Question

In subsea oil and natural gas production, hydrocarbon fluids may leave the reservoir with a temperature of \(70^{\circ} \mathrm{C}\) and flow in subsea surrounding of \(5^{\circ} \mathrm{C}\). As a result of the temperature difference between the reservoir and the subsea surrounding, the knowledge of heat transfer is critical to prevent gas hydrate and wax deposition blockages. Consider a subsea pipeline with inner diameter of \(0.5 \mathrm{~m}\) and wall thickness of \(8 \mathrm{~mm}\) is used for transporting liquid hydrocarbon at an average temperature of \(70^{\circ} \mathrm{C}\), and the average convection heat transfer coefficient on the inner pipeline surface is estimated to be \(250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The subsea surrounding has a temperature of \(5^{\circ} \mathrm{C}\) and the average convection heat transfer coefficient on the outer pipeline surface is estimated to be \(150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the pipeline is made of material with thermal conductivity of \(60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), by using the heat conduction equation (a) obtain the temperature variation in the pipeline wall, \((b)\) determine the inner surface temperature of the pipeline, \((c)\) obtain the mathematical expression for the rate of heat loss from the liquid hydrocarbon in the pipeline, and \((d)\) determine the heat flux through the outer pipeline surface.

Short answer

A1: The formula for inner radius (ri) and outer radius (ro) can be calculated as: \(r_\mathrm{i} = \dfrac{d_\mathrm{i}}{2}\) \(r_\mathrm{o} = r_\mathrm{i} + t_\mathrm{w}\) Q2: What is the formula for the thermal resistance of the pipeline? A2: The thermal resistance for conduction through the pipe wall (Rc), and the convective resistance on the inner (Ri) and outer (Ro) surfaces can be calculated as: \(R_\mathrm{c} = \dfrac{\ln(r_\mathrm{o} / r_\mathrm{i})}{2 \pi L k}\), \(R_\mathrm{i} = \dfrac{1}{h_\mathrm{i} 2 \pi r_\mathrm{i} L}\), \(R_\mathrm{o} = \dfrac{1}{h_\mathrm{o} 2 \pi r_\mathrm{o} L}\) Q3: How can we find the temperature variation in the pipeline wall? A3: To find the temperature variation in the pipeline wall, the total thermal resistance (Rtotal) must be calculated by summing the thermal resistances for conduction and convection. The temperature difference (ΔT) between the inner and outer surfaces of the pipeline wall can be obtained using the formula: \(\Delta T = T_\mathrm{r} - T_\mathrm{s} = Q \cdot R_\mathrm{total}\) Q4: How can we calculate the inner surface temperature of the pipeline? A4: The inner surface temperature (Ti) can be found using the following equation: \(T_\mathrm{i} = T_\mathrm{r} - Q \cdot R_\mathrm{i}\) Q5: What is the formula for the rate of heat loss from the hydrocarbon liquid? A5: The rate of heat loss from the hydrocarbon liquid (dQ/dL) can be obtained by deriving Q with respect to the length L as: \(\dfrac{dQ}{dL} = \dfrac{d (\dfrac{\Delta T}{R_\mathrm{total}})}{dL}\) Q6: How can we calculate the heat flux through the outer pipeline surface? A6: The heat flux through the outer pipeline surface (qo) can be found by multiplying the heat transfer coefficient for the outer surface (ho) by the temperature difference between the outer surface temperature (To) and the subsea surrounding temperature (Ts) as: \(q_\mathrm{o} = h_\mathrm{o} (T_\mathrm{o} - T_\mathrm{s})\)

Step-by-step solution

Find the inner and outer radii of the pipeline

First, we have to determine the inner and outer radii of the pipeline. We are given the inner diameter of 0.5 m and the wall thickness of 8 mm, which can be converted into meters like this: 8 mm * 0.001 m/mm = 0.008 m. The inner and outer radii can be calculated as follows: \(r_\mathrm{i} = \dfrac{d_\mathrm{i}}{2} = \dfrac{0.5}{2} = 0.25 \mathrm{~m}\) and \(r_\mathrm{o} = r_\mathrm{i} + t_\mathrm{w} = 0.25 + 0.008 = 0.258 \mathrm{~m}\).

Calculate the thermal resistance

To find the temperature variation in the pipeline wall, we need to calculate the thermal resistance of the pipeline. The thermal resistance for the conduction through the pipe wall \((R_\mathrm{c})\) and the convective resistance on the inner \((R_\mathrm{i})\) and outer \((R_\mathrm{o})\) surfaces can be calculated as: \(R_\mathrm{c} = \dfrac{\ln(r_\mathrm{o} / r_\mathrm{i})}{2 \pi L k}\), \(R_\mathrm{i} = \dfrac{1}{h_\mathrm{i} 2 \pi r_\mathrm{i} L}\), \(R_\mathrm{o} = \dfrac{1}{h_\mathrm{o} 2 \pi r_\mathrm{o} L}\), where L is the length of the pipeline, \(h_\mathrm{i}\) and \(h_\mathrm{o}\) are the inner and outer surface heat transfer coefficients, and k is the thermal conductivity of the pipe material.

