Heat Flux and Heat Transfer Coefficient

Abstract

Heat flux and heat transfer coefficient may either be measured by a single measurement process, or they may require the measurement of several quantities before they may be estimated. Of course, it is best if the required quantity is directly measurable. We will give as much detail as possible, kee** in mind the target reader of the book. Many techniques are beyond the scope of the book and require direct reference to the appropriate research literature.

Exercise II

1.1 II.1   Temperature Measurement

 

Ex II.1::

Use library resources and write a note on International Practical Temperature Scale of 1968 (IPTS68) and International Temperature Scale of 1990 (ITS90). Bring out the major differences between these two.

Ex II.2::

Temperature of a gas of known composition may be measured by acoustic thermometry. Search the recent literature and prepare a short bibliography on this topic.

Discuss any one of the references from this bibliography in full detail.

Ex II.3::

The following data pertains to the measurement of the temperature of a certain freezing metal using a constant volume gas thermometer. The pressure at triple point and the corresponding pressure at the freezing point of the metal are tabulated below.

$p_{tp},\; \text {mm}$ Hg	15.6	45.7	112.3
$p_t,\; \text {mm}$ Hg	112.5	358.0	890.2

Determine the freezing point of the metal according to the ideal gas scale. Make a plot to indicate how the intercept is estimated from a quadratic fit to data in the form of pressure ratio versus pressure at the triple point.

Ex II.4::

The following readings were taken with a medium precision constant volume gas thermometer. The bulb of the thermometer was immersed in hot water at a constant temperature for these measurements. The amount of gas in the bulb was systematically varied to gather data.

$p_{tp},\; \text {mm}$ Hg	40	60	80	101	120
$p_t,\; \text {mm}$ Hg	51.2	76.7	102.1	128.7	152.6

Estimate the temperature of water bath using the above data.

Hint: Fit a straight line to data of pressure ratios. Use the precision of slope to specify an error bar to the temperature derived by you.

Ex II.5::

We would like to use a K type thermocouple to measure temperature in the range 30–1000$^{\,\circ } \text {C}$ with the reference junction at the ice point. It is desired that the resolution should be ±1$^{\,\circ } \text {C}$. Specify the range and least count of a voltmeter for this purpose.

Make use of thermocouple reference tables for answering this problem.

Ex II.6::

A K type thermocouple is used to measure the temperature of a system that is known to be $850^{\,\circ } \text {C}$. The potentiometer binding posts known to be at $31^{\,\circ } \text {C}$ act as the reference junction. What is the $\text {mV}$ reading indicated by the potentiometer? If an observer takes the $\text {mV}$ reading and calculates the temperature assuming the reference temperature to be the ice point, what will be the temperature error?

Make use of thermocouple reference tables for answering this problem.

Ex II.7::

Figure 6.19 shows the way connections have been made for measuring temperature using K type thermocouple. What is the correct temperature if the output is 16 mV?

Ex II.8::

A millivolt recorder is available with a range of 0–10 mV and an accuracy of 0.5% of full scale. A copper–constantan thermocouple pair with a reference junction at the ice point is used with the recorder. Determine the temperature range and the accuracy of the installation.

Note: For a copper–constantan thermocouple, 10 mV corresponds approximately to 213$^{\,\circ } \text {C}$ and the Seebeck coefficient is approximately $40\,\upmu \text {V}/^{\,\circ } \text {C}$.

Ex II.9::

For a certain thermocouple, the following table gives the thermoelectric output as a function of temperature $t^{\,\circ } \text {C}$. The reference junction is at the ice point.

$t,^\circ \text {C}$	10	20	30	40	50
$E,\; \text {mV}$	0.397	0.798	1.203	1.611	2.022
$t,^\circ \text {C}$	60	70	80	90	100
$E,\; \text {mV}$	2.436	2.850	3.266	3.681	4.095

The manufacturer of the thermocouple recommends that the following interpolation formula should be used for calculating E as a function of T:

$$\begin{aligned} E(t) = 0.03995t+5.8954\times 10^{-6}t^2-4.2015\times 10^{-9}t^3+1.3917\times 10^{-13}t^4 \end{aligned}$$

How good is the polynomial as a fit to data? Base your answer on all the statistical parameters and tests you can think of. Obtain the Seebeck coefficient at 50 and 100 $^{\,\circ } \text {C}$.

Ex II.10::

The table below gives data for a T type thermocouple. It has been decided to approximate the thermocouple Seebeck emf by the relation $E = 0.03865t+4.15018E-5 t^2$. What is the error in use of the approximation at the lowest and highest temperatures in the range shown in the table?

