Comparative Analysis of Heat Transfer in a Counter-Current Moving Bed

Transcription

1 Comparative Analysis of Heat Transfer in a Counter-Current Moving Bed Karinna Saxby, Trevor D. Hadley *, José Orellana, Yuhua Pan and Kok-Seng Lim CSIRO Process Science and Engineering Bayview Avenue, Clayton, 3168, Victoria, Australia * trevor.hadley@csiro.au Abstract There is a strong drive to reduce the carbon footprint of many processes. Mining and metallurgical processes are energy intensive and there is opportunity to improve their overall energy efficiency. One such way is via waste heat recovery from hot solids or gases. CSIRO is currently investigating the recovery of waste heat from the second stage of the Dry Slag Granulation process by countercurrent direct-contact cooling of the slag granules with air in a moving bed. One of the factors hindering the design and scale up of a counter-current moving bed, for the cooling of slag granules, is the uncertainty in the degree of heat transfer from the solids to the air at the granular level. Therefore an appropriate model to determine and predict air temperature variation along the length of the moving bed was investigated. Keywords- Moving bed; Heat transfer; Heat recovery; Heat exchange I. INTRODUCTION In the steel manufacturing process, for every 1 tonne of iron produced, approximately 0.3 tonnes of molten hot slag is generated (and again a further 0.1 tonnes downstream). Upon separation from the iron, the hot slag is still upwards of 1500 C. Currently the slag is either being sent straight to landfill or it is cooled and granulated with water for use in the cement industry, in a process called wet slag granulation. The wet slag granulation process is however environmentally unfriendly. Not only are large amounts of fresh water required, but hydrogen sulphide emissions are produced and the quenched slag-water mixture then requires further drying (more energy input) before it can be used in cement production. Additionally this system does not recover any of the heat from the hot slag. As Australia is extremely conscious of the need to conserve water, and in order to recover the heat from the hot slag, CSIRO has developed a novel dry slag granulation process, where the slag is granulated by using a spinning disc and cooled via direct contact with air in a two stage heat recovery process. In the first heat recovery stage the molten slag is atomised under centrifugal forces exerted by the spinning disc. The hot slag is simultaneously quenched and solidified using air to recover the heat, thereby cooling the slag below 900 C [1]. In the secondary heat recovery stage, the hot solids are cooled via heat exchange with air to around 100 C. It s (a) (b) Figure 1. (a) Direct counter-current heat exchange principle, and (b) heat transfer mechanisms for direct counter-contact heat exchange in a moving bed

2 proposed to conduct the secondary heat recovery in a countercurrent moving bed heat-exchanger, using air as the coolant. Direct contact of the coolant with the hot solids (instead of indirect contact using coolant gas or liquid in tubes) was chosen for a number of reasons, including: It is of a simpler design with no internals, and; effective heat transfer through intimate contact with the air and solids can be achieved. The moving bed process is similar to contacting the hot slag counter-currently with a gas in a packed bed. However, utilising a moving bed approach permits continuous instead of batch operation. Due to difficulties with solids handling, heat exchange in a moving bed reactor is seldom utilised in industry, yet continuous operation can provide higher temperature driving forces than other packed bed configurations [2] and this investigation in a moving bed experimental rig will attempt to affirm its efficacy. One of the factors hindering the design and scale up of a counter-current moving bed, for the cooling of slag granules, is the uncertainty in the degree of heat transfer from the solids to the air at the granular level. Therefore an appropriate model to determine and predict air temperature variation along the length of the moving bed is required. This paper describes an investigation into appropriate heat transfer models and convective heat transfer correlations by using experimental data for validation. II. HEAT TRANSFER MODELS The heat