c WW

\

*'"»eAu o*

Q NBS TECHNICAL NOTE 999

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

A Study of the Dynamic

Flue-Gas Temperature and

Period Mass Flow Rate of a

Residential Gas-Fired Furnace

jyC

100

J5753

^0.999

1979

:.2
NATIONAL BUREAU OF STANDARDS

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Bureau's overall goal is to strengthen and advance the Nation's science and technology and

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and the Institute for Computer Sciences and Technology.

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1

The National Bureau of Standards was reorganized, effective April 9, 1978.
JUL 1 3 B79

A Study of the Dynamic Flue-Gas 7J6^^5

Temperature and Off-Period Mass Flow Rate

of a Residential Gas-Fired Furnace

Cheol Park

William J. Mulroy

George E. Kelly

Center for Building Technology

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

Sponsored by

Department of Energy

20th & Massachusetts Ave., NW

Washington, DC 20545

^<

CO o

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued July 1979
National Bureau of Standards Technical Note 999

Nat. Bur. Stand. (U.S.), Tech. Note 999, 41 pages (July 1979) CODEN:NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1979

For sale by the Superintendent of Documents, U.S. Govemnient Printing Office, Washington, D.C. 20402

Stock No. 003-003-02092-3 Price $2

(Add 25 percent additional for other than U.S. mailing).

J
A STUDY OF THE DYNAMIC FLUE-GAS TEMPERATURE AND OFF-PERIOD

MASS FLOW RATE OF A RESIDENTIAL GAS-FIRED FURNACE

Abstract

The flue-gas temperature and mass flow rate through a gas-fired

furnace were studied in the laboratory. Temperature profiles were

measured under cycling conditions and compared with profiles predicted

mathematically using data obtained while the furnace was cooling down

from steady-state operation and warming up from equilibrium. The mass

flow rates at various flue-gas temperatures were measured using both

a vane anemometer and a tracer-gas technique, and these results are compared with the mass flow rate predicted by the theoretical equations. The effect on the off-period flow rate of automatic stack

dampers having different sized damper openings was experimentally

determined. Theoretical equations are presented for predicting the

effectiveness of a stack damper as a function of the ratio of the area of the damper to the area of the stack and a system friction

factor.

Key Words: Automatic stack damper; flue-gas temperature profile; gasfired furnace; off-period mass flow rate; part-load performance; seasonal efficiency.

iii
ACKNOWLEDGMENTS

The authors are indebted to the Department of Energy for funding for

this study. They also wish to acknowledge the valuable contribution

of Mrs. Mary Reppert of the Center for Building Technology, National

Bureau of Standards, for editing this report.

IV
Contents

Page

Abstract ill

Acknowledgments iv

List of Figures vl

Nomenclature vli

Introcfuctlon 1

Experimental Set-up 1

Experimental Procedures 2

Temperature Profiles 8

Mass Flow Rate 17

Damper Effectiveness 25

Effects of Temperature Probe Locations on Calculated Furnace

Efficiencies 29

Conclusions 29

References 31
List of Figures

Page

Figure 1. Set-up for a gas furnace testing 3

Figure 2. Thermocouples in the test plane 4

Figure 3. Flow control system for infrared absorption

analyzer 5

Figure 4. Flue-gas temperature at cool-down test 11

Figure 5, Flue-gas temperature at heat-up testing 12

Figure 6. A relation between raonotonic and cyclic patterns .. 14

Figure 7. Cyclic Pattern of flue-gas temperature at the test

plane 18

Figure 8, Cyclic pattern of flue-gas temperature at the heat

exchanger passage outlets 19

Figure 9. Mass flow rate of flue-gas through the stack of the

furnace 24

Figure 10. A square-edged orifice in a circular duct 27

Figure 11. Variation of stack flow with damper area 30 1

vi
NOMENCLATURE

A/F mass ratio of stoichiometric air to fuel

2 A orifice area, ft

2 Ac stack cross-sectional area, ft

^air specific heat of air, Btu/lb-°F

C loss coefficient

Cx. Qpp correction factor for the effect of cycling during cooldown period

C^ ON corrrection factor for the effect of cycling during

* heat-up period

D diameter of the duct, ft

Dp off-period flue-gas draft factor

Dq stack damper effectiveness

Dg off-cycle stack-gas draft factor

f friction factor

g gravitational acceleration, ft/ sec

h height of the furnace-flue system, ft

H^ dynamic pressure loss, Ibf/ft

Rf friction head loss, Ibf/ft

HHV higher heating value, Btu/lb

H^ total head loss, Ibf/ft^

z vertical distance, ft

k system friction factor

Lg Qpp off-period sensible heat loss in percent

•fflp flue-gas mass flow rate, Ib/min

mg Qpp mass flow rate of the stack gas during off-period, Ib/min

vii
fflg QM mass flow rate of the stack gas during on-period, Ib/min

p pressure, Ibf/ft

9 p' pressure difference from ambient condition, Ibf/ft

P ambient pressure, Ibf/ft

AP pressure drop, Ibf/ft

Qp volumetric flow rate, ft /min

Qtj^ fuel input rate, Btu/min

Qc Qpp sensible heat loss during the off-period, Btu

Q total heat input, Btu

R Reynolds number

I

ratio of actual combustion air to stoichiometric combustion

air

t time, min

tQpp duration of off-cycle, min

tQjq duration of on-cycle, min

At time interval, min

Tp flue-gas temperature, °F ("R)

