Heat Transfer

Abstract

Thermodynamics deals with the transfer of heat to and from a working fluid and the performance of work by that fluid. Since the transfer of heat to a working fluid is central to thermodynamics, a short excursion into the technology of heat transfer is useful to tie thermodynamics to real world devices. Heat transfer processes are never ideal and a study of the technology of heat transfer will develop an understanding of the trade offs in the design of the devices that actually accomplish the heat transfer. Heat transfer technology provides the basis on which heat exchangers are designed to accomplish the actual transfer of thermal energy.

Problems

Problem 12.1: A brick wall 20 cm thick with thermal conductivity 1.4 W/(m. °C) is maintained at 40 °C at one face 240 °C at the other face. Calculate the heat transfer rate across 6-m² surface area of the wall.

Problem 12.2: The heat flow rate across an insulating material of thickness 4 cm with thermal conductivity 1.0 W/(m. °C) is.250 W/m². If the hot surface temperature is 180 °C, what is the temperature of the cold surface?

Problem 12.3: The heat flow rate through a 4-cm thick wood board for a temperature difference of 30 °C between the inner and outer surface is 80 W/m². What is the thermal conductivity of the wood?

Problem 12.4: A temperature of 550 °C is applied across a block of 10 cm thick with thermal conductivity 1.0 W/(m. °C). Calculate the heat transfer rate per square meter area.

Problem 12.5: By conduction 1000 W is transferred through a 0.5-m² section of a 5-cm thick insulating material. Determine the temperature difference across the insulating layer if the thermal conductivity is 0.1 W/(m. °C).

Problem 12.6: Using Fig. 12.18, consider an insulated rod of length L and constant cross section area A in which a steady state has been reached between two fixed ends of temperature T ₁ and T ₂ that are separated by this rod. Compute the rate of heat transfer through the rod.

Fig. 12.18

Conduction of heat through an insulated conducting bar

Problem 12.7: A compound slab is shown in figure below which is consisting of two different materials having two separate thicknesses as L₁ and L₂ respectively with different conductivity k₁ and k₂ accordingly. If the temperatures of the outer surfaces are kept at T₂ and T₁, calculate the heat transfer through the compound slab in a steady state situation (Fig. 12.19).

Fig. 12.19

Conduction of heat through two layers of matter with different thermal conductivities

Problem 12.8: A thin metal plate 0.2 by 0.2 m is placed in an evacuated container whose walls are kept at 400 °K. The bottom surface of the plate is insulated, and the top surface is maintained at 600 °K because of electric heating. If the emissivity of the surface of the plate is $\varepsilon= 0.1$, what is the rate of heat exchange between the plate and the walls of the container? Take Boltzmann constant to be $\sigma= 5.67\times {10^{-8}}{\text{W}}/({{\text{m}}^2}\cdot {{\text{K}}^4})$.

Problem 12.9: Two large parallel plates, one at a uniform temperature 600 °K and the other at 1000 °K, are separated by a nonparticipating gas. Assume that the surfaces of the plates are perfect emitters and that the convection is negligible; determine the rate of heat exchange between the surfaces per square meter. Take Boltzmann constant to be $\sigma= 5.67 \times{10^{-8}}{\text{W}}/({{\text{m}}^2}\cdot {{\text{K}}^4})$.

Problem 12.10: A thin metal sheet separates two large parallel plates, one at a uniform temperature of 1200 °K and the other 600 °K. Blackbody conditions can be assumed for all surfaces, and heat transfer can be assumed to be by radiation only. Calculate the temperature of the separating sheet.

Problem 12.11: Two very large, perfectly black parallel plates, one maintained at temperature 1000 °K and the other at 500 °K, exchanging heat by radiation (i.e. convection is negligible). Determine the heat transfer rate per 1-m² surface. Take Boltzmann constant to be $\sigma= 5.67 \times{10^{-8}}{\text{W}}/({{\text{m}}^2}\cdot {{\text{K}}^4})$.

