Heat Transfer to Copper Coolers in Freeze Lined Furnaces: The Role of Radiation and the Influence of Slag Liquidus

Abstract

Prediction of the thermal duties of copper coolers in freeze lined furnaces is challenging. Where copper coolers are immersed in liquid slag the heat transfer is often defined by a natural convection cell immediately adjacent to the wall, in the absence of other factors producing forced convection. Slag coolers are typically not fully immersed, slag temperature , liquidus and superheat can vary, and bath levels may change significantly with batch tapping, leading to substantial variations in radiation heat fluxes. Radiation heat transfer is discussed with reference to the non-obvious impact of slag liquidus on radiation losses in the exposed cooler sections. Reference is made to literature and the results of analytical and FEM models.

Introduction

Slag and refractory are so chemically similar, that it is difficult to arrest the long-term erosion of refractory without a coating of frozen slag on the refractory hot-face, as shown in Fig. 1 [1]; hence, forced liquid cooled structures are increasingly common in modern furnace designs, even ones of modest intensity, e.g. slag cleaning furnaces. In many cases, copper cooling elements have become a default solution today, due to their ability to improve campaign life, robustness in a thermal crisis and other advantages, such as immunity to refractory chemical hydration, i.e. MgO + H₂O = Mg(OH)₂.

../images/468727_1_En_51_Chapter/468727_1_En_51_Fig1_HTML.gif — Fig. 1
[1]. (Left) Advanced copper cooler with cast-in-pipes and a patterned hot-face. Freeze lining and ‘natural’ convective slag flow zone at the hot-face shown. (Right) Photograph of a cooler with a hot-face pattern [20]

The operating principles for the immersed portions of cooling elements have been established by industry, including:

(a)
careful control of slag superheat to prevent excessive sidewall heat fluxes, and
(b)
the minimization of forced convection or direct impingement of hot slag on a wall, e.g. by correct furnace geometry and use of adequate wall feeding or ‘dressing’.

However, the contribution of radiation on the exposed portion of the cooling elements to overall cooler heat losses has not been well established, and based on the experience of the authors, can become a significant factor in some circumstances.

In most cases, radiation to cooling elements is not considered significant or is assumed to be of the same order of magnitude as the convective heat flux ; therefore, the ‘design’ convective heat flux is often applied over the full area when determining the thermal load of the system, e.g. to select the required coolant flow rate or to determine the expected cooling system duty. Furnace designers often lack operational data regarding the role of radiation , but often possess anecdotal evidence, e.g. minimal heat flux variation before/after tapping. Furnaces with dusty atmospheres [2], solid feed/reductant present on the surface of the bath (so-called ‘black top’ operation), good wall dressing or low slag superheat will tend to show little variation in sidewall heat flux with changes in exposed cooler area. For other cases, particularly for furnaces operating with an open-bath , radiation can be an important factor in determining overall duty on cooling elements. In addition, variations in bath level, chemical composition (liquidus ), operational control (electrode position/energy intensity/temperature /superheat), etc., can lead to highly varying radiative fluxes on exposed portions of cooling elements. This paper will explore the role of both absolute temperature and superheat with regard to their impact on radiative heat flux to exposed areas of copper coolers.

Cooling Methods and Key Equations

A variety of cooling methods are applied in order to develop a freeze lining, e.g. from simple systems using high thermal conductivity refractory and external shell water cooling [3, 4], to high-intensity deep water cooled copper elements [5] such as the one shown in Fig. 1. Superheat and convective stirring both tend to rise with increases in process intensity [6] leading to higher overall cooling duty and a need for more intense cooling . Different cooling methods suitable for different duties are listed in Table 1 [6] for reference.

