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New Mexico State University

Mechanical & Aerospace Engineering Department

Experimental Methods II

ME 445

LAB Exercise-4

TIME FOR BRIDGEWIRE BREAK

4.1 Objective

To apply the principles of heat transfer to estimate the break time of a resistive wire through which a constant electric current is flowing.

 Through this experiment, students will theoretically estimate the time using energy balance equations.

 Apply linear regression to fit manufacturer’s data with the model to deduce unknown heat transfer parameters.

 Predictions will be verified or contradicted by experimental measurement.

4.2 Theoretical Background The physical representation of the problem is shown in the following figure:

Figure 4.0.1: Physical representation of the wire

A wire of length L and diameter D is considered. Due to the passage of electric current through the wire, heat is generated internally. If radiation and convective heat loss are presumed as the principal heat loss mechanisms, the energy balance for this problem, based on lumped mass for the wire and infinite length, can be written: Rate of change of Internal Energy (Qstored)

= Rate of Internal Energy Generation (Qgenerated) – Rate of Heat Loss (Qloss) In the above equation, note that the heat input is not considered since no heat is being supplied to the wire from its boundaries. Symbolically, we can write the energy equation as:

    

 TThATTAi dt

dT mc

wireswires

442 

(4.1)

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where, m = mass of the wire = density of the wire * volume of the wire = ρV Twire = Surface temperature of the wire

i = Current R = Resistance of the wire σ = Stefan-Boltzmann constant = 5.67 X 10

8 W/m

-2 K

-4

c = Specific heat capacity of the wire material ε = Emissivity of the wire As = Surface area of the wire

For present purposes the assumption is made that convection around the horizontal wire is fully

developed. The quantities m, c, I,  , σ, As and Ts are presumed known. However, uncertainty exists in the emissivity of the wire because of oxidized state is not precisely known, and the convection coefficient is known to vary somewhat with size, and mean temperature across the thermal boundary layer. For the case of the wire which is to be used in this experiment, the manufacturer has provided temperature versus current data for steady state. Hence, by using a multi-variable linear regression, it is possible to use this data along with the steady state energy equation, to obtain estimates for h and ε. However, when such an approach is taken, it is found that the value of ε exceeds unity, an impossible condition. In order to resolve this problem, one can deduce that the linear regression knows nothing about the laws of thermodynamics, but it is simply assigning a coefficient to the unknown which multiplies the highest power of temperature. Hence, the temperature dependence of the convection coefficient must be examined. To do this, consider free convection from a long horizontal cylinder.

m

DD CRa

k

Dh Nu 

where C and m are constants which take on different values for different ranges of the Raleigh number. By using the thermophysical properties of air over the temperature range expected for this experiment, i.e., room temperature up to wire melt, it can be demonstrated that h can be represented by the form

BTAh  or for convenience later,

  

 TTCCh wire21

where 

 TCCA 21

and 2

CB  . By using this functional relationship where there are now 3

fit parameters (C1, C2, ), it is possible to use data for the steady wire temperature versus

current to evaluate these constants and obtain a fit for <1. i.e., the three constants might vary slightly for different wire temperatures, hence assuming steady state, we will estimate these constants at different temperature intervals assuming the wire temperature is initially at room temperature and finally reaches the melting point of the wire. It should be noted that the resistance of the wire also changes with the wire temperature. From the data contained in the

(4.2)

(4.3)

(4.4)

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following picture from the manufacturer’s catalog for nichrome wire, for each wire size (AWG), a set of data is given which shows wire temperature versus the current that the wire conducts. This same information also shows that there is a slight change in resistivity of the wire as temperature increases. Therefore, the model predictions can use an average of wire resistivity for slightly improved predictions.

Figure 4.0.2: Information from Omega Electric Heaters Handbook (23)

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Figure 4.0.3: Information from Omega Electric Heaters Handbook (23)

To set up the linear regression, consider the steady state form for equation 4.1

    

 TThATTAi wireswires

442  (4.5)

Now, if the convection coefficient is replaced by equation 3.4, then the expression can be written

      

 TTATTCCTTAi wireswirewires

][ 21

442  (4.6)

Or      2 21

442

  TTACTTACTTAi

wireswireswires  (4.7)

which is of the form: y = m1x1+m2 x2 +m3 x3 where m1 can be identified as the emissivity, m2 can be identified as C1 and m3 can be identified as C2 . Therefore, LINEST function in Excel can be used, and the arrays y, x1, x2, and x3 can be generated from the data in the table. Once we find the constants for each wire temperature, the heat loss for each wire temperature, the energy equation can be used to find the following two times: Δt1 = Time for the wire to reach its melting temperature (Tmelt) from room temperature (Tœ) Δt2 = Time required to melt the wire completely when it is at its melting temperature.