Problem 12.12: One surface of a thin plate is exposed to a uniform heat flux of 500 W/m², and the other side dissipates heat by radiation to an environment at ${T_\infty }$ = − 10 °C. Determine the temperature of the plate ${T_p}$. Assume a blackbody conditions for radiation and take Boltzmann constant to be $\sigma= 5.67\times {10^{-8}}{\text{W}}/({{\text{m}}^2}\cdot {{\text{K}}^4})$ and that the convection is negligible.

Problem 12.13: In many practical situations, a surface loses or receives heat by convection and radiation simultaneously where the two methods act in parallel to determine the total heat transfer. Assume a horizontal steel pipe having an outer diameter of 80 mm is maintained at a temperature of 60 °C in a large room where the air and wall temperature are at 20 °C. The average free convection heat transfer coefficient between the outer surface of the pipe and the surrounding air is 6.5 W/(m² K), and the surface emissivity of steel is 0.8. Calculate the total heat loss by the pipe per length. Use the following figure to have some concept of the idea of combined convection and radiation heat transfer (Fig. 12.20).

Fig. 12.20

The system

Problem 12.14: Using the figure below and writing an energy balance for a differential volume element, derive the one-dimensional time-dependent heat conduction equation with internal energy generation $g$ and variable thermal conductivity in the rectangular coordinate system for the $x$ variable (Fig. 12.21).

Fig. 12.21

One-dimensional layout in Cartesian

Problem 12.15: Using the figure below and writing an energy balance for a differential cylindrical volume element $r$ variable, derive the one-dimensional time-dependent heat conduction equation with internal heat generation $g$ and variable thermal conductivity in the cylindrical coordinate system for the $r$ variable (Fig. 12.22).

Fig. 12.22

One-dimensional layout in cylindrical coordinate

Problem 12.16: The local drag coefficient ${c_x}$ can be determined by the following relationship;

$$ {c_x} = {\left.{\frac{{2v}}{{u_\infty^2}}\frac{{\partial u(x,y)}}{{\partial y}}}\right|_{y = 0}}$$

(a)

If the velocity profile $u(x,y)$ for boundary layer flow over a flat plate is given by;

$$ \frac{{u(x,y)}}{{{u_\infty }}}= \frac{3}{2}\left[{\frac{y}{{\delta (x)}}}\right]-\frac{1}{2}{\left[{\frac{y}{{\delta (x)}}}\right]^3} $$

(b)

Where the boundary-layer thickness $\delta (x)$ is

$$ \delta (x) = \sqrt {\frac{{280}}{{13}}\frac{{vx}}{{{u_\infty }}}}$$

(c)

And assume that average drag coefficient ${c_m}$ over a distance $0\le x\le L$ is also given by the following equation;

$$ {c_m} = \frac{1}{L}\int_{x = 0}^L {{c_x}dx}$$

(d)

Develop an expression for the local drag coefficient ${c_x}$.

Develop an expression for the average drag coefficient ${c_m}$ over a distance $x = L$ from the leading edge of the plate.

Problem 12.17: The exact expression for the local drag coefficient ${c_x}$ for laminar flow over a flat plate is given by the following relation and $\operatorname{Re}_x^{1/2}$ is square root of Reynolds number

$$ {c_x} = \frac{{0.664}}{{\operatorname{Re}_x^{1/2}}}$$

knowing that the mean value of the drag coefficient ${c_m}$ over $x = 0$ to $x = L$ is defined as

$$ {c_m} = \frac{1}{L}\int_{x = 0}^L {{c_x}dx}$$

And drag force acting on the same plate from $x = 0$ to $x = L$ for width can be found by

$$ F = wL{c_m}\frac{{\rho u_\infty^2}}{2} $$

Air at atmospheric pressure and ${T_\infty } = 300 {\text{K}}$ flows with a velocity of ${u_\infty } = 1.5 {\text{m}}/{\text{s}}$ along the plate. Determine the distance from the leading edge of the plate where transition begins from laminar to turbulent flow. Calculate the drag force $F$ acting per 1-m width of the plate over the distance from $x = 0$ to where the transition starts.

