Property Modeling | Contents | Bubble Pump Performance

Cycle Thermodynamic Model

 

CYCLE THERMODYNAMIC MODEL

To create a thermodynamic model of the Einstein refrigeration cycle, conservation of mass and the first and second laws of thermodynamics are applied to each component. In this study, the cycle is divided into the following five components: an evaporator, a combined condenser/absorber, a generator, a bubble pump, and a pre-cooler (Figure 3-1). For the bubble pump, conservation of momentum and Bernoulli's equation are also required. Since the entire cycle operates at a constant pressure, the pressure in each component is nearly the same and will be referred to as the system pressure.

Figure 3-1: The Einstein Refrigeration Cycle

The Evaporator

In the evaporator, the refrigerant and pressure equalizing gas both arrive from the pre-cooler in nearly pure form (Figure 3-2, state points 4 and 2). In the cycle described by the Einstein-Szilard patent the refrigerant evaporates in the presence of the pressure equalizing gas due to a lowering of the refrigerant's partial pressure. However, since the pressure equalizing gas is close to condensation in the evaporator, the actual process is more complex.

Figure 3-2: The Evaporator

First, nearly pure saturated liquid butane flows in from the condenser/absorber at the condenser/absorber temperature (state 2 in Figure 3-2). Simultaneously, saturated vapor ammonia (state 4 in Figure 3-2) is bubbled into the liquid butane. The presence of the ammonia vapor reduces the partial pressure of the butane causing it to evaporate. As it evaporates into the ammonia vapor, the butane cools itself, the ammonia vapor, and produces external cooling. A small amount of ammonia vapor is also absorbed into the liquid butane producing some heat of absorption which is also removed by the evaporating butane. For a given system pressure, the temperature in the evaporator depends upon the relative butane-ammonia flow rates. To prevent a temperature glide, which increases the evaporator temperature, the design condition is taken to be at the azeotrope.

Figure 3-3: T-x-x-y Diagram for Ammonia-i-Butane at 3.1 bar

This design condition produces a minimum temperature in the evaporator at the three phase flash (VLLE) temperature at the system pressure. In Figure 3-3, the three phase flash occurs at 257.4 K.

Mass conservation must be satisfied in the evaporator for both ammonia and butane. The evaporator is assumed to operate under steady state conditions with fixed inlet and exit velocities and cross sectional areas. Under these conditions, conservation of mass for the control volume in Figure 3-2 yields the following equations for ammonia and butane respectively:

(3-1)

(3-2)

In equation 3-1, ya,3 represents the vapor mass fraction of ammonia at state point 3 while xa,2 represents the liquid mass fraction of ammonia at state point 2. Similarly, in equation 3-2, yb,3 represents the vapor mass fraction of butane at state point 3 while xb,2 represents the liquid mass fraction of butane at state point 2

The conservation of energy for the evaporator is:

(3-3)

In this study heat transferred to a control volume is considered positive.

Finally, the entropy generation for the evaporator is:

(3-4)

where Tevaporator is considered uniform.

Equations 3-1 through 3-3 provide sufficient information to model the evaporator for zeotropic vapor-liquid fluids. With such fluids (e.g. butane and hydrogen chloride) the evaporator's temperature will be somewhere between the pure component saturation temperatures of the two components at the system pressure. However, if a minimum boiling azeotrope exists (e.g. ammonia and butane at pressures above 15 bar), the evaporator's temperature can be lower than both pure component saturation temperatures at the system pressure. This may also be the case when VLLE exists.

Since obtaining the lowest possible evaporator temperature is of interest, predicting the state for which this temperature exists is a useful addition to the thermodynamic analysis of the evaporator. In the case of a minimum boiling azeotrope, the minimum evaporator temperature (under VLE conditions) occurs when the compositions in the vapor and liquid phases are equal.

(3-5)

For the case of VLLE (e.g. ammonia and butane below 15 bar), the minimum evaporator temperature state is fixed by the system pressure alone and is determined by satisfying equation 2-47.

The Pre-Cooler

Although the vapor mixture leaving the evaporator at state 3 is not cold enough to provide additional refrigeration, it is cold relative to the fluids coming to the evaporator. The pre-cooler allows the cooling of the two streams entering the evaporator via the heating of the stream leaving the evaporator (Figure 3-4).

