Making the most use of the heat energy from the sun that has been collected and/or saved to achieve cooking is a major desire for the manufacturers of solar cookers. Different methods of cooking with energy from the sun, as well as different materials, have been described by different researchers, ranging from cooking directly with solar irradiation to indirectly with transferred heat. Another method is to use copper tubes and, most recently, to use heat pipes (Esen, 2004; Khalifa, 1986; Mathioulakis & Belessiotis, 2002). While some of these are still being developed, some are quite simple and others appear complicated.
In selecting an appropriate method to transfer the heat from the storage to the cooking side, three questions were asked: Is it effective? Is it ambiguous? Is it economic? The ultimate desire was to develop an appropriate and low-cost technology for which the materials are readily available. Therefore, a spiral-type copper cook head was made and, in order to achieve an accurate theoretical analysis of this type of solar cooking section, the first assumption made was that a constant heat is supplied to the pot. If the cooking is done in a shaded place or in a place where there is no direct exposure of the cooking pot to the sun, the energy from the storage tank is modelled as the only energy supplied to the cooking plate. However, if the cooking is done outside, an additional direct irradiation from the sun should be added to the energy input as the secondary irradiation falling on the pot (Craig & Dobson, 2015). These simple facts were used to analyse all the work done by the energy supplied to the cooking section, including all the losses. Figure 4-11 shows how the energy transferred to the cooking pot from the spiral cooper head 𝑄
̇
1 was utilised52
Figure 4-12: Heat transfer in cooking section
The model for the analysis of the solar cooking pot presented by El-Kassaby (1991) and modified by Amer (2003) and Kimambo (2007), who worked backed from the energy supplied to the food to analyse their systems. In this system, however, 𝑄̇𝑓𝑑 represents the energy supplied to whatever is being heated up in the pot (food, pancake, water, etc.). The work that this energy performs can be divided into two. First, it has to raise the internal energy of the food, 𝑈𝑓𝑑,𝑖, and second, it has to overcome the convective losses through the air top of the pot, as 𝑄̇𝑡𝑜𝑝.
𝑄̇𝑓𝑑= 𝑄̇𝑡𝑜𝑝+ 𝑈𝑓𝑑,𝑖 (4.63) Writing Equation 4.63 explicitly in terms of the instantaneous temperature of the food 𝑑𝑇𝑓𝑑
𝑑𝑡 , the mass of food that is cooked, 𝑚𝑓𝑑, the specific heat capacity of the food and the heat transfer coefficient between the inside of the cooking pot and the food being cooked are shown in Equation 4.64.
𝑄̇𝑓𝑑 = 𝑄̇𝑡𝑜𝑝+ 𝑚𝑓𝑑𝑐𝑓𝑑 𝑑𝑇𝑓𝑑
𝑑𝑡 = ℎ𝑓𝑑𝐴𝑓𝑑(𝑇𝑝𝑜𝑡− 𝑇𝑓𝑑) (4.64) The heat energy from the tank is transferred to the cooking pot through the copper tube cooker head, represented as Q̇1, and this energy is transferred to the food as 𝑄̇𝑓𝑑, which is expressed in Equation 4.63 and 4.64 to perform the two functions identified in the previous paragraph. The energy lost through convection from the side walls of the cooking pot is represented as 𝑄̇𝑝,𝑠, and that lost from the bottom of the pot to the surroundings is 𝑄̇𝑝,𝑏. The radiation from the pot bottom to the and from the sides to the cover is represented by 𝑄̇𝑟,𝑏 and 𝑄̇𝑟,𝑠 respectively in Figure 4-11. These energies can all be combined in an energy balance equation, presented as
𝑄̇1= 𝑚𝑝𝑜𝑡 𝑐𝑝𝑜𝑡 𝑑𝑇𝑝𝑜𝑡
𝑑𝑡 + 𝑄̇𝑓𝑑+ 𝑄̇𝑝,𝑠 + 𝑄̇𝑝,𝑏+ 𝑄̇𝑟,𝑠+ 𝑄̇𝑟,𝑏 (4.65) where
53
