Fundamentals of Temperature Control

By William K. Roots

Fundamentals of Temperature Control focuses on theoretical foundations and principles involved in temperature control. The book first offers information on thermal-process representation and response. Discussions focus on response to damped harmonic inputs, principle of superposition, bode diagrams, ramp, step, and impulse functions, harmonic response, electrical analogs, basic equations, and thermal conductivity. The text then examines common thermal elements and open-loop temperature control. The publication ponders on closed-loop temperature control and the dynamics of discontinuous temperature control. Topics include dynamics in the phase plane and time domain, dynamic analysis, closed-loop control, secondary feedback, and cooling processes. The manuscript then examines quasi-continuous and continuous temperature control, as well as quasi-continuously controlled process behavior in the time domain and quasi-continuously controlled process behavior in the phase plane. The text is a vital source of data for researchers interested in the fundamentals of temperature control.

• Language: English
• Publisher: Academic Press
• Release date: Jun 28, 2014
• ISBN: 9781483275758

Book preview

FUNDAMENTALS OF TEMPERATURE CONTROL

William K. Roots, Ph.D.,

DEPARTMENT OF ELECTRICAL ENGINEERING, UNIVERSITY OF WINDSOR, WINDSOR, ONTARIO

FORMERLY: PROFESSOR OF ELECTRICAL ENGINEERING, POLYTECHNIC INSTITUTE OF BROOKLYN, NEW YORK

New York and London

Table of Contents

Cover image

Title page

Copyright

Dedication

PREFACE

NOMENCLATURE

Chapter 1: THERMAL-PROCESS REPRESENTATION

Publisher Summary

MATHEMATICAL MODELS

SIMPLE PROCESSES

ELECTRICAL ANALOGS

BASIC EQUATIONS

THERMAL CONDUCTIVITY

EXPONENTIAL LAG

ANALOGOUS ELECTRICAL AND THERMAL NETWORKS

Chapter 2: THERMAL-PROCESS RESPONSE

Publisher Summary

2.1 Ramp Functions

2.2 Step Functions

2.3 Impulse Functions

2.4 Harmonic Response

2.5 Bode Diagrams

2.6 Nyquist Diagrams

2.7 Principle of Superposition

2.8 Response to Damped Harmonic Inputs

2.9 The s Plane and the Laplace Transform

Chapter 3: COMMON THERMAL ELEMENTS

Publisher Summary

3.1 Transit Delays

3.2 Cascade Lags

3.3 Thermometer Examples

3.4 Distributed Parameters

3.5 Liquid Heating (or Cooling) Processes

3.6 Common Thermal Processes

Chapter 4: OPEN-LOOP TEMPERATURE CONTROL

Publisher Summary

Chapter 5: CLOSED-LOOP TEMPERATURE CONTROL

Publisher Summary

5.1 Closed-Loop Control

5.3 Cooling Processes

5.4 Overriding Commands

Chapter 6: DYNAMICS OF DISCONTINUOUS TEMPERATURE CONTROL

Publisher Summary

6.1 Dynamics in the Phase Plane

6.2 Dynamics in the Time Domain

6.3 Dynamic Analysis

6.4 Heating–Cooling Processes

Chapter 7: QUASI-CONTINUOUS AND CONTINUOUS TEMPERATURE CONTROL

Publisher Summary

QUASI-CONTINUOUS CONTROL

QUASI-CONTINUOUS SYSTEMS

CONTINUOUS CONTROL

QUASI-CONTINUOUSLY CONTROLLED PROCESS BEHAVIOR IN THE TIME DOMAIN

QUASI-CONTINUOUSLY CONTROLLED PROCESS BEHAVIOR IN THE PHASE PLANE

THE ERROR (yp) CAUSED BY THE CALIBRATION INTERVAL (pn)

NONLINEARITIES IN THE CONTROL-ELEMENT PROFILE

PREDICTING SYSTEM PERFORMANCE

APPENDICES

BIBLIOGRAPHY

AUTHOR INDEX

SUBJECT INDEX

Copyright

COPYRIGHT © 1969, BY ACADEMIC PRESS, INC.

