Predictive Modelling of Grate Combustion and Boiler Dynamics

A hybrid boiler consists of a combustion chamber surrounded by a water tube heat transfer panel and a boiler drum with a set of convection heat transfer tubes as shown in Figure 1. Hot flue gas generated in the combustion chamber transfers heat to the water wall panel and enters the convection tubes placed inside the boiler drum.

The water tube heat transfer panel receives water from the drum through the downcomer. Riser tubes of furnace heat transfer panel receive heat from flue gas to generate water and steam mixture, which rises to the steam drum due to buoyancy effect. Drum water also receives heat from flue gas through the fire tube convection heat transfer surface. The boiler drum receives feed water and supplies steam to process.

1.png

Figure 1. Constructional details of the coal-fired hybrid boiler

2.1 Combustion modelling

Solid fuel combustion passes through various stages like drying, volatile generation, volatile combustion, and char combustion. Figure 1 shows the reciprocating grate as a combustor, where fuel is pushed in different zones by the reciprocating action of the grate. As it has a significant delay in combustion, it is difficult to correlate the rate of fuel firing and heat generation. Combustion delay is quite high in fuel like coal with a lower percentage of the volatile and higher percentage of char, as char combustion is a very slow process. A good combustion model involves heat transfer, evaporation, devolatilization, and char combustion. This makes the model quite complex. It also reduces the usability of the model due to its strong correlation with fuel ingredients and expected variation in fuel. A data-driven model can be used for the prediction of the combustion behaviour of the fuel.

If a fuel particle spends ‘t’ second over the grate before getting transferred to the ash removal system, cumulative heat generation can be approximated as follows.

$Q_{\text {cgen }}=m_{p} L H V\left(a-b e^{-c t}\right)$           (1)

where, m_p is the mass of the fuel particle and LHV is the low heat value of the fuel. a, b and c are the coefficients.

Eq. (1) is modified by dividing the total time in multiple time steps and cumulative fractional heat generation by a mass of fuel after i_th time step can be expressed as follows.

$Y_{i}=W_{1}-W_{2} e^{-W_{3} i}$           (2)

where W₁, W₂ and W₃ are coefficients and can be calculated by using regression analysis. Cumulative fractional heat generation is the ratio of cumulative heat generation and the product of lower heat value and mass of the fuel.

Fractional heat generation for the first time step can be calculated as follows.

$\Delta Y_{1}=W_{1}-W_{2} e^{-W_{3}}$            (3)

Fractional heat generation for the other time steps can be calculated as the difference of cumulative fractional heat generation of i_th time step and cumulative fractional heat generation of i-1_th time step.

$\Delta Y_{i}=Y_{i}-Y_{i-1}$           (4)

If any fuel particle spends ‘M’ time steps over a reciprocating grate, a grate contains fuels of each time step and total heat generation can be expressed as a summation of heat generation of fuels of different time steps.

$Q_{g e n}=\sum_{i}^{M} m f_{i} \Delta Y_{i} L H V$           (5)

where mf_i represent the mass of the fuel, which has spent i_th time over the grate.

2.2 Heat transfer analysis

Total heat generated in the boiler is not transferred to the water for steam generation, a significant portion of the total heat is lost to the atmosphere as stack loss. Heat transfer to water and steam mixture through heat transfer surface is expressed as follows.

$Q_{T}=W_{4} Q_{g e n}$         (6)

where W₄ represents the heat transfer efficiency.

Total heat transfer in the boiler is distributed between the furnace and the convective tube bank. Heat transfer distribution among furnace and convective tube banks is an important parameter for the analysis of boiler dynamics. Fractional heat transfer in the furnace can be expressed as a function of load with a polynomial equation.

$X_{F}=W_{5}+W_{6}$ Load $+W_{7}$ Load $^{2}$        (7)

where, W₅, W₆, and W₇ are the coefficients and can be calculated by using regression analysis.

Heat transfer in the furnace is calculated as follows.

$Q_{F}=X_{F} Q_{T}$       (8)

Similarly, heat transfer in the convective bank can be calculated as follows.

