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An analytic solution to the electric analog simulation of the regenerative heat exchanger with time-varying fluid inlet temperatures

Eine analytische Lösung der elektrischen Analogie des regenerativen Wärmetauschers bei zeitlich veränderlichen Einlaßtemperaturen des Fluids

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Abstract

Direct simulation on an electric analog of heat transfer problems, has been extensively used, for obtaining approximate solutions. The electric simulation is based on discrete space continuous time differential equations, using electric capacitances and resistances with special devices for simulating convective terms and boundary conditions. (See ref. [16] in the text). In this paper, an analytic solution of the discretized space continuous time differential equations, describing the performance of the periodic-flow regenerative heat exchanger, is presented. With this method the cyclic periodic operation and the starting-up periods of the regenerator, having arbitrary fluid inlet and initial matrix temperatures, can be determined. Several cases with constant inlet fluid temperatures are given and the results are in a good agreement with those previously reported in the literature; the computations for the case of a starting-up period having linearly varying fluid inlet temperature are also presented, which compares well with the results given in a previous paper. Using the present method Hausen's simplified theory is tested and it is found to be justifiable.

Zusammenfassung

Die direkte Simulation von Wärmeübergangsproblemen in einem elektrischen Analoggerät wurde häufig benutzt, um Näherungslösungen zu erhalten. Die elektrische Simulation beruht auf Differentialgleichungen mit diskreten Raumelementen und kontinuierlicher Zeit, wobei Kapazitäten und Widerstände sowie besondere Einrichtungen zur Simulation konvektiver Terme und der Grenzbedingungen benutzt werden (vgl. Referenz [16]). In dieser Arbeit wird eine analytische Lösung einer Differentialgleichung mit diskretisiertem Raum und kontinuierlicher Zeit mitgeteilt, die das Verhalten eines periodisch betriebenen regenerativen Wärmetauschers beschreibt. Hiermit können sowohl die zyklische Arbeitsperiode als auch der Anlaufbereich beschreiben werden und zwar mit beliebigen Einlauf-und anfänglichen Matrixtemperaturen. Einige hier angegebene Fälle mit konstanten Einlauftemperaturen stimmen gut mit früheren Rechnungen aus der Literatur überein. Rechnungen im Anlaufbereich mit einer linear veränderlichen Einlauftemperatur stimmen gut mit Ergebnissen einer früheren Arbeit überein. Ein Vergleich mit der vereinfachten Theorie von Hausen zeigt, daß diese berechtigt ist.

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Abbreviations

A:

total heat transfer area [m2]

An, Aj :

constants

a:

Δξ/1 + Δξ dimensionless

b:

1/1 + Δξ dimensionless

C:

flow stream capacity [Cal/hr°K]

c:

specific heat of matrix material [Cal/Kgm-°K]

E:

regenerator effectiveness for one period

Er :

regenerator effectiveness

f(τ):

exp (τ) U0(τ) dimensionless temperature

G:

dimensionless parameter defined by the Eq.(77)

h:

corrected heat transfer coefficient [Cal/hr-m2-°K]

h :

heat transfer coefficient [Cal/hr-m2-°K]

K:

overall heat transfer coefficient [Cal/hr-m2-°K]

Kn :

general solution of the Eq. (20)

K0 :

overall heat transfer coefficient for infinitely long periods [Cal/hr-m2-°K]

k:

thermal conductivity of matrix material [Cal/hr-m-°K]

L:

length of the matrix [m]

M:

total mass of the matrix [Kgm]

N:

total number of nodes

P:

period [hr]

Q:

total heat transfered per period [Cal/period]

Rn :

particular solution of Eq. (20)

S rn :

quantity defined by the Eq. (45)

T:

matrix temperature [°K]

Tj :

fluid temperature [°K]

t:

time [hr]

U:

dimensionless fluid temperature

V:

dimensionless matrix temperature

w:

thickness of the matrix [m]

Z, Z1 :

dimensionless temperatures

z:

length along the flow [m]

αj,n :

function of τ, defined by the Eq. (23)

γ :

ratio of CP/C′P′

η:

= hAt/cM dimensionless time

Λ:

dimensionless length of the regenerator

ξ:

= hAz/CL dimensionless length

Π:

dimensionless period

τ:

=η dimensionless time

n:

refer to the nth node

0:

refer to the entrance condition

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  1. Department of Applied Sciences, College of Staten Island, Cuny St. George Campus, 10301, 130 Stuyvesant Place Staten Island, N. Y., USA

    P. Razelos

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Razelos, P. An analytic solution to the electric analog simulation of the regenerative heat exchanger with time-varying fluid inlet temperatures. Wärme- und Stoffübertragung 12, 59–71 (1979). https://doi.org/10.1007/BF01672442

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