Table 1

[6]: Typical operational heat fluxes and system design limits for various furnace sidewall cooling technologies

Cooling technology	Typical kW/m²	Design kW/m²	References
Natural air convection	1	4.5	[3]
Forced air	2	6	[3]
Ripple cooling	3	10	[3]
Spray cooling	5	20	[3, 4]
Vertical chamber (atm)	5	20	[3]
Horizontal channels (2–4 bar)	10	40	[3]
Vertical channels (2–4 bar)	20	80	[3]
Finger coolers	15	40	[5]
Copper plates	20	100	[3, 5]
Vertical copper coolers with a hot-face pattern	30	>1000	[5]

The heat flux through the immersed section of the liquid cooled copper elements can be estimated by Eq. (1) and can be equated to the heat flux through the freeze thickness at steady state. Assuming superheats of 50, 150 and 250 °C, and a heat transfer coefficient of 250 W/m² · K, yields immersed heat fluxes of 12.5, 37.5 and 62.5 kW/m² from Eq. (1). Assuming the freeze lining to be the limiting resistance with a thermal conductivity, k_{freeze-lining} of 2 W/m·K, and a thermal driving force of 1400 °C, equilibrium freeze thicknesses, d_{freeze-lining} of: 224, 75 and 45 mm can be found using a simplified version of Eq. (2) considering only the freeze lining resistance for the previously indicated fluxes. Equations (1) and (2) can therefore be used to estimate steady state freeze lining thicknesses and thermal boundary conditions for the immersed sections of the coolers for use with Finite Element Modelling (FEM).

$\frac{{Q_{c} }}{{A_{c} }} = h_{slag - wall} \left( {T_{bath} - T_{hotface} } \right)$

(1)

where Q_c is the heat flow [W] and A_c is the cooled element area [m²], h_slag-wall is typically the natural convection heat transfer coefficient between the molten slag and the hot-face of the freeze lining [W/m²·K], T_bath is the temperature of the bulk bath material arriving into the natural convection zone near the wall [K] and T_hotface is the hot-face temperature of the furnace lining in front of the cooler [K]. The hot-face temperature has often been assumed to be the liquidus , although it has been suggested to take an average of the liquidus and solidus [7], which has subsequently been shown experimentally to be closer to the true physical situation [8]. The traditional assumption is used in this analysis, for reasons which will become apparent in the discussion section.

$d_{freeze - lining} = k_{freeze - lining} \left[ {\frac{{(T_{hotface} - T_{coolant} )}}{{\frac{{Q_{c} }}{{A_{c} }}}} - \frac{{d_{refractory} }}{{k_{refractory} }} - \frac{{d_{copper} }}{{k_{copper} }} - \frac{1}{h}} \right]$

(2)

For simplicity, a constant and generally conservative h_slag-wall is assumed with a magnitude of 250 in this article; however, it is acknowledged that h_slag-wall can have values ranging from <100 in low intensity smelting or for highly viscous slag , to >500 in situations where forced convection is encountered, e.g. in or near tap holes and adjacent to electrodes or at extremely high superheat (low viscosity ). Again it is noted that h_slag-wall tends to increase somewhat with process intensity/superheat [6], but this effect has been ignored in the following analysis to emphasize the impact of liquidus and superheat on the freeboard fluxes.

Radiative Heat Transfer Theory

A simplified theoretical discussion will be given here to help clarify the FEM assumptions and chosen boundary conditions, and improve the understanding of the model results to follow. The general equation for radiation between surfaces is shown as Eq. (3) [9]:

$Q_{1 - 2} = \frac{{\sigma \left( {T_{1}^{4} - T_{2}^{4} } \right)}}{{\frac{{1 - \varepsilon_{1} }}{{\varepsilon_{1} A_{1} }} + \frac{1}{{A_{1} F_{1 - 2} }} + \frac{{1 - \varepsilon_{2} }}{{\varepsilon_{2} A_{2} }}}}$

(3)

where Q_1-2 is the radiation flow from surface 1 to surface 2 [W], σ is the Stefan-Boltzmann constant 5.670367 × 10⁻⁸ [W/m² · K⁴], T_i is the absolute temperature of surface i [K], ε_i is the emissivity of surface i, A_i is the surface area of surface i [m²], and F_1-2 is the view factor between surface 1 and 2, i.e. the fraction of the radiation from surface 1 arriving at surface 2.

