An electric motor operating at steady state draws a current...

An electric motor operating at steady state draws a current of 10 amp with a voltage of $220 \mathrm{~V}$. The output shaft rotates at 1000 RPM with a torque of $16 \mathrm{~N} \cdot \mathrm{m}$ applied to an external load. The rate of heat transfer from the motor to its surroundings is related to the surface temperature $T_{\mathrm{b}}$ and the ambient temperature $T_{0}$ by $\mathrm{hA}\left(T_{\mathrm{b}}-T_{0}\right)$, where $\mathrm{h}=100 \mathrm{~W} / \mathrm{m}^{2}$. $\mathrm{K}, \mathrm{A}=0.195 \mathrm{~m}^{2}$, and $T_{0}=293 \mathrm{~K}$. Energy transfers are considered positive in the directions indicated by the arrows on Fig. P6.51. (a) Determine the temperature $T_{\mathrm{b}}$, in $\mathrm{K}$. (b) For the motor as the system, determine the rate of entropy production, in $\mathrm{kW} / \mathrm{K}$. (c) If the system boundary is located to take in enough of the nearby surroundings for heat transfer to take place at temperature $T_{0}$, determine the rate of entropy production, in $\mathrm{kW} / \mathrm{K}$, for the enlarged system.

An electric motor operating at steady state draws a current of 10 amp with a voltage of $220 \mathrm{~V}$. The output shaft rotates at 1000 RPM with a torque of $16 \mathrm{~N} \cdot \mathrm{m}$ applied to an external load. The rate of heat transfer from the motor to its surroundings is related to the surface temperature $T_{\mathrm{b}}$ and the ambient temperature $T_{0}$ by $\mathrm{hA}\left(T_{\mathrm{b}}-T_{0}\right)$, where $\mathrm{h}=100 \mathrm{~W} / \mathrm{m}^{2}$. $\mathrm{K}, \mathrm{A}=0.195 \mathrm{~m}^{2}$, and $T_{0}=293 \mathrm{~K}$. Energy transfers are considered positive in the directions indicated by the arrows on Fig. P6.51.
(a) Determine the temperature $T_{\mathrm{b}}$, in $\mathrm{K}$.
(b) For the motor as the system, determine the rate of entropy production, in $\mathrm{kW} / \mathrm{K}$.
(c) If the system boundary is located to take in enough of the nearby surroundings for heat transfer to take place at temperature $T_{0}$, determine the rate of entropy production, in $\mathrm{kW} / \mathrm{K}$, for the enlarged system.

Key Concepts

Energy Balance in Steady State Systems

In a steady state system, the energy input into the system equals the energy output. For a motor, this means that the electrical power supplied is converted into mechanical work and heat losses, with no accumulation of energy in the system. This concept is central to by analyzing energy flows and balances in all steady state thermodynamic systems.

Heat Transfer by Convection

Heat transfer through convection is modeled by the product of a heat transfer coefficient, the area through which the heat is being transferred, and the temperature difference between the surface and the surrounding fluid. This concept quantifies the rate of energy loss due to thermal gradients, and is widely used to calculate heat dissipation in various engineering applications.

Entropy Production in Irreversible Processes

Entropy production is a measure of the irreversibility in a process. In any real system, processes such as mechanical friction, electrical resistance, or thermal gradients lead to the generation of entropy. This concept is critical in thermodynamics as it helps in assessing the efficiency and thermodynamic performance of systems by accounting for losses associated with irreversible processes.

System Boundary and Control Volume

Defining a system boundary is essential for performing energy and entropy analyses because it determines which energy transfers are treated as inputs or outputs. Changing the system boundary, for example by enlarging the control volume to include parts of the surroundings, can affect the calculated entropy production. This concept emphasizes the importance of careful system definition in thermodynamic analyses.

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