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International Journal of Thermal Sciences 102 (2016) 210e227

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

A mathematical model of a solar air thermosyphon integrated withbuilding envelope

Himanshu DehraEgis India Consulting Engineers Pvt. Ltd, Egis Tower, Plot No. 66, Sector-32, Gurgaon, Haryana 122003, India

a r t i c l e i n f o

Article history:Received 5 October 2014Received in revised form23 October 2015Accepted 16 November 2015Available online xxx

Keywords:Solar thermosyphonHeat equationHeat transportControl volumeThermal networkMathematical model

E-mail address: anshu_dehra@hotmail.com.

http://dx.doi.org/10.1016/j.ijthermalsci.2015.11.0221290-0729/© 2015 Elsevier Masson SAS. All rights re

a b s t r a c t

Amathematical model comprising of analytical and two dimensional network procedures is proposed forpredicting steady heat flow and heat transport in a solar heated thermosyphon integrated with buildingenvelope. The detail mathematical solution method is presented for solving partial differential heat flowand transport equations by performing two dimensional energy balances at surface and air nodesthrough conjugate heat exchange and heat transport analysis of a solar thermosyphon. The mathematicalsolution procedure is devised for conduction and radiation heat exchange between surface nodes toimprove the accuracy of traditional analytical solution for predicting buoyancy-induced mass flow rate ofair flowing through a solar thermosyphon. The matrix inversion solution of mathematical model isunconditionally stable, with accuracy dependent on magnitude of conductance terms and number ofnodes in the grid. Only Dy is chosen as aspect ratios (Dx ¼ L, L/H and W/H) are defined by the geometryof the thermosyphon. The conduction and convection conductance terms are based on discretisationheight Dy, thermal capacity conductance (mcp) is based on air-gap length Dx, whilst integrated radiationconductance terms are based on both height Dy and width Dx of the grid. The proposed model hascompared the results obtained from analytical method, two dimensional network method for a single setof environmental condition with given geometry of thermosyphon. The proposed model is validatedwith experimental results obtained from outdoor experimental setup comprising of thermosyphonbased photovoltaic solar wall system.

© 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction

The present state of work for developing model for a thermo-syphon is available for design of industrial thermosyphon with noexposure to solar heat flux. Most of the work in the literature isrestricted to geometries based on experimental work of labora-tories. The motivation of the paper is to present a detailed math-ematical model for a steady state analysis of rectangular solarheated thermosyphon connected to a building. The mathematicalmodel is interpreted from the view of physical thermal and fluidflow phenomenon occurring in the thermosyphon. In literaturemost of work is done on Trombe wall [1]. In the work on Trombewall, the theoretical calculations for the temperature distribution ofa Trombe wall are obtained by using a thermal network andcompared the results with the experimental data [1]. Work is alsoperformed on system modelling and obtaining operation charac-teristics [2]. Trombe wall with PV cells is also modelled [3].

served.

The mathematical models in various scientific applications aresolved by simultaneous partial differential equations. The numer-ical solutions are based on solving techniques such asGausseJordan elimination in which large system of equations arereduced to obtain the simplified solution [5]. The solution methodsfor solving system of algebraic nodal equations are simplified withmatrix computations. The unknown temperatures at nodes ondiscretised surfaces for a given system are determined with directmatrix solution methods. Despite availability of various numericalmethods for solution of linear algebraic equations, there are manyfactors involved for pursuing the search for the analytical solutions.The analytical solutions on the other hand are useful in validation ofnumerical analysis because many parameters are kept constant inmaking assumptions during the development of a numericalmethod. Apart from this factor, analytical solutions provide sensi-tivity analysis so as to take into account the effect of parameters nottaken into consideration during numerical analysis. However thenon-availability of the closed-form analytical solutions for thepartial differential equations has developed the need of numericalmethods based on the discretisation principle [6].

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227 211

2. Operation of a solar air thermosyphon integrated withbuilding envelope

The rectangular solar heated thermosyphon is integrated withthe building through its well-insulated inner wall. The outer wall ofa solar thermosyphon is exposed to uniform quasi steady-statesolar heat flux. The cold fluid source in a solar thermosyphon isambient air moving into the building hot-space sink through a solarthermosyphon. The schematic of a solar heated thermosyphon isillustrated in Fig. 1. The three different interacting temperaturevariables with only two space coordinates are used in heat flowanalysis of a solar thermosyphon. The heat flow is a vector quantityassociated with kinetic energy {mk(vk)2/2} of molecules in solid orfluid with motion from higher temperature region to lower tem-perature region. The origin, direction and magnitude of heat flowvectors are obtained by solution of simultaneous heat equations ofthermal network for a given system. The Fourier's conduction lawgives negligible heat flow value towards negative y-direction for airflowing through a solar thermosyphon because of its very lowthermal conductivity. The procedure is useful to eliminate onetemperature dependent conduction term in performing energybalance on air nodes.

The partial differential heat equation may contain up to fourindependent variables-three space coordinates and time. It is alinear partial differential equation in temperature variable for thecase of constant thermo-physical properties. The boundary condi-tion of radiation heat exchange at the enclosure surface of the wallsis associated with long range distance phenomenon and conser-vation of energy is not applicable with in the control volume, but isapplied in between all the composite surfaces observing net radi-ation exchange. The unique characteristics of the improved nu-merical solution method are: (i) inclusion of radiation exchangecalculations using radiosity-irradiation method by assumingenclosure between outer and inner walls of a solar thermosyphon;and (ii) inclusion of conduction heat flow along y-direction forouter and inner walls of a solar thermosyphon.

The thermal network for a solar heated thermosyphon inconjunctionwith the building thermal system is associated with-(i)establishing heat travel and exchange paths; (ii) thermal designdata-composition of thermal air and equipment/building con-struction materials-their equivalent heat capacities and conduc-tances; (iii) environmental and climatic variables-prevailing winddirection and speed, solar irradiation, humidity, pressure and drybulb temperature; (iv) feedback thermal set point control; and (v)

Outlet Damper

Building Zone

Inner Wall (Insulated)

Ambient Zone

Y-axis

System Boundary

X-axisInlet Damper

Outer Wall

Air PassageS

L

H

ti

to

Fig. 1. Schematic of a solar heated thermosyphon-open rectangular type.

load variation with periodicity and thermal storage. Simultaneouseffect of above variables on the model is a complex phenomenon.Thermal network with large number of inter-dependent variablesleads to arbitrary conditions solvable through stochastic models.

The affect of conjugate heat exchange and heat transport ontemperature distribution in a solar thermosyphon is considered soas to improve the accuracy of numerical solution. The climatic andthermal design data is kept constant in steady heat flow analysis ofa solar thermosyphon. The single climatic variable of ambient airtemperature, solar irradiation and building zone air temperaturesare known constants in the analysis.

3. Mathematical physics, analysis and geometry

The paper has presented the detailed mathematical physics,analysis and geometry for a solar heated thermosyphon connectedto a building as illustrated in Fig. 2. The heat exchange analysis iscarried out on the geometry of a solar thermosyphon with dis-cretisation of its total covered volume into surface and air nodeslocated by formulation of control volumes. A geometrical method isillustrated for conduction and radiation heat exchange betweensurface nodes to improve the accuracy of traditional analytical so-lution for predicting buoyancy-induced mass flow rate of airflowing through a thermosyphon. As is illustrated in Fig. 2, a solarthermosyphon is placed along the y-axis with y ¼ 0 near the bot-tom end of the system boundary and y ¼ H near the top end of thesystem boundary. The solar thermosyphon is rectangular in cross-section with width W in z-direction and air-gap length, L in x-di-rection. The thermal conductivities of outer wall and inner wall areassumed to be constant along with their dimensions-L, W and H.The inner wall is well-insulated with thermal conductance ui. Theouter wall is of good thermal conductance (uo) for conducting heatflux of solar irradiation. The heat transfer between building spaceandwell-insulated inner wall is assumed to be nil. The heat transferbetween side walls of length L, and height H and surrounding zoneis also assumed to be nil. The air passage for a solar thermosyphonis connected with the building space through a damper operatingsystem. The physical domain of a solar thermosyphon is analysed asa parallel-plate channel.

The mathematical analysis for the heat flow consists of dis-cretisation of volume of thermosyphon into the small volumeelement V between four cross-sectional planes P1x, P2x, P1y and P2y.The planes P1x, P2x are perpendicular to the y-axis with plane P1xlocated at y and plane P2x located at y þ dy. The planes P1y, P2y areperpendicular to the x-axis with plane P1y located at x ¼ to and P2yis located at x ¼ to þ L. Further the planes P1x and P2x are having

Plane P

Plane P

Plane P Plane P

P ((t + L + t ), 0)

Building Space

Y-axis

X-axis

TAmbient Zone T

P (k , t , u , T ) P (mc ,T )

P (k , t , u , T )

P (0,0)

P ((t + L + t ), y)

P ((t + L + t ), y + dy)

W

H

(to+L+ti)

L

Fig. 2. The system coordinates: Nodes and cell faces in a control volume.

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227212

volume V consists two solid-air control volumes dVo and dVi formedbetween outer wall-air and air-inner wall respectively. The airpassage as well as air nodes are shared between the two controlvolumes. In Fig. 2, the surface and air nodes are representing uni-form thermo-physical properties in control volumes.

3.1. Assumptions and initial conditions

The key assumptions and initial conditions used in carrying outmathematical analysis are:

￮ The outer wall is thin, light weight and good conductor of heat.￮ The net solar heat flux, qo on the outer wall is quasi steady-stateand distributed uniformly over the surface.

￮ The inner wall is light weight and good insulator for heat.￮ The temperature variation only along y-ordinate, being taken aslumped in x and z-coordinates.

￮ The heat conduction (diffusion) equation term with negligiblevalue for air is not included in the energy balance.

￮ The heat transfer between the side walls/inner wall of the solarthermosyphon and the surrounding environment is negligible.

￮ The temperatures of ambient air (Ta) and single air zone (Ts) ofbuilding are specified.

