Heat transfer modes and the energy balance

Thermodynamics tells us how much energy moves when a system changes state. It says nothing about how fast that energy moves, or what size of equipment it takes to move it. Heat transfer answers those questions. It explains why a bare steam pipe wastes thousands of dollars of fuel a year, why a district heating pipe can be buried for fifty years with only a modest loss, how large a heat exchanger must be, and how long a stopped water line can sit in frozen ground before it freezes. This course follows the expansion study at Northgate District Energy in Minneapolis, where the team that analyzed the plant's power and refrigeration cycles in Thermodynamics II now has to size the hardware: pipe insulation, an economizer, a new substation heat exchanger for Riverside General Hospital, and the controls that keep everything safe through a prairie winter.

Elena Varga, the plant's lead engineer, has asked Tomás Reyes, the returning co-op student, to start with a walk through the boiler hall with an infrared camera. Graham Okoro, the operations manager, knows where the trouble spots are: a twelve-meter run of the 1 MPa steam main whose insulation was removed for a valve repair and never replaced, a hot boiler casing, and a wall that frosts on the inside in January. Kofi Mensah, the consultant, wants numbers for each one. The first step is to recognize which of the three modes of heat transfer is at work, and to write an energy balance that ties them together.

Heat, temperature and rate

Heat is energy in transit because of a temperature difference. It always flows from high temperature to low temperature, as the second law requires, and the rate at which it flows is measured in watts. We will use \(q\) for a heat rate (W), \(q'\) for a heat rate per unit length (W/m), useful for pipes, and \(q''\) for a heat flux, the rate per unit area (W/m²). The total energy transferred over a time is the integral of the rate, measured in joules or, for annual plant figures, gigajoules.

Every heat transfer problem has the same structure. A driving force, the temperature difference, pushes heat through a resistance that depends on the geometry, the materials and the motion of any fluids. Most of this course is about finding that resistance, or its inverse, a conductance, for real configurations. Once it is known, the rate follows directly.

Conduction

Conduction is the transfer of energy through a substance from more energetic particles to less energetic ones, by molecular collisions in gases and liquids, and by lattice vibrations and free electrons in solids. It needs no bulk motion. For steady one-dimensional conduction, the rate is given by Fourier's law:

The minus sign reflects the fact that heat flows down the temperature gradient. The property \(k\) is the thermal conductivity, in W/m·K. It ranges over five orders of magnitude: about 400 W/m·K for copper, 45 to 60 for carbon steel, 15 for stainless steel, about 1 to 1.5 for concrete and moist soil, 0.6 for water, 0.026 for still air, and roughly 0.02 to 0.03 for the polyurethane foam used in pre-insulated pipes. Insulation works by trapping gas in small cells, so the material's conductivity approaches that of the gas while the cells are too small for the gas to circulate.

Convection

Convection is heat transfer between a surface and a moving fluid. At the surface itself the fluid is at rest and energy passes by conduction, but the moving fluid then carries the energy away, which makes convection far more effective than conduction through a still fluid. When the motion is driven by a fan, pump or wind, we speak of forced convection; when it is driven by buoyancy, because heated fluid is less dense and rises, we speak of natural or free convection. Boiling and condensation are convection processes with a phase change, and they produce the largest coefficients of all. Convection is described by Newton's law of cooling:

Here \(h\) is the convection heat transfer coefficient, in W/m²·K. Newton's law looks simple, but all the physics is hidden in \(h\), which depends on the fluid's properties, its velocity, the geometry and whether the flow is laminar or turbulent. Lessons 6 to 9 develop the correlations used to find it. Typical ranges are worth memorizing as a check on any calculation.

Radiation

Every surface above absolute zero emits electromagnetic radiation. Unlike conduction and convection, radiation needs no medium; it crosses a vacuum and, in practice, passes through air almost unchanged. The maximum possible emission from a surface, that of a blackbody, is given by the Stefan-Boltzmann law, \(E_b = \sigma T^4\), with \(\sigma = 5.67 \times 10^{-8}\) W/m²·K⁴ and \(T\) in kelvin. Real surfaces emit a fraction \(\varepsilon\), the emissivity, of this amount. For a small surface completely surrounded by much larger surroundings at \(T_{\text{sur}}\), such as a pipe in a large hall, the net exchange is

Because of the fourth power, radiation becomes dominant at high temperatures: it is the main mode in a boiler furnace. Even near room temperature it is rarely negligible. A dark, oxidized steel pipe at 175 °C loses roughly as much by radiation as by natural convection. A shiny aluminum jacket, with an emissivity near 0.1, radiates far less, which is one reason jacketed insulation runs cooler to the touch than painted surfaces. It is often convenient to write radiation like convection, using a radiation coefficient \(h_r = \varepsilon\sigma (T_s + T_{\text{sur}})(T_s^2 + T_{\text{sur}}^2)\), so that \(q_{\text{rad}} = h_r A (T_s - T_{\text{sur}})\) and the two can be added when the air and the surroundings are at the same temperature.

The energy balance

Heat transfer problems are solved by combining the rate equations with the first law of thermodynamics. For a control volume, the rate form of the energy balance is

where \(\dot E_g\) is energy generated inside the volume, by electrical heating, a chemical reaction or nuclear fission for example, and \(dE_{\text{st}}/dt\) is the rate of change of stored energy. At steady state the right side is zero. For a flowing stream with negligible changes in kinetic and potential energy, the energy carried is \(\dot m c_p (T_{\text{out}} - T_{\text{in}})\), which is how a heat exchanger duty or a pipe's temperature drop is linked to its heat loss.

