1 Introduction

The biomass cookstove usage is as old as human civilization. Majority of the world population (55%) lives in rural areas, and this situation will remain until 2050 [1] who depend primarily on biomass. Even today, the biomass cookstove is the first choice of rural people in many developing countries. India is one such country, where 70% of its population still lives in rural areas and about 66% and 32% of its rural and urban population still depend on traditional biomass, as a source of total primary energy need, as per the report of Ministry of renewable energy (MNRE), India. The traditional biomass cookstoves suffered with inefficient design and more emissions, causing environmental and health hazard issues. Modernization of the biomass cookstoves, in India, started in the 1940s in terms of cookstove design, alternative fuel used and to reduce indoor air pollution (IAP) [2]. The first improved model of biomass cookstove developed in the late 1940s was Magan Chulha, reported in the literature. However, the success was limited in terms of performance and efficiency. During the 1950s, the initiative was taken by the western countries in developing the improved biomass cookstoves for the rural people of third world country [3]. The oil crisis during 1970 also helped in generating the wave for improvement in biomass cookstoves. It also dragged the total attention towards environmental issues and conservation measures associated with biomass cookstoves. The scientific progress in the design of biomass cookstove started in the 1980s. Figures 1 and 2 show the progress in the field of biomass cookstove in terms of research papers published year-wise and country-wise, respectively [4]. In the present time, a wide variety of developed cookstoves are available for usage by the locals in the different parts of the world [5,6,7,8,9,10]. India launched the first programme on improved Chulhas (or biomass cookstoves), National programme on improved Chulha (NPIC) during 1985–86 [11]. Baldwin [5] introduced general semi-empirical guidelines to design and improve the performance of cookstove. In 1954, the first laboratory test was conducted by Theodorovic in Egypt. The design aspects of the stove were shifted to the monitoring of exhaust coming out of the stove for measuring its performance during the early 1980s. The various methods were used to measure the performance parameters of the stove. Different testing protocols for the working of cookstove were also developed. The testing methods help to compare different stoves (Chulhas), considering various parameters. In 1980, Intermediate Technology Development Group (ITDG) came up with the procedure for testing of a cookstove in the laboratory as well as in the field [11]. Subsequently, in 1982 Volunteers in Technical Assistance (VITA) carried out further work on the above to release a draft protocol as a provisional international standard [12]. These standard procedures were reviewed and accepted in 1985 by several groups working on cookstoves. Therefore, the three testing protocols, namely water boiling test (WBT), kitchen performance test (KPT) and controlled cooking test (CCT), are the most popular testing methods used by the researchers. In 1991, Bureau of Indian Standards (BIS) adopted the water boiling test as its own standard to test the biomass cookstoves, which also included the measurement of CO/CO2 ratio as a separate test. The summary of different testing protocols is well explained in research articles [7, 8].

Fig. 1
figure 1

Source: Web of science

Publication with years worldwide

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Fig. 2
figure 2

Source: Web of science

Country-wise publications

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The current investigations are mostly focused on the reduction of pollutants coming out from the traditional cookstoves by introducing the “improved single-pot” cookstoves [13,14,15,16]. Many parameters, influencing the performance of cookstoves, were identified and studied by researchers [17,18,19,20,21,22,23,24,25,26]. The effect of moisture content [21, 22], different types of fuels [21, 23, 24] and different design aspects [5, 25, 26] are investigated. Few researchers [27, 28, 30] also conducted the field surveys and evaluation of many improved “single-pot” cookstoves. However, the majority of work so far deals with the single-pot cookstoves, and very few work has been reported on “multipot” natural draft biomass cookstoves. Most of the houses in rural areas in India use a multipot biomass cookstove [29].

The recent research on the performance investigations with the experiment on the multipot cookstoves lack geometrical design parameters consideration. Moreover, these experiments were conducted on a trial and error basis. Till date, no contemporary research provides a mathematical model on the multipot cookstove. There are few investigations on the multipot cookstove: Oanh et al. [22], in his research work, presented the results of 12 selected wood-burning cookstove used in Asia. Out of the 12, metallic and ceramic made Nepalese 2-pots were the multipot cookstoves used. The overall performance of the stove was poor with thermal efficiency lower than 20%. The emission of particulate matter and polyaromatic hydrocarbon was highest among the other stoves. Bhattacharya et al. [26] compared the emission of the Nepalese cookstoves used by Oanh [22] with wood and charcoal. The thermal efficiencies were below 15%, and the emission factors were also not appreciable. The emission factors for ceramic and metallic pot cookstoves were around 113 and 45 g/kg of fuel used, respectively. Honkalaskar et al. [30] performed experiments on two-pot and three-pot cookstoves by inserting the twisted tape device. The experimental results showed a substantial decrease in the fuel consumption and soot accumulation. Also, the cooking time was reduced by around 18.5%. However, the making of twisted tape is not an easy task for the local artisans/blacksmith; it involves cutting, drilling, heating, twisting and welding operations. Though the authors claim to show some improvement in the performance of cookstove, no detailed analysis was presented in the paper. Joshi and Srivastava [31] performed experiments on three-pot cookstove and compared the results with the traditional mud stove. The paper failed to present the detailed analysis of WBT results. MaCarty et al. [32] presented the results of fifty cooking stoves in the laboratory, and few of the stoves were multipot cookstove. They used two-pot cookstoves like Uganda 2-pot cookstove, Onil stove and Justa stove along with the other stoves. All the stoves were provided with the chimney and used sunken pots. The efficiencies for Onil and Justa stoves are below 25% with the highest fuel consumption, among the fifty cookstoves. The highest efficiency of 35% was obtained with Uganda 2-pot; however, the diameter of the vessel used for operation was less than that of the cookstove hole, restricting the users to use specific vessels for the stove. The chimney also needed continuous monitoring to keep the “indoor air” pollutants free.