$Q_{C}=X_{C} Q_{T}$         (9)

where, X_C is the fractional heat generation in convective tube bank and can be calculated as follows.

$X_{C}=1-X_{F}$            (10)

2.3 Boiler dynamics

For simplification, the boiler can be divided into three parts: water circulation circuit consisting of downcomer and riser tubes of furnace water tube panel, drum below the water level and drum above the water level. Unlike a water tube boiler, complete heat transfer does not take place in riser tubes and heat transfer distribution between furnace and convection tube section is calculated for drum modelling.

2.3.1 Water circulation circuit

The water circulation circuit consists of downcomer and evaporator tubes. Downcomer receives the saturated water from steam drum and the saturated water enters the evaporator tubes, where it receives the heat from the flue gas and the evaporation of water takes place. Water and steam mixture in evaporator rises due to buoyancy force and enters to the steam drum. Mass, energy, and momentum conservation equations are required to understand and analyse the behaviour of the natural circulation circuit.

Mass conservation equation of natural circulation circuit is represented by the following equation.

$\frac{d}{d t}\left(\rho_{s} \overline{\alpha_{v}} V_{r}+\rho_{w}\left(1-\overline{\alpha_{v}}\right) V_{r}+\rho_{w} V_{d c}\right)=q_{d c}-q_{r}$       (11)

where, q_dc is the saturated water flow from steam drum to downcomer and q_r is the flow of water and steam mixture from riser to steam drum. V_r and V_dc are the volumes of risers and downcomers.

Mass conservation equation of natural circulation circuit can be further simplified as follows.

$\begin{aligned} {\left[V_{r} \overline{\alpha_{v}} \frac{d \rho_{s}}{d p}+\{(1-\right.} &\left.\left.\overline{\alpha_{v}}\right) V_{r}+V_{d c}\right\} \frac{d \rho_{w}}{d p} \left.+\left(\rho_{s} V_{r}-\rho_{w} V_{r}\right) \frac{d \overline{\alpha_{v}}}{d p}\right] \frac{d p}{d t}+\left(\rho_{s} V_{r}-\rho_{w} V_{r}\right) \frac{d \overline{\alpha_{v}}}{d \alpha_{r}} \frac{d \alpha_{r}}{d t}+q_{r} =q_{d c} \end{aligned}$          (12)

$\overline{\alpha_{v}}$ is the average void fraction in the riser and is calculated by the following [5].

$\overline{\alpha_{v}}=\frac{\rho_{w}}{\rho_{w}-\rho_{s}}\left[1-\frac{\rho_{s}}{\alpha_{r}\left(\rho_{w}-\rho_{s}\right)} \ln \left(1+\frac{\rho_{w}-\rho_{s}}{\rho_{s}} \alpha_{r}\right)\right]$        (13)

where α_r is the dryness fraction at the outlet of riser tubes.

The following equation represents the energy conservation equation for the natural circulation circuit.

$\frac{d}{d t}\left\{\rho_{s} \cdot h_{s} \overline{\alpha_{v}} \cdot V_{r}+\rho_{w} h_{w}\left(\left(1-\overline{\alpha_{v}}\right) V_{r}+V_{d c}\right)-p V_{r}\right\}=Q_{F}+q_{d c} h_{w}-\left(\alpha_{r} h_{c}+h_{w}\right) q_{r}$         (14)

where h_c is the difference between saturated vapour and saturated liquid enthalpy and is referred to as condensation enthalpy.

The energy conservation equation of the natural circulation circuit can be simplified as follows.

