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Response lag in thermal radiation calorimeters
SotarEnergy, Vol. 13,pp. 277-281.
P~gamonPress, 1971.
PrintedinGreatBriUtin
TECHNICAL
NOTE
Response Lag in Thermal Radiation Calorimeters J. M. DAVIES, R. J. G O F F and P. H. PETER*
(Received 22 May 1970) IN A SMALL copper disk calorimeter, the peak response tends to lag behind the incident radiation. A large part of the lag is due to the relatively high resistance to heat flow in the solder used to fasten the thermocouple wires to the disk; it can be reduced significantly by reducing the amount of solder. The calorimeters used to measure the radiation from the U.S. A r m y Natick Laboratories' solar furnace[l] and other sources have been described by Cotton and Levine[2]. They are similar but not identical to the devices used at the U.S. Naval Radiological Defense Laboratory [3, 4] and at the U.S. Naval Applied Science Laboratory. The essential part of the device is a small copper disk, 1-13 cm in diameter (area = I cm 2) or larger, and 0.2 cm thick. Two thermocouple wires are inserted through the back surface and soldered to the interior of the disk. The wire at the center is constantan; the one nearer the edge is copper. The front surface is blackened; usually the surface is cupric oxide produced by treatment with Ebonoi C Special, and for that material the a v e r a ~ reflectance for the Sun's radiation is ~ 0-025[5]. The disk is mounted on three steel ~ held by a brass ring. T h e edges are bevelled so that converging radiation strikes only the front surface. T h e complete calorimeter is shown in Fig. 1 and the cross section o f the disk in Fig. 2. The device has a fairly quick response, and correction for heat loss during exposure can be made simply and presumably is fairly accurate. The calorimeter is convenient to use and reliable but there have been minor difficulties. F o r example, Cotton and Levine found the output e.m.f, continued to rise after the radiation stopped. F o r pulses as short as i sec, the delay in the peak response was ~ 2 sec and the rise was ~ 2½ per cent above the level at the end of the pulse; the delay was < ¼ sec for a 5-sec pulse. This response is illustrated in Fig. 3A, for a disk calorimeter exposed to the radiation from a 5000-W incandescent lamp with an ellipsoidal mirror system[6]. The exposure time was ~ 0.8 sec. The thermocouple e.m.f, was measured with a Sanborn model 350 preamplifier with a connecting network to a model 150 recording galvanometer; the overall response time was ~ 0-005 sec. In Fig. 3A the delay was about 2 sec. Calculations by Cotton and Levine indicated that flow of heat in the disk should cause a delay of no longer than 0-01 sec. This report discusses the causes of the delay and indicates some remedies. Various possibilities can be suggested to account for the delay, l f t h e thin fast acting shutters get hot before the exposure, heat could be transferred from them to the calorimeter after the exposure. This effect was kept to a minimum by effectively shielding the shutters. Possibly, the black coating could get much hotter than the body of the disk because of high thermal resistance in the coating or between the coating and the copper. Transfer of this heat could occur over a relatively long time, but it is difficult to see how sufficient energy could he stored in the very thin coating to account for a rise of this magnitude. The major effect seems to arise from the disturbance caused by the thermocouple wire. The wire and the solder have lower thermal conductivity than the copper disk; the values for copper, constantan and solder are 0.94, 0.055 and 0.08 to 0-11 cal cm -1 °C -1 see -1, respectively [7]. Because of the lower conductivity of the wire and solder, on exposure the part of the disk immediately in front of them will become hotter than the rest of the disk. This excess heat will be distributed throughout the disk fairly rapidly, *Pioneering Research Laboratory, U.S. Army Natick Laboratories, Natick, Mass. 01760, U.S.A. 277 S.E. Vol 13 No 2.1
278
DAVIES, R. J. G O F F and P. H. PETER
J.M.
\
,~----A--A'
/
I
o.fo2
0.031~ 0-079
4
=
I
Section A - - A
Section
0.025
A'm~
Fig. 2. Cross section of the disk (dimensions in cm).
