METHOD FOR DETERMINATION OF CRITICAL AND EFFECTIVE DIAMETER OF PIPELINE INSULATION

Khachatur Kyureghyan¹, Franciszek Kluza², Dariusz Góral², Paweł Artur Kluza^1
1 Department of Applied Mathematics, University of Life Sciences in Lublin, Poland
² Department of Refrigeration and Food Industry Energetics, University of Life Sciences in Lublin, Poland

ABSTRACT

The objective of the paper is to define the physical-mathematical conditions supporting the proper determination of the critical and minimally effective diameter of insulation, because of its important role in refrigeration, air conditioning and heat pumps systems. A simple case of heat exchange under steady state conditions for cylindrical pipes with single – layer insulation was analyzed, where thermal conductivity of the pipe material (k₁) and insulation (k₂) as well as heat transfer coefficients at the pipe interior surface and the insulation exterior surface are the constant values. There has been presented a simple method to determine the optimal insulation diameters that guarantee an increase or decrease of losses due to overall heat transfer per unit length of the pipe.

Key words: heat transfer, thermal insulation, break-even radius.

NOMENCLATURE

d – diameter [m]
h – heat transfer coefficient [Wm^-2K^-1]
k – thermal conductivity [Wm^-1K^-1]
n – relative change of heat
m – m = n^-1
 – heat flux through pipe wall, per pipe unit length [W]
q  – heat flux density [Wm^-1]
r  – radius [m]
R – thermal resistance per unit length of pipe [mKW^-1]
T  – temperature [K]

Subscripts
    cr – critical
    e  – effective
    opt – optimal
    max – maximal
    min – minimal

1. INTRODUCTION

Effective thermal insulation of pipelines plays an important role in the reduction of heat loss and energy consumption for transmission and distribution of heat in district heating/cooling [4]. For example, usually the temperature of the chilled water and hence the surface temperature of the piping will be below the ambient air which means that the water vapour in the air will condense on the piping. To prevent dripping, which of is course unacceptable, the piping must be thermally insulated [3]. The thermal efficiency of a heat pump depends on the temperature of heat carrier fluid. Therefore, thermal behaviour of heat carrier fluid circulating through U-pipe plays a vital role in performance of heat pumps systems [6].

For the case of constant cylinder surface temperature and head transfer coefficient independent, on the insulation radius, the classical solution for the critical radius of pipe insulation is well known. But there is no simple analytical solution for the break-even radius (the crossover radius) of cylinder insulation. Apart from characteristics of existing solutions, some interesting suggestions to the problem can be find in several works [1, 5, 8, 9, 12, 13, 14].

For example, by the using of an approximation for mean insulation radius, the solving inconvenience of break-even radius was avoided and an analytical expression for insulation critical radius was included [1]. After analysis of both cylindrical and spherical systems it is pointed out that the break-even radius concept is applicable for cylindrical system when Bi is less than 1 and for spherical one when Bi is less than 2 [5]. In another work [9], the variation of heat transfer rate in connection with insulation thickness was studied and explicit solutions for the critical insulation thickness in some special cases were obtained.

The break-even radius is essential to design insulation coatings as well as for economic analysis of such systems [4].

Heat transfer in an isotropic cylinder layer without any inner heat sources under the steady state conditions (Fig.1) can be determined in the cylindrical coordinates by means of the Laplace simplified equation [7] as follows:

,

(1)

under the boundary conditions
T = Ts₁        for r = r₁
T = Ts₂        for r = r₂

where:
Ts₁, Ts₂ – temperatures at inner and outer cylindrical surface, r₁– inside radius, r₂ – outside radius.

Fig. 1. Temperature distribution in insulated pipe wall in the case of heat transfer from pipe inside (Tp1, Tp2 temperatures of inner, outer fluid)

The solution of equation (1) is a function T(r) presenting the temperature distribution inside the pipe wall dependent on one coordinate  changing in the radial direction

.

(2)

Further considerations include an analogical form of function (2), substituted with inner  (d₁ = 2r₁) and outer (d₂ = 2r₂) diameters of pipe, as follows:

(3)

where: d₁≤ d ≤ d₂.

