One-dimensional analysis of compressible flow in solar chimney power plants

Solar Energy 135 (2016) 810–820

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Solar Energy journal homepage: www.elsevier.com/locate/solener

One-dimensional analysis of compressible flow in solar chimney power plants Aleksandar S. C´oc´ic´ a,⇑, Vladan D. Djordjevic´ b a b

University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Belgrade, Serbia Serbian Academy of Sciences and Arts, Knez Mihailova 35, 11001 Belgrade, Serbia

a r t i c l e

i n f o

Article history: Received 16 March 2015 Received in revised form 28 December 2015 Accepted 20 June 2016

Keywords: Compressible flow Solar chimney power plant Collector Chimney

a b s t r a c t A novel theoretical approach for the calculation of the buoyancy driven air flow in all constitutive parts (entrance to collector, collector, turbines, collector-to-chimney transition section and chimney) of a solar chimney power plant is presented in the paper. It consists in the use of one-dimensional model of flow. The flow in the collector and the chimney is considered as compressible, while the flow in entrance to collector, turbines and collector-to-chimney transition section is treated as incompressible. Differential equations that describe the flow in the collector and in the chimney, together with algebraic equations that describe the flow in other parts of the plant are simultaneously solved. As a result, distribution of basic physical quantities, like velocity, temperature, pressure and density, in the collector and the chimney are obtained. The model is tested on two solar chimney power plants: well known Manzanares plant and Enviromission plant. The obtained results are in good agreement with measured results from Manzanares plant known in literature, together with predicted values of turbine power and turbine pressure drop for Enviromission plant. In addition, dimensional analysis of the model equations is performed and the results for mass flow rate, available turbine power, chimney height, etc. are presented. These results can be used as reliable prediction of the performance of solar chimney power plants. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Development of energy systems which can provide clean and sustainable energy is one of the most important tasks nowadays. One example of generating this kind of energy is solar chimney power plant (SCPP) which relies on natural driving force, i.e. energy from the Sun. The solar energy transforms first into mechanical energy and then into electrical one. The process of producing energy in solar chimney power plant has been a topic of intensive research in previous decades, especially in the last decade. Schematic representation of solar chimney power plant is shown in Fig. 1. It has three essential elements: collector, chimney and turbines. The collector consists of a transparent roof and the ground on the collector floor. The roof of the collector is directly exposed to solar radiation. Since it is made of transparent material, radiation heats the ground. This creates the greenhouse effect, so the air under the collector roof is heated. Due to the buoyancy effect, continuous air flow in the collector is established, directed from the perimeter to the center of the collector where chimney is located. This thermal and kinetic energy of the air is used to gen⇑ Corresponding author. E-mail address: [email protected] (A.S. C´oc´ic´). http://dx.doi.org/10.1016/j.solener.2016.06.051 0038-092X/Ó 2016 Elsevier Ltd. All rights reserved.

erate electricity via one or several turbines. Turbines can be placed horizontally at the base of the chimney, or vertically. Detailed analysis about performance of turbines in different configurations can be found in Fluri and Von Backström (2008). There have been a lot of theoretical research of air flow in solar chimney power plants. Pretorius (2004), Pretorius and Kröger (2006) and Pretorius (2007) present one-dimensional conservation equations for air flow in collector and chimney derived from conservation principles for elementary control volume. Solving these equations simultaneously with draught equations, Pretorius (2007) evaluates the performance of a large scale solar chimney power plant. Bernardes et al. (2003) develop comprehensive algebraic model which describes the performance of solar chimneys. Authors used their model to estimate power output of solar chimney power plant as well as to examine the effect of various ambient conditions and structural dimensions on the power output. Koonsrisuk and Chitsomboon (2013) also develop algebraic model to estimate the performance of solar chimney power plants, and they investigate the optimum ratio between the turbine extraction pressure and available driving pressure. They also use dimensional analysis to evaluate turbine power output. The flow inside the chimney is treated analytically by Von Backström and Gannon (2000) and by Von Backström (2003), by

´ oc´ic´, V.D. Djordjevic´ / Solar Energy 135 (2016) 810–820 A.S. C

811

Nomenclature Acoll e D e1 D e H e S d D1 D2 F f fg fr g H H1 H4 H5 I Km M m NG P p p0 pa Dpt Pm R r S

area of collect or (m2 ) dimensionless diameter of collector diameter to height ratio of collector dimensionless collector height characteristic dimensionless parameter of collector chimney diameter (m) collector outer diameter (m) collector inner diameter (m) Froude number Darcy friction factor in collector Darcy friction factor for collector ground Darcy friction factor for collector roof gravitational acceleration ðg ¼ 9:81 m=s2 x) collector height at arbitrary position (m) height of collector inlet cross-section (m) height of chimney inlet cross-section (m) height of chimney outlet cross-section (m) heat from solar irradiation ðW=m2 Þ overall efficiency parameter of solar chimney power plant Mach number standard atmosphere lapse rate ðm ¼ 6:5  103 K=mÞ turbine output power (W) dimensionless pressure local pressure (Pa) total pressure (Pa) atmospheric pressure at zero level in still air (Pa) turbine pressure drop (Pa) turbine hydraulic power (W) gas constant of air ðR ¼ 287:15 J=ðkg KÞÞ radial coordinate (m) total amount of heat received by air in collector ðW=m2 Þ

s T T0 Ta V v X Ym Z _ m

entropy per unit mass (J/kg) local temperature (K) total temperature (K) ambient temperature at zero level in still air (K) dimensionless velocity local velocity (m/s) dimensionless coordinate turbine work per unit mass (J/kg) dimensionless coordinate mass flow rate (kg/s)

