Oct 13, 2014

E-Cat長期試験報告 007

(訳注 ここからの翻訳は、最新原稿(綴り間違いの訂正があったとされる版でelforsk社サイトに保管)

4.2 Convection


In order to calculate the heat dissipated by convection, two different kinds of surfaces must be taken into consideration, the smooth cylindrical surfaces of the rods and reactor caps, and the ridged cylinder of the reactor body.


If one identifies both the rods and the reactor caps as cylinders immersed in air,


one may, for each of them, calculate the heat Q emitted by convection per time unit by means of Newton’s relation.


If Ta indicates air temperature, A the surface area of a cylinder, and Ts the cylinder’s temperature, we have:


Q = hA(Ts–Ta) = hAΔT [W]      (2)

where h defines the thermal exchange coefficient [W/m²K].

ここで、hは熱交換係数を定義する [W/m²K]

Calculating h is the fundamental problem of thermal convection calculation, and has been tackled by various authors more or less empirically (See f.i. [6], [7], and [8]).

 h を計算することは、熱対流計算の基本的な問題であり、さらに、多かれ少なかれ経験的に、さまざまな著者によって取り組まれています(f.i.参照[6]、[7]及び[8])。

In the specific case of cylindrical surfaces, one of the more commonly used expressions is the following one:


h = (kCRaⁿ) / D [W/m²K]   (3)

where k indicates the coefficient of thermal conductivity of air [W/mK],

ここで、kは、空気の熱伝導率を示す [W/mK],

C and n are two constants, Ra is Rayleigh’s number, and D the diameter of the cylinder.


Rayleigh’s number is a dimensionless parameter given by the following expression:


Ra = (gβ(Ts – Ta)D³) / να     (4)

where g [m/s2] is gravitational acceleration,

ここで、g [m/ S ^ 2] は、重力加速度、

β[K–1] is the volumetric thermal expansion coefficient,

β[K ^ -1]は、体積熱膨張係数である、

which, for an ideal gas (applied here to air for simplicity) is= 1/T;

これについ、理想気体に対して(簡単にするために空気にここで適用) = 1/T である、

next, ν [m²/s] is kinematic viscosity, and α [m²/s] is thermal diffusivity.

次に、ν[㎡/ s]は、動粘度である、さらに α[㎡/ s]は熱拡散率である。

Coefficients β, k, α, and ν are all temperature-dependent,


and must be calculated at the so-called “film temperature” Tf = (Ts + Ta) / 2.

いわゆる「フィルム温度」で計算されなければならない Tf = (Ts + Ta) / 2.

Plots 2, 3, and 4 express these trends for a range of temperatures from 100 to 1000 K

プロット2、3、さらに4は、100から1000 Kまでの温度範囲のためにこれらの傾向を表現する、

and have been taken from the data reported in Appendix A of [9].


(訳注 ここでpage10が終わり)

Plots 2, 3, and 4.


Trends of thermal conductivity k [W/mK], kinematic viscosity ν [m²/s], and thermal diffusivity α [m²/s] of air in function of temperature, reproduced from data found in the literature [9]. 

熱伝導率の動向 k [W/mK]、動粘性率ν [m²/s]、及び、温度の関数内の空気の熱拡散率 α [m²/s]、文献[9]に見たデータから再生。

The convention used to present numerical values of the properties is illustrated by this example: 


for T = 300 [K] we have k ∙ 103= 26.3 [W/mK],ν ∙ 106= 15.9 [m²/s], and α ∙ 106= 22.5 [m²/s]; therefore k = 0.026 [W/mK], ν = 0.000016 and α = 0.000023.

 T = 300 [K] に対して、k ∙ 103= 26.3 [W/mK] であり、 ν ∙ 106= 15.9 [m²/s]、さらに、α ∙ 106= 22.5 [m²/s]である、それ故、  k = 0.026 [W/mK]、ν = 0.000016 さらに α = 0.000023 である。

(訳注 ここで11ページの終わり)

The Rayleigh number expresses the ratio of buoyancy forces to viscous forces,


and its value is indicative of the laminar-turbulent transition, which occurs when Ra >109.

