污泥薄層干化熱通量估算.

1、HEAT FLUX ESTIMATION IN THIN-LAYER DRYING薄層干化熱通量估算ABSTRACTA method is developed in order to determine the drying curves for sludge thin-layer drying. Since mass balance is very difficult to measure for thin-layer drying, these curves are obtained through energy balance. An experimental device is bui

2、lt up, in order both to provide and to estimate the heat flux density at the interface between a hot metallic plate and the drying sample. An analytical direct model is made using the quadripole formalism, and the system transfer function is calculated. The inverse problem is solved using the beck&#

3、39;s sequential function specification method, and the corresponding drying curve is deduced by a simple energy balance. Real experiment results are presented.摘要開(kāi)發(fā)了一種方法，以確定污泥薄層干燥的干燥曲線。 由于薄層干燥的質(zhì)量平衡很難測(cè)量，所以這些曲線是通過(guò)能量平衡獲得的。 建立了一個(gè)實(shí)驗(yàn)裝置，以便提供和估計(jì)熱金屬板和干燥樣品之間的界面處的熱通量密度。 使用四極形式來(lái)進(jìn)行分析直接模型，并計(jì)算系統(tǒng)傳遞函數(shù)。 使用beck的順序函數(shù)規(guī)范方

4、法解決了逆問(wèn)題，并通過(guò)簡(jiǎn)單的能量平衡推導(dǎo)出相應(yīng)的干燥曲線。 介紹了實(shí)驗(yàn)結(jié)果。INTRODUCTION緒論Among contact drying technologies, drum drying is widely used in the food industry, to treat heat-sensitive products. It also presents a great interest for sludge drying. The product is sprinkled or coated on a hot rotating cylinder. The wall tempe

5、rature is above boiling temperature. Absence of mixture and agitation constitutes an advantage for viscous products like sludge, enabling a good control of residence time, averaged water content and thermal efficiency. For contact dryers design, it is important to predict drying rates, but this rate

6、 is limited by internal conductive transfers. It is then necessary to find a aximum ideal reference, not driven by internal transfers, but by the intrinsic products thin film boiling rate : this is the main scope of this paper. The present studys application is for sludge. 在接觸干燥技術(shù)中，滾筒干燥廣泛用于食品工業(yè)，用于處理

7、熱敏產(chǎn)品。 它也對(duì)污泥干燥有很大的興趣。 將產(chǎn)品灑在或涂在熱旋轉(zhuǎn)圓筒上。 壁溫高于沸點(diǎn)溫度。 缺乏混合和攪拌是粘性產(chǎn)品如污泥的優(yōu)點(diǎn)，可以很好地控制停留時(shí)間，平均含水量和熱效率。 對(duì)于接觸式干燥器的設(shè)計(jì)，重要的是預(yù)測(cè)干燥速率，但是這個(gè)速率受到內(nèi)部傳導(dǎo)傳遞的限制。 因此，需要找到一個(gè)最佳的理想?yún)⒖贾?，而不是由?nèi)部轉(zhuǎn)移驅(qū)動(dòng)，而是通過(guò)本征產(chǎn)物的薄膜沸騰速率：這是本文的主要范圍。 本研究的應(yīng)用是污泥。In a previous work, it was shown by Vasseur (1983) that very high heat flux are exchanged during the fi

8、rst instants for viscous foodstuff thin film drying, due to the low products thickness.NOMENCLATURE術(shù)語(yǔ)A, B, C, DQuadripole elements：四級(jí)桿原件F(s)- Transfer Function：傳遞函數(shù)X- Moisture content：水分含量Y- Measured temperature：測(cè)量溫度M- Dry matter load：干物質(zhì)負(fù)荷f(t)- Inverse transfer function：反向傳遞函數(shù)dt- Time step：時(shí)間階q- In

