International Journal of Heat and Mass Transfer 75 (2014) 450–461

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Pressure and power potential of sloped-collector solar updraft towerpower plant

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.03.0250017-9310/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +86 27 87543838.E-mail addresses: xpzhou08@hust.edu.cn (X. Zhou), songbao@hust.edu.cn (B. Song).

Xinping Zhou a,⇑, Yangyang Xu a, Shuo Yuan a, Ranchi Chen a, Bao Song b

a School of Civil Engineering and Mechanics, Huazhong University of Science and Technology (HUST), Wuhan 430074, Chinab School of Mechanical Science and Engineering, Huazhong University of Science and Technology (HUST), Wuhan 430074, China

a r t i c l e i n f o

Article history:Received 23 July 2013Received in revised form 7 November 2013Accepted 11 March 2014Available online 26 April 2014

Keywords:Sloped collectorSolar updraft towerPower generationTemperature riseHumidityPressure potential

a b s t r a c t

The expressions containing no integral and considering the gravitational effect for predicting the pressurepotential of a sloped-collector solar updraft tower power plant (SCSUTPP) under various conditions aredeveloped based on a compressible fluid model. Based on the mathematical model of a sloped collectorpresented by Zhou et al. (2013) [8] and the expressions of the pressure potential developed, the perfor-mance of SCSUTPP is comprehensively studied under various conditions by assuming a steady compress-ible flow. The pressure potential for the vertical tower estimated using the relative humidity of the airbased on the temperature at the tower inlet due to higher temperature is shown to be over-predictive,as compared to that based on the temperature at the collector inlet, when the effect of the collectorheight on the pressure potential has not been considered. The results for SCSUTPP using the pressurepotential expression for dry air are basically shown to be in good agreement with those using the essen-tial expression. The pressure potential and power output are shown to decrease with the increase in thesame relative humidity of the air inside and outside the SCSUTPP. This variation trend is opposite to thatobtained for dry air outside the SCSUTPP. The pressure potential and power output are found to graduallyincrease with the increase in the ambient pressure, the collector height, or the tower height, or thedecrease of the ambient temperature. The increase in the temperature due to the latent heat releasedfrom the potential vapor condensation is large enough to stop the vapor to continue condensation. Thiswork lays a good foundation for the predication of the potential power production from SCSUTPP.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Solar updraft tower power plant (SUTPP) is an interesting solarthermal power generating plant [1], which consists of a solarcollector, a solar updraft tower (SUT, also known as solar chimney),and turbine generators. In the solar collector, solar radiation isreceived and heat is generated. In the SUT, the difference betweenthe density of warm air inside a high SUT and the density of theambient air produces a buoyancy (Fig. 1), which drives the aircurrent to drive the turbine generators to generate electricity.The buoyancy force (also known as driving force, and pressurepotential) determines the inflow velocity, and then power output[2]. The pressure potential of the SUTPP mainly resulting fromSUT effect and collector temperature rise is in essence equal tothe ambient atmospheric pressure difference between the collectorinlet (point 1) level and the SUT outlet (point 4) level minus the

gravitational force of the whole air inside the plant. The essentialexpression for the pressure potential of the SUTPP is therefore [3]:

Dppoten ¼ gZ Heff

0ðq1 � qÞdh ð1Þ

where q1 and q are the densities of the static atmosphere and theair inside the SUTPP, respectively, at a height h above the collectorinlet (point 1), both of which are calculated using the stateequation, g is the gravitational acceleration, and Heff is the effectivevertical height of the SUTPP. The essential expression for thepressure potential is inconveniently used for predicting the poweroutput of SUTPP. The conventional solar collector is horizontallyconstructed on the ground, and the corresponding SUTPP is calledhorizontal-collector SUTPP (HCSUTPP). An expression containingno integral for the pressure potential of the conventional HCSUTPPwas developed by Kröger and Blaine during relatively quiet (nosignificant ambient winds) periods based on a compressible fluidmodel [4]. Their expression is given in Eq. (2).

Nomenclature

A cross-sectional area (m2)Ar collector area from outlet to position with a distance r

(m2)b roof shape exponentB upright distance from the roof to the mountain side (m)cp specific heat capacity (J/kg K)Cpo pressure coefficient at outlet of a cooling tower (in our

case SUT)D diameter (m)f relative humidityg gravitational acceleration, 9.81 (m/s2)h height (m)H SUT or collector height (m)I solar radiation intensity (W/m2)L cross width of collector passage (m)_m mass flow rate (kg/s)

p pressure (Pa)P power (W)q heat of 1 kg air (J/kg)Q heat flux (W)r distance along the mountainside from the collector out-

let to a random position (m)R specific gas constant for air (J/kg K)Rcoll length of collector from inlet to outlet (m)T temperature (�C)v velocity (m/s)w humidity ratio (kg/kg dry air)x ratio of turbine pressure drop to total pressure potential

Greek symbolsa mountainside slope (�)e pressure loss coefficientc specific heat ratiog energy conversion efficiency (%)q density (kg/m3)s shear stress (Pa)

nT temperature gradientD difference or variation

Subscriptsacc accelerating axial airflowavg average between points 2 and 3br internal bracingcoll collectorcross cross flowd dry aireff effectivef wall friction, or relative humiditygro groundi inlet or initialk kinetic energyL latent heatload turbine load conditionloss losspoten potentialroof collector roofs saturationsup supportsut solar updraft towerstatic static states1, s2 any two states 1 and 2t throattg turbine generatorstot totalturb turbinev vapor1 atmosphere1 collector inlet2 collector outlet3 SUT inlet4 SUT outlet

Fig. 1. Perspective of HCSUTPP and its reference typical positions labeled.

X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461 451

Dppoten ¼ p11 � p3 ¼ p11 1�1� gHeff

cpdT11

1� gHeff

cpdT3

0@

1A

cd= cd�1ð Þ0B@

1CA ð2Þ

where cd is the specific heat ratio of dry air, and cpd is the specificheat capacity of dry air. The expression can directly and conve-niently be used to calculate the pressure potential once the temper-ature at the SUT inlet T3 is known besides the given values of T11

and Heff. An expression containing no integral for predicting thepressure potential of the HCSUTPP influenced by the atmosphericcross flow was also developed by Zhou et al. [5]:

Dppoten ¼ p11 1 �1�

gHeffcpdT11

1�gHeffcpdT3

!cd= cd�1ð Þ0@

1A þ Cpo � 1

� �� 1

2 qcrossv2cross

1� 1� gHeff

cpdT3

� ��cd= cd�1ð Þ� �

where Cpo is the pressure coefficient at the

outlet of a cooling tower (in our case SUT), and qcross and vcross arerespectively the density and the velocity of the cross flow at theSUT outlet.

Bilgen and Rheault [6] proposed a novel sloped-collector SUTPP(SCSUTPP) with a sloped collector constructed on a suitable moun-tainside and a short vertical SUT built on the collector top (Fig. 2).

