Maritime University of Szczecin
Akademia Morska w Szczecinie
2010, 23(95) pp. 5–9 2010, 23(95) s. 5–9
Application of the similarity theory in model investigations
of a ship’s fluidized bed boiler
Zastosowanie teorii podobieństwa w badaniach modelowych
okrętowego kotła fluidalnego
Andrzej Adamkiewicz1, Wojciech Zeńczak2
1 Maritime University of Szczecin, Faculty of Mechanical Engineering, Institute of Ship Power Plant Operation
Akademia Morska w Szczecinie, Wydział Mechaniczny, Instytut Technicznej Eksploatacji Siłowni Okrętowych 70-500 Szczecin, ul. Wały Chrobrego 1–2, e-mail: a.adamkiewicz@am.szczecin.pl
2 West Pomeranian University of Technology, Faculty of Maritime Technology
Department of Engines and Marine Power Plants
Zachodniopomorski Uniwersytet Technologiczny, Wydział Techniki Morskiej Zakład Silników i Siłowni Okrętowych
71-065 Szczecin, al. Piastów 41, e-mail: wojciech.zenczak@zut.edu.pl Key words: theory of similarity, heat transfer coefficient, fluidized bed, ship boiler Abstract
In model investigation of ship’s fluidised bed boiler the similarity theory has been applied, which permits to transfer the investigation results from the model to the real object. For setting the non- dimensional numbers in similar phenomena, the analysis of differential equations describing the examined phenomenon or the dimensional analysis is applied. The article presents an example of the search for the non-dimensional numbers characterising the phenomenon of heat exchange in ship’s fluidised bed boiler.
Słowa kluczowe: teoria podobieństwa, współczynnik przejmowanie ciepła, warstwa fluidalna, kocioł okrętowy
Abstrakt
W badaniach modelowych okrętowego kotła fluidalnego wykorzystano teorię podobieństwa pozwalającą na przenoszenie wyników badań z modelu na obiekt rzeczywisty. Dla ustalenia modułów bezwymiarowych w zjawiskach podobnych stosuje się analizę równań różniczkowych, opisujących rozpatrywane zjawisko lub też analizę wymiarową. W artykule przedstawiono przykład poszukiwania modułów bezwymiarowych cha-rakteryzujących zjawisko wymiany ciepła w okrętowym kotle fluidalnym.
Introduction
One of the practical issues within the heat exchange that appear in the investigations of the boilers is the determination of the heat transfer coefficient value. This problem can be solved in the analytical manner in many cases, however at times this is likely to prove difficult or impossible to implement without the application for the mathe-matical description of the assumptions that simplify the course of the actual phenomena [1]. In such situations the alternative becomes to conduct the
experiment on the real object or its physical model. The investigations conducted by use of the model are significantly cheaper and less complicated than the same conducted on the real object. They, however, require the compliance with some specific conditions so as to allow to transfer the model investigation results to the calculations of the real object. The conditions include factors such as the appropriate construction of the equipment model – the boiler in this case, knowledge of the limits the results applicability as well as the method of conducting the experiment permitting to take into
consideration the influence of some significant factors. The investigation of the heat exchange in the ship’s fluidised bed boiler are particularly complex due to ship’s rolling on waves affecting the behaviour of the fluidised bed in such condi-tions. In such situation the construction of the physical model of the fluidised bed boiler is especially justified.
For the purpose of dealing with many issues in shipbuilding it is sufficient to consider only the simple motion, i.e. of one degree of freedom. This motion takes place in the ship’s centre line or in the midship section plane. Such approach applies most commonly to the rolling motion which due to its intensity, because of the direct impact on the stabi-lity safety as well as the effectiveness of the opera-tion of some systems and equipment, play a special role [2].
The investigation stand allowing to conduct the examination of the processes occurring in the boilers with the bubbling fluidised bed and in the boilers with circulating fluidised bed in the condi-tions simulating the ship’s rolling has been pre-sented amongst others in the works of the authors [3, 4, 5]. In order to increase the applicability scope of the boiler model another stand version has been executed where the fluidising column construction has undergone some significant modifications.
The construction of the fluidising column
In comparison to the construction of the first column of circular section, referred to in the studies [3, 4], the new one takes the rectangular section corresponding in a larger degree to the actual boiler construction. Moreover, provision has been made for the possibility of placing the heating element (heating probe) at various heights of the column. The heating element may be, similar as in the first version of the stand, miniature cylinder probe made of copper pipe with the electric spiral positioned inside. Alternatively for the examination of the average heat transfer coefficient a ball-shaped probe or heat flow density meter [6] could be applied. The fluidising column has been made of perspex. The heating probe is fixed to the transpa-rent cap which might located in one of the openings of the column side wall. The remaining openings of the column are blanked by use of the transparent caps that guarantee the smoothness of the column internal surface.