transfer models presented here are derived from energy balances, where a primary assumption is that all of the heat lost from the solids (see Fig. 1) is gained by the air as follows: Q s = - Q g ṁ s Cp s ΔT s = ṁ g Cp g ΔT g ΔT g = T g T g1 T g = T g1 + ṁ s Cp s ΔT s /(ṁ g Cp g ) (1) However, these models do not account for heat losses. An additional constraint for consideration is the interfacial heat transfer from solids to gas according to: Q = haδt lm (2) where: A is the generic heat transfer area available ΔT lm is the log mean temperature difference between the solids and gas, and h is the convective gas-solid heat transfer coefficient. A number of heat transfer models have been sourced from literature and are summarised in Tab. I. These models aim to predict the air temperature variation along the length of the moving bed. By analysing their differences and similarities, we aim to provide a better basis for future experimental work and modelling. III. CONVECTIVE HEAT TRANSFER COEFFICIENTS The convective heat transfer coefficient, h, appears in each of the heat transfer models and dictates the amount of heat being transferred from the solid to the fluid per unit time and unit temperature difference across a given unit area. Thus, the convective heat transfer rate is dependent on a number of factors, namely, the temperature driving force, the available heat transfer area, and the fluid and gas properties. One of the ways in which the convective heat transfer coefficient can be evaluated is from the dimensionless Nusselt Number, Nu, where: Nu = hd p /k g (3) As Nu is empirically determined, there is a host of correlations published in literature. Many of these are expressed as functions of the dimensionless Reynolds (Re) and Prandtl (Pr) Numbers, where: Re = ρ g u g d p /µ g (4) Pr = Cp g µ g /k g. (5) A number of convective heat transfer correlations were sourced from literature. Tab. II contains a refined set of correlations which was chosen based on their applicability to either the moving bed experimental rig, or for application to future scaling possibilities. IV. EXPERIMENTAL SET-UP Fig. 2 contains the process flow diagram of the CSIRO experimental rig (total height of 4 m) used to investigate direct contact counter-current cooling of slag granules with air. TABLE I. HEAT TRANSFER MODELS TO OBTAIN THE GAS TEMPERATURE ALONG THE LENGTH OF A MOVING BED Model Equation and comments Fogler [5] T g = T s (T s2-t g1) exp[-haz/(ρ gµ gcp g)] (6) a is broadly defined as the heat exchange area per unit volume of the reactor. Derived assuming that T s is known and constant over the height, z, examined. Incropera et al [3] T g = T s (T s2-t g1) exp[-h(1-ε)/(ρ gµ gcp g)] (7) Model derived for packed beds, with variation used to approximate air outlet temperature from packed bed of uniform temperature, T s Contains no term for bed height - may be difficult to predict future scale up. Rhodes [6] T g = T s + (T g1-t s2) exp[-haz/(ρ gµ gcp g)] (8) Analogous to the Fogler [4] Model a, but applied to a fluidised bed The specific heat transfer area, a, is defined as 6(1-ϵ)/d p Pan [4] T g = C 0 + (1/C 2 KC) exp (C 2 z + C 1) (9) Constants, gas and solid properties evaluated at average temperature along the bed height: C = 1 [K/m] K = G scp s / ah C 0 = T s2 1/C 2 exp(c 1) C 1 = ln [C 2(T s1 - T s2)/(exp(c 2 H) - 1) ] C 2 = ah [1/G scp s 1/G gcp g] a.this model is also presented in Geankoplis [24] and Wakao & Kageui [16]

3 TABLE II. REFINED SET OF CONVECTIVE HEAT TRANSFER CORRELATIONS Correlation Equation and comments Gupta & Thodos Nu = (2.06 / ε) Re Pr (10) [15] Used fixed beds of spherical particles. Model can be extended to different shapes Conditions for use are 0.6 < Pr < 60 and 90 < Re < 4,000 Kramers [8] Nu = Pr Pr 0.31 Re 0.15 (11) Determined empirically using steel spheres (d p cm) in air, water and oil Conditions for use are 0.7 < Pr < 400 and Re < 10 6 Kuwahara et al [11] Wakao & Kaguei [16] Littman & Sliva [9] Bird et al. [14] Autorenkollektiv [18] Ranz & Marshall [10] Nu = [ (1-ε) / ε ] (1- ε) 0.5 Re 0.6 Pr (12) Correlation based on numerical modelling and simulation using CONVEX software Conditions for use are 0.20 ϵ 0.90 and Re Nu = Re 0.6 Pr (13) Based on data from steady and non-steady state conditions and widely used for forced convection in a packed bed 0.89 Re < Re < 13 Nu = 1.75 Re 