$t^{\,\circ } \text {C}$	50	55	60	65	70	75
$E,\;\text {mV}$	2.036	2.251	2.468	2.687	2.909	3.132
$t^{\,\circ } \text {C}$	80	85	90	95	100
$E,\;\text {mV}$	3.358	3.585	3.814	4.046	4.279

The above thermocouple is used as a differential thermocouple to measure the temperature difference between the inlet and the outlet to a small heat exchanger. The differential thermocouple gave an output of $0.125\; \text {mV}$. What is the temperature difference? Base your answer on the Seebeck coefficient at $75^{\,\circ } \text {C}$. What is the Peltier emf for the T type thermocouple in the above problem at $t= 75^{\,\circ } \text {C}$? A thermopile consisting of 40 cold and 40 hot junctions is made using T type thermocouple wires. The hot junctions are all maintained at $75^{\,\circ } \text {C}$ while the cold junctions are all at a temperature of $50^{\,\circ } \text {C}$. What will be the output of the thermopile in $\text {mV}$?

Ex II.11::

In a certain calibration experiment, the following data was obtained:

$t^{\,\circ } \text {C}$ (standard)	22.5	46.8	64.7	78.8	93.3	103.5	112
$t^{\,\circ } \text {C}$ (candidate)	22.8	46.5	64.2	79.2	91.9	102.9	113.2

Comment on the distribution of error between the candidate and the standard. Can you specify a suitable error bar within which the candidate is well represented by the standard?

Ex II.12::

Write a note on the constructional details of wire wound and film type Platinum Resistance Thermometers. Comment on their relative merits.

Ex II.13::

The resistance of a certain sensor varies with temperature as $R(t) = 500(1+0.005 t-2\times 10^{-5}t^2)$, where the resistance R(t) is in $\Omega$ and the temperature t is in $^\circ \text {C}$. Determine the resistance and the sensitivity at 100$^{\,\circ } \text {C}$.

Ex II.14::

A four-wire PRT100 is fed a current of $5\; \text {mA}$ by suitable external power connected using two of the four lead wires. A digital voltmeter is connected across the RTD by using the other two lead wires. The RTD is exposed to an ambient at $65^{\,\circ } \text {C}$. What will be the voltage reading? If the RTD has a dissipation constant of $0.001\; \text {W}/^{\circ } \text {C}$, what will be the voltage reading? Base your answers taking into account the Callendar correction.

Ex II.15::

A thermistor code named 20 A is in the form of a glass-coated bead 0.75 mm diameter. Its resistance at three different temperatures are given below.

$t, ^\circ \text {C}$	0	5.6	50
$R,\; \Omega$	8800	3100	1270

Determine $\beta$ for this thermistor. If resistance is measured with a possible error of $\pm 100\;\Omega$, what is the error in estimated t when the nominal value of resistance is 1$500\;\Omega$?

Redo the problem using the Steinhart–Hart model.

Ex II.16::

A thermistor has a resistance of $100\;\Omega$ at $20^{\,\circ } \text {C}$ and a resistance of $3.3\;\Omega$ at $100^{\,\circ } \text {C}$. Determine the value of $\beta$ for this thermistor. Determine also the ratio of the resistance of the thermistor when the temperature changes from $0\; \text {to}\; 50^{\,\circ } \text {C}$. What are the sensitivities of the thermistor at these two temperatures. Based on the above, make suitable comments, in general, on the use of thermistors.

Ex II.17::

A typical thermistor has the following specifications as given by the manufacturer.

Nominal Resistance at $25^{\,\circ } \text {C}$	$10\;\text {k}\Omega$
$\beta$ value ($25< t <85^{\,\circ } \text {C}$)	$3480\;\text {K}$
Dissipation Constant	$0.4\; \text {mW/K}$
Heat capacity	$1.3\; \text {mJ/K}$
Thermistor Diameter	$0.8\pm 0.1\; \text {mm}$
Thermistor Length	$1.4\pm 0.4\;\text {mm}$

Determine: (a) Ratio of resistance of thermistor $R_{25}/R_{85}$ and (b) Temperature error when the thermistor carries a current of $1\;\text {mA}$ and has a resistance value of $1500\;\Omega$.

Ex II.18::

The resistance of a certain temperature detector is given by the data given below.

