Question

The tube in a heat exchanger has a 2-in. inner diameter and a 3 -in. outer diameter. The thermal conductivity of the tube material is \(0.5 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\), while the inner surface heat transfer coefficient is \(50 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\) and the outer surface heat heat transfer coefficients based on the outer and inner surfaces.

Short answer

Based on the given information about the heat exchanger tube with given inner and outer diameters, thermal conductivity of the tube material, and the inner surface heat transfer coefficient, describe the process to find the outer surface heat transfer coefficient and explain how the overall heat transfer coefficient relates to it.

Step-by-step solution

Calculate the thermal resistance of the tube material

The thermal resistance can be calculated using the following formula: \(R_{tube} = \frac{\ln(\frac{D_o}{D_i})}{2 \pi k}\) where: \(R_{tube}\) is the thermal resistance of the tube material, \(D_i = 2\,\text{in.}\) is the inner diameter, \(D_o = 3\,\text{in.}\) is the outer diameter, \(k = 0.5 \, \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\) is the thermal conductivity of the tube material. We first convert the inner and outer diameters from inches to feet: \(D_i = \frac{2\,\text{in.}}{12\,\text{in./ft}} = \frac{1}{6}\,\text{ft}\) \(D_o = \frac{3\,\text{in.}}{12\,\text{in./ft}} = \frac{1}{4}\,\text{ft}\) Now, we can calculate the thermal resistance: \(R_{tube} = \frac{\ln(\frac{1/4}{1/6})}{2 \pi (0.5)}\)

Calculate the overall heat transfer coefficient (U) for the tube

The overall heat transfer coefficient (U) can be calculated as the reciprocal of the sum of the thermal resistances of the tube material and the inner surface heat transfer coefficient. The formula is: \(U = \frac{1}{\frac{1}{h_i} + R_{tube}}\) where: \(U\) is the overall heat transfer coefficient, \(h_i = 50 \, \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\) is the inner surface heat transfer coefficient, \(R_{tube}\) is the thermal resistance of the tube material. We plug in the values and compute U: \(U = \frac{1}{\frac{1}{50} + R_{tube}}\)

Determine the outer surface heat transfer coefficient (h_o)

Using the overall heat transfer coefficient (U) and the tube's geometry, we can determine the outer surface heat transfer coefficient (h_o) as follows: \(h_o = \frac{U}{1 - \frac{U}{h_i}}\) We substitute the U value from Step 2 and calculate h_o: \(h_o = \frac{U}{1 - \frac{U}{50}}\) In conclusion, the outer surface heat transfer coefficient can be found using the steps outlined above – first, calculating the thermal resistance of the tube material, then finding the overall heat transfer coefficient, and finally determining the outer surface heat transfer coefficient using the overall heat transfer coefficient and the tube's geometry.

Key concepts

Understanding Thermal Resistance Calculation

The concept of thermal resistance is often used as an analogy to electrical resistance in the context of heat transfer, where it quantifies how much a material or surface resists the flow of heat. Calculating thermal resistance is crucial in designing systems like heat exchangers, where you want to predict how well heat will move across materials.

Let's delve into the example of the heat exchanger tube given in the exercise. The formula to calculate the thermal resistance of a cylindrical material is \( R_{tube} = \frac{\ln(\frac{D_o}{D_i})}{2 \pi k} \). Here, \(D_i\) and \(D_o\) represent the inner and outer diameters of the tube, respectively, and \(k\) is the thermal conductivity of the material — essentially a measure of how easily heat can flow through it.

For our tube, converting the diameters from inches to feet is the initial step, since thermal conductivity is given in units involving feet. After computing the natural logarithm of the diameter ratio, we divide by the product of \(2\pi k\) to find the thermal resistance of the tube. With a lower thermal resistance, heat transfers more efficiently, which is preferable in most heat exchanger applications.

Grasping Overall Heat Transfer Coefficient

The overall heat transfer coefficient (\(U\)) is a measure that encompasses all the different layers of materials and air gaps that could potentially resist heat flow between two fluids in a heat exchanger. A higher \(U\) value indicates a more effective transfer of heat. To compute \(U\), you'd inversely sum up all individual thermal resistances, which includes both the tube material and the resistances from the heat transfer coefficients on the inside and outside surfaces of the tube.

Per the exercise, after we have the tube's thermal resistance, \(U\) can be found using the formula \(U = \frac{1}{\frac{1}{h_i} + R_{tube}}\). The inner surface heat transfer coefficient (\(h_i\)) is given, so we incorporate that into our calculations. All thermal resistances are added in their reciprocal form, and then we take the reciprocal of that total to determine \(U\). The resulting \(U\) value is critical for analyzing the heat exchanger's performance and is essential for subsequent calculations such as determining the heat transfer coefficient for the tube's outer surface.

Maximizing Heat Exchanger Efficiency

Heat exchanger efficiency is a reflection of how well a heat exchanger performs its function — transferring heat from one medium to another with minimum energy loss. This is often assessed by looking at the heat transfer coefficients and the overall heat transfer coefficient. The outer surface heat transfer coefficient (\(h_o\)) is directly tied to the efficiency, as it affects how heat is dissipated into the surrounding environment or received by another fluid.

In our textbook example, calculating \(h_o\) uses the relationship between \(U\) and \(h_i\) shown in the formula \(h_o = \frac{U}{1 - \frac{U}{h_i}}\). A higher \(h_o\) suggests that the outer surface is transferring heat effectively, which is generally advantageous.

It's essential to optimize both \(h_i\) and \(h_o\) in the design of heat exchangers to achieve high efficiency. By targeting materials with low thermal resistance and ensuring the heat transfer coefficients are favorable, designers can maximize the exchanger's ability to transfer heat between fluids, thereby optimizing energy consumption and system performance.