Internal Flows Models

Abstract

It is common for engineering devices to be designed with pipes or ducts in which fluids (gas or liquid) flow. The fluid enters at an inlet temperature and the walls of the duct are at a different temperature. This chapter addresses internal flows in which a single ducted stream is involved. The next chapter considers heat exchangers, where one stream transfers heat to another. When mass flows into a control volume, it carries the energy associated with it. This mechanism of energy transfer is termed advection and will be modeled as a fluid temperature-dependent current source. There are also momentum considerations, as the flows are driven by pressure (or gravity) and opposed by friction, so that pumps or compressors are generally needed to move the fluid. Several 1-node models are developed (thermally long, thermally short, average temperature, and well mixed) and contrasted. The choice of a model in practice depends on whether temperatures or flow rates are considered to be input parameters.

Appendices

Workshop 11.1. Finned Pipe

Water at temperature T₁, pressure P₁ enters an aluminum-finned steel pipe (dimensions defined in sketch) with a mass flow rate of ṁ and flows with negligible pressure drop. The pipe is exposed to air at T_∞ with a cross-flow velocity (not shown in sketch) of V_air. For the parameter values given, determine the heat transfer rate ($\dot{Q}$), length of pipe (L), and number of fins (n_fins) if the exit is a saturated vapor. Treat the external flow as a forced convection over a flat plate of length w_fin and neglect radiation effects. Treat the internal flow as forced convection with fully developed flow.

Default parameter values

Quantity	Symbol	Value	Unit
Mass flow rate	ṁ	0.02	kg/s
Ambient temperature	T∞	25	°C
Inlet temperature	T1	150	°C
Internal fluid pressure	P1	1.0	Atm
Air cross-flow velocity	Vair	10	m/s
Pipe thermal conductivity	kpipe	43	W/m/°C
Fin thermal conductivity	kfin	212	W/m/°C
Pipe outer diameter	Dpipe	0.05	m
Pipe thickness	tpipe	0.00655	m
Fin thickness	tfin	0.00159	m
Fin width	wfin	0.19	m
Fin spacing	Δx	0.01	m

Results

Quantity	Symbol	Value	Unit
Overall heat transfer coefficient (based on inner area)	Ui		W/m2/K
Total heat transfer rate	$\dot{Q}$		Watts
Pipe length	L		m
Number of fins (round up)	nfins		–

A detailed model development follows. It is a good idea to think about this problem and tackle it first before reading on.

Model Development

Use a 1-node fin resistance model (Fig. 11.14) and a 1-node overall flow model (Fig. 11.15). Since the inlet and exit temperature are specified (the exit is a saturated vapor), the average temperature can be set equal to the geometric average for the most accuracy. A single closed-form expression can be obtained that includes all the fins acting in parallel, or a more brute force computer simulation that steps through the pipe fin by fin until the final desired temperature is achieved.

Fig. 11.14

Section detail. Note: the average fluid temperature is the fluid temperature at this section, which varies with distance along the pipe

Fig. 11.15

Overall equivalent circuit. Note: the average fluid temperature represents the average between inlet and outlet of the pipe

Workshop 11.2. Solar Pipe in Continuous Mode: 2-Node Model

Water at temperature T₁ = 15 °C flows into a copper pipe (k_pipe = 400 W/m/K) of outer diameter D = 0.0155 m, thickness t_pipe = 0.002 m, length L = 1.35 m and is exposed to solar radiation (concentrated with a parabolic mirror) with an insolation value of I₀ = 500 W/m² and a solar collection width to rod diameter ratio of n = 19.1. An optical efficiency η_opt = 0.9 (defined as the ratio of the solar radiation that strikes the pipe to the collected radiation) accounts for collected solar energy not absorbed by the pipe. The angle between the incoming solar rays and the pipe is θ = 20°. The absorption coefficient of incident radiation is α = 0.85. The rod is exposed to air at a temperature T_∞ = 25 °C with a natural convection coefficient (from a Nusselt number correlation) that has a mild temperature dependency given by:

$$ {h}_{c, outer}=1.32{\left(\frac{T-{T}_{\infty }}{D}\right)}^{1/4}\ \mathrm{W}/{\mathrm{m}}^2/\mathrm{K} $$

where T is the surface temperature of the pipe. The emissivity of the rod surface is ε = 0.9. The average temperature of the walls with which the rod exchanges thermal radiation is T_w∞ = 30 °C. The convection coefficient on the inside of the pipe can be approximated using the larger of Dittus/Boelter correlation or laminar flow lower limit. That is, the Nusselt number (SI units) is the higher of:

$$ N{u}_D=\frac{h_{inner}\left(D-2t\right)}{k}=0.023{\left(\frac{\rho V\left(D-2t\right)}{\mu}\right)}^{0.8}{\left(\frac{\mu {c}_p}{k}\right)}^{0.4}\kern2em OR\kern2em N{u}_D=3.66 $$