452 X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461

The sloped solar collector also acts as an additional ‘‘SUT’’ due to itsgradual ascent along the mountainside. Compared to the conven-tional solar collector constructed on the horizontal ground, the col-lector on an appropriate slope at mid and high latitudes will reducethe incidence angle, increase insolation received by the collector,and show higher thermal performance, thus effectively improvingthe power output of the SCSUTPP. The SCSUTPP pressure potentialis considered to be equal to the ambient atmospheric pressure dif-ference between the collector inlet (point 1) level and the SUT out-let (point 4) level minus the gravitational force of the whole air inthe sloped collector and the SUT. It is difficult to use a theoreticalexpression containing no integral to describe the variation of pres-sure and density of the air inside the sloped collector. Therefore, anexpression containing no integral similar to the two above for con-ventional HCSUTPP cannot be obtained for calculating the pressurepotential of SCSUTPP regardless of whether the influence of theatmospheric cross flow is considered or not. Bilgen and Rheault[6] initially developed a form of simple expression containing nointegral for predicting the pressure potential of the SCSUTPP basedon an incompressible fluid model by neglecting the change of theatmosphere density with heights and the change of difference ofthe atmospheric density and the density of the current inside theshort SUT with heights:

Dppoten ¼ q11 � q2ð Þg Hsut þHcoll

2

� �ð3Þ

where Hsut is the height of the SUT above its inlet (point 3), and Hcoll

is the vertical height of the collector above its inlet (point 1). In thisexpression (Eq. (3)), only the collector outlet current density q2 wasobtained from their model of sloped solar collector. In their modelof solar collector, the loss of heat transferred to gravitational poten-tial energy of the current and the pressure drop by the effect of thegravitational force have not been considered. If their model is used,these features will result in higher temperature T2 and larger staticpressure p2 for the collector outlet current, and thus induce inaccu-rate predictions of the density q2, the pressure potential Dppoten andthe power output. Later, Koonsrisuk [7] proposed another simpleexpression containing no integral for SCSUTPP pressure potential(p1 � p4) � [(p1 � p2) + (p3 � p4)], which is equal to (p2 � p3). Theexpression would return negligible pressure under no load condi-tions, and would be the pressure drop at the turbine, which is thepressure potential times the turbine pressure drop factor. Basedon the mathematical model of the SCSUTPP developed, Zhou et al.[8] found that the simple expression containing no integral(Eq. (3)) for predicting the SCSUTPP pressure potential is inaccurateas compared to the essential pressure potential expression (Eq. (1)).

Fig. 2. A schematic diagram of an SCSUTPP [6]

Recently, compared to other proposed SUTPP concepts [9–18],the SCSUTPP performance has been an active study area. Caoet al. [19] established the mathematical model for SCSUTPP basedon the simple expression containing no integral (Eq. (3)), and tooka 5 MW SCSUTPP designed in Lanzhou, China as an example tostudy the SCSUTPP performance based on their model. Later, they[20] compared the performance of conventional HCSUTPPs andSCSUTPPs proposed in China based on their simulation. They alsoperformed economic analysis of SCSUTPPs proposed in NorthwestChina [21]. Panse et al. [22] developed a model based on Bernoulli’sequation by assuming a steady compressible flow to investigatethe performance of the SCSUTPP without a short SUT at the collec-tor top. Koonsrisuk [7] studied the effects of a near-unity ratio ofthe collector inlet area and its outlet area, the assumption thatthe density differences in the collector and that in the chimneyare approximately equal, the chimney height, and the collectorarea on the SCSUTPP performance based on a mathematical modeldeveloped. Later he compared the HCSUTPPs and SCSUTPPs usingsecond law analysis [23]. Zhou et al. [8] developed a mathematicalmodel to study the performance of the SCSUTPP and particularlyanalyzed the properties (i.e., the static pressure, the temperature,the density, and the velocity) of the indoor dry air from the inletof the sloped collector to the SUT outlet. Kalash et al. [24] designedan SCSUTPP setup, and performed experimental investigation ofthe temperature field in the sloped solar collector. However, thework on reliable pressure potential expressions containing no inte-gral for conveniently predicting the power output of the SCSUTPPunder various conditions (for example, under the wet air condi-tions) has not much been reported. The humidity contained inthe wet air will influence the air density and then pressure poten-tial, thus affect the air flow field [4,15,25]. In this paper, theSCSUTPP pressure potential expressions containing no integralunder various conditions are proposed based on a compressiblefluid model. Based on the model with the pressure potentialexpressions developed, the SCSUTPP performance under variousconditions: (1) for dry air, (2) for unsaturated wet air, and (3) forsaturated wet air, is predicted, by considering the change of theatmosphere density with heights and the change of difference ofthe atmospheric density and the density of the current inside theSUT with heights based on a compressible fluid model where thegravitational effect is present.

2. Mathematical model for SCSUTPP

A schematic diagram of an SCSUTPP and its typical referencepositions are shown in Fig. 2. The typical reference positions

and its reference typical positions labeled.

X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461 453

denote the collector inlet, the collector outlet, the SUT inlet, andthe SUT outlet which are labeled by the numbers 1, 2, 3, and 4,respectively.

2.1. Flow and heat transfer in sloped solar collector

For an SCSUTPP as shown in Fig. 2, the trapezoidal-shape slopedcollector was designed by Zhou et al. [8]. The passage width L isdesigned to decrease linearly from the collector inlet (point 1) tooutlet (point 2), and the passage width L2 is designed to be equalto the diameter Dsut of the vertical SUT located at the outlet (point2) of the sloped collector. The passage width L is defined accordingto:

L ¼ Dsut þL1 � Dsut

Rcoll

� �r ð4Þ

where Rcoll is the collector length, i.e., the normal distance from thecollector inlet (point 1) to its outlet (point 2), and r is the distancealong the mountainside from the collector outlet (point 2) to arandom position. The upright distance B is assumed to increaseaccording to the exponential function of the distance r [26]:

B ¼ B2r2

r2 þ r

� �b

ð5Þ

where, b is the roof shape exponent, and is given a value of 0.5 [8].The passage cross-sectional area A is equal to the product of L and B,that is, A = L � B.

At a random position, the vertical height h above the collectorinlet (point 1) can be expressed as a function of r and themountainside slope b:

h ¼ Hcoll � r sin b ð6Þ

The effective ‘‘SUT’’ height Heff for the SCSUTPP is defined to beequal to the sum of the vertical height of sloped collector, Hcoll,and the SUT height, Hsut:

Heff ¼ Hcoll þ Hsut ð7Þ

The constant-pressure specific heat capacity of wet air cp can beexpressed with a function of the humidity ratio w, and the con-stant-pressure specific heat capacities of dry air and vapor, cpd

and cpv. The latter two parameters have a relationship accordingto cpv � 1.9cpd � cp therefore can be written as:

cp ¼ cpd þwcpv � ð1þ 1:9wÞcpd ð8Þ

where the humidity ratio w, denoting the vapor mass and 1 kg ofdry air coexist in the wet air, is

w ¼ qvqd¼ 0:622

pvp� pv

; or pv ¼w

0:622þwp ð9Þ

in which the vapor partial pressure, pv, is equal to the vapor partialpressure of the saturated wet air pvs multiplied by the relativehumidity f:

pv ¼ pvs f ð10Þ

The saturation pressure of vapor pvs within temperature range from0 �C to 40 �C can be approximated by [4]:

pvs ¼ M expð�N=TÞ ð11Þ

where M = 2.36874 � 1011, and N = 5406.1915. This empirical equa-tion is also suitable for the temperatures between 40 �C and 50 �C[27].