The sum of these two times will give us the time to break the wire, providing the wire does not break until the wire is completely melted. Now, the energy equation can be integrated to yield T=f(t) up to the melt temperature. For heating, the following equation can be used:

      ][][ 1,,1

2

21

44

1

2

 

twiretwirewireswireswires TTmctTTACTTACTTAti  (4.8)

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The lower limit of integration will be room temperature, and the upper limit will be melt temperature. After melt temperature has been reached, equation (3.8) is no longer valid because temperature will become constant and the excess energy (generation-loss) will go into phase change.

      sfmeltsmeltsmwlts

mhtTTACTTACTTAti   2

2

21

44

2

2 ][ (4.9)

In the above equation, hsf is the latent heat of fusion for the wire. In order to “scope” the effect of heat losses, this can be done for: 1) the adiabatic case (no heat loss), 2) the case of radiant heat loss only, 3) the case of both radiant and convection heat loss, and d) the case of radiant and conductive (to surrounding air) heat loss. The time for bridgewire break is then the sum of the time required for the wire to heat from its initial temperature to melt temperature plus the time required to melt. Schematically, the three cases should appear something like

Figure 4.0.4: Schematic of total time to melt for 3 cases of heat loss

Note that when the wire reaches the fully melted state, i.e. liquid, surface tension will cause the melted cylindrical shape to form into spheres and then begin falling due to gravity. When the spheres form, the electrical continuity will be broken. This implies that the wire may break before the wire melts completely. This is what we observe in the experiment. If we represent the fraction of the wire which has melted as ψ, then the rate of phase change can be written

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lossheatofRatei dt

d mh

sf 

2

During melt, temperature will remain constant. If an additional assumption is made that the wire resistance does not change as the wire melts (conjecture), then this equation can be integrated between ψ = 0 and ψ = 1 to yield the time interval required for melting. As an example, plot the temperature-time history of a 30 AWG copper wire that is passing 25 amps of current, for 5 seconds. The spreadsheet set-up is indicated in the table below. The following figure shows the computed temperature of the wire as a function of time.

Table 4.1 Spreadsheet set up for a 30 AWG copper wire that is passing 25 amps of current, for 5 seconds.

(4.10)

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Figure 4.0.5: Example of heat loss case with conduction and radiation.

4.3 Experiment

The experiment is conducted with two different wires and the wire break time will be calculated in both prelab and full lab.

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Figure 4.6: Bridgewire Break setup

4.4 Procedure

An electrically resistive wire will increase in temperature with the passage of electric current. If the current is sufficiently high, the wire will heat to melt and break electrical continuity. It is the objective of this experiment to analytically model the process for a given current and wire size, and from this information, to predict break time. This will be completed during the prelab exercise. Three cases for heat loss to the surroundings should be considered: 1) adiabatic case, which will yield the shortest time, but is most analytically tractable, 2) radiation exchange with surroundings only, 3) radiation to surroundings plus convection to air. If there is interest, a fourth case can be considered; radiation to surroundings plus conduction to the air. There are two conditions which need to be considered: 1)time to heat from initial temperature to melt, and 2) condition (1) plus the time required for the wire to pass from incipient melt to complete melt. To carry out this prediction, it will be necessary to estimate values for convection coefficient and emissivity of the wire from temperature versus current data for a similar wire size (from Omega Electric Heaters Handbook) along with a multivariable linear regression (LINEST in EXCEL). By using this handbook data and the steady state heat balance for the wire, the power generated can be considered the dependent variable, with three independent variables. Then during the laboratory experiment, a series of physical experiments will be conducted to measure the actual time for bridgewire break. Statistical methods will be applied to the results. The experimental value will then be compared to the predicted time. Discrepancies between predictions and measurements will be discussed as to reasonable cause.

4.5 Laboratory exercise Theoretical Calculations

 From energy conservation principles, write a differential equation which describes the system thermal inertia, the energy generation and the energy dissipation terms assuming both radiative and convective heat losses.

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 For different wire temperatures estimate the emissivity and the convection coefficient.

 Using the constants found at each time interval, run your model find both the heating and melt times. The break time should then be the total of these two times. In your model, you should consider three cases: 1) no heat loss from the wire, 2) only radiation heat loss, 3) radiation and convection heat loss.

 Assuming an average C1, C2 and emissivity based on the values obtained in excel, estimate Δt1 assuming Twire,t = Tmelt and Twire,t-1 = Tœ. Compare it with the times you estimated

 Plot the transient temperature versus time for the three cases listed above and indicate your estimated break time on each plot. (Ultimately, your experimental results will be compared to these results and then you should be able to make conclusions as to which heat dissipation terms are important.) All of this work should be included in your report.

Experimental Procedure

 Connect data recorder (either computer data acquisition system or oscilloscope) and power supplies to the apparatus and familiarize yourself with the set-up.

 Adjust the current supply to the specified amperage.

 Configure recorder to suit your experiment (i.e. sweep time, voltage range, etc.).

 Turn power off and connect bridgewire to posts.

 Conduct experiment and when wire has broken, stop recorder.

 Measure break time from record.

 Unlock scope and turn off power supply.

 Repeat steps 4-7 for 10 trials. Once the calculations are complete and the results are plotted, show them to your TA and get his/her signature.

4.6 Report

 Provide the procedure you used to estimate the break time for different heat loss conditions used. Which condition best represents our experiment? Why?

 Calculate the standard deviation and mean reading for your data set obtained in the experiments.

 Comment on the uncertainties associated with the experimental data and the calculated results.

 Assuming your theoretical model is accurate, estimate mass of wire that is melted.

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 Make sure that your report provides all pertinent data, including wire size and electric current and discusses the analysis and results which were used in the pre-lab to make break time predictions.

4.7 Safety & Cleaning

 Please follow all the safety procedure inside the LAB

 Do the experiments with caution

 Clean the set up area when you are done with the experiment

 Put the tools back to the original location from where did you bring those