Tp

gg flue-gas temperature at steady state, °F ("R)

Tj^ room air temperature, °F (°R)

Tg

gg steady-state stack gas temperature, °F (°R)

V average velocity, ft/sec

Vq velocity at the orifice, ft/sec

Gp temperature difference for heat-up period, °F

viii
9p . actual temperature difference for cycling condition during

heat-up period, "F

e_

p predicted temperature difference for cycling condition

during heat-up period, °F

®F X predicted temperature difference at the beginning of the

heat-up period, °F

AG difference in 9, °F

Up dynamic viscosity coefficient of the flue-gas, Ib/ft-min

p' density difference from ambient condition, lb/ft

Pp flue-gas density, lb/ft

o P ambient density, lb/ft

o P stack-gas density, lb/ft

^OFF cool-down time constant

Tqjj heat-up time constant

'I'p temperature difference for cool-down period, °F

H'p . actual temperature difference for cycling condition during

cool-down period, °F

'I'p p predicted temperature difference for cycling condition

during cool-down period, °F

I'p Q y predicted temperature difference at the beginning of cooldown period, °F

A* difference in *, °F

IX

A STUDY OF THE DYNAMIC FLUE-GAS TEMPERATURE AND OFF-PERIOD

MASS FLOW RATE OF A RESIDENTIAL GAS-FIRED FURNACE

by

Cheol Park, William J, Mulroy, and George E, Kelly

Introduction

In order to conserve fuel in heating, it is necessary for industry to manufacture more efficient furnaces and boilers. Test and

calculation prcrcedures must be developed to make this possible

through the identification of high-efficiency systems. Test and calculation procedures for gas- and oil-fired furnaces and boilers have been

recommended by Kelly et al [1]. The purpose of this study is to provide background information about the equations used by Kelly et al

in their calculation procedure.

Flue-gas temperature profiles with respect to time for on-period

heat-up and off-period cool-down of the furnace will be described.

Detailed derivations of mass flow rate equations will be made. For

cyclic patterns of heat-up and cool-down, correction factors C^,qvj and

^t»OFF ^^^^ ^^ derived.

These theoretically derived equations will be compared with cor- responding experimentally measured values, using data from a gas-fired

furnace tested at the National Bureau of Standards (NBS). In addition,

experimental set-up and test procedures will be explained and measure- ments and a theoretical analysis of the stack damper effectiveness will

be described.

Experimental Set-up

The up-flow forced warm air gas furnace employed in the experimental study had a 132,000 Btu/hr (38.68 kW) rated bonnet capacity and a 165,000 Btu/hr (48.35 kW) rated input capacity. This furnace had six

burners with each burner having a separate heat exchange passage

through the furnace. The gas valve switch of the furnace was operated

manually instead of by a thermostat for these tests, and the blower

was operated by a manual switch with override by the high temperature

limit control. A 6-inch (15.24 cm) diameter, 5-foot (1.52 m) high test

stack was attached to the furnace throughout the rest of the study,

although in a few tests the stack height was increased to 10 feet

(3,05 m) , The upper portion of the furnace, the warm air supply duct,

and the stack were covered by a layer of R7 fiberglass insulation faced

with aluminum foil in order to reduce heat loss. The integral draftdiverter relief opening could be open or sealed depending upon the

test condition. However, most tests were conducted with the integral

draf t-diverter relief opening sealed.
Warm air was exhausted into the room, where temperature was maintained at a constant level by a high capacity air-conditioner. Inlet

air temperature was kept constant within + S^F (+ 2.8°C), The overall

set-up is shown in Figure 1. Thermocouples, installed at various locations, were used to make temperature measurements, A shielded 24-gage

K-type thermocouple was inserted approximately one inch into the outlet

of each of the six heat exchange passages. Nine thermocouples, wired

in parallel, were installed on a horizontal test plane located one

foot from the inlet of the test stack. Hereafter this plane will be

referred to as the test plane. Figure 2 shows the locations of thermocouples in the test plane. Electromotive force created by each thermocouple due to temperature deviation from the reference point was channeled into a data acquisition system. There was also a thermocouple

near the test plane, connected to a low speed strip-chart recorder,

to continuously monitor the variation of temperature during the course

of each test.

Volumetric ratio of carbon dioxide concentration was measured by a non-dispersive infrared analyzer [2], A sample gas was taken through a copper tubing connected to the stack at the test plane. Sampling was

also made in the heat exchanger outlets upsteam of the integral draft

diverter. Figure 3 shows a schematic diagram of a flow control system

for gas concentration measurement. Stack flow rate was determined by

using the methane tracer gas method. Methane gas was introduced into

each of six heat exchange passages through a supply line incorporating

a rotameter and a six-branch feeding manifold. Methane concentration

at the top of the furnace stack was measured by an infrared absorption

analyzer. During some tests using the methane tracer gas method, electric heaters were installed in place of the gas burners, A vane

anemometer was mounted 2 inches (5,1 cm) above the top of the stack

for stack flow rate measurement during some cool-down tests. The

anemometer had been calibrated under a geometrically similar flow condition. In carrying out this calibration, the air velocity, at the top

of a circular duct with the same diameter and height as the test stack, was measured at the test room temperature, A high capacity dry gas

meter was installed between the air supply line and this duct to measure the mass flow rate. The anemometer used here was 4 inches

(10,2 cm) in diameter with an eight-vane rotor. Its effective cross- sectional area was 9,42 in (60,8 cm )

,

For tests of the effectiveness of automatic stack dampers,

manually operated damper plates having various size openings were installed in the stack approximately 18 inches above the inlet of the

stack.

