Example: 3.9.1) Calculate the useful gain, exit fluid temperature and collection efficiency for a cylindrical parabolic concentrator system of 2m width and 8m length. The absorning cylinder has a diameter of 6cm and the transparent cover has a diameter of 9cm optical properties are estimated as \(\rho = 0.85, (\tau,\alpha)=0.77, \gamma=0.94\).Heat transfer coefficient from fluid inside to surrroundings, U0 = 5.2 kcal/hr m2 0C(6.04 W/m2 0C),Heat transfer cofficient from absorber cover surface to surroundings, UL = 6.0 kcal/hr m2 0C(6.98 W/m2 0C). The incident beam radiation o the aperture of the collector is 600 kcal/hr m2(698 W/m2) and the ambient temperature is 25 0C. The collector is designed to heat a fluid entering the absorber at 1500C, at a flow rate of 400 kg/hr. The fluid has Cp =0.30 kcal/kg 0C (1.256 kJ/kg 0C).
A_r = area of the receiver pipe = \(\pi.D_0.L\)
A_a = aperture area of the concentrator = (W-dco)* L
– Where W = width of concentrator
d_co = diameter of transparent cover
L = length of concentrator
Collector efficiency factor, \[ F' = \frac{U_0}{U_L}\]
Heat removed factor, \[ F_R = \frac{\dot{m}.C_p}{A_r. U_L.F'}[1-e^{-\frac{A_r.U_L.F'}{\dot{m}.C_p}}]\]
The absorbed solar energy is,
\[ S = H~b~.R~b~.\rho.\gamma.(\tau.\alpha) \] The values of F’ and FR will be same n any unit, since they are factors(dimensionless).
Useful gain, (In MKS unit) \[ Q_u = A_a.F_R. [S-\frac{A_r.U_L}{A_a}(T_fi-T_a)] \]
Note: The exit fluid temperature can be obtained from,
\[ Q_u = \dot{m}.C_p(t_co-t_ci) \] Where - tci = fluidinlet temperature - tco = collector fluid temperature at outlet
#1,2) Calculation of A_r = area of the receiver pipe and A_a = aperture area of the concentr"ator
d_0<-0.06 # diameter of cylinder 6cm
L<-8 # length L = 8m
d_co<-0.09 # diameter of transparent cover 9cm
W<-2 # width of concentrator is 2m
A_r<-round(pi*d_0*L,digit=2)
sprintf("1) The area of the receiver pipe,A_r = %s sq.m",A_r)## [1] "1) The area of the receiver pipe,A_r = 1.51 sq.m"A_a<-round((W-d_co)*L,digit=2)
sprintf("2) The aperture area of the concentrator,A_a = %s sq.m",A_a)## [1] "2) The aperture area of the concentrator,A_a = 15.28 sq.m"# 3,4) Calculation of collector efficiency factor,F' and heat removed factor,F_R
# Given:
U_0<-5.2
U_L<-6.0
m<-400 #flow rate 400kg/hr
C_p<-0.30
F_dash = round(U_0/U_L,digit=2)
x<-((m*C_p)/(A_r*U_L*F_dash))
y<- ((A_r*U_L*F_dash)/(m*C_p))
F_R<-round(x*(1-exp(-y)),digit=2)
#F_R1<- round(((m*C_p)/(A_r*U_L*F_dash))*(1-exp(-(1/((m*C_p)/(A_r*U_L*F_dash))))),digit=2)
sprintf("3) The collector efficiency factor F' is %s", F_dash)## [1] "3) The collector efficiency factor F' is 0.87"sprintf("4) The heat removed factor F_R is %s", F_R)## [1] "4) The heat removed factor F_R is 0.97"# 5) Calculation of Absorbed solar energy
#Given:
rho <- 0.85
tau.alpha <- 0.77
gamma <- 0.94
Hb_Rb <- 600 #incident beam radiation on the aperture of the collector is 600kcal/hr sq.m
S<-round(Hb_Rb*rho*gamma*(tau.alpha),digit=0)
sprintf("5) The absorbed solar energy S is %s kcal/ hr sq.m", S)## [1] "5) The absorbed solar energy S is 369 kcal/ hr sq.m"#6) calculation of useful gain
#Given:
T_fi<-150
T_a<-25
#A_a,F_R,S,A_r,U_L:calculated earlier
Q_u<- round(A_a*F_R*(S-((A_r*U_L*(T_fi-T_a))/A_a)),digit=0)
sprintf("6) The useful gain Q_u is %s kcal.hr", Q_u)## [1] "6) The useful gain Q_u is 4371 kcal.hr"# 7) Calculation of exit fluid temperature
# The exit fluid temperature can be obtained from Q_u = m.c_p*(t_co-t_ci)
#t_co=collector fluid temperature at outlet
#t_ci=fluid inlet temperature
t_ci<-150
t_co<-t_ci+(Q_u/(m*C_p))
sprintf("7) The useful gain t_co is %s ", t_co)## [1] "7) The useful gain t_co is 186.425 "# 8) Calculation of collector efficiency
eta_collector = round((Q_u/(A_a*Hb_Rb)*100),digit=2)
sprintf("8) The useful gain eta_collector is %s percentage", eta_collector)## [1] "8) The useful gain eta_collector is 47.68 percentage"