and electrical potential through the anode coal in a Hallâ. Héroult cell. ...... J. Gibbs: The Scientific Papers of J. Willard Gibbs, Dover Publishing,. New York, NY ...
Peltier Effects in Electrode Carbon ELLEN MARIE HANSEN, ESPEN EGNER, and SIGNE KJELSTRUP The thermoelectric power of a cell with platinum electrodes and a carbon conductor was determined. The electromotive force (emf) was measured as a function of the temperature difference between the electrodes at temperatures varying from 310 7C to 970 7C. From these measurements, the transported entropy of electric charge in carbon was found to vary from 21.7 to 21.9 J/(K mole) at temperatures around 300 7C, from 22.0 to 22.3 J/(K mole) at temperatures around 550 7C, and from 23.4 to 23.7 J/(K mole) at temperatures around 950 7C. This transported entropy had not before been determined for temperatures above 550 7C. Also, it is shown how the previously neglected surface properties can be taken into account to interpret the measurements. In the Hall–He´roult cell, the anode is made of a similar kind of carbon. Hence, the transported entropy found above can be used to describe the often neglected coupling between transport of heat and electric charge in this electrode. It is shown that the calculated electric potential profile through a coal sample will change significantly if the coupling is neglected, but the calculated temperature profile is independent of whether the coupling is neglected. New equations are also developed that can be used to evaluate the importance of the coupling in other systems.
I.
INTRODUCTION
MOST aluminum production is made in Hall–He´roult cells. According to Grjotheim et al.,[1] a typical value for the total energy consumption in the cell is 14 kWh/kgAl. In contrast, the theoretical minimum energy requirement is 7 kWh/kgAl. The aluminum industry is, hence, interested in minimizing the excess energy consumption. A method to describe this excess energy consumption is nonequilibrium thermodynamics (NT), as described by de Groot and Mazur,[2] and later by Førland et al.[3] In this work, NT will be used to calculate profiles of temperature and electrical potential through the anode coal in a Hall– He´roult cell. Once this is known, the excess energy consumption, or dissipated energy, in the anode coal can be calculated. This knowledge of the locations and sources of the dissipated energy is the first step toward reducing it. To calculate the profiles of temperature and electrical potential, a coupling coefficient between these two kinds of transport is needed. It is called the Peltier effect. If the Peltier effect is divided by temperature, the transported entropy of electric charge results. The transported entropy has been determined in several materials before, such as in salt melts, in solid salts, or in metals. In salts, the electric charge is carried by ions, and the transported entropy is typically around 100 J/(K mole). One example is the transported entropy of copper ions in liquid CuCl, determined to be 109 5 1 J/(K mole) at 900 K by Pezzati et al.[4] The transported entropy in solid salts are of the same magnitude as in molten salts, one example being the transported entropy of copELLEN MARIE HANSEN, formerly Research Assistant, Department of Physical Chemistry, The Norwegian University of Science and Technology, is Project Engineer, Prediktor, P.O. Box 296, N-1601 Fredrikstad, Norway. ESPEN EGNER, formerly Master’s Student, Department of Physical Chemistry, The Norwegian University of Science and Technology, is Project Engineer, Saga Petroleum ASA, N-1301 Sandvika, Norway. SIGNE KJELSTRUP, Professor, is with the Department of Physical Chemistry, The Norwegian University of Science and Technology, N-7034 Trondheim, Norway. Manuscript submitted November 7, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS B
per ions in solid CuCl at 573 K, which was determined to be 136 5 2 J/(K mole) by Mogilevski and Usmanov.[5] R. Haase[6] and J. Agar[7] have tabulated transported entropies in several metallic conductors. One example is the transported entropy in solid silver, which was found to be 20.20 J/(K mole) at 800 K. The values for several other electronic conductors, like lead, copper, and platinum, are of the same magnitude. The determination of the transported entropy of electric charge in carbon was done in a cell where carbon was the conductor, and the electrodes were made of platinum. The Peltier effect is related to the processes taking place at the electrode-conductor interfaces, so a special form of NT (NT for surfaces) is used.[8] A preliminary form of parts of this work, relating to carbon as an electrode, was presented earlier.[9]
II.
