Posted on Categories:Heat Transfer, 传热学, 物理代写

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## 物理代写|传热学代写Heat Transfer代考|3-D Heat Conduction Problem

Sometimes we need to solve 3-D heat conduction problems in Cartesian (rectangular) coordinates as shown in Figure 3.6. Basically, we first convert the 3-D into the 2-D heat conduction problem and then solve the 2-D problem by using the separation of variables method discussed previously. The following is a brief outline on how to solve this type of problem.
The steady-state 3-D heat conduction equation without heat generation is

$$\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}=0$$
Let $\theta=T-T_1$.
The above governing equation and the associated $\mathrm{BCs}$ become
$$\frac{\partial^2 \theta}{\partial x^2}+\frac{\partial^2 \theta}{\partial y^2}+\frac{\partial^2 \theta}{\partial z^2}=0$$
$x=0, \theta=0$, or $\partial \theta / \partial x=0 ; x=a, \theta=0$, two homogeneous BCs,
$y=0, \theta=0$, or $\partial \theta / \partial y=0 ; y=b, \theta=0$, two homogeneous $\mathrm{BCs}$,
$z=0, \theta=\theta_0 ; z=c, \theta=0$, one nonhomogeneous $\mathrm{BC}$.

## 物理代写|传热学代写Heat Transfer代考|Nonhomogeneous Heat Conduction Problem

The problem of the steady-state 2-D heat conduction with uniform heat generation can be divided into two problems shown below, $\theta(x, y)=\psi(x, y)+\phi(x)$. We already know how to solve these two problems.
$$\begin{array}{r} \frac{\partial^2 \theta}{\partial x^2}+\frac{\partial^2 \theta}{\partial y^2}+\frac{q}{k}=0 \ \frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}=0 \end{array}$$
where
$$\psi=X(x) Y(y)=\left(c_1 \cos \lambda x+c_2 \sin \lambda x\right) \cdot\left(c_3 e^{-\lambda y}+c_4 e^{\lambda y}\right)$$
Equation (3.20) can be solved using the procedure shown in Section $2.2$ of Chapter $2 .$

## 物理代写|传热学代写Heat Transfer代考|3-D Heat Conduction Problem

$$\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}=0$$
$$\text { 让 } \theta=T-T_1 \text {. }$$

$$\frac{\partial^2 \theta}{\partial x^2}+\frac{\partial^2 \theta}{\partial y^2}+\frac{\partial^2 \theta}{\partial z^2}=0$$
$x=0, \theta=0 ，$ 或者 $\partial \theta / \partial x=0 ; x=a, \theta=0$, 两个同质 $\mathrm{BC}$ ，
$y=0, \theta=0$ ，或者 $\partial \theta / \partial y=0 ; y=b, \theta=0$ ，两个齐次 $\mathrm{BCs}$ ， $z=0, \theta=\theta_0 ; z=c, \theta=0$, 个非齐次 $\mathrm{BC}$.

## 物理代写|传热学代写Heat Transfer代考|Nonhomogeneous Heat Conduction Problem

$$\frac{\partial^2 \theta}{\partial x^2}+\frac{\partial^2 \theta}{\partial y^2}+\frac{q}{k}=0 \frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}=0$$

$$\psi=X(x) Y(y)=\left(c_1 \cos \lambda x+c_2 \sin \lambda x\right) \cdot\left(c_3 e^{-\lambda y}+c_4 e^{\lambda y}\right)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 物理代写|传热学代写Heat Transfer代考|CONDUCTION THROUGH FINS WITH UNIFORM CROSS-SECTIONAL AREA

From Newton’s Law of Cooling, the heat transfer rate can be increased by either increasing the temperature difference between the surface and fluid, the heat transfer coefficient, or the surface area. For a given problem, the temperature difference between the surface and fluid may be fixed, and increasing the heat transfer coefficient may result in more pumping power. One popular way to increase the heat transfer rate is to increase the surface area by adding fins to the heated surface. This is particularly true when the heat transfer coefficient is relatively low such as the air-side of heat exchangers (e.g., the car radiators) and air-cooled electronic components. The heat transfer rate can increase dramatically by increasing the surface area many times with the addition of numerous fins. Therefore, heat is conducted from the base surface into the fins and dissipated into the cooling fluid. However, as heat is conducted through the fins, the surface temperature decreases due to a finite thermal conductivity of the fins and the convective heat loss to the cooling fluid. This means the fin temperature is not the same as the base surface temperature and the temperature difference between the fin surface and the cooling fluid reduces along the length of the fins. It is our job to determine the fin temperature in order to calculate the heat loss from the fins to the cooling fluid.
In general, the heat transfer rate will increase with the number of fins. However, there is a limitation on the number of fins. The heat transfer coefficient will reduce if the fins are packed too close. In addition, the heat transfer rate will increase with thin fins that have a high thermal conductivity. Again, there is limitation on the thickness of thin fins due to manufacturing concerns. At this point, we are not interested in optimizing the fin dimensions but in determining the local fin temperature for a given geometry and working conditions. We assume that heat conduction through the fin is 1-D steady state because the fin is thin. The temperature gradient in the other two dimensions is neglected. We will begin with the constant cross-sectional area fins and then consider variable cross-sectional area fins. The following is the energy balance of a small control volume of the fin with heat conduction through the fin and heat dissipation into cooling fluid, as shown in Figure 2.7. The resulting temperature distributions through fins of different materials can be seen from Figure $2.8$.