さらに、その値は、層流-乱流遷移を示す、それは、Ra >109 のとき発生する。

Constants C and n are dependent on the value of Ra, according to what is expressed by Table 1 [9].


Table 1.Values of the constants C and n corresponding to variations of the Rayleigh number.


Thermal flow emitted by the body of the reactor by natural convection may be in turn calculated by an expression suitable to objects having circular fins,


to which our ridges may be compared for simplicity’s sake.


Figure 8 shows a single circular fin, triangular in profile.


This shape is the closest possible to the reactor’s ridges, and is appropriate to represent them.


Figure 8. Representation of a circular fin having triangular profile. Its shape is very similar to that of the reactor ridges, and was used as a model to calculate natural convection.

図8。 三角形の断面を有する円形フィンの表現。その形状は、反応器の隆起部ものと非常に類似している、および自然対流を計算するためのモデルとして使用された。

Let us then approximate the body of the reactor to that of a cylinder having N fins,


each one having surface Af.

それぞれのものは、表面 Af を有する。

If we take At as the its total surface, we have:

私たちは、At を その全表面とする場合は、次のようになります

At = NAf     (5)

The length of the reactor body is given by L = 200 mm, and that of the base of each ridge is given by δb = 3.25 mm.

反応器本体の長さは、L = 200 mm によって与えられる、各稜線の基部のそれは δb = 3.25 mm で与えられる

If we compare this to a finned cylinder having no space between fins,


the number of ridges/fins along it is = N = L / δb ≈ 61. For the area of each fin, we have:

それに沿った稜線/フィンの数は、 = N = L / δb ≈ 61。 各フィンの面積については、こうなります、

Af= 2π(ra² – rb²) = 3.22 ∙ 10 ^ –4 [m²]   (6)

where ra is the distance between the axis of the cylinder and the tip of a fin, = 1.23 ∙ 10–2[m],

ここで、ra は、円筒の軸とフィンの先端間の距離 = 1.23 ∙ 10–2[m] であり、

while rb is the radius of the cylinder = 1.0∙10–2[m] (Figure 8).

一方、 rbは、円筒の半径  = 1.0 * 10 ^ –2[m] である(図8)。

Note how this formula for the area is actually fit for fins having a rectangular, not triangular, profile;


(訳注 ここで 12ページの終わり)

this approximation is however commonly used, as one may see f.i.in [10].

この近似は、しかし一般的に使用され、[10]のf.i. に見ることができます。

We may calculate the total thermal power emitted by convection by the reactor body in the following manner [9]:


Q = NηhAf(Ts–Ta) [W]     (7)

assuming that coefficient h is equivalent of what one would have for a finless surface.

その係数h は、フィン無し面に関してそういう物が持っているであろうものと同じであるとと仮定すします。

This coefficient is therefore calculated as in (3), referring to a cylinder having the size of the reactor but completely devoid of fins (see here [9]).

この係数は、したがって、(3)のように計算される、ある反応器のある大きさを有するシリンダを参照することで、ただしフィンを完全に欠いた物であるが、 (ここ[9]を見よ)。

Parameter η represents here the efficiency of each fin, and is an index of its thermal performance.


Since the driving potential for convection is expressed by the difference in temperatures between a body and its exchange fluid,


it is obvious that the maximum thermal flow for a fin would be had if its entire surface were at the same temperature as its base.


However, as each fin is characterized by a finite resistance to thermal conduction,


there will always be a thermal gradient along it, and the condition given above is a mere idealization.


Therefore, the efficiency for a fin is defined as the ratio of heat actually exchanged with air to its the maximum ideal amount.


In the case of a fin having triangular profile, one may calculate the trend of η as a function of a dimensionless parameter m, equal to:


m = b(2h / k δb) ^ 0.5  with b = r a- r b = 2.3 * 10^–3 [m]; k [W/mK]: 

thermal conductivity of the cylinder  (8)
シリンダの熱伝導率  (8)

This trend may be seen in Figure 9; for calculation details see [10].

Figure 9. A plot showing the efficiency of a circular fin having triangular profile. From [10].

図9。 三角形の断面を有する円形フィンの効率を示すプロット。 [10]から。

(訳注 4.2(13ページの中部まで)はここで終わり)