Figure 3-4: The Pre-Cooler

The pre-cooler is assumed to operate under steady state conditions with no fluid friction and is insulated so that the only heat transfer occurs between the entering streams and exiting streams. Since the conservation of mass will be satisfied between the evaporator and condenser/absorber, the only equation necessary for the pre-cooler is the conservation of energy.

(3-6)

The entropy generated by the pre-cooler due to heat transfer across a finite temperature difference is:

(3-7)

In this study, the effect of non-ideal heat exchange is accommodated simply with the use of pinch points. In a perfect, counter-current flow, two stream heat exchanger, the entering stream and exiting stream on one side would approach the same temperature (the case when a heat exchanger has infinite area). In reality, the streams maintain a temperature difference which, in this study, is defined as the pinch point. Since the pre-cooler transfers heat between three streams it requires two pinch points. The first is defined between the vapor mixture entering at state point 3 and the liquid leaving at state point 2. The second is defined between the vapor mixture leaving at state point 6 and the liquid entering at state point 5.

(3-8)

(3-9)

The Condenser/Absorber

In the Einstein refrigeration cycle, the condenser and absorber are combined into a single component where both processes occur simultaneously as shown in Figure 3-5. When the vapor mixture leaving the pre-cooler at state point 6 enters the condenser/absorber, it encounters a large surface area created by a falling film of sub-cooled liquid water weak in ammonia. The water film, which enters the condenser/absorber at state point 9, readily absorbs the ammonia from the vapor mixture. This increases the concentration of butane in the vapor and hence the partial pressure on the butane in the vapor. Now the butane can condense at its saturation temperature for this system pressure which is well above the temperature at which it evaporated earlier in the evaporator.

Figure 3-5: The Condenser/Absorber

The liquid water, now rich in ammonia, and butane descend the walls of the condenser/absorber. Since the ammonia-water solution is immiscible with the butane, and is more dense, it sinks to the bottom of the condenser and flows out at point 7. The light, immiscible butane floats atop the solution and exits at point 1. The condenser operates at steady state and the liquids leaving the condenser are assumed to be in thermal and vapor-liquid equilibrium at the temperature of the condenser.

Since , conservation of mass for the control volume in Figure 3-5 yields the following equations:

(3-10)

(3-11)

Conserving energy for the control volume in Figure 3-5 yields the heat transfer from the condenser:

(3-12)

Finally, the entropy generation for the control volume in Figure 3-5 is:

(3-13)

The Generator

In the generator, shown in Figure 3-6, ammonia rich water arriving from the condenser is heated. This generates vapor ammonia (5) which then flows to the evaporator. The remaining water, containing less ammonia, drops to the bottom of the generator where it flows into the bubble pump and is returned to the condenser. Before it is returned to the condenser, this hot ammonia-water solution (8f) transfers its heat to the cooler ammonia-water solution (7) arriving from the condenser.

Conserving mass and energy flow for the generator's control volume yields:

(3-14)

(3-15)

(3-16)

The entropy generated by the generator is:

(3-17)

Figure 3-6: The Generator

To account for the generator's internal heat exchanger, another control volume is necessary (Figure 3-7).

Figure 3-7: Inside the Generator

For the control volume shown in Figure 3-7, conservation of mass and energy yield the following equations:

(3-18)

(3-19)

(3-20)

The entropy generated by the generator's internal heat exchanger is accounted for with equation 3-17.

The internal heat exchanger in the generator also requires a pinch point.

(3-21)

The Bubble Pump

To circulate the working fluids without a mechanical pump, the Einstein cycle relies on a bubble pump. A bubble pump is a heated tube (length L and diameter d) communicating with two reservoirs, one higher than the other (see Figure 3-8). The liquid in the lower reservoir initially fills the tube to the same level (h). Heat is applied at the bottom of the tube at a rate sufficient to evaporate some of the liquid in the tube. The resulting vapor bubbles rise in the tube carrying the liquid above them to the higher reservoir. The bulk density of the fluid in the tube is reduced relative to the liquid in the lower reservoir, thereby creating an overall buoyancy lift.

