Question

An ammonia-water absorption refrigeration unit operates its absorber at \(0^{\circ} \mathrm{C}\) and its generator at \(46^{\circ} \mathrm{C}\). The vapor mixture in the generator and absorber is to have an ammonia mole fraction of 96 percent. Assuming ideal behavior, determine the operating pressure in the (a) generator and \((b)\) absorber. Also determine the mole fraction of the ammonia in the \((c)\) strong liquid mixture being pumped from the absorber and the \((d)\) weak liquid solution being drained from the generator. The saturation pressure of ammonia at \(0^{\circ} \mathrm{C}\) is \(430.6 \mathrm{kPa},\) and at \(46^{\circ} \mathrm{C}\) it is \(1830.2 \mathrm{kPa}\).

Short answer

Question: Determine the operating pressures of the absorber and generator in an ammonia-water absorption refrigeration unit at the given conditions. Also, find the mole fractions of ammonia in the strong liquid mixture coming from the absorber and the weak liquid solution drained from the generator. Solution: Step 1: Absorber operating pressure (\(P_{abs}\)): Using the equation, \(P_{abs} = \frac{P_{NH_3(abs)}}{x_{NH_3}}\) and substituting the values, \(P_{abs} = \frac{430.6 \mathrm{kPa}}{0.96}\) \(P_{abs} = 448.54 \mathrm{kPa}\) Step 2: Generator operating pressure (\(P_{gen}\)): Using the equation, \(P_{gen} = \frac{P_{NH_3(gen)}}{x_{NH_3}}\) and substituting the values, \(P_{gen} = \frac{1830.2 \mathrm{kPa}}{0.96}\) \(P_{gen} = 1906.25 \mathrm{kPa}\) Step 3: Mole fraction of ammonia in the strong liquid mixture (\(x_{NH_3 (strong)}\)): Using the equation, \(x_{NH_3 (strong)} = \frac{P_{NH_3(abs)}}{P_{sat, 0^{\circ}C}}\) and substituting the values, \(x_{NH_3 (strong)} = \frac{430.6 \mathrm{kPa}}{430.6 \mathrm{kPa}}\) \(x_{NH_3 (strong)} = 1\) Step 4: Mole fraction of ammonia in the weak liquid solution (\(x_{NH_3 (weak)}\)): Using the equation, \(x_{NH_3 (weak)} = \frac{P_{NH_3(gen)}}{P_{sat, 46^{\circ}C}}\) and substituting the values, \(x_{NH_3 (weak)} = \frac{1830.2 \mathrm{kPa}}{1830.2 \mathrm{kPa}}\) \(x_{NH_3 (weak)} = 1\) Results: The operating pressures of the absorber and generator are 448.54 kPa and 1906.25 kPa, respectively. The mole fractions of ammonia in the strong liquid mixture and weak liquid solution are both 1.

Step-by-step solution

Determining Absorber Operating Pressure

As we are following ideal behavior, the partial pressure of ammonia in the vapor mixture is equal to its mole fraction times the total pressure. Since the ammonia mole fraction in the vapor mixture is 96%, the partial pressure of ammonia in the absorber can be determined using: \(P_{NH_3(abs)} = x_{NH_3} \times P_{abs}\), where \(P_{NH_3(abs)}\) is the partial pressure of ammonia in the absorber, \( x_{NH_3} \) is the ammonia mole fraction (0.96), and \(P_{abs}\) is the absorber operating pressure. As ammonia is saturated at \(0^{\circ} \mathrm{C}\), its saturation pressure should equal its partial pressure in the absorber: \(P_{NH_3(abs)} = P_{sat, 0^{\circ}C} = 430.6 \mathrm{kPa}\) We can now solve for the absorber operating pressure: \(P_{abs} = \frac{P_{NH_3(abs)}}{x_{NH_3}}\)

