Thermal Performance Evaluation of a Single-Mouth Improved Cookstove: Theoretical Approach Compared with Experimental Data

Abstract

This work aims to address the knowledge gap in the thermal efficiency performance of a locally made cookstove in Mali. Despite the fact that the thermal efficiency of cookstoves is a crucial aspect of cooking, the performance of commercially produced cookstoves in Mali has not been thoroughly studied. In this context, the thermal efficiency of a single-mouth biomass stove has been investigated using a theoretical and experimental approach. First, the fundamental principles of physics for the three forms of heat transfer were applied. Then, the theoretical thermal efficiency of the stove was calculated based on the percentage share of energy gains and losses for the respective heat transfer modes. This analysis shows that the highest energy gain is achieved by radiation heat transfer from the flame and the fuel bed, followed by convection heat transfer to the bottom and sides of the pot, respectively. In order to validate the findings, the theoretical results have been compared with the experimental data at a case study site in Katibougou, Mali. Accordingly, the experimental thermal efficiency is slightly lower than the theoretical value, with a measured value of 27% compared to the theoretical value of 31.45%. The theoretical thermal efficiency can be closer to the experimental efficiency if the combustion losses caused by incomplete combustion of the fuel are taken into account.

1. Introduction

From the dawn of human history, biomass has been utilized for cooking and heating purposes. Currently, around 2.7 billion people use biomass to meet their daily energy needs [1]. The demand for firewood has led to global deforestation, soil degradation, and erosion. In addition, burning firewood emits dangerous chemicals and greenhouse gas emissions into the atmosphere, thereby adversely affecting both human health and the global climate. In 2021, biomass accounted for 10% of the world’s energy supply and 13% of energy consumption [2]. A large proportion of this is used by rural families in developing countries, who depend on biomass for 90% of their domestic energy needs. Most of these individuals use inefficient traditional cookstoves to prepare their meals, leading to high levels of local/indoor air pollution and increased firewood consumption. Therefore, the widespread use of efficient cookstoves can be a significant step to improve rural livelihood in these countries.

In the last four decades, numerous global organizations have made efforts to offer individuals improved and environmentally friendly cooking technologies. In addition, great progress has been made in improving cookstoves in recent decades [3,4,5,6]. Although there have been notable advancements in cookstove design, numerous households in developing countries continue to utilize inefficient cookstoves as they cannot afford the upgraded models or there is a lack of awareness of and technology for improved cookstoves (ICSs).

More than 94% of the urban Malian population rely on solid fuels for cooking, and only about 6% have access to clean energy sources. Indeed, less than 0.5% of the population use improved cookstoves, which combust firewood or charcoal with a higher efficiency [7]. Panels a–f in Figure 1 indicate some samples of traditional wood and charcoal cookstoves used by Malian households.

Hence, a successful approach for providing individuals with better cooking facilities is to identify potential enhancements and turn them into valued assets. To achieve this, heat transfer has been recognized as a research field that has the greatest potential to improve stove performance and allow more people to take advantage of improved cooking systems. The present work has been carried out in view of developing a theoretical model for understanding the basic physics underlying the three forms of heat transfer, i.e., conduction, convection, and radiation. The objective here is to examine each form of heat transfer separately and identify the theoretical thermal efficiency of a single-mouth improved cookstove, which is locally made in Mali. Once the efficiency is calculated and analyzed, the manufacturer could use the results to improve efficiency further by changing the geometry, materials, etc,, as well as the awareness in those villages, by comparing the low efficiency and health hazards of traditional open mouth stoves to the adoption of improved, higher efficiency (and thereby requiring less firewood) and less polluting (indoor health environment) cookstoves. To the authors’ knowledge, those locally built stoves in Katibougou have not yet been analyzed for their thermal performance. Therefore, this study aims to bridge the gap by investigating the thermal efficiency of a locally produced biomass stove using a combined theoretical and experimental approach.

In the past, several models have been reported for the performance analysis of cookstoves. Initial attempts at creating models for cookstoves date back to the 1980s at Eindhoven University when Prasad et al. introduced preliminary heat transfer models for both open and shielded fires in 1981 [8]. Following that, De Lepeleire et al. introduced basic design principles aimed at improving cookstove performances [9]. In their other work, De Lepeleire et al. made significant progress in the advancement of contemporary heat transfer modeling by introducing a mathematical model coupling convective heat transfer and a simple flue-gas flow [10]. In 1985, Prasad et al. derived the combustion chamber wall losses for three distinct materials—dried clay, ceramics, and metal—assuming one-dimensional transient heat conduction within a flat wall [11]. A theoretical framework carried out in 1988 by the Centre for Technology Alternatives for Rural Areas (CTARA) in India aimed to predict the efficiency and excess-air factors for various geometric and operational parameters of a wood-burning stove [12]. These studies provided an initial basis for developing better mathematical models later.

From the 1990s to the early 2000s, the researchers in this field concentrated on using CFD analysis. Although a CFD approach can have a high level of accuracy compared to a simple model, it requires higher computational and operational costs as well as a properly skilled operator, thereby restricting its application to only high-value engineering fields. Therefore, in the traditional cookstove sector, conducting a parametric variation study using CFD has not been a viable choice [13]. In 2011, Agenbroad et al. created a simplified steady-state model to estimate bulk flow rate, temperature, and excess air ratio based on the stove geometry and firepower. In their work, the steady-state prediction results showed a satisfactory level of agreement with temporally averaged validation data obtained from typical stove operation [14]. Later in 2015, Kshirsagar et al. [13] presented a computationally inexpensive mathematical tool for the performance prediction of ‘rocket’ stoves with an unshielded pot. This work explored a new operating parameter called the ‘inlet area ratio’ and investigated the impact of this parameter on cookstove performance. It was concluded that the newly introduced parameter has a significant impact on the performance of a natural draft stove. In 2019, Parajuli et al. [15] formulated a mathematical framework for a dual-pot mud biomass stove, considering transient heat transfer, combustion chemistry, and fluid flow. The model is able to modify the operating and design parameters. Recently, in 2022, Sadiki and their team examined how varying levels of firepower affect the flow dynamics in biomass natural convection-driven cooking stoves. They developed a basic analytical model for the rate of entropy generation in the flow field and validated the model with experiments [16].