THEORY
A. Introduction to Nonequilibrium Thermodynamics for Surfaces According to NT, the transport equations for a system with transport of heat (Jq) and of electric charge (j) are a set of so-called coupled flux equations, giving the fluxes in terms of the thermodynamic forces: ¹T 2 Lq j ¹f T ¹T j 5 2Ljq 2 Ljj ¹f T
Jq 5 2Lqq
[1] [2]
with Lik being the phenomenological coefficients, T the temperature, and f the electrical potential. According to Onsager,[10,11] the cross-coefficients are equal, in this case: Lq j 5 Ljq
[3]
In terms of the fluxes (Ji) and forces (Xi), the dissipated energy (D) of a system of volume V is given by VOLUME 29B, FEBRUARY 1998—69
flux (Jq) and the electric current density (j). All fluxes and gradients are assumed to be one-dimensional, and the flux equations are for the bulk electrodes and the bulk conductor i Jiq 5 2Lqq
j 5 2Lijq
j 5 2Lsljm Fig. 2—The experimental system.
i
(Tsl 2 Tml) (Tcl 2 Tsl) sl 2 Lcc ml T Tcl 2 Lslcj (fcl 2 fml)
[8]
(Tsl 2 Tml) (Tcl 2 Tsl) 2 Ljcsl ml T Tcl 2 Lsljj (fcl 2 fml)
[9]
and for the right electrode surface, [4]
Nonequilibrium thermodynamics assumes linear relations between fluxes and forces. An NT theory can be defined for two-dimensional systems (surfaces) in essentially the same manner as for bulk systems. The conservation laws differ, however. The excess properties of the surface are central in this context. Gibbs[12] has already introduced this concept (Figure 1). Here, the excess surface concentration (G) is represented by the shaded area. The excess concentration is the surplus of what would have been at the surface if the bulk phases a and b had been continued all through the surface. A typicallength scale for the figure is 1 nm, making the excess surfae concentration appear as a discontinuity on a scale of, e.g., 1 mm. In this work, the excess surface concept will also be used for electric and thermal properties. B. Interpretation of the Measurements A sketch of the experimental system is shown in Figure 2. The system consists of a variable source of electrical potential, two platinum wires (the electrodes) and a carbon sample (the conductor). At the potential source, both electrodes have the same temperature (T0). Close to the electrode-conductor interface, the electrodes have the temperatures Tml and Tmr, respectively. These are the measured temperatures. To consider the relation between the measured temperatures and the measured electromotive force (emf), the system is divided into five subsystems; the bulk left electrode, the left electrode surface, the bulk conductor, the right electrode surface, and the bulk right electrode. In all subsystems, the only fluxes present are the heat 70—VOLUME 29B, FEBRUARY 1998
[6]
[7]
sl Jclqn 5 2Lcm
i i
1 dT df z 2 Ljji T dx dx
(Tsl 2 Tml) (Tcl 2 Tsl) sl 2 Lmc ml T Tcl 2 Lslmj (fcl 2 fml)
sl Jml qn 5 2Lmm
* ~Σ X J ! dV
[5]
The superscript i can be either l (for the left electrode), c (for the conductor), or r (for the right electrode). For the left electrode surface, the linear laws are
Fig. 1—Explanation of excess surface property.
D5
1 dT df z 2 Lqi j T dx dx
(Tmr 2 Tsr) (Tsr 2 Tcr) sr 2 Lmc mr T Tcr 2 Lsrmj (fmr 2 fcr)
[10]
(Tmr 2 Tsr) (Tsr 2 Tcr) sr 2 Lcc mr T Tcr 2 Lsrcj (fmr 2 fcr)
[11]
(Tmr 2 Tsr) (Tsr 2 Tcr) 2 Lsrjc mr T Tcr 2 Lsrjj (fmr 2 fcr)
[12]
sr Jmr qn 5 2Lmm
sr Jcrqn 5 2Lcm
j 5 2Lsrjm
The subscript n means that the heat fluxes are normal to the interface, and m refers to the electrodes. When the electric current density is zero, the emf can be found from Eqs. [6], [9], and [12]: Liq j dT df 5 2 i Ljj T
