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外文原文：
Theory of fluid properties
We will concentrate mainly on three fluid properties in this chapter:
? The density which leads to mass and hence to hydraulic inertia effects.
? The viscosity which leads to the hydraulic friction effects.
? The compressibility and thus the bulk modulus which leads to the hydraulic system stiffness. Notice that the compressibility effect can be modified by air release, cavitation phenomena and by expansion of a pipe, hose or chamber containing the hydraulic fluid.
1 Density and compressibility coefficient
The density is the mass of a substance per unit volume:
Density has dimensions of [M/L] and is expressed in kilograms per cubic meter [kg/m]. As mentioned previously the density is a function of the pressure and the temperature:
This function can be approximated by the first three terms of a Taylor series:
This can also be expressed as:
With
And
This equation is the linearized state equation for a liquid. Using the definition of the density, the two coefficients α and B can also be expressed as:
B is known as the isothermal bulk modulus or for simplicity the bulk modulus and α is known as the cubical expansion coefficient. Since fluid density varies with the applied pressure, this implies that a given mass of fluid submitted to a pressure change changes its volume. This phenomenon leads to the definition of the compressibility coefficient β:
where β is expressed in units Pa (or m/N). Considering the relation for a closed hydraulic circuit the mass is constant, and hence:
it follows that
Using the definition of the compressibility coefficient β we obtain:
More usually we use the bulk modulus B also known as the volumetric elasticity modulus:
The relation between ρ and B implies mass conservation. This relation must be RIGOROUSLY RESPECTED in the calculations. In the modeling and simulation context of fluid energy systems, disregarding the relation between ρ and B leads to abnormal evolutions of pressure in the closed circuit submitted to compression and expansion cycles. This phenomenon is strongly accentuated if aeration occurs in the circuit (when dissolved air in the fluid reappears in the form of bubbles). We shall approach this point by examining the phenomena of aeration and cavitation. The air can also have adverse consequences on a fluid compressibility. In liquid air can be present in two forms: entrapped and dissolved.
Entrapped air
When the return pipe is not submersed in the tank the liquid jet can entrain some air bubbles in the tank. Another phenomenon that affects the quantity of air in liquid is the leakage.
Figure 1: Liquid leakage
Figure 2: Air is entrained
This air stays in the liquid as cavities and can modify the fluid compressibility. In this context we talk about effective bulk modulus. Figure 3 shows the bulk modulus of a diesel fuel at 40 °C with 0, 0.01, 0.1, 1, 10% air. The plot is obtained using the system shown. The model of the diesel fuel properties is based on accurate ex-perimental measurements and are designed for use with injection system which are very fast acting. For this reason air is assumed to be entrained rather than dissolved.
Figure 3
Dissolved air
Air can also be dissolved in a liquid. A certain amount of air molecule can be part of the liquid. In this case the dissolved air does not significantly change the fluid properties.
2 Air release and cavitation
Air can be dissolved or entrained in liquids and it is possible for air to change from one of these two forms to the other depending on the conditions to which the fluid is subjected.
Suppose the fluid is in equilibrium with a certain percentage of dissolved gas (usually air: nitrogen and oxygen). Lowering the pressure above a critical value called the saturation pressure induces aeration. This is the process where the dissolved gas forms air bubbles in the liquid until all the dissolved gases or air are free.
The exact point where all the dissolved gas has come out of solution is difficult to pin-point because it depends on the chemical composition and behavior of the gas. This is a non-symmetrical dynamic process: the growing process does not have the same dynamics as when air bubbles disappear. In consequence the total amount of bubbles created when the pressure drops may or may not be redissolved in the liquid when it rises again.
If the pressure is dropped further and above another critical value called the vapor pressure, the fluid itself starts to vaporize. It corresponds to a liquid phase change. At some point only fluid vapor and gas exist. In liquid systems the term cavitation usually refers to the formation and collapse of cavities in the liquid even if cavities contain air or liquid vapor.
To summarize with a sketch what we have introduced see above:
Figure 4: Air release and cavitation
The development of a cavity is now recognized as being associated with a nucleation center such as microscopic gas particles, wear or wall asperities. When the liquid is subjected to a tensile stress, cavities do not form as a result of liquid rupture but are caused by the rapid growth of these nuclei.
