According to Fig.4.10, it can be stated that the temperature of the fuelbed is about 900 K when the excess-air factor is between 1 and 2. In order to calculate the radiative energy transfer from the fuelbed to the pan, we assume that the gases and flames between the pan and fuelbed do not absorb or emit any radiation. The view factor F between fuelbed and pan is equal to the view factor of two concentric parallel disks and can be found in many textbooks (see, for example, Sparrow and Cess 1970).

In order to check the estimated temperatures of the fuelbed and thus the radiative hat transfer to the pan, experiments were done by Herwijn with a radiative heat flux meter above a woodfire. The estimates of the fuelbed temperatures and the measured temperatures are presented in Table 5.1. The experimental results are obtained for a fire on a grate. The power density of the fire is 18 W/cm". In order to simulate a steady fire, the wood is added in small blocks. With such a procedure almost no distinction can be made between the burning of volatiles and charcoal. The temperatures are calculated from the measurements according to the following formula:

Here

is the total radiosity arriving at the instrument.

the fuelbed surface, $F_{b-r}$ the view factor from fuelbed to the radiation meter,

the temperature of the fuelbed, and

the temperature of the radiation meter. The estimated temperatures are calculated according to Eq.(4.15); for these calculations the excess air factor $\lambda$ is taken to be 1. The differences in the estimated temperatures are due to the variation of the power density on the grate.

In general, the measurements and estimated temperatures compare well. Although the measured temperatures with eccentric displacement of the meter are lower compared to the temperatures with a concentric situation, the deviation is within 15%.

**Table 1:** Results of the measurement with the Radiative Heat Flux Meter and Estimated Fuelbed Temperatures
Sl. No.	P (kW)	b (cm)	H (cm)	e (cm)	G
						Measured	Estimated
1.	3.5	18	15	0	10	910	880
2.	5.9	20	15	0	15	960	950
3.	7.0	22.5	15	0	19	985	940
4.	8.8	25	15	0	23	995	940
5.	5.9	20	10	0	21	930	950
6.	5.9	20	10	8	7.1	775	950
7.	5.9	20	15	8	7.8	880	950
8.	8.8	25	15	8	8.7	900	940

Using the above information we obtain the results for the radiant heat transfer to the whole pan as shown in Table 2. The calculations are based on a pan of 100 ° and the measured fuelbed temperatures calculated form Eq.(1). The view factor is taken for a pan of 28-cm diameter placed 15 cm above the fuelbed. The diameter of the fuelbed varies with the power of the fire such that the power density is constant at 18 $%% \mathrm{W/Cm^2}$ . From the table it appears that 12% of the wood heat is radiated to the pan.

**Table 2:** Radiative Heat Transfer to the Pan, Calculated On The Basis of Results Of The First Four Tests in Table 1
Sl. No.	P (kW)	T_b (K)
				(kW)	(%)
1.	3.5	910	0.42	0.41	11.7
2.	5.9	960	0.41	0.62	10.5
3.	7.0	985	0.39	0.82	11.8
4.	8.8	995	0.38	1.04	11.8

If we compare this to the efficiency achieved through common practice in developing countries - an open-fire efficiency between 10 and 15% - it can be stated improved stoves can only show a higher efficiency due to optimal use of the convective heat transfer.

Radiation from the Flames

One of the factors that determines the radiation from the flames is the emissivity. The total emissivity of the flames is a superposition of the contribution from the soot particles and the gases.

Emission of the soot particles

The emissivity of the soot particles is dependent on two quantities, the flame height and the volume fraction of the soot particles in the flames. When using an open fire, it is common observation that soot will deposit on the cold pan. From experiments it appears that on the average about 5 g of soot deposits on the pan when burning 1 kg of wood. With an excess-air factor 3 we find a volume fraction for the soot particles of about 2.10 $%% ^{-7}$ . This agrees reasonably well with experiments done in a closed brick stove, where a soot concentration between 5 and 10 $\mathrm{mg/m^{3}}$ of the combustion products was found (Claus et al. 1981).

Using the work of Felske and Tien (1973) to calculate the emissivity of the soot we find a value between 0.025 and 0.036 for a fire power between 3 and 8 kW.

Emissivity of the gases

Of the gases commonly present in a fire, only $\mathrm{CO_2}$ and $\mathrm{%% H_2O}$ contribute to the emissivity. The other gases ( $\mathrm{O_2}$ and $%% \mathrm{N_2}$ ) consist of symmetrical molecules that emit and absorb radiation only at much higher temperatures. In order to make an estimate of the emissivity of the gases $\mathrm{CO_2}$ and $\mathrm{H_2O}$ we will assume that no incomplete combustion will occur and that the total excess-air factor is equal to 2.

