Table Of ContentFUNDAMENTALS
OF TEMPERATURE CONTROL
William K. Roots, Ph.D.
CHAIRMAN, DEPARTMENT OF ELECTRICAL ENGINEERING
UNIVERSITY OF WINDSOR, WINDSOR, ONTARIO
FORMERLY: PROFESSOR OF ELECTRICAL ENGINEERING
POLYTECHNIC INSTITUTE OF BROOKLYN, NEW YORK
1969
ACADEMIC PRESS New York and London
COPYRIGHT © 1969, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
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United Kingdom Edition published by
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LIBRARY OF CONGRESS CATALOG CARD NUMBER: 69-13476
PRINTED IN THE UNITED STATES OF AMERICA
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 con
venient 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 pre
viously 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
ix
X PREFACE
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 Hey wood, 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 justi
fication 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).
PREFACE xi
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]
b Secondary feedback [p. 100]
2
e Actuating signal (volts or p.s.i.g., or mm of motion, etc.)
[P- 92]
e , e specific values of e [p. 95]
x 2
/ General symbol for a function, i.e., f(t) is a function of time
{/nW}n=i.2.....Q Ensemble of functions of time [p. 37]
/ Frequency (Hz) [p. 64]
f DEC frequency (Hz) [p. 135]
Q
{gn(t)}n=i.2 Q Ensemble of responses to {f (t)} [p. 37]
n
h General symbol for heat flow (Btu/sec) [p. 4]
Ämax maximum heat flow (Btu/sec) [p. 70]
hi Heat loss at conveyor apertures (Btu/sec) [p. 76]
hs Heat loss at stack (Btu/sec) [p. 76]
h(i) Unit step function [p. 17]
i General symbol for an input [p. 28]
i Amplitude of a periodic input, i.e., i = i sin cut [p. 28]
0 0
j v-i
m Manipulated variable (numeric) [p. 70]
m , m y m Manipulated variable in the heating and cooling functions of
h e e2
heating-cooling process control [pp. 116, 117]
m Mean value of m(t) in DEC (numeric) [p. 135]
m Where 1 < n < Q. Intermediate value of manipulated
n
variable in a Q position system, i.e., 0 < m < +1 (numeric)
n
[p. 165]
mû(= 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]
XIV NOMENCLATURE
q Magnitude of the lost motion in the characteristic of a dis
continuous control element (units of e) [p. 100]
q' ( = q/Ηχ) Magnitude of lost motion rated in units of the controlled
variable Θ (units of Θ) [p. 106]
8 Small increment in the lost-motion magnitude (units of Θ)
Q
[p. 104]
Reference input (units of e) [p. 92]
Laplace complex variable (= ζ + jœ) (sec-1) [p. 41]
General symbol for time (usually, sec) Note: θ(ί) indicates
Θ as a time-varying function; similarly, m{t), etc.
Time of switching operation (sec) [p. 23]
Cooling time (sec) [p. 134]
Elapsed time since last switching operation (sec) [p. 71]
t Auxiliary cycling time variables in a quasicontinuously
eQ
controlled process (sec) [p. 171]
Heating time (sec) [p. 134]
On time (sec) [p. 134]
Off time (sec) [p. 134]
Periodic time (sec) [p. 135]
Minimum value of t (sec) [p. 145]
q
Time corresponding to temperature 6 (sec) [p. 83]
y
At-Atf, etc. Arbitrary increments in t that correspond with increments
x y
AJ-Aß, etc. in Θ (sec) [p. 83]
At Overshoot recovery time (sec) [p. 134]
0
At Undershoot recovery time (sec) [p. 134]
u
u Disturbance input (units of Θ) [p. 79]
v ( = ωΤ) 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 ( = 0 — Θ) Error in the control of a process (units of Θ) [p. 107]
r
y ( = 0 _ Θ) Mean value of y in DEC [p. 124]
r
y Error due to overcompensated secondary feedback (jb > q).
d 2
A component of y (units of Θ) [p. 114]
y Error due to deadspace magnitude p (units of Θ) [p. 180]
v
z Arbitrary constant [p. 18]
z[ = (rh + l)/2] Switching condition for bang-bang controllers (numeric)
[p. 159]
Zi Incoming quality level [p. 163]
z Outgoing quality level [p. 163]
0
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.]
