Optimum heating of cylindrical pressure vessels

Optimum heating of a long hollow cylinder with free ends will be analyzed. Three thermal stress components, radial σ _{T, r} , circumferential σ _T,φ , and axial σ _{T, z} are given by [14, 15]

where the mean temperatures T _{r, m} (r, t) and T _m (t) are given by

The stress components in a thick walled cylinder under internal pressure loading are [14, 18]

Equations (1–8) were derived assuming the elastic state of stress and constant physical properties which are independent of the position and temperature. The cylinder wall temperature varies only in the radial direction while in the axial and circumferential directions remains constant. For the flow of steam, hot water or other fluids changes of temperature in the axial direction are small compared to the radial direction. Non-uniformity of the temperature on the perimeter of the cylindrical vessel occurs only for steam condensation or stratified flow in the horizontal vessels. The equivalent stress at the inner surface of the cylinder is calculated according to the Maxwell–Huber–Hencky–von Mises theory as follows

Substituting Equations: (1), (6), (2), and (7) for r = r _in into (9) gives

If fluid temperature f(t) is time dependent, the temperature T(r _in , t) and the circumferential stress σ _T,φ (r _in , t) at the cylinder inner surface will be calculated using the Duhamel integral [16, 17]

where f(t) = T _f (t) − T ₀ designates time dependent fluid excess temperature. The initial temperature T ₀ , is assumed to be uniform.

Equations (11) and (12) are based on the superposition principle, which in turn can be used for solving linear problems. This means that the material properties must be independent of temperature.

The function u _T (r _in , t) is a solution of the initial boundary problem at the surface r  = r _in , for a unit step function, f(t) = 1, t  > 0. This function is also called an influence function. The function u _S (r _in , t) represents time changes of the circumferential thermal stress component at the inner surface r = r _in caused by the unit stepwise increase in the fluid temperature for t  > 0.

Time changes in the fluid temperature are approximated by a stepwise function (Fig. 1)

Coordinates of the points in time θ _i are given by

where θ ₀  = 0.

If time steps ∆θ _i are constant and equal \(\Delta \theta ={{t}_{M}}/M\) then

Using the method of rectangles, the integral in the formula (1) for temperature at time t _M gives

where \({{\theta }_{0}}=0\quad \text{and}\quad u\left( {{r}_{in}},{{t}_{M}}-{{\theta }_{M}} \right)=u\left( {{r}_{in}},0 \right)=0.\)

The symbol T(r _in , t _M ) designates the wall temperature excess above the initial temperature T ₀ which is constant through the thickness of the cylinder wall. Circumferential stresses σ _T,φ (r _in , t _M ) are calculated in the same way as the temperature T(r _in , t _M ). To determine the influence function u _S (r _in , t _M ) for circumferential thermal stresses, the influence function u _T (r _in , t _M ) for the wall temperature, must be found at first. If we assume that the cylinder outer surface is thermally insulated (Fig. 2), then the influence function u _T (r, t) is given by:

for the following boundary and initial conditions

Here \(Bi=\alpha \,{{r}_{in}}/\lambda \) denotes the Biot number, \({{k}_{o}}={{r}_{o}}/{{r}_{in}}\) is the ratio of the outer to inner radius, \(Fo=at/r_{in}^{2}\) the Fourier number, and \(R=r/{{r}_{in}}\).

The solution of the initial-boundary problem (17–20) is [16‐19]

where

The roots\({{\mu }_{n}}\) are determined from the solution of the following characteristic equation

where J ₀ and J ₁ are the zero and first order Bessel functions of the first kind, Y ₀ and Y ₁ are the zero and first order Bessel functions of the second kind. The first ten roots of the characteristic equations (24) and (25) for selected values of the ratio k ₀ are shown in Tables 1 and 2 respectively. The roots listed in Tables 1 and 2 were obtained by solving transcendental equations by the Müller root-finding algorithm [20, 21]. The circumferential stress in the hollow cylinder with free ends due to unit stepwise increase in fluid temperature is given by

where \({{\bar{u}}_{T}}\) is the average temperature over the wall thickness which is defined as

Substituting expression (21) into equations (27) and (26) gives

If the ends of the hollow cylinder are free, axial and circumferential stresses at the inner as well as at the outer surface are equal. The previous optimization studies show [10, 11] that the optimum fluid temperature changes T _f (t) obtained from the solution of the Volterra integral equation of the first kind, can be well approximated

where: T ₀ —initial fluid temperature, a, b, c,—constants, t—time.

At first, the optimum fluid temperature changes are approximated by the ramp function T _f (t) (Fig. 2)

which can easily be carried out in practice. The symbols in Eq. (30) stand for: a—initial stepwise temperature increase, b—constant rate of fluid temperature changes. The optimum values of parameters a, b and c appearing in the function (29) or the parameters a and b in the function (30) will be determined from the condition

where: \({{\sigma }_{e}}\left( {{r}_{in}},{{t}_{i}} \right)\)—the equivalent von Misses stress at the inner surface, \({{r}_{in}}\)—inner radius of the cylinder, t _i —i-th time point, n _t —number of time points in the analyzed period of time, σ _a —allowable stress determined using current standards.

The overdetermined system of nonlinear algebraic eqations (31) was solved using the least squares method. Optimum values of searched parameters a and b or a, b, and c were determined from the condition:

by the Levenberg Marquardt method [21].

If the pressure p inside the cylinder is equal to the ambient pressure p _o , condition (9) simplifies to the form

Optimum fluid temperature changes during heating of the hollow cylinder will be calculated for p = p _o and p > p _o .

The solution steps are depicted in the flow chart (Fig. 3).