Greek symbols gcoll collector efficiency gt turbine efficiency c ratio of specific heat capacities (c ¼ 1:4 for air) k Darcy friction coefficient in chimney q local density ðkg=m3 Þ q0 total density ðkg=m3 Þ entrance loss coefficient in collector f1 fm intake loss coefficient of bell mouth Abbreviations SCPP solar chimney power plant 1-D one-dimensional Subscripts 1, 2, 3, 4, 5 cross-section denotations a atmospheric conditions at zero level ch chimney coll collector

cuss the possible effect of cross-section area variation upon the flow in the chimney, and in particular upon the energy loss at the exit cross-section. Zhou et al. (2009) develop numerical model including three-dimensional Navier-Stokes equations to evaluate

employing one-dimensional model of flow. They express logarithmic differentials of characteristic physical quantities in terms of the local value of the Mach number and draw important conclusions concerned with the effect of compressibility. They also dis-

z 5

5

d

D1

H5 D2 4

4 I

H4

3

3

2 H(r)

3

3

2

0

1 H1 1

0 r

Fig. 1. Axi-symmetric view of characteristic geometrical parameters and sections of the flow, in which turbines are placed horizontally.

812

´ oc´ic´, V.D. Djordjevic´ / Solar Energy 135 (2016) 810–820 A.S. C

the performance of compressible flow through a solar chimney. They validate the model using results given in Von Backström and Gannon (2000) and find good agreement. Additional computations with commercial CFD software Fluent is also performed, and the authors find that both Boussinesq and full-buoyancy model give poor results in prediction of the flow parameters inside the chimney. Additional information about the state-of-art research in SCPP can be found in comprehensive review paper by Zhou et al. (2010). In this paper, a one-dimensional (1-D) theoretical model is developed that includes the flow details in all constitutive parts of a solar chimney power plant (entrance region, collector, turbines, collector-to-chimney transition section and chimney). The premise for use of 1-D model of flow lies in fact that the flow in SCPP is primarily axi-symmetric, with radial velocity component prevailing in the collector, and axial velocity component prevailing in the chimney. Thus, there is a preferable flow direction in both of these two parts of a SCPP, which justifies the employment of this model of flow. Due to large dimensions of the collector and the chimney, and intensive solar irradiation which heats the flow in the collector, the flow in these parts of the SCPP is followed by density variations which cannot be neglected. Thus, the flow in the collector and the chimney is treated as compressible. In contrast, the flow in the entrance region, collector-to-chimney transition section and turbines is treated as incompressible, because the density differences in these part of the SCPP are not appreciable, as already noticed by Gannon (2002), Gannon and Von Backström (2003), Fluri (2008), Kirstein (2004) and Kirstein and Von Backström (2006). Differential equations for the flow in the collector and the chimney, together with algebraic equations for the flow in the entrance region, turbines and collector-to-chimney transition section form a general mathematical model of SCPP. These equations are solved by ‘‘marching” in the flow direction, starting from the entrance to collector and ending with the exit cross-section of the chimney, where the pressure is equal to the ambient one. As a result, distribution of different physical quantities inside the collector and the chimney are obtained, and particular attention has been paid to the complex flow with heat exchange in the collector. To the best of our knowledge, this flow was not treated in so much detail in the literature previously. Obtained results also offer the possibility to analyze the performance of the complete SCPP. At that, some useful results, ready for use by designer, are obtained, as for example the dependence of the mass flow rate on the chimney height, and others.

2. Modeling and governing equations The flow of air in whole system is assumed as steady and onedimensional. The sketch of characteristic cross-sections and geometrical quantities is shown in Fig. 1, where turbines are placed horizontally. It is assumed that the collector roof is sloped, so the distance from the ground to the collector roof changes in radial direction. It is to be noted that one-dimensional character of the flow is perturbed by turbines and by collector-to-chimney transition section, and is reestablished at the entrance into the chimney at height H4 (see Fig. 1). The air flow through SCPP is divided into five distinct parts. The flow between cross-sections 0 and 1 represents the flow from distant still air to the inlet of the collector. The flow between 1 and 2 is the flow in collector, between 2 and 3 is the flow through turbines, between 3 and 4 is the flow through collector-to-chimney transition section, and finally the flow between 4 and 5 corresponds to the flow through the chimney.

2.1. Entrance to collector Since the air speed is low in this region, it can be assumed that the flow is incompressible, i.e. q0 ¼ q1 ¼ const. The pressure in cross-section 0 is equal to atmospheric pressure pa , while the velocity is zero. The temperature in this section is T a , so assuming that the air is perfect gas, we can write

q0  qa ¼

pa ; RT a

ð1Þ

The pressure in Section 1 can be determined from modified Bernoulli equation which can be written in following form, Djordjevic´ (2004):

  1 þ f1 v 21 p1 ¼ pa 1  2 RT a

ð2Þ

Value for f1 is determined according to Idelchik (2005). It is assumed that collector roof at the entrance cross-section has a sharp edge, and our estimation, based on similar cases from Idelchik (2005) yields the value f1 ¼ 0:25. The velocity v 1 is directly related to the mass flow rate, while the temperature T 1 is determined and equation of state,

v1 ¼

_ m

qa D1 pH1

;

T1 ¼

p1

ð3Þ

q1 R

Mach number at entrance cross-section 1 is defined as the ratio of local velocity v 1 and the speed of sound in that section, or

v1

M1 ¼ pffiffiffiffiffiffiffiffiffiffiffi cRT 1

ð4Þ

In this paper we will confine ourselves to low Mach number flow only. However, the flow in both collector and chimney will be treated as compressible. Thus, introduction of Mach number is necessary and important. 2.2. Flow in collector As already noticed, the flow in the collector is treated as compressible. In order to describe this flow mathematically we use conservation equations of continuity, momentum and energy in the form with logarithmic differentials of dependent and independent variables. This form of the equations can be found in several textbooks, see White (2003). However, the form presented in here is taken directly from Djordjevic´ (2004), and applies for an 1-D axisymmetric flow.

