Energy Systems

, Volume 10, Issue 1, pp 141–161 | Cite as

A novel stochastic energy analysis of a solar air heater: case study in solar radiation uncertainty

  • Hamed Johnny Sarnavi
  • Ali M. NikbakhtEmail author
  • Ali Hasanpour
  • Feyzollah Shahbazi
  • Niccolo Aste
  • Fabrizio Leonforte
Original Paper


There is a growing recognition of the fact that solar energy utilization plans cannot be carried out without explicitly accounting for the uncertainty presented in the received solar irradiation. This may be expressed as an uncertainty quantification problem. A novel stochastic energy analysis is introduced to study the transient heat transfer problem of a typical flat plate solar air heater, based on the polynomial chaos expansion approach. The constructed model was equipped with the numerical finite difference method and the Galerkin projection scheme in the random space. The numerical model was verified against the available exact analytical solutions. The results of polynomial chaos method was compared to corresponding basic Monte Carlo sampling results. Finally, a case study with realistic solar irradiance data of Urmia, a cold climate city in Iran, was conducted for a typical solar air heater. Afterward, the outlet temperature of the air heater was tracked in a probabilistic framework, to find the reliable hours for extracting solar energy stably during a typical summery day. These hours were found between 11 am to 5 pm.The proposed approach could be highly worthwhile in the designing and contriving control plans taking into consideration the non-negligible uncertainty of solar radiation.


Solar energy Air heater Stochastic modeling Radiation uncertainty 



Distance from collector inlet (m)

\(\Delta {{\varvec{x}}}\)

mesh length in numerical solution (m)


Length of collector (m)


Dimensionless distance from collector’s inlet \(\left( X=\frac{x}{L}\right) \)


Width of collector (m)


time started from sunrise (h)

\(\Delta {{\varvec{t}}}\)

Time step in numerical solution (h)

\( \bar{t} \)

\( \bar{t} = \left\{ {\begin{array}{*{20}l} {0,} &{} {\quad t \le t_{{ss}} } \\ {t - t_{{ss}} ,} &{} {\quad t > t_{{ss}} } \\ \end{array} } \right. \)


Solar irradiance (W/m2)


Maximum solar irradiance during a day (W/m\(^2\))

\(\mathbb {E}\)

First statistical moment (mean)

\(\mathbb {VAR}\)

Second central statistical moment (variance)

\({{\varvec{N}}}( {-\infty ,+\infty } )\)

Normal distribution


Independent variable defined in Eq. 4


Independent variable defined in Eq. 4

\({\varvec{\mu }}\)

Summation numerator in Eq. 4


Summation numerator in Eq. 4

\({\varvec{\delta }} \)

Kronecker delta


Order of \(R_{\left( t \right) } \) polynomial representation

\({\varvec{\theta }} \)

weighting coefficient of Crank–Nicholson backward difference scheme in Eq. 6


Time mesh superscript in numerical solution


Space mesh subscript in numerical solution

\({{\varvec{n}}}_{{\varvec{{i}}}} \)

Number of space mesh in numerical solution

\({{\varvec{n}}}_{{\varvec{{j}}}} \)

Number of timesteps in numerical solution

\({\upxi }\)

Random outcome with normal distribution which can take values between \(-\infty \) and \(+\infty \).


Solar irradiance intensity ratio (dimensionless)

\({{\varvec{R}}}_{{\varvec{{( t )}}}} \)

Solar irradiance intensity ratio as a function of time \(R_{\left( t \right) } =\left\{ {\begin{array}{*{20}l} {\sum \nolimits _{{k = 0}}^{n} {a_{k} } t^{k} ,} &{} {\quad 0 \le t \le t_{{ss}} } \\ {0,} &{} {\quad t > t_{{ss}} } \\ \end{array} } \right. \)

\(\mathbf{R}_{( \mathbf{t};{\upxi } )} \)

Uncertain solar irradiance Intensity Ratio

\({\varvec{\tau }}\)

Temperature (C)


Dimensionless temperature


the overall collector heat loss coefficient (W/m2C)

\({{\varvec{c}}}_{{\varvec{p}}} \)

Absorbing plate specific heat (J/kgC)

\({{\varvec{c}}}_{{\varvec{f}}} \)

fluid (blowing air) specific heat (J/kgC)


Air mass flow rate (kg/s)

\({{\varvec{m}}}_{{\varvec{p}}} \)

mass of plate per unit length (kg/m)

\({{\varvec{N}}}_{{\varvec{C}}} \)

Convection number

\({{\varvec{N}}}_{{\varvec{L}}} \)

Heat loss number

\({\varvec{\gamma }}\)

\(\gamma =UW/m_p c_p \)

\({{\varvec{I}}}_0 \)

First order Bessel function

\({{\varvec{w}}}_{( {\varvec{\xi }} )} \)

weighting function of normal distribution \(w_{\left( \xi \right) } =\frac{1}{\sqrt{2\pi }}e^{-\frac{\xi ^{2}}{2}}\)

\({{\varvec{H}}}_{( {\varvec{\xi }} )}^{\varvec{\kappa }} \)

Hermite polynomials of \(\kappa \)-th order (Table 2)

\({\varvec{\iota }} \)

Order of Hermite polynomials

\({\varvec{\kappa }} \)

Order of Hermite polynomials


Parameter defined in Eq. 13

\({\varvec{\epsilon }}\)

Number of random dimensions (defined in Eq. 13)


The highest degree of polynomial (defined in Eq. 13)

\( \hat{{\varvec{T}}}_{{\varvec{f}}}^{*} \)

Dimensionless outlet temperature at noon


Dimensionless outlet temperature



Absorbing plate


Agent fluid (blowing air)






Along collector length



Flat plate solar air heaters


Polynomial chaos method


Monte Carlo method


Probability density functions


Solar irradiance intensity


Finite difference


Collector time constant


Partial differential equation


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Hamed Johnny Sarnavi
    • 1
  • Ali M. Nikbakht
    • 1
    Email author
  • Ali Hasanpour
    • 1
  • Feyzollah Shahbazi
    • 2
  • Niccolo Aste
    • 3
  • Fabrizio Leonforte
    • 3
  1. 1.Department of Mechanical Engineering of BiosystemsUrmia UniversityUrmiaIran
  2. 2.Department of Mechanical Engineering of BiosystemsLorestan UniversityKhorramabadIran
  3. 3.Department of Architecture, Built Environment and Construction EngineeringPolitecnico di Milano UniversityMilanItaly

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