When considering a vessel with holes, the method proposed in the paper may also be used for determining the optimum temperature of the fluid. In that case, the sum of squared differences of the calculated circumferential stress: σ _Ψ  = σ _T,Ψ  + σ _p and allowable stress σ _a for the selected n _t time points at the point P of the maximum stress concentration on the hole edge should be minimum

Fluid excess temperature f (θ) = T _f  − T ₀ in the sum (34) can be assumed a function (29) or (30).

To illustrate the effectiveness of the proposed method test calculations have been carried out for a water separator of the supercritical boiler and for a drum installed in the boiler with natural circulation.

The following data were adopted for the water separator: r _in  = 0.255 m, r _o  = 0.315 m, λ = 29.45 W/(mK); c = 513.75 J/(kgK); ρ = 7782.5 kg/m³; E = 2.07·10¹¹ N/m²; β = 1.13·10⁻⁵ 1/K, and ν = 0.3. The heat transfer coefficient on the inner surface of the water separator was assumed to be: α = 2000 W/(m²K). The pressure inside the vessel is equal to atmospheric pressure i.e. p = p _o . The first 12 roots of the characteristic equation (24) are: μ ₁  = 5.12868, μ ₂  = 16.71547, μ ₃  = 28.98302, μ ₄  = 41.72671, μ ₅  = 54.71176, μ ₆  = 67.82378, μ ₇  = 81.00823, μ ₈  = 94.23722, μ ₉  = 107.49529, μ ₁₀  = 120.77334, μ ₁₁  = 134.06564, μ ₁₂  = 147.36846. A comparison of the influence function calculated analytically and using the FEM for the heat transfer coefficient equal to α = 2000 W/(m²K) is shown in Fig. 4. The compatibility of the results is very good. Small differences between the results may be observed at the beginning of heating. This difference is mainly due to insufficient accuracy of the analytical solution (28) for small values of the time, in spite of the 12 terms in the series.

In order to assess the accuracy of the method, first the circumferential stress on the inner surface of the water separator was calculated using the FEM. The fluid temperature T _f is given by the following formula

The calculated circumferential stress is shown in Fig. 5. Thus this circumferential stress is considered as a time dependent allowable stress σ _a . The allowable stress \({{\sigma }_{a}}\left( {{t}_{i}} \right)\) in (33) was given at 50 time points: \({{t}_{i}}=4+40.61\left( i-1 \right),\quad i=1,...,50\). The allowable stresses are indicated in Fig. 5 by means of hollow circles. The optimum temperature determined from the condition (32) is illustrated in Fig. 5. The optimum fluid temperature

coincides with the temperature (35) adopted for the solution of the direct problem (Fig. 5). In the next test calculation the optimum fluid temperature was determined from the condition (32) approximating fluid temperature changes by the function (29) or (30). In addition, the optimum fluid temperature was determined from the solution of the integral equation

where the circumferential stress is given by the Duhamel integral (12). The method of rectangles was used to solve sequentially (37) [11]. The allowable stress σ _a  = − 253.3 MPa was determined using the European Standard EN 12952-3 assuming 2000 start-ups of the boiler from the cold state. The comparison of the results ilustrates Fig. 6. Larger differences between the optimum temperature runs occur only at the beginning of the heating process.

Next, optimum fluid temperature changes were estimated for a thick walled boiler drum. The dimensions of the drum are: inner radius r _in  = 850 mm and wall thickness s = 90 mm. The following properties of steel were adopted for the calculation: λ = 47.3 W/(mK); c = 511 J/(kgK); ρ = 7775 kg/m³; E = 2.07·10¹¹ N/m²; β = 1.13·10⁻⁵ 1/K, and ν = 0.3. Allowable stress is: σ _a  = − 68 MPa. To calculate the influence function u _N (t) for various Biot numbers the first twelve roots of (24) were determined (Table 3). Time changes of the influence function for various Biot numbers obtained using the analytical solution (26) and the Finite Element Method (FEM) are compared in Fig. 7. The heat transfer coefficient on the inner surface of the drum was assumed to be: α = 1000 W/(m²K). First, optimum fluid temperature were estimated by solving overdetermined system of nonlinear algebraic equations (33). At every time step, the circumferential stress at the inner surface was calculated using discrete form of the Duhamel integral. The unknown parameters a, b, and c in function (29) or parameters a and b in function (30) were determined by the least squares method applying the Levenberg- Marquardt method. In the second approach thermal stresses were computed using the Ansys software [22] without introducing the influence function. Thermal stresses at the cylinder inner surface were calculated for given time points using the FEM method at every iteration step of the Levenberg-Marqardt algorithm. The optimum fluid temperature changes are shown in Figs. 8 and 9. In adition, the optimum parameter values appearing in the functions (29) and (30) are given in Table 4. The analysis of the results shown in Figs. 8 and 9 indicates that optimum temperature changes determined by the both described methods differ slightly. During optimum heating the circumferential thermal stresses at the inner surface are equal approximately the allowable value (Figs. 10 and 11). If the optimum temperature is approximated by the function (29) containing two unknown parameters then the agreement between the calculated and the allowable stress is better (Fig. 10) in comparison with two parameter curve given by the function (30). The discrepancy between the calculated and allowable stress is larger at the beginning of the heating process (Fig. 11). The form of the optimum temperature change given by (30) is to simple to make calculated circumferential stress equal to the allowable value. However, the fluid time temperature changes resulting from (30) can be easily conducted in the practice. Fig. 12 illustrates the influence of the inner pressure on optimum fluid temperature changes. With increasing pressure the initial stepwise temperature increase, as well as, the heating rate v _T goes up.