Question

Water flows through a pipe at an average temperature of \(T_{\infty}=90^{\circ} \mathrm{C}\). The inner and outer radii of the pipe are \(r_{1}=\) \(6 \mathrm{~cm}\) and \(r_{2}=6.5 \mathrm{~cm}\), respectively. The outer surface of the pipe is wrapped with a thin electric heater that consumes \(400 \mathrm{~W}\) per \(\mathrm{m}\) length of the pipe. The exposed surface of the heater is heavily insulated so that the entire heat generated in the heater is transferred to the pipe. Heat is transferred from the inner surface of the pipe to the water by convection with a heat transfer coefficient of \(h=85 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming constant thermal conductivity and one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of the heat conduction in the pipe during steady operation. Do not solve.

Short answer

#Answer# The mathematical formulation of the heat conduction in the pipe during steady operation consists of the following equation and boundary conditions: 1. Temperature equation: \(T(r) = C_{1}\int \frac{dr}{k} + C_{2}\) 2. Constant \(C_{1}\) representing heat flux at the interface of pipe and heater: \(C_{1} = -\frac{400 \mathrm{~W/m}}{2 \pi r_2}\) 3. Boundary Condition 1 at the interface of pipe and heater (\(r_{2}\)): \(q_{r_2} = -k \frac{dT}{dr} \vert_{r=r_2} = \frac{400 \mathrm{~W/m}}{2 \pi r_2}\) 4. Boundary Condition 2 at the interface of pipe and water (\(r_{1}\)): \(h (T_{\infty} - T(r_1)) = -k \frac{dT}{dr} \vert_{r=r_1}\) By applying the second boundary condition, we can determine the expression for constant \(C_{2}\). Then, by substituting the expressions for \(C_{1}\) and \(C_{2}\) into the temperature equation, we can completely define the temperature distribution throughout the pipe.

Step-by-step solution

Heat Conduction Equation

First, let's write the general heat conduction equation for cylindrical coordinates. In steady-state and one-dimensional conditions (where heat transfer in only the radial direction), the general heat conduction equation is given by: \(\frac{d}{dr}\left( k \frac{dT}{dr} \right) = 0\) where \(k\) is the thermal conductivity, \(T\) is the temperature, and \(r\) is the radial coordinate.

Integrating the Heat Conduction Equation

Since we are asked to express the mathematical formulation of the heat conduction in the pipe without solving it, we will now integrate the heat conduction equation: Integrating once with respect to \(r\): \(k \frac{dT}{dr} = C_{1}\) Integrating one more time: \(T(r) = C_{1}\int \frac{dr}{k} + C_{2}\) Now we will apply the boundary conditions based on the information given in the problem.

Boundary Condition 1 – Interface of Pipe and Heater

At the interface between the pipe and the heater (i.e., at \(r = r_2\)), the heat generated by the heater is transferred to the pipe. So, the heat flux at the outer surface of the pipe must be equal to the rate at which heat is generated by the electric heater: \(q_{r_2} = -k \frac{dT}{dr} \vert_{r=r_2} = \frac{400 \mathrm{~W/m}}{2 \pi r_2}\) This is our first boundary condition. This equation can be substituted into the equation TResulting from the integration of the heat conduction equation: \(C_{1} = -\frac{400 \mathrm{~W/m}}{2 \pi r_2}\)

Boundary Condition 2 – Interface of Pipe and Water

At the interface between the pipe and the water (i.e., at \(r = r_1\)), heat is transferred from the inner surface of the pipe to the water by convection. According to Newton's law of cooling, the heat flux at the surface of the pipe is: \(q_{r_1} = h (T_{\infty} - T(r_1))\) Where \(T_{\infty}\) is the temperature of the water, and \(h\) is the heat transfer coefficient. The heat flux can also be expressed as: \(q_{r_1} = -k \frac{dT}{dr} \vert_{r=r_1}\) Equating the two expressions for the heat flux: \(h (T_{\infty} - T(r_1)) = -k \frac{dT}{dr} \vert_{r=r_1}\) Now we can substitute the expression for \(T(r)\) and solve for \(C_{2}\). This will give us the second boundary condition. At this point, the mathematical formulation of the heat conduction in the pipe during steady operation includes the differential equation for temperature (after integrating the heat conduction equation), expressions for \(C_{1}\) and \(C_{2}\) (heat flux at the interface of pipe and heater, and pipe and water), and boundary conditions at the interfaces.