Radiation is a surface phenomenon; hence, small changes in surface properties can have a significant impact on radiation heat transfer behavior. In general, it is assumed that all surfaces in a slag containing furnace will become coated with slag , accretions of feed materials and/or condensates with slag -like properties and that the radiative properties will therefore tend towards those of slag . Slag is assumed to be a ‘grey body’ radiator, i.e. possessing a frequency independent emissivity, ε, of less than 1.0, and of the order of 0.7–0.9 according to the authors’ experiences. A ‘typical’ temperature independent value of 0.8 has therefore been selected, which varies slightly from the 0.7 recommended by others [2]. It is further noted that the true emissivity may decrease with increasing temperature , as it does for refractories [10] and many other materials [11]. A sensitivity analysis has been performed to explore the impact of various emissivity values on the modelling results.

If an idealized geometry is assumed, e.g. 2 parallel grey-body infinite plates, the view factor F_1-2 is equal to 1, A₁ = A₂ and Eq. (3) can be simplified to [9]:

$\frac{{Q_{1 - 2} }}{A} = \frac{{\sigma \left( {T_{1}^{4} - T_{2}^{4} } \right)}}{{\frac{1}{{\varepsilon_{1} }} + \frac{1}{{\varepsilon_{2} }} - 1}}$

(4)

If surfaces 1 and 2 are both assumed to have emissivities of 0.8, and surface 1 is assumed to be the radiating surface of the slag , and surface 2 the coldest heat sink in the system, e.g. the hot-face of the cooled copper elements, then Eq. (4) can be solved to give order of magnitude estimates of the heat fluxes for different combinations of slag and cooler hot-face temperature as shown in Fig. 2. Also shown in Fig. 2 are the slag bath heat fluxes for the natural convection region, using the same assumed slag and hot-face temperatures. The theoretical estimates shown in Fig. 2, indicate that radiation fluxes could greatly exceed convective fluxes, particularly for large temperature differences between the exposed portions of the cooling elements and the surface of the bath.

../images/468727_1_En_51_Chapter/468727_1_En_51_Fig2_HTML.gif — Fig. 2
Radiation heat flux versus slag bath and hot-face temperature in degrees Celsius for both immersed and exposed cooler sections. Equal areas and an ε of 0.8 for bath and wall with a view factor of 1 assumed for discussion purposes only, see Eq. (4)

Figure 2 indicates that: (1) some cold solid material on the slag surface, (2) a very high degree of radiation attenuation in the gas freeboard, or (3) a very low effective view factor would be required to have similar magnitudes between exposed and immersed cooler sections in an open-bath furnace . Radiation attenuation has been ignored in the FEM results to follow, view factors and the role of indirect radiation from the furnace freeboard have been estimated internally by the commercial software package applied (COMSOL^®). However, the impact of both attenuation, freeboard geometry and view factor are discussed in a treatment of radiation theory in open-bath furnaces, which also contains interesting pilot/demonstration scale data and numerical modelling results [2]. The reader is also directed to a related paper on scale-up of such furnaces [12].

Finite Element Modelling

It is assumed that the following thermal analysis applies at least qualitatively to any open-bath slag containing furnace , having either an arc or immersed electrodes. The majority of the power of an arc is present in the thermal energy of the plasma [13], and this energy is directed down upon the slag surface, which becomes the primary radiation source even for arc furnaces [14]. In addition, it has been shown that the freeboard radiation does not vary over the short term with arc-on or arc-off [2], supporting the assumption that the slag bath is the primary radiator for open-bath slag containing arc furnaces.

A furnace with an open-bath and substantially ‘transparent’ freeboard, e.g. a slag cleaning furnace , is considered in the present analysis. In the freeboard of such furnace , the heat transfer is dominated by radiation , and convective heat transfer has therefore been neglected in the present model . No participating media have been assumed, either gas or dust. Typically the diatomic gases present in most electric furnaces can be ignored in terms of their impacts on radiation , while dust may have an impact depending on size, physical properties and ‘loading’ [2]. Due to the presence of dust and condensates on the freeboard surfaces, they are assumed to be ‘diffuse’, which means that the angle of reflected radiation is independent of the angle of incidence of the incoming radiation [2] and have identical emissivities, which are assumed to be 0.8 for the base case.