A
Fig. 3. Response curves: A, a typical early model. B, a later model with less thermal resistance from disk to thermocouple wire (a, shutter opened; b, shutter closed).
Response lag in thermal radiation calorimeters
279
in less than 0.07 sec, as will be evident later. Heat will flow to the thermocouple junction at a rate determined by the thermal resistance at the contact area and in the wire. The error caused by this disturbance in estimating surface temperatures, was calculated by Masters and Stein[8] and Beck and Hurwicz[9], taking into account the reduced heat capacity near the surface of the body but not the reduced heat flow into and in the couple. We are concerned with the error in estimating the average temperature of the disk. A n effort was made to improve the contact between the wire and the copper by making the wire fit tightly into the hole and using a minimum of solder. The small holes were drilled with a no. 83 drill, ~ 0.012 in. (0.030 cm) in diameter. The constantan wire in the center hole is 0.010 in. (0.025 cm) in diameter. A n attempt was made to grind the ends of the wires flat and drill the holes with a flat bottom, but the latter effort was not successful. As an alternate method, the hole was drilled, as shown in Fig. 2, with the usual drill angle and the wire was tapered to fit this hole by machining in a jeweler's lathe. T h e wire was tinned with a very thin coating of solder and inserted in the hole with the solder and disk hot, forcing excess solder out. With this minimum amount of solder, a typical response curve is shown in Fig. 3B. There was no rise in e.m.f, after the shutter closed, but the response was not ideal since the e.m.f, did not start to decrease immediately. It took about 2 sec to attain the steady rate of decrease; obviously some heat was still being transferred to the junction after the shutter closed. Other small disk systems were made, taking all of the above precautions except the wires were not tapered. The response curves were somewhat similar to that in Fig. 3B; there was only a slight rise, about J of that in Fig. 3A. These resuRs indicate that a major part of the lag was due to rather high resistance to thermal flow into the thermocouple wire, and much of it can be overcome by reducing this resistance; there still remains a small but significant lag. 0.565 h
h
h
h
)'68 2h
h
h
h
1 I
h
"h
h
h
h
h
h
h
I
l
l
I l
0-2
0'4.97
I i
!
i
0"034 h
0.034~
h
ill
n
n
h
h
n
h
~'~
0-531
hi-
0,531 z - 0.497 z 2(0"531) 0"2
Fig. 4. Diagram of considerations involved in the use of Eq. (2).
280
J . M . DAVIES, R. J. G O F F and P. H. PETER
In looking for other causes, it seemed possible that the bevelled edge might be responsible. The situation is represented in Fig. 4. In Fig. 4 A the absorbed power per unit area h due to irradiance H (h = a l l , where a is the absorptance) is uniform over the face of the disk. The ring of triangular cross section will momentarily get hotter than the area of uniform thickness; the rise in temperature will be about twice that of the rest of the disk. The situation is about the same as in Fig. 4B where the triangular section is replaced by a rectangular section of half the width, having the same heat capacity, and the absorbed power is uniform and equal to h for 0 < r < ( a - - w) and is equal to 2h for (a - w) < r < ( a - - w/2), where r is the distance from the center, a the radius to the edge of the bevel and w the width of the bevel. The temperature rise will be very nearly the same as in Fig. 4C, where the excess power over the face of the small ring is replaced by an equivalent hypothetical radial power over the edge of the disk, given by ht =~ h (w/2d) ( 1 - w/4a) in which d is the thickness of the disk. The temperature rise can now be considered as the sum of two effects: (1) that from uniform power input h over the face, and (2) that from uniform input power over the area of the edge. F r o m the former, the temperature will rise quickly, with a lag of no more than 0.01 sec. F r o m the latter, the rise may be somewhat slower. It can be calculated from an equation given by Carslaw and Jaeger[10]. At r = 0.