9、terface heat flux density：界面熱通量密度T- Computed Temperature：計(jì)算溫度a- Thermal diffusivity：熱擴(kuò)散系數(shù)lv- water latent heat：水潛熱r- Future time steps number：未來(lái)時(shí)間步數(shù)s- Laplace variable：拉普拉斯變量t- Time：時(shí)間b- Thermal effusivity：熱效率e1- Sensor location：傳感器位置e- Plate thickness：板厚e2- e e1l- Thermal conductivity：導(dǎo)熱系數(shù)j- Heat f

10、lux density：熱通量密度f(wàn)- Laplace Tr. heat flux：拉普拉斯轉(zhuǎn)換熱通量q- Laplace Tr. Temperature：拉普拉斯轉(zhuǎn)換溫度rcp-Volumetric thermal capacity：比熱容Mass rate drying, when involved in a vaporization thin film process, is very difficult to measure :當(dāng)涉及蒸發(fā)薄膜工藝時(shí)，質(zhì)量干燥是非常難以測(cè)量的：l Boiling transfer is a rather stochastic and violent ph

11、enomena, and highly fast for thin film boiling。沸騰傳熱是一種相當(dāng)隨機(jī)和劇烈的現(xiàn)象，對(duì)于薄膜沸騰來(lái)說(shuō)是非?？斓摹 Mass losses are difficult to measure, because very little material is coated on the wall。質(zhì)量損失是難以衡量的，因?yàn)楹苌俚牟牧贤扛苍诒谏稀 Products sampling is made impossible, because drying rate is too high. 因?yàn)楦稍锼俣忍?，所以產(chǎn)品的取樣是不可能的。Since mass

12、balance cant be directly known, a way to estimate the drying rate is doing an energy balance, assuming that the amount of energy involved in vaporization drying is known. It is thus necessary to obtain reliable values of the interface heat flux between the drying product and the heated surface.由于質(zhì)量平

13、衡不能直接得知，所以估計(jì)干燥速率的一種方法是做出能量平衡，假設(shè)蒸發(fā)干燥所涉及的能量已知。 因此，需要獲得干燥產(chǎn)物和加熱表面之間的界面熱通量的可靠值。The experimental device, developed for this study, is presented in next section.本研究開(kāi)發(fā)的實(shí)驗(yàn)裝置將在下一節(jié)中介紹。1. Experimental device實(shí)驗(yàn)裝置The first objective of this experimental device is to produce thermal conditions for thin film vapori

14、zation contact drying.The product's thin-layer (about 0.7 mm thickness) is coated on a hot metallic plate. The plate must be thick enough in order to store the energy amount necessary for complete drying (58 mm thickness). It must be highly diffusive in order to ensure high heat flux densities a

15、t the interface. The heat is stored at a temperature above boiling temperature and suddenly discharged when the product is coated. Initial temperature is provided by placing the plate in a drying loop.該實(shí)驗(yàn)裝置的第一個(gè)目標(biāo)是產(chǎn)生用于薄膜蒸發(fā)接觸干燥的熱條件。將產(chǎn)品的薄層（約0.7mm厚）涂覆在熱金屬板上。 該板必須足夠厚以便儲(chǔ)存完全干燥所需的能量（58毫米厚度）。 為了確保界面處的高熱通量密度，

16、必須高度擴(kuò)散。 將熱量?jī)?chǔ)存在高于沸點(diǎn)溫度的溫度下，并在產(chǎn)品被涂覆時(shí)突然排出。 通過(guò)將板放置在干燥回路中來(lái)提供初始溫度。The second main goal is to estimate the interface heat flux between the drying product and the heated surface. It must be emphasized that no direct surface temperature can be made : a sensor located on the front side would perturb the materi

17、al coating, as well as heat transferThus wall temperature and flux must be estimated using interior location temperature and an inverse heat conduction method.第二個(gè)主要目標(biāo)是估計(jì)干燥產(chǎn)品和被加熱表面之間的界面熱通量。 必須強(qiáng)調(diào)的是，不能形成直接的表面溫度：位于前側(cè)的傳感器將擾亂材料涂層以及傳熱.因此壁溫和通量必須使用內(nèi)部位置溫度和反向熱傳導(dǎo)方法。A thin thermocouple with separated contacts is

