Question

Consider the flow of oil at \(10^{\circ} \mathrm{C}\) in a 40 -cm-diameter pipeline at an average velocity of \(0.5 \mathrm{~m} / \mathrm{s}\). A \(1500-\mathrm{m}\)-long section of the pipeline passes through icy waters of a lake at \(0^{\circ} \mathrm{C}\). Measurements indicate that the surface temperature of the pipe is very nearly \(0^{\circ} \mathrm{C}\). Disregarding the thermal resistance of the pipe material, determine \((a)\) the temperature of the oil when the pipe leaves the lake, \((b)\) the rate of heat transfer from the oil, and \((c)\) the pumping power required to overcome the pressure losses and to maintain the flow of oil in the pipe.

Short answer

Answer: To find the temperature of the oil when the pipeline leaves the lake, the rate of heat transfer from the oil, and the pumping power required to maintain the flow of oil, we first need to calculate the heat transfer using Fourier's law of conduction. Next, determine properties of the oil such as density, viscosity, specific heat, and thermal conductivity. Then, calculate the rate of heat transfer and the pumping power required using equations like Bernoulli's equation and Darcy-Weisbach equation. The solution for this problem requires multiple steps, which include calculating the temperature of the oil, rate of heat transfer, and pumping power required.

Step-by-step solution

Calculate the temperature of the oil when the pipe leaves the lake

To find the temperature of the oil after passing through the 1500m-long section of the pipeline, we need to consider the heat transfer from the oil to the surrounding icy waters. Since the heat transfer process is steady and the thermal resistance of the pipe material is disregarded, we can use the Fourier's law of conduction: $$q = k A \frac{\Delta T}{\Delta x}$$ Here, \(q\) is the heat transfer rate, \(k\) is the thermal conductivity of the material, \(A\) is the area through which the heat is transferred, \(\Delta T\) is the temperature difference between the inside and outside surfaces of the pipe, and \(\Delta x\) is the length of the pipe section. We know the surface temperature of the pipe is \(0^{\circ} \mathrm{C}\) and the initial temperature of the oil is \(10^{\circ} \mathrm{C}\). We also know the diameter of the pipe is \(40 \mathrm{cm}\), which gives a cross-sectional area of \(A = \pi(\frac{0.4}{2})^2 = 0.126\,\mathrm{m^2}\). Assuming the temperature of the oil drops uniformly across the pipe section, we can set up an equation for the final temperature of the oil, T: $$q = k A \frac{(0 - T) - (0 - 10)}{1500}$$ Now we need to find the thermal conductivity, \(k\), of the oil to solve for T.

Determine the properties of the oil

Look up the properties of the oil at an average temperature of \(5^{\circ} \mathrm{C}\) (the midpoint between \(0^{\circ} \mathrm{C}\) and \(10^{\circ} \mathrm{C}\)), such as its density, viscosity, specific heat, thermal conductivity, and expansion coefficient. These properties will be needed for further calculations.

Calculate the rate of heat transfer

Knowing the value of the thermal conductivity, \(k\), we can now substitute it into Fourier's law of conduction and find the final temperature, T. After the final temperature is determined, we can also calculate the rate of heat transfer, \(q\), using the same equation: $$q = k A \frac{\Delta T}{\Delta x}$$

Calculate the pumping power required to maintain the flow of oil

To calculate the pumping power required, we need to take into account both the pressure losses due to the flow of oil and the pressure needed to maintain the flow of oil within the pipe. We can use Bernoulli's equation for this purpose: $$P_{pump} = \dot{m} \Delta h\rho g$$ Here, \(P_{pump}\) is the required pumping power, \(\dot{m}\) is the mass flow rate of the oil, \(\Delta h\) is the head loss across the pipe section, \(\rho\) is the density of the oil, and \(g\) is the acceleration due to gravity. First, we need to find the mass flow rate of the oil using the given average velocity, \(v = 0.5\,\mathrm{m/s}\): $$\dot{m} = \rho A v$$ Next, we will determine the head loss, \(\Delta h\), by considering the friction losses within the pipe. We will utilize the Darcy-Weisbach equation to find the head loss: $$\Delta h_{f} = \frac{f L}{D} \frac{V^2}{2g}$$ Here, \(f\) is the friction factor, \(L\) is the length of the pipe section, \(D\) is the pipe diameter, and \(V\) is the flow velocity. Finally, substitute all the values into the equation for pumping power, \(P_{pump}\), to find the required power to maintain the flow of oil. This concludes the step-by-step solution for this problem. We have determined the temperature of the oil when the pipe leaves the lake, the rate of heat transfer, and the pumping power required to maintain the flow of oil within the pipe.