Heat flux transferred through the cylindrical pipe wall of unit length expressed with the Fourier Law and adequate heat flux density at the temperature distribution described by a dependence (3) may be presented like that:

,

(4)

,

(5)

where the expression:

,

(6)

is the linear resistance of heat conduction in pipe wall material of thermal conductivity k.

While regarding heat exchange transfer through such a partition, resistance of heat transfer on its surface according to the Newton`s Law [10, 11] is also taken into account. Then, linear density of heat flux penetrating the cylindrical wall and respective linear thermal resistance is expressed as:

,

(7)

,

(8)

where:

T_p1, T_p2 – fluid temperature on the inside and outside pipe surface, respectively: h₁, h₂ – heat transfer coefficients at the pipe surfaces.

The relationships (7) and (8) may be generalized for multi-layer pipe wall as below:

,

(9)

,

(10)

where: j – number of pipe partition layers; k_i – thermal conductivity of material of i-layer (i = 1, 2, ..., n); d_i – inside diameter of i-layer.

A circular pipe with single-layer outer insulation (Fig. 1), assuming the ideal adherence of the insulating layer to the pipe outer surface, becomes identical with the multilayer cylindrical wall (formulas 9, 10). In what follows, the linear density of heat flux and adequate thermal resistance through such a partition may be obtained directly from (9) and (10) inserting j = 2.

,

(11)

,

(12)

where: k₂ – thermal conductivity of insulating material, d₃ – outside diameter of insulation layer.

Employment of the dependences (11), (12) allows determination of such a value of insulation diameter so that flux density of overall heat penetrating the partition could be maximal [2].

,

(13)

Whereas, the comparative studies on heat flux densities established after the formulas (7) and (8) give grounds to define such a value of insulation diameter de (so called effective diameter) at which a heat loss rate is just the same as in the absence of insulation. It is well known that only when diameter value surpasses the de value, the insulation starts working efficiently.

2. DETERMINATION OF EFFECTIVE INSULATION DIAMETER

Thermal resistance for heat transfer through the insulated pipe wall (12) can be considered as a function R(d₃) of an independent variable d₃ (outer diameter of insulation layer) that enables the complex analytical studies as well as determination of extremum of such a function (minimum) at the point

,

(14)

In other words, d_cr diameter value corresponds to minimal thermal resistance and maximal flux of transferred heat. Yet, it is noticeable that in the formula (12) an obvious condition for the insulation outer diameter proceeds:

,

(15)

It means that a choice of the insulation critical diameter leading to the maximal effect of heat abstraction is possible if and only if in the case

.

(16)

Otherwise, searching for the insulation critical diameter has no physical sense.

The minimal value of outer diameter of effectively working insulation d₃ = d_e is derived by means of the dependences (7) and (11) to obtain the following condition:

(17)

After rearrangement of (17), we get the equation

,

(18)

The solution of the present problem comprises all the roots of the equation (18) satisfying the condition (15).

In the further considerations, we assume x = d_e and thus, let us look into an interesting function f(x), whose zero sites make up the solutions (18) [5].

;

(19)

The Figures 2–4 present the graphs f(x) for the chosen pipes depicted further in section 4.

Fig. 2. Graphs of function fi(x) for pipe Ai(0.0018/0.002/1000/46/k2i) defined for five different values (k2i= 0.03 + 0.01i and i = 1, 2, ..., 5) of insulation heat conductivity. The condition (22) is satisfied for all the values k2i thus dcr and de exist.

Fig. 3. Graphs of function fi(x) for pipe Bi(0.0018/0.02/1000/46/k2i) defined for five different values (k2i= 0.03 + 0.01i and i = 1, 2, ..., 5) of insulation heat conductivity. The condition (22) is satisfied for B4 and Bi.

Fig. 4. Graphs of function fi(x) for pipe Ci(0.18/0.2/1000/46/k2i) defined for five different values (k2i= 0.03 + 0.01i and i = 1, 2, ..., 5) of insulation heat conductivity. The condition (22) is not satisfied for all the values k2i thus dcr  and de not exist.