Experimental Procedures

Test procedures for steady-state, cool-down and warm-up followed

closely the test procedures prepared by Kelly et al [1], The furnace

used natural gas as its fuel for all tests. Room air temperature was
I

I

Venturi tube

o

Gas

furnace

f

Exhaust to room

n CX

View B-B

(not scaled)

^

^

Section A-A

Supply air duct

r

Heat exchanger

passage outlets -

Room air

Stack

Test plane

Jpuouaudp— Integrated

draft

diverter

relief

opening

Return air duct

Figure 1. Set-up for a gas furnace testing
Type K thermocouples (24 gauge)

Figure 2. Thermocouples in the test plane
I

Sample

valve

Coarse

filter

Fresh

Tair (zero gas)

Vapor

trap

6

Sampling

probe

Exhaust -^

Exhaust

Shut-off

valve L^ Pressure

regulator

Span gas

Infrared absorption analyzer

Figure 3. Flow control system for infrared absorption analyzer
maintained at 75°F (24°C) with less than 5°F (2.8°C) deviation. A room exhaust fan was operated to withdraw stack gas from the room. All

instruments were warmed up before conducting each test. Flue-gas temperature changes were monitored by a thermocouple located near the

test plane, and continuously recorded on a chart recorder during each

test.

Steady-state testing was performed by operating the furnace with

the burners and the blower turned on until stack, temperature variation

in three successive readings taken 15 minutes apart was not more than

S^F (1.7°C), The integral draft diverter was open during this test.

When steady state had been reached, room temperature and steady-state

gas temperature were measured by nine thermocouples located in the

stack. The average value of nine thermocouple outputs was taken as stack-gas temperature. A sample of stack gas was pumped out at a rate

of 3 ft per hour (85 L/hr) through a sampling copper tube with its

opening at the test plane. The sample gas was analyzed by the infrared

analyzer to determine the concentration by volume of carbon dioxide

present in dry stack gas. After taking the value of carbon dioxide

concentration of stack gas, a sample of flue gas was obtained by moving

a sampling probe around in the heat exchanger passage outlets into

the draft diverter. The average value of carbon dioxide concentration

in the flue gas was taken. The furnace was then temporarily turned

off to seal the draf t-diverter relief opening with a layer of R7 fiberglass insulation backed with aluminum foil. The unit was then turned

on again and when the steady-state condition had been re-achieved, the

stack outlet was progressively restricted with a piece of metal sheet

until the carbon dioxide concentration in the flue gas measured at

the test plane agreed within + 0.2 percentage points of the previously

determined average value measured at the heat exchanger passage outlets.

The average temperature then occurring at the measurement plane was assumed to be the representative flue-gas temperature. During steadystate tests, temperature readings from each of the thermocouples in

the heat exchanger passages were also recorded.

Cool-down testing was executed after steady-state testing was com- pleted. As soon as the burner was turned off, temperatures at the

test plane and at the heat exchanger were measured in various time

intervals. Since the decay rate of the temperature during the early

stages of cool-down period was great, the interval in taking data was

set to 10 seconds. Near equilibrium, the decay rate was very small,

and the data interval was set to 5 minutes. The blower was turned

off 3 minutes after the burners shut off, as recommended by the test

procedures in [1]. The burners remained off until equilibrium was attained, as indicated by variation in the flue-gas temperature of not more than 3°F (1.7°C) between three successive readings taken 15 minutes apart. It took about 55 minutes in a typical test condition for

the unit to reach a state of equilibrium.
After an equilibrium condition was attained, a heat-up test was conducted. The furnace was turned on and flue-gas temperature was measured in a similar way as in cool-down testing. As recommended by

the test procedures in [1], the blower was turned on 1.5 minutes after

burners had been turned on and temperature measurement was continued

until steady state was reached.

Tests were also performed under the condition when the pattern of

temperature profile with respect to time is cyclic and the temperature

of flue gas reaches neither equilibrium nor steady state. A typical

test was made with the duration of the on-cycle chosen as 8 minutes

and the duration of off-cycle as 2 minutes. The blower remained on during all of these tests, and temperature data were taken every 10

seconds until three complete cycles of warm-up and cool-down were

achieved.

Measurements of the flow rate of stack gas through the insulated

circular test stacks as a function of temperature were made by means

of a vane anemometer and the tracer-gas method, with the integral

draft diverter sealed. Prior to making a test for stack flow measurement, the vane anemometer was calibrated. Compressed air from a supply

line at room temperature was passed through a dry gas meter before it was allowed to flow into a geometrically similar calibration duct. The

anemometer was placed 2 inches (5,1 cm) above the end of the duct and

centered. Air flow velocity was determined by reading the distance

indicated on the dial of the vane anemometer for a given interval of

time, and flow rate was computed based on air speed and the cross sec- tional area of the duct. Results were plotted against the calibrated

flow rate indicated on the dry gas meter in a given time interval. In

order to calculate the mass flow rate through the stack of the furnace,

the air speed of stack gas was measured during a cool-down test period,

with the vane anemometer installed on the furnace stack in the same manner as was done during its calibration procedures. Flue-gas temperatures both at the test plane and in the heat exchange passage outlets were obtained every 30 seconds from steady state to equilibrium.