3.2. Heat equations in two coordinates of space

The principles of physics are applied for obtaining solutions ofheat equations for the inner and outer walls of a solar thermosy-phon. The outer wall is having good conductance, and inner wall isof good insulation. Since specific heat capacity is related to density,materials selected are light weight. Some examples are aluminiumbased alloy for outer wall and polystyrene filled plywood board forinner wall. The steady-state heat equation for the outer wall withheat generation is written with temperature gradients vTo/vx andvTo/vy in x and y-ordinates, which are proportional to their partialderivatives in these directions. The proportionality constant iscalled thermal conductivity and partial differential equation isderived as Poisson's heat equation:

v2

vx2To þ v2

vy2To þ qo

ko¼ 0 (1)

Similarly, Laplace's heat equation for the inner wall withoutinternal heat generation is written as:

v2

vx2Ti þ

v2

vy2Ti ¼ 0 (2)

3.3. Convection heat exchange and heat transport

With a parallel-plate channel flow, the air enters the thermo-syphon at y ¼ 0 with a mean air flow velocity v. At quasi steady-state value of solar irradiation, the net heat flux on outer wall of asolar thermosyphon is qo and the temperature of air for the dis-cretised control volume is its bulk air temperature, Tf. Over thesurface face area of the outer wall, heat flux is distributed evenlyand is dissipated from the outer wall by convection to the air andradiation exchange with the inner wall of a solar thermosyphon.The inner surfaces of the outer and inner walls of a solar thermo-syphon are assumed to be gray, diffusively emitting and diffusivelyreflecting, with a uniform surface emissivity of ε. The convectiveheat transfer coefficient, h between outer and inner walls and air is

assumed to be constant in control volume. With this simplifiedmathematical analysis, employing an energy balance on controlvolume V ¼ W L dy gives: Enthalpy flux of air in at y þ convectiveflux of airesurface in over dy ¼ enthalpy flux of air out at y þ dy.

In mathematical form, this is expressed as:

mðyÞ,cp,Tf ðyÞ þ h,W,

�To,ðyÞþTi,ðyÞ

2� TfðyÞ

�,dy

¼ mðyÞ,cp,Tf,ðyþ dyÞ (3)

where, mass flow rate is function of density and for simplification itis expressed as m(y) ¼ r(y)v(y)LW ¼ m for constant density andvelocity within the control volume.

This is expanded to:

m,cp,Tf ðyÞ þ h,�To,ðyÞ þ Ti,ðyÞ

2� Tf,ðyÞ

�,W,dy

¼ m,cp,�Tf,ðyÞ þ

ddy

TmðyÞ,dy�

(4)

Equation (4) is reduced to:

m,cp,�

ddy

Tf ðyÞ�

¼ h,W,

�To,ðyÞ þ Ti,ðyÞ

2� Tf,ðyÞ

�(5)

For simplified mathematical representation, the mean walltemperature for outer and inner wall is written as:

TwðyÞ ¼ ToðyÞ þ TiðyÞ2

(6)

Equation (5) is having unknown variable of mean wall temper-ature and solution is obtained by inlet condition for the half controlvolume (Fig. 6):

Tf ¼ Ta at y ¼ Dy2

(7)

3.4. Radiation heat exchange

The radiation exchange analysis is simplified by assuming idealsurface properties. All the surface elements of a solar thermosy-phon are assumed to be gray and diffuse emitters, absorbers andreflectors. The Kirchhoff's law of emissivity is applied for the sur-face elements of the walls i.e. each surface element of outer andinner walls emits precisely as much energy as it absorbs. With theabove assumptions, energy balance on the surface element of outerwall yields net volumetric heat flux across unit length:

qo ¼ ðqe þ qc þ qrÞ � qa (8)

where qe is emissive power for the given surface element, qr isreflective power from the given surface element, qc is convectiveheat loss from the given surface element to the air and qa (¼aS) isshort wave solar radiation absorbed on the outer wall. qe and qa aredirectional quantities and directions are to be assumed as per thesign notation. According to this definition net heat flux, qo is pos-itive if the heat is coming from inside of the wall, by conductionheat or other means (qo > 0), and negative if going from the airenclosure into the walls (qo < 0). This is expressed as:

qo ¼ qout � qin ¼ ðqe þ qc þ qrÞ � a S (9)

where a is the short-wave absorbtance of surface element on outerwall and S is the irradiation on the surface element. Surface

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227 213

radiosity qout is sum of emissive power and reflective power leavingthe surface element. Equation (9) is written as:

qo ¼ ðqe þ qc þ SÞ � a S (10)

The net heat flux qo on the outer wall of the thermosyphon isdissipated towards the flow passage by convection heat exchangeand emitted between various exchanging paths between the innersurfaces of the walls of a solar thermosyphon. In equation form thisis written as:

qo ¼ h�TwðyÞ � Tf ðyÞ

�þ ε

1� ε

hsðTwðyÞÞ4 � JðyÞ

i(11)

In Equation (11), the radiation term on right hand side is writtenas [9]:

qradiative ¼ ε

1� ε

hsðTwðyÞÞ4 � JðyÞ

i(12)

The Equations (11) and (12) are applicable with uniform emis-sivity of outer and inner walls of thermosyphon. The analysis is notapplicable for composite surfaces with different emissivities andreflectivity for surfaces elements. Equation (11) is also not appli-cable to distant surfaces with view factors for radiation exchange.Moreover, Equation (12) makes radiation analysis complex due tonon-linear temperature variables T0(y) and Ti(y) in the mean walltemperature Tw(y) and unknown radiosity variable J(y). To over-come these difficulties enclosure analysis is assumed withradiosity-irradiation formulation.

3.5. Enclosure analysis for a solar thermosyphon

The enclosure analysis for a solar thermosyphon is performedfor evaluating surface radiation calculations. The radiosity-irradiation formulation is applied for determination of the radio-sities of all the surface elements on the wall. The general radiosityequation for the oth radiosity element on the outer wall of thesurface element is obtained as:

Jo ¼ εosðToÞ4 þ ð1� εoÞXni¼1

�Fo;i � qi

(13)

where, Equation (13) is radiosity of the oth element due to reflec-tance and emittance from the ith wall elements. The surfaceelement ith on the inner wall are loosing radiosity power (reflectiveand emissive) with the oth surface element by a view factor of Fo,isummed up to n elements of ith surface, with qi as the net heat lossdue to radiation from the ith surface element on the inner wall.

The net radiation heat loss qi from a typical ith surface elementon the inner wall is the difference between the emitted radiationand the absorbed portion of the radiation and is obtained as:

qi ¼ ai �hεosðToÞ4 � aiSi

i(14)

In Equation (14), Si is irradiation on the ith surface element andai is the face area of ith surface element. For constant temperaturewith in the discretised surface elements on the walls for obtainingthe total net rate of heat loss for n surface elements from the othsurface element, the Equation (14) is approximated to:

qi ¼Xni¼1

zo;i � ai � s�hðToÞ4 � ðTiÞ4

i(15)

where, ai is the face area of the ith surface element on the innerwall. And zo,i is written as:

zo;i ¼xo;i � εo � εi

ri(16)

In script factor matrix zo,i, xo,i is calculated by inverse of co,imatrix, which is defined by do,i the Kronecker's delta function givenby following set of equations:

xo;i ¼�co;i�1 (17)

co;i ¼ do;i � ro � Fo;i for i � o (18a)

co;i ¼ do;i � Fo;i for i>o (18b)

do;i ¼ 1 for i ¼ o (18c)

do;i ¼ 0 for iso (18d)

where do,i is the Kronecker's delta function for matrix co,i. In matrixco,i for diagonal elements, Equations (18a) and (18c) are applicable,whereas for non-diagonal elements Equations (18b) and (18d) areapplicable. In Equation (17) matrix xo,i is independent of tempera-ture profile of surface elements and Fo,i is view factor of oth surfaceelement to ith surface element. The view factor Fo,i is derived for theenclosure of a solar thermosyphon using Hottel's crossed-stringmethod described [8] for the radiation exchange between surfaceelements of outer and inner walls:

Fo;i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�L

nDy

�2s

� LnDy

�Xi�1

i¼1

Fo;i�1 (19)

where, view factor Fo,i is calculated for the oth surface element withrespect to the ith surface element.

PFo,i�1 is the sum of view

factors up to the (i � 1)th surface element on the inner wall. InEquation (19) n has value between 1 to ith surface element forwhich view factor is obtained. L is the perpendicular distance be-tween parallel outer and inner walls and is also air-gap lengthwidth between outer and inner walls.

In Equations (12) and (15), two expressions for the radiative heatloss from surfaces are obtained. Equation (12) assumes uniformsurface properties, whereas for Equation (15) properties of twosurfaces i and o with n different surface elements are chosen.Equation (15) is more suitable to represent radiative exchange be-tween composite surface nodes of the outer and inner walls of asolar thermosyphon. The Equation (11) is re-written as:

qo ¼ h�TwðyÞ�Tf ðyÞ

�þXni¼1

zo;i� ai�s�hðToÞ4�ðTiÞ4

i(20)

3.6. Boundary conditions on walls

The boundary conditions on walls are obtained by applyingenergy balance on the surface nodes. The boundary condition atx ¼ to give convection and radiation heat exchange give the Fou-rier's heat equation for the outer wall of a solar thermosyphon:

�ko

�v

vxTo

�¼ ho

�Tf � To

�þXni¼1

hro;iðTi � ToÞ (21)

where,P

hro,i is the linearised radiation exchange factor (REF)obtained from Equation (15) as:

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227214

Xni¼1

hro;i ¼Xni¼1

4zo;is�Tmso;i

3 (22)

where zo,i is given by Equation (16) and Tmso,i mean surface tem-perature between surface nodes of outer and inner wall of a solarthermosyphon. The Equation (22) is applied in each discretisedsurface element to obtain the summation of radiation heat ex-change. For inner wall, boundary condition at x ¼ to þ L is writtenas:

�ki

�v

vxTi

�¼ hi

�Tf � Ti

�þXno¼1

hro;iðTo � TiÞ (23)

whereP

hro,i is the linearised radiation exchange factor (REF) forthe inner wall obtained from Equation (15) as:

Xno¼1

hri;o ¼Xno¼1

4zi;os�Tmsi;o

3 (24)

3.7. Ideal-air motion analysis

The ideal-air motion is assumed for obtaining the buoyancy-induced mass flow rate through a solar thermosyphon. The motionof air in a solar thermosyphon is assumed to be due to buoyancy.ASHRAE fundamentals handbook [9] has provided useful expres-sions for obtaining stack effect in the ducts due to thermal gravity.The lawof conservation ofmass is applied to the airmotion in a solarthermosyphon. The integral continuity equation applicable to acontrol volume is used to establish lawof conservationofmass.Withnowind pressure, buoyancy effect with still air is the critical case formathematical analysis. The constant density of air is assumed ineach control volume, i.e. conservative form of integral continuityequation is used to establish conservation of mass and obtainingBernoulli's equation along the streamline. The integral form ofsteady flow continuity equation over a control surface is written as:Z

r vdA ¼ 0 (25)

