The analytical model for determining the temperature distribution in transitory regime in piping walls.

Moroianu, Corneliu ; Samoilescu, Gheorghe ; Patrichi, Ilie 等

1. INTRODUCTION

Within the post-processing calculations of pipe networks there are a series of magnitudes such are: linear temperature gradient, linear temperature gradient, thermal discontinuity. These magnitudes appear as part of a thermal transition of the fluid contained in the analyzed pipes. The temperatures transition of the fluid can be of step type (madden jump of temperature), of ramp type (linear rise of temperature), or a succession of combinations of these two types of temperature (Welty, 1974). According to these mentioned above, we propose a thermal analysis which should put at our disposal the temperature distribution in the pipe walls, the average temperature on the pipe thickness, the linear gradient, the nonlinear gradient (Frank & David, 20001).

2. THE HEAT EQUATION SOLVING

The analytical model for determining the temperature distribution in the transitory regime in the wall of an enclosure is based on the following calculus hypotheses (Drake, 1982):

--the ratio between the wall thickness and the enclosure diameter is small enough to justify the neglecting of the pipe bending. As a result, this will be represented as a plane plate.

--to calculate the temperature gradients in the wall, the flow of radial heat is only essential.

--the initial temperature of the enclosure is the same on the whole thickness and length of it on the given distance.

--the thermal properties of the enclosure and the fluid considered in the calculus are those proper to the average temperature of the fluid during the heat transition. Notation:

[theta] = T - Tf;

where,

[T.sub.f]--the fluid temperature on the "t" moment;

[T.sub.i]--the initial temperature of the plate wall;

T = T(x,t)--the wall temperature on the "t" moment and at the "x" depth;

h--film heat transfer coefficient;

[alpha] = k * c / [rho] * C * c]; a--the thermal diffusivity of the wall;

[K.sub.c]--the thermal conductivity of the plate wall the specific heat of the plate;

r--the density of the plate material.

It is considered the cooling/heating of an infinite plate of an even thickness "a". The plate is considered at the initial, uniform temperature [T.sub.i]. At the moment t = 0 the plate is suddenly wetted by the fluid with the temperature [T.sub.f].

The plate is exposed to this heat source with T = [T.sub.f] for all the times when t > 0 and the time coefficient "h" from the fluid to the wall is constant during the heating ([T.sub.f] > T) or cooling ([T.sub.f] < T).

The temperature distribution on the plate thickness, [T.sub.(x, t)] meets the Fourier heat transfer equation (Drake, 1982):

[[partial derivative].sup.2] [theta] / [partial derivative] [x.sup.2] = 1 / [alpha] [partial derivative][theta] / [partial derivative] (1)

--with the initial condition:

a. [[theta].sub.i] (x,t) = [T.sbu.i] - [T.sub.f], for t = 0;

b. [partial derivative][theta](x,t) / [partial derivative]x = 0 for x = 0 (there are no heat losses on the plate face);

c. [partial derivative][theta](x,t) / [partial derivative]x = h / kc [theta]([alpha], t) for x = a (the heat quantity transferred by the fluid is equal to that by the wall).

To integrate the quadratic partial differential equation we assume the general solution [[theta].sub.(x,t)] in the form of (Brun & Irvine, 1996):

[theta](x,t) = X(x)Y(t); (2)

Substituting into the relation (1) and separating the variables, we shall obtain:

1/x [[partial derivative].sup.2 x / [[partial derivative][x.sup.2] = 1 / [alpha]y [partial derivative]y / [partial derivative], (3)

the above equation becomes:

[[partial derivative].sup.2]x / [partial derivative][x.sup.2] + [lambda] x = 0 (4)

[partial derivative]y / [partial derivative] + [alpha][lambda]y = 0

with the solutions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The general solution of the equation (3) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (6)

where, if [lambda] is substituted for [lambda] / [square root of [alpha]], we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (7)

To meet the limiting conditions (b) it results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (8)

where [C.sub.2] = 0.

The limiting condition (c) implies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (9)

After the multiplication through "a", we shall obtain:

[[lambda].sub.a] / [square root of [alpha]] tan ([[lambda].sub.a] / [square root of [alpha]] = [h.sub.a] / [k.sub.c] (10)

The constant:

- [h.sub.a] / [k.sub.c] = [B.sub.i]-the number of Biet.

