Fluid Flow, Heat Transfer, and Mass Transport

     化工   流体流动、传热和传质 

An Introduction to Fluid Flow, Heat Transfer, and Mass Transport

The subject of transport phenomena describes the transport of momentum, energy, and mass in the form of mathematical relations[1]. The basis for these descriptions is found in the laws for conservation of momentum, energy, and mass in combination with the constitutive relations that describe the fluxes of the conserved quantities[2]. The most accurate way to express these conservation laws and constitutive relations in continuum mechanics is to use differential equations[3].

Solving the equations that describe transport phenomena and interpreting the results is an efficient way to understand the systems being studied. This methodology is successfully used for studying fluid flow, heat transfer, and chemical species transport in many fields, including:

  • Engineering sciences
  • Biology
  • Chemistry
  • Environmental sciences
  • Geology
  • Material science
  • Medicine
  • Meteorology
  • Physics
 

动量、能量和质量传递的类比

可以从简单的原理中推导出不同传递量(动量、能量和质量)的守恒定律。假定一个矢量 j = (jx, jy, jz) 给出了某个系统的通量,那么就以其中的一个守恒量 Φ 为例。通过尺寸为 Δx、Δy 和 Δz 的每个小体积单元可以得到量 Φ 的平衡,其中的通量 j 是单位面积上、单位时间内的物理量。生成项或消耗项 Rs 是单位体积、单位时间内的物理量。

The geometry of an example for analyzing different transported quantities. The geometry of an example for analyzing different transported quantities.

根据上图,物理量达到平衡的方程如下:

其中,jx,x 表示 x-方向上 x 位置处的通量,jx,x+Δx 则是 x 方向上 x+Δx 位置处的通量矢量。下标 tt+Δt 指相应时间时的状态。这个方程表明,如果没有生成或消耗(Rs = 0),流入或流出体积单元的净通量必须被该物理量在一段时间内的累加或损耗所平衡掉。此外,如果在一段时间内没有累加或损耗,那么进入该体积单元的通量必须准确地平衡流出该体积单元的通量,从而使总进出该体积单元的总通量为零。

将上面的方程除以体积 ΔxΔyΔz,并使 Δx、Δy 和 Δz 都趋近于零,就得到了物理量 Φ 的以下方程:

让 Δt 趋近 0,可得:

为了遵循守恒定律,所研究系统中的某个连续体(如某一流体或固体)中的每一细小单元都必须满足这一平衡方程。在建模与传递现象的仿真过程中,这个偏微分方程——“部分”得益于其被表示为随着某一时刻上的一个自变量而发生变化的形式(x、y、zt 都是自变量)——能够描述出动量、能量和质量的守恒定律。

在动量守恒之中,守恒的物理量是一个矢量,而通量项则被表示为张量的形式,其中包含所谓的应力张量。结合动量守恒,通过动量通量的本构方程以及不可压缩牛顿流体的质量守恒可得纳维-斯托克斯方程。这些方程是流体流动建模(CFD)的基础,它们的解描述了流动流体的速度和压力场。如果守恒量为能量,则该系统中的传热方程可以从上述守恒方程中导出。

最后再来看一下质量传递。假设要研究的是某个流体的成分组成,且该流体存在着传递和反应。然后,我们就可以对该流体中的每种物质的质量守恒方程进行定义和求解。每种物质 i 的浓度 ci 为守恒量,其通量由 Ni 表示。通过上述守恒方程,每种物质都有以下方程:

如果同时假定通量由扩散给出,并且菲克定律定义了该通量的本构关系(如某一溶液中的一种溶质),那么我们还可以得到一个扩散-反应方程,该方程常被用于对流体流动可忽略的反应系统进行建模:

在这个方程中,Di 表示溶液中物质 i 的扩散系数。

A model of a diffusion-reaction process. A model of a diffusion-reaction process.

Diffusion in water surrounding a zebra fish embryo and diffusion and reaction of oxygen in the fish embryo's body (reproduced from [4]). Note the high concentration of oxygen in the yolk, where there is hardly any metabolism taking place, only energy storage. Also, insects "breathe" by diffusion. Diffusion-reaction processes are widely used to describe biological systems.