[13]
sl Lmj Lslcj (T cl 2 T sl ) (T sl 2 T ml ) (f cl 2 f ml ) 5 2 sl 2 Ljj T ml Lsljj T cl
[14]
Lsrmj (T mr 2 T sr) Lsrcj (T sr 2 T cr) (f mr 2 f cr ) 5 2 sr 2 Ljj T mr Ljjsr T cr
[15]
Before these equations can be used, an interpretation of the coefficients must be given. For the bulk phases, we consider the heat flux coupled to the electric current density when all driving forces, except the gradient in electrical potential, are zero. By dividing Eq. [5] by Eq. [6], we get the Peltier heat for the bulk materials:
~ ! J iq j
dT/dx 5 0
5
Liq j Lijj
[16]
METALLURGICAL AND MATERIALS TRANSACTIONS B
Table I. Property Ash content Baking temperature Graphitization ratio Real density Apparent density Thermal conductivity Specific electrical resistance
Data for the Carbon Types Used
Unit
5BGN
BN
HC10
pct 7C — g/cm3 g/cm3 W/(m K) Vmm
0.3[13] 3000[14] 0.26[14] 2.14[13] 1.65[13] 46[13] 14[13]
0.3[13] 2300[14] — 2.20[13] 1.62[13] 110[13] 11[13]
1[25] 2300[14] 0.23[14] 2.14[25] 1.64[25] 30[25] 20[25]
By introducing Eqs. [17] through [20] into Eqs. [13] through [15], integrating, and rearranging, we obtain F(f ml 2 f 0l ) 5 2S *Pt(T ml 2 T 0)
[21]
F(f 2 f ) 5 2S *Pt(T 2 T ) 2 S *C (T 2 T )
[22]
F(f 2 f ) 5 2S *C (T 2 T )
[23]
F(f mr 2 f cr) 5 2S *Pt (T mr 2 T sr) 2 S *C (T sr 2 T cr)
[24]
F(f 0r 2 f mr) 5 2S *Pt (T 0 2 T mr)
[25]
cl
ml
cr
sl
cl
ml
cr
cl
sl
cl
where f 2 f is the measured electrical potential difference between the positions where the temperature at both electrodes is T0 (Figure 2). An expression for the total emf of the cell is obtained by adding Eqs. [21] through [25]: 0r
0l
F(f 0r 2 f 0l ) 5 (S *Pt 2 S *C ) (T sr 2 T sl )
[26]
It is seen that if the interface temperatures (T ) are equal to the measured temperatures (Tmk), we get the classical expression, which can be found from Førland et al.:[3] sk
F(f 0r 2 f 0l ) 5 (S *Pt 2 S *C ) (T r 2 T l ) where T 5 T sk
Fig. 3—The Kanthal oven used in the experiments.
Looking at Figure 2, we see that in the electrodes, the energy, or heat, transported by the electric current is given by the transported entropy in platinum (S*Pt). In the conductor it is given by the transported entropy in carbon (S*C ).
~ !
5
Lkq j S *Pt T k 5 k Ljj F
[17]
~ !
5
Lcq j S *C T c 5 c Ljj F
[18]
J kq j J cq j
dT/dx 5 0
dT/dx 5 0
The superscript k can be either r (for the right electrode), or l (for the left electrode). Similar considerations can be made at the surfaces, and the results are the surface Peltier heats:
~Jj !
5
sk Lmj S *Pt T mk 5 Lskjj F
[19]
~Jj !
5
Lcjsk S *C T ck 5 Lskjj F
[20]
m qn
DT m 5 DT c 5 0
c qn
DT m 5 DT c 5 0
where F is the Faraday’s constant, and DTm or DTc is the temperature difference across the electrode or conductor side of the interface, respectively. The transported entropies are here assumed to be independent of temperature, because the differences involved are rather small (0 to 15 K). METALLURGICAL AND MATERIALS TRANSACTIONS B
mk
5T
k
[27] [28]
Equation [27] shall be used to evaluate the measurements, as too little is known about the surface properties to make good assumptions concerning the temperature differences Tsk 2 Tmk. In Section IV–B, calculations are performed to show how sensitive the results for S*C are to the surface properties. To do this, equations for the temperature difference across the interfaces are needed. These are given in Appendix A. III.