Table 1. Selected list of studies and their system configuration.

Studies	System Configuration	Highlights of the Study
Prasad et al., 1981 [8]	Open fires, shielded fires, and heavy stoves	Development of a preliminary heat transfer model.
De Lepeleire et al., 1981 [9]	Various portable and fixed stoves	Preparation of a technical panel for firewood and charcoal stoves.
De Lepeleire et al., 1983 [10]	Wood-burning stove	Demonstrating a correlation between the stove geometry and its performance.
Prasad et al., 1985 [11]	Shielded fire	Modeling of a transient heat transfer considering only heat conduction.
Date, 1988 [12]	CTARA wood-burning stove	Predicting the magnitudes of efficiency and excess air factors for different geometrical and operating parameters.
Agenbroad et al., 2011 [14]	Insulated rocket elbow stove	Development of a simplified model for predicting bulk flow rate, temperature, and air access ratio.
Kshirsagar et al., 2015 [13]	Rocket stove with unshielded pot	Development of a mathematical tool for the performance prediction of ‘rocket’ stoves.
Parajuli et al., 2019 [15]	Two-pot enclosed mud cookstove	Development of a mathematical model for a two-pot enclosed mud cookstove combined with transient heat transfer, combustion chemistry, and fluid flow.
Augustin et al., 2022 [16]	Biomass cookstove driven by natural convection.	Development of a simplified analytical model of the entropy generation rate within the flow field and model validation with experiments.

shows selected relevant literature that studied the type of stoves using different models.

2. Materials and Methods

The cookstove used in this work is illustrated in panel (f) of Figure 1. It is a single-mouth ICS produced by local manufacturers, which is equipped with a chimney, and the stove structure’s is made of fired clay, brick, cement, and metal. In addition, it is designed for burning firewood, branches, and briquettes. The important geometric parameters for the given stove configuration are listed in

Table 2. Geometric parameters of single-mouth ICS with chimney.

Parameter	Value	Symbol	Unit
Cookstove dimensions	1120 × 750 × 710	L × H × W	mm
Length of combustion chamber in lower portion	230	Llow	mm
Length of combustion chamber in upper portion	270	Lup	mm
Diameter of the combustion chamber in lower portion	230	dlow	mm
Diameter of the combustion chamber in upper portion	550	dup	mm
Chimney diameter	100	dchimney	mm
Chimney height	2250	hchimney	mm
Pot diameter	550	dpot	mm
Thickness of pot	4	tpot	mm
Height of air gap between pot and drip pan	50	hpot_gap	mm
Fuel bed area	196,250	Afuel_bed	mm2
Air flow area	78,000	Aair_flow	mm2

.

This work aims to develop a theoretical approach to analyze three modes of heat transfer for the given single-mouth cookstove. For this purpose, fundamental principles of physics were applied to all parts of the stove and pot. The key components of the study consist of heat transfer principles, first law of thermodynamics, surface boundary layer behavior, and impinging jet flow. In order to determine the corresponding heat transfer parameters, the stove geometry and the firepower specified by the stove manufacturer were used as input values. To derive the output parameters, the mass flow rate and flue gas temperature were initially calculated. This was accomplished by setting the flow parameters and adjusting them based on the theoretical value of loss coefficient [17]. The fuel bed, flame, and surface temperatures as well as the calculation of the steady-state flue gas temperatures for different parts of the stove were derived from the methodology described in [17,18].

2.1. Theoretical Framework and Assumptions

As can be seen from Equation (1), the total thermal efficiency of the stove (η_t) depends on both combustion (η_c) and heat transfer efficiency (η_h) [18].

η_t = η_c × η_h

(1)

The combustion efficiency measures how much chemical energy stored in fuel is converted into thermal energy. The studies show that heat transfer efficiency is the most influential parameter for improving stove performance compared to the combustion efficiency. Therefore, if the heat transfer efficiency increases, this can potentially have a positive effect on the overall heat efficiency of the stove [18,19]. In this study, the combustion aspects of the process were ignored for the sake of simplicity and we assumed that the entire chemical energy of the fuel is converted into thermal energy (η_c = 1).

Figure 2 shows a schematic view of the single-mouth ICS with a chimney, along with the corresponding heat transfer modes.

Equation (2) expresses the rate of energy transfer through mass flow into or out of a system [18].

$$Rate of energy flow=\dot{m}\times \theta$$

(2)

where $\dot{m}$ is the bulk mass flow rate of the fluid ($\mathrm{k}\mathrm{g}/\mathrm{s}$) and $\theta$ is the total energy of the flowing fluid per unit of mass ($\mathrm{J}/\mathrm{k}\mathrm{g}$). The overall energy of passing air/flue gas per unit of mass can be further broken down into its basic components as given in Equation (3).

$$\theta =h+ke+pe$$

(3)

where h is enthalpy (J/kg); ke and pe are kinetic energy and potential energy (J/kg), respectively.