ALL RIGHTS RESERVED.

NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by

ACADEMIC PRESS, INC. (LONDON) LTD.

Berkeley Square House, London W.1

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 69-13476

PRINTED IN THE UNITED STATES OF AMERICA

Dedication

Dedicated to

the late Professor Frederick Walker, Ph.D.

whose untimely death prevented him

from contributing to this book; and to

P. N. Robinson, F.I.E.E.

who established the scholarship

that enabled the author to change

from industry to academic life

PREFACE

… there have been, in recent years, many attempts to evolve an analytical theory of temperature control. None, however, appears to be completely satisfying as yet. While the theoretical foundations are old and the fundamental principles can be found in classical texts on the sciences, analytical methods have not yet penetrated properly into the field of temperature control.

R. Griffiths, Thermostats and temperature regulating instruments, 3rd ed., p. 201.

Griffin, London, 1951.

This quotation summarizes the purpose of this book. Until now the engineer, physicist, chemist, and metallurgist has had no convenient reference work on the basic theory of thermal systems analysis and temperature control. Consequently, these workers have lacked the necessary foundation for studying the large number of papers on temperature control that have appeared in the last five years in Transactions of the Institute of Electrical and Electronics Engineers, The Proceedings of the Institution of Electrical Engineers (London), Journal of the Institute of Fuel, The Review of Scientific Instruments, Journal of Scientific Instruments, Archiv fuer Technisches Messen und Industrielle Messtechnik, etc. This book aims at filling this gap by packaging these basic fundamentals into one slim reference volume. The mathematics involved have been deliberately kept as simple as possible without losing the analytic approach. Thus the intention is to provide a reference work equally useful to both the practicing engineer and scientist, as well as the new graduate student not previously exposed to temperature control.

In this work emphasis has been placed on discontinuous temperature control. This is because discontinuous control techniques are not only the least understood, but also because they form the bulk of the temperature-control systems used in industrial, laboratory, domestic, and weapons applications. Continuous control theory has been well covered elsewhere, and the interested reader is referred to the books by D. P. Eckman, Automatic process control. Wiley, New York, 1966; C. R. Webb, Automatic control. McGraw-Hill, New York, 1964; O. I. Elgerd, Control systems theory. McGraw-Hill, New York, 1967; all of which provide a foundation for studying the continuous temperature-control problem. In passing, the author acknowledges his own debt to these authors for the elementary concepts that, consciously or unconsciously, appear in the first chapter.

No descriptions of temperature-control hardware have been incorporated because these are adequately provided by M. Kutz, in Temperature control. Wiley, New York, 1968; by W. F. Coxon, in Temperature measurement and control. Macmillan, New York, and Heywood, London, 1962; and in the collations periodically compiled by the American Institute of Physics and published by Van Nostrand(Reinhold), Princeton, New Jersey.

It is unfortunate that many recent books use control as a mere mathematical plaything and concentrate upon esoteric topics. By contrast this book concentrates on a common real-world problem that has universal applications. Such a practical approach may not meet with the approval of many control theoreticians, but its justification is found in the words of two control authorities:

We have all seen examples of papers published in our professional journals that are unintelligible to any engineer. It is sad to relate that quite often when the confused engineer takes this sort of thing to the mathematician, he is told that it is nonsense to the mathematician as well! The control engineer must stop trying to supplant the mathematician. It is not the control engineer’s job to generate theorems, lemmas and proofs; it is sufficient to understand and apply them.

There is a great need now—and for at least the next five years—to learn how to apply some of the new theory.

My feeling is that if we restrict our attention to those problems that are well defined from the point of view of the mathematician, we are engaging in a process of self-sterilisation.

J. E. Gibson, From control engineering to control science, IEEE Spectrum 2(5), 69–71 (1965).

The development of the theory of control has reached a stage at which further progress requires some departure from complete generality. By considering particular classes of system, in which characteristic combinations of considerations restrict the design, progress may be made towards the development of more definite procedures, which take account of the particular restrictions and bring to light possibilities of improvement.

