Abstract
The paper presents an efficient analytical method of solving the problem of a rotary heat regenerator taking into account longitudinal heat conduction in the matrix. The small parameter method, Laplace transform as well as one of the spline functions have been applied for approximation of an initial condition in the reversion time. In the application part, the solution for a model in analysis of an influence of longitudinal conduction in the matrix on effectiveness of rotary heat regenerator in a wide range of dimensionless parameters as well as for the particular matrix applied in air-conditioning was used.
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Abbreviations
- A :
-
heat exchange area (m2)
- A nk :
-
the elements of the matrix
- Bes n :
-
special functions
- Bs n :
-
special functions
- c :
-
specific heat (J/kg K)
- C :
-
dimensionless number defining capacity of the heat regenerator
- c p :
-
specific heat at constant pressure (J/kg K)
- D j :
-
special functions
- E j :
-
generalized dimensionless numbers of the heat regenerator
- H :
-
dimensionless number
- H (η):
-
heaviside function
- i :
-
natural number
- K s :
-
dimensionless parameter taking into account longitudinal heat conduction in the matrix
- k L :
-
dimensionless parameter taking into account longitudinal conduction in the matrix, k L = K s/Λ
- L :
-
length of the regenerator (m)
- \({\dot{m}_{\rm f}}\) :
-
fluid mass stream
- M n :
-
elements of the column matrix
- N :
-
natural number
- n :
-
natural number
- p :
-
pressure (Pa)
- p :
-
co-ordinate in the area of the complex variable
- Q :
-
energy (J)
- R jn :
-
special functions
- R j :
-
special functions series
- S :
-
specific surface of matrix heat exchange (m2/m3)
- T :
-
dimensionless temperature
- t :
-
temperature (°C)
- w :
-
velocity of the air inlet in front of the matrix, w = εw f (m/s)
- w f :
-
velocity of the air in the matrix ducts (m/s)
- x :
-
geometric co-ordinate (m)
- α:
-
coefficient of heat transfer inside the matrix (W/m2 K)
- α0 :
-
coefficient of heat transfer at the boundary surface (W/m2 K)
- δ(η):
-
Dirac δ distribution
- ε:
-
matrix porosity
- ε:
-
elementary interval of polygon function
- ρ:
-
density (kg/m3)
- η:
-
dimensionless time
- Λ:
-
dimensionless length of regenerator
- λ:
-
thermal conductivity coefficient (W/m K)
- ξ:
-
dimensionless geometric co-ordinate in the direction of fluid flow
- Π:
-
dimensionless parameter
- τ:
-
time (s)
- τ0 :
-
characteristic time (s)
- Φt :
-
effectiveness of the thermal energy recovery
- σΦt :
-
relative drop of the thermal energy recovery effectiveness as a consequence of the longitudinal conductivity in the matrix
- ω:
-
angular speed of rotor of regenerator (1/s)
- c:
-
total volume of fluid and matrix
- i:
-
inlet
- f:
-
fluid
- s:
-
matrix, matrix surface
References
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Appendix
Appendix
-
1.
Dirac δ distribution
$$\delta {\left( \xi \right)} = \left\{ {\begin{array}{*{20}c} {{0\quad \rm for\quad \xi \ne 0}} \\ {{\infty \quad \rm for\quad \xi = 0}} \\ \end{array} } \right.\quad \hbox{and}\quad {\int\limits_{- \infty}^{+ \infty} {\delta (\xi){\rm d}\xi = 1}}$$(81) -
2.
Heaviside function
$$H{\left( \eta \right)} = \left\{ {\begin{array}{*{20}c} {{0\quad \rm for\quad \eta < 0}} \\ {{1\quad \rm for\quad \eta \geq 0}} \\ \end{array} } \right.$$(82) -
3.
Convolutions of functions
-
Single convolution of the function:
$$f(\eta)*g(\eta) = {\int\limits_0^\eta {f(\eta ^{'})g(\eta - \eta ^{'}){\rm d}\eta ^{'}}}$$(83) -
Double convolution of the function:
$$f(\xi, \eta){\mathop *\limits_\xi}{\mathop *\limits_\eta}g(\xi, \eta) = \int\limits_0^\xi \int\limits_0^\eta f(\xi ^{'}, \eta ^{'})g[(\xi - \xi ^{'}),(\eta - \eta ^{'})]{\rm d}\xi ^{'} {\rm d}\eta ^{'}$$(84)
-
-
4.
