Influence of longitudinal conduction in the matrix on effectiveness of rotary heat regenerator used in air-conditioning

Mieczysław Porowski; Edward Szczechowiak

doi:10.1007/s00231-006-0205-8

Heat and Mass Transfer

September 2007, Volume 43, Issue 11, pp 1185–1200 | Cite as

Influence of longitudinal conduction in the matrix on effectiveness of rotary heat regenerator used in air-conditioning

Authors
Authors and affiliations

Mieczysław PorowskiEmail author
Edward Szczechowiak

Original

First Online: 17 November 2006

Received: 21 March 2006

Accepted: 12 October 2006

126 Downloads
3 Citations

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.

Keywords

Aluminium Matrix Energy Recovery Quasi Steady State Longitudinal Conduction Fixed Matrix

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

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)

Greek symbols

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)
Φ_t: effectiveness of the thermal energy recovery
σΦ_t: 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)

Subscripts

c: total volume of fluid and matrix
i: inlet
f: fluid
s: matrix, matrix surface

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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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Authors and Affiliations

Mieczysław Porowski
- 1
Email author
Edward Szczechowiak
- 1

1.Institute of Environmental EngineeringPoznań University of TechnologyPoznańPoland

Influence of longitudinal conduction in the matrix on effectiveness of rotary heat regenerator used in air-conditioning

Abstract

Keywords

List of symbols

Greek symbols

Subscripts

References

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