A. Tustin, et al., The design of systems for the automatic control of the position of massive objects, IEE Proc. Suppl. 1 C105, 2 (1958).

WILLIAM K. ROOTS,     Birmingham, England

April 1969

NOMENCLATURE

a General symbol for an arbitrary constant

a Symbol for a cross-sectional area [p. 7]

b Symbol for a dimension, usually thickness [p. 7]

b Primary feedback (units of e) [p. 92]

b2 Secondary feedback [p. 100]

e Actuating signal (volts or p.s.i.g., or mm of motion, etc.) [p. 92]

e1, e2 specific values of e [p. 95]

f General symbol for a function, i.e., f(t) is a function of time

{fn(t)}n=1,2,…. Q Ensemble of functions of time [p. 37]

f Frequency (Hz) [p. 64]

fq DEC frequency (Hz) [p. 135]

{gn(t)}n=1,2,… Q Ensemble of responses to {fn(t)} [p. 37]

h General symbol for heat flow (Btu/sec) [p. 4]

hmax maximum heat flow (Btu/sec) [p. 70]

hL Heat loss at conveyor apertures (Btu/sec) [p. 76]

hS Heat loss at stack (Btu/sec) [p. 76]

h(t) Unit step function [p. 17]

i General symbol for an input [p. 28]

io Amplitude of a periodic input, i.e., i = io sin ωt [p. 28]

j

m Manipulated variable (numeric) [p. 70]

mh, mc, mc2 Manipulated variable in the heating and cooling functions of heating-cooling process control [pp. 116, 117]

Mean value of m(t) in DEC (numeric) [p. 135]

mn Where 1 < n < Q. Intermediate value of manipulated variable in a Q position system, i.e., 0 < mn < +1 (numeric) [p. 165]

mQ(=0) Lowest value of manipulated variable in a Q position system (numeric) [p. 180]

n The general term in an ensemble

p Magnitude of the deadspace in a multiposition controller (units of e) [p. 153]

q Quantity of heat (Btu) [p. 4]

q Magnitude of the lost motion in the characteristic of a discontinuous control element (units of e) [p. 100]

q′ (=q/H1) Magnitude of lost motion rated in units of the controlled variable θ (units of θ) [p. 106]

δq Small increment in the lost-motion magnitude (units of θ) [p. 104]

r Reference input (units of e) [p. 92]

s Laplace complex variable (= ζ + jω) (sec−1) [p. 41]

t General symbol for time (usually, sec) Note: θ(t) indicates θ as a time-varying function; similarly, m(t), etc.

ta Time of switching operation (sec) [p. 23]

tc Cooling time (sec) [p. 134]

te Elapsed time since last switching operation (sec) [p. 71]

teo, teq Auxiliary cycling time variables in a quasicontinuously controlled process (sec) [p. 171]

th Heating time (sec) [p. 134]

to On time (sec) [p. 134]

tp Off time (sec) [p. 134]

tq Periodic time (sec) [p. 135]

tq Minimum value of tq (sec) [p. 145]

ty Time corresponding to temperature θy (sec) [p. 83]

Δ1t−Δ4t, etc. Arbitrary increments in t that correspond with increments

Δ1θ−Δ4θ, etc. in θ (sec) [p. 83]

Δot Overshoot recovery time (sec) [p. 134]

Δut Undershoot recovery time (sec) [p. 134]

u Disturbance input (units of θ) [p. 79]

v (=ωT) Algebraically convenient variable for frequency of a periodic input [p. 55]

x General symbol for a dimension [pp. 64, 65]

x An arbitrary constant [p. 84]

y (= θr− θ) Error in the control of a process (units of θ) [p. 107]

Mean value of y in DEC [p. 124]

yd Error due to overcompensated secondary feedback (b2 > q). A component of y (units of θ) [p. 114]

yp Error due to deadspace magnitude p (units of θ) [p. 180]

z Arbitrary constant [p. 18]

Switching condition for bang-bang controllers (numeric) [p. 159]

zi Incoming quality level [p. 163]

zo Outgoing quality level [p. 163]

A Amplitude of a phasor [p. 29]

A RIE sensitivity (units of e and units of θ) [p. 91]

A Conveyor aperture area (ft²) [p. 76]

AS Actuating signal comparator [p. 92]

B Arbitrary constant [p. 67]

Btu British thermal unit

C Thermal capacitance (Btu/°F) [p. 4 et seq.]

