### 数学代写|计算方法代写computational method代考|Heat conduction

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## 数学代写|计算方法代写computational method代考|Heat conduction

Steady state potential flow problems are among the physical phenomena that can be modeled as scalar elliptic boundary value problems. In this section the formulation of a mathematical problem that models heat flow by conduction in solid bodies is described.

Mathematical models of heat conduction are based on two fundamental relationships: the conservation law and Fourier’s law of heat conduction described in the following.

1. The conservation law states that the quantity of heat entering any volume element of the conducting medium equals the quantity of heat exiting the volume element plus the quantity of heat retained in the volume element. The heat retained causes a change in temperature in the volume element which is proportional to the specific heat of the conducting medium $c$ (in $\mathrm{J} /\left(\mathrm{kg} \mathrm{K}\right.$ ) units) multiplied by the density $\rho$ (in $\mathrm{kg} / \mathrm{m}^{3}$ units). The temperature will be denoted by $u(x, y, z, t)$ where $t$ is time.
The heat flow rate across a unit area is represented by a vector quantity called heat flux. The heat flux is in $\mathrm{W} / \mathrm{m}^{2}$ units, or equivalent, and will be denoted by $\mathbf{q}=\mathbf{q}(x, y, z, t)=$ $\left{q_{x}(x, y, z, t) q_{y}(x, y, z, t) q_{z}(x, y, z, t)\right}^{T}$. In addition to heat flux entering and leaving the volume element, heat may be generated within the volume element, for example from chemical reactions. The heat generated per unit volume and unit time will be denoted by $Q$ (in $\mathrm{W} / \mathrm{m}^{3}$ units).
Applying the conservation law to the volume element shown in Fig. 2.1, we have:
\begin{aligned} &\Delta t\left[q_{x} \Delta y \Delta z-\left(q_{x}+\Delta q_{x}\right) \Delta y \Delta z+q_{y} \Delta x \Delta z-\left(q_{y}+\Delta q_{y}\right) \Delta x \Delta z+\right. \ &\left.q_{z} \Delta x \Delta y-\left(q_{z}+\Delta q_{z}\right) \Delta x \Delta y+Q \Delta x \Delta y \Delta z\right]=c \rho \Delta u \Delta x \Delta y \Delta z . \end{aligned}
Assuming that $u$ and $\mathbf{q}$ are continuous and differentiable and neglecting terms that go to zero faster than $\Delta x, \Delta y, \Delta z, \Delta t$, we have:
$$\Delta q_{x}=\frac{\partial q_{x}}{\partial x} \Delta x, \quad \Delta q_{y}=\frac{\partial q_{y}}{\partial y} \Delta y, \quad \Delta q_{z}=\frac{\partial q_{z}}{\partial z} \Delta z, \quad \Delta u=\frac{\partial u}{\partial t} \Delta t$$
On factoring $\Delta x \Delta y \Delta z \Delta t$ the conservation law is obtained:
$$-\frac{\partial q_{x}}{\partial x}-\frac{\partial q_{y}}{\partial y}-\frac{\partial q_{z}}{\partial z}+Q=c \rho \frac{\partial u}{\partial t}$$
In index notation:
$$-q_{i, i}+Q=c \rho \frac{\partial u}{\partial t} .$$

## 数学代写|计算方法代写computational method代考|The differential equation

Combining equations (2.16) through (2.20), we have:
\begin{aligned} &\frac{\partial}{\partial x}\left(k_{x} \frac{\partial u}{\partial x}+k_{x y} \frac{\partial u}{\partial y}+k_{x z} \frac{\partial u}{\partial z}\right)+\frac{\partial}{\partial y}\left(k_{y x} \frac{\partial u}{\partial x}+k_{y} \frac{\partial u}{\partial y}+k_{y z} \frac{\partial u}{\partial z}\right)+ \ &\frac{\partial}{\partial z}\left(k_{z x} \frac{\partial u}{\partial x}+k_{z y} \frac{\partial u}{\partial y}+k_{z} \frac{\partial u}{\partial z}\right)+Q=c \rho \frac{\partial u}{\partial t} \end{aligned}
which can be written in the following compact form:
$$\operatorname{div}([K] \operatorname{grad} u)+Q=c \rho \frac{\partial u}{\partial t}$$
or in index notation:
$$\left(k_{i j} u_{j}\right)_{i}+Q=c \rho \frac{\partial u}{\partial t} .$$
In many practical problems $u$ is independent of time. Such problems are called stationary or steady state problems. The solution of a stationary problem can be viewed as the solution of some time-dependent problem, with time-independent boundary conditions, at $t=\infty$.

