A Nichrome Element Settling

Q. A heating element using nichrome connected to a 230 V supply draws an initial current of 3.2 A which settles after a few seconds to a steady value of 2.8 A. The room temperature is 27.0 °C, and the temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10⁻⁴ °C⁻¹. What is the steady temperature of the heating element, in °C? expects 867 °C

What the draft establishes

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The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

Q. A heating element using nichrome $α_{h} = α_{c}$ , $R_{0,h} = R_{0,c}$ connected to a 230 V $V_{h} = 230$ V supply draws an initial current $V_{c} = V_{h}$ , $T_{c} = T_{0}$ of 3.2 A $I_{c} = 3.2$ A which settles after a few seconds to a steady value of 2.8 A $I_{h} = 2.8$ A. The room temperature is 27.0 °C $T_{0} = 27.0$ °C, and the temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10⁻⁴ °C⁻¹ $α_{c} = 1.70 \times 10^{−4}$ °C⁻¹. What is the steady temperature of the heating element, in °C?

The playground

Inputs

supply

V_{h}

230 V

initial

I_{c}

3.2 A

steady

I_{h}

2.8 A

room

T_{0}

27 °C

coefficient

α_{c}

0.00017 °C⁻¹

Outputs

temperature

T_{h}

The solution

Solved with TemperatureDependenceOfResistance

Given

$V_{h} = 230$ V
$I_{c} = 3.2$ A
$I_{h} = 2.8$ A
$T_{0} = 27.0$ °C
$α_{c} = 1.70 \times 10^{−4}$ °C⁻¹
$α_{h} = α_{c}$ from “A heating element using nichrome”
$R_{0,h} = R_{0,c}$ from “A heating element using nichrome”
$V_{c} = V_{h}$ from “draws an initial current”
$T_{c} = T_{0}$ from “draws an initial current”

Find $T_{h}$ .

Working

$T_{0} = 27.0$ °C $= 300.15$ K
$α_{c} = 1.70 \times 10^{−4}$ °C⁻¹ $= 1.7 \times 10^{−4}$ K⁻¹
ohms law(hot) $V_{h} = I_{h} \cdot R_{h}$
$230 = 2.8 \cdot R_{h}$
$R_{h} = 82.1428571$ Ω
draws an initial current (voltage of cold) $V_{c} = V_{h}$
$V_{c} = 230$
$V_{c} = 230$ V
ohms law(cold) $V_{c} = I_{c} \cdot R_{c}$
$230 = 3.2 \cdot R_{c}$
$R_{c} = 71.875$ Ω
draws an initial current (temperature of cold) $T_{c} = T_{0}$
$T_{c} = 300.15$
$T_{c} = 300.15$ K
resistance grows with temperature(cold) $R_{c} = R_{0,c} \cdot (1 + α_{c} \cdot (T_{c} - T_{0}))$
$71.875 = R_{0,c} \cdot (1 + 1.7 \times 10^{−4} \cdot (300.15 - 300.15))$
$R_{0,c} = 71.875$ Ω
A heating element using nichrome (reference resistance of hot) $R_{0,h} = R_{0,c}$
$R_{0,h} = 71.875$
$R_{0,h} = 71.875$ Ω
A heating element using nichrome (temperature coefficient of hot) $α_{h} = α_{c}$
$α_{h} = 1.7 \times 10^{−4}$
$α_{h} = 1.7 \times 10^{−4}$ K⁻¹
resistance grows with temperature(hot) $R_{h} = R_{0,h} \cdot (1 + α_{h} \cdot (T_{h} - T_{0}))$
$82.1428571 = 71.875 \cdot (1 + 1.7 \times 10^{−4} \cdot (T_{h} - 300.15))$
$T_{h} = 867.3$ °C
$T_{h} = 1140.48613$ K $= 867.336134$ °C
$T_{h} \approx 867$ °Cto 3 significant figures

Conclusion

At first the element is at room temperature and passes more current; as it heats up its resistance grows until the current settles, and that growth tells how hot it is.

The Ruby

1require_relative "../physics"23question :a_nichrome_element_settling do4  kind :numerical5  describes "A heating element using nichrome connected to a %{supply} supply draws an initial current of " \6            "%{initial} which settles after a few seconds to a steady value of %{steady}. The room temperature is " \7            "%{room}, and the temperature coefficient of resistance of nichrome averaged over the temperature range " \8            "involved is %{coefficient}. %{temperature}, in °C?"9  given supply: "230 V"10  given initial: "3.2 A"11  given steady: "2.8 A"12  given room: "27.0 °C"13  given coefficient: "1.70e-4 °C-1"14  asks temperature: "What is the steady temperature of the heating element"15end