To understand this, think of beer (or champagne if you prefer) in a bottle, when it is closed you see no air bubbles and the liquid does not look fizzy. The pressure in the bottle is above the saturation pressure of the gas in the liquid. When you open the bottle suddenly bubbles appear and so the dissolved gas (molecules of gas held in the liquid) starts to appear as gas.
In fact the liquid is gas saturated and the atmospheric pressure is less than the saturation pressure of the liquid. This phenomenon is clearly not cavitation but air release (aeration). Considering nuclei effects, bubbles form only at particular places in your glass: around the glass (due to small asperities) and round any particles present in the liquid. Theoretically, if your liquid was perfectly pure and the wall of the system perfectly regular, air release or cavitation would occur with great difficulty!
The key point about cavitation is that it is a phase change: the liquid changes to vapor. A comparison can be made between cavitation and boiling. If we look at the phase diagram below:
Figure 5: Cavitation and boiling
Boiling is a phase change at constant pressure and variable temperature and cavitation is a phase change at constant temperature and variable pressure.
In any system air release starts first and if the pressure decreases further, cavitation may occur. This means that, sometimes, people talk about cavitation when the real phenomenon is air release. Both phenomena can lead to destruction of the material or component.
In both cases it is entrained gas that causes the troubles. When cavities encounter high pressure in the downstream circuit, these bubbles or cavities can be unstable and can collapse implosively. The pressure developed at collapse can be large enough to cause severe mechanical damage in the containing vessel. It is well-known that hydraulic pumps and pipework can be badly damaged by cavitaton and air release.
In all classical hydraulic systems air release and cavitation must be avoided to prevent material destruction but sometimes it is required like for injection systems to prepare the spray formation.
3 Viscosity
Viscosity is a measure of the resistance of the fluid to flow. This characteristic has both positive and negative effects on fluid power systems. A low viscosity leads to oil leaks in the dead zone formed between the mechanical parts in movement, and a high viscosity will lead to loss of pressure in hydraulic ducts.
Viscosity is a characteristic of liquids and gases and is manifested in motion through internal damping. Viscosity results from an exchange of momentum by molecular diffusion between two layers of fluid with different velocities. In this sense, the viscosity is a fluid property and not a flow property.
Figure 6: Viscosity
Figure 6 shows the relation between shearing constraint and difference of flow velocity between two layers .
The definition of viscosity was first given by Newton. Between two layers of distance dy, the exerted force between these two layers is given by:
where U(y) is the velocity depending on the radial position y and dU/dy the velocity gradient. This proportionality expresses the notion of Newtonian fluid and allows the introduction of μ defined as the dynamic viscosity or the absolute viscosity.
The dimension of μ is [MLT ] and the SI unit is kg/m/s or Pa s. The older unit is the Poise, P, which is 0.1 kg/m/s. However, this is very small and hence the milli Poise, mP, is the common unit which is 10-4 kg/m/s.
The dynamic viscosity is the constant of proportionality between a stress and the intensity of shearing between two neighboring layers:
However the absolute viscosity is not very often used in fundamental equations. For example the dynamics of the elementary volume between the two layers is expressed as:
and thus using the shear stress calculation:
In other formulas (e.g. Navier Stokes) the ratio between the absolute viscosity and the density occurs so often that a new parameter called the kinematic viscosity ν is introduced .
of dimension [L T ] and so the SI unit is the m/s. The older unit of kinematic viscosity is the Stoke, St, which is 10 m/s. However, even this is a very small unit and hence the centistoke cSt is the common unit with 1 cSt = 10 m/s. This parameter is easily measured with viscometers.
Note that the viscosity varies significantly with the fluid temperature.
Figure 7: Viscosity against temperature
Normally in absence of air release and cavitation the variation with pressure is not great unless the pressure is very extreme.
Figure 8: Variation with pressure
Viscosity influence on the flow
Another important aspect of the viscosity is its influence on the flow conditions of the fluid. We can distinguish two types of flow conditions:
? Laminar flow for which the flow lines are parallel and shearing forces create a pressure drop.
? Turbulent flow for which the fluid particles have a disordered, random movement leading to a loss of pressure.