For water properties, use ρ = 1,000 kg/m³, c_p = 4,200 J/kg/K, k = 0.6 W/m/K, and μ = 5.0(10)⁻⁴ kg/m/s. Assume the system is pressurized sufficiently to prevent water from changing phase.

Calculate, as a function of volumetric flow rate (AV):

• The steady-state average temperature of the pipe (T_pipe, in °C).

• The exit temperature of the water (T₂, in °C).

• The minimum pressure required to prevent phase change.

• The rate of heat transfer to the water (in Watts).

• The thermal efficiency, defined as the ratio of heat transfer rate to the water divided by the collected solar flux (solar radiation rate entering top plane).

Think about and tackle yourself before reading on…

2.1 2-Node Well-Mixed Flow Model

The thermal resistance network shown for this system defines an absorbing surface node (the outer surface of the pipe, at an average temperature T_pipe), which receives solar radiation (modeled as a current source). This surface is in thermal contact with the ambient air surrounding it (at temperature T_∞) and the surrounding walls (at T_w∞). Note that there are two very different radiation modes at play; the absorption of solar radiation (modeled as a current source), and thermal radiation of the pipe surrounded by the local environment (modeled as a resistance). The absorbing surface is in thermal contact with the fluid flowing through it (at an average temperature ${\overline{T}}_w$) through a conductive resistance across the pipe in series with an inner convective resistance. The energy flux associated with the water flow is modeled as current sources entering and leaving. A nodal energy balance on the absorbing surface node is:

$$ 0={\dot{Q}}_{abs}+\frac{T_{\infty }-{T}_{pipe}}{R_{h\infty }}+\frac{T_{w\infty }-{T}_{pipe}}{R_{r\infty }}+\frac{{\overline{T}}_w-{T}_{pipe}}{R_{pipe}+{R}_{h, inner}} $$

In the well-mixed flow model (which is numerically stable for all cases, and yields correct high and low flow limits), the average water temperature for the heat transfer term is set equal to the exit temperature (${\overline{T}}_w={T}_2$).

Using Newton’s Law of Cooling and the Stefan–Boltzmann law for radiation (with σ = 5.669(10)⁻⁸ W/m²/K⁴), the thermal resistances are defined as:

$$ {R}_{h\infty }=\frac{1}{h_{c, outer}\pi DL} $$

$$ {R}_{r\infty }=\frac{1}{h_r\pi DL}=\frac{1}{\varepsilon \sigma \left[{\left({T}_{pipe}+273\right)}^2+{\left({T}_{w\infty }+273\right)}^2\right]\left({T}_{pipe}+273+{T}_{w\infty }+273\right)\pi DL} $$

$$ {R}_{hinner}=\frac{1}{h_{c, inner}\pi \left(D-2{t}_{pipe}\right)L} $$

${R}_{pipe}=\frac{t_{pipe}}{k_{pipe}\pi \left(D-{t}_{pipe}\right)L}$ (thin-wall approximation, which is valid here)

The rate of heat absorption is defined by:

$$ {\dot{Q}}_{abs}=\alpha {\eta}_{opt}nDL{I}_0 \cos \theta $$

The nodal equation can be rearranged into a form suitable for the method of successive substitution, with a “k” superscript to indicate iteration number, noting that the thermal resistances between the outer pipe wall and ambient are functions of temperature:

$$ {T}_{pipe}^{\left(k+1\right)}=\frac{{\dot{Q}}_{abs}+\frac{T_{\infty }}{R_{h\infty}^{(k)}}+\frac{T_{w\infty }}{R_{r\infty}^{(k)}}+\frac{T_2^{(k)}}{R_{pipe}+{R}_{h, inner}}}{\frac{1}{R_{h\infty}^{(k)}}+\frac{1}{R_{r\infty}^{(k)}}+\frac{1}{R_{pipe}+{R}_{h, inner}}} $$