We note:

[m.sub.n] = [[lambda].sub.n] a / [square root of [alpha]]--as being the root of "n" order of the transcendental equation:

[m.sub.n] tan [m.sub.n] = [B.sub.i]; (11)

Because the differential equation is linear the solution can be written:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (12)

For the limiting and checking condition (a), we have:

[theta](x, 0) = [T.sub.i] - [T.sub.f] = [[summation].sup.[infinity].sub.n=1] [C.sub.n] x cos [m.sub.n]x / a; (13)

We multiply both members through the expression cos [m.sub.n]x / a and we integrate on the plate thickness:

([T.sub.i] - [T.sub.f]) [[integral].sup.a.sub.0] cos [m.sub.n] x / a dx = [[summation].sup.[infinity].sub.n=1] [C.sub.n] x [[integral].sup.a.sub.0] cos [m.sub.n]x / a x cos [m.sub.n]x / a x cos [m.sub.n]x / a dx; (14)

For m [not equal to] n, the integral is 0, and for m = n we obtain:

([T.sub.i] - [T.sub.f]) a / [m.sub.n] sin [m.sub.n] = [c.sub.n] (a / 2 + a sin 2[m.sub.n] / 4[m.sub.n]; (15)

results:

[C.sub.n] = 4([T.sub.i] - [T.sub.f]) sin [m.sub.n] / 2[m.sub.n] + sin 2[m.sub.n];

Noting with:

[F.sub.0] = [alpha]t / [a.sup.2]--the Fourier number.

The complete solution for determining the temperature in a section of the enclosure wall analyzed at the "x" distance transversally (radial) measured at a given moment "t" is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (16)

When the fluid temperature varies linearly the above expression is integrated in relation to the time (from "0" to "t").

The resulted solution is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

The last two relations consider that at the initial moment, the temperature of the wall is identical on its thickness. For the general situation, when at the initial moment t = 0 there is a temperature distribution in the enclosure wall, the initial condition (a) becomes:

[theta]([x.sub.1], 0) = [b.sub.1]

[theta]([x.sub.2], 0) = [b.sub.2]

[theta]([x.sub.n], 0) = [b.sub.n] (18)

In this case, the coefficients Cn of the general solution are determined from the system of N linear equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

The number N of the terms [C.sub.n] mast be high enough (to provide the accuracy of the solution) and corresponds to the number of points where the temperatures on the thickness are determined. In practice, N is considered sufficient when is included between 30 and 40 (Moroianu, 2001).

3. CONCLUSION

As a result of the thermal calculus method presented, the temperature distributions at different intervals of time will result in the enclosure wall.

In the most cases, the temperature distribution on the thickness at a certain given moment is non-linear.

Therefore, the temperature distribution mast be linearization, decomposing to three components separately calculated:

--a constant component with non-zero mean:

[T.sub.med] = 1 /1 [[integral].sup.1.sub.0] [T.sub.(x,t)]dx; (20)

--a linear component with zero mean:

[T.sub.1] = 6[T.sub.med] - 12 / [1.sup.2] [[integral].sup.1.sub.0] x[T.sub.(x,t)] dx (21)

--and a non-linear component:

[T.sub.2] = Max{[[T.sub.(a/2) - [T.sub.med]] - [absolute value of [DELTA][T.sub.1] / 2]; [[T.sub.(a/2) - [T.sub.med]] - [absolute value of [T.sub.1]] / 2; 0}; (22)

4. REFERENCES

Brun, E.; Irvine, C., (1975), Modern research laboratories for heat and mass transfer, Michigan University, ISBN 9231011030, Michigan

Drake, R .M. (1959). Analysis the heat and mass transfer, McGraw-Hill, New York

Frank P.; David P. (20001), Fundamentals of heat end mass transfer, Publisher Wiley, ISBN 978-0471386506, New York

Moroianu, C., (2001), The liquid fuel burning in the naval propulsion systems Editura Academiei Navale "Mircea cel Batran" ISBN 973-8303-04-4, Constanta

Welty, J.R. (1974). Engineering heat transfer, Jon Wiley, New York