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 of a small hot metal object as it cools in an environment using the given information: mass, specific heat, initial temperature, surrounding temperature, and heat transfer coefficient of the colorless gas. Answer: \(\frac{dT(t)}{dt} + \frac{h}{mc}(a)T(t) = \frac{h}{mc}(a)T_{\infty}\), where \(T(t)\) is the temperature at time \(t\), \(m\) is the mass, \(c\) is the specific heat, \(h\) is the heat transfer coefficient, \(a\) is a constant representing the surface area, and \(T_{\infty}\) is the surrounding temperature.

Step-by-step solution

Write the energy balance equation for the metal object

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: \(mc\frac{dT(t)}{dt} = -hA(t)(T(t)-T_{\infty})\)

Simplify the energy balance equation

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 \(mc\): \(\frac{dT(t)}{dt} = -\frac{h}{mc} (a)(T(t)-T_{\infty})\)

Derive the differential equation for temperature

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.

Key concepts

Differential Equation

The concept of a differential equation is central in understanding many physical phenomena, including heat transfer. A differential equation, in simple terms, is an equation that contains one or more derivatives of an unknown function. In the context of this exercise, the function is the temperature of the metal object, and the derivative represents how this temperature changes over time.

More specifically, the derivative, denoted as \(\frac{dT(t)}{dt}\), represents the rate of change of temperature \(T\) with respect to time \(t\). Here, the objective is to set up a relation that illustrates the balance between internal energy change and heat transfer. This balance is what transforms into a first-order linear differential equation. Understanding this setup is crucial because it forms the basis for many engineering problems that involve temperature change over time.

Convection

Convection plays a vital role in the transfer of heat between the hot metal object and its cooler surroundings. It refers to the mode of heat transfer that occurs through a fluid medium, which can be a gas or liquid, circulating over the surface of a solid.

In this exercise, convection is quantified by the heat transfer coefficient \(h\), which signifies the efficiency with which heat is transferred. The formula for the rate of heat transfer via convection is given by \(-hA(t)(T(t)-T_{\infty})\). Here, \(A(t)\) represents the surface area of the object, \(T(t)\) is the object's temperature at time \(t\), and \(T_{\infty}\) is the ambient temperature.

The negative sign is crucial as it indicates a loss of heat from the object. This loss results in the cooling of the object over time and is what we aim to model with our differential equation. By understanding convection, we grasp how efficiently the object can shed its heat to the environment.

Energy Balance

Energy balance is an essential principle used to derive the differential equation for the cooling of the metal object. It involves equating the rate of internal energy change to the rate of heat transfer due to convection. This setup follows the law of conservation of energy, ensuring that all energy entering or leaving a system is accounted for.

The rate of internal energy change is formalized as \(mc\frac{dT(t)}{dt}\), where \(m\) is the mass, \(c\) is the specific heat, and \(\frac{dT(t)}{dt}\) is the temperature's rate of change. Meanwhile, the rate of heat transfer due to convection is defined as \(-hA(t)(T(t)-T_{\infty})\).

By aligning these two expressions, we form the foundation of our differential equation. Simplifying by assuming constant parameters allows us to understand the dynamic nature of how the object's temperature responds to its environment. This balance is fundamental not only in this exercise but also in various applications of thermodynamics and fluid mechanics.

Temperature Variation

Understanding temperature variation is key to analyzing how the metal object cools over time. In this problem, the temperature of the object changes consistently, assuming uniform cooling. Thus, the variation of temperature is described by a linear relationship with respect to time, which simplifies the process of deriving our equation.

By employing the simplified form \(A(t) = a\), where \(a\) denotes a constant surface area unaffected by time, we can directly connect temperature variation to convection. This direct relation surfaces in the differential equation \(\frac{dT(t)}{dt} + \frac{h}{mc}(a)T(t) = \frac{h}{mc}(a)T_{\infty}\), which succinctly captures how the object's temperature drops as it sheds heat to the environment.

This observation tells us that temperature change is predictable and consistent if the conditions remain constant, thus aiding in the practical application of heat transfer models in real-world scenarios. Understanding temperature variation helps improve forecasts in engineering tasks involving controlled cooling processes.