A special case is the surface energy balance. A surface has no volume, so it can neither store energy nor generate it, and what arrives must leave. For the outside of a sunlit roof membrane, absorbed sunshine is balanced by convection to the air and radiation to the sky. Writing \(\alpha G = h(T_s - T_\infty) + \varepsilon\sigma(T_s^4 - T_{\text{sky}}^4)\) gives one equation in one unknown, the surface temperature, which must be solved by iteration because of the fourth power. Many practical problems, from thermocouple errors to frost on windows, come down to a surface energy balance like this one.

# Heat transfer modes at Northgate District Energy: a bare steam pipe in the boiler hall
import math
sigma = 5.670e-8                       # Stefan-Boltzmann constant, W/m2.K4

# 1. Conduction through the boiler-hall wall (plane wall, steady)
k_wall, L_wall, A_wall = 0.72, 0.20, 180.0          # brick-concrete wall, W/m.K, m, m2
T_in, T_out = 24.0, -30.0
q_wall = k_wall * A_wall * (T_in - T_out) / L_wall
print(f"wall conduction: q = {q_wall/1000:.1f} kW  (flux {q_wall/A_wall:.0f} W/m2)")

# 2. A bare section of the 1 MPa steam main: convection plus radiation from its surface
D, Lp = 0.2191, 12.0                   # DN200 pipe outside diameter (m), exposed length (m)
Ts, Tair, Tsurr = 175.0, 24.0, 24.0    # surface, air and wall temperatures (C)
h_conv, eps = 9.0, 0.80                # natural convection coefficient (W/m2.K), oxidized steel emissivity
A = math.pi * D * Lp
q_conv = h_conv * A * (Ts - Tair)
q_rad = eps * sigma * A * ((Ts + 273.15)**4 - (Tsurr + 273.15)**4)
h_rad = q_rad / (A * (Ts - Tsurr))
print(f"bare pipe: area {A:.2f} m2, convection {q_conv/1000:.2f} kW, radiation {q_rad/1000:.2f} kW, total {(q_conv+q_rad)/1000:.2f} kW")
print(f"           radiation coefficient h_rad = {h_rad:.1f} W/m2.K, share of radiation {q_rad/(q_conv+q_rad):.0%}")

# 3. What the loss costs over a year of operation
hours, boiler_eff, gas_price = 8000, 0.85, 4.0      # h/yr, -, currency per GJ of fuel (assumed)
E_GJ = (q_conv + q_rad) * hours * 3600 / 1e9
print(f"annual loss {E_GJ:.0f} GJ of steam -> {E_GJ/boiler_eff:.0f} GJ of fuel, cost about {E_GJ/boiler_eff*gas_price:,.0f} per year")

# 4. Surface energy balance: temperature of a sunlit, dark roof membrane on a calm summer day
alpha, eps_r, G_sun, h_out, T_amb = 0.90, 0.90, 850.0, 12.0, 30.0
T = T_amb
for _ in range(60):                    # solve alpha*G = h(T - Tamb) + eps*sigma*(T^4 - Tsky^4), sky at 10 C
    f = alpha*G_sun - h_out*(T - T_amb) - eps_r*sigma*((T+273.15)**4 - (10+273.15)**4)
    df = -h_out - 4*eps_r*sigma*(T+273.15)**3
    T -= f/df
print(f"sunlit roof membrane settles near {T:.1f} C")

wall conduction: q = 35.0 kW  (flux 194 W/m2)
bare pipe: area 8.26 m2, convection 11.23 kW, radiation 12.19 kW, total 23.42 kW
           radiation coefficient h_rad = 9.8 W/m2.K, share of radiation 52%
annual loss 674 GJ of steam -> 793 GJ of fuel, cost about 3,174 per year
sunlit roof membrane settles near 65.3 C

US practice still mixes units. Heat rates are often given in Btu/h (1 W = 3.412 Btu/h), coefficients in Btu/h·ft²·°F (1 W/m²·K = 0.1761 Btu/h·ft²·°F), conductivities in Btu/h·ft·°F (1 W/m·K = 0.5778 Btu/h·ft·°F), and insulation in R-values in ft²·°F·h/Btu, where R-1 equals 0.176 m²·K/W. The bare pipe's 23.4 kW is about 80,000 Btu/h. Building energy requirements are set by state and local adoption of the International Energy Conservation Code or ASHRAE Standard 90.1.

Choosing the modes that matter

Real problems rarely involve one mode alone, and the skill lies in deciding which modes matter and which can be ignored. A useful habit is to estimate each resistance or each coefficient first. Radiation from a shiny surface at room temperature is small; radiation from a dark surface above about 100 °C is comparable to natural convection; radiation in a furnace dominates. Conduction through a thin metal wall is almost always negligible compared with convection on either side, but conduction through insulation usually controls everything. Forced convection overwhelms natural convection once the velocity exceeds a few tenths of a meter per second. These judgments are checked by calculation, but making them first keeps the calculation focused.

Tomás's camera survey turns up one more effect: the bare pipe looks hotter on the screen than the painted pipes beside it, while a stainless flange looks almost cold even though it is touching the same steam. Both effects are about emissivity, not temperature, and Lesson 10 explains how to correct for them.