Question

Consider a tube for transporting steam that is not centered properly in a cylindrical insulation material $(k=$ $0.73\mathrm{W}/\mathrm{m}\cdot \mathrm{K})$. The tube diameter is $D_1=20\mathrm{cm}$ and the insulation diameter is $D_2=40\mathrm{cm}$. The distance between the center of the tube and the center of the insulation is $z=5\mathrm{mm}$. If the surface of the tube maintains a temperature of ${100}^{\circ }\mathrm{C}$ and the outer surface temperature of the insulation is constant at ${30}^{\circ }\mathrm{C}$, determine the rate of heat transfer per unit length of the tube through the insulation.

Short answer

Answer: The rate of heat transfer per unit length of the tube through the insulation is approximately $466.67\mathrm{W}/\mathrm{m}$.

Step-by-step solution

Calculate the temperature drop through the insulation

The temperature difference between the inner surface of the tube and the outer surface of the insulation is the driving force for heat transfer. We can calculate it as follows: $\mathrm{\Delta }T=T_1-T_2={100}^{\circ }\mathrm{C}-{30}^{\circ }\mathrm{C}={70}^{\circ }\mathrm{C}$

Calculate the equivalent thermal resistance of the insulation

The insulation is not centered on the pipe; thus, the heat flow in the insulation is not radial. However, since its diameter ($D_2$) is large compared to the pipe diameter ($D_1$), it is reasonable to approximate the flow as radial. We can calculate the equivalent thermal resistance ($R_{\text{th}}$) of the insulation using the equation: $R_{\text{th}}=\frac{\ln (\frac{D_2}{D_1})}{2\pi k}$ Using the given values, we get: $R_{\text{th}}=\frac{\ln (\frac{0.4\mathrm{m}}{0.2\mathrm{m}})}{2\pi (0.73\mathrm{W}/\mathrm{m}\cdot \mathrm{K})}=\frac{\ln (2)}{4.581}=0.150\mathrm{m}\cdot \mathrm{K}/\mathrm{W}$

Calculate the rate of heat transfer per unit length

Now that we have the equivalent thermal resistance and the temperature drop, we can calculate the rate of heat transfer per unit length ($q^{\prime }$) using Fourier's Law: $q^{\prime }=\frac{\mathrm{\Delta }T}{R_{\text{th}}}$ Substituting the values, we get: $q^{\prime }=\frac{{70}^{\circ }\mathrm{C}}{0.150\mathrm{m}\cdot \mathrm{K}/\mathrm{W}}=466.67\mathrm{W}/\mathrm{m}$ Thus, the rate of heat transfer per unit length of the tube through the insulation is approximately $466.67\mathrm{W}/\mathrm{m}$.

Key concepts

Fourier's Law

Fourier's Law is the fundamental principle that governs heat conduction. It states that the rate at which heat energy is transferred through a material is directly proportional to the negative of the temperature gradient and the cross-sectional area through which heat is flowing, and is inversely proportional to the material's thickness. In simpler terms, heat flows from hot regions to cold ones, and the flow rate increases with larger temperature differences and cross-sectional areas, but decreases with greater distances.

Mathematically, Fourier's Law for one-dimensional heat flow can be expressed as: $$q=-kA\frac{dT}{dx}$$ where $q$ is the heat transfer rate, $k$ is the thermal conductivity of the material, $A$ is the cross-sectional area perpendicular to the heat flow, and $\frac{dT}{dx}$ is the temperature gradient along the direction of heat flow. In the context of the textbook exercise, this law helps us calculate the heat transfer rate per unit length once we know the thermal resistance and temperature difference.

Thermal Resistance

Thermal resistance, analogous to electrical resistance, is a measure of a material's ability to resist the flow of heat. When considering heat transfer through a material, thermal resistance describes how well the material insulates against the movement of heat. The higher the thermal resistance, the slower the heat flows through the material, and vice versa.

Thermal resistance can be calculated using the formula: $$R_{\text{th}}=\frac{\mathrm{\Delta }x}{kA}$$ where $R_{\text{th}}$ is the thermal resistance, $\mathrm{\Delta }x$ is the thickness of the material, $k$ is the thermal conductivity, and $A$ is the cross-sectional area. In practice, when dealing with non-uniform geometries like the insulation in the exercise, we make reasonable approximations to simplify calculations. Here, a radial heat flow assumption allows us to represent the thermal resistance of cylindrical insulation as an equivalent resistance, which is a crucial step in determining the heat transfer rate.

Radial Heat Flow

Radial heat flow refers to the movement of heat in a radial direction outward or inward through a cylindrical or spherical object. This type of heat flow is prevalent in systems where there is a non-uniform geometry such as pipes or tubes.

In the case of radial heat flow through a cylindrical object, the thermal resistance is different from that in one-dimensional heat flow situations, due to the varying cross-sectional area as heat moves radially. The formula to calculate thermal resistance for radial heat flow in a cylinder is: $$R_{\text{th}}=\frac{\ln (\frac{r_2}{r_1})}{2\pi kL}$$ where $r_1$ and $r_2$ are the inner and outer radii of the cylindrical layer through which heat is flowing, $k$ is the thermal conductivity, and $L$ is the length of the cylinder. This concept is crucial in the step-by-step solution provided, where approximating the insulation as a radially conducting layer allows us to calculate the heat transfer rate using the modified equation for thermal resistance in cylindrical coordinates.