulation consists of the so-called Desuperheater, which is used for intermediate cooling between the individual high-pressure turbine stages. The required intermediate cooling is accomplished by the injection of water into the vapor stream. As a cooler, a so-called Spray cooler is preferred because of its fast response.

For the control of the temperature in this example, the scheme relies on a comparison between the freshwater temperature and the freshwater regulator to reduce the heat flux. This is done through a Spray cooler injection and Freshwater temperature regulator as in the final control element.

From the comparison of p_zul and p_ist in the temperature regulator, the input variable to the I-element becomes established.

Figure 22 — Block Diagram Freshwater Temperature Control

The block diagram shows the structure of the freshwater temperature control circuit: including the Turbine, Spray cooler, Temperature sensor, PI regulator, and final control element (injection valve), all arranged in a closed-loop structure.

Figure 23 — Computing Circuit for Freshwater Temperature Control

The detailed analog computing circuit implements the block diagram using integrators, potentiometers, and summing amplifiers to represent each functional block of the control system.

Figure 24 — Computing Circuit for Fresh-Water-Temperature Control

This circuit shows an alternate representation that includes additional potentiometers for coefficient-setting of the PI controller parameters (integration time T_i, proportional gain V).

Computing Circuit for an Autoclave (chemical reactor)

For the chemical reaction, the computation is the direct digital control process, which is designed for an Autoclave Reactor. As per the direct digital regulation process, which provides an Autoclave:

In the simplest case the function of a computing element is not different from that of an analog computing element.

Figure 25 — Computing Circuit for an Autoclave Reactor

The computing circuit for the autoclave reactor includes representations of temperature, concentration, and reaction rate state variables integrated as feedback loops with the appropriate nonlinear function elements.

[page 16]

In order to be able to optimize process-engineering procedures in this way, i.e., to find in each case the best quality and the largest quantity of end products as well as the best utilization of energy, one must go above and beyond pure laboratory-and test-bench investigation and make use of mathematical simulation.

Chemical reaction processes are described by differential equations whose rate of reaction dC/dt is proportional to the concentration of the substance:

−dC/dt = k · C

where the minus sign indicates a consumption of the concentration. For bimolecular reactions, the rate of reaction is proportional to the concentration change of one of the reacting substances, that is, proportional to the product of the concentrations of both substances:

−dC₁/dt = k · C₁ · C₂

The constant k is in turn not wholly independent of stationary conditions, but is rather a function of temperature and, e.g., of the nature and amount of catalysts present, so that in the general case the so-called Arrhenius equation applies, with k being a temperature-dependent quantity. For the pressure dependence, the exponential law also applies. The regulation of the technical behavior of the process can be described beforehand; the regulatory behavior of the technical equipment is set forth in the solution of the differential equation system.

Analog Computing Circuit for a Monomolecular Reaction Process

The computing circuit for this system is shown in the accompanying figure. It should be noted that the uncontrolled reaction kinetics of chemical processes are in many cases not mathematically accessible in closed form; that is, the measurement data available for a given endpoint condition (e.g., improved yield of reaction) allow the verification of an assumed model.

The system is described using a set of ordinary differential equations as a Substitute (approximating model). Overloading in a system of ordinary differential equations then leads to the requirement for a partial differential equation. For this the analog computer can be used, for example in connection with a magnetic drum storage unit, to compute one profile section at a time, with the results serving as the input to the next computing step.

Thus the differential equation system is solved only once at each time step on the analog computer. The given input values (e.g., read from a magnetic drum) serve as the input values for the next computing step.

By this method of so-called iterative computing, two analog computers or two computing levels within one computer can be connected in an iterative procedure. The treatment of partial differential equations is naturally not only a field of chemistry but also concerns hydraulics (computational fluid dynamics), which can be treated by the analog computer, with an external memory. Together with the RA 800 Hybrid system, the necessary memory-integrators and control elements are already on-board. One can also connect via two table-top analog computers to carry out a step-by-step computation.

Figure 27 — Optimization of a Chemical Process

Our figure describes another important application of analog computers: the control and simultaneous optimization of processes. In this case the computer goes directly into an on-line operation with a process, i.e., it handles an online input/output situation without an attendant operator and without intermediate manual data entry. In off-line operation the computer is connected directly to information sources about the process state (e.g., data on temperatures, pressures, flow rates, concentrations, etc.) and processes these immediately. In on-line operation, the regulatory inputs are fed directly back to the process.

For the optimization of the chemical process shown here, the optimization of a process occurs in regard to a minimization of a short possible computing path. The example shows a so-called process optimization for Hydrogenation by Nickel catalyst, used in an autoclave under specific conditions by a flow of hydrogen in molten wax at elevated temperature. The starting product is beeswax, mixed with Nickel catalyst and in an autoclave under specific conditions is treated with hydrogen under pressure.

For the optimization of the chemical process in terms of the shortest possible computing time:

Because of the variable composition of the output products, the varying efficiency of the catalyst, the varying quality of the input products, and the costs of the process, a mathematical simulation must be used to find optimum values for the flow rate and material inputs, and the analogue computer receives analogue measurements of the process directly as input.

Along with these already-mentioned advantages of the analog computer, its light programmability and fast solution capability make it the best way for engineers and scientists to solve new problems quickly. The analog computer enables the engineer, by a comparatively modest computing effort, increasingly to handle the everyday operations of a practical engineer.

Literature:

G.W. Götz and P. Lauber — Analogrechner, Springer-Verlag, Berlin 1965
VDI-Zeitschrift, No. 26, October 1963 — Article on the use of analog computers and their application to the analog computer
G.W. Götz — Analogrechner in der Ingenieurpraxis, VDI-Verlag, Düsseldorf 1964

The figures in the drawings 7, 8, 9, 10, 12, 13, 15, and 16 were taken with the friendly permission of Springer-Verlag from the above work.