Consider a small hot metal object of mass \(m\) and specific heat \(c\) that is initially at a temperature of \(T_{i}\). Now the object is allowed to cool in an environment at \(T_{\infty}\) by convection with a heat transfer coefficient of \(h\). The temperature of the metal object is observed to vary uniformly with time during cooling. Writing an energy balance on the entire metal object, derive the differential equation that describes the variation of temperature of the ball with time, \(T(t)\). Assume constant thermal conductivity and no heat generation in the object. Do not solve.

An energy balance equation can be described as the rate of change of energy storage (internal energy) equal to the rate of heat transfer (convection). For the metal object, the rate of change of internal energy is given by: Rate of internal energy change = \(mc\frac{dT(t)}{dt}\) Where \(m\) is the mass, \(c\) is the specific heat, and \(\frac{dT(t)}{dt}\) is the rate of change of temperature with respect to time. Next, the rate of heat transfer due to convection can be described as: Rate of heat transfer (convection) = \(-hA(t)(T(t)-T_{\infty})\) Where \(h\) is the heat transfer coefficient, \(A(t)\) is the surface area of the metal object, \(T(t)\) is the temperature at time \(t\), and \(T_{\infty}\) is the surrounding temperature. The negative sign indicates that heat is being lost from the metal object to the surrounding environment. Now, let's equate both sides of the energy balance equation: \(mc\frac{dT(t)}{dt} = -hA(t)(T(t)-T_{\infty})\)

As \(A(t)\) is a function of time, it will be easier to solve the problem if we eliminate \(A(t)\). Since it is given that the temperature of the metal object is observed to vary uniformly with time, the rate at which the surface area changes can be assumed constant over time. Therefore, \(A(t) = a\), where \(a\) is a constant. Substitute \(A(t)\) with the constant \(a\) in the energy balance equation: \(mc\frac{dT(t)}{dt} = -h (a)(T(t)-T_{\infty})\) Alternatively, divide both sides by \(mc\): \(\frac{dT(t)}{dt} = -\frac{h}{mc} (a)(T(t)-T_{\infty})\)

Now, the resulting energy balance equation can be written as the following first-order linear differential equation: \(\frac{dT(t)}{dt} + \frac{h}{mc}(a)T(t) = \frac{h}{mc}(a)T_{\infty}\) This is the desired differential equation that describes the variation of temperature with time (\(T(t)\)) for the given metal object. No need to solve the equation as the problem stated.