AT = ( htatlk ) { 2a t/a12 --¼ -- 2 ~ exp (-- a'y2t/al z) /(?,2Jo (y,) }
( 1)
where k = conductivity, ~x= diffusivity, ax is the effective radius and y, are the positive roots of Jl (3') =~ 0. With the values for copper and for at and ht, A T ----0"0098 {8"298t--¼-- 2 [exp (-- 60.9155t) / 14.6820 Jo (3"8317) + exp (-- 204.2074 t)/19.2185 Jo (7.0156) + exp ( - 429.4192 t)/103-4995 3"o(10" 1734) ] }.
(2)
The series converges fairly rapidly and no further terms are needed. F o r a I-sec pulse, assuming no heat loss, this calculation shows a slight temperature rise after the shutter closes but it is within 0-03 per cent of the steady value after 0.07 sec; with heat loss, the temperature will start to decrease at that time or sooner. The time lag will be even less than indicated because the very hot edge of the bevel will lose heat rather rapidly and the increase in temperature at the center will be less than that given by Eq. (2). The calculated time delay is very short compared to the effects of interest here, i.e. the effect of the bevel edge is not important in accounting for the observed lag. The thermal resistance between the copper disk and the constantan wire seems to be responsible for much of the difficulty. Considering the dimensions involved, probably with extreme care, the delay can be reduced still further but it is likely that an appreciable amount will remain. The significance of these results is two fold. Firstly, the decreased lag reduces the uncertainty in the determination of the loss during exposure. It has been assumed that the rate of loss of heat during exposure is the same as after exposure. The latter is small but not negligible, as is evident in Fig. 2. The rate of loss is a complicated function of temperature, but within reasonable limits, after the lag, the loss is linear with time. The time to reach the constant rate is rather long, and Cotton and Levine recommended using the slope for 5 < t < 10 sec for extrapolation of the half-exposure time. The slope in Fig. 3B is about the same as in Fig. 3A, but the steady rate is reached sooner. The extrapolation can be made using the slope for 2 < t < 7 and this reduces the uncertainty significantly. Secondly, and perhaps equally important, knowing the factors involved in the lag gives a better sense of satisfaction about the reliability of the measurement. Something more needs to be said about the rate of heat loss. This loss includes at least three mechanisms; re-radiation, convection and flow through the supporting needles. The relative magnitudes are uncertain but, as shown by Cotton and Levine, the radiation loss is small. The convection loss may be ~ to ½of the total. It has been assumed that the rate ofioss/°C is about the same during exposure as after. Klei[11] has shown that the convection loss varies greatly with time; it is very large at first and then decreases rapidly to a nearly steady state condition. F o r the calorimeter disk during exposure there is no steady state; the temperature is steadily increas-
Response lag in thermal radiation calorimeters
281
ing. Whether the rate of loss is high or low is not known. Since the convection loss is a large part of the total, the rate of loss during exposure may be much greater than the indicated value after exposure. REFERENCES [1] E. S. Cotton, W. P. Lynch, W. Zagieboylo and John M. Davies, United Nations Conference on New Sources of Energy, Rome, 5 May 1961. [2] E. S. Cotton and A. Levine, Quartermaster Research and Engineering Center, Pioneering Research Division, Natick, Mass., Report T-28, 29 August 1960. [3] A. Broido and A. B. Willoughby, U.S. Naval Radiological Defense Laboratory, TR-35, 1 February 1955. [4] Private communications. [5] J. M. Davies, R. J. Goffand P. H. Peter, Appl. Opticsg, 1473 (1970). [6] J. M. Davies, Rev. Scient. Instrum. 41, 1040 (1970). [7] Metals Handbook, American Society of Metals, Cleveland, 1948, pp. 903, 1062,964. [8] J. I. Masters and S. Stein, RS127, 1065 0956). [9] J. V. Beck and H. Hurwicz.J. Heat Transfer82, 27 0960). [10] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edn, p. 328, IIl. Clarendon Press, Oxford 0959). [11] H. E. Klei, B.S. Thesis, 1957, Massachusetts Institute of Technology. A Study of Unsteady State Natural Convection for a Vertical Plate.
