Question

The cavitation number is defined as \(C a=\frac{p-p_{\text {vap }}}{(1 / 2) \rho \mathrm{v}^{2}}\). Now, for ethanol produced by sugar fermentation, a company has installed a pump for the transport of the liquid product, which is assumed to be pure. The ambient pressure is \(1020 \mathrm{hPa}\) and the pump is situated \(1 \mathrm{~m}\) below a vessel from which the product is pumped through a duct of \(10 \mathrm{~cm}\) diameter with a mass flow rate of \(25 \mathrm{t} \cdot \mathrm{h}^{-1}\). The temperature at the pump suction side is \(20^{\circ} \mathrm{C}\). At this temperature, the vapor pressure is \(5.7 \times 10^{3} \mathrm{~Pa}\). The density of ethanol is 789 \(\mathrm{kg} \cdot \mathrm{m}^{-3}\). What is the background of cavitation? Does it occur in this situation? Which assumption(s) have you made? When will there be a possibility for this phenomenon to occur?

Short answer

Answer: No, cavitation does not occur in the given situation.

Step-by-step solution

Calculate the flow velocity

Using the mass flow rate and density, we can find the flow velocity (v). The formula for flow rate (Q) is: \(Q=\rho \cdot A \cdot v\), where A is the cross-sectional area of the duct. Let's calculate A and v. For the duct of 10 cm diameter, the radius (r) is 5 cm or 0.05m. Therefore, the cross-sectional area (A) is given by: \(A=\pi r^{2}=\pi (0.05)^{2}=0.00785 \mathrm{~m}^2\) Now, rearrange the equation of flow rate: \(v=\frac{Q}{\rho \cdot A}\) The mass flow rate is given as 25 t·h⁻¹, which needs to be converted into kg·s⁻¹: Mass flow rate \((\mathrm{kg} \cdot \mathrm{s}^{-1}) = \frac{25000 \ \mathrm{kg} \cdot \mathrm{h}^{-1}}{3600 \ \mathrm{s}^{-1}} = 6.944 \mathrm{kg} \cdot \mathrm{s}^{-1}\) Now, calculate the flow velocity (v): \(v=\frac{6.944}{789 \cdot 0.00785}=1.129 \mathrm{~m} \cdot \mathrm{s}^{-1}\)

Calculate the ambient pressure in Pascals

The ambient pressure is given in hectopascals (hPa) and needs to be converted into Pascals (Pa): \(p = 1020 \ \mathrm{hPa} \cdot \frac{100 \ \mathrm{Pa}}{1 \ \mathrm{hPa}} = 102000 \ \mathrm{Pa}\)

Calculate the cavitation number

Now, use the formula of the cavitation number (Ca) to find its value: $C a=\frac{p-p_{\text {vap }}}{(1 / 2) \rho \mathrm{v}^{2}}$ Plug the given values and calculated flow velocity into the formula: $C a=\frac{102000-5.7 \times 10^{3}}{(1 / 2) \cdot 789 \cdot (1.129)^{2}}=5.76$

Determine the possibility of cavitation

Cavitation occurs when the cavitation number is less than or equal to 1. In this case, we have a cavitation number of 5.76, which means that cavitation does not occur in this situation.

Discuss assumptions and possibility of occurrence

The main assumption made here is that the liquid product is pure ethanol and the flow is steady and continuous. This assumption might not hold true in real situations, as the product could contain impurities or the flow might not be steady. In general, the possibility of cavitation occurring increases when the pressure drops below the vapor pressure, which can happen due to the presence of impurities, increase in flow rate, or a decrease in ambient pressure. In summary, cavitation does not occur in this situation, but it could potentially occur if any of the above-discussed conditions are met.

Key concepts

Cavitation Number

Cavitation is a phenomenon in fluid dynamics where vapor bubbles form in a liquid because the pressure of the fluid falls below its vapor pressure and then rapidly collapse back into the liquid when they reach an area of higher pressure. This can lead to significant damage in machinery, such as pumps and propellers.

The cavitation number, denoted by Ca, is a dimensionless quantity used to predict the likelihood of cavitation happening in a fluid flow. It's calculated by the equation \(Ca=\frac{p-p_{\text{vap}}}{(1/2) \rho v^{2}}\), where \(p\) is the ambient pressure, \(p_{\text{vap}}\) the vapor pressure of the fluid, \(\rho\) the density, and \(v\) the velocity of the fluid flow. A cavitation number greater than 1 indicates a low risk of cavitation; the higher the number, the less likely cavitation will occur. The exercise provided an example with a cavitation number well above 1, suggesting that cavitation was not an issue for the operating conditions given.

Ethanol Properties

Ethanol, known widely as a form of alcohol found in alcoholic beverages, is also a commonly used chemical in various industrial applications. One of its uses is as a solvent, due to its capacity to dissolve both water-soluble and fat-soluble substances. It has a relatively low boiling point of 78.37 °C at atmospheric pressure which is important for its role in applications like thermometers, as a fuel, and in this instance, ethanol's properties as a fluid.

With a density of 789 kg/m³ at 20°C, which affects its flow characteristics, it is lighter than water. Knowing the physical properties of ethanol, like its density and vapor pressure, assists in calculating parameters necessary for ensuring proper flow and avoiding issues such as cavitation in pumps and pipelines as was mentioned in the workout example.

Fluid Flow Rate

In the context of fluid dynamics, the fluid flow rate represents the volume of fluid that passes through a given area within a particular time frame. It's an essential parameter in engineering tasks that involve fluid transport, such as piping systems or pumps. Determining the correct flow rate is critical to ensure efficient and safe operations of such systems.

The flow velocity (v) can be calculated if the fluid's density (\(\rho\)) and the cross-sectional area (A) are known through the formula: \(v=\frac{Q}{\rho \cdot A}\), where Q stands for the mass flow rate (mass of fluid passing per unit time). In the exercise example, converting and using the given mass flow rate and the density of ethanol allowed for the calculation of the flow velocity, which is integral to assessing both the performance of the pump and the likelihood of cavitation.

Vapor Pressure

Vapor pressure is a critical property of fluids that indicates the pressure exerted by a vapor in thermodynamic equilibrium with its liquid phase at a given temperature. It is a measure of a liquid's tendency to evaporate. When the vapor pressure equals the ambient pressure, the liquid will boil.

The vapor pressure of a fluid can influence the formation of cavitation in fluid systems. High vapor pressures at a certain temperature suggest a fluid will more readily form vapor bubbles at lower ambient pressures. For the practical example involving ethanol, knowing its vapor pressure at the operational temperature was essential for calculating the cavitation number and determining whether cavitation could occur or not. When the ambient pressure drops below the vapor pressure, or if the fluid temperature increases, conditions become favorable for cavitation, which can create operational challenges.