The air enters plane P2x and leaves plane P1x (Fig. 2) assumingthe air velocity v is normal to all the planes where air crosses,Equation (25) is written as:Z

r1v1dA�Z

r2v2dA ¼ 0 (26)

If the air density and velocities are constant within the controlvolumes, Equation (26) is written as:

r2A2v2 ¼ r1A1v1 (27)

The Bernoulli's equation is developed with the assumption ofconstant density of air with in the control volume. The Bernoulliequation is developed by equating the forces on planes P1x and P2xassuming frictionless air flow to the rate of momentum change ofair motion from planes P1x and P2x. On integration over a controlvolume between planes P1 and P2 for the steady-state flow [9]:

v2

2þZ

1ra

dPa þ gY ¼ constant (28)

where, Pa is absolute pressure, g is acceleration due to gravity and Yis elevation of control volume from datum. Assuming constant airdensity within the control volume, Equation (28) reduces to:

v2

2þ Pa

raþ gY ¼ constant (29)

The Equation (29) is for steady, frictionless air flow along thestreamline. It is applied to a solar thermosyphon assuming ductflow. The air motion resistance between planes P1x and P2x in termsof pressure is written as:

r1v212

þ P1 þ gr1Y1 ¼ r2v222

þ P2 þ gr2Y2 þ DPd;1�2 (30)

In Equation (30), v1 and v2 are average velocities (replaced bylocal velocity v), because loss coefficients result in error in calcu-lating velocity pressure (rv2/2) with use of local velocities acrossstreamlines [9]. DPd,1-2 is total pressure loss due to friction anddynamic losses between control volume planes P1x and P2x. Equa-tion (30) is used to obtain DPd,1-2 is total pressure loss in terms ofstatic pressure, velocity pressure and buoyancy induced pressureas:

DPd;1�2 ¼"Ps1 þ r

ðv1Þ22

#�"Ps2 þ r

ðv2Þ22

#

þ gðro � rÞðY2 � Y1Þ (31)

The Equation (31) is obtained by assuming r1 ¼ r2 ¼ r (i.e.constant density of air in the control volume) and using definitionof gauge pressure to obtain static gauge pressures Ps1 and Ps2 atplanes P1x and P2x respectively. It is re-written in terms of buoyancyinduced pressure and total pressure as:

DPd;1�2 ¼ DPd þ DPb (32)

where in Equation (32) buoyancy induced pressure DPb is:

DPb ¼ g�ra � rf

�ðY2 � Y1Þ (33)

The analysis is performed for total pressure change (DPd) due tofriction, buoyancy, loss coefficients for fittings at inlet and exit suchas dampers to control the air flow. Once total pressure change DPdis obtained, buoyancy induced velocity (v) is determined over allthe control volumes from Equations (31)e(33). Assuming air as anideal gas, equation of state for air to determine absolute pressure atinlet of the thermosyphon is written as:

Pa ¼ raR Ta (34)

When moisture is not taking into account i.e. for dry air, Equa-tion (33) is re-written with use of Equation (34) as:

Dr ¼�ra � rf

�¼ 1

RPaTf � PfTa

TaTf(35)

Equation (35) is reduced to:

Dr ¼ raDTTa

�1� TaDP

DTPa

�¼ raDT

Tað1� CÞ (36)

The value of correction constant C is smaller than 0.01 [4]. Theabove expression is reduced to following expression with an errorsmaller than 1% [4]:

Dr ¼ raDTTa

¼ rDTTa

(37)

The moisture effect in dry air is added by introducing correctionfactor in Equation (36) for C, which is 0.08 in summer and 0.02 inwinter [4]. With known mass flow rate due to buoyancy induced

Neutral Plane

Outlet Damper

Inlet Damper

Poexit

Pexit

Pinlet

Poinlet

InsideOutside

dPexit

dPinlet

Vexit

Vinlet

Yshift

P dP V

Hout

Hin

H

Yshift Yshift

Fig. 3. Air motion pattern in a solar thermosyphon.

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227 215

velocity and known ambient air temperature, enthalpy change ofair flow from the ambient to the thermosyphon is determined bythe inlet condition. The enthalpy change of air based on cold airinlet condition in the control volume is written in partial differ-ential form:

v

vTaEa ¼ racpVa (38)

where Va is the cold air volumetric flow rate entering into thecontrol volume. The conservation of energy is applied on the con-trol volume for determining temperature difference in terms ofdissipated heat from the walls of solar thermosyphon, Equation(38) is written as:

DT ¼ EaracpVa

¼ Efrf cpVf

(39)

where, Ef is the change in enthalpy of air due to temperature dif-ference between fth air control volume and (f � 1)th air controlvolume. Vf is the hot air volumetric flow rate leaving the controlvolume. The correlation for the volume flow rate in terms of densitydifference is written for inlet flow condition as:

Va ¼ ck;inAin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Dr g Yn

ra

s(40)

where, ck,in is the discharge coefficient at inlet, Yn is the verticaldistance between neutral pressure plane and inlet datum plane. Interms of temperature difference Equation (40) is written as:

Va ¼ ck;inAin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DT g Yn

Ta

s(41)

The air velocity is also derived in terms in terms of inlet or outletcondition. Air velocity in terms of temperature difference at theinlet flow condition is:

va ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 DTg Yn

Ta cf ;in

s(42)

where in Equation (42) cf,in is the flow coefficient at the inlet. Interms of enthalpy change of air, the velocity expression based oninlet condition is written as [4]:

va ¼ 0:037�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEaYn

ck;inAin

3

s ffiffiffiffiffiffiffiffiffi1

cf ;in

s(43)

The neutral height Yn is calculated as:

Yn ¼ Y1þ n2 (44)

where n2 is derived as:

n2 ¼ TfTa

cf ;outcf ;in

24 ck;in

ck;out

!2�AinAout

�235 (45)

If the discharge and flowcoefficients at inlet and outlet are equali.e., ck,in ¼ ck,out and cf,in ¼ cf,out and as well as inlet and outlet areasare same, then Equation (45) is approximated to:

n2 ¼ TfTa

(46)

As illustrated in Fig. 3, the outside atmospheric pressure andpressure inside the air passage of the thermosyphon are equal atneutral height Hin [4]. Up to neutral height Hin there is negativepressure inside the air passage of the thermosyphon and above theneutral height Hin, the pressure is positive so that air flows out ofthe thermosyphon through opening at the outlet at a distance Houtfrom neutral pressure height Hin. There is shift in Y-axis as sym-metrical axis for the air motion analysis. The analysis is alsoapplicable to solar water heaters using thermosyphon mode forheat transport and is useful for eliminating reverse circulation.

4. Analytical solution method

The solution method is devised as analytical because of itsapplicability at each node in domain of the problem [7]. The partialdifferential equations are re-written in standard form and initialconditions and assumptions are applied to formulate the problem.Equation (5) is re-written as:

qðyÞ � v

vyTf ;y þ Tf ;y �

To;y þ Ti;y2

¼ 0 (47)

where parameter q is defined as:

qðyÞ ¼ q ¼ vðyÞrðyÞL cphcðyÞ (48)

where, q(y) is function of g1(y) as velocity v(y), density r(y) andhc(y) are dependent on another function g2(y). As functions g1(y)and g2(y) are constant within the control volume therefore for agiven control volume it is expressed as q.

4.1. Initial boundary value problem (IBVP) formulation

With the assumptions and initial conditions, IBVP is formulatedfor obtaining the analytical solution. The boundary conditions areestablished in the control volume as per heat travel paths illus-trated in Fig. 4.

For the outer wall with uniform heat flux, Equation (1) is writtenwith boundary conditions as:

v2

vx2To þ v2

vy2To þ qo

ko¼ 0 in 0< x< to 0<y<H (49)

��

v

vxTo

�¼ a S� haðTo � TaÞ

koat x ¼ 0 (50)

(∂To/∂x)=qo (∂Ti/∂x)=0

(∂Ea/∂Tf)=ρfcpVf

To=Po/βo

Ti=Pi/βi

y

x

F

H

Tf=Ta

Fig. 5. The boundary conditions applicable to a physical domain.

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227216

v

vxTo ¼

ho

�To � Tf

�þ hrðTo � TiÞko

at x ¼ to (51)

For the inner wall with insulation, Equation (2) with boundaryconditions is:

v

vx2Ti þ

v2

vy2Ti ¼ 0 in L þ to < L þ to þ ti 0<y<H (52)

��

v

vxTi

�¼

hi

�Ti � Tf

�þ hrðTi � ToÞki

at x ¼ L þ to (53)

v

vxTi ¼

hsðTi � TsÞki

¼ 0 at x ¼ L þ to þ ti (54)

Equation (47) is re-written (say in l-form) with its boundaryvalue as:

qðyÞ�v

vyTf ðyÞ

�þ Tf ðyÞ �

�ToðyÞ þ TiðyÞ

2

�¼ 0 in 0< y<H

(55)

Tf ðyÞ ¼ Tf ðk� 1Þ at y ¼�kþ 1

2

�,Hn

k ¼ 0…ðn� 1Þ (56)

In Equation (56), k varies from 0 to (n� 1), where n is number ofnodes in y-ordinate. Graphical representation of physical domainutilising effective boundary conditions given by set of Equations(49)e(56) is illustrated in Fig. 5.

Equations (50) and (51) are simplified after rearranging termsas:

��

v

vxTo

�x¼0

þ b1oTo � P1o ¼ 0 (57)

�v

vxTo

�x¼to

þ b2oTo � P2o ¼ 0 (58)

where parameters b1o, b2o and P1o, P2o are written as:

(i+1)

(i)

(i-1)(o-1)

(o)

(o+1)

(f)

(f-1)

(f+1)

X-axis

Y-axis

qo

qo

qo

Fig. 4. Generalised heat travel paths between the control volumes.

b1o ¼ ha

kob2o ¼ ho þ hr

koP1o ¼ a Sþ haTa

ko

P2o ¼ hoTf þ hrTiko

(59)

Parameters b1o, b2o and P1o, P2o are summed up as:

bo þ b1o þ b2o ¼ ho þ hrþ ha

koPo ¼ P1o þ P2o

¼ haTa þ hoTf þ hrTi þ a Sko

(60)

Similarly Equations (53) and (54) are re-arranged as:

��

v

vxTi

�x¼toþL

þ b1iTi � P1i ¼ 0 (61)

�v

vxTi

�x¼toþLþti

þ b2iTi � P2i ¼ 0 (62)

where parameters b1i, b2i and P1i, P2i are written as:

b1i ¼hi þ hr

kib2i ¼

hs

kiP1i ¼

hiTf þ hr Toki

P2o ¼ hs

ki(63)

The parameters b1i, b2i and P1i, P2i are summed up as:

bi ¼ b1i þ b2i ¼hi þ hrþ hs

kiPi ¼ P1i þ P2i

¼ hsTs þ hiTf þ hr Toki

(64)

where, hr is constant radiation heat exchange coefficient betweenouter and inner walls of the thermosyphon.