2 Need of study

The use of firewood in rural areas is still predominant, since it is often the only acceptable, accessible and affordable fuel in the region. Acceptability of firewood is very high since ancient times and has therefore shaped cooking habits accordingly. Accessibility of firewood is a crucial factor as it is available closely to their homestead for cooking purpose, especially in rural areas where LPG is not easily available. Also, the biomass is available year-round and not susceptible to heavy seasonal fluctuations. Affordability plays a decisive role in the use of firewood for cooking, as many households can collect firewood for free, so it remains the cheapest energy source for cooking and heating. However, it must be recognized that firewood would be extremely expensive if the additional cost of labour done by women and children collecting firewood were considered and the negative impacts on health and the environment are internalized.

Around 3–4 million people prematurely die every year, and many more are affected by morbidity due to indoor air pollution (IAP). Speaking about India, the reports from the India State-Level Disease Burden Initiative report 2017 [34], around 12.4 lakh deaths occurred due to air pollution, out of which 4.8 lakh deaths were attributed to IAP. The root cause for IAP is the exhaust coming out of the traditional biomass stoves. The toxic gases like carbon monoxide, polycyclic aromatic hydrocarbons and soot particles contribute to IAP [23, 33,34,35]. The health-damaging carbon monoxide and particulate matter cause chronic obstructive lung disease, respiratory infections and many such fatal diseases [36]. According to a World Health Organization report [37], smoke from biomass cookstove inhaled by the operator is equivalent to smoking 400 cigarettes in an hour and causes severe respiratory and other diseases. Out of the million deaths globally, 27% are due to pneumonia, 18% from stroke, 27% from ischemic heart disease, 20% from chronic obstructive pulmonary disease (COPD), and 8% from lung cancer. Majority of the deaths occur among the person handling (who happen to be females) the cookstove, and hence, awareness programmes regarding the adverse effect of IAP should be conducted. In spite of such fatal consequences, people have not stopped using cookstoves, especially in rural areas. However, in urban areas, firewood for cooking has been dominantly replaced by LPG or other modern fuels [38, 39]. The main reason is the free availability of fuel and the least maintenance cost of cookstoves. Every house in a rural place uses a single- or multipot biomass cookstove.

The use of biomass cookstove has also contributed to deforestation over past years in developing countries. The fuel consumption is high due to very low thermal efficiency of traditional biomass cookstove. So it is very necessary to know the dependency of different parameters to enhance the thermal performance of the cookstove. The major hindrance is the measurement of the performance on the field, as the result differ quickly. Therefore, a mathematical model is proposed to save time and energy resource. Hence, the present work is an attempt to minimize the efforts required for the experimentation process and avoid the costly instruments to predict the performance of the cookstove. The work comprises mathematical modelling, experimentation and computational approach to determine the performance of cookstoves.

2.1 Methodology

2.1.1 Mathematical model

The modified geometry is selected from the two-pot cookstoves used in the rural parts of Maharashtra by doing the survey [29]. The model geometry consists of a single feeding zone, a primary combustion chamber and a secondary zone where second pot can be placed.

The model is shown in Fig. 3 . The cookstove is divided into different zones as shown in Fig. 3.

Fig. 3
figure 3

Modified two-pot cookstove model

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  • Zone 1: a primary zone where the combustion takes place

  • Zone 2: an intermediate zone which connects zone 1 and zone 3

  • Zone 3: a secondary zone where flame from zone 1 propagates

  • Zone 4: pot 2 bottom

  • Zone 5: pot 2 gap

  • Zone 6: sides of pot 2

2.1.2 Assumptions

The cookstove involves a complex combustion phenomena. The heat transfer occurs through conduction, convection and radiation. To simplify such complex phenomena, certain assumptions were made without neglecting the important aspects of cookstove operation. The assumptions are as follows:

  1. 1.

    The temperature and the emissivity of burning char are taken as 1100 K and 0.85, respectively [20, 40,41,42].

  2. 2.

    The emissivity of the stove inner surface and pot bottom is 1. (The few trials before the final reading turn both the surfaces black due to soot deposition).

  3. 3.

    The pressure drop across the combustion bed is negligible.

  4. 4.

    There is no pot gap between zone 1 and pot 1; hence, the mass of flue will remain constant throughout all the zones till exit.

The correlations used to model the two-pot cookstove are as follows:

$$\mu = 0.3501 \times T^{0.69} \times 10^{ - 6}$$
(1)
$$k = 8 \times 10^{ - 5} \times T^{1.0152}$$
(2)
$$\rho = \frac{359.7}{{T^{1.003} }}$$
(3)
$$C_{p} = 415.6 \times T^{0.1591}$$
(4)
$$\vartheta = 0.0009 \times T^{1.7097} \times 10^{ - 6}$$
(5)
$$h_{s} = 0.00032273 \times T^{2} + 1.64833661388 \times T + 2021.531847$$
(6)

2.1.3 Mathematical equations

Zone 1

Combustion chamber

The fuel combustion takes place in this zone. It is the feeding zone, through which the fuel input is metered. Thus, the amount of air entering is controlled by changing the inlet area ratio (IAR) [20]. The losses like char radiation, flame radiation, heat losses through the walls and feed door (inlet opening) are taken into consideration. In addition, sensible heat loss of hydrogen and moisture present in fuel are substantial and are included in the modelling.

$${\text{IAR}} = \frac{{{\text{Area}}\;{\text{unoccupied}}\;{\text{by}}\;{\text{the}}\;{\text{feed}}\;{\text{door }}}}{{{\text{cross - sectional}}\;{\text{area}}\;{\text{of}}\;{\text{the}}\;{\text{chimney}}}}$$
(7)

Applying heat balance for Zone 1,

$$Q_{1} = Q_{2} - \left( {Q_{3} + Q_{4} + Q_{5} + Q_{6} + Q_{7} + Q_{8} } \right)$$
(8)

where

Q1 is the heat gain by the flue gases and Q2 is the actual amount of heat supplied by the fuel, while Q3, Q4, Q5, Q6, Q7 and Q8 are different heat transfer losses in zone 1 as shown in Fig. 4.