Temperature, $^\circ \text {C}$	0	12.7	24.1	46.2	63.8	76.9	98.6
Resistance, $\Omega$	2757	2556	2354	1955	1691	1464	1054

Estimate the temperature when the resistance is (a) 1500, (b) 2000, and (c) 2600 $\Omega$. You are expected to develop a relation between the temperature and resistance and use it to get the answers.

Ex II.19::

A thermistor is connected in series with a variable resistance $R_s$ such that the current through the thermistor is limited to 1 mA when the battery voltage is 9 V and the thermistor is at a room temperature of 20$^{\,\circ } \text {C}$ (see Fig. 6.20). The thermistor is characterized by a resistance of 2 k$\Omega$ at 20$^{\,\circ } \text {C}$ and a $\beta$ value of 3000 $\text {K}$. Determine the value of the series resistance when the thermistor is at 20$^{\,\circ } \text {C}$. The thermistor is exposed to an environment at 40$^{\,\circ } \text {C}$. The variable resistance is to be adjusted such that the current is again 1 mA. What is the series resistance?

Ex II.20::

For a certain thermistor, $\beta = 3100$ K and the resistance at $70^{\,\circ } \text {C}$ is $998\pm 5\,\Omega$. The thermistor resistance is $1995\pm 4\,\Omega$ when exposed to a temperature environment that is to be estimated. Estimate the temperature and its uncertainty.

Ex II.21::

A thermistor manufacturer has provided the following data on a special order thermistor.

(a) Resistance at ice point: $R_0 = 2050\; \Omega$.

(b) “Beta” value for the thermistor: $\beta = 3027\; \text {K}$.

It was decided to verify this data by conducting a carefully conducted “calibration” experiment using a precision constant temperature bath capable of providing an environment within $\pm 0.02^{\,\circ } \text {C}$ of the set value. The following data was obtained from this experiment:

Temperature $^\circ \text {C}$	Resistance $\Omega$
22.3	890
46.7	404

The measured resistance values are within an error band of ${\pm } 0.5\,\Omega$. Based on the experimental data, comment on the “outcome” of the calibration experiment.

Ex II.22::

The thermistor of Example II.21 is connected in series with a standard resistor of $1000\;\Omega$. A well-regulated DC power of $9\; \text {V}$ is connected across the resistor–thermistor combination. The thermistor is subject to change in temperature in the range $5 {-} 80 ^{\,\circ } \text {C}$. Plot the output across the standard resistor in this range.

Ex II.23::

A thermistor is in the from of a glass-coated bead $0.75\;\text {mm}$ in diameter. Its resistance at two different temperatures is given as $R(t=0^{\,\circ } \text {C})=8800\;\Omega$ and $R(t=50^{\,\circ } \text {C})=1270\;\Omega$. Determine $\beta$ for this thermistor. If resistance is measured with an error of ${\pm }10\;\Omega$, what is the error in temperature when the nominal value of R is $1600\;\Omega$? What is the sensitivity of the thermistor at this temperature?

Ex II.24::

The resistance of a certain semiconductor material varies with temperature according to the relation $R(T) = 1200\text {e}^{\left[ 3100\left( \frac{1}{T}- \frac{1}{273.15}\right) \right] }$ where the resistance is in $\Omega$ and the temperature is specified in K. At a certain temperature, the resistance has been measured as $600\pm 10\;\Omega$. Determine the temperature and its uncertainty. Make a plot of R versus T in the range $273.15< T < 500$ K.

Ex II.25::

A thermistor follows the resistance temperature relationship given by $R(T) = 1500\text {e}^{\left[ 3000\left( \frac{1}{T}-\frac{1}{273.15}\right) \right] }$ where the resistance is in $\Omega$ and temperature is in K. This thermistor is connected in series with a resistance of $500\;\Omega$. A DC power supply of $1.5\; \text {V}$ is connected across the resistance–thermistor combination. A voltmeter measures the voltage across the series resistor. What is the voltage reading when the thermistor is at the ice point? What is the indicated voltage when the thermistor is at room temperature of $30^{\,\circ } \text {C}$? If the voltmeter used for measuring the voltage across the fixed resistance has a resolution of $1\;\text {mV}$, what is the resolution in temperature at $30^{\,\circ } \text {C}$?

Ex II.26::

An RTD with $\alpha = 0.0034/^{\,\circ } \text {C}$ and $R= 100\;\Omega$ at the ice point has a dissipation constant of $0.04\;\text {W}/^{\circ } \text {C}$. This RTD is used to indicate the temperature of a bath at $50^{\,\circ } \text {C}$. The RTD is part of a bridge circuit with all other resistances of $100\;\Omega$ each. The battery connected to the bridge provides $6\;\text {V}$. What voltage will appear across the RTD? Take into account the self-heating of the RTD due to the current that flows through it. When is the effect of self-heating more significant, at $t = 0^{\,\circ } \text {C}$ or at $t= 100^{\,\circ } \text {C}$? Explain.

