Plot as a function
5. Plot as a function of (Tf, in – Tambient). Is it possible to estimate the values of the heat removal factor (FR) and the overall global conductance Uc for the collector? If so, then estimate both, if not comment on why not.
Figure 9. Efficiency (ƞ_c) as a function of (Tf, in – Tamb) (closed loop)
Figure 10. Efficiency (ƞ_c) as a function of (Tf, in – Tamb) (open loop)
Figure 9 and Figure 10 show the instantaneous efficiency plot against the adjusted difference in temperature. Figure 9 shows that efficiency decreases with larger temperature differences. The curve of efficiency vs time (Figure 5) and efficiency vs temperature (Figure 10) are very similar for the closed loop. So, the closed loop does not exhibit significant difference. Since the conditions under which the experiment was carried out were not perfect, the curve does not exhibit the exact shape that one would expect. If the experiment would have been carried out under ideal conditions, the efficiency line would be going down linearly. However, a linear trendline was added to the graph and it can be noticed that it does follow the path of an ideal curve. This indicates us that efficiency and (Tf, in – Tamb) are inversely proportional to each other.
On the other hand, for the open loop, the curve does not exhibit the expected shape. One of the reasons for this is that the data was gathered during a very small period so at the end very few points were used to plot the efficiency as a function of the temperature difference. However, a trendline was also added to this graph, and this trendline once again follows the same path as the curve of efficiency for an experiment carried out under ideal conditions.
From the graph of vs it would be difficult to estimate the values of the heat removal factor (Fr) and the overall conductance (Uc) for the collector. The reason for this is that there are equivalent equations used to estimate these two parameters and each equation includes a variable that we were not asked to use to plot the efficiency curve.
Equation 1 defined the collector efficiency (η) as the ratio of useful energy gain (Qu) to the incident solar energy over a period of time. Furthermore, the useful energy gain can be written in terms of certain parameters such as absorptivity (α) and transmissivity (τ) as shown in equation 5 below.
If it is assumed that FR, τ, α and UC are constants for a given collector and flow rate, then the efficiency is a linear function of the three parameters defining the operating condition: Solar irradiance (Ic), Fluid inlet temperature (Tf, in) and Ambient air temperature (Tamb). Then, the performance of a Flat-Plate Collector can be approximated by measuring these three parameters in experiments.
Since the efficiency curve for this specific question only asks to plot the efficiency (as a function of temperature but not Ic, I cannot accurately estimate the values of Fr and Uc. I need the three parameters mentioned before to be able to approximate these values.
However, since I have data for IC, I can make a plot of efficiency vs and use it to estimate the two parameters. In fact, the slope of the vs curve is given by Equation 6 below.
m = -FR*Uc Equation 6
Also, the y-intercept of the vs curve is related to absorptivity, transmissivity and heat removal factor by Equation 7 below.
y-int = Equation 7
If we assume values for , the two relations given in Equation 6 and Equation 7 can be used to estimate values for FR and Uc.
The equation of a line indicates that
mx + b Equation 8
Where m is the slope and b the y-intercept.
For the efficiency curve shown below in Figure 11, we can get the equation of the line and compare it to get the value of the slope, m, and the y-intercept, b.
Figure 11. Efficiency of the curve as a function of (Tf, in – Ta)/IC
The respective equation of the line for the efficiency curve is given by
From this equation we see that that the slope -FR*Uc = -5.9 and y-int = 134.6.
Using Equation 7, we can plug the values assumed for and solve for FR. From this calculation we get an estimate value of FR = 166.17. Furthermore, using Equation 6 and solving for Uc, we get and estimate value of Uc = 0.034 W/m^2 K. These estimations do not make sense because they are way off from theoretical values. The reason for this can be errors in the calculations or the fact that data is not gathered under ideal conditions.
Another way of estimating values for Fr and Uc is to use the fact that Uc can be estimated to be 8 W/m^2 K for 1 glass cover. Hence, using this value in Equation 6 and solving for FR gives us a more accurate estimation of FR = 0.7.
6. With the new operational parameters and data gathered in the experiment, update the Flat Plate Collector ‘calculator’ that you will develop (instructions will be provided in class). Use it to estimate the values of the heat removal factor (FR) and the overall global conductance (Uc) for the collector, report those values and comment on what you did.
After developing the collector calculator, values for the heat removal factor (Fr) and overall global conductance (Uc) were estimated. The results indicated that the estimation of the heat removal factor was very accurate. In fact, the result obtained was 0.862. During class, the value obtained was 0.845. The difference is minimum, so we can assume that the calculator works properly when trying to estimate Fr.
On the other hand, the overall global conductance was estimated to be 3.5488. This parameter was not estimated during class, so I cannot compare the value obtained with the calculator. However, the typical value for Uc when using 1 glass cover is 8 W/m^2 K. The value estimated with the calculator is not the same to the theoretical value, but it is still acceptable considering that some of the parameters used vary with time and day.
01020304050607080101214161820Collector Efficiency (ƞ)(Tf, in –Tambient)Efficiency (ƞ_c) as a function of (Tf, in –Ta) (Closed Loop) ƞ Linear (ƞ )
01020304050607080-8.04-8.02-8-7.98-7.96-7.94-7.92Collector Efficiency (ƞ)(Tf, in –Ta)Efficiency (ƞ_c) as a function of (Tf, in –Ta) (Open Loop)ƞLinear (ƞ)
y = -5.9069x + 134.5901020304050607080101214161820Collector Efficiency (ƞ)(Tf, in –Tambient)/ICEfficiency (ƞ_c) vs (Tf, in –Ta)/IC (Closed Loop) ƞ Linear (ƞ )