These two conditions can be distinguished using the Reynolds number which is defined as follows:
With
U: average fluid velocity
d: diameter of the duct (hydraulic diameter for others geometries)
ρ: density
μ: dynamic viscosity
ν: kinematic viscosity
The transition between laminar to turbulent flow occurs at the critical Reynolds number. This is not well defined, there exists always a transition region. In a hydraulic line, the critical Reynolds number is generally between 1500 to 2000. For uneven geometries (thin-walled orifices), the critical Reynolds number can be lower than 100.
For non-circular cross sections, the hydraulic diameter can be used to determine the Reynolds number. Hydraulic diameter is defined as follows:
We now give one example:
? Circular orifice of diameter:
Flow through orifices
Orifices (also called restrictions) can be fixed or variable and occur in huge numbers in fluid systems. Not surprisingly in Engineering courses a mathematical description is presented. This is usually based on Bernoulli’s equation and leads to the form
where Cq is the flow coefficient. This is variously described as typically 0.7 or varying with orifice geometry and Reynolds number.
The second alternative is obviously more correct. If we do take a constant value, we are forced to have the gradient of Q against infinity at the origin! This cannot be and if you try to implement it is a numerical disaster! Clearly the flow is laminar for sufficiently small pressure drops which means that Cq is certainly not constant. One solution is to perform detailed experiments and compute Cq against Reynold’s number. In the context of the orifice (not necessarily circular) the Reynold’s number is
where U is a mean velocity and dh the hydraulic diameter. If we take U=Q/A, we end up with the form Cq =f(Q) and ultimately with
It is possible to work with an implicit relationship like this but we would prefer an explicit formula.
This is provided by introducing another dimensionless number known as the flow number and denoted by λ. This is defined as