$\begin{aligned}\left\{\overline{\alpha_{v}} V_{r}\left(h_{s} \frac{d \rho_{s}}{d p}+\right.\right.&\left.\rho_{s} \frac{d h_{s}}{d p}\right) +\left(\left(1-\overline{\alpha_{v}}\right) V_{r}+V_{d c}\right)\left(h_{w} \frac{d \rho_{w}}{d p}\right.\left.+\rho_{w} \frac{d h_{w}}{d p}\right) +\left(\rho_{s} h_{s} V_{r}-\rho_{w} h_{w} V_{r}\right) \frac{d \overline{\alpha_{v}}}{d p} \left.-V_{r}\right\} \frac{d p}{d t} +\left(\rho_{s} h_{s} V_{r}-\rho_{w} h_{w} V_{r}\right) \frac{d \overline{\alpha_{v}}}{d \alpha_{r}} \frac{d \alpha_{r}}{d t} +\left(\alpha_{r} h_{c}+h_{w}\right) q_{r}=Q_{F}+q_{d c} h_{w} \end{aligned}$                 (15)

In a natural circulation loop, water flow in the downcomer takes place due to density differences in the downcomer and riser column. This is calculated by using the momentum balance equation of the natural circulation loop. Elgandelwar et al. [15] present a steady-state model using a momentum balance equation for the flow distribution analysis in natural circulation. The following equation represents the momentum balance equation of a natural circulation loop in a water tube or hybrid boiler [6].

$\left(\rho_{w}-\rho_{s}\right) \overline{\alpha_{v}} V_{r} g-\frac{k_{d c} q_{d c}{ }^{2}}{2 \rho_{w} A_{d c}}=0$                 (16)

where k_dc is a dimensionless number, which depends on the friction factor and geometry of the boiler and A_dc is the area of the downcomer.

Normally a constant value of k_dc is used for the dynamic simulation of a boiler but the dryness fraction of the riser has a great influence on this dimensionless number. It can be expressed as a function of dryness fraction at the outlet of the riser tube as follows.

$k_{d c}=W_{8}+W_{9} \alpha_{r}+W_{10} \alpha_{r}^{2}$          (17)

where W₈, W₉ and W₁₀ are coefficients and calculated by regression analysis.

2.3.2 Drum below water level

For simplification, the drum is divided into two volumes: drum below the water level and drum above the water level. Drum volume below the water level receives feed water and supplies steam to the upper section of the drum. It also receives water and steam mixture from riser tubes and supplies saturated water to the downcomer pipes. If the steam generation in the drum is higher than the steam required in the process, pressure and the corresponding saturation temperature of the drum increases receiving heat from the steam and causing the condensation of steam. If the steam generation in the drum is less than the process requirement, pressure and corresponding saturation temperature decreases rejecting heat to the drum water and causing evaporation of the steam. To model this phenomenon, q_cd term is introduced to represent the rate of condensation in the steam drum. A negative rate of condensation indicates the flashing of steam. To understand and analyse the phenomenon in the drum below the water level, two separate mass conservation equations for water and steam are required in addition to a combined energy equation.

The following equation represents the steam mass conservation equation in the steam drum below the water level.

$\frac{d}{d t}\left(\rho_{s} V_{d w} \varphi\right)=\alpha_{r} q_{r}-q_{s d}-q_{c d}$             (18)

where, V_dw is total drum volume below the water level and ф is the volume fraction occupied by steam. q_cd is the rate of steam condensation and q_sd is the steam flow to the upper section of the drum. Steam mass conservation equation of steam drum below water level can be further simplified as follows. 

$V_{d w} \varphi \frac{d \rho_{s}}{d t}+\rho_{s} \varphi \frac{d V_{d w}}{d t}+\rho_{s} V_{d w} \frac{d \varphi}{d t}=\alpha_{r} q_{r}-q_{s d}-q_{c d}$          (19)

$V_{d w} \varphi \frac{d \rho_{s}}{d p} \frac{d p}{d t}+\rho_{s} \varphi \frac{d V_{d w}}{d t}+\rho_{s} V_{d w} \frac{d \varphi}{d t}-\alpha_{r} q_{r}+q_{s d}+q_{c d}=0$          (20)

Similar to the steam mass conservation equation, water mass conservation can be also written for the drum below the water level.

$\frac{d}{d t}\left\{\rho_{w} V_{d w}(1-\varphi)\right\}=\left(1-\alpha_{r}\right) q_{r}+q_{c d}+q_{f}-q_{d c}$          (21)

q_f is the mass of feedwater flow rate to the boiler drum. The water mass conservation equation can be simplified as follows.

