Choose the correct statement.The curve for unsteady state heating and cooling of bodies with Bi

Question

Choose the correct statement.The curve for unsteady state heating and cooling of bodies with Bi < 0.1 is

Prepp Team · Accepted Answer

Understanding Unsteady State Heating and Cooling with Bi < 0.1

When a body undergoes unsteady state heating or cooling, its temperature changes with time. The shape of the temperature-time curve depends on several factors, including the properties of the body and the surrounding fluid, and the heat transfer coefficient. A key parameter in determining the nature of this curve is the Biot number (Bi).

What is the Biot Number?

The Biot number is a dimensionless quantity that compares the thermal resistance within a body to the thermal resistance at the surface of the body. It is defined as:

$ \text{Bi} = \frac{hL_c}{k} $

Where:

$h$ is the convective heat transfer coefficient
$L_c$ is the characteristic length of the body
$k$ is the thermal conductivity of the body material

Lumped Capacitance Analysis (Bi < 0.1)

When the Biot number is less than 0.1 (Bi < 0.1), it signifies that the internal thermal resistance of the body is much smaller than the convective thermal resistance at the surface. In such cases, the temperature within the body can be considered approximately uniform at any given time. This simplification allows us to use the lumped capacitance method to analyze the unsteady heat transfer process.

Temperature Variation with Time

For a body initially at temperature $T_0$ suddenly exposed to a fluid at temperature $T_{\infty}$, the temperature $T(t)$ at any time $t$ is given by the equation derived from the lumped capacitance analysis:

$ \frac{T(t) - T_{\infty}}{T_0 - T_{\infty}} = e^{-\frac{hA}{\rho V c_p} t} $

Where:

$A$ is the surface area
$V$ is the volume
$\rho$ is the density
$c_p$ is the specific heat capacity

Let's define the temperature difference $\theta(t) = T(t) - T_{\infty}$. The initial temperature difference is $\theta_0 = T_0 - T_{\infty}$. The term $\frac{hA}{\rho V c_p}$ is a constant, let's call it $K$. The equation can be rewritten as:

$ \frac{\theta(t)}{\theta_0} = e^{-Kt} $

Or

$ \theta(t) = \theta_0 e^{-Kt} $

Analyzing the Curve Shape

The equation $\theta(t) = \theta_0 e^{-Kt}$ represents an exponential decay function of time $t$.

At $t = 0$, $\theta(0) = \theta_0 e^0 = \theta_0$. The curve starts at the initial temperature difference.
As $t \to \infty$, $e^{-Kt} \to 0$, so $\theta(t) \to 0$. This means the temperature difference approaches zero as time goes to infinity.

Plotting $\theta(t)$ versus $t$ results in a curve that starts at $\theta_0$ and decays exponentially, approaching the time axis ($\theta=0$) as $t$ increases indefinitely. This indicates that the curve is asymptotic to the time axis.

If we plot the temperature $T(t)$ versus $t$, we have $T(t) = T_{\infty} + \theta_0 e^{-Kt}$. This curve starts at $T_0$ and exponentially approaches $T_{\infty}$ as $t \to \infty$. This curve is asymptotic to the line $T = T_{\infty}$, which is parallel to the time axis.

Considering the options provided and the standard interpretation of such curves in the context of unsteady heat transfer towards an ambient temperature, the temperature or temperature difference follows an exponential path. The exponential curve for the temperature difference $\theta(t)$ is asymptotic to the time axis. The exponential curve for the temperature $T(t)$ is asymptotic to a line parallel to the time axis ($T=T_{\infty}$). The phrase "asymptotic to time axis" in the options usually refers to the variable approaching zero or the curve approaching the horizontal axis as time goes to infinity.

Evaluating the Options

Option 1: Hyperbolic curve and asymptotic to both time and temperature axis. This is incorrect. The curve is exponential, not hyperbolic.
Option 2: Exponential curve and asymptotic to both time and temperature axis. The curve is exponential, but it's asymptotic to the time axis (or a line parallel to it, representing $T_{\infty}$), not generally to both the time and temperature axes (unless $T_{\infty}=0$, which is a specific case).
Option 3: Exponential curve and asymptotic to time axis. This aligns with the exponential nature of the temperature decay/rise and its asymptotic behavior as time tends to infinity, approaching the final temperature (or zero temperature difference).
Option 4: Parabolic curve and asymptotic to time axis. This is incorrect. The curve is exponential, not parabolic.

Based on the lumped capacitance analysis for Bi < 0.1, the temperature variation with time is exponential, and the curve approaches the final temperature asymptotically as time tends to infinity. This approach corresponds to the temperature difference approaching zero, which graphically appears as the curve becoming asymptotic to the time axis.