In formulating eq. $(2.23)$ we assumed that $k_{i j}$ are differentiable functions. In many practical problems the solution domain is comprised of subdomains $\Omega_{i}$ that have different material properties. In such cases eq. (2.23) is valid on each subdomain. On the boundaries of adjoining subdomains continuous temperature and flux are prescribed.

To complete the definition of a mathematical model, initial and boundary conditions have to be specified. This is discussed in the following section.

## 数学代写|计算方法代写computational method代考|Boundary and initial conditions

The solution domain will be denoted by $\Omega$ and its boundary by $\partial \Omega$. We will consider three kinds of boundary conditions:

1. Prescribed temperature (Dirichlet condition): The temperature $u=\hat{u}$ is prescribed on boundary region $\partial \Omega_{u}$.
2. Prescribed flux (Neumann condition): The flux vector component normal to the boundary, denoted by $q_{n}$, is prescribed on the boundary region $\partial \Omega_{q}$. By definition;
$$q_{n} \stackrel{\text { def }}{=} \mathbf{q} \cdot \mathbf{n} \equiv-([K] \operatorname{grad} u) \cdot \mathbf{n} \equiv-k_{i j} u_{j} n_{i}$$
where $\mathbf{n} \equiv n_{i}$ is the (outward) unit normal to the boundary. The prescribed flux on $\partial \Omega_{q}$ will be denoted by $\hat{q}_{n}$.
3. Convection (Robin condition): On boundary region $\partial \Omega_{c}$ the flux vector component $q_{n}$ is proportional to the difference between the temperature of the boundary and the temperature of a convective medium:
$$q_{n}=h_{c}\left(u-u_{c}\right), \quad(x, y, z) \in \partial \Omega_{c}$$
where $h_{c}$ is the coefficient of convective heat transfer in $\mathrm{W} /\left(\mathrm{m}^{2} \mathrm{~K}\right)$ units and $u_{\mathrm{c}}$ is the (known) temperature of the convective medium.

The sets $\partial \Omega_{u}, \partial \Omega_{q}$ and $\partial \Omega_{c}$ are non-overlapping and collectively cover the entire boundary. Any of the sets may be empty.

The boundary conditions may be time-dependent. For time-dependent problems an initial condition has to be prescribed on $\Omega: u(x, y, z, 0)=U(x, y, z)$.

It is possible to show that eq. (2.23), subject to the enumerated boundary conditions, has a unique solution. Stationary problems also have unique solutions, subject to the condition that when flux is prescribed over the entire boundary $\partial \Omega$ then the following condition must be satisfied:
$$\int_{\Omega} Q d V=\int_{\partial \Omega} q_{n} d S .$$
This is easily seen by integrating
$$\left(k_{i j} u_{j}\right){, i}+Q=0$$ on $\Omega$ and using the divergence theorem, eq. (2.2) and the definition (2.26). Note that if $u{i}$ is a solution of eq. (2.29) then $u_{i}+C$ is also a solution, where $C$ is an arbitrary constant. Therefore the solution is unique up to an arbitrary constant.

In addition to the three types of boundary conditions discussed in this section, radiation may have to be considered. When two bodies exchange heat by radiation then the flux is proportional to the difference of the fourth power of their absolute temperatures, therefore radiation is a non-linear boundary condition. The boundary region subject to radiation, denoted by $\partial \Omega_{r}$, may overlap $\partial \Omega_{c^{*}}$. Radiation is discussed in Section 9.1.1.