The results of the investigation of the average values of heat transfer coefficients in the circulating fluidised beds, as published by other authors point out at its significant dependence on the probe
distance from the grid [7, 8]. The investigations do not take into account the cyclical deviations of the columns from the vertical (swinging motion) which are likely to occur in the operation conditions of the fluidised bed boiler on board the ship. The pro-posed new column construction shall permit among others to conduct the investigation of the average heat transfer coefficient in the circulating fluidised bed in relation to the probe positioning heights over the grid during the cyclical column swinging mo-tion. The arrangement ensuring the column motion has been made on the stand specified in [3]. The diagram of the column is shown in figure 1. The examination of the heat transfer coefficient at the said stand has been done by the use of the stabilised heat flow method [6].
Fig. 1. The diagram of the fluidising column with the flui-dising bed material return system
Rys. 1. Schemat kolumny fluidyzującej z systemem odzyski-wania materiału fluidyzującego
The application of the dimensional analysis for the determination of the similarity modules
The dimensional analysis which is a part of the theory of similarity allows to find non-dimensional numbers of similarity pursuant to the analysis of the
air outlet cyclone a b c d A fluidising air inlet auxiliary air distributor A A – A probe
a, b, d – blanking caps for the openings in column for the alternative positioning of the cap c/w probe
dimensions describing given phenomenon [9]. Its application is particularly recommended, when it is difficult to formulate the differential equations characterising this phenomenon, whereas the para-meters influencing the searched figure are known.
Many factors influence the course of heat ex-change in the fluidised bed. The parameters that heat transfer coefficient is subject to can be divided into three groups. The first group comprises the fluid physical parameters (fluidised bed) exchang-ing the heat with the surface, the second – motion parameters, and the third – geometrical parameters. The major parameters include thus: the fluidising medium flow rate and its heat condition, concentra-tion of the loose material contained within the fluidised bed, its physical properties and the geo-metrical parameters of the fluidising chamber with heat exchanger. As the investigation indicates the value of the transfer coefficient is very strongly influenced by the column swinging motion [4].The fluidising bed temperatures less than 1273 K give the grounds to disregard the radiation effect.
Basing on the observation of the phenomenon, mental analysis and literature data it has been as-sumed that the α heat transfer coefficient depends on 16 independent variables which can be put in the general form as:
w c d c D d H T g l z h
f p,p,p, p,p, , s,s, z, r, , , , , , (1) where: wp, vp, λp, cp, ρp – flow rate, kinematic viscosity, heat transfer coefficient, specific heat capacity and density for air, respectively; d – flui-dised bed material grain diameter; cs, ρp – specific heat capacity and fluidised bed material density, respectively; Dz – column internal section substitute diameter; dr – outer diameter of the heating pipe;H – fluidised bed rest height; T – column motion
period; g – gravitational acceleration; l – length of the heating pipe (probe); z – the height of heating pipe positioning over the air distributor; h – height of the fluidising column.
In case of the channels of non-circular section the substitute (equivalent) diameter Dz determined from the relation [10] is taken as the characteristic linear dimension:
P A
Dz 4 (2)
where: A – channel sectional area, P – the length of perimeter of the transverse section of the channel moistened by the fluid.
The new element adopted for the investigation, resulting from the ship’s rolling, is relating the heat transfer coefficient to the period of the swinging motion of the column T. The amount of the loose material is determined and expressed by the flui-dised bed height as measured at rest H.