Table 1Physical domain for a solar thermosyphon.

Property Value

Environment conditionsSolar irradiation 600 W m�2

Ambient heat transfer coefficient 13.5 W m�2 �C�1

Ambient air temperature �5 �CBuilding space temperature 20 �COuter wall (aluminium alloy)Height 3.0 mWidth 1.0 mThickness of outer wall 0.002 mAbsorbtance of outer wall with flat black paint 0.95Thermal conductivity (AleSi) 137 W m�1 �C�1

Inner wallRSI value of outer wall 1.0 m2 �C W�1

Thickness of outer wall 0.04 mAirWidth of air gap 0.05 mThermal conductivity 0.02624 W m�1 �C�1

Specific heat of air (cp) 1000 J/kg �CDensity of air 1.1174 kg/m3

Kinematic viscosity of air 15.69 � 10�6 m2/sPrandtl number of air 0.708Buoyancy-induced velocity from analytical solution 0.48 m/sMass flow rate obtained from analytical solution 0.027 kg/sBuoyancy-induced velocity from numerical solution 0.45 m/sMass flow rate obtained from numerical solution 0.025 kg/sSurfaceStefan Boltzmann constant 5.67 � 10�8 W/m2 K4

Emissivity of back surface of thermosyphon walls 0.95Grid sizeNumber of nodes in x e direction Nx ¼ 3Number of nodes in y e direction Ny ¼ 10

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227 217

The temperatures of outer and inner walls are obtained byadding Equations (59) and (60) and Equations (61) and (62) withassumption of lumped temperature along x-axis:

To ¼ Pobo

for�

v

vxTo

�x¼0

¼�

v

vxTo

�x¼to

(65)

Ti ¼Pibi

for�

v

vxTi

�x¼toþL

¼�

v

vxTi

�x¼toþLþti

(66)

4.2. IBVP solution of l-form

For obtaining the solution of Equation (55), method of separa-tion of variables is applied and Equation (55) is re-written as:

v

vyTf ðyÞ ¼

1qðyÞ

�ToðyÞ þ TiðyÞ

2� Tf ðyÞ

�(67)

The general solution of Equation (67) is:

ln�ToðyÞ þ TiðyÞ

2� Tf ðyÞ

�¼ y

qðyÞ þ c (68)

At y¼ [(kþ 1/2)H/n], Tf,(y)¼ Tf(k� 1) substituting general formof initial boundary value to Equation (68), the general form ofconstant c is obtained as:

c ¼ ln�ToðyÞ þ TiðyÞ

2� Tf ðk� 1Þ

�(69)

After back substituting value of c in Equation (69) and simpli-fication, the particular solution of IBVP of l-form is obtained:

Tf,ðyÞ ¼To,ðyÞ þ Ti,ðyÞ

2þ�Tf ,ðk� 1Þ

� To,ðyÞ þ Ti,ðyÞ2

�,e�2, Dy

qðyÞ (70)

where, Dy ¼ H/n is discretisation height of the control volume, andq(y) ¼ q is constant within the control volume at steady flowconditions. Equation (70) is applicable with in the control volumeand it predicts particular solution for each Dy from the values ofTf(y) at previous air node. The exponential solution of Equation (70)is analytical in nature because of its applicability for the nodes within the physical domain.

4.3. Analytical solution procedure

Step 1. Initial conditions and assumptions used in the solutionare provided in Table 1.Step 2. Initial temperatures for outer wall and inner wall areassumed.Step 3. Approximate mass flow rate is calculated to obtaintemperature profile of the air with height of thermosyphonfrom Equation (70).Step 4. From Equations (65) and (66), temperature profiles of theouter and inner walls are obtained.Step 5. The buoyancy-induced mass flow rate is calculated frominitial temperatures obtained from analytical solutions.Step 6. The convective heat transfer coefficients and radiationheat exchange factors are calculated assuming constant tem-peratures for the outer wall, air passage and inner wall.Step 7. The value of buoyancy-induced mass flow rate is iteratedfew times unless its value agrees closely with its previous value

and is used to give final temperatures of outer wall, air and innerwall.

5. Numerical analysis of partial differential heat equations

The analytical solution is based on energy balance on surfacenodes with heat exchange between walls and air in the thermo-syphon. The heat exchange paths are in x-ordinate, whilst heat flowis in positive y-ordinate as observed from Equations (65), (66) and(70) and illustrated in Fig. 4. After examining IBVP, difficulties wereinvolved in Section 3 in finding solution of partial differentialequations. There is no closed form solution for IBVP or any methodfor evaluation of coefficients of the general solution. The boundaryconditions are non-homogeneous, more ever there is requirementof at least three homogeneous boundary conditions each forEquations (49)e(56) for finding solution with available methods ofsolution. Moreover, if three homogeneous boundary conditions arecreated then available methods are unable to adopt simpler solu-tions for the second order heat equation. Only with the reasonsstated above, the assumptions as mentioned in Section 3.1 areapplied on physical domain to obtain the traditional analytical so-lutions used in practice.

The numerical solutions are obtained by creation of additionalheat exchange paths in the computational grid [7]. The numericalanalysis involves [7] (i) construction of nodal networks; (ii) energybalance on the surface nodes located at solid-air edges of the walls;(iii) energy balance on control volume for air passage; and (iv)computer solution of system of algebraic equations. As illustrated inFig. 6, nodal or lattice points are created in the rectangular mesh atwhich temperatures are to be approximated. Nodal points arecreated after dividing the thermosyphon system into control vol-umes. The distance between control volume nodes on xey plane isDxo ¼ (to þ L)/2, Dxi ¼ (ti þ L)/2 for outer wall and inner wall in x-

Outer wall

Air Nodes

Inner wall

X-axis

Y-axis

1

2

3

4

5

6

7

8

9

10

tito

Vertical Grid Lines

Horizontal Grid Lines

Half Solid-Air Control Volume

Solid-Air Control Volume

L

Surface Nodes

dy

Air

Fig. 6. Grid representation for nodes in discretised control volumes.

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227218

ordinate and Dy in y-ordinate. The control volumes are lumped subsystem, in which temperature represented at the node representthe average temperature of the volume. The nodal network isdeveloped by drawing five vertical construction lines at distancex¼ 0, to, (to þ L/2), (to þ L), and (to þ ti þ L) apart and ten horizontalconstruction lines at Dy distance apart starting from y ¼ Dy/2. Thenodes are located at all the intersections of the construction lines.The control volumes are formed by drawing horizontal and verticallines that exist midway between adjoining construction lines. Thecontrol volumes formulated are solid up towidth of the outer or theinner wall and continuedwithmade up of air of width (L/2). Surfacenodes are locatedmidway and air nodes are located on the edges ofthe control volume. Air-nodes are common to the two adjoiningsolid-air and air-solid control volumes.

5.1. Energy balance on control volume

The two types of nodal equations that can be developed onnodes are surface node and control volume nodal equationsdepending on location of the node at edge or interior surface of thecontrol volume [6]. The law of conservation of energy is applied inorder to balance incoming and outgoing conductances calculated atthe nodes located in the control volume. In order to illustrate en-ergy balance on control volume, Equation (1) is written for anyinternal solid node (m,n) in the outer wall:

DEx þ DEy þ E,DV ¼ DExþDx þ DEyþDy (71)

where DV ¼ dDxDy, is control volume for any internal node (m,n).Using second order Taylor's series finite-difference approximationfor the Fourier's law of conduction, Equation (71) is written as:

�kodDy�Tm;n�Tm�1;n

Dx

��kodDx

�Tm;n�Tm;n�1

Dy

�þqoðdDxDyÞ

::¼�kodDy�Tmþ1;n�Tm;n

Dx

��ko dDx

�Tm;nþ1�Tm;n

Dy

�(72)

The assumption of lumped temperature parameter imposesinterior surface elements to behave like solid-air edge by formu-lation of solid-air control volume. The assumption is validated bythe fact that outer wall is thin and good thermal conductor andinner wall is good insulator and light weight.

Equation (72) represents energy balance for the interior node(m,n) on the outer wall and after applying assumption of lumpedtemperature in x-ordinate and re-arranging the terms, it is reducedto:

� ko,d,Dx,�Tm;n � Tm;n�1

Dy

�þ qo,ðd,Dx,DyÞ

þ ko d Dx�Tm;nþ1 � Tm;n

Dy

�¼ 0

(73)

With these assumptions on the numerical solution, energybalance on nodes located in control volumes is written for outerwall, inner wall and air passage.

Therefore energy balance on surface node for an outer wall iswritten as:

DEðambÞx;o þ DEðcondÞy;o þ E DVþ DEðconvÞx;oþ DEðcondÞyþDy;o þ

Xi

EðradÞxy;o

¼ 0

(74)

where DE(amb) is the heat loss to the surrounding environment,DE(cond) is heat flow by conduction, DE(conv) is convection heatloss to the air and DE(rad) is the radiation heat loss to the surfaceelement. Similarly energy balance on surface node for the innerwall is written as:

DEðzoneÞxþDx;i þ DEðcondÞy;i þ DEðconvÞx;i þ DEðcondÞyþDy;i

þXo

EðradÞxy;i¼ 0

(75)

where DE(zone) is heat transfer between the node and the buildingzone is assumed later in the analysis to be negligible. Energy bal-ance for the air node is written as:

DEðenthÞy;f þ DEðconvÞx;f þ DEðconvÞxþDx;f ¼ 0 (76)

where DE(enth) is the enthalpy change of the air w.r.t. temperatureof air at the inlet condition.