The specific gas constant for wet air R also can be expressedwith a function of the humidity ratio w, and the specific gas con-stants of dry air and vapor, Rd and Rv. The latter two parameters

have a relationship according to Rd/Rv = 287.08/461.52 = 0.622. Rtherefore can be written as:

R ¼ Rd þwRv ¼ ð1þw=0:622ÞRd ð12Þ

The constant-volume pressure specific heat capacity of wet air cv

can be given by

cv ¼ cp � R ¼ ð1þ 1:9wÞcpd � ð1þw=0:622ÞRd ð13Þ

The specific heat ratio of wet air c is then calculated by

c¼ cp=cv ¼ ð1þ1:9wÞcpd=ðð1þ1:9wÞcpd�ð1þw=0:622ÞRdÞ ð14Þ

According to the density concept, the expression of the densityof wet air that is also the state equation of the compressible flow inthe SUTPP can be written as:

q ¼ ð1þwÞqd ¼ ð1þwÞ pd

RdT¼ ð1þwÞp� pv

RdT

¼ ð1þwÞ 1� w0:622þw

� �p

RdTð15Þ

The steady compressible flow in a sloped collector passage canbe described using a one-dimensional mathematical model withgoverning equations including state equation (Eq. (15)), continuityequation, momentum equation, and energy equations. The conti-nuity, momentum, and energy equations are respectively definedas follows [8]:

Continuity equation :

ddrðqvBLÞ ¼ 0 ð16Þ

Momentum equation:

�qv dvdr¼ dp

drþ qg

dhdrþ sroof þ sgro þ ssup

Bð17Þ

Energy equation:

dqdr¼ �q00

_mdAr

dr¼ cp

dTdrþ v dv

drþ g

dhdr

ð18Þ

where Ar is the collector area from the collector outlet to theposition with a distance r, and the mass flow rate _m is given by

_m ¼ qvA ð19Þ

where A is the cross sectional area of the passage in the SUTPP. Thetemperature of the air at the collector inlet T1 is assumed to equalthat of the ambient air at the ground level T11, and the static pres-sure at the collector inlet p1 is equal to the ambient atmosphericpressure at the ground level p11 minus the dynamic pressure andpressure loss at the collector inlet:

p1 ¼ p11 �12q1v2

1 � ei12q1v2

1 ð20Þ

with the collector inlet loss coefficient recommended as ei = 1 [26].

2.2. Properties of air inside SUT and atmosphere

The air experiences an isentropic change in state inside the SUTand outside the SCSUTPP according to [4]:

pqc ¼ constant ð21Þ

ps1

ps2¼ Ts1

Ts2

� � cc�1

ð22Þ

qs1

qs2¼ Ts1

Ts2

� � 1c�1

ð23Þ

454 X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461

where subscripts s1 and s2 denote any two states, and c is the spe-cific heat ratio of air. The pressure gradient in gravity field is:

dpdh¼ �qg ð24Þ

Substituting Eq. (15) into Eq. (21), differentiating, and substitutingEqs. (24), (14), and (15) yields

dTdh¼ c�1

cTp

dpdh¼�c�1

cTpqg

¼� 1�1þRdð1þw=0:622Þcpdð1þ1:9wÞ

� �Tpð1þwÞ 1� w

0:622þw

� �p

RdTg

¼� gcpd

1þw1þ1:9w

ð25Þ

The temperature of the air inside the SUT at any height h abovethe collector inlet (point 1) can be expressed as a function of thetemperature of the air at the SUT inlet (point 3) in the followingway:

TT3¼ 1� 1þw

1þ 1:9wgðh� HcollÞ

cpdT3ð26Þ

T4 can be calculated using Eq. (26) when h is equal to the effective‘‘SUT’’ height Heff . The static pressure of the air current inside theSUT therefore can be given by

pp3¼ 1� 1þw

1þ 1:9wgðh� HcollÞ

cpdT3

� �c=ðc�1Þ

ð27Þ

where cc�1 ¼ 0:622 cpd

Rd

1þ1:9wwþ0:622 after substituting Eq. (14). Similar to

Eqs. (26) and (27) for the air inside the SUT, the temperature andstatic pressure of the ambient atmosphere changing vertically withheights h above the collector inlet (point 1) can be expressed as

T1T11¼ 1� 1þw

1þ 1:9wgh

cpdT11ð28Þ

with T11 being the temperature of the ambient air at the groundlevel, and

p1 ¼ p11 1� 1þw1þ 1:9w

ghcpdT11

� �c=ðc�1Þ

ð29Þ

where p14 can be calculated using Eq. (29) when h = Heff. Whenw = 0, Eqs. (26) and (27) will correspond to dry air in the SUT, andEqs. (28) and (29) will correspond to dry ambient atmosphere out-side the SUTPP [5,28,29].

The static pressure at the SUT outlet p4 approximates the staticpressure p14 of ambient air at the level of the SUT top duringrelatively quiet (no significant ambient winds) periods [5,26]. Thestatic pressure at the SUT inlet p3 therefore can be expressed as

p3 ¼ p4 1� 1þw1þ 1:9w

gHsut

cpdT3

� ��c=ðc�1Þ

¼ p11

1� 1þw1þ1:9w

gHeff

cpdT11

1� 1þw1þ1:9w

gHsutcpdT3

0@

1A

c=ðc�1Þ

ð30Þ

2.3. Pressure potential caused by sloped solar collector and SUT

The pressure potential of the SUTPP is used to drive the currentin the plant from the collector inlet to the SUT outlet. It will pro-duce the maximum flow when no any turbine is installed andthe pressure losses are not considered. The pressure potential ofthe SUTPP can also be equal to the dynamic pressure at the collec-tor outlet or the SUT inlet under no load conditions by neglectingthe pressure losses when air current flows through the plant:

Dppoten ¼12q3v2

3 ð31Þ

The pressure potential of the SUTPP was considered by Koonsrisukand Chitsomboon [30] to be equal to the difference of the staticpressures at the collector inlet (point 1) and outlet (point 2) equiv-alently at the SUT inlet (point 3) under no load conditions, that is,(p1 � p3). The momentum equation (which may be derived fromEq. (17) by removing the gravitational force term) without consid-ering the shear stress items for conventional HCSUTPP horizontalcollector can be given by