In this study, an isobaric steady-flow condition was assumed, in which the mass flow in the stove remains constant. Moreover, kinetic and potential energy in the entire control volume from the inlet to the outlet is constant. Furthermore, ideal gas behavior and no mechanical work was assumed. Equations (4)–(8) provide the basis for evaluating the energy balance of the stove according to these assumptions [18].

$$\dot{E}i =\dot{E}o$$

(4)

$$\dot{Q}i+\dot{W}i+{\sum }_i\dot{m}\theta =\dot{Q}o+\dot{W}o+{\sum }_o\dot{m}\theta$$

(5)

$$\dot{Q}i-\dot{Q}o=\dot{m}(\theta o-\theta i)$$

(6)

$$\dot{Q}i-\dot{Q}o=\dot{m}(ho-hi)$$

(7)

$$\dot{Q}i-\dot{Q}o=\dot{m}Cp,avg (To-Ti)$$

(8)

where subscripts i and o are the inlet and outlet variables, respectively; $\dot{Q}$ is the rate of heat energy transfer (J/s); $\dot{W}$ is the rate of work energy transfer (J/s); Cp,avg is the average constant pressure specific heat of air/flue gas (J/kg K); and T is the gas temperature (K).

Equation (8) shows the relationship between mass flow rate, temperature, and heat transfer inside and outside of the cookstove. The steady-state energy balance provides a key perspective for assessing each form of energy transfer. As can be seen in Figure 3, all heat transfer modes (conduction, convection, radiation) are divided into two distinct groups: losses and gains. The heat losses are connected to the heat transferred to the stove body or to the surroundings, whereas the heat gains are related to the heat that is transferred to the pot. The steady-state energy balance with the corresponding heat transfer forms will be explained in the following sections.

2.1.1. Conduction

Heat is transferred by conduction, from the bottom of the stove floor to the ground, from the body of the stove to the surroundings, and the thickness of the pot. It was assumed that the stove walls were perfectly insulated; thus, no conduction heat transfer to the surroundings and pot was taken into account. Thus, the sole heat conduction loss considered in the overall energy balance is the transfer of heat through the stove floor to the ground, as shown in Figure 2. The steady-state heat transfer rate is defined by Equation (9) [18].

$$Q_{\mathit{conduct}}=K\times A\times {\displaystyle \frac{\Delta T}{ L}}$$

(9)

where Q_conduct is the conduction heat transfer (kW); K is the thermal conductivity (W/m K); A is the object cross-sectional area (m²); ΔT is the temperature difference (K); and L is the object thickness (m).

2.1.2. Convection

The high temperature gasses interact across two surfaces: the inside surface of the cookstove through the upper and lower parts of the combustion chamber (heat losses) and the outside surface of the pot (heat gains). Equation (10) defines the relative convection contribution in the steady condition [18].

Q_{conv_total} = Q_{conv_low} + Q_{conv_up} + Q_{conv_pot bottom} + Q_{conv_sides}

(10)

where subscripts “low” and “up” are the variables in lower and upper portion; Q_{conv_total} is the total convection heat transfer (kW); Q_{conv_low} is the convection from gasses to lower combustion chamber walls (kW); Q_{conv_up} is the convection from gasses to upper chamber walls (kW); Q_{conv_pot bottom} is the convection to the pot bottom (kW); and Q_{conv_sides} is the convection to the pot sides (kW).

This study considered the lower and upper sections (Figure 3) as two separate cylindrical channels, each having fully developed internal flow patterns with a constant mass flow rate and gas temperature [18]. In order to estimate a rough convection coefficient, flow parameters were calculated for each section in the transition zone between laminar and turbulent. Equations (11) and (12) explain the convection heat transfer across the lower and upper cylindrical channels, respectively [18].

$$Q_{\mathit{conv}\text{\_}\mathit{low}}=h_{\mathit{low}}\times A_{\mathit{low}}\times (T_{\mathit{gas}}-T_{\mathit{low}})$$

(11)

$$Q_{\mathit{conv}\text{\_}\mathit{up}}=h_{\mathit{up}}\times A_{\mathit{up}}\times (T_{\mathit{gas}}-T_{\mathit{up}})$$

(12)

where h is the convection coefficient (W/m²K); A is the object cross-sectional area (m²); and T is the temperature of the object (K).

The total convection gain originates from two primary sources, specifically impingement of the combustion gasses on the bottom surface of the pot and the scraping of the corresponding gasses on the pot sides. In this study, both sources were analyzed independently. The convection coefficient was calculated to determine the heat transfer into the pot under the assumption of an isothermal, non-reacting, impinging jet flow, using the method as described in the literature [18]. The convection of gasses to the pot bottom and sides is expressed by Equations (13) and (14), respectively [18].

$$Q_{\mathit{conv}\text{\_}\mathit{pot}}=h_{\mathit{conv}\text{\_}\mathit{pot}}\times A_{\mathit{pot}}\times (T_{\mathit{gas}}-T_{\mathit{pot}})$$

(13)

$$Q_{\mathit{conv}\text{\_}\mathit{sides}}=h_{\mathit{conv}\text{\_}\mathit{sides}}\times A_{\mathit{sides}}\times (T_{\mathit{gas}}-T_{\mathit{pot}})$$

(14)

where h_{conv_pot} and h_{conv_sides} are convection coefficients at the pot bottom and pot sides (W/m²K), respectively; A_pot is the pot bottom surface area (m²); A_sides is the pot side surface area (m²); T_gas is the gas temperature (K); and T_pot is the pot temperature (K).

2.1.3. Radiation

The radiant heat from the fuel bed or the flames is transferred to the pot, the stove body, and the surrounding. Equation (15) represents the total radiation energy balance [18].

Q_{rad_total} = Q_{rad_low_ab} + Q_{rad_up_ab} + Q_{rad_flame_ab} + Q_{rad_pot_ab} + Q_{rad_pot sides}

(15)

where Q_{rad_total} is the total radiation heat transfer (kW); Q_{rad_low_ab} is the radiation from the flames/fuel bed to the lower combustion chamber (kW); Q_{rad_up_ab} is the radiation from the flames to the upper combustion chamber (kW); Q_{rad_flame_ab} is the radiation to the pot from the high flames (kW); Q_{rad_pot_ab} is the radiation transfer to the pot from the fuel bed (kW); and Q_{rad_pot sides} is the radiation lost by the outer pot surface to the ambient surroundings (kW).