Ex II.27::

A thermistor has a resistance of $1000\;\Omega$ at $20^{\,\circ } \text {C}$ and a resistance of $33\;\Omega$ at $100^{\,\circ } \text {C}$. Determine the value of $\beta$ for this thermistor. Determine also the ratio of resistance of the thermistor when the temperature changes from $0\;\text {to}\; 50^{\,\circ } \text {C}$. The above thermistor is connected in parallel with a resistance of $500\;\Omega$ and the parallel combination is connected in series with a resistance of $500\;\Omega$. A DC power supply of $9\; \text {V}$ is connected across the combination. If the thermistor is exposed to a process space at $40^{\,\circ } \text {C}$, what is the voltage across it? What is the temperature error if the dissipation constant is known to be $10\; \text {mW}/^{\circ } \text {C}$?

Ex II.28::

A certain incandescent solid is at a temperature of $1200\pm 5\;\text {K}$. The total emissivity of the object is known to be $0.85\pm$0.02. Estimate along with the uncertainty the brightness temperature of the object.

Ex II.29::

Estimate the actual temperature of an object whose brightness temperature has been measured using a vanishing filament pyrometer to be 1000$^{\,\circ } \text {C}$. The spectral emissivity of the object at 0.665 $\upmu \text {m}$ is 0.55. If the emissivity is subject to an error of 0.5%, what is the corresponding error in the estimated actual temperature?

Ex II.30::

The actual, brightness, and color temperatures of a target are, respectively, given by $1185\;\text {K}$, $1136\;\text {K}$ and $1212\;\text {K}$. The brightness temperature is at an effective wavelength of $0.66\;\upmu \text {m}$. The color temperature is based on the second wavelength of $0.45\;\upmu \text {m}$. Determine the emissivities at the two wavelengths.

Ex II.31::

Estimate the actual temperature of an object whose brightness temperature has been measured using a vanishing filament pyrometer to be $850^{\,\circ } \text {C}$. The transmittance of the pyrometer optics is known to be $0.95\pm 0.01$. The spectral emissivity of the object at $0.66\,\upmu \text {m}$ is 0.58. If the emissivity is subject to an error of ${\pm }1.5\%$, what is the corresponding error in the estimated actual temperature?

Ex II.32::

The brightness temperature of a metal block is given as $950^{\,\circ } \text {C}$. A thermocouple embedded in the block reads $1032^{\,\circ } \text {C}$. What is the emissivity of the surface? The pyrometer used in the above measurement is a vanishing filament type with an effective $\lambda$ of $0.65\;\upmu \text {m}$. Assuming that the thermocouple reading is susceptible to an error of ${\pm }5^{\,\circ } \text {C}$ while the brightness temperature is error-free, determine an error bar on the emissivity determined above.

Ex II.33::

The emissive power of a black body in $\text {W/m}^2$ is given by the formula $E_b(T) = 5.67\times 10^{-8}T^4$ where T is in Kelvin. A certain black body has the temperature measured at $1100\pm 25\;\text {K}$. Specify the emissive power along with its uncertainty. Express the uncertainty in physical units as well as in percentage form. If the uncertainty in the temperature is reduced by a factor of two, what is the corresponding reduction in the uncertainty in the emissive power?

Ex II.34::

Brightness temperature of a target is estimated, using a vanishing filament pyrometer operating at $0.66\;\upmu \text {m}$ wavelength, to be $1245\pm 10\; \text {K}$. Target temperature was measured independently using an embedded precision thermocouple as $1400\pm 2\; \text {K}$. What is the effective target emissivity and its uncertainty?

Ex II.35::

Total emissivity of a certain surface varies with temperature according to the relation $\varepsilon (T) = 0.068+0.00012\; \text {T}$ where $300<T<1200\; \text {K}$. The brightness temperature of such a surface has been measured using a thermal detector as $T_B=700\; \text {K}$. What is the actual temperature of the surface?

