Question

Water enters a circular tube whose walls are maintained at constant temperature at a specified flow rate and temperature. For fully developed turbulent flow, the Nusselt number can be determined from \(\mathrm{Nu}=0.023 \mathrm{Re}^{0.8} \mathrm{Pr}^{0.4}\). The correct temperature difference to use in Newton s law of cooling in this case is (a) The difference between the inlet and outlet water bulk temperature. (b) The difference between the inlet water bulk temperature and the tube wall temperature. (c) The log mean temperature difference. (d) The difference between the average water bulk temperature and the tube temperature. (e) None of the above.

Short answer

Answer: The correct temperature difference to use in Newton's law of cooling in this case is the log mean temperature difference.

Step-by-step solution

Understanding Newton's Law of Cooling

Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. In this case, the body is the fluid flowing through the circular tube, and the surrounding is the tube wall which is maintained at a constant temperature.

What is the Nusselt number \(\mathrm{Nu}\)?

The Nusselt number is a dimensionless number that indicates the relative significance of conduction and convection in a system. In this exercise, the Nusselt number can be determined from the given formula: \(\mathrm{Nu}=0.023\mathrm{Re}^{0.8}\mathrm{Pr}^{0.4}\)

What are the Reynolds number \(\mathrm{Re}\) and Prandtl number \(\mathrm{Pr}\)?

The Reynolds number is the dimensionless quantity that represents the ratio of inertial forces to viscous forces in a fluid, and it helps predict the onset of turbulence. The Prandtl number is the dimensionless quantity that represents the ratio of momentum diffusivity to thermal diffusivity. These two numbers are essential to determine the Nusselt number.

Analyzing the available choices

We will assess each option and determine which one is the correct temperature difference for Newton's law of cooling: (a) The difference between the inlet and outlet water bulk temperature. - This option does not consider the wall temperature and focuses only on the fluid's temperature changes, so it is not appropriate. (b) The difference between the inlet water bulk temperature and the tube wall temperature. - This option considers the temperature difference between the fluid and the tube wall but does not account for the change in temperature as the fluid flows through the tube. (c) The log mean temperature difference. - This option is appropriate because it accounts for the temperature variations as the fluid flows through the tube while also considering the temperature of the tube walls. (d) The difference between the average water bulk temperature and the tube temperature. - This option neither considers the inlet and outlet temperature differences nor the log mean temperature difference, so it is not appropriate. (e) None of the above. - We have found an appropriate option above, so this statement is not correct.

Conclusion

The correct temperature difference to use in Newton's law of cooling in this case is the log mean temperature difference (option c).

Key concepts

Newton's Law of Cooling

Newton's Law of Cooling is a fundamental principle in heat transfer. It proposes that the rate of heat exchange between a body and its surroundings is proportional to the temperature difference between them.
This implies that the greater the difference in temperature, the more rapid the heat transfer.
In practical terms, for an object like a fluid flowing in a tube, the surrounding environment is the tube wall. This wall is maintained at a constant temperature. Thus, the temperature difference between the fluid and the tube wall drives the heat transfer.
Understanding this concept helps in calculating the heat transfer rate in various applications, which is crucial for designing efficient heating or cooling systems.

Nusselt Number

The Nusselt number is an essential concept when studying convection heat transfer. This dimensionless number indicates the efficiency of the convective heat transfer compared to the conductive heat transfer across a boundary.
Mathematically, it is expressed as the ratio between the convective and conductive heat transfer: \[ Nu = \frac{hL}{k} \]where \(h\) stands for convective heat transfer coefficient, \(L\) for the characteristic length, and \(k\) for thermal conductivity.
In the given scenario, the Nusselt number is derived using the expression \(\mathrm{Nu}=0.023\mathrm{Re}^{0.8}\mathrm{Pr}^{0.4}\). These calculations provide insights into the behavior of heat transfer in a system.

Reynolds Number

The Reynolds number, a key parameter in fluid mechanics, helps predict the transition from laminar to turbulent flow in fluid motion.
It is calculated using the formula: \[ Re = \frac{\rho v L}{\mu} \]where \(\rho\) is the fluid density, \(v\) is the flow velocity, \(L\) is the characteristic length, and \(\mu\) is the dynamic viscosity.
In simpler terms, the Reynolds number is a ratio of inertial forces to viscous forces. A higher Reynolds number typically indicates turbulent flow, which enhances heat transfer due to increased mixing. As part of the formula for the Nusselt number, it significantly affects the heat transfer rate in systems like the circular tube discussed in the exercise.

Prandtl Number

The Prandtl number is another critical dimensionless number in fluid dynamics, especially in heat transfer analyses.
Defined by the equation: \[ \text{Pr} = \frac{u}{\alpha} = \frac{\mu c_p}{k} \]where \(u\) is kinematic viscosity, \(\alpha\) is thermal diffusivity, \(c_p\) is specific heat, and \(k\) is thermal conductivity, the Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity.
This ratio indicates how effectively a fluid transfers heat relative to momentum. Understanding the Prandtl number helps in predicting how heat might move through a fluid, aiding in designing and optimizing thermal systems.

Log Mean Temperature Difference

In heat exchanger calculations, the log mean temperature difference (LMTD) is a pivotal concept. It gives a more accurate average temperature difference between the hot and cold fluids across a heat exchanger.
The LMTD is defined mathematically as: \[ \Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln \left( \frac{\Delta T_1}{\Delta T_2} \right)} \]where \(\Delta T_1\) and \(\Delta T_2\) are the temperature differences at each end of the heat exchanger.
This concept is particularly crucial in systems maintaining constant wall temperatures, like the tube in the exercise, ensuring that heat exchanger design adequately accounts for varying temperature differences along the flow path.