18、 dulled inside the plate, near the surface, laid in a direction parallel to isothermal curves. With this particular method, the sensor location is precisely known (fig. 1). Another sensor is laid out on the plate's bottom side (T2 on fig. 2).具有分離接觸件的薄熱電偶在板內(nèi)部，在表面附近鈍化，平行于等溫曲線。利用這種特定的方法，傳感器位置是精確的（圖

19、1）。另一個(gè)傳感器布置在板的底部（圖2中的T2）。圖一傳感器布置2. Direct mode直接模式A direct model is built in order to calculate the sensor temperature evolution, when a given heat flux is applied to the plates surface. The system is shown on fig.2. An analytical one dimensional conductive model is built, assuming that the plates b

20、ottom is insulated. The quadripole formalism (Degiovanni, 1988) is used, and the transfer function F1(s) between the interface heat flux ö and temperature T1 is calculated.建立一個(gè)直接模型，以便在給定的熱通量施加到板的表面時(shí)計(jì)算傳感器溫度演化。 系統(tǒng)如圖2所示。 假設(shè)板的底部是絕緣的，建立了分析一維導(dǎo)電模型。 使用四極形式（Degiovanni，1988），計(jì)算界面熱通量f和溫度T1之間的傳遞函數(shù)F1（s）。圖二

21、實(shí)驗(yàn)示意圖Heat transfer is here assumed to be purely conductive and the sample homogeneous :這里假設(shè)傳熱是純導(dǎo)電的，并且樣品是均勻的with a known heat flux density at the plate surface x=0 and insulated bottom at x=e. 在板表面x = 0處具有已知的熱通量密度，并且在x = e處具有絕緣底部。Initially, the whole system is assumed to be at uniform temperature T0.

22、 A new variable is then considered such as T=T*- T0. 最初，假設(shè)整個(gè)系統(tǒng)處于均勻的溫度T0。 然后考慮一個(gè)新的變量，如T = T* - T0To write the previous system in a quadripole form, Laplace transform is applied. Equation (1) becomes an ordinary differential equation : 要以四極形式寫(xiě)入先前的系統(tǒng)，應(yīng)用拉普拉斯變換。方程式（1）成為一個(gè)常微分方程：with heat flux definition s

23、uch as具有熱通量定義如Expressions (2) and (3) are then equivalent to a quadripole presentation (Batsale, 1994) such as然后，表達(dá)式（2）和（3）等同于四極呈現(xiàn)（Batsale，1994），例如where (i=1,2) :其中（i = 1,2）：It is then easy to determine the transfer functions Fi (s) between the Laplace transformed superficial heat flux , and the tem

24、peratures i to be calculated by因此，很容易確定拉普拉斯變換的表面熱通量和由下式計(jì)算的溫度i之間的傳遞函數(shù)FiInverse method allows to calculate the superficial heat flux density q(t) from the measured temperatures Y1(t) at sensor location, and corresponding temperature T1(t) from the direct model. Inverse transfer function fi(t) are obta

25、ined from Fi(s) through a numerical Laplace inversion (Stehfest, 1970).逆方法允許從傳感器位置的測(cè)量溫度Y1（t）和直接模型的相應(yīng)溫度T1（t）計(jì)算表面熱通量密度q（t）。 通過(guò)數(shù)值拉普拉斯反演（Stehfest，1970）從Fi（s）獲得反向傳遞函數(shù)fi（t）。The temperature Ti(t) is then calculated by a convolution product :然后通過(guò)卷積乘積計(jì)算溫度Ti（t）：Of particular interest is the transfer function

