The intake manifold of an automobile engine can be...

The intake manifold of an automobile engine can be approximated as a $4-\mathrm{cm}-\mathrm{ID}$ tube, $30 \mathrm{~cm}$ in length. Air at a bulk temperature of $20^{\circ} \mathrm{C}$ enters the manifold at a flow rate of $0.01 \mathrm{~kg} / \mathrm{s}$. The manifold is a heavy aluminum casting and is at a uniform temperature of $40^{\circ} \mathrm{C}$. Determine the temperature of the air at the end of the manifold.

Key Concepts

Thermal Characteristics of Conduction and Convection in Ducts

This concept encompasses the analysis of heat transfer in conduits where conduction through the wall and convection between the wall and the fluid interact. In systems where the wall temperature is maintained uniformly (as in a heavy or high thermal mass casting), the temperature profile of the fluid along the duct can be determined by balancing the heat gained from the wall through convection against the thermal capacity of the moving fluid.

Dimensionless Analysis in Heat Transfer

Dimensionless numbers such as Reynolds, Prandtl, and Nusselt are crucial for characterizing the convective heat transfer process in internal flows. The Reynolds number indicates the flow regime (laminar or turbulent), the Prandtl number relates the momentum diffusivity to thermal diffusivity, and the Nusselt number correlates the convective to conductive heat transfer, all of which are used to estimate the heat transfer coefficient.

Forced Convection

Forced convection refers to heat transfer in a fluid system where an external force, such as a pump or fan, drives the fluid flow, enhancing the mixing and heat exchange between the fluid and surrounding surfaces. It is characterized by convective heat transfer coefficients, which are often estimated using correlations involving dimensionless numbers like the Reynolds, Prandtl, and Nusselt numbers.

Energy Balance in Fluid Systems

This concept involves applying the first law of thermodynamics to a flowing fluid, where the change in the fluid’s energy (in this case, its temperature) is determined by the net heat gain or loss along its path. In heat exchange problems, an energy balance is used to relate the heat transfer from the surroundings (or walls) to the fluid’s specific heat capacity and mass flow rate, which ultimately allows for the calculation of the fluid’s outlet temperature.

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