$\begin{aligned} V_{d w}(1-\varphi) \frac{d \rho_{w}}{d p} & \frac{d p}{d t}+\rho_{w}(1-\varphi) \frac{d V_{d w}}{d t}-\rho_{w} V_{d w} \frac{d \varphi}{d t}-\left(1-\alpha_{r}\right) q_{r}-q_{c d} =q_{f}-q_{d c} \end{aligned}$        (22)

Energy equation can be combined for water and steam in the drum below the water level, as it involves phase change and heat transfer across the phase. The following equation represents the energy equation for drum volume below water level.

$\begin{aligned} \frac{d}{d t}\left\{\rho_{s} V_{d w} \varphi h_{s}+\right.& \rho_{w} V_{d w}(1-\varphi) h_{w}+m_{d r u m} C_{m} T_{s} \left.-P V_{d w}\right\} =Q_{C}+\left(\alpha_{r} q_{r}-q_{s d}\right) h_{s} +\left\{\left(1-\alpha_{r}\right) q_{r}-q_{d c}\right\} h_{w}+q_{f} h_{f} \end{aligned}$          (23)

Similar to other equations, this equation can be expanded and the final equation can be written as follows.

$\begin{aligned}\left\{V_{d w} \varphi\left(h_{s} \frac{d \rho_{s}}{d p}+\right.\right.&\left.\rho_{s} \frac{d h_{s}}{d p}\right) +V_{d w}(1-\varphi)\left(h_{w} \frac{d \rho_{w}}{d p}\right.\left.+\rho_{w} \frac{d h_{w}}{d p}\right)+m_{d r u m} C_{m} \frac{d T_{s}}{d p}\left.-V_{d w}\right\} \frac{d p}{d t}+\left\{\rho_{s} h_{s} \varphi+\rho_{w} h_{w}(1-\varphi)\right.\\ &-p\} \frac{d V_{d w}}{d t}+\left\{\rho_{s} h_{s} V_{d w}-\rho_{w} V_{d w} h_{w}\right\} \frac{d \varphi}{d t}-\left\{\alpha_{r} h_{s}+\left(1-\alpha_{r}\right) h_{w}\right\} q_{r} +h_{s} q_{s d}=Q_{C}-q_{d c} h_{w}+q_{f} h_{f} \end{aligned}$         (24)

2.3.3 Steam mass flow

Steam flows from the lower section of the drum to the upper section of the drum because of the buoyancy effect. Many different models have been attempted for the calculation of steam mass flow rate q_sd. Astrom et al. [6] has presented an empirical model for the calculation. Oritz [10] has also attempted an empirical model for the simulation of a fire tube boiler. Kim et al. [8] and Adam et al. [7] use the drift velocity formula for the calculation of steam separation velocity. The following equation [16] is used for the calculation of drift velocity in the boiler drum.

$u_{s}=1.41\left\{\frac{\eta g\left(\rho_{w}-\rho_{s}\right)}{\rho_{w}{ }^{2}}\right\}^{1 / 4}$        (25)

A_sep represents the steam separation area. As the complete steam separation area is not used for the steam separation. It can be multiplied by ф to consider the area of steam separation surface occupied by the steam, as ф represents the fraction of volume in the drum below water level occupied by steam.

$q_{s d}=A_{\operatorname{sep}} \varphi u_{s} \rho_{s}$          (26)

2.3.4 Steam drum above water level

As it contains single-phase steam, it requires only one mass conservation equation for steam to represent the phenomenon. The following equation represents the mass conservation equation in the drum volume above water level.

$\frac{d}{d t}\left(\rho_{s} V_{d s}\right)=q_{s d}-q_{s}$            (27)

where V_ds is the volume of the drum above water level and q_s is the steam flow to the process. This can be expanded as follows.

$V_{d s} \frac{d \rho_{s}}{d t}+\rho_{s} \frac{d V_{d s}}{d t}=q_{s d}-q_{s}$         (28)

As the total volume of the drum is constant, it can be written as follows.