Cn(n = 1, 2, …) Thermal capacitance at specified locations (Btu/°F) [p. 59]

Cc Capacitance of coating [p. 61]

Cm Capacitance of mercury [p. 61]

CE Control element [p. 92]

D(d/dt) Heaviside operator [p. 4 et seq.]

D Block-diagram symbol for a control element’s characteristic [p. 92]

DEC Dynamic equilibrium cycling [p. 111 et seq.]

G General symbol for a forward transfer function [p. 56]

G(D) Transfer operator [p. 35]

G(s) Transform of transfer function [p. 42]

G(jω) Transfer function [p. 35]

H General symbol for feedback transfer function

H1 PFE sensitivity (units of e and units of θ) [p. 92]

H2 SFE sensitivity (units of θ) [p. 100]

J Imaginary component [p. 35]

K1, K2 Arbitrary constants [pp. 77, 86 et seq.]

L Arbitrary dimension [p. 75]

Laplace transform [p. 43]

M Mass (lb) [p. 4 et seq.]

N Disturbance-input element attenuation (numeric) [pp. 76, 79]

{Nn}n=1,2, …., Q An ensemble in N of magnitude Q [p. 79]

PFE Primary feedback element [p. 92]

Q General symbol for the magnitude of a quantity [p. 19] or an ensemble [p. 37], i.e., 1, 2, …, Q

Q Quantity of flow (lb/sec) [p. 69]

R Thermal resistance [°F/(Btu)(sec)−1] [p. 4]

Rc Coating resistance [p. 61]

Re Equivalent thermal resistance of a parallel ensemble Re = [Σ(1/Rn)]−1 [p. 79]

Rg Glass resistance [p. 61]

{Rn} (n = 1, 2, …) R at specific locations [p. 59 et seq.]

Real component [p. 35]

RIE Reference input element [p. 92]

S Specific heat [Btu/(lb)(°F)] [p. 4]

Sa, Sb, etc. SFE’s in a quasicontinuous temperature controller [p. 168]

S1, S2, etc. et seq.]

SCR Silicon controlled rectifier [p. 95]

SFE Secondary feedback element [p. 100]

SPDT Single pole, double throw [p. 94]

T Thermal time constant (sec) [p. 4]

U Heat loss coefficient [Btu/(ft²)(°F−1)(sec−1)] [p. 76]

V General symbol for velocity (ft/sec) [p. 76]

VL Velocity of light [p. 64]

X Arbitrary constant [p. 18]

Y Arbitrary constant [p. 18]

Z Arbitrary constant [p. 18]

Z Constant of integration [p. 76]

α Arbitrary constant [p. 66]

β Arbitrary constant [p. 67]

γ Inverse of thermal diffusivity (sec/ft²) [p. 66]

γ[=1 − exp(−τ/T)] Index of process time parameters [pp. 132, 134]

γh, γc γ index for heating and cooling functions of a heating−cooling process [p. 161]

δ Small increment i.e., δθ

δ(t) Unit impulse function [p. 24]

ɛ Base of natural logarithms

ζ Damping factor (s = ζ + jω) [p. 37]

θ Controlled variable of a thermal process (typically °F, but could equally be °C, °K, etc.) [p. 92]

θa, θb Temperatures at specific locations (a, b, etc.)

θd Amplitude of DEC (units of θ) [p. 14]

θg Auxiliary variable in a quasicontinuous temperature controller (units of θ) [p. 168]

θi Input temperature (units of θ) [p. 15]

θm Maximum value of θ(t) in DEC (units of θ) [p. 125]

θn Minimum value