�qv dvdr¼ dp

drð32Þ

After integration Eq. (32) can be written asZ 3

1qvdv ¼ 1

2

Z 3

1qdv2 ¼ p1 � p3 ð33Þ

There is almost constant pressure in the horizontal collector, wherethe density of the air current changes slightly and therefore, can beconsidered to be a constant. The total pressure in the horizontal col-lector is equal to the sum of the static pressure and dynamic pres-sure at the collector inlet (point 1), and is also equal to the sum ofthe static pressure and dynamic pressure at the collector outlet(point 2), or equivalently at the SUT inlet (point 3) under no loadconditions. Based on this, the pressure difference (p1 � p3) in theEq. (33) can be expressed as: ðp1 � p3Þ ¼ 1

2 q3v23 � 1

2 q1v21

� �. In com-

parison to Eq. (31), the pressure difference (p1 � p3) is not suitablefor the pressure potential of the SUTPP for a collector with a smallinlet area (corresponding to small inlet height or small collectordiameter) leading to a large velocity of air current at the inlet, butfor that with a large inlet area leading to a small velocity of air cur-rent at the inlet when the dynamic pressure is negligible. In fact, thepressure potential of the HCSUTPP should be considered to be equalto the difference of the ambient pressure at the collector inlet (point1) level and the static pressure at the collector outlet (point 3), thatis, (p11 � p3). Just based on this, Eq. (2) was deduced [4].

From the momentum equation (Eq. (17)), as the indoor air risesfrom the sloped collector inlet to outlet, the indoor air currentshould be reasonably accelerated, and its gravitational potentialenergy increases, but the static pressure gradually decreases. Grav-itational force takes a significant effect on the movement of the aircurrent up the sloped collector. The no integral expression foraccurately predicting the pressure potential of SCSUTPP is there-fore difficult to be derived like for the horizontal collector SUTPP(similar to Eq. (2)). After integrating, the momentum equation(Eq. (17)) without considering the shear stress items for the slopedcollector also can be expressed asZ 3

1qvdv ¼ 1

2

Z 3

1qdv2 ¼ p1 � p3 � g

Z 3

1qdh ð34Þ

In order to remove the integral signs in Eq. (34) to obtain an expres-sion containing no integral for the SCSUTPP pressure potential, themean density of the air inside the sloped collector is introduced intoEq. (34). Eq. (34) can be recasted as

12

�qðv23 � v2

1Þ ¼ p1 � p3 � �qgHcoll ð35Þ

As shown in Ref. [8], the density of the indoor dry air does notchange linearly but decreases exponentially with heights from theinlet (point 1) to the outlet (point 2, equivalently the SUT inlet,point 3, under no load conditions) of the sloped solar collector,however, the mean density of indoor air for the actual exponentialdecrease seems to be just slightly larger than the mean density ofindoor air for the linear variation. For the sake of simplicity ofmodel, the latter is used as �q, which can be given by

X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461 455

�q ¼ ðq1 þ q3Þ=2 ð36Þ

By substituting Eq. (36) into Eq. (35), one obtains:

12q3v2

3 �12q1v2

1

� �� 1

2ðq3 � q1Þ

v21 þ v2

3

2¼ p1 � p3 � �qgHcoll ð37Þ

In Eq. (37), the term 12 ðq3 � q1Þ

v21þv2

32 is small enough and negligible

compared to the term 12 q3v2

3 � 12 q1v2

1

� �. Eq. (37) therefore can be

simplified in an approximate form, and the expression (Eq. (31))of the SCSUTPP pressure potential Dppoten can be further written as

Dppoten ¼12q3v2

3 ¼ p1 þ12q1v2

1 � p3 � �qgHcoll

¼ p11 � p3 �q1 þ q3

2gHcoll ð38Þ

Because the density of the air at the collector inlet approximatesthat of the ambient air at the ground level, Eq. (38) therefore canapproximate the expression given by

Dppoten ¼ p11 � p3 �q11 þ q3

2gHcoll ð39Þ

2.3.1. Pressure potential for unsaturated wet airBy substituting Eqs. (15) and (30) into Eq. (39), we obtain the

expression of the SCSUTPP pressure potential when the humidityratio of the indoor air is equal to that of the ambient. Thisexpression is:

Dppoten ¼ p11 � p3 � ð1þwÞ 1� w0:622þw

� �1

2Rd

p11

T11þ p3

T3

� �gHcoll

¼ p11 1�1� 1þw

1þ1:9wgHeff

cpdT11

1� 1þw1þ1:9w

gHsutcpdT3

0@

1A

c=ðc�1Þ0B@

1CA

� 0:311 � gHcoll1þw

0:622þwp11

RdT11

� 1þ T11

T3

1� 1þw1þ1:9w

gHeff

cpdT11

1� 1þw1þ1:9w

gHsutcpdT3

0@

1A

c=ðc�1Þ0B@

1CA ð40Þ

When the humidity ratio of the indoor air is not equal to that ofthe ambient, based on the deduction of Eq. (40), the pressurepotential can be given by:

Dppoten ¼ p11 1�1� 1þw1

1þ1:9w1

gHeff

cpdT11

� �c1=ðc1�1Þ

1� 1þw1þ1:9w

gHsutcpdT3

� �c=ðc�1Þ

0B@

1CA�0:311 �gHcoll

� p11

RdT11

1þw10:622þw1

þ 1þw0:622þw

T11

T3

1� 1þw11þ1:9w1

gHeff

cpdT11

� �c1=ðc1�1Þ

1� 1þw1þ1:9w

gHsutcpdT3

� �c=ðc�1Þ

0B@

1CA

ð41Þ

Especially, when only the ambient air is considered to be dry(w1 = 0) but the air in the plant is wet, Eq. (41) becomes

Dppoten ¼ p11 1�1� gHeff

cpdT11

� �cd=ðcd�1Þ

1� 1þw1þ1:9w

gHsutcpdT3

� �c=ðc�1Þ

0B@

1CA�1

2gHcoll

� p11

RdT111þ0:622

1þw0:622þw

T11

T3

1� gHeff

cpdT11

� �cd=ðcd�1Þ

1� 1þw1þ1:9w

gHsutcpdT3

� �c=ðc�1Þ

0B@

1CA ð42Þ

The cases discussed above suitably using Eq. (41) or (42) can befound for the SUTPP integrated with an open water source (e.g.,the open seawater [15], the open wet soil for vegetation [31], and

the open crop drying [32,33]) in the solar collector, or an openwater source (e.g., from solar pond [14]) around the SUT inlet ofthe SCSUTPP without solar collector.