𝑄̇𝑝,𝑠= ℎ𝑔 𝐴𝑠𝑖𝑑𝑒 (𝑇𝑝𝑜𝑡− 𝑇𝑎) (4.66) 𝑄̇𝑝,𝑏 = ℎ𝑎 𝐴𝑡𝑜𝑝(𝑇𝑡𝑜𝑝− 𝑇𝑎) (4.67) 𝑄̇𝑟,𝑏 = 𝜎 𝜀𝑝𝑜𝑡 𝐴𝑡𝑜𝑝 (𝑇𝑝𝑜𝑡4− 𝑇𝑎4) (4.68) 𝑄̇𝑟,𝑠= 𝜎 𝐴𝑠𝑖𝑑𝑒 (𝑇𝑝𝑜𝑡4− 𝑇𝑎4)(1 𝜀⁄ 𝑝𝑜𝑡− 1 𝜀⁄ 𝑐𝑜𝑣𝑒𝑟− 1) (4.69) The air trapped between the tank cover and the inside part of the pot section was also modelled, because it was expected to transfer the heat to and/or get the heat from both the pot and the pot cover. This can be expressed as
𝑄̇𝑝,𝑠+ 𝑄̇𝑡𝑜𝑝+ 𝑄̇𝑝,𝑠2= 𝑚𝑎𝑐𝑎 𝑑𝑇𝑔
𝑑𝑡 (4.70) where the newly introduced convection term, 𝑄̇𝑝,𝑠2, for the air trapped in the cooking section is defined as follows:
𝑄̇𝑝,𝑠2= ℎ𝑔𝐴𝑐𝑜𝑣(𝑇𝑐𝑜𝑣− 𝑇𝑔) (4.71) Two types of cooking can be performed with the experimental setup: the first is cooking under direct sunshine, where the top of the cooking section is left exposed to direct sunshine, and the second is when the top part of the cooking section is covered with insulators (this was used to represent indoor cooking). When cooking was done in sunshine and the top of the cooking section in the storage tank was not covered with insulation, to allow additional direct heating from sunshine, it was necessary to model the exposed cover into the set of defined equations, because the cover of the pot was also expected to receive direct irradiation (DNI) from the sun and to gain heat from the inner part of the pot through convection 𝑄̇𝑝,𝑠2. It was also expected to store some heat as latent internal energy, and to lose heat to the atmosphere through radiation (𝑄̇𝑟) and convection (𝑄̇𝑐𝑜𝑣,𝑐𝑜𝑛𝑣). The energy balance for this expression is given below, where 𝑄̇2 is the secondary DNI falling on the cooking section: 𝑄̇2+ 𝑄̇𝑟𝑠 = 𝑄̇𝑟+ 𝑄̇𝑝,𝑠2+ 𝑄̇𝑐𝑜𝑣,𝑐𝑜𝑛𝑣+ 𝑚𝑐𝑜𝑣 𝑐𝑐𝑜𝑣 𝑑𝑇𝑐𝑜𝑣 𝑑𝑡 (4.72)
where
𝑄̇2 = 𝐴𝑐𝑜𝑣𝛼𝑐𝑜𝑣 𝐼 (4.73) 𝑄̇𝑟 = 𝜎 𝜀𝑐𝑜𝑣 𝐴𝑐𝑜𝑣(𝑇𝑐𝑜𝑣4− 𝑇𝑎4) (4.74) 𝑄̇𝑐𝑜𝑣,𝑐𝑜𝑛𝑣= ℎ𝑎 𝐴𝑐𝑜𝑣 (𝑇𝑐𝑜𝑣4− 𝑇𝑎4) (4.75)Rearranging the above equations to form a first-order differential equation that can be solved numerically gives the following sets of equations:
𝑑𝑇𝑓𝑑
𝑑𝑡 = 𝐹𝑓𝑑 (𝑇𝑓𝑑, 𝑇𝑝𝑜𝑡, 𝑡) (4.76)
𝑑𝑇𝑝𝑜𝑡
54 𝑑𝑇𝑔
𝑑𝑡 = 𝐹𝑔 (𝑇𝑓𝑑, 𝑇𝑝𝑜𝑡, 𝑇𝑔, 𝑇𝑐𝑜𝑣, 𝑡) (4.78) 𝑑𝑇𝑐𝑜𝑣
𝑑𝑡 = 𝐹𝑐𝑜𝑣 (𝑇𝑓𝑑, 𝑇𝑝𝑜𝑡, 𝑇𝑐𝑜𝑣, 𝑡) (4.79) The above sets of equations can be solved using the fourth-order Runge-Kutta method for numerical solutions of first-order differential equations. The Chapra and Canale (2003) model was used to generate the results. The time was taken in minutes, as 𝑇𝑓𝑑, 𝑇𝑝𝑜𝑡, 𝑇𝑔 and 𝑇𝑐𝑜𝑣, with all temperatures in ℃, and can be used to compare the theoretical and experimental results. However, when the second type of cooking was performed and the top part of the cooking section was covered with insulation, Equation 4.79 was excluded from the analysis because there was no direct exposure of the cover of the top to the DNI, and Equations 4.76 to 4.78 are used with the temperature time derivative independent of 𝑇𝑐𝑜𝑣.
An alternative way to solve the heat transfer in the cooking section is by using the thermal resistance method. Figure 4-12 shows an alternative method to analyse the heat flow in the cooking section by using the concept of thermal resistance. This resolves heat transfer as the basic electricity equation, 𝑉 = 𝐼𝑅, with V the electrical potential in analogy to temperature difference, the current, I, analogical to the rate of heat transfer, and the resistance R relating to thermal resistance, as shown in Figure 4-12 below
55
Figure 4-14: Thermal resistance diagram of the cooking section
The analogy in Figure 4-12 can be used to solve the energy equations using the resistances shown in Figure 4-13. The various heat transfer modes with resistance analysis are shown in the sets of equations below:
𝑅𝑐𝑜𝑛𝑣 = 1 ℎ𝐴 (°C W⁄ ) (4.80) 𝑅𝑐𝑜𝑛𝑑 =∆𝑥 𝑘𝐴 (°C W⁄ ) (4.81) 𝑅𝑟𝑎𝑑= 1 𝜀𝜎𝐴(𝑇𝑎2+ 𝑇𝑏2) (°C W⁄ ) (4.82)