26、between q(t) and the computed temperature T1(t) at sensor location :特別感興趣的是傳感器位置處的q（t）和計(jì)算溫度T1（t）之間的傳遞函數(shù)：The insulated boundary condition will be validated from back side measurements Y 2 (T 2 location). Nevertheless, it would be possible to integrate these values in the boundary conditions knowledge

27、.絕緣邊界條件將從背面測(cè)量Y2（T2位置）進(jìn)行驗(yàn)證。然而，可以將這些值整合到邊界條件的知識(shí)中。3. Inverse method反方法In the case of one dimensional problem with fast and high heat flux changes, a sequential method is adapted. The function specification method, with a sequential constant heat flux functional form is used (Beck, 1970).在具有快速和高熱通量變化的一維

28、問(wèn)題的情況下，適用順序方法。 使用具有順序恒定熱通量函數(shù)形式的函數(shù)規(guī)范方法（Beck，1970）。For that method (Beck, 1985) , q(t) is approximated in a discrete form : 對(duì)于該方法（Beck，1985），q(t)以離散形式近似：q = q1 q2 qi qi+1 qn (13)where qi = q(ti) = q(i.dt). Equation (11) can be expressed in a discrete form :其中qi = q（ti）= q（i.dt）。 等式（11）可以以離散形式表示：with f

29、j = f1( tj) .其中 fj = f1( tj) 。Assuming that q1 -qi and T1 -Ti are known, qi+1 is searched by an ordinary least square procedure using the next Yi+1 -Yi+r measurements, assuming a constant heat flux qi+1 during the r future time steps (Fig.3) :假設(shè)q1 - qi和T1 -Ti都是已知的，則使用下一個(gè)Yi+1 -Yi+r通過(guò)普通最小二乘法來(lái)搜索qi + 1，

30、假設(shè)在未來(lái)時(shí)間恒定的熱通量qi + 1（圖3）：qi = qi+1 = qi+2 = = qr-1 = qr (15)圖3 - 時(shí)間步長(zhǎng)ti時(shí)間離散化The least squares procedure minimizes the functional form J with respect to q i+1 :最小二乘法使功能形式J相對(duì)于q i + 1最小化：Equation (16) is differentiated with respect to q i+1 and set equal to zero :等式（16）相對(duì)于q i + 1而不同，并且設(shè)置為等于零：The heat fl

31、ux increment during the time step is searched :時(shí)間增加期間的熱通量增量：qi+1=qi+dq (18)Thanks to Eq. (15), the r future computed temperatures Ti+j(qi+1 ) can be written as :感謝方程式 （15）, r的計(jì)算溫度Ti + j（qi + 1）可以寫(xiě)為：and for the same reason (qi = qi+1 ), the sensitivity coefficients in Eq. (17) and Eq. (19) are equal

32、:由于同樣的原因（qi = qi + 1），方程式 (17) 和 (19)精確度相同。Equations (17 20) yield to the heat flux increment q for q temporarily assumed constant (Eq. 21). This equation is used in a sequential manner by increasing i by one each time step.等式（17-20）產(chǎn)生的熱通量增量q為q暫時(shí)假定常數(shù)（等式21）。 通過(guò)在每個(gè)時(shí)間步長(zhǎng)中增加i，以順序方式使用該等式。The sensitivity c

33、oefficients defined in Eq. (21) are known from Eq. (14) with i replaced by i+1 to i+r. For example, the temperature Ti+j at time ti+j , with assumption of constant heat flux after time ti, can be expressed as:靈敏度系數(shù)定義在等式（21）從等式（14）用i+1替換為i+r。例如，假設(shè)在時(shí)間t1之后的恒定熱通量的時(shí)間t1+j處的溫度Ti+j可以表示為：Assuming Eq. (15) yi

34、elds to the sensitivity coefficients,with j = 1 to r :假設(shè)方程 （15）產(chǎn)生靈敏度系數(shù)，j = 1到r：It is shown the great interest that have a direct model with a convolution such as Eq. (11) coupled with the specification function method : the sensitivity coefficients are constant with time step, and only the first r+1