Hottel (1954) presents graphs from which the emissivity of the carbon dioxide and the water vapor can be determined. Hottel also gives a method to determine the mean beam length, depending on the volume, to be used with the graphs. For our calculations we assume that the reference volume is a circular cylinder with a radius equal to the radius of the flames at their tips. The emissivity for the total gas radiation is listed in Table 3 as a function of the power of the fire. In this table also the emissivity of the soot particles is given.

**Table 3:** The emissivities of the luminous flames and the radiative energy corresponding to these emissivities
P (kW)	h (cm)	(cm)				(m²)	(W)	(%)
1	8	25	0.016	0.035	0.050	0.016	61	6.1
2.	10.5	3.2	0.021	0.041	0.061	0.028	128	6.4
3.	12.3	3.8	0.025	0.045	0.069	0.038	200	6.7
4.	13.8	4.3	0.028	0.049	0.076	0.049	280	7.0
5.	15.1	4.7	0.031	0.051	0.080	0.059	348	7.0
6.	16.2	5.0	0.032	0.053	0.083	0.067	417	6.9
7.	17.2	5.4	0.035	0.055	0.088	0.076	503	7.2
8.	18.2	5.6	0.036	0.057	0.091	0.084	575	7.2
9.	19.0	5.9	0.038	0.059	0.095	0.092	659	7.3
10.	19.8	6.1	0.039	0.060	0.097	0.099	724	7.2

The total emissivity of the flames

In order to calculate the total emissivity of the flames one has to determine the effect of mutual absorption of the radiation from the gases and soot. Tien et al. (1972) give a simple formula to calculate the total emissivity $\epsilon _{t}$ when both soot emissivity $\epsilon _{s}$ and gas emissivity $\epsilon _{g}$ are known. This formula is

In Table 3 the total emissivity is listed; also the total luminous radiative power $\phi _{f}$ is given. We see that the flame radiation is about 7% of the total power of the fire. For these calculations an excess-air factor for the volatiles is taken to be 3. According to Fig.1 the average gas temperature for this situation is about 1400 K. Now that we know the part of the power of the fire that is converted to the radiation of the flames, we can revise the solution of Eq. (5.17). This is approximated by

where $q_{mw}$ and $b_{w}$ are the mass flow and the combustion value of the wood respectively. The temperature of the gases becomes lower due to this correction. The result is given in Fig.1. The (small) decrease in temperature of the gases does not influence the emissivity factors listed in Table 3.

In experimental work the part of the flame radiation detected by the radiative heat flux meter has to be estimated in order to check whether a correction should be applied for this contribution. For this correction we consider the source of the flame radiation to be circular, with equal power in the total column and the same diameter. This source is placed in the middle between fuelbed and meter. The part of the flame radiation impinging on the radiation meter is determined by the view factor. Because this view factor is for the given situation, of the order of 10 $^{-4}$ , no correction is needed for the temperatures of the fuelbed.

The contribution of the flame radiation to the heat transfer to the pan can be estimated in the same way. In this situation the view factor is 0.4, which means that due to flame radiation 3% of the total power impinges on the pan bottom.

Convective Heat Transfer to the Pan

Three factors determine the convective heat transfer to the pan on an open fire in a wind-free environment:

The efficiency of the heat transfer process can be defined in terms of $%% T_{e}$ , the exit temperature of the combustion gases (the average gas temperature at the top plane of the pan), and $T_{p}$ , the pan temperature. When $T_{e}>T_{p}$ , heat is carried away unutilized and when $T_{e}<T_{p}$ , the pan loses heat. A loss in efficiency is the result in either case. Figure depicts six hypothetical cases schematically corresponding to the three factors mentioned above. For a well-matched power level of the fire, the height of the pan above the fuelbed and the pan geometry $T_{e}$ will be within a few degrees of $T_{p}$ .

In order to make an estimate of the convective heat transfer from the hot gases to the pan one has to know the temperature of the combustion gases as well as the convective heat transfer coefficient. In general, the convective heat transfer $\phi_c$ is given by the relation

where

is the heat transfer coefficient,

the concerned surface, and

and

the gas and pan temperature, respectively. Before we present estimates of temperature and heat transfer coefficients we first divide the area around the pan into several subregions, each with its own flow characteristic and a general method to estimate the gas temperature.