From a modeling point of view λ contains quantities we know. Using λ we have
and provided we have
,
we have an explicit relationship which is easy to evaluate. There are no more problems to obtain measurements for
than for
and so the flow number form has many advantages.
References ：
[1] McCloy D, Discharge Characteristics of Servo Valve Orifices, 1968 Fluid International Conference.
[2] R.C. Binder, “Fluid Mechanics”. 3rd Edition, 3rd Printing. Prentice-Hall, Inc., Englewood Cliffs,NJ. 1956.
譯文：
液壓油理論
我們將在本章主要討論液壓油的三個(gè)特性：
?密度（使油液具有質(zhì)量和液感效應(yīng)）；
?粘性（使油液具有液阻效應(yīng)）；
?可壓縮性和體積彈性模量（使油液具有容性效應(yīng)），值得提醒的是容性效應(yīng)會(huì)受油液中析出的空氣、氣穴現(xiàn)象和裝有油液的的管道、軟管或油腔的影響。
1密度和壓縮系數(shù)
密度是單位體積的物質(zhì)的質(zhì)量：
密度的單位是千克每立方米[kg/m]。和先前提到的密度一樣，這里的密度也是壓力和溫度的函數(shù)：
這個(gè)函數(shù)可以大致用泰勒級(jí)的前三項(xiàng)來表示：
這個(gè)式子也可表達(dá)成：
同時(shí)
并且
這個(gè)方程是對液體線性化處理情況下的方程。利用對密度的定義，系數(shù)α和B同樣可以表達(dá)如下：
B是通常大家所知的等溫體積彈性模量或者對非專業(yè)人士來說體積彈性模量和α就是在空間方向上的膨脹系數(shù)。既然液體的密度是隨著作用的壓力而變化的，那么這就意味著給定質(zhì)量的液體在壓力的變化下其體積也會(huì)發(fā)生相應(yīng)的變化。這個(gè)現(xiàn)象導(dǎo)致壓縮系數(shù)β出現(xiàn)：
這里的β是用單位Pa 或者m/N表達(dá)的?？紤]到對一個(gè)封閉的液壓回路來說，其油液的質(zhì)量是不變的，因此有：
所以有
利用壓縮系數(shù)β的定義我們可以得到：
更為常見的是我們通常用體積彈性模量B（也常稱作the volumetric elasticity modulus）：
ρ 和 B的關(guān)系暗示著質(zhì)量守恒，這個(gè)關(guān)系在計(jì)算中必須嚴(yán)格地遵守。在液壓系統(tǒng)的建模和仿真過程中，忽視ρ 和 B的關(guān)系將會(huì)導(dǎo)致閉環(huán)回路的壓強(qiáng)的仿真不正常。如果摻氣發(fā)生了的話，這個(gè)現(xiàn)象尤其應(yīng)著重強(qiáng)調(diào)（當(dāng)油液中溶解的空氣以氣泡的形式出現(xiàn)時(shí)），我們將通過檢查摻氣和氣穴現(xiàn)象來解決這個(gè)問題。由于液體的可壓縮性，空氣也將會(huì)出現(xiàn)相反的結(jié)果。在液體中空氣可以有兩種存在的形式：以泡沫的形式存在于液體中和溶解于液體中。
誘陷的空氣
當(dāng)回油管沒有完全淹在油箱中的油液里，液流將會(huì)帶進(jìn)一些氣泡到油箱中去，另外一個(gè)導(dǎo)致一定數(shù)量的氣進(jìn)入液體中的現(xiàn)象的因素是泄漏。
Figure 1: Liquid leakage
Figure 2: Air is entrained
這種以空穴的形式存留于液體中的空氣可以改變液體的可壓縮性。在這個(gè)段落我們談?wù)摰氖怯行w積彈性模量。圖3顯示是溶有0, 0.01, 0.1, 1, 10%的空氣的柴油在四十度的溫度下的體積彈性模量值。這個(gè)曲線是通過該軟件的系統(tǒng)自帶的數(shù)據(jù)得到的。柴油的模型的性質(zhì)是基于精確的測量實(shí)驗(yàn)的，并且這個(gè)實(shí)驗(yàn)是被設(shè)計(jì)用于高速動(dòng)作的注射系統(tǒng)的。因此是假定帶入的空氣比溶入的多些。
Figure 3
溶解的空氣
空氣也可以溶于液體中。一定量的空氣分子能夠成為液體的一部分。在這種情況下，溶解的空氣不會(huì)對液體的性質(zhì)產(chǎn)生多大的影響。
2 空氣的釋放和氣穴現(xiàn)象
空氣可以溶解或著存留于液體中并且空氣可以由這兩種形式轉(zhuǎn)換成另一種依賴于液體的屬性的形式。
