Question

In a solar pond, the absorption of solar energy can be modeled as heat generation and can be approximated by $\dot{e}_{\text {gen }}=\dot{e}_{0} \dot{e}^{-b x}\(, where \)\dot{e}_{0}$ is the rate of heat absorption at the top surface per unit volume and \(b\) is a constant. Obtain a relation for the total rate of heat generation in a water layer of surface area \(A\) and thickness \(L\) at the top of the pond.

Short answer

Answer: The relation for the total rate of heat generation in a water layer of surface area A and thickness L at the top of the pond is given by the formula \(E_\text{total} = A\left(\frac{\dot{e}_{0}}{b}\right) \left(1 - \mathrm{e}^{-b L}\right)\).

Step-by-step solution

Understand the heat absorption equation

The equation \(\dot{e}_{\text {gen }}=\dot{e}_{0}\dot{e}^{-b x}\) represents the instantaneous heat generation per unit volume at a distance x from the top surface of the layer. \(\dot{e}_{0}\) is the initial rate of heat absorption at x = 0 (top surface) and b is a constant.

Integrate the equation to find the total heat generation

In order to find the total heat generation in the layer of thickness L, we will integrate the equation with respect to the thickness x from 0 to L: $$ E_\text{gen} = \int_0^L \dot{e}_{\text {gen }} dx = \int_0^L \dot{e}_{0} \cdot \mathrm{e}^{-b x} dx $$

Evaluate the integral

Now we will evaluate the integral using integration by substitution: $$ \begin{aligned} E_\text{gen} &= \frac{\dot{e}_{0}}{-b} \cdot \mathrm{e}^{-b x} \Big |_{0}^{L} \\ &= \frac{\dot{e}_{0}}{-b} \left[\mathrm{e}^{-b L} - \mathrm{e}^{0}\right] \\ &= \frac{\dot{e}_{0}}{b} \left(1 - \mathrm{e}^{-b L}\right) \end{aligned} $$ At this point, we have found the total heat generation per unit volume of the water layer with thickness L.

Relate the result to the surface area A

In order to find the total rate of heat generation in a water layer of surface area A, we need to multiply the total heat generation per unit volume (derived in Step 3) by the surface area A: $$ E_\text{total} = A \cdot E_\text{gen} = A\left(\frac{\dot{e}_{0}}{b}\right) \left(1 - \mathrm{e}^{-b L}\right) $$

Final result

The relation for the total rate of heat generation in a water layer of surface area A and thickness L at the top of the pond is: $$ E_\text{total} = A\left(\frac{\dot{e}_{0}}{b}\right) \left(1 - \mathrm{e}^{-b L}\right) $$