Structural Design and Analysis of Hyperboloid with Tower Assembly of Solar-Thermal Plant

Abstract

A 2MWe solar plant is proposed to be set-up. A 40.8 m diameter hyperboloid supported on three towers (60 m height), 118 heliostats of dimension 10 m × 10 m, 228 heliostats of dimension 5 m × 5 m and an 8 m diameter receiver are the major components of the proposed plant. Hyperboloid with tower assembly is one of the key components of the concentrated solar power plants based on central receiver concept. Hyperboloid reflects the reflected rays from the field heliostats to the receiver mounted at the ground. It consists of hyperboloid frame on to which aluminium-based reflector is fixed. Hyperboloid frame is made of carbon steel and aluminium tubes. Aluminium reflector (mirror) is made of 0.8 mm aluminium sheet. Hyperboloid structure along with reflectors is supported by tower structures. Towers are made of carbon steel tubes. Hyperboloid is fixed with three towers by bolting arrangement. Hyperboloid with tower assembly is designed against wind load of 39 m/s basic wind speed. Finite element analyses have been performed to optimize the design and estimate the deflections and stresses due to dead weight, imposed loads, earthquake loads and wind loads. Stresses due to various loads and their combinations have been checked against the allowable stresses as per IS 800. Buckling checks have been performed as per IS 800.

1 Introduction

India is a tropical country where sunshine is available for longer hours per day and in great intensity. Several technologies are available for large-scale solar power deployment. Solar-thermal plant based on central receiver technology with molten salt as the working fluid is one such concept. Solar power tower type plant has been planned employing field heliostats, hyperboloid with tower, molten salt-based receiver with steam in secondary side, turbine, etc. The proposed plant uses central receiver concept wherein the incident solar radiation concentrated to heat up the molten nitrate salt in receiver. Molten salt is a mixture of KNO₃ and NaNO₃ in a proportion of 40:60 w/w. In the steam generating system, the heated salt transfers its energy to water to generate steam. The steam is then passed on to a steam turbine for electricity generation.

Hyperboloid with tower assembly is one of the key components of the concentrated solar power plants based on central receiver concept. Hyperboloid reflects the reflected rays from the field heliostats to the receiver mounted at the ground. Hyperboloid reflector is mounted 60 m above the ground at the top of three lattice towers. Hyperboloid is 40.8 m in diameter. It consists of hyperboloid frame on to which aluminium-based reflector is fixed. Hyperboloid frame is made of carbon steel and aluminium tubes. Aluminium reflector (mirror) is made of 0.8 mm aluminium sheet. Towers are made of carbon steel tubes. Hyperboloid is fixed with tower assembly by bolting arrangement.

Size of hyperboloid is calculated based on optimum heliostat layout for 5250 kWth power in molten salt. The area of the hyperbolic mirror is estimated to be approx. 1403 m² for a tower height of 60 m. The diameter and depth of the hyperboloid is 40.8 m and 6.5 m, respectively. The thermal radiation incident on the reflector is 6.46 MWth, which generates a radiant heat flux of 3.9 kWth/m². Temperature rise of secondary hyperboloid reflector due to incident solar radiation is calculated to be 55 °C. Hyperboloid with tower assembly is designed against wind load of 39 m/s basic wind speed. Approximate weight of the assembly is 220 tonnes. Figure 1 shows the hyperboloid with tower assembly.

Fig. 1

Hyperboloid with tower assembly

2 Material Properties

Structural tubes are available in various grades. YSt 310 grade has been taken conforming to IS: 1161:1998 (steel tubes for structural purposes). Aluminium tubes have also been used in hyperboloid. The mechanical properties of YSt 310 grade steel and aluminium are given in Table 1 [1].

Table 1 Mechanical properties

3 Design Specifications

3.1 Geometrical Specifications

• 1. Height of towers = 60 m

• 2. Diameter of hyperboloid = 40.8 m

• 3. Depth of the hyperboloid = 6.5 m

3.2 Types of Loads

Hyperboloid with tower assembly has been designed as a truss structure as per IS 800. Loads which need to be considered as per IS 800 are as follows [2, 3]:

• 1. Dead loads

• 2. Imposed loads

• 3. Wind loads (as per IS 875 Part 3)

• 4. Earthquake loads (as per IS 1893 Part 1).

3.3 Wind Loads and Operational Limits

• 1. Operating wind speed is 40 kmph (user’s requirement)

• 2. Survival wind speed is 140.4 kmph (as per IS 875 Part 3).

4 Loads Considered in the Analysis

4.1 Wind Load on Hyperboloid Surface and Towers

Wind loads on hyperboloid and towers have been taken as per IS 875 Part 3. IS 875 Part 3 provides force coefficient for most structural shapes [4,5,6].

Wind load on any object is given by

$$F = C_{{\text{f}}} \times A_{{\text{e}}} \times p_{{\text{d}}}$$

where

C _f :

force coefficient;

A _e :

effective area of the object normal to the wind direction, m²;

p _d :

design wind pressure, N/m².

4.1.1 Design Wind Speed

The basic wind speed for any site shall be obtained from IS 875 Part 3 and shall be modified to include the following effects to get design wind speed, V_Z at any height, Z.