Determining Generator Operating Pressure

Similarly, the partial pressure of ammonia in the generator can be determined using: \(P_{NH_3(gen)} = x_{NH_3} \times P_{gen}\), where \(P_{NH_3(gen)}\) is the partial pressure of ammonia in the generator, and \(P_{gen}\) is the generator operating pressure. As ammonia is saturated at \(46^{\circ} \mathrm{C}\), its saturation pressure should equal its partial pressure in the generator: \(P_{NH_3(gen)} = P_{sat, 46^{\circ}C} = 1830.2 \mathrm{kPa}\) We can now solve for the generator operating pressure: \(P_{gen} = \frac{P_{NH_3(gen)}}{x_{NH_3}}\)

Determining Mole Fraction of Ammonia in Strong Liquid Mixture

The strong liquid mixture being pumped from the absorber is in equilibrium with the vapor mixture in the absorber. Therefore, the mole fraction of ammonia in the strong liquid mixture can be determined using the equilibrium relation for an ideal solution: \(x_{NH_3 (strong)} = \frac{P_{NH_3(abs)}}{P_{sat, 0^{\circ}C}}\)

Determining Mole Fraction of Ammonia in Weak Liquid Solution

The weak liquid solution drained from the generator is in equilibrium with the vapor mixture in the generator. Therefore, the mole fraction of ammonia in the weak liquid solution can be determined using the equilibrium relation for an ideal solution: \(x_{NH_3 (weak)} = \frac{P_{NH_3(gen)}}{P_{sat, 46^{\circ}C}}\) Now, we can solve all the equations to obtain the desired results.

Key concepts

Operating Pressure

Understanding the operating pressure within an absorption refrigeration system is critical for ensuring its efficient performance. In this type of system, different components operate at varying pressures. The generator, which heats a solution causing the refrigerant (often ammonia) to vaporize, and the absorber, where the refrigerant is absorbed and condensed, operate at distinct pressures specified by the temperature and concentration of the refrigerant.

To find the operating pressures, you would apply the concept of partial pressures and the relationship with the mole fraction of ammonia in the vapor phase. For example, the absorber operating pressure is found by dividing the saturation pressure of ammonia at the operating temperature by its mole fraction in the vapor phase. This calculation allows us to understand how much pressure is needed to maintain absorption at a certain temperature and ammonia concentration. Similarly, we calculate the generator's operating pressure, reflecting the system design's requirements for efficient cycle operation.

Ammonia-Water Refrigeration

In ammonia-water absorption refrigeration systems, the ammonia serves as the refrigerant, and water acts as the absorbent. The operation of these systems is driven by a solution's ability to absorb and release ammonia. The main advantage of using ammonia is its high efficiency and effectiveness as a refrigerant. It has a greater refrigeration effect per volume than other common refrigerants and is well-suited for large industrial applications.

These systems operate by harnessing the heat energy to drive the refrigeration cycle, making them useful in situations where heat is readily available or electricity costs are high. Understanding how the solution's concentration and temperature affect absorption and vaporization can lead to designing more efficient refrigeration systems, where control of ammonia concentration and pressure is key to the system's overall performance.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a particular substance to the total moles of all substances present. In the context of absorption refrigeration systems, the mole fraction of ammonia is crucial because it affects not only the operating pressure of the system but also the energy efficiency of the refrigeration process.

Knowing the mole fraction allows us to determine the partial pressures of components in a mixture. In our exercise example, the ammonia mole fraction in the vapor mixture directly impacts both the absorber and generator pressures. Additionally, by understanding the equilibrium between the liquid and vapor phases, we can calculate the ammonia mole fraction in the strong liquid leaving the absorber and the weak liquid leaving the generator, which are essential for the system’s operation.

Saturation Pressure

Saturation pressure refers to the pressure at which a substance changes from one phase to another at a particular temperature. For refrigerants like ammonia, the saturation pressure is critical because it defines the conditions under which the refrigerant will evaporate (boil) or condense. This property is temperature dependent and integral to the design and operation of refrigeration systems.

In our exercise, the saturation pressures at 0°C and 46°C are given. These values are used to determine the actual pressures at which the absorption and the generation (evaporation) processes occur within the system. By matching the saturation pressure with the partial pressure of ammonia in both the generator and absorber, we can ascertain the correct operating pressures necessary for the refrigeration cycle to function as intended, thus ensuring that the system operates under the ideal conditions for energy efficiency and effectiveness.