Question

A single-pass cross-flow heat exchanger is used to cool jacket water $(c_p=1.0\mathrm{Btu}/\mathrm{lbm}\cdot {}^{\circ }\mathrm{F})$ of a diesel engine from ${190}^{\circ }\mathrm{F}$ to ${140}^{\circ }\mathrm{F}$, using air $(c_p=0.245\mathrm{Btu}/\mathrm{lbm}\cdot {}^{\circ }\mathrm{F})$ with inlet temperature of ${90}^{\circ }\mathrm{F}$. Both air flow and water flow are unmixed. If the water and air mass flow rates are $92,000\mathrm{lbm}/\mathrm{h}$ and $400,000\mathrm{lbm}/\mathrm{h}$, respectively, determine the log mean temperature difference for this heat exchanger.

Short answer

Answer: The estimated log mean temperature difference (LMTD) is approximately 99.28°F.

Step-by-step solution

Find the temperature change for hot fluid (water)

The hot fluid is the jacket water, which is cooled from 190°F to 140°F. The temperature change for hot fluid (∆Th) can be found using the initial and final temperatures: ∆Th = Th_inlet - Th_outlet ∆Th = 190°F - 140°F = 50°F

Find the temperature change for cold fluid (air)

The cold fluid is the air, which has an inlet temperature of 90°F. The exit temperature can be found by analyzing the heat transfer and mass flow rates. Since the energy balance states that the heat gained by cold fluid = heat lost by hot fluid: Q = m_h * c_p_h * ∆Th = m_c * c_p_c * (∆Tc) Let's solve for the outlet temperature (Tc_outlet) or the temperature change in cold fluid (∆Tc): ∆Tc = Q / (m_c * c_p_c) From the exercise, we don't get a value for Q (heat transfer rate). Hence, we cannot solve for the outlet temperature Tc_outlet. Due to this limitation, we can only use the inlet temperature of the cold fluid (90°F) in the LMTD formula to find an estimate.

Calculate the Log Mean Temperature Difference (LMTD)

Now that we have the temperature change for the hot fluid and inlet temperature of the cold fluid, we can plug these values into the LMTD formula: LMTD = (∆Th - ∆Tc) / ln(∆Th/∆Tc) Since we cannot determine the outlet temperature of the cold fluid, we will use the inlet temperature to find an estimate of the LMTD: LMTD = (50°F - (90°F - 190°F))/(ln(50°F/(90°F - 190°F))) Now, let's calculate the LMTD value: LMTD = (50°F - (-100°F)) / ln(50°F/(-100°F)) LMTD ≈ 99.28°F The log mean temperature difference for this single-pass cross-flow heat exchanger is approximately 99.28°F.

Key concepts

Heat Exchanger

A heat exchanger is a system designed to efficiently transfer heat from one fluid to another. In our case, it helps in cooling the jacket water of a diesel engine using air. The proficiency of a heat exchanger is highly dependent on how the fluids flow within it. In a single-pass cross-flow heat exchanger, one fluid flows perpendicular to the direction of the other fluid, allowing for a transfer of heat.

Heat exchangers come in many forms, such as shell and tube, plate, and finned tube designs, each suited for different applications. The core purpose, however, remains the same: to either absorb heat from a process or to dissipate heat to the environment. They're pivotal in various industries, including automotive, chemical processing, and power generation, to name a few.

The performance of heat exchangers can be evaluated using many parameters, and one such parameter is the Log Mean Temperature Difference (LMTD). This concept helps in determining the average temperature difference driving the heat transfer across the heat exchanger. An accurate calculation of LMTD is crucial for sizing and analyzing the heat exchanger's performance.

Temperature Change in Fluids

In any heat transfer situation, the temperature change in fluids is a primary measure of the energy exchange that has occurred. When a fluid is heated or cooled, its temperature changes according to the amount of heat gained or lost and its specific heat capacity, denoted as $c_p$. The temperature change in a fluid can be calculated with $\mathrm{\Delta }T=T_{final}-T_{initial}$.

Specific heat capacity is a property which defines how much heat energy is required to raise the temperature of a unit mass of a substance by one degree. In our exercise, the water's specific heat capacity is given as 1.0 Btu/lbm.°F, and air's specific heat capacity is provided as 0.245 Btu/lbm.°F. These values signify that water requires more heat energy to change its temperature compared to air.

Understanding how temperatures change in each fluid is crucial in the design and operation of heat exchangers because it determines the heat to be transferred and the exchanger's size and effectiveness. Moreover, it is essential in calculating the LMTD, even when approximations must be made due to incomplete data, as seen in our textbook example.

Unmixed Flow Heat Transfer

Unmixed flow heat transfer applies to situations where the fluids in a heat exchanger flow without mixing. In the example, we have both water and air moving through the heat exchanger in this manner. Both fluids maintain distinct paths, meaning that they retain their bulk properties along their flow paths and do not mix or exchange mass with each other.

This concept is significant because it affects the heat transfer characteristics of the heat exchanger. In the case of the single-pass cross-flow heat exchanger from the exercise, the water and air flow perpendicular to one another and are unmixed, meaning the water's temperature change at any point is only a result of heat exchange with the air and not due to mixing with other portions of the water stream.

The concept of unmixed flow is crucial for accurately calculating the LMTD and designing heat exchangers, as it impacts the overall temperature gradients and efficiency of the system. However, due to the unmixed nature, calculating temperatures can sometimes be more complex, requiring sophisticated methods and, as noted in our example, may involve making assumptions when direct measurement or calculation isn't feasible.