Fig. 4
figure 4

Sankey diagram for heat balance (for FP—3.119 kW)

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$$Q_{1} = m_{\text{actual}} \times C_{p\;fg1} \times \left( {T_{fg1} - T_{\text{amb}} } \right)$$
(9)
$$m_{\text{actual}} = C_{\text{d}} \times \frac{359.7}{{T_{fg1}^{1.003} }} \times A_{\text{cross}} \sqrt {2 \times g \times \left( {H' - D_{2} - t} \right)\left( {\frac{{T_{fg1} }}{{T_{\text{amb}} }} - 1} \right)}$$
(10)

Cd is the coefficient of discharge which accounts for losses occurring in the cookstove due to viscous effects and flow distribution of combustion process. The value of Cd varies from 0 to 1 for the different types of chimneys [17, 43, 44]

$$Q_{2} = \dot{m}_{f} \times {\text{NCV}}$$
(11)
$$\dot{m}_{f} = \frac{{\text{FP}}}{\text{NCV}}$$
(12)

Using the radiation network method [45], char bed radiative heat transfer and flame radiative heat transfer (Q3 and Q4) are given as:

$$Q_{3} = \frac{{\sigma \times A_{\text{cross}} \times \left( {T_{\text{char}}^{4} - T_{p1}^{4} } \right)}}{{\frac{{1 - \varepsilon_{c} }}{{\varepsilon_{c} }} + \frac{2}{{1 + F_{c - p} }}}}$$
(13)
$$Q_{4} = \sigma A_{1} \left( {\varepsilon_{1} T_{fg1}^{4} - \alpha_{1} T_{p1}^{4} } \right)$$
(14)

The standard expressions for emissivity and absorptivity in terms of temperature [46] are given as,

$$\varepsilon_{1} = e^{{A + B\ln \left( {0.2*3.6\frac{{V_{\text{pm}} }}{{A_{\text{pm}} }}} \right)}}$$
(15)
$$\alpha_{1} = \varepsilon_{1} \times \left( {\frac{{T_{fg} }}{{T_{s} }}} \right)^{0.5}$$
(16)
$$A = 0.848 + 9.02 \times 10^{ - 4} T_{fg1}$$
(17)
$$B = 0.9589 + 4.8 \times 10^{ - 6} T_{fg1}$$
(18)

where

$$V_{pm} = \frac{\pi }{4} \times D_{1}^{2} H'$$
(19)
$$A_{pm} = \frac{\pi }{4} \times D_{1}^{2}$$
(20)

The heat loss from flame to outer surrounding in zone 1 is given as,

$$Q_{5} = \frac{{T_{fg1} - T_{\text{amb}} }}{{\frac{1}{{(h_{ci1} + h_{rfl1} + h_{rch1} )A_{i1} }} + \frac{{\ln \left( {\frac{{D_{o1} }}{{D_{1} }}} \right)}}{{2\pi H'_{1} K_{\text{insu}} }} + \frac{1}{{\left( {h_{co1 + } h_{ro1} } \right)A_{o1} }}}}$$
(21)

The convective heat transfer coefficient of inside combustion chamber is estimated using the Dittus–Boelter equation as given as [47]:

$$h_{ci1} = \frac{{k_{1} }}{{D_{1} }} \times 0.023 \times Re_{{D_{1} }}^{0.8} \times Pr^{0.4}$$
(22)

The convective heat transfer coefficient for an outer wall is derived for all values of Gr × Pr, considering constant heat flux.

$$h_{{{\text{co}}1}} = \frac{{k_{{{\text{flue}}\;{\text{gas}}}} }}{{H'_{1} }} \times \left[ {0.825 + \frac{{0.387\left( {Gr \times Pr} \right)^{0.167} }}{{\left\{ {1 + \left[ {\frac{0.437}{Pr}} \right]^{0.5625} } \right\}^{0.296} }}} \right]^{2}$$
(23)
$$h_{{{\text{ro}}1}} = \sigma \varepsilon_{2} \left( {T_{{{\text{wo}}1}}^{2} + T_{\text{amb}}^{2} } \right)\left( {T_{{{\text{wo}}1}} + T_{\text{amb}} } \right)$$
(24)
$$h_{{{\text{flr}}1}} = \frac{{\sigma \left( {\varepsilon_{1} T_{fg1}^{4} - \alpha_{1} T_{wi1}^{4} } \right)}}{{T_{fg1} - T_{wi1} }}$$
(25)
$$h_{\text{chr}} = \frac{{\sigma \left( {T_{\text{char}}^{2} + T_{wi1}^{2} } \right) \times \left( {T_{\text{char}} + T_{wi1} } \right)}}{{\frac{{1 - \smallint_{c} }}{{\varepsilon_{c} }} + \frac{{F_{c - w} + F_{c - p} }}{{F_{c - w} \left( {F_{c - w} + 2 \times F_{c - p} } \right)}}}}$$
(26)

The shape factor [46] and configuration factors [48, 49] are calculated using

$$F_{c - p} = \frac{{\left( {D_{1}^{2} + 2\left( {H'_{1} + W_{i1} } \right)^{2} - 2\left( {H'_{1} + W_{i1} } \right)\sqrt {D^{2} + \left( {H'_{1} + W_{i1} } \right)^{2} } } \right)}}{{D^{2} }}$$
(27)
$$F_{c - w} = 1 - F_{c - p} - F_{c - d}$$
(28)

