Question

The bulk or mixed temperature of a fluid flowing through a pipe or duct is defined as (a) \(T_{b}=\frac{1}{A_{c}} \int_{A_{c}} T d A_{c}\) (b) \(T_{b}=\frac{1}{\dot{m}} \int_{A_{c}} T \rho V d A_{c}\) (c) \(T_{b}=\frac{1}{\dot{m}} \int_{A_{c}} h \rho V d A_{c}\) (d) \(T_{b}=\frac{1}{A_{c}} \int_{A_{c}} h d A_{c}\) (e) \(T_{b}=\frac{1}{\dot{V}} \int_{A_{c}} T \rho V d A_{c}\)

Short answer

Answer: \(T_{b}=\frac{1}{\dot{m}} \int_{A_{c}} T \rho V d A_{c}\)

Step-by-step solution

Option (a) - \(T_{b}=\frac{1}{A_{c}} \int_{A_{c}} T d A_{c}\)

This option involves integrating the temperature \(T\) over the cross-sectional area \(A_c\) and dividing the result by \(A_c\). Physically, this would mean that we're taking the average of the temperatures in different parts.cross the pipe, to calculate the bulk temperature. This equation does not account for factors such as density, velocity, or enthalpy of fluid flow.

Option (b) - \(T_{b}=\frac{1}{\dot{m}} \int_{A_{c}} T \rho V d A_{c}\)

This equation involves integrating the product of density (\(\rho\)), velocity (\(V\)), and temperature (\(T\)) over the cross-sectional area (\(A_c\)) and dividing the result by the mass flow rate (\(\dot{m}\)). This definition takes into account the different speeds and densities of fluid particles across the cross-section area when calculating the bulk temperature, making this equation physically meaningful.

Option (c) - \(T_{b}=\frac{1}{\dot{m}} \int_{A_{c}} h \rho V d A_{c}\)

In this option, the bulk temperature is represented as the integration of the product of enthalpy (\(h\)), density (\(\rho\)), and velocity (\(V\)) across the cross-sectional area (\(A_c\)), divided by the mass flow rate (\(\dot{m}\)). The presence of enthalpy instead of temperature in the equation doesn't fulfill the requirement of calculating the bulk temperature.

Option (d) - \(T_{b}=\frac{1}{A_{c}} \int_{A_{c}} h d A_{c}\)

This option is very similar to option (a), but with enthalpy (\(h\)) instead of temperature (\(T\)). Again, this equation does not correctly represent the definition of the bulk temperature as the formula calculates the bulk enthalpy.

Option (e) - \(T_{b}=\frac{1}{\dot{V}} \int_{A_{c}} T \rho V d A_{c}\)

This option integrates the product of density (\(\rho\)), velocity (\(V\)), and the temperature (\(T\)) across the cross-sectional area (\(A_c\)) and divides by the volume flow rate (\(\dot{V}\)). Dividing by the volume flow rate does not provide a meaningful representation of the bulk temperature. After analyzing all five options, option (b) provides the most accurate and meaningful definition for the bulk temperature of a fluid flowing through a pipe or duct.

Key concepts

Fluid Mechanics

Fluid mechanics is a crucial branch of physics involved in the study of the movement of fluids, whether liquid or gas. It encompasses the behavior of these fluids under various forces and in different environmental conditions. When discussing fluid flow through a pipe or duct, an integral concept is the distribution of temperature within the fluid.

This distribution is not uniform; the fluid closer to the pipe's wall is often cooler due to heat loss, whereas the fluid in the center may be warmer. Comprehensive knowledge of fluid mechanics enables us to understand 'bulk temperature,' which refers to an average temperature that accounts for this uneven distribution. It is calculated by considering both the velocity and the cross-sectional area, ensuring that each element of the fluid's volume contributes to the final temperature estimate according to its flow speed and proximity to heat transfer surfaces.

Heat Transfer

Heat transfer in fluid mechanics pertains to how thermal energy moves within the fluid and between the fluid and its surroundings. It's a fundamental aspect when understanding the temperature distribution across different sections of the fluid's flow.

There are three modes of heat transfer: conduction, convection, and radiation, but in the context of fluid flowing through a pipe, conduction and convection are most relevant. Convection occurs as the fluid moves, carrying energy with it. When considering the temperature at any point in the fluid, we must integrate the effects of these heat transfer modes to find the bulk temperature. Unlike a straightforward average, the bulk temperature takes into account the varying flow velocities and properties of the fluid, which could affect heat convection rates and result in a more accurate representation of the fluid's thermal state.

Mass Flow Rate

The mass flow rate, denoted by \(\dot{m}\), represents the amount of mass passing through a given cross-sectional area of a pipe or duct per unit time. It is a vital parameter in fluid dynamics and thermodynamics as it characterizes the quantity of fluid moving and its potential to transport energy.

In the equation for bulk temperature, the mass flow rate becomes particularly important when integrating the temperature across the cross-sectional area. It helps in normalizing the temperature distribution by the amount of mass flow, giving a realistic and precise value that considers variations in speed and density within the flowing fluid. The accurate calculation of bulk temperature hinges on recognizing the mass flow rate, a measure that bridges fluid mechanics and thermodynamics to provide insights into the energy characteristics of the system.

Thermodynamics

Thermodynamics is another foundational concept that intersects with fluid mechanics in the analysis of bulk temperature. It involves the study of heat and temperature and their connection to energy and work. The principles of thermodynamics are crucial in understanding how energy transfer is related to the physical properties of the fluid and how these properties change as the fluid flows and undergoes temperature variations.

For fluid flow in a pipe, thermodynamics tells us that the temperature field will be affected by energy transfer processes such as work done by or on the fluid and heat exchange with the pipe's walls. The equation for bulk temperature is a thermodynamic application that aims to summarize a complex temperature landscape into a single, effective temperature value, accounting for the various dynamic energy interactions within a moving fluid.