When we observe an impinging jet on a flat plate held normal to the jet, we see a stagnation point region along the line of symmetry where the boundary layer is expected to have constant thickness. Of course, this may not be true in the presence of combustion. Beyond this region, over the outer half of the pan bottom, the flow could be treated as an axisymmetric wall jet. At the pan corner the gases turn again over an angle of $1/2\pi$ due to the buoyancy forces and will rise up along the pan wall. Figure 3 shows the several regions; the bending of the flow around the corner of the bottom is treated as a separate region and the flow along the pan wall is divided into regions where combustion occurs and where no combustion occurs. For each region a heat balance can be made. It will have the general form

I.	$\phi_{\mathrm{out1}}-\phi_{\mathrm{in1}}$	$\phi_{g1} - \phi_{p1} - \phi_{e1}$
II.	$\phi_{\mathrm{out2}}-\phi_{\mathrm{in2}}$	$\phi_{g2} - \phi_{p2} - \phi_{e2}$
III.	$\phi_{\mathrm{out3}}-\phi_{\mathrm{in3}}$	$%% \phi_{g3} - \phi_{p3} - \phi_{e3}$
IV.	$\phi_{\mathrm{out4}}-\phi_{\mathrm{in4}}$	$\phi_{g4} - \phi_{p4} - \phi_{e4}$
V.	$\phi_{\mathrm{out5}}-\phi_{\mathrm{in5}}$	$\phi_{g5} - \phi_{p5} - \phi_{e5}$

In words this means that the difference in energy content of the gas leaving a region ( $\phi _{oj}$ ) and entering a region ( $\phi _{ij}$ ) is balanced by the heat production due to combustion ( $\phi _{gj}$ ) minus losses to the pan ( $\phi _{pj}$ ) and losses to the environment ( $\phi _{ej}$ ) whether due to radiation or due to dilution by entrainment of air. In general, the combustion stops in region IV, which means that in region IV the contribution of $\phi _{gj}$ is zero. In region III the heat losses to the pan can be assumed zero because the pan surface exposed here is negligible. The inlet temperature of each region is of course equal to the outlet temperature of the previous region. As of now such a calculation, which requires a complete numerical scheme, has never been attempted. There have been several attempts to provide approximate results for the heat transfer to the pan. The first one was due to Krishna Prasad et al. (1985) and a much more refined version was provided by Bussmann (1988). We shall describe the latter work in detail.

Flow field

A schematic view of the fuelbed-flame-pan configuration is shown in Fig. 4 . The figure indicates the four regions of interest;

Available literature on modelling flow characteristics of impinging jets uses the diameter of the nozzle exit ( ${2R_1}$ ) as the characteristic length. For woodfires the value of ${2R_1}$ is chosen to be the diameter of the hot gas plume at the top of the flame. However when the distance between the pan and the fuelbed (

) is smaller than the flame height, ${2R_1}$ is set equal to the calculated diameter of the fire at the pan height.

In this region it is also assumed that no ambient air is entrained. This assumption is based on the work of Era and Saima (1976) who investigated the flow characteristics of an axisymmetric wall jet created by gases issuing from a nozzle and impinging on a flat plate held close to the nozzle tip. In the axisymmetric wall region there is an increase in the mass flow rate due to entrainment of air from the ambient and is estimated from the semi-empirical relation (Hertel 1962)

$\displaystyle {\frac{m}{m_{1}}}$	$\displaystyle =$	$\displaystyle 0.877{\frac{r}{2R_{1}}},$	(6)
$\displaystyle r$	$\displaystyle \geq$	$\displaystyle 2.28R_{1}$	(7)

The flow field at the pan corner is difficult to describe. Thus Bussmann assumed that no air entrains at the corner and the flow develops into a two-dimensional wall jet as soon as it comes into contact with the pan side.

Temperature

The next step in the model is to estimate the temperatures prevailing in the different flow regions described above. The stagnation point region is divided into two regions.

is the temperature at the pan height calculated from the flame model presented in 5.2. According to this model $T_{2}$ is not a function of

and thus is constant.

The heat balance provides temperature as a function of radius and accounts for heat loss from the gas stream to the pan. $h_{2}$ is the heat transfer coefficient in this region.

The temperature in the axial wall jet region

, where

is the radius of the pan, is obtained from the quasi-empirical relation provided by Hertel (Eq.5.21) and the following heat balance equation.