35、 values of the inverse transfer function are to be calculated. The sum of inverse transfer function values in Eq. (23) also imply a growing behavior for these sensitivity coefficients. This fact improves the future time steps stabilization effect in Eq. (21), because each future time step number inc

36、rement will contribute with a sensitivity coefficient growth. Sensitivity analysis is to be detailed in next section.顯示出具有卷積的直接模型的極大興趣，如方程式（11）與規(guī)范函數(shù)方法相結(jié)合：靈敏度系數(shù)隨時(shí)間步長(zhǎng)恒定，僅計(jì)算反向傳遞函數(shù)的第一個(gè)r+1值。式中的反向傳遞函數(shù)值之和（23）也意味著這些敏感系數(shù)的增長(zhǎng)行為。這個(gè)事實(shí)改善了公式中未來(lái)的時(shí)間步長(zhǎng)穩(wěn)定效應(yīng)。（21），因?yàn)槊總€(gè)未來(lái)的時(shí)間步長(zhǎng)數(shù)增加將有助于靈敏度系數(shù)增長(zhǎng)。靈敏度分析將在下一節(jié)中詳細(xì)介紹。4. Sensitivity

37、 analysis靈敏度分析The direct model, described in section 2, is used with a heat flux functional form similar to experimental results. Reduced sensitivity coefficients to the main parameters are plotted on fig.(4). It is shown that sensitivity of T 1 to e 1 is very low. This is due to the sensor location

38、, near to the surface.第2部分描述的直接模型與實(shí)驗(yàn)結(jié)果類(lèi)似的熱通量功能形式使用。 圖（4）繪制了主要參數(shù)靈敏度系數(shù)的降低。顯示T1對(duì)e1的靈敏度非常低。這是由于傳感器的位置，靠近表面。Sensitivity coefficients to thermal conductivity and thermal diffusivity are the same order than sensitivity to mean heat flux gain 0. It is then important to get good knowledge of the plates therm

39、al properties.對(duì)熱導(dǎo)率和熱擴(kuò)散率的敏感度系數(shù)與對(duì)平均熱通量增益0的靈敏度相同。了解板材的熱性能很重要。Equation (14) can be written in a matrix form, in order to express the sensitivity matrix X to the unknown heat flux components :等式（14）可以寫(xiě)成矩陣形式，以便將未知熱通量分量的靈敏度矩陣X表示：T=Xq （24）Figure 4 Reduced sensitivity coefficients圖4 - 靈敏度系數(shù)的降低A whole time do

40、main estimation procedure would be difficult to regularize : The qi component at time ti can be estimated only very close to time ti .整個(gè)時(shí)域估計(jì)程序?qū)㈦y以規(guī)范化：時(shí)間ti的qi分量可以估計(jì)得非常接近時(shí)間ti。The dirac form of sensitivity coefficients to q can be understood with Eq. (25), and is due to the deep exponential decrease of

41、the transfer function f(t).靈敏度系數(shù)對(duì)q的狄拉克形式可以用公式（25），并且是由于傳遞函數(shù)f（t）的深指數(shù)下降。Sensitivity coefficients, obtained from Eq. (24 - 25) are presented on a map by fig. (5). It is shown that the sensitivity coefficient is non zero only for a very few points very close to each heat flux component qi at time ti。靈敏度

42、系數(shù)，從等式（24-25）顯示在圖（5）上。顯示了對(duì)于在時(shí)間t1處非常接近每個(gè)熱通量分量qi的非常少的點(diǎn)，靈敏度系數(shù)不為零。圖5 - 靈敏度系數(shù)圖The comparison between fig. (5) and fig. (6) makes clear the present algorithm interest, and the future time steps stabilization effect, since the sensitivity coefficients curve shown by fig. (6) is a growing function with the