$V_{d w}+V_{d s}=c$          (29)

After differentiation, it yields to the following equation.

$\frac{d V_{d w}}{d t}=-\frac{d V_{d s}}{d t}$            (30)

Eq. (30) is used for the substitution of $\frac{d V_{d s}}{d t}$ by $\frac{d V_{d w}}{d t}$ in Eq. (28). This results in the following equation.

$V_{d s} \frac{d \rho_{s}}{d p} \frac{d p}{d t}-\rho_{s} \frac{d V_{d w}}{d t}-q_{s d}=-q_{s}$         (31)

2.3.5 Integrated Model for drum dynamics of boiler

Eqns. (12), (15), (20), (22), (24), (26) & (31) represent the final equations in terms of desired variables $\frac{d p}{d t}, \frac{d V_{w}}{d t}, \frac{d \varphi}{d t}, \frac{d \alpha_{r}}{d t}, q_{r}, q_{s d}, q_{c d}$ for model development. These equations are simultaneously solved to predict the boiler drum dynamics. These equations and variables are numbered from 1 to 7. Coefficients of the different variables are specified as C_ij, where i represents equation number and j represents a variable number. The left side constant of the equation is specified as di, where i represents the equation number.

Eq. (12) represent the mass conservation equation for the natural circulation circuit and can be specified as equation number 1. Eq. (12) can be written as follows.

$\begin{aligned} {\left[V_{r} \overline{\alpha_{v}} \frac{d \rho_{s}}{d p}+\{(1\right.} &\left.\left.-\overline{\alpha_{v}}\right) V_{r}+V_{d c}\right\} \frac{d \rho_{w}}{d p}\left.+\left(\rho_{s} V_{r}-\rho_{w} V_{r}\right) \frac{d \overline{\alpha_{v}}}{d p}\right] \frac{d p}{d t}+\left(\rho_{s} V_{r}-\rho_{w} V_{r}\right) \frac{d \overline{\alpha_{v}}}{d \alpha_{r}} \frac{d \alpha_{r}}{d t}+q_{r} =q_{d c} \end{aligned}$

This equation can be rewritten as follows.

$C_{11} \frac{d p}{d t}+C_{12} \frac{d V_{d w}}{d t}+C_{13} \frac{d \varphi}{d t}+C_{14} \frac{d \alpha_{r}}{d t}+C_{15} q_{r}+C_{16} q_{s d}+C_{17} q_{c d}=d_{i}$              (32)

where all the coefficients can be calculated as follows.

$C_{11}=V_{r} \overline{\alpha_{v}} \frac{d \rho_{s}}{d p}+\left(V_{r}\left(1-\overline{\alpha_{v}}\right)+V_{d c}\right) \frac{d \rho_{w}}{d p}+\left(\rho_{s} V_{r}-\rho_{w} V_{r}\right) \frac{d \overline{\alpha_{v}}}{d p}$ 

$C_{12}=0$

$C_{13}=0$

$C_{14}=\left(\rho_{s} V_{r}-\rho_{w} V_{r}\right) \frac{d \overline{\alpha_{v}}}{d \alpha_{r}}$

$C_{15}=1$

$C_{16}=0$

$C_{17}=0$

$d_{1}=q_{d c}$

Downcomer flow (q_dc) is calculated by using Eq. (16). Other equations can be also rewritten in the same form and a generalized equation can be deduced as follows.

$C_{i 1} \frac{d p}{d t}+C_{i 2} \frac{d V_{d w}}{d t}+C_{i 3} \frac{d \varphi}{d t}+C_{i 4} \frac{d \alpha_{r}}{d t}+C_{i 5} q_{r}+C_{i 6} q_{s d}+C_{i 7} q_{c d}=d_{i}$            (33)

where, i represent the equation number. A system of the equation can be developed and arranged in a matrix as follows.