Design wind speed (V_Z)

$$V_{Z} = V_{{\text{b}}} \times k_{1} \times k_{2} \times k_{3} \times k_{4}$$

where

V _Z :

design wind speed at any height z, m/s;

V _b :

basic wind speed, m/s;

k ₁ :

probability factor (risk coefficient);

k ₂ :

terrain, height and structure size factor;

k ₃ :

topography factor;

k ₄ :

importance factor for the cyclonic region.

Various parameters used in the calculation of design wind speed have been listed in Table 2.

Table 2 Parameters used in the calculation of design wind speed

4.1.2 Design Wind Pressure

The wind pressure at any height above mean ground level shall be obtained by the following relationship between wind pressure and wind speed.

Design wind pressure (p_Z)

$$p_{Z} = 0.6V_{Z}^{2}$$

Table 3 gives the design wind speed and pressure at different heights of the structure.

Table 3 Design wind speed and pressure at different heights

4.2 Earthquake Load on Hyperboloid Surface and Towers

Earthquake loads have been considered as per IS 1893 Part 1. For the purpose of determining seismic forces, the country is classified into four seismic zones. The design horizontal seismic coefficient A_h for a structure shall be determined by the following expression [7]:

$$A_{{\text{h}}} = \frac{Z}{2} \times \frac{I}{R} \times \frac{{S_{{\text{a}}} }}{g}$$

where

Z :

zone factor

R :

response reduction factor

I :

importance factor

S _a /g :

average response acceleration coefficient for rock or soil sites.

Site is located in seismic zone III. Hyperboloid with tower assembly is welded steel structure, damping for design basis earthquake has been considered as 2%. Response spectra used in the analysis for 2% damping is shown in Fig. 2.

Fig. 2

Response spectra for 2% damping

5 Finite Element Analysis

Finite element analyses have been performed to optimize the design and estimate the deflections and stresses due to dead weight, imposed loads, earthquake loads and wind loads. Beam elements have been used to model the structural members. Lumped mass elements have been used to model imposed loads. Hyperboloid is connected to tower at 18 locations and to star delta at 24 locations with the help of link elements. All degrees of freedom have been fixed at the bottom nodes of the tower for analyses. Figure 3 shows the finite element model of hyperboloid with tower assembly.

Fig. 3

Finite element model of hyperboloid with tower assembly

Static analyses have been performed to estimate the stresses and deflections due to dead weight, imposed loads and wind loads. Response spectrum analysis has been performed to estimate the stresses and deflections due to earthquake loads. The maximum deflections due to various loads and their combinations have been listed in Table 4. Deflections due to dead loads, imposed loads and operating wind loads are within the allowable limit (35 mm). Figure 4 shows the deflected shape due to dead weight and imposed loads of reflecting surface.

Table 4 Maximum deflections due to various loads and their combinations

Fig. 4

Deflected shape (dead loads + imposed loads)

6 Design Checks

6.1 Permissible Stresses as Per IS 800

6.1.1 Axial Stress in Tension

The permissible stress in axial tension on the net cross-sectional area of hollow sections shall not exceed the values of σ_at.

$$\sigma_{{{\text{at}}}} = 0.6f_{{\text{y}}}$$

where

f _y :

minimum yield stress, MPa.

6.1.2 Axial Stress in Compression

The direct stress in compression on the gross cross-sectional area of axially loaded steel hollow sections shall not exceed 0.6 f_y nor the permissible stress σ_ac, calculated using the following formula:

$$\sigma_{{{\text{ac}}}} = 0.6\frac{{f_{{{\text{cc}}}} \times f_{{\text{y}}} }}{{\left[ {\left( {f_{{{\text{cc}}}} } \right)^{n} + \left( {f_{{\text{y}}} } \right)^{n} } \right]^{1/n} }}$$

where

σ _ac :

permissible stress in axial compression, MPa

f _cc :

elastic critical stress in compression  =  $\pi^{2} E/\lambda^{2}$ MPa

λ:

l/r = ratio of the effective length of the member and the radius of gyration

f _y :

minimum yield stress, MPa

E :

modulus of elasticity, MPa

n :

a factor assumed as 1.4.

6.1.3 Bending Stresses

In hollow sections, the tensile bending stress and the compressive bending stress in the extreme fibres shall not exceed the values of σ_bt.

$$\sigma_{{{\text{bt}}}} = 0.66f_{{\text{y}}}$$

where

f _y :

minimum yield stress, MPa.

Table 5 gives the allowable stresses for tension and bending for structural steel and aluminium for different load combinations. Table 6 gives the permissible stresses in compression based on slenderness ratios for steel and aluminium members for different load combinations.