EXPERIMENTAL
Measurements were performed on three different carbon materials (5BGN, BN, and HC10), all of which are generally used as electrode materials in industrial cells. The HC10 contained some salt after having been used, whereas the 5BGN and BN were new. Table I shows some data for the different carbon types. Note that carbon in the form used here is a semiconductor. Carbon-type 5BGN is graphitic,[13] whereas type BN is graphitized,[13] and type HC10 is semigraphitic.[14] Figure 3 shows a sketch of the oven used, a normal Kanthal oven for use up to 1300 7C made as described by Motzfeldt.[15] In the figure, (1) is a cooling system for the oven (a thin copper pipe with water flowing through it), (2) is the insulating material, (3) is a Kanthal Al heating wire, (4) is a carbon stick (the conductor), (5) is an extra heating element with Kanthal Al heating wire, (6) is a type-S thermoelement VOLUME 29B, FEBRUARY 1998—71
connected to the control panel (PAPST Type 8550 N), and (7) is a type-S thermoelement for measurement of Tmr and Tml, with platinum electrodes for measurement of Df. Argon gas was carefully added from the bottom of the oven to prevent oxidizing of the carbon samples at high temperatures. The flow rate was controlled by a flowmeter (SHO-RATE 1355/D2A4AIC5F000). Carbon sticks of 15-cm length and 2-cm diameter were prepared. In each stick, two small holes were drilled close to the ends. The thermocouples were put into the small holes. There were copper wires between the cooling bath and the multimeter. Alundum cement was used to fasten the thermoelements to the carbon stick. As a reference for the measurements, a mixture of ice and water in a vacuum bottle was used, placed at the connection between the thermoelement threads and the copper threads. A Hewlett-Packard 3457A multimeter was used as a voltmeter. It was connected to a personal computer, which registered measurements every minute. The platinum threads of the thermocouples were also used as electrodes for the cell. IV.
Table II.
Measured and Calculated Values with T mr Just Above 300 7C; S *Pt Is 21.1 J/(K Mole)
Type of Carbon
7.2 5 1.4 6.6 5 1.3 8.5 5 1.8 10 4.5
5BGN BN HC10 Loebner[18] La Rosa[17]
Table III.
Df/DT (mV/K)
T mc (7C) 310 315 320 300 320
S *C [J/(K mole)]
55 55 5 10 5 10
21.8 21.7 21.9 22.0 21.5
5 5 5 5 5
0.1 0.1 0.2 0.1 0.1
Measured and Calculated Values with T mr Just Below 550 7C; S *Pt Is 21.4 J/(K Mole)
Type of Carbon
Df/DT (mV/K) 8.8 5 1.4 7.1 5 0.9 5.8 5 0.8 7.6
5BGN BN HC10 La Rosa[17]
T mc (7C) 545 540 550 535
5 5 5 5
S *C [J/(K mole)] 5 5 5 25
22.1 22.3 22.0 22.1
5 5 5 5
0.1 0.1 0.1 0.1
RESULTS AND CALCULATIONS
A. Transported Entropy in Carbon Values for Df/DT were found by linear regression, and the values are given in Tables II through IV. The tables also contain values for S*C calculated using values for S*Pt from Moore and Graves.[16] Table II shows the results from measurements with Tmr just above 300 7C. The mean temperatures are rounded off to the nearest 5 7C. The numbers after 5 in the column for Tmr give the temperature intervals in which Tmr was varied. The two lower lines in the table contain values calculated from literature data. Table III shows the results from the measurements where Tmr was somewhat below 550 7C. The last line contains values obtained from La Rosa.[17] Loebner[18] did no experiments at such high temperatures. Table IV shows the results from measurements with Tmr around 950 7C. The results show that the transported entropy of electric charge in carbon (S* C ) has a negative value. It is also seen that the absolute value of S*C increases with increasing temperature, as can be expected. Comparison between the samples HC10 and BN or 5BGN does not show any significant influence from the type of carbon used. We shall, therefore, also use these values for anode carbon in Hall–He´roult cells. B. Sensitivity of the Results to Surface Properties To test the sensitivity of the results toward the surface properties, the transported entropy of carbon was calculated with different assumptions concerning the thermal conductivities at the interfaces; refer to the appendix for details. This was done for one carbon type at each temperature interval. The results demonstrated that, with reasonable assumptions about the conductivities at the surfaces, the error from neglecting the special properties of the interfaces is smaller than the experimental uncertainties reported here. In doing these calculations, we assumed the thermal conductivity in the electrodes to be 78.2 J/(m s K),[19] the ther72—VOLUME 29B, FEBRUARY 1998
Table IV. Measured and Calculated Values with T mr Around 950 7C; S *Pt Is 22.0 J/(K Mole) Type of Carbon 5BGN BN
Df/DT (mV/K)
T mc (7C)
S *C [J/(K mole)]
14.5 5 4.7 17.8 5 5.4
970 5 15 940 5 10
23.4 5 0.5 23.7 5 0.5
mal conductivity of the conductor to be 46 J/(m s K),[13] and the corresponding surface conductivities to be between 1027 and 1 times these values. It can be seen from Eqs. [52] through [56], which follow, that the result is caused by large thermal conductivities in the electrodes. As a consequence, the difference between the temperature in the electrode close to the electrode-conductor interface and the temperature of the interface is negligible, independent of the properties of the interface. C. Profiles in the Anode of the Hall–He´roult Cell The distribution of temperature and electric potential in a coal sample with a certain heat and charge flux was calculated. Only one dimension was considered, and the calculations were performed by both neglecting and including the Peltier effect. The difference between the two sets of results is a measure of the importance of the Peltier effect. To accomplish the calculations, typical values for the heat flux, electric current density, thermal conductivity, electrical conductivity, and temperature at one end were chosen. Approximate values are shown as follows: heat flux 1000 W/m2 current density 1000 A/m2 thermal conductivity 10 W/(m K) electrical conductivity 20,000 V21m21 temperature 20 7C The basis for the calculations are Eqs. [1] to [2], while the coefficients are found from Eqs. [29] through [31]. METALLURGICAL AND MATERIALS TRANSACTIONS B
~
S * jx exp ~ 1 ! S* j Fl ! S* j Jq
T 5 T0 2
f 2 f0 5
C
C
~SF*lJ C
q
2
Jq
[34]
C
!
