Question

Obtain a relation for the time required for a lumped system to reach the average temperature \(\frac{1}{2}\left(T_{i}+T_{\infty}\right)\), where \(T_{i}\) is the initial temperature and \(T_{\infty}\) is the temperature of the environment.

Short answer

Answer: The time required for a lumped system to reach the average temperature can be calculated using the derived relation: $$t = \frac{1}{-hA} \ln \left(\frac{\frac{1}{2}(T_i + T_\infty) - T_\infty}{T_i - T_\infty}\right)$$, where \(h\) is the heat transfer coefficient, \(A\) is the surface area of the object, \(T_i\) is the initial temperature of the object, and \(T_\infty\) is the surrounding temperature.

Step-by-step solution

Write down Newton's Law of Cooling

Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between the object's temperature \(T(t)\) and the surrounding temperature \(T_\infty\). Mathematically, it can be written as: $$\frac{dT}{dt} = -hA(T - T_\infty)$$ where \(h\) is the heat transfer coefficient, \(A\) is the surface area of the object, and \(T(t)\) is the temperature of the object at time \(t\).

Separate variables

We need to separate the variables of the equation to solve it. We do this by dividing both sides by \((T - T_\infty)\) and multiplying both sides by \(dt\): $$\frac{dT}{T - T_\infty}= -hA dt$$

Integrate both sides

Integrate both sides of the equation with respect to their respective variables. The left side will be integrated with respect to \(T\), while the right side with respect to \(t\). Note that \(h\) and \(A\) are constants. $$\int_{T_i}^{T_f} \frac{dT}{T - T_\infty} = -hA \int_{0}^{t} dt$$

Evaluate the integrals

Evaluate both integrals. The left side will give us the natural logarithm due to its form, while the right side will give us a linear function. $$\ln \left(\frac{T_f - T_\infty}{T_i - T_\infty}\right) = -hAt$$

Solve for time \(t\)

We are interested in finding the time required for the average temperature to be reached, i.e., \(T_f = \frac{1}{2}(T_i + T_\infty)\). Plug this into our equation and solve for \(t\): $$\ln \left(\frac{\frac{1}{2}(T_i + T_\infty) - T_\infty}{T_i - T_\infty}\right) = -hAt$$ To isolate \(t\), take the exponential of both sides and then divide by \(-hA\): $$t = \frac{1}{-hA} \ln \left(\frac{\frac{1}{2}(T_i + T_\infty) - T_\infty}{T_i - T_\infty}\right)$$ This derived relation gives us the time required for a lumped system to reach the average temperature \(\frac{1}{2}(T_i + T_\infty)\).