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Load CalculationsHeat Balance Method - Theory
Prof. Jeffrey D. SpitlerSchool of Mechanical and Aerospace Engineering, Oklahoma State University
Tonight
The heat balance method – theoryThe heat balance method – applicationDemonstrationOverview of the RTS method and other simplified methodsStrengths, weaknesses and limitations of simplified methods.
Abbreviations
“MPS 171” refers to the textbook (McQuiston, Parker and Spitler, 5th
Edition) p. 171“ASHRAE HOF 30.40” refers to the 2001 ASHRAE Handbook of Fundamentals, page 30.40
Important Announcement
After installing software from CD-ROM, download latest version from web site:http://www.mae.okstate.edu/hvac
Important Definitions
Zone: room(s) for which loads are det’dassumed to have uniform temperature. (i.e. controlled by a single thermostat and for multiple rooms, they should all have similar load profiles.) Boundaries are slightly vague, depending on method – usually drawn at inside surface of walls, windows, ceiling, floor, but includes heat storage of building materials.
Important Definitions
Heat Gains: Instantaneous rate of heat transfer or heat gain into the zone. Includes convective and radiative heat transfer. (Heat may be conducted to zone boundary, but is then convected or radiated into zone.)Cooling load: heat transfer rate required to maintain constant zone air temperatureHeat extraction rate: actual heat transfer rate of system, with zone air temperature changing (e.g., due to thermostat setback, limited system capacity or proportional control)
Heat Gain, Cooling Load, Heat Extraction Rate
Instantaneousheat gain
Furnishings,structure,
variable heatstorage
Instantaneouscooling load
Convective component
Radiativecomponent
Convection(with time-delay)
Heat extractionby equipment
Overview
Discuss “heat balance method” first –all other methods are approximations of the heat balance method, or approximations of approximations (e.g. the CLTD/SCL/CLF method approximates the transfer function method, which approximates the heat balance method)
Heat Balance Method
Based on heat balances for exterior zone surfaces, interior zone surfaces, and zone air. Assures conservation of energy (which is not guaranteed for approximate methods).Pardon the equations!
ABSORBEDINCIDENT
SOLAR
CONVECTIONTO OUTSIDE
AIR
LWRADIATION
OUTSIDE FACEHEAT BALANCE
THROUGH THEWALL CONDUCTION
INSIDE FACEHEAT BALANCE
SW RADIATIONFROM LIGHTS
LW RADIATIONFROM INTERNAL
SOURCES
TRANSMITTEDSOLAR CONVECTION
TO ZONEAIR
LW RADIATIONEXCHANGE WITH
OTHERSURFACES
AIR HEATBALANCE
INFILTRATIONVENTILATIONEXHAUST AIR
CONVECTION FROM INTERNAL
SOURCES
HVAC SYSTEMAIR
Heat balance - zone surface
Wall
Interiorsurface
Exteriorsurface
q"solar,out,j,θ
q"convection,out,j,θ
q" radiat
ion,ou
t,j,θ
q"conduction,out,j,θ q"conduction,in,j,θ q"convection,in,j,θ
q"radiation,in,j,θ
totos,j,θ
tis,j,θ
ti
q" solar
,in,j,θ
Transient conduction heat transfer – conduction transfer functions
∑∑ ∑=
−= =
−− Φ+++−−=Nq
nnjinconductionn
Nz
n
Ny
nnjosnjosonjisnjisojinconduction qtYtYtZtZq
1,,,
1 1,,,,,,,,,,, "" δθδθθδθθθ
∑ ∑∑= =
−−−=
Φ+++−−=Nx
n
Nq
nnjoutconductionnnjosnjosonjis
Ny
nnjisojoutconduction qtXtXtYtYq
1 1,,,,,,,,,
1,,,,, "" δθδθθδθθθ
∑ ∑∑= =
−−−=
Φ++−=Nx
n
Nq
nnjoutconductionnnjosnnjis
Ny
nnjout qtXtYH
1 1,,,,,,,
1,, " δθδθδθθ
θθθθ ,,,,,,,,," joutjosojisojoutconduction HtXtYq ++−=
Exterior Surface Heat Balance
θ,,, joutconductionq ′′ θ,,, joutsolarq ′′= θ,,, joutconvectionq ′′+θ,,, joutradiationq ′′+
tjoutsolar Gq αθ =′′ ,,,
)( ,,,,, θθ josocjoutconvection tthq −=′′
)()( ,,,,,,,, θθθ josskyskyrjosggrjoutradiation tthtthq −+−=′′ −
skyrgrco
skyskyrggroctjoutjisojos hhhX
thththGHtYt
−−
−−
+++++++−
=αθθ
θ,,,,
,,
Heat Balance - Fenestration
Window InteriorpaneExterior
pane
q"solar,out,j,θ
q"convection,out,j,θ
q" radiation,out,j,θ
q"convection,in,j,θ
q"radiation,in,j,θ
to tos,j,θ tis,j,θ ti
q"solar,in,j,θ
Rc
Rr
Interior surface heat balance
Similar to exterior surface, accounts for Radiation heat transfer between surfaces with detailed model (e.g. MRT/balance)Solar radiation transmitted through windows, absorbed onto interior surfacesRadiation heat transfer from internal equipment, lighting, and peopleConvective heat transfer from zone air.