(Hint: Iterative solution is required)

Ex II.36::

Repeated measurements, using a vanishing filament pyrometer, by different operators indicate a mean brightness temperature $900\;\text {K}$ with a standard deviation of $5\;\text {K}$ while an embedded thermocouple gives a reading of $946\;\text {K}$ with an uncertainty of $\pm 2\;\text {K}$. The transmittance of the pyrometer is known to be 0.95. If the effective wavelength of the pyrometer is $0.655\;\upmu \text {m}$, what is the emissivity of the target? What is its uncertainty?

Ex II.37::

If the volume of mercury in the bulb of a thermometer is $200\;\text {ml}$ and the diameter of the capillary is $0.08\;\text {mm}$, how much will the mercury column move for a $1^{\,\circ } \text {C}$ change in the temperature of the bulb? The difference in the cubical expansion of mercury and glass is given as $0.0002^{\,\circ } \text {C}^{-1}$.

Ex II.38::

A bimetallic strip $40\; \text {mm}$ long is made of yellow brass and Monel bonded together at $30^{\,\circ } \text {C}$. The thickness of both the materials is $0.3\; \text {mm}$. The bimetallic strip is in the form of a cantilever beam and is used for “on–off” control with a set point of $75 ^{\,\circ } \text {C}$. Calculate the deflection at this temperature of the free end for a temperature increase of $0.5^{\,\circ } \text {C}$ above the set point.

 

Fig. 6.19

Thermocouple circuit for Exercise II.7

Fig. 6.20

Thermistor circuit for Exercise II.19

1.2 II.2   Transient Temperature Measurement

 

Ex II.39::

An experiment is conducted to determine the time constant of a K type thermocouple as below. We prepare a beaker of boiling water, insert the thermocouple into it, and allow it to reach equilibrium with the boiling water. Then we quickly remove it to a stand so that it is cooled by air moving across the junction by convection. The emf generated by the thermocouple is amplified with a DC amplifier with a gain of 100 and is then recorded, on a chart recorder at a speed of $25\;\text {mm/s}$ during the cooling period. From this record, the following data has been gathered.

$T^{\,\circ } \text {C}$	85	75	65	55	45	35
$t\;s$	0.350	0.600	0.937	1.438	2.175	3.250

Determine the time constant of the system. Also estimate the initial temperature of the thermocouple. Comment on this value. Assume that air temperature remains constant at $25^{\,\circ } \text {C}$.

Ex II.40::

A thermocouple is attached to a thin rectangular sheet of copper of thickness $0.5\; \text {mm}$ and extent $50\times 150\;\text {mm}$. The copper sheet is initially at a temperature of $75^{\,\circ } \text {C}$ and is suddenly exposed to still air at $15^{\,\circ } \text {C}$. How long should one wait for the thermocouple to read $50^{\,\circ } \text {C}$? It may be assumed that the entire plate cools uniformly with time and the heat loss is by natural convection. Assume that the $150\;\text {mm}$ edge is vertical. The heat transfer coefficient may be based on an average plate temperature of $62.5^{\,\circ } \text {C}$. Use suitable natural convection correlation from a book on Heat Transfer.

Ex II.41::

A first-order system is subjected to a ramp input given by $T_f$ = 20 + Rt where $R = 1,2,5, 10^{\,\circ } \text {C/s}$. The initial temperature of the probe is 30 $^\circ \text {C}$. Consider two cases with $\tau = 2$ s and 10 s, and for these two cases draw graphs showing input as well as output responses of the system.

Ex II.42::

Consider a first-order system subjected to a periodic input. The initial temperature of the system is $T_0$ =30 $^{\,\circ } \text {C/s}$. The amplitude of the periodic input is $T_a$ = 10 $^{\,\circ } \text {C/s}$, with $\omega \tau = 1,2,5$. Plot the response of the system as a function of $t/\tau$ for all these cases.

Ex II.43::

A first-order system is subject to a ramp input given by $T_\infty = 20 + 5t$ where temperature is in $^\circ \text {C}$ and t is in s. The initial temperature of the probe is $30^{\,\circ } \text {C}$ and the time constant of the probe is $\tau = 2\;\text {s}$. Determine the probe temperature at $t= 25\;\text {s}$. What is the response of the probe if $t\gg \tau$? Explain your answer.

Ex II.44::

The time constant of a first-order thermal system is given as $30\;\text {s}$. The uncertainty in the value of the time constant is given to be ${\pm }0.5\;\text {s}$. The initial temperature excess of the system over and above the ambient temperature is $25^{\,\circ } \text {C}$. It is desired to determine the system temperature excess and its uncertainty at the end of 30 s from the start.

Hint: System temperature excess variation is exponential.