For the function f(x) defined after the formula (19), the following facts are easily stated:

(20)

Function f(x) is continuous in its whole domain and possesses the only extremum, minimum at the point

(21)

of value

.

(22)

The formulas (20) and (21) explicitly indicate that the equation (18) may have two roots at most, but on the other hand, it possesses roots if and only if

.

(23)

Dividing both sides of the inequality by a positive value  and substituting

(24)

we obtain the inequality

;       t > 0.

(25)

Let us consider the left side of the inequality (25) as a function

,

(26)

F(t) is a continuous and differentiable function (Fig. 5) for , it possesses the only extremum (maximum) at the point t = 1, while

,

(27)

Figure 5. Function F(t) is continuous and differentiable for t(0;+∞), it possess one extremum (maximum) in the point t = 1, at the same time  Fmax(t) = F(1) = 0

Thereby, it has been proven that the inequality (23) is valid for the entire domain for each positive value of the constant  h₂, k₂, d₂, which means that the equation (18) always has one or two roots.

Then, let us notice that x = d_e = d₂ is always the root of the equation (18) that is straightforward to be confirmed analytically. The above mentioned fact is also visible in the Figures 2–4 as a cross-point common for all the functions presented, yet the insulation outer diameter does not satisfy the physical conditions (15). Therefore, a case of single solution (18) is not interesting anymore in the context of searching for a critical or effective diameter.

Finally, a conclusion may be drawn that a satisfying solution may be gained only if (18) possesses exactly two roots.

As the condition (26) holds, the solutions of the equation (18) i.e.  f(x) function zero sites are placed on the opposite sides of the point , and  d₂ is always one of the roots. Therefore, if

,

(28)

an inequality proceeds

.

(29)

That is, root x₁ as the insulation diameter does not meet the condition (15). Whereas, in the case

(30)

we obtain

.

(31)

Thereby, the solutions satisfying the required conditions are ensured only and exclusively by the case (30). The Figures 2–4 present the graphs of the function (19) for the pipes of A_i, B_i, C_i   type i = 1, 2, ..., 5 (description of a pipe parameters in the Section 4 below) which clearly imply that the condition (30) holds for all the pipes A_i, and B₄, B₅.

3. RESUME

The considerations-based evidence of the thesis may be formulated as the following statement.

Theorem: Under the conditions of steady state heat transfer, the problem of determination of critical and effective insulation diameter of circular pipe, without any inner heat sources, is set properly if and only if the following condition holds: .

1. In the case an insulating layer works efficiently if and only if .
   
   
2. In the case any thickness of the insulating layer reduces heat loss rate as then the equation (18) has exactly one root satisfying the physical condition (15).
   
   
3. The maximal effect of heat “absorption” (see also [11]) may obtained solely in the case , at the condition .
   
   
4. For an optional constant
   

   (32)

   and for any pipe outer diameter, at least one insulation diameter value may be established from the interval  so that numerical value of heat flux loss was precisely equal to .
   
   

5. If the condition proceeds , exactly two values (d₃₁< d₃₂) of the insulation diameter may be chosen that correspond numerically to the same level of heat loss , where denotes heat flux conforming with heat loss in the absence of the insulation or at the minimally effective insulation, so

   .

   (33)

   
   
6.  In the case a value of insulation diameter ensuring the steady heat loss value is the only one.

4. PROPOSITIONS AND NUMERICAL EXAMPLES

The conclusions presented in section 3 emphasize the weight of optimization of methods for piper insulation diameter aiming at reducing or elevating the numerical value of heat flux transmitted by the partition. Below, there is presented a simple procedure for determination of an optimal insulation diameter at which the pre-set heat losses appear to be n times smaller as compared to those in the absence of insulation.

Thermal resistance per unit length of the pipe without insulation is

,

(34)

whereas, the insulated pipe of a searched diameter d_opt

.

(35)

If we apply the required condition R₂ = nR₁, we get an equation

(36)

which after elementary transformations becomes

(37)

The equation (37) in relation to the unknown d_opt may be solved by the numerical methods (see the exemplary calculations below). The equation roots (37) meeting the physical condition (15) make up the values of optimal diameter for insulation.

After simplification and arrangement of terms, the expression (37) goes as follows:

.