5.2. Thermal network nodal equations

The energy balance equations for the N nodes are written andassembled in square U-matrix [N � N] and written in matrixequation form as [7]:

UN;N � TN;1 ¼ Q1;N N ¼ oþ f þ i (77)

where, subscripts o, f and i are number of nodes in the grid for outerwall, air and inner wall respectively. Thermal network nodalequations are formed with Equation (77). It involves formulation of(UN,N)-matrix with conductances and heat source elements (Q1,N).Conductances describe entropy flux over the discretised area (inW/K units) at the node. Inverse of U-matrix is multiplied with heatsource matrix to give temperature solution of the thermal network.In writing nodal equations in matrix form, sign notation is adoptedfor automatic formulation of U-matrixwith unknown temperaturesand heat source elements. Sum of all incoming heat source ele-ments and U-matrix conductances multiplied with temperaturedifference with respect to the unknown temperatures at othernodes are equal to zero. The definition of sign notation is observed

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227 219

from Equations (74)e(76), which are written as per law of conser-vation of energy. The energy balance is written in equation form forany general node (m,n) [7]:

XNn¼1

�Um;n � DTm;n

þXNn¼1

Qm;n ¼ 0 (78)

where Um,n is the conductance at node (m,n),DTm,n is the differencebetween unknown temperature at the node (m,n) and unknowntemperature at surrounding heat exchange node. QN is heat sourceterm at the node (m,n).

Based on the sign notation of Equation (78), nodal equation forthe surface node on the outer wall as per energy balance of Equa-tion (74) is written as:

ðTo � TaÞUo;a þ qoDAyþ ðTo � Toþ1ÞUo;oþ1 þ ðTo � To�1ÞUo;o�1

þ�To � Tf

�Uo;f þ

Xi

Uo;iðTo � TiÞ

¼ 0

(79)

where, Uo,a, Uo,oþ1, Uo,o-1, Uo,f and Uo,i are conductances entering theoth node, with temperature To. Similarly thermal network equationfor the ith node on the inner wall as per energy balance of Equation(76) is written as:

ðTi � TsÞUi;s þ ðTi � Tiþ1ÞUi;iþ1 þ ðTi � Ti�1ÞUi;i�1

þ�Ti � Tf

�Ui;f þ

Xo

Ui;oðTi � ToÞ

¼ 0

(80)

where, Ui,s, Ui,iþ1, Ui,i�1, Ui,f and Ui,o are conductances entering the inode, with temperature Ti. Based on Equation (76) thermal networkequation for the node of air control volume is expressed as:

Uf ;f�1

�Tf � Tf�1

�þ Uf ;i

�Tf � Ti

�þ Uf ;o

�Tf � To

�¼ 0 (81)

where, Uf,f-1, Uf,I and Uf,o are conductances entering the f node, withtemperature Tf.

5.3. Constitutive relations

The conductance terms at the nodes are determined usingconstitutive relations for conjugate heat exchange and heat trans-port [7]. For the air nodes in the control volume, constitutive re-lations for conductances are:

Uf ;f�1 ¼ m� cp Uf ;i ¼ hf ;i � DAy Uf ;o ¼ hf ;o � DAy (82)

Constitutive relations for surface nodes of the outer wall are:

Uo;oþ1 ¼ Uo;o�1 ¼ uo � DAy Uo;f ¼ ho;f � DAy Uo;i

¼Xi

hro;i � DAy (83)

Similarly constitutive relations for surface nodes of the innerwall are written as:

Ui;iþ1 ¼ Ui;i�1 ¼ ui � DAy Ui;f ¼ hi;f � DAy Ui;o

¼Xo

hri;o � DAy (84)

The heat source matrix elements of Equation (84) are onlycalculated for nodes with known temperatures. The ambient air

and building zone temperatures are known. Therefore heat sourceelements for the node located at the outer wall are written ingeneral form as:

QoðoÞ ¼ Ua � Ta þ qo � DAy (85)

The inner wall of thermosyphon is well insulated with buildingfaçade. Therefore, heat source elements of inner wall are assumedto be negligible in heat source matrix because of negligible heatexchange between building zone and inner wall. However for air asair medium entering from cold source of ambient air into thethermosyphon has heat source term of enthalpy only for 1st nodeof the half control volume of the air (Fig. 6). The heat source termfor 1st air node with inlet condition is written as:

Qf ðfÞ ¼ m� cp � Ta at y ¼ Dy2

(86)

5.4. Formulation of U-matrix

The conductance entries in U-matrix are formulated automati-cally by writing diagonal and off-diagonal elements with propersign notation [7]. From Equations (79)e(81), Q's are separated andwritten as per expressions in Equations (85) and (86). The equa-tions are re-arranged to obtain diagonal and off-diagonal elementsof U-matrix. As per definition of constitutive relations described byEquations (82)e(84) and writing the conductances with propersign notation will result in off-diagonal elements. Diagonal ele-ments are obtained by adding all the conductances entering thenode with unknown temperature. As per the sign notation off-diagonal elements are negative in U-matrix and diagonal ele-ments in the U-matrix are positive. The off-diagonal and diagonalelements in the U-matrix for the node at the outer wall are writtenas:

Uðo; oþ 1Þ ¼ �Uo;oþ1 Uðo; o� 1Þ ¼ �Uo;o�1 Uðo; fÞ¼ �Uo;f Uðo; iÞ ¼ �Uo;i (87)

where, U(o,o þ 1) is the U-matrix entry in row o, column o þ 1;U(o,o� 1) is the U-matrix entry in row o, column o� 1; U(o,f) is theU-matrix entry in row o, column f; and U(o,i) is the U-matrix entryin row o, column i. Whereas diagonal element for the outer wall iswritten after summing up all the conductances for element (o,o) inU-matrix:

Uðo; oÞ ¼ Uo;o�1 þ Uo;oþ1 þ Uo;f þ Uo;i (88)

Similarly off-diagonal and diagonal elements for the inner wallare written in U-matrix as:

Uði; i ¼ 1Þ ¼ �Ui;iþ1 Uði; i� 1Þ ¼ �Ui;i�1 Uði; fÞ¼ �Ui;f Uði;oÞ ¼ �Ui;o (89)

Uði; iÞ ¼ Ui;i�1 þ Ui;iþ1 þ Ui;f þ Ui;o (90)

where, U(i,i þ 1) is the U-matrix entry in row i, column i þ 1;U(i,i� 1) is the U-matrix entry in row i, column i� 1; U(i,f) is the U-matrix entry in row i, column f; and U(i,o) is the U-matrix entry inrow i, column o.

General form of off-diagonal elements for the air nodes in U-matrix is written as:

Uðf ; f � 1Þ ¼ �Uf ;f�1 Uðf ; iÞ ¼ �Uf ;i Uðf ; oÞ ¼ �Uf ;o (91)

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227220

The diagonal elements are summed for conductances enteringthe thermal network air node (f,f) in U-matrix are written as:

Uðf ; fÞ ¼ Uf ;f�1 þ Uf ;i þ Uf ;o (92)

where, U(f,f � 1) is the U-matrix entry in row f, column f-1, but forthe air control volumes, U(f� 1, f)¼ 0. In the case of thermosyphoninternal flow, i.e. thermal capacity conductance for the f node iscalculated from its value at (f � 1) node. The thermal capacityconductance participates in the upstream direction of heat trans-port. In the downstream direction of the heat transport, thermalcapacity conductances are zero as observed from Equation (92) andU-matrix is asymmetrical. The systematic approach based on heatexchanger analogy for obtaining thermal capacity resistance forvarious flow conditions is described in Appendix A.

5.5. Grid size and accuracy

The discretisation or truncation error induced through finite-difference expressions is proportional to control volume size. Forsteady heat equation, it decreases to zero as control volume size DV(¼dDxDy) approaches infinitesimal small value, where d is depth ofcontrol volume in z-ordinate. As temperature functions for theouter wall, inner wall and air are lumped in x-ordinate, thereforeDx2 does not participate in the error. The error associated withEquation (73) or (74) is proportional toDy2. However for calculationof thermal capacity conductance terms (mcp), truncation errordepends on Dx (¼L, air-gap width in the thermosyphon). Thegreater the air gap width, the larger is the error induced, due toincrease in the control volume size. The number of nodal algebraicequations depends on the number of nodes N in the grid. Thegreater the number of algebraic equations, truncation error de-creases, but increases the time of computation. The other approachfor analysing the problem is to fix the grid size i.e. fix the number ofnodal equations but vary the length Dy by varying height (H) ofsystem boundary. The thermosyphon problem is analysed forN ¼ 30, with 10 control volumes for the outer wall, air flow andinner wall. The value for Dy ¼ 0.3 m is represented to assumeconstant temperature of nodes in the control volume with systemboundary height of 3.0 m. As direct explicit method of matrixinversion is applied, the numerical solutions are unconditionallystable.

Outer Wall Nodes

Air Nodes

Inner Wall Nodes

X-axis

sixa-Ysixa-Y

X-axis

Inner Wall Nodes

Outer Wall Nodes

Conduction resistance

Thermal capacity resistance

Convection resistance

Radiation resistance matrix (20X20)

Conduction resistance

Convection resistance

Fig. 7. Thermal network for nodes located on discretised control volumes.

5.6. Numerical solution procedure

Step 1. The input values given in Table 1 are used to initialise thenumerical solution.Step 2. The conductance values are calculated from the consti-tutive relations.Step 3. The corrected iterated value of the mass flow rate asdepicted in Table 1 is obtained from the numerical solution andis used for obtaining thermal capacity conductance values.Step 4. The heat transfer coefficients are calculated using tem-peratures obtained from the analytical solution.Step 5. The effect of integrated radiation heat exchange betweensurface nodes of outer and inner wall is considered withradiosity-irradiation method assuming enclosure analysis.Step 6. The radiation heat exchange factors are calculated foreach node using script factor matrix of size (20 � 20). Usingradiation heat exchange factors radiation conductance valuesare calculated, which also form matrix (20 � 20).Step 7. The values of convective heat transfer coefficients ob-tained from analytical solution are also used in the numericalsolution.

Step 8. Once conjugate heat exchange conductance values for 30nodes are calculated, U-matrix of size (30� 30) is formulated. U-matrix is formulated after obtaining off-diagonal and diagonalentries.Step 9. The inverse of U-matrix (30� 30) is multiplied with heatsource element matrix (1 � 30), to obtain temperatures at 30nodes (30 � 1).