假定油液溶有一定百分?jǐn)?shù)的氣體（通常所說的空氣：氮?dú)夂脱鯕猓┎⑻幱谄胶鉅顟B(tài)。降低油液的壓力到低于其臨界值（稱作飽合壓力值）將會(huì)導(dǎo)致?lián)綒猬F(xiàn)象的產(chǎn)生。這就是溶解了的空氣在液體中形成氣泡直到所有溶解掉的氣體或空氣變得自由的過程
這精確的臨界值是難以確定的，因?yàn)檫@是取決于其化學(xué)組成和氣體的行為。這是一個(gè)不對稱的動(dòng)態(tài)過程：這個(gè)漸進(jìn)的過程不是和氣泡消失時(shí)具有一樣的動(dòng)力學(xué)，因此當(dāng)壓力下降時(shí)產(chǎn)生的氣泡在壓力上升時(shí)可能不會(huì)被再次溶解掉。
當(dāng)壓力再進(jìn)一步下降并且臨近另一個(gè)臨界值（稱作氣化點(diǎn)）時(shí)，液體本身會(huì)開始?xì)饣?。這是相應(yīng)的液體過渡階段。在某個(gè)壓力值時(shí)僅僅只有氣體和水蒸氣存在。在流體界，術(shù)語氣穴現(xiàn)象通常涉及到液體中空穴的形成與消亡，甚至空穴中有空氣或蒸氣存在。
下面用一幅圖來總結(jié)一下以上我們所介紹的：
Figure 4: Air release and cavitation
現(xiàn)在氣穴的形成被認(rèn)為與晶核點(diǎn)有關(guān)，比如精微的氣體粒子、容器壁上的微刺。當(dāng)液體受控于一個(gè)可變的壓力時(shí)，空穴不會(huì)形成但會(huì)由于急速增長的晶核點(diǎn)導(dǎo)致氣穴的產(chǎn)生。
為了理解這個(gè)，試想一瓶啤酒（或者香檳），當(dāng)它未被打開時(shí)，我們看不到氣泡，并且液體看起來不是沸騰的。瓶子里的壓力是高于液體中氣體的飽合壓力的。當(dāng)你打開瓶子的時(shí)候，氣泡就會(huì)迅速地出現(xiàn)并且溶解掉的氣體（液體中的氣體分子）開始以氣體的形式出現(xiàn)。
事實(shí)上液體就是飽合的氣體，并且大氣壓是低于液體的飽合壓力值的。這個(gè)現(xiàn)象明顯不是氣穴現(xiàn)象但是是氣體釋放現(xiàn)象?？紤]到核子的影響，氣泡僅僅形成于你的玻璃杯中特定的位置：在杯子的周圍和任何液體中的粒子處。理論上講，如果你的液體是絕對的純凈并且系統(tǒng)的壁子是絕對的光滑，那么氣體釋放和氣穴現(xiàn)象是非常難產(chǎn)生的！
氣穴現(xiàn)象的關(guān)鍵點(diǎn)是它是一個(gè)液體轉(zhuǎn)變?yōu)闅怏w的過程。我們可以做一個(gè)之氣穴現(xiàn)象和沸騰現(xiàn)象之間的區(qū)別實(shí)驗(yàn)。如果我們看到如下的圖表的話：
Figure 5: Cavitation and boiling
沸騰就是一個(gè)在不變的壓力和可變的溫度下的相變過程，而氣穴現(xiàn)象是一個(gè)在不變的溫度和可變的壓力下的相變過程。
在任何系統(tǒng)中如果壓力進(jìn)一步下降，空氣就會(huì)首選釋放出來，這時(shí)氣穴現(xiàn)象就有可能產(chǎn)生。這就意味著，有時(shí)人們所說的氣穴現(xiàn)象就是氣體釋放現(xiàn)象。兩種現(xiàn)象都會(huì)導(dǎo)致材料或元件的損壞。
在這兩種情況下帶入的氣體就會(huì)導(dǎo)致麻煩產(chǎn)生。當(dāng)空穴在回路中遇到高壓的情況時(shí)，氣泡或者空穴就會(huì)不穩(wěn)定，就會(huì)一下子消失。壓力在崩潰的時(shí)候會(huì)足以在容器中導(dǎo)致嚴(yán)重的機(jī)械事故。大家都知道液壓泵和管道工程會(huì)被氣穴現(xiàn)象和氣體釋放嚴(yán)重破壞。
在所有的傳統(tǒng)的液壓系統(tǒng)中，為了使材料免受損傷會(huì)采取方法避免氣體釋放和氣穴現(xiàn)象的產(chǎn)生，但是有的時(shí)候在噴射系統(tǒng)中為了產(chǎn)生噴霧這些現(xiàn)象又是需要利用的。
3 粘性
粘性是用來衡量液流的阻尼性質(zhì)的。這種特性對液壓系統(tǒng)來說有好的一面也有消極的一面。一種低粘度的油液會(huì)在運(yùn)動(dòng)中機(jī)械部件中形成的閉死區(qū)導(dǎo)致泄漏，并且高粘度會(huì)導(dǎo)致管道中的沿程壓力損失。
粘度是液體和氣體的特性并且可由衰減的阻尼運(yùn)動(dòng)得到證實(shí)。粘性是液流中兩層相鄰層中的分子不同的運(yùn)動(dòng)速度造成的，在這個(gè)意義上講，粘性是一種液體的特性而非流體的特性。
Figure 6: Viscosity
圖六表明了剪切力與兩層間液流速度的微分之間的關(guān)系。