Continuity :

Momentum :

Energy :

dq

q

þ

dr dH dv þ þ ¼0 r H v

v dv þ

dp

q

¼f

v 2 dr H

  dT dp SR dr R : ¼  f cRM 2 þ cp T p pv H |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

ð5Þ

ð6Þ

ð7Þ

ds

In addition, in order to close this system of equations, we use the equation of state in its differential form as

dp dq dT  ¼ 0:  q T p As well known, the definition of friction factor reads

f ¼

2sw qv 2

ð8Þ

´ oc´ic´, V.D. Djordjevic´ / Solar Energy 135 (2016) 810–820 A.S. C

Here sw is the sum of the wall shear stresses on both, ground and roof of the collector, which are generally not the same. In the process of deriving the momentum equation, these two stresses appear as summed up,

fr

v 2 dr 2H

þ fg

v 2 dr 2H

:

so the overall friction factor in collector can be defined as

fr þ fg f ¼ : 2

ð9Þ

Thus, the expression for total friction force becomes as given on the right-hand side of Eq. (6). Both ground and roof surfaces are treated as rough, together with large values of Reynolds number, so that f r and f g are taken as constants. Their numerical values used in this paper will be stated later. Before we start with the numerical treatment of system (5)–(8) we will introduce the local value of Mach number and solve the system for unknown physical quantities:

dT dp dv , T;v T

and

dq

q,

in order

to make a qualitative analysis of the flow in collector. We get:

    S dr 1  M dT dr dH dr ¼ M2 þ þ f cM 2  1  cM 2 r H H pv H c1 T

dT 0 c1 2 S dr ¼ T0 c 2 þ ðc  1ÞM2 pv H

ð16Þ

" # dp0 c1 2 S dr ¼ cM2 f þ 2c 2 þ ðc  1ÞM2 pv H p0

ð17Þ

  S dr ds ¼ R f cM2 þ pv H

ð18Þ

Thus, Mach number increases in the flow direction, together with total temperature and entropy, while total pressure decreases. It is to be noted that the behavior of T 0 ; p0 and s is not affected by whether M < 1 or M > 1. Eqs. (10)–(13) represent the system of ordinary nonlinear differential equations, which can be solved numerically. In order to perform efficient numerical integration, the equations are transformed into a dimensionless form. The dimensionless coordinate X is defined as

X¼

2

ð1  M 2 Þ

2

ð1  M Þ ð1  M 2 Þ

 dp dr dH dr ¼ cM 2 þ þ ½1 þ ðc  1ÞM 2  f p r H H 2 S dr þ ðc  1ÞM pv H dv

v

dq

q

  dr dH dr c  1 S dr ¼ þ  f cM2  r H H c pv H ¼ M2

  dr dH dr c  1 S dr þ þ þ fc r H H c pv H

ð10Þ

ð12Þ

dp > 0; dr

dq > 0; dr

ð14Þ

which means that velocity increases in the flow direction ðdr < 0Þ, while the pressure and the density decrease. Strictly speaking, the behavior of the temperature in the flow direction remains unclear, because the right hand side of Eq. (10) has one positive and one negative term. But having in mind that Mach number is usually very small, and that ð1  cM 2 Þ ¼ Oð1Þ, we can claim that second term on the right hand side of (10) is larger than the first term. This means that the temperature increases in the flow direction, as expected, and as our numerical calculations clearly show. It is also interesting to analyze the behavior of some other characteristic quantities, like Mach number, total pressure, total temperature and entropy. Starting from Eqs. (10)–(13), after performing some additional algebraic operations, the following equations are obtained:

ð1  M 2 Þ

   dM c  1 2 dr dH dr ¼ 1þ M þ þ f cM2 M 2 r H H 

c1 S dr ð1 þ cM2 Þ 2c pv H

p ; p1

ð15Þ

0 6 X 6 1;

ð19Þ

V¼

v ; v1

e ¼ H : H H1

ð20Þ

Introducing these variables into the system (10)–(13), the following equations are obtained: 2 e e D dðM 2 Þ M 2 ½2 þ ðc  1ÞM2  1 dH e1 M  ¼ þ f cD 2 e e e dX dX 1M 2H 1  DX H

þ

ð13Þ

Obviously, Mach number, as usual in gas dynamics, plays a very important role in that subsonic ðM < 1Þ and supersonic ðM > 1Þ flows differ qualitatively. In our case, the flow is always subsonic. We also suppose that collector always gathers the heat from the Sun ðS > 0Þ. Also, the slope of the collector roof is very low, so that the term dH in system (10)–(13) can be neglected in comparison to H other terms. Taking into account these assumptions, the following conclusions can be made:

dv < 0; dr

D1  2r ; D1  D2

while the dimensionless pressure, velocity and height are defined with

P¼ ð11Þ

813

c  1 eSð1 þ cM2 ÞM2 c PV

ð21Þ

2 e e D dV V 1 dH e1 M  ¼ þ f cD e e e dX 1  M2 1  DX dX H H



! þ

c1 c

e S

ð22Þ

e HPð1  M2 Þ

e e D dP cPV 1 dH  ¼ 2 e e dX dX 1  M 1  DX H 

!

