Chap4 STS Till 5Jan12

7/29/2019 Chap4 STS Till 5Jan12

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Department of Mechanical EngineeringHITEC University Taxila

1

Thermal Analysis of Flat Plate Solar Collectors

Basic Flat Plate Energy Equation

When a certain amount of solar radiation falls on the surface of a collector, most of it is

absorbed and delivered to the transport fluid, and it is carried away as useful energy

As in all thermal systems, Heat Losses to the environment by various modes of heat

transfer are inevitable

Glazing

Casing

Back and side

Insulation

Absorber Plate with

Selective Coating

Heat Loss to Ambient

Reflection

Fluid Carrying

passage

Incident radiation

(visible range)

Emitted radiation by absorber(Infrared range)


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Assumptions to simplify the Numerical Modeling and Calculations

Collector is in a steady state

Collector is of the header and riser type

fixed on a sheet with parallel tubes

Headers cover only a small area of thecollector and can be neglected

Heaters provide uniform flow to the riser

tubes

Flow through the back insulation is one

dimensional

Temperature gradients around tubes are neglected

Properties of materials are independent of temperature

Heat flow through the cover is one dimensional


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Assumptions to simplify the Numerical Modeling and Calculations

Temperature drop through the cover is

negligible

Covers are opaque to infrared radiation

Same ambient temperature exists at thefront and back of the collector

Dust effects on the cover are negligible

There is no shading of the absorber plate


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Energy Absorbed By The Collectorper unit area of absorberS is equal to the differencebetween the incident solar radiation and the optical losses, as defined by the following Eq.

= + + +

4.2

Thermal performance of a collector can be calculated from a First-law Energy Balance.according to the first law of thermodynamics, for a simple flat-plate collector an

instantaneous steady-state energy balance is

Useful energy = Energy Absorbed Heat Loss To

Gain (Qu) By The Collector Surroundings4.1



Thermal Energy Lost To Surroundings by

conduction, convection, and infrared radiation

can be represented as the product of a heattransfer coefficient UL times the differencebetween mean absorber plate temperatureTpmand the ambient TemperatureTa

QL = UL (Tpm - Ta) 4.3


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In steady state the useful energy output Qu of a collector of area Ac is the differencebetween absorbed solar radiation and thermal loss: combining Eqs. (4.1, 4.2 and 4.3):

Qu = Ac [S - UL (Tpm - Ta)] 4.4

Problem with Eq. (4.4) is that the mean absorber temperatureTpm is difficult to calculate ormeasure since it is a function of the collector design, incident solar radiation, and the

entering fluid conditions. The Eq. will be reformulated so that useful energy gain can beexpressed in terms of fluid temperatureTf.

Collector efficiency is defined as the ratio of the useful gain over some specified time

period to the incident solar energy over the same time period

4.5


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Collector Energy Losses (U-value Estimation)

As in all thermal systems, heat losses to the environment by various modes of heat

transfer are inevitable

Thermal network is drawn for a

single-cover, flat-plate collector

in terms of conduction,

convection, and radiation and in

terms of the Resistance

Between Plates

Temperature of the plate is Tp,the collector back temperature is

Tb, and the absorbed solarradiation is S

Fig. 4.1

Various thermal losses from the

collector can be combined into a

simple resistance, RLso that theenergy losses from the collector

4.6


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4.6

UL= Overall Heat Loss Coefficientbased on collector areaAc(W/m2-K)

Tp = Plate Temperature (C)

UL is a complicated function ofthe collector construction and

its operating conditions, given

by:

4.7

Ut = Top loss coefficient (W/m2-K)Ub= Bottom heat loss coefficient

(W/m2-K)Ue= Heat loss coefficient form the

collector edges (W/m2-K)

Fig. 4.1


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Top Heat Loss/Top loss coefficient, Ut (W/m2

-K)

Fig. 4.1

Heat transfer upward from the absorber plate

at temperatureTpto the glass cover atTgandfrom the glass cover at Tg to ambient at Ta isby convection and infrared radiation

Heat loss from absorber plate to glass is givenby

Ac = collector area (m2)

hc, p-g = convection heat transfer coefficient between

the absorber plate and glass cover (W/m2-K)

hr, p-g = radiation heat transfer coefficient between

the absorber plate and glass cover (W/m2-K)

Qt,p-g

qt,p-g 4.8


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Fig. 4.1

=

+

.

(.).