0.49 Pr < Re < 180 (14) 1.03 Re 0.59 Pr Re > 180 Validated using packed beds of glass and steel spheres [22] Specific surface area a = 6(1-ϵ)/d p used for modelling in fixed beds [22] hd p/(k(1-ε)φ) = 2.19(Re Pr) Re < 13 (15) Nu = 1.10 Re Pr / (Re ) Re > 13 Correlations proposed for moderate to high Re for particles in packed beds φ is a shape correction factor, and the specific heat transfer area is a = 6 (1-ϵ)/d p Nu = Re / ε [Re 0.35 ( )] -1 (16) Variation of Gupta & Thodos [14] Model, and an average of 30 correlations Condition for use is Re > 60 Nu = Re 0.5 Pr (17) Correlation is for particle-fluid heat transfer in a multi-particle system Condition for use is Re > 50 Kuni & Levenspiel [12] Nu = 0.03 Re < Re < 100 (18) 2 + ( ) Re 0.5 Pr Re > 100 Correlations based on a fluidised bed in plug flow Whitaker [19] Nu = (0.5 Re Re ) Pr (19) Correlation developed for packed beds and suitable for a wide range of packings Condition for use is 10 < Re < 10 4, where Re = ρ g u g d p/(μ g(1-ϵ)) Baskakov et al. [7] Nu = 0.009Ar 0.5 Pr (20) Correlation developed for particles in a fluidised bed with 0.16 mm < d p < 4 mm Baskakov et al. Nu = 0.4 (Re / ε ) Pr (21) [22] Correlation developed for gas to solid heat transfer in a fluidised bed Yoshida et al [17] Nu = ε Pr Re (22) Correlation developed for glass spherical particles, ϵ = 0.36 and d p 1 mm Appropriate for relatively low Re with 0 Re 35 McAdams [20] Nu = 0.37 Re 0.6 (23) Correlation useful for high gas flow rates, 17 Re 70,000 Pfeffer [21] Nu = 1.2 [1-(1-ε) /W] (d pcp gg g/k g) (24) Correlation based on forced convective flow in a packed bed, applicable to high Peclet (Pe = Re Pr) numbers and low Re Wp = 2-3γ + 5 γ 5-2 γ 6, where γ = (1-ε) Kunii & Suzuki [13] Nu = [ϕ s (1-φ)/6/(1-ε)] [d pcp gg g/k g] (25) φ = (T g1 - T s2)/(t g2 - T s1), and specific heat transfer area is defined as a = 6(1-ϵ)/d p Slag granules are gravity-fed from a hopper into a heated section. They are heated to the required temperature (maximum of 750 C) via an externally-heated furnace, where the temperature is maintained via feedback control. The heated solids pass into the heat exchange zone where the solids are cooled with counter-current flow of cold air. The cooled solids are collected below in the product pot. The air flow rate is regulated using a mass flow controller and the flow of solids is controlled by an extraction screw located just above the product pot. The temperature profile in the heat exchange section is monitored using a number of thermocouples, as shown in Fig. 3. It should be noted that the thermocouples were located in the bed of granular material and therefore measured a combination of the solid and gas temperature at that particular point in the bed. The two exceptions were the location of the air inlet thermocouple (inside the air distributor, at the bottom of the heat exchange zone) and the location of the air outlet thermocouple (at the interface between the granular material and surrounding gas at the top of the heat exchange zone).

4 Figure 2. Simplified block flow diagram of moving bed experimental rig Figure 3. Diagram showing thermocouple placements (red) in the heat exchange zone (highlighted) The experimental flow ranges were 2-7 kg/h for air and 5-31 kg/h for slag granules. Slag granules were sieved to mm with d 50 of 1.5 mm and had a bulk density of 1,695 kg/m 3 and particle density of 3,100 kg/m 3. A thermal capacity of 830 J/kg.K and thermal conductivity of 1.55 W/m.K was used for the slag granules. V. NUMBERICAL METHOD The heat transfer models, summarised in Tab. I, were applied to experimental data in order to establish the most appropriate models for a moving bed for cooling of slag granules with air. Additionally, variations/combinations in the specific heat transfer area (see Section VII for derivations) and the convective heat transfer coefficient were applied to each heat transfer model. These comparisons were conducted by constructing a template with macros in Microsoft Excel. The inputs to the comparison included experimental data for flow rates, steady-state temperature profiles along the heat exchange zone and air inlet and outlet temperatures. It should be noted that the measured bed temperature, was actually a combination of the solids and gas temperature. However, for ease of modelling, with the exception of the Pan