It has further been assumed that the primary radiator, i.e. the source of the net energy input to the freeboard, is the surface of the slag bath [2, 12]. The slag bath has been assumed to be free of feed and well mixed, having a single characteristic temperature both across the surface and vertically. A homogeneous horizontal temperature is an approximation, given that hot zones are expected near any immersed electrode [15] or an arc attachment spot in arc-mode [16], and cooler natural convection zones will exist near immersed cooling elements [17]. The assumption of limited vertical variation is supported by previous measurements by the authors and limited information available in the literature [18]. The metal layer has been assumed to be liquid iron with a representative thermal conductivity of 35 W/m² · K [19]. Convection within both slag and metal have been neglected in the model formulation; however, it is the presence of good mixing in the bulk slag , that minimizes the vertical and horizontal temperature gradients, i.e. the greater the magnitude of the mixing of the slag , the better the quality of the model assumptions of uniform temperature . Convective flows in the slag phase typically do not result in mixing of the metal phase, due to the large surface tension existing between these immiscible phases. No assumptions have been made regarding the metal temperature distribution , which are the result of the calculated fluxes through hearth and wall, and the assumption of stagnant metal.

A simplified geometry has been assumed, having a flat roof and hearth, and single layers of refractory on all insulated surfaces. The number and location of electrodes have been ignored to give a fully ‘open-bath ’ and preserve axial symmetry. The presence of steel cladding on the outer surface has been ignored, as having negligible heat transfer resistance. The resulting geometry, shown in Fig. 3, has been solved using COMSOL^® version 5.1 using 2-D axial symmetric modelling. 8 layers of mesh have been applied at all boundaries to more precisely estimate steep gradients, and 21373 elements have been used in total. The densest mesh was in the zone of the copper coolers and the cooling channels. The impact of much tighter meshing was tested and found to produce no significant effect on the results, other than a much longer computational time.

../images/468727_1_En_51_Chapter/468727_1_En_51_Fig3_HTML.gif — Fig. 3
Geometry and temperature distribution for 1700 °C slag bulk temperature , 1450 °C slag liquidus (d_{freeze-lining} = 45 mm), 1450 °C exposed cooler hot face temperature (17 mm of slag coating), average immersed flux of 62.5 kW/m² and exposed radiative flux of 161.9 kW/m²

Simple temperature independent thermal conductivities have been assumed for the model work. k_slag, k_refractory and k_{freeze-lining} are all assumed to be 2 W/m² · K, while the hearth has been assumed to have a weighted average thermal conductivity of 5. The outer surface of the roof and upper walls are assumed water-cooled, and along with the surface of the cooling element channels, are assumed to be at 35 °C. The air-cooled outer surface of the copper cooling elements and the insulated lower vertical wall and side of the hearth are assumed to have an effective combined radiation +convection heat transfer coefficient of 40 W/m² · K, while the lower side of the hearth has been assumed to have 25 W/m² · K. Key model assumptions and boundary conditions are summarized in the Appendix.

It should be noted that in reality, frozen slag in the freeboard may have a different liquidus than the current slag in the bulk slag pool. In the current analysis, it is assumed that the slag freeze lining has the same liquidus in both exposed and immersed sections.

The exact dimensions of the geometry are not important for the discussion of principles. Modelling results should be substantially independent of scale, provided geometrical similarity is preserved between model and actual furnace design [2].

Results and Discussion

A key premise in the present FEM analysis, is that the hot-face of the frozen slag coating in either the freeboard or immersed zones cannot have a surface temperature exceeding the slag ’s liquidus at steady state. In the immersed sections, thin freeze linings produced by periodic temperature excursions can be ‘healed’ by the presence of a large volume of slag , once superheat decreases. In the freeboard, no such reservoir of slag exists. If excessive radiation causes the temperature of the hot-face to exceed the liquidus , the freeze lining on the exposed cooler will melt and liquid slag will flow into the bulk slag pool below. At high temperatures, this will result in thin coatings in exposed areas for slag linings having a low liquidus . The resulting temperatures and heat fluxes have been shown in Fig. 3 for 250 °C of superheat, indicating an equilibrium flux 2.6 times higher and an equilibrium freeze thickness thinner by a factor of 2.6 in the radiation zone compared to the immersed zone of the cooler .