Besides the influence of moisture contained in the atmosphereair, other factors, e.g., the interaction between solar radiation andthe atmosphere, can influence the properties of the actual atmo-sphere. This leads to a possible certain deviation between the tem-perature lapse in the atmosphere and the dry adiabatic lapse rate(DALR) which is often close to the temperature lapse rate in aridareas [4] where the commercial SUTPPs are suitably constructed.An International Standard Atmosphere (ISA) approximates theatmospheric conditions prevailing for most of the year in temper-ate latitudes defined as a mean sea level pressure of 101325 Pa, acorresponding temperature of 15 �C, with a mean lapse rate of0.0065 �C/m to a height of 11 km, and the atmosphere air isassumed to be a clean, dry, perfect gas mixture. If the dry ISAassumption is used outside the SCSUTPP, the pressure potentialcan be given by:

Dppoten¼ p11 1�1�0:0065 Heff

T11

� �5:255

1� 1þw1þ1:9w

gHsutcpdT3

� �c=ðc�1Þ

0B@

1CA�1

2gHcoll

� p11

RdT111þ0:622

1þw0:622þw

T11

T3

1�0:0065 Heff

T11

� �5:255

1� 1þw1þ1:9w

gHsutcpdT3

� �c=ðc�1Þ

0B@

1CAð43Þ

2.3.2. Pressure potential for dry airFrom Eq. (40) when w = 0, the expression of the SCSUTPP

pressure potential for dry air can be derived:

Dppoten ¼ p11 1�1� gHeff

cpdT11

1� gHsutcpdT3

0@

1A

cd= cd�1ð Þ0B@

1CA

� gHcollp11

2RdT111þ T11

T3

1� gHeff

cpdT11

1� gHsutcpdT3

0@

1A

cd= cd�1ð Þ0B@

1CA ð44Þ

This equation can be used when the DALR is applicable for the airinside the SCSUTPP and the atmosphere outside the plant.

2.3.3. Pressure potential for saturated wet airThe humidity ratio of saturated wet air is calculated by

ws ¼ 0:622pvs=ðp� pvsÞ ð45Þ

The critical saturation height hsi is defined as a height above the SUTinlet where the wet air initially becomes saturated and condensa-tion is assumed to commence. This height can be calculated basedon Eqs. (9)–(11) by:

pvs ¼w

0:622þwp ¼ M expð�N=TÞ

¼ M exp �N= T3 �1þw

1þ 1:9wghsi

cpd

� �� �ð46Þ

By substituting Eq. (27) into Eq. (46), one obtains

w0:622þw

p3 1� 1þw1þ 1:9w

ghsi

cpdT3

� �c=ðc�1Þ

¼ M expð�Ncpdð1þ 1:9wÞ=ðcpdð1þ 1:9wÞT3 � ð1þwÞghsiÞÞð47Þ

The critical saturation height hsi is obtained from Eq. (47):

456 X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461

hsi¼cpd

g1þ1:9w

1þw

� T3þN lnw

0:622þwp3

M1� 1þw

1þ1:9wghsi

cpdT3

� �c=ðc�1Þ !, !

ð48Þ

Then the temperature Tsi and the static pressure psi also can beobtained at the saturation height hsi and the maximum (critical)humidity ratio wsi is equal to w. Once the air initially becomessaturated, the condensation is assumed to commence. Thecondensed vapor mass can be found from:

Dws ¼ w�ws ð49Þ

So, the latent heat of vapor condensation is:

DQ ¼ D _mvsqL ¼ _mdðw�wsÞqL ¼ _mw�ws

1þwqL ð50Þ

where specific latent heat qL is given by

qL ¼ 2502535:259� 212:56384T ð51Þ

A small quantity of liquid water Dws may be carried by the updraftto flow up the SUT. The latent heat is assumed to heat all the currentwith mass flux _m by a temperature rise DTs:

DTs ¼DQcp _m

¼_m w�ws

1þw qL

cpdð1þ 1:9wÞ _m¼ qLðw�wsÞ

cpdð1þ 1:9wÞð1þwÞ ð52Þ

In this case, the temperature of the indoor current above the criticalsaturation height can be given by:

Ts ¼ T3 �ð1þwÞghs

cpdð1þ 1:9wÞ þ DTs

¼ T3 �ð1þwÞghs

cpdð1þ 1:9wÞ þqLðw�wsÞ

cpdð1þ 1:9wÞð1þwÞ ð53Þ

Ref. [4] proposed the approximate pressure above the critical satu-ration height hsi as

ps � psið1þ nT hs=TsiÞ�0:622gð1þwsÞ=ðRnT ðwsþ0:622ÞÞ

¼ psið1þ nT hs=TsiÞ�0:021233gð1þwsÞ=ðRnT ðwsþ0:622ÞÞ ð54Þ

where nT is the temperature gradient, which can be assumed to beapproximately constant [4].

2.4. Turbine and power extraction

When turbines are installed between the collector outlet (point2) and the SUT inlet (point 3) of the system, the temperature andstatic pressure at point 3 are respectively given by,

T3 ¼ T2 �DpturbAsutvavg

cp _mð55Þ

p3 ¼ p2 � Dpturb;i � Dpturb ð56Þ

where, Dpturb,i is the turbine inlet pressure loss, and Dpturb is thepressure drop at the turbine, which can be expressed as

Table 1Parameters for a 1500 m high SUT in Ref. [4] and this paper when DALR is applicable out

Ref. [4] This paper

Hsut (m) 1500 1500p11 (kPa) 90 90T11 or T1 (�C) 30 30T3 (�C) 50 50f (%) 0 0w (kg/kg dry air) 0 0Pressure potential (Pa) 981.3 982.6

Dpturb ¼ xDppoten ð57Þ

where the turbine pressure drop factor x is the ratio of the pressuredrop at the turbine to the total pressure potential proposed bySchlaich et al. [34] as an operational parameter and recommendeda constant value of 0.8 for maximizing the power output. After then,much work on the ratio has been reported [35–38]. Based on theprevious work, the ratio x is selected to be 0.8 in this work. Theaverage velocity vavg can be calculated with the average pressureand the average temperature between points 2 and 3 according to:

vavg ¼_m

Asutqavgð58Þ

where the average density qavg is given by

qavg ¼ ð1þwÞ ðp2 � pv2Þ þ ðp3 � pv3ÞRdðT2 þ T3Þ

ð59Þ

Eq. (59) can be used for dry air current in the SUTPP when w = 0.The total pressure losses in the SUTPP are the pressure potential

minus Dpturb:

Dploss ¼ Dppoten � Dpturb

¼ Dpcoll þ Dpacc þ Dpturb;i þ Dpbr þ Dpf þ Dpk ð60Þ

with

Dpcoll ¼ Dpcoll;along þ ei12q1v2

1

Dpturb;i ¼ eturb;i12q2v2

2

Dpacc ¼ eacc12q3v2

3

Dpbr ¼ ebr12q3v2

3

Dpf ¼ fsutHsut

Dsut

12q3v2

3

Dpk ¼ ek12q4v2

4

ð61Þ

The turbine inlet loss, the inside wall friction, the internal bracingwheel drag forces, the vertical accelerating axial airflow, and theexit kinetic energy loss all contribute to a pressure drop over theheight of the SUT. The parameters related to the pressure lossesare: the turbine inlet pressure loss coefficient of the SCSUTPPeturb,i = 0.14, the SUT wall friction factor fsut = 0.008428, the coeffi-cient of pressure loss due to the internal bracing wheel drag forcesebr = 0.25, the exit kinetic energy coefficient ek = 1.26, and the verti-cal accelerating axial airflow pressure loss is neglected by flaringthe SUT to keep the through-flow velocity constant, so v4 = v3 [8].The pressure loss produced by the horizontal accelerating radial air-flow is calculated using the momentum equation, i.e., Eq. (17). Thepressure loss at the collector inlet is considered in Eq. (20). Thepressure drop made by the inside friction with ground and roofand supports will take a very small proportion of the pressurepotential, and are assumed to be negligible and therefore ignoredin this work. Dploss can be further expressed as

side SUT.