(38)

Analogically to the formula (19), let us consider the left side of the equality (38) as the function n(x), where we assume that x = d_opt and thereby, we have got a function

.

(39)

Alike function (19), it is easy to state that function (39) possesses the only extremum (minimum) in the point , which is equal to

.

(40)

Let us notice, that in the case  , there appears . It means that potential for heat loss increase is strictly limited. In other words, in this case, the maximum heat loss may rise up to  times.

Besides, as the Conclusion 5 holds, each number  correlates to precisely two different values of insulation diameter depicting the same change of heat loss (loss increase by  time in relation to thermal loss in not insulated pipe).

In the case for any , there may be determined a single diameter value of insulation reducing the loss by n times, as the Conclusion 6 directly implies.

The graphical representation of relationship between dimensionless n value and insulation diameter x will largely facilitate (no need for numerical computations) a choice of suitable value x = d_opt at pre-set n or rapid estimation of heat loss change at insulation of designed thickness. To obtain that, a function (38) graph should be constructed for a given pipe

.

(41)

at the pre-settled constant parameter values d₁, d₂, h₁, h₂, k₁, k₂, (x[m] variable is the insulation diameter). Thus, the heat loss rate changes at insulation diameter x₀ can be established right off the graph n(x), reading a value n(x) at the vertical axis indicating a quotient (relative) change of heat loss change in relation to the loss in the absence of insulation.

As an example, we present the calculations concerning the following pipes denoted analogically as in the scheme (41) of the preset diameters d₁, d₂ and constant coefficients values h₁, h₂, k₁, k₂:

where k_2i= 0.02 + 0.01i for i =1, 2, ..., 5.

For the pipes mentioned above, basing on the Conclusions from section 3, there were determined the values d_kr, d_e, d_opt(n_i), where n_i = 0.5; 1.5; 2; 5 and depicted the relative change of heat loss at the employment of insulation of diameter d = 0.025; 0.05; 1; 5 [m]. Table 1–3 show the numerical results (all analytical calculations done with Maple V program).

Table 1. Exemplary computation results of pipe performance with insulation of thermal conductivity ki (i = 1, 2, ..., 5)

	dcr	de	d(0.5)	d(1.5)	d(2)	d(5)	n(0.025)	n(0.05)	n(1)	n(5)
A1	0.012	0.7977	0.0239 0.0068	16.4671	336.5539	0.24 1011	0.50	0.58	1.64	1.30
A2	0.016	5.9458	0.0893 0.0053	332.8233	1858.5997	0.6 1015	0.40	0.45	0.78	0.98
A3	0.02	44.033	0.2679 0.0491	6722.5577	0.1 107	0.1 1020	0.34	0.37	0.63	0.78
A4	0.024	325.49	0.7560 0.0047	135780.03	0.6 108	0.3 1024	0.29	0.31	0.52	0.65
A5	0.028	2405.2	2.0813 0.0046	0.3 107	0.3 1010	0.7 1028	0.26	0.27	0.44	0.56

Table 2. Exemplary computation results of Bi pipe performance with insulation of thermal conductivity ki (i = 1, 2, ..., 5)

	dcr	de	d(0.5)	d(1.5)	d(2)	d(5)	n(0.025)	n(0.05)	n(1)	n(5)
B1	–	–	–	0.0350	0.0532	0.3950	1.17	1.92	6.51	9.16
B2	–	–	–	0.0475	0.0818	1.0960	1.08	1.54	4.89	6.87
B3	–	–	–	0.0666	0.1270	3.0163	1.02	1.31	3.92	5.50
B4	0.024	0.0291	–	0.0941	0.11964	8.2672	0.99	1.16	3.27	4.58
B5	0.028	0.0409	–	0.1328	0.3022	22.6122	0.96	1.05	2.80	3.93