The level of radiation emitted by loss or gain is highly dependent on the temperature difference between two elements, the emitting surface area, the emissivity of each element, and the view factor [18]. The emissivity values of combustion flames, the fuel, and the stove body surface were obtained from the literature [20,21,22,23]. In terms of radiative losses to the body of the stove, the upper and lower chambers were studied individually; both were assumed to be fully closed, infinitely long concentric cylinders, with the fuel/flame bed representing the inner chamber geometry, as describe in the literature [18].

In order to develop the assumption of infinitely long concentric cylinders, the dimensions of the stove body (rectangular wall enclosure) were first converted into a circular wall enclosure as expressed in Equation (16) [24,25]. This assumption is made in order to reduce the complexity of the geometry and to simplify the calculation, as the lateral surface can be described by a constant value of the radius [26]. Therefore, the inner and outer parts of the combustion chamber were considered as two concentric cylinders, as shown in Figure 3. The inner and outer parts are identified by the indices “a” and “b”, respectively.

$$R_b=\sqrt{{\displaystyle \frac{LW}{\pi }}}$$

(16)

where R_b is radius of the outer wall (m); L and W are the length and width of the rectangular enclosure wall (m), respectively.

The steady-state radiation losses at the walls of the lower and upper combustion chamber is described by Equations (17) and (18), respectively [18]

$$Q_{\mathit{radiation}\text{\_}\mathit{low}\text{\_}\mathit{ab}}={\displaystyle \frac{{\sigma }_{\mathit{Stefan}}\times A_{\mathit{low}\text{\_}\mathit{a}}\times ({T_{low\_a}}^4\text{\_}{T_{low\_b}}^4) }{\frac{1}{{\epsilon }_{\mathit{low}\text{\_}a}}+\frac{1-{\epsilon }_{\mathit{low}\text{\_}b}}{{\epsilon }_{\mathit{low}\text{\_}b}}\times \left[\frac{r_{\mathit{low}\text{\_}a}}{r_{\mathit{low}\text{\_}b}}\right]}}$$

(17)

$$Q_{\mathit{radiation}\text{\_}\mathit{up}\text{\_}\mathit{ab}}={\displaystyle \frac{{\sigma }_{\mathit{Stefan}}\times A_{\mathit{up}\text{\_}a}\times \left({T_{up\_a}}^4\text{\_}{T_{up\_b}}^4\right)}{\frac{1}{{\epsilon }_{\mathit{up}\text{\_}a}}+\frac{1-{\epsilon }_{\mathit{up}\text{\_}b}}{{\epsilon }_{\mathit{up}\text{\_}b}}\times \left[\frac{r_{\mathit{up}\text{\_}a}}{r_{\mathit{up}\text{\_}b}}\right]}}$$

(18)

where σ_Stefan is Stefan Boltzmann’s constant (W/m² K⁴); A is the object’s cross-sectional area at the lower and upper portions (m²); T is the temperature of the object (K); ε_{low_a} and ε_{up_a} are the average emissivity value between the flames and fuel bed surface in the lower and upper portions; ε_{low_b} and ε_{up_b} are the emissivity of the clay in the lower and upper portions; and r is the radius of the object in the lower and upper portion (m).

Equation (19) specifies the total radiation heat transferred from the fuel bed to the bottom of the pot. It is assumed that there is no radiation from the flames, the pot is a black body, and the fuel bed’s reflectivity is negligible. Equation (20) defines the total radiation heat transfer supplied by the flame sheet to the pot bottom, assuming that the pot is a black body, and the reflectivity of the flames is negligible [18].

$$Q_{\mathit{rad}\text{\_}\mathit{pot}\text{\_}\mathit{ab}}=A_{\mathit{pot}\text{\_}a}\times F_{\mathit{ab}}\times {\sigma }_{\mathit{Stefan}}\times ({\epsilon }_{\mathit{fuel}\text{\_}\mathit{bed}}\times {T_{\mathit{pot}\text{\_}a}}^4-{T_{\mathit{pot}\text{\_}b}}^4)$$

(19)

$$Q_{\mathit{rad}\text{\_}\mathit{flame}\text{\_}\mathit{ab}}=A_{\mathit{flame}\text{\_}a}\times F_{\mathit{flame}\text{\_}\mathit{ab}}\times {\sigma }_{\mathit{Stefan}}\times ({\epsilon }_{\mathit{flame}}\times {T_{\mathit{flame}\text{\_}a}}^4-{T_{\mathit{pot}\text{\_}b}}^4)$$

(20)

where A_{pot_a} is the effective emitting area on the pot (m²); ε_{fuel_bed} is the emissivity value of the metal fuel bed surface; T_{pot_a} is the average temperature of the emitting area (K); T_{pot_b} is the average temperature of the pot surface (K); A_{flame_a} is the effective emitting flame area (m²); ε_flame is the emissivity value of the diffusion flame; and T_{flame_a} is the average temperature of the flame (K). The relative areas of the respective surfaces and the distance between them have been modified using view factors, which result from Equations (21) and (22) [18].

$${\mathrm{S}}_{\mathrm{view} }=1+{\displaystyle \frac{1+{\left(\frac{{\mathrm{r}}_{\mathrm{p}\mathrm{r}\mathrm{o}}}{{\mathrm{L}}_{\mathrm{pot} }}\right)}^2}{{\left(\frac{{\mathrm{r}}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{{\mathrm{L}}_{\mathrm{pot} }}\right)}^2}}$$

(21)

$$\left.{\mathrm{F}}_{\mathrm{a}\mathrm{b}}={\displaystyle \frac{1}{2}}\times \left[{\mathrm{S}}_{\mathrm{view} }-{\left[{\mathrm{S}}_{\mathrm{view} }^2-4{\left({\displaystyle \frac{{\mathrm{r}}_{\mathrm{p}\mathrm{r}\mathrm{o}}}{{\mathrm{r}}_{\mathrm{e}\mathrm{f}\mathrm{f}}}}\right)}^2\right]}^{{\displaystyle \frac{1}{2}}}\right.\right]$$

(22)

where S_view is the variable simplification for the view factor calculation; r_eff is the radius of the effective emitting area (m); r_pro is the radius of the projected emitted area on the pot (m); and L_pot is the separation distance between the two effective areas (m).