5.7. Physical interpretation of thermal network

The thermal network for control volume representation as perschematic of Fig. 6 is illustrated in Fig. 7. The conjugate heat ex-changes of conduction, convection, radiation and thermal capacityresistances are used in the thermal network. The radiation resis-tance network is shown separately in Fig. 7. Though thermosyphonis a simple geometry case, but use of inhomogeneous materials andthe non-linear variation in boundary conditions makes difficultchoice for the solution method. Mass flow balance network is usedto determine the mass flow rate distributions. The number ofequations used for describing each mass flow network depends onthe number of inlets or infiltration sources. With buoyancy effect,the mass flow balance network of the system of equations is tem-perature dependent. Therefore some of the multi-functionalnetwork elements are temperature as well as mass flow ratedependent and are determined by the local temperature of the airin the discretised air control volume. Whereas the temperaturevariation of the air flowing through the thermosyphon is modelledby use of local temperature distribution elements at the heat ex-change surfaces at solid-air edges which are termed as surfacenodes. They are classified as modifying network elements, as theyare responsible for change in temperature of air [10].

The temperature variation of the air flowing through the ther-mosyphon heat transferring surface is described by the heat flowbalance network part of the model. This is performed by theapplication of mass flow dependent dual function elements of theheat flow balance network. In this case, the heat flow balance partof the model is a modifying network element of the model. Thefinal result of the thermal analysis is given by heat flow balancenetwork equations of the simultaneous thermo-hydro-dynamicalnetwork [10]. At quasi steady state with constant heat flux input,the temperature of air varies as a function of the space in y-ordi-nate. The temperature variation is determined by the local massflow rate and the local heat flux distribution towards the air flowingthrough the thermosyphon. The local temperature of the air andthe heat flux exert an effect on the local heat transfer conditions. Atthe same quasi steady-state, the temperature variations of the air

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227 221

also affect the values of the mass flow rate of the air. Therefore ifmass flow rates are validated beforehand, complete solution of thecoupled thermal airflow network is possible. Simultaneousnetwork model is build up from mass flow network and heat flownetwork models to take into account thermal and air flowinteractions.

The heat flow balance network is build by viewing the space asdivided into discrete parts, and each discretised space or cell isrepresented by the node temperature, unique for the surface or aircell in the control volume. The number of discretised cells or con-trol volumes is chosen to meet the required accuracy and easinessof the computation [6]. The representation of temperatures at eachdiscretised cell i.e. thermal performance is analysed by conjugateheat exchange conductance terms in the direction of the assumedentropy fluxes (in W/K units) e.g. convection heat conductanceassumed from air to surface (heating of surface or cooling of air) aswell as from surface to air (cooling of surface or heating of air) isable to generate bi-directional thermal resistances. The heat ex-change participation with convection, conduction and thermal ca-pacity resistances is with in the solid-air control volume. Forradiation heat exchange, resistances exchange heat throughnetwork of discrete paths summed for all the surface nodesexchanging radiation.

6. Results and comparisons

The explicit solutions are initialised by buoyancy-induced massflow rate, ambient air temperature, building space temperature andsolar radiation absorbed by the outer wall. The analytical and nu-merical solutions for temperature variations of outer wall, air andinner wall with the height of the system boundary are illustrated inFigs. 8(a), 9(a) and 10(a). The percentage deviations of analyticalsolutions from numerical solutions are also represented inFigs. 8(b), 9(b) and 10(b). The explicit solutions are obtained inaccordance with designed thermo-physical properties for a solarthermosyphon components are provided in Table 1.

6.1. Comparison between analytical and numerical solutions

6.1.1. Guiding principlesThe guiding principles are applied to represent the system

boundary of thermosyphon with overall balance between ease ofcomputation and physical realistic behaviour [6]. The discretisedcontrol volume distance Dy between the nodes is chosen to matchthe practical limitations that may involve in modelling tall buildingstructures. The physical realism is applied to obtain boundary

181920212223242526272829

0.15 0.45 0.75 1.05 1.35 1.65 1.95 2.25 2.55

Height (m)

Tem

pera

ture

( o C

)

Temperature of Outer Wall-Numerical SolutionTemperature of Outer Wall-Semi-analytical Solution

(a) Temperature Fig. 8. Comparison of temperature for outer wall ob

conditions compatible with the physical conditions [6]. The carte-sian coordinates are only used in the mathematical analysis torepresent solar thermosyphon integrated with a building. Thecomparison of the numerical solutionwith the analytical solution isperformed with the aim of determining the accuracy of the nu-merical solution and validating the assumptions used to build thephysical domain.

The analytical solutions obtained from Equations (66), (67) and(71) do not consider heat flow conductance in all directions to-wards x, y and z axes. The error induced by not considering con-duction heat flow along height of thermosyphon (y-ordinate) issignificant. Moreover analytical solution does not take into affect ofintegrated radiation heat exchange in between composite surfacenodes of thermosyphon. The error induced by neglecting inte-grated radiation heat exchange is small, but small increase ordecrease in surface temperature causes exponential increase ordecrease in temperature of air. Appendix B gives procedure forobtaining heat storage and thermal storage capacities of thermo-syphon and concluded that non-consideration of heat flow along y-ordinate imposes significant error in temperatures obtained fromanalytical solution.

6.1.2. Discretisation principlesThe discretisation concept [6] is applied to obtain control vol-

ume formulation. The grid spacing Dy both for analytical and nu-merical solutions is kept same for the comparison purposes. Systemof linear partial differential equations with non-linear boundaryconditions of combined convection and radiation is simplified bylumped temperatures in x and z-ordinates and then employingenergy balance on each node located in the control volumes. Thediscretisation process into control volumes is common to bothanalytical and numerical solutions. The heat storage capacity isneglected in both analytical and numerical solutions. Theassumption of lumped parameter applicability in x-direction im-poses a very small percentage of error in the numerical solution.Moreover, the error induced by neglecting heat storage capacity issmall because of control volume analysis. As the induced error dueto non consideration of heat capacity is dependent on grid size,therefore with increased number of nodes in the grid, the inducederror with not considering heat storage capacity is negligible. Asillustrated in Fig. 6, the xey grid is created by placing 30 nodes with10 nodes each for outer wall, air and inner wall of the duct withDy ¼ 0.30 m. In obtaining analytical solutions, Equation (71) isapplied to each air node, thus obtaining cell temperatures unique toeach air node placed in a cell or control volume. The air

Percentage Deviation of Semi-Analytical Solution from Numerical Solution-Outer Wall

0123456789

101112

0.15 0.45 0.75 1.05 1.35 1.65 1.95 2.25 2.55 2.85

Height (m)

Per

cent

age

Ove

r Pr

edic

tion

(%)

2.85

(b) Percentage over prediction tained from numerical and analytical solutions.

Percentage Deviation of Semi-Analytical Solution from Numerical Solution-Air

1520253035404550556065707580

0.15 0.45 0.75 1.05 1.35 1.65 1.95 2.25 2.55 2.85

Height (m)

Per

cent

age

Ove

r Pr

edic

tion

(%)

-4-202468

101214161820

0.15 0.45 0.75 1.05 1.35 1.65 1.95 2.25 2.55 2.85

Height (m)

Tem

pera

ture

( o C

)

Temperature of Air-Numerical SolutionTemperature of Air-Semi-analytical solution

(a) Temperature (b) Percentage over prediction

Fig. 9. Comparison of temperature for air obtained from numerical and analytical solutions.

68

1012141618202224

0.15 0.45 0.75 1.05 1.35 1.65 1.95 2.25 2.55 2.85

Height (m)

Tem

pera

ture

( o C

)

Temperature of Inner Wall-Numerical SolutionTemperature of Inner Wall-Semi-analytical solution

Percentage Deviation of Semi-Analytical Solution from Numerical Solution-Inner Wall

05

101520253035404550

0.15 0.45 0.75 1.05 1.35 1.65 1.95 2.25 2.55 2.85

Height (m)

Per

cent

age

Ove

r Pr

edic

tion

(%)

(a) Temperature (b) Percentage over prediction

Fig. 10. Comparison of temperature for inner wall obtained from numerical and analytical solutions.

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227222

temperature is thus used again in Equations (66) and (67) to obtaintemperatures at the surface nodes.

6.1.3. Entropy generationThe constitutive relations for obtaining conductance terms are

calculated over discretised control areas in yez plane for conduc-tion heat flow, radiation and convection heat exchange. Whilst,heat transport conductance terms are calculated from mass flowrate crossing the control volume in xez plane assuming no leakageor infiltration sources in the thermosyphon. The numerical solu-tions have considered the effect of thermal storage by incorporatingconduction heat flow factors in y-direction for outer and inner wallsof thermosyphon. The error induced by not considering heat stor-age capacity is negligible in comparison with the error induced bynot considering thermal storage in the walls of thermosyphon. Thealuminium alloy based outer wall has low specific heat in com-parisonwith air and inner wall with building insulation. This resultin small value of heat capacity for the metallic outer wall in com-parison with inner wall made of plywood board filled with poly-styrene. As discussed in Appendix B, plywood board filled withpolystyrene insulation has high specific heats. The polystyrene isalso considered as highly inflammable during any fire hazards. Thelow thermal conductivities of plywood board and polystyrene givepoor thermal storage for the inner wall.

The counter-effects of good heat storage and poor thermalstorage in inner wall whilst poor heat storage and good thermalstorage for outer wall is balanced by considering the conductionheat flow along y-direction for inner wall and outer wall of ther-mosyphon. This is because heat storage capacity signifies thevulnerability of material towards heat, which is not as importantas significance for thermal storage in performing thermal analysis.Therefore, it is definitely not essential to perform two-dimensionalsolution of steady heat conduction Equations (1) and (2) in com-parison with the necessity of performing transient analysis for thecase where inner wall is massive and has high heat storagecapacity.

The results of Figs. 8(b), 9(b) and 10(b) show that temperaturesolutions obtained from the analytical solutions are over-predictedby an average value of 7.5%, 25.5% and 34% for outer wall, inner walland air respectively in comparison with numerical solutions. Thevalues are higher for air temperature because of increasing errordue to: (i) finite difference upwind approximation in numericalsolutions; and (ii) counter effects of all the conjugate heat exchangeconductances and its exponential behaviour in analytical solutions.

The critical cases of thermal capacity resistance (mcp) forvarious flow configurations of thermosyphon are presented inAppendix A. The critical analysis for considering the effect of heatstorage capacities and thermal storage capacities of solar heated

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227 223

thermosyphon is described in Tables B1 and B2 respectively ofAppendix B.