$\displaystyle {\frac{\mathrm{d}(mC_pT_3) }{\mathrm{d} r}} = - 2 \pi r h_3 (T_3 -373) + (C_pT_a + {\frac{B_v }{\lambda_v \psi_v}} {\frac{\mathrm{d}m }{{rm d}r}}$

(9)

The first term on the right represents the heat absorbed by the pan. The second term comes in two parts both associated with the entrainment of outside air. The first part simply accounts for the heat content of the entrained gases. The second part accounts for the heat liberation by combustion. Note that this term was absent in the stagnation point region. This is primarily due to the manner in which the flame model was implemented. That model is always computed on the basis of a chosen amount of excess air factor at every height. Since there is no additional air entrained in the stagnation region our flame model will not yield any heat release. The inclusion of heat release in this region is also questionable in the sense it is outside the calculation possibility of our flame model since it is only one-dimensional (that is, heat release occurs only as a function of height in the flame model; in the region below the pan the height is constant). Thus both cases - with and without combustion - was investigated by Bussmann. There is yet another consideration - volatiles coming out of the fuel bed should not all have been consumed before the gases arrive at the pan. Thus

$\nu$ here represents the volatile fraction and $\lambda$ is the excess air factor. The last zone where temperature has to be estimated is in region 4 mentioned earlier. Bussmann assumed that the thickness of the hot gas layer along the vertical side of the pan to be small in comparison to the radius of the pan. This assumption permits the use of a relation proposed by Seban and Back (1961) for a two-dimensional wall jet

$\displaystyle {\frac{T }{T_4}} = 7.7 {\frac{\rho_4 }{\rho_a}}{\left( {\frac{1 }{\delta_4}} + 12 \right)}^{-0.6}$

(11)

in the above is taken as the temperature at the bottom corner of the pan and is consistent with the assumption that there is no air entrained in the corner. This probably will lead to an overestimate of

. $\delta_4$ , the thickness of the wall jet at the bottom corner, is calculated from

where

is taken to be the velocity at the pan bottom corner. From the work of Era and Saima, cited earlier, we obtain for

the following relation.

Heat transfer Coefficients

In order to evaluate the various temperatures above and to estimate the total heat transfer to the pan, we need to know the heat transfer coefficients. These are the same as the ones used in Bussmann et al (1983) and reproduced here.

Results and discussion

We will now present the results obtained from the above analysis. Fig. 5 presents the results of gas temperatures under the pan as a function of the radial distance.The three pairs of solid and dashed curves correspond to temperatures obtained for three distances between fuelbed and pan bottom - 0.1, 0.15 and 0.2m.

Fig.5. Distribution in the gas under the pan for different pan-fuelbed disaxial wall jet without combustion in axial wall jet

The pan radius for these calculations is 0.14m. The solid and dotted lines denote respectively the cases with and without combustion being considered in the axial wall jet. The discontinuity at

is due to the fact that a correction has been incorporated for heat absorption in the central region of the pan bottom. The discontinuity at

is due to the fact that the axial wall jet relationship is applicable from this point onwards. The result for the largest distance shows that there is no combustion along the pan bottom at all due to the fact the volatiles have been exhausted by the time the gases arrive at the pan bottom. Furthermore at the lowest distance, the temperature at the bottom corner of the pan drops to 500K in the absence of combustion. This is unrealistic. At this height, even if we include combustion the temperatures at the bottom corner does not even reach the values at the axis for the largest distance. Thus considerable amount of combustion is likely to occur on the side of the pan. According to our heuristic discussion in connection with Fig.2 this is undesirable.

Figure 6 presents the average heat transfer coefficients over the entire pan as a function of pan-grate distance. These were computed on the basis of an average temperature difference between the gas and the pan. The figure shows an increasing heat transfer coefficient with increasing pan-grate distance. The experimental results of Herwijn on the contrary show a rather constant value. The difference must be attributed to the conditions for which the relations (5.28) - (5.30) were derived. For example the Schlunder - Gnielinski formula was derived for a turbulent jet with a distance between the nozzle and the plate being between the nozzle diameter and 12 times the value. The Reynolds numbers in the fires of interest here can be much smaller and the flow is probably not laminar but cannot be fully developed turbulent either.

What is more the distance between the grate and pan bottom corresponds to the lower end of the limit prescribed by Schlunder and Gnielinski. Thus in calculating efficiencies a constant value of the heat transfer coefficient is adopted, but it will be varied parametrically.

The next three figures (7 - 10) show some of the more significant results obtained from the heat transfer calculation procedure presented above. Fig.7 presents efficiency as a function of the pan-fuelbed distance for three values of heat transfer coefficients, namely, 5, 10 and 15 $\mathrm{W/m^2K}$ . The flame parameters used in these calculations are: the excess air factor $\lambda = 2$ ; the entrainment constant, $\alpha = 0.08$ ; and volatiles fraction, $\nu = 0.8$ . The fuelbed diameter is held constant at 0.18m, the pan diameter is 0.28m and the combustion value of wood is taken to be 18.7MJ/kg.