$\left[\begin{array}{lllllll}C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} & C_{17} \\ C_{21} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} & C_{27} \\ C_{31} & C_{32} & C_{33} & C_{34} & C_{35} & C_{36} & C_{37} \\ C_{41} & C_{42} & C_{43} & C_{44} & C_{45} & C_{46} & C_{47} \\ C_{51} & C_{52} & C_{53} & C_{54} & C_{55} & C_{56} & C_{57} \\ C_{61} & C_{62} & C_{63} & C_{64} & C_{65} & C_{66} & C_{67} \\ C_{71} & C_{72} & C_{73} & C_{74} & C_{75} & C_{76} & C_{77}\end{array}\right]\left[\begin{array}{c}d p / d t \\ d V_{d w} / d t \\ d \varphi / d t \\ d \alpha_{r} / d t \\ q_{r} \\ q_{s d} \\ q_{c d}\end{array}\right] =\left[\begin{array}{l}d_{1} \\ d_{2} \\ d_{3} \\ d_{4} \\ d_{5} \\ d_{6} \\ d_{7}\end{array}\right]$

2.3.6 Integrated model

An integrated model for drum dynamics requires input like water side heat transfer in the furnace, waterside heat transfer in convection tube bank and downcomer flow. A set of empirical equations are used for the calculation of heat generation (Eqns. (2) to (5)), waterside heat transfer in the furnace (Eqns. (6) to (10)), waterside heat transfer in convection bank (Eqns. (6) to (10)), the dimensionless parameter for the calculation of downcomer flow (Eq. (17)). These equations are represented as a function of 10 coefficients (W₁ to W₁₀) as variables. These equations are used for the calculation of required parameters like water side heat transfer in the furnace, waterside heat transfer in convection bank and downcomer flow for the integrated boiler dynamics model. Integrated boiler dynamics is used for the prediction of pressure and water level at different time steps. Predicted and actual values of pressure and water level are compared to estimate errors. The objective of this model is to minimise errors by identifying the optimum set of variables.

The following equations are used for the calculation of the errors in pressure and water level estimation.

$\operatorname{ErP}_{i}=p a_{i}-p m_{i}$               (34)

$E r L_{i}=L a_{i}-L m_{i}$             (35)

where ErP_i and ErL_i are errors in pressure and water level. Pa_i and Pm_i are actual and predicted values of pressure respectively. La_i and Lm_i are actual and predicted values of water level respectively.

This is the case of multi-objective optimisation as both errors in pressure and water level need to be minimised. Weighted sum method and least square technique are used, where both objectives are combined in a single objective function by adding them after multiplying the square of errors and weightage as follows.

$\operatorname{Er}=X_{p} \sum_{i}^{N} \operatorname{Er} P_{i}^{2}+X_{L} \sum_{i}^{N} \operatorname{Er} L_{i}{ }^{2}$              (36)

where, X_P and X_L are weightage of pressure and level errors.

The batch gradient descent method can be used for the minimisation of the error function (Er). Initial values of the various coefficient variables (W₁ to W₁₀) are assumed. The rate of combustion, heat transfer, distribution of heat transfer and downcomer flow are calculated. The boiler dynamics model is used to predict pressure and water level. Model output is compared with the experimental data to compare errors in pressure and water level for each time step. The final error function is calculated by using Eq. (36). A small change in each variable is introduced to calculate the new value of the error function and gradients are calculated. The new set of coefficient variables is calculated by using the gradient descent algorithm. The stepwise calculation has been explained by flowcharts in Figure 2a and Figure 2b.

2a.png

(a)

2b.png

(b)

Figure 2. (a)  Flow chart for the calculation of error function; (b) Flow chart for the minimisation of the error function

2.4 Extension of the integrated model for the prediction of flue gas composition

The air to fuel ratio in grate combustion is controlled by multiple air dampers. Damper position and fuel firing rate in a grate boiler are controlled by a control table, where damper positions and fuel firing are tabulated as a function of boiler load. It is quite difficult to operate coal-fired grate boiler at optimum air to fuel ratio due to significant combustion lag in coal combustion. If the integrated model is used for the prediction of the rate of combustion, it is possible to analyse the characteristics of air damper and predict the air quantity and oxygen level in the flue gas. In the test boiler, two primary air dampers and one secondary air damper are used.