Model, this measured temperature was taken to be the temperature of the solids. Utilising the measured steady-state temperature profile for each experiment, the gas temperature at different heights along the bed was predicted for the Fogler, Rhodes, Incropera et al., and Pan Models. These were solved vertically, travelling up the column, with the measured gas inlet temperature and solid outlet temperature applied (at z = 0 m) as boundary conditions. The incremental height was defined as the distance between the thermocouples (i.e. height between measured temperatures) positioned throughout the vertical length of the heat exchange zone of the bed (for this case dz = m). For each increment it was also assumed that the axial heat transfer, interparticle heat transfer as well as heat loss to the environment were negligible. The prediction of the gas temperature at a particular bed height is dependent on the heat capacity, density, thermal conductivity, and viscosity of the gas. Given that these properties were also dependent on temperature, an iterative solution was required. These properties were evaluated at each height increment and then substituted into the Prandtl number and the corresponding convective heat transfer correlation for Nusselt number and h. The resultant values for h, using the various convective heat transfer correlations, summarised in Tab. II, were then incorporated into the Fogler, Rhodes, Incropera et al. and Pan Models. The Fogler, Rhodes and Pan Models were also applied with variations in the specific heat transfer area, a. However, the Incropera et al. Model does not contain this term. In the Pan Model, the temperature of the solids and gas were solved incrementally from the temperature of the bed. Additionally, each of the h correlations were evaluated at each incremental height and then averaged. This averaged h value was then used in the specific heat transfer model. In order to compare each of the convective heat transfer correlations, the results were graphed and tabulated. Furthermore, the deviation of the predicted outlet air temperature from the measured outlet air temperature was used to compare heat transfer model and convective heat transfer correlation combinations. This deviation was defined utilising the following equation:

5 Figure 4. Comparison of predicted and measured gas temperatures from experiment 30214, using the Fogler/Rhodes (2) Models with a = 1/dz. (a) predicted gas temperatures for each convective heat transfer correlation, (b) showing best fit using the Littman & Sliva [9] correlation. %T g2.diff = abs[(t g2 T g2.m )/( T g2 + T g2.m )].200 (26) The convective heat transfer correlation which achieved the lowest deviation for that particular heat transfer model and experiment was deemed the best fit. The most accurate models overall were then determined by summing the deviation from the measured values calculated in each experiment. The lowest summation was therefore designated the best model, across the span of experiments. VI. APPLICATION OF MODELS AND COEFFICIENTS TO EXPERIMENTAL DATA The Incropera [3], Pan [4], and three variations of the Fogler [5] and Rhodes [6] Models were applied to the steadystate temperature profile obtained from 22 experiments, using various correlations for h for each heat transfer model investigated. The Fogler and Rhodes variations were defined (see derivations in Section VII) as follows:

6 Fogler/Rhodes (1) for a specific heat transfer area definition of a = (1-ε)/H, (27) derived from Eqn 32 with dz = H. Fogler/Rhodes (2) for a specific heat transfer area definition of a = 1/dz, (28) with dz as the height difference between thermocouples (75 mm) Fogler/Rhodes (3) for a specific heat transfer area definition of a = 6(1-ε)/d p. (29) Therefore, for four heat transfer models, and (where applicable) varying definitions for h and the specific heat transfer area, a total of 100 combinations were compared for each experimental result. Numerical solutions were evaluated, using Microsoft Excel, for each model starting at the bottom of the heat exchange zone and calculating the gas temperature upwards towards the exit. A comparison was then made between the measured and calculated gas temperatures at the exit, T g2. As an example, Fig. 4 shows the application of the Fogler and Rhodes (2) Model with a = 1/dz using various