Question

Oil is being cooled from \(180^{\circ} \mathrm{F}\) to \(120^{\circ} \mathrm{F}\) in a 1 -shell and 2-tube heat exchanger with an overall heat transfer coefficient of \(40 \mathrm{Btu} / \mathrm{h} \mathrm{ft} 2{ }^{\circ} \mathrm{F}\). Water \(\left(c_{p c}=1.0 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right)\) enters at \(80^{\circ} \mathrm{F}\) and exits at \(100^{\circ} \mathrm{F}\) with a mass flow rate of \(20,000 \mathrm{lbm} / \mathrm{h}\), determine (a) the log mean temperature difference and \((b)\) the surface area of the heat exchanger.

Short answer

Answer: The log mean temperature difference (LMTD) for this system is approximately \(48.27^{\circ}\mathrm{F}\), and the surface area of the heat exchanger is approximately \(207.6\ \mathrm{ft}^2\).

Step-by-step solution

Calculate the shell-side mass flow rate

Using the energy balance equation for water, we can calculate the shell-side mass flow rate: \(Q = m_c \cdot c_{pc} \cdot (T_{c,out} - T_{c,in})\), where \(Q\) is the heat transfer rate, \(m_c\) is the mass flow rate of water (cold fluid), \(c_{pc}\) is the specific heat capacity of water, and \(T_{c,out}\) and \(T_{c,in}\) represent the outlet and inlet water temperatures, respectively. Plug in the given values: \[ Q = (20000\ \mathrm{lbm}/\mathrm{h}) \cdot (1.0\ \mathrm{Btu}/\mathrm{lbm}{\cdot}^{\circ}\mathrm{F}) \cdot (100^{\circ}\mathrm{F} - 80^{\circ}\mathrm{F}) \] \[ Q = 400\,000\ \mathrm{Btu/h} \]

Calculate temperature differences at both ends of the heat exchanger

Calculate the temperature differences for hot and cold fluids at both ends of the heat exchanger: 1. Temperature difference at the inlet, \(\Delta T_1 = T_{h,in} - T_{c,in} = 180^{\circ}\mathrm{F} - 80^{\circ}\mathrm{F} = 100^{\circ}\mathrm{F}\). 2. Temperature difference at the outlet, \(\Delta T_2 = T_{h,out} - T_{c,out} = 120^{\circ}\mathrm{F} - 100^{\circ}\mathrm{F} = 20^{\circ}\mathrm{F}\).

Calculate the LMTD

Calculate the LMTD using the temperature differences obtained in step 2: \[ \text{LMTD} = \frac{(\Delta T_1 - \Delta T_2)}{\ln{(\Delta T_1/\Delta T_2)}} \] \[ \text{LMTD} = \frac{(100 - 20)}{\ln{(100/20)}} \] \[ \text{LMTD} \approx 48.27^{\circ}\mathrm{F} \]

Calculate the heat transfer rate using given parameters

According to LMTD, heat transfer rate can be expressed as follows: \[ Q = U \cdot A \cdot \text{LMTD} \], where \(U\) is the overall heat transfer coefficient, and \(A\) is the surface area of the heat exchanger. We have the values for \(Q\) and LMTD from previous steps, and the value of overall heat transfer coefficient is given as \(40\ \mathrm{Btu}/\mathrm{h}\mathrm{ft}^2{ }^{\circ}\mathrm{F}\).

Calculate the surface area of the heat exchanger

Rearranging the equation from step 4, we can solve for surface area \(A\): \[ A = \frac{Q}{U \cdot \text{LMTD}} \] \[ A = \frac{400\,000\ \mathrm{Btu/h}}{(40\ \mathrm{Btu}/\mathrm{h}\mathrm{ft}^2{ }^{\circ}\mathrm{F}) \cdot 48.27^{\circ}\mathrm{F}} \] \[ A \approx 207.6\ \mathrm{ft}^2 \] So, (a) the log mean temperature difference is approximately \(48.27^{\circ}\mathrm{F}\) and (b) the surface area of the heat exchanger is approximately \(207.6\ \mathrm{ft}^2\).

Key concepts

Overall Heat Transfer Coefficient

The overall heat transfer coefficient, denoted as U, is a measure of a heat exchanger's ability to transfer heat between two fluids that are at different temperatures. It encapsulates how well heat is being transferred through the walls of the heat exchanger and accounts for the conductive and convective heat transfers on both sides (hot and cold) of the heat exchanger.

For example, a high value of U indicates that the heat exchanger is efficient at transferring heat. When calculating the heat transfer rate, the coefficient is crucial as it represents the area-averaged heat transfer rate per unit area per unit temperature difference. The formula to calculate the heat transfer rate using the coefficient is: \[ Q = U \cdot A \cdot \text{LMTD}, \]
where Q is the heat transfer rate, A is the surface area, and LMTD stands for Log Mean Temperature Difference. The units of U are typically Btu/(h ft² °F) in the U.S. customary system or W/(m² K) in the Metric system.

Log Mean Temperature Difference (LMTD)

Log Mean Temperature Difference, or LMTD, is a logarithmic average of the temperature difference between the hot and cold fluids at each end of the heat exchanger. It provides a consistent temperature driving force for heat transfer when the temperature differences at each end vary.

The LMTD is a crucial component in the design and analysis of heat exchangers, particularly when the process involves parallel or counter-flow arrangements. The expression to calculate LMTD is given by:\[ \text{LMTD} = \frac{\Delta T_1 - \Delta T_2}{\ln{(\Delta T_1/\Delta T_2)}} \]
where \(\Delta T_1\) and \(\Delta T_2\) are the temperature differences between the fluids at the two ends of the heat exchanger. It is important to use the natural logarithm (ln) when calculating this value. In our exercise, the LMTD was found to be approximately 48.27°F, which represents the average temperature difference for the heat transfer process.

Heat Transfer Rate

The heat transfer rate, often denoted as Q, represents the amount of heat that is transferred per unit time in a heat exchange process. In a heat exchanger, it is essential to understand how quickly energy is moving from the hot fluid to the cold fluid. The rate can vary depending on the overall heat transfer coefficient, the surface area for heat exchange, and the temperature difference.

Following the formula mentioned earlier, the heat transfer rate can be quantified as:
\[ Q = U \cdot A \cdot \text{LMTD} \]
In the given example, we calculated the heat transfer rate to be 400,000 Btu/h. This value was then used to determine the necessary surface area of the heat exchanger to ensure efficient operation.

Specific Heat Capacity

Specific heat capacity, which has a symbol c_p, is a property that represents the amount of heat required to raise the temperature of one unit of mass of a substance by one degree of temperature (typically in °F or °C). In the context of a heat exchanger, specific heat capacity is significant because it affects how much heat a fluid can transport.

In our exercise, the specific heat capacity of water was given as 1.0 Btu/(lbm·°F), meaning each pound mass of water needs 1.0 Btu of energy to increase its temperature by 1°F. This value, along with the mass flow rate and temperature change, provided us with the heat transfer rate for water, which is crucial for sizing the heat exchanger:
\[ Q = m_c \cdot c_{pc} \cdot (T_{c,out} - T_{c,in}) \]
Understanding specific heat capacity is necessary not only for determining heat transfer rates but also for thermal energy calculations in various engineering applications.