2.1.4. Chimney Effect

In addition to investigating the gas temperature and mass flow rate impacts on heat transfer, it is essential to understand how these variables are affected by the geometry of the gas path, which differs depending on the stove geometry and the interface between the stove and the pot. The height of the chimney is a key geometric parameter that specifies the vertical distance between the heat source and the hot flue gasses discharged into the surroundings.

The heat generated by the stove produces hot gasses that have a lower density than the surrounding air located outside of the stove mouth. Therefore, the pressure gradient is caused by the buoyancy effect, resulting from the density gradient, as described in Equation (23) [18].

$$\Delta P_{\mathit{induced}}=g{h}_{\mathit{chim}}({\rho }_{\mathit{ambient}}-{\rho }_{\mathit{hot}})=g{h}_{\mathit{chim}}{\rho }_{\mathit{ambient}}(1-{\displaystyle \frac{T_{\mathit{ambient}}}{T_{\mathit{hot}}}})$$

(23)

where ΔP_induced is the pressure difference (Pa); ρ_ambient is the ambient density (kg/m³); ρ_hot is the density of hot gasses (kg/m³); g is the gravitational acceleration (m/s²); h_chim is the chimney height (m); T_ambient is the ambient temperature (K); and T_hot is the temperature of the hot gasses (K).

The pressure gradient in Equation (23) can be expressed in the form of mass flow rate across the stove, which depends on the chimney height, chimney cross-sectional area, the loss coefficient, and the temperature of the combustion gasses and ambient surroundings [17]. This relationship is represented by Equation (24).

$$\dot{m}=LC\times A\left({\displaystyle \frac{{\Delta \mathrm{P}}_{induced}}{R_s T_{hot}}}\right)\sqrt{2g{h}_{chim}\left({\displaystyle \frac{T_{hot}-T_{ambient}}{T_{ambient}}}\right)}$$

(24)

where $\dot{m}$ is the mass flow rate (kg/s); LC is the loss coefficient that reflects efficiency losses related to the friction and viscous losses of the flow caused by stove geometry and the heat lost at the chimney walls; A is the flow area (m²); and R_s is the universal gas constant (J/K·mol).

Assuming that the exhaust gas behaves as an ideal gas, the following correlations were defined for the exhaust gas properties ranging from 300 to 1500 K at 1 atm [13,27].

$$\mu =0.0447\times {10}^{-5}\times T^{0.7775}$$

(25)

$$k=0.00031847\times T^{0.7775}$$

(26)

$$C_p=0.9362+0.0002\times T$$

(27)

$$\rho =353/T$$

(28)

$$v={\displaystyle \frac{\dot{m}}{\rho \times A}}$$

(29)

$$Re={\displaystyle \frac{\rho vd}{\mu }}$$

(30)

Pr = 0.685 (Constant)

(31)

Nu = 0.565 + Pr^0.5 Re^0.5

(32)

$$h_{\mathrm{conv}}={\displaystyle \frac{Nu\times k}{d}}$$

(33)

where μ is the dynamic viscosity of the flue gas (kg/m s); T is the temperature of the exhaust gas (K); k is the thermal conductivity of the exhaust flue gas (W/m K); C_p is the specific heat of the flue gas (kJ/kg K); Re is the Reynolds number; $\rho$ is the density of the exhaust gas (kg/m³); v is the flue gas velocity (m/s); $\dot{m}$ is the mass flow rate (kg/s); A is the flow area (m²); d is the diameter of the flow path (m); Pr is the Prandtl number; Nu is the Nusselt number; and h_conv is the convection coefficient (W/m² K). The Reynolds number for impingement flow is interpreted as follows [18]:

100 < Re < 10³ laminar flow
10³ < Re < 10⁴ transition to turbulence
10⁴ < Re turbulent flow

(34)

The input parameters required for calculating the flow gas parameters and energy balance of all heat transfer modes are summarized in

Table 3. Input parameters for single-mouth ICS with chimney.

Parameters	Symbol	Value	Unit	Remark
Input parameters (operational)
Firepower	P	25	kW	Determined by manufacturer
Ambient temperature	Tambient	305	K
Fuel bed temperature	Tfuel _bed	1127	K	Estimated based on [18]
Average temperature of the flame	$T_{\mathit{flame}}$	819	K	Estimated based on [18]
Average surface pot temperature	$T_{\mathit{pot}}$	325	K
Input parameters (physical)
Effective radius of inner cylinder in lower portion	rlow_a	115	mm
Effective radius of outer cylinder in lower portion	rlow_b	501	mm
Length of channel in lower portion	Llow	230	mm
Length of channel in upper portion	Lup	270	mm
Effective radius of inner cylinder in upper portion	rup_a	275	mm
Effective radius of outer cylinder in upper portion	rup_b	501	mm
Chimney diameter	dchimney	100	mm
Chimney height	hchimney	2250	mm
Pot diameter	dpot	550	mm
Pot height	hpot	350	mm
Separation distance between two effective areas	Lpot	280	mm
Height of air gap between pot and drip pan	hpot_gap	50	mm
Fuel bed area	Afuel_bed	196,250	mm2
Thickness concrete floor	$L$	50	mm
Intermediate parameters (predicted)
Emissivity value of metal fuel bed surface	${\epsilon }_{\mathit{fuel}\text{\_}\mathit{bed}}$	0.83	-
Emissivity value of diffusion flame	${\epsilon }_{\mathit{flame}}$	0.72	-
Average emissivity value between flames and fuel bed surface in lower portion	${\epsilon }_{\mathit{low}\text{\_}a}$	0.797	-
Emissivity of the clay in lower part	${\epsilon }_{\mathit{low}\text{\_}b}$	0.75	-
Stefan Boltzmann constant	${\sigma }_{\mathit{Stefan}}$	5.67 × 10−8	W/m2K4
Average emissivity value between flames and fuel bed surface in upper portion	${\epsilon }_{\mathit{up}\text{\_}a}$	0.753	-
Emissivity of the clay in upper part	${\epsilon }_{\mathit{up}\text{\_}b}$	0.75	-
Thermal conductivity of concrete	Kconc	0.8	w/mK