+

+ ( ) +

For tilt angles () up to 75, the convective

heat transfer coefficient, hc,p-g is calculatedfrom Nusselt No. correlation, which is given

by:

4.9

Plus sign represents positive values only

Rayleigh value, Ra, is

4.10

g = gravitational constant, = 9.81 m2/s= volumetric coefficient of expansion; for ideal gas,

= 1/TPr = Prandtl numberL =absorber to glass cover distance (m)= kinetic viscosity (m2/s)


-K)


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Fluid properties in Eq. (4.10) are evaluated at

the mean gap temperature (Tp + Tg)/2

4.10

Radiation Heat Transfer Coefficient term in Eq.

(4.8) can be linearized to give:

Fig. 4.1

4.11

Eq. 4.11 is for radiation heat exchange between two

flat and equal surfaces which is derived from the

general case of radiation heat exchange (Eq.4.12)b/w any two surfaces:

4.12

p= Infrared emissivity of absorber plateg= Infrared emissivity of glass cover



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heat loss from glass cover to ambient is given

by:


Fig. 4.1

4.14

Qt,g-a

4.15qt,g-a

hc,g-a= convection heat transfer coefficient

between the the glass cover and

ambient (W/m2-K)

hr, g-a = radiation heat transfer coefficient

between the glass cover and ambient

(W/m2-K)



-K)


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hc,g-a = hw= Convective heat transfer coefficientfor wind blowing over the

collector.(W/m2.oC) = 5.7+3.8Vair


Fig. 4.1


Vair = Average wind speed (m/sec)

4.17

4.16


-K)

aggp

t

RRU

11

4.18

4.19


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In some cases, collectors are constructed

with two glass covers in an attempt to lower

heat losses




-K)

Another resistance Rg1-g2 will be added to thesystem shown to account for the heat transfer

from the lower to upper glass covers

By following a similar analysis, the heat

transfer from the lower glass at Tg2 to theupper glass atTg1is given by

Qt,g1-g2qt, g1-g2 4.20

Fig. 4.1


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Fig. 4.1



-K)

The convection heat transfer coefficient hc,g2-g1can be obtained by Eqs. 4.9 and 4.10

Radiation heat transfer coefficient can be

obtained again from Eq. (4.11) and is given

by

4.21

4.22

Procedure of solving for the Ut using Eqs.(4.8) through (4.20) is an iterative process

A guess is made for the unknown cover

temperature Tg


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Fig. 4.1



-K)A guess is made for the unknown cover

temperature Tg, from which the convectiveand radiative heat transfer coefficients b/w

parallel surfaces are calculated

Since the energy exchange between pates

must be equal to the overall heat loss, a new

set of cover temperatures can be calculated:

jirhjich

aTpTtU

iT

jT

,,

)(4.23

i and j refer to two adjacent flat

surfaces. E.g. absorber plate and glass

cover or glass cover-1 and 2. Iterative Process is repeated until the cover

temperatures do not change significantly

b/w successive iterations


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-K)Example 4.1:

Calculate the top loss coefficient for an absorber with a single glass cover having

following specifications:

o Plate to cover spacing: 25 mm

o Plate Emittance: 0.95

o Ambient Air Temperature = 10 oC

o Wind Speed = 3 m/sec

o Collector tilt = 45o

o Glass Emittance = 0.88

= + . (.).

+

+ ( ) +

jirhjich

aTpTtU

iT

jT

,,

)(


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-K)

As the iterations required are tedious and time consuming, especially for the case of

multiple-cover systems, straightforward evaluation of Ut is given by the followingempirical equation developed by Klein (1975) with sufficient accuracy for design purposes

4.23

Where;


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-K)Example 4.2:

Repeat Example 4.1 using the empirical Eq. 4.23 and compare the results.

Example 4.3:

Estimate the top heat loss coefficient of a collector that has the following specifications:

Collector slope = 35, Number of glass covers = 2,

Thickness of each glass cover = 4 mm, Thickness of absorbing plate = 0.5 mm, Space

between glass covers = 20 mm, Space between inner glass cover and absorber = 40 mm

Thickness of back insulation = 50 mm,

Back insulation thermal conductivity = 0.05 W/m-K.

Mean absorber temperature,Tp = 80C = 353 K, Ambient air temperature = 15C = 288 K.Absorber plate emissivity, p = 0.10, Glass emissivity, g = 0.88.Wind velocity = 2.5 m/s.