Heat Balance – Zone Air
ti
Ajq"convection,in,j,θtis,j,θ
Ajq"convection,in,j,θ
A jq"convection,in,j,θ
Aj q"convection,in,j,θ
q infiltration/ventilation
q systemq
internal,conv,θ
Heat Balance – Zone Air
Air temperature is assumed uniform throughout space. (This is sometimes called the well-stirred model.)Because of this, air leaving the space does not affect the energy balance directly.Radiation only affects surface heat-balances -air is assumed to be non-participating media
Heat Balance – Zone Air
Can be formulated to solve for cooling load directly, assuming zone air temperature is constant
onv,θinternal,cioptiona,infiltraijisjic
N
jjsystem qttcmtthAq &&& −−−−−= ∑
=
)()( ,,,,1
, θθ
Heat Balance – Zone Air
Or, can be formulated to solve for zone temperature when there is no system contribution (e.g., where space temperature is allowed to float):
ptiona,infiltrajic
N
jj
onv,θinternal,coptiona,infiltrajisjic
N
jj
i
cmhA
qtcmthAt
&
&&
+
++=
∑
∑
=
=
,,1
,,,,1
)( θ
Heat Balance – Zone Air
Or, can be formulated to solve for zone air temperature when the system is on:
ptiona,infiltrajic
N
jj
onv,θinternal,coptiona,infiltrajisjic
N
jj
i
cmhAb
qtcmthAat
&
&&
++−
+++=
∑
∑
=
=
,,1
,,,,1
)( θ
Heat Balance – Zone AirRelies on a piecewise linear representation of system capacity qsys=a+bTz
Qsys (W)
-10000-8000-6000-4000-2000
0200040006000
10.0 15.0 20.0 25.0 30.0
Zone Air Temperature (C)
Hea
ting
(+) /
Coo
ling
(-)
Simple Steady-State Example of Heat Balance for a Roof (1)
Consider a horizontal roof at 36°N, 84°W:June 21, 1 p.m. CDT, 317 Btu/hr-ft2 total incident solar radiation, α=0.8No thermal mass; U=0.2 Btu/hr-ft2 (surface-to-surface)Outside air dry bulb temperature = 85°FWind speed = 12 mph, hc=1.896 Btu/hr-ft2°FSky temperature = 74.2°F; hr,sky=1.183 Btu/hr-ft2°FInside surface temperature, tis= 72°F“contrived” – tis, hc , hr,sky typically not known a priori
Steady-state Example (2)
Wall
Interiorsurface
Exteriorsurface
q"solar
q"convection,out
q" radiat
ion,ou
tto tos
tis72 F
q"conduction
85 F
Tsky74.2 F
Steady-state Example (3)
q”solar+q”convection,out +q”radiation,out=q”conduction
)()()( isososkyrosoct ttUtthtthG −=−+−+α
UhhtUththG
trc
isskyroctos ++
+++=
)()()(α
In other words, a weighted average.
Steady-State Example (4)
2.0183.1896.1)72(2.0)2.74(183.1)85(896.1)317(8.0
+++++
=ost
Ftos °= 65.157
)7265.157(2.0)(−=
−=′′ isosconduction ttUq
=17.1 Btu/(hr-ft2)
Questions?