.

2.2. Experimental Validation

In this study, a water boiling test (WBT) was carried out to validate the theoretical thermal efficiency of the given cookstove. The water boiling test is a comprehensive assessment that evaluates the performance of a cooking stove by measuring several parameters including water mass loss, wood burned, and, optionally, pollutant data. The WBT uses these variables to calculate various performance metrics such as thermal efficiency (η_th), and is conducted in three different operational phases [28], which are as follows:

• Phase 1: High power (cold start)

At the beginning of the WBT, two-thirds of the pot was filled with tap water as specified by the procedure, and the initial condition of the WBT was recorded as summarized in

Table 4. The initial condition of the WBT.

Parameters	Value	Unit	Remarks
Ambient temperature	305	K
Air relative humidity	18	%	Reported in March [29]
Local boiling point	363.04	K	Test location: Katibougou, Mali
Density of ambient air	1.157	kg/m3
Wind condition	-	-	No wind
Fuel dimensions	300 × 50	mm
Equilibrium moisture content of the wood	3.95	%	Calculated [34]
Dry weight of the pot	9.93	kg
Pre-weighed firewood	13.64	kg
Lower heating value of firewood (LHV)	1.828 × 107	J/kg

. In this regard, the equilibrium moisture content of the firewood was calculated, using the air relative humidity reported in March [29] and the recorded ambient temperature at the Katibougo site, Mali. According to the studies, the moisture content of firewood can vary significantly depending on the specific type of wood, climate, and storage conditions [30,31,32,33]. For the WBT, local Malian firewood (Pterocarpus erinaceus) that was stored outdoors and exposed to the direct sunlight was provided from a local market.

The water boiling test was started with a high-power cold start, in which the pre-weighed firewood was used to boil the water using the cookstove under study. The fuel was fed into the stove and ignited, and then the temperature of the water in the pot was measured with a thermometer at regular 2 min intervals until the water reached the boiling point (Figure 4). Afterwards, the following approaches were achieved:

(a)

The time at which the water inside the pot reached the local boiling point was recorded along with the temperature.

(b)

All fuel samples were carefully extracted from the cookstove and the flame was extinguished by gently blowing on the ends to remove loose ash, which was then collected and placed in a container for subsequent weighing and analysis.

The mass of unburnt firewood extracted from the cookstove was identified and recorded.

(c)

The weight of the pot containing hot water was measured and the weight recorded.

(d)

The ash collected from the stove was measured and the weight recorded.

• Phase 2: High power (cold start)

The high-power hot start phase test was conducted directly following the high-power cold start phase to assess the performance of the stove in cold and warm conditions. The test methodology used for the high-power cold start phase was duplicated for the high-power hot start phase.

• Phase 3: Low Power (Simmering)

The final test in the WBT was the simmering phase test. In this step, the quantity of fuel needed to heat a certain amount of water to just below boiling point was evaluated. The fire was regulated to keep the water temperature slightly below the local boiling point (max. 6 degrees) for 45 min.

Table 5. Water boiling test parameters.

Parameters	Cold Start Phase		Hot Start Phase		Simmer Phase
	Start	Finish	Start	Finish	Start	Finish
Water temperature of the pot (K)	295.75	363.85	297.25	363.10	363.25	364.15
Weight of Pot with water (kg)	47.64	46.50	45.92	45.60	45.60	41.83
Weight of wood (kg)	9.58	6.31	6.31	4.18	4.18	3.01

shows the water boiling test parameters for the fixed ICS model. The corresponding operational parameters of the given fixed ICS were determined using water boiling test data [28].

3. Results and Discussion

In this work, the heat transfer characteristics of the single-mouth cookstove were analyzed theoretically and experimentally. First, a theoretical approach for three modes of heat transfer was applied as a tool to evaluate the overall thermal efficiency of the stove, and then the results were validated with experiments.

All heat transfer modes analyzed in this work are influenced by the flow characteristics of the combustion gasses through the stove geometry, as presented in

Table 6. Flow characteristics of the combustion gasses through the stove geometry.