6.1.4. Model validationThe proposed model is validated through outdoor experimental

setup in an outdoor room comprising of solar thermosyphon basedphotovoltaic solar wall. The comparison between experimentalresults and proposed model results is presented in Fig. 11.

7. Conclusions

The analytical solution is obtained by solution of the IBVP in thephysical domain. It is initialised by the assumed surface

14

18

22

26

30

34

38

42

46

50

14 18 22 26 30

Exp

erim

enta

l res

ults

( °C

)

Proposed mod

Solar ThermosypComparisons of PV module (out

(a). PV module (outer wall)

1215182124273033363942

14 17 20 23

Exp

erim

enta

l res

ults

( °C

)

Proposed

Solar ThermosyphoComparisons of inner wall t

(b) Inner wall

1416182022242628303234

14 16 18 20 22 24

Exp

erim

enta

l res

ults

( °C

)

Proposed m

Solar Thermosyphon:Comparisons of air tempera

(c) Air

Fig. 11. Comparison of Experimental Results with Proposed M

temperatures to obtain temperatures at the air nodes located on theedges of the control volumes. The air temperatures are then used toestablish surface node temperatures for the outer and inner wallsby exchanging data through heat balance model of the IBVP. Theheat balance model is obtained by applying lumped temperatureassumptions and initial conditions to the solid-air edges in thecontrol volumes. Control volumes formulated are combination ofsolid and air sub-volumes as illustrated in Fig. 6. The electricnetwork analogy is used to represent thermal networks of U-matrixconductances, which represent heat exchange among nodesthrough conduction, convection, radiation and enthalpy flow. En-ergy balance is applied on the nodes associated with air andadjoining surface nodes of thermosyphonwalls to formulate the U-

R² = 0.9878

34 38 42 46 50

el results ( °C)

hon:er wall) temperature

R² = 0.9906

26 29 32 35 38

model results ( °C)

n:emperature

R² = 0.9955

26 28 30 32 34 36 38

odel results ( °C)

ture

odel: (a) PV module (outer wall); (b) Inner wall; (c) Air.

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227224

matrix. The numerical solution of the thermal network is obtainedby the use of the asymmetrical conductance U-matrix. The diagonaland off-diagonal elements are automatically formulated by use ofthe sign notation for writing the nodal equation. The U-matrix isconstituted with diagonal and of-diagonal elements formulated byconductance, convective, radiative and thermal capacity U-matrixconductances.

The heat exchange paths for thermal networks are in twodimensions, allowing heat transfer between nodes of surface-eair/airesurface (convective heat exchange), surfaceesurface(conductive heat and radiative heat exchange), and aireair(enthalpy flow). Surfaceesurface nodes with radiative conduc-tances are constituted by assuming enclosure analysis betweenthe walls of a solar thermosyphon. The Hottel's crossed stringmethod is used to derive the view factor matrix describingradiative exchange pattern between surface nodes. Air to airthermal capacity conductances in between the air nodes areconstituted by upwind scheme [6], in which thermal capacityconductances in the upstream direction of heat flow are takeninto consideration. The enthalpy change of air is the dissipatedheat by convection heat exchange between surface and airnodes in the control volume. The heat transfer from the outerwall and integrated radiation heat exchange over all the surfacenodes is eventually convection heat loss taken up by air to in-crease its enthalpy content. As energy balance is applied on thenode at solid-air edge, conductivity or conductance terms getcancelled in the heat balance equation as observed in Equations(65) and (66).

The heat conductances which describe heat exchange in be-tween thermal nodes are represented by single conductance U-matrix (30 � 30). The temperature solution is obtained by multi-plying inverse of U-conductance matrix with the heat source ma-trix elements. Equation (83) is based on upwind finite-differenceapproximation to give exponential temperature variation for the airnodes. The two dimensional numerical solution is initialised byassumed surface temperatures and inlet air temperature. Theambient air temperature and quasi steady solar irradiation are usedfor obtaining heat source matrix. The steady buoyancy-inducedvelocity is used in calculating the mass flow rate of air. The appli-cable correlations for the Nusselt number for mixed convectionheat transfer situation as available from the literature are used inthe modelling work [11]. The ambient heat transfer coefficient isused in conjugation with below normal wind situation is todescribe heat exchange between ambient air and outer wall of asolar thermosyphon. The surface radiation properties are designedfor maximum heat transfer from adjoining surfaces to the airflowing through a solar thermosyphon.

The mathematical model is a useful simulation tool for pre-dicting thermal performance of thermosyphon based ventilationsystems [7]. The methodology is also useful for accurately pre-dicting thermal performance of heat exchangers, motor compo-nents with internal heat generation and other HVAC devices withknown mass flow rates. It is also useful for simulating solar heatedbuilding structures such as ventilated façades, airflow windows,and thermal modelling of renewable energy systems such asventilated ducts of building integrated photovoltaic modules andfuel cells [12,14]. The mathematical model was validated bycomparing the obtained numerical results with measurement datafor a configuration of a photovoltaic solar wall system connected toa building under different operating conditions [15]. The sampleexperimental datawas collected to validate and simulate the modelunder single set of realistic environmental conditions of an outdoorexperimental setup [16].

Appendix A

Thermal capacity resistance (1/mcp)

The thermal capacity resistance models the heat transportphenomenon associated with the air flowing through the ther-mosyphon. It is dual function element of the heat flow balancenetwork [10]. The two possible cases for representing thermal ca-pacity resistances are external and internal ventilation network ofresistances. The analogy of heat exchanger thermal analysis basedon heat transport expressions is used to represent network ofcontrol volumes with thermal capacity resistances [13]. Theexternal ventilation network of thermal capacity resistances isrepresented by double stream-series flow arrangement of networkof parallel flow heat exchangers. The internal ventilation can beeither open or closed loop. It is represented by a single streamnetwork of heat exchangers. Each discretised control volume or cellis represented by heat exchanger with hot and cold air streamsentering together or leaving each other depending on parallel orcounter flow arrangement. Fig. A1 has illustrated three possiblecases of series arrangement of control volumes with counter flow,internal flow and parallel flow conditions.

A1.1. Thermal analysis of air control volumeThe heat exchanger transport equation is written in partial dif-

ferential form as:

v

vAsq� uexðTh � TcÞ ¼ 0 (93)

where vAs ¼ p.dy for one-dimensional arrangement involvingparallel flowand counter flow, and dAs¼ dx.dy for two-dimensionalarrangement involving cross flow. Analogy of Equation (94) appliedto air control volumes. uex is overall heat loss coefficient betweenhot and cold streams. Cold air stream from previous control volumeis coming into hot air stream control volume. Control volumes of airexchange heat at the cell faces. Assuming constant mass transfer atcell face, the uex is non-diffusive i.e., conductionediffusion equationdoes not participate in the energy balance and uex is reduced tothermal capacity conductance (mcp) for the air control volume.

A1.2. One-dimensional arrangement of series network of air controlvolumes

The energy balance is applied on differential control volumesdVh ¼ Ah.dy and dVc ¼ Ac.dy and equation for hot and cold streamstakes the form:

v

vHhqþ 1 ¼ 0 (94)

v

vHcq� 1 ¼ 0 (95)

where, vHh and vHc are changes in the bulk-stream enthalpy ratesrelative to the direction of the flow of the individual air streams.Heat transfer rate vq from hot to cold air results in decrease in Hh

and increase in Hc in stream wise directions and downstream di-rections respectively. The bulk-stream enthalpy rates and bulk-stream temperatures are related as:

v

vThHh � �m cp

h ¼ 0 (96)

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227 225

v

vTcHc �

�m cp

c ¼ 0 (97)

where, mh andmc are mass flow rates in the streamwise directions.Using Equations (96) and (97) to eliminate vHh and vHc with vThand vTc in Equations (94) and (95), the bulk stream energy equa-tions reduces to:

v

vThqþ �m cp

h ¼ 0 (98)

v

vTcq� �m cp

c ¼ 0 (99)

Because of vTh and vTc (or vHh and vHc) represent the changes inbulk-stream temperature or enthalpy rates relative to the directionof flow of the individual streams. The arrangement of air controlvolumeswith hot bulk stream in one direction and cold bulk streamin downstream direction, represent internal ventilation network ofthermal capacity resistances both in upward and downward di-rections, inwhich net energy exchange between streams is zero e.g.internal circulation of air in cavities of walls.

Fig. A1. Series arrangement of network of control volumes-(a) counter flow mixedstreams with internal ventilation-closed loop; (b) internal flow-open loop; (c) parallelflow mixed streams with external ventilation-infiltration.

A1.3. Two-dimensional arrangement of series network of air controlvolumes

Because of cross-flow arrangement, the differential air controlvolumes used in thermal analysis involve two differential lengthsdx and dy. The mass flow rates entering the control volumes aregiven by:

v

vymh ¼ ðr� v� dÞh ¼ mh

W(100)

v

vxmc ¼ ðr� v� dÞc ¼

mc

L(101)

where, vh and vc are velocities of the air in hot and cold streams ofunit distance dh and dc respectively in y and x-ordinates. Mass flowrates are written as:

mh ¼ ðr� v� dÞh �W (102)

mc ¼ ðr� v� dÞc � L (103)

Control volumes are written as:

dVh ¼ ðdhdzÞdx dVc ¼ ðdc dxÞdy (104)

Energy balance equations are obtained as:

v

vHhqþ 1 ¼ 0 (105)

v

vHcq� 1 ¼ 0 (106)

It is similar to the form of Equations (95) and (96) but is 2-Dwith x and y-ordinates. Where, vHh and vHc are changes in thebulk-stream enthalpy rates relative to the direction of flow of theindividual air streams. Equations are applied to the control vol-ume with mixedemixed arrangement with common passage forthe moving air stream. The change in enthalpy rates in twodimensional mixedemixed arrangement with single passage formoving air stream with both stream wise and lateral spatialchanges is:

v

vThHh � �m cp

h ¼ 0 (107)

v

vTcHc ¼

�m cp

c ¼ 0 (108)

By substituting Equations (107) and (108) in Equations (105) and(106) gives bulk-stream energy equations for cross-flow arrange-ments of the form:

v

vThqþ �m cp

h ¼ 0 (109)

v

vTcq� �m cp

c ¼ 0 (110)

For the two-dimensional solution, Equations (109) and (110) areapplied at each air node in the control volume in stream-wise di-rection with mass flow rate calculated using cross-stream wisedirection.