j x k
[35]
From Eq. [35], it can be seen that the impact of the Peltier heat on the distribution of electric potential is determined by the ratio of S *C Jq j to Fl k In this example, S *C Jqk Fjl
5 0.056
Regarding the temperature profile, the situation is more complicated (Eq. [34]). To simplify the expression, the approximation
Fig. 4—Electric potential profile through the coal sample.
ey ' 1 1 y
[36]
valid for small y may be used. Equation [34] can then be written
~
T 5 T0 1 1
!
Jq S *C j x 2 x Fl l
[37]
From this, it can be seen that the Peltier heat is negligible if S *C jx ,, 1 Fl In this example, max
Fig. 5—Example of a plot of Df against DT.
~Jj !
5
~¹TJ !
5 Lqq 2
q
¹T 5 0
q
j 5 0
~¹jf!
Jq 5 0
Lq j S *CT 5 Ljj F Lq2 j 5 lT Ljj
L2q j 5 Ljj 2 5k Lqq
[29] [30]
Jq x l
j f 2 f0 5 2 x k The general result is METALLURGICAL AND MATERIALS TRANSACTIONS B
C
A graphical presentation of the results is given in Figure 4. The impact of the Peltier effect on the temperature was negligible. On the other hand, a change in the electrical potential profile is observed when the Peltier effect is neglected. V.
DISCUSSION
A. Experimental Results [31]
where l is the thermal conductivity and k is the electrical conductivity. When Eqs. [1] and [2] are rearranged and integrated, the following equations for the distribution of temperature and electric potential result when the Peltier effect is neglected: T 5 T0 2
~SF*ljx! 5 0.003
[32] [33]
Uncertainties for the results were given in Tables II through IV. One of the reasons for the changing standard deviation from temperature to temperature is that the number of measurements was not the same at all temperatures, this being for practical reasons. The standard deviations are larger at higher temperatures, indicating structural changes in the platinum, as discussed subsequently. The plot of Df against DT did not cross the origin (Figure 5). We believe that this is due to other emF sources than the temperature difference, e.g. impurities or the contact potentials between the materials. The error will not change the slope of the line Df vs DT, but merely move it parallel to the ordinate. The phase diagram of platinum and carbon, e.g., that provided by Massalski,[20] shows that carbon will dissolve VOLUME 29B, FEBRUARY 1998—73
in platinum at high temperatures. Loebner[18] reported increased brittleness of the parts of his platinum threads that had been close to the carbon samples. This was observed when the overall temperature was 800 K for prolonged intervals of time, and he ascribed it to diffusion of carbon into the platinum threads. We observed that the part of the platinum thread which had been in direct contact with the carbon sample was more brittle after use than the other parts of the thread. On the other hand, the thorough investigations of Tammann and Scho¨nert[21] into diffusion of carbon into metals showed that carbon would not diffuse into platinum at temperatures below 980 7C. This was true for experiments that lasted 4 to 5 hours. The values for S*Pt were taken from Moore and Graves.[16] They reported very small uncertainties, and even by taking into account the large uncertainty in the temperatures, the resulting uncertainties in S* Pt are smaller than 50.05 J/(K mole). Hence, these values are given without uncertainties. Loebner[18] reported an uncertainty of less than 2 pct, whereas La Rosa[17] did not give any relevant uncertainties. One cannot draw conclusions as to how the value for S* C is influenced by whether the electrode carbon has been used in an electrolysis cell. B. Comparison to Literature Values The measurements by Haase[6] and Agar[7] show that the transported entropy and, thereby, the Peltier heat, are of the same magnitude as for silver, i.e., S* ' (50.2)J/(K mole), for several electronic conductors (lead, copper, and platinum). For salts, the Peltier heat has a larger value. Our value, around 22 J/(K mole), which agrees with Loebner[18] and La Rosa,[17] is a likely result, considering that carbon is not as good a conductor of electrons as metals are. The transport of electric current through carbon is a combination of a flux of negative electrons and a flux of positive holes or defects. For the materials studied here, the electrons are the dominant current carriers, evident from the negative values for S*C . The values are rather small, compared to molar entropies, indicating that the dominance is not a big one. Hence, impurities or structural changes are apt to have a significant influence on S* C by shifting the balance between the current carriers somewhat. Still, we did not observe differences between the carbon materials studied. This may be because the carbon types used as electrodes are quite similar. The increase in absolute value by increasing temperature is due to the increased thermal energy, leading to increased movement of the electrons, which again leads to more possible positions for the electrons. The new values for S* C are comparable to those obtained by Loebner[18] and La Rosa.