Problem 12.18: Air at atmospheric pressure and 100 °F (37.8 °C) temperature flows with a velocity of ${u_\infty }$ = 3 ft/s (0.915 m/s) along a flat plate. Determine the boundary-layer thickness $\delta (x)$ and the local-drag coefficient ${c_x}$ at a distance $x = 2$ ft (0.61 m) from the leading edge of the plate. What is the mean drag coefficient over the length $x = 0$ to 2 ft, and the drag force acting on the plate over the length $x = 0$ to 2 ft per foot width of the plate? Use the exact solution for boundary layer Thickness and the local drag coefficient for laminar flow along a flat plate as $\delta (x) =(4.96x)/\sqrt {{{\operatorname{Re} }_x}}$ and ${c_x} =(0.664)/\sqrt {{{\operatorname{Re} }_x}}$ respectively. Assume the mean the mean value of the drag coefficient ${c_{m, L}}= 2{c_x}$ in this case and drag force $F$ acting on the plate over given length is $x = 0$ to $x = L$ and width $w$ described as;

$$ F = wL{c_{m, L}}\frac{{\rho u_\infty^2}}{{2{g_c}}}l{b_f}{\text{or}}(N) $$

Problem 12.19: Air at atmospheric pressure and at a temperature 150 °F (65.6 °C) flows with a velocity of ${u_\infty }$ = 3 ft/s (0.915 m/s) along a flat plate which is kept at a uniform temperature 250 °F (121.1 °C). Determine the local heat transfer coefficient $h(x)$ at a distance $x = 2$ ft (0.61 m) from the leading edge of the plate and the average heat transfer coefficient ${h_m}$ over the length $x = 0$ to 2 ft (0.61 m). Calculate the total heat transfer rate from the plate to the air over the region $x = 0$ to 2 ft per foot width of the plate. Use solution that is also provided by Pohlhausen as${\text{N}}{{\text{u}}_x} = \frac{{h(x)x}}{k} = 0.332{\Pr^{1/3}}\operatorname{Re}_x^{1/2}$.

Problem 12.20: Air at atmospheric pressure and at a temperature 24.6 °C flows with a velocity of ${u_\infty }$ = 10 m/s along a flat plate L = 4 m which is kept at a uniform temperature 130 °C. Assume ${\operatorname{Re}_c} = 2.0\times {10^5}$. Using Figure below and show that the flow is Turbulent and use experimental correlation for turbulent boundary layer along a flat plate as (Fig. 12.23)

$$ {\text{N}}{{\text{u}}_x} = \frac{{h(x)x}}{k} = 0.029{\Pr^{0.43}}\operatorname{Re}_x^{0.8} $$

Fig. 12.23

Flow over a flat plate

• a. Calculate the local heat transfer coefficient at $x = 2, 3$, and 4 m from the leading edge of the plate. Assume Reynolds number is ${\operatorname{Re}_x} = \frac{{{u_\infty }L}}{\nu }$

• b. Find the average heat transfer coefficient over L = 4 m. Assume that, $\frac{{{h_m}L}}{k}{\text{ = N}}{{\text{u}}_m} = 0.036{\Pr^{0.43}}({\operatorname{Re}_L^{0.8}-9200}){\left({\frac{{{\mu_\infty }}}{{{\mu_W}}}}\right)^{0.25}}$ and neglect the viscosity correction and it is equal to unity.

• c. Determine the heat transfer rate from the plate to the air per meter width of the plate.