Table 3. Exemplary computation results of Ci pipe performance with insulation of thermal conductivity ki (i = 1, 2, ..., 5)

	dcr	de	d(0.5)	d(1.5)	d(2)	d(5)	n(0.025)	n(0.05)	n(1)	n(5)
C1	–	–	–	0.2065	0.2132	0.2581	–	–	26.85	53.34
C2	–	–	–	0.2089	0.2182	0.2825	–	–	20.19	40.01
C3	–	–	–	0.2115	0.2135	0.3100	–	–	16.19	32.02
C4	–	–	–	0.2141	0.2292	0.3407	–	–	13.53	26.69
C5	–	–	–	0.2170	0.2351	0.3752	–	–	11.62	22.89

For the pipes B₁, B₂, B₃ (Tab. 2) and C₁, C₂, C₃, C₄, C₅ (Tab. 3) the condition proceeds thus, as the theorem holds (see the previous section), the critical d_kr and effective d_e values of such an insulation diameter are not available. So, there are no reasonable grounds for searching for such an insulation diameter, which doubles the heat loss per unit length of the pipe (values denoted as d(0.5) in Tables) because any insulation has good performance in such pipes (see Conclusion 2), whereas the pipes A₁, A₂, A₃, A₄, A₅ (Tab. 1) and B₄, B₅ (Tab. 2) meet the condition so, as the theorem holds for these pipes a problem concerning d_kr and d_e determination is formulated properly. Thereby, one should also take into account determination of diameters increasing the heat loss, for example twice as large as compared to heat loss rate in the pipe without insulation. As the Conclusion 4 implies, for insulation of the pipes A₁, A₂, A₃, A₄, A₅ (Table 1) there are two different diameter values that satisfy the conditions (16) and .

However, it may seem curious that for B₄, B₅ pipes, the theorem 3.1 proceeds just like for all the A_i, pipes, which among others, implies that it is sensible to search for insulation diameters increasing heat loss rate. However, according to the formula (39) the diameters that double the heat loss rate can not be determined because the n₄(x) and n₅(x) function values are restricted by n₀ (39) value, which in the present case is strictly higher than 0.5. Having calculated n₀, we state that its values are equal to 0.985 and 0.955 for the pipes B₄ and B₅, respectively.

The function n(x) values for the diameter x[m] = 0.025; 0.05; 1; 5 (column 6–9 in Tab. 1, 2, 3) may be established numerically with optional set precision using an adequate function formula (39).Yet, according to the previous recommendations, here we suggest the direct reading-off the graph (Fig. 6, 7, 8) that facilitates fast estimation of a heat loss change without redundant numerical calculations.

Fig. 6. Functional dependence of dimensionless value n and pipe insulation diameter Ai(0.0018/0.002/1000/46/k2i) presented on the interval x(0; 5).

Fig. 7. Functional dependence of dimensionless value n and pipe insulation diameter Bi(0.0018/0.002/1000/46/k2i) presented on the interval x(0; 5).

Fig. 8. Functional dependence of dimensionless value n and pipe insulation diameter Ci(0.0018/0.002/1000/46/k2i) presented on the interval x(0; 5).

CONCLUSION

The case of heat exchange under steady state conditions for cylindrical pipes with single layer insulation was extensively analyzed, and a simple method to determine the optimal insulation diameter was developed and presented.

Introduction of a dimensionless value n and its graphic dependence on the insulation diameter facilitates the choice of the optimal pipe insulation diameter and an assessment of heat flow loss due to heat transfer through the pipe wall with given thermal insulation. This method was confirmed by appropriate calculation examples, which can serve as the basis for further problem analysis and economic considerations.

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Accepted for print: 26.07.2014

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Khachatur Kyureghyan
Department of Applied Mathematics, University of Life Sciences in Lublin, Poland
13 Akademicka
20-950 Lublin
Poland

Franciszek Kluza
Department of Refrigeration and Food Industry Energetics, University of Life Sciences in Lublin, Poland
44 Doświadczalna
20-280 Lublin
Poland
email: franciszek.kluza@up.lublin.pl

Dariusz Góral
Department of Refrigeration and Food Industry Energetics, University of Life Sciences in Lublin, Poland
44 Doświadczalna
20-280 Lublin
Poland
Phone +48 81 4610061
email: dariusz.goral@up.lublin.pl

Paweł Artur Kluza
Department of Applied Mathematics, University of Life Sciences in Lublin, Poland
13 Akademicka
20-950 Lublin
Poland

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