Question

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.

Short answer

Question: Derive a differential equation for the temperature variation of a hot metal object with mass \(m\), specific heat \(c\), initial temperature \(T_{i}\), environmental temperature \(T_{\infty}\), and heat transfer coefficient \(h\) as it cools down in an environment. Answer: The temperature variation of the hot metal object with time can be described by the following differential equation: $$ \frac{dT(t)}{dt} = -\frac{hA}{mc}(T(t) - T_{\infty}) $$

Step-by-step solution

Define the given parameters and properties

We are given the mass of the hot metal object \(m\), its specific heat \(c\), the initial temperature \(T_{i}\), the environmental temperature \(T_{\infty}\), and the heat transfer coefficient \(h\). We want to find an expression for the temperature \(T(t)\) of the object at any time \(t\).

Set up the energy balance equation

The amount of heat energy gained by an environment is equal to the amount of heat energy lost by the hot metal object. The rate of heat transfer is proportional to the temperature difference between the object and the environment and it depends on the heat transfer coefficient \(h\). Therefore, the rate of heat transfer can be written as: $$ -\frac{dQ}{dt} = hA(T(t) - T_{\infty}) $$ where \(A\) is the surface area of the hot metal object and \(-\frac{dQ}{dt}\) denotes the rate of heat transfer (with negative sign indicating loss of heat).

Find the heat capacity of the object and its connection to temperature

The heat capacity of the hot metal object is defined as the product of its mass and specific heat: $$ C = mc $$ The amount of heat energy stored in the hot metal object at any given time t can be related to its temperature using the heat capacity: $$ Q(t) = C(T(t) - T_{\infty}) $$

Differentiate the heat energy stored in the object with respect to time

To establish a connection between the heat flow rate and temperature changes, we need to differentiate the above equation with respect to time: $$ \frac{dQ}{dt} = C\frac{dT(t)}{dt} $$

Combine both expressions involving the rate of heat transfer and derived differential equation

Now, we equate the above differential equation expression with the expression for the rate of heat transfer: $$ -C\frac{dT(t)}{dt} = hA(T(t) - T_{\infty}) $$

Simplify the differential equation

Finally, we can write the differential equation describing the temperature variation of the object with time as: $$ \frac{dT(t)}{dt} = -\frac{hA}{mc}(T(t) - T_{\infty}) $$ Thus, we have derived the differential equation for the temperature of the metal object as a function of time.