## 物理代写|传热学代写Heat Transfer代考|Fin Performance

Most often, before adding the fins, we would like to know whether it is worthwhile to add fins to the smooth, heated surface. In this case, we define the fin effectiveness. The fin effectiveness is defined as the ratio of the heat transfer rate through the fin surface to that without the fin (i.e., convection from the fin base area).

$$\eta_{\varepsilon}=\frac{q_{\text {with fin }}}{q_{\text {without fin }}}$$
The fin effectiveness must be greater than unity in order to justify using the fins. Normally, it should be greater than 2 in order to include the material and manufacturing costs. In general, the fin effectiveness is greater than 5 for most of the effective fin applications. For example, for the long fins (case 4 fin tip boundary conditions), the fin effectiveness is
$$\eta_{\varepsilon}=\frac{q_{\text {with fin }}}{q_{\text {without fin }}}=\frac{\theta_b \sqrt{h P k A_c}}{h \theta_b A_c}=\sqrt{\frac{k P}{h A_c}}>1-5$$

## 物理代写|传热学代写Heat Transfer代考|Fin Performance

$$\eta_{\varepsilon}=\frac{q_{\text {with fin }}}{q_{\text {without fin }}}$$

$$\eta_{\varepsilon}=\frac{q_{\text {with fin }}}{q_{\text {without fin }}}=\frac{\theta_b \sqrt{h P k A_c}}{h \theta_b A_c}=\sqrt{\frac{k P}{h A_c}}>1-5$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Heat Transfer, 传热学, 物理代写

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 物理代写|传热学代写Heat Transfer代考|INTRODUCTION: CONDUCTION, CONVECTION, AND RADIATION

Conduction is caused by the temperature gradient through a solid material. For example, Figure $1.1$ shows that heat is conducted through a wall of a building or a container from the high-temperature side to the low-temperature side. This is a onedimensional (1-D), steady-state, heat conduction problem if $T_1$ and $T_2$ are uniform. According to Fourier’s conduction law, the temperature profile is linear through the plane wall.

Fourier’s Conduction Law
$$q^{\prime \prime}=-k \frac{\mathrm{d} T}{\mathrm{~d} x}=k \frac{T_1-T_2}{L}$$
and
$$q^{\prime \prime} \equiv \frac{q}{A_{\mathrm{c}}} \text { or } q=q^{\prime \prime} A_{\mathrm{c}}$$
where $q^{\prime \prime}$ is the heat flux $\left(\mathrm{W} / \mathrm{m}^2\right), q$ is the heat rate $(\mathrm{W}$ or $\mathrm{J} / \mathrm{s}), k$ is the thermal conductivity of solid material $(\mathrm{W} / \mathrm{m} \mathrm{K}), A_{\mathrm{c}}$ is the cross-sectional area for conduction, perpendicular to heat flow $\left(\mathrm{m}^2\right)$, and $L$ is the conduction length $(\mathrm{m})$.

One can predict heat rate or heat loss through the plane wall by knowing $T_1, T_2, k$, $L$, and $A_c$. This is the simple 1-D steady-state problem. However, in actual applications, there are many two-dimensional (2-D) or three-dimensional (3-D) steady-state heat conduction problems; there are cases where heat generation occurs in the solid material during heat conduction. Also, transient heat conduction problems take place in many engineering applications. In addition, some special applications involve heat conduction with a moving boundary. These more complicated heat conduction problems will be discussed in the following chapters.