Figure 3- 8: Bubble Pump Schematic

There are four flow regimes for two phase up flow in a fixed diameter vertical pipe. For low vapor flow rates, small, finely dispersed vapor bubbles will rise in a continuous liquid phase. This is the bubble flow regime. Increasing the vapor flow causes the vapor bubbles to coalesce into bullet shaped slugs of vapor which rise in the liquid phase. This is the slug flow regime, and a bubble pump operates most efficiently here. Further increase of vapor flow causes a highly oscillatory flow with a tendency for each phase alternatively to fill the tube. This is the churn flow regime. The last flow regime, reached by even further increase of vapor flow, is the annular flow regime in which the liquid forms a film around the pipe wall and the vapor rises up the core (Chisholm, 1983).

A bubble pump operates most efficiently in the slug flow regime. The maximum diameter tube in which slug flow occurs is given by the following equation (Chisholm, 1983):

(3-22)

where vf and vg are the specific volumes of the liquid and vapor respectively, and is the surface tension. Note, for a given fluid in a tube of diameter greater than that predicted by 3-22, slug flow will never occur.

This study assumes slug flow in the bubble pump. The air lift pump analytical model of Stenning and Martin was used as a starting point for modeling the bubble pump (Stenning and Martin, 1968). Point 1 in Figure 3-8 represents the inlet of the bubble pump. Applying Bernoulli's equation between the surface of the lower reservoir and point 1 yields:

(3-23)

Next, continuity of mass is applied to the control volume to which heat is applied, CV2. Assuming that the mixture of vapor bubbles and the liquid exit this control volume at a mixture velocity, V2, continuity of mass yields:

(3-24)

or,

(3-25)

rearranging yields,

(3-26)

The specific volume at point 2 is assumed to be the specific volume of a vapor-liquid mixture with a quality, x, where:

(3-27)

and,

(3-28)

Combining equations 3-26, 3-27, and 3-28 yields:

(3-29)

Now, the mass flow rate of vapor, , is assumed negligible relative to the mass flow rate of liquid, , and the specific volume of the liquid, , is assumed negligible relative to the specific volume of the vapor, . Equation 3-29 then becomes:

(3-30)

where  and are the volume flow rates of the vapor and liquid respectively.

Next, conservation of momentum is applied to CV2 in Figure 3-8. Neglecting friction pressure losses over this short distance yields:

(3-31)

substituting equation 3-30 into equation 3-31 yeilds:

(3-32)

Now, equations 3-22 and 3-32 are combined to yield:

(3-33)

Next, the conservation of momentum is applied to the pipe connecting the lower and upper reservoir.

(3-34)

In equation 3-34, b is the perimeter of the pipe, and W is the weight of fluid in the pipe. The friction factor, f, is calculated assuming liquid flow through the pipe alone. Substituting equation 3-30 for V2 yields:

(3-35)

The weight of fluid in the pipe is the combined weight of the liquid and vapor in the pipe:

(3-36)

where Af is a superficial area through which the liquid flows and Ag is a superficial area through which the vapor flows. Earlier, it was assumed that the mass flow rate of the vapor was negligible relative to the mass flow rate of the liquid. Since the velocity of the vapor, Vg, is on the order of the velocity of the liquid, Vf, equation 3-36 is simplified by assuming that  is negligible relative to , and by the substituting the following equations:

(3-37)

(3-38)

(3-39)

Now,

(3-40)

where  (3-41)

In the slug flow regime, s will be between 1.5 and 2.5 (Griffith, 1961).

Equations 3-40 is now substituted into 3-35 to give

(3-42)

Finally, equation 3-42 is equated with equation 3-33 to complete the momentum model of the bubble pump:

(3-43)

where,

(3-44)

K can be an adjustable parameter to account for losses other than friction in the tube. Pipe elbows and entrance effects may be accounted for by increasing the value of K. Furthermore, K may also be adjusted to match experimental data since losses are sometimes difficult to quantify analytically. In this study, the flow is always laminar. The friction factor for laminar flow is (White, 1986):

(3-45)

where,

(3-46)

In equation 3-43, the vapor flow rate, , is produced by the addition of heat to the lower portion of the pump tube. Assuming the fluid in the lower reservoir and tube to be saturated, and no heat transfer over the length of the pump tube, the heat required to produce  is:

(3-47)

Now, the liquid flow produced by the bubble pump, , may be calculated as a function of heat input from equations 3-43 through 3-47.


Property Modeling | Contents | Bubble Pump Performance