In addition to the vane anemometer, the tracer-gas method was also

used in flow rate measurement for some tests of air flow through the

furnace as a function of the flue-gas temperature. In these tests, the

draft diverter was blocked and electric heaters were installed in the

furnace in place of the gas burners. The furnace was then operated at

different steady-state flue-gas temperatures by varying the voltage to

these electric resistance heaters. Methane gas was released into each

heat exchanger passage at a predetermined constant flow rate, while a flue-gas sample taken at the top of the flue was continuously monitored

on an infrared methane gas analyzer. The flow rate through the furnace was then calculated from methane flow rate and its dilution.

Experiments to determine the effectiveness of automatic stack dampers at blocking flow during the off-period were conducted using
manually operated dampers installed in the furnace test stack. The

draft diverter was blocked and the furnace was operated at a full-load

condition for approximately one hour with its damper open. The furnace was then shut off, the damper closed, and a constant methane flow rate

of 5.3 ft per hour (150 L/hr) established. The furnace was allowed to

cool naturally and flue-gas temperature and stack methane concentration

were measured continuously. The cool-down period usually lasted for one hour. After each test the area of the damper plate was reduced by

drilling holes one inch or less in diameter in the damper plate and the

steady-state and cool-down tests were repeated. Tests were also made

with a damper with a single large hole in it, in order to compare its

performance with the damper having a number of small holes. In these

comparison tests, a 10-foot insulated test stack was employed instead

of the 5-foot stack. Although blocking the draf t-diverter relief opening of the furnace results in the stack damper being tested as a flue

damper, the effectiveness of the damper was determined at relatively

low flue-gas temperature and thus the results should satisfactorily

describe the performance of stack dampers.

Temperature Profiles

By examining experimental data, it was possible to develop equations to predict the flue-gas temperature during the heat-up and cooldown periods [1,3]. When the unit is allowed to cool down from steady

state to equilibrium, the temperature of flue gas in the furnace decays

exponentially. Temperature profile with respect to time may be

expressed as:

"^OFF

'*'F^t) = *F,0,X ^ ' (1)

where ^^it) = Tp(t) - Tp(<»), Tp ( °°) denotes flue-gas temperature at

equilibrium, and 'I'p q v and fQpp are constants to be determined from

known conditions.

A similar expression can be written for flue-gas temperature during

the heat-up period.

^F^t) = ^F,0,X ^ > (2)

where 9p(t) - Tp gc ~ Tp(t), Tp gg denotes flue-gas temperature at

steady state, anS FOX ^^'^ ^ON ^^^ constants.

1
If the flue-gas temperatures at time t^ and t2 are experimentally

known, the time constant during the heat-up period, Tqjj can be obtained

as follows:

ep-Ctj^) - Tp^gg TpCtj^) - ep^o.x ^ 'ON

at t = t-j^.

and

^F^H^ = Tp^gg - Tp(t2) ^F,0,X ^ 'ON

at t = t.

Solving both equations simultaneously gives:

TON

h - H

'^F,SS '^F^^l^

."^F.SS "'^F^'^2^-1

(3)

Similarly if flue-gas temperature during the cool-down period is

measured at times to and t^ after the furnace is turned off, "^Qpp can be obtained using

*F(t3) = iPp^o.X

'OFF

at t = t3.

and

^(t^) = i|^F^o,X ^ 'OFF

at t = t/

These two equations, when solved simultaneously, yield

"^OFF =

t4 - t3

in

Tp(t3) - Tp(co)

-

TpCt^) - Tp(«>)

(4)
The initial constant values, ^p q X

^'^^ %" X» ^^^ then be defined by

using the measured flue-gas temperatures a£ i^ and to, respectively,

to obtain

H H

^F,0,X = Qp^ti) e = [Tp^ss - Tp(ti)] e , (5)

and

^3 '3

"^OFF ^OFF

*F,o,x = V^s) ^ = tV^a) - Tf(")] e . (6)

Figures 4 and 5 show typical experimental results obtained for

the flue-gas temperature as the furnace cooled down from steady-state

operation and heated up from equilibrium conditions. Data were taken

at the test plane as well as at the heat exchanger passage outlets.

The predicted flue-gas temperature cool-down and heat-up profiles are shown by dotted lines. The following values of tj^ , t2 , t^ , and t^,

which were recommended by Kelly et al in [l],were used.

ti = 0.5 min to = 2.5 min

to = 1.5 min t^ = 9.0 min.

A good agreement between measured values and predicted values is found

except near the beginning of the time period.

When the cool-down and heat-up periods are finite, the pattern of

the flue-gas temperature changes with respect to time. Due to the

finite length of time for on- and off-cycle periods, the initial values

of the exponential function, ^p q v and ^p q y become smaller due to

the fact that the unit may never Aeat up to steady state or cool down

to equilibrium. It is therefore necessary to introduce correction factors such that for the heat-up period.

"^ON 'F^t> = C.^Oj^e^^Q^^ e , (7)

and for the cool-down period,

10
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12
"^OFF

*F(t) = C^^oFF*F,0,X ^ • ^^)

The quantities C^^ q^ and C^^ OFF* which are correction factors for

the on- and off-periods respectively, can be determined as follows.

