3 Steady-state Conduction

STEADY-STATE HEAT CONDUCTION

Topic outline: Steady heat conduction in plane walls

Thermal contact resistance

Generalized thermal resistance network

Heat conduction in cylinders and spheres

Critical radius of insulation

STEADY HEAT CONDUCTION IN PLANE WALLS


The energy balance for the wall canbe expressed as:

dEwall /dt = 0 for steady operation,since there is no change in thetemperature of the wall with time atany point.

Therefore the rate of heat transferthrough the wall must be constant,Qcond, wall = constant


For one-dimensional steady heatconduction through the wall, the Fourierslaw of heat conduction can be expressedas:

The rate of heat conduction through aplane wall is proportional to the averagethermal conductivity k, the wall area A,and the temperature difference T2-T1, butis inversely proportional to the wallthickness L.

If more than one material is present,

The heat flow may be written as

Solving simultaneously, we get


THERMAL RESISTANCE CONCEPT

The heat transfer rate may be considered as a flow, and thecombination of thermal conductivity, thickness of material, and areaas a resistance to this flow.

The expression for heat conduction through a plane wall can bearranged as

where

is the thermal resistance of the wall against heat conduction or simplythe conduction resistance of the wall.




Analogy between thermal andelectrical resistance concepts.

The use of electrical analog islimited to systems through whichthe rate of heat transfer remainsconstant, i.e. to systems involvingsteady heat transfer with no heatgeneration.



For convection heat transfer from a solid surface at Ts to a fluid whosetemperature sufficiently far from the surface is T, Newtons law ofcooling for convection heat transfer rate is expressed as

It can be rearranged as

where

is the thermal resistance of the surface

against heat convection, or simply the

convection resistance of the surface.

The rate of radiation heat transfer between a surface of emissivity and area As at temperature Ts and the surrounding surfaces at someaverage temperature Tsurr can be expressed as

where

is the thermal resistance of a surface against radiation, or the radiationresistance, and

is the radiation heat transfer coefficient



A surface exposed to the surrounding airinvolves convection and radiationsimultaneously.

The total heat transfer at the surface isdetermined by adding (or subtracting, if inthe opposite direction) the radiation andconvection components.

When Tsurr T, the radiation effect canproperly be accounted for by replacing h inthe convection resistance relation by

Modelling and Solving Heater Transfer Problem

1. Normally the systems in our world can be modeled using a rectangular, a

cylindrical, and a spherical coordinate because of their geometries.

2. Once the system is modeled, and the initial and boundary conditions are

determined. The rest is solving the mathematical equations.

3. When the geometry of the system or the conditions are more complex,

numerical simulation is required.

Wall made of concrete

The wall thickness is 20cm

AC

AC maintains the temperature

of the room at 25C

Out side temperature is at 36C

What is the temperature profile across the wall thickness?

The problem is a 1-D steady

state problem. 25C 36C


Consider a 0.8-m-high and 1.5-m-wideglass window with a thickness of 8 mmand a thermal conductivity of k=0.78W/m oC. Determine the steady rate ofheat transfer through this glass windowand the temperature of its innersurface for a day during which the roomis maintained at 20oC while thetemperature of the outdoors is -10oC.Take the heat transfer coefficients onthe inner and outer surfaces of thewindow to be h1=10 W/m

2 oC and h2=40W/m2 oC which includes the effects ofradiation.


The rate of steady heat transfer between two surfaces is equal tothe temperature difference divided by the total thermalresistance between those two surfaces.

The ratio of the temperature drop to the thermal resistanceacross any layer is constant, and thus the temperature dropacross any layer is proportional to the thermal resistance of thelayer.


General Thermal Resistance Networks

The thermal resistance concept or the electrical analogy can alsobe used to solve steady heat transfer problems that involveparallel layers or combined series-parallel arrangements.

Approximate solutions can be obtained by assuming one-dimensional heat transfer and using the thermal resistancenetwork



Heat loss through a composite wall

STEADY HEAT CONDUCTION IN CYLINDERS

Consider a long cylindrical layer (suchas a circular pipe) of inner radius r1,outer radius r2, length L, and averagethermal conductivity k.

two surfaces of the cylindrical layer aremaintained at constant temperatures T1and T2

no heat generation in the layer

thermal conductivity is constant.


Fouriers law of heatconduction for heat transferthrough the cylindrical layercan be expressed as

For spheres, we simply repeat the analysis for cylinders and applyit to spherical layer by taking A= 4r2 and performing theintegrations.

The result can be expressed as where

STEADY HEAT CONDUCTION IN SPHERES

STEADY HEAT CONDUCTION IN CYLINDERS AND SPHERES

Consider steady one-dimensionalheat flow through a cylindrical orspherical layer that is exposed toconvection on both sides to fluids attemperatures T1 and T2 with heattransfer coefficients h1 and h2,respectively, as shown.

The rate of steady heat transfer canbe expresses as:

Cylinder

Sphere

STEADY HEAT CONDUCTION IN CYLINDERS AND SPHERES

For multilayered cylinder:

STEADY STATE HEAT CONDUCTION

Heat transfer through a medium is sometimes more conveniently expressed in terms of the overall heat transfer coefficient, U.

CRITICAL RADIUS OF INSULATION

Adding more insulation to a wall always decreases heattransfer. The thicker the insulation, the lower the heat transferrate because of the increase of the thermal resistance of thewall.

However, adding insulation to a cylindrical pipe or a sphericalshell, increases the conduction resistance of the insulationlayer but decreases the convection resistance of the surfacebecause of the increase in the outer surface area forconvection.

The heat transfer from the pipe may increase or decrease,depending on which effect dominates.

Note that the critical radius of insulation depends on the thermal conductivity of the insulation k and the external convection heat transfer coefficient h.

The rate of heat transfer from the cylinder increases with the addition of insulation for r2 < rcr, reaches a maximum when r2 = rcr, and starts to decrease for r2 > rcr.


THERMAL CONTACT RESISTANCE

When two surfaces are pressed against each other, an interface willcontain numerous air gaps of varying sizes that act as insulation becauseof the low thermal conductivity of air. Thus, an interface offers someresistance to heat transfer, and this resistance per unit interface area iscalled the thermal contact resistance, Rc.

The value of thermal contact resistance depends on the surfaceroughness and the material properties as well as the temperature andpressure at the interface and the type of fluid trapped at the interface.

Thermal contact resistance is observed to decrease with decreasingsurface roughness and increasing interface pressure.

The thermal contact resistance can be minimized by applying a thermallyconducting liquid called a thermal grease such as silicon oil on the surfacesbefore they are pressed against each other

3 Steady-state Conduction

Documents

Transcript of 3 Steady-state Conduction