Static Characteristics of a Liquid Cooling System as a Controlled Object

L. V. Nechaev

(Presented by Prof. Dr. V. K. Nechaev)

Many years of experience in the operation of internal-combustion engines of various types and purposes, theoretical considerations, and specially conducted experiments show that the principal characteristics of an engine—power, economy, and service life with respect to wear—depend to a significant degree on the thermal regime, determined mainly by the temperatures of the oil and cooling water in the cooling system \[1, 2, 3, 4, 5, 6, 7\].

These and other analogous works have established that for each engine there is a certain definite (so-called optimum) temperature in the cooling system which ensures the attainment of the greatest values of effective power, economy, and service life at given cyclic fuel delivery and rotational speed. In the opinion of some authors \[7, 9\], for a given engine the optimum temperature is almost the same for all speed and load conditions. Therefore it is desirable to design the cooling system so that the optimum temperature, nearly constant in magnitude, is maintained in it under various operating conditions.

It is known that an ordinary radiator-type closed liquid cooling system with forced circulation of the cooling liquid does not ensure constancy of temperature in the cooling system when the operating condition of the engine changes \[1, 2, 5, 9, 10\]. The thermostats, shutters, and insulating covers used up to the present time contribute to a certain degree to stabilizing the water temperature, but in themselves they have a number of substantial shortcomings; therefore their use is not a successful solution to the problem of maintaining the most advantageous temperature regime.

In recent years a number of firms (Porsche, Saurer—Cummins, General Motors, and others), our domestic plants (Gorky, Ural, Minsk automobile plants), and scientific-research institutes (NAMI, NATI) have been giving serious attention to the development of devices that automatically change the heat-dissipating capacity of the radiator by periodically switching it off, smoothly changing the rotational speed, or changing the setting angle of the fan blades \[8\]. At the same time, a reduction in the power expenditure on the fan is achieved, especially noticeable

during engine operation in winter and at partial loads [2, 5, 8, 11].

However, the creation of such devices in most cases proceeds “by feel.” In textbooks and specialized literature, the questions of automatic control in the cooling system are not given proper attention; there are almost no data on how to determine the characteristics of the controlled object—the cooling system; what requirements the regulator must satisfy; and how and on what the controlled parameter—the temperature of the cooling liquid—depends.

It is known from the general theory of control that, in order to select a control system and the type of regulator and to impose definite requirements on it, one must have the static and dynamic characteristics of the controlled object. In our case these characteristics represent the dependence of the controlled parameter—the temperature of the cooling water (most often the temperature of the cooling water \(t_2\) at the engine outlet [8, 9])—on the principal factors that change this temperature and characterize the engine operating conditions, i.e., the load \(p_e\), the crankshaft speed \(n\), and the temperature of the air \(t_{в1}\) cooling the radiator, i.e.,

\[ t_w = f(p_e, n, t_{в1}), \]

or

\[ t_2 = f(p_e, n, t_{в1}), \]

where \(t_w\) is the mean temperature in the cooling system.

The static characteristics determine these dependences for the case of an infinitely slow change in the quantities \(p_e\), \(n\), \(t_{в1}\). The dynamic characteristics determine the nature of the change in temperature \(t_w\) (or \(t_2\)) with time for a prescribed law of variation of \(p_e\), \(n\), or \(t_{в1}\) (usually a stepwise change in these quantities is specified).

It should be noted that knowledge of the static characteristics of the cooling system will be useful for solving many other problems not associated with automation in the cooling system—questions of increasing the effectiveness of the cooling system, reducing the dimensions of the radiator, as well as questions of heat transfer, economy, service life, detonation, etc., under operating conditions.

At present there is very little factual material on the static characteristics of the cooling system [for example, 2, 9, 12], and an analytical generalization of experimental data, with the exception of the work of E. A. Kanarchuk [12], carried out for the engine of the GAZ-MM automobile, is lacking.

It is clear that static characteristics can be determined experimentally and, from experimental data, empirical dependences can be derived:

\[ t_w = f_1(p_e) \quad \text{for } n = \mathrm{const} \text{ and } t_{в1} = \mathrm{const}, \]

\[ t_w = f_2(n) \quad \text{for } p_e = \mathrm{const} \text{ and } t_{в1} = \mathrm{const}, \]

\[ t_w = f_3(t_{в1}) \quad \text{for } p_e = \mathrm{const} \text{ and } n = \mathrm{const}. \]

This is precisely the work carried out by E. A. Kanarchuk [12]. The character of the curves for different engines under identical operating conditions, as well as for a single engine under different operating conditions, may be very different. Naturally, such a method of determining static characteristics is associated with a large expenditure of time for conducting numerous experiments and with the need to have an already finished engine.