43、future time steps number.由于圖（6）所示的靈敏度系數(shù)曲線是未來(lái)時(shí)間步長(zhǎng)數(shù)的增長(zhǎng)函數(shù)，所以圖（5）和圖（6）之間的比較清楚了本算法的興趣和未來(lái)的時(shí)間步長(zhǎng)穩(wěn)定效應(yīng)。未來(lái)時(shí)間步數(shù)Figure 6 Future time step sensitivity coefficients for the sequential method: Si+j , j=1 to r (Eq. 23) （圖6 - 順序方法的未來(lái)時(shí)間階躍靈敏度系數(shù)）5. Heat flux estimation results熱通量估算結(jié)果Some methods validation tests have be

44、en implemented with “numerical experiments”, and have proved its efficiency,but the corresponding results are not presented herein. In this section, only real experiment results are shown.一些方法的驗(yàn)證測(cè)試已經(jīng)用“數(shù)值實(shí)驗(yàn)”來(lái)實(shí)現(xiàn)，并且已經(jīng)證明了它的效率，但是這里沒(méi)有給出相應(yīng)的結(jié)果。 在本節(jié)中，僅顯示實(shí)際的實(shí)驗(yàn)結(jié)果。The plate is placed inside a drying loop, and a

45、 stable initial temperature above the products boiling temperature is reached before the experiment is started. At time 0, the product is quickly coated on the plate, while Y 1 and Y 2 acquisition goes on.將板放置在干燥回路內(nèi)，在實(shí)驗(yàn)開(kāi)始之前達(dá)到高于產(chǎn)品沸點(diǎn)溫度的穩(wěn)定的初始溫度。 在時(shí)間0，產(chǎn)品快速涂在板上，而Y 1和Y 2采集繼續(xù)。Although a lot of experiments

46、have been made, the goal of the present paper is only to present the general work and method, and the results are exhibited only for one case, with initial temperature T0 equal to 138 °C.雖然已經(jīng)做了大量的實(shí)驗(yàn)，但是本文的目的只是提出一般的工作和方法，只有一種情況才能顯示出結(jié)果，初始溫度T0等于138。Time step is dt = 0.01 s., the estimated parameter

47、s number is 2518, and the future time steps number r is 5.時(shí)間步長(zhǎng)為dt = 0.01 s，估計(jì)參數(shù)為2518，未來(lái)時(shí)間步長(zhǎng)r為5。Measured and computed temperatures evolution are plotted on fig. (7). The measured sensor temperature Y1 is perfectly fitted with T1 , but this is directly due to the inverse method, very close to an exact

48、 matching algorithm. More important is the perfect fit between back side measurement Y2 and computed temperature T2, because this means a good concordance between the experiment and the model, and proves the validity of back side insulated boundary condition assumption. Wall temperature T and T1 are

49、 almost identical, due to the low distance e1 and the coppers high thermal conductivity.測(cè)量和計(jì)算的溫度演變?nèi)鐖D（7）所示。測(cè)量的傳感器溫度Y1完全適合T1，但這是直接歸因于逆方法，非常接近精確匹配算法。更重要的是背面測(cè)量Y2和計(jì)算溫度T2之間的完美匹配，因?yàn)檫@意味著實(shí)驗(yàn)和模型之間的良好一致性，并證明了背面絕緣邊界條件假設(shè)的有效性。由于距離e1和銅的高導(dǎo)熱率，壁溫T和T1幾乎相同。圖7 - 計(jì)算和測(cè)量的溫度Y1 is submitted to a straight diminution due to sud

50、den boiling phenomena, and the corresponding superficial high heat flux density. The decrease is stopped when the heat flux intensity falls down while thermal diffusion inside the copper plate tends to its homogenization. When q(t) is near to zero,front side and back side temperatures tend to the sa

51、me value,corresponding to a quasi-permanent state.Y1由于突然沸騰現(xiàn)象以及相應(yīng)的表面高熱通量密度而直接減小。 當(dāng)熱通量強(qiáng)度下降時(shí)，銅板內(nèi)的熱擴(kuò)散趨于均勻化，停止減少。當(dāng)q（t）接近零時(shí)，正面和背面溫度趨于相同的值，對(duì)應(yīng)于準(zhǔn)永久性狀態(tài)。The wall temperature T is always above product's boiling temperature. Boiling transfer stops probably due to the product structure evolution : a high exte

