Question

Consider a hot semi-infinite solid at an initial temperature of \(T_{i}\) that is exposed to convection to a cooler medium at a constant temperature of \(T_{\infty}\), with a heat transfer coefficient of \(h\). Explain how you can determine the total amount of heat transfer from the solid up to a specified time \(t_{o}\).

Short answer

Question: Given a semi-infinite solid with an initial temperature of \(T_i\), and a cooler medium with constant temperature \(T_\infty\) surrounding the solid. The heat transfer coefficient between the solid and the medium is \(h\). Determine the total amount of heat transfer from the solid to the medium up to a specified time \(t_o\). Answer: To find the total amount of heat transfer from the solid up to the time \(t_o\), first compute the heat flux at the surface of the solid up to \(t_o\) using Fourier's law and the temperature profile solution. Then, integrate the heat flux with respect to time from \(0\) to \(t_o\) to find the total heat transfer per unit area: $$Q(t_o) = \int_0^{t_o} q_x(t) dt= \frac{h\sqrt{\alpha} (T_i - T_{\infty})}{\sqrt{\pi}} \int_0^{t_o} \sqrt{t} (1-e^{-h\sqrt{t/\alpha}}) dt$$ You can use numerical methods, such as Simpson's rule or Gaussian quadrature, to estimate the integral and obtain the total amount of heat transfer from the solid up to time \(t_o\).

Step-by-step solution

Understand the temperature profile

Set up a coordinate system with \(x=0\) at the surface of the solid, and the \(x\)-axis directed into the solid. We will consider the temperature profile of the solid as a function of position, \(x\), and time, \(t\): \(T(x,t)\).

Apply Fourier's law

According to Fourier's law, the heat flux (rate of heat transfer per unit area) in the \(x\)-direction is proportional to the temperature gradient: $$q_x = -k\frac{\partial T}{\partial x}$$ \(k\) is the thermal conductivity of the solid and is a material property.

Apply the energy balance equation

Use the energy balance equation for an infinitesimal control volume in the solid: $$\rho c_p\frac{\partial T}{\partial t} = k\frac{\partial^2 T}{\partial x^2}$$ Where \(\rho\) is the density of the solid and \(c_p\) is its specific heat. Rearrange the equation: $$\frac{\partial T}{\partial t} = \alpha\frac{\partial^2 T}{\partial x^2}$$ Where \(\alpha\) is the thermal diffusivity and is given by \(\alpha = \frac{k}{\rho c_p}\).

Boundary conditions

For this problem, we are given the following boundary conditions at \(x=0\): 1. Temperature initially: \(T(x,0) = T_i\) 2. Convection boundary condition: \(-k\frac{\partial T}{\partial x}|_{x=0} = h(T(0,t) - T_\infty)\)

Solve for temperature profile

To find the temperature profile of the semi-infinite solid, solve the equation from Step 3 subject to the boundary conditions from Step 4. This involves using advanced mathematical techniques (like separation of variables and Laplace transforms) which are beyond the scope of this explanation. However, the temperature profile of a semi-infinite solid under convection is a well-known solution in Heat Transfer textbooks and is given by: $$T(x,t) = T_{\infty} + (T_i - T_{\infty})\operatorname{erf}\left(\frac{x}{2\sqrt{\alpha t}}\right)$$ Where the error function, erf, is defined as: $$\operatorname{erf}(z) = \frac{2}{\sqrt{\pi}}\int_0^z e^{-z^2}dz$$

Determine heat transfer up to time \(t_o\)

To find the total amount of heat transfer from the solid up to the time \(t_o\), first find the heat flux at the surface of the solid up to \(t_o\). Use the temperature profile solution from Step 5 to compute the derivative at \(x=0\), and plug the result into Fourier's law: $$q_x(t) = -k\frac{\partial T}{\partial x}|_{x=0} = \frac{h\sqrt{\alpha t}}{\sqrt{\pi}}(T_i - T_{\infty})(1-e^{-hx\sqrt{t/\alpha}})$$ Now integrate \(q_x(t)\) with respect to time from \(0\) to \(t_o\) to find the total heat transfer per unit area: $$Q(t_o) = \int_0^{t_o} q_x(t) dt= \frac{h\sqrt{\alpha} (T_i - T_{\infty})}{\sqrt{\pi}} \int_0^{t_o} \sqrt{t} (1-e^{-h\sqrt{t/\alpha}}) dt$$ Solving this integral analytically might not be possible. In that case, you can use numerical methods, such as the Simpson's rule or Gaussian quadrature, to estimate the integral and obtain the total amount of heat transfer from the solid up to time \(t_o\).