The main assumptions are:

®F,P _ Qf,A at the start of the heat-up period,

and

*F,P *F,A

*F,0,X '^F.SS " "^F^")

at the start of the cool-down period,

where the subscripts P and A refer to the predicted and actual value of

ep(t) and l'p(t) under cycling conditions.

Referring to Figure 6 and defining A9 to be the difference between

the predicted value of Gp at the beginning of heat-up from cycling con- dition and the predicted value of 9p at the beginning of heat-up from

equilibrium condition ( 9p q y )

, the assumed relation may be expressed

as:

6f,0,X " ^9 "^F.SS "

"^F^"^ " *F^^OFF^ ^^^

= " » 9f,0,X ^F,SS ~ ^F ^"^

where tQpp is the duration of the off-cycle. The difference A9 may thus

be written as:

^ (^ ^9

Le=^ OFF^ F,0,X (10)

^f,ss - ^f(")

Similarly, defining aiJj as the difference between the predicted value of

i|)p at the beginning of cool-down from cycling condition and the predicted value of \|;p at the beginnning of cool-down from steady-state

condition ( % q X^ results in

13
I

;sS -a 1 I =5

-s .^ -c .^ a V a V

tS • tS £ < a. < a. @@e<§)C =5

09

«rf I u e

-s z ee ca e E eE 09 V n % M

tS >- >• (B T3 09

IB t; *« u Ee W u B Ea k u ^ ^>i ^B

09

u •

-s

G « t;

a =S

a ^ t 09 C 0> < o. < a. ee@@

14
F

'^ ON'' F,0,X

Ae = , (11)

where Iq^ is the duration of the on-cycle, A sketch of the heat-up

and cool-down curves from steady state and equilibrium conditions as well as under cycling condition is shown in Figure 6,

Boundary conditions for the equations predicting the heat-up and

cool-down flue-gas temperature profiles under cyclic condition can be

imposed such that the predicted temperature at the end of each on- and off-period approximately coincides with the actual values. The

imposed condition causes a discontinuity in the temperature at the

beginning of each on- and off-period which violates the physical

intuition, but gives results which are in good agreement with measured

data, except near the beginning of each period. These boundary conditions are:

V^On) = t^F,O.X- ^Q] -

°''

. (12)

and

tr _ _'OFF

•'F^'-OFF^ - i '^F,0,X " ^OFF

%(t^vv) = [*!. o y

- A*] e . (13)

Substitution of Eqs. (10) and (11) into Eqs. (12) and (13), respectively, yields:

t t ON _ ON

"^ON ^ON , , F,0,X ^ „.„^._„ ^ = fi„ „ „ . (14) QpCtoN) + 1^ :>!:7^>, e ^pCtOFF ) = 9f,0,X e ^F,SS ^F*^ ^

»

and

t t OFF _ OFF

* ^OFF ^OFF

^^-"^^ e QFCtoN) + *F(toFF) = 'i'F,0,X e . ^^^^

'^F,SS Tp(»)

15
In order to find the unknown variables QpCtQw) and 'I'pCtQpp), the system

of equations consisting of Eqs. (14) and (15; must be solved simultaneously. This results in the following equations, as introduced by

Chi and Kelly [3]:

ep(t) = e^^Q e

'ON

(16)

and

v^> '*t,0

OFF

where Q^^q = C^^q^ ^¥,0,X »

and ipp^Q ^t,OFF ''V,0,X .

(17)

(18a)

(18 b)

The correction factors are expressed as

^ 1 - F_^O^X

^OFF

OFF

Tf,ss - TpC-)

't,ON

1 - Qf,o,x%,o,x

tTp.SS - Tf(«)]'

-ON

^ON

toFF^ *

'OFF

(19 a)

and

1 - .!f^x -

-ON

^ON

Tf,ss - V-)

't.OFF

I _ ^F.O.X '^F.O.X

tTp.SS - Tf(«)]'

^ON ^ '^OFF

ON 'OFF

(19b)

16
Figure 7 shows a set of curves obtained by plotting the experimental flue-gas temperature profile measured at the test plane

previously described and the predicted flue-gas temperature profile

given by

e ON

and

pCt) - Tp(=o) + C^^Qpp 9f,0,X

^

"^OFF

for on-cycle,

for off-cycle,

with the values of C^ qj, and C^^ Qpp obtained from equations (19a) and

(19b), respectively.

A curve of the average flue-gas temperature measured at the six

heat exchanger passage outlets is shown in Figure 8, along with the

predicted flue-gas temperature profile. The results presented in both

figures show that the predicted flue-gas temperature profiles at both

locations closely approximate the measured heat-up and cool-down profile although there exist discontinuities at the beginning of the

heat-up and cool-down periods.*

Mass Flow Rate

The mass flow rate through a furnace or a boiler during the offperiod can be obtained from a consideration of hydrostatic pressure

difference between flue gas and ambient air and the equations describing turbulent flow through a circular pipe having a finite length.

The differential equation governing the pressure inside the fur- nace may be expressed as |^+ P'(z)g=0. "^^ (20)

where z is the vertical distance inside the furnace above the point

where combustion takes place,

* It would be possible to construct a set of equations having no discontinunities , but this would require a considerably more complex

mathematical treatment.