correlations of h, to the experimental data of experiment For this experiment ṁ g was 3.59 kg/h and ṁ s was 5.96 kg/h, T g1 and T g2 were 55 C and 350 C respectively, and T s1 and T s2 were 388 C and 184 C respectively. The best fit was obtained with the Littman & Sliva [9] correlation, with a deviation of 1.21% from the measured gas outlet temperature. Tab. III summarises the best convective heat transfer correlations for each heat transfer model applied to experiment It also shows the deviation of the predicted and measured outlet gas temperatures from each other. For this experiment, the best model-correlation fit overall was by using the Pan Model and Littman & Sliva correlation, with a specific heat transfer area, a, of 1/dz. Utilising the Fogler/Rhodes (3) Model, with a = 6(1-ε)/d p resulted in an instantaneous solution whereby the predicted gas temperature equaled the measured solids temperature at any given height in the moving bed. Subsequently, the predicted exit gas temperature deviated by 2.52% from the measured value irrespective of the h chosen, and therefore no appropriate correlation for h could be determined. TABLE III. SUMMARY OF BEST CORRELATIONS FOR EACH HEAT TRANSFER MODEL CONSIDERED, FOR EXPERIMENT Model Best Correlation Deviation (%) h (W/m 2.K) Fogler/ Rhodes (1) Kramers [7] Fogler/ Rhodes (2) Littman & Sliva [9] Fogler/ Rhodes (3) N/A 2.15 N/A Incropera Kramers [7] Pan Littman & Sliva [9] The same procedure was followed for all experimental data, with the results given in Tab. IV. The models resulting in a local thermal equilibrium solution (i.e. those with a = 6(1-ε)/d p ) have been omitted. From Tab. IV it can be seen that the calculated h varies between 5 and 280 W/m 2.K and is highly dependent on the convective heat transfer correlation and heat transfer model used; i.e. the model-correlation pairing. VII. DISCUSSION A. Fogler/Rhodes (1) and Incropera Models By fixing a = (1-ε)/H, the Fogler/Rhodes (1) Model achieves the same results as the Incropera Model. This is because the Fogler/Rhodes (1) Model and the Incropera Model are the same when a = (1-ε)/H (see Eqs 6 and 7) at the exit conditions. The Fogler/Rhodes (1) and Incropera Models also approach similar values for the average convective heat transfer coefficient, with all of the values exceeding 100 W/m 2.K, with the exception of the Baskakov et al. correlation (~50 W/m 2.K). The Ranz & Marshall [10] and the Kuwahara et al. [11] correlations are the most appropriate. However, the Ranz & Marshall correlation was obtained for Re > 50 and incorporates a general 2 term, which could potentially be dominating the h. For these reasons, Kuwahara et al. is deemed best. This correlation achieves large values for the convective heat transfer coefficient, averaging at ~240 W/m 2.K, but was shown to be appropriate for a wide range of solids and gas flow rates. B. Pan Model Unlike Fogler/Rhodes (1) and Incropera Models which calculate air temperatures from the known (measured) solid temperatures along the height of the bed, the Pan Model [4] calculates both the air temperature and the solid temperature as functions of the bed height. When fixing a = 1/dz, the Pan Model approaches the outlet air temperature solution within a very small bed height. The Littman & Sliva expression (eq. 14) was found most appropriate, achieving the best fit in two thirds of all conducted experiments. This expression is however dependent only on Re, suggesting that Pr, due to its limited variation range ( ) for an air temperature range of C, potentially has minimal effect on this particular model at low fluid flow rates. C. Fogler/Rhodes (2) and Pan Models For these models, a was defined as 1/dz and both achieved similar values for the convective heat transfer coefficient, typically ranging between 30 and 50 W/m 2.K. Furthermore, the Littman & Sliva correlation appears to be appropriate for use in both models, suggesting that the specific heat transfer area and the correlation choice for h are inter-connected. The Baskakov et al. and Littman & Sliva correlations are the most appropriate over a wide range of flows. These convective heat transfer correlations are described for fluidised beds and packed beds respectively and obtain an average h value of 40 W/m 2.K. Baskakov et al. is independent of Re while Littman & Sliva is purely a function of Re.