Establishing some generalized regularity for different engines and operating conditions will be difficult, since in doing so the mechanism of heat transfer from the gases to the water in the engine and from the water to the air in the radiator proves to be concealed. It is therefore clear that static characteristics obtained experimentally, as well as the regularity of their course established empirically from experimental data, will have a particular character and can be used for other engines only with very great difficulty.

The establishment of general regularities in the course of static characteristics must be based on an analysis of the mechanism of heat transfer in the engine and the radiator.

In the present article, equations are derived analytically that relate the temperature in the cooling system to the operating conditions or to certain characteristics of the cooling system that depend directly or indirectly on these conditions.

It is known that, in calculations and investigations of cooling systems, the Newton equation for convective heat exchange is widely used in engine-building practice; it provides quite sufficient practical accuracy [2, 10].

Newton’s equation in elementary form is

\[ dQ_{wd} = kF\,(T - t_w)\,d\tau, \]

where

\(dQ_{wd}\) — the amount of heat given up by the gases through the cylinder walls to the water, in kcal;

\(k\) — the instantaneous value of the heat-transfer coefficient (determined from the equations of N. R. Brilling, Nusselt, Eichelberg, Woschni, etc.), in kcal/\(m^2\)·h·deg;

\(T\) — the instantaneous value of the temperature of the gases in the cylinder, in \(^\circ\)C;

\(t_w\) — the mean water temperature, in \(^\circ\)C;

\(d\tau\) — an elementary interval of time.

The quantities \(k\), \(F\), and \(T\) in a real engine vary with time. The amount of heat transferred to the water per hour is proportional to the amount of heat per cycle and to the number of cycles

\[ Q_{wd} = Z \int_{\tau_{\text{cycle}}} dQ_{wd}, \]

where \(Z\) is the number of cycles per hour.

It is known that the heat transfer \(Q_{wd}\) can be represented in the form of Newton’s equation

\[ Q_{wd} = k_{dw}\,F_{dw}\,(T_e - t_w), \quad \frac{\text{kcal}}{\text{h}}, \tag{1} \]

where

\(Q_{wd}\) — the amount of heat removed from the gases to the water in the engine per hour;

\(k_{dw}\) — the effective coefficient of heat transfer from the gases through the cylinder wall to the coolant, in kcal/\(m^2\)·h·deg;

\(F_{dw}\) — the effective heat-exchange surface area (constant in time) for all cylinders, in \(m^2\);

\(T_e\) — the effective temperature of the gases in the cylinder, in \(^\circ\)C;

\(t_w\) — the mean water temperature in the engine, in \(^\circ\)C, equal to

\[ t_w = \frac{t_1 + t_2}{2}, \]

where \(t_1\) is the temperature of the water entering the engine, in \(^\circ\)C;

\(t_2\) — temperature of the water leaving the engine, in \({}^{\circ}\mathrm{C}\).

By the effective values \(k_{de}, T_{de}\), and \(T_e\) one should understand such time-constant (under given operating conditions) values of the heat-transfer coefficient, heat-exchange area, and gas temperature that provide, over one hour, the same heat transfer as in the real engine with variable \(k\), \(F\), and \(T\). These effective values \(k_{de}, T_{de}\), and \(T_e\) can be determined from experiment by appropriate processing of the results of temperature measurements during heating and cooling of the engine.

Newton’s equation for heat transfer in the radiator has the form

\[ Q_p = k_p F_p (t_w - t_v),\ \frac{\mathrm{kcal}}{\mathrm{h}}, \tag{2} \]

where

\(Q_p\) is the quantity of heat removed from the water to the air in the radiator, in \(\mathrm{kcal}\);

\(k_p\) is the heat-transfer coefficient from water to air in the radiator, in \(\mathrm{kcal}/\mathrm{m}^2\,\mathrm{h}\,\mathrm{deg}\);

\(F_p\) is the effective cooling area of the radiator, in \(\mathrm{m}^2\);

\(t_v\) is the mean air temperature in the radiator, in \({}^{\circ}\mathrm{C}\), equal to

\[ t_v = \frac{t_{v1} + t_{v2}}{2}; \]

\(t_{v1}\) and \(t_{v2}\) are the air temperatures before and after the radiator, in \({}^{\circ}\mathrm{C}\).