Special functions Bes n (ξ, η) and Bs n (ξ, η)
-
The functions defined by ach and Pieczka [10, 11] have form of series:
$$\hbox{Bs}_{n} (\Lambda \xi, \Pi \eta) = {\sum\limits_{m = \max (0,n)}^\infty {\frac{{(\Lambda \xi)^{m} }}{{m!}}{\sum\limits_{k = 0}^{m - n} {\frac{{(\Pi \eta )^{k}}}{{k!}}}}}}$$(85)$$\hbox{Bes}_{n} (\Lambda \xi, \Pi \eta) = {\sum\limits_{m = \max (- n,0)}^\infty {\frac{{(\Lambda \xi)^{{m + n}} (\Pi \eta)^{m}}}{{(m + n)!m!}}}}$$(86)
-
-
5.
Special functions \(D_{j}(\Lambda \xi, \Pi \eta)\)
$$D_{0} (\Lambda \xi, \Pi \eta) = \delta (\xi)\exp (- \Pi \eta) + \Lambda \exp (- \Lambda \xi - \Pi \eta)\hbox{Bes}_{{- 1}} (\Lambda \xi, \Pi \eta)$$(87)$$D_{1} (\Lambda \xi, \Pi \eta) = \exp (- \Lambda \xi - \Pi \eta)\hbox{Bes}_{0} (\Lambda \xi, \Pi \eta)$$(88)$$D_{2} (\Lambda \xi, \Pi \eta) = \frac{1}{\Lambda }\exp (- \Lambda \xi - \Pi \eta)\hbox{Bes}_{1} (\Lambda \xi, \Pi \eta)$$(89) -
6.
Special functions R jn
$$R_{{00}} (\Lambda \xi, \Pi \eta) = \exp (- \Lambda \xi - \Pi \eta)\hbox{Bs}_{0} (\Lambda \xi, \Pi \eta)$$(90)$$R_{{10}} (\Lambda \xi, \Pi \eta) = \frac{1}{\Lambda }\exp (- \Lambda \xi - \Pi \eta)[\Lambda \xi \hbox{Bs}_{0} (\Lambda \xi, \Pi \eta) - \Pi \eta \hbox{Bs}_{2} (\Lambda \xi, \Pi \eta)]$$(91)$$R_{{20}} (\Lambda \xi, \Pi \eta) = \frac{1}{\Lambda }\exp (- \Lambda \xi - \Pi \eta){\rm Bs}_{1} (\Lambda \xi, \Pi \eta)$$(92)$$R_{{30}} (\Lambda \xi, \Pi \eta) = \frac{1}{{\Lambda ^{2}}}\exp (- \Lambda \xi - \Pi \eta){\sum\limits_{i = 1}^\infty {\hbox{Bs}_{{i + 1}} (\Lambda \xi, \Pi \eta}})$$(93)$$R_{{01}} (\Lambda \xi, \Pi \eta) = \exp (- \Lambda \xi - \Pi \eta)\left[ - \hbox{Bes}_{{- 2}} (\Lambda \xi, \Pi \eta) + (1 - \Lambda \xi)\hbox{Bes}_{{- 3}} (\Lambda \xi, \Pi \eta) + \Lambda \xi \hbox{Bes}_{{- 4}} (\Lambda \xi, \Pi \eta)\right]$$(94)$$R_{{11}} (\Lambda \xi, \Pi \eta) = \frac{1}{\Lambda }\exp (- \Lambda \xi - \Pi \eta)\left[\hbox{Bes}_{{- 2}} (\Lambda \xi, \Pi \eta) + \Lambda \xi \hbox{Bes}_{{- 3}} (\Lambda \xi, \Pi \eta)\right]$$(95)$$R_{{21}} (\Lambda \xi, \Pi \eta) = \Lambda \xi \exp (- \Lambda \xi - \Pi \eta)\left[\hbox{Bes}_{{- 3}} (\Lambda \xi, \Pi \eta) - \hbox{Bes}_{{- 2}} (\Lambda \xi, \Pi \eta)\right]$$(96)$$R_{{31}} (\Lambda \xi, \Pi \eta) = \xi \exp (- \Lambda \xi - \Pi \eta)\hbox{Bes}_{{- 2}} (\Lambda \xi, \Pi \eta)]$$(97)$$\begin{aligned} R_{{02}} (\Lambda \xi, \Pi \eta) &= \exp (- \Lambda \xi - \Pi \eta)\left\{\vphantom{\frac{1}{2}}- \hbox{Bes}_{{- 3}} (\Lambda \xi, \Pi \eta) - (2\Lambda \xi - 5)\hbox{Bes}_{{- 4}} (\Lambda \xi,\Pi \eta) \right.