Ex II.45::

A certain first-order system has the following specifications:

 

• Material: copper shell of wall thickness $1\;\text {mm}$, outer radius $6\;\text {mm}$;

• Fluid: Air at $30^{\,\circ } \text {C}$;

• Initial temperature of shell: $50^{\,\circ } \text {C}$.

How long should one wait for the temperature of the shell to reach $40^{\,\circ } \text {C}$? Assume that heat transfer is by free convection. Use suitable correlation from a heat transfer text to solve the problem.  

Ex II.46::

In order to determine the time constant of a temperature sensor, an experiment was conducted by subjecting it to a step input. The ambient temperature remained constant at $40^{\,\circ } \text {C}$ throughout the experiment. Temperature time data was collected by a suitable recorder at $\frac{1}{4}$ s intervals as shown in the table.

$t\;s$	0	0.25	0.5	0.75	1	1.25	1.5	1.75	2
$T^{\,\circ } \text {C}$	75.28	69.65	64.82	60.90	57.73	55.54	53.44	51.08	49.31
$t\;s$	2.25	2.5	2.75	3	3.25	3.5	3.75	4
$T^{\,\circ } \text {C}$	47.67	46.87	46.05	44.39	43.89	43.40	42.61	43.40

What is the time constant of the sensor?

 

1.3 II.3    Thermometric Error

 

Ex II.47::

A temperature sensor is in the form of a long cylinder of diameter $2\;\text {mm}$. It is placed in a duct carrying air at a temperature of $300^{\,\circ } \text {C}$ moving with a speed of $2\;\text {m/s}$ such that the flow is normal to the axis of the cylinder. The duct walls are known to be at a temperature of $200^{\,\circ } \text {C}$. What is the error in the indicated temperature if the emissivity of the sensor surface is 0.65 and conduction along the sensor is ignored?

If the sensor has an effective thermal conductivity of $12\;\text {W/m}^{\,\circ } \text {C}$ and is immersed to a length of $25\;\text {mm}$, what will be the thermometer error when the conduction error is included? What is the minimum depth of immersion if the thermometric error should not exceed $1^{\,\circ } \text {C}$?

Make a plot of thermometric error as a function of surface emissivity of the sensor.

Ex II.48::

A temperature sensor is in the form of a long cylinder of diameter $1.5\;\text {mm}$. It is placed in a duct carrying air at a temperature of $200^{\,\circ } \text {C}$ moving with a speed of $5\;\text {m/s}$ such that the flow is normal to the axis of the cylinder. The duct walls are known to be at a temperature of $180^{\,\circ } \text {C}$. What is the error in the indicated temperature if the emissivity of the sensor surface is 0.80 and conduction along the sensor is ignored? Air properties given below may be made use of: $\text {Density}=0.6423\;\text {kg/m}^3$, $\text {Dynamic viscosity}=2.848\;\times 10^{-5}\;\text {kg/m}\; \text {s}$,

$\text {Thermal conductivity} = 0.0436\;\text {W/m}^{\,\circ } \text {C}$, and $\text {Prandtl number}=0.680$.

Ex II.49::

(a) A thermocouple pair consists of $0.25\;\text {mm}$ diameter wires of copper and constantan placed $0.75\;\text {mm}$ apart by encapsulating the two wires in a sheath material of thermal conductivity equal to $1\;\text {W/m}^{\,\circ } \text {C}$. The external dimensions of the sheath are $L_1= 1.25\;\text {mm}$ and $L_2 = 0.75\;\text {mm}$. The thermocouple is inserted in a solid up to a depth of $15\;\text {mm}$ and is so installed that there is perfect contact between the sheath and the surface of the hole. The temperature indicated by the thermocouple is $T_t = 125^{\,\circ } \text {C}$ when the exposed leads of the thermocouple are convectively cooled by ambient medium at $T_\infty = 35^{\,\circ } \text {C}$ via a heat transfer coefficient of $h= 4.5\;\text {W/m}^{2\,\circ }\text {C}$. Estimate the temperature of the solid.

(b) If the solid temperature is maintained at the value you have determined in the above part, what should be the depth at which the thermocouple is to be inserted if the thermometer error should be less than $0.5^{\,\circ } \text {C}$?

