1. A steel plate initially at 21.2°C is placed in a furnace at 785°C for 58.0 minutes. While in the furnace the temperature of the plate increases at a rate of 4.18% per minute. After 58.0 minutes, the plate is removed from the furnace and placed on a concrete slab to cool. While on the slab, it cools exponentially at a rate of 1.54% per minute.
a) Calculate the temperature of the steel plate immediately after being removed from the furnace.
b) Determine the temperature of the steel plate 1.60 hours after being removed from the furnace.
c) How long will it take the plate to cool to its initial temperature?
a) temperature after 58 minutes
= 21.2(1.0418)^58 = 227.94°
b)
temperature after 1.6 hrs or 96 minutes
= 227
94(.9846)^96 = 51.38°
c) to go from 227.94 back to 21.2 ...
227.94(.9846)^t = 21.2
.9846^t = .093007
t log .9846 = log .093007
t = log .093007/log .9846 = 153.035 minutes
= appr 2 hrs. 33 minutes
To solve this problem, we will use the concept of exponential growth and decay. We'll need to calculate the temperature at each step to find the answers to the given questions.
Given:
Initial temperature of the steel plate (t=0): T0 = 21.2°C
Temperature of the furnace: Tf = 785°C
Time in the furnace: t1 = 58.0 minutes
Rate of temperature increase in the furnace: r1 = 4.18% per minute
Rate of temperature decrease on the slab: r2 = 1.54% per minute
a) To calculate the temperature of the steel plate immediately after being removed from the furnace, we need to find the temperature at t = t1.
Step 1: Calculate the temperature after being in the furnace for t1 minutes.
To find the temperature at t1, we use the formula for exponential growth:
T1 = Tf * (1 + r1)^t1
Substituting the given values:
T1 = 785 * (1 + 0.0418)^58
Using a calculator, evaluate T1 = 1908.88°C
Therefore, the temperature of the steel plate immediately after being removed from the furnace is approximately 1908.88°C.
b) To determine the temperature of the steel plate 1.60 hours (96 minutes) after being removed from the furnace, we need to find the temperature at t = t1 + 96.
Step 1: Calculate the temperature after being in the furnace for t1 minutes.
We have already calculated the temperature at t1 in Step 1(a).
T1 = 1908.88°C
Step 2: Calculate the temperature after cooling on the slab for 96 minutes.
To find the temperature at t1 + 96, we use the formula for exponential decay:
T2 = T1 * (1 - r2)^(t1 + 96)
Substituting the given values:
T2 = 1908.88 * (1 - 0.0154)^(58 + 96)
Using a calculator, evaluate T2 = 112.03°C
Therefore, the temperature of the steel plate 1.60 hours after being removed from the furnace is approximately 112.03°C.
c) To find out how long it will take the plate to cool to its initial temperature (21.2°C), we need to find the time when the temperature equals T0.
Step 1: Calculate the temperature after being in the furnace for t1 minutes.
We have already calculated the temperature at t1 in Step 1(a).
T1 = 1908.88°C
Step 2: Calculate the time it takes for T2 to reach T0.
To find the time when the temperature reaches T0, we rearrange the exponential decay formula:
t2 = ln(T0 / T1)/ln(1 - r2)
Substituting the given values:
t2 = ln(21.2 / 1908.88) / ln(1 - 0.0154)
Using a calculator, evaluate t2 ≈ 152.11 minutes
Therefore, it will take approximately 152.11 minutes for the plate to cool to its initial temperature.
NOTE: Make sure to double-check all the calculations, as rounding errors may occur.