17
A

r--^ iM < 1

,1

— ^ co — \ — - co1

93 Q. ^ c m

09E

93 ou X

liJ

Ji ( 1 1 * 1 1 — I ceD — \

S^^ r ;^ J I r—

^ 3 )rk ( 1 . I,

Ei=] CQ

J— 1 1 1 1 1 . 1 ^ ' 1

CM

en

CSi

CNJ

CM

(O

03E c

CB

09

CO

Q»

03a.E

03

re

CM

09nu>O

093 — ^

CO

(do) Ji

rg

18
09E

09

OS

CDM

V>K

cen

09

09E

09

(A

reen

09

09 ti

reu

>- O

ed

093

(Jo) ^i

eg

19
p' (z) = p (z) - Pq,

and p" (z) = p (z) -

p o The quantities p(z) and p(z) are the pressure and density, respectively, inside the stack at height z; p and p are the pressure and

density, respectively, outside the stack and may be assumed independent

of z for the purpose of this calculation; and g is the gravitational

acceleration.

Making use of the ideal gas law results in the following relationship

between p^ and the flue-gas temperature, Tp(z) in absolute units:

where Ttw is the room air temperature.

Inserting this expressing into Eq. (20) and assuming that the variation of the flue-gas temperature with z can be ignored yields:

AP - Pogh

Tp / (22)

where h is height of the furnace-flue system, and Ap is the pressure

drop between the inlet and outlet of the furnace-flue system.

The pressure drop between the inlet and outlet of the furnaceflue system should be balanced by the friction forces created by the

turbulent flow on the surface of the circular duct. If the inner

diameter of the duct is D and the average velocity of steady flow

is V, the force balance equation is given by Bird et al [4]:

(.Dh) (I p^ . ^) f = AP {^) , (23)

where f is a friction factor.

2 Using the mass flow rate expression: iDp = ppv/ frP \ and

solving for the pressure drop, Ap, Eq, (23) becomes

,2

32 hfrny^

AP = „ ^ ^ . (24)

20
The empirical equation for the friction factor in turbulent pipe flow

is given by a simple expression [5]:

f = 0^ (25)

Re

•

in the range of 30,000<R^<10^.*

The Reynolds number, R , is based upon flue-gas density, average gas

velocity, diameter of the duct and the dynamic viscosity coefficient

of the flu^ gas, Pp, and is given by

Re = ^— . (26)

The dynamic viscosity of air varies with temperature and is determined from the table provided by Ekert and Drake [6] as being approximately proportional to the 0.75th power of Tp.

Setting Eq. (22) equal to Eq. (24) and substituting the above relationship for f and Up given by Eqs. (25) and (26), a final expression for

the mass flow rate of flue gas through the furnace is obtained.

(Tp _ Tr,)0.56 ^ ^

^F

where K is a constant.

Under steady-state condition this becomes;

(t t ^0.56 ^ = K.

. '-^F,SS - ^RA-* (28)

F,SS T, 1.19

^F,SS

* Note that the well-known Blasius formula is

f ^ 0.0791

I

'^e ,0.25

in the range of 2.1 x 10^<Rg<10^.

21
Combining Eqs. (27) and (28) results in the following equation:

F,SS

,'^F,SS " '^RA

0.56

•F,SS

1.19

(29)

Using Eq. (29), the ratio of mass flow rate during off-cycle to the mass flow rate during on-cycle is easily obtained.

mF,OFF

F,ON

\ "^F.SS ~ "^RA/ \ "^F.OFF

.1.19

(30)

where Dp is an off-period flue-gas draft factor, which has been introduced to account for the effect of power burners or flue dampers. It

should be noted that temperature has absolute units of "R (°K). This

expression was introduced by Chi and Kelly [3] with a brief explanation.

Under full-load condition, mp ^^ is the sum of the fuel flow rate

and the combustion air flow rate:

F,ON

^

[1 + Rt f(A/F)]

HHV T,F^ ' ^J (31)

where Q-tk. is fuel input rate, HHV the higher heating value of the fuel,

Rrp p the ratio of actual combustion air to stoichiometric combustion air,

ancl A/F the stoichiometric air-to-fuel ratio. The quantity R^ p is

determined from the carbon dioxide concentration under steady-state

operation with a particular fuel. Eq. (31) is also assumed by Chi and

Kelly [3] to adequately describe the flue-gas mass flow rate during the

heat-up period, since for most furnaces and boilers Tp rises quite rapidly

and variation in mp can usually be ignored.

It is also possible to derive another form of Eq. (29) for the mass flow rate using an assumption that the flue-gas density is the same as that of air at the same temperature. The following formula

for calculating the flow rate induced in the stack, is provided in the

ASHRAE Equipment Handbook [7]:

22
Q = K d2 /Ie^ . (32)

F 1 / k p^

where Qp is the volumetric flow rate, k is a system friction factor,

and Ki is a constant. Using the quantity K with a subscript to

indicate constants having different values in the equations presented below, and employing the fact that the theoretical draft is due

to pressure differences Ap gives [7]:

Ap = K2 Po ^(^- ;p-V ^^d (33)

Qp = K3D2 / ^(Tp _ T^)

^ (34)

From Eq. (21) and Eq. (34), the mass flow rate expression may be

written as

T \ 2 / h(T - T )

F '^F F \ o T^y 3 / k Tj^

The mass flow rate at a flue-gas temperature Tp divided by the mass

flow rate at the steady-state temperature, Tp gc , is then given by

m

p gg

\Tf,SS %A / \ ^F /

It is interesting to see that Eq . (29) and Eq. (35) are almost identical in form except for different values of power. However, the difference has a negligible effect on the ratio mp/mp,gg, as shown in

Figure 9.

