The objective of this work is to numerically determine the thermal performance of the parabolic cylinder cooker commonly “blazing tube”. These performances were determined by establishing heat balances at the different levels of the system. The equations obtained have been discretized; simplifying assumptions have been made to facilitate their resolution. We adopted Gauss Seidel ’ s method using MATLAB software to solve these equations. The temperatures of the coolant, the glass and the absorber were determined as a function of time and along the tube. The thermal efficiency was also determined. It emerged that the different temperatures evolve linearly as a function of the length of the tube. Yield and temperatures depend on the amount of sunshine.
The foods consumed by men are mostly cooked beforehand. The source of energy necessary for this cooking is generally fossil fuel. This increasingly rare, perishable and expensive source of energy is also polluting. It should be noted above all that in African countries in general, the most widely used source of energy is wood. These reasons lead researchers to turn to renewable energies, including solar. The literature shows that several solar cookers have been developed and studied. Box type cookers are the most common. This type of cooker consists of an insulated box with a transparent glass cover [
The device on which the simulation study will be carried out is the parabolic cylindrical solar cooker called “blazing tube”, an indirect solar cooker. It is made up of:
· an aluminum reflector,
· a transparent pyrex glass tube under vacuum,
· a copper absorber tube covered with black paint containing a heat transfer fluid which is vegetable oil,
· a hearth or cooking box where the container (kitchen utensil) is located.
It is also equipped with wheels to facilitate its mobility.
The solar cooker concentrates the solar rays through the reflector on the absorber tube filled with oil (linear concentration). The heated oil rises by thermosiphon to the cooking hearth where the pot is located [
Before performing the heat balances, we made some simplifying assumptions.
Hypotheses
➢ the heat transfer fluid is incompressible;
➢ the ambient temperature Ta around the receiver is uniform;
➢ the solar flux at the level of the absorber is uniformly distributed;
➢ a transient regime for each heat transfer medium;
➢ One-dimensional heat transfer.
The heat transfer is done according to the three modes of transfer that is to say the convection, the conduction and the radiation. During this heat exchange we observe gains but also losses.
The glass exchanges heat with the external environment but also with the absorber.
· Thermal balance of exchanges at the level of the external wall of the glass:
m v C p v d T v , e 2 d t = h c o n d , v e − v i S v ( T v i − T v e ) − h c o n v , v e − a S v ( T v e − T a ) − h r a y , v e − c S v ( T v e − T c ) + P v (1)
· Thermal balance of exchanges at the level of the internal wall of the glass:
m v C p , v d T v , i 2 d t = h r a y , a b s − v i S e x t , a b s ( T a b s − T v i ) + h c o n v , v i − a b s S e x t , a b s ( T a b s − T v i ) − h c o n d , v i − v e S v ( T v i − T v e ) (2)
· Thermal balance of the external wall of the absorber:
m a b s C p , a b s d T a b s , e x t 2 d t = φ c o n d , a b s . e − a b s . i − φ r a y , a b s − v i − φ c o n v , a b s − v i + P a b s (3)
· Thermal balance of the internal wall of the absorber:
m a b s C p , a b s d T a b s 2 d t = − φ c o n v , a b s − f − φ c o n d , a b s . i − a b s . e (4)
· Overall thermal balance at the level of the absorber:
m a b s C p , a b s d T a b s d t = h r a y , a b s − v i S a b s ( T v i − T a b s ) − h c o n v , a b s − v i S a b s ( T a b s − T v i ) − h c o n v , a b s − f S a b s ( T a b s − T f ) + P a b s (5)
m f C p , f d T f d t = h c o n v , f − a b s S a b s ( T f p − T f ) + m ˙ C p , f d T f d z (6)
Conditions to the limits:
T f ( j , 0 ) = T a m b
T f ( 0 , n ) = T f e
· convection between the heat transfer fluid and the absorber
Heat transfer between the absorber tube and the heat transfer fluid (oil in our case) takes place by natural convection with a regime that can be laminar or turbulent.
In natural convection we have:
N u ¯ = 0.86 × [ R a sin ( θ ) ] 0.25 [
with R a = g β D h Δ T α ν (8)
β : 1 T f
T f = T p + T ∞ 2
h f − a b s = N u λ f D h (9)
· Convection between the absorber and the glass
Heat transfer between the absorber tube and the glass (where air is trapped between them) takes place by natural convection. The heat transfer coefficient is determined using the following relationships: [
h c o n v a b s − v i = 2 π k e f f ln ( D i n t , v D e x t , a b s ) (10)
where
k e f f k f = 0.386 ( P r a i r 0.861 + P r a i r ) 1 / 4 × R a c 1 / 4 (11)
and
R a c = ln ( D i n t , v D e x t , a b s ) 4 L 3 ( D e x t , a b s 1 + D i n t , v − 3 / 5 ) 5 × R a (12)
L = D i n t , v − D e x t , a b s 2
The Rayleigh number is defined as follows:
R a = g β a i r ρ a i r L 3 ( T r − T 0 ) α a i r μ a i r (13)
· Convection between the glass and the ambient
The heat exchange between the glass and the surrounding environment takes place by convection.
It can be determined by several linear relationships depending on the wind speed:
Hottel and Woertz relationship
h c o n v , v e = 5.67 + 3.86 V v [
Watmuff et al. relationship
h c o n v , v e = 2.8 + 3.3 V v [
For our work we have opted for the relation of Hottel and Woertz which is more rigorous and more used.