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Fig. 4.1



Bottom Heat Loss Coefficient, Ut (W/m2

-K)

Energy loss from the bottom of the collector

is first conducted through the insulation and

then by a combined convection and infrared

radiation transfer to the surrounding

ambient air

1

Magnitudes ofRp-b and Rb-a are such that it isusually possible to assume Rb-a is zero andall resistance to heat flow is due to the

insulationBack loss coefficient is:

tb = thickness of back insulation (m)kb = conductivity of back insulation (W/m-K)

heat loss from the back of the plate rarely

exceeds 10% of the upward loss

Typical values of the back surface heat loss

coefficient are 0.30.6 W/m2-K

4.24


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Edge Heat Loss Coefficient, Ue (W/m2

-K)Heat transfer coefficient for the heat loss from the collector edges can be obtained from

4.25

Where,te= thickness of edge insulation (m)ke= conductivity of edge insulation (W/m-K)hc,e-a= convection heat loss coefficient from edge to ambient (W/m

2-K)Typical values of the edge heat loss coefficient are 1.52.0 (W/m2-K)

Evaluation of edge losses is complicated, therefore in a well-designed system, the edge

loss should be small so that it is not necessary to predict it with great accuracy

Losses through the edge should be referenced to the collector area, If the edge loss

coefficient area product is (UA)edge then edge loss coefficient

c

edge

eA

UAU

)( 4.25A


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Overall Heat Loss Coefficient, Ue (W/m2

-K)Collectors overall heat loss coefficient is:

U= Ut +Ub + Ue 4.25B


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Fig. 4.3


Temperature Distribution in Flat-Plate

Collectors and Collector Efficiency Factor

Energy transferred to fluid will heat the fluid, causing a temperature gradient to exist in the

direction of flow


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Temperature difference b/w tubes can be derived if we assume the temperature gradient in

the flow direction is negligible




Analysis can be performed by considering the sheet-tube configuration, where the

distance between the tubes is W, the tube diameter is D, and the sheet thickness is tab

Region between the center line separating the tubes and the tube base can be considered

as a classical Fin Problem

An elemental region of width, dx, and length L in the flow direction are shown

Length of the fin is (W-D)/2

Fig. 4.4

dx

tabtab


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x

(W-D)/2

Tb

tab

qx qx+dx

dqc

dxFig. 4.5




4.26

By solving energy balance on this element

Eq. (4.26) gives the

temperature distribution in thex direction at any given y

Energy conducted to the region

of tube per unit length in the flow

direction is:

4.27

or with the help ofFin Efficiency, F

4.28

F = Standard Fin Efficiency for straight fins with arectangular profile


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tab

Useful gain of the collector

also includes the energy

collected above the tube

region:

This energy ultimately must be transferred to the fluid, which can be expressed in terms of

two resistances as

Cb

is the bond conductance, which can be estimated from

knowledge of the bond thermal conductivity, kb, theaverage bond thickness, , and the bond width, b

useful energy gain per unit length

in the direction of the fluid flow is:

4.29

4.30

4.31

hfi= heat transfer coefficient between the fluid and the tube wall


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oF represents the ratio of the actual useful energy gain to the useful energy gain that

would result if the collector absorbing surface had been at the local fluid temperatureoFrepresents the effect of the temperature drop between the absorber plateTp and the

fluid in the pipeTf

oThe numerator of Eq. (4.32) is the heat transfer resistance from absorber plate to

ambient and denominator is the heat transfer resistance from the fluid to the ambient

air.

Solving Eq. (4.31) forTb, substituting it into Eq. (4.30), and solving the resultant equationfor the useful Heat Gain of the Collectorin terms of local fluid temperature is:

4.33

where Fis the Collector Efficiency Factor, given by

4.32


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oFconsiders the heat transfer resistance from absorber to fluid in the pipe

due to fin conduction,

due to the conduction through the contact bond between absorber and pipe, and

due to the forced convection between the pipe inner wall and the flowing fluid




oFdecreases with increased tube center-to-center distances and increases with increasein both material thicknesses and thermal conductivity

4.34


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Example 4.4:

For a collector having the following characteristics and ignoring the bond resistance,

calculate the fin efficiency and the collector efficiency factor:

Overall loss coefficient = 6.9 W/m

2

-CTube spacing = 120 mm

Tube outside diameter = 15 mm

Tube inside diameter = 13.5 mm

Plate thickness = 0.4 mm

Plate material = copper

Heat transfer coefficient inside the tubes = 320 W/m2-C


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Temperature Distribution in Flow Direction

Consider an infinitesimal length y of the tube as shown in Figure. Useful energydelivered to the fluid is quy

Fluid enters the collector at temperatureTfi and increases in temperature until at the exit itisTfo

By solving energy balance on this element

Fig. 4.6

4.33B

p

L

Lafi

Laf

CmynWFU

USTTUSTT

.