如果存在对流(这意味着整个溶液发生净传递),那么我们就可以得到一个传递方程,该方程常被用于存在流体流动的反应系统:

在这个方程中,u 表示速度矢量。如果有一个电场 E 作用在溶液和离子之上,那么我们就能得到电化学系统中使用的 Nernst-Planck 方程:

在这个方程中,zi 表示物质 i 的化合价,而 ui 表示物质 i 的迁移率。通过 Nernst-Planck 方程,迁移率与扩散系数就被直接关联起来了。通量矢量种的第三项被称为迁移项

在通量的本构关系中,也存在着类似于传递物理量的守恒。例如,从分子性质导出的传递属性使得动量传递中有:由牛顿流体定律给出的粘滞项、由傅里叶传热定律给出的传导项、由菲克扩散定律给出的扩散项。迁移项与电场的线性关系符合欧姆定律(该定律源于金属中电子的传递属性)。

在气体中,粘度、导热系数和扩散系数的传递属性源于碰撞、布朗运动和分子间相互作用。在液体中,这一理论的普适性有所降低,但所给到的某个流体中的分子动量、能量以及质量传递属性依然具有相关性。

总之,定义模型方程所用的原则简单明了。具体就是对守恒定律进行定义,以及找出引起通量的那些关系。对于给到的一个系统,在不同的条件下一遍又一遍地对这些方程进行求解,然后对结果进行研究,就可以对该系统中的传递现象有所了解。

 

Finally, let us look at mass transport. Assume that we want to study the composition of a fluid where transport and reactions are present. We can then define and solve the conservation equations for the mass of each species in the fluid. The concentration ci of each species i is the conserved quantity and its flux is denoted by Ni.. Using the conservation equation above gives us the following equation for each species:

If we also assume that the flux is given by diffusion and that Fick’s law defines the constitutive relation for the flux (for a solute in a solvent, for example), we obtain the diffusion-reaction equation often used to model reacting systems with negligible fluid flow:

In this equation, Di denotes the diffusion coefficient of species i in the solution.

A model of a diffusion-reaction process. A model of a diffusion-reaction process.

Diffusion in water surrounding a zebra fish embryo and diffusion and reaction of oxygen in the fish embryo's body (reproduced from [4]). Note the high concentration of oxygen in the yolk, where there is hardly any metabolism taking place, only energy storage. Also, insects "breathe" by diffusion. Diffusion-reaction processes are widely used to describe biological systems.

If there is advection, which means that there is a net transport of the whole solution, then we get the transport equation often used in reacting systems where fluid flow is present:

In this equation, u denotes the velocity vector. If there is an electric field E applied on the solution and ions are present, then we obtain the Nernst-Planck equations used in electrochemical systems:

In this equation, zi denotes the valence of species i and ui denotes the mobility of speciesi. The mobility is directly related to diffusivity through the Nernst-Einstein relation. The third term in the flux vector is called the migration term.

The analogy in the conservation of transported quantities is also present in the constitutive relations for the fluxes. For example, transport properties derived from molecular properties yield the viscous term in momentum transport, which is given by Newton's law for fluids; the conduction term in heat transfer, given by Fourier's law for heat transfer; and the diffusion term in mass transport, given by Fick's law for diffusion The linear relation of the migration term to the electric field parallels Ohm's law, which arises from the transport properties of electrons in metals.

In gases, the transport properties for viscosity, thermal conductivity, and diffusivity are derived from collisions, Brownian motion, and molecular interactions. In liquids, the theory is less general, but still relates molecular momentum, energy, and mass transport properties for a given fluid.

In conclusion, the principle of defining the model equations is straightforward. It is a matter of defining the conservation laws and the relations for how flux can be provoked. It is by solving these equations for a given system over and over again under different conditions, and then studying the results, that we get an understanding of the transport phenomena in the system.

References

  • R.B. Bird , W.E. Stewart , and E.N. Lightfoot , Transport Phenomena, 2nd edition, John Wiley & Sons, Inc., 2007.
  • "Transport Phenomena", Wikipedia.
  • R.P. Feynman , R.B. Leighton , and M. Sands , The Feynman Lectures on Physics, Vol II, p. 2-1, The California Institute of Technology, 1989.
  • S. Kranenbarg , Oxygen Diffusion in Fish Embryos, Doctoral Thesis, Experimental Zoology, Wageningen University, The Netherlands, 2002.