A soft drink (mostly water) flows in a pipe at a beverage plant with a mass flow rate that would fill 220 0.355-L cans per minute. At point 2 in the pipe, the gauge pressure is 152 kPa and the cross-sectional area is 8.00 cm². At point 1, 1.35 m above point 2, the cross-sectional area is 2.00 cm². Find the (b) volume flow rate. (c) flow speeds at points 1 and 2.

Verified step by step guidance

1. To find the volume flow rate, first calculate the total volume of soft drink flowing per minute. Since 220 cans are filled per minute and each can has a volume of 0.355 L, multiply these values to get the total volume per minute: \( V = 220 \times 0.355 \text{ L} \).
2. Convert the volume flow rate from liters per minute to cubic meters per second. Use the conversion factor \( 1 \text{ L} = 0.001 \text{ m}^3 \) and \( 1 \text{ min} = 60 \text{ s} \).
3. To find the flow speeds at points 1 and 2, use the continuity equation, which states that the volume flow rate \( Q \) is constant throughout the pipe: \( Q = A_1 v_1 = A_2 v_2 \). Here, \( A_1 \) and \( A_2 \) are the cross-sectional areas at points 1 and 2, respectively, and \( v_1 \) and \( v_2 \) are the flow speeds.
4. Calculate the flow speed at point 2 using the formula \( v_2 = \frac{Q}{A_2} \), where \( A_2 = 8.00 \text{ cm}^2 \). Convert \( A_2 \) to square meters by using the conversion factor \( 1 \text{ cm}^2 = 0.0001 \text{ m}^2 \).
5. Calculate the flow speed at point 1 using the formula \( v_1 = \frac{Q}{A_1} \), where \( A_1 = 2.00 \text{ cm}^2 \). Again, convert \( A_1 \) to square meters using the same conversion factor.

Key Concepts

Continuity Equation

The continuity equation in fluid dynamics states that the mass flow rate must remain constant from one cross-section of a pipe to another, assuming incompressible flow. It is expressed as A1V1 = A2V2, where A is the cross-sectional area and V is the flow speed. This principle helps determine the flow speeds at different points in the pipe.

Bernoulli's Principle

Bernoulli's principle relates the pressure, velocity, and height in a flowing fluid, stating that an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or potential energy. It is crucial for understanding how pressure and velocity change between two points in the pipe, especially when elevation changes are involved.

Volume Flow Rate

Volume flow rate is the volume of fluid that passes through a given surface per unit time, typically measured in cubic meters per second (m³/s). It can be calculated using the mass flow rate and the density of the fluid, or directly from the product of cross-sectional area and flow speed. This concept is essential for determining how much fluid is moving through the pipe.