!  ½1 þ ðc  1ÞM2 

e e 1 PM2 f cD SM 2  ð c  1Þ e  M2 Þ e Hð1 HVð1  M2 Þ

ð23Þ

where

e ¼ D1  D2 ; D D1

e 1 ¼ D1  D2 ; D H1

and

D1 pSðD1  D2 Þ e S¼ : _ 1 2mRT

ð24Þ

System of Eqs. (21)–(23), together with boundary conditions:

X¼0:

M ¼ M1 ; V ¼ 1; P ¼ 1

ð25Þ

is now used for numerical integration. Having in mind that the slope of the collector roof is small, as already noted, we will now e e ¼ 1. neglect the term ddXH in (21)–(23) and take H 2.3. Flow through turbine Within one-dimensional theory it is not possible to go into details of flow inside a flow machine (pump or turbine), but just to use modified Bernoulli equation in which the work exchanged between the fluid and machine is included. If the energy per unit mass received by the turbine is denoted by Y m , the hydraulic power

´ oc´ic´, V.D. Djordjevic´ / Solar Energy 135 (2016) 810–820 A.S. C

814

_ m . For simplicity, we will treat the of the turbine will be P m ¼ mY flow in the turbine as incompressible ðq2  q3 Þ. Really, in several papers available in literature this flow is treated as incompressible, and it was shown that this model is accurate enough for practical purposes. For details see Gannon (2002) and Gannon and Von Backström (2003). It is also assumed that areas of cross-sections 2 and 3 are approximately equal, and from the continuity equation it follows that v 2  v 3 . Now, from modified Bernoulli equation for incompressible flow we can relate the pressure in cross-section 3 with pressure in cross-section 2 as

p3 ¼ p2  q2 Y m  q2 fm

v 22 2

;

ð26Þ

dq

Continuity :

q

dv

v

¼0

ð30Þ

  dp dv 2 ck þ cM2 dz ¼ M2 1 þ 2 p v kF 2d

Momentum :

Energy :

þ

c dT dp ck  ¼ M2 dz c1 T p 2d

 dp M2 2 ck 2 ¼ dz 1 þ ð c  1ÞM þ p kF 2 2d 1  M2

fm ¼ 0:5 e14:114r=dh ;

dq

where r is bell mouth radius, and dh is hydraulic radius of the duct. Fluri and Von Backström (2008) assumed the value r=dh to be 0.12, so the value of fm is 0.09. Now, with pressure determined from Eq. (26), the temperature in cross-section 3 reads T 3 ¼ p3 =ðq3 RÞ. 2.4. Flow through collector-to-chimney transition section In the collector-to-chimney transition section the flow can be also considered as incompressible. This is due to the fact that changes of the pressure, caused by velocity changes, are not appreciable in this section of SCPP. Thus, it can be assumed that q4 ¼ q3 . Continuity equation then yields:

v4 ¼

4D2 H3 2

d

v3

ð28Þ

Applying modified Bernoulli equation between cross-sections 3 and 4 (see Fig. 1) we can write:

1 1 þ f4 p4 ¼ p3 þ q3 v 33  q3 v 24 2 2

ð29Þ

The value for the loss coefficient f4 can be found in Idelchik (2005) for this section. The effect of inlet guide vanes (IGV) on values of f4 in cases when the turbine is placed vertically at the entrance of the chimney is investigated in detail in Kirstein (2004) and Kirstein and Von Backström (2006). With the pressure p4 determined, it is possible to calculate temperature at cross-section 4, using equation of state for perfect gas, T 4 ¼ p4 =ðq3 RÞ. 2.5. Flow in the chimney Chimney is considered as vertical pipe with constant diameter d. Its height is relatively large and thus the effect of gravity on the chimney flow must be taken into account. In addition, inertial and viscous forces also affect the chimney fluid flow and cannot be neglected. These influences on the adiabatic compressible flow was studied by Djordjevic´ and Milanovic´ (1995), where the authors analyze adiabatic flow in long, arbitrarily sloped pipelines with large height differences. As well known in Fluid Mechanics, the ratio between inertial and gravity forces is characterized by the Froude number. In writing the conservation equations for the flow in the chimney the local value of this number will naturally appear. It is denoted by pffiffiffiffiffiffi F and its definition is: F ¼ v = gd. Approach similar to the one applied by Djordjevic´ and Milanovic´ (1995) is used here also for writing the governing equations that describe the flow in the chimney:

ð32Þ

where k ¼ 4f is friction coefficient in the chimney. Additionally, Eq. (8) is used for the closure of this system. After solving the system (30)–(32) for unknown logarithmic differentials, we get:

where fm is intake loss coefficient due to the wall-mounted bell mouth. This loss coefficient can be estimated using expression given in Idelchik (2005):

ð27Þ

ð31Þ

q

¼

M2 1M

2

1þ

!