The solid lines give the efficiencies obtained by including combustion in the axial wall jet region while the dotted line shows the result without combustion for the case with heat transfer coefficient of $\mathrm{10W/m^2K}$ . The calculations incorporate corrections for the fuelbed thickness and the so-called averaged peak power of the volatiles. The nature of these concepts are discussed in connection with the experimental results to be presented in the next section. The figure also shows the comparison of the theory with experiment. It is easy to see the excellent agreement between theory and experiment when one uses a heat transfer coefficient of 10W/m"K. In addition it is also seen that it is important to account for the combustion in the axi-symmetric wall jet region under the pan. This result confirms our comment earlier on this question.

The next figure (Fig.8) compares the efficiencies obtained from experiments carried out with different wood species and the theoretical results. In the experiments the nominal power for definition; also the next section) was held constant.

A function of pan-fuelbed distance. The points are from experiment
the dotted line is without combustion for h = 10W/m^2K.

Fig.7.A function of pan-fuelbed distance. The points are from experiment the dotted line is without combustion for h = 10W/m²K.

However corrections have been introduced in the calculated efficiencies for the measured differences in the averaged peak power. For this reason the results are shown in terms of power output. The calculations have all been performed by assuming a uniform fuelbed thickness of 50mm. It is seen that calculated values are in good agreement with the experiments. The somewhat larger deviations from theory seen for species 2 and 3 must be attributed to the inevitable variations in the fuel bed thickness in the experiments (see next section).

Finally Fig.9 shows a break-up of efficiency in terms of heat transfer by radiation and convection as reported by Krishna Prasad et al. (1985). The convective heat transfer is further divided into heat transfer to the bottom of the pan and the pan side. The heat transfer to the pan bottom above an open fire was measured with a heat flow meter by Herwijn (1984). This meter was placed such that the detecting surface lies in the plane of the pan bottom. The heat flow meter measures the radiant as well as the convective heat flux to the surface. Tests were performed for two powers of the fire, 6 and 8.7 kW; two distances between pan and fuelbed, 10 and 15 cm; and two eccentricities of the heat flow meter in the pan, 0 and 8 cm (the pan was always placed concentric with the fire). Table 5.4 gives a more detailed presentation of the measurements. The radiative contribution to the heat transfer found earlier can be subtracted from the total measured heat transfer. This gives an estimate of the convective heat transfer. These values are also listed in the table. The total, radiative, and convective heat transfer are related to the power of the fire in order to get the efficiency contribution of the various terms. The efficiency obtained for the convective contribution is obtained as before by dividing the pan bottom into two regions: the stagnation point region and the axisymmetric wall jet region. The radius of the boundary between the two regions is taken to be equal to the plume diameter calculated in the flame height model.

Fig.9. Efficiencies according to heat transfer by radiation and convection. Points are from experiments of Herwijn (1984)

In Fig. 9 the estimated efficiencies for a 6-kW fire are shown. Although the measured radiative contribution agrees well with the model, as already was shown in Table 2, the convective contribution to the pan bottom does not agree at all. The main reason for this disagreement is the assumption that the combustion stops at the pan height. This has been corrected in the work of Bussmann.

**Table 4:** Results of the measurements with the Heat Flow Meter
Sl. No.	P (kW)	(cm)	(cm)	(cm)	(kW/m²)	(kW/m²)	(kW/m²)
1.	6.0	15	20	0	32.4	17.5	14.9
2.	5.8	15	20	8	25.8	17.9	7.9
3.	8.8	15	25	0	37.1	14.4	22.8
4.	8.6	15	25	8	28.2	19.5	8.7
5.	5.9	10	20	0	37.9	21.0	16.9
6.	6.0	10	20	8	32.5	25.5	7.0

The point about this earlier work is it clearly establishes the relative importance of radiative and convective heat transfer to the pan. At the low height the radiative heat transfer contributes as much as one-third of the total heat input to the pan. At larger heights the importance of radiation goes down, A second point of importance that emerges from this figure is the fact that the convective heat transfer to the bottom and side of the pan are of equal orders of magnitude. Thus many a multi-pan improved stove with a chimney do not provide for heat transfer to the pan sides. Of course they do provide additional heat transfer surface by adding more pans. But that makes the stove bulky. Hence it is important to clearly delineate the merits of such stoves in comparison to single pan stoves that provide for adequate heat transfer surface.

Heat transfer

Radiation from the Fuelbed