A slab of a thermal insulator with a cross-sectional area of 115 cm2 is 4.00 cm thick. It is made out of red brick with thermal conductivity 0.600 J/s∙m•C°. The temperature difference between opposite faces is 95.0 C°. Calculate the following:
(a) The heat current through the slab
(b) The amount of heat flowing through the slab in two days
(c) The temperature gradient through the slab (unit: C°/cm)
To calculate the answers to these questions, we will use the formula for heat conduction:
Q = k * A * (ΔT / d)
where Q is the heat current, k is the thermal conductivity, A is the cross-sectional area of the slab, ΔT is the temperature difference, and d is the thickness of the slab.
Let's break down each question and find the answers step by step:
(a) The heat current through the slab:
We can use the formula Q = k * A * (ΔT / d). Given that k = 0.600 J/s∙m•C°, A = 115 cm2, ΔT = 95.0 C°, and d = 4.00 cm, we can substitute these values into the formula:
Q = (0.600 J/s∙m•C°) * (115 cm2) * (95.0 C° / 4.00 cm)
Now, we need to convert the units so they are consistent. The unit of the thermal conductivity is J/s∙m•C°, so we need to convert the area and thickness to meters:
1 cm = 0.01 m, so 115 cm2 = 115 * (0.01 m)^2 = 0.0115 m^2
4.00 cm = 4.00 * 0.01 m = 0.04 m
Substituting these values into our equation, we have:
Q = (0.600 J/s∙m•C°) * (0.0115 m^2) * (95.0 C° / 0.04 m)
Simplifying this expression will give us the heat current through the slab.
(b) The amount of heat flowing through the slab in two days:
To calculate the amount of heat flowing through the slab in two days, we need to multiply the heat current (from question a) by the time period of two days.
To convert two days to seconds, we need to multiply by the number of seconds in a day:
1 day = 24 hours * 60 minutes * 60 seconds = 86,400 seconds
So, two days would be 2 * 86,400 seconds = 172,800 seconds.
Multiply the heat current by the time period to get the total amount of heat flowing through the slab in two days.
(c) The temperature gradient through the slab (unit: C°/cm):
The temperature gradient is the change in temperature per unit length. In this case, the unit of length is centimeters (cm).
To calculate the temperature gradient, we divide the temperature difference (ΔT) by the thickness of the slab (d) in centimeters.
Temperature gradient = ΔT / d
Substitute the values of ΔT and d to find the temperature gradient through the slab.
To calculate the required values, we need to use the formula for heat current, amount of heat, and temperature gradient.
(a) The heat current can be calculated using the formula:
Heat Current = (Thermal Conductivity * Cross-Sectional Area * Temperature Difference) / Thickness
Given:
Thermal Conductivity (k) = 0.600 J/s·m·C°
Cross-Sectional Area (A) = 115 cm^2 = 115 * 10^-4 m^2
Temperature Difference (ΔT) = 95.0 °C
Thickness (L) = 4.00 cm = 0.04 m
Substituting these values into the formula, we get:
Heat Current = (0.600 * 115 * 10^-4 * 95.0) / 0.04
Calculating the above expression, we get:
Heat Current = 163.875 J/s
Therefore, the heat current through the slab is 163.875 J/s.
(b) The amount of heat flowing through the slab in two days can be calculated using the formula:
Amount of Heat = Heat Current * Time
Given:
Time (t) = 2 days = 2 * 24 * 60 * 60 seconds (converting days to seconds)
Substituting the time value, we get:
Amount of Heat = 163.875 * (2 * 24 * 60 * 60)
Calculating the expression above, we get:
Amount of Heat = 282,672,000 J
Therefore, the amount of heat flowing through the slab in two days is 282,672,000 J.
(c) The temperature gradient through the slab can be calculated using the formula:
Temperature Gradient = Temperature Difference / Thickness
Given:
Temperature Difference (ΔT) = 95.0 °C
Thickness (L) = 4.00 cm = 0.04 m
Substituting the values, we get:
Temperature Gradient = 95.0 / 0.04
Calculating the expression above, we get:
Temperature Gradient = 2375 °C/m
Therefore, the temperature gradient through the slab is 2375 °C/m.