Download Transient Conduction

ME 4331: Thermal Energy Engineering Laboratory
Transient Conduction Experiment
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Objectives
Determine the transient heat conduction characteristics of a copper bar.
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Apparatus
2.1
Equipment/Materials List
1. Copper bar
2. Type K thermocouples
3. Agilent Dual Output DC Power Supply
4. Agilent data acquisition unit
5. 10 W film heater
2.2
Transient Conduction experiment Apparatus
The transient conduction experiment apparatus is depicted in figure 1. An electric
current is applied to a film heater that is clamped to the end of a copper bar with
a C-clamp. The temperature at 8 different positions (each thermocouple is spaced
by 0.75 in and the first thermocouple is 0.375 in from the heated end of the bar) of
the copper bar is acquired with type K thermocouples which are connected to a data
acquisition unit. The temperatures are written to a data file using Labview.
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Figure 1: Transient conduction experiment apparatus
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3.1
Principals and Background
Transient conduction background
Many heat transfer problems are time dependent, i.e. transient. Consider a hot
metal billet that is removed from a furnace and exposed to a cool airstream. Energy
transfer by conduction occurs from the interior of the metal to the surface, and
the temperature at each point in the billet decreases until a stead-state condition is
reached. Such time-dependent effects occur in many industrial heating and cooling
processes.
3.2
Theory: Finite-Difference Method
Analytical solutions to transient problems are restricted to simple geometries and
boundary conditions. In most cases, it is much more convenient to solve the heat
equation using an iterative finite-difference approach. While this approach does not
yield an exact solution, performing these numerical computations with a computer
allows for reduced time steps and nodal sizes which result in approximate solutions
with negligible error.
Assume that the rectangular copper bar used in this lab is a one-dimensional rod,
i.e. the temperature gradients in the non-axial directions are negligible. Under transient conditions with constant properties and no internal generation, the appropriate
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form of the heat equation is
1 ∂T
∂ 2T
=
α ∂t
∂x2
where T is the temperature, α is the thermal diffusivity, t is the time, and x is
position in the axial direction. The bar can be discretized into nodes where
subscript of each temperature denotes a given node (see figure 2). By equating
(1)
the
the
the
Figure 2: Example of discretizing bar with a nodal energy balance
change in energy of an internal element to the energy transfer rates to and from the
element, the following finite difference equation can be derived in terms of Ti at time
t + ∆t
∆
k
(Ti+1 (t) + Ti−1 (t) − 2Ti (t))
Ti (t + ∆t) =
ρcp (∆x)2
2h (∆y + ∆z)
+
(T∞ (t) − Ti (t)) + Ti (t) (2)
∆y∆z
where ∆t is the time step per iteration, ρ is density, cp is the specific heat, k is
the thermal conductivity, h is the convective heat transfer coefficient, T∞ (t) is the
ambient temperature at time t, ∆x is the distance between nodes, and ∆y and ∆z
are the width and height of each element. The first term on the right side of equation
2 accounts for the heat in and out of the element due to conduction. The second
term on the right side accounts for heat transfer to the environment due to natural
convection.
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To solve equation 2 for all internal elements for different time points, initial and
boundary conditions need to be applied. The initial temperatures of each node
must be known. The node closest to the film heater has heat coming in equal
to the electrical power being supplied to the heater, i.e. the product of voltage
and current. Therefore, equation 2 must be modified for this node to include this
boundary condition. The last node is open to atmosphere, and either an adiabatic
or convective tip condition may be applied.
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Procedure
The objective of the experiment is to observe the transient conduction characteristics
of copper bar. This will be done by measuring the temperature at 8 different positions
of the copper bar while heating occurs at one end.
4.1
Calibration of initial temperature
To account for any bias errors or deviations of the initial nodal temperatures relative
to the ambient temperature, the initial temperatures will be acquired prior to the
heating of the bar.
• Open up ‘ME 4331 Transient Lab’ Labview file on the desktop.
• Turn on Agilent data acquisition unit.
• In Labview:
– Refresh I/O resource name
– 101:108 temperature channel list
– 3 seconds between reading
– Set file path as ‘Trans Cond Baseline’ (or anything relevent/unique to this
lab/section)
• Run the Labview file by clicking run button.
• Acquire approximately 20 samples by clicking the ‘Record Data’ button on the
front panel and acquire for about one minute.
• Click the ‘Record Data’ button again and the ‘End Program’ button to stop
the program.
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4.2
Heating of the copper bar
To characterize the transient conduction behavior of the copper bar, a heating element must be attached to one end of the copper bar. A circuit is set up to supply
the heating element with a known voltage and current.
• Change file name to ‘Trans Cond Exp’ (or anything besides the path used
previously).
• Run the program again.
• Turn on DC power Supply. Press ‘Output on/off’ button. Press ‘<’ to switch
to 10 digital display screen.
• Increase the voltage to 28 V. At the same time, start acquiring data.
• Record voltage and current of the DC Power Supply.
• Exit program once the first thermocouple (the one nearest to heating element)
reaches 35 ◦C.
• Save your data to a USB drive.
• Turn off DC Power Supply and keep Agilent Data Acquisition unit on.
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Data Analysis
• Calculate the average initial temperatures for each node.
• Using a Nusselt number correlation for natural convection, estimate the convective heat transfer coefficient for the copper bar.
• Using these initial temperatures, perform a finite-difference analysis of the
copper bar with and without a convection loss term. Is there a significant
difference between the two cases?
• Compare the finite-difference solution with the experimental solution by plotting temperature vs time for the different nodal positions and temperature vs
nodal position for different time points.
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References
[1] F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer.
Wiley, 5th ed., 2001.
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