\\ &\quad- \left[\frac{1}{2}(\Lambda \xi)^{2} - 8\Lambda \xi + 7\right]\hbox{Bes}_{{- 5}} (\Lambda \xi, \Pi \eta)\\ &\quad-\left[\frac{3}{2}(\Lambda \xi)^{2} - 10\Lambda \xi + 3\right]\hbox{Bes}_{{- 6}} (\Lambda \xi, \Pi \eta) - \Lambda \xi (\frac{3}{2}\Lambda \xi - 4)\hbox{Bes}_{{- 7}} (\Lambda \xi, \Pi \eta) \\ &\left.\quad+ \frac{1}{2}(\Lambda \xi)^{2} \hbox{Bes}_{{- 8}}(\Lambda \xi, \Pi \eta)\right\} \end{aligned}$$(98)$$\begin{aligned} R_{{12}} (\Lambda \xi, \Pi \eta) &= \frac{1}{\Lambda }\exp (- \Lambda \xi - \Pi \eta)\left\{\vphantom{\frac{1}{2}}\hbox{Bes}_{{- 3}} (\Lambda \xi, \Pi \eta) + 2(\Lambda \xi - 2)\hbox{Bes}_{{- 4}} (\Lambda \xi, \Pi \eta)\right.\\ &\quad + \left[\frac{1}{2}(\Lambda \xi)^{2} - 6\Lambda \xi + 3\right]\hbox{Bes}_{{- 5}} (\Lambda \xi, \Pi \eta) - \Lambda \xi (\Lambda \xi - 4)\hbox{Bes}_{{- 6}} (\Lambda \xi, \Pi \eta) \\ &\left.\quad+ \frac{1}{2}(\Lambda \xi)^{2} \hbox{Bes}_{{- 7}} (\Lambda \xi, \Pi \eta)\right\} \end{aligned}$$(99)$$\begin{aligned} R_{{22}} (\Lambda \xi, \Pi \eta) &= \exp (- \Lambda \xi - \Pi \eta)\left\{- \Lambda \xi \hbox{Bes}_{{- 3}} (\Lambda \xi, \Pi \eta) - \Lambda \xi \left(\frac{1}{2} \Lambda \xi - 5\right)\hbox{Bes}_{{- 4}} (\Lambda \xi,\Pi \eta) \right.\\ &\quad+ \Lambda \xi \left(\frac{3}{2}\Lambda \xi - 7\right)\hbox{Bes}_{{- 5}} (\Lambda \xi, \Pi \eta) - \Lambda \xi \left(\frac{3}{2}\Lambda \xi - 3\right)\hbox{Bes}_{{- 6}} (\Lambda \xi, \Pi \eta) \\ & \left.\quad + \frac{1}{2}(\Lambda \xi)^{2} \hbox{Bes}_{{- 7}} (\Lambda \xi, \Pi \eta)\right\} \end{aligned}$$(100)$$\begin{aligned} R_{{32}} (\Lambda \xi, \Pi \eta) &= \frac{1}{\Lambda }\exp (- \Lambda \xi - \Pi \eta)\left\{\vphantom{\frac{1}{2}}\Lambda \xi \hbox{Bes}_{{- 3}} (\Lambda \xi, \Pi \eta) + \Lambda \xi \left(\frac{1}{2}\Lambda \xi - 4\right)\hbox{Bes}_{{- 4}} (\Lambda \xi, \Pi \eta)\right.\\ &\quad\left.