Figure 9 shows actual data points measured by a vane anemometer and by

the tracer-gas method for mass flow rate through a furnace. It also

shows two theoretical curves given by Eq. (29) and Eq. (35). In calculating the flow rate using the vane anemometer measurements, the air

speed of the flue gas at flue temperature Tp was corrected for the

fact that the vane anemometer was calibrated at temperature T by means

of the following formula [8]:

23
o

u

CD

a»

<

oc

M

03

09

O

CO

a?

u

en

C0 CM

SS'J Ul lU

2A
o 1 o

Figure 9 indicates that both Eq, (29) and Eq. (35) adequately describe

the mass flow rate through the particular furnace tested. The slight

deviation of the experimental data from the theoretical curves for

Tp - Ttja in the mid-range temperature could reflect the fact that it

is not rigorously correct to make the assumption that Tp is independent

of z.

Damper Effectiveness

During the cool-down period, heat loss from a furnace or boiler

occurs mainly due to the flow of air through the furnace draft control

device and up the stack. For units which use indoor air for combustion,

the off-period sensible heat loss is given, in percent, by the following relation:

Qs,OFF

^S.OFF ~ ^ '^^ » Qt

where Qg Qpp is the sensible heat loss during the off-period and Qrp is

the total heat input equal to the product of the fuel input rate Q-rv, and the time during which the burners are turned on, Cq^t, The sensible

heat loss during the off-period in a certain time interval, At, may

be expressed as

'^S,OFF ="^air TMS,OFF^t ^T , (37)

where C • is the specific heat of air, m^ Qpp the mass flow rate of

the stack gas, and At = Tg,Qpp - T^^^ is tfie temperature difference

between the stack gas and room air. The mass flow rate through the

stack can be shown to be given by the equation:

m /t T \0.56 / t \ 1.19=d; ( S,QFF - Tra \

/ ^S,SS \ (38) S,OFF S S,ON (^Ts^ss -

-^kA ) \ ^s.OFF i

by carrying out an analysis similar to the one used to derive Eq. (30)

25
In the above equation, Dg is the off-cycle stack gas draft factor which

accounts for the effect of a stack damper, Tg go the steady-state stackgas temperature, ^and m^. Qjq the mass flow rate' through the stack during

the on-period, mg ^^i^ the product of nip»QM and the ratio of the stackgas mass flow rate* to the flue-gas mass flow rate, S/F,

During the cool-down period the total sensible heat lost up the stack

(ignoring the infiltration loss) may be obtained by integrating

Eq. (37) with respect to time to obtain

^S,OFF =^air^S^,ON^^^^^J \t_,-T "/ VVoff/ -S.OFF'V^^'

(39)

where tQpp is the length of the off-period. The factor Dg depends

upon the type of system under tQ«t, For furnaces and boilers with stack

dampers, and draft diverters on draft heads, Dg is the same as the

stack damper effectiveness, Dq, (Refer to Table 1 in [1].)

In order to determine the value of stack damper effectiveness factor, D , an equation is first theoretically derived, then compared with

experimental data.

For a system employing a stack damper, the gas flow through the

damper may be considered similar to the flow through a square-edged

orifice in a duct (Figure 10). Due to contraction of area of the duct

in this type of flow, there exists considerable dynamic pressure loss.

This dynamic pressure loss, H^, can be expressed in terms of the stackgas density, P , and the gas velocity at the orifice, v^, using

H = C i P v^ , (40)

d o 2 so

where C is a loss coefficient.

By means of the equation for mass continuity it is possible to rewrite

this equation as

A. H = C ( -J-] i p V

2 . 2

d o\Aq/ 2ss (41)

26
i

> <

7-

/

/

/

/

/

/

/

/

/

/

/

//

v^z^ ^77Z^.

e

<

/

/

/

/

/

/

/

/

//

/

/

/

/

/

/

/

/A

O

IIe

<

CM M

JO3U

a»

C3)

<

V

S

<

27
where v^ is the velocity in the large cross-sectional area of the

stack, Ag is the stack cross-sectional area, and A the orifice area,

For the flow through a square- edged orifice, the coefficient C can be

obtained from the values tabulated in the ASHRAE Handbook of Fundamentals [9] and expressed approximately (within an error of ± 5%) by

1.58

On the other hand, the friction loss through the stack may be

written by

H = k i p V , , f 2 ^s s , (43)

where the friction factor, k, is a diraensionless constant containing

a scale factor for the size of the stack. The total loss of pressure

head, H^ , through a stack containing a square-edged orifice is then

the sura of the dynamic pressure loss and friction loss and is given by

Ht = Hd + Hf = Co ( ^"j + k 1 p^v^ . (44)

For a stack without the orifice, total loss is just equal of the friction loss and may be written as

1 2 Ht = Hf = k i. p v . (45)

2 s s

Defining the damper effectiveness, D , as the ratio of the flow rate

with the damper, to that of a flow rate without the damper at the same pressure differential, D is obtained by dividing the square root of

Eq. (45) by the square root of Eq. (44) as

Do = / /A„V (^6) ^^ Co [^

28
After substituting Eq. (42) into Eq, (46) and expressing the result in

terms of the damper area, A^, the equation for the effectiveness of the

stack damper at off-period stack, flow becomes

1 (47) k

o

1 •^ "^

2.6

V^s/

y 1 -

2

Figure 11 contains experimental results obtained using the tracer-gas

method of flow measurement during cool-down from steady state to equilibrium condition. Data were obtained for 5-foot-high and 10-foot-high

test stacks as described in the section entitled Experimental Procedures. The values of D as given in Eq. (47) are also plotted against

Ajj/Aq for k = 5 and k = 10. A reasonable agreement is achieved between

experimental and theoretical results for K=5 . D = corresponds to

a completely effective damper, while D = 1.0 corresponds to a damper

that does not restrict the flow at all during the off-period.

