On the Dynamic Characteristics of Heat Exchangers
Abstract Generated abstract
This paper examines the dynamic characteristics of continuously operating heat exchangers whose external action, such as heating-medium temperature or heat flux, is constant along the flow coordinate. It proposes an analytical approach that relates distributed partial differential models to lumped mathematical models, separates the main external-action channels, and derives transfer-function expressions for one- and two-capacity heat-exchanger models under standard simplifying assumptions. The approach supports classification of heat exchangers by dynamic properties, construction of tables of exact analytical characteristics, simplified calculation formulas based on ordinary differential equations, and an experimental-analytical method for identifying dynamic characteristics from response curves.
Full Text
ON THE DYNAMIC CHARACTERISTICS OF HEAT EXCHANGERS
P. A. ANDRIYANOV
(Presented by the scientific final conference of the Faculty of Chemical Technology)
Knowledge of the dynamic characteristics of heat-exchange apparatuses makes it possible to calculate automatic-control systems at the design stage. The dynamic properties of heat exchangers—typical representatives of continuously operating apparatuses—are described by partial differential equations. The approach proposed below to the consideration of dynamic characteristics, connected with the use of the concept of a lumped mathematical model, makes it possible to reveal regularities in the construction of analytical expressions and provides the possibility of classifying heat exchangers according to their dynamic properties, on the basis of which a table of their dynamic characteristics can be compiled; to investigate the influence of design and technological parameters on the dynamics; to obtain simplified calculation formulas using an ordinary differential equation to describe the dynamic properties of the distributed model of a heat exchanger; to develop a method of mathematical modeling on analog computers and calculation algorithms for electronic digital computers; and to obtain an experimental-analytical method for determining dynamic characteristics from experimental dynamic data (response curves and frequency characteristics).
The model considered here covers a large group of heat-exchange apparatuses united by one common feature—the constancy of the external action, the temperature of the heating heat-transfer medium or the heat flux along the spatial coordinate $x$, taken in the direction of motion of the flow of the heated heat-transfer medium.
These include:
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Heat exchangers with a constant temperature of the heating heat-transfer medium, for example, under conditions of phase transformation of the latter (condensation, boiling) at constant pressure; steam preheaters, condensers-coolers, etc.;
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Heat exchangers with a constant heat flux supplied by the heating heat-transfer medium and independent of the temperature of the heated heat-transfer medium, for example, in radiative heat exchange, heat transfer from a high-potential source (steam superheaters, etc.);
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Heat exchangers with internal heat sources with heat generation uniform along the length (in the case of electric heating, nuclear reactors, etc.).
In the distributed model of a heat exchanger, three channels of external actions are distinguished: “\(t_0—t(l,\tau)\),” “\(f—t(l,\tau)\),” and “\(v—t(l,\tau)\),” with the corresponding transfer functions \(W_p(p)\), \(W_f(p)\), and \(W_v(p)\). Then the transfer function of the heat exchanger is
\[ W(p)=W_p(p)+W_f(p)+W_v(p). \tag{2} \]
The dynamic properties of the multicapacity models of heat-exchange apparatus under consideration, with the usually adopted assumptions (independence of the physical properties of the heat carriers from temperature; high turbulence of the flow of the heated heat carrier; unlimited thermal conductivity of the walls in the direction perpendicular to its motion; constancy of the heat-transfer coefficients; neglect of the axial heat flux), are described by a linear nonhomogeneous partial differential equation of first order with respect to the spatial coordinate \(x\) and of arbitrary order with respect to the variable time, which, after applying a double Laplace transform with respect to \(x\) and \(\tau\), takes the form
\[ B(p)\bar{v}t(s,p)+A(p)\bar{t}(s,p)=\bar{f}(s,p)+\bar{t}(0,p). \tag{2} \]
If, from the distributed model to which equation (2) corresponds, one isolates a lumped mathematical model, with the corresponding ordinary differential equation
\[ A(p)\bar{t}(p)=\bar{f}(p), \tag{3} \]
then the transfer function (1) for \(f(x)=\mathrm{const}\) takes the form
\[ W(p)=W_p(p)+W_c(p)\left[1-W_p(p)\right]+W_v(p), \tag{4} \]
where
\[ W_f(p)=W_c(p)\left[1-W_p(p)\right]. \tag{5} \]
In turn,
\[ W_p(p)=\exp\left(-\frac{z_0}{T}\right)\exp(-z_0p)\prod_{n=1}^{n-1}\exp\frac{\lambda z_0}{p+\beta}, \tag{6} \]
where the elements with transfer function
\[ \exp\frac{\lambda z_0}{p+\beta} \tag{7} \]
appear in two- and multicapacity models of heat-exchange apparatus.
On the basis of the approach set forth for the analysis of dynamic characteristics, and in accordance with the resulting classification of heat-exchange apparatus by the nature of their dynamic properties in the main regulating channel, a table of exact analytical expressions has been constructed for determining the dynamic characteristics of one- and two-capacity models, taking into account the thermal capacities of the heated heat carrier and the wall of heat-exchange apparatus with an external action constant in \(x\). The external action is: the wall temperature for a one-capacity model, or the temperature of the heating heat carrier for a two-capacity model; heat released in the heated heat carrier or entering it from the wall for a one-capacity model, or heat released in the wall or entering through its outer surface for two-capacity models. This includes a large group of heat-exchange apparatus: steam heaters, condensers-coolers, electric heaters, steam superheaters, thermal unheated pipes, etc. [1].
On the basis of an analysis of the quantitative influence of the terms of the analytical expressions, a calculation method using simplified formulas is proposed, with extensive use of the ordinary differential equation of a lumped mathematical model [2].
The description of the transient response in the initial time interval by the solution of an ordinary differential equation, together with the expression for the steady-state value, is used to determine the dynamic characteristics of distributed models from experimental acceleration curves; this forms the basis of the proposed experimental–analytical method for determining the dynamic characteristics of heat-exchange apparatuses.
In the following articles, algorithms are given for calculating transient responses by exact computational formulas.
REFERENCES
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P. A. Andrianov, I. M. Maslennikov. Methods for the analytical determination and analysis of the dynamic characteristics of continuously operating apparatuses. Heat and Mass Transfer, vol. 6, 421–433, Minsk, 1966.
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P. A. Andrianov. Candidate dissertation, Moscow, MIKhM, 1964.
Notation
$t(l,\tau)$ — change in the temperature of the heated heat-transfer medium at the outlet of the heat exchanger;
$t_0 \equiv t(0,\tau)$ — temperature at the inlet to the heat exchanger;
$v$ — velocity of motion of the heated heat-transfer medium;
$\tau_0$ — residence time of the liquid during the heating process in the apparatus;
$T, \lambda, \beta$ — constant coefficients depending on design and technological parameters;
$f(x,\tau)$ — change in the external disturbance (the temperature of the heating heat-transfer medium or of the heat flux);
$p, s$ — variables of the Laplace transform with respect to $\tau$ and $x$, respectively.
$A(p), B(p)$ — polynomials in $p$.
$W(p)$ — transfer function.