Ref. [4] This paper

1500 150090 9030 3050 5090 90Between 0.0895 and 0.09 0.092Around 1800 1706.2

Fig. 3. Pressure potential and humidity ratio for reference 1500 m high SUT versusrelative humidity of the air current calculated using the temperatures at points 1and 3, respectively, without including the equations of solar collector, when DALR isapplicable outside the SUT.

X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461 457

Dploss ¼ eturb;iq3

q2þ ebr þ fsut

Hsut

Dsut

� �12q3v2

3 þ ek12q4v2

3 ð62Þ

The power extracted from the turbine generators under aturbine load condition, Pload, can also be expressed as

Pload ¼ gtgDpturbAsutvavg ð63Þ

where gtg is the mechanical efficiency of the turbine generators,selected to be 77% [6].

The total efficiency of the system under a turbine load condi-tion, gload, can be calculated by

gload ¼Pload

AcollIð64Þ

where I is the solar radiation intensity.In a horizontal solar collector, the heat gains from solar radia-

tion are used to heat and accelerate the indoor air current. While,in the sloped solar collector, besides heating and accelerating theindoor air current, the heat gains from solar radiation are trans-ferred to gravitational potential energy of the indoor current. Thetotal energy conversation efficiency (i.e., the utilizing efficiencyof solar radiation) of the solar collector gcoll,tot can be calculatedusing the following expression

gcoll;tot ¼cp _mðT2 � T1Þ þ 0:5 _m v2

2 � v21

� �þ _mgHcoll

AcollIð65Þ

The heat energy and the kinetic energy of the indoor air current arethe two factors that determine the flowing potential of the air in theSUTPP. The effective thermal efficiency of the solar collector to heatand accelerate the indoor air current, gcoll,eff, can be calculated by

gcoll;eff ¼cp _mðT2 � T1Þ þ 0:5 _m v2

2 � v21

� �AcollI

ð66Þ

The effective thermal efficiency of the solar collector gcoll,eff is there-fore smaller than the total energy conversation efficiency of thesolar collector gcoll,tot. In this work, gcoll,tot is given a value of 56% [6].

The above equations are solved by using the Runge–Kuttanumerical method with MATLAB programming language to deter-mine all the variables. The convergence criterion for the iterativescheme is specified to be 1 � 10�10. The main flowchart for thesolving procedures of the Matlab code developed was the sameas that presented in Ref. [8].

3. Results and discussion

Based on the reference of a 1500 m high SUT for a commercialSUTPP proposed [4], Kröger and Blaine examined the effect of thehumidity ratio calculated on a basis of relative humidity of theair current at the SUT inlet on the pressure potential. The pressurep11 and temperature T11 of static atmosphere were selected to be90 kPa and 30 �C, respectively. The temperature T3 of the air cur-rent at the SUT inlet (point 3) was assumed to be 50 �C. In orderto validate the calculated results, Table 1 compared the humidityratios and pressure potentials calculated in our model and thosecalculated in Ref. [4] for the 1500 m high SUT, without includingthe equations of horizontal or sloped solar collector, in two caseswhen DALR is applicable inside and outside the SUT and whenDALR is applicable outside the SUT and the relative humidity ofthe indoor air current at point 3 is assumed to be 90%. In this case,the pressure potential expressions for conventional SUT presentedin Ref. [4] are used. As presented in Table 1, the pressure potentialsare shown to be almost equal for the case of the dry air currentinside the SUT, and close for the case of 90% of the relative humid-ity of the indoor air current at point 3. For the case of 90% of therelative humidity of the indoor air current at point 3, the humidityratio in this paper is a little higher than that in Ref. [4].

Since T3 is 20 �C higher than T11, the humidity ratio of the aircurrent at point 3 is larger than that at point 1 (the collector inlet)using Eqs. (9)–(11) based on the same relative humidity and theassumed same pressures of p11 = p13 = 90 kPa. In fact, when noadditional vapor evaporates from an open water source (e.g., theopen seawater [15], the open wet soil for vegetation [31], andthe open crop drying [32,33]) in the solar collector, or an openwater source (e.g., from solar pond [14]) around point 3 of theSCSUTPP without solar collector, the humidity ratio of the airinside the collector that is almost at a constant pressure is nothigher than that of the entering air at point 1 (which is equal tothe humidity ratio of the atmosphere at the ground level). In thehorizontal solar collector, the temperature gradually increasesand thus the relative humidity gradually decreases from point 1to point 2 (the collector outlet). Therefore, the performance shouldnot be evaluated on the basis of the relative humidity and humid-ity ratio of the air current at point 3, but on the basis of those of theentering air at point 1 (i.e., those of the atmosphere at the groundlevel). Fig. 3 shows the variations of the pressure potential andhumidity ratio for the reference 1500 m high SUT with the relativehumidities of the air current calculated using the temperatures atpoints 1 and 3, respectively, without including the equations ofhorizontal or sloped solar collector, when DALR is applicable out-side the SUT. In the figure, with an increase in relative humidity,the humidity ratio and pressure potential increase, and the increas-ing ratio using the temperature T3 at point 3 is higher than thatusing the temperature T11 at point 1. The humidity ratio usingT3 is far higher than using T11. They reach 0.1039 kg/kg dry airand 0.031 kg/kg dry air, respectively, for 100% of relative humidityof the air at points 3 and 1. The latter is between those of 30% and40% of relative humidity of the air at point 3 with T3. Therefore,higher relative humidity for the air at point 3 is difficult to achievewhen no additional vapor evaporates from the open water sourcesunder the collector roof. The pressure potential using T3 is over-predicted as compared to using T11. For example, the pressurepotential using T3 and T11 reach 1786 kPa and 1250.4 kPa, respec-tively, for 100% of relative humidity of the air at points 3 and 1. Useof the relative humidity of the air current estimated based on thetemperature at point 3 to predict the SUTPP pressure potentialand power output is not a good choice, which would amplify thecalculated results to a large extent. In the following parts, the dis-cussion and analyses will be performed by using the relativehumidity of the air current estimated based on the temperatureat point 1.

