Abstract
This work presents an analytical solution for the periodic heat transfer problem of regenerators used in air conditioning, which are operating at low regeneration temperatures and mass flow rates. These types of regenerators are characterized by NTU/Cr ∗<1. The partial differential equations for hot and cold airflows as well as the regenerator matrix were solved using a successive transformation of variables. They were reduced to the ordinary Bessel differential equation of the type xf″+f′−xf=0. The conventional initial and reversal boundary conditions were used in this work. The solution gives a correlation for the prediction of the regenerator effectiveness. Besides the effectiveness, the solution facilitates the calculation of the matrix temperature distribution and exit airflow temperatures. The result is compared with the available numerical and analytical solutions from literature. The result of this analysis reveals that the consideration of a non-linear matrix temperature distribution as in some previous work for low temperature regenerators just complicates the solution procedure with no significant improvement in the accuracy within the parameter space typical for air conditioning applications.
Zusammenfassung
In diesem Beitrag wird eine analytische Lösung zur Beschreibung des periodisch instationären Wärmeübergangs in Regeneratoren vorgeschlagen. Diese Lösung gilt in einem eingeschränkten Parameterbereich, der durch NTU/Cr ∗<1 gekennzeichnet ist und niedrige Temperaturniveaus und niedrige Massenströme charakterisiert. Die zugrunde liegenden partiellen Differentialgleichungen werden durch sukzessive Transformation auf eine Bessel-Funktion zurückgeführt. Die Lösung dieser Gleichungen führt auf eine Beziehung zur Vorhersage des Regenerator-Wirkungsgrades. Zudem ermöglicht das Verfahren die Berechnung der Temperaturverteilung in der rotierenden Matrix und im Austritts-Luftstrom. Das vorgestellte Verfahren wird mit literaturbekannten numerischen und analytischen Lösungen verglichen, wobei der Vorteil des vereinfachten Ansatzes für den in der Raumklimatisierung typischen Anwendungsbereich deutlich wird.
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Abbreviations
- a :
-
Constant
- A :
-
Heat transfer area, Constant
- b :
-
Constant
- B :
-
Constant
- C :
-
Constant
- c p :
-
Specific heat capacity
- C:
-
Heat capacity
- I :
-
Modified Bessel function of first kind
- L :
-
Length
- J :
-
Integral function
- K :
-
Modified Bessel function of second kind
- M :
-
Mass flow rate
- N :
-
Rotational speed
- t :
-
Time
- T :
-
Temperature
- u :
-
Velocity
- V :
-
Matrix volume
- x :
-
Axial coordinate
- α :
-
Heat transfer coefficient
- λ :
-
Thermal conductivity, arbitrary variable
- ε :
-
Effectiveness
- ρ :
-
Density
- φ :
-
Correction factor
- θ :
-
Normalized temperature
- C :
-
Gases heat capacity ratio
- Cr :
-
Matrix to the gas heat capacity ratio
- NTU :
-
Number of transfer units
- ζ:
-
Dimensionless length
- η :
-
Dimensionless time
- c :
-
Cold
- i :
-
Inlet
- h :
-
Hot
- o :
-
Outlet, initial condition
- °:
-
Degree
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The authors acknowledge the Alexander von Humboldt Stiftung research fellowship to A.A. Rabah.
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Rabah, A.A., Kabelac, S. A simplified solution of the regenerator periodic problem: the case for air conditioning. Forsch Ingenieurwes 74, 207–214 (2010). https://doi.org/10.1007/s10010-010-0126-z
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DOI: https://doi.org/10.1007/s10010-010-0126-z