A COMSOL^® model has been solved for a number of different slag superheats and for a slag liquidus of 1450 °C, and the results are summarized in Fig. 4. Results in Fig. 4 indicate that the equilibrium freeze thickness decreases quickly at higher superheats, while the heat fluxes increase dramatically in the exposed cooler areas. The equilibrium freeze thickness in the freeboard section was found to vary inversely to superheat to the power of 1.7, with an R² approaching 1. Results for the immersed and exposed portions of the coolers are compared in Fig. 5, showing much higher heat fluxes in the exposed portions, which is substantially in agreement with the simplified theoretical results presented in Fig. 2 at high superheat.

../images/468727_1_En_51_Chapter/468727_1_En_51_Fig4_HTML.gif — Fig. 4
Average hot-face temperatures and radiation heat flux as a function of instantaneous freeze thickness in the freeboard, for slag liquidus of 1450 °C and superheats from 50–250 °C

../images/468727_1_En_51_Chapter/468727_1_En_51_Fig5_HTML.gif — Fig. 5
Equilibrium freeze thickness and average heat fluxes for both immersed and exposed cooler sections at steady state for k_{freeze-lining} of 2 W/m · K, h_slag-wall of 250 W/m²K and a constant slag liquidus of 1450 °C. Average radiation flux at a constant 50 °C superheat is shown for comparison

In Fig. 5 the impact of bulk slag temperature versus the impact of superheat on radiation fluxes are examined by both increasing bulk temperature at a constant 50 °C superheat and by increasing bulk temperature at a constant slag liquidus (1450 °C), i.e. steadily increasing superheat. Figure 5 shows that radiation heat fluxes increase approximately linearly for the immersed and exposed cooler sections for all cases, but that the magnitudes are more than 8 times as high for 250 °C superheat at 1700 °C bulk slag temperature , when compared to 50 °C superheat at the same temperature . The thickness of the freeze coating in the freeboard also becomes dangerously thin at high superheat. It is the experience of the authors that thicknesses less than 25 mm can spontaneously spall under rapidly changing heat transfer conditions.

In the absence of solid feed, extensive carbothermic reduction or air infiltration, the major heat flows will be through insulated portions of the furnace , and particularly via the immersed and exposed portions of copper coolers. Under these circumstances, the magnitude of the total heat flux from the open surface of the slag must be substantially equal to the sum of the conductive losses through the insulated walls and roof, and through the residual refractory and/or freeze lining in front of any exposed copper cooling element. Except for the radiation zone of the coolers in the absence of refractory , these losses will increase linearly with thermal driving forces. Even the radiation losses driven by T_bath⁴- T_liquidus⁴ tend to appear rather linear for typical ranges of operation for T_bath and T_liquidus. It is therefore not surprising that the net radiative losses from open-bath furnaces have already been shown to vary nearly linearly with bath temperature [2, 12], and not by temperature to the 4th power as might be assumed based on casual observation of Eq. (3). This result is confirmed by the results shown in Figs. 5 and 6, showing fluxes and freeboard temperatures increasing nearly linearly with bulk slag temperature over quite wide temperature ranges.

../images/468727_1_En_51_Chapter/468727_1_En_51_Fig6_HTML.gif — Fig. 6
Comparison of average upper wall and roof temperatures at constant liquidus (1450 °C) and constant superheat (50 °C) for bulk slag temperatures between 1500 and 1700 °C

In Fig. 6, it can also be seen that roof and wall temperatures increase more for lower superheats, as less of the radiated bath energy is lost through the freeze lining of the copper coolers. The frozen layer on the copper coolers is thicker at lower superheat, as shown in Figs. 4 and 5, and more of the bath radiation is available to heat the walls and roof to higher temperature .