The following equations can be used for the estimation of air quantity for each damper.

$m a_{1}=A_{1}+A_{2} D_{1}+A_{3} D_{1}^{2}+A_{4} D_{1}^{3}$                   (37)

$m a_{2}=A_{5}+A_{6} D_{2}+A_{7} D_{2}^{2}+A_{8} D_{2}^{3}$                 (38)

$m a_{3}=A_{9}+A_{10} D_{3}+A_{11} D_{3}^{2}+A_{12} D_{3}^{3}$                 (39)

where, D₁, D₂ and D₃ are the damper position and A₁ to A₁₂ are the coefficients of the polynomial equations.

Total air quantity is calculated as the sum of the air quantity for each damper.

$m a=m a_{1}+m a_{2}+m a_{3}$               (40)

Eq. (5) is used for the rate of heat generation and the rate of combustion can be calculated by dividing heat generation by LHV. Flue gas composition is calculated by using total air quantity and rate of combustion. Oxygen percentage in dry flue gas is calculated and compared with the measured value of oxygen. The following equation is used for the calculation of error in oxygen.

$\operatorname{Ero}_{i}=o a_{i}-o m_{i}$             (41)

Mean squared error in prediction in oxygen level is expressed as follows.

$M S E r o=\frac{1}{N} \sum_{i}^{N} \operatorname{Ero}_{i}^{2}$             (42)

Mean squared errors in oxygen prediction is minimised by using the Batch Gradient Descent algorithm for the calculation of the optimum set of coefficient variables. Figure 3a and Figure 3b present flowcharts for the calculation and minimisation of the error function.

3a.png

(a)

3b.png

(b)

Figure 3. (a) Flow chart for the calculation of error function; (b) Flow chart for the minimisation of the error function

The boiler dynamics of a hybrid boiler is quite different from a water tube boiler as heat transfer is distributed in the water tube furnace and fire tube convection surface. As heat transfer takes place in the boiler drum, it makes the hydrodynamic analysis more complex in comparison with the water tube boiler. It is easy to estimate heat generation and heat transfer in an oil and gas fired boiler, as combustion is instantaneous and directly calculated from the fuel firing rate. In a solid fuel fired boiler, a fuel particle passes through the different stages of combustion and releases heat differently at different times depending on the stage of combustion. It requires a complex model for the prediction of heat generation. The predictability of this model is quite less due to variability in fuel properties and composition. A simplified data-driven model is used for the prediction of heat generation, heat transfer and distribution of heat transfer among furnace and convective tubes. This model can capture the variability of fuel and its effect on heat release patterns. This model can also predict the variation in heat transfer due to slagging and fouling of the heat transfer surface. This model is integrated with the hydrodynamic model. The hydrodynamic model divides the boiler into three zones: riser downcomer circuit for a furnace water wall, drum below water level and drum above water level. Mass and energy conservation equations are developed for these three zones and a separate equation for steam separation has been added to the model. 

An integrated combustion, heat transfer and hydrodynamic model is used to predict the pressure and water level in the boiler. The predicted value is compared with the experimental value to identify errors and the multi-objective function optimisation strategy is used for the minimisation of errors. Model is developed for 60 minutes to identify the optimum set of variables and used to test the accuracy of the model for the next 30 minutes of data. The root mean absolute error method presents a simplified approach for the estimation of the accuracy of the model. It shows the RMAE value as 13.05% and 8.72% in the estimation pressure and water level respectively. This model can be extended for the control of pressure and water level of a coal-fired reciprocating grate boiler in extremely fluctuating load conditions.

The model has been extended for the prediction of oxygen level, where the mass flow rate of combustion air is calculated and the predicted value of the rate of combustion is used for the estimation of oxygen level in the flue gas. It shows an RMAE of 6.15% for the test data. Oxygen level indicates the air to fuel ratio and plays a significant role in efficient fuel combustion and emission control. This model can be extended for the control of pressure, water level and air to fuel ratio for the efficient and safe operation of a coal-fired boiler.