Output Parameter
Parameters	Value	Unit	Remark
Average flue gas temperature in lower channel	1000	K	calculated based on [18]
Average flue gas temperature in upper channel	942	K	calculated based on [18]
Density of flue gasses in lower portion of the channel	0.353	kg/m3
Flue gas temperature in chimney	700	K	calculated based on [18]
Density of flue gasses in chimney	0.504	kg/m3
Overall mass flow rate through the stove	8.83 × 10−3	kg/s	assuming LC = 0.5
Gas velocity of the flu in channel	0.264	m/s
Gas velocity of the flue in chimney	2.23	m/s
Gas viscosity in channel	9.61 × 10−5	kg/m⋅s
Gas viscosity in chimney	7.28 × 10−5	kg/m⋅s
Thermal conductivity of gas in channel	0.068	W/m⋅K
Reynolds number in lower channel flow	233	-
Nusselt number in channel	13.19	-
Convection coefficient for lower channel flow	4.3	W/m2K
Convection coefficient for upper channel flow	1.832	W/m2K
Convection coefficient for pot bottom	11.98	W/m2K
Reynolds number between pot and drip pan	107	-
Reynolds number in chimney	1543	-
Nusselt number in chimney	33.07	-
Pressure difference due to chimney effect	14.39	Pa
Head loss in channel	1.12 × 10−3	m	calculated based on [18]
Head loss in chimney	0.101	m	calculated based on [18]
Pressure loss in channel	0.019	Pa	calculated based on [18]
Pressure loss in chimney	0.49	Pa	calculated based on [18]
Pressure loss at the elbow (chimney bottom)	0.75	Pa	calculated based on [18] Resistance coefficient for 90-degree bend, K90 = 0.3
Total pressure drop through the stove	1.259	Pa

. The parameters are determined by three dimensionless numbers, the Reynolds number (Re), Prandtl number (Pr), and Nusselt number (Nu), as well as properties that are dependent on the gas temperature. The Reynolds number is used to determine the flow behavior of the gas, whether it is laminar or turbulent, and to calculate the heat transfer coefficient in the combustion chamber and chimney. It requires knowledge of the gas velocity, gas density, length characteristics such as dimeter of the flow path, and dynamic viscosity of the flue gas. For this purpose, the average gas velocity required to calculate the Reynolds number was derived from the mass flow rate that is influenced by the pressure gradient generated by the chimney effect. The results show that the pressure difference due to the chimney effect is 14.39 Pa. To calculate the mass flow rate accurately, the friction and viscous losses of the flow caused by the stove geometry and heat lost at the chimney walls were calculated using the loss coefficient, as described in [18]. In addition, the pressure losses in the combustion chamber, chimney, and elbow at the chimney bottom were estimated separately, which can be considered for future design and operation optimization of the stove. The results indicate that the maximum pressure loss is caused by the elbow at the bottom of the chimney (0.75 Pa), followed by the pressure loss at the chimney (0.49 Pa) and the pressure loss at the channel (0.019 Pa).

The Prandtl number is another important dimensionless quantity that was used to calculate the Nusselt number and the subsequent convective heat transfer required to analyze the convective heat transfer mode. Afterwards, the contributions of the convective, conductive, and radiative heat transfer modes for the given cookstove were analyzed. Based on the theoretical results summarized in

Table 7. Energy balance of the entire stove for all terms of heat transfer contributions.

Parameters	Symbol	Value (Kw)	Relative Energy Contribution (%)
Energy gain
Radiation transfer to pot from fuel bed	Qrad_pot_ab	1.922	7.759
Radiation to pot from high flames	Qrad_flame_ab	2.801	11.307
Convection to the pot bottom	$Q_{\mathit{conv}\text{\_}\mathit{pot}}$	1.751	7.069
Convection to the pot sides	$Q_{\mathit{conv}\text{\_}\mathit{sides}}$	1.317	5.316
Total heat transfer to the pot (thermal efficiency)		7.791	31.451
Energy loss
Radiation from flames/fuel bed to lower combustion chamber	Qrad_low_ab	2.995	12.10
Radiation from flames to upper combustion chamber	Qrad_up_ab	5.889	23.774
Radiation lost by outer pot surface to ambient	Qrad_ sides	2.100	8.477
Conduction heat transfer through the concrete floor	$Q_{\mathit{conduct}}$	1.290	5.207
Convection from gasses to lower combustion chamber walls	$Q_{\mathit{conv}\text{\_}\mathit{low}}$	0.946	3.819
Convection from gasses to upper chamber walls	$Q_{\mathit{conv}\text{\_}\mathit{up}}$	0.099	0.399
Wasted heat out the chimney	$Q_{\mathit{waste}}$	3.66	14.773
Total energy accounted		24.77	100.00

, the highest energy gain was derived by the radiation to the pot from the high flames (11.30%) and the radiation transfer to the pot from the fuel bed (7.76%) followed by convection to the pot bottom (7.06%) and convection to the pot sides (5.32%), respectively. Accordingly, the theoretical efficiency of the stove was calculated by adding the convection and radiation gains together, resulting in an overall efficiency of 31.45%.

The theoretical model developed in this study not only quantifies the thermal efficiency of the biomass cookstove but also illustrates how various heat transfer mechanisms contribute to energy dynamics and provides several key insights that advance the understanding of the thermal efficiency of single-mouth biomass stoves. Firstly, it highlights the significance of radiation heat transfer in terms of energy gain achieved from the flame and fuel bed to the pot. This finding suggests that optimizing the stove design to improve the radiation heat transfer could significantly improve the thermal efficiency. Secondly, the theoretical model indicates that convection heat transfer to the pot bottom and pot sides is also an important contributor to energy gain. This has implications for stove design, because it implies that optimizing the pot design and placement within the cookstove could further enhance the thermal efficiency. Moreover, using pots made from materials with higher thermal conductivity can better facilitate heat transfer, thereby improving the overall cooking performance. Finally, the theoretical model allows us to obtain insight into the energy losses, which is a crucial factor for improving the stove design.