這個(gè)粘度的定義是首先由牛頓提出來的。在兩層之間的距離dy和兩層間的外力的關(guān)系是由下式給定的：
式中U(y)是依賴于矢徑y(tǒng)的速度，而dU/dy是速度的梯度。這個(gè)比例表明了牛頓液流說的觀點(diǎn)并且允許引入一個(gè)μ（定義為動(dòng)力粘度或絕對粘度）。
μ的尺度是[MLT ]并且其國際標(biāo)準(zhǔn)單位是kg/m/s 或 Pa s。舊單位是泊，P（0.1 kg/m/s）。但是，這太小了因此毫泊是常用的單位，其值為10kg/m/s。
動(dòng)力粘度是壓力和相鄰兩層間剪切強(qiáng)度的比值：
但是絕對粘度不會(huì)在基本方程中經(jīng)常使用。例如基本的動(dòng)力學(xué)方程表示如下：
因此用剪切強(qiáng)度進(jìn)行計(jì)算：
在其它公式中絕對粘度與密度的比值經(jīng)常出現(xiàn)因此一個(gè)新的稱為運(yùn)動(dòng)粘度的參數(shù)ν被引用進(jìn)來了。
其國際單位是m/s 舊單位是沱，其值是10 m/s。但是，這是一個(gè)非常小的單位
因此厘沱是常用的單位，1厘沱=10 m/s.這個(gè)值很容易用粘度計(jì)測得。
值得提醒的是粘度會(huì)明顯得隨著溫度的變化而變化。
Figure 7: Viscosity against temperature
通常情況下，如果不存在氣體釋放和氣穴現(xiàn)象的話，粘度隨壓力的變化不是很大，除非其壓力值很大。
Figure 8: Variation with pressure
粘度對液流的影響
粘度的另外一個(gè)方面是影響液體的流態(tài)。我們可以區(qū)分兩種不同的流態(tài)：
?層流（其流線是相互平行的并且其剪切力會(huì)產(chǎn)生壓降）；
?紊流（其液體分子在作不規(guī)則的運(yùn)動(dòng)從而引起壓力損失）。
這兩種流態(tài)可以用雷諾數(shù)來界定，其定義如下：
式中
U：平均液流速度；
d:管道的直徑（其它幾何形狀的液力學(xué)直徑）；
ρ:密度；
μ:動(dòng)力粘度；
ν:運(yùn)動(dòng)粘度。
層流轉(zhuǎn)變?yōu)槲闪靼l(fā)生于液流處于臨界雷諾數(shù)的值的時(shí)候。這不是很好區(qū)分的，這存在一個(gè)轉(zhuǎn)換區(qū)間。在液壓管線中，其臨界雷諾數(shù)通常在1500到2000之間。對于那些不均勻的幾何圖形來說（薄壁孔），其臨界雷諾數(shù)要少100。
對于非圓和連結(jié)部分，其液壓學(xué)直徑可以用來確定雷諾數(shù)，液壓學(xué)直徑定義如下：
下面我們給出一個(gè)例子:
?圓孔的直徑：
液流通過節(jié)流孔
節(jié)流孔（通常稱作阻尼孔）有不可變和可變之分，大量地出現(xiàn)在液壓系統(tǒng)中。在工程學(xué)中有這樣的數(shù)學(xué)描述是不奇怪的。這通常是基于伯努力方程，推導(dǎo)出如下形式：
式中Cq是流動(dòng)系數(shù)。它通常取0.7或者是隨著孔的幾何形狀和雷諾數(shù)的變化而變化的。
第二種選擇明顯是更為合理的。如果我們實(shí)在是要取一個(gè)恒值，那么我們就不得不使Q的梯度值在原點(diǎn)處取無窮大！如果你要這樣做的話，這將導(dǎo)致數(shù)字災(zāi)難！在足夠小的壓力降的的作用下液流明顯呈層流流態(tài)，也就意味著Cq當(dāng)然不是一個(gè)定值。有一個(gè)解決方法就是詳細(xì)地列出Cq與雷諾數(shù)。在本文中孔（不一定要是圓的）的雷諾數(shù)是：
式中U是平均速度，dh是液壓學(xué)直徑。如果我們?nèi)=Q/A，最終可得到形如Cq =f(Q)的式子，具體如下式：
在這樣的公式中使隱函數(shù)關(guān)系也是可能的，但是我們更喜歡用明顯的公式。
這個(gè)是由另一個(gè)稱為流動(dòng)數(shù)的無量綱數(shù)提供的并且用λ表示，定義如下：
從一個(gè)模型點(diǎn)λ所包含的數(shù)量我們可以知道，使用λ我們可以得出：
同時(shí)我們有
,
我們有一個(gè)易于求值的顯式關(guān)系，對于式來說在尋找一個(gè)測量法不會(huì)比式更難，而且流動(dòng)數(shù)的這個(gè)形式還有很多優(yōu)點(diǎn)。
參考資料：
[1] McCloy D, Discharge Characteristics of Servo Valve Orifices, 1968 Fluid International Conference.
[2] R.C. Binder, “Fluid Mechanics”. 3rd Edition, 3rd Printing. Prentice-Hall, Inc., Englewood Cliffs,NJ. 1956.