A1.4. Entries in U-matrixIn a closed loop ventilation network of thermal capacity re-

sistances, the heat transport is taken up by the air in the closedsystem e.g. air cavities in walls and windows. A precondition forapplying thermal capacity resistances in the closed system is that,when writing the nodal equation, the resistances in the upstreamand downstream direction of heat transport are of opposite sign ofthe assumed direction of heat transport. As a result there is nomasstransfer from the system and symmetrical U-matrix does not havethermal capacity resistances. The schematic of closed loop venti-lation is illustrated in Fig. A1(a).

The schematic for open loop ventilation is illustrated inFig. A1(b). The flow enters from one end and leaves the system fromother end e.g. thermosyphon, collectors for mass and heat trans-port. In this case thermal capacity resistance only in the upstreamdirection of heat transport is considered i.e. conductance matrix isasymmetrical. The interpretation of the above equation in networkmodelling is that the conductance is positive in the upstream di-rection of heat transport and 0 in the downstream direction of heattransport. The fluid going through the control volume is notaffected by the downstream control volume.

As illustrated in Fig. A1(c), external ventilation network issimilar to an infiltration or an induced flow in both directions. In

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227226

this case, precondition for forming thermal capacity resistances isthat, whenwriting the nodal equation, thermal capacity resistancesin both the upward and downward directions of heat transport aretaken into consideration. The conductance U-matrix is symmetricalin this case.

Appendix B

Heat storage capacity

The procedure for calculating heat storage capacities for tem-perature differences in y and x-directions of solar heated thermo-syphon is presented in Table B1. The thermo-physical properties ofair and outer wall were assumed constant along all directions i.e. x,y, and z-ordinates and were evaluated from any standard heattransfer text book. The temperature differences along y-ordinateare obtained from the numerical solutions. The temperature dif-ferences along x-direction are obtained by assuming same tem-perature difference per unit thickness of material along x and y-ordinates. The total heat storage capacity in vertical direction (y-ordinate) due to absorbed solar radiation on outer wall of ther-mosyphon with building insulation is 1.99 MJ. The heat storagecapacity for temperature differences across x-direction is 1.25% ofthe heat storage capacity for temperature differences across y-di-rection. Therefore temperatures are assumed uniform and lumpedin x-direction.

Thermal storage capacity

The procedure for obtaining thermal storage capacities of solarheated thermosyphon is described in Table B2. The thermal storagecapacities are obtained from individual thermal conductivities ofouter wall, air and inner wall. The time constant (T ¼ rdCpddd/hd)for each component of thermosyphon is calculated from individualheat capacities and assumed heat transfer film coefficients alongheight and thicknesses for various components of solar thermo-syphon. The sum of thermal storage capacities of various compo-nents of thermosyphon along y-ordinate is 1.46 MJ. The thermalstorage capacity of thermosyphon along x-ordinate is nil (0.0003%)in comparison with its thermal storage capacity in y-direction.Therefore it is necessary to consider conduction heat flow analysisin the y-direction.

Table B1Heat storage capacity.

Component rd kg m�3 Cpd J Kg�1 K�1 dd m H m W

Aluminium 2659 867 0.002 3.0 1.0Air 1.1174 1000 0.05 3.0 1.0Plywood 550 1750 0.007 3.0 1.0Polystyrene 1050 1200 0.026 3.0 1.0Plywood 550 1750 0.007 3.0 1.0Total e e e

Table B2Thermal storage capacity.

Component kd W m�1 K�1 rdCpddd J m�2K�1 hd W m�2 K�1

Aluminium 137 4611 10.0Air 0.02624 55.6 10.0Plywood 0.0835 6737.5 10.0Polystyrene 0.02821 32,760 1.0Plywood 0.0835 6737.5 10.0Total e e

Nomenclature

GeneralDy Distance between discretised surface nodes in y direction

(m)Dx Air-gap width of the thermosyphon, L (m)DAy Discretised element with area {Dy � W} (m2)A Face area of outer wall (m2)ai Face area of ith surface element (m)cp Specific heat of air at constant pressure (J kg�1 K�1)Fo,I View factor between oth and ith discretised finite

element surfaceG Acceleration due to gravity (m s�2)H Convective heat transfer coefficient (W m�2 K�1)H Height of thermosyphon (m)hi Convective heat transfer coefficient for inner wall

(W m�2 K�1)ho Convective heat transfer coefficient for outer wall

(W m�2 K�1)hr Linearised radiation heat exchange factor (W m�2 K�1)Si Irradiation on the ith surface element (W m�2)Jo Radiosity at the oth discretised surface (W m�2)k Variation of y-ordinate from 0 to (n-1)ki Thermal conductivity of the inner wall (W m�1 K�1)ko Thermal conductivity of the outer wall (W m�1 K�1)L Air gap width between outer and inner walls of the

thermosyphon (m)m Mass flow rate through the thermosyphon (kg s�1)mcp Thermal capacity resistance (W K�1)mk mass of molecule in solid-air control volume (kg)n Number of discretised elements in y-ordinateN Total number of nodes in the gridQ1,30 Heat source matrix for discretised nodes (1 � 30)qa Absorbed short wave solar radiation on the surface

(W m�2)qo Net heat flux at the surface (W m�2)S Solar irradiation on thermosyphon (W m�2)T30,1 Unknown temperature matrix at discretised nodes

(30 � 1)Tf Temperature variable for the air (�C)Ti Temperature variable for the inner wall (�C)ti Thickness of inner wall (m)Tmso,i Mean-surface temperature between surface elements (K)

m rdCpddd J m�2K�1 DTV K DTH K IV kJ IH kJ

4611 7.5 0.005 103.75 0.0755.6 17.5 0.3 2.9 0.05

6737.5 13.6 0.18 274.90 3.632,760 13.6 0.18 1336.6 17.76737.5 13.6 0.18 274.90 3.6e 1993.05 25.02

T sec DTV K DTH K QV kJ QH J

461.1 7.5 0.005 1421.34 0.6315.56 17.5 0.3 0.007 0.002

674 13.6 0.18 2.3 0.02232,760 13.6 0.18 37.70 4.33674 13.6 0.18 2.3 0.022

1463.647 5.007

H. Dehra / International Journal of Thermal Sciences 102 (2016) 210e227 227

To Temperature variable for the outer wall (�C)to Thickness of outer wall (m)Ts Building space temperature (�C)U Conductance at nodal element (W K�1)uex Overall heat loss coefficient between hot and cold streams

(W m�2 K�1)U(m,n) Entry in conductance matrix in rowm, column n (W K�1)U30,30 Conductance matrix for discretised nodes (30 � 30)Um,n Conductance between nodes m and n (direction m to n)

(W K�1)v Air velocity through the thermosyphon (m s�1)vk Velocity of the molecules in the solid or air (m s�1)dVo Solid-air control volume with the outer walldVi Air-solid control volume with the inner wall

Greek lettersa Short-wave absorption of the solar irradiation on the

outer wallai Absorbtance at ith surfacedj,m Kronecker's delta functionε Integrated emissivity of the absorbed heat flux on the

surfaceεi Emissivity of ith discretised surface element nodeεo Emissivity of oth discretised surface element nodezo,i Script factor matrix (20 � 20) in radiosity-irradiation

formulationxo,i Inverse of co,i matrix (20 � 20)ra Ambient air density (kg m�3)ri Reflectivity of back surface of the inner walls StefaneBoltzmann constant (5.67 � 10�8 W m�2 k�4)co,i Matrix defined by view factor, reflectivity and Kronecker's

delta function

Subscriptsa Ambient airf Airi Inner wall

m Position of row in U-matrixn Position of column in U-matrixo Outer wallp Specific heat of air at constant pressures Building space

References

[1] W. Smolec, A. Thomas, Some aspects of Trombe Wall heat transfer models,Energy Convers. Manag. 32 (1991) 269e277.

[2] G.L. Morrison, J.E. Braun, System modeling and operation characteristics ofthermosyphon solar system, Sol. Energy 30 (1985) 341e350.

[3] Ji Jie, Yi Hua, He Wei, Pei Gang, Lu Jianping, Jiang Bin, Modeling of a novelTrombe wall with PV cells, Build. Environ. 42 (2007) 1544e1552.

[4] K.T. Anderson, Theoretical considerations on natural ventilation by thermalbuoyancy, ASHRAE Trans. 101 (2) (1995) 1103e1117.

[5] David S. Watkins, Fundamentals of Matrix Computations, Wiley-Interscience,NY, USA, 2002.

[6] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Wash-ington D.C., USA, 1980.

[7] H. Dehra, Mathematical analysis of a solar thermosyphon, in: InternationalConference on Advances in Energy Research, Department of Energy Scienceand Engineering, Indian Institute of Technology, Bombay, 12-14 December2007, ISBN 023063432X, pp. 218e223. Macmillan, USA.

[8] M.F. Modest, Radiative Heat Transfer, McGraw Hill Inc, NY, USA, 1993.[9] ASHRAE, ASHRAE Handbook-fundamentals, American Society of Heating,

Refrigerating, and Air-Conditioning Engineers, Inc., Atlanta, USA, 1997.[10] L. Imre, L.I. Kiss, Analysis of the transient thermal performances of composite

devices by network methods, in: R.H. Lewis, K. Morgan, B.A. Schrefler (Eds.),Numerical Methods in Heat Transfer, vol. II, JohnWiley & Sons, NY, USA, 1983,pp. 395e445.

[11] F.P. Incropera, D.P. Dewitt, Introduction to Heat Transfer, John Wiley & Sons,NY, USA, 1990.

[12] R. Kothari, D. Buddhi, R.L. Sawhney, Studies on the effect of temperature of theelectrolytes on the rate of production of hydrogen, Int. J. Hydrogen Energy 30(3) (2005) 261e263.

[13] Lindon C. Thomas, Heat Transfer-professional Version, Capstone PublishingCorporation, TN, USA, 1999.

[14] H. Dehra, The entropy matrix generated exergy model for a photovoltaic heatexchanger under critical operating conditions, Int. J. Exergy 5 (2) (2008)132e149.

[15] H. Dehra, Solar energy absorbers, in: Reccab Manyala (Ed.), Solar Collectorsand Panels, Theory and Applications, Sciyo Publication, 2010, pp. 111e134.

[16] H. Dehra, A two dimensional thermal network model for a photovoltaic solarwall, Sol. Energy 83 (11) (2009) 1933e1942.