P~gamonPress, 1971.
PrintedinGreatBriUtin
TECHNICAL
NOTE
Response Lag in Thermal Radiation Calorimeters J. M. DAVIES, R. J. G O F F and P. H. PETER*
(Received 22 May 1970) IN A SMALL copper disk calorimeter, the peak response tends to lag behind the incident radiation. A large part of the lag is due to the relatively high resistance to heat flow in the solder used to fasten the thermocouple wires to the disk; it can be reduced significantly by reducing the amount of solder. The calorimeters used to measure the radiation from the U.S. A r m y Natick Laboratories' solar furnace[l] and other sources have been described by Cotton and Levine[2]. They are similar but not identical to the devices used at the U.S. Naval Radiological Defense Laboratory [3, 4] and at the U.S. Naval Applied Science Laboratory. The essential part of the device is a small copper disk, 1-13 cm in diameter (area = I cm 2) or larger, and 0.2 cm thick. Two thermocouple wires are inserted through the back surface and soldered to the interior of the disk. The wire at the center is constantan; the one nearer the edge is copper. The front surface is blackened; usually the surface is cupric oxide produced by treatment with Ebonoi C Special, and for that material the a v e r a ~ reflectance for the Sun's radiation is ~ 0-025[5]. The disk is mounted on three steel ~ held by a brass ring. T h e edges are bevelled so that converging radiation strikes only the front surface. T h e complete calorimeter is shown in Fig. 1 and the cross section o f the disk in Fig. 2. The device has a fairly quick response, and correction for heat loss during exposure can be made simply and presumably is fairly accurate. The calorimeter is convenient to use and reliable but there have been minor difficulties. F o r example, Cotton and Levine found the output e.m.f, continued to rise after the radiation stopped. F o r pulses as short as i sec, the delay in the peak response was ~ 2 sec and the rise was ~ 2½ per cent above the level at the end of the pulse; the delay was < ¼ sec for a 5-sec pulse. This response is illustrated in Fig. 3A, for a disk calorimeter exposed to the radiation from a 5000-W incandescent lamp with an ellipsoidal mirror system[6]. The exposure time was ~ 0.8 sec. The thermocouple e.m.f, was measured with a Sanborn model 350 preamplifier with a connecting network to a model 150 recording galvanometer; the overall response time was ~ 0-005 sec. In Fig. 3A the delay was about 2 sec. Calculations by Cotton and Levine indicated that flow of heat in the disk should cause a delay of no longer than 0-01 sec. This report discusses the causes of the delay and indicates some remedies. Various possibilities can be suggested to account for the delay, l f t h e thin fast acting shutters get hot before the exposure, heat could be transferred from them to the calorimeter after the exposure. This effect was kept to a minimum by effectively shielding the shutters. Possibly, the black coating could get much hotter than the body of the disk because of high thermal resistance in the coating or between the coating and the copper. Transfer of this heat could occur over a relatively long time, but it is difficult to see how sufficient energy could he stored in the very thin coating to account for a rise of this magnitude. The major effect seems to arise from the disturbance caused by the thermocouple wire. The wire and the solder have lower thermal conductivity than the copper disk; the values for copper, constantan and solder are 0.94, 0.055 and 0.08 to 0-11 cal cm -1 °C -1 see -1, respectively [7]. Because of the lower conductivity of the wire and solder, on exposure the part of the disk immediately in front of them will become hotter than the rest of the disk. This excess heat will be distributed throughout the disk fairly rapidly, *Pioneering Research Laboratory, U.S. Army Natick Laboratories, Natick, Mass. 01760, U.S.A. 277 S.E. Vol 13 No 2.1
278
DAVIES, R. J. G O F F and P. H. PETER
J.M.
\
,~----A--A'
/
I
o.fo2
0.031~ 0-079
4
=
I
Section A - - A
Section
0.025
A'm~
Fig. 2. Cross section of the disk (dimensions in cm).