· Radiation between the glass and the celestial vault
h r a y , v e = σ ε ( T v + T c ) ( T v 2 + T c 2 ) (16)
· Radiation between the absorber and the glass
h r a y , a b s − v i = σ 1 ε a b s + 1 − ε v ε v ( D e x t , a b s D i n t , v ) (17)
· Conduction between the glass walls
h c o n d , v e − v i = λ v ln ( D e x t , v D i n t , v ) (18)
· Conduction between the walls of the absorber
h c o n d , a b s . i − a b s . e = λ a b s ln ( D e x t , a b s D i n t , a b s ) (19)
· Power received by the collector
P c = S c I (20)
· Power absorbed by the glass
P v = I α v ρ c C g S v (21)
· Power absorbed by the absorber
P a b = I α a b τ v ρ c C g S a b s (22)
Le rendement thermique est le rapport entre la puissance thermique gagnée par le fluide caloporteur et la puissance solaire reçue par le capteur. Il est donné par la relation suivante:
The thermal efficiency is the ratio between the thermal power gained by the heat transfer fluid and the solar power received by the collector. It is given by the following relationship:
η = m ˙ f c p f ( T s , f − T e , f ) I ( A c × C g + 1 2 S a b s ) (23)
The global radiation RG is expressed by the relationship:
R G = R D I R + R D I F (24)
RDIF is diffuse radiation
R D I F = 1 2 ( 1 + cos ( p i 180 β ) ) 2 S R D I F H + ( 1 − 1 2 cos ( p i 180 β ) S A ( I 0 sin ( p i 180 h ) ) ) + R D I F H
RDIFH is diffuse radiation on a horizontal surface
R D I F H = I 0 C ( sin ( p i 180 h ) ) ( 0.271 − 0.2939 exp ( − B / sin ( p i 180 h ) ) )
RDIR is direct radiation
R D I R = I 0 C A exp ( − B / sin ( p i 180 h ) ) ( cos ( p i 180 I ) )
The equations were discretized by the finite difference method. They are then solved by the Gauss Seidel method in a computer program using MATLAB software. This program made it possible to obtain the curves below.
It can be seen that the overall sunshine increases (8 a.m. to 12 p.m.) then decreases as a function of time. The curve is “bell-shaped”. It reaches its maximum around 12 noon with a value of approximately 920 W/m2. Direct sunlight reaches a maximum value of approximately 860 W/m2.
It can be seen that all the curves have the same shape and that the temperatures evolve in an increasing way according to the length of the tube. We also note that the temperature of the absorber is higher than that of the fluid which is higher than that of the glass. We find the same profiles in the simulation results of Chekirou et al. [
Temperatures change during the day or time. This means that the temperatures
are a function of the sunshine. During the day the temperature of the absorber is the highest and that of the glass the lowest throughout the day, the temperature of the fluid is then between the two. The same profiles are obtained in the work of Chekirou et al. [
The thermal efficiency evolves in an increasing way until its maximum (0.40 to 0.33) around 1 p.m. The thermal efficiency curve has a bell shape, like that of sunshine. The efficiency depends on the solar power.
An algorithm developed in this study allowed us to determine the performance of the cylindrical parabolic solar cooker. Thus, it was possible to evaluate the temperatures of the various components of the system and of the heat transfer fluid. There is an increasing evolution of these temperatures along the length of the tube. Also the sunshine directly influences the evolution of the fluid temperature as well as on all the components.
The authors are grateful to international Science Program (ISP), Uppsala university for supporting their research group BUF 01.
The authors declare no conflicts of interest regarding the publication of this paper.
Dianda, B., Ki, M., Ouédraogo, W.P.G., Bado, N., Ky, S.M.T., Korgo, B., Kam, S. and Bathiebo, D.J. (2022) Numerical Study of a Cylindro-Parabolic Cooker “Blazing Tube”. Open Journal of Applied Sciences, 12, 1783-1795. https://doi.org/10.4236/ojapps.2022.1211123
Symbols Designation Unity
A Atmospheric haze factor
Ac useful surface of the collector m2
B Atmospheric haze factor
C correction factor
Cg geometric concentration - -
Cpspecific heat J/kg.K
D Diameter m
Dh hydraulic diameter m
f collector efficiency factor -
Gr Grashoff number -
h height of the sun (˚) degré
hcondheat exchange coefficient by conduction W/m2∙K
hconvheat exchange coefficient by convection W/m2∙K
hrayradiation heat exchange coefficient W/m2∙K
I solar radiation W/m2
I0Solar constant W/m2
K overall heat exchange coefficient W/m2∙K
Keffeffective thermal conductivity of fluid W/mK
l collector length m
m mass kg
m ˙ f mass flow of the heat transfer fluid kg/s
Nu Nusselt number -
P Power W
PrPrandtl number -
Ra Rayleigh number -
Racsimplified Rayleigh number
S concentrator opening area m2
T temperature K
V wind speed m/s
Greek letters Designation Unity
α Absorption coefficient -
ε Emissivity -
η Thermal efficiency -
λ Thermal conductivity W/m.K
ρ Reflection coefficient -
σ Stephan-Boltzman constant W/m2∙K4
τ Transmission coefficient -
φ Amount of heat exchanged W
Indices Designation
ab absorbed
abs absorbed
air air
c collector
e External or input
ext Outside
f fluid
i internal
int Interior
r receiver
s exit
v glass