The losses of energy in feeding zone, i.e. from the combustion chamber bed to feeding door, are as follows:

$$Q_{6} = F_{c - d} \times \sigma \times A_{\text{cross}} \times \left( {\left( {T_{\text{char}} } \right)^{4} - \left( {T_{\text{amb}} } \right)^{4} } \right)$$
(29)
$$Q_{7} = \left( {h_{s} - 2547} \right)*0.09*H_{2} *\dot{m}_{f} \times 1000$$
(30)
$$Q_{8} = \left( {h_{s} - 4.187 \times \left( {T_{\text{amb}} - 273} \right) \times \left( {\frac{M}{100 - M}} \right) \times \dot{m}_{f} } \right) \times 1000$$
(31)

Also,

Heat transfer from the flame to inner wall = heat transfer from inner wall to outer wall = heat transfer from outer wall to the surrounding,

$$Q_{9} = \frac{{\left( {T_{fg1} - T_{wi1} } \right)}}{{\frac{1}{{\left( {h_{ci1} + h_{flr1} + h_{\text{chr}} } \right) \times A_{i1} }}}} = Q_{10} = \frac{{\left( {T_{wi1} - T_{wo1} } \right)}}{{\frac{{{ \ln }\left( {\frac{{D_{o1} }}{{D_{1} }}} \right)}}{{2\pi \times H' \times K_{\text{ins}} }}}} = Q_{11} = \frac{{\left( {T_{wo1} - T_{\text{amb}} } \right)}}{{\frac{1}{{\left( {h_{co1} + h_{ro1} } \right) \times A_{o1} }}}}$$
(32)

Solving Eq. (8) and Eq. (32), we get the values of Tfg1, Twi1 and Two1.

Zone 2

The flue gas temperature in zone 2 (Tfg2) is estimated by averaging the temperature Tcentral1 at the entrance of the zone 2 and Tcentral2 at the exit. The equations used to determine the value of Tcentral1 are as follows:

$$Q_{\text{pot1}} = \left( {h_{\text{conv}} + h_{\text{rad}} } \right) \times A_{\text{pot1}} \times \left( {\frac{{T_{fg1} + T_{\text{central1}} }}{2} - T_{p1} } \right)$$
(33)
$$h_{\text{rad}} = \in_{1} \times \sigma \left( {\frac{{T_{fg1} + T_{{{\text{central}}1}} }}{2} + T_{wi} } \right)\left( {\left( {\frac{{T_{fg1} + T_{{{\text{central}}1}} }}{2}} \right)^{2} + T_{wi}^{2} } \right)$$
(34)
$$Q_{1} - Q_{{{\text{pot}}1}} = m_{\text{actual}} \times C_{p\;fg2} \times \left( {T_{{{\text{central}}1}} - T_{\text{amb}} } \right)$$
(35)

The equations used to determine the value of Tcentral2 are as follows:

$$m_{\text{actual}} \times C_{p\;fg1 - 2} \left( {T_{{{\text{central}}1}} - T_{\text{amb}} } \right) - Q_{12} = m_{\text{actual}} \times C_{p\;fg2 - 3} \left( {T_{{{\text{central}} 2}} - T_{\text{amb}} } \right)$$
(36)
$$Q_{12} = \frac{{T_{\text{central}} - T_{\text{amb}} }}{{\frac{1}{{(h_{ci2} + h_{flr2} )A_{i2} }} + \frac{{ln\left( {\frac{{D_{o2} }}{{D_{2} }}} \right)}}{{2\pi H'K_{\text{insu}} }} + \frac{1}{{\left( {h_{co2 + } h_{ro2} } \right)A_{o2} }}}}$$
(37)

The convective heat transfer coefficient for the inner flame in zone 2 is given as:

$$h_{ci2} = \frac{{k_{{{\text{flue}}\;{\text{gas}}}} }}{L} \times \left[ {4.82 + 0.0185 \times \left( {Re_{D} \times Pr} \right)^{0.827} } \right]$$
(38)
$$h_{ro2} = \sigma \varepsilon_{5} \left( {\left( {T_{wo1} } \right)^{2} + \left( {T_{\text{amb}} } \right)^{2} } \right)\left( {T_{wo1} + T_{\text{amb}} } \right)$$
(39)
$$h_{co2} = \frac{{k_{{{\text{flue}}\;{\text{gas}}}} }}{L} \times \left\{ {0.6 + 0.387\left[ {\frac{{Gr_{D2} \times Pr}}{{\left\{ {1 + \left( {\frac{0.559}{Pr}} \right)^{0.5625} } \right\}^{0.296} }}} \right]^{0.167} } \right\}^{2}$$
(40)

The convective heat transfer coefficient (hco2) for an outer wall is obtained at film temperature.

Again, from the analogy of multimode heat transfer for the wall in zone 2, we have,

$$Q_{13} = \frac{{T_{\text{central}} - T_{wi2} }}{{\frac{1}{{\left( {h_{ci2} + h_{rfl2} } \right) \times A_{i2} }}}} = Q_{14} = \frac{{T_{wi2} - T_{wo2} }}{{\frac{{\ln \left( {\frac{{D_{o2} }}{{D_{2} }}} \right)}}{{2 \times \pi \times L \times K_{\text{ins}} }}}} = Q_{15} = \frac{{T_{wo2} - T_{\text{amb}} }}{{\frac{1}{{\left( {h_{co2} + h_{ro2} } \right) \times A_{o2} }}}} = Q_{12}$$
(41)
$$T_{\text{central}} = \frac{{T_{{{\text{central }}1}} + T_{{{\text{central}} 2}} }}{2} = T_{fg2}$$
(42)

Solving Eqs. (35), (36) and (41), we get Tcentral 2, Twi2 and Two2.