Ex II.50::

A thermocouple may be idealized as a single wire of diameter $0.5\;\text {mm}$ of effective thermal conductivity of $25\;\text {W/m}^{\,\circ } \text {C}$. The insulation may be idealized as a layer of thickness $0.5\;\text {mm}$ and thermal conductivity of $0.4\;\text {W/m}^{\,\circ } \text {C}$. The thermocouple is attached to a disk of $12\;\text {mm}$ diameter and is exposed to a flowing fluid at a temperature of $300\;\text {W/m}^{\,\circ } \text {C}$ and a heat transfer coefficient of $67\;{\text {W/m}^2}^{\,\circ } \text {C}$. The disk has a surface emissivity of 0.2. The disk is also able to see a background which is at $250^{\,\circ } \text {C}$. The thermocouple lead is taken out of the back of the disk and is exposed to an ambient at $30^{\,\circ } \text {C}$ and a convective heat transfer coefficient of $7.5\;{\text {W/m}^2}^{\,\circ }\text {C}$. Assuming that the lead wire is very long, determine the thermocouple reading.

Ex II.51::

Air is flowing in a tube of diameter $D = 100\; \text {mm}$ with a Reynolds number based on a tube diameter of $Re = 1.5\times 10^5$. In order to measure the temperature of the air, a thermometer well is installed in the pipe, normal to its axis. The thermometer well material has a thermal conductivity of $45\; \text {W/m}^{\,\circ } \text {C}$, has an ID of $3\; \text {mm}$ and an OD of $4.5\; \text {mm}$, and has a depth of immersion of $50\; \text {mm}$. Determine the true air temperature if a thermocouple attached to the well bottom indicates a temperature of $77^{\,\circ } \text {C}$ while a thermocouple attached to the pipe wall indicates a temperature of $56^{\,\circ } \text {C}$. Assume that the thermal conductivity of air is $0.03\, \text {W/m}^{\,\circ } \text {C}$ and the Prandtl number of air is 0.7.

Ex II.52::

A certain temperature measurement system has the following specifications:

Material: copper shell of wall thickness $1\;\text {mm}$ and outer radius $6\;\text {mm}$. A thermocouple is attached to the shell and is made of wires of very small radii such that lead wire conduction may be ignored.

Fluid: Air at $65^{\,\circ } \text {C}$ surrounds the spherical shell. The surface of the shell has an emissivity of 0.65.

Determine the temperature indicated by the thermocouple if the shell interacts radiatively with a background at $25^{\,\circ } \text {C}$.

Assume that heat transfer from air to the shell is by free convection. Use suitable correlation from a heat transfer text to solve the problem.

Ex II.53::

A thermometer well has an OD of $6\; \text {mm}$ and an effective area conductivity product of $6\times 10^{-5}\; \text {W m}/^{\,\circ } \text {C}$. The well is attached to the walls of the duct at $90^{\,\circ } \text {C}$. Air at a temperature of $100^{\,\circ } \text {C}$ is flowing through the duct at an average speed of $1\; \text {m/s}$. The axis of the thermometer well is normal to the flow direction. The surface of the well is a polished metal surface of emissivity 0.05. What is the thermometer error if the well is $50\; \text {mm}$ long?

Ex II.54::

A temperature sensor is in the form of a long cylinder of equivalent diameter $2\;\text {mm}$. It is placed in a duct carrying air at a temperature of $300^{\,\circ } \text {C}$ moving with a speed of $2\; \text {m/s}$ such that the flow is normal to the axis of the cylinder. The duct walls are known to be at a temperature of $280^{\,\circ } \text {C}$. Assume that the conductivity of the cylinder is $14\; \text {W/m}\; \text {K}$.

(a) What should be the length of immersion if the temperature error should be less than $0.5^{\,\circ } \text {C}$?

Air properties given below may be made use of:

Density $=0.6423\; \text {kg/m}^3$, Dynamic viscosity $=2.848\times 10^{-5}\; \text {kg/m}\;\text {s}$, Thermal conductivity $= 0.0436\; \text {W/m}\;\text {K}$, and Prandtl number $=0.680$.

(b) Assume that the depth of immersion has been chosen as above. The emissivity of the sensor surface is known to be 0.05? How will it affect the thermometric error?

Ex II.55::

Mercury in a glass thermometer has a diameter of $6\,\text {mm}$ and is placed in a vertical position in a large room to measure the air temperature. The room walls are exposed to the sun and are at $45^{\,\circ } \text {C}$. The heat transfer coefficient for heat exchange between the room air and the thermometer is $5.5 \,\text {W/m}^{2\,\circ } \text {C}$. The thermometer may be assumed a gray body with an emissivity of 0.65. What is the true air temperature if the thermometer reads $35^{\,\circ } \text {C}$?