In the following it will be assumed that the coefficients of thermal conduction, the flux prescribed on $\Omega_{q}$ and the coefficient $h_{c}$ prescribed on $\Omega_{c}$ are independent of the temperature. This assumption can be justified on the basis of empirical data in a narrow range of temperatures only.

## 数学代写|计算方法代写computational method代考|Heat conduction

1. 守恒定律规定，进入传导介质的任何体积元的热量等于离开体积元的热量加上保持在体积元中的热量。保留的热量会导致体积元素中的温度变化，该变化与传导介质的比热成正比C（在Ĵ/(ķGķ) 单位) 乘以密度ρ（在ķG/米3单位）。温度将表示为在(X,是,和,吨)在哪里吨是时间。
单位面积上的热流率由称为热通量的矢量表示。热通量在在/米2单位，或等价物，并将由q=q(X,是,和,吨)= \left{q_{x}(x, y, z, t) q_{y}(x, y, z, t) q_{z}(x, y, z, t)\right}^{T}\left{q_{x}(x, y, z, t) q_{y}(x, y, z, t) q_{z}(x, y, z, t)\right}^{T}. 除了进入和离开体积元素的热通量之外，热量可能在体积元素内产生，例如来自化学反应。单位体积和单位时间产生的热量记为问（在在/米3单位）。
将守恒定律应用于图 2.1 所示的体积元，我们有：
Δ吨[qXΔ是Δ和−(qX+ΔqX)Δ是Δ和+q是ΔXΔ和−(q是+Δq是)ΔXΔ和+ q和ΔXΔ是−(q和+Δq和)ΔXΔ是+问ΔXΔ是Δ和]=CρΔ在ΔXΔ是Δ和.
假如说在和q是连续的、可微分的和忽略的项，它们比ΔX,Δ是,Δ和,Δ吨， 我们有：
ΔqX=∂qX∂XΔX,Δq是=∂q是∂是Δ是,Δq和=∂q和∂和Δ和,Δ在=∂在∂吨Δ吨
关于保理ΔXΔ是Δ和Δ吨得到守恒定律：
−∂qX∂X−∂q是∂是−∂q和∂和+问=Cρ∂在∂吨
在索引符号中：
−q一世,一世+问=Cρ∂在∂吨.

## 数学代写|计算方法代写computational method代考|The differential equation

∂∂X(ķX∂在∂X+ķX是∂在∂是+ķX和∂在∂和)+∂∂是(ķ是X∂在∂X+ķ是∂在∂是+ķ是和∂在∂和)+ ∂∂和(ķ和X∂在∂X+ķ和是∂在∂是+ķ和∂在∂和)+问=Cρ∂在∂吨

div⁡([ķ]毕业⁡在)+问=Cρ∂在∂吨

(ķ一世j在j)一世+问=Cρ∂在∂吨.

## 数学代写|计算方法代写computational method代考|Boundary and initial conditions

1. 规定温度（狄利克雷条件）：温度在=在^规定在边界区域∂Ω在.
2. 规定通量（诺依曼条件）：垂直于边界的通量矢量分量，表示为qn, 规定在边界区域∂Ωq. 根据定义；
qn= 定义 q⋅n≡−([ķ]毕业⁡在)⋅n≡−ķ一世j在jn一世
在哪里n≡n一世是垂直于边界的（向外）单位。规定的通量∂Ωq将表示为q^n.
3. 对流（Robin 条件）：在边界区域∂ΩC通量矢量分量qn与边界温度和对流介质温度之间的差成正比：
qn=HC(在−在C),(X,是,和)∈∂ΩC
在哪里HC是对流传热系数在/(米2 ķ)单位和在C是对流介质的（已知）温度。

∫Ω问d在=∫∂Ωqnd小号.

(ķ一世j在j),一世+问=0在Ω并使用散度定理，等式。（2.2）和定义（2.26）。请注意，如果在一世是 eq 的解。(2.29) 那么在一世+C也是一个解决方案，其中C是一个任意常数。因此，对于任意常数，解都是唯一的。

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