Question

A patent application describes a device for chilling water. At steady state, the device receives energy by heat transfer at a location on its surface where the temperature is \(540^{\circ} \mathrm{F}\) and discharges energy by heat transfer to the surroundings at another location on its surface where the temperature is \(100^{\circ} \mathrm{F}\). A warm liquid water stream enters at \(100^{\circ} \mathrm{F}, 1 \mathrm{~atm}\) and a cool stream exits at temperature \(T\) and \(1 \mathrm{~atm}\). The device requires no power input to operate, there are no significant effects of kinetic and potential energy, and the water can be modeled as incompressible. Plot the minimum theoretical heat addition required, in Btu per \(\mathrm{lb}\) of cool water exiting the device, versus \(T\) ranging from 60 to \(100^{\circ} \mathrm{F}\).

Short answer

The minimum theoretical heat addition ranges from 0 Btu/lb at 100°F to 40 Btu/lb at 60°F and can be plotted linearly.

Step-by-step solution

Understand the Problem

The device transfers energy by heat from one part of its surface at 540°F to another part of its surface at 100°F while cooling a water stream. We need to calculate the minimum theoretical heat addition required per pound of cool water exiting the device across a range of exit temperatures.

Identify the Given Data

Steady-state conditions, temperatures and pressures are provided: Entrance temperature of water, \[ T_{\text{in}} = 100^{\text{o}}\text{F} \] Exit temperature of water, \[ T_{\text{out}} = T \ \text{Range: } 60^{\text{o}}\text{F} \text{ to } 100^{\text{o}}\text{F} \] Heat source temperature: \[ T_{\text{source}} = 540^{\text{o}}\text{F} \] Heat sink temperature: \[ T_{\text{sink}} = 100^{\text{o}}\text{F} \]

Convert Temperatures to Rankine

First, convert all temperatures from Fahrenheit to Rankine: \[ T_{\text{R}} = T_{\text{F}} + 459.67 \] So, \[ T_{\text{source}} = 540 + 459.67 = 999.67 \text{R} \] \[ T_{\text{sink}} = 100 + 459.67 = 559.67 \text{R} \] \[ T_{\text{in}} = 100 + 459.67 = 559.67 \text{R} \]

Energy Balance on the Device

For minimum theoretical heat addition, consider avoiding any external work: \[ Q_{\text{in}} - Q_{\text{out}} = m \times c_p \times (T_{\text{in}} - T) \] where: \( m \) = mass flow rate, \( c_p \) = specific heat of water Suppose \( m = 1 \text{ lb} \text{ of water} \). The specific heat of liquid water is \( c_p \text{ = 1 \text{ Btu/lb-°F}} \). Thus, \[ Q_{\text{in}} = Q_{\text{out}} + 1 \times 1 \times (100 - T_{\text{out}}) \]

Calculate the Minimum Heat Addition

Using the formula: \[ Q_{\text{min}} = m \times c_p \times (T_{\text{in}} - T) \], which simplifies to \[ Q_{\text{min}} = 100 - T \], rememebering the range of the theoretical minimum heat addition required would be from \( 100 - 100 = 0 \text{ Btu/lb} \text{ to } 100 - 60 = 40 \text{ Btu/lb} \) respectively in a temperature span from 60 to 100°F.

Plot the Result

Create a graph with the x-axis representing the exit temperature range (60°F to 100°F) and the y-axis representing the minimum heat addition (in Btu per lb). The plot will have a linear relationship, decreasing as the exit temperature increases.

Key concepts

heat transfer

Heat transfer involves the movement of thermal energy from one place to another due to temperature differences. In this exercise, the device transfers heat from a high-temperature area at 540°F to a low-temperature area at 100°F.
The efficiency of the device relies on the heat flow, which cools the water stream. Heat naturally flows from a hotter to a cooler region until thermal equilibrium is achieved.

steady-state conditions

The term 'steady-state' means that the system's properties do not change over time. In steady-state conditions, the energy entering and leaving the system remains constant.
This implies the device operates continuously without an accumulation of energy in the system. This simplifies the calculations because we don't need to consider the variations in kinetic or potential energy.

specific heat of water

The specific heat capacity is a material’s heat storage capability, defined as the heat required to raise the temperature of a unit mass by one-degree temperature. For water, this value is 1 Btu/(lb°F).
This value means that 1 Btu of heat is needed to raise the temperature of 1 pound of water by 1°F, making it straightforward to calculate the energy balance for water in this device.

energy balance

An energy balance ensures the input energy equals the output energy plus any changes within the system. For this device, no work is done, and no potential or kinetic energy changes need to be considered.
The energy balance equation is: \[ Q_{in} - Q_{out} = m \times c_p \times (T_{in} - T) \]Here, we need to balance the heat added and removed to calculate the minimum heat needed to cool the water.

temperature conversion

Temperature conversion between Fahrenheit and Rankine scales ensures consistent units during calculations. To convert from Fahrenheit (°F) to Rankine (°R), add 459.67 to the Fahrenheit value.
For example:\[ T_{source} = 540 + 459.67 = 999.67 \text{R} \]\[ T_{sink} = 100 + 459.67 = 559.67 \text{R} \]This conversion simplifies working with thermodynamic equations where absolute temperatures are needed.