[17] Loebner measured Df/DT for a cell similar to the one used here as a function of the baking temperature of the carbon samples. He found a clear correlation, and explained this by structural changes in carbon just above 2000 7C. Carbon samples treated at this temperature gave higher Df/DT than samples treated at both higher and lower temperatures. The values from Loebner[18] cited in Table II are for soft carbon which has undergone heat treatment at 1400 7C. These were chosen as they were the best-documented values. The samples BN and 5BGN were both baked at 2300 7C, and should, hence, according to Loebner, show larger 74—VOLUME 29B, FEBRUARY 1998
values for Df/DT. The opposite tendency is seen in Table II, but this is not statistically significant. If the samples had been treated at temperatures as high as 3000 7C, Loebner predicted lower values. La Rosa’s samples were treated at 800 7C, which according to Loebner should lead to lower values for Df/DT than for the samples treated at 1400 7C or 2300 7C. This tendency can indeed be seen in Tables II and III. A question is whether some of this effect may be caused by impurities which were removed by use of La Rosa’s procedure, but not by Loebner’s. Loebner investigated this, and concluded that impurities in ordinary graphitized or graphitic materials could not be seen to have any significant effect on the measurements. C. Comments on the Profiles in Anode Carbon The calculations presented in Figure 4 were performed using the average transported entropy of carbon from Table II. The baking temperature of anode coal is typically 900 7C to 1300 7C. Due to the small total number of measurements, and the fact that Loebner’s predictions are partly contradicted by the measurements presented here, all five values are taken into account. The uncertainties in the average values of S* C at 300 7C and 550 7C, calculated from Tables II and III, do not give reason to state that the transported entropy is larger at 550 7C than at 300 7C. Hence, the transported entropy was kept constant in the calculations. The thermal and electrical conductivities are also kept constant, mostly to not complicate the equations unnecessarily. Calculations were also performed with conductivities varying with temperature, as given by Log[22] and Delhae`s and Carmona.[23] The resulting temperature profile was then somewhat less steep, but still, the effect of including or omitting the Peltier effect was negligible. The resulting profile in electrical potential was about twice as steep as in Figure 4, and the influence of the Peltier effect was somewhat less pronounced, though still significant. It is seen from Eq. [35] that the gradient in electrical potential will decrease if the transported entropy of electric current can somehow be made smaller. This will decrease the dissipated energy. The experiments show that the transported entropy of electric current does not change observably between different carbon types. Hence, in a search for better carbon as anode material, the Peltier effect can be neglected. Still, the dissipated energy can be lowered by using materials with high conductivities. If materials other than carbon are considered, the transported entropy should also be taken into account, and a material with low transported entropy of electric current will tend to give less excess energy consumption. Today, the dissipated energy associated with the Peltier effect in anode carbon is about 0.05 kWh/kgAl, out of a total excess energy consumption of about 7 kWh/kgAl, i.e., 0.7 pct. The term is directly proportional to the transported entropy of electric charge, so finding a material with a transported entropy of electric charge of about 21 J/(K mole) would reduce this contribution to below 0.4 pct. This is indeed a small contribution, compared to what could be gained by, e.g., changing the thermal conductivities. Still, the dissipated energy is today approaching a lower, theoretical limit, so contributions of a magnitude of 1 pct of the total are getting increased attention. METALLURGICAL AND MATERIALS TRANSACTIONS B
VI.