1require_relative "../questions/a_nichrome_element_settling"2require_relative "../science/current_electricity/temperature_dependence_of_resistance"34solve :a_nichrome_element_settling, using: TemperatureDependenceOfResistance do5  objects :cold, :hot67  take supply: hot.voltage8  take initial: cold.current9  take steady: hot.current10  take room: T_011  take coefficient: cold.temperature_coefficient12  deduce_from "A heating element using nichrome", hot.temperature_coefficient => cold.temperature_coefficient,13                                                 hot.reference_resistance => cold.reference_resistance1415  deduce_from "draws an initial current", cold.voltage => hot.voltage, cold.temperature => T_01617  temperature = hot.temperature18  answer temperature:, in: "°C", figures: 31920  conclusion "At first the element is at room temperature and passes more current; as it heats up its " \21             "resistance grows until the current settles, and that growth tells how hot it is."22end

1require_relative "../solutions/a_nichrome_element_settling"23play_with :a_nichrome_element_settling do4  title "A Nichrome Element Settling"5end

The laws

quantity	written	units	within	from
voltage	$V$	V	—	OhmsLaw
current	$I$	A	—	OhmsLaw
resistance	$R$	Ω	0 … ∞	OhmsLaw
length of the conductor	$l$	m	—	OhmsLaw
cross section	$A$	m²	—	OhmsLaw
resistivity	$ρ$	Ω m	—	OhmsLaw
temperature coefficient	$α$	K⁻¹	—	TemperatureDependenceOfResistance
reference resistance	$R_{0}$	Ω	—	TemperatureDependenceOfResistance
reference temperature	$T_{0}$	K	—	TemperatureDependenceOfResistance
temperature	$T$	K	0 … ∞	TemperatureDependenceOfResistance
voltage of ground	$V_{g}$	V	—
current of ground	$I_{g}$	A	—
resistance of ground	$R_{g}$	Ω	0 … ∞
voltage of cold	$V_{c}$	V	—
current of cold	$I_{c}$	A	—
resistance of cold	$R_{c}$	Ω	0 … ∞
voltage of hot	$V_{h}$	V	—
current of hot	$I_{h}$	A	—
resistance of hot	$R_{h}$	Ω	0 … ∞
resistivity of ground	$ρ_{g}$	Ω m	—
length of the conductor of ground	$l_{g}$	m	—
cross section of ground	$A_{g}$	m²	—
resistivity of cold	$ρ_{c}$	Ω m	—
length of the conductor of cold	$l_{c}$	m	—
cross section of cold	$A_{c}$	m²	—
resistivity of hot	$ρ_{h}$	Ω m	—
length of the conductor of hot	$l_{h}$	m	—
cross section of hot	$A_{h}$	m²	—
reference resistance of ground	$R_{0,g}$	Ω	—
temperature coefficient of ground	$α_{g}$	K⁻¹	—
temperature of ground	$T_{g}$	K	0 … ∞
reference resistance of cold	$R_{0,c}$	Ω	—
temperature coefficient of cold	$α_{c}$	K⁻¹	—
temperature of cold	$T_{c}$	K	0 … ∞
reference resistance of hot	$R_{0,h}$	Ω	—
temperature coefficient of hot	$α_{h}$	K⁻¹	—
temperature of hot	$T_{h}$	K	0 … ∞

V_{c} = I_{c} \cdot R_{c}

ohms law

R_{c} = \frac{ρ_{c} \cdot l_{c}}{A_{c}}

resistance from resistivity

R_{c} = R_{0,c} \cdot (1 + α_{c} \cdot (T_{c} - T_{0}))

resistance grows with temperature

V_{g} = I_{g} \cdot R_{g}

ohms law(ground)

V_{c} = I_{c} \cdot R_{c}

ohms law(cold)

V_{h} = I_{h} \cdot R_{h}

ohms law(hot)

R_{g} = \frac{ρ_{g} \cdot l_{g}}{A_{g}}

resistance from resistivity(ground)

R_{c} = \frac{ρ_{c} \cdot l_{c}}{A_{c}}

resistance from resistivity(cold)

R_{h} = \frac{ρ_{h} \cdot l_{h}}{A_{h}}

resistance from resistivity(hot)

R_{g} = R_{0,g} \cdot (1 + α_{g} \cdot (T_{g} - T_{0}))

resistance grows with temperature(ground)

R_{c} = R_{0,c} \cdot (1 + α_{c} \cdot (T_{c} - T_{0}))

resistance grows with temperature(cold)

R_{h} = R_{0,h} \cdot (1 + α_{h} \cdot (T_{h} - T_{0}))

resistance grows with temperature(hot)

α_{h} = α_{c}

A heating element using nichrome (temperature coefficient of hot)

R_{0,h} = R_{0,c}

A heating element using nichrome (reference resistance of hot)

V_{c} = V_{h}

draws an initial current (voltage of cold)

T_{c} = T_{0}

draws an initial current (temperature of cold)