- \Lambda \xi (\Lambda \xi - 3)\hbox{Bes}_{{- 5}} (\Lambda \xi, \Pi \eta) + \frac{1}{2}(\Lambda \xi)^{2} \hbox{Bes}_{{- 6}} (\Lambda \xi, \Pi \eta)\right\} \end{aligned}$$(101)$$\begin{aligned} R_{{03}} (\Lambda \xi, \Pi \eta) &= \exp (- \Lambda \xi - \Pi \eta)\left\{\vphantom{\frac{1}{2}}- \hbox{Bes}_{{- 4}} (\Lambda \xi, \Pi \eta) - (3\Lambda \xi - 11)\hbox{Bes}_{{- 5}} (\Lambda \xi, \Pi \eta)\right.\\ &\quad- \left[\frac{3}{2}(\Lambda \xi)^{2} - 25\Lambda \xi + 39\right]\hbox{Bes}_{{- 6}} (\Lambda \xi, \Pi \eta)\\ &\quad - \left[\frac{1}{6}(\Lambda \xi)^{3} - 9\frac{1}{2}(\Lambda \xi)^{2} + 75\Lambda \xi - 61\right]\hbox{Bes}_{{- 7}} (\Lambda \xi, \Pi \eta) \\ &\quad +\left[\frac{5}{6}(\Lambda \xi)^{3} - 23(\Lambda \xi)^{2} + 105\Lambda \xi - 44\right]\hbox{Bes}_{{- 8}} (\Lambda \xi, \Pi\eta) \\ &\quad- \left[\frac{5}{3}(\Lambda \xi)^{3} - 27(\Lambda \xi)^{2} + 70\Lambda \xi - 12\right]\hbox{Bes}_{{- 9}} (\Lambda \xi, \Pi \eta) \\ &\quad + \Lambda \xi \left[\frac{5}{3}(\Lambda \xi )^{2} - 15\frac{1}{2}\Lambda \xi + 18\right]\hbox{Bes}_{{- 10}} (\Lambda \xi, \Pi \eta) \\ &\left.\quad- (\Lambda \xi)^{2} \left[\frac{5}{6}\Lambda \xi - 3\frac{1}{2}\right]\hbox{Bes}_{{- 11}} (\Lambda \xi, \Pi \eta) + \frac{1}{6}(\Lambda \xi)^{3} \hbox{Bes}_{{- 12}} (\Lambda \xi, \Pi \eta)\right\} \end{aligned}$$(102)$$\begin{aligned} R_{{13}} (\Lambda \xi, \Pi \eta) &= \frac{1}{\Lambda }\exp (- \Lambda \xi - \Pi \eta)\left\{\vphantom{\frac{1}{2}}- \hbox{Bes}_{{- 4}} (\Lambda \xi, \Pi \eta) + (3\Lambda \xi - 10)\hbox{Bes}_{{- 5}} (\Lambda \xi, \Pi \eta)\right. \\ &\quad+ \left[\frac{3}{2}(\Lambda \xi)^{2} - 22\Lambda \xi + 29\right]\hbox{Bes}_{{- 6}} (\Lambda \xi, \Pi \eta) \\ &\quad+ \left[\frac{1}{6}(\Lambda \xi)^{3} - 8(\Lambda \xi)^{2} + 53\Lambda \xi - 32\right]\hbox{Bes}_{{- 7}} (\Lambda \xi, \Pi \eta) \\ &\quad - \left[\frac{2}{3}(\Lambda \xi)^{3} - 15(\Lambda \xi)^{2} + 52\Lambda \xi - 12\right]\hbox{Bes}_{{- 8}} (\Lambda \xi, \Pi \eta)\\ &\quad+ \Lambda \xi\left[(\Lambda \xi)^{2} - 12\Lambda \xi + 18\right] \hbox{Bes}_{{- 9}} (\Lambda \xi, \Pi \eta) \\ &\left.