CONCLUSIONS
In the search for electrode materials other than carbon, the Peltier effect should be taken into account. A material with low Peltier effect will lead to less excess energy consumption. It is also shown how values for the transported entropy of electric charge can be used to predict profiles of temperature and electric potential in anode carbon, used in the aluminum electrolysis. The coupling coefficient between transport of heat and electric charge in carbon was determined experimentally. The transported entropy varied from 21.7 to 21.9 J/(K mole) at temperatures just above 300 7C, from 22.0 to 22.3 J/(K mole) at temperatures around 550 7C, and from 23.4 to 23.7 J/(K mole) at temperatures around 950 7C. In the interpretation of the experiment, special properties of the electrode-conductor interfaces can safely be neglected.
One of the authors (EMH) is grateful to the EXPOMAT program and the Norwegian aluminum industry for a research grant.
Temperatures at the conductor-electrode interfaces The measurements were performed at steady state, with zero electric current, in which case energy is conserved, and cl c cr mr r J lq 5 J ml qn 5 J qn 5 J q 5 J qn 5 J qn 5 J q 5 Jq
[38]
The difference or gradient in electrical potential can now be eliminated from Eqs. [5], [7], [8], [10], and [11] by the help of Eqs. [6], [9], and [12]. When j is set equal to zero, the results are: l cqq cr Jq 5 2 (T 2 T cl ) Dx sl 5 2l slmm (T sl 2 T ml ) 2 l mc (T cl 2 T sl ) 5 2l
(T 2 T ) 2 l (T 2 T ) ml
sl cc
cl
sl
[39] [40] [41]
sr 5 2l srmm (T mr 2 T sr) 2 l mc (T sr 2 T cr)
[42]
5 2l
[43]
sr cm
(T
mr
2 T ) 2 l (T 2 T ) sr
where lik 5 Lik 2
[46]
Lsurf
[47]
L5
ka abulk z 1 2 ka d
[48]
The thermal conductivities are assumed to be measured in the situation of no electric current.
APPENDIX
sl
abulk d asurf ka z abulk 5 5 d d
Lbulk 5
Finally, Eqs. [46] and [47] can be inserted into Eq. [45] to get a new expression for the excess surface coefficient L:
ACKNOWLEDGMENT
sl cm
coefficient would have if the surface was equal to the bulk, regarding the property of interest. The L coefficients can be described in more detail by relating them to some conductivity function a, which can be the thermal conductivity times temperature, some diffusivity function, etc., and to d, the surface thickness. The total surface conductivity function is assumed to be proportional to the corresponding bulk value, with a proportionality factor ka. A k value of zero would mean that the surface acts as a total barrier toward transport of this kind, whereas a value of unity would mean that the surface is equal to the bulk in this respect.
sr cc
sr
Lij Ljk Ljj
cr
[44]
ki li z 5 l skii 1 2 ki d
[49]
c lc 5 l cc
[50]
Index k can be either l (for the left electrode) or r (for the right electrode), and index i can be either m (for the electrodes) or c (for the conductor). The factor ki represents the fraction of the surface conductivity to the bulk conductivity, and both interfaces are assumed to differ from the bulk phases in the same way. Assuming that l smc 5 0
[51]
i.e., no coupling of heat transport across the interface, we may rearrange Eqs. [39] through [43] to get Jq T mr 2 T sr 5 2 f m k m l /d
[52]
Jq T sr 2 T cr 5 2 f c k c l /d
[53]
Jq T cl 2 T cr 5 2 c l /Dx Jq T sl 2 T cl 5 2 f c k c l /d
[54]
and Dx is the length of the conductor. To use these equations, an interpretation of the coefficients is needed. The meaning of excess surface coefficients can be explained by the following equations.[24] First, it is stated that the resistance toward a certain kind of transport can be seen as a sum of the contribution the bulk phase would have given and the excess surface contribution. Solved for the excess surface contribution, the result is
Jq T ml 2 T sl 5 2 f m k m l /d
1 1 1 5 surf 2 bulk L L L
These equations may be added and further rearranged to get an expression for Jq:
[45]
Here, L represents excess surface coefficients; Lsurf is the total surface coefficient; and Lbulk is the value the surface METALLURGICAL AND MATERIALS TRANSACTIONS B
where k fi 5
ki 1 2 ki
T mr 2 T ml Jq 5 2 f m (2d)/(k m l ) 1 (2d)/(k cf lc) 1 Dx/le
[55] [56]
[57]
[58]
VOLUME 29B, FEBRUARY 1998—75
U a d k l p f D F J L S* T V X j l x
NOMENCLATURE produced entropy, J/(K m3) conductivity function surface thickness, m electrical conductivity, 1/(V m) thermal conductivity, J/(K m s) Peltier heat, J/mole electric potential, J/C dissipated energy, J/(m3s) Faraday’s constant, C/mole flux Onsager coefficient transported entropy, J/(K mole) temperature, K volume, m3 driving force electric current density, C/(s m2) coefficient position, m
Dx
conductor length, m
Subscripts and Superscripts c j l n q r s 0
conductor electric left electrode normal heat right electrode surface initial, starting point REFERENCES
1. K. Grjotheim, C. Krohn, M. Malinovsky´, K. Matiasˇovsky´, and J. Thonstad: Aluminium Electrolysis, Fundamentals of the Hall–He´roult Process, 2nd ed., Aluminium-Verlag, Du¨sseldorf, 1982, p. 112.