Based on the 5 MW SCSUTPP proposed in Ottawa [6], Zhou et al.[8] designed the collector structure of SCSUTPP (the designed

458 X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461

SCSUTPP is called SC1 for short) and investigated its performanceas compared to that of conventional HCSUTPP. In the paper, thepressure potential of the SCSUTPP is estimated based on SC1 [8].The parameters related to SC1 are presented in Table 2. The pres-sure p11 and temperature T11 of static atmosphere, are selectedto be 90 kPa, and 30 �C (i.e., 303.2 K), respectively, except whenthe ISA is assumed for the atmosphere. The temperature is differ-ent from 293 K proposed in Ref. [6], which is not suitable for thecondition under strong solar radiation I = 1000 W/m2.

In Table 3, when DALR is applicable inside and outside the SC1,the parametric results calculated with the pressure potentialexpression developed (Eq. (44)) are presented and compared withthose by using the essential pressure potential expression (Eq. (1))that takes into consideration the change of the atmosphere densitywith heights and the change with heights for the difference of theatmospheric density and the density of the current inside theSCSUTPP [8], based on this SCSUTPP model, in which the effect ofthe gravitational force and the loss of heat transferred to gravita-tional potential energy of the current are considered. The slopedcollector temperature rise is shown to increase by 1.65 �C, andthe pressure potential decreases by 56.75 Pa for Eq. (44) as com-pared to those for Eq. (1), thus resulting in the reduction of thevelocity of the air current at point 2 by 0.68 m/s. The calculatedpower output decreases by 1.27 MW based on Eq. (63). The calcu-lated results are shown to be close to the results obtained by usingthe accurate SCSUTPP model. Although some differences exist, theexpressions containing no integral for the pressure potential in thispaper provide a convenient and acceptable means to predict theperformance of the SCSUTPP engineering.

The humidity inside the SUTPP usually comes from the atmo-sphere at the ground level. When the humidity inside the SUTPPis considered for the prediction of SUTPP performance, the DALRassumption is not applicable outside the SUTPP. In order to exam-ine the effect of the DALR assumption outside the SUTPP on theSCSUTPP performance, the studies of the SC1 performance forvarious relative humidities of the entering air current at point 1

Table 2Design parameters for SC1 [8].

Collector shape Trapezoidal shape

b (�) 38.4Rcoll (m) 1365.2L1 (m) 1337.7L2 (m) 54B1 (m) 5.906B2 (m) 42.412b 0.5Acoll (m2) 950,000Hcoll (m) 848Hsut (m) 123Dsut (m) 54I (W/m2) 1000gcoll,tot 56%gtg 77%

Table 3Predicted parametric results for SC1 by using accurate model [8] and in this work forDALR assumption inside and outside the plant.

Using accurate model [8] In this work

T2 � T1 (�C) 14.93 16.58v2 (m/s) 11.06 10.38gcoll,eff (%) 36.08 37.39Dppoten (Pa) 456.4 399.65Pload (MW) 7.13 5.86

f1 are done using our model with and without the DALR assump-tion and with the ISA assumption outside the plant. The resultsare shown in Fig. 4. In this figure, with an increase in relativehumidity of the static atmosphere at the ground level, comparedto the gradual increase in the pressure potential for DALR assump-tion outside the SC1, the pressure potential gradually decreases byconsidering the effect of the humidity on the static pressure of theatmosphere. This leads to the reduction of the velocity of the aircurrent inside the SUT, and thus the reduction of the power outputby 4.61% from 5.86 MW for the DALR assumption outside the plantto 5.59 MW for the relative humidity of the entering air current atpoint 1 at 100%. However, the collector temperature rise slowlydecreases due to the increase in specific heat capacity of wet airwith the increase in its relative humidity. For the relative humidityfrom 0 to 100% the temperature at point 2 reaches around 46 �C,which is close to a constant point 3 temperature given 50 �C forestimating the 1500 m high SUT pressure potential. However, theSC1 pressure potentials are far smaller than those for the HCSUTPPwith a 1500 m high SUT presented in Ref. [4] as shown in Fig. 3.The reasons can be presented as follows: (1) the effective ‘‘SUT’’vertical height of 971 m for SC1 is much lower than the height ofthe 1500 m high SUT for the HCSUTPP; and (2) the mean temper-ature of the sloped collector where the air temperature increasesfrom the ambient temperature at point 1 to the temperature atpoint 2 is much lower than the temperature at point 2 or 3 ofthe HCSUTPP. In contrast with the results with the DALR assump-tion outside the plant, the humidity has a small negative effect onthe pressure potential and power output when the effect of thehumidity on the static pressure of the atmosphere is considered.With the increase in the relative humidity of the ambient air atthe ground level, the parametric results for the ISA assumptionoutside the plant have the same change tendency as those forthe DALR assumption outside the plant, but the pressure potential,velocity, and power output are just a little lower, and the collectortemperature rise is a little higher for the former than the latter.

In order to demonstrate the effects of the weather conditionsother than the humidity and the sizes on the SCSUTPP perfor-mance, the SC1 performance for the relative humidity of the atmo-sphere at the ground level f11 at 50% is studied by changing theparameters including ambient pressure and temperature, and thevertical heights of the sloped collector and SUT, when the effectof the humidity on the static pressure of the atmosphere is consid-ered. The variations of the pressure potential and power outputwith ambient pressure and temperature are shown in Fig. 5, whilethose with the vertical height of sloped collector and SUT areshown in Fig. 6. Similar to the results for the 1500 m high SUT withDALR assumptions inside and outside the SUT [4], the pressurepotential gradually increases with the ambient pressure increasingor the ambient temperature decreasing, thus leading to theincrease in the power output. With an increase in the ambient tem-perature from 0 �C to 50 �C, the power output gradually decreasesfrom 6.49 MW to 5.11 MW. With an increase in the ambient pres-sure from 80 kPa to 110 kPa, the power output slowly increasesfrom 5.71 MW to 5.75 MW. The increase in the power output isslight, because with the increase in the ambient pressure, the den-sity of the air inside the SC1 increases, which induces the temper-ature and velocity of the indoor air to decrease for a given effectivesolar heat input. Therefore, the site with low humidity, low ambi-ent temperature, and high ambient pressure should be selected inorder to obtain a high power output for an SCSUTPP installation. Asshown in Fig. 6, as expected, the pressure potential and power out-put increase with the increase in the vertical height of sloped col-lector and the SUT height. The power output increases from 0 MWto 27.74 MW with the sloped collector height increasing from 0 mto 2000 m. With an increase in the SUT height from 0 m to 1500 m,the power output increases from 4.4 MW to 20.73 MW. The

Fig. 4. Performance for SC1 versus relative humidity f1including equations of sloped collector.

Fig. 5. Power output for SC1 versus ambient pressure and ambient temperature for relative humidity f11 of 50%.

Fig. 6. Power output for SC1 versus sloped collector height Hcoll and SUT height Hsut for relative humidity f11 of 50%.

X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461 459

increasing amplitude for the former is larger than for the latterbecause sloped collector actually acts both as an air heater heatingthe indoor air current and an additional ‘‘SUT’’ driving them.Larger-scale power generation is therefore preferable by a combi-nation of economic analysis.