In compliance with the Buckingham Π theorem every homogenous equation can be represented by means of the function constructed of n – r non- -dimensional numbers, where r is the number of the basic dimensions and n is the number of the dimen-sioned parameters. In the equation (1) there are four (r = 4) basic dimensions: temperature t, length l, mass m and time . Since in the equation (1) the number of the dimensioned parameters n = 17, then the smallest number of the non-dimensional mo-dules is n – r = 13. According to the principles the dimensional analysis the general relation (1) has been presented in the form of the power function:
j k l m n p q r i z h s g s f e p d p c p b p a p h z l g T H d D c d c w C (3) where: C – constant value.Upon replacing of the individual figures by their basic dimension the following has been obtained:
i j k l
m
n p q h g f e d c b a l l l l l l l ml t l l ml t l t ml l l C t m 2 3 1 2 2 3 1 2 2 1 3 1 2 1 1 3 (4)If the function is homogenous, then the dimen-sions on both sides of the equation mark must be capable of reduction which means that the sum of indices of each dimensioned basic figure must be equal to zero. Thus the system of four equations has been obtained: for m: 1 = c + e + h for τ: –3 = – a – b – 3c – 2d – 2g – 2m + l (5) for t: –1 = – c – d – g for l: 0 = a + 2b + c + 2d – 3e + f + 2g – + 3h + i + j+ k + m + n +p + q
The above equation system can be solved in re-lation for any 12 indices, eg:
c = 1 – e – h
a = – b + e + h – 2m – l (6)
d = e + h – g
f = – b + e + h + m – l – 1 – i – j – k – n – p – q
Eliminating the power indices c, a, d and f, and substituting the remaining to formula (3) the fol-lowing has been obtained:
q p n m l k j r i z h s g s k q p n j i l b h m e e p g p h p e p h p e p p b p m p b p l p h p e p h z l g T H d D c d d d d d d d dd d d d c c c w w w w w C 2 (7)
Upon grouping of the dimensioned parameters of the same power indices and adding multipliers
h p h p n p n p e p e p ,
, , whose value is equal to 1, there has
been obtained: k k g g p p n n j j r i i z g p g s h p h p h p h p h p h h p l l l p m p m m e p e p e p e p e p e e p b b p b p p d H d h d z d l d d d D c c c d w d T w w g d c d w d w C d 2 (8)
By replacing the individual expressions by the numbers known from the theory of similarity, namely Nusselt, Reynold’s, Prandtl, Froude’s num-bers [9], i.e.: p d Nu , p pd w Re , p p p p λ ρ ν c Pr , Fr 2 p w gd (9)
the equation has been obtained consisting of 13 non-dimensional modules: q p n k j r i z h p s g p s l p m h e h e b d h d z d l d H d d d D c c d T w C Fr Pr Re Nu (10)
The Froude’s number plays a significant role in the natural convection caused by the gravity. In the forced flow which occurs in the examined case, it is practically of no major importance, therefore it can be skipped [9]. If the experiment is conducted in the same thermal conditions, it applies to the same fluidised bed material, and the only items under-going any significant changes are the air rate, flui-dised bed height, period of the column swinging motion and dimensions of heater and its height over the distributor, then the relation (10) can be reduced further to the following form:
F E r D z B p A h z d l D H d T w C Re Nu 1 (11)
In the experiment conducted with the immov-able column always with the same heater in the
equation (11) there can be disregarded the non- -dimensional modules related with the period of swinging motion and heater dimensions (T = idem,
l = idem, dr = idem). In the effect the further reduc-tion is obtainable down to:
1 1 1 Re Nu 2 F D z A h z D H C (12)
And correspondingly in the investigations con-ducted during the column motion with the same heater, but with the constant fluidised bed height at rest (H = idem), the equation (11) gets the fol-lowing form: 2 1 2 Re Nu 3 F B p A h z d T w C (13) where C1, C2, C3 – constant.
The simplexes of the geometrical dimensions (Dz / d) and (H / d) – occurring in the equation (10) have been replaced in the equations (11) and (12) by the simplex (H / Dz). And similarly the sim-plexes of the geometrical dimensions (z / d) and (h / d) have been replaced by the simplex (z / h), whereas the simplexes (dr / d) and (l / d) – by the simplex (l / dr).
The numerical values of the constant C1, C2, C3 and the indices, in the equations (11), (12) and (13) depending on the adopted investigation programme, can be determined on the basis of the experimental data by means of an adequate software for analysis, e.g. Statistica.
Summary
Apart from the previously conducted investiga-tions of the fluidised combustion technology in the special conditions on board sea-going ships, owing to their motion on waves, on the prototype stand, referred to inter alia in [4, 5], the planned investiga-tions shall contribute to the further development of the knowledge in this field. The proposed general relations obtained on the basis of the theory of simi-larity shall allow – upon the completion of the series of measurements – to determine the heat transfer coefficient in the bubbling or circulating fluidised bed for the rectangular-section columns. The provision is made also for the quantitative determination of the distribution of the average heat transfer coefficient on the surface of the heating element depending on its height over the air dis-tributor. The novelty implemented in the investiga-tions is considering the heat transfer coefficient subject to the average period of column swinging motion.
In the further perspective the investigation shall be focussed chiefly on the circulating fluidised beds. The plans are made for the experiments with heating probes of different shapes. It would neces-sitate finding of other similarity modules adequate for the probe shapes.
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The study financed from the means for the education within 2009–2012 as own research project No. N N509 404536. Recenzent: dr hab. inż. Marek Dzida, prof. PG Politechnika Gdańska