76—VOLUME 29B, FEBRUARY 1998
2. S. de Groot and P. Mazur: Non-Equilibrium Thermodynamics, NorthHolland, Amsterdam, 1962, p. 57. 3. K. Førland, T. Førland, and S. Kjelstrup Ratkje: Irreversible Thermodynamics, Theory and Applications, John Wiley & Sons, Chichester, 1988, pp. 40 and 58. 4. E. Pezzati, C. Margheritris, and A. Schiraldi: Z. Naturforsch., 1975, vol. 30, p. 388. 5. B. Mogilevski and O. Usmanov: E´lektrokhimiya, 1967, vol. 3, p. 1124. 6. R. Haase: Thermodynamics of Irreversible Processes, AddisonWesley, Reading, MA, 1969, p. 344. 7. J. Agar: in Advances in Electrochemistry and Electrochemical Engineering, P. Delahay, ed., Interscience Publishers, New York, 1963, vol. 3, p. 70. 8. D. Bedeaux: Advances in Chemical Physics, 1986, vol. 64, pp. 47109. 9. E. Hansen, S. Kjelstrup Ratkje, B. Flem, and D. Bedeaux: in Light Metals, W. Hale, ed., TMS, Warrendale, PA, 1996, pp. 211-18. 10. L. Onsager: Phys. Rev., 1931, vol. 37, pp. 405-26. 11. L. Onsager: Phys. Rev., 1931, vol. 38, pp. 2265-79. 12. J. Gibbs: The Scientific Papers of J. Willard Gibbs, Dover Publishing, New York, NY, 1961, pp. 96 and 224. 13. S.C. Group: Cathode Blocks of Carbon and Graphite for the Aluminium Industry, product information, 1995. 14. I. Nyga˚rd and S. Kjelstrup Ratkje: Mechanisms of Bottom Heave in Carbon Cathodes in Aluminium Electrolysis, Technical Report, The Norwegian Institute of Technology, Trondheim, Norway, 1994. 15. K. Motzfeldt: in Physiochemical Measurements at High Temperatures, J. Bockris, J. White, and J. Mackenzie, eds., Butterworth and Co., London, 1960, pp. 47-86. 16. J. Moore and R. Graves: J. Appl. Phys., 1973, vol. 44, p. 1174. 17. M. La Rosa: Nuovo Cimento, 1916, vol. 12, p. 284. 18. E. Loebner: Phys. Rev., 1956, vol. 102, pp. 46-57. 19. C. Lide: CRC Handbook of Chemistry and Physics, 66th ed., CRC Press, Boca Raton, FL, 1988, p. E9. 20. T.B. Massalski: Binary Alloy Phase Diagrams, AMS INTERNATIONAL, Metals Park, OH, 1990, p. 874. 21. G. Tammann and K. Scho¨nert: Z. Anorg. Chem., 1922, vol. 122, pp. 27-43. 22. T. Log: Ph.D. Thesis, The Norwegian Institute of Technology, Trondheim, Norway, 1989, p. 97. 23. P. Delhae`s and F. Carmona: Chem. Phys. Carbon, 1981, vol. 17, pp. 89-174. 24. E. Hansen and S. Kjelstrup: Light Metals, TMS, Warrendale, PA, 1997, pp. 355-60. 25. Carbon Savoie: Aluminium Cell Linings, product information, 1990.
METALLURGICAL AND MATERIALS TRANSACTIONS B