\quad - (\Lambda \xi)^{2} \left[\frac{2}{3}\Lambda \xi - 3\frac{1}{2}\right]\hbox{Bes}_{{- 10}} (\Lambda \xi, \Pi \eta) + \frac{1}{6}(\Lambda \xi)^{3} \hbox{Bes}_{{- 11}} (\Lambda \xi, \Pi \eta)\right\} \end{aligned}$$(103)$$\begin{aligned} R_{{23}} (\Lambda \xi, \Pi \eta) &= \exp (- \Lambda \xi - \Pi \eta)\left\{\vphantom{\frac{1}{2}}- \Lambda \xi \hbox{Bes}_{{- 4}} (\Lambda \xi, \Pi \eta) - \Lambda \xi (\Lambda \xi - 11)\hbox{Bes}_{{- 5}} (\Lambda \xi, \Pi \eta)\right. \\ &\quad- \Lambda \xi \left[\frac{1}{6}(\Lambda \xi)^{2} - 7\Lambda \xi + 39\right]\hbox{Bes}_{{- 6}} (\Lambda \xi, \Pi \eta) \\ &\quad+ \Lambda \xi \left[\frac{5}{6}(\Lambda \xi)^{2} - 18\Lambda \xi + 61\right]\hbox{Bes}_{{- 7}} (\Lambda \xi, \Pi \eta) \\ &\quad- \Lambda \xi \left[\frac{5}{3}(\Lambda \xi)^{2} - 21\Lambda \xi + 44\right]\hbox{Bes}_{{- 8}} (\Lambda \xi, \Pi \eta) \\ &\quad+ \Lambda \xi \left[\frac{5}{3}(\Lambda \xi)^{2} - 13\Lambda \xi + 12\right]\hbox{Bes}_{{- 9}} (\Lambda \xi, \Pi \eta) \\ &\left.\quad - (\Lambda \xi)^{2} \left[\frac{5}{3}\Lambda \xi - 3\right]\hbox{Bes}_{{- 10}} (\Lambda \xi, \Pi \eta) + \frac{1}{6}(\Lambda \xi)^{3} \hbox{Bes}_{{- 11}} (\Lambda \xi, \Pi \eta)\right\} \end{aligned}$$(104)$$\begin{aligned} R_{{33}} (\Lambda \xi, \Pi \eta) &= \exp (- \Lambda \xi - \Pi \eta)\left\{\vphantom{\frac{1}{2}}\Lambda \xi \hbox{Bes}_{{- 4}}(\Lambda \xi, \Pi \eta) \right.\\ &\quad+ \Lambda \xi (\Lambda \xi - 10)\hbox{Bes}_{{- 5}} (\Lambda \xi, \Pi \eta) + \Lambda \xi \left[\frac{1}{6}(\Lambda \xi)^{2} - 6\Lambda \xi + 29\right]\hbox{Bes}_{{- 6}} (\Lambda \xi, \Pi \eta) \\ &\quad- \Lambda \xi \left[\frac{2}{3}(\Lambda \xi)^{2} - 12\Lambda \xi + 32\right]\hbox{Bes}_{{- 7}} (\Lambda \xi, \Pi \eta) \\ &\quad+ \Lambda \xi \left[(\Lambda \xi)^{2} - 10\Lambda \xi + 12\right]\hbox{Bes}_{{- 8}} (\Lambda \xi, \Pi \eta) \\ &\left.\quad- (\Lambda \xi)^{2} \left(\frac{2}{3}\Lambda \xi - 3\right)\hbox{Bes}_{{- 9}} (\Lambda \xi, \Pi \eta) + \frac{1}{6}(\Lambda \xi)^{3} \hbox{Bes}_{{- 10}} (\Lambda \xi, \Pi \eta)\right\} \end{aligned}$$(105)
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Porowski, M., Szczechowiak, E. Influence of longitudinal conduction in the matrix on effectiveness of rotary heat regenerator used in air-conditioning. Heat Mass Transfer 43, 1185–1200 (2007). https://doi.org/10.1007/s00231-006-0205-8
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DOI: https://doi.org/10.1007/s00231-006-0205-8