Abstract
In this chapter, linear systems analysis is described in detail using a representative example. We consider a semi-infinite soil thermal field whose fundamental equation is an unsteady-state heat transfer equation. First, we discuss the physical implications of the fundamental equation, the concept of discretization, and how continuum space is discretized. Next, system state equations based on vector matrix notation are derived. This book describing time discretization for system state equations will help the readers understand the processes of numerical analysis. In the second half, giving examples of changes in single room temperature and thermal load computation, specific programming methods are described in detail. We also discuss the stability of discretized numerical solutions, and introduce the finite element method (FEM) commonly used for space discretization.
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- 1.
The diffusivity or diffusion coefficient has the same unit [m2/s] as for other transport phenomena other than heat (e.g., the molecular diffusivity coefficient). In any phenomenon, the diffusivity coefficient (diffusivity) essentially represents the transport efficiency, which is the constant of proportionality in terms of the relationship between the transferred flux and the concentration gradient imposed by a potential difference (described later). This universal relationship is referred to as Fick’s law.
- 2.
Many textbooks show the transpose symbol at the upper right of the vector or matrix, but this convention may be easily confused with powers; hence, in this book, it is shown at the upper left.
- 3.
An inverse is any quantity that when multiplied by its original quantity, yields an identity. In scalars, the unit element is 1; in matrices, it is the unit matrix E. If a unit is multiplied by a quantity, it yields the same quantity.
- 4.
No reverse can be defined for the element (1,1) at the Earth’s surface temperature node because they are zero. For this reason, here explanation is made in general terms.
- 5.
This argument derives from the fact that the time evolution of the error between the numerical solution and the explicit solution obeys the original equation.
- 6.
Solar radiation absorption varies greatly with color of the wall surfaces. Perfectly black bodies absorb all radiation (absorption = 1), while white surfaces can absorb as little as 0.5 of incoming radiation. However, emissivity is insensitive to surface color and is typically around 0.9.
- 7.
Humid air is a mixture of dry air (DA) and humidity (moisture). In physical terms, the most appropriate humidity parameter for moisture concentration is the specific humidity[g/g]or mass ratio of water vapour to humid air. In contrast, the absolute humidity [g/g(DA)](or [g/g′]) is the mass ratio of water vapour to dry air. Although inconsistent with the true definition of concentration, specific humidity is a standard thermodynamic function. In this book, we also introduce the hygroscopic equation, whose potential function is absolute humidity.
- 8.
The system state equation may also be computed by incorporating Eq. (2.73.2), without making this approximation (more specifically, C o θ o may be incorporated). This assumption was introduced to simplify the explanation.
- 9.
These weather data consist of hourly temperature and hourly solar radiation applied with excess frequency ratio 2.5 % (so-called TAC 2.5 [Technical Advisory Committee of ASHRAE]; meaning top 2.5 % highest temperature and radiation rate in last 10 years as the statistical samples), which, if used in cooling design, will overestimate the device capacity of refrigerators and air-conditioning units. This occurs because the time-series constructed from the weather data distorts the real-life events.
- 10.
Despite this, Eq. (2.92), as a general expression, interpolates using the highest linear function of the temperature at the edge nodes in the finite element. “Smooth” is within that possible in a linear approximation.
- 11.
Gauss Divergence Theorem
The divergence theorem states that the integration over volume V of the divergence of vector u is equivalent to surface integration of the normal component of u over the boundary curve S surrounding V (this should make sense physically). Figure 2.34 shows this. Mathematically, this is expressed \( \begin{array}{l}{\displaystyle \underset{V}{\int } div\mathbf{u} dV}\left(={\displaystyle \underset{V}{\int}\nabla \bullet \mathbf{u} dV}\right)={\displaystyle \underset{S}{\int}\mathbf{u}\bullet \mathbf{n} dS}\hfill \\ {}\iff {\displaystyle \underset{V}{\int}\frac{\partial {u}_i}{\partial {x}_i} dV}={\displaystyle \underset{S}{\int }{u}_i\cdot {n}_i dS},\hfill \end{array} \)
where \( div\mathbf{u}=\frac{\partial {u}_x}{\partial x}+\frac{\partial {u}_y}{\partial y}+\frac{\partial {u}_z}{\partial z}=\nabla \bullet \mathbf{u} \).
Substituting u ≡ vw, Gauss’ divergence theorem is expressed as
$$ \begin{array}{l}{\displaystyle \underset{V}{\int}\frac{\partial }{\partial {x}_i}{(vw)}_i dV}={\displaystyle \underset{S}{\int }{(vw)}_i\cdot {n}_i dS}\hfill \\ {}\iff {\displaystyle \underset{V}{\int}\frac{\partial {v}_i}{\partial {x}_i}{w}_i dV}={\displaystyle \underset{S}{\int }{(vw)}_i\cdot {n}_i dS}-{\displaystyle \underset{V}{\int }{v}_i\frac{\partial {w}_i}{\partial {x}_i} dV.}\hfill \end{array} $$On the right side of the equivalence sign, the formula for the derivative of an integral, (f ⋅ g)′ = f′ ⋅ g + f ⋅ g′ is used. The partial integration formula learned at senior high school, ∫ (f ⋅ g)′ = ∫ (f′ ⋅ g + f ⋅ g′) ⇔ ∫ f ⋅ g′ = f ⋅ g − ∫ f′ ⋅ g, is basically equivalent to Gauss’ divergence theorem.
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Tanimoto, J. (2014). Linear Systems Analysis Methods. In: Mathematical Analysis of Environmental System. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54622-1_2
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DOI: https://doi.org/10.1007/978-4-431-54622-1_2
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