Table 5 Allowable stresses for tension and bending

Table 6 Permissible stresses in axial compression in MPa

6.1.4 Combined Stresses

Combined stresses in axial compression and bending should satisfy Eq. (1).

$$\frac{{\sigma_{{{\text{ac}}}}, {\text{cal}}.}}{{\sigma_{{{\text{ac}}}} }} + \frac{{C_{{{\text{mx}}}} \times \sigma_{{{\text{bcx}}}} ,{\text{cal}}.}}{{\left\{ {1 - \frac{{\sigma_{{{\text{ac}}}} ,{\text{cal}}.}}{{0.60f_{{{\text{ccx}}}} }}} \right\}\sigma_{{{\text{bcx}}}} }} + \frac{{C_{{{\text{my}}}} \times \sigma_{{{\text{bcy}}}} ,{\text{cal}}.}}{{\left\{ {1 - \frac{{\sigma_{{{\text{ac}}}} ,{\text{cal}}.}}{{0.60f_{{{\text{ccy}}}} }}} \right\}\sigma_{{{\text{bcy}}}} }} \le 1.0$$

(1)

Similarly combined stresses in axial tension and bending should satisfy the Eq. (2).

$$\frac{{\sigma_{{{\text{at}}}} ,{\text{cal}}.}}{{0.60f_{{\text{y}}} }} + \frac{{\sigma_{{{\text{btx}}}} ,{\text{cal}}.}}{{0.66f_{{\text{y}}} }} + \frac{{\sigma_{{{\text{bty}}}} ,{\text{cal}}.}}{{0.66f_{{\text{y}}} }} \le 1.0$$

(2)

where

$\sigma_{{{\text{ac}}}} ,{\text{cal}}.$ :

calculated average axial compressive stress

$\sigma_{{{\text{at}}}} ,{\text{cal}}.$ :

calculated average axial tensile stress

$\sigma_{{{\text{bc}}}} ,{\text{cal}}.$ :

calculated bending compressive stress in extreme fibre

$\sigma_{{{\text{bt}}}} ,{\text{cal}}.$ :

calculated bending tensile stress in extreme fibre

$\sigma_{ac} \,$ :

permissible axial comp. stress in the member subject to axial comp. load only

$\sigma_{{{\text{at}}}}$ :

permissible axial tensile stress in the member subject to axial tensile load only

$\sigma_{{{\text{bc}}}}$ :

permissible bending compressive stress in extreme fibre

$\sigma_{{{\text{bt}}}}$ :

permissible bending tensile stress in extreme fibre

$f_{{{\text{cc}}}}$ :

elastic critical stress in compression

$C_{{\text{m}}}$ :

a coefficient.

6.2 Maximum Stresses in Structural Tubes Due to Various Loadings

Structural tubes of different sizes have been used in the design of hyperboloid with tower assembly. Stresses in axial tension, axial compression and bending are listed in Tables 7, 8, 9 and 10 for different load combinations.

Table 7 Dead loads + imposed loads

Table 8 Dead loads + imposed loads + wind loads in X direction (survival)

Table 9 Dead loads + imposed loads + wind loads in Y direction (survival)

Table 10 Dead loads + imposed loads + earthquake loads

The stresses listed in the above tables for various loads and their combinations have been checked against the allowable stresses as per IS 800 and are found within the allowable limit. Considering as fixed–fixed end conditions, buckling checks have been performed as per IS 800 and found safe.

Combined stresses in axial compression and bending, and axial tension and bending are listed in Tables 11, 12, 13 and 14 for different load combinations. The stress ratios listed in Tables 11, 12, 13 and 14 are less than 1, hence the structure is safe.

Table 11 Dead loads + imposed loads

Table 12 Dead loads + imposed loads + wind loads in X direction (survival)

Table 13 Dead loads + imposed loads + wind loads in Y direction (survival)

Table 14 Dead loads + imposed loads + earthquake loads

7 Vortex Shedding Frequency

Vortex shedding frequency of a structure is determined by the following formula:

$$f_{{\text{s}}} = \frac{{S_{{\text{t}}} \times \overline{V}_{z,H} }}{b} = \frac{0.25 \times 40.44}{{40.8}} = 0.25\,{\text{Hz}}$$

where

S _t :

Strouhal number;

$\overline{V}_{z,H}$ :

hourly mean wind speed at height z;

b :

breadth of structure normal to the wind direction in the horizontal plane.

Vortex shedding frequency of the hyperboloid with tower assembly is 0.25 Hz which is lesser than the fundamental frequency (1.1 Hz) obtained from modal analysis of the structure. Frequencies and mass participations obtained from modal analysis of the structure have been listed in Table 15.

Table 15 Frequencies and percentage mass participations

8 Conclusions

Hyperboloid with tower assembly has been analyzed and design checks have been performed as per IS 800. Based on the above analyses, following conclusions can be made:

• 1. Hyperboloid with tower assembly has been analyzed to estimate the stresses due to dead weight, imposed loads, wind loads and earthquake loads.

• 2. Wind loads on hyperboloid with tower assembly have been considered as per IS 875 and earthquake loads as per IS 1893.

• 3. Fundamental frequency of the structure is 1.1 Hz. IS 875 recommends frequency more than 1.0 Hz to avoid wind oscillations.

• 4. Deflections due to dead loads and operating wind loads are within the allowable limit (35 mm).

• 5. Stresses due to various loads and their combinations have been checked against the allowable stresses as per IS 800 and are found within the allowable limits.

• 6. Buckling checks have been performed as per IS 800 and found safe.