Question

Liquid ethanol is a flammable fluid that has a flashpoint at \(16.6^{\circ} \mathrm{C}\). At temperatures above the flashpoint, ethanol can release vapors that form explosive mixtures, which could ignite when source of ignition is present. In a chemical plant, liquid ethanol is being transported in a pipe \((k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) with an inside diameter of \(3 \mathrm{~cm}\) and a wall thickness of \(3 \mathrm{~mm}\). The pipe passes through areas where occasional presence of ignition source can occur, and the pipe's outer surface is subjected to a heat flux of \(1 \mathrm{~kW} / \mathrm{m}^{2}\). The ethanol flowing in the pipe has an average temperature of \(10^{\circ} \mathrm{C}\) with an average convection heat transfer coefficient of \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Your task as an engineer is to ensure that the ethanol is transported safely and prevent fire hazard. Determine the variation of temperature in the pipe wall and the temperatures of the inner and outer surfaces of the pipe. Are both surface temperatures safely below the flashpoint of liquid ethanol?

Short answer

Based on the step by step solution given above, here is a short answer: To determine the radial temperature distribution in the pipe wall and ensure the temperatures are safely below the flashpoint of ethanol, we can follow these steps: 1. Derive the radial temperature distribution equation for the pipe wall using cylindrical heat conduction equation. 2. Integrate the simplified equation to obtain the temperature distribution equation. 3. Apply boundary conditions at the pipe's inner and outer surfaces to find the constants of integration. 4. Calculate the inner and outer surface temperatures using the given material properties, flow conditions, and boundary conditions. 5. Check whether the calculated temperatures are below the flashpoint of ethanol to ensure safe transport. By obtaining the temperature distribution and checking the safety against ethanol's flashpoint, we can prevent potential fire hazards in the chemical plant.

Step-by-step solution

Derive the radial temperature distribution equation in the pipe wall

We will start with the general heat conduction equation in cylindrical coordinates, but simplifying it since there is no heat generation within the pipe wall and the steady-state temperature distribution only varies in the radial direction. The simplified equation will be: $$\frac{d}{dr}\left(k\frac{dT}{dr}\right) = 0$$

Integrate to obtain the temperature distribution equation

Integrate the simplified equation twice with respect to the radial distance, r, to get the temperature distribution equation in the pipe wall: $$T(r) = -\frac{C_{1}}{k}r^{2}+C_{2}$$ Where \(C_{1}\) and \(C_{2}\) are constants of integration.

Apply boundary conditions to find the constants of integration

Apply the two boundary conditions for this problem: 1. At the inner surface of the pipe (r = \(R_{1}\)): heat transfer by convection between the ethanol and the inner surface of the pipe (Newton's law of cooling) $$-k\left.\frac{dT}{dr}\right|_{r=R1} = h(T_{1}- T_\mathrm{ethanol})$$ 2. At the outer surface of the pipe (r = \(R_{2}\)): heat transfer by conduction is equal to the heat flux $$-k\left.\frac{dT}{dr}\right|_{r=R2} = q_{\text{flux}}$$ Where \(T_{1}\) is the inner surface temperature, \(T_{2}\) is the outer surface temperature, \(R_{1}\) is the inner radius of the pipe wall, \(R_{2}\) is the outer radius of the pipe wall, and \(h\) is the convection heat transfer coefficient between the ethanol and the pipe wall.

Calculate temperatures

Use the given values of \(k\), \(q_{\text{flux}}\), \(R_{1}\), \(R_{2}\), \(h\), and \(T_\mathrm{ethanol}\), and substitute them into the boundary conditions in step 3 to calculate the inner surface temperature \(T_{1}\) and the outer surface temperature \(T_{2}\). After obtaining the values of \(C_{1}\) and \(C_{2}\), we can also write down the temperature distribution in the pipe wall as: $$T(r) = -\frac{C_{1}}{k}r^{2}+C_{2}$$ Then, check if the inner and outer surface temperatures are safely below the flashpoint of ethanol, which is \(16.6^{\circ} \mathrm{C}\). Following these steps will allow you to determine the temperature distribution in the pipe wall and ensure the safe transport of ethanol through the chemical plant, preventing fire hazards.

Key concepts

Heat Conduction Equation

Understanding the heat conduction equation is crucial for predicting temperature variations within a material. In our classroom discussion, we talked about how heat travels through a substance from high to low temperature regions. The simplified heat conduction equation in cylindrical coordinates, which we used in the exercise, captures this physics for a radially symmetric pipe wall with no internal heat generation and a steady-state condition.

Let's put it into simpler terms. Imagine heat spreading out from the center of a cinnamon roll. It moves outward in all directions evenly. In our case, the 'cinnamon roll' is the pipe, and the heat moves from the inner surface of the pipe to the outer surface. The mathematical expression for this scenario is given by the differential equation \[\frac{d}{dr}\left(k\frac{dT}{dr}\right) = 0\] which, upon integration, provides the temperature distribution as a function of the radial distance, or how far you are from the center of the roll, represented by \(r\). Integrating this gives us a relationship between temperature \(T\) and position \(r\), shaped by physical constants that describe the pipe's material properties.

Convection Heat Transfer Coefficient

When discussing how heat is transferred between a fluid and a surface, the convection heat transfer coefficient, denoted by \(h\), plays a starring role. In our ethanol transport problem, this coefficient is a measure of how well heat moves from the liquid ethanol to the pipe wall or vice versa.

Why is \(h\) so important?

Think of \(h\) as a number that tells us how effective that 'handshake' between the ethanol and the pipe wall is in terms of exchanging heat. A higher value of \(h\) means they are really good at exchanging heat—like a firm, snappy handshake. A lower value indicates a weak heat exchange—imagine a limp, lifeless handshake.

In this exercise, using the value of \(h\) along with Newton's law of cooling, which relates the rate of heat transfer to the temperature difference between the fluid and the surface, lets us set a boundary condition for the heat conduction equation. This boundary condition is the recipe needed to cook up the constants in our temperature distribution 'recipe'.

Boundary Conditions in Heat Transfer

Boundary conditions in heat transfer are the 'rules' that the temperature must obey at the edges of a material. Think of these as the 'do's and 'don'ts' at a pool. How you enter the pool (jump, dive, or tiptoe) affects everything that happens inside the pool. Similarly, how heat enters or leaves the surfaces of a material defines how it behaves inside.

Our exercise showcases two types of boundary conditions: one at the inner surface where the pipe wall meets the flowing ethanol (like where your toe first touches the water) and one at the outer surface where the pipe is exposed to external heat flux (like where your body rises to the surface of the water).

These conditions can be expressed as equations that connect temperature and heat transfer at those precise points. They are essential for solving heat conduction problems because they allow us to find the mystery constants \(C_{1}\) and \(C_{2}\) for our temperature equation. Once these constants are determined, one can conclude about the overall temperature profile, just like knowing the entry points into a pool can help predict the ripples and waves inside.