A great amount of tiny granules in the updraft originating fromthe ground can be used as effective condensation nuclei of mois-ture [15,25,39,40]. If the wet air current becomes saturated inthe SUT, the vapor may condense [15] even to form a cloud system

in the SUT [40], thus influencing the plant performance. In order toexamine whether the wet air current would become saturated inthe SUT, Fig. 7 compares the humidity ratio of the atmosphere atthe ground level and that of the saturated wet air assumed at theSUT outlet (point 4) for the relative humidities of the atmosphereat the ground level at 80% and 90% by changing the SUT height Hsut

from points 3 to 4 for the SCSUTPP whose design parameters arethe same as those of SC1 except the SUT height. In this figure,the humidity ratio of the saturated wet air current at point 4

460 X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461

decreases, which is determined by the static pressure and the sat-urated vapor pressure at point 4 (the saturated vapor pressure atpoint 4 is determined by the static temperature at point 4), as pre-sented in Eq. (45). The humidity ratios of the indoor air current aresmaller than those of the saturated wet air at point 4 for the SUTwhose height Hsut varying from 0 m to 1500 m for the relativehumidity of 80%, whereas for the relative humidity of 90% thehumidity ratio of the indoor air current is shown higher than thatof the saturated wet air at point 4 for the height SUT Hsut is over1409 m. Fig. 8 shows the humidity ratio of the potential saturatedwet air for two cases with and without considering the latent heateffect at various heights h above point 1 in the SCSUTPP, whosedesign parameters are the same as those of SC1 except the SUTheight of 1500 m, with a sloped collector of 848 m height and a1500 m high SUT for the relative humidity of the atmosphere atthe ground level at 90%. In the sloped collector, with an increasein the height, the static pressure decreases up the sloped collector,the temperature gradually increases and then decreases for a shortsegment around point 2 [8], and the vapor partial pressure willhave the same variation as the temperature variation based onEqs. (10) and (11). The effect of the variation of the vapor partialpressure on the humidity ratio is greater than that of the staticpressure. This results in the same variation of the humidity ratioof the saturated wet air as that of the air current temperatureand the vapor partial pressure. The air current flows through theturbine with a decrease of the temperature and static pressure,

Fig. 8. Humidity ratio of potential saturated wet air for two cases of with andwithout considering latent heat effect at various effective heights h above point 1 inSCSUTPP with a sloped collector of 848 m height and a 1500 m high SUT for relativehumidity f11 of 90%.

Fig. 7. Humidity ratio of atmosphere at the ground level (point 1) and that of thesaturated wet air at the outlet of SUTs (point 4) with various heights Hsut ofSCSUTPPs with a sloped collector of 848 m height for relative humidities f11 of 80%and 90%.

producing a sudden drop of the humidity ratio of the saturatedwet air. In the SUT, the temperature and static pressure graduallybecome smaller due to the effect of the gravitational force. Thetemperature drop causes the saturated vapor pressure to decrease,producing the reduction of the humidity ratio of the saturated wetair, though the static pressure drop can slow down the reduction. Itis found from Fig. 7 that in the SUT 1500 m high the wet air currentwith a relative humidity f11 above 90% would become saturated ata height below the SUT outlet. However, the critical saturationheight hsi above point 3 in the SUT cannot be obtained fromFig. 7, but can be calculated with the help of Eq. (48), to be1368.9 m for the 1500 m high SUT as shown in Fig. 8, which islower than the value of 1388 m for the 1450 m high SUT for air rel-ative humidity of 90%. Based on the assumption made in themodel, the condensation will take place above the critical satura-tion height hsi in the SUT. However, the latent heat is releasedaccompanying the vapor condensation and heat the current itselfinside the SUT whose reinforced concrete wall acts as a good ther-mal resistance to transfer heat outside. Total potential increase inthe temperature would reach about 3.9 �C due to vapor condensa-tion. Once the vapor starts to condense above the critical conden-sation height, due to the latent heat released, the indoor airtemperature would increase significantly, the saturation vaporpartial pressure in the air would increase, and the humidity ratioof the saturated wet air would increase to a value larger than thatof the actual air current at the same height. It is concluded that thelatent heat accompanying the vapor condensation will stop thevapor from condensing. The work presented in section 2.2.3 issometimes unrealistically used unless the liquid water in theSUTPP is carried by the strong current after entering into the plant,e. g., by spraying water fogs in the plant.

4. Conclusions

The expressions containing no integral for the pressure poten-tial of an SCSUTPP under various conditions have been presented.Taking the SCSUTPP case (SC1) designed by Zhou et al. [8] as anexample, based on the mathematical model of a sloped collectorby assuming a steady compressible flow and the expressions ofthe pressure potential developed, the SCSUTPP performance hasbeen comprehensively studied under various conditions. Withoutthe effect of the collector on the pressure potential, the pressurepotential for the vertical SUT using the temperature at the SUTinlet has been compared with that using the temperature at thecollector inlet. The parametric results of SCSUTPP using the expres-sions developed have been compared with those using the essen-tial expression of the pressure potential. The conclusions aredrawn as follows:

(1) The pressure potential for the SUT should be estimated usingthe relative humidity of the air based on the temperature atthe collector inlet. The pressure potential for the SUT basedon the temperature at the SUT inlet due to higher tempera-ture is shown to be over-predictive, as compared to thatbased on the temperature at the collector inlet.

(2) The pressure potential and power output decrease with theincrease in relative humidity of the air inside the SCSUTPP,which is assumed to be equal to that of the atmosphere atthe ground level. This variation trend is opposite to that whenthe air outside the SCSUTPP is assumed to be DALR or ISA.

(3) The expressions containing no integral and considering thegravitational effect are available for predicting the pressurepotential of SCSUTPP under various conditions based on asteady compressible fluid model when an average densityof the air current inside sloped collector is assumed.

X. Zhou et al. / International Journal of Heat and Mass Transfer 75 (2014) 450–461 461

(4) The pressure potential gradually increases with the ambientpressure, the collector height, and the SUT height increasingor the ambient temperature and the ambient humiditydecreasing, leading to the increase in power output. There-fore, in order to obtain a high power output, the site selec-tion with low humidity, low ambient temperature, andhigh ambient pressure should be done, and larger-scalepower generation is preferable by a combination of eco-nomic analysis.

(5) The potential temperature increase due to the latent heatreleased from the vapor condensation is so large that thehumidity ratio of the potential saturated wet air is largerthan that of the actual saturated wet air above the criticalsaturation height. This will decline the vapor to continuecondensation. It is concluded that the latent heat accompa-nying the vapor will stop the vapor from condensing.

The work lays a good foundation for the prediction of the poten-tial power production from SCSUTPP.

Conflict of Interest

None.

Acknowledgements

This research has been partially supported by the NationalNatural Science Foundation of China (No. 50908094), the Ph.D. Pro-grams Foundation of Ministry of Education of China (No.20100142120071), and the Fundamental Research Funds for theCentral Universities, HUST (No. 2013NY015).

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