Find the temperature variation

The total thermal resistance \((R_\mathrm{total})\) can be found by summing the thermal resistances for conduction and convection: \(R_\mathrm{total} = R_\mathrm{c} + R_\mathrm{i} + R_\mathrm{o}\). The temperature difference \((\Delta T)\) between the inner and outer surfaces of the pipeline wall can be obtained using the formula: \(\Delta T = T_\mathrm{r} - T_\mathrm{s} = Q \cdot R_\mathrm{total}\), where \(T_\mathrm{r}\) is the reservoir temperature (70°C), \(T_\mathrm{s}\) is the subsea surrounding temperature (5°C), and Q is the heat transfer rate per unit length.

Calculate the inner surface temperature

To find the inner surface temperature \((T_\mathrm{i})\), we need to rearrange the temperature difference equation: \(T_\mathrm{i} = T_\mathrm{r} - Q \cdot R_\mathrm{i}\).

Obtain the rate of heat loss from the hydrocarbon liquid

The rate of heat loss from the hydrocarbon liquid \((\dfrac{dQ}{dL})\) can be obtained by deriving Q with respect to the length L. This can be done by expressing the heat transfer rates in terms of the thermal resistance and temperature difference, and then taking the derivative with respect to L: \(\dfrac{dQ}{dL} = \dfrac{d \left(\dfrac{\Delta T}{R_\mathrm{total}}\right)}{dL}\).

Calculate the heat flux through the outer pipeline surface

To find the heat flux through the outer pipeline surface \((q_\mathrm{o})\), we need to multiply the heat transfer coefficient for the outer surface \((h_\mathrm{o})\) by the temperature difference between the outer surface temperature \((T_\mathrm{o})\) and the subsea surrounding temperature \((T_\mathrm{s})\): \(q_\mathrm{o} = h_\mathrm{o} (T_\mathrm{o} - T_\mathrm{s})\).

Key concepts

Thermal Resistance

Thermal resistance is a crucial concept when analyzing heat transfer in subsea pipelines. It acts like an obstacle to heat flow, much like resistance in an electrical circuit. In the case of our pipeline, thermal resistance determines how easily heat can leave the warm hydrocarbon fluid inside the pipe and travel to the much cooler subsea environment outside. The total thermal resistance of the pipeline is a combination of three parts: heat conduction through the pipe wall, and convection heat transfer on both the inner and outer pipe surfaces.

Heat conduction through the pipe wall has its resistance calculated using the formula:\[ R_c = \frac{\ln(r_o / r_i)}{2 \pi L k} \]where \(r_o\) and \(r_i\) are the outer and inner radii, \(L\) is the length, and \(k\) is the pipeline's thermal conductivity. Convection heat transfer resistance on the surfaces is given by:\[ R_i = \frac{1}{h_i 2 \pi r_i L} \] and\[ R_o = \frac{1}{h_o 2 \pi r_o L} \]These equations show how the thermal properties and geometry of the pipeline influence how effectively it can manage the temperatures between its contents and the surroundings.

Convection Heat Transfer Coefficient

The convection heat transfer coefficient, often denoted by \(h\), is a measure of the heat transfer capacity between a fluid and a surface through convection. It's crucial for subsea pipelines where the inner fluid and the outer seawater influence how heat is transferred.

In our example, the heat transfer coefficient on the inner surface of the pipeline is \(250 \, \mathrm{W/m^2 \cdot K}\), representing how well the pipeline conducts heat from the hydrocarbon fluid to the inner wall. Similarly, the outer surface coefficient is \(150 \, \mathrm{W/m^2 \cdot K}\), dictating the heat transfer from the outer wall to the cold subsea water.

These coefficients depend on factors like fluid velocity, density, viscosity, and the specific properties of the pipe material and the surrounding fluid. Higher coefficients generally indicate better heat transfer, which plays a vital role in preventing pipeline issues like wax deposition and hydrate formation.

Heat Conduction Equation

The heat conduction equation illustrates the flow of thermal energy through a solid medium, such as the wall of a pipeline. The equation used in this scenario helps us calculate how temperature changes occur across the pipeline wall, demonstrating the transformation of heat from the inner surface to the outer surroundings.

The equation involves the material's thermal conductivity, which is the rate at which heat passes through a material. It's essential for determining how effectively a pipeline retains or sheds heat. This specific heat conduction equation for a cylindrical pipe is:\[ \frac{dT}{dr} = -\frac{Q}{2\pi r L k} \]where \(r\) is the radius, \(T\) is temperature, \(Q\) is the heat transfer rate, \(L\) is the length of the pipe, and \(k\) is the material's thermal conductivity. By understanding this equation, one can determine temperature gradients across the pipeline and ensure the integrity of the system.

Pipeline Thermal Conductivity

Pipeline thermal conductivity, denoted as \(k\), describes the capability of the pipeline material to conduct heat. A pipeline with higher thermal conductivity will transfer heat more rapidly than one with lower conductivity. This property is vital for managing how much heat the warm fluid inside can lose to the cold seawater outside.

In our case, the pipeline material has a thermal conductivity value of \(60 \, \mathrm{W/m \cdot K}\). This measure indicates a good ability to transfer heat out of the pipe, crucial for maintaining temperature and preventing blockage issues like hydrate or wax formation. Materials with high thermal conductivity ensure efficient heat distribution, balancing the inner temperature and avoiding dramatic temperature variations.

Knowing the thermal conductivity helps engineers predict how the pipeline will perform in various environmental conditions, ensuring that the pipeline optimally maintains fluid temperature and operational integrity.