Ch 19: Work, Heat, and the First Law of Thermodynamics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 74

Chapter 19, Problem 74

A flow-through electric water heater has a 20 kW electric heater inside an insulated 2.0-cm-diameter pipe so that water flowing through the pipe will have good thermal contact with the heater. Assume that all the heat energy is transferred to the water. Suppose the inlet water temperature is 12°C and the flow rate is 8.0 L/min (about that of a standard shower head). What is the outlet temperature?

Verified step by step guidance

Step 1: Start by identifying the key variables in the problem. The power of the heater is $ P = 20 \; \text{kW} = 20,000 \; \text{W} $, the inlet water temperature is $ T_{\text{in}} = 12 \; ^{\circ}\text{C} $, the flow rate is $ Q = 8.0 \; \text{L/min} = 8.0 \times 10^{-3} \; \text{m}^3/\text{min} $, and the specific heat capacity of water is $ c = 4186 \; \text{J/(kg·°C)} $. The density of water is approximately $ \rho = 1000 \; \text{kg/m}^3 $.

Step 2: Convert the flow rate from $ \text{m}^3/\text{min} $ to $ \text{m}^3/\text{s} $. Since there are 60 seconds in a minute, divide the flow rate by 60: $ Q = \frac{8.0 \times 10^{-3}}{60} \; \text{m}^3/\text{s} $.

Step 3: Calculate the mass flow rate of water, $ \dot{m} $, using the formula $ \dot{m} = \rho \cdot Q $, where $ \rho $ is the density of water and $ Q $ is the volumetric flow rate in $ \text{m}^3/\text{s} $. Substitute the values to find $ \dot{m} $ in $ \text{kg/s} $.

Step 4: Use the formula for heat transfer to determine the temperature change of the water: $ P = \dot{m} \cdot c \cdot \Delta T $, where $ \Delta T = T_{\text{out}} - T_{\text{in}} $. Rearrange the formula to solve for $ \Delta T $: $ \Delta T = \frac{P}{\dot{m} \cdot c} $. Substitute the known values for $ P $, $ \dot{m} $, and $ c $ to calculate $ \Delta T $.

Step 5: Finally, calculate the outlet temperature $ T_{\text{out}} $ using the relationship $ T_{\text{out}} = T_{\text{in}} + \Delta T $. Add the temperature change $ \Delta T $ to the inlet temperature $ T_{\text{in}} $ to find the outlet temperature.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Heat Transfer

Heat transfer is the process by which thermal energy moves from a hotter object to a cooler one. In this scenario, the electric heater transfers heat to the water flowing through the pipe. Understanding the mechanisms of conduction, convection, and radiation is essential for analyzing how effectively heat is transferred to the water.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For water, this value is approximately 4.18 J/g°C. This concept is crucial for calculating the temperature change of the water as it absorbs heat from the heater, allowing us to determine the outlet temperature.

Flow Rate

Flow rate is the volume of fluid that passes through a given surface per unit time, typically measured in liters per minute (L/min). In this problem, the flow rate of 8.0 L/min indicates how quickly water moves through the heater. This rate, combined with the heat transfer from the heater, influences the final temperature of the water exiting the system.

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