2

ckF

2

ck dv dz ¼  2d v

! dT ðc  1ÞM2 2 ck 2 M þ ¼ dz T 1  M2 ckF 2 2d

ð33Þ

ð34Þ

ð35Þ

Thus, in subsonic flow case pressure, density and temperature decrease with the chimney height, while velocity increases. Governing equations in this, or in a slightly different form, can be found also in Von Backström and Gannon (2000) and Von Backström (2003). Equations describing some other characteristic physical quantities, like total values of temperature and pressure, and entropy can be derived from (33)–(35) as:

dT 0 4ðc  1ÞM 2 ck dz ¼ 2 T0 ckF ½2 þ ðc  1ÞM2  2d

ð36Þ

( ) dp0 4c ck 2 dz ¼ M 1 þ p0 ckF 2 ½2 þ ðc  1ÞM2  2d

ð37Þ

ds ¼ RM2

ck dz 2d

ð38Þ

Again, total temperature and total pressure decrease, while entropy increases in flow direction, for both subsonic and supersonic flow. Now, these equations are written in dimensionless form by introducing M; F and P as dependent variables and

Z¼

z  H4 ; H5  H4

0 6 Z 6 1;

ð39Þ

as independent one. The following dimensionless equations are obtained:

! 2 dðM 2 Þ ðM 2 Þ c1 2 cþ1 H  H4 1þ ck 5 ¼ M þ 2 2 dZ 2 d 1M ckF

ð40Þ

! dðF 2 Þ F 2 M2 2 H  H4 ¼ 1 þ ck 5 dZ d 1  M2 ckF 2

ð41Þ

" # dðP 2 Þ P2 M2 2 H  H4 2 1 þ ð c  1ÞM þ ck 5 ¼ dZ d 1  M2 cF 2

ð42Þ

where P ¼ p=p4 is dimensionless pressure. Using the boundary conditions

Z¼0:

M ¼ M4 ; F ¼ F 4 ; P ¼ 1

ð43Þ

´ oc´ic´, V.D. Djordjevic´ / Solar Energy 135 (2016) 810–820 A.S. C

the system of Eqs. (40)–(42) can be integrated numerically. The values of M4 and F 4 are obtained from calculations performed for region between cross-sections 3 and 4. Still, additional boundary condition needs to be satisfied, which ensures that pressure at the chimney exit (cross-section 5) is equal to ambient pressure at that height,

Z¼1:

 g p mH5 mR P5 ¼ a 1  : p4 Ta

815

taking into account the height differences. This equation has the following form:

p3 ¼ p2 þ q2

v 22  v 23 2

 q2 f4

v 23 2

 q2 gðHturb  0:5Hcoll Þ

The value of loss coefficient f4 for this geometry is, according to Idelchik (2005), equal to 0.268. The friction coefficient k in the chimney is assumed to be constant. The chimney in case of Manzanares plant was made of iron plating, so the value of average roughness of  ¼ 0:15 mm is chosen, as given by White (2003). Value of the Reynolds number for typical average air velocities in the chimney of 7 m/s is around

ð44Þ

Using Eqs. (1)–(3), (21)–(29) and (39)–(44) it is possible to compute the flow throughout the whole solar chimney power plant, starting from known conditions in the atmosphere and geometrical parameters of the plant, and provided the value of one of the governing parameters like mass flow rate, chimney height or turbine power is presumed. All these equations are transferred to computer code written in Python programming language, in which numerical solutions for the flow in collector and chimney are obtained using fourth-order Runge-Kutta method. Eq. (44) is used for correction of initially assumed parameter. This correction is stopped when difference between calculated pressure at chimney exit and the ambient pressure at that height is less than 1 Pa.

5  106 . Now, using the formula given by Haaland (1983) or White (2003), the value k ¼ 0:0098 is obtained. Chosen values of friction factors in collector, given in Eq. (9) are f r ¼ 0:0044 and f g ¼ 0:0052, based on considerations given by Pretorius (2007). Value of diameter D2 (see Fig. 2) is adopted as D2 ¼ 12 m. Now, the model is tested for four different operational points, for which experimental results can be found in literature. These experimental data are originally presented by Haaf (1984), but they can be also found in papers by Schlaich et al. (2005) or Koonsrisuk and Chitsomboon (2013), for example. They are given in Table 1. Data given in Fig. 2 and Table 1 are used as the input data for calculations. The total amount of energy received by the air in collector, designated with S in Section 2.2, is calculated with simple formula, Koonsrisuk and Chitsomboon (2013)

3. Results and discussion 3.1. Validation of the model In order to validate the model, calculations are first performed for Manzanares SCPP. Details about this plant can be found in papers by Haaf et al. (1983), Haaf (1984) and Schlaich et al. (2005). It should be mentioned here that in case of Manzanares plant the turbine was installed at the entrance of the chimney. However, that does not alter the principal method of calculation presented in this paper. The only difference is that flow in collector-to-chimney transition section precedes the flow through turbine. The sketch for Manzanares prototype with its technical data is shown in Fig. 2. The turbine was placed at nine meters above the ground, and the value of H4 ¼ 11 m was chosen for the position of inlet cross-section for the chimney. This way we assume that at this value of height H4 the 1-D character of the flow is reestablished after passing through turbine. The pressure in cross-section 3 is calculated from the modified Bernoulli equation

S ¼ gcoll I;

ð46Þ

while the hydraulic power of the turbine is calculated as

Pm ¼

NG

gt

:

ð47Þ

According to Haaf et al. (1983) the designed efficiency for Manzanares turbine was gt ¼ 0:83, but the actual measurements have shown lower value of efficiency. By taking into the account the actual measured power output given by Haaf (1984), the value of gt ¼ 0:75 is adopted. This value of efficiency corresponds to the operating point at midday. The operating point of turbine changes during the day, due to change in mass flow rate, but the efficiency curve of the turbine built in Manzanares cannot be found in the

z 5

5 Collector diameter Average collector height Chimney height Chimney diameter Position of cross-section 4

d

D1 = 244 m H = 1.85 m H5 = 194.6 m d = 10.16 m H4 = 11 m

D1

H5 D2

H4

4

4

3

3

I 2

1 H

2

ð45Þ

H1 1

r

Fig. 2. Axi-symmetric view of Manzanares prototype plant and its geometrical parameters, obtained from Haaf et al. (1983).

´ oc´ic´, V.D. Djordjevic´ / Solar Energy 135 (2016) 810–820 A.S. C

816

Table 1 Measured values of T a ; I; gcoll and N g at Manzaneres plant at particular time during the day on September 2nd, 1982. Measured atmospheric pressure was pa ¼ 93; 900 Pa. Data obtained from Haaf (1984). Time of the day Ambient temperature Solar irradiation Collector efficiency Measured power output

T a ðKÞ I ðW=m2 Þ gcoll ð%Þ N G ðkWÞ

10:00

12:00

14:00

16:00

294.1 744.4 24.3 26.5

296.3 850 27.1 36.33

299.6 755.6 25.7 27.83

300.7 455.6 23.6 16.33

Table 2 Calculated mass flow rates at particular times during the day on September 2nd 1982. Time of the day _ ðkg=sÞ m Dpt ðPaÞ

Mass flow rate Turbine pressure drop

10:00

12:00

14:00

16:00

592.2 63.28

594.7 84.7

594.5 64.82

479.4 47.77

100

Δ pt [Pa]

80

60

40

20 Measured Calculated 0 8

10

12

14

16

Time during the day Fig. 3. Comparison of measured and computed results for pressure drop in turbine for Manzanares SCPP on September 2nd 1982.

literature. In fact, most authors use the constant value of gt , as presented by Zhou et al. (2010). In calculations presented in this part of the paper constant value gt ¼ 0:75 is used. The results for the calculated mass flow rate and the pressure drop in the turbine are given in Table 2. As expected, the mass flow rate is directly related to intensity of solar radiation, and it is lowest at last operational point. Obtained values of pressure drop and power output in the turbine are in relatively good agreement with measured values given by Haaf et al. (1983). The comparison between measurement and calculated results is shown in Fig. 3. The largest difference is about 15 Pa for the first and the third operational point. At operational point at midday that difference is only 2 Pa, or 2.5%. The distribution of velocity, temperature and pressure in the collector are given in Fig. 4(a). It can be seen that values of maximum velocities in the collector are in range of 6.5–8.2 m/s, and that practically velocity profiles for first three operational points coincide. This is due to the fact that values of solar irradiation and ambient temperature do not change that much in these operational points and that computed values of mass flow rate are very similar. The temperature rise in collector goes from Dt ¼ 10:5  C in working regime at 16:00 h, to 18  C in the working regime at 10:00 h, while at midday predicted temperature rise in collector is 15 °C. which is in good agreement with results by Haaf et al. (1983). On the other hand, the pressure variations in collector are very small as clearly seen in Fig. 4(a). Distributions of velocity, temperature and pressure inside the chimney are shown in Fig. 4

(b). It is obvious that velocity increases with the height, while temperature and pressure decrease. All these dependencies are approximately linear, with very small variations along the chimney. Velocities in the chimney are in the range 5.7–7.2 m/s, which is in fairly good agreement with the data given by Haaf et al. (1983) (velocity 7.8 m/s at midday in exit cross-section of the chimney). The distributions of physical quantities in the chimney shown in Fig. 4(b) obtained from the solution of system (40)–(42) where we get local values of Mach, Froude number and dimensionless pressure inside the chimney. Velocity distribution is obtained directly from the Froude number. When velocity is known, the temperature follows from the definition of the Mach number. In 5 the variations of Mach and Froude number for the operational point at midday are shown. Both Mach and Froude numbers vary approximately linearly with respect to Z. On the other hand, the slope of the lines is very small. Distribution for Mach and Froude number for other operational points is similar to distributions shown in Fig. 5. Model is also tested on large scale SCPP proposed by Enviromission, for which the geometrical parameters are found by Schlaich et al. (2005) and Danzomo et al. (2012). For this SCPP turbines are placed horizontally, at the collector end and its geometrical parameters are: collector diameter D1 ¼ 7000 m, collector height H1 ¼ 20 m, while the height and diameter of the tower are H5 ¼ 1000 m and d ¼ 120 m, respectively. Expected power output for this huge power plant is around 200 MW, for the mass flow rate of 215  103 kg=s. At this flow rate the velocity at collector entrance cross-section is 0.407 m/s. Taking all these parameters as input data, together with S ¼ 250 W=m2 ; pa ¼ 99; 900 Pa; T a ¼ 303 K, and estimated constant values of f1 ¼ 0:25; f ¼ 0:0048; k ¼ 0:015; fm ¼ 0:1 and f4 ¼ 0:4, the turbine hydraulic power is varied until the condition (44) is satisfied. The obtained results for turbine hydraulic power and turbine pressure drop are Pm ¼ 200:55 MW and Dpt ¼ 1:247 kPa, respectively. This is in accordance with the turbine power estimated by Schlaich et al. (2005). The distribution of physical quantities inside the collector and the chimney for this SCPP are similar to the one obtained for Manzanares SCPP. For example, distribution of pressure and temperature inside the collector in case of Enviromission power plant are given in Fig. 6. As expected, the temperature rise in collector is higher then for Manzanares SCPP. On the other hand, the pressure distribution practically coincides with the one observed in Manzanares SCPP (Fig. 4(a)). Velocity at the exit of the collector is 14.3 m/s, which is much higher. Distributions of velocity and temperature inside the chimney are given in Fig. 7. Like in case of Manzanares SCPP, all physical quantities change approximately linearly in the chimney. For chimney of this scale, temperature and pressure drops are higher in comparison to the chimney of Manzanares plant. From the results presented in this section, it is obvious that solutions for two different SCPP have very similar characteristics. The main reason for differences between the values of their various parameters are directly related to the different geometrical data of these plants. Thus, it is convenient to present results for whole solar chimney power plant using suitably defined dimensionless quantities. That approach is presented in following section. 3.2. Analysis of the dependence between dimensionless parameters We have shown, that in writing our governing equations in dimensionless form, several mutually independent dimensionless parameters, pertinent to geometry of SCPP, intensity of solar irradiation, characteristics of installed turbines, etc., appear.