A
Fig. 3. Response curves: A, a typical early model. B, a later model with less thermal resistance from disk to thermocouple wire (a, shutter opened; b, shutter closed).
Response lag in thermal radiation calorimeters
279
in less than 0.07 sec, as will be evident later. Heat will flow to the thermocouple junction at a rate determined by the thermal resistance at the contact area and in the wire. The error caused by this disturbance in estimating surface temperatures, was calculated by Masters and Stein[8] and Beck and Hurwicz[9], taking into account the reduced heat capacity near the surface of the body but not the reduced heat flow into and in the couple. We are concerned with the error in estimating the average temperature of the disk. A n effort was made to improve the contact between the wire and the copper by making the wire fit tightly into the hole and using a minimum of solder. The small holes were drilled with a no. 83 drill, ~ 0.012 in. (0.030 cm) in diameter. The constantan wire in the center hole is 0.010 in. (0.025 cm) in diameter. A n attempt was made to grind the ends of the wires flat and drill the holes with a flat bottom, but the latter effort was not successful. As an alternate method, the hole was drilled, as shown in Fig. 2, with the usual drill angle and the wire was tapered to fit this hole by machining in a jeweler's lathe. T h e wire was tinned with a very thin coating of solder and inserted in the hole with the solder and disk hot, forcing excess solder out. With this minimum amount of solder, a typical response curve is shown in Fig. 3B. There was no rise in e.m.f, after the shutter closed, but the response was not ideal since the e.m.f, did not start to decrease immediately. It took about 2 sec to attain the steady rate of decrease; obviously some heat was still being transferred to the junction after the shutter closed. Other small disk systems were made, taking all of the above precautions except the wires were not tapered. The response curves were somewhat similar to that in Fig. 3B; there was only a slight rise, about J of that in Fig. 3A. These resuRs indicate that a major part of the lag was due to rather high resistance to thermal flow into the thermocouple wire, and much of it can be overcome by reducing this resistance; there still remains a small but significant lag. 0.565 h
h
h
h
)'68 2h
h
h
h
1 I
h
"h
h
h
h
h
h
h
I
l
l
I l
0-2
0'4.97
I i
!
i
0"034 h
0.034~
h
ill
n
n
h
h
n
h
~'~
0-531
hi-
0,531 z - 0.497 z 2(0"531) 0"2
Fig. 4. Diagram of considerations involved in the use of Eq. (2).
280
J . M . DAVIES, R. J. G O F F and P. H. PETER
In looking for other causes, it seemed possible that the bevelled edge might be responsible. The situation is represented in Fig. 4. In Fig. 4 A the absorbed power per unit area h due to irradiance H (h = a l l , where a is the absorptance) is uniform over the face of the disk. The ring of triangular cross section will momentarily get hotter than the area of uniform thickness; the rise in temperature will be about twice that of the rest of the disk. The situation is about the same as in Fig. 4B where the triangular section is replaced by a rectangular section of half the width, having the same heat capacity, and the absorbed power is uniform and equal to h for 0 < r < ( a - - w) and is equal to 2h for (a - w) < r < ( a - - w/2), where r is the distance from the center, a the radius to the edge of the bevel and w the width of the bevel. The temperature rise will be very nearly the same as in Fig. 4C, where the excess power over the face of the small ring is replaced by an equivalent hypothetical radial power over the edge of the disk, given by ht =~ h (w/2d) ( 1 - w/4a) in which d is the thickness of the disk. The temperature rise can now be considered as the sum of two effects: (1) that from uniform power input h over the face, and (2) that from uniform input power over the area of the edge. F r o m the former, the temperature will rise quickly, with a lag of no more than 0.01 sec. F r o m the latter, the rise may be somewhat slower. It can be calculated from an equation given by Carslaw and Jaeger[10]. At r = 0.