Problem 12.21: Helium at 1 atm, ${u_\infty }$ = 30 m/s, and 300 °K flows over a flat plate L = 5 m long and W = 1 m wide which is maintained at a uniform temperature of 600 °K. Calculate the average heat transfer coefficient and the total heat rate. Use the following figure and assume ${\operatorname{Re}_c} = 2\times {10^5}$ further assume that, $\frac{{{h_m}L}}{k}{\text{ = N}}{{\text{u}}_m} = 0.036{\Pr^{0.43}}({\operatorname{Re}_L^{0.8}-9200}){\left({\frac{{{\mu_\infty }}}{{{\mu_W}}}}\right)^{0.25}}$ and neglect the viscosity correction and it is equal to unity (Fig. 12.24).

Fig. 12.24

Flow over a flat plate

Problem 12.22: A fluid at 27 °C flows with a velocity of 10 m/s across a 5-cm OD tube whose surface is kept a uniform temperature of 120 °C. Determine the average heat transfer coefficients and the heat transfer rates per meter length of the tube for;

• a. Air at atmospheric pressure. Use given correlation for part (c) and ignore viscosity correction part.

• b. Water. Use general correlation for the average heat transfer coefficient ${h_m}$ for flow across a single cylinder as;

$$ {\text{N}}{{\text{u}}_{\text{m}}}= 0.3+\frac{{0.62{{\operatorname{Re} }^{1/2}}{{\Pr }^{1/3}}}}{{{{[{1+{{(0.4/\Pr)}^{2/3}}}]}^{1/4}}}}{\left[{1+{{\left({\frac{{\operatorname{Re} }}{{282000}}}\right)}^{5/8}}}\right]^{4/5}}$$

• c. Ethylene glycol. Use general correlation for the average heat transfer coefficient ${h_m}$ for flow across a single cylinder as below with viscosity correction from Appendix Table for given conditions (Fig. 12.25);

$$ {\text{N}}{{\text{u}}_{\text{m}}}= \frac{{{h_m}D}}{k} =(0.4{\operatorname{Re}^{0.5}}+0.06{\operatorname{Re}^{2/3}}){\Pr^{0.4}}{\left({\frac{{{\mu_\infty }}}{{{\mu_W}}}}\right)^{9.25}}$$

Fig. 12.25

Flow across a Single Cylinder

Problem 12.23: A very long, 10 mm diameter copper rod (k = 370 W/(m. °K) is exposed to an environment at 20 °C. The base temperature of the rod is maintained at 120 °C. The heat transfer coefficient between the rod and the surrounding air is 10 W/(m2. °K).

• a. Determine the heat loss at the end, and us the following relationship for the rate of loss from the fin as

$$ Q = \sqrt {hPkA} {\theta_b}\frac{{\sinh (mL)-(h/mk)\cosh (mL)}}{{\cosh (mL) + (h/mk)\sinh (mL)}}$$

• b. Compare the results with that of an infinitely long fin whose tip temperature equals the environment temperature of 20 °C. For an inifite long rod use the following heat transfer equation

$$\begin{array}{*{20}{l}} Q&{ = - kA{{\left( {\frac{{dT}}{{dx}}} \right)}_{x = 0}}}\\ {}&{ = kAm({T_b} - {T_\infty })}\\ {}&{ = kAm{\theta _b}}\\ {}&{ = \sqrt {hPkA} ({T_b} - {T_\infty })} \end{array}$$

where ${\theta_b} = {T_b}-{T_\infty }$

Problem 12.24: In a specific application, a stack (see figure below) that is 300 mm wide and 200 mm deep contains 60 fins each of length L = 12 mm. The entire stack is made of aluminum which is everywhere 1.0 mm thick. The temperature limitations associated with electrical components joined to opposite plates dictate the maximum allowable plate temperature of ${T_b}$ = 400 °K and ${T_L}$ = 350 °K. Determine the rate of heat loss from the plate at 400 °K, give $h$ = 150 W/(m². °K) and ${T_\infty }$ = 300 °K. Take k _Aluminum = 230 W/(m². °K). Use the rate of heat loss from the fin can be determined by making use of the following equation (Fig. 12.26).