An energy balance on the water node is:

$$ \dot{m}{c}_p\left({T}_2-{T}_1\right)=\frac{T_{pipe}-{\overline{T}}_w}{R_{pipe}+{R}_{h, inner}} $$

where the mass flow is $\dot{m}=\rho AV$. Invoking the well-mixed model (${\overline{T}}_w={T}_2$), and rearranging for the exit water temperature in a form suitable for Method of Successive Substitution (note that the Jacobi method uses “k” and Gauss–Seidel uses “k + 1”):

$$ {T}_2^{\left(k+1\right)}=\frac{\frac{T_1}{R_{flow}}+\frac{T_{pipe}^{(k)\kern0.5em or\kern0.5em \left(k+1\right)}}{R_{pipe}+{R}_{h, inner}}}{\frac{1}{R_{flow}}+\frac{1}{R_{pipe}+{R}_{h, inner}}} $$

where a new type of thermal resistance has been introduced: ${R}_{flow}=\frac{1}{\dot{m}{c}_p}$. This thermal resistance only works in this way for the well-mixed model.

The thermal efficiency of the panel is defined as the heat transferred to the water divided by the incident solar flux entering the top:

$$ {\eta}_{thermal}=\frac{\mathrm{Heat}\ \mathrm{t}\mathrm{o}\ \mathrm{Fluid}}{\mathrm{Heat}\ \mathrm{Collected}}=\frac{\dot{m}{c}_p\left({T}_2-{T}_1\right)}{nDL{I}_0 \cos \theta } $$

2.2 Workshop

In teams of three or four, write program code with:

• A commented title section.

• A commented “INPUT PARAMETER” section in which all parameters are defined and enter values from the problem statement.

• A commented “DERIVED PARAMETERS” in which quantities that do not depend on the flow rate or unknown pipe temperature are calculated (i.e., R_pipe).

• A commented “ITERATION LOOP” section that uses an outer FOR loop to step through AV (flow rate), and an inner WHILE loop in which the method of successive substitution is implemented, and tested for convergence.

• A commented “RESULTS” section that plots pipe outer temperature, pipe inner temperature, and fluid exit temperature vs. flow rate (report in units of liters per hour), thermal efficiency vs. flow rate, and thermal efficiency vs. exit temperature.

Workshop 11.3. Solar Pipe in Continuous Mode: Multi-node Model

The problem statement is the same as Workshop 11.2. A multi-node mode is developed and applied.

3.1 Multi-node Well-Mixed Flow Model

A more complex model is required to account for spatial variations in the pipe surface temperature. On the one hand, such a variation would seem to be important for flow rates, where there is a substantial change in water temperature with position. On the other hand, conduction axially in the pipe would tend to maintain a constant wall temperature, especially for highly conductive pipes (like copper).

Consider the thermal resistance network shown (larger versions shown subsequently), in which the pipe and fluid spaces are broken into discrete elements (represented by nodes) and connected thermally by appropriate thermal resistances. Each pipe surface node absorbs solar radiation, exchanges heat with the environment (by convection and radiation acting in parallel), with the adjacent inner fluid (by a conductive resistance across the pipe in series with a convective resistance inside the pipe), and with the neighboring pipe nodes (by a conductive resistance through the pipe wall). The end segments (T_p1, T_pN, T₁, and T_N) differ from the interior nodes (T_p2…T_pN and T₂…T_N) in that they communicate with only one neighbor. Each fluid node is in thermal contact with the pipe outer surface through a series thermal resistance across the pipe wall and convection on the inner surface. There is an inlet flow and outlet flow to each fluid node and the well-mixed model sets the average temperature equal to the outlet temperature. Finally, heat is conducted forward and backward through an axial conductive resistance (in parallel with the flow). This latter mechanism can be important for highly conductive pipes like copper. The forward and backward conduction in the fluid is in parallel with the flow and is a mechanism whose relative importance will be shown to depend on the size of the spatial grid.