52、rnal thermal resistance is found between the dried product and the plate interface.壁溫T總是高于產(chǎn)品的沸騰溫度。 沸騰轉(zhuǎn)移停止可能是由于產(chǎn)品結(jié)構(gòu)演變：在干燥產(chǎn)品和板接口之間發(fā)現(xiàn)高的外部熱阻。Figure (8) indicates superficial heat flux density q(t) estimated from Y1 measurements and inverse procedure. A high value, about 106 W.m-2 is raised after a very

53、short time smaller than one second, due to boiling phenomena. The flux decrease occurs when boiling drying stops, and is followed by an evaporative drying period, involving smallest heat transfer (about 103 W.m-2 ).圖（8）表示從Y1測(cè)量和逆程序估計(jì)的表面熱通量密度q（t）。由于沸騰現(xiàn)象，在非常短的時(shí)間內(nèi)，由于沸騰現(xiàn)象，高值約為106 W.m-2。 當(dāng)沸騰干燥停止時(shí)，發(fā)生通量減少，隨

54、后是蒸發(fā)干燥期，涉及最小的熱傳遞（約103W·m-2）。The main objective of thin-layer drying is to dry most of product with boiling drying. The transient period between boiling and evaporation drying is more or less defined. When initial temperature T0 is near to the product's boiling temperature, this transient pe

55、riod is large.薄層干燥的主要目的是干燥大多數(shù)產(chǎn)品。沸騰和蒸發(fā)干燥之間的瞬態(tài)周期或多或少地被定義。 當(dāng)初始溫度T0接近產(chǎn)品的沸騰溫度時(shí)，此瞬態(tài)時(shí)間大。Figure 8 Estimated superficial heat flux density q(t)圖8 - 估計(jì)的表面熱通量密度q（t）Assuming the classical statistical description of temperature measurement errors (Beck, 1977), it is possible to estimate the corresponding standar

56、d deviations. For the present experiments,the standard deviation T for temperature measurements is found to be about 0.07 °C. Estimated heat flux standard deviation q is plotted on fig. (9) as a function of the future time steps number r. The decay is due to the stabilization effect of growing

57、r (see on fig. 6).假設(shè)溫度測(cè)量誤差的經(jīng)典統(tǒng)計(jì)描述（Beck，1977），可以估計(jì)相應(yīng)的標(biāo)準(zhǔn)偏差。對(duì)于本實(shí)驗(yàn)，發(fā)現(xiàn)溫度測(cè)量的標(biāo)準(zhǔn)偏差T為約0.07。 估計(jì)的熱通量標(biāo)準(zhǔn)差q繪制在圖（9）中。作為未來(lái)時(shí)間步數(shù)r的函數(shù)。衰變是由于生長(zhǎng)r的穩(wěn)定作用（見(jiàn)圖6）。Figure 9 Estimated heat flux standard deviation q as a function of the future time steps number r.圖9 - 作為未來(lái)時(shí)間步長(zhǎng)數(shù)r的函數(shù)的估計(jì)熱通量標(biāo)準(zhǔn)差q。6. Energy balance and drying curves能量平衡

58、和干燥曲線Superficial heat flux knowledge can be used to calculate the energy density E lost through the surface :表面熱通量知識(shí)可用于計(jì)算通過(guò)表面損失的能量密度E：The global energy density lost by the copper plate after a new quasi-permanent state is reached (when front side and back side temperatures are equal to Tf ) is :在達(dá)到新的準(zhǔn)永久性狀態(tài)之后，銅板損失的整體能量密度（當(dāng)前側(cè)和后側(cè)溫度等于Tf時(shí)）為：E is found to be quite close to