Zone 3 and zone 4 (pot 2)

The different heat zones for pot 2 have been solved for the various parameters using the heat balance equations

$$m_{\text{actual}} \times C_{p4} \times \left( {T_{{{\text{central }}2}} - T_{\text{amb}} } \right) - Q_{16 - 27} - \left( {m_{\text{actual}} \times C_{p5} \times \left( {T_{e3} - T_{\text{amb}} } \right)} \right) - Q_{{{\text{pot}}2}} = m_{\text{actual}} \times c_{p} \left( {T_{e3} - T_{\text{amb}} } \right)$$
(43)

The heat loss from the wall of stove body in zone 3 of pot 2 is given as

$$Q_{16} = \frac{{T_{fg3} - T_{\text{amb}} }}{{\frac{1}{{(h_{ci3} + h_{flr3} )A_{i3} }} + \frac{{\ln \left( {\frac{{D_{o3} }}{{D_{3} }}} \right)}}{{2\pi H'_{2} K_{\text{insu}} }} + \frac{1}{{\left( {h_{co3 + } h_{ro3} } \right)A_{o3} }}}}$$
(44)

The different heat transfer coefficients for pot 2 are determined in the similar fashion as that of pot 1

$$h_{ci3} = \frac{{k_{{{\text{flue}}\;{\text{gas}}}} }}{{D_{3} }} \times 0.023 \times Re_{{D_{1} }}^{0.8} \times Pr^{0.4}$$
(45)
$$h_{co3 } = \frac{{k_{{{\text{flue}}\;{\text{gas}}}} }}{{\left( {D_{2} + 2 \times t} \right)}} \times \left[ {0.825 + \frac{{0.387\left( {Gr \times Pr} \right)^{0.167} }}{{\left\{ {1 + \left[ {\frac{0.437}{Pr}} \right]^{0.5625} } \right\}^{0.296} }}} \right]^{2}$$
(46)
$$h_{ro3} = \sigma \times \varepsilon_{6} \times \left( {T_{wo3}^{2} + T_{\text{amb}}^{2} } \right)\left( {T_{wo3} + T_{\text{amb}} } \right)$$
(47)

Using the multimode heat transfer through walls of zone 3, we have,

$$Q_{17} = \frac{{T_{fg3} - T_{wi3} }}{{\frac{1}{{\left( {h_{ci3} + h_{flr3} } \right)A_{i3} }}}} = Q_{18} = \frac{{T_{wi3} - T_{wo3} }}{{\frac{{\ln \left( {\frac{{D_{o3} }}{{D_{3} }}} \right)}}{{2\pi \times \left( {D_{2} + t} \right) \times K_{\text{ins}} }}}} = Q_{19} = \frac{{T_{wo3} - T_{\text{amb}} }}{{\frac{1}{{\left( {h_{co3} + h_{ro3} } \right)A_{o3} }}}} = Q_{16}$$
(48)
$$Q_{{{\text{pot}}2}} = \left( {h_{{{\text{conv}}3}} + h_{{{\text{rad}}3}} } \right) \times A_{{{\text{pg}}1}} \times \left( {\frac{{T_{fg3} + T_{e3} }}{2} - T_{p2} } \right)$$
(49)
$$h_{{{\text{conv}}3}} = \frac{{k_{\text{bottom}} }}{{D_{3} }} \times 0.726 \times Re_{D2}^{0.53} \times \left( {\frac{{W_{i} }}{{D_{2} }}} \right)^{ - 0.191}$$
(50)
$$h_{{{\text{rad}}3}} = \sigma \times \varepsilon_{7} \left( {\frac{{\left( {T_{fg3} + T_{e3} } \right)}}{2} + T_{wi3} } \right)\left[ {\left( {\frac{{T_{fg3} + T_{e3} }}{2}} \right)^{2} + \left( {T_{wi3} } \right)^{2} } \right]$$
(51)

The change in enthalpy of flue gas leaving zone 3 is,

$$Q_{20} = m_{\text{actual}} \times \left( {h_{fg3} - h_{e3} } \right) \times 1000$$
(52)

The heat gain by flue gases leaving zone 3 is taken by pot bottom,

$$Q_{21} = h_{{{\text{conv}}3}} \times A_{3} \times \left( {\frac{{T_{fg3} + T_{e3} }}{2} - T_{p2} } \right)$$
(53)

Solving the above equations, we can get the values of Te3 and Tfg3, Twi3, Two3

Zone 5 (for pot 2)

Pot 2 gap

The heat transfer on pot bottom surface using heat balance is given as

$$Q_{22} = m_{\text{actual}} \times \left( {h_{e3} - h_{c3} } \right) \times 1000$$
(54)
$$Q_{23} = h_{{{\text{pot}}\;{\text{gap}}}} A_{\text{pg}} \times \left( {\frac{{T_{c3} + T_{e3} }}{2} - T_{p2} } \right) + \sigma \times A_{\text{pg}} \times \left( {\varepsilon_{8} \left( {\frac{{T_{c3} + T_{e3} }}{2}} \right)^{4} - \alpha_{4} \left( {T_{p2} } \right)^{4} } \right)$$
(55)

And, the heat transfer coefficient is evaluated by,

$$h_{{{\text{pot}}\;{\text{gap}}}} = \frac{{k_{\text{pg}} }}{{D_{3} }} \times 0.726 \times Re_{D2}^{0.53} \times \left( {\frac{{W_{i} }}{{D_{3} }}} \right)^{ - 0.191}$$
(56)

Solving Eqs. (54) and (55), we get Tc3

Pot 2 side (zone 6)

The convective and radiative heat transfer from flue gas coming out through zone 3 increases efficiency of the cookstove. The heat losses from the sides of the pot to the surroundings are also considered in this zone.