Effects of Temperature Probe Locations on Calculated Furnace Efficiencies

As mentioned earlier, measurements of temperature were performed

at two different locations, the heat exchanger passage outlets and the

test plane. The average temperature measured at heat exchanger passage outlets had a higher value during heat-up periods and a lower

value during cool-down periods. After conducting a series of tests and

computing furnace annual efficiencies, using the procedures described

by Kelly et al [1], it was found that there was less than 1% difference

on the average between the annual efficiency calculated from data

obtained at the heat exchanger outlets and those obtained at the test

plane for the particular furnace tested. The same result was also

found for the seasonal efficiencies calculated using data from the two

different temperature measurement locations. Thus, as long as one's

interest is in annual or seasonal efficiency computation, the use of

temperature measurements at the test plane described in the section

in Experimental Set-up appears to give satisfactory results.

Conclusions

There exists, in general, good agreement between experimental and

predicted results for flue-gas temperature profiles and mass flow rates

under cycling conditions for a typical gas-fired furnace. Two forms

of the equation for predicting the flue-gas mass flow rate during the

29
CMIe V

in

II

II >c iC <ec »- I8nM

30
off-period are presented which give almost identical results. It is

shown that the effectiveness of a stack damper is dependent upon the

ratio of the damper area to the stack area and upon a system friction

factor, k. For a furnace or boiler installation with an unknown system

friction factor, it is suggested that a value of k = 5 be used.

Seasonal efficiencies calculated using the flue-gas temperature mea- sured in an insulated test stack, one foot from the furnace outlet, were found to agree within 1% with values calculated using temperature

data obtained at the heat exchanger outlets.

References

1. Kelly, George E., Chi, Joseph, and Kuklewicz, Mark, Recommended

Testing and Calculation Procedures for Determining the Seasonal

Performance of Residential Central Furnaces and Boilers, NBSIR 78-1543,

March 1978.

2. Verdin, A., Gas Analysis Instrumentation , John Wiley & Sons, 1973.

3. Chi, Joseph, and Kelly, George E., A Method for Estimating the

Seasonal Performance of Residential Gas and Oil-fired Heating

Systems. (Presented at the sixth International Heat Transfer

Conference, Toronto, Canada, August 7-11, 1977.)

4. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport

Phenomena , John Wiley & Sons, 1960.

5. Kays, W.M., Convective Heat and Mass Transfer , McGraw-Hill, 1966.

6. Eckert, E.R.G., and Drake, Jr., R.M. , Heat and Mass Transfer , McGraw-Hill, 1959.

7. ASHRAE Equipment Handbook , ASHRAE, 1975.

8. Ower, E., and Pankhurst, R.C., The Measurement of Air Flow , 5th ed.,

Pergamon Press, 1977.

9. ASHRAE Handbook of Fundamentals, ASHRAE, 1972.

31
NBS>114A iREv. s-7ei

U.S. DEPT. OF COMM. 2.0ov't Accession NiPI

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN 999

tx-ife<i-.-«,«.- iosiiiifmti-. ,j;-st.

4. TITLE AND SUBTITLE

A STUDY OF THE DYNAMIC FLUE-GAS TEMPERATURE AND

OFF-PERIOD MASS FLOW RATE OF A RESIDENTIAL GAS-FIRED

FURNACE

5. Publication Date July 1979

7. AUTHOR(S)

Cheol Park, William J. Mulroy, and George E, Kelly

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

18. f^|ec^5ast(/Wof>( ifeH No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS fSfreet, city, st^te. ziP)

Department of Energy

20th & Massachusetts Ave. , NW

Washington, DC 20545

13. Type of Report & Period Covered Final

14. %)eii»»1fls Ag«iscy Coci«

15. SUPPLEMENTARY NOTES

I I Document describes a corii><uter program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or teas {actual aummary ot moat atgnilicartt information. It document includes a aignilicant bibliography or

literature aurvey, mention it here.)

The flue-gas temperature and mass flow rate through a gas-fired furnace were studied in the laboratory. Temperature profiles were measured under cycling

conditions and compared with profiles predicted mathematically using dat? obtained

while the furnace was cooling down from steady-state operation and warming up from

equilibrium. The mass flow rates at various flue-gas temperatures were measured

using both a vane anemometer and a tracer-gas technique, and these results are compared with the mass flow rate predicted by the theorectical equations. The

effect on the off-period flow rate of automatic stack dampers having different

sized damper openings was experimentally determined. Theorectical equations are presented for predicting the effectiveness of a stack damper as a function of the

ratio of the area of the damper to the area of the stack and a system friction

factor.

17. KEY WORDS (aix to twelve entries; alphabetical order; capitalize only the first letter ot the first key word unless a proper name;

separated by aemicolona) . _.. *- . .. <- . -i Automatic stack damper; flue-gas temperature profile; gas-fxred

furnace; off-period mass flow rate; part-load performance; seasonal efficiency.

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