As can be seen from Table 7, the major energy losses are caused by the radiation of the flames into the upper combustion chamber (23.77%), followed by the energy loss from the chimney (14.77%); the radiation of the flames/fuel bed into the lower combustion chamber (12.10%); and the radiation lost by the outer pot surface to the ambient environment (8.47%), which all contribute to a lower thermal efficiency. Therefore, minimizing such energy losses can potentially increase the thermal efficiency of the stove. The minimum radiation loss can be achieved by improving the stove surface, insulating around the stove and optimizing the position and size of the pot. For example, a smooth surface can be created by applying a ceramic coating or using a metal with a high reflectivity. Additionally, insulation around the cookstove can reduce heat loss by reducing the temperature difference between the cookstove and the surrounding environment. Furthermore, optimization of the position and size of the pot can minimize radiation losses by adjusting the surface area exposed to the environment. Energy loss through the chimney is also a significant contributor to the overall efficiency of cookstove. In this context, insulation and optimization of the chimney diameter will reduce heat losses considerably by optimizing the surface area exposed to the surrounding environment. By implementing these approaches, a higher overall efficiency can be gained for the given stove.

In order to validate the theoretical thermal efficiency, a water boiling test was carried out and the theoretical findings were compared with the experiments. As can be seen from

Table 8. Derived parameters from WBT and data calculation sheet.

Parameters	Cold Start Phase	Hot Start Phase	Simmer Phase	Unit	Remarks
Wood consumed	3.27	2.13	1.17	kg
Water vaporized from pot	1.14	0.32	3.77	kg
Time to boil	50.00	26.00	46.00	min	Simmer test should be ~45 min
Thermal efficiency	24.00	30	42.00	%	** Overall thermal efficiency is 27%
Burning rate	0.062	0.078	0.025	kg/min
Firepower	19.16	23.80	7.62	kW

** The overall thermal efficiency of the stove corresponds to the average efficiency of the cold start high-power phase and hot start high-power phase only (i.e., not including the simmer phase) [35].

, the stove achieves a thermal efficiency of 24% in the cold start phase and 30% in the hot start phase, corresponding to an overall efficiency of 27%. Therefore, the experimental thermal efficiency is slightly lower than the theoretical thermal efficiency. In addition, a comparison of the theoretical and experimental results with the literature shows that the results are in the range in which the studies indicate to be the typical heat transfer efficiency of stoves, which is frequently in the range of 10 to 40% [19].

Despite the water boiling test being a widely accepted standard for evaluating stove performance, it has limitations in reflecting real-world cooking practices because it is conducted in a laboratory based-setup with trained technicians that may not accurately reflect how stoves are used in everyday life. To obtain a more accurate view of stove performance, especially with local foods and local cooking practices, it is necessary to use the controlled cooking test (CCT) in conjunction with the water boiling test [28]. The CCT not only captures the variability of actual cooking scenarios but it also accounts for user behavior, which plays a significant role in the effective use of the stove. Additionally, combining both testing methods can provide a more comprehensive understanding of energy efficiency under realistic cooking conditions, ultimately leading to stove designs that are better suited for the specific needs and practices of users in Malian cultural contexts.

It is obvious that the theoretical thermal efficiency is slightly higher than the experimental one. The reason lies in the simplifications of the model, as the theoretical thermal efficiency of the stove is calculated based on the steady-state energy balance for each mode of heat transfer, and combustion losses due to incomplete combustion of the fuel are unaccounted in this work. Based on the studies, typical combustion efficiencies for cooking stoves and even sometimes three-stone fires can be more than 90% [19]. Therefore, the theoretical thermal efficiency of the stove compared to the experimental thermal efficiency can be modified by taking the combustion losses into account.

However, the assumption of 100% combustion efficiency could have the following implications for the findings and potential real-world applications. Firstly, this assumption may overestimate the actual performance of the stove, resulting in an inaccurate representation of the stove’s ability to convert fuel into usable heat. Secondly, the results may not be directly comparable to other studies that have taken combustion efficiency into account. In addition, it might lead to conclusions that are not representative of real-world scenarios, where combustion efficiency is typically lower. Additionally, assuming a combustion efficiency of 100% may lead to designs that are not optimized for real-world applications or may underestimate emissions from the stove in real operations, leading to inaccurate estimates of environmental impact.

In view of the above points, it is crucial to consider incorporating parameters related to combustion efficiency in future studies. Doing so would enhance the accuracy of the results, ensuring that the findings better reflect the complexities of real-world cooking practices and contribute positively to stove design improvements and environmental sustainability.

An increase in the efficiency of a biomass stove can bring potentially three major benefits to households: (1) a reduction in firewood consumption resulting in economic and time-related benefits and increasing the sustainable use of the natural resources; (2) a reduction in human exposure to harmful air pollutants; and (3) a reduction in greenhouse gas emissions, which are expected to increase the risk of global climate change [36]. In general, an efficient cookstove can save a family around 700 kg of firewood annually, while reducing CO₂ emissions by 161 kg at the same time [37].

4. Conclusions

This study has focused on three modes of heat transfer as key parameters for determining the thermal efficiency of a single-mouth cookstove. In this context, a theoretical approach was used to identify the energy gains and losses of each heat transfer mode, and then the theoretical thermal efficiency of the stove was calculated based on the convection and radiation gains. In order to validate the theoretical thermal efficiency, a water boiling test was performed in the laboratory and the theoretical finding was compared with the experimental values. The main conclusions of the study are summarized below:

• The maximum energy gain was achieved by radiation to the pot from the high flames and radiation transfer to the pot from the fuel bed followed by convection to the pot bottom and convection to the pot sides, respectively.
• The thermal efficiency calculated on basis of the theoretical approach agrees closely with the experimental values.
• By considering the combustion losses, the theoretical thermal efficiency of the stove can be modified compared to the experimental values.

Although the theoretical approach can estimate the heat transfer and thermal efficiency of the single-mouth cookstove, it is not able to assess the combustion-related factors for CO, CO₂, and combustion efficiency. Thus, research into combustion parameters can further enhance the results of this study.

To achieve a more accurate assessment of stove performance, it is crucial to integrate the controlled cooking test (CCT) with the water boiling test, as this combination allows for a comprehensive evaluation of stove performance under conditions that reflect local food choices and cooking practices.