The convective and the radiative heat transfer in this zone is calculated using the following equations

$$Q_{24} = h_{{{\text{side}}3}} \times A_{{{\text{side}}1}} \times \left( {\frac{{T_{c3} + T_{o3} }}{2} - T_{p2} } \right)$$
(57)
$$Q_{25} = \sigma \times A_{{{\text{side}}1}} + \left\{ {\varepsilon_{9} \left( {\frac{{T_{c3} + T_{o3} }}{2}} \right)^{4} - \alpha_{5} \times T_{p2}^{4} } \right\}$$
(58)

The heat loss from pot side to the surrounding is calculated by assuming the combined convective and radiative heat transfer coefficient to be 10 W/m2K.

$$Q_{{{\text{side}}\;{\text{loss}}}} = 10 \times A_{{{\text{side}}1}} \times \left( {\frac{{T_{c3} + T_{o3} }}{2} - T_{\text{amb}} } \right)$$
(59)
$$h_{\text{side}} = \frac{{k_{\text{side}} }}{{h_{w} }} \times 0.664 \times \left( {Re_{\text{hw}} } \right)^{0.5} \times Pr^{0.333}$$
(60)
$$Q_{27} = m_{\text{actual}} \times \left( {h_{c3} - h_{o3} } \right) \times 1000$$
(61)
$$Q_{26} = Q_{24} + Q_{25} + Q_{{{\text{side}}\;{\text{loss}}}}$$
(62)

Solving Eq. (62) gives To3

Thus, the overall efficiency can be calculated as,

$$\eta_{o} = \frac{{Q_{{{\text{pot}} 1}} + Q_{{{\text{pot}} 2}} + Q_{24} + Q_{25} + 0.67Q_{3} }}{\text{FP}} \times 100$$
(63)

2.2 Performance evaluation criterion

The geometrical parameters were selected by considering the thermal efficiency and excess air ratio (EAR). The values of thermal efficiency for natural draft biomass cookstove should be above 25% as per the MNRE, India. Also, as per the guidelines for evaluating the performance of cookstove given by international workshop agreement (ISO 2012), the stove is rated between five-tiered (0–5) rating system for fuel use and emissions. For the stove to lie above Tier 2, the efficiency of the stove should be greater than 25%. Since the model fails to present the exhaust, an indirect method of the evaluation of combustion is applied. As excess air has some advantages and disadvantages, it should have some safe limit. Excess air increases the turbulence and promotes mixing in the combustion chamber ensuring complete fuel combustion. It converts the harmful gases of CO into CO2. It also lowers the formation of unburned hydrocarbons. However, the excess air reduces the thermal efficiency and combustion temperature and increases the value of NOx. Hence, there has to be some limit for the value of EAR. Hence, there has to be some limit of excess air ratio (EAR). Prasad et al. [50] suggested that the excess air factor should lie between 1.5 and 2.5 for safe working of stove. Liu et al. [51] claimed that the optimize value for EAR is 2. Hasen et al. [52] suggested the best wood combustion occurs when the value ranges between 1.4 and 1.6. Carvalho et al. [53] recommended the EAR value of 1.6–2.2 for minimum CO emissions. Obaidullaha et al. [54] performed experiments on wood stove with capacity 10 kW and 20 kW and found the EAR values of 1.76 and 2.05 for minimum CO emissions for the 10 kW and 20 kW, respectively. Kshirsagar and Kalamkar [34] suggested the region of good combustion which is bounded by limiting value of the excess air ratio between 1.95 and 5.98. Ingwald et al. [55] claimed that for gas burner the excess air factor is around 1.2 for almost complete gas burnout regarding CO and organic compounds. Jian Sun et al. [56] performed experimentation on wood-burning stove and measured the value of exhaust. It was found that the optimized EAR is 2.5. Hence, considering the above literature, the limiting value of EAR was set to be between 1 and 6 for safe working of stove.

2.3 Experimental method

The experimental set-up is shown in Fig. 5. The insulation was provided with ceramic wool to prevent the heat losses. Babool wood was used as fuel. The recorded quantities were mass of fuel required to boil water from pot 1, water temperature in both the pots, velocity of flue, time taken to boil, amount of char left and exhaust emissions, i.e. CO, CO2, O2 and particulate matter(PM). Table 1 shows the specifications of the instrument used. The duct with the cross section of 65 × 65 cm2 was used to collect the exhaust gas. An exhaust fan was connected at the end of the duct, and the velocity of the fan was controlled by a dimmer. The anemometer probe was placed exactly at the centre of the duct. Care was taken to control the velocity such that the combustion flame should not get disturbed. The probe for measuring the PM and a K-type thermocouple to measure the flue gas temperature was placed inside the hood. Thus, calculating the dilution ratio the values of PM2.5 were obtained in μg/m3. The time taken to boil water from pot 1 was noted down by a stopwatch. The weight of the charcoal left was recorded to measure the exact dry fuel consumed. The firepower (FP) values obtained through three sets of water boiling test (WBT) were used as input for the proposed mathematical model. The flame temperature, thermal efficiency and exhaust obtained through the experimentation were compared with the mathematical model results.

Fig. 5
figure 5

Schematic of experimental set-up

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Table 1 Instrumentation used in experimentation
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2.4 Computational

The mathematical model has no provision to calculate the values of exhaust gases, and hence, using the mass flow rate of air from the mathematical model as an input to the computational model the exhaust coming out of the stove is calculated. Along with the exhaust values, the temperatures at different zones were calculated and validated with the experimental data. ANSYS Fluent 16.0 was used to simulate the combustion. The assumptions used are:

  1. 1.

    Since the average parameters over a length of operational time are more important, the operation of cookstove is assumed to be at steady state for a particular inlet area ratio and firepower [4, 46, 57,58,59].

  2. 2.

    A two-dimensional (2-D) model is considered for solving the computational domain to save the computational time [58, 60,61,62].

  3. 3.

    The exhaust generated in the domain does not undergo any further chemical change.