´ oc´ic´, V.D. Djordjevic´ / Solar Energy 135 (2016) 810–820 A.S. C

817

7.5 8 7

v [m/s]

v [m/s]

6

4

6.5

6

2

5.5

0 320

315 315 313

T [K]

T [K]

310 305

309

300 295

307

290

305

93.6

p [kPa]

93.95

p [kPa]

311

93.9

93.85

92.8

92

91.2 10:00

93.8 0

12:00 0.2

14:00

0.4

0.6

16:00

10:00

0.8

1

0

12:00 0.2

14:00

0.4

0.6

X

Z

(a)

(b)

16:00 0.8

1

Fig. 4. Distribution of velocity, temperature and dimensionless pressure at particular times during the day on 2nd September 1982 for Manzanares plant. (a) The collector and (b) the chimney.

Froude number 0.8

0.021

0.75

0.02

F

M

Mach number 0.022

0.019

0.018

0.7

0.65

0

0.2

0.4

0.6

0.8

1

0.6

0

0.2

0.4

0.6

Z

Z

(a)

(b)

0.8

1

Fig. 5. Changes of Mach and Froude number with the respect to Z for operational point at midday on 2nd September 1982.

Application of the boundary condition at the exit cross-section of the chimney (44) now leads to a relation between them, that enables one to express one parameter as a function of the others.

Dimensionless parameters in collector are defined in (24). For some typical values of S ¼ 250 W=m2 and f ¼ 0:0048, their values e ¼ 0:951, cf D e 1 ¼ 0:843 and e are D S ¼ 0:41 in case of Manzanares

´ oc´ic´, V.D. Djordjevic´ / Solar Energy 135 (2016) 810–820 A.S. C

350

1.001

340

1.0005

330

1

p/p1

T [K]

818

320

0.9995

310

0.999

300

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

X

X

(a)

(b)

0.8

Fig. 6. Distribution of temperature and pressure inside the collector for Enviromission SCPP.

346 20.5

344

20

v [m/s]

T [K]

342 340 338

19.5

336 19 334 332

18.5 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

Z

Z

(a)

(b)

0.8

1

Fig. 7. Distribution of temperature and velocity inside the chimney for Enviromission SCPP.

plant, while in case of Enviromission plant they are: e ¼ 0:965; cf D e 1 ¼ 2:345 and e D S ¼ 1:058. Dimensionless parameter which appears in the equations for the chimney, (40)–(42), is defined as

ð48Þ

and basically it represents the ratio of chimney height and diameter. One of the most important parameters of a solar chimney power plant is the turbine hydraulic power. We can relate this power with the energy rate received by the collector, and define a new parameter K m as follows

Pm Km ¼ ; SAcoll

0

8

4

16

150

ð49Þ

where Acoll is the area of the collector. Using results and estimation from the literature, calculated values of K m for Manzanares and Enviromission SCPP are 0.004 and 0.02, respectively. This parameter actually represents overall efficiency of SCPP. e and cf D e 1 , it is possible to make Now, for specific values of D e ch  e diagrams which will show dependence H S for various values of K m . Using algorithm described at the end of Section 2.5 the following numerical results are obtained. e ¼ 0:951 and cf D e 1 ¼ 0:878 (Manzanares SCPP) they are For D e ch increases with K m . presented in Fig. 8. For fixed value of e S; H

~ Hch

e ch H

H5  H4 ¼ : d

Km⋅103

200

~ ~ D = 0.951, γ f D1 = 0.878

100

50

0

0.2

0.4

0.6

0.8

1

1.2

~ S e on e Fig. 8. Dependence of dimensionless parameter H S for different values of K m , e ¼ 0:951 and cf D e 1 ¼ 0:878 (Manzanares SCPP). for D

Thus, more turbine power requires higher chimney, and the chimney height can be evaluated from Fig. 8. However, in practice construction of a very tall chimney is difficult. It is noticed that family of curves seem to have both horizontal and vertical asymptotes, e ch . which means that there is a kind of restriction for e S and H This can be seen from the diagrams in Fig. 9 where the e ch and K m over e dependencies of 1=e S over H S are shown. Since the