7 D. Effect of Gas to Solid Ratio on Modelling Results The results obtained in Tab. IV were further analysed, specifically by increasing the ratio of solids-to-gas flow rate, ṁ s /ṁ g. In order to attain maximum heat transfer, it is desired to maintain a high ratio to increase the gas residence time thereby achieving the highest possible air outlet temperature. However, the results in Tab. IV show that there is no clear bias for a particular convective heat transfer correlation choice based on the ratio of solids-to-gas flow rate. This suggests that, for our experimental data, a range of flow conditions are appropriate for most fitted correlations. E. 2 Factor in Nu Correlations Many Nu correlations incorporate a 2 constant (Eqs 11, 13, 17, 18, 22). For our particular set-up, Re is low ( ) and this constant was shown to dominate the expression on occasion and therefore yield Nu in the order of 2, independent of process conditions. This limitation was also noted by Kramers [8], who compared a wide range of flows and noted an over-estimation of the Nusselt Number for Nu < 10. Hence evaluating the convective heat transfer coefficient using correlations with the 2 constant may not be appropriate for the particular experimental rig. Comparison of the two correlations proposed by Kunii & Levenspiel (Eq. 18) [12] where the 2 term is incorporated for Re > 100 further supports this hypothesis. The 2 term generally stems from fundamental heat transfer and flow calculations pertaining to spheres completely surrounded by a fluid. Thus the equations with the 2 term present are possibly more suitable for fluidised beds than for packed beds. F. Specific Heat Transfer Area Some of the proposed convective heat transfer correlations indicate that they are only appropriate for specific definitions of the available specific heat transfer areas (see Kunii & Suzuki [13]). However, unfortunately few authors elaborate on the definition and application of the associated specific heat transfer area, a, values. The definition of a is nonetheless an important consideration in the heat transfer models. As represented in Tab. IV, increasing the value of a reduces the calculated corresponding h in order to match the experimental results. As the convective heat transfer coefficient is a measure of the amount of heat transfer from the solids to gas over a given area, the h and a terms are linked when modelling. Additionally, using the definition of 6(1- ε)/d p for a leads to an instantaneous solution whereby the predicted gas temperature equaled the measured solids temperature at any given height in the moving bed. Some publications have defined the available heat transfer area as the area of the solids available in the bed itself, such that the particle surface area is equal to: where: A p = (1- ε) A c (30) A c is the superficial cross sectional area of the bed. Incropera et al. avoid defining the specific heat transfer area in their expression by dividing the area of the particles by the area of the channel (superficial cross sectional area) of the bed, A p /A c and thus obtained the expression A p A c = (1- ε). Rhodes and Pan however define the specific heat transfer area in the solid fraction of packed and moving beds to be the surface area of the particles divided by the volume of the particles (assuming spherical particles): a = πd p 2 (1- ε)/(πd p 3 /6) = 6(1- ε)/d p (31) This definition however assumes a complete wetting of the particles by a fluid/gas and therefore obtains a significantly large value for a, particularly with small particle diameters, and hence tends to solve instantaneously with the temperature of the air rapidly converging to the solids temperature in minimal distance. However, the measured outlet gas temperatures suggest that this is not the case, and therefore another definition for the specific surface area must be employed. A definition proposed by Fogler for the specific heat transfer area for a tubular heat exchanger is: a = A/V = A p /V = [(1- ε)πd 2 /4]/[πD 2 dz/4] = (1- ε)/dz and for square and rectangular heat exchangers: a = A/V = A p /V = DW(1- ε)/dwdz = (1- ε)/dz (32) Another definition for a is the superficial area divided by the volume of an increment of the moving bed column, hence a = 1/dz. However, this does not take into account the porosity of the bed [5]. The convective heat transfer correlations proposed by Gupta & Thodos [15], Kramers [8], Kuwahara et al. [11], Wakao & Kaguei [16], Ranz & Marshall [10], and Yoshida et al. [17] all obtained high values for h whereas the correlations by Autorenkollektiv [18], Littman & Sliva [9], Kunii & Levenspiel [12], Kunii & Suzuki [13], Whitaker [19], Baskakov et al. [7], McAdams [20], and Pfeffer [21] achieved lower values. The proposed convective heat transfer correlations hence represent significant deviations in calculated h values. Ultimately with these deviations, an appropriate correlation can be fitted to experimental data based on selection of the associated specific heat transfer area. Hence inspection into an unambiguous definition for a, whether through experimental determination or modelling, should be the next step. G. Prandtl Number Several convective heat transfer correlations also negate the effect of Pr at low Re. However, as the exponential term (to the power of two thirds) contained in Pr is based upon boundary layer theory established for both steady laminar and turbulent flows [14], and as Pr was present in most of the correlations from our refined list, it is suggested to be an important inclusion in the choice of convective heat transfer correlation. Nevertheless, the main influence on the heat transfer modelling however appears to arise from the flow rate of the gas and the particular specific heat transfer area.

8 TABLE IV. COMPARISON OF ALL EXPERIMENTAL DATA. BEST CONVECTIVE HEAT TRANSFER CORRELATIONS ARE SHOWN FOR EACH HEAT TRANSFER MODEL CONSIDERED. Foger/Rhodes (1) Fogler/Rhodes (2) Exp. # ṁ g, ṁ s Best Correlation T g2 % deviation h (W/m 2.K) Best Correlation T g2 % deviation h (W/m 2.K) , 25.8 Kuwahara Baskakov , 6.0 Kramers Littman , 24.5 Gupta Baskakov , 23.1 Ranz Littman , 6.1 Ranz Littman , 6.7 Ranz Littman , 25.4 Gupta Baskakov , 40.3 Kuwahara Baskakov , 6.7 Wakao Baskakov , 34.9 Kuwahara Baskakov , 6.4 Ranz Kunii , 34.4 Kuwahara Baskakov , 6.3 Wakao Littman , 35.5 Kuwahara Baskakov , 6.8 Gupta Baskakov , 30.5 Ranz Littman , 5.8 Baskakov Littman , 48.9 Gupta Baskakov , 11.4 Kuwahara Baskakov , 31.0 Ranz Littman , 5.5 Baskakov Littman , 12.2 Wakao Baskakov Incropera Pan (a = 1/dz) , 25.8 Kuwahara Baskakov , 6.0 Kramers Littman , 24.5 Gupta Littman , 23.1 Ranz Littman , 6.1 Ranz Littman , 6.7 Ranz Littman , 25.4 Gupta Baskakov , 40.3 Kuwahara Baskakov , 6.7 Wakao Whitaker , 34.9 Kuwahara Littman , 6.4 Ranz Whitaker , 34.4 Kuwahara Littman , 6.3 Wakao Whitaker , 35.5 Kuwahara Littman , 6.8 Gupta Baskakov , 30.5 Ranz Littman , 5.8 Baskakov Littman , 48.9 Gupta Littman , 11.4 Kuwahara Whitaker , 31.0 Ranz Littman , 5.5 Baskakov Littman , 12.2 Wakao Littman CONCLUSIONS A number of heat transfer models and convective heat transfer correlations have been obtained from literature and have been prioritised based on likely application to heat recovery via counter-current cooling of slag granules with air in a moving bed. The refined list of models and correlations were applied to experimental data obtained from a moving bed heat exchange rig and were shown to be effective in analysing gas temperature variation along the height of the moving bed. Analysis of a wide range of convective heat transfer correlations also showed that, at low Re, correlations with incorporation of a 2 term may be inappropriate for determining the convective heat transfer coefficient under the conditions of the experimental apparatus. The most promising model-correlation pairing results, for observing the temperature profile, have been obtained from application of the Fogler and Rhodes Models, using the Kuwahara et al. correlation for the convective heat transfer

9 coefficient (h between 210 and 250 W/m 2.K), and a specific heat transfer area definition of a = (1-ε)/H. The correlations for the convective heat transfer coefficient (h between 30 and 50 W/m 2.K) proposed by Littman & Sliva and Baskakov et al. were found to be appropriate when using the Pan and Fogler/Rhodes (2) Models with a specific heat transfer area definition of a = 1/dz. Ultimately, it was found that correct model-correlation pairing is important in fitting experimental data to predictions. The convective heat transfer correlation postulated by Kuwahara et al. in the Fogler/Rhodes (1) Model was shown to achieve the greatest accuracy throughout the comparative study. It is therefore postulated to be the best current foundation for future scale up and affirmation of heat exchange efficacy in a moving bed of this type. It is however suggested that an unambiguous definition for a should be investigated to establish an appropriate model-correlation pairing for application to the dry slag-air system. NOMENCLATURE a surface area per unit volume m 2 /m 3 A generic heat transfer area m 2 Ar Archimedes Number - C 0, C 1, C 2 constants in Pan Model [4] K, -, K -1 Cp heat capacity J/kg.K d p particle diameter m dz incremental height used in evaluation m D diameter of a tubular heat exchanger/ or depth in a rectangular heat exchanger m G mass flux kg/m 2.s h convective heat transfer coefficient W/m 2.K H total height contact in heat exchange zone m k thermal conductivity W/m.K ṁ mass flow rate kg/s Nu Nusselt Number - Pr Prandtl Number - Q heat transfer rate W Re Reynolds Number - Sc Schmidt Number - T temperature K u superficial velocity m/s V volume m 3 W internal width of experimental rig m W p factor used in correlation of Pfeffer [21] - z height from the bottom of the moving bed m Greek γ factor used in correlation of Pfeffer [21] - ε voidage of the packed/moving bed - µ viscosity Pa.s ρ density kg/m 3 φ shape correction factor - ϕ sphericity factor by Kunii - Subscripts 1 conditions at the entry 2 conditions at the exit c g lm m s for unit cell under consideration gas log mean measured solid ACKNOWLEDGMENT The authors would like to acknowledge CSIRO Minerals Down Under (MDU) Flagship for their support of this work. 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