 

1.4 II.4   Heat Flux Measurement

 

Ex II.56::

A Gardon gauge is made with a foil of radius $R= 2.3\;\text {mm}$ and thickness $\delta = 20\;\upmu \text {m}$. The foil material has a thermal conductivity of $k= 19.5\;\text {W/m}^{\,\circ } \text {C}$, specific heat of $c = 390 \;\text {J/kg}^{\,\circ } \text {C}$, and density of $\rho = 8900\; \text {kg/m}^3$. Determine the gauge constant as well as the time constant. What will be the temperature difference between the center of the foil and the periphery when radiant flux of $1000\; \text {W/m}^2$ is incident on the foil?

Ex II.57::

A Gardon gauge uses a constantan foil of $3\;\text {mm}$ diameter and $40\;\upmu \text {m}$ thickness. The thermal conductivity of the foil material may be taken as $22\;\text {W/m}^{\,\circ } \text {C}$. Determine the gauge constant assuming that the Seebeck coefficient for Copper–Constantan pair is $40\;\upmu \text {V}/^{\,\circ } \text {C}$. It is desired to make the gauge twice sensitive as compared to the above by suitable choice of the foil diameter–thickness combination. Mention what would be your choice of the diameter–thickness combination? What is the differential temperature generated in the two cases if the incident heat flux is $10^5\;\text {W/m}^2$?

Ex II.58::

A Gardon gauge of diameter $2.7\; \text {mm}$ is $20\;\upmu \text {m}$ thick. It is exposed to an incident heat flux of $1.2\; \text {W/cm}^2$. What is the temperature difference developed by the gauge? If the diameter of the gauge is uncertain to an extent of 2% and the thickness is uncertain to an extent of 5%, what will be the uncertainty in the output when exposed to the heat flux mentioned above?

Ex II.59::

A Gardon gauge of diameter $2.7\;\text {mm}$ is $20\;\upmu \text {m}$ thick. It is exposed to an incident heat flux of $1.2\;\text {W/cm}^2$. What is the temperature difference developed by the gauge? What is the time constant of the gauge? If the diameter of the gauge is uncertain to an extent of 2% and the thickness is uncertain to an extent of 5%, what will be the uncertainty in the output when exposed to the heat flux mentioned above, if the heat flux itself is uncertain to the extent of 2.5%? What is the uncertainty in the time constant of the gauge?

Ex II.60::

A slug type sensor has a slug of mass $0.005\;\text {kg}$ made of a material of specific heat equal to $300\;\text {J/kg}^{\,\circ } \text {C}$. It is in the form of a short cylinder with a flat frontal area exposed to the incident flux of $1\;\text {cm}^2$. The exposed surface of the slug which has an emissivity of 0.86 receives incident radiation flux of $10^5\,\text {W/m}^2$. How long can the slug be exposed to the incident flux if the temperature of the slug is not to exceed $150^{\,\circ } \text {C}$? Take into account only the heat loss by radiation from the exposed surface of the slug.

Ex II.61::

A cylindrical heat transfer probe has a diameter of $12.5\,\text {mm}$. The main heater is $20\,\text {mm}$ wide. Suitable guard heaters are provided to prevent heat transfer along the axial direction. In a certain cold experiment, the heat input to the main heater was 2 W and the temperature of the sensor was noted to be at $60^{\,\circ } \text {C}$. The temperature of the ambient medium was recorded at $32^{\,\circ } \text {C}$. Estimate the heat transfer coefficient. What is the uncertainty in the estimated value of heat transfer coefficient if the temperatures are subject to errors of ${\pm } 0.5^{\,\circ } \text {C}$ and the main heater input has an uncertainty of $2\%$?

Ex II.62::

An axial conduction heat flux probe has the main probe exposed to a heat flux of $10^5\; \text {W/m}^2$. The frontal area of the main probe is $8\; \text {mm}$. The probe is made of an alloy having a thermal conductivity of $45\; \text {W/m}^{\,\circ } \text {C}$. Two thermocouples are used to sense temperatures within the probe body with an axial separation of $4\,\text {mm}$. The temperature difference indicated is $12^{\,\circ } \text {C}$. Determine the amount of heat that has to be removed by the cooling system, assuming that the guard is an annulus with an ID of $8.75\; \text {mm}$ and an OD of $12.5\; \text {mm}$. Suggest a cooling arrangement for this probe.

Ex II.63::

Consult the paper “The Plate Thermometer—A Simple Instrument for Reaching Harmonized Fire Resistance Tests” by Ulf Wickström.³ Based on this paper, explain the principle of a plate thermometer and what it measures.