For a numerical choice of N segments (2N nodes), the grid spacing is given by:

$$ \Delta x=\frac{L}{N} $$

All the resistance values and solar absorption term are the same as that in the 2-Node model, except that the dimension L is replaced with the grid spacing, Δx. The axial pipe resistance and axial resistance for the inner fluid are new to this model and are given by:

$$ {R}_{p, axial}=\frac{\Delta x}{k_{pipe}\left(\pi /4\right)\left({D}^2-{\left(D-2{t}_{pipe}\right)}^2\right)}=\frac{\Delta x}{k_{pipe}\pi D{t}_{pipe}\left(1-{t}_{pipe}/D\right)} $$

$$ {R}_{axial}=\frac{\Delta x}{k_f\pi {\left(D-2t\right)}^2/4} $$

A nodal energy balance on each of the 2N unknown temperatures, written in the form suitable for a Gauss–Seidel iteration, results in the following system of algebraic equations (nonlinear when the outer convection and radiation resistances are functions of temperature):

For the first pipe node:

$$ {T}_{p1}=\frac{\Delta {\dot{Q}}_{abs}+\frac{T_{\infty }}{R_{\infty }}+\frac{T_{w\infty }}{R_{w\infty }}+\frac{T_{p2}}{R_{p, axial}}+\frac{T_1}{R_{pipe}+{R}_{h,i}}}{\frac{1}{R_{\infty }}+\frac{1}{R_{w\infty }}+\frac{1}{R_{p, axial}}+\frac{1}{R_{pipe}+{R}_{h,i}}} $$

For the interior pipe nodes (i = 2 to N − 1):

$$ {T}_{pi}=\frac{\Delta {\dot{Q}}_{abs}+\frac{T_{\infty }}{R_{\infty }}+\frac{T_{w\infty }}{R_{w\infty }}+\frac{T_{p\left(i-1\right)}}{R_{p, axial}}+\frac{T_{p\left(i+1\right)}}{R_{p, axial}}+\frac{T_i}{R_{pi pe}+{R}_{h,i}}}{\frac{1}{R_{\infty }}+\frac{1}{R_{w\infty }}+\frac{2}{R_{p, axial}}+\frac{1}{R_{pi pe}+{R}_{h,i}}} $$

For the last pipe node:

$$ {T}_{pN}=\frac{\Delta {\dot{Q}}_{abs}+\frac{T_{\infty }}{R_{\infty }}+\frac{T_{w\infty }}{R_{w\infty }}+\frac{T_{p\left(N-1\right)}}{R_{p, axial}}+\frac{T_N}{R_{pipe}+{R}_{h,i}}}{\frac{1}{R_{\infty }}+\frac{1}{R_{w\infty }}+\frac{1}{R_{p, axial}}+\frac{1}{R_{pipe}+{R}_{h,i}}} $$

An energy balance on the first fluid node, rearranging into a form suitable for MOSS, and invoking the well-mixed flow model, with ${R}_{flow}=1/\dot{m}{c}_p$ and:

$$ {T}_1=\frac{\frac{T_{IN}}{R_{flow}}+\frac{T_{p1}}{R_{pipe}+{R}_{h,i}}+\frac{T_2}{R_{axial}}}{\frac{1}{R_{flow}}+\frac{1}{R_{pipe}+{R}_{h,i}}+\frac{1}{R_{axial}}} $$

For interior fluid nodes (i = 2 to N − 1):

$$ {T}_i=\frac{\frac{T_{i-1}}{R_{flow}}+\frac{T_{pi}}{R_{pi pe}+{R}_{h,i}}+\frac{T_{i-1}}{R_{axial}}+\frac{T_{i+1}}{R_{axial}}}{\frac{1}{R_{flow}}+\frac{1}{R_{pi pe}+{R}_{h,i}}+\frac{2}{R_{axial}}} $$

For the last (exit) fluid node:

$$ {T}_N=\frac{\frac{T_{N-1}}{R_{flow}}+\frac{T_{pN}}{R_{pipe}+{R}_{h,i}}+\frac{T_{N-1}}{R_{axial}}}{\frac{1}{R_{flow}}+\frac{1}{R_{pipe}+{R}_{h,i}}+\frac{1}{R_{axial}}} $$

This system of equations can be programmed into an iterative loop to solve for all nodal temperatures, and then any desired quantity.

Consider the relative importance of the axial conduction and the axial advection:

$$ \frac{R_{axial}}{R_{Flow}}=\frac{\frac{\Delta x}{k_f\pi {\left(D-2t\right)}^2/4}}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\dot{m}{c}_p$}\right.}=\frac{4\dot{m}{c}_p\Delta x}{\pi {k}_f{D}_i^2} $$

The axial conductive mechanism becomes more important than the advective flow resistance when the spacing between nodes becomes sufficiently small.