2.4.1 Grid independence test

It is important to have a grid independence test before solving any computational problem. It helps to save the computational time required to solve the given problem. Two temperature points T1 and T2 in the computational domain were monitored. Table 2 shows the independence test done for firepower 2.99 kW. The number of nodes was varied from 117,051 to 253,748. The values of T1 and T2, at 170,410, 209,340 and 253,748 nodes were having the difference of less < 1%, so the simulation was carried out with 209,340 nodes.

Table 2 Grid independence test
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The computational domain with the boundary conditions is shown in Fig. 6. The values of mass flow rates for different firepowers are given in Table 3. The effect of radiative and convective heat transfer is taken into account though walls and pot bottoms of the cookstove. At the exit, the pressure outlet condition is given.

Fig. 6
figure 6

Computational domain with the boundary conditions

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Table 3 Mass flow rate at different firepowers
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2.4.2 Governing equations

The continuity, momentum and energy equations were solved in the non-premixed combustion model. The combustion chemistry of fuel was solved by generating probability density function table (PDF). The transport equation for conserved scalar mixture fraction is taken from Biswas and Eswaran [63].

$$\frac{ \partial \rho }{\partial t} + \nabla .\left( {\rho v} \right) = 0$$
(64)
$$\frac{{\partial \left( {\rho v} \right)}}{\partial t} + \nabla .\left( {\rho vv} \right) = - \nabla p - \nabla \tau + \rho g$$
(65)

In Eq. (2), the first and second right hand side terms are the pressure gradient and molecular transport due to viscosity, respectively, where τ is the viscous stress tensor.

$$\tau = - \mu \left[ {2S - \frac{2}{3}\nabla v} \right]$$
(66)
$$S = \frac{1}{2}\left( {\nabla v + \nabla v^{T} } \right)$$
(67)

where ∇\(v\)T is the transpose of the velocity gradient and µ is the dynamic viscosity

$$\frac{{\partial \left( {\rho H} \right)}}{\partial t} + \nabla .\left( {\rho vH} \right) = \nabla .\left( {\frac{{k_{t} }}{{C_{p} }}\nabla H} \right) + S_{h}$$
(68)

Equation (5) represents the energy equation.

where the total enthalpy H is defined as the sum of the mass fraction and enthalpy for the jth species.

$$H = \mathop \sum \limits_{j} Y_{j} H_{j}$$
(69)

where Hj is defined as

$$H_{j} = \mathop \int \limits_{{T_{{{\text{ref}},j}} }}^{T} C_{p,j} {\text{d}}T + h_{j}^{o} \left( {T_{{{\text{ref}},j}} } \right)$$
(70)

where h o j (Tref, j) is the enthalpy of formation of species j at the jth reference temperature.

2.4.3 The PDF transport equation model

The non-premixed combustion probability density function (PDF) of the mixture fraction is selected for modelling the sub-grid scale mixing. The transport equation for conserved scalar mixture fraction is written as:

$$\begin{aligned} & \frac{\partial (\rho P)}{\partial t} + \nabla .(\rho \nu P) + (\rho g - \nabla \overline{p} ).\nabla_{v} P + \sum\limits_{i = 1}^{n} {\frac{\partial }{{\partial \psi_{i} }}} \left[ {w_{i} P} \right] \\ & = \nabla_{v} .\left[ { - \nabla .\tau + \nabla (p^{'} |v,\psi )P} \right] - \sum\limits_{i = 1}^{n} {\frac{\partial }{{\partial \psi_{i} }}\left[ {(\nabla .(\rho D\nabla \psi_{i} )|v,\psi )P} \right]} \\ \end{aligned}$$
(71)

The first two left hand side (LHS) terms of Eq. (71) are the local change and convection of the PDF in physical space. The third term represents transport in velocity space by gravity and the mean pressure gradient. The last term on the LHS contains the chemical source terms. In the right hand side (RHS) of the transport equation, there are two terms that contain gradients of quantities conditioned on the values of velocity and composition. Therefore, if gradients are not included as sample space variables in the PDF equation, these terms occur in unclosed form and have to be modelled. The first unclosed term on the RHS describes transport of the probability density function in velocity space induced by the viscous stresses and the fluctuating pressure gradient. The second term represents transport in reactive scalar space by molecular fluxes. This term represents molecular mixing. Many combustion simulations tend to ignore the effect of radiation in the calculations. This is because the governing radiative transfer equation is of integral–differential nature, which makes the analysis difficult and computationally expensive. The method and model selection was done by using ANSYS Fluent user manual guide.

3 Results and discussion

The equations were solved iteratively using a MATLAB® script, and the iterative solution yields 13 temperatures values. The other output parameters like mass flow rates, thermal efficiency, different heat encountered, enthalpies, etc., were then determined from these temperatures values and flue gas properties. The calculation for firepower (FP) = 3.119 kW for selected geometry is shown in Table 4.

Table 4 Temperatures at different zones
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The energy balance for the entire stove is given in Table 5.

Table 5 Energy balance of the stove
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Three sets of WBT were performed on the model stove. The average performance parameters with standard deviations are given in Table 6 for high- and low-power phases. The parameters include thermal efficiency of the stove, specific fuel consumption, firepower, time taken to boil water, dry fuel consumed and the exhaust coming out of the stove. Few parameters like temperature, exhaust, mass flow rate are used to validate the mathematical and computational models. The relative error percentage which is the ratio of the difference between the experimental and model predicted to the experimental values was calculated using Eq. [64].