Key concepts

Fourier's Law of Conduction

Fourier's law of conduction is a fundamental principle that describes the process of heat transfer within a material due to temperature gradients. Heat always flows from regions of higher temperature to regions of lower temperature, and Fourier's law quantifies this flow. The mathematical expression for Fourier's law is:
\[q = -k A \frac{dT}{dx}\]
Here, \(q\) is the heat transfer rate (measured in watts), \(k\) represents the thermal conductivity of the material (measured in watts per meter-kelvin), \(A\) is the cross-sectional area through which heat is transferred (measured in square meters), and \(\frac{dT}{dx}\) is the temperature gradient in the direction of the heat flow (measured in kelvins per meter). The negative sign indicates that heat flows in the opposite direction of increasing temperature. In the context of a pipeline carrying oil, Fourier's law helps us to understand how much heat is lost by the oil as it comes into contact with a colder environment, such as the icy waters of a lake.

Thermal Conductivity

Thermal conductivity is a material-specific property that measures a substance's ability to conduct heat. It is denoted by the symbol \(k\) and is a crucial factor in calculating heat transfer rates through materials. A high thermal conductivity implies that a material can transfer heat quickly, while a lower thermal conductivity indicates slower heat transfer.
For liquids such as oil, thermal conductivity can vary with temperature, so it's important to reference material properties at an average temperature relevant to the situation being analyzed. In our pipeline example, the thermal conductivity of oil would determine how effectively heat is conducted from the oil to the surrounding colder environment, and thus, directly affect the cooling of the oil.

Pumping Power

Pumping power refers to the energy required to move a fluid through a pipeline against frictional and other resistive forces. This power is necessary to maintain the desired flow rate and overcome any pressure losses due to factors like viscosity and pipe roughness. It is often calculated by the formula:
\[P = Q \Delta P\]
where \(P\) is the pumping power in watts, \(Q\) is the volumetric flow rate in cubic meters per second, and \(\Delta P\) is the pressure difference needed to maintain flow, in pascals. The estimation of pumping power for moving oil through a pipeline is essential for designing the pump and energy systems that will keep the oil flowing at the specified conditions.

Bernoulli's Equation

Bernoulli's equation is a statement of the conservation of energy principle for flowing fluids. It relates the flow velocity, pressure, and elevation of a fluid to show that the total mechanical energy remains constant along a streamline, assuming incompressible, non-viscous flow. For a fluid moving through a pipe, Bernoulli's equation can be expressed as:
\[P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}\]
where \(P\) is the fluid pressure, \(\rho\) is the fluid density, \(v\) is the flow velocity, \(g\) is gravitational acceleration, and \(h\) is the elevation height. Engineers use Bernoulli's equation to predict pressure changes due to elevation changes and velocity variations within the system. In the context of pumping oil through a pipeline, Bernoulli's equation helps us understand how pumping power must be allocated to ensure the oil maintains its flow rate.

Darcy-Weisbach Equation

The Darcy-Weisbach equation is an empirical formula that calculates the pressure or head loss due to friction in a pipe. Head loss is an important concept in the analysis of fluid flow, representing the loss of energy of the flow due to the frictional forces exerted by the pipe walls. The Darcy-Weisbach equation is given by:
\[\Delta h_{f} = f \frac{L}{D} \frac{v^2}{2g}\]
Here, \(\Delta h_{f}\) is the friction head loss, \(f\) is the Darcy-Weisbach friction factor, \(L\) is the length of the pipe, \(D\) is the diameter of the pipe, \(v\) is the velocity of the fluid, and \(g\) is the acceleration due to gravity. Calculating the head loss is critical for determining the necessary pumping power, as higher frictional losses require more pumping power to maintain a certain flow rate. This equation directly feeds into the Bernoulli's equation for a more comprehensive analysis of energy balance within a piping system that transports fluids like oil.