AT = ( htatlk ) { 2a t/a12 --¼ -- 2 ~ exp (-- a'y2t/al z) /(?,2Jo (y,) }
( 1)
where k = conductivity, ~x= diffusivity, ax is the effective radius and y, are the positive roots of Jl (3') =~ 0. With the values for copper and for at and ht, A T ----0"0098 {8"298t--¼-- 2 [exp (-- 60.9155t) / 14.6820 Jo (3"8317) + exp (-- 204.2074 t)/19.2185 Jo (7.0156) + exp ( - 429.4192 t)/103-4995 3"o(10" 1734) ] }.
(2)
The series converges fairly rapidly and no further terms are needed. F o r a I-sec pulse, assuming no heat loss, this calculation shows a slight temperature rise after the shutter closes but it is within 0-03 per cent of the steady value after 0.07 sec; with heat loss, the temperature will start to decrease at that time or sooner. The time lag will be even less than indicated because the very hot edge of the bevel will lose heat rather rapidly and the increase in temperature at the center will be less than that given by Eq. (2). The calculated time delay is very short compared to the effects of interest here, i.e. the effect of the bevel edge is not important in accounting for the observed lag. The thermal resistance between the copper disk and the constantan wire seems to be responsible for much of the difficulty. Considering the dimensions involved, probably with extreme care, the delay can be reduced still further but it is likely that an appreciable amount will remain. The significance of these results is two fold. Firstly, the decreased lag reduces the uncertainty in the determination of the loss during exposure. It has been assumed that the rate of loss of heat during exposure is the same as after exposure. The latter is small but not negligible, as is evident in Fig. 2. The rate of loss is a complicated function of temperature, but within reasonable limits, after the lag, the loss is linear with time. The time to reach the constant rate is rather long, and Cotton and Levine recommended using the slope for 5 < t < 10 sec for extrapolation of the half-exposure time. The slope in Fig. 3B is about the same as in Fig. 3A, but the steady rate is reached sooner. The extrapolation can be made using the slope for 2 < t < 7 and this reduces the uncertainty significantly. Secondly, and perhaps equally important, knowing the factors involved in the lag gives a better sense of satisfaction about the reliability of the measurement. Something more needs to be said about the rate of heat loss. This loss includes at least three mechanisms; re-radiation, convection and flow through the supporting needles. The relative magnitudes are uncertain but, as shown by Cotton and Levine, the radiation loss is small. The convection loss may be ~ to ½of the total. It has been assumed that the rate ofioss/°C is about the same during exposure as after. Klei[11] has shown that the convection loss varies greatly with time; it is very large at first and then decreases rapidly to a nearly steady state condition. F o r the calorimeter disk during exposure there is no steady state; the temperature is steadily increas-
Response lag in thermal radiation calorimeters
281
ing. Whether the rate of loss is high or low is not known. Since the convection loss is a large part of the total, the rate of loss during exposure may be much greater than the indicated value after exposure. REFERENCES [1] E. S. Cotton, W. P. Lynch, W. Zagieboylo and John M. Davies, United Nations Conference on New Sources of Energy, Rome, 5 May 1961. [2] E. S. Cotton and A. Levine, Quartermaster Research and Engineering Center, Pioneering Research Division, Natick, Mass., Report T-28, 29 August 1960. [3] A. Broido and A. B. Willoughby, U.S. Naval Radiological Defense Laboratory, TR-35, 1 February 1955. [4] Private communications. [5] J. M. Davies, R. J. Goffand P. H. Peter, Appl. Opticsg, 1473 (1970). [6] J. M. Davies, Rev. Scient. Instrum. 41, 1040 (1970). [7] Metals Handbook, American Society of Metals, Cleveland, 1948, pp. 903, 1062,964. [8] J. I. Masters and S. Stein, RS127, 1065 0956). [9] J. V. Beck and H. Hurwicz.J. Heat Transfer82, 27 0960). [10] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edn, p. 328, IIl. Clarendon Press, Oxford 0959). [11] H. E. Klei, B.S. Thesis, 1957, Massachusetts Institute of Technology. A Study of Unsteady State Natural Convection for a Vertical Plate.