Key concepts

Heat Transfer Coefficient

The heat transfer coefficient, denoted as 'h', plays a crucial role in the exchange of heat between a solid surface and a fluid in motion. It quantifies the heat transfer rate per unit area and per unit temperature difference between the solid surface and the fluid. The heat transfer coefficient is measured in watts per square meter per Kelvin (\(W/m^2\text{K}\))

When water flows through a pipe, like in the given exercise, heat is transferred from the inner surface of the pipe to the water predominantly by convection. Calculating or correctly assuming the heat transfer coefficient is vital for predicting how effectively the pipe can transfer heat to the water. An accurate 'h' value ensures that any thermal analysis or design is reliable, impacting systems ranging from simple heating setups to complex heat exchangers in industrial applications.

The exercise implies the use of 'h' to establish the boundary condition at the interface where the heat is carried away by the water. Essentially, Newton's law of cooling leverages 'h' to relate the heat flux from the pipe's surface to the temperature difference between the surface and the flowing water.

Thermal Conductivity

Thermal conductivity, symbolized by 'k', is a material property that indicates the ability of the material to conduct heat. It is defined as the rate at which heat is transferred through a unit thickness of a material, over a unit surface area, under a unit temperature gradient. The SI units for thermal conductivity are watts per meter per Kelvin (\(W/m\text{K}\)).

In the context of the exercise, where heat needs to be transferred through the pipe's wall thickness, the thermal conductivity of the pipe material directly influences how the heat from the electric heater is distributed across the pipe. Materials with higher 'k' values are better conductors of heat and are preferable for applications requiring efficient thermal transport.

In the step-by-step solution, 'k' appears in the differential heat conduction equation and is fundamental in determining the rate at which temperature changes with respect to the radial position in the pipe. This information is crucial for engineers when choosing appropriate materials for heat transfer applications.

Steady-State Heat Conduction

Steady-state heat conduction refers to a condition where the temperature field within a body does not change with time. In other words, any point within the material experiences a constant temperature, even though heat may be flowing through it. This is an important simplification assumption in many heat transfer problems as it restricts the consideration to time-independent solutions.

The assumption of steady-state heat conduction, as made in the given exercise, means that the heat input from the electric heater and the heat removed by the water flowing through the pipe have reached a balance over time. The temperature distribution within the pipe wall will not fluctuate and remains static, allowing the use of the simple form of the heat conduction equation without any time-derivative terms.

This condition largely simplifies the analysis, as it discounts the complexities introduced by transient temperature variations. It's particularly useful in many industrial applications where operations are meant to reach a steady operating condition.

Boundary Conditions

Boundary conditions in heat transfer problems are constraints that define how heat is transferred at the boundaries of a domain. They are essential parameters for solving heat conduction equations as they specify the thermal behavior at the surfaces where the material interfaces with other media or materials.

In the exercise at hand, two types of boundary conditions are described: the first one is at the outer surface of the pipe where it contacts the heater, and the second one at the inner surface where the pipe contacts the water. At the outer surface, a set amount of heat is added per unit length from the heater. On the inner surface, heat is transferred to the water via convection. These conditions dictate the values of the constants found when integrating the heat conduction equation.

By applying the boundary conditions, one can tailor the general solution of a differential equation to fit the specific scenario of interest. This process transforms a general mathematical model into a precise representation of a physical system.

Cylindrical Coordinates Heat Transfer

When analyzing heat transfer in geometries that are inherently cylindrical, such as pipes, using cylindrical coordinates provides a natural and efficient framework. This system of coordinates aligns one of the dimensions with the axis of the cylinder (the radial coordinate 'r'), simplifying the representation of the heat transfer phenomena that is predominantly radial in such cases.

In the problem provided, the radial direction is where heat conduction through the pipe's wall is occurring. The cylindrical coordinates enable the breaking down of the heat conduction equation into a radial part, making it easier to solve for the temperature distribution within the pipe. This format not only aligns with the physical setup but also reduces complexity by focusing only on the relevant dimension, which is particularly helpful in steady-state conditions where the temperature gradient exists predominantly in the radial direction and not in the angular or axial dimensions.