´ oc´ic´, V.D. Djordjevic´ / Solar Energy 135 (2016) 810–820 A.S. C

819

20

6

Km⋅103

5 15

3

~ H

~ 1/S

4 Km⋅103

2

0

8

4

16

20

16

30

10

5

1 0

5

0

50

100

150

200

0

~

0.6

Hch

0.8

1

1.2

1.4

~ S

(a)

(a)

0.01 2

0.009 0.008 0.007

1.5

0.005

~ 1/S

Km

0.006

0.004 0.003

Km⋅103

Hch

0.002 0.001 0 0.2

1

0.3

0.4

0.5

0.6

0.7

0.8

15

25

20

30

0.9

1

1.1

0.5

1.2

~ S

parameter 1=e S is directly proportional to the mass flow rate e ch to chimney height, it can be concluded through the plant, and H that increase in chimney height will increase mass flow rate, but only to some extent (Fig. 9(a)). When this is reached, further increase in chimney height will affect the mass flow rate only negligibly. In Fig. 9(b) the dependence K m  e S, for different values of

e ch is shown. The character of the curves are similar to the ones H presented in Figs. 8 and 9(b), with similar conclusions. What is notable for both figures is that values of K m are relatively small. e ch , the values of K m For some physically acceptable values of H are less than 0.01, which means that overall efficiency of solar chimney plant in this case is less than 1%. More precisely, for Manzaneres SCPP measured data show that this efficiency is 0.4%. e and cf D e 1 , also increases Increasing the values of parameters D

e ch  e the value of parameter K m . In Fig. 10 the dependence H S for parameters that corresponds to Enviromission plant is shown. It is evident that behavior is similar as in previous case, but the values of characteristic parameters are different. Important thing is that the values of K m are higher. This means that the solar chimney power plant of larger scale has better efficiency, which is in agreement to the results of Mullet (1987), who deduces that solar chimney power plants have low overall efficiencies, making large scale ventures the only economically feasible option. In this case, which corresponds to Enviromission plant proposed by Schlaich et al. (2005), using diagrams shown in Fig. 10 we get the overall efficiency between 2% and 3%.

20

16

30

0 0

5

10

15

20

~ Hch

(b) e ¼ 0:951 and Fig. 9. Dependencies of characteristic dimensionless parameters for D e 1 ¼ 0:878. (Manzanares SCPP). cf D

5

(b) e ¼ 0:965 and Fig. 10. Dependencies of characteristic dimensionless parameter for D e 1 ¼ 2:345. (Enviromission SCPP). cf D

Diagrams presented in this section can be used as benchmark in design of solar chimney power plants. Using them a designer of SCPP can have a deeper insight into the power which can be obtained from a solar chimney plant, and its efficiency. 4. Conclusions Now, the following conclusions can be drawn.  1-D theory applied in this paper for the calculation of buoyancy driven air flow in all constitutive parts of a SCPP turns out as very efficient and accurate, as well as simple enough.  At that the flow in relatively long SCPP sections (collector and chimney) was treated as compressible, while in remaining constitutive parts (entrance to collector, turbine(s) and collectorto-chimney transition section) was treated as incompressible. Neglecting the compressibility in these parts of SCPP did not affect the accuracy of calculations in a more serious way.  Compressible flow in the collector was treated as the flow with friction and heat exchange with environment, while the compressible flow in the chimney was treated as adiabatic flow with friction. As far as we know, for the first time in literature we use local values of the Mach and the Froude number as dependent variables. Equations that describe the flow in collector, which follow from 1-D theory, represent for theirselves certain contribution to the analysis of this flow.

820

´ oc´ic´, V.D. Djordjevic´ / Solar Energy 135 (2016) 810–820 A.S. C

 Numerous parameters appearing in governing equations, geometrical and physical ones, were grouped together on the occasion of writing the governing equations in dimensionless form, within five mutually independent parameters. Among them the quantity e S deserves special attention. It comprises both solar irradiation and mass flow through collector. As far as we know, such dimensionless number did not appear in relevant literature until today. Numerical integration of governing equations with the boundary condition at the exit cross-section of the chimney included into the procedure, leads to establishment of a certain relation between these parameters. This relation enables reliable evaluation of some very important global characteristics of SCPP flow like the mass flow rate, available power in the turbine(s), chimney height, etc. These flow characteristics are very important for successful design of the SCPP. Perhaps, the most important result of this kind is the dependence of the mass flow rate on the height on the chimney for fixed values of other parameters. This dependence reveals that increase in chimney height causes the mass flow rate to increase, but only to some extent. Also, these results enable one to make a reliable estimate of an overall efficiency of the plant. Another conclusion is that global efficiency can be improved with increase of global geometrical parameters of the plant.  We compared our results with the measurements conducted on the Manzanares pilot plant, and with the available results for the proposed large scale Enviromission plant. Our results showed very good agreement.

Acknowledgements This work is supported by the Ministry of Education, Science and Technological Development, Republic of Serbia through project TR 35046 and by Serbian Academy of Science and Arts through project F 196. References dos Bernardes, M.A., Voß, A., Weinrebe, G., 2003. Thermal and technical analyses of solar chimneys. Sol. Energy 75, 511–524. Danzomo, B., Jibrin, S., Moksin, M., 2012. Similitude model design and performance evaluations of solar tower systems. ARPN J. Eng. Appl. Sci. 7 (4), 461–466.

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