Key concepts

Fourier's Law

When exploring heat transfer, Fourier's law is fundamental, as it describes the relationship between the heat flux and the temperature gradient within a material. Specifically, the law states that the rate of heat transfer per unit area, commonly represented by the symbol \(q_x\), is directly proportional to the negative gradient of the temperature. In mathematical terms, this is expressed as: \[ q_x = -k\frac{\partial T}{\partial x} \] where \(k\) is the thermal conductivity of the material—a measure of its ability to conduct heat.

Why the 'negative' sign, you might ask? It signifies the direction of heat flow—from hotter to cooler regions. Hence, as per Fourier's law, heat moves down the temperature gradient, and the thermal conductivity \(k\) quantifies how efficiently this occurs. For students trying to visualize, imagine a hot semi-infinite solid: heat will flow from its initially high-temperature side to the cooler ambient temperature, demonstrating Fourier's law in action.

Convection

Convection is one of the primary modes of heat transfer and plays a vital role when the heat exchange occurs between a solid boundary and a fluid in motion. This fluid motion could either be caused naturally by density differences, referred to as 'natural convection,' or be induced by external means such as a fan or pump, known as 'forced convection.'

The concept in this scenario is encapsulated by the heat transfer coefficient, denoted by \(h\). The relationship that governs convective heat transfer from the solid's surface is articulated as: \[ -k\frac{\partial T}{\partial x}|_{x=0} = h(T(0,t) - T_{\infty}) \] Here \(T(0,t)\) represents the temperature at the solid's surface, while \(T_{\infty}\) is the temperature of the surrounding fluid. The heat transfer coefficient \(h\) encapsulates the convection phenomenon, summarizing the complex interplay between fluid properties, flow characteristics, and temperature differences.

Thermal Diffusivity

Thermal diffusivity is a material property that combines the thermal conductivity, density, and specific heat into a single quantity representing the ability to conduct thermal energy relative to the ability to store thermal energy. It's denoted by \(\alpha\) and described by the equation: \[ \alpha = \frac{k}{\rho c_p} \] where \(\rho\) is the density, and \(c_p\) is the specific heat capacity at constant pressure.

Thermal diffusivity plays a critical role since it appears in the heat conduction equation, which in turn helps solve for the temperature distribution within solids during heat transfer processes. A high diffusivity means that the material can quickly adjust its temperature to changes in its surroundings, whereas a low diffusivity indicates a sluggish temperature response. In our exercise, understanding thermal diffusivity is essential for predicting how heat travels through the solid over time.

Boundary Conditions

Boundary conditions are the constraints that define how a physical problem behaves at the limits of the domain being studied. They are crucial for the mathematical modeling of physical systems. In the context of heat transfer, we often encounter two types of boundary conditions: a specified temperature (Dirichlet condition) or a specified heat flux (Neumann condition).

In our particular heat transfer problem, we are dealing with the latter—where the heat flux at the boundary is defined by the convective exchange of the solid with the environment. This is mathematically expressed at the surface \(x=0\) as: \[ -k\frac{\partial T}{\partial x}|_{x=0} = h(T(0,t) - T_{\infty}) \] This equation accounts for the semi-infinite nature of the solid and the convection from its surface. It's one of the key pieces of information necessary to solve for the temperature distribution within the solid.

Error Function

The error function, often denoted as \(\operatorname{erf}\), is a special non-elementary function that arises frequently in probability, statistics, and partial differential equations related to heat transfer and diffusion processes. It's defined by the integral: \[ \operatorname{erf}(z) = \frac{2}{\sqrt{\pi}}\int_0^z e^{-z^2}dz \]

In the realm of heat transfer, the error function appears when solving for the temperature distribution in problems involving semi-infinite domains, such as the one we're examining. The temperature \(T(x,t)\) within the semi-infinite solid can be expressed using the error function as: \[ T(x,t) = T_{\infty} + (T_i - T_{\infty})\operatorname{erf}\left(\frac{x}{2\sqrt{\alpha t}}\right) \] This relationship encapsulates the spatial and temporal evolution of the temperature within the solid, making it possible to determine the heat transfer away from its surface over time, as required for our exercise.