Table 6 Average parameters of WBT test with standard deviations (n = 3)
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3.1 Flame temperature

The values of flame temperature were monitored by placing R-type thermocouple in each zone. Two thermocouples were placed in the zone 1 and zone 3, while one more was placed in zone 2. The average temperatures in the zones were validated with the modelling results. Figures 7, 8 and 9 compare the variation of flame temperatures in different zones. The average temperature variations with the mathematical modelling for zone 1, zone 2 and zone 3 are 10%, 3% and 8%, respectively. Also, the average temperature variation with the computational modelling for zone 1, zone 2 and zone 3 is 7%, 12% and 13%, respectively. It can be seen that the values obtained through experimentation are less than those obtained through modelling. The possible reasons can be due to the placement of the thermocouples, due to the non-uniformity of flame temperatures in the radial directions, the assumption of a 2-D model and unaccounted losses.

Fig. 7
figure 7

Variation of flame temperature in zone 1 with firepower

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Fig. 8
figure 8

Variation of flame temperature in zone 2 with firepower

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Fig. 9
figure 9

Variation of flame temperature in zone 3 with firepower

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Mass flow rate: The variation of mass flow rate of air with firepower is shown in Fig. 10. The model-predicted values are in good agreement with the experimental values. The average deviation is ± 10.88%. Six different values of firepower of firepower of hot start test are used to validate the mathematical model. The values of the air flow rate are calculated considering all the losses involved in the stove. Losses like fuel bed resistance, friction losses in the chimney, sudden contraction or expansion, sudden bend in the flow, losses in the entry as well as the exit are considered while calculating the flow rate mathematically.

Fig. 10
figure 10

Variation of mass flow rate with firepower

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3.2 Thermal efficiency

The value of thermal efficiency of mathematical model is validated with the experimental results of high-power hot start test. The experimental and model values are found to be in good agreement. The average deviation is estimated to be 7%. Table 7 shows the value of thermal efficiency for different firepowers. The average value of thermal efficiency is just above Tier 3 [65].

Table 7 Thermal efficiency for different firepowers
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3.3 Exhaust

3.3.1 Carbon monoxide (CO emissions)

The output of the mathematical model is given as input for the computational model to understand the working of the cookstove. Figure 11 shows the variation of CO emissions in parts per million (ppm) with the firepower. The average deviation of experimental and computational values is 9.09%, and also the values predicted by the computational model are on the higher sides, so it is safe to design cookstove with the help of a mathematical and computational model. The exhaust was also calculated by WBT standard in g/MJd. The average value of CO in g/MJd for high-power test was 10.59. Figure 12 shows the comparison of CO emissions for different stoves available in the literature [5, 29]. The stoves like Uganda 2-pot, Onil stove and Justa stove which are sunken pot stoves can have a good comparison with the present model. The advantage with the current design is that no specified pots are required for the operation of stove likewise in the other two-pot stoves mentioned above. The values of CO are considerably low compared to traditional multipot biomass and 3 stone fire available on the field.

Fig. 11
figure 11

Variation of CO with firepower

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Fig. 12
figure 12

Comparison of CO emissions with stoves available in the literature

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3.3.2 Combustion efficiency

The value of modified combustion is given as,

$${\text{MCE}} = \frac{{{\text{CO}}_{ 2} }}{{{\text{CO}}_{ 2} + {\text{CO}}}}$$

The variation of experimental and computational modified combustion efficiencies is shown in Table 8. The computational values are in good agreement with the experimental results.

Table 8 Variation of modified combustion efficiency with firepower
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The value of modified combustion efficiency decreases with increasing firepower; it means the combustion deteriorates as we increase the firepower keeping the same IAR [66]. The same is observed in the experimental as well as computational results.

3.3.3 Percentage of O2 variation

Variation of O2% with firepower is shown in Fig. 13. The trend of O2 is same as predicted by the other researchers [20, 40]. The values of O2% decrease with increase in firepower. As firepower is directly proportional to mass flow rate, after increasing the fuel input, the amount of air decreases, i.e. by inserting more sticks in the feeding zone, we restrict the flow of air into the combustion zone. The same is observed in the case of modelling as well as experimentation. The average variation of O2% between experimental and computational values is 18%.

Fig. 13
figure 13

Variation of O2 with firepower

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3.4 Flow structures

Figure 14a shows the temperature contours for different values of firepower. CFD helps in predicting the flow structure with the increase in the firepower. The flame propagation increases with the increase in firepower. The shifting of the higher-temperature zone can be seen through the temperature contours. Figure 14b shows the streamlines which help in detecting the flow pattern in the selected geometry, and also, it determines the wake region and gives a clear picture of the flow at the exit. The distribution of the flow at the bottom can be seen through the streamlines. Also, CFD helps in determining other parameters like velocity, concentration of species, etc.

Fig. 14
figure 14

a Temperature contours and b Flow streamlines for different firepowers

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4 Model limitations and future scope

The mathematical model is applied where the mass of flue remains constant throughout the working zone of the cookstove. Also, the model holds good only for the natural draft cookstove and is not applicable for a forced draft or gasifier stove. The limitation of exhaust value determination by the mathematical model is overcome by using the computational model. In future, the mathematical as well as the computational model can be used to obtain the results much closer to the experimental values. The model generates a large number of solutions for different geometrical and operational inputs. This will help to optimize the design, which is the task to be taken up in future.

5 Conclusions

The mathematical approach to design the cookstove serves to be a better solution to develop a new biomass cookstove. The model will help to predict the performance of cookstove for various geometrical inputs. The parameters like thermal efficiency, mass flow rates and flame temperatures at different locations can be predicted by the model. The non-premixed combustion model along with the PDF gives a better picture of the working of a cookstoves. This will not only save energy resources but will help in experimenting with different parameters of the cookstove. A detailed parametric analysis may result in finding a new parameter which may directly or indirectly affect the performance of cookstove. The model-predicted values of temperature, as well as exhaust, are varying with the maximum average deviation of 13 and 16%, respectively, from the experimental results. The cost of the instruments required to determine the values of CO and PM2.5 is too high and is not always available with the researcher. Hence, the combination of mathematical and computational modelling can be of great help to design cookstove with good accuracy.