For any steady operating condition the heat balance in the cooling system has the form

\[ Q_{wd} = Q_p \beta, \tag{3} \]

where \(\beta\) is a coefficient greater than 1, allowing for heat transfer by the side surface of the engine. Usually in calculations one takes

\[ \beta = 1. \]

If one uses the equations available in the literature [2, 10, 12, 13] that relate the quantity of heat \(Q_{wd}\) to the engine design and operating conditions, then, on the basis of equations (2) and (3), assigning from radiator-test data the values \(k_p\) and \(F_p\) for the particular radiator, one can determine the thermal head in the radiator, equal to

\[ \Delta t = t_w - t_v = \frac{Q_{wd}}{k_p F_p}. \]

In this equation we replace the mean temperature values \(t_w\) and \(t_v\) by the initial and final values according to the equations

\[ t_2 = 2t_w - t_1;\quad t_{v2} = 2t_v - t_{v1}, \]

and we also use the equations for the quantity of heat given up by the water in the radiator

\[ Q_{wp} = G_w c_w (t_2 - t_1)\ \frac{\mathrm{kcal}}{\mathrm{h}} \tag{4} \]

and received by the air

\[ Q_v = G_v c_v (t_{v2} - t_{v1})\ \frac{\mathrm{kcal}}{\mathrm{h}}, \tag{5} \]

where \(G_w\) and \(G_v\) are the hourly flow rates of water and air, in kg/hour; \(c_w\) and \(c_v\) are the heat capacities of water and air, in kcal/kg deg, and let us write two equations

\[ \Delta t = t_w - t_v = \frac{t_2 + t_1}{2} - \frac{t_{v2} + t_{v1}}{2} = \frac{Q_{wo}}{k_p F_p}, \]

\[ G_w c_w (t_2 - t_1) = G_v c_v (t_{v2} - t_{v1}). \]

From these equations one can determine the unknown quantities \(t_2, t_1\), or \(t_{v2}\). For example,

\[ t_2 = \frac{Q_{wo}}{k_p F_p} + \frac{t_{v2}+t_{v1}}{2} + \frac{1}{2}\frac{G_v c_v}{G_w c_w}(t_{v2}-t_{v1}). \]

On the basis of the equation of the heat balance of the entire system, which for the steady-state regime has the form

\[ Q_{wo}=Q_{wp}=Q_p=Q_v, \tag{6} \]

and equation (5), one can finally determine the temperature \(t_2\),

\[ t_2 = \frac{Q_{wo}}{k_p F_p} + \frac{1}{2}Q_{wo} \left( \frac{1}{G_v \cdot c_v} + \frac{1}{G_w \cdot c_w} \right) + t_{v1}. \tag{7} \]

Equation (7) is an analytical expression for the static characteristic of the cooling system. The inconvenience of this equation lies in the fact that it includes the quantity \(Q_{wo}\), which itself depends on the temperature of the cooling water \(t_w\) [10, 12, 13]. We, however, wish to use the equations of the static characteristics precisely to determine the temperature of the cooling water. This circumstance greatly complicates the problem.

Let us use another method to determine the relation between the water temperatures \(t_2\) and \(t_1\) and the operating conditions, which in our equations (1), (2), (4), and (5) are taken into account indirectly or directly in the form of the air temperature \(t_{v1}\) and the values corresponding to the given operating regime \(k_{dv}, F_{dv}, T_e, k_p, G_v, G_w\).

Rewrite equations (1), (2), (4), and (5) in the following form:

\[ Q_{wo} = k_{dv} F_{dv} \left( T_e - t_1 - \frac{\Delta t_w}{2} \right), \tag{1a} \]

\[ Q_p = k_p F_p \left( t_1 + \frac{\Delta t_w}{2} - t_{v1} - \frac{\Delta t_v}{2} \right), \tag{2a} \]

\[ Q_{wp}=G_w c_w \Delta t_w, \tag{4a} \]

\[ Q_v=G_v c_v \Delta t_v. \tag{5a} \]

In these equations,

\[ \Delta t_w = t_2 - t_1 \]

is the heating of the water in the engine, equal in magnitude to the cooling of the water in the radiator (for the steady-state operating regime), in °C; \(\Delta t_v = t_{v2} - t_{v1}\) is the heating of the air in the radiator, in °C.