NOMENCLATURE XV
C(n = 1, 2,...) Thermal capacitance at specified locations (Btu/°F) [p. 59]
n
C Capacitance of coating [p. 61]
e
C Capacitance of mercury [p. 61]
m
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. Ill 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(ju)) Transfer function [p. 35]
H General symbol for feedback transfer function
Ηχ PFE sensitivity (units of e and units of Θ) [p. 92]
H SFE sensitivity (units of 0) [p. 100]
2
J Imaginary component [p. 35]
K, K Arbitrary constants [pp. 77, 86 et seq.]
x 2
L Arbitrary dimension [p. 75]
J£? Laplace transform [p. 43]
M Mass (lb) [p. 4 et seq.]
N Disturbance-input element attenuation (numeric) [pp. 76,
79]
{Nn}n-i,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,..., 0
Q Quantity of flow (lb/sec) [p. 69]
R Thermal resistance [T/tBtuXsec)"1] [p. 4]
R Coating resistance [p. 61]
e
R Equivalent thermal resistance of a parallel ensemble
e
Re = [Σ d/i?»)]-1 [p. 79]
R Glass resistance [p. 61]
g
{R} (n = 1, 2,...) R at specific locations [p. 59 et seq.]
n
& Real component [p. 35]
RIE Reference input element [p. 92]
S Specific heat [Btu/(lb)(°F)] [p. 4]
»S , S, etc. SFE's in a quasicontinuous temperature controller [p. 168
0 b
Si , S, etc. et seq.]
2
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/tft'OCF^Xsec-1)] [p. 76]
V General symbol for velocity (ft/sec) [p. 76]
V Velocity of light [p. 64]
L
X Arbitrary constant [p. 18]
Y Arbitrary constant [p. 18]
Z Arbitrary constant [p. 18]
XVI NOMENCLATURE
Z Constant of integration [p. 76]
a Arbitrary constant [p. 66]
j8 Arbitrary constant [p. 67]
y Inverse of thermal difrusivity (sec/ft2) [p. 66]
y[=l — exp( —τ/Τ)] Index of process time parameters [pp. 132, 134]
y , y y index for heating and cooling functions of a heating—cooling
k e
process [p. 161]
δ Small increment i.e., ΒΘ
8(t) Unit impulse function [p. 24]
e Base of natural logarithms
ζ Damping factor (s = ζ + j<a) [p. 37]
Θ Controlled variable of a thermal process (typically °F, but
could equally be °C, °K, etc.) [p. 92]
θ , 9 Temperatures at specific locations (a, b, etc.)
α b
d Amplitude of DEC (units of 0) [p. 14]
d
9 Auxiliary variable in a quasicontinuous temperature controller
g
(units of Θ) [p. 168]
6 Input temperature (units of Θ) [p. 15]
t
B Maximum value of d(t) in DEC (units of 0) [p. 125]
m
0 Minimum value of 0(i) in DEC (units of 0) [p. 125]
n
0 Amplitude of a periodic output, i.e., 0 = 0 sin ωί [p. 28]
O O
θ Overriding command (units of Θ) [p. 117]
ρ
0 Command (units of Θ) [p. 88]
r
e« Steady-state value of Θ [p. 37]
o. Disturbance (units of Θ) [p. 77]
Disturbance ensemble (units of Θ) [p. 77]
Θ, Arbitrary value of Θ corresponding to time t [p. 82]
y
«L ( = *, - u) Process loading (units of Θ) (p. 134)
e (=r- ■w Process reserve (units of Θ) [p. 134]
s
#1 , 02 Temperatures at specific locations [p. 3] Temperature at
points 1, 2, etc. [p. 123]
8Θ Departure of θ(ί) from a quasicontinuous controller's profile
(units of Θ) [p. 180]
ΔΘ Arbitrary increment of Θ (units of Θ) [p. 83]
Δ^β-Δ^θ Increments in Θ corresponding to times Δ1-Δ$ (units of Θ)
χ
[p. 83]
ΔΘ Overshoot (units of Θ) [p. 134]
0
Δθ Undershoot (units of Θ) [p. 134]
η
λ Index of command { = &S!^L) [p. 133]
λ Wavelength [pp. 64, 65]
p Thermal conductivity [p. 7]
T Transit delay (sec) [p. 47]
T Transit delay between heating—cooling apparatus and con
g
trolled variable (sec) [p. 77]
T Transit delay between disturbance and controlled variable
U
(sec) [p. 77]
φ Phase angle of a phasor (deg) [p. 28]
