Flat Plate Collector - Physical Model

A flat plate collector (FPC) converts solar energy into thermal energy which is transferred into a heat transfer fluid. Solar collectors usually operate with low and variable energy and do not track the sun, thus requiring minimal maintenance. They can deliver moderate temperatures upto 100 \(^o\text{C}\). The following model uses physical equations to predict the useful heat generated by an FPC as a function of its design, heat transfer fluid and irradiance. This flat plate model supports steady-state only.

Degrees of Freedom/Variables

The model has 7 degrees of freedom that should be fixed for the unit to be fully specified. Typically, the following 5 variables define the input stream and the output state variables.

Variables	Variable name	Symbol	Unit
Inlet volume flow rate	inlet.flow_mass_phase_comp['Liq','H2O']	\(m_{l}\)	\(\text{m}^3 / \text{s}\)
Inlet volume flow rate	inlet.flow_mass_phase_comp['Vap','H2O']	\(m_{v}\)	\(\text{m}^3 / \text{s}\)
Inlet temperature	inlet.temperature	\(T_{f}\)	\(\text{K}\)
Inlet pressure	inlet.pressure	\(P_{in}\)	\(\text{Pa}\)
Outlet pressure	outlet.pressure	\(P_{out}\)	\(\text{Pa}\)

The following variables must be fixed by the user for a fully defined model.

Variables	Variable name	Symbol	Unit
Collector area	collector_area	\(A_{c}\)	\(\text{m}^2\)
Total irradiance	total_irradiance	\(G_{total}\)	\(\text{W}/\text{m}^2\)

Model Structure

This flat plate collector model consists of 2 StateBlocks assigned to the inlet and outlet ports of the heating fluid.

• Inlet

• Outlet

The flow rate and temperature at the inlet port must be specified by the user.

Sets

The model consists of the phase set included in the property package.

Description	Symbol	Indices
Time	\(t\)	[0]
Phases	\(p\)	[‘Liq’, ‘Vap’]

Parameters

The following parameters are used and are mutable (except for the test conditions).

Description	Parameter Name	Symbol	Value	Units
Number of collectors	number_collectors	\({n}_{c}\)	1	\(\text{dimensionless}\)
Product of cover transmittance (t), and shortwave absorptivity of absorber (a)	trans_absorb_prod	\(\tau\alpha\)	1	\(\text{dimensionless}\)
Product of collector heat removal factor (FR), cover transmittance (t), and shortwave absorptivity of absorber (a)	FR_ta	\({F}_{R}\tau\alpha\)	0.689	\(\text{dimensionless}\)
Product of collector heat removal factor (FR) and overall heat loss coeff. of collector (UL)	FR_UL	\({F}_{R}{U}_{L}\)	3.85	\(\text{W}/\text{m}^2-\text{K}\)
Mass flow rate of fluid during characterization test (fixed)	mdot_test	\(\dot{m}_{test}\)	1	\(\text{kg} / \text{s}\)
Specific heat capacity of fluid during characterization test (fixed)	cp_test	\({c}_{ptest}\)	4184	\(\text{J}/\text{kg}-\text{K}\)
Specific heat capacity of fluid being used for heat transfer in operation	cp_use	\({c}_{use}\)	4184	\(\text{J}/\text{kg}-\text{K}\)
Pump power	pump_power	\({P}_{pump}\)	1	\(\text{W}\)
Pump efficiency	pump_eff	\(\eta_{pump}\)	1	\(\text{dimensionless}\)
Ambient temperature	temperature_ambient	\({T}_{amb}\)	303.15	\(\text{K}\)
Maximum irradiance at the location	max_irradiance	\({G}_{max}\)	1000	\(\text{W} / \text{m}^2\)
Influent minus ambient temperature	factor_delta_T	\(\Delta T\)	0.03	\(\text{K}\)

Equations

The following equations calculate the variables used in estimating heat transfer in a flat plate collector.

Description	Variable Name	Equation	Units
Product of collector efficiency factor and overall heat loss coefficient at test conditions	Fprime_UL	\(F^{'}U_{L} = -(\dot{m}_{test}*{c}_{ptest})/A_{c}* log(1-{F}_{R}{U}_{L}*A_{c}/(\dot{m}_{test}*{c}_{ptest}))\)
Ratio of FRta_use to FRta_test	ratio_FRta	\(r = [m_{l}*{c}_{ptest}/A_{c}]*[1 - exp(-A_{c}*F^{'}U_{L}/m_{l}*{c}_{ptest})]/F_{R}U_{L}|_{test}\)
Useful net heat gain	net_heat_gain	\(h_{gain} = {n}_{c}*A_{c}*r*(F_{R}\tau\alpha*G_{total}*\tau\alpha - F_{R}U_{L}*(T_{f}-{T}_{amb}))\)	\(\text{W}\)
Rated plant heat capacity	heat_load	\(h_{load} = {n}_{c}*A_{c}*(F_{R}\tau\alpha*G_{total}*\tau\alpha - F_{R}U_{L}*\Delta T )\)	\(\text{MW}\)

Costing

The FPC capital cost includes direct costs, indirect costs and sales tax. The direct costs include cost of the collector and a contingency factor. The indirect costs are a fraction of the direct cost and include cost of land. The land area is assumed to be the total collector area. A fixed operating cost is calculated as a linear function of the FPC heat load.

Description	Variable Name	Equation
Direct capital costs	direct_capital_cost	\(Capital Cost_{direct} = ({n}_{c}*A_{c}*\text{Collector Cost per }m^{2})*(1 + \text{Contingency fraction})\)
Indirect capital costs	indirect_capital_cost	\(Capital Cost_{indirect} = Capital Cost_{direct}*\text{indirect capital cost fraction} + \text{Land area}*\text{Cost per acre}\)

References

[1] Solar Engineering of Thermal Processes, Duffie and Beckman, 4th ed.