Applying the method of solving a system of equations by means of determinants, it can be shown that this system of equations has a unique solution for the unknowns \(t_1, \Delta t_w, \Delta t_v, Q\), where, on the basis of equality (6),

\[ Q = Q_{wo}=Q_{wp}=Q_p=Q_v. \]

Denote

\[ k_{dv} T_{dv}=a,\qquad k_p F_p=b,\qquad G_w c_w=c,\qquad G_v c_v=d. \]

Using equality (6) and the adopted notation, we solve the system of equations (1a), (2a), (4a), and (5a) with respect to the sought unknowns \(t_1\), \(\Delta t_w\), \(\Delta t_v\), and \(Q\):

\[ t_1 = T_e \left[ 1 - \frac{bd(2c+a)}{c(2bd+2ad+ab)} \right] + \]

\[ {}+ t_{v1}\frac{bd(2c+a)}{c(2bd+2ad+ab)}, \tag{7} \]

\[ \Delta t_w = (T_e - t_{v1})\,\frac{2abd}{c(2bd+2ad+ab)}, \tag{8} \]

\[ \Delta t_v = (T_e - t_{v1})\,\frac{2ab}{2bd+2ad+ab}, \tag{9} \]

\[ Q = (T_e - t_{v1})\,\frac{2abd}{2bd+2ad+ab}. \tag{10} \]

On the basis of equations (7) and (8) it can be found that the temperature \(t_2\) at the engine outlet is equal to

\[ t_2 = T_e \left[ 1 - \frac{bd(2c-a)}{c(2bd+2ad+ab)} \right] + t_{v1}\frac{bd(2c-a)}{c(2bd+2ad+ab)}. \tag{11} \]

The derived relationships make it possible to determine analytically the character of the change in any parameter (i.e., \(t_1\), \(t_2\), \(\Delta t_w\), or \(\Delta t_v\)) when one or several characteristics of the system change. By characteristics of the system we shall mean the quantities \(a\), \(b\), \(c\), \(d\), and \(T_e\).

As is evident from equations (7) and (11), the water temperatures \(t_1\) and \(t_2\) depend linearly on the air temperature \(t_{v1}\) (with the other characteristics constant). The experiments of E. A. Kanarchuk [12] show that, in practice, the dependence of the mean water temperature on the air temperature is very close to linear.

If, on the basis of equations (7) and (11), one constructs the graphical dependence of \(t_2\) and \(t_1\) on \(a = k_{dv}F_{dv}\), then curves are obtained similar to those shown in Fig. 1. Analyzing equations (7) and (11) and the graphs in Fig. 1, one may conclude that the steady-state water temperatures at the engine inlet and outlet vary with \(a\) according to a curvilinear law, asymptotically approaching certain limiting values

\[ \lim_{a \to \infty} t_1 = T_e - \frac{(T_e-t_{v1})bd}{2dc+bc}, \]

\[ \lim_{a \to \infty} t_2 = T_e + \frac{(T_e-t_{v1})bd}{2dc+bc}, \]

and the limit of the mean temperature \(t_w\) is equal to

\[ \lim_{a \to \infty} t_w = \frac{\lim t_1 + \lim t_2}{2} = T_e. \]

It should be noted that the expression for \(\lim_{a \to \infty} t_2\) has no physical meaning, since the temperature of the system cannot be higher than the tem-

…of the gas temperature \(T_{\mathrm{e}}\); therefore the mean temperature of the water for any values of \(a\) and \(b \ne 0\) will be lower than the gas temperature \(T_{\mathrm{e}}\).

If equations (7) and (11) are used to determine the mean temperature of the water, it can be found that its value does not depend on the velocity of the water. This circumstance considerably simplifies the problem of analytically determining the mean temperature in the cooling system under an unsteady thermal regime.

The study of the dependences of the parameters \(t_1\), \(t_2\), \(\Delta t_w\), \(\Delta t_v\) on the system characteristics according to equations (7)—(11) should, naturally, be begun for the case of an idealized engine scheme, assuming that

Fig. 1.

the characteristics of the system do not depend on one another. By an idealized scheme, for which the equations under consideration have been drawn up, we shall understand an engine in which:

a) the heat-transfer surface \(F_{dv}\) between the gas and the coolant is invariant with time;

b) the volume of the cylinders with surface \(F_{dv}\) is filled with gas at the constant temperature \(T_{\mathrm{e}}\);

c) the heat-transfer coefficient \(k_{dv}\) from the gases to the water through the cylinder wall is constant in time and over the surface \(F_{dv}\).

It was indicated above that in a real engine, because of the cyclic nature of its operation, the heat-transfer coefficient \(k\), the temperature \(T\), and the heat-transfer surface \(F\) vary with time. The effective quantities \(k_{dv}\), \(F_{dv}\), \(T_{\mathrm{e}}\) are related to the instantaneous values \(k\), \(F\), \(T\) by the following relations:

\[ F_{dv}= \frac{\displaystyle \int_{\tau=0}^{\tau_{\text{cycle}}} F\,d\tau} {\tau_{\text{cycle}}}, \]

\[ k_{dv}= \frac{\displaystyle \int_{\tau=0}^{\tau_{\text{cycle}}} kF\,d\tau} {\displaystyle \int_{\tau=0}^{\tau_{\text{cycle}}} F\,d\tau}, \]

\[ T_{\mathrm{e}}= \frac{\displaystyle \int_{\tau=0}^{\text{cycle}} kFT\,d\tau} {\displaystyle \int_{\tau=0}^{\text{cycle}} kF\,d\tau}. \]

In an actual engine, the characteristics of the system \(a\), \(b\), \(c\), \(d\), and \(T\) are interrelated and depend, indirectly or directly, on the operating conditions, i.e., on the load \(p_c\), the crankshaft speed \(n\), and atmospheric conditions. In a number of cases these dependences are either sufficiently simple or have been studied to a considerable extent (for example, the dependence of fan capacity on its rotational speed, the dependence of the radiator heat-transfer coefficient on the mass velocity of the air and, consequently, on the fan capacity). In other cases, the interrelation among the characteristics and their dependence on operating conditions are complex and can be established only on the basis of additional investigations of various engines (for example, the dependence of the temperature \(T_{\mathrm{e}}\) on load or rotational speed, the interrelation between the characteristics \(a\) and \(T_{\mathrm{e}}\), and the dependence of the water-pump capacity on impeller speed at various temperatures \(t_1\)).

The author intends to discuss the questions of the interrelation of the system characteristics and their dependence on operating conditions, studies of the static characteristics of an idealized engine, the question of the dynamic characteristics of a liquid cooling system, and the method for determining the effective values \(k_{\mathrm{av}}\), \(E_{\mathrm{av}}\), and \(T_{\mathrm{e}}\) in subsequent articles.

REFERENCES

1. Levin M. I. Optimal temperature conditions in engine cooling systems and requirements for automatic temperature control. Collection TsNIDI, No. 26, Mashgiz, 1954.

2. Silaev A. A. Liquid Cooling Systems of Engines. Mashgiz, 1948.

3. Druzgal’skii V. V. The influence of cylinder-liner temperature on the intensity of their corrosive wear. Journal “Mechanization and Electrification of Socialist Agriculture,” No. 4, 1961.

4. Kugel’ R. V. Durability of Automobiles. Mashgiz, 1961.

5. Kuvshinov Ya. Improving the thermal regime of tractor engines. Journal “Technology in Agriculture,” No. 7, 1961.

6. Gurvich I. B. Wear of Automobile Engines. From the experience of the Gorky Automobile Plant, Mashgiz, 1961.

7. Salimov A. U. The influence of the thermal state of an engine on its power and economy. Proceedings of the Central Asian Polytechnic Institute, issue 15, 1961.

8. Minkin M. L. Automatic control of the cooling-system fan. Journal “Automobile Industry,” No. 10, 1958.

9. Levin M. I., Tsirkin M. I. Systems for the automatic regulation of temperatures in marine diesel installations. Sudpromgiz, 1959.

10. Design and Calculation of Automotive and Tractor Engines, ed. Prof. Yu. A. Stepanov. Mashgiz, 1957.

11. Gavrilov F. Investigation of the influence of radiator-fan operation on the thermal state and economy of an automotive tractor engine. Proceedings of the Siberian Automobile and Road Institute, collection 6, Omsk, 1957.

12. Kanarchuk E. A. Investigation of the influence of operating conditions on the temperature regime of an automobile-engine cooling system. Candidate dissertation, Kiev, 1950.

13. Rakhmanovich A. N. The influence of cooling temperature on diesel operation. Proceedings of TsIAM, collection No. 1, 1936.