2.1. Given Data and Initial Calculations

The crude oil of viscosity 0.9 poise and specific gravity 0.8 is flowing through a horizontal circular pipe of diameter 80 millimeters and length of 15 meter. Calculate the difference of pressure at the 2 ends of the pipe, if 50 kilograms of oil is collected in a tank in 15 seconds. The diameter is 80 millimeters and the length is 15 meters.

The problem introduces crude oil flowing through a circular pipe, specifying its viscosity and specific gravity. Viscosity, given in poise, measures a fluid's resistance to flow, while specific gravity compares the fluid's density to that of water. The task is to determine the pressure difference at the ends of the pipe based on the flow characteristics and the mass of oil collected in a given time.

Imagine a garden hose delivering water. The thickness and pressure determine how fast water can flow through. If you squeeze the hose, making it narrower (similar to how a narrower pipe would behave), the pressure in the hose increases, reducing the speed of the water out. Similarly, this pipe carries crude oil with certain properties, and we will explore how those affect the flow.

Given that the specific gravity (S) is 0.8, density can be calculated as follows: Assuming the density of water is 1000 kg/m³, the density of the oil becomes 800 kg/m³. The volume of oil collected in 15 seconds is determined by dividing the mass of oil by its density, resulting in a volume of 0.0625 m³.

Using specific gravity, we can derive the density of the oil. Since specific gravity is defined as the ratio of a fluid's density to the density of water, we calculate the density of the oil to be 800 kg/m³. Then, to find the volume of the oil collected, we divide the total mass (50 kg) by this density, leading us to find that 0.0625 cubic meters of oil has been collected in the tank over the time interval of 15 seconds.

Think of measuring how much syrup you pour into a jar. If you know how heavy it is (mass) and how thick it is (density), you can calculate the amount of room it takes up inside the jar (volume). A similar calculation helps us understand how crude oil behaves in the pipe.

The discharge (Q) can be found by dividing the volume by time: Q = 0.0625 m³ / 15 s = 4.17 × 10⁻³ m³/s. The area (A) of the pipe can be calculated as A = πD²/4, which gives A = 5.026 × 10⁻³ m².

Discharge measures the volume flow rate of the fluid, which tells us how fast the fluid is moving through the pipe. By dividing the volume of oil collected by the time, we calculate the discharge rate as approximately 0.00417 m³/s. Following this, we compute the area of the circular pipe using the formula for the area of a circle, yielding an area of around 0.005026 m², which is vital for further calculations.

Imagine you're filling up a bucket with water using a small hose. The hose's diameter will determine how fast you can fill the bucket (discharge). The area of the hose itself tells you how much water can flow through at once.

The average velocity of the oil is computed as: V_avg = Q / A = 0.83 m/s. The Reynolds number (Re) is then calculated using the formula: Re = (ρ * V_avg * D) / μ, resulting in Re = 590, indicating laminar flow.

The average velocity indicates the speed at which the liquid travels through the pipe, computed by dividing the discharge by the pipe area. In this case, the average velocity works out to be 0.83 m/s. The Reynolds number helps us assess whether the flow is laminar or turbulent. Since Reynolds number here is below the threshold of 2000, the flow is deemed laminar, which is characterized by smooth and orderly movements.

Picture a calm river (laminar) flowing steadily versus a fast white-water river (turbulent). The speed of flow and the conditions of flow can change the overall behavior of the liquid, just as the calculated Reynolds number does for the oil in our pipe.

Using the derived formula for laminar flow, Q = -8π/(8μ) * (dp/dx) * R⁴, we obtain: dp/dx = -373.32 N/m²/m. Over 15 meters, this results in a pressure difference of P2 - P1 = -5599 N/m².

To find the pressure difference between the two ends of the pipe, we apply the corresponding formula for laminar flow. After substituting values into the equation, including the average velocity and radius of the pipe, we find the pressure gradient per meter. By multiplying this pressure difference by the total length of the pipe, we calculate the total pressure drop as approximately 5599 N/m².

Imagine water flowing through a long pipe that's getting pinched at one end. The pressure at the end with pinch will be significantly lower than at the other. This pressure drop is akin to the difference we've calculated, resulting from the oil flowing through our circular pipe.

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