Question

In a thermal system, water enters a \(25-\mathrm{mm}\)-diameter and \(23-\mathrm{m}\)-long circular tube with a mass flow rate of \(0.1 \mathrm{~kg} / \mathrm{s}\) at \(25^{\circ} \mathrm{C}\). The heat transfer from the tube surface to the water can be expressed in terms of heat flux as \(\dot{q}_{s}(x)=a x\). The coefficient \(a\) is \(400 \mathrm{~W} / \mathrm{m}^{3}\), and the axial distance from the tube inlet is \(x\) measured in meters. Determine \((a)\) an expression for the mean temperature \(T_{m}(x)\) of the water, \((b)\) the outlet mean temperature of the water, and \((c)\) the value of a uniform heat flux \(\dot{q}_{s}\) on the tube surface that would result in the same outlet mean temperature calculated in part (b). Evaluate water properties at \(35^{\circ} \mathrm{C}\).

Short answer

Answer: To find the expression for the mean temperature T_m(x) of the water, we need to first determine the total heat transfer rate from the tube surface to the water as a function of x, integrate the heat transfer rate expression, find the temperature change, and write an expression for the mean temperature T_m(x).

Step-by-step solution

Find an expression for the mean temperature T_m(x) of the water

First, we need to determine the total heat transfer rate from the tube surface to the water as a function of x. This can be represented by \(\dot{Q}(x) = \pi D L \dot{q}_{s}(x)\), where D = 25 mm and L = 23 m. Next, we can integrate the heat transfer rate expression, find the temperature change, and write an expression for the mean temperature T_m(x).

Calculate the outlet mean temperature T_m(L)

Now that we have an expression for the mean temperature T_m(x), we can find the outlet mean temperature T_m(L) by substituting the tube length L = 23 m into the T_m(x) expression.

Determine the uniform heat flux q_s that would produce the same outlet mean temperature

To find the uniform heat flux q_s that would result in the same outlet mean temperature, we can write the total heat transfer rate in terms of the uniform heat flux: \(\dot{Q}_{uniform} = \pi D L \dot{q}_{s, uniform}\). Then, we can equate this expression with the previously calculated total heat transfer rate \(\dot{Q}(L)\) and solve for the uniform heat flux \(\dot{q}_{s, uniform}\).

Key concepts

Mean Temperature Calculation

Understanding the mean temperature of a fluid within a thermal system, such as water flowing through a tube, is crucial for analyzing heat transfer processes. The mean temperature, denoted as \( T_m(x) \), is the average temperature of the fluid at a given point along the flow path. It is influenced by the energy added or removed from the fluid and can vary along the length of the tube.

To calculate the mean temperature at any cross-section along the tube, we integrate the heat added from the tube's surface up to that point. With the given linear heat flux \( \dot{q}_{s}(x) = a x \), we need to take into account the variable heat input depending on the axial position \( x \). The mean temperature is then found by integrating this heat flux and dividing by the mass flow rate and specific heat capacity of water. By integrating from the tube inlet to any given point \( x \), we can determine how the temperature of the water changes as it travels through the tube.

Thermal System Analysis

A thermal system analysis involves examining the heat transfer characteristics of a system to understand how energy is being transferred. This encompasses studying the rate of heat transfer, the temperature distribution within the system, and any factors that could affect these parameters, such as the physical properties of the fluid or the geometry of the system.

For instance, by analyzing the given thermal system where water flows through a tube, we take into account factors such as the diameter and length of the tube, the mass flow rate of water, and the heat flux. These help us establish equations that model how the water's temperature will evolve from the inlet to the outlet of the tube. To perform a thorough analysis, we also need to consider the properties of water, such as density and specific heat, which determine how much energy is required to raise the temperature of the water. Overall, such an analysis helps in predicting the system's performance under various operating conditions and is essential for the design and optimization of thermal systems.

Uniform Heat Flux

Uniform heat flux, represented as \( \dot{q}_{s, uniform} \), is a scenario where the heat applied per unit area over the surface of a tube is constant along the length of the tube. This simplifies the analysis as every cross-section of the fluid receives the same amount of energy per unit length. Even though the actual heat flux might vary with the position, such as \( \dot{q}_{s}(x)=a x \) in our exercise, finding an equivalent uniform heat flux can provide a useful comparison for the analysis of the system's behavior.

In practice, determining a uniform heat flux that yields the same outlet temperature as a varying flux involves integrating the varying heat flux over the length of the tube and equating it to the total heat transferred by the uniform heat flux. This approach ensures identical energy input and thus allows for direct comparison between different heating strategies.

Heat Transfer Rate

The heat transfer rate, indicated as \( \dot{Q} \), represents the amount of thermal energy transferred per unit time. It is a central concept in heat transfer analyses and has direct implications for the temperature changes observed in a fluid or object. The rate is commonly expressed in watts (W) and can vary along a system depending on spatial changes in heat flux, surface area, or temperature differences.

In our tube system where water is being heated, the overall heat transfer rate can be related to the local heat flux by integrating the flux over the surface area through which the heat is being transferred. The total rate of heat gained by the water as it flows can then be linked to the spatial variations in temperature. This is crucial for determining how quickly and uniformly a fluid may be heated in a process, which is of particular interest in industrial applications such as heating systems, reactors, or heat exchangers.