$$ Q = \sqrt {hPkA} {\theta_b}\frac{{\cosh (mL)-({\theta_L}/{\theta_b})}}{{\sinh (mL)}}$$

Fig. 12.26

A stack containing fins as explained in the problem

Problem 12.25: Saturated water at ${T_{sat}}$ = 100 °C is boiled inside a copper pan having a heating surface A = 5 × 10⁻² m² which is maintained at a uniform ${T_w}$ = 100 °C. Calculate

• a. The surface heat flux and

• b. The rate of evaporation,

Problem 12.26: Repeat Problem 12.25 assuming for a pan made of brass

Problem 12.27: Water at atmospheric pressure and saturation temperature is boiled by using an electrically heated, circular disk of diameter D = 20 cm with the heated surface facing up. The surface of the element is maintained at a uniform temperature ${T_w}$ = 110 °C. Calculate

• a. The surface heat flux

• b. The rate of evaporation, and

• c. The peak heat flux

Problem 12.27: Saturated water at T _v = 100 °C is boiled with a copper heating element having a heating surface A = 4 × 10⁻² m² which is maintained at a uniform temperature T _w = 115 °C. Calculate the surface heat flux and the rate of evaporation.

Problem 12.28: In problem 12.25, if the heating element were made of brass instead of copper, what would be the heat flux at the surface of the heater?

Solution 12.28: The problem is exactly the same as that in Problem 12.25, except that C _sf  = 0.013 should be replaced by C _sf  = 0.006 according to Table 12.5. Then from Eq. 12.51 we write

$$ \frac{{{q_{{\text{water - brass}}}}}}{{{q_{{\text{water - copper}}}}}}= {\left({\frac{{{C_{sf,{\text{water - brass}}}}}}{{{C_{sf,{\text{water - copper}}}}}}}\right)^3} $$

Substituting the numerical values and taking result of Problem 12.25 for $q$ as $4.84\times {10^5} {\text{W/}}{{\text{m}}^{\text{2}}}$, we have;

$$ {q_{{\text{water - brass}}}}=(4.84\times {10^5})({\text{W/}}{{\text{m}}^{\text{2}}}){\left({\frac{{0.013}}{{0.006}}}\right)^3} = 4.93\times {10^5} {\text{W/}}{{\text{m}}^{\text{2}}}$$

Problem 12.29: Water at saturation temperature and atmospheric pressure is boiled with an electrically heated, horizontal platinum wire of D₀ = 0.127 cm diameter. Compute the boiling heat transfer coefficient ${h_m}$ and the heat flux for a temperature difference ${T_w}-{T_{sat}}= 650 {}^\circ {\text{C}}$. Assume no radiation and then consider radiation effects using the following formulas

$$ {h_r} = \frac{1}{{1/\varepsilon+ 1/\alpha -1}}\frac{{\sigma (T_w^4-T_{sat}^4)}}{{{T_w}-{T_{sat}}}}$$

You may also assume that ${h_m}$ is given by empirical equation as follow;

$$ {h_m} = {h_0} + \frac{3}{4}{h_r} $$

Hint: You may also assume that $\varepsilon= \alpha \simeq 1$.

Problem 12.30: Water at saturation temperature and atmospheric pressure is boiled with an electrically heated, horizontal platinum wire of D₀ = 0.2 cm diameter, $\varepsilon$ = 1 in stable film boiling regime with a temperature difference ${T_w}-{T_{sat}}= 654{^0}C$. Compute the film boiling heat transfer coefficient ${h_m}$ and the heat flux. Assume no radiation and then consider radiation effects using the following formulas

$$ {h_r} = \frac{1}{{1/\varepsilon+ 1/\alpha -1}}\frac{{\sigma (T_w^4-T_{sat}^4)}}{{{T_w}-{T_{sat}}}}$$

You may also assume that ${h_m}$ is given by empirical equation as follow;

$$ {h_m} = {h_0} + \frac{3}{4}{h_r} $$

Hint: You may also assume that $\varepsilon= \alpha \simeq 1$.