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Chapter 4: Problem 2

Liquid propane enters an initially empty cylindrical storage tank at a mass flow rate of \(10 \mathrm{~kg} / \mathrm{s}\). The tank is 25 -m long and has a 4-m diameter. The density of the liquid propane is \(450 \mathrm{~kg} / \mathrm{m}^{3}\). Determine the time, in \(\mathrm{h}\), to fill the tank.

Short Answer

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Step by step solution

01

- Calculate the volume of the tank

The volume of a cylinder is given by the formula: \[ V = \text{st} \times r^2 \times \text{height} \] where \( r \) is the radius of the base and \( \text{height} \) is the height (or length) of the cylinder. Given radius = 4/2 = 2 meters and height (length) = 25 meters, we have: \[ V = \text{pi} \times (2 \text{m})^2 \times 25 \text{m} = \text{pi} \times 4 \text{m}^2 \times 25 \text{m} = 314.16 \text{m}^3 \].
02

- Calculate the mass required to fill the tank

Mass can be determined using the formula: \[ m = \rho \times V \] where \( \rho \) is the density. Given \( \rho = 450 \frac{\text{kg}}{\text{m}^3} \) and volume \( V = 314.16 \text{m}^3 \), we have: \[ m = 450 \frac{\text{kg}}{\text{m}^3} \times 314.16 \text{m}^3 = 141372 \text{kg} \].
03

- Determine the time to fill the tank

Time can be calculated using the formula: \[ \text{time} = \frac{\text{mass}}{\text{mass flow rate}} \] Given the mass flow rate is \( 10 \text{kg/s} \) and total mass \( m = 141372 \text{kg} \), we find: \[ \text{time} = \frac{141372 \text{kg}}{10 \text{kg/s}} = 14137.2 \text{s} \]. Convert seconds to hours by dividing by 3600: \[ \text{time} = \frac{14137.2 \text{s}}{3600 \text{seconds/hour}} = 3.93 \text{hours} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cylinder volume calculation
To calculate the volume of a cylindrical storage tank, we start with the formula for the volume of a cylinder: The formula is \[ V = \pi \times r^2 \times h \] \( V \) stands for volume, \( \pi \) is a constant (approximately 3.14), \( r \) is the radius of the base, and \( h \) is the height/length of the cylinder. Given a propane storage tank with a diameter of 4m, the radius \( r \) is half of the diameter: \( r = \frac{4\,\text{m}}{2} = 2\,\text{m} \). The tank's height or length is 25m. Plug these values into the formula: \[ V = \pi \times (2\,\text{m})^2 \times 25\,\text{m} \] Simplifying further: \[ V = 3.14 \times 4\,\text{m}^2 \times 25\,\text{m} \approx 314.16\,\text{m}^3 \] So, the volume of the tank is approximately 314.16 cubic meters.
density and mass relationship
To find the mass of the propane required to fill the tank, we use the relation between density, volume, and mass. The formula is: \[ m = \rho \times V \] where \( m \) is mass, \( \rho \) is density, and \( V \) is volume. Given the density \( \rho \) of liquid propane is 450 kg/m鲁 and the volume \( V \) is 314.16 m鲁, we calculate: \[ m = 450 \frac{\text{kg}}{\text{m}^3} \times 314.16\,\text{m}^3 \approx 141372\,\text{kg} \] Thus, the mass of propane needed to fill the tank is approximately 141,372 kilograms.
mass flow rate
The mass flow rate is the amount of mass passing through a given point per unit time. Here, it tells us how fast the propane is flowing into the tank. The formula to find the time to fill the tank using the mass flow rate is: \[ \text{time} = \frac{m}{\text{mass flow rate}} \] Given that the mass flow rate is 10 kg/s and the total mass required is 141372 kg, we can find the time: \[ \text{time} = \frac{141372\,\text{kg}}{10\,\text{kg/s}} = 14137.2\,\text{s} \] This tells us it takes 14,137.2 seconds for the tank to fill up with propane.
unit conversion
Finally, we need to convert the fill time from seconds to hours for easier interpretation. Since there are 3600 seconds in an hour, we convert seconds to hours by dividing by 3600: \[ \text{time in hours} = \frac{14137.2\,\text{s}}{3600\,\text{s per hour}} \approx 3.93\,\text{hours} \] Therefore, it will take approximately 3.93 hours to fill the propane storage tank. To summarize:
  • Calculate the volume of the cylinder
  • Use the density to find the mass
  • Determine the time using the mass flow rate
  • Convert the time to the desired units
Together, these steps help solve the problem of determining the time it takes to fill the propane storage tank.

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Most popular questions from this chapter

A \(0.5 \mathrm{~m}^{3}\) tank initially contains air at \(300 \mathrm{kPa}, 350 \mathrm{~K}\). Air slowly escapes from the tank until the pressure drops to \(100 \mathrm{kPa}\). The air that remains in the tank undergoes a process described by \(p v^{1.3}=\) constant. For a control volume enclosing the tank, determine the heat transfer, in kJ. Assume ideal gas behavior with constant specific heats.

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Forced-air warming systems involving inflatable thermal blankets are commonly used to prevent subnormal body temperature (hypothermia) during and following surgery. A. heater and blower provide a stream of warmed air to the blanket. While the air temperature leaving the heater/blower is monitored by a temperature sensor, the temperature of the air providing warming to patients can vary widely, causing in some instances overheating and localized burning of patients. The object of this project is to develop costeffective modifications of existing thermal blankets that would control the air temperature and eliminate injurious "hot spots." The modifications must conform to standards governing the safety of systems involving heating in medical applications. Summarize your conclusions in a memorandum.

Ammonia enters a control volume operating at steady state at \(p_{1}=14\) bar, \(T_{1}=28^{\circ} \mathrm{C}\), with a mass flow rate of \(0.5 \mathrm{~kg} / \mathrm{s}\). Saturated vapor at 4 bar leaves through one exit, with a volumetric flow rate of \(1.036 \mathrm{~m}^{3} / \mathrm{min}\), and saturated liquid at 4 bar leaves through a second exit. Determine (a) the minimum diameter of the inlet pipe, in \(\mathrm{cm}\), so the ammonia velocity does not exceed \(20 \mathrm{~m} / \mathrm{s}\). (b) the volumetric flow rate of the second exit stream, in \(\mathrm{m}^{3} / \mathrm{min} .\)
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