## 物理代写|传热学代写Heat Transfer代考|Convection

Convection is caused by fluid flow motion over a solid surface. For example, Figure $1.2$ shows that heat is removed from a heated solid surface to cooling fluid. This is a 2-D boundary-layer flow and heat transfer problem. According to Newton, the heat removal rate from the heated surface is proportional to the temperature difference between the heated wall and the cooling fluid. The proportionality constant is known as the heat transfer coefficient; the same heat rate from the heated surface can be determined by applying Fourier’s Conduction Law to the cooling fluid.
1.1.2.1 Newton’s Cooling Law
$$q^{\prime \prime}=-\left.k_f \frac{d T}{d y}\right|{\text {at wall }}=h\left(T_s-T{\infty}\right)$$
Also,
$$h=\frac{q^{\prime \prime}}{T_s-T_{\infty}}=\frac{-\left.k_f \frac{d T}{d y}\right|{y=0}}{T_s-T{\infty}}$$
and
$$q^{\prime \prime}=\frac{q}{A_s} \quad \text { or } \quad q=q^{\prime \prime} A_s$$

where $T_s$ is the surface temperature $\left({ }^{\circ} \mathrm{C}\right.$ or $\left.\mathrm{K}\right), T_{\infty}$ is the fluid temperature $\left({ }^{\circ} \mathrm{C}\right.$ or $\left.\mathrm{K}\right)$, $h$ is the heat transfer coefficient $\left(\mathrm{W} / \mathrm{m}^2 \mathrm{~K}\right), k_f$ is the thermal conductivity of fluid $(\mathrm{W} / \mathrm{mK}), A_s$ is the surface area for convection, exposed to the fluid $\left(\mathrm{m}^2\right)$.

It is noted that the heat transfer coefficient depends on fluid properties (such as air or water as the coolant), flow conditions (i.e., laminar or turbulent flows), surface configurations (such as flat surface or circular tube), and so on. The heat transfer coefficient can be determined experimentally or analytically. This textbook focuses on analytical solutions. From Equation (1.3), the heat transfer coefficient can be determined by knowing the temperature profile in the cooling fluid during convection. With this analytical profile, the temperature gradient near the wall, $d T / d y$, can be used to determine the heat transfer coefficient. However, this requires solving the 2-D boundary-layer equations and will be the subject of the following chapters. Before solving 2-D boundary-layer equations, one needs the heat transfer coefficient as the convection boundary condition $(\mathrm{BC})$ in order to solve the heat conduction problem. Therefore, Table $1.1$ provides some typical values of heat transfer coefficient in many convection problems. As can be seen, in general, forced convection provides more heat transfer than natural convection; water as a coolant removes much more heat than air; and boiling or condensation, involving a phase change, has a much higher heat transfer coefficient than single-phase convection.

## 物理代写|传热学代写热传导代考|简介:传导，对流，和辐射

$$q^{\prime \prime}=-k \frac{\mathrm{d} T}{\mathrm{~d} x}=k \frac{T_1-T_2}{L}$$

$$q^{\prime \prime} \equiv \frac{q}{A_{\mathrm{c}}} \text { or } q=q^{\prime \prime} A_{\mathrm{c}}$$
where $q^{\prime \prime}$ 是热流密度 $\left(\mathrm{W} / \mathrm{m}^2\right), q$ 是热率 $(\mathrm{W}$ 或 $\mathrm{J} / \mathrm{s}), k$ 固体材料的导热系数是多少 $(\mathrm{W} / \mathrm{m} \mathrm{K}), A_{\mathrm{c}}$ 传导的截面积是否垂直于热流 $\left(\mathrm{m}^2\right)$，以及 $L$ 为导通长度 $(\mathrm{m})$.

## 物理代写|传热学代写Heat Transfer代考|对流

1.1.2.1牛顿冷却定律
$$q^{\prime \prime}=-\left.k_f \frac{d T}{d y}\right|{\text {at wall }}=h\left(T_s-T{\infty}\right)$$

$$h=\frac{q^{\prime \prime}}{T_s-T_{\infty}}=\frac{-\left.k_f \frac{d T}{d y}\right|{y=0}}{T_s-T{\infty}}$$

$$q^{\prime \prime}=\frac{q}{A_s} \quad \text { or } \quad q=q^{\prime \prime} A_s$$

where $T_s$ 是表面温度 $\left({ }^{\circ} \mathrm{C}\right.$ 或 $\left.\mathrm{K}\right), T_{\infty}$ 是流体温度 $\left({ }^{\circ} \mathrm{C}\right.$ 或 $\left.\mathrm{K}\right)$， $h$ 换热系数是多少 $\left(\mathrm{W} / \mathrm{m}^2 \mathrm{~K}\right), k_f$ 流体的导热系数是多少 $(\mathrm{W} / \mathrm{mK}), A_s$ 对流的表面积是否暴露在流体中 $\left(\mathrm{m}^2\right)$.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。