'

exp// 4.33A

h l l f l l l ll


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Collector Heat Removal Factor, Flow Factor , and Thermal

Efficiency

Heat Removal Factor FR represents the ratio of the actual useful energy gain that wouldresult if the collector-absorbing surface had been at the inlet fluid temperature,Tfi

It is usually desirable to express the collector total useful energy gain Qu in terms of thefluid inlet temperature,Tfi

4.34

4.34

Rearranging yields:

4.35

h l l i f l l S l C ll


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Collector Heat Removal Factor, Flow Factor, and Thermal

Efficiency

Another parameter usually used in the analysis of collectors is the Flow FactorF,whichis defined as the ratio ofFR to F

4.36

FR is equivalent to the Effectiveness of a Conventional Heat Exchangerwhich is defined asthe ratio of the actual heat transfer to the maximum possible heat transfer

Maximum possible useful energy gain in a solar collector occurs when the whole collector

is at the inlet fluid temperature,Tfi; heat losses to the ambient are then at a minimum

4.35

collector flow factor F is a function of only a single variable, the dimensionlesscollector capacitance rate, mcp /AcULF

Th l A l i f Fl Pl S l C ll


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FR times this maximum possible useful energy is the actual useful energy:

Qu = Ac FR[S - UL (Tfi - Ta)]

oThis is same as Eq. (4.4), with the difference that the inlet fluid temperature (Tfi)replaces the average plate temperature (Tp) with the use of the FR

4.37

oEq. (4.37), is a convenient representation when analyzing the solar energy systems,

since the inlet fluid temperature (Tfi) is usually known

Qu = Ac [S - UL (Tpm - Ta)] 4.4


Collector Heat Removal Factor, Flow Factor, and Thermal

Efficiency

Th l A l i f Fl t Pl t S l C ll t


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Calculate the useful gain and efficiency of an

array of 10 solar collector modules installed

in parallel, at a slope of 60o and a surface

azimuth of 0o. The hourly radiation on the

plane of the plane of the collector IT, thehourly radiation absorbed by the absorber

plate S, and the hourly ambient temperatureTa, are given in the table. For the collectorassume UL to be 8.0 W/m

2 oC, and the plate

efficiency factorF to be 0.841. The water flow

rate through each 1 2-m collector panel is0.03 kg/s and the inlet water temperature

remains constant at 40 oC. Assume a

controller turns off the water flow whenever

the outlet temperature is less than the inlet

temperature.


Collector Heat Removal Factor, Flow Factor, and Thermal Efficiency

Example 4.5:

Qu = Ac FR[S - UL (Tfi - Ta)]



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Example 4.5:--contd--



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To evaluate Collector performance, It is necessary to know ULand hfi

Mean Fluid and Plate Temperature

Both UL and hfi are to some degree functions of temperatures

Mean Fluid Temperature can be found by integrating Eq. 4.33A from zero to L

4.38

Solving this integration and substituting FR from Eq. (4.35),Tfm is:

4.39

Solving Eqs. (4.37) and (4.4), for the mean Plate Temp. (Tpm):

Qu = Ac FR[S - UL (Tfi - Ta)] Qu = Ac [S - UL (Tpm - Ta)]

4.40



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Liquid Heater Plate Geometries

Here only one basic collector design Sheet and tube solar water heater with parallel tubes

on the back of the plate is analyzed

There are many designs ofFlat-plate Collectors

Fortunately, it is not necessary to develop a completely new analysis for each situation

Generalized relationships for the tube and sheet case apply to most collector designs

It is necessary to derive the appropriate form of the Collector Efficiency FactorFand Eqs.(4.35 - 4.37) then can be used to predict the thermal performance

Fig. 4.7a

Under some circumstances, ULwillhave to be modified slightly



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Liquid Heater Plate Geometries

Fig. 4.7b Fig. 4.7c

Fig. 4.7d

Fig. 4.7e

In Figs. 4.7d, e, narrow, flat absorbers

are mounted inside evacuated glass

tubes

Configuration of Fig. 4.7d is similar to

type a but with a single riser Type e collecor is down and back

with a U-tube joining the two conduits



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D t t f M h i l E i i 39


Collector Heat Removal Factor, Flow Factor , and Thermal

Efficiency

Radiation level where the absorbed solar radiation and loss term are equal

Critical Radiation Level

This is obtained by setting the term in the right-hand side of Eq. (4.37) equal to 0

Qu = Ac FR[GT() - UL (Tfi - Ta)] 4.37

Critical Radiation Level, Gtc, is given by

4.38

Collector can provide useful output only when the

available radiation Gavis higher than the criticalone Gtc

Qu = Ac FR ()av(GT GTc)+ 4.39

Collector output can be written in terms of the critical radiation level:

o Eq. 4.39 indicate that for the collector to produce useful out put, i.e, Qu> 0, absorbedradiation must exceed the thermal losses and GT must be greater than GTC

oThis implies that there is a Controlleron the collector that shutts off the flow of fluid

when the value in parentheses is not positive

Chap4 STS Till 5Jan12

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