In Fig. 7, emissivity has been varied from the 0.8 base case to values from 0.6 to 0.9, and it can be seen that the copper cooler heat losses, as well as the total heat losses from the freeboard, increase with increased emissivity. Equilibrium freeze thickness correspondingly decreases with increased emissivity.

../images/468727_1_En_51_Chapter/468727_1_En_51_Fig7_HTML.gif — Fig. 7
Radiation zone freeze lining thickness and average heat fluxes at 1700 °C with 1450 °C liquidus (250 °C superheat). Results are shown as ratios against the base case emissivity values of 0.8. The impact on total ‘liquid cooled’ copper cooler and freeboard heat losses are also shown

Conclusions

Worst-case bath to copper cooler radiation heat flux estimates for design purposes can be made using either numerical methods (FEM) or simplified approaches, e.g. Equation (4) by assuming that the hot face of the cooler is at the liquidus temperature and the surface of the slag is at the nominal bulk temperature .

Both FEM and analytical results indicated that radiative heat fluxes to the exposed zones of copper coolers can exceed immersed heat fluxes by large factors, particularly for cases where there is a significant difference between the liquidus of the freeze lining and the surface temperature of the open slag bath.

If the slag superheat is maintained within the typically recommended range of 50–150 °C compared to both the frozen slag in the exposed and immersed sections of the coolers, the radiation fluxes will not exceed the convective fluxes by more than 150% based on the FEM results.

Applying Eq. (4) with Eq. (2) allows estimates of radiative heat flux and freeze lining thickness to be made. Results agree with the FEM estimates within ~50% for cases with ≥150 K difference between the slag surface and the liquidus of the exposed freeze lining, i.e. cases where radiative heat transfer is likely to be large and the freeze lining dangerously thin.

FEM model results indicate that radiation fluxes and total furnace freeboard energy losses increase approximately linearly with both bulk slag temperature and slag emissivity for open-bath furnaces over the range of values examined (1500–1700 °C and ε = 0.6–0.9).

Both designers and operators of potentially open-bath furnaces, should consider the impacts of slag liquidus , temperature , and level variation, on radiation losses and overall furnace thermal efficiency . Where possible, variations in level should be minimized and copper coolers kept either fully immersed, protected by high melting point refractory , or ‘wall dressed’ with solid feed in exposed sections to minimize radiation losses.

Appendix

Modelling assumptions and boundary conditions

Item	Value	Location	Comments
Emissivity	0.8 for base case	Assumed equal for slag , wall and roof, and independent of temperature	Varied 0.6–0.9 for sensitivity analysis
Thermal conductivity, W/m · K	2	For freeze lining, wall and roof refractory	Temperature dependence ignored
Thermal conductivity, W/m · K	5	Weighted average value for hearth	Temperature dependence ignored
Solid copper thermal conductivity, W/m · K	334	Copper coolers	Warm, P-deoxidized copper , value assumed constant
Liquid iron thermal conductivity, W/m · K	35	Metal pool	Assumed constant
Vertical wall combined convection and radiation heat transfer coefficient, W/m² · K	40	For non-liquid cooled portions of lower sidewall and side of the hearth	Simplifying assumption to give realistic behavior, with low impact on results
Hearth combined convection and radiation heat transfer coefficient, W/m² · K	25	For bottom surface of the hearth	Simplifying assumption to give realistic behavior
Water cooling temperature , °C	35	Assumed at cooling channels, upper wall and roof outer surface	Assumed constant for all cases (high liquid phase h)
Assumptions
Single layers of refractory (simplification) and no steel shell (negligible resistance)
No participating media in the freeboard. Not true if there is a lot of dust or tri-atomic gases
No convective heat transfer in the freeboard
Diffuse radiation
Well mixed slag with negligible vertical and horizontal temperature gradients
Stagnant metal layer
No solid feed on the surface of the slag bath or on the exposed sections of the copper coolers

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