SESSION 14.2

MODELLING OF THE DYNAMICS OF A LOW SPEED GAS-LIQUID HEAT ENGINE

J. A. R. PARISE and C. M. P. CUNHA

PONTIFfCIA UNIVERSIDADE CAT6LICA DO RIO DE JANEIRO DEPARTMENT OF MECHANICAL ENGINEERING

Abstract

The present paper deals with the simulation model of a gas-liquid heat engine which is characterized by very low speeds (1-3 rpm) and relatively high torque. The engine operates according to the *Mint0 Thermal Wheel”. It is based on the conversion of thermal energy from the heat source, through gas expansion, into mechanical work, by means of the fall of a mass of liquid. A prototype has already been constructed showing great ability to operate at very low tem- perature differences between the heat source and heat sink. This makes the engine quite suitable to the utilisation of low temperature heat sources such as solar energy and waste heat. On the other hand, the number of moving parts is kept to a minimum, since the piston of traditional positive displacement engines (PDE) is now replaced sim- ply by a mass of liquid. The mathematical model consists of applying the energy equation, in its time-derivative form, to representative en- gine control volumes, resulting in a set of linear ordinary diferential equations. Their integration provides the time variation of pressure and temperature of the working fluid. The engine performance can thus be predicted as a function of engine operating conditions and geometric characteristics. In the present paper, the engine dynam- ics (ie, variable angular speed) has been taken into account, as well as heat losses in the engine structure. Results and further design considerations are discussed.

1. Introduction

The present paper is concerned with the development of a simulation model for a low-speed gas-liquid heat engine. This engine, depicted in Figure 1, differs from conventional prime-movers, such as the recip- rocating internal combustion engine, the gas turbine or the Stirling engine, in the sense that it is capable of operating at a much lower heat source to heat sink temperature difference. It is known as the ” M i t o engine”, named after its inventor \I].

Figure 2 shows, schematically, the engine principle of operation, with the sequence of events that are assumed to occur within it. It con- sists of two chambers, A and B, connected by a tube. The whole system, called module (M), is completely sealed and rotates contin- uously. Initially, state 1, all liquid is in chamber (A). The remaining of the module (residualvolume of chamber (A), connecting tube and chamber (B)) is filled with gas. If the gas is the liquid own vapour, so that either vaporization or condensation are allowed to occur, the engine is said to be of the ‘wet’ type; otherwise it is regarded as ‘dry”. The heat source is located at the bottom of the engine and the heat sink at the top. Initially, the engine is placed horizontally, with chamber (A) in contact with the hot side and chamber (B) with the cold one.

As the engine rotates, anti-clockwise, heat transfer from the heat source makes the gas in chamber (A) expand, “pumping” the liquid, through the column, to chamber (B), process 1-2. At the same time, in chamber (B), the gas is being cooled by the heat sink. Thus, the process of liquid raise is controlled by the heat transfer rates

in both upper and lower chambers. As it happens, an increasing pressure difference, due to the liquid column in the connecting tube, is set between the two chambers. The transfer of liquid from (A) to (B) has meant that thermal energy was partially converted into gravitational potential energy, moving the engine centre of gravity. The unbalance that has been created corresponds to the available torque at the engine shaft. Basically, the amount of liquid that has been pumped and the distance between chambers will determine the amount of torque that is produced.

After half turn, state 2, the liquid, that has not been pumped to (B), drains by gravity into (B). After that, state 3, gas pressure tends to become uniform throughout the module. Positions are now inverted, process 3-4, with (B) in the hot side and (A) in the cold one. The cycle, therefore, corresponds to a full turn of the model. By associating a number of modules, a ‘wheel” can be formed, as shown in Figure 1. It allows a more steady movement of the wheel as well as less variation on the net torque output.

A programme has been under way at the Department of Mechan- ical Engineering of, PUC - RJ for the development of the engine. A first prototype was constructed and tested [2]. It consisted of a four-module ’wet’ unit, running with Refrigerant-11. Hot water, with temperatures ranging from 40°C to BOOC, was used as the heat source. The prototype was tested against a specially built dynamome- ter, with power output, shaft torque, thermal efficiency and heat

ENGINE

Figure 1 - The Minto wheel (four-module configuration)

losses being measured. The main attributes of the engine were found to be high torque, constructional simplicity and the capacity to op- erate at very low heat source to heat sink temperature differences. Thermal efficiency was inherently low making, the engine more suit- able to work with alternative nondepleting heat sources such as solar energy or waste heat.

46

Concomitantly, work has been under way in the development of a simulation model. The use of a computer code, simulating the be- haviour of the wheel, helps explore the influence of a number of engine parameters (wheel diameter, volume of the chambers, gas pressure, mass of liquid, etc) on its performance. A 6rst description of the simulation model is provided in 131. The main objective of that p k per was to present the basic approach for simulating such an engine. Being a positive dipalcement device (ie, with mechanical work be- ing obtained by means of volume variation of a working gas) it was demonstrated that it could be simulated by the same control-volume lumped-parameter approach that was used successfully in the simu- lation of reciprocating compressors and internal combustion engines.

6

P

STATE 3 e -tw e = 180’

ss 3 - 4 065600 e = 360°

Figure 2 - Sequence of events of the engine

In the model representative control volumes of the engine are se- lected and the energy equation, in its time-derivative form, is applied to each of them. Other equations, relating the properties involved in the energy equation, are also necessary. The result is a set of ordi- nary differential equations which, when integrated, provide the time variation of the properties of the working fluid within the control volumes.

It was stated [3] that the model was still far from providing a realistic prediction of the engine behaviour, basically for not considering a number of aspects of the engine. Two of them, observed during tests with the prototype [2], have now been taken into account. They are:

1) the non-uniformity of the shaft speed and 2) the heat losses due to the thermal storage capacity of the engine metallic structure (“regenerator” effect).

The modifications to the model, to account for these aspects, are developed in the presented paper. Results, showing their effect on the engine predicted performance, are presented.

2. Mathematical Model

Six processes are identified in the cycle. They can be described, for chamber A, as follows.

1- 1’: expansion and heating of the gas; liquid pumping to the other chamber; the column is partially filled with liquid.

1’-2: expansion and heating of the gas; liquid pumping to the chamber; the column is completely filled with liquid.

2 -3: the liquid flows, by the force of gravity, to the other cham- ber; the gas quantities in both chambers come into contact.

3 -3’: compression and cooling of the gas; liquid pumping from the other chamber; the column is partially filled with liquid.

3’-4: compression and cooling of the gas; liquid pumping from the other chamber; the column is completely filled withh liquid.

4 -1: the liquid flows, by the force of gravity, out from the cham- ber. Two volumes of control are defined for the gas, Figure 3a. The f i s t comprises the volume occupied by the gas in chamber (A). The sec- ond includes the volume occupied by the gas in chamber (B) and, eventually, in the connecting tube. Two other control volumes are required, Figure 3b, for the walls of each chamber. The following assumDtions are made:

Gas conditions are uniform within each of the control volumes; The gas behaves as a perfect gas; The walls of both chambers and connecting tube are rigid; There is no heat transfer between the liquid and the gas; Processes 2-3 and 4 1 are instantaneous; Processes 2-3 and 4 1 are adiabatic; Following processes 2-3 and 41, gas properties are uniform thr- oughout the module; Wall temperature is uniform throughout the entire chamber.

n w 0

/---

Th l T c

Figure 3 - Volumes of control

For process 1-1’-2, with assumptions (a), (b), (c) and (a), the energy equation can be written for each control volume as follows.

where the first term of the LKS refers to the rate of heat transfer between the gas and the wall of the chamber. The second term is the rate of work done, due to variation of the control volume. The right hand side is the rate of change of the gas internal energy. The gas mass, temperature, pressure, volume and specific heat at constant volume are represented by m, T , P , V and c , , respectively; h is the heat transfer coefficient and A is the heat transfer area. Subscripts

47

h, c, A and B refer to the heating and cooling media and to chamber (A) and (B), respectively. The state equation for a perfect gas with constant R is,

At any position of the wheel, during proceslr l-l'-2, gas pressures in chamber A and B are related by the hydrostatic equation as follows,

PA = PB + pLghsin0 (5)

where 0 is the crank angle. Liquid density is given by p ~ , the gravi- tational acceleration is g and h is the length of the column occupied by the liquid. For process 112, when the liquid has completely filled the connecting tube, h = D. The total volume of the module is given by,

VT = VL + VA + VB (6) h m assumption (c) it follows that,

n A f l B - = -- dt dt

Taking equations (3), (4) and (7) into (1) and (2),

(7)

(h&A(TWA - T A ) - mAR-- TA = m (8) VA dt

TB UA (hA)wB(TwB - TB) + m R - - VE = mBcv% (9)

Also, by taking the time-derivative of the hydrostatic equation,

If the column is only partially filled with liquid, then,

And equation (10) must be re-written to give,

A heat balance (energy equation), applied to the chambers walls, Figure Sb, gives,

\ --.

I P C d

Figure 4 - fiee-body diagram of the wheel.

C M = I $ ; C M = M ~ + M , - M ~ (14,151

ML is the shaft torque produced by the unbalance in the two cham- bers and Me accounts for the unbalance produced in the connecting tube, when it is partially filled.

(16) D 2 ML =pLgCOSe-(vB-vA)

The mass moment of inertia of the wheel, I, has three components: one due to the engine structure, Iw, and the other two relative to the amount of liquid in chambers, I,, and in the connecting tube, IL.,

I = I, + IL + Z L ~ (18)

Finally, the rotational speed is equal to the time-derivative of the shaft angle B .

dB dt

U ) = -

Equations (8), (9), (lo), (12), (U), (14) and (20) form a system of seven ordinary differential equations with seven unknowns: TA, TB, VA, T w ~ , T,,,B, w and B. Time, t , is the independent variable.

The system can be written schematically as,

(214 Equations (12) any (13) result from assumption (h), that the inter- nal thermal resistance of the wall is negligible in comparison with dt dt dt the external (wall to gas and wall to heat source/sink) resistances. Metallic walls with a small thikneea, together with a low rotational

Figure 4 shows the free-body diagram of the wheel. The equation of motion will provide the rate of variation of the rotational speed as a function of the resistive torque, MR, and the instantaneous amount of liquid in each chamber, as well as in the connecting tube. Thus, takiing the sum of moments about the centre of rotation, 0,

031- dTA + a3a- dTB +as- d v ~ = b3

dB (214 Cl f, 9)

dTwB dw -b4; -= dt bs; x = b s ; -=br dt n w A speed, make this assumption acceptable. --

dt

where,

a11 = II~AC. i ala = 0 ; ala = PA ; -1 = 0 (22,23,24,25)

48

bi = (hA)wA(Twr - TA) ; ba = (~A) ,B(T ,B - TB)

bs = pLgDw cos 0

b7 = w

If the liquid has not completely filled the column, process 1-1' (h < D), then the following equations apply,

For process 2-3, being adiibatic, assumption (f), and bovolumetric, (c) and (e), it follows that the fluid total energy remains unchanged,

Ea = Es (40)

This gives, with assumption (b), the final temperature, Ts,

S. Numerical Solution

First, the linear system of algebraic equations in dTa/dt, dTB/dt and dVa/dt, eqns. (2la,b,c), is solved. Then, the seven time- derivatives are numerically integrated to provide the values of TA,TB,VA,T,A,T,B, w and 8. The Euler method of integration was used. Integration starts at 8 = 0" (engine horizontally positioned) and pr+ gresses at variable time intervals, so as to give a prescribed angle increment of 0.05 degrees. Initial values for each of the seven vari- ables have to be estimated. After a number of cycles of integration convergence is achieved. Due to the high thermal inertia of the wheel structure, T,A and T,B are the dominant variables for convergence. In some analyses the value of Z,, in comparison with ZL; and Z,, may be large enough to allow for the constant speed hypothesis to be used. In this case the value of be in made equal to zero.

4. Results

A computer code was developed i d utilised in the analysis of the engine. In this section, some of the results obtained are presented. Basic input data for the analysis was as follows [3] .

engine type: heat sink temperature: heat source temperature: wheel diameter: connecting tube diameter: volume of chamber: volume of liquid per module: mass of air charge: number of engine modules: overall heat conductances:

'dry", air-water 3 0 w 8 O O C 1.025 m 0.014 m 50 litres 5 litres 0.1 kg 1 50 W/m2 K

4.1 Influence of Engine Diameter and Shaft Speeed

In this first set of results the constant speed thin-walled (ie, no ther- mal capacity for the chamber walls) model was employed. The objec- tive was to study the effect of D and w on the engine performance. The wheel diameter was made to vary from 0.5 m to 3 m, and the shaft speed, from 0.5 rpm to 3 rpm.

Two parameters were chosen for the analysis: the engine indicated mean torque and the thermal efficiency. The 6rst is calculated by,

2% - Mi = M(0)dB

where M(0) is the net torque at each crank-angle.

J 2 3

SHAFT SPEED (rpm)

Figure 6 - Predicted mean indicated torque as a function of wheel diameter and shaft speed.

Figur-5 shows the variation of zi with D and w. It can be seen that Mi increases with D and w. Whilst wheels WithJarger diame- ters are expected to develop more torque, increasing Mi with speed contradicts the experimental results of Burjaili and Parise 121. In that paper it has been shown that, for the prototype tested, there was a torque increase for lower speeds. The fact, celebrated as an important characteristic of the engine, as it allowed for a stable operation, was explained in the light that a lower speed would provide longer periods of liquid pumping from A to B. The fact that the engine tested was of the wet type, in opposition to the dry type of the present simulation, should not account for the change in the trend. Yet, careful exami- nation of the model results revealed that liquid pumping occurred in the first degrees of rotation, with the engine nearly horizontal, tak- ing advmtage of the low pressure difference (pLgDsin8) between the volumes. After this first period, liquid transfer from A to B stopped, only returning at 150-170 degrees. In the particular example of the present study, a large amount of liquid was transferred in the first stage. This had an adverse effect on the net torque, as any amount

49

of liquid in the upper chamber, before the engine reaching ita verc tical position (ea = eoo), contributes to negative torque. Fignre 6 shows the amount of liquid pumped to chamber B before 0 equals 90 degrees. Considering that the total volume of liquid in the module was 5 litres, it can be seen that the values plotted in Figure 6 domi nated the resulting torque. This explains the results of Figure 5. In the prototype 121, either operating and charging conditbm were such that this did not occur, or, most probably, chamber geometry did not allow for liquid pumping not until ca = 20" - 30". Ideally, liquid transfer should only start after 90'.

.

SHAFT SPEED (rpm)

Figure 6 - Predicted amount of liquid pumped to the upper chamber, before ea = 900 as a fhction of diameter and shaft speed

I

0.2 O 3 I

O.'t

A

0 1 - 1 I 1 1 I 0 I 2 3

SHAFT SPEED (rpm)

Figure 7 - Predicted thermal efficiency against wheel diameter and shaft speed

The thermal efficiency was defined in the usual manner, as the ratio between the net work output and the amount of heat added to the engine. In Figure 6 it is plotted against D and W. It seems clear that the predicted effect of premature pumping of liquid also affected the thermal efficiency, as far as shaft speed is concerned. Again, the predicted results suggest the use of wheels of larger diameters. An expected, predicted values of 9 were low. Comparatively, *th the temperatures involved (3OOC and 80°C), the Carnot efficiency would be 0.142.

4.2 Influence of wall thickness and gae charge

One important aspect of the engine to be taken into account in the fact that, during one complete rotation, the wheel structure is sub mitted, intermitently, to both heating and cooling, in the hot and cold regions. Due to ita thermal capacity, a certain amount of heat in transferred from the heat so- to the heat sink, through the strue- tore, with no participation of the working fluid. The wheel structure thus behaves an a rotary regenerator. The amount of heat it carries is lost, this loan being called the .regenerator effect'.

The constant speed model wan employed to ansesn the effect of the wall thickness on the engine performance. Figure 8 shows the variation of the thermal efficiency as a function of the wall thicluresa and the gas charge. The thermal efficiency is still dehned as the ratio of the net work output to the heat transferred from the heat source to the engine, including ita structure. Thus.,

(43)

where r is the time taken by half turn of the wheeL

For the reaulta in Figure 8 the chamber walln were a~umed to be made of steel (po = 7800kg/m3 and cw = 0.47kJlkgK) and the thickness wan made to vary from 0.2 to 1 mm. For sero thickness (thin walled model), resulta from the previous. analysis were taken. The simulation was carried out for two valuen for the total amount of gas: 0.1 and 0.2 kg. The gas charge han a direct effect on the operating prewure of the engine and, comequently, on the required wall thickness. Figure 8 shown that the incream on the wall thickness has a considerable effect on the engine thermal efficiency. More gas charge and, consequently, higher operating pressure, result in higher thermal efficiency. The point of optimum efficiency for a given engine will derive from the trade-off between gas charge and wall thicknem. In a "wet' engine [2] the vapour prewure of the working fluid will have to be taken into account as well. For instance, Bujaili and Parise 121 chocse Refrigerant - 11 as the working fluid for ita moderate saturatmn premure at near ambient temperatures.

CHARGE OF GAS - 0.1 kg --- 0.2kg y- .

o a2 0.4 m o a I WALL THICKNESS (mm)

Figure 8 - Influence of wall thicknw and gan charge on engine thermal efficiency

4.8 Variable speed Simulation In this final set of results the variable speed thin-walled model was utilised to study the dynamics of the engine. The moment of inertia

I

E ff. 1

3t zi

Figure 9

3

2 -

1 -

0-

- -.-.-.-.- . - _ _ . C'

/

I I I ' ' ' I

of the structure was assumed to be equal to 65 kg ma. By knowing the details of the chamber geometry this value could be easily calculated.

Figure 9 shows the variation of PA, Pa, TA, TB, w and M with the shaft angle, for three different combinations of the resistive torque and the heat source temperature: (8OoC, 1 Nm), (8OoC, 0.5 Nm) and (60'C, 1 Nm). The simulation was initiated with the gas in both chambers at the heat sink temperature (3OoC), representing a "cold- start'. The cycles displayed in Figure 9 correspond to the situation of periodical operation, achieved after a number of cycles. One observes three distinct phases. In the first degrees both pressures decrease, due to the cooling of the gas in chamber B. Eventhough the gas also undergoes simultaneously a heating process in A, the decreasing trend is maintained mainly because the larger mass of gas in B governs the process. As the wheel rotates, the pressure difference between A and B starts to increase. The temperatures follow the same trend of the pressure curves. Torque is positive in this phase.

In the second stage, ranging approximately from 10' to 140' there is no liquid transfer to the upper chamber. Shaft torque decreases and becomes negative after 90'. With the relatively large period of time available (angular speeds settled around 1 and 3 rpm) the gas temperature in both chambers tended to the corresponding tempem tures of the heat sink and heat source. When the liquid reaches again the top of the connecting tube, one observes, in Figure 9, that the temperature remains almost constant, indicating that process 1'-2 can in a simplified analysis, be regarded as isothermal. As the wheel rotates, the effective of liquid (pLghsinO), which establishes the pres- sure difference between chambers, tends to rero, bringing PA and PB together. One observes, in Figure 9, the changes in the engine be- haviour due to different operating conditions. The greatest effect has been observed in the engine speed which dropped by nearly 53%, with the reduction of the heat source temperature from 80'C to 60'C.

Finally, it has been observed that a further increase in the heat source temperature would result in a complete liquid transfer from A to B, before O equals 90'. As result, the wheel would oscillate without performing a complete turn. For instance, for the heat source at QO'C, the oscillation remained within a range of 40 degrees.

5. Concluding Remarks A mathematical model was derived to predict the performance of the Minto Wheel heat engine. The model provided a number of important considerations about the engine that will help develop a more efficient device. Amongst the major conclusions drawn from the predicted results one could mention:

1) due to the regenerator effect the total maas of the wheel structure should be kept as low as possible;

2) liquid should be allowed to be transferred to the upper chamber only after the connecting tube reaches the vertical position:

3) for typical configurations the resulting shaft speed is slow and non-unifonn;

4) thermal efficiency is low making the engine more suitable to run from alternative energy sources (solar, for example);

5) the engine is capable to operate at small heat source to heat sink temperature differences.

Since no prototype of the air-water type was constructed, it was not possible to validate the model with experimental data. Development of phase change engine simulation model, taking into account heat and friction losses, is under way.

6. References

[l] Lindsley, E.F., Wally Minto's Wonder Wheel, Popular Science, 208, n 3, p. 79, September 1976.

[2] Burjaili, M.M. and Parise, J.A.R., Experimental Ana& of a Low-Speed Heat Engine, Experimental Thermal and Fluid Sci- ence, 2: 45-50, 1989.

131 J.A.R. Parise, Mathematical